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Math ??■ '''-■/

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[ Whole Number 167]

BUREAU OF EDUCATION
CniCTJLAR OF INrORMATION NO. 3, 1890

THE

TEACHING AND HISTORY

MATHEMATICS

THE UNITED STATES

FLOBIAN CAJORI, It S. (University of Wisconsiii)

FORBOCRLT PrOPKSSOR OP APPLIED MATHEMATICS IN THE TULANB UNIVERSITY OF

Louisiana, now Professor op Physics in Colorado College,

#■♦»•

WASHIITGTON"

GOVERNMENT PRINTING OFFIOH

1890

I WltAt Kmtlbtr VS\

BUREAU OF EDUCATION
CmCTCrLAR OF INFORMATION NO. Z, \h'4<i

THE

TEACHING AXD niSTOKT

MATHEMATICS

THE UXTTED ST.iTj;-

XtfCIiSASU. JfiCTW Fl.'.»MBVi: iff ynXhi*^ if ' ..

|Y\aJt^ ^2,^0.4

\* \ ^.i .*" *> ^< A -J

Depahtment of the Interior,

Bureau of Education,
Washingtonj D. C, February 19, 1889.

Sir: I have the honor to transmit herewith the manuscript of a His-
tory of Mathematical Teaching in the United States, by Prof. Florian
Cajori, a graduate of the University of Wisconsin, student at Johns
Hopkins University, and recently professor of applied mathematics in
Talane University of Louisiana — a work prepared with your approval,
under the direction of this Office.

The table of contents indicates the wide scope of the work and the
variety of subjects treated, but scarcely more than suggests the pains-
taking labor involved in its preparation. Professor Cajori's researches
have extended through several years, and have been pursued in the
libraries of Baltimore, Philadelphia, and Washington. He has person-
ally conducted a large correspondence with alumni, and past and pres-
ent instructors in the higher educational institutions, and has been
aided by 1,000 circulars of inquiry sent from this Office relating to the
present condition of mathematical teaching in schools of all grades.

I am convinced that this monograph will prove of great value to all
teachers and students of mathematics, and will be not without interest
to any person engaged in the work of education. I therefore respect-
fully recommend its publication.

I have the honor to be, sir, very respectfully, your obedient servant,

K H. E. Dawson,

Comnmsioner,

Hon. Wm. F. Vilas,

) Secrctafy of the Interior ^ Washington^ D. 0.

Depahtment op the Intetmos,

WashingtoUj D, C\, April 11, 1889.

The Commissioner of Education:

Sir : 1 acknowledge the receipt of your letter of February 19, 1889,
in which you recommend the publication of a monograph, a history oi
Mathematical teaching in the United States.

Authority is hereby given for the publication of the monoferaph, pro-
vided there are funds in sufficient amount available for such x)urpose
Very respectfully,

John W. Koble,

Secretary.

CONTENTS.

Pago.

I. Colonial Times 9

(a) Elementary Schools 9

(b) Colleges 18

Harvard College 18

Yale College 28

William and Mary College 33

University of Pennsylvania 36

(o) Self-taaght Mathematicians 37

II. Influx of English Mathematics 44

(a) Elementary Schools • 45

(6) Colleges ...i 55

Harvard College 57

Yale College 61

University of Pehnsylvania 65

College of New Jersey (Princeton) 71

Dartmouth College 73

Bowdoin College • 75

Georgetown College • 77

University of North Carolina 77*

University of South Carolina 81

Kentucky University 83

United States Military Academy 84

(o) Self-taaght Mathematicians 86

(d) Surveying of Government Lands 92

{e) MathematicalJournals 94

III. Influx of French Mathematics 98

(a) Elementary Schools 106

(b) Colleges — United States Military Academy 114

Harvard College 127

Yale College 151

College of New Jersey 160

Dartmouth College 165

Bowdoin College 170

Georgetown College 173

Cornell University 176

Virginia Military Institute 188

University of Virginia 191

University of North Carolina > 204

University of South Carolina 208

University of Alabama 214

University of Mississippi 219

Kentucky University 225

University of Tennessee .....•« 227

'1
1

8 CONTENTS.

Page.

m. Ikvlux of French Mathematics— Colleges— Continued.

Talane University of Louisiana 231

University of Texas !236

Washington University 239

University of Michigan • 244

University of Wisconsin • 253

Johns Hopkins University 261

(o) MathematicalJonmals 277

(d) U. 8. Coast and Geodetic Survey 286

IV. Thk Mathematical Teaching at the Present Time 293

V. Historical Essays:

(a) History of Infinite Series 361

(6) On Parallel Lines and Allied Subjects 376

(d) Difference between Napier's and Natural Logarithms 388

(e) Circle Squarers 391

APPENDIX.

BiSUOQBAPHY OF FLUXIONS AND THE CALCULUS. 395

THE TEACHING AND HISTORY OP MATHEMATICS IN

THE ONITED STATES.

COLONIAL TIMES.

Elementary Schools.

On the study of mathematics in elementary schools of the American
colonies but little can be said. In early ccTlonial days schools did not
exist except in towns and in the more densely settled districts ; and
even where schools were kept, the study of mathematics was often not
pursued at all, or consisted simply in learning to count and to perform
the fundamental operations with integral numbers. Thus, in Hamp-
stead, N. H., in 1750, it was voted *' to hire a school-master for six mouths
in ye summer season to teach ye children to read and write.^ Arith-
metic had not yet been introduced there. As late as the beginning of -
this century there were schools in country districts in which arithmetic
was not taught at all. Bronson Alcott, the prominent educator, born in
Massachusetts in 1799, in describing the schools of his boyhood, says:
•'Until within a few years no studies have been permitted in the day
school but spelling, reading, and writing. Arithmetic was taught by a
few instructors one or two evenings in a week. But in spite of the most
determined opposition arithmetic is now permitted in the day school.''
This was in Massachusetts at the beginning of this century.

In secondary schools, " ciphering" was taught during colonial times,
which consisted generally in drilling students in the manipulation of'
integral numbers. He was an exceptional teacher who possessed a fair i
knowledge of " fractions" and the " rule of three," and if some pupil of
rare genius managed to master fractions, or even pass beyond the " rule
of three," then he was judged a finished mathematician.

The best teachers of those days were college students or college
graduates who engaged in teaching as a stepping-stone to something
better. An example of this class of teachers was John Adams, after-
wards President of the United States. Immediately after graduating
at Harvard and before entering upon the study of law, he presided, for
a few years, over the grammar school at Worcester. From a letter

10 TEACHING AND HISTOBV OP MATHEMATICS.

written by him at Worcester, September 2, 1755, we clip the following
description of the teacher's daily work :

As a haaghty monarch ascends liis thronei the pedagogue monnts his awfal great
chaiKj and dispenses right justice through his whole empire. His ohsequious subjects
execute the imperial mandates with cheerfulness, and think it their high happiness
to he employed in the service of the emperor. Sometimes paper, sometimes his pen-
knife, now birch, now arithmetic, now a ferule, then ABC, then scolding, then flatter-
ing, then thwacking, calls for the pedagogue's attention.

School appliances in those days were wholly wanting (excepting the
ferule and birch rods). Slates were entirely unknown for school use
until some years after the Eevolution ; blackboards were introduced
much later. Paper was costly in colonial days, and we are told that
birch bark was sometimes used in schools in teaching children to write
and figure. Thirty-six years ago a writer in one of our 'magazines*
wrote as follows :

** There are probably men now living who learned to write on birch
and beech bark, with ink made out of maple bark and copperas." But
more generally *^ ciphering"" was done on paper. Dr. L. P. Brockett
says that on account of its dearness and scarcity, *Hhe backs of old let-
ters, the blank leaves of ledgers and day-books, and even the primer
books were eagerly made use of by the young arithmeticians."

Since few or none of the pupils had text-books it became necessary
for the teacher to dictate the '^ sums." As in the colleges of that time,
so in elementary schools, manuscript booJcs were used whenever printed
ones were not accessible. To advanced boys the teacher would give
exercises from his manuscript or " ciphering-book," in which the prob-
lems and their solutions had been previously recorded. " With a book
of his own the pupil solved the problems contained in it in their proper
order, working hard or taking it easy as pleased him, showed the solu-
tions to the master, and if found correct generally copied them in a blank-
book provided for the purpose. • • • Some of these old manuscript
ciphering-books, the best, one may suppose, having come down through
several generations, are still preserved among old family records, bear-
ing testimony to the fair writing and the careful copying, if not to the
arithmetical knowled ge, of those who prepared them. When a pupil was
unable to solve a problem he had recourse to the master, who solved it
for him. It sometimes happened that a dozen or twenty pupils stood
at one time in a crowd around the master's desk waiting with • • •
problems to be solved. There were no classes in arithmetic, no explana
tions of processes either by master or pupil, no demonstrations of princi-
ples either asked for or given. The problems were solved, the answers
obtained, the solutions copied, and the work was considered complete.
That -some persons did obtain a good knowledge of arithmetic under
such teaching must be admitted, but this result was clearly due rather
to native talent or hard personal labor than to wise direction."! Those

• North Carolina University Magazine, Raleigh, 1853, Vol. II, p. 452.
« t History of Education in Pennsylvania '^ ' " Pyle Wickersham, p. 205.

COLONIAL TIMES. 11

teachers who were the fortunate possessors of a printed arithmetic used
it as a guide in place of the old *< ciphering-book.''

In the early schools, arithmetic was hardly ever taught to girls. Eev.
William Woodbridge says that in Connecticut, just before the Kevolu-
tion, he has <' known boys that could do something in the first four rules
of arithmetic. Girls were never taught if* In the two " charity
schools" in Philadelphia, which before the Revolution were the most
celebrated schools in Pennsylvania, boys were taught reading, writing,
arithmetic; girls, reading, writing, sewing. Thus, sewing was made
to take the place of arithmetic. Warren Burton, in his book entitled^
" The District School as it was, by one who went to it,'^ says that, among
girls, arithmetic was neglected. The female portion " generally ex-
pected to obtain husbands to perform whatever arithmetical operations
they might' need beyond the counting of fingers." Occasionally women
were employed in summer schools as teachers, but they did not teach
arithmetic. A school-mistress ''would as soon have expected to teach
the Arabic language as the numerical science."

The early school-books in Kew England and in all other English set-
tlements were much the same as those of Old England. John Locke, in
his Thoughts concerning Education (1690), says that the method of teach-
ing children to read in England has been to adhere to " the ordinary
road of horn-book, primer, psalter, testament, and bible." This same
road was followed in New England. We are told that books of this
kind were sold to the people by John Pynchon, of Springfield, from
1656 to 1672 and after.t Eegular arithmetics were a great rarity in
this country in the seventeenth century. The horn-book has been raised
by some to the dignified name of a " primer" for teaching reading and
imparting religious instruction. If this be permissible, then why should
we not also speak of it as an arithmetical primer ? For, what was the
horn-book ? It consisted of one sheet of paper about the size of an
ordinary primer, containing a cross (called "criss-cross"), the alphabet
in large and small letters, followed by a small regiment of monosyllables;
then came a form of exorcism and the Lord's Prayer, and, finally, the
Roman numerals. The leaf was mounted on wood, and protected with
transparent horn,

"To save from fingers wet the lettbrs fair."

It is on the strength of the Eoman numerals that we venture to pro- \
pose the horn-book as a candidate for the honor of being the first math- , •
ematical primer used in this country. Hornbooks were quite common
in England and in the English colonies in America down to the time of
George II. They disappeared entirely in this country before the Revo-
lution. In early days the common remark expressive of ignorance was
"he does not know his hprn-book," This is equivalent to the more
modern saying, " he does not know his letters."

* Reminiscenses of Female Education, in Barnard's Journal of Education, 1864, p.
137.

» t Barnard's Journal of Education, Vol. XXVII. •

12 TEACHING AND HISTORY OF MATHEMATICS.

George Fox, the founder of the Society of Friends, published in 1674,
in England, a primer or spelling-book, which was republished at Phila-
delphia in 1701, at Boston in 1743, and at Newport, R. I., in 1769.*
Wickersham describes this little book as containing the alphabet, les-
sons in spelling and reading, explanations of scripture names, Roman
numef alSf lessons in the fundamental rules of arithmetic andtceights and
measures, a perpetual almanac, and catechism with the doctrine of the
Friends. It may be imagined that a mere primer, covering such a wide
range of subjects, could contain only a very few of the simplest rudi-
ments of a subject like arithmetic. Fox's book was used little outside
of the Society of Friends.

Wickersham (p. 201) speaks of another book which is of interest as
illustrating the book-making of those old times. It is entitled, " The
American Instructor, or Young Man's Best Companion, containing Spell-
ing, Reading, Writing, Arithmetic, in an Easier Way than any yet Pub'
lished, and how to Qualify any Person for Business without the Help
of a Master." It was written by George Fisher, and printed in Phila-
delphia, in 1748, by Franklin and Hall. This work never attained any
popularity.

Dr. Brockett says that in New Jersey and, perhaps, also in Virginia,
a book resembling the " New England Primer," but as intensely Roy-
alist and High Church in religion as the New England Primer was
Puritan and Independent, was in use in schools. It was called "A
Guide for the Child and Youth, in two parts ; the First for Children, •
* • the second for Youth : Teaching to write, cast accounts and read
more perfectly ; with several other varieties, both pleasant and profit-
able. ByT. H.,M. A., Teacher of a Private School, London, 1762."
It does not appear that this book was reprinted here.

Wickersham gives another book of similar stamp but of much later
date. '^ Ludwig Hooker's Rechenbiichlein was published at Ephrata
[Pennsylvania] in 1786. The Ephrata publication is an exceedingly
curious compound of religious exercises and exercises in arithmetic.
The creed, the Lord's Prayer, hymns, and texts of scripture, are strangely
intermixed with x)roblems and calculations in the simpler parts of arith-
metic."!

One of the earliest purely arithmetical books used in this country was
' the arithmetic of James Hodder. It may possibly have fallen into the
hands of as early a teacher as Ezekiel Cheever, " the father of Con-
necticut school -masters, the pioneer and patriarch of elementary classi-
cal culture in New England."} In a history of schools at Salem,. Mass.,
we are told that '* among our earliest arithmetics was James Hodder's.'^

• History of Eclacation in PeDnsylvaniu, by James Pyle Wickersliam, p. 194.

t Ibid,, p. 200.

t After having been a faithfal Bcbool-master for seventy years, he died in 1708, at
the ago ofninety-foari having *'held hia abilities in an anuaaal degree to the very
last.''

COLONIAL TIMES. "13

Hodder was a famons English teacher of the seventeenth century.
Later writers have borrowed largely from his arithmetic of which the
first edition, entitled '' Hodder's Arithmetick, or that necessary art
made most easy,'' appeared iu London in 1G61. An American edition
from the twenty-fifth English edition was published in Boston in 1719.
This is the first purely arithmetical book known to have been printed •
in this country.

In New York the Dutch teachers of the seventeenth century im-
ported from Holland an arithmetic called the " Goffer Konst," written
by Pieter Yenema, a Dutch school-master, who died about 1G12. So
popular was the book that an English translation of it was published
in New York in 1730. Venema's appeared to be the second oldest arith-
metic printed in America.

An English work almost as old as Hodder's, which met with a limited
circulation in this country, is Cocker's Arithmetic. The first edition
appeared in England after the death of Cocker, in 1677. According to
its title page it was " perused and published by John Hawkins, • • •
by the author's correct copy." De Morgan is perfectly satisfied that
" Cocker's Arithmetic was a forgery of Hawkins's, with some assistance,
it may be, from Cocker's papers." Eegarding the book itself, De Mor-
gan says :* "Cocker's Arithmetic was the first which entirely excluded '
all demonstration and reasoning, and confined itself to commercial
questions only. This was the secret of its extensive circulation. There
is no need of describing it; for so closely have nine out of ten of the
subsequent school treatises been modelled upon it, that a large propor-
tion of our readers would be able immediately to turn to any rule in
Cocker, and to guess pretty nearly what they would find there. Every
method since his time has been "according to Cocker." This book was
found here and there in the colonies at an early date. Thus we read
in Benjamin Franklin's Autobiography that (at about the age of six-
teen ; L e.j about 1722) " having one day been put to the blush for my
ignorance in the art of calculation which I had twice failed to learn
while at school, I took Cocker's treiNtise on arithmetic and went throngu
it by myself with the utmost ease." An American edition of the work
appeared in Philadelphia in 1779. It contains the rude portrait of the
author, " which might be taken for a caricature," and also the following
poetical recommendation :

Ingenioas Cocker, now to Rest thou 'rt gone,
Ko Art can show thee fallji bat thine own;
Thy rare Arithmetick alone can show
Th' vast Thanks we for thy labours owe.

Wickershamt mentions Daniel Fenning's Der Oesohwinde Bechneras
having been published by Sower in 1774.

* Article, ** Cocker," Penny Cyclopaedia.

t History of Education in Pennsylvania, p. 200.

14 TEACHING AND HISTORY OP MATHEMATICS.

The fu*st arithmetic written by an American author and printed here
' was that of Prof. Isaac Greenwood of Harvard College, in 1729. The
book was probably used by the author in his classes at Harvard. We
have nowhere seen it mentioned except in a biographical sketch of its
author.* So far as we know, there are only three copies of Greenwood's
Arithmetic in existence, two in the Harvard library and one in the Con-
gressional Library. Prof. J. M. Greenwood, superintendent of schools
in Kansas City, sends the writer the following description of it:

The book is a small duodecimo volume of 158 pages^ exclusive of an
advertisement (4 pages) prefixed, and the table of contents (4 pages)
put at the end. The following is a transcript of the title-page : ''Arith-
metick, Vulgar and Decimal: with the Application theredf to a variety
of Cases in Trade and Commerce. (Vignette.) Boston : N. E., Printed
by S. Kneeland and T. Green, for T. Hancock at the Sign of the Bible
and Three Crowns, in Ann Street, MDCCXXIX.''

The headings of chapters are as follows : The introduction ; chapter
1, Numeration ; chapter 2, Addition ; chapter 3, Subtraction j chapter
4, Maltiplication ; chapter 5, Division ; chapter 6, Keduction ; chapter
7, Vulgar Fractions ; chapter 8, Decimal Fractions ; chapter 9, Eoots
and Powers; chapter 10, Continued Proportion ; chapter 11, Disjunct
Proportion; chapter 12, Practice; chapter 13, Rules relating to Trade
and Commerce.

From the preface : ** The Author^s Design in the following Treatise is to give a very
concise Account of such RaleS| as are of the easiest practice in all the Parts of Vulgar
and Decimal Arithmetick and to illustrate each with such examples, as may be suf-
ficient to lead the Learner to the full Use thereof in all other Instances.''

" The Reader will observe that the Anthor has inserted under all those Rules, where
it was proper, Examples with Blanks for his Practice. This was a Principal End to
the Undertaking ; that such persons as were desirous thereof might have a compre-
hensive Collection of aU the best Rules in the Art of Numbering, with Examples
wrought by themselves. And that nothing might be wanting to favour this Design,
the Impression is made upon several of the best sorts of Paper. This method is en-
tirely new, * * *."

f The paper nsed in the book is thick, the type large. Words and
phrases to which the anthor desires to call special attention are printed
in italic characters, and as more than half the book is, in the anther's
eyes, important, more than half the book is printed in italics*

In 1788, when Nicholas Pike published his arithmetic, Greenwood's
book was entirely unknown, and Pike's was believed to be the first arith-
metic written and printed in America.

> The first arithmetic which enjoyed general popularity and reached an
extended circulation in the colonies was the School-master^s Assistant,
* by Thomas Dilworth. The first edition of this was published in London
in 1744 or '45. According to Wickersham, there appeared a reprint of
this in Philadelphia in 1769. Other American editions were brought
out at Hartford in 1786, iTew York in 1793 and 1806, Brooklyn in 1807,

* Appleton's Dictionary of American Siography. '

COLONIAL TIMEI^. 1 5

New liondon 1797, and Albany 1824. At the beginning of the Eevolu-
tiou this was the most popular arithmetic^ and it continued in use long
after.

We have now enumerated all the arithmetics which were used to our
knowledge in the American colonies. It may be instructive to give the
last book which we have mentioned a closer examination ; for Dil worth's
School-master's Assistant was the most n6ted arithmetic of its time . As^
an arithmetician Dil worth belonged to the school founded by Cocker,
which scrupulous! j'excIudeS ail demonstration. and rea^jiing. The
School-master's Assistant gives all rules and definitions in the form of
questions and answers. Let us turn to page 44. of the twenty-second
London edition, 1784, and examine his mode of explaining i>roportion,
or, as the subject was then called, the ^^ Eule of Three."

OF THE SINGLE RULE OF THREB.

Q, How many Parts are tliere in the Rule of Three t

A, Two : Single or Simple, and Doable or Compound.

Q, By what is the single Rule of Three known f

A, By three Terms, which are always given in the Question, to find a fourth.

Q. Are any of the terms given to be reduced from one Denomination to another?

A. If any of the given terms be of several denominations, they must be reduced into

the lowest Denomination mentioned.
Q. What do you observe concerning the first and third Terms t
A. They must be of the same Name and Kind.
Q, What do you observe concerning the fourth Term t
A. It must be of the same Name and Kind as the second.
Q, What do you observe of the three given Terms taken together t
A, That the two first are a Supposition, the last is a Demand.
Q, How is the third Term known t

A, It is known by these, or the like Words, What coat fi How many f How much t
Q, How many Sorts of Proportion are there 1
A, Two : Direct and Inverse.

And so on. We have quoted enough to give an idea of the book. It
is not easy to see how a pupil beginning the subject of proportion could
get clear notions from reading the abcjve. Nor can we see how a boy
who had never before heard of fractions could get any idea whatever
of a fraction from Dilworth's definition, which is (p. Ill) : A fraction
^^ is a broken number and signifies the part or parts of a whole num-
ber." *

A closer examination of this arithmetic discloses many other strange
things. It consists really of three parts, more or less complete in them-
selves, namely: Parti, on whole numbers j Part II, on vulgar fractions;
Part III, on decimal fractions. In Part I, the student is carried through
the elementary rules, and through interest, fellowship, exchange, double
rule of three, alligation, single and double position, geometrical pro-
gression, and permutations. He is carried through all these without
Laving as yet even heard effractions. The advanced and comparatively
unimportant subjects, such as alligation and progressions, are made

16 TEACHING AND HISTORY OF MATHEMATICS.

to precede so important and fundamental a subject as fractions. The
teaching of decimal fractions after interest is illogical, to say the least.
In Part II, after fractions have been explained, the rule of three is taken
up a second time; and in Part III, under decimal fractions, it is re-
sumed a third time.* Thus this rule is explained three times ; the first
time with whole numbers, the second time with common fractious, the
third time with decimal fractions — thus leaving the impression that the
rule is different in each one of the three case^

The whole book is nothing but a Pandora's box of disconnected rules.
It appeals to memory exclusively and completely ignores the existence
of reasoning powers in the mind of the learner. Noticeable is the fact
that in the treatment of common fractions, the process of** cancellation,"
which may be made to shorten operations so much, is not even men-
tioned. The book abounds in unnecessary and perplexing technical
terms, such as "practice,'^ ** conjoined proportion,'' "alligation medial,"
** alligation alternate," '* comparative arithmetic," "biquadrate roots,'^
"sursolids," "square cubes," " second sursolids," "biquadrates squared,"
."third sursolids," and "square cubes squared." Under the head of
duodecimals are given rules like these: " Feet multiplied by feet give
feet ; " " feet multiplied by inches give inches," etc. These rules, taken
literally, are absurd. We can no more multiply feet by feet than we
can multiply umbrellas by umbrellas. These rules are in opposition to
the fundamental ideas of multiplication in arithmetic. A concrete
number can not be multiplied by a concrete number. It seems strange
that so gross an error should not have been corrected in later editions
of the book ; but still more strange is the fact that nearly all arithme-
tics down to the present day should have persisted in making this
mistake.

As an instance of the confusion of ideas to which it gives rise, I quote
the following from an article "Early School Days " in Indiana, con-
tributed by Barnabas 0. Hobbs.* A law had just been passed requir-
ing that teachers' examinations should be conducted by three county
examiners instead of the township trustees, as had been the practice
before. "I shall not forget," says Hobbs, " my first experience under
the new system. The only question asked me at my first examination
was, ^ What is the product of 25 cents by 25 cents t' ♦ ^ • • We
were not as exact then as people are now. We had only Pike's Arith-
metic, which gave the sums and the rules. These were considered
enough at that day. How could I tell the product of 25 cents by 25
cents, when such a problem could not be found in the book ? The ex-
aminer thought it was 6 J cents, but was not sure. I thought just as he
did, but this looked too small to both of us. We discussed its merits
for an hour or more, when he decided that he was sure I was qualified
to teach school, and a first-class certificate was given me."

' The Indiana Schoolsy by Jamos II. bmart| 1876.

COLONIAL TIMES. 17

We have spoken of Dilwortb's School- master's Assistant at some
length, because from it we can see what sort of arithmetics we inherited
from the English. All arithmetics of that time were much alike. The
criticisms upon one will therefore apply to all.

Before proceeding to another subject we shall examine briefly the
** Short Collection of Pleasant and Diverting Questions" in Dilworth.
We shall meet there with a company of familiar friends. Who has not
heard of the farmer, who, having a fox, a goose, and a peck of corn,
and wishing to cross a river, but being able to carry but one at a time,
was confounded as to how he should carry them across so that the fox
should not devour the goose, nor the goose the corn f Who has not heard
of the perplexing problem of how three jealous husbands with their
wives may cross a river in a boat holding only two, so that none of the
three wives shall be found in company of one or two men, unless her hus-
band be present ? Many of us, no doubt, have also been asked to place
the nine digits in a quadrangular form in such a way that any three fig-
ures in a line may make just 15 f When these pleasing problems were
first proposed to us, they came like the morning breeze, with exhilarat-
ing freshness. We little suspected that these apparently now-born
creatures of fancy were in reality of considerable antiquity ; that they
were found in an arithmetic used in this country one hundred years ago.
Still greater is our surprise when we learn that at the time they were
published in Dilworth's School-master's Assistant some of these ques-
tions for amusement had already seen as many as one thousand birth-
days. The oldest record bearing upon this subject is found in a manu-
script entitled Propositiones ad acuendos juveyies. The authorship of
this paper has been generally attributed to Alcuin, whose years of great-
est activity were spent in France, in the court of the great Charlemagne,
and who was one of the most learned scholars and celebrated teachers
of the eighth century. The MS. attributed to him contains the puzzle
about the wolf, goat, and cabbage, which in the modeln version is known
as the " fox, goose, and peck of corn'' puzzle.

In a MS. coming from the thirteenth century, two learned German
youths, named Firri and Tyrri, are made to propose to each other prob-
lems and puzzles. Firri takes among others the hard nut of Alcuin
about the*wolf, goat, and cabbage head, and lays it before Tyrri in the
modified and improved version of the three wives and the three jealous
husbands. This same document contains also the following: "Firri
says : There were three brothers in Cologne, having nine vessels of wine.
The first vessel contained 1 quart (amam), the second 2, the third 3, the
fourth 4, the fifth 5, the sixth C, the seventh 7, the eighth 8, the ninth 0.
Divide the wine equally among the three brothers, without mixing the
contents of the vessels.'^

This problem admits of more than one solution, and is closely related
to the last problem we quoted from Dilworth's collection. It is of spe-
881—^^0. 3 2

TEACHING AND HI8T0BY OF MATHEMATICS.

cial interest, Bince it gives rise to tlie followiiig magic square, in which
any three figures in a straight line have 15 for their sum.

Thehistory of magic squares is a rich field for investigation. The
Germans were by no means the origiuators of them. This honor must
be given to the Brahmins in India. Later on the study of these curi-
ous problems was zealously pursued by the Arabs, who transmitted the
fruits of their study to the Europeans.

Had we the time, we would attempt to trace the history of some
other familiar puzzles. But enough has been said to show that many
of them possess great antiquity. Nevertheless, when they were first
proposed to us, they betrayed no signs of old age. May they continue
perpetually in their youth, and may they delight the minds of men
for numberless centuries to come I

Colleges,
habyabd college.

As early as 1636 the people of Massachusetts stamped their approval
upon the cause of higher education by the founding of Harvard College.
The nature of the early instruction given at this oldest of American
colleges is of special interest to us. The earliest record bearing on the
history of the rise of mathematical studies at Harvard is a tract en-
titledi ''Few England's First Fruits.'^ It was originally published in
1643, or five years after the college had opened, and contained the cur-
riculum of studies then pursued. Whoever expects to find in it an ex-
tended course of mathematical studies resembling that in our colleges
of to-day will be much disappointed.

In the first place, a student applying for admission to Harvard in
1643 was not confronted and embarrassed by any entrance examinations
in mathematics. The main requirement for admission was Latin. Con-
trary to the practice of to-day, Latin was then taught as a spoken lan-
guage. *' So much Latin as was sufficient to understand Tully, or any
like classical author, and to make and speak true Latin in prose or verse,
and so much Greek as was included in declining perfectly the para-
digms of the Greek nouns and verbs,'' «vere the necessary requisites for
admission ; bat in mathematics applicants were required to know not
even the multiplication table.

When we come to examine the college course, which extended origi-
nally through only three years, we meet with other surprises. Boys did

COLONIAL TIMES. 19

not receive that thorough " grinding'' in the elements daring the first
years of college that they do now ; on the contrary, no mathematics at
all was taught except during the last year. The mathematical course
began in the Seilior year, and consisted of arithmetic and geometry
during the first three-quarters of the year, and astronomy daring the
last quarter. Algebra was then an unknown science in the New World.
It is interesting to notice that, in this original curricnlum, the atten-
tion of each class was concentrated for a whole day upon only ouq or
two subjects. Thus, Mondays and Tuesdays were devoted by the third
year students exclusively to mathematics or astronomy, Wednesdays to
Greek, Thursdays to "^Eastern tongues," and so on. The importance
attached to mathematical studies, as compared with other branches of
discipline, may be inferred from the fact that ten hours per week were
devoted tq philosophy, seven to Greek, six to Rhetoric, four to Oriental
languages, but only two to mathematics. According to these figures,
Oriental languages were considered twice as important as mathematics.
But we must remember that this course was laid out for students who
were supposed to choose the clerical profession. For that reason, phil-
osophical, linguistic, and theological studies were allowed to monopolize
nearly the whole time, while mathematics was excluded almost en-
tirely.

In what precedes we have measured the college work done in 1643
by the standards of 1889. Let us now compare it with the contempo-
raneous work in English universities. We may here premise that in the
middle of the seventeenth century rapid progress was made in the
mathematical sciences. In 1643, Galileo had just passed away ; Gav-
alieri, Torricelli, Pascal, Fermat, Boberval, and Descartes were at the
zenith of their scientific activity; John Wallis was a young man of
twenty-seven, Isaac Barrow a youth of thirteen, while Isaac Newton
was an infant feeding from his mother's breast. Though much original
work was being done, especially by French and Italian mathematicians,
the enthusiasm for mathematical study had hardly reached the univer-
sities. Some idea of the state of mathematics at Cambridge, England,
previous to the appearance of Kewton, may be gathered from a dis-
course by Isaac Barrow, delivered in Latin, probably in 1C54, or
eighteen years after the founding of Harvard College. In it occurs the
following passage: ''The once horrid names of Euclid, Archimedes,
Ptolemy, and Diophantus, many of us no longer hear with trembling
ears. Why should I mention the fact that by the aid of arithmetic, we
have now learned, with easy and instantaneous work, to compute ac-
curately the number of the very sands (themselves). * • • And in-
deed that horrible monster that men call algebra many of us brave men
(that we are) have oviercome, put to flight, and (fairly) triumphed over;
(while) very many (of us) have dared, with straight-along glance, to
look into optics; and others (still), with intellectual rays unbroken,
have dared to pierce (their way) into the still subtler and highly useful
doctrine of dioptrics,'^

20 TEACHING AND HISTORY OF MATHEMATICS.

From this it would seem that mathematical studies had been intro-
duced into old Cambridge only a short tim'e before Barrow delivered
his speech. It thus appears that about 1636, when new Cambridge was
founded in the wilds of the west, old Cambridge waanot mathematical
at all. In further support of this view we quote from the Penny Gy-
clopaedia, article " Wallis," the following statement: ''There were no
mathematical studies at that time [when Wallis entered Emmanuel
College in 1632] at Cambridge, and none to give even so much as advice
what books to read. The best mathematicians were in Loudon, and the
science was esteemed no better than mechanical. This account is con.
firmed by his [Wallis's] contemporary, Horrocks, who was also at Em-
manuel and whose works Wallis afterwards editfed." In a biography of
Seth Ward, an English divine and astronomer, we meet with similar
testimony.* He entered Sidney Sussex College, Cambridge, in 1632.
'' In the college library he found, by chance, some books that treated
of the mathematics, and they being wholly new to him, he inquired all
the college over for.a guide to instruct him that way, but all his search
was in vain ; these books were Greek, I mean unintelligible, to all the
fellows in the college."

If so little was done at old Cambridge, then we neednot wonder at
the fact that new Cambridge failed to be mathematical from the start.
The fountain could not rise higher than its source. It was not until the
latter half of the seventeenth century that mathematical studies at
old Cambridge rose into prominence. Impelled by the genius of Sir
Isaac Newton, old Cambridge advanced with such rapid strides that
the youthful college in the west became almost invisible in the distant
rear.

The mathematical course at Harvardremained apparently the same
till the beginning of the eighteenth century. Arithmetic and a little
geometry and astronomy constituted the sum total of the college in-
struction in the exact sciences. Applicants for the master's degree
had only to go over the same ground more thoroughly. Says Cotton
Mather: '* Every scholar that giveth up in writing a system or synop-
sis or sum of logic, natural and moral philosophy, arithmetic, geometry,
and astronomy, and is ready to defend his theses or positions, withal
skilled in the originals, as above said, and of godly life * • * is fit
to be dignified with the second degree."!

These few unsatisfactory data are the only fragments of information
that we could find on the mathematical course at Harvard during the
seventeenth century. The following note on the nature of the instruc-
tion given in physics is not without interest: Mr. Abraham Pierson,
jr. (first rector of Yale College), graduated at Harvard in 16G8. The
college (Yale) possesses several of his MSS., '* containing notes made by

• Lif« of Ri^bt Reverend Seth, Lord Bishop of Salisbury, by Walter Pope. Loo-
don, 1G97, p. 9.
t Magnalia, Book IV, 128th ed., 1702.

COLONIAL TIMES. 21

him daring his student life at Harvard on logic, theology, and physics,
and BO throwing light on the probable compass of the manuscript text*
book on physics compiled by him, which was handed down from one
college generation' to another for some twenty-five years, until super-
seded by Clarke's- Latin translation of Eohault's Trait6 de Physique.
The Harvard notes on physics seem (from an inscription attaQheil) to
to have been derived in like manner from the teachings of the Kev.
Jonathan Mitchel (HaFvard College, 1647); they are rather metaphysi-
cal than mathematical in form, and it is even difficult to determine
what theories of physical astronomy the writer held. Suffice it to say
that he ranged himself somewhere in the wide interval between the
Ptolemaic theory (generally abandoned one hundred years earlier) and
the Newtonian theory (hardly known to any one in this part of the world
until the eighteenth century). In other words, while recognizing that
the earth is round, and that there is such a force as gravity, there is no
proof that he had got beyond Copernicus to Kepler and Galileo.'' *

In this extract our attention is also called to the common practice
among successive generations of students at that time of copying manu-
script text books. As another instance of this we mention the manu-
script works, a System of liogic and a Compendium Physical, by Eev.
Charles Morton, which (about 1692) were received as text-books at
Harvard, " the students being required to copy them."t

We shall frequently have occasion to observe that astronomical pur-
suits have always been followed with zeal and held in high estimation
by the American people. As early as 1651 a !N"ew England writer, in
naiming the ** first fruits of the college,'' speaks of the " godly Mr. Sam
Danforth, who hath not only studied divinity, but also astronomy ; he
put forth man}' almanacs," and " was one of the fellows of the college."
Another fellow of Harvard was John Sherman. He was a popular
preacher, an " eminent mathematician," and delivered lectures at the
college for many years. Ho published several almanacs, to which he
appended pious retiections. The ability of making almanacs was then
considered proof of profound erudition. A somewhat stronger evidence
of the interest taken in astronomy was the publication at Cambridge
of a set of astronomical calculations by Uriah Oakes. Cakes, at that
time a young man, had graduated at Harvard in 1049, and in 16S0 bo-
came president pro tern, and afterwards president of Harvard College.
In allusion to his size, he attached to his calculations the motto,
^^Parvumparva decent^ sed ijiest sua gratia parvis.'^^ (Small things befit
the small, yet have a chara:i their own.)

The preceding is an account of the mathematical and physical studies
at Harvard during the seventeenth century. We now proceed to the
eighteenth century. It appears that in 1700 algebra had not yet be-

♦ Yale Biographies and Annals, 1701-1745, by Franklin Bowditch Dexter, p. 6L
tQuincy's History of Harvard Universityi Vol. 1, p. 70.

22 TEACHING AND HISTORY OP MATHEMATICS.

come a college study. The Autobiography of Rev. John Barnard ♦
throws some light oa this subject. Barnard took his first degree at
Harvard in 170Q, then returned to his father's house, where he betook
himself to studying. " While I continued at my father's I prosecuted
my studies and looked something into the mathematics, though I gained
but little, our advantages therefor being noways equal to what they
have who now have the great Sir Isaac Newton and Dr. Halley and
sotue other mathematicians for their guides. About this time I made
a visit to the college, as I generally did once or twice a year, where I
remember the conversation turning upon the mathematics, one of the
company, who was a considerable proficient in them, observing my ig-
norance, said to me he would give me a question, which if I answered
in a month's close application he should account me an apt scholsur.
He gave me the question. I, who was ashamed of the reproach cast
upon me, set myself hard to work, and in a fortnight's time returned
him a solution of the question, both by trigonometry and geometry,
with a canon by which to resolve all questions of the like nature.
When I showed it to him he was surprised, said it was right, and
owned he knew no other \ray of resolving it but by algebra, which I
was an utter stranger to." Though a graduate of Harvard, he was an
utter stranger to algebra. From this we may safely conclude that in
1700 algebra was not yet a part of the college curriculum.

What, then, constituted the mathematical instruction at that time f
Was it any different from the course given in 1643 ? Until about 1655,
the entire college course extended through only three years; at this
time it was lengthened to four years. We might have supposed that
the mathematics formerly taught in the third year would have been
retained as a study for the third or Junior year, but this was not the
case. In the four-years' course, mathematics was taught during the
last, or Senior year. Quincy, in his history of Harvard University (Vol.
I, p. 441), quotes from Wad worth's Diary the list of studies for the year
1726. The Freshmen recited in TuUy, Virgil, Greek testament, rheto^
ric, Greek catechism; the Sophomores in logic, natural philosophy,
classic authors, Heerebord's Meletemata, Wollebins's Divinity; the
" Junior sophisters" in Heerebord's Meletemata, physics,' ethics, geogra-
phy, metaphysics ; while the ^' Senior sophisters, besides arithmetic,
recite Alsted's Geometry, Gassendi's Astronomy in the morning; go
over ihe arts towards the latter end of the year, Ames's Medulla on
Saturdays, and dispute once a week. " This quotation establishes the
fact that ninety years after the founding of Harvard, the mathematical
course was essentially the same as at the beginning. Arithmetic,
geometry, and astronomy still constituted the entire course. Mathe-
matics continued to be considered the crowning pinnacle instead of a
corner-stone of college education ; natural philosophy and physics were

* CoUecUons of the Maas. Hist. Soo., Third Series, Vol. V, pp. 177-243.

COLONIAL TIMES, 23

still taught before aritlitnetio and geometry. Bat we must observe that,
inl726,jt?ri/ite^ treatises were used as text-books in geometry and astron-
omy. We are not informed at what time these printed books were in-
troduced. They may. have been used as text-books much earlier than
the above date. The authors of these books were in their day scholars
of wide reputation. Johann Heinrich Alsted (1558-163S), the author of
the Geometry, was a German Protestant divine, a professor of philoso-
phy and divinity at Herborn in Nassau, and afterwards in Carlsburg in
Transylvania. In one of his books he maintained that the millenium
was to come in 1694.

Pierre Gassendi (1592-1655), whose little astronomy of one hundred and
fifty pages* was used as a class-book at Harvard, was a contemporary of
Descartes and one of the most distinguished naturalists, mathematicians,
and philosophers of France. He was for a time professor of mathe-
matics at the Oollfege Eoyal of Paris. What seems very strange to us
is that nearly a century after the first publication of these books they
should have been still in use and apparently looked upon as the best of
their kind. Forty years after the publication of Newton's Principia
an astronomy was being studied at Harvard whose author died before
the name of Newton had become known to science. The wide chasm
between the theories of Newton and those of Gassendi is brought to full
view by the following quotation from WhewelPs History of the Induc-
tive Sciences (Third edition, VoL I, p. 392) : " Gassendi's own views of
the causes of the motions of the heavenly bodies are not very clear.
• • * In a chapter headed 'Quae sit motrix siderum causa,' he
reviews several opinions ; but the one which he seems to adopti is that
which ascribes the motion of the celestial globes to certain fibers,
of which the action is similar to that of the muscles of animals. It
does not appear therefore that he had distinctly apprehended, either
the continuation of the movements of the planets by the first law of
motion, or their deflection by the second law."

The year 1726 is memorable in the annals of Harvard for the estab-
lishing of the Hollis professorship of mathematics. Thomas HoUis,
a kind-hearted friend of the college, transmitted to the treasurer of the
coUege the then munificent sum of twelve hundred pounds sterling,
and direcijed that the funds should be applied to " the instituting and
settling a professor of mathematics and experimental philosophy in
Harvard College." To the same benefactor Harvard was indebted
for the establishment of the professorship of divinity, Down to the
commencement of the nineteenth century only one additional professor
was appointed in the undergraduate department, namely, the Hancock
professor of Hebrew, in 1765. Hence, it follows that almost all regular
instruction was given by tutors. Previous to the establishment of the
Hollis professorship the mathematical instruction was entirely in the
hands of tutors. Since almost any minister was considered competent
to teach mathematics, and since tutors held their place sometimes for

24 TEACHING AND HISTOEY OF MATHEMATICS.

only one year, we may imagine that the teaching was not of a very high
order.

The first appoiRtment to the Hollis pj^ofessorship of mathematics and
natural philosophy was that of Isaac Greenwood. He was the first to
occapy a collegiate chair of mathematics in New England, but not the
first in America, as is sometimes stated. This honor belongs to a pro-
fessor at William and Mary Oollege. Oreenwood graduated at Harvard
in 1721, then engaged in the study of divinity, visited England, ^and be-
gan to preach in London with some approbation.* He also attended
lectures delivered in that metropolis on experimental philosophy and
mathematics. In 1727 he entered upon his duties at Harvard. ^' In
scientific attainments Greenwood seems to have been well qualified for
his professorship." He made astronomical contributions to the Philo-
sophical Transactions of 1728, and published in 1729 an arithmetic.
That seems to have been the earliest arithmetic from the pen of an
American author. This is all we know of Greenwood as a mathema-
tician and teacher. Unfortunately he did not prove himself worthy of
his place. We regret to say that the earliest professor of mathematics
in the oldest American college was *' guilty of many acts of gross iq-
temperance, to the dishonor of God and the great hurt and reproach of
the society." His intemperance brought about his removal from his
chair in 1738.

On the dismissal of Greenwood, Nathaniel Prince, who had been tutor
for thirteen years, aspired to the professorship. He was, says Elliot,
superior " to any man in New En£:land in mathematics and natural
philosophy." But his habits being notoriously irregular, John Win-
throp of Boston, was appointed in his stead. Winthrop graduated at
Harvard in 1732, and was only twenty-six years old when he was chosen
professor of mathematics and natural philosophy. He filled this chair
for over forty years (until 1779) with marked ability. In mathematical
science he came to be regarded by many the first in America.

If we could turn the wheel of time backward through one hundred
and twenty revolutions, and then enter the lecturo-room of Professor
Winthrop and listen to his instruction, what a chapter in the his-
tory of mathematical teaching would be uncovered I But as it is, this
history is hidden from us. We know only that during the early part
of his career as professor, " and probably many years before," the text-
books were the following : Ward's Mathematics, Gravesande^s Philos-
ophy, and Euclid's Geometry ; besides this, lectures were delivered by
the professors of divinity and mathematics.!

From this we see that some time between the years 1726 and 173S,
Ward's Mathematics had been introduced, and Alsted's old Geometry
had given place to the still older but ever standard work of Euclid.
This is the first mention of Euclid as a text-book at Harvard. The in-

* Qaiaoy's History of Haryftrd Uniyenity, Vol. II, p. 14.
t Peirce's History of Harvard, p. S37.

COliONUL TIMES. 25

troduction of Graveaande's Philosophy is another indication of progress.
Gravesaude was for a time professor of mathematics and astronomy at
the University of Leyden. He was the first who on the continent of
Europe publicly taught the philosophy of Newton, and he thus con-
tributed to bring about a revolution in the physical sciences. By the
adoption of his philosophy as a text-book at Harvard we see that the
teachings of Newton had at last secured a firm footing there. Ward's
Mathematics continued for a long time to be a favorite text-book.*

It is probable that with the introduction of Ward's Mathematics, alge-
bra began to be studied at Harvard. The second part of the Young
Mathematiciao's Guide consists of a rudimentary treatise on this subject.
It is possible^ then, that the teaching of algebra at Cambridge may have
begun some time between 1726 and 1738. But I have found no direct
evidence to show that algebra actually was in the college curriculum
previous to 1786.

Since Ward's Mathematics were used, to our knowledge, not only at
Harvard, but also at Yale, Brown, and Dartmouth, and as a book of
reference at the University of Pennsylvania, a description of the Young
Mathematician's Guide may not be out of placet

The first part treats of arithmetic (143 pages). Though very deficient
according to modern notions, the presentation of this subject is superior
to that in Dilworth's School-master^s Assistant. It is less obscure.

' According to ex- President D. Woolsey, the author of this book was the Ward
who had been " president of Trinity College, Cambridge, and bishop of Exetej*/'
(Yale College ; A Sketch of its History, William L. Kingsley, Vol. II, p. 499.) Now, '
the only individaal answering to this description is Seth Ward, the astronomer,
whose time of activity preceded the epoch of Newton. We shall show that the book
in question was not written by Seth Ward, but by John Ward, who flourished half
a century later than Seth Ward and whose Young Mathematician's Guide was for a •
long time a popular elementary text-book in England. Wherever we have seen
Ward's book mentioned in the curricula of American colleges it was always called
"Ward's Mathematics." The baptismal name of the author was never given. Ttis
shows that there was only one Ward (either Seth or John) whose mathematical books
were known and used in our colleges. Now, Benjamin West, professor of mathematics
in Brown University from 1786 to 1799, published in the first volume of the American
Academy of Arts and Sciences a paper *^ On the extraction of roots," in which he
offers improvements on "Ward's" method. Now, I have seen a copy of Seth Ward's
Astronomia Geometrica, but have found nothing in it on root extraction. One would
hardly expect to find anything on it in Seth's "Trigonometry" or "Proportion."
John Ward, on the other hand, treats of roots in his "Guide," and gives a "general
method of extracting roots of all single powers." West takes two examples (two
numbers, one of 14, the other of 18 digits) from "Ward," and shows how the reauired
roots can be extracted by his method. But both these examples are given in John
Ward's Young Mathematician's Guide. This evidence in favor of John Ward's book
maybe considered conclusive. Further information on "Ward's Mathematics" will
be found in an article by the writer in the Papers of the Colorado College Scientific
Society, Vol. I.

tThe copy which the writer has before him (Twelfth edition, London, 1771), was
kindly lent him by Dr. Artemas Martin, of the U. S. Coast Survey, who has for years
been making a collection of old axld rare books on mathematios.

26 . V TEACHING AND HISTOEY OF MATHEMATICS.

Like all books of that time, it contains rules, but no reasoning. What
seems strange to us is the fact that subjects of no value to the begin-
ner, such as arithmetical and geometrical proportion (i. a., progression),
alligation, square root, cube root, biquadrate root, sursolid root, etc.,
are given almost as much space and attention as common and decimal
fractions.

The second part (140 pages) is devoted to algebra. Ward had pub-
lished a small book on algebra in 1698, but that, he says, was only " a
compendium of that which is here fully handled at large." Like Har-
riot, he speaks of his algebra as "Arithmetick in species.'' This name
is appropriate, inasmuch as he does not (at least at the beginning) rec-
ognize the existence of negative quantities, but speaks of the mintis sign
always as meaning only subtraction, as in arithmetic. A little further
on, however, Ije brings in, by stealth, "affirmative" and " negative"
quantities. The knowledge of algebra to be gotten from this book is
exceedingly meagre. Factoring is not touched upon. The rule of signs
in multiplication is proved, but further on all rules are given without
proof. The author develops a rule showing how binomials can be
raised "to what height you please without the trouble of continued in-
volution." He then says : " I proposed this method of raising powers
in my Compendium of Algebra, p. 57, as wholly new (viz, as much of it
as was then useful), having then (I profess) neither seen the way of
doing it, nor so much as heard of its being done. But since the writing
of that tract, I find in Doctor Wallis's History of Algebra, pp. 319 and
331, that the learned Sir Isaac Newton had discovered it long before.''
The subject of " interest " is taught in the book algebraically, by the use
of equations.

Part III (78 pages) treats of geometry. In point of precision and
scientific rigor, this is quite inferior. After the definitions follow
twenty problems, intended for the excellent purpose of exercising the
<< young practitioner," and bringing <' his hand to the right manage-
ment of a ruler and compass, wherein I would advise him to be very
ready and exact." Then follows a collection of twenty-four << most use-
ful theorems in plane geometry demonstrated." This part is semi-em-
pirical and semi-demonstrative. A few theorems are assumed and the
rest proved by means of these. The theorem, " If a right line cut two
parallel lines, it will make the opposite (i. 6., alternate interior) angles
equal to each other," is proved by aid of the theorem, that " If two
lines intersect each other, the opposite angles will be equal." The proof
is based on the idea that <' parallel lines are, as it were, but oue broad
line," and that by moving one parallel toward the other, the figure for
the former theorem reduces to that of the latter. The next chapter
contains the algebraical solution of twenty geometrical problems.

Part IV, on conic sections (36 pages), gives a semi-empirical treat-
ment of the subject. Starting with the definition of a cone, it shows
how the three sections are obtained from it, and then gives some of
their principal properties.

COLONIAL TIMES. 27

Part V (36 pages) is on the arithmetic of infinites. Judging from
this part of the book, its author knew nothing Of fluxions. The first
editiou appeared in 1707, after Newton had published the first edition
of his Principia, in 1687, but his Method of Fluxions was not published
till 1736, though written in 1671. Ward employs the method of inte-
gration bj series of Oavalieri, Eoberval, and John Wallis, and, thereby, "'
finds the superficial and solid contents of solid figures. It does not ap-
pear that this part of the book was ever studied in American colleges.

Ward^s book met with favor in England. In the preface to the
twelfth edition he says: "I believe I may truly say (without vanity)
this treatise hath proved a very helpful guide to near five thousand per-
sons, • • • and not only so, but it hath been very well received
amongst the learned, and (I have been often told) so well approved
on at the universities, in England, Scotland, and Ireland, that it is
ordered to be publicly read to their pupils."

In former times all professors of mathematics in American colleges
gave instruction, not merely in pure mathematics, but also in natural
philosophy and astronomy ; and it appears that as a general rule these
professors took more real interest and made more frequent attempts at
original research in the fields of astronomy and natural philosophy than
in pure mathematics. The main reason for this lies probably in the
fact that the study of pure mathematics met with no appreciation and
encouragement. Original work in abstract mathematics would have
been looked upon as useless speculations of idle dreamers. The scien-
tific activity of John Winthrop was directed principally to astronomy.
His reputation abroad as a scientist was due to his work in that line.
In 1740 he made observations on the transit of Mercury, which were
printed in the Transactions of the Koyal Society. In 1761 there was a
transit of Venus over the sun's disk, and as Newfoundland was the most
western part of the earth whert> the end of the transit could be ob-
served, the "province'' sloop was fitted out a* the public expense to con-
vey Winthrop and party to the place of observation.* He took with
him two pupils who had made progress in mathematical studies. One
of these, Samuel Williams, became later his successor at Harvard. In
1769 Winthrop had another chance for observing the transit of Venus,
at Cambridge. "As it was the last opportunity that generation could
be favored with, he was desirous to arrest the attention of the peo-
ple. He read two lectures upon the subject in the college chapel,
which the students requested him to publish. The professor put this
motto upon the title page: Agite, mortales! et oculos in spectaculum ver-
tite, quod hucusque spectaverunt perpaucissimi ; spectaturi iterum sunt
mdli.^ (Come, mortals! and turn your eyes upon a sight which, to this
day, but few have seen, and which not one of us will ever see again.)
The transit of 1769 was also observed in Philadelphia by David Rit-
teuhouse, and in Providence by Benjamin West. These observations

I j„, _ , . r-i---f ■■! T-llll--- ■ 1 "BT— n TT^

• "John Winthrop," in the Biographical Dictionary by John Eliot, 1809.

28 TEACHING AND HISTORY OP MATHEMATICS.

were an important aid in determining the sun's parallax. Most grati-
fying to us is the interest in astronomical pursuits manifested in those
early times. Expeditions fitted out at public expense, and private mu-
nificence in the purchase of suitable instruments, bear honorable testi-
mony to the enlightened zeal which animated the friends of science.

In 1767 John Winthrop wrote his Cogita de Cometis, which he dedi-
cated to the Eoyal Society, of which he had been elected a member.
This was reprinted in London the next year, and gave him an extensive
literary reputation.

In 1764 a calamity befell Harvard College. The library and philo-
sophical apparatus — the collections of over a century — were destroyed
by fire. Among the books recorded as having been lost are the follow-
ing : " The Transactions of the Eoyal Society, Academy of Sciences in
France, Acta Eruditorum, Miscellanea Curiosa, the works of Boyle
and Kewton, with a great variety of other mathematical and philo-
sophical treatises."* It is seen from this that, before the fire, books of
referenceiin higher mathematics had not been entirely wanting.

John Winthrop died in 1779, and the robe of the departing prophet
fell upon his former disciple, the Eev. Samuel Williams. Williams filled
the mathematical chair for eight years. Having inherited from his mas-
ter a love of astronomy, he frequently published observations and no-
tices of extraordinary natural phenomena in the memoirs of the Ameri-
can Academy of Arts and Sciences. He occupied the mathematical chair
at Harvard until 1788. Then he lectured at the University of Vermont
on astronomy and natural philosophy for two years, and was subse-
quently minister at Eutland and Burlington, Vermont.

YALE OOLLEaE.

Yale, the second oldest New England college, was founded in 1701,
or sixty-three years after the opening of Harvard. During the first
fifteen years it maintained a sortof nomadic existence. Previous to 1816
instruction seems to have been given partly at Saybrook and partly at
Killingworth and Milford. Its course of instruction was then very
limited. The mathematical teaching during the first years of its exist-
ence was even more scanty than in the early years at Harvard. Benja-
min Lord, a Yale graduate of 1714, wrote in 1779 as follows in reply to
inquiries by President Stiles: "As for mathematics, we recited and
studied but little more than the rudiments of it, some of the plainest
things in it. Our advantages in that way were too low for any to rise
high in any branch of literature." t Doctor Johnson, of the same class,
says : " Oommon Arithmetick and a little surveying were the ne phis
ultra of mathematical acquirements." It appears from this that sur-
veying was taken sit Yale, instead of the geometry which formed part

*Vi<le Quincy's History of Harvard UniverBity, Vol. II, p. 481.
tYalc Biographiea aud AnnalB, 1701-45| by Franklin BowditcU Dexter, pp. 115
and 116.

COLONIAL TIMES. 29

of the course at Harvard. In a new and only partially settled country
some knowledge of surveying was a great desideratum. But the study
of surveying without a preliminary course in geometry and trigonometry
is truly characteristic of the purely practical teadencies-of the times.
Men took eager interest in the applications of science, but cared nothing
for science itself. The little mathematics studied was evidently not
pursued for its own sake, nor for the mental discipline which it afforded,
but simply for the pecuniary profit which it would afterwards bring.

As at Harvard, so at Yale, the mathematics were studied, at that
time, during the last year of the college course and after the study of
physics had been completed.* During the next six or seven years, the
course at Yale was extended somewhat. In 1720 it was identical with
the Harvard course of 1726. In 1719, when Jonathan Edwards was a
member of the Junior class at New Haven, he wrote as follows to his
father: " I have enquired of Mr. Cutter, what books we shall have need
of the next year. He answered he would have me get against that time,
Alsteds' Geometry and Gasseudi's Astronomy." t

At this time progress was also made in the teaching of physics. The
earliest guide in this study were the manuscript lectures by Eector
Pierson, which were a repetition of lectures he had heard while a
student at Harvard College. They were metaphysical rather than
mathematical, ^^ recognizing the Copernican theory, but knowing nothing
of Kepler and Galileo, and much less of Newton.'^t

During the first seventeen years at Yale the doctrines of the school-
men in logic, metaphysics, and ethics still held sway. Descartes,
Boyle, Locke, Bacon, and Newton were regarded as innovators from
whom no good could be expected. It is pleasing to think that the in.'
troduction of Newtonian ideas and the rise of mathematical studies at
Yale was partly due to an act of charity by the great Sir Isaac Newton
himself. In the year 1715 a collection of books made in Englandby
Mr. Drummer, the agent of the colony, amounting to about eight
hundred volumes, was sent over to the college. The collection con-
sisted of donations by well-spirited gentlemen in Britain. " Sir Isaac
Newton gives the second edition of his Principia (which appeared in
1713) " ; '' Doctor Halley sends his edition of Apollonius.''§ But these
and many other donations would have been barren of results had there
not been young men of talent and energy to master the cpntents of
these precious volumes. Such a man was Samuel Johnson. He gradu-
ated in 1714 and was appointed tutor a few years later. Drummer's
collection furnished him with a '^ feast of fat things." To use his own
words: *^He seemed to himself like a person suddenly emerging out of
the glimmer of twilight into the full sunshine of open day." He and

* Yale College ; a sketch of its history, William L. Kingsley, Vol. II, p. 496.

t Edwards' Works, Vol. I, p. 30.

X Ey-President D. Woolsey, in Yale Book, Vol. II, y. 499.

(Yale Bio^aphies and Annals, 1701-45; by Franklin Bowditoh Dexter, p. 141.

30 TEACHING AND HISTORY OP MATHEMATICS.

Mr. Brown, another young tutor, exerted themselves to the utmost for
the improvement of the students under their charge. Imbued with the
grand ideas of Newton, they extended the mathematical course for the
understanding of the Newtonito system, and then taught this system
in place of the older. There was at that time much contention as to
the place where the college should be permanently located. This was
a fortunate circumstance for the young tutors, since these troubles
without withdrew public attention from the innovations within.* In
1722 Johnson and Brown resigned their tutorships and sailed for Eng-
land to receive ordination from an English bishop. Johnson became
later president of King's (now Columbia) College in New York.

Soon after this the Physics of Eohault was introduced at Yale as a
text-book. Eohault (1620-75) was a French philosopher and an im-
plicit follower of the Cartesian theory. The edition used was that by
the celebrated Samuel Clarke, who had taken the rugged Latin version
of the treatise of Eohault (then used as a text-book at the University
of Cambridge, England), and published it in better Latin, together
with numerous critical notes, which he had added with a view of
bringing the Cartesian system into disrepute by exposing its fallacies.
This disguised Newtonian treatise maintained its place at Yale until
1743, when it was superseded by the work of Gravesande.

During President Clap's time, Martin's Philosophy, inthrcQ volumes,
was the text-book in this science; when this work came to be out of
print. President Stiles procured Enfield's Philosophy, which was the
first introduction into American colleges of that now obsolete work.

It is worthy of remark that, in 1749, Benjamin Franklin presented to
the college an electric machipe, and that, a few years later, Ezra Stiles,
then tutor at the college, began to make experiments with it. These
are supposed to have been the earliest of the kind made in New England.

It appears that in 1733, Euclid was being used as a text-book in
geometry. The earliest mention of Euclid at Harvard is in 1737. In
1733, Dr. John Hubbard of New Haven, who had received the honorary
degree of master of arts three years previously, testified his gratitude
by writing a i)anegyric, "The Benefactors of Yale College."

He introduced a recent gift of mathematical books by Joseph Thomp-
son, of London, with the following stanza:

"The Mathematicks too oar thoHs employ,
Which nobly elevate the Student's joy :
The little Enclids round the table sot
And at their rigid demonstrations sweat.'' t

This same Joseph Thompson donated to the college also "a complete
set of surveying instruments, valued at £21." *' A reflecting telescope,
a microscope, a barometer, and other mathematical instruments — valued

at £37, were bought by a subscription from the trustees and other8."{

'■- " '«■■■ ■ ' ■ - — .1 , .., . . I . . , - , ■

• Barnard's Journal. Vol. XXVII, 1877 ; Article ; '* Samuel Johnson.'*
t Yale Biographies and Aunals, 1701-45, by F. B. I^exter, p. 473.
I Ibid.f p. 5Sil.

COLONIAL TIMES. 31

■

In 1742, elementary mathemaitics came to be removed from its angast
position in the corriculnm as a senior study, and to be assigned an
humbler but more befitting place nearer the beginning of the coarse.
In 1742 the rector of the college advised the students to pursue a regular
coarse of academic studies in the following order : '^ In the first year to
study principally the tongues, arithmetic, and algebra; the second,
lo^c, rhetoric, and geometry; the third, mathematics, and natural
philosophy ; and the fourth, ethics and divinity.'^*

That these changes were not made earlier than 1742 is evident from
a passage in the memoir of Samuel Hopkins, who graduated in 1741,
stating that then <^ metaphysics and mathematics found their place in
in the fourth year, being in their turn the subject of study and recita-
tion for the first four days of every week.^t

At what time this dethronement of elementary mathematics as a
senior study took place at Harvard, we are not able to state. It will
be noticed that, at Yale, mathematics and natural philosophy had at
this period exchanged places, the former now preceding the latter.
From the above it is also evident that algebra was studied at Yale in
1742. The earliest mention of algebra at Oambridge is in 1786, though
it doubtless began to be taught there much earlier. What branch of
matheiAatics constituted the study for the third or Junior year remains
a matter of conjecture. The << mathematics " spoken, of in the extract
probably referred to trigonometry, possibly together with some other
branches.

A strong impetus to the study of mathematics at Yale was given
during President Olap's administration. Thomas Clap graduated at
Harvard in 1722. Doctor Stiles, his successor in the presidency at
Yale, says that Clap studied the higher branches of mathematics, and
was one of the first philosophers America has produced, ^^ that he was
equalled by no man, except the most learned Professor Winthrop.'' In
his history of Yale, written in 1766, the year of his resignation. President
Clap gives the following account of the studies pursued by students at
the college :

*^ In the first year they learn Hebrew, and principally pursue the
study of the languages, and make a beginning in logic and some parts
of the mathematics. In the second year they study the languages, but
principally recite logic, rhetoric, oratory, geography, and natural phi-
losophy ; and some of them make good proficiency in trigonometry and
algebra. In the third year they will pursue the study of natural phi-
losophy and most branches of mathematics. Many of them well under-
stand surveying, navigation, and the calculation of eclipses ; and some
of them are considerable proficients in conic sections and fluxions. In
the fourth year they principally study and recite metaphysics, ethics,
and divinity .''J

*Yale Biographies and Annals, 1701-45, p. 724*

tNew Englander, August, 1852, p. 452: Professor Park's Memoir of Hopkins.

X Yale College ; a Sketch of its History, by Win. L. Kingsley, Vol. II, pp. 497 and 498.

82 TEACHING AND HISTORY OP MATHEMATICS.

The mathematicar coarse in the above cnrricnlilm is indeed one that
Tale had reason to be prond of. It shows that not only algebra and
geometry, but also trigonometry, and even conic sections and flexions,
were studied at Yale previous to the year 1766. This is the earliest
distinct mention of conic sections and fluxions as college studies in
America.

Mathematics seem to have come to occupy some of the time which
was given at first to logic. President Clap does not enumerate the
text-books employed, but his successor, Doctor Stiles, in his diary for
November 9, 1779, mentions a list of books recited in the several classes
at his accession to the presidency, in 1777. The mathematical books
are, for the Freshman class. Ward's Arithmetic ; Sophomore class, Ham-
mond's Algebra, Ward's Geometry (Saturday), Ward's Mathematics ;
Junior class, Ward's Trigonometry, Atkinson and Wilson's Trigonom-
etry.

On comparing this mathematical course with that given by President
Clap eleven years previous we observe some changes. The study of
conic sections and fluxions had been apparently discontinued. This
waning of mathematical enthusiasm was probably due to the departure
of President Glap, and also to the political disturbances and confusions
of the times. It would seem that duriug Clap's administration not all
the students took higher mathematics, but only those who were partic-
ularly fond of them. Clap says, ''■ Many of them well understand sur-
veying, navigation, and the calculation of eclipses ; and some of them
are considerable proficients in the fionic sections and fluxions."

That optional studies were then pursued occasionally is evident from
a statement by President Stiles that he began instructing a class in He-
brew and Oriental languages, which he " selected out of all other
classes, as they voluntarily offered themselves." The extent to which
each of these branches was studied may probably be correctly inferred
from the contents of Ward's Young Mathematician's Guide. This con-
sists of five parts : arithmetic, algebra, geometry, conic sections, and
arithmetic of infinites. Students that were mathematically inclined
went through the entire work it would seem, excepting the algebra,
which was studied from Hammond's book.

The year 1770 is memorable for the creation of the chair of " mathe-
matics and natural philosophy" at Yale. This was done apparently to
fill the gap caused by the departure of President Clap, who was uncom-
monly skilled in those sciences. The first occupant of this chair was
Nehemiah Strong, who kept it eleven years. He belonged to the class
of 1755 at Yale, and was tutor there from 1757 to '60. Before entering
upon the duties of his chair, he had been pastor. After his resigna-
tion of his chair, he entered upon the study and practice of law. He
published an "Astronomy Improved " (New Haven, 1784). President
T. Dwight speaks of him as " a man of vigorous understanding."

COLONIAL TIMES. 33

WILLIAM AND MABY OOLLEOB.

William and Mary is next to Harvard the oldest of American col-
leges. From 1688, the year of its organization at Williamsburg, Ya.,
until the inauguration of the University of Virginia, it was the leading
educational institution in the South. Owing to the repeated destruc-
tion by fire of the college buildings and records, not even the succes-
sion of the professors has been preserved. The early courses were in
all probability much the same as the contemporaneous courses at Haf-
vard. According to Campbell, 5 professorships were provided for by
the charter, namely, those of Greek and Latin, mathematics, mor^
philosophy, and two of divinity. In speaking of the early course of
study, Howison says that it embraced also a '< natural philosophy
which was just beginning to believe that the earth revolved round the
sun, rather than the sun round the earth."

The earliest mathematical professor at William and Mary whose name
has come down to us, was Bev. Hugh Jones. The college had a pro-
fessorship of mathematics from its very beginning, and at a date when
mathematical teaching at Harvard was still in the hands only of tutors.
The names of the predecessor or predecessors of Hugh Jones are not
known. He is the earliest jpro/a^sor of mathematics in America whose
name has been handed down to us. He ^as an Englishman of univer-
sity education ; came to Maryland in 1696 ; was for a time pastor of
a church; and then was appointed to the chair of mathematics at
William and Mary. He was a man of broad, scholarly attainments, and
endeared himself to the student of history quite as much as to the
mathematician, by writing his invaluable book on The Present State of
Virginia (1724). Says Dr. Herbert B. Adams : " His monograph is
acknowledged to be one of the best sources of information respecting
Virginia in the early part of the eighteenth century." The following
quotations from it (p. 44) may be of interest: "They (the Virginians)
are more inclinable to read men by business and conversation than to
dive into books, and are for the most part only desirous of learning
what is absolutely necessary in the shortest and best method."

" Having this knowledge of their capacities and inclination from suf-
ficient experience, I have composed on purpose some short treatises
adapted with my best judgment to a course of education for the gentle-
men of the plantations, consisting In a short English Orammarj an Ac-
cidence of Christianity, an Accidence to the Mathematick in aU its parts
and applications. Algebra^ Oeometry, Surveying of Land^ and Naviga-
iioiiJ^

" These are the most useful branches of learning for themj and such
as they willingly and readily master, if taught in a plain and short
method, truly applicable to their genius ; which I have endeavored to
ilo, for the use of them and all others of their temper and parts."

We are not to understand by the above that his "Accidence to the
881— No. 3 3

34 TEACHING AND HISTORY OF MATHEMATICS.

Matheiuatick " and the other books mentioned were actually printed ;
they existed only in manuscript copies. From the above it appears that
about 1724 the mathematical course at William and Mary was quite
equal to that in either of the two New England colleges. We must, of
course, guard ourselves against the impression that full and exhaustive
courses were given in algebra, geometry, surveying, and navigation.
As is pointed out by the author himself, the merest rudiments only
were imparted.

Eevereud Jones was succeeded by Alexander Irvine, and he in turn
by Joshua Fry. Fry was educated at Oxford, and, after coming to this
country, was made master of the grammar school connected with Will-
iam and Mary, and later, professor of mathematics in the college. In
company with Peter Jefferson, the father of Thomas Jefferson, he made
a map of Virginia. He also served on a commission appointed to deter-
mine the Virginia and North Oarolina boundary line. He was suc-
ceeded in 1758 by William Small.

A fisw years before the outbreak of the Eevolutionary War William
and Mary College had among her students several who afterwards rose
to prominence ; she had four who became signers of the Declaration of
Independence, and also the illustrious Thomas Jefferson, who became
the author of this great document. At William and Mary, Jefferson
was a passionate student of mathematics. The college long exercised
the duties of the oflSce of surveyor-general of the Colony of Virginia.
Thomas Jefferson's father was a practical surveyor, who had been
chosen in 1747 with Joshua Fry, then professor of mathematics at Will-
am and Mary, to continue the boundary line between Virginia and
North Carolina.

When Thomas Jefferson, at the age of seventeen, entered the Junior
class, he came into intimaterelation with Dr. William Small, of Scotland,
who was then the professor of mathematics. As an instructor he had
the happy gift of making the road of knowledge both easy and profit-
able. In his Autobiography Jefferson says : <^ It was my great good
fortune, and what probably fixed the destinies of my life, that Dr. Will-
iam Small, of Scotland, was then professor of mathematics, a man pro-
found in most of the useful branches of science, with a happy talent of
communication, correct and gentlemanly manners, and an enlarged and
liberal mind. He, most happily for me, became soon attached to me,
and made me his daily companion when not engaged in the school ; and
from his conversation I got my first views of the expansion of science,
and of the system of things in which we are placed."

In 1773 Thomas Jefferson was appointed surveyor of the county of
Albemarle. But the college of Williamsburg left it« stamp upon Jef-
ferson, not merely as a qualified surveyor, but also as a statesman, phi-
losopher, economist, and educator. We dwell with special interest upon
his association at college with Dr. Small, because in later years, when
filling the ofiBce of President of the United States, we shall marvel at
the rich fruits his early association with a lover of exact science brought

COLONIAL TIMES. 35

forth. It was daring Jefferson^a administration that a systematic plan
of conducting the Government surveys of the great North- West Terri-
tory was initiated ^ it was daring his administration that the great work
of the U. S. Coast Survey was first inaugurated. He took also great
interest in the enlargement of the U. 8. Military Academy. In these
great movements the personal interest and enlightened zeal of Jefferson
himself were the primary motive power. His biographers tell us that
he was the first discoverer of the formula for constructing the mould-
board of a plow on mathematical principles. He wrote to Jonathan
Williams on this subject, July 3, 1796 : " I have a little matter to com-
municate, and will do it ere long. It is the form of a mould board of
least resistcmce. I had some years ago conceived the principles of it, and
1 explained them to Mr. Eittenhouse.'^ We quote the following to show
that even in his old age he still loved the favorite study of his youth.
Said he in a letter to Col. William Duane, dated October, 1812, " When
I was young, mathematics was the passion of my life. The same pas-
sion has returned upon me, but with unequal powers. Processes which
I then read off with the facility of common discourse, now cost me labor
and time, and slow investigation.'' Of interest are also certain pas-
sages in a course of legal study which he drew up for a young friend :
" Before you enter on the study of law a sufficient groundwork must
be laid. • • • Mathematics and natural philosophy are so useful
in the most familiar occurrences of life and are so peculiarly engag-
ing and delightful as would induce every one to wish an acquaint-
ance with them. Besides this, the faculties of the mind, like the mem-
bers of the body, are strengthened and improved by exercise. Mathe-
matical reasoning and deductions are, therefore, a fine preparation for
investigating the abstruse speculations of the law." Among the books
in mathematics recommended by Jefferson to his young friend are,
Bezout's Cours de MathSmatique — the best for a student ever published ;
Montucla, or Bossut, Histoire des Math^matiques ; Astronomy — Fergu-
son, and Le Monnier or De Lalande.

It should not be left unmentioned here that George Washington once
applied to the College of William and Mary for an elective course in
land surveying, and that he received his first commission as county
surveyor from the faculty of the college. In this connection we can not
refrain quoting a passage from the excellent monograph by Dr. Herbert
B. Adams on the College of William and Mary.* '« It is interesting,'^
says he, "to trace the evolution of men as well as of institutions. It is
generally known that Washington began his public life as a county
surveyor, but, in all probability, few persons have ever thought of his
service in that office as the historical and economic germ of his political
greatness. Most people regard this early work as a passing incident
in his career, and not as a determining cause, and yet it is possible to
show that Washin^on's entire public life was but the natural out-

.^^.

* Circular of Information of the Bureau of Education, No. 1, 1887, p. 30.

36 TEACHINa Am) HISTOBT OF BIATHEMATICS.

growth of that original appointment given him in 1749, at the age of
seventeen, by the College of William and Mary. That appointment, in
the colonial days of Virginia, was the equivalent of a degree in civil
engineering, and it is interesting to observe what a pecnliar bias it
gave to Washington's life before and after the Bevolntion."

Professor Small's successors in the mathematical chair at William
and Mary were Bev. Thomas Gwatkin, George Blackbnm, Ferdinand
8. Campbell, Bobert Saunders, Beivjamin S. Ewell, and Thomas T. L.
Snead.

UNIVERSITY OP PENNSYLVANIA.

The University of Pennsylvania was chartered in 1756, and was
known before the Bevolution as the College, Academy, and Charitable
School of Philadelphia. The celebrated Dr. William Smith, D, D., was
the first provost. He was a man of great leariiiug aod superior execu-
tive ability. Under his administration, previous to the outbreak of the
Bevolution, the college made marvellous progress. The teachers were
men of well-established reputation throughout the colonies. Dr.
Smith, who was very fond of mathematical studies, gave lectures on
mathematics, natural philosophy, astronomy, and rhetoric. In 1769 he
appears as one of the founders of the American Philosophical So-
ciety. The first volume of the transactions of that society contains ac-
curate observations by Bittenhouse and himself of the transits of
Venus and Mercury. Associated with him at the college as professor
of mathematics, from 1760 to 1763, was Hugh Williams. He was a
graduate of the institution, and a minister. Afterward he studied
medicine abroad and then practiced in Philadelphia. He took great in.
terest in astronomy, and observed the transit of Venus and Mercury
for the Philosophical Society.

Theophilus Grew is also mentioned as a mathematical instructor.
Bev. Ebenezer Kinnersley, Franklin's assistant in his electrical experi-
ments, gave instruction in physics. ^' In this institution," says Dr.
Smith, '< there is a good apparatus for experiments in natural philoso-
phy, done in England by the best hands and brought over from thence
in different parcels. There is also in the experiment- room an electrical
apparatus, chiefly the invention of one of the professors, Mr. Kinners-
ley, and perhaps the completest of the kind now in the world." The
courses of study mapped out by Dr. Smith are preserved in his
works.* According to this, the mathematical and physical instruction
during the three years at college was as follows (in 1758) :

First year. — Common and decimal arithmetic reviewed, including fractions and the

extraction of roots ; algebra through simple and quadratic equations, and log*

arithmical arithmetic; first six hooks of Euclid.
Second year.— Plane and spherical trigonometry ; surveying, dialing, navigation ;

eleventh and twelfth books of Euclid; conic sections ; fluxions; architecture,

with fortification ; physics.
^irdyear, — ^Light and color, optics^ perspective, astronomy.

•William Smith's Works, 1803, p. 238.

COLONIAL TIMES. 37

There is given, in addition to this, the following list of ** books recom-
mended for improving the yonth in the various branches.'^

First year, — Barrow's Lectarea, Pardie's Geometry, Maclaurin's Algebra, Ward's

Ma^ematics, KeiPs Trigonometry.
Second ^ear.— Patoan's Navigation, Gregory's Geometry aod Fortification ; Simson's

Conic Sections; Maclaarin's and Emerson^s Fluxions.
Tfiirdyear, — Helsham's Lectures ; Gravesande; Cote's Hydrostatics; Desagnliers;

MuBchenbroec ; Keil's Introduction; Martin's Philosophy, Maclaurin's View of

Sir Isaac Newton's Philosophy, Rohault per Clarke.

It appears that the instrnction was given by lectures, the books of
which the above is a partial list, were (says Dr. Smith) ^^ to be con-
sulted occasionally in the lectures, for the illustrations of any particular
part ; and to be read afterwards, for completing the whole." How
closely this advancjpd curriculum of Dr. Smith was adhered to, and how
nearly his ideal scheme came tjo be realized in the actual work of the
college, we have no means of determining. This much is certain, that
before the Eevolution the institution attracted a large number of stu-
dents. According to Dr. Smith, the attendance in the college alone
went as high as one hundred, while the total attendance, including the
pupils of the academy and charity schools, surpassed three hundred. .
Of the course of study which he planned for the institution, it has been
said by competent judges that '^no such comprehensive scheme of edu-
cation then existed in the American colonies."

But there followed a reaction. Political troubles at the beginning of
the Revolutionary War broke up the institution. The authorities of
the college were accused of disloyalty, and in 1779 the charter was an-
nulled by the Provincial Assembly, and the college estate vested in a
new board. Dr. Smith was ejected, and in 1791 there was organized
the ''University of Pennsylvania." Many years elapsed before the
institution regained the popularity it enjoyed before the war.

SBLF-TAUaHT MATHEMATICIANS.

The mathematicians mentioned in the previous pages were all men
engaged in the profession of teaching. But, strange as it may seem,
the most noted mathematician and astronomer of early times was not a
professor in a college, nor had he been trained within college walls.
We have reference to David Bittonhouse. He was bom near German-
town, Pa., in 1732. Until about his eighteenth year, he was employed
on his father's farm. The advantages for obtaining an education in
rural districts were then exceedingly limited, but the elasticity of his
genius was superior to the pressure of adverse fortune. At the age of
twelve he came in possession of a chest of carpenter's tools, belonging
to an uncle of his, who had died some years previously. This chest
contained, besides the implements of trade, several elementary books
treating of arithmetic and geometry. This humble cofFer was to him an
invaluable treasure, for the tools afforded him some means of exercising

38 'TEACHIKG AKD HlSTOBlT OF ICATHEMATlCd.

the bent of bis genias toward mechanics, while the books early led his
mind to those pnrsnits for which it was pre-eminently fitted. While a
boy he is said to have covered the fences and plows on his father's farm
with geometrical figures. At the age of seventeen he constructed a
wooden clock.

The delicacy of his constitution and the irresistible bent of his genius
induced his parents to yield to his oft-repeated wish of giving up farm-
ing, and to procure for him the tools of a clock and mathematical instru-
ment maker. Bittenhonse now worked diligently with his tools during
the day, and at night spent a portion of his time which should have
been passed in taking repose in the prosecution of his studies. His
success seems to have been extraordinary, for his biographers assert
that before the age of twenty he was able to read the Principia, and
that he had discovered the method of fluxions without beiug aware that
this had already been done by Newton and Leibnitz. In Sparks's
American Biography we read that since Kewton in his Principia
"follows the synthetic method of demonstration and gives no clue to
the analytic process by which the truth of this proposition was first dis-
covered by him, • • • Bittenhouse began to search for the instru-
ment which might be applied to the purpose of similar discoveries, and
in his researches attained the principles of the method of flnxions."

Dr. Bush, in his eulogy on Bittenhouse, says in the same way: "It
was during the residence of our iugenious philosopher with his father
in the country that he made himself master of Sir Isaac Newton's Prin-
cipia, which he read in the English translation of Mr. Motte. It was
here, likewise, he became acquainted with the science of fluxions; of
which sublime invention he believed himself, for a while, to be the
author, nor did he know for some years afterwards that a contest had
been carried on between Sir Isaac Newton and Leibnitz for the honor
of the great and useful discovery. What a mind was here ! Without
literary friends or society, and with but two or three books, he became,
before he had reached his four and twentieth year, the rival of two of
the greatest mathematicians in Europe."

Our information concerning the studies of our young philosopher is
so scanty, that we find it impossible to determine the exact range of his
thoughts or the consequences that flowed from them. Not the slight-
est information as to the exact nature of his alleged invention has been
preserved. He himself seems to have attached no weight to it. We
are of the opinion that his invention, whatever it may have been, was
not of sufficient importance to deserve the name of an " invention of
fluxions." If Bittenhouse actually made an invention of such trans-
cending magnitude before the age of twenty, and at a time when he
had hardly begun his scientific studies, how is it that he made not the
slightest approach to any similar discovery during the forty-four years
of his maturer life Y Though always a passionate lover of scientific
pursuits^ he made no original contributions whatever to the science of

COLONIAL TIMES. 39

pare mathematics. Science is iDdebteded to him chiefly for his orreries
and the observations of the transit of Venns. We are, therefore, of
the opinion that the alleged invention of fluxions was little more than
a ^^ rnmor set afloat by idle gossip.'' It serves to show as, however,
in what unbounded admiration he was held by his countrymen.

At the age of nineteen Bittenhouse made the acquaintance of Thomas
Barton, a talented young clergyman who bad been a student at the
University of Dublin. An intimate friendship grew up between them,
which proved advantageous to the mental improvement of both. Bar-
ton was able to furnish Bittenhouse with some books saitable for his
instruction. The burning zeal with which our young scientist pursued
bis studies appears from the following extract of a letter he wrote to
Barton on September 20, 1756, at the age of twenty four: '*I have no
health for a soldier [the country was then engaged in war], and as I
have no expectation of serving my country in that way, I am spending
my time in the old trifling manner, and am so taken with optics, that I
do not know whether, if the enemy should invade this part of the coun-
try, as Archimedes was slain while making geometrical figures on the
sand, sol should die making a telescope."

As a mechanic, Bittenhouse became celebrated for the extreme ex-
actness and finish of his workmanship. Especially celebrated were his
chronometer clocks. It was while thus engaged in the manufacture of
clocks that he planned and executed an instrument which brought into
play both his mechanical and mathematical skill. This instrument was
the orrery. Concerning this wonderful mechanism, he wrote to Barton
January 28, 1767, as follows: ^^I do not design a machine which will
give the ignorant in astronomy a just view of the solar system, but would
rather astonish the skilful and curious observer by a most accurate cor-
respondence between the situations and motions of our little represent-
atives of our heavenly bodies and the situations and motions of those
bodies themselves. I would have my orrery really useful by making it
capable of informing us truly of the astronomical phenomena for any
particular point of time, which I do not find that any orrery yet made can
do." It was, indeed, intended to be a sort of a perpetual astronom-
ical almanac, in which the results, instead of being exhibited in tables,
were to be actually exhibited to the eye. His orrery greatly exceeded
all others in precision. It attracted very general attention among well-
informed persons, and the Legislature of Pennsylvania, in appreciation
of the talents of Bittenhouse, voted that the sum of three hundred
pounds be given to him.

There arose a lively competition between different colleges in this
country for the possession of this orrery. While the College of Phila-
delphia was negotiating for its purchase, a committee from the College
of New Jersey went to examine it, and concluded to buy it at once ; and
thus, much to the chagrin of Dr. William Smith, Princeton bore off the
palm from Philadelphia in obtaining possession of the first orrery con-

40 TEACHINa AND H^OBT OF MATHEMATICS.

stracted by Bittenhonse. He afterwards made another one for the Phil-
adelphia College. The author of The Vision of Golombas, a poem first
published in 1787, allades to the Eittenhouse orrery m Philadelphia and
the mass of people crowding to the college hall to see it, in the ibllowiu^f
lines (Book Vn):

See the sage Hittenhonse, witli ardent eye,

Lift the long tobe and pierce the starry sky;

Clear in his view the circling systems roU,

And broader splendonrs gild the central pole.

He marks what laws th* eccentric wand'rers bind^ »

Cop.ies Creation in his forming mind,

And bids, beneath his hand, in semblance rise,

With mimic orbs, the labours of the skies.

There wondering crowds with raptnr'd eye behold

The spangled heavens their mystic maze unfold ;

While each glad sage his splendid hall shall grace,

With all the spheres that cleave th' ethereal space.

In August, 1768, Rittenhouse was appointed by the American Philo-
sophical Society in Philadelphia as one of a committee to observe the
transit of Venus on June 3d of the following year. A temporary ob-
servatory was built by him for the purpose near his residence at Norri-
ton. Dr. William Smith aided him in procuring suitable instruments,
and the preliminary arrangements were made with most scrupulous
care. The approaching phenomenon was one of great scientific impor-
tance. Only two transits of Venus had been observed before his time,
and of these, the first, in 1639, had been seen by only two persons.
These transits happen so seldom that there cannot be more than two
in one century, and in some centuries none at all. But the transits of
Venus are the best means we have for determining the parallax of the
sun. At the approach of the transit, Bittenhonse and his assistants in
this observation, Dr. William Smith and Mr. Lukens, then surveyor-
general of Pennsylvania, awaited the contacts in silence and anxiety.
The observations were a success, and established for Bittenhonse tbe
reputation of an exact and careful astronomer. The transit was ob-
served in Boston by Professor Winthrop, and in Providence by Benja-
min West, at almost all the observatories in Europe, and in various
other parts of the globe. During the transit Bittenhonse saw one
phenomenon which escaped the notice of all other astronomers. When
the planet had advanced about half of its diameter upon the sun,
that part of the edge of the planet which was off the sun's disc appeared
illuminated, so that the outline of the entire planet could be seen. But •
a complete circle of light aroand Venus would indicate that more than
half of Venus is illuminated. This can happen, as far as we know,
only when the rays of light are refracted by an atmosphere. Hence,
it would follow from the observations of Rittenhouse that Venus is sur-
rounded, like the earth, by an atmosphere. But this appearance of a
ling of light was not oonflrmed by other astronomersi and the state-

COLONIAL TIMES. ' 41

ment of Bittenhouse excited no attention for nearly a century, until his
observation was, at last, confirmed by other astronomers.

An important invention made by Bittenhouse is that of the " colli-
mator,'^ " a device for obtaining a meridian mark without going far away ;
it has lately come back from Germany, where it was re-invented."*

The reputation which Bittenhouse had now acquired as an astronomer
attracted the attention of the Government, and he was employed in
several important geodetic operations. In 1779 he was named one of
the commissioners for adjusting a territorial dispute between the States
of Bennsylvania and Yirginia; in 1786 he was employed in fixing the
line which separates Pennsylvania from the State of New York, and in
the following year he assisted in determining the boundary between
New York and Massachusetts. In 1791 he was chosen successor of Dr.
Franklin in the presidency of the American Philosophical Society. All
his scientific communications were made to that society and published
in its Transactions.

Bittenhouse came to be looked up to by his countrymen as an as-
tronomer equalled by few and surpassed by none of his contemporaries.
Listen, if you please, to Thomas Jefferson's estimateof him. In answer to
the assertion of Abb6 Baynal that ^'America had not yet produced one
good poet, one able mathematician, one man of genius in a single art or a
single science," Jefferson says : ^^When we shall have existed as a people
as long as the Greeks did before they produced a Homer, the Bomans a
Yirgil, the French a Bacine and Yoltaire, the English a Shakespeare
and Milton, should this reproach be still true, we will inquire from what
unfriendly causes it has proceeded, that the other countries of Europe
and quarters of the earth shall not have inscribed any name in the roll
of poets. • • • In war we have produced a Washington, whose
memory will be adored while liberty shall have votaries, whose name
shall triumph over time, and will in future ages assume its just station
among the most celebrated worthies of the world. • • • In physics
we have produced a Franklin, than whom no one of the present age has
made more important discoveries, nor has enriched philosophy with
more, or more ingenious, solutions of the phenomena of nature. We
have supposed Mr, Eittenhouse second to no astronomer living; that in genius
lie must be the firstj because he is self-taught. As an artist he has ex-
hibited as great a proof of mechanical genius as the world has ever pro-
duced. He has not indeed made a world : but he has by imitation
approached nearer its Maker than any man who has lived from the
creation to this day.^'t

Such was Jefferson's estimate of Bittenhouse. James Eenwick says
that " he [Bittenhouse] had shown himself the equal in point of learn-
ing and skill as an observer to any practical astronomer then living. ''
Dr. Bush, in his eulogy, exclaims: "What a mind was herel Without

*Tlio Developmontof Astronomy in the United States, by Prof. T. H. Safbrd, p. 8.
\ Jefferson's Notes on Virginia.

42 TEACHING AND HISTORY OP MATHEMATICS.

literary friends or society, and with bat two or three books, he became*
before he had reached his four-and-twentieth year, the rival of two of
the greatest mathematicians in Europe I ^

If we* estimate Bittenhouse by what he might have done had he had a
more ragged physical constitation and better facilities for self-develop-
ment ; had he had an observatory at his disposal sach as those of his
great contemporaries, Maskelyne and William Herschel in England,
Lalande and Count Gassinl in France, Tobias Mayer in Germany, then
ttie above estimates may be correct. Bat if oar astronomer be judged
by the original coutribntions which, under existing adverse ciropm-
stances, he actually did make to astronomy and mathematics, then it
mast be admitted that he can not be placed in the foremost rank of as-
tronomers then living. Friends will judge him by what he might have
done; the world at large will judge him by what he actually acoom-
pliahed. Our greatest indebtedness to Bittenhouse lies not in the origi-
nal contributions he made to science, but rather in the interest which
he aroused in astronomical pursuits, and in the diffusion of scientific
knowledge in the New World which resulted from his efforts.

One who enjoyed, in his day, the reputation of being a *< great mathe-
matician," was Thomas Godfrey, of Philadelphia. He was a glazier by
trade. Having met accidentally with a mathematical book, he became
so delighted with the study that by his own unaided perseverance he
mastered every book he could get on the subject. He pursued the study
of Latin in order that he might read Newton's Principia. Optics and
astronomy became his favorite studies, and the exercise of his thoughts
led him in 1730 to conceive an improvement of the quadrant. In 1732
a description of his invention was sent to Dr. Hadley in England.
Meantime, in 1731, Hadley had made a communication to the Boyal
Society of Loudon, describing an improvement of the quadrant similar
to that of Oodfrey. The claims of both parties were afterwards inves-
tigated by the Boyal Society, and both were entitled to the honor of in-
vention. The Instrument is still called << Hadley's quadrant," though of
the two Oodfrey was the first inventor. Afterwards it appeared that
both had been anticipated in their invention by Newton.

Some of the personal characteristics of Godfrey are known to us
through the writings of Benjamin Franklin. "I continued to board
with Godfrey, who lived in part of my house with his wife and children,
and had one side of the shop for his glazier's business, though he worked
but little, being always absorbed in mathematics.'' In the autumn of
1727 Franklin formed most of his ingenious acquaintances into a club
for mutual improvement, called Junto. It met Friday evenings. " One
of the first members of our Junto," says Franklin, " was Thomas God-
frey, a self-taught mathematician, great in his way, and afterwards in-
ventor of what is now called Hadley's Quadrant. But he knew little
out of hifl way, and was not a pleasing companion, as, like most great
mathematicians I have met with, he expected universal precision in

COLONIAL TIMES. 43

everytbing said, and was forever denying and distinguishing upon
trifles, to the disturbance of all conversation."

This assertion of Franklin that all mathematicians he had met were
insufferable from their trifling and captious spirit, has been extensively
quoted by opponents of the mathematical sciences. It was quoted by
Goethe, and afterwards by Sir William Hamilton, the metaphysician,
when he was engaged in a controversy with Whewell, the celebrated
author of the History of the Inductive Sciences, on the educational
value of mathematical studies. Hamilton attempted to prove the start-
ling proposition that the study of mathematics not only possessed no
educational value, but was actually injurious to the mind. He must
have experienced exquisite pleasure in finding that Franklin, the great-
est physical philosopher of America, had made a statement to the effect
that all mathematicians he had met were ^* forever denying and dis-
tinguishing upon trifles."

We shall not speak of this controversy, except to protest against any
general conclusion being drawn from Franklin's experience of the
captiousness of mathematicians. Take, for examples, David Bitten-
house and Nathaniel Bowditch, who were early American mathemati-
cians, and, like Godfrey, self-taught men. Though Franklin's state-
ment may be true in case of Thomas Godfrey, it is most positively
unjust and false when applied to the other two scholars. The biogra-
phers of David Bittenhouse are unanimous and explicit in their asseition
that in private and social life he exhibited all those mild and amiable
virtues by which it is adorned. As to Kathaniel Bowditch, of whom we
shall speak at length later on, we have the reliable testimony of numer-
ous writers that he was a man remarkable for his social virtues,
modest and attractive manners, and Franklinian common sense.

Mention should be made here of Benjamin Banneker, the self-taught
'* negro astronomer and philosopher," born in Maryland, who became
noted in his neighborhood as an expert in the solution of difficult prob-
lems, and who, with the use of Mayer's Tables, Ferguson's Astronomy,
and Leadbeater's Lunar Tables, made creditable progress in astronomy,
and calculated several almanacs. His first almanac was for the year 1792.
The publishers speak of it as *' having met the approbation of several
of the most distinguished astronomers in America, particularly the cele-
brated Eitteuhouse." Banneker sent a copy to Mr. Jefferson, then Sec-
retary of State, who said in his reply, ** I have taken the liberty of send-
ing your almanac to Monsieur de Condorcet, secretary of the Academy
of Sciences at Paris, and member of the Philanthropic Society, because
I considered it a document to which your whole color had a right for
their justification against the doubts which have been entertained of
them."*. Banneker was invited by Andrew Ellicott to accompany " the
Commissioners to run the lines of the District of Columbia " upon their
mission.

* History of the Nefiro Race in America, by George W. Williams, p. 386.

INFLUX OF ENGLISH MATHEMATICS, 1776-1820.

The Revolutionary War bore down so heavily upon the educational
work in both elementary and higher institutions, that many of them,
for a time, actually closed their doors. The majority of students and
professors of Harvard and Yale were in the Army, or were in some
other way rendering aid to the national cause. The buildings of Nas-
sau (Princeton) College were for a time used as barracks. The business
of Columbia College in New York was almost entirely broken up. The
professors and students of Batgers College at New Brunswick, N. J.,
were sometimes compelled by the presence of the enemy to pursue their
academical studies at a distance from New Brunswick. The operations
of Brown University in Providence, R. L, were discontinued during
part of the war, the college building being occupied by the militia and
the troops of Boohambean. At William and Mary College the exer-
cises were suspended in 1781 for about a year, and the building was oc-
cupied at different times by both British and American troops. The
walls of the college were '^ alternately shaken by the thunder of the can-
non at Yorktown and by the triumphant shouts of the noble bands w^ho
had fought and conquered in the name of American Independence."
Academies and primary schools were either deserted or taught by wo-
men and white-haired men too old to fight. That the philosophic pur-
suits of scientific societies should have sunk very low is not surprising.
Fifteen years elapsed between the publication of the first and second
Tolumes of the Transactions of the American Philosophical Society in
Philadelphia.

In spite of the financial depression and poverty which existed imme-
diately after the war, much attention was paid to education. While in
* 1776 there existed in the colonies only seven colleges, the number was
increased to nineteen before the close of the eighteenth century. Acade-
mies and grammar schools were established, and a large number of
text-books were pat through the press. Even during the war the print-
ing-press sent out an occasional school-book. Thus, in 1778, while the
war was raging most fiercely, an edition of Dil worth's spelling-book was
printed, which contained in its preface the following patriotic passage :
"At the beginning of the contest between the Tyrant and the States,
it was boasted by our unnatural enemy, that, if nothing more, they
could at least shut up oar ports by their navy and prevent the importa-

INFLUX OF ENGLISH HATHEHATIOS. 45

tion of books and paper, so that in a few years we should sink down
into barbarity and ignorance, and be fit companions for the Indians,
our neighbors to the westward." These words, printed at the darkest
period of the Bevolutionary War, disclose a spirit far from submissive.
The colonists were not quite ready to sink down into barbarity and
ignorance. During the twenty-five years after the Declaration of In-
dependence, more real progress was made in education than in the
entire century preceding. Between 1776 and 1815 a large number of
books on elementary and a few on higher mathematics were published
in America. Many of them were reprints of English works, while
others were compilations by American writers, modelled after English
patterns. French and German authors were almost unknown. We
may therefore call this the period of the ^^ Influx of English Mathe.
matics " into the United States. What little mathematics was studied
in the colonies before the Be volution was, to be sure, gotten chiefly
from English sources, but the scientific currents thither were then so
very feeble and slow that we can hardly speak of an << influx.'^

Elementary Schools.

It is a significant fact that of the arithmetics used before the Bevoln-
tion, but one work in the English language was written by an American
author. It is equally significant that with the close of the great strug-
gle for liberty, there began a period of activity in the prodaction of new
school-books. The second book devoted exclusively to arithmetic, com-
piled by an American author, and printed in the English language, was
the New and Complete System of Arithmetic by Nicholas Pike, (New-,
buryport, 1788. )•

Nicholas Pike (1743-1819) was a native of New Hampshire, graduated
at Harvard College in 1766, and was for many years a teacher and after-
ward a magistrate at Newburyport in Massachusetts. His arithmetic
received the approbation of the presidents and professors of the leading
New England colleges. A recommendation from Harvard professors
contains the following timely remark: << We are happy to see so use-
ful an American production, which, if it should meet with the encour
agement it deserves, among the inhabitants of the United States, will
save much money in the country, which would otherwise be sent to

* It appears that Greenwood's Arithmetio, pablished nearly sixty years previously,
was at this time not known to exist. Pike's Arithmetio was called the ^«< American
work of its kind. Dr. Artemas Martin has sent the writer the American Antiqua-
rian, (Vol. IV, No. 12, New York, May, 1888) giving an account of Pike's book. It
gives a letter written by George Washington at Mount Vernon, June 20, 1788, to
Nicholas Pike, in which the former politely acknowledges the receipt of a copy of
Pike's Arithmetic. We quote from the letter the following passage :

''Its merits being established by the approbation of competent judges, I flatter
myself that the idea of its being an American production &nd the first of the kind which
has appeared, will induce every patriotic and liberal character to give it all the coun-
tenance and patronage in his power. "

46 TEACHING AND HISTOEY OP MATHEMATICS.

Europey for publications of this kind." Pik^s arithmetic passed
•thron^li many editions, was long the standard mathematical manoal in
Kew England schools, and formed the basis for other arithmetics. It
was a very extensive and complete book for that time. A large

, proportion of the rules were given without demonstration, while some
were proved algebraically. In addition to the subjects ordinarily found
in arithmetics, it contained logarithms, trigonometry, algebra, and
conic sections, but these latter subjects were so briefly treated as to pos-

' sess little value. After the appearance of Webber's, Day's, and Farrar's
Mathematics for colleges, which elaborated these subjects at greater
length, they were finally omitted in the fourth edition of Pike's Arith-

* metic, in 1822.

In 1788, when the first edition appeared, English money was still the
prevalent medium of exchange in the United States. To be sure, Ped»
eral money was adopted by Congress as early as 1786, but previous to
1794 there was no United States coin of the denomination of a dollar. It
was merely the money of account, based upon the Spanish dollar,
which had long been in use in this country. Congress passed a law
organizing a mint in 1792, but permitting the circulation of foreign
coins for three years, by which time it was believed the new coinage
would be ready in sufficient amount. When dollars and cents began
to replace pounds and shillings, it became desirable that the Federal
currency be explained in arithmetics and taught in schools. In conse-
quence of this, the sterling notation was changed to Federal in the
third edition of Pike's arithmetic, which was brought out in Boston in
1808 by Nathaniel Lord. Similar changes were made in other arith-
metics.*
Down to the year 1800, the only arithmetic written by an American,

^ which enjoyed wide-spread and prolonged popular favor, was the one ef
Nicholas Pike. In 1800 appeared a second successful arithmetic, The
School-master's Assistant, by Nathan DaboU, a teacher in New London

* Conteinporaneonsly with Pike's Arithmetio there appeared in Philadelphia the
Elementary Principles of ArithmetiCi by Thomas Saijeant. This l^ook, as weU as the
Federal Arithmetio, or the Science of Numbers (Philadelphia, 1793), by the same
author, had only an ephemeral reputation. John Gough's Treatise on Arithmetic in
Theory and Practice, edited by Benjamin Workman (Boston, 1789), as well as
Gough's American Accountant, or School-master's New Assistant, abridged by Benja-
min Workman and revised by Patterson (Philadelphia, 1796), had a rather limited
circulation. Nor did John Vinall's Arithmetio (Boston, 1792), enjoy better success.
After having been a teacher in Newburyport for seventeen years, Vinall at last be-
came writing-master in a school in Boston, his native city. lie is said to have been
coarse in speech and, like his book, unpopular. Gordon Johnson wrote an arithme-
tic (Springfield, 1792), which never had more than a passiug local reputation. Some-
what more successful was the Introduction to Arithmetic (Norwich, Conn., 1793), by
Erastus Boot, a graduate of Dartmouth, for several years a teacher, and afterward
an active politician and member of Congress.

Our list of arithmetics printed previously to the year If^ includes the names of
several other "quaint and onrioaa volumes,'^ which, after an ephemeral repntatiooi

INFLUX OP ENGLISH MATHEMATICS. 47

(bom 1750 and died 1813). This work passed through namerous edi-
tions. Though DaboU had to compete with Pike's Abridged Arithmetic
and with the celebrated Scholar's Arithmetic of Daniel Adams, it nev-
ertheless acquired an extensive popularity. The expression, ^< accord-
ing to Daboll," came to be a synonym for *^ mathematical correctness."
It pushed aside the less favorite works. The main element of popularity
of DaboU's School-master's Assistant lay in the fact that it introduced
Federal money immediately after the addition of whole numbers, and
showed how to find the value of goods therein immediately after simple
multiplication. This arrangement, says the author, may be of great
advantage to many who perhaps will not have an opportunity to learn
firactions. Decimal fractions were wisely made to precede vulgar frac-
tions. In the *' Eecollections " by Peter Parley, of the town of Eidge-
field, Conn., are found the following interesting remarks: ^< We were
taught arithmetic in DaboU, then a new book, and which, being adapted
to our measures of length, weight, and currency, was a prodigious leap
over the head of poor old Dilworth, whose rules and examples were
modelled upon English customs. In consequence of the general use of
Dilworth in our schools for perhaps a century, pounds, shillings, and
pence were classical, and dollars and cents vulgar for several succeed-
ing generations. ^I would not give a penny for it' was genteel ; *I
would not give a cent for it' was plebeian."

Since the adherence to pounds and shillings came to be offensive to
the people of the young republic, Mr. Hawley, in 1803, undertook to
revise the work and alter all the problems to Federal currency. He
called the new work "Dil worth's Federal Calculator," but after this
change the book was so completely different firom the original that the
use of Dilworth's name in the title seemed hardly justifiable. Be that
as it may, the Federal Calculator was not a success.

At the beginning of the nineteenth century there were three ^* great
arithmeticians" in America, namely, Nicholas Pike, Nathan DaboU, and

passed into forgetfulness, never to be resurrected to memory except by the cariosity
of some inquiring lovers of "forgotten lore." To the above names we add: The
American Tntor's Assistant, by John Todd, Philadelphia, third edition, 1797; Arith-
metic by Zachariah Jess, Philadelphia, 1797 ; American Arithmetic, by David Cook,
New Haven, Conn., 1799; The Usher, comprising arithmetic in whole numbers,
Federal money, mensuration, surveying, etc., by Ezekiel Little, Exeter, 1799; Usher's
Arithmetic, abridged, by Ezekiel Little, 1804 ; The American Accountant, by Chaun-
oey Lee, Lansingburg, 1797 ; The American Accountant, by William Milne, New
York, 1797, in which^ instead of the answers to the problems, which were usually
given, the author gave the remainders, after casting out the nines from the answers.
A curious little volume is the following : *' The Young Gentleman's and Lady's Assist-
ant, containing Gteography, Natural Philosophy, Rhetoric, Miscellaneous, to which is
added a short and complete system of Practical Arithmetic, wherein the money of
the United States of America is rendered easy to the perception of youth. The
wholid divided into small sections for the convenience of sdioolB, by Donald Fr&8er,
author of the Colombian Monitor, New York, 170C.''

48 TEACmNG ASD HI8T0BT OF MATHEMATICS.

. Daniel Adams. Having noticed the first two, we shall briefly speak of
the third. Daniel Adams published in 1801 the Scholar's Arithmetic,
a work which in point of merit towers far above the mass of contempo-
rary school-books. Adams was a native of Massachnoetts^ graduated
at Dartmouth College in 1797, and then became teacher, physician, and
editor. He taught school in Boston from 1806 to 1813, then removed to
"Nqw Hampshire, where he afterward served as State Senator. Though
engaged in various lines of thought, arithmetical studies were his favor-
ite. He furnished the school-boys' satchels not only with the Scholars'
Arithmetic, but ako with the Primary Arithmetic, and, in 1827, with
the l^ew Arithmetic, which passed through numerous editions. The

, Kew Arithmetic differed from the Scholars' Arithmetic in being ana-
lytic instead of synthetic in treatment The analytic or inductive
method of teaching, introduced into Switzerland by Pestalozzi, was

' gaining ground rapidly in this country at the beginning of the second
quarter of the present century.

Between 1800 and 1820, a large number of arithmetics sprang into
existence. Most of them enjoyed only a mushroom popularity. Among
the more successful of the new aspirants to arithmetical fame were the
following : Jacob Willetts, of Poughkeepsie, K. T. , the author of the
Scholar's Arithmetic, 1812^ William Eanne, of Maine, who graduated
at Yale in 1804, and subsequently became teacher in his native
State and the author of A Short System of Practical Arithmetic,

• Hallo well, second edition, 1807 ; Michael Walsh, the author of A New
System of Mercantile Arithmetic, adapted to the commerce of the
United States, Newbury port, 1801; Stephen Pike, whose arithmetic
was published in Philadelphia in 1813 ; Samuel Webber's Arithmetic,
1810, which was used chiefly by students preparing for Harvard.*

Our list of arithmetics published during the first twenty years of this
century is doubtless very imperfect. Of the larger number of publica-
tions, the great majority had only an ephemeral reputation. Excepting
those of Pike, Adams, and Daboll, hardly any have survived the recol-
lection even of the aged.

* Less widely nsed were the following books: Jonathan Gront's Qoide to Practical
Arithmetic, 1802 ; Caleb Alexander's New System of Arithmetic, Albany, 1802 ; W.
M. Finlay's Arithmetical Magazine, or Mercantile Accountant, New York, 1803 ;
James Noyes's Federal Arithmetic, 1804 ; the American School-master's Assistant, by
Jesne Guthrie, of Kentucky, Lexington. 1804 ; Samuel Temple's System of Arithmetic
in Federal Currency, Boston, 1804 ; the Youth's Arithmetical Guide, byTandon Ad-
dinjjtoA and VSTatson, Philadelphia, 1805 ; Mathematical Manual for the Use of St
Mary's College of Baltimore, containing arithmetic and algebra, by L. L M. Cheyigne,
Baltimore, 1806 ; Kimber's Arithmetic Made Easy for Children, second edition, 1807 ;
Ballard's Ganging Unmasked, 1806 ; Robert Patterson's Treatise on Arithmetic, Phil-
adelphia, 1819; Daniel Staniford's Practical Arithmetic, Boston, 1818; George Fen-
wick's Arithmetical Essay, Alexandria, 1810 ; Compeodium of Practical Arithmetic,
by Osgood Carleton, Boston, 1810 ; the American Arithmetic, by Oliver Welch of New
Hampshire ; The Teacher's and Pupil's Assistant, by Dale Tweed of northern Now
York, 1820; the Arithmetic of Leonard Loom is ; The Columbian Tutor's Assistant, by
D. MoCurdy, V^aahington, 1819 ; The First Lines of Arithmetic, 1818, by De Wolf and

INf*LUX OF ENGLISH MATHEMATICS. 49

Between 1815 and 1820 a reform in mathematical teaching was in-
aagurated in this country. Foremost among the leaders in this new
movement was John Farrar of Harvard, who translated into English
for the nse of colleges a number of French works. The French books
of that time were far in advance of the English. This reform in the
teaching of the more advanced mathematics was accompanied by a
similar reform in arithmetical teaching. The new ideas of Pestalozzi
were vigorously forcing their way from Switzerland to all parts of the
civilized world. Among the* earliest fruits they bore in this country
were the First Lessons in Arithmetic, by Warren Oolburn, 1821. This
primer contained points of great excellence, and it had a sale such as
no other arithmetic ever had before. •

• We do not now speak of these reforms, except simply for the purpose
of marking the end of an old period and the beginning of a new one.

Having enumerated the text- books published in the United States
after the Revolution and preceding the year 1820, we shall now briefly
examine their contents. The leading characteristics which we observed
in Dilworth's School-master's Assistant, are found to exist in the books
of this period. The arithmetics of this time were little more than Pan-
dora's boxes of ill-formed rules to be committed to memory. Reason-
ing was exiled from the realm of arithmetic, and memory was made to
rule supreme. A science chiefly intended to cultivate the understand-
ing was offered to the exercise merely of memory.

This banishment of demonstration and worship of memory did not,
I am glad to say, originate in this country. As already remarked, it
came from England. About the middle of the seventeenth century
there arose in England the commercial school of arithmeticians. To
this school, says De Morgan, "we owe the destruction of demonstrative
arithmetic in this country, or rather the prevention of its growth. It
never was much the habit of arithmeticians to prove their rules ^ and
the very word 'proofs in that science, never came to mean more than a

Brown, teachers in Hartford, Conn. ; the Arithmetic of Zacharlah Jess of Delaware(f ) ;
The Scholar's Guide to Arithmetic, by Plinehas Merrill of New Hampshire ; Collec-
tion of Arithmetical Tables, Hartford, 1812; Arithmetic Simplified, 18m, by John J^
White; and The Youth's Guide, by Mordecai Stewart, Baltimore, 1818; Rev. John
White's Mental Arithmetic, Philadelphia, 1818 ; '^ The American Youth : being a New
and Complete <]lourse of Introductory Mathematics: designed for the use of Priyate
* Students, by Consider and John Sterry." Vol. 1, Providence.

Besides these American works there^ere a number of foreign books republished
in this country. Among these are. The Tutor's Guide, by Charles Vyse, London, 1770,
which reached the thirteenth American edition in Philadelphia in 1806; A Complete
Treatise on Arithmetic, by Charles Hutton, Edinburgh, 1802, first American edition,
New York, 1810 ; A System of Practical Arithmetic, by Rev. J. Joice, London, 1816,
and adapted to tHe commerce of the United States by J. Walker, Baltimore, 1819 ;
The Scholar's Guide to Arithmetic, by John Bonnycastle, London, 1786, Philadelphia,
1818. These English books can hardly be said to have excelled oar American arith-
metics ; nor did they attain to any remarkable success in the New World.
881— No. 3 4

50 . TEACmNO AND HISTOET OP MATHEMATICS.

test of the correctness of a particular operation, by reversing the pro-
cess, casting oat the nines, or the like. As soon as attention was fairly
diverted to arithmetic for commercial purposes alone, such rational ap-
plication as had been handed down from the writers of the Sixteenth
century began to disappear, and was finally extinct in the work of
Cocker or Hawkins, as I think I have shown reason for supposing it
should be oalled. From this time began the finished school of teachers,
whose pupils ask, when a question is given, what rule it is in, and run
away, when they grow up, from any numerical statement, with the dec-
laration that anything may be proved by figures — as it may, to them."*
Such is the history of the commercial school of arithmeticians in Eng-
land. In America this school 'became firmly established wherever
arithmetic was taught. Thus the sins of tfie early English pedagogues
were visited upon the children in England and America unto the third
and fourth generations. As late as 1818, one of our American compilers
of text-books, Daniel Staniford, actually stated in the preface^ as a
recommendation to his book, that ''all mathematical demonstrations
.are purposely omitted, to clear illustrations of the rules by easy exam-
ples and such as tend to prepare the scholar for business." Not only
was this method adopted in practice, but even advocated in theory. In
both English and American arithmetics the Tales were ill-arranged and
disconnected. The pupil had to learn a dozen rules which might have
been reduced conveniently to two or three principles. The continuity
in the reasoning upon quantities expressed by integers and those ex-
pressed in common or decimal fractions was often so completely dis-
guised that it became necessary to repeat the rules. Thus Dilworth
and Bonnycastle give in their arithmetics three distinct ruiesi as fol-
lows:

Eule of three for integers.

Bule of three for vulgar fractions.

Bule of three for decimal fractions.

Nicholas Pike, Daniel Staniford, and John Yinall eaoh give '^Bules of
interest," and later on again "Eules of interest by decimals.'^ The re-
sult of tbis cumbrous rule-system is that the scholar acquires the art of
solving problems, provided he knows what rule it falls under, which is
net always sure to be the case, for the first practical problem which will
arise may be one requiring not one rale, but a combination of rules,
which can therefore not be solved directly by the rules in his book. '
And here he is fairly aground, for he has no mastery of principles, but is
the abject slave of rules. Such a system of arithmetic has been very ap-
propriately called ciphering^ since intellect goes tor nothing throughout.
. Among other features which characterize old American arithmetics
are the following:

(1) The total absence of exercises in mental arithmetic.

* Arithmetical Books, p. xxi, •

INFLUX OF ENGLISH MATHEMATICS. 51

(2) The meagre treatment of firactions. The numMr of exercises was
so very limited that it was impossible for the student to acquire a mas- -
tery of fractions without additional drill.

(3) The process of << cancellation," which shortens calculations so .
much, was entirely unknown. Strange as it may seem, it is less than
fifty years since cancellation was introduced into our arithmetics. One
of the first books containing this process was published in 1840 by G. *
Tracy, in New Haren, entitled <* A IS'ew System of Arithmetic, in which

is explained and applied to practical purposes ^ ^ ^ the principle
of cancelling. • • •»

(4) The system of numeration in early American arithmetics was not
the French now generally used, but the English, in which the digits of
a number are distributed in periods of six, and consequently proceed by
millions. This method was first adopted by the Italians. Lucas de
Burgo gives it in a work written in 1494. The method of reckoning by
three places, as used in this country and on the Oontinent, seems to
have originated with the Spanish.

(5) The subject which we now call proportion was then called the '
*< Bule of Three." It was taught as a mere rule. The principle under-
lying it was ignored completely. That a proportion is the expression
of the equality of two ratios was then not even hinted at. This fact
goes to explain a point which otherwise would seem mysterious. If pro-
portion is the equality of ratios, then why was not the usual symbol
used to express that equality 1 Why were four dots used instead of the
two horizontal lines f • The answer seems to be that arithmeticians were
not in the habit of thinking of a proportion as the equality of two ratios.
A ratio was expressed by two dots, and the four dots were placed be-
tween the ratios simply to disjoint the terms, and to show that the sec-
ond and third terms of the proportion were not in the same relation to
each other as the first and second or third and fourth.

(6) The old arithmetics contain two methods of solving problems
which are but rarely given in modern arithmetics. The methods I re-
fer to were first used by the Hindoos on the far off banks of the Ganges,
then borrowed from the Hindoos by the Arabs, who transmitted them
to the Europeans. They are called the methods of '< single position "
and " double position." They teach us how to resolve questions by
making one or two suppositions of false numbers, and then making
corrections for the resulting errors.

We have seen that previous to the year 1820 a large number of arith-
metics were published in this country; counting both American and
foreign, there were to our knowledge over sixty different authors.
^Notwithstanding this fact, the majority of schools had an inadequate
supply of arithmetics. In country schools especially, books were scarce
and of a rather miscellaneous character, such as had been in families
perhaps half a century. Johnnie Smith would, perhaps, bring to school
a dilapidated copy of Dil worth's Arithmetic, which had been used once

52 TEACHINa AND HISTOBY OF MATHEMATICS.

by his father. His classmate, Billy Brown, would carry in his satchel
a copy of Nicholas Pike's Abridged Arithmetic; the curly-headed
Jimmy Jones would express his preference for Baboll's Schpol-maslers'
Assistant, while the majority of the boys had no books at all. When-
ever the sapply of arithmetics was insufficient, manuscript books were
resorted to. Arithmetics were sometimes covered with sheepskin, in
the economical expectation that they might be made to lead not only
one boy, but also his younger brothers in succession, to the golden sci-
ence of numbers.

We have now spoken of the most popular arithmetics once used in
this country. We have also briefly examined their contents. Our next
task will be to ascertain, as far as possible, the manner in which arith-
metic was taught in the school-room.

One of the first inquiries in this connection is regarding the quality
*of the teachers. The best teachers in elementary schools that our fore-
fathers knew were young students who taught school for money to
finish their courses in theology, medicine, or law. But this class of
. school-teachers was not large in early days. The representative school-
' masters of by- gone times were the itinerant school-masters. They were
mostly foreigners. Their qualifications seemed to be the inability to
earn anything in any other way. They were generally without families
and had no fixed residence; they kept school first in one place and
then in another, and wandered about homeless. Many were given to
drinking and gambling. As a class, their knowledge was limited to
the merest elements. We are told that as late as 1822, in a town in
the State of Connecticut, six out of fifteen applicants for positions as
teachers were rejected because they did not understand notation and
numeration of numbers. And yet these candidates came well recom-
mended as having taught school acceptably in other towns for one, two,
or three winters. If this story be true, then it will not seem strange to
hear that it was a common practice for teachers in those early days to
have their scholars " skip" fractions. This omission was justified on
the ground that ^* fractions were rarely used in business," but there were
generally other good and untold reasons for " skipping ^ the subject.
There were few schools that carried the students beyond the rule of
three or proportion.

We have seen the great defects in the old arithmetics. The state-
ment of rules took the place of explanations and reasoning. If the
school-masters had been competent and well trained, then the defects
of bad books might have been remedied by skillful teaching, but the
teaching was generally of the poorest kind. The truth of this asser-
tion will be strikingly illustrated by a few examples. Joseph T. Buck-
ingham tells us how, in 1790 or 1791, when he was about twelve years
old, he began to learn arithmetic. I quote his exact words : " I told
the master I wanted to learn to cipher. He set me a sum in simple ad-

INFLUX OP ENGLISH MATHEMATICS. 53

dition, five columns of figures and six figures in each column. All the
instruction he gave me was — add the figures in the first column^ carry
one for every ten, and set the overplus down under the column. I sup-
posed he meant by the first column the left-hand eolumn, but what he
meant by carrying one for every ten was as much a mystery as Sam-
son's riddle was to the Philistines. I worried my brains an hour or
two, and showed the master the figures I had made. You may judge
what the amount was when the columns were added from left to right.
The master frowned and repeated his former instruction — add up the
column on the rights carry one for every ten, and set down the remainder.
Two or three afternoons (I did not go to school in the morning) were
spent in this way when I begged to be excused from learning to cipher,
an'U the old gentleman with whom I lived thought it was time wasted ;
and if I attended the school any further at that time reading, spelling,
and a little writing were all that was taught." l^ezt winter a more
communicative teacher had the school, <^ and under him some progress
was made in arithmetic, and I made tolerable acquisition in the first
four rules, according to Dilworth's School-master's Assistant, of which
the teacher and one of the oldest boys had a copy."

An experience similar to that of the writer just quoted was that of
Warren Burton. In his book entitled ^^ The District School as it was,
by one who went to it," he says that simple addition was easy; ^^but
there was one thing I could not understand — that carrying of tens. It
was absolutely necessary, I perceived, in order to get the right answer,
yet it was a mystery which that arithmetical oracle, our school-master,
did not see fit to explain. It is possible that it was a mystery to him.
Then came subtraction ; the borrowing of ten was another unaccount-
able operation. The reason seemed to be then at the very bottom of
the well of science ', and there it remained for that winter, for no friendly
bucket brought it up to my reach."

Mr. William B. Fowle gives an interesting account of John Tileston,
who was chief writing-master in a reading school in Boston about the
year 1790. It illustrates both the modes of teaching and the compe-
tency of teachers. One regulation of that school required the writing-
master to teach << writing, arithmetic, and the branches usually taught
in town schools, including vulgar and decimal fractions." Mr. Fowle
speaks of Tileston as follows : *

" He loved routine. • • • Printed arithmetics were not used in
the Boston schools until after the writer left them, and the custom was
for the master to write a problem or two in the manuscript of the pupil
every other day. No boy was allowed to cipher till he was eleven years
old, and writing and ciphering were never performed on the same day.
Master Tileston had thus been taught by Master Proctor [his predeces-
sor], and all the sums he set for his pupils were copied exactly from his

* Barnard's Joamal, Vol. V, p. 336.

54 TEACHINa AHD mSTOBT OF MATHEMATICS.

old mannscript. Any boy coald copy the work from the manuscript of
any other farther advanced than himself, and the writer never heard of
any explanation of any principle of arithmetic while he was at school.
Indeed, the pupils believed that the master could not do the sums he
set for them. * * * It is said that a boy who had done the sum
set for him by Master Tileston carried it up, as usnal, for examination.
The old gentleman, as usual, took out his manuscript, compared the
slate with it, and pronounced it wrong. The boy went to his seat and
reviewed his work, but finding no error in it, returned to the desk, and
asked Mr. Tileston to be good enough to examine the work, for he could
find no error in it. This was too much to require of him. He growled,
as his habit was when displeased, but he compared the sums again, and
at last, with a triumphant smile, exclaimed, < See here, yonnurly (gnarly)
wretch, you have got it, ^^ If four tons of hay cost so much, what will
seven tons cost t ^ when it should be, <^ If four tons of English hay
cost so and so." Now go and do it all over again.'" In this story, it
may be remarked, some allowance must doubtless be made for the
genius of the narrator.

The illustrations which have been given of the incompetency of
teachers may appear to be exaggerations, and we certainly wish that
for the good name of our early educators they were exaggerations.
But the more we inquire into this subject and the more evidence we
accumulate, the stronger the conviction becomes that most of them are
not exaggerations, but fair samples of the teaching done by the aver-
age school-master in elementary schools eighty or one hundred years
ago.

' In view of these facts, the most obstinate pessimist will be forced to
admit that, within the last one hundred years, progress has been made.
We have better books and abler teachers. Our methods of teaching
arithmetic, though still imperfect, are a prodigious leap in advance of
those of olden times. We boast of our material progress, and we
oertainly have great reason for doing so, but the progress in intellect-
ual fields, and in education in particular, though less ostentatious, is
none the less instructive.

Not without interest are the following two stanzas of a poem, en*
titled ^< A Country School," which was anonymously contributed to the
New Hampshire Spy, and preserved in E. H. Smith's collection of
American poems (1793) :

Will pray Sir Master mend my pen f

Say, Master, tliat'H enouj^h. Here, Ben,

Is this your copy t Can't you tell t

Set all your letters parallel.

Fve done my $vkm — Hiajmt a groat —

Let's see it. Master ^ m' / g'out f

Yes, bring some wood in. What's that noise f

It iin'i I, $irf it'$ them boy$.

INFLUX OF ENGLISH HATHEMATIOS. 5S

Gome Billy, read. What's that f That's A— •

Sir, Jim has snatched my rule away —

Return it, James. Here rule with this—

Billy, read on— that's crooked S.

Read in the spelling-hook. Begin—

The hoy 8 are out. Then call them in^

My noie hleed8f maynH I get some tee.

And hold it in my breeches f Yes.

John, keep your seat. My sum is more-^

Then do't again — divide by four,

By twelve, and twenty— mind the rule.

Now speak, Manasseh, and spell tool.

I can't. Well try. T, W, L.

Not wash'd your hands yet, booby, ha f ,

You had your orders yesterday.

Give me the ferule, hold your hand,

Oh ! Oh ! there — mind my next command.

Colleges.

Before proceeding to the history of mathematics in higlier institutions,
we shall speak of American reprints of English mathematical works for
colleges. First of all comes that good old Greek geometry of Euclid,
of which the English made excellent translations. An edition of Euclid
appeared in Worcester in 1784. This seems to be the earliest American
edition. After the beginning of this century numerouis editions of it
were published. In 1803, Thomas and George Palmer, in Philadelphia,
published Robert Simson's Euclid, together with the book of Euclid's
Data, and the Elements of Plane and Spherical Trigonometry. The book
was sold ^^ at the book-stores in Philadelphia, Baltimore, Washington,
Petersburg, and I^orfolk.''

Prof. S. Webber says, in his " Mathematics,'' that a good American
edition of Playfair's Elements of Euclid, containing the first six books,
with two books on the geometry of solids, was given by Mr. P. Nicholls,
of Philadelphia, 1808. John D. Craig, teacher of mathematics in Balti-
more, brought out an edition of Euclid in 1818. Bobert Simson's Eu-
clid was republished in Philadelphia in 1821 ; Playfair's in New York
in 1819 and 1824, and in Philadelphia in 1826. In 1822 appeared the
following work : " Euclid's Elements of Geometry, the first six books, to
which are added the Elements of Plane and Spherical Trigonometry, a
System of Conic Sections, Elements of Natural Philosophy as far as it
relates to Astronomy, according to the Newtonian System, and Elements
of Astronomy, with notes by Eev. John Allen, A. M., professor of mathe-
matics at the University of Maryland." John D. Craig, in a notice of
this book, says that Newton's work at this day is << almost a locked
treasure among us," owing to the " scarcity of tracts giving the necessary
preparatory knowledge." The object of this volume was to supply that
want.

An English mathematician, whose works found their way across the
ocean, was John Bounycastle, professor of mathematics at the Eoyal

56 TEACHING AND HISTORY OP MATHEMATICS.

Military Academy, Woolwich. His Introduction to Algebra (London,
1782) was revised and edited in tliis country by James Eyan in 1822.
Bonnycastle was a teacher bf rules rather than principles.

An English author well known in this country was Thomas Simpson.
His Treatise on Algebra was published in Philadelphia in 1809. The
Second American from . the eighth English edition, revised by David
McClure,. teacher of mathematics, came out in Philadelphia in 1821.
As was frequently the case in those days, all demonstrations are here
given by themselves in the manner of notes placed below a horizontal
line on the page. ' They could be taken or omitted by teacher and pupil
at pleasure, and were generally omitted. The author's explanations
and demonstrations wanted simplicity, and wo need not wonder if they
were " looked upon, by some, as rather tending to throw new difficulties
in the way of the learner than to the facilitating of his progress."

Another English algebra reprinted here was that of 6. Bridge, fellow
of 8t. Peter's College, Cambridge (second American edition from
eleventh London, Philadelphia, 1839). We are informed that this work
was introduced into the University of Pennsylvania, the Western Uni-
versity, Pittsburg, in Qummere's School at Burlington, tlie Friends^
College at Haverford, and <' a great number of the best schools in the
United States.'' The Three Conic Sections, by the same author, was
also patronized by some of our colleges. This subject was here treated
.purely synthetically, as was the case also in Bobert Simson's Conic
Sections, reprinted here in 1809 (f ), and in all other English treatises of
.that time. Analytic methods, which proved so powerful in the hands
. of mathematicians on the Continent, were still underrated in England.
The exclusive adherence to the synthetic method was due to an exces-
sive worship of the views of Newton, who favored synthesis and em-
ployed it throughout his Principia.

We next mention Kev. Samuel Vince's Fluxions, printed in Philadel-
phia in 1812, or about twenty years after its first appearance in Eng-
land. This seems to be the only work devoted exclusively to fluxions
which was ever published in this country. Before the introduction of
the Leibnitzian notation it was the text-book most generally used in
our colleges, whenever fluxions were taught. An edition of Viuce's
Astronomy came out in Philadelphia in 1817. Of his other works, his
'* Conic Sections, as preparatory to the reading of Newton's Princii)ia,"
was best known in America. Vince held the position of Plumiau pro-
fessor of astronomy and experimental philosophy in the University of
Cambridge, England. His works generally lacked elegance, and failed
to teach the more modern and improved forms of the mathematical
science.

More prominent than any of the English authors here mentioned was
Charles Hutton. His Course of Mathematics was edited in America
by Bobert Adraio, and will be spoken of again later.

INFLUX OF ENGLISH MATHEMATICS. 57

HABYABD GOLLEGE.

It has already been stated that the chair of mathematics and natural
philosophy at Harvard was occupied from 1779 to 1788 by Rev. Samuel
Williams, a pupil of John Winthrop. Williams wrote manuscript books
on astronomy, mathematics,. and philosophy. His mathematical mai^-
scripts were probably studied in place of Ward's Mathematics, which
had been used by his predecessor, John Winthrop. We possess hardly
any information on the mathematical instruction during his time. The
following is taken from the diary of a student who, in 1786, applied for
admission to the third term of the Junior year : " Mr. Williams asked
me if I had studied Euclid and arithmetic.''* This question having
been answered, apparently, in the affirmative, he was admitted. Prom
this it would seem that at that time Euclid and arithmetic were the
only mathematical studies pursued previous to the close of the Junior
year. The fourth year, says Amory, seems to have been principally
occupied in the study of mathematics. From a quotation given by
Amory, we infer that algebra was a college study at this time. It had
probably been so during the last fifty years, but we possess no data
from which this could be positively affirmed.

A ray of light upon the inner workings of the college is thrown by
quotations from the diary of a student who was at Harvard in 1786.
They show that the tutors of the college failed to command the esteem
and respect of the students. Complaints were made that the Greek
tutor was too young. "Before he took his second degree, which was
last commencement, he was chosen a tutor of mathematics^ in which he
betrayed his ignorance often." Of another tutor it is remarked : " We re-
cite this week to our own tutor in Qravesand's Experimental Philosophy.
This gentleman is not much more popular than the rest of the tutors.''
Whoever has observed the freedom with which college boys speak of
their instructors, knows that statements like these must be taken with
some allowance. But the practice alluded to above, of selecting grad-
uates who had excelled mainly in clatssical studies as tutors in math-
ematics^ seems absurd. And yet it is well known that this custom was
continued even in some of our best colleges down to a comparatively
recent date. The objections to the custom which existed at Harvard
previous to 1767 are still more obvious. In the early days of Harvard
each tutor taught all branches to the class assigned to him, throughout
the whole collegiate course. But in 1767 the rule was introduced that
one tutor should not teach all the subjects, but only one subject, such
as Greek 5 another tutor should have Latin; another mathematics,
physics, natural philosophy, etc.

In 1789 Samuel Williams was succeeded in the professorship of
mathematics and natural philosophy by Samuel Webber. Webber en-
gaged, while a boy, in agricultural pursuits, and at the advanced age of

* " Old Cambridge and New,'' by Amory, in North American Review, Vol. 114, 1872.

68 MJACHINO AND HISTORY OP MATHEMATICS.

twenty, in 1780, he entered Harvard College. After graduating he re-
mained two years at the college, studyiag theology. He then held a
tutorship till his appointment to the HoUis professorship, in which
office he spent seventeen of the most important years of his life. In
1806 he was elected president of Harvard. He occupied this position
titl his death, in 1810. Henry Ware, in his eulogy of Webber, says:
<< As a scholar his attainments were substantial, embracing various
branches of learning, but, mathematical science being most congenial
to his taste and habits, he quitted his professorship for the presidency
with reluctance. In communicating instruction, he united patience and
facility with a thorough acquaintance with his subject.'^ Edward Ever-
ett, who was a student at Harvard in Webber's time, makes a some-
what different estimate of him, saying that Webber was " reputed a
sound mathematician of the old school, but rather too much given to
routine."* In another place, Everett speaks of him in the same way as
" a person of tradition and routine." Judge Story says of him, " Pro-
fessor Webber was modest, mild, and quiet, but unconquerably reserved
and staid." t

In 1787, just before Samuel Webber was elected professor, the course
of studies at Harvard was revised, with a view of raising the standard
of learning. According to the new scheme, the classics << formed the
principal study during the first three college years. The Freshmen
were instructed, also, in rhetoric, the art of speaking, and arithmetic 'j
the Sophomores in algebra, and other branches of mathematics; the
Juniors in Livy, Doddridge's Lectures, and once a week, the Greek Tes-
tament ; the Seniors in logic, metaphysics, and ethics." j:

The elementary mathematics were now studied in the first half of the
college course instead of the second half. The Freshmen and Sopho-
mores now began taking mathematics, though, we fear, only in ineffect-
ual homoeopathic doses. If arithmetic was begun in the Freshman year,
then we may be sure that no very extensive course could have been
given before the close of the Sophomore year. According to Judge
Story, Saunderson's Algebra was used in 1795 or 1796. The original
work of this blind mathematician was extensive, and in two volumes.
The book used at Harvard consisted most likely of << Selected parts of
Professor Saunderson's Elements of Algebra," published in one vol-
ume.§

W. Williams, a classmate of Channing (class of 1798), says: "The
Sophomore year gave us Euclid to measure our strength. Many halted
at the ^pone asinorum? But Channing could go over clear at the first
trial, as could some twelve or fifteen of us. This fact is stated to show
that he had a mind able to comprehend the abstrusities of mathematics,

• Old Cftmbridge and New, Vol. IV, p. 199.

t Memoir of W. E. ChanniDg, by W. F. CbanniDg, fourtb ed., 1850, Vol. I, p. 47.

X Qainoy's History of Harvard University, Vol. II, p. 279.

f Tbird edition, London, 1771.

INFLUX OP ENGLISH MATHEMATICS. 59

though, to my apprehension, he excelled more decidedly in the Latin
and Greek classics, and had a stronger inclination to polite literature."

We are, moreover, told of Channing : " He delighted, too, in geom-
etry, and felt so rare a pleasure in the perception of its demonstrations
that he took the fifth book of Euclid with him as an entertainment dur-
ing one vacation." Such experiences are frequent with a student Af
advanced mathematics, but, unfortunately, too rare with pupils study-
ing the elements.

If the course given by Quincy be supposed complete, then no mathe^
maticd was studied after the Sophomore year. This was probably not
^rue ; it certainly was not true ten years later. In 1797, at least some
of the students pursued the more advanced mathematics during the
latter part of the college course. A glimpse of light on this subject is
thrown by the following quotation from the eulogy of John Pickering:
^' Great as was his enthusiasm for classical learning, he had in college
as real a love for the study of mathematics, and highly distinguished
himself in this department. Near the close of his Senior year he re-
ceived the iionor of a mathematical part, which appeared to give him
more pleasure than all his other college honors. It afforded him an
opportunity to manifest his profound scholarship in a manner most
agreeable to his feelings. When he had delivered the corporation and
overseers this part, containing solutions of problems by fluxions, he
had the rare satisfaction to be told that one of them was more elegant
than the solution of the grea); Simpson, who wrote a treatise on flux-
ions, in which the same problem was solved by him.'' It follows from
this that provisions were made for the study of fluxions, at least for
students who may have desired to study them.

Of the mathematical theses, written by Juniors and Seniors, which
have been deposited in the Harvard Library, one hundred and thirty-
three were written during the period from 1781 to 1 807. Of these, the
great majority are on the calculation and projection of eclipses. Sur-
veying and the algebraic solution of problems receive also a large
amount of attention. Of the one hundred and thirty-three theses only
seven show by their titles that they contain "fluxionary problems.'' Their
dates are, 1796, 1803 (two), 1804, and 1806 (three). After 1807, theses
containing solutions of problems on fluxions are quite numerous. It
may be of interest to state that John Parrar, the future professor of
mathematics at Harvard, wrote in 1803 a thesis on the ^' Calculation
and Projection of a Solar Eclipse." James Savage, the great authority
on American genealogy, furnished a colored view of churches and col-
lege buildings ; Everett, the diplomatist, a colored " TempU Episcopalis
Delineatio Perspectiva,^^ One thesis is on the " Calculation and Pro-
jection of a Solar Eclipse which took place in the year of the Cruci-
fixion."*

* Biographical Contributions of the Library of Harvard University, No, 32.

60 TEACHING AND HISTORY OF MATHEMATICS.

In 1S02 the standard for admission to Harvard Oollcge was raised.
In mathematics a knowledge of Arithmetic to the *' Rule of Three " was
required. Thas, in 1803, for the first time had it become necessary, ac-
cording to regalations, for a boy to know something about arithmetic
before he could enter Harvard. We surmise, however, that the require-
ments in arithmetic were very light, for we know from the diary of a
student in the Freshman class in 1807 that arithmetic continued to be
taught during the first year at college.* After 1816 the whole of arith-
metic was required for admission.

From the beginning of the nineteenth century on, we can get- more
definite informatlom regarding the extent to which the mathematical
ptudies were pursued. We need only -examine the college text-books
which began then to be printed in this country. The jsarliest mathematical
text-books for colleges, written by an American author, are those of Sam-
uel Webber. In 1801 were published in two volumes his ^* Mathematics,
compiled from the best authors and intended to be a text-book of the
course of private lectures on these sciences in the University of* Gam-
bridge." A second edition appeared 1808. These works jwere for a
time almost exclusively used in New England colleges, but they finally
gave place to translations from French works, executed by John Far-
rar, the successor of Webber in the professorship of mathematics.

Within two volumes, each of 460 pages and in large print, are em-
braced the following subjects : arithmetic, logarithms, algebra, geome-
try, plane trigonometry, mensuration of surfaces, mensuration of solids,
gauging, heights and distances, surveying, navigation, conic sections,
dialing, spherical geometry, and spherical trigonometry. Some idea of
the extent to which each branch of mathematics was carried may be ob-
tained, if we state that in Webber's works 124 pages were given to
algebra, while Newcomb's Algebra, for instance, numbers 545 pages.
The subject of conic sections was disposed of by Webber within 68 short
pages, while Went worth's Analytic (Geometry covers 273 crowded pages.
Comparatively much space was given by Webber to the applications of
mathematics, such as gauging, heights and distai^s, surveying, navi-
gation, and dialing. These practical subjects received much more at-
tention then than they are now receiving in the academic department in
the majority of our colleges. Webber devotes only 47 pages to th6 im-
portant and extensive subject of geometry, and gives solutions of geo-
metrical problems, but no theorems. This apparent neglect of the oldest
and most beautiful of mathematical sciences is explained by Webber in
tlie second edition of his work. In a foot-note (p. 339) he says that <^A
tutor teaches, in Harvard College, Playfair's Elements of Geometry, con-
taining the first six books of Euclid, with two books on the geometry of
soTids. Of this work Mr. F. Kichols, of Philadelphia, has given a good
American edition ^ (1806). Webber's chapter on geometry was, there-
fore, intended simply as a book with problems to accompany or

• « Old Cftmbridf e and New," by Aniory, North Americftn Review, Vol. 114, p. 118.

INFLUX OP EKGLISH MATHEMATICS. * 61

follow Eaclid. If the coarse in elementary geometry was taught as here
indicated, then it can hardly be said that this subject was neglected.
Wherever Euclid is diligently studied, there geometry is not slighted.

Of John Farrar, the successor of Webber in the chair of mathematics
and natural philosophy, we shall speak when we consider the influx pf
French mathematics into the United States.

JBefore leaving Harvard College we shall quote two short passages
taken from the Harvard Lyceum. This journal was a publication by the
students, and was the earliest of that kind at this college. The quota-
tions about to be given disclose an effort to arouse interest among stu-
dents in mathematical studies. In the first number, which appeared
July 14, 1810, we read as follows : ^^ The dry field of mathematics has
brought forth most ingenious and elegant essays, most curious and eu-
tertaining problems. It is our wish to construct or select such ques-
tions in their various branches as may exercise the skill of our corres-
pondents in their solution." This promise was not strictly kept.
Mathematical enthusiasm could not be aroused quite so easily. There
id to be found, to be sure, an ingenious essay on mathematical learning,
presumably written by a Sophomore, in which we read: "Perhaps no
science has been so universally decried by the overweeningly dull as the
mathematics. Superficial dabblers in science, contented to float in
doubts and chimeras, and unable to see the advantage of demonstra-
ble truth, turn back before they have passed the narrow path which
leads to the firm ground of mathematical certainty, and not willing to
have others n)ore successful than themselves, like the Jewish spies,
they endeavor to deter them from the way by horrid stories of giant
spectres in the promised land of demonstration, and scarcely a Caleb
is found to render a true account of its beauties." But the Jewish spies
were too eloquent, and there was no Galeb to furnish curious and enter-
taining problems.*

TALE COLLEGE.

The chair of mathematics and natural philosophy at Yale College
was established in 1770. Its first incumbent wa9 Nehemiah Strong,
who occupied it till 1781. In 1794 Josias Meigs was appointed to the
position. Meigs was graduated at Yale in 1778, and served as tutor in
'mathematics, natural philosophy, and astronomy, from 1781 to 1784.
In 1783 he was admitted to the bar, and some years later practiced in
Bermuda. He appeared as a defender of American vessels that were
captured by British privateers, and was, in consequence, tried fur
treason. He was professor at Yale until 1801, teaching mathematics,
natural philosophy, and. chemistry.t He then became president of the

' The Harvard Book, by F.^O. ValUe and H. A. Clark, Vol. II, p. 174.

fProf. BoDJamin Silliraan '(class of 1796) says, that on November 4,1795, '^Mr.
Meigs heard the class recite at noon, as Dr. Dwight is ont of town. Although Mr.
Meigs is a very sensible man and very well calculated for the office which he now
fills, still it is very easy to make a contrast between him and the president ; but I am

62 TEACHING AND HISTORY OP MATHEMATICS.

University of Georgia. In 1812 lie was appointed surveyor-general, and
two years later, Commissioner of the General Land 01£ice of the United
States.

An important event at this period was the growth, among students,
of a love for rhetoric and literature. English literature had hitherto
been quite neglected. A taste for this study was excited by two youn^
men, John Trumbull and Timothy Dwight, who were elected tutors in
1771. John Trumbull published, during the first year of his tutorship,
a poem entitled the ^^ Progress Qf Dulness," a satire, intended to expose
the absurdities then prevailing in the system of college instruction.
Ancient languages, mathematics, logic, and divinity received, in his
opinion, an altogether disproportionate amount of time. In his poems,
he introduces ^< Dick Hairbrain," a town tbp, ridiculous in dress and
empty of knowledge, and speaks of him as follows :

'' What though in algebra, his station
Was negative in each equation;
Though in astronomy surveyed,
His constant course was retrograde;
O'er Newton's system though he sleeps,
And finds his wits in dark eclipse!
His talents proved of highest price
At all the arts of card and dice ;
His genius turn'd with greatest skiU,
To whist, loo, cribbage, and quadrille.
And taught, to every rival's shame,
Each nice distinction of the game."

Timothy Dwight's love for literature did not entirely displace his in-
terest for mathematics. On the contrary, we read in a life of him by
his son that, ^' In addition to the customary mathematical studies, he
carried them [his students] through spherics and fluxions, and went as
far as any of them would accompany him into the Principia of Kew-
ton." << This, however, must have been a very rare thing," says Presi-
dent Woolsey. Dwight was tutor at the college for six years. To ex-
hibit his continued interest in mathematics during that time we quote
from the biography of him the following passage : ^^ At a subsequent
period, during his residence in college as a tutor, he engaged deeply in
the study of the higher branches of the mathematics. Among the
treatises on this science to which his attention was directed, was New-
ton's Principia, which he studied with the utmost care and attention,
and demonstrated, in course, all but two of the propositions in that
profound and elaborate work. This diflScult but delightful science, in
which the mind is always guided by certainty in its discovery of truth,
so fully engrossed his attention and his thoughts that, for a time, he

doubtful whether the comparison is not a false one, because the president is one of
those characters which we very seldom meet with in the world, and who form its
greatest ornaments." (Barnard's Educational Journal, vol. 26, 1876, International
Series, vol. 1, p. 230.)

INFLUX OF ENQLISH MATHEMATICS. * 63

lost even his relish for poetry ; and it was not without difficulty that
his fondness for it was recovered.''

It will be remembered that the Mathematics of Ward had been intro-
duced before the Bevolntlon. In 17S8 Nicholas Pike's arithmetic was
adopted. Soon after 1801 Samuel Webber's Mathematics displaced the
works previously used, excepting Euclid, which was presumably used
during this whole period as the text-book in geometry. At about th^
beginning of this century the mathematical course was as follows:
>• Freshmen, Webber's Mathematics ; Sophomores, Webber's Mathematics

and Euclid's Elements ; Juniors^ Enfield's JSfatural Philosophy and As-
tronomy, and Vince's Fluxions ; Seniors, natural philosophy and as-
tronomy.

A strong impetus to the study of mathematics in American colleges
was given by Jeremiah Day. He graduated from Yale in 1795, became
tutor there in 1798, and was elected professor of mathematics and nat-
ural philosophy in 1801. Feeble health prevented him from entering
upon the duties of his professorship till 1803 ; but after that he con-
tinued in them till 1817, when he succeeded President D wight in the
presidency of the college. The chair was then given to Alexander Met-
calf Fisher.

At the beginning of this century the great want of this country in
the department of pure mathematics was adequate text-book^ Pro-
I fessor Webber, of Harvard, was the first who attempted tosupi)ly this

want* In those colleges in which a single system of mathematics had
been adopted, preference was generally given to the ^< Mathematics " of
Webber. But his compilation was rather imperfectly adapted to the
purposes for which it was made. It was not sufficiently copious. Many
topics, though strictly elementary and practically important, were
passed over in silence. The method of treatment was too involved and
the style not sufficiently dear to make the subject attractive to the
young studtnt. Accordingly Professor Day set himself to work to
write a series of books which should supply more adequately the needs
of American colleges. In 1814 appeared his Algebra, and his Mensura-
tion of Superficies and Solids, in 1815 his Plane Trigonometry, and in
1817 his Navigation and Surveying. It was the original intention of the
author to prepare also elementary treatises on conic sections, spherics,
and fluxions, but on his elevation to the presidency he abandoned this
design. His Algebra passed through numerous editions, the latest of
which was issued in 1852, by the joint labors of himself and Professor
Stanley.

Day's books are very elementary, and introduce tbe student by easy
and gradual steps to tbe first principles of tbe respective branches.
To us of to day, they appear too elementary for college use, but it must
not be forgotten that at the time they were prepared, they were just
what was needed to meet the demands of the times. Students apply-
ing for matriculation in those days had received very defective prepara

64 TEACHING AND HISTOBT OF MATHEMATICS.

tory traioiDg, especially in mathematics. With such unwron^ht mate-
rial before him, it was nataral for the teacher to show his preference
for a text-book in which every process of development and reasoning
was worked out patiently and minutely through all its snccessive steps.
Day took for a model the dififase manner of Euler and Lacroix, rather
than the concise and abridged mode of the English writers. The great
danger in this course is that no obstacles are left to be removed by the
student through his own exertion. In the opinion of some teachers,
Day has laid himself open to criticism by carrying the principle of mak-
ing mathematics easy somewhat too far. It is no little praise for a book
written at that time to say that, unlike most books of that period,
Day's mathematics did not encourage the cramming of rules or the per-
forming of operations blindly. On the contrary, the diligent student
acquired from them a rational understanding of the subject.

Day's mathematics were at once everywhere received with eager-
ness. They were introduced in nearly all our colleges. Even at
the end of a period of fifty years they still held their place in many
of our schools. In view of these facts, *< it may safely be said that the
value of what their author did by means of them for the college and for
the country at large, while holding the office of professor from 1S03 to
1817, the time when he succeeded Dr. Dwight, was not surpassed by
anytliing in science and literature which he did subsequently during
his long term of office as president of the college."*

As a teacher and writer, President Day was distinguished for the
simplicity and clearness of his methods of illustration. His kind-
heartedness and urbanity of. demeanor secured the love and respect
both of friends and pupils.

Ho was succeeded in the chair of mathematics and nataral philosophy
by Alexander Metcalf Fisher, who held it until his death by drowui^ig
in 1822, at the shipwreck of the Albion^ off the Irish coast. Fisher
possessed extraordinary natural aptitudes for learning. He had pre-
pared a tali course of lectures in nataral philosophy, both theoretical
and experimental, which were marked for their copiousness and their
exact adaptation to the purpose of instruction. His clear conception
of what a text-book should be is well shown in his review of Enfield's
Philosophy.t

Regarding the course in natural philosophy at Yale, it may be re-
marked, that in 1788 Martin's Philosophy, which had gone out of print,
was succeeded by Enfield's Katural Philosophy, first published in 1783.
William Enfield was a prominent English dissenter. He preached in

• Yalo College: A Sketch of ito History, by William Kingsley, Vol. I, p. 115.

t American Jourual of Science, Vol. Ill, Itttl, p. 125. In Vol. V, p. 83, of the same
jouruaJ, is an article by him, **0n Maxima and Minima of Fnnctions of Two Variable
Quantities.'' He contributed solutious to questions in the American Monthly Maga-
Eiut% and in Ley bourn's Mathematical Repository (London). The fonrth volume of
t!io Memoirs of the American Academy of Arts and Sciences contains observations by
him on the comet of 1819 and caloolations of its orbit.

INFLUX OF ENGLISH MATHEMATICS. 65

TTnitarian chnrcbes and published several volumes of sermons. Being
engaged chiefly in theological studies, comparatively little attention
was paid by him to the exact sciences. ^Nevertheless, he succeeded in
compiling a work on natural philosophy which possessed elements of
popularity and was used in our American colleges for four decennia. In
1820 appeared the third American edition of this work, which was
then used by nearly all the seminaries of learning in INew England,
notwithstanding the fact that, excepting in electricity and magnetism
I and a few particulars in astronomy, it presented hardly any idea of

the progress made in the different branches of philosophy since the
period of Newton.

TJNIVEESITY OF PENNSYLVANIA.

The University of Pennsylvania, which had such a remarkable growth
nnder the administration of Dr. William Smith, before the Revolution-
ary War, had a comparatively small attendance of students after the
war, and the college department is said to have been quite inferior to
that of the leading American colleges of that time.

An educator who was long and prominently connected with this in-
stitution and whose activity was directed towards maintaining and rais-
ing its standard, was Robert Patterson, the elder. He was born in
1743 in Ireland, and at an early age showed a fondness for mathematics.
* In 1768 he emigrated to Philadelphia. He first taught school in Buck-
ingham, and one of his first scholars was Andrew EDicott, who after-
ward became celebrated for his mathematical knowledge displayed in
the service of the United States.

About this time Maskelyne, the astronomer royal of England, com-
piled and published regularly the Kautical Almanac. This turned the
attention of the principal navigators in American ports to the calcula-
tions of longitude from lunar observations, in which they were eager to
obtain instruction. Patterson removed to Philadelphia, began giving
instruction on this subject, and soon had for his scholars the most dis-
tinguished commanders who sailed from this port. Afteward he be-
came principal of the Wilmington Academy, Delaware, and in 1779
was appointed professor of mathematics and natural philosophy at the
University of Pennsylvania, which post he filled for thirty-five years.
He was also elected vice-provost of that institution.*

Robert Patterson communicated several scientific papers to the Phil-
osophical Transactions (Vols. II, III, and IV), and was a frequent con-
1 tributor of problems and solutions to mathematical journals. He ed-
ited James Ferguson's Lectures on Mechanics (1806), and also Fergu-
son's ^< Astronomy explained upon Sir Isaac Newton's principles and
made easy to those who have not studied mathematics ^ (1809). Fer-

* Transaotions of American Philosophical Society, VoL 11, New Series, Obituary
Notice of Bobert PattorsoD, LL. D., late President of the American Pliilosopbioal So-
doty.

881— Ko. 3- — 5

66 TEAOHING AND HIST0B7 OF MATHEMATICS.

gnson was a celebrated lecturer on astronomy and mechanics in Eng<
land, who contribated more than perhaps any other man there to the.
extension of physical science among all classes of society, but especially
among that largest class whose circumstances preclude them from a
regular course of scientific instruction. His influence was strongly felt
even in this country, as is seen from the American editions by Bobert
Patterson of his astronomy and mechanics. Patterson wrote a small
astronomy, entitled the Kewtouian System, which was published in
1808. Ten years later he published an arithmetic, elaborated from his
own written compends, previously used in the University. Though
lucid and ingenious, this arithmetic was rather difficult for beginners,
and never reached an extended circulation.

It is believed by many that mathematicians generally possess a strong
memory for numbers. This was certainly not true of Patterson, for
we are told that he could not remember even the number of his own
house. He met this dilemma by devising a mnemoniOy which was indeed
worthy of a mathematician. The number of his residence was 285,
which answered to the following conditions : ^^ The second digit is the
cube of the first, and the third the mean of the first two.'' It is to
be wondered that, during some fit of intense abstraction, the learned
professor did not pronounce 111 to be the number of his house, instead
of 285 ; for 111 is a number satisfying the above conditions quite aa
well as 285.

When Eobert Patterson resigned his position at the University of
Pennsylvania in 1814, he was succeeded by his son, Robert M. Patter-
son. The latter was graduated at the University in 1804. After reoeiv*
ing the degree of M. D., in 1808, he pursued professional studies in
Paris and London. In 1814 he was appointed professor of mathematics
and natural philosophy, which office he filled until 1828, when he ac-
cepted the chair of natural philosophy at the University of Virginia.
Bobert M. Patterson published no mathematical books.

From 1828 to 1834 the chair of mathematics and natural philosophy
was filled by Prof. Bobert Adrain. The days of greatest activity of
this most prominent teacher of mathematics were spent at other in-
stitutions, but we take this opportunity of introducing a sketch of his
life.*

Bobert Adrain was bom in Ireland. At the age of fifteen he lost both
his parents, and thenceforward he supported himself by teaching. At
the end of an old arithmetic he found the signs used in algebra. His
curiosity becoming greatly excited to discover their meaning, he gave
himself no rest until at last he found out what they meant. In a
short time he was able to resolve any sum in the arithmetic by algebra*
Thenceforth he devoted himself with enthusiastic ardor to mathematics.
He took part in the Irish rebellion of 1708, received a severe wound, and

• ThlB iketoh it eztraoted from an artiole in the Deiuooratio Beview, 1844, VoL

xiy.

niFLUX OP .BITGLISH MATHEMATICS 67

estt^^ed to America. Immediately after his arrival he began teaching
in New Jersey. After two or three years he became principal of the
York Ootinty Academy in Pennsylvania. He then began contributing
.problems and solutions to the Mathematical Oorrespondent, a journal
published in Kew York. This was the means of bringing his mathe-
matioal talents before the public. He obtained several prize medals,
awarded for the best solutions.

In 1805 he moved to Beading, Pa., to take charge of the acad/yny of
that place. He started there a mathematical journal called the Analyst.
The first number was published in Beading, but its typographical exe-
cution disappointed him so much that he employed a publisher at Phila-
delphia and incurred the extra expense of a republication. We shall
speak of this journal again later.

In 1810 he was called to the professorship of mathematics and natural
philosophy at Queen's (now Butgers) College; and, in 1813, to the pro-
fessorship of mathematics at Columbia Oollege. In Kew York he be-
came the center of attraction to those pursuing mathematical studies.
A mathematical club was established, in which he -shone as the great
luminary among lesser lights. As a teacher, he had a most happy
&culty of imparting instruction.

In 1826 the delicate state of his wife's health induced him to leave
Oolumbia Oollege in Kew York and to remove to the pure air and
healthfnl breezes of the country near New Brunswick. About two
years later he was induced to accept the professorship of mathematics
at the University of Pennsylvania, a position which had been held at
the beginning of the century by the well-known Bobert Patterson.
Adrain became also vice-provost of this institution.

He resigned this position in 1834 and returned to his country seat
near New Brunswick, intending to pass his time with his family and
in study. But he did not remain there long, for the habit of teaching
had become too strong easily to be resisted. He moved to New York
and taught in the grammar school connected with Columbia College
until within three years of his death. At this time his mental faculties
began very perceptibly to fail. He greatly lamented their decay, and,
one day when a friend called in to see him, be had a volume of La Place
on his lap endeavoring to read it. ^' Ah," said he, in a melancholy tone
of voice, '^ this is a dead language to me now; once I could read La
Place, but that time has gone by." He died in 1843.

Among American mathematicians of bis day, Bobert Adrain was ex-
celled only by Nathaniel Bowditch. Of his many contributions to
mathematical journals, one of the earliest was an essay published in
1804 in the Mathematical Correspondent on Diophantine analysis. This
was the earliest attempt to introduce this analysis in America.

In 1808 Adrain began editing and publishing the Analyst, or Mathe-
matical Museum, At that time he had not yet entered upon his career
as college professor. The above periodical contained chiefly solutions

68 TEACHINQ AND HISTORY OF MATHEMATICS.

to mathematical questions proposed by the varioas contribators. It
a small, modest publicatioD, which had only a very limited cireolatioa in
this country, and was unknown to foreign mathematicians. It lived,
moreover, only a very short time, for only five numbers ever appeared.
And yet, this apparently insignificant little journal, edited by a^ teacher
at an ordinary academy, contained one article which was an original
contribution of great value to mathematical science. It was, in fact,
the firfrf} original work of any importance in pure mathematics that had
been done in the United States. I refer to Robert Adrain's deduction
of the Law of Probability of Error in Observation. The honor of the first
statement in printed form of this law, commonly known as the Principle of
Least Squares, is due to the celebrated French mathematician Legendre,
who proposed it in 1805 as an advantageous method of adjusting obser*
vations. But upon Eobert Adrain falls the honor of being the first to
publish a demonstration of this law. He does not use the term ^^ least,
squares," and seems to have Ijeen entirely unacquainted with the
writings of Legendre. It follows, therefore, that not only the two de-
ductions of this principle given by Adrain were original with him, but
also the very principle itself.

We now give the histoVy of this discovery by Adrain. Bobert Patter*
son, of the University of Pennsylvania, proposed in the Analyst the
following prize question: ^<In order to find the content of apiece of
ground, • • • I measured with a common circumferentor and chain
the bearings and lengths of its several sides, • • • but upon cast-
ing up the difference of the latitude and departure, I discovered * • •
that some error had been contracted in taking the dimensions. Kow, it
is required to compute the area of this inclosnre an the fnost probable
supposition of this error." This was proposed in Ko. II of the Analyst,
and after being a second time renewed as a prize question in Ko. Ill^it
was at length, in TSo. lY, solved by a course of special reasoning by
Nathaniel Bowditoh, to whom Adrain awarded the prize of ten dollars.
Immediately following Bowditch's special solution, the editor, Adrain,
added his own solution of the following more difficult general problem:
<^ Besearch concerning the probabilities of the errors which happen in
making observations." * This paper is of great historical interest, as
containing the first deduction of the law of facility of error.

<p {x) being the probability of any error ar, and o and h quantities de-

* Analyst, No. IV, pp. 93-97. Copies of this Jonnial are very rare. Ko. IV is to be
found in the Congressional Library in Washington \ No. Ill and No. IV are in the
Library of the American Philosophical Society, Philadelphia. Adrain's first proof of
the Principle of Least Squares was re-published by Cleveland Abbe in the American
Journal of Science and Arts, third series, 1871, pp. 411-415. Adrain's second proof
was re-published by Mansfield Merriman in the Transactions of the Connecticut Acad-
emy, Vol. IV, 1887, p. 164 ; also in the Analyst (edited aud published by J. £. Hen-
dricks, Pes Moines, Iowa), YoL lY, No. II, p. 33.

INFLUX OF ENGLISH MATHEMATICS. 69

pending on the precision of the measurement. Adrain gives two proofs
of this law. The first proof depends upon the ^^ self-evident principle^"
as he calls it, that the true errors of measured quantities are propor-
tional to the quantities themselves. The arbitrary nature of this as-
sumption is pointed out by J. W. L. Glaisher in the Memoirs of the
Boyal Astronomical Society, Part II, vol. 39, 1871-72. '< This," says
Glaisher, ^^ seems very tar from being evident, not to say very far from
being true, generally. One would expect a less relative error in a
» greater distance." Glaisher raises other objections toAdrain's first

proof, and then pronounces it entirely inconclusive. Adrain'if second
proof, which is essentially the same as that given later by John Her-
schel, and usually called Herschel's proof, is likewise defective, as has
been pointed out by Prof. Mansfield Merriman.

In order to place these criticisms on Adrain's two demonstrations in
the proper light, it should be remarked here that the subject of which
they treat is one of great difBcnlty. There has been great difference
of opinion among leading mathematicians as to what assumptions re-
garding the nature of errors can be safely and legitimately made, and
taken a« a basis upon which to construct demonstrations and what oiies
should be rejected as being false or as demanding demonstration.

Subsequently to Adrain's paper, proofs were published by Gauss, La
Place, Bessel, Ivory, John Herschel, Tait, Donkin, and others. Alto-
« gether, there appeared over a do2(en distinct proofs, but all of these
<< contain, to say the least, some point of difficulty" (Glaisher). If,
therefore, it be said that Adrain's proofs are inconclusive, we must re-
member that all other proofs hitherto given possess to a greater or less
degree the same defect.

The number of the Analyst which gives Adrain's two proofs contains
also the following applications of this method : (1) To find the most
probable value of any quantity of which a number of direct measure-
ments is given; (2) to find the most probable position of a point in
space; (3) to correct dead-reckoning at sea; to correct the bearings
and distances of a field survey.

At the close of the article he says : << I have applied the principles
of this essay to the determination of the most probable value of the
earth's ellipticity, etc., but want of room will not permit me to give the
investigation at this time." It was published nine years later in Vol-
ume I, new series, of the Transactions of the American Philosophical
Society (papers No. IV and XXVII). In the first paper he finds the
^ earth's ellipticity to be ^ instead of ^4t? ^s was erroneously given by

La Place (La M^canique O^leste, Vol. III). In the second paper Adrain
applies his rule to the evaluation of the mean diameter of the earth,
;rhich he finds to be 7,918.7.

His rule for correcting dead-reckoning at sea was adopted by Dr.
Bowditch in his last edition of his Practical Navigator. Adrain's rule
for correcting a survey is referred to by John Gummere in his Survey-

70 TEACHINO £SD HISTORY OF MATHEMATICS.

ing, as having been given an'd demonstrated by Bowditch and Adrain
in ttie Analyst

It thus appears that these rales of Adrain were made nse of by at least
some of the contemporary American mathematicians, but the principles
£rom which these rules were deduced and the demonstrations of these
principles appear to have excited little attention, and to have been Bo<m
forgotten. Foreign mathematicians never heard of Adrain's investiga-
tions on the subject of least squares until Adrain's first proof and ex-
tracts from other papers were reprinted by Oleveland Abbe in the Amer-
icaq Journal of Science and Arts in 1871, or sixty years after their first
publication in the Analyst. By a very strange oversight Oleveland
Abbe does not even mention Adrain's second proof. The existence of
this proof was pointed out, however, a few yeacs later by Prof. Manaftdd
Merriman.

It is not much to the credit of American mathematicians that thqr
should have permitted theoretical investigations of such great value to
remain so long in obscurity. Let justice be done to Adrain, and let him
be credited ^^ with the independent invention and application of the
most valuable arithmetical process that has been invoked to aid the
progress of the exact sciences."

By the numerous elegant solutions which Adrain contributed to math-
ematical journals in this country, by his labors as teacher at Bntgers
College, Columbia College, and tilie University of Pennsylvania, by his
editions of Button's Mathematics, he contributed powerfhlly to the
progress of mathematical studies in the United States. His first edition
of Button's Mathematics was followed in 1812 by a second edition, and
in 1822 by a third edition, in which he added an elementary treatise of
sixty pages on descriptive geometry, '^ in which the principles and funda-
mental problems are given in a simple and easy manner." Other edi*
tions came out later. Adrain's edition of Button contained improve-
ments in method and important c(»iections, the value of which was reo>
ognised by Mr. Button himself.

It may be well to call to mind at this place that Charles Button was
professcNT of mathematics at the Boyal Military Academy at Woolwich
from 1773 to 1805. Bis course of mathematics was published in Lon-
don, 1798-1801. In its day this work was doubtless the best of its
kind in the English language. But at that time the English were far
behind the French in the cultivation and teaching of matiiematics.
Button's course was plain and simple, but defective both in eztient and
analysis. The English works of that day generally ecmtained rules
without principles, and were decidedly inferior to the explanatory trear
tises of Lacroiz and Bourdon, then used in France. Button's mathe-
matics were used once at our own Military Academy at West Point,
but were soon exchanged for the more analytical and copious treatises
by French authors.

Wo close our remarks on Robert Adrain with the following quotatkm

IKFLUX OF ENGLISH MATHEMATICS. 71

from the Democratic Beview of 1844, Y(d. XIY : << He pablisbed little,
because he was. too severe a critic upon his own writings* He wouU\
revise and re-revise. It is said that while at Golambia, he had a trea*
tise on the differential and integral calculus all written out and ready
for the press } but upon giving it a further revision he became dissat-
isfied with some parts of it, and committed the whole to the flames.''
He left a number of manuscripts with commentaries on the M^canique
Analjtique of Lagrange and the M^canique Celeste of La Place.* ^

* COLLEGE OP NEW JEESEY (PEINOETON).

The College of Ifew Jersey first opened at Elizabethtown, in 1746.
Soon after, it was transferred to ^Newark, and in 1756 to Princeton.
About seventy students moved from Newark to Princeton. The first pres-
ident died after having been in office less than a year. His successor^
Aaron Burr, the elder, held the post for ten years. He was an incessant
worker and toiler. Though he was assisted by two tutors, he was him-
self teacher in Oreek, logic, ontology, natural philosophy, amdin thecal'
eulation of eoKpsesA The courses in physics were illustrated by appa-
ratus which had been obtained from Philadelphia. Popular lectures
were delivered on the new electricity, and both president and students
repeated Franklin's experiment on the influence of pressure on the boil-
ing point, with glass globes of their own.
\ At first the college had no professors with fixed functions and perma-

nent salaries. The instruction in classics and mathematics was com-
mitted to tutors who had lately graduated and were preparing for the
ministry.^ They taught generally for but few years.

The first professor of mathematics and natural philosophy was Will-
iam Ohurchill Houston. In early manhood he entered Princeton College,
taught in the college grammar school, and was graduated in 1768. He
was then appointed tutor, and, in 1771, elected professor. At the be-
. ginning of the Revolutionary War, he and Dr. Witherspoon were the
only professors in the college. When Princeton was invaded in 1776,
and the college was closed, he took active part in the war. As soon as
quiet was restored at Princeton, he resumed his college duties. Soon
after be was sent as a representative to Congress. He resigned his chair
in 1783. In midst of his many duties, he had acquired a sufficient
knowledge of law to be admitted to the bar. As a lawyer he soon ac-
quired great reputation.

John Adams visited Princeton College in the opening days of the
Revolution, when he was on his way to the Continental Congress. In
his diary of August 26, 1774, he says: " Mr. Euston,| the professor of
mathematics and natural philosophy, showed us the library ; it is not
large, but has some good books. He then lead us into the apparatus

*For want of the necessary material, our sketch of the University of Penosylyft-
nia will not be continaed.
t The College Book, by Charles F. Richardson and Henry A. Clark, 1878, p. 97.
t Mr. Houston was probably intended.

72 TEACHING AND HISTOBY OF MATHEMATICS.

room ; here we saw a most beautiful machine — an orrery or planetariam,
constructed by Mr. Kittenhouse, of Philadelphia." It will be remembered
that both the University of Pennsylvania and Princeton GoUege had
been negotiating for the first orrery made by Bittenhouse, and that
Princeton carried it off, much to the chagrin of Dr. William Smith, the
president of the University of Pennsylvania.

The chair of mathematics and natural philosophy was filled two years
iP>fter the resignation of Houston by the appointment of Ashbel Greexu
He was a native of New Jersey, and was graduated at Princeton Gol-
Jege in 1783. He entered the ministry, and was professor for the two
years succeeding 1785. Later, he became president of the institution.

Green's successor was Dr. Walter Minto, a Scotch mathematician
of eminence. He was graduated at the University of Edinburgh, and
then became tutor to the sons of George Johnstone, a member of Parlia-
ment. With them he travelled over much of Europe. In Pisa he be-
came acquainted with Dr. Slop, the astronomer, and through him with
the then novel application of the higher analysis to planetary motion*
After returning to Scotland he became teacher of mathematics at
Edinburgh. He came to the United States in 1786, and one year after
became connected with Princeton College. Before coming to America he
published a Demonstration of the Path of the New Planet; Besearches
into Some Parts of the Theory of Planets } and (with Lord Buchan) an
Account' of the Life, Writings, and Inventions of Napier of Merchiston.

While at Princeton, he delivered on the evening preceding the annual
commencement of the year 1788 <<an inaugural oration on the Progress
and Importance of the Mathematical Sciences." In this address he
traces the history of mathematics down to the time of Newton, then
directs his remarks to the students and trustees, emphasizing the im-
portance of mathematical study. ^^ The genius of Newton," says he,
^< had he been born among the Indians, instead of discovering the laws of
the universe, would have been limited to the improvement of the in-
struments of hunting, or to the construction of commodious wigwams."
At the time when this address was delivered he had been professor at
Princeton about a year. Near the close of his oration he says : *^ It gives
me a deal of pleasure, gentlemen, to have occasion to observe, in this
public manner, that a considerable majority of those of you who have
studied the mathematics under my direction have acquitted yourselves
even better than my expectations, which, believe me, were very san-
guine." This inaugural address is his only publication while he was
connected with Princeton College, but the college library contains some
careful and curious MSS. on mathematical analysis written by him.
Doctor Minto died at Princeton in 1796.

The mathematical duties were now assigned to Dr. John Maclean, a
native of Scotland. In his day he was one of the most distinguished
professors at Princeton, ^* the soul of the faculty." His specialty was
chemistry, which he had studied in Paris. He is said to have been one

INFLUX 0]P ENGLISH MATHEMATICS. 73

of the first to reprodacein America the views of the new French school
in* chemistry. Daring seven years he was professor not only of chem-
istryi bat also of natural history, mathematics, and natural philosophy;
and after a short interval of four years, during which he was relieved
from mathematical instruction by the appointment of Dr. Andrew
Hunter to those duties, he again assumed charge of all the scientific
instruction given to the students. He died in 1814.

From 1812 to 1817, Elijah Slack, a graduate of Priuceton and a min-
ister, was professor of natural philosophy and chemistry. He taught
also mathematics. He was afterwards president of Oincinnati College.
Henry Yethake taught mathematics from 1817 to 1821. In 1823, Mr.
John Maclean, a young man of only twenty-three years, was made pro-
fessor of mathematics.

It may here be remarked that in the library of Princeton Oollege there
is a folio volume of great interest and value, containing a copy of the
first printed edition of Euclid^s Elements in Greek (Basle, 1583); the
commentary of Proclus on the First Book of Euclid (Basle, 1533) ; a
twofold Latin translation (Basle, 1558) — one the Adelard-Oampanus
version, from the Arabic, the other the first translation into Latin from
the Greek from Theon's Revision. This folio was once the property of
Henry Billingsley, who three hundred years ago made the first tnins-
lation of Euclid into English. By the examination of this folio. Dr. Gl
B. Halsted was able to show that the first English translation was made
from the Greek, and not, as was formerly supposed, from any of the
Arabic-Latin versions.*

DARTMOUTH OGLLEaE.

Dartmouth College, at Hanover, was chartered in 1769. Wheelock
w^ the first president, and his first associate in instruction as tutor
was Bezaleel Woodward, who had graduated at Yale in 1764, during
the presidency of Clap, of whom it was said that in mathematics and
natural philosophy ^^ he was not equalled by more than one man in
America."

Three of Dartmouth's first class were prepared for college at the
"Indian Charity School" in Lebanon, and passed their first three years
at Tale.

The facilities for acquiring a classical and scientific edncation appear
to have been substantially the same at Dartmouth, at the outset, as in
other American colleges of that period.t Some notion regarding the
mathematical course at this college may be drawn from a letter written
in 1770 by Nathan Teasdale, a learned and indefatigable teacher in
eastern Connecticut, to Dr. Wheelock, the president of the college,
introducing one of Teasdale's pupils, who applied for admission to the
Senior year. The young man is described as "a g'enius somewhat

*Kote on the First English Enclid, American Journal of Mathematics, Vol. II, 1879.
t History of Dartmouth College, by B. P. Smith, p. 58.

74 TEACHIKG A3SiD HIST0B7 OF MATHEMATICS.

better than common,'^ who ^^ had made excellent progress.^ ^^In arith-
metic, vulgar and decimal, he is well versed. I have likewise taaght
him trigonometry', altimetry, longimetry, navigation, surveying, dial-
ing, and ganging." ^^ He likewise studied Whiston's Astronomy, all
except the calculations." We are, probably, not far from the truth, if
we conclude that the studies here enumerated were, in substance, the
mathematics then studied at Dartmouth during the l^st three years*

The flrst twelve of thirteen years were years of very great trial for
Dartmouth. The funds of the college were small and the students few.
The Revolutionary War, though it did not interrupt the college exercises
and disperse the 8tudent>s, must have diminished their number and
affected their spirits. As in other localities, so in Few Hampshire, the
means of fitting for college were very imperfect and many of the college
studies were inadequately pursued* <^ I remember," says Samuel Gil-
man Brown,* ^< hearing one of the older graduates say that the first
lesson of his class in mathematics was twenty pages in Euclid, the
instructor remarking that he should require only the captions of the
propositions, but if any doubted the truth of them he might read the
demonstrations, though for his part his mind was perfectly satisfied."
In stories like this, however, we must allow something for the genius
of the narrator. This story, if not true, is certainly of the ben tromto
sort. The requirements for admission to American colleger in those
days were low, and the system of choosing the tutors, to whose care
the Freshmen and Sophomore classes were entirely committed, was
enough to destroy any chances of rectifying the errors of bad and
insufficient preparation. Not unfrequently a fresh graduate who had
excelled in classics alone, with very little taste for mathematics, would
be chosen to fill a tutorial vacancy requiring him to teach mathe-
matics, and vice vetsa. The bad consequences of such a system need
not be dwelt upon here.^

We see from the above that Euclid's geometry had been introduced
in the early day» of the college.

In 1700 the studies in college were as follows : t

<(The Freshmen study the learned languages, the rules of speaking,
and the elements of mathematics. The Sophomores attend to the lan-
guages, geography, logic,. and mathematics. The Junior Sophisters,
beside the languages, enter on natural and moral philosophy and com-
position. The Senior class compose in English and Latin; study meta-
physics and the elements of natural and political law.

^* The books used by the students are Lowth's English Grammar,
Perry's Dictionary, Pikers Arithmetic^ Guthrie's Geography, WardPi
Mathematical Atkinson's Epitome, Hammond^s Algebra^ Martinis and
EnfieWe Natural Philosophy^ Ferguson?s Astronomyj Locke's Essay,

'Address before the Society of Alamni of Dartmoath College, 1855, p. 17.
t Barnard's Journal of Edacation, Vol. 26, 1876, lotemational Series, VoL I, p. 878^
quoted hj Judge Crosby from Belknap's History of New Hampshire, p. 2d6.

INFLUX OF ENGLISH MATHEMATICS. 75

Montesqaien's Spirit of Laws, and Barlamaqni's ITatnral and Political
Law." Hammond's Algebra was, we believe, an English work. In a
oatalogue of old, second-hand books we find, ^^ Hammond^ N., Elements
of Algebra in a new and easy method, etc, 8vo. calf, 1742." ><

Of the early graduates of Dartmouth we would mention Daniel
Adams, of the class of 1797, who furnished the schoolboy's satchel with
the Scholar's Arithmetic, one of the best and most popular books of the
time.

Another graduate somewhat distinguished in the mathematical line
was John Hubbard, of the class of 1785. After studying theology, he
became preceptor of the New Ipswich and Deerfield academies in Mas-
sachusetts. Afterwards he was judge of probate of Oheshire County,
N. H. In 1804 he succeeded B. Woodward in the chairof mathematics
and natural philosophy at Dartmouth, and filled it till his death in 1810.
He published an Oration, Budiments of Oeography, The American
Beader, and an essay on Music, but nothing on mathematics.

For twenty-three years, beginning in 1810, Ebenezer Adams was pro-
fessor of mathematics and natural philosophy. In 1833 lie was made
professor emeritus.

For 1824 the mathematical studies as indicated in the catalogue of
the college, was as follows : The Freshmen reviewed ^^ arithmetick " and
then studied algebra during the third term. 'So mathematical studies
are given for the first two terms. The Sophomores were put down for
six books of Euclid during the first term; plane trigonometry and its
usual applications during the second term, and the completion of Euclid
during the third term. The Juniors studied ^^ conick " sections, and
'^ spherick" geometry and trigonometry during the first term. The rest
of the year was given to natural philosophy and astronomy. "So mathe-
matics in the last year.

In 1838 the college course was the same, but algebra to the end of
simple equations was added to the terms for admission.

BOWDOIN COLLEGE.*

When Bowdoin College was first organized, in 1802, the requirements
for admission were an acquaintance with the *' fundamental rules of
arithmetic." Later, the expression ^^ well- versed in arithmetic " is used.
The first definite increase in the requirements did not occur till 1834,
when part of algebra was added.

During the first three years of its existence the college had no regular
professor of mathematics. But in 1805 the faculty was reinforced by
the arrival of Parker Gleaveland, who six years before had graduated
at Harvard first in his class and had been tutor in the university. The
department of mathematics and natural philosophy was assigned to the

* For the greater part of the material used in writing this sketohi the writer is in-
debted to Prof. George T. Little of fiowdoin GoUege.

76 TEACHING AND HI8T0BY OF MATHEMATICS.

youthful instractor. He remainod at Bowdoin till his death in 1868
and earned for himself the enviable reputation of *' Father of American
mineralogy." Gleaveland was professor of mathematics from 1805 till
1835. One of the first books nsed was Michael Walsh's arithmetic,
published at Newburyport in 1801. Webber^s mathematics were taught
for many years, until they wete displaced by Farrar's ^' Oambridge
mathematics."

The course in mathematics at the beginning and JTor twenty years
after, was, in the Freshman year, arithmetic ^ Sophomore year, algebra,
geometry, plane trigonometry, mensuration of surfaces and solids^ Jun-
ior year, heights and distances, surveying, navigation, conic sections ;
Senior year, spherical geometry and trigonometry with application to
astronomy. Algebra was gradually forced back to the Freshman year,
but a part of the first term of this year was given to arithmetic as late
as 1850.

In regard to the instraction in mathematics during the professorship
of Parker Gleaveland, Professor Little sends us a copy of a letter from
their oldest living graduate, the Bev. Dr. T. T. Stone, of the class of
1820. Says he: *< Until near the close of our college life we had but
one professor with the president and two tutors. Professor Gleaveland
added to his duties as teacher of the natural sciences, in particular
chemistry, mineralogy, and such as were contained in Enfield's Natural
Philosophy, those of instructor in mathematics ; although, I think, in
the latter, that is, in mathematics, one of the tutors took part. Of the
tutors who had most to do with this department, I remember Joseph
Huntington Jones, afterward a Presbyterian minister in Philadelphia;
Samuel Greene, well known in later days as minister of— I think, the
Essex Street Ghurch, Boston ; and Asa Gummings, minister soon aft^
of the First Ghurch in North Yarmouth (the North since dropped ofi: ),
and, later still, editor for many years of the Ghristian Mirror, and, if I
am not mistaken, other tutors sometimes assisted in the department, as
Mr. Newman, who Arom tutor became professor of the ancient languages
in the spring of 1820, and afterward professor of rhetoric. It was he,
unless my memory fails me, who took our class out to survey a piece of
land to the north or south-west of what was then the college grounds,
including probably the place where he and Professor Smyth and Pro-
fessor Packard afterward lived — the only thing connected with the
mathematics which I now remember outside of the recitations and the
preparation, such as they were, for the regular exercises.

^^ Of the books then used, the first, and that which went with us, I
am not sure but through the whole course from the Fresbman year to
the Senior, was Webber's Mathematics. The only other book of pure
mathematics was Playfair's edition of Euclid, in which we went through
BO much, I now forget how much, as we had time of the six books which
comprised a large part of the work. Added to this, Enfield's Philoso-
phy took in, with its natural science, not a little of mathematical illus-
tration.

INFLUX OF ENGLISH MATHEMATICS. 77

^< Of the methods of instraction, I have already stated that the only
exception I remember to simple recitation was a single slight piece of
surveying. We were required to study the prescribed lesson in the
book, then to repeat it, not of coarse word for word, bat distinctly, to
the professor or tutor at th0 recitation ; that was alL"

GEORGETOWN COLLEGE.*

Shortly after the close of the American Revolution the idea of es-
tablishing a college in Maryland, then the chief seat of the Catholic
religion in this country, presented itself to the Bev. John Carroll,
afterwards first Archbishop of Baltimore. Buildings were erected for
this purpose in 1789, and a school first opened two ^ears later. It
rapidly grew into favor. Great attention was .then paid to the classic
languages, but only little to mathematics. Until 1806, when the col*
lege came into the hands of the Jesuits, the school was rather of
preparatory grade. At this time a regular college course was arranged.

In 1807 Fr. James Wallace came to Georgetown. He had the classes
here for two years. He was then sent to New York, where he taught in
the *«New York Literary Institution," an offshoot of Georgetown. While
in New York he published a work on the Use of the Globes (New York,
1812 ). He returned to Georgetown in 1813 or 1814, and remained there
until 1818, when he removed to Charleston, S. C. In 1821 his connec-
tion with the Society of Jesus was severed. After leaving Georgetown,
he was for several years professor of mathematics in the South Caro-
lina College. During his second stay at Georgetown be solved a prob-
lem proposed by the French Academy ; as a reward they sent him
many fine mathematical works. Professor Wallace was a man of ability,
and a most x^atient and successful teacher.

Bev. Thomas C. Levins, born March 15, 1791, taught here from 1822
till 1825. He studied at Edinburgh, under Leslie, and then* taught at
Stonyhurst College, England. In 1825 he went to New York. Dr.
Shea, in his Catholic Church in the United States (p. 403), states that
Fr. Levins was one of the engineers of the Croton Aqueduct. He died in
New York, May 6, 1843.

We have not been able to obtain more definite information on the
early mathematical teaching at this college.

UNIVEESITT OF NORTH OAEOLINA.

The first impulse towards the establishment of a university in North
Carolina came, it seems, from the Scotch-Irish element occupying the
midland belt of the State. ^' The early emigrants and settlers of this
people brought their preachers, who also filled the office of teachers for
the young. Tradition informs us that the most, popular and best sus-
tained of these nurseries of the young were located in the influential

* The above information is drawn mainly from a letter of Prof. J. F. Dawson, 8.
J., professor of phyBics and mechanics at Georgetown College.

78 TEACHINO AND HISTORY OF MATHEMATICS.

oonntiM of Iredell, Mecklenbnrgh, Oailford, and Orange. It was from
these narseries came the desire for higher education that formulated
the article that decreed a State university. Doubtless the granting of
a charter for William and Marjr and for Harvard by the royal preroga-
tive of the mother country, and the refusal of a like charter to Queen^s
Gollege at Charlotte, in Mecklenburgh, during the colonial govemment,
angered the hametSj fired the resentment of the Bevolutionary pa-
triots, and quickened their action in the blessings of liberty under the
shield of the new- bom Bepublic.''*

The doors of the university first opened for the admission of students
in 1795. It was organized after the model of Princeton OoUege, which^
in turn, was patterned after the Scottish universities. Shortly afiber the
University of Korth Oarolina had begun, Oharles W. Harris, a graduate
of Princeton OoUege, was elected to the professorship of mathematics.
He occupied this chair for only one year. It had been his original pur-
pose to study law, and after one year's experience in teaching he re-
signed in order to .enter the legal profession. He was regarded as a
man of considerable ability, but he died at the age of 33.

He was succeeded by Bev. Jo8ei)h Caldwell, who was also a graduate
of Princeton and a native of New Jersey. He had been one year tutor
at his alma mater. This remarkable man gave for nearly forty years
his best energy to the interests of the university. In 1804 he was
elected president, which office he retained till his death in 1835, with
the exception of four years, from 1812 to 1816, during which period he re-
tired voluntarily to the professorship of mathematics so as to secure
more time for the study of theology.

At first the faculty was very small. In 1814 it consisted of ^^ President
Caldwell, Professor Bingbam, and Tutor Henderson. Their college
titles were <01d Joe,' 'Old Slick,' and ^Little Dick.' < Old Joe,' how-
ever, was only thirty years of age, and possessed * * * a formi*
dable share of youthful activity ."t

It is not generally known that Dr. Caldwell, in August, 1832, com*
pleted the first college observatory built in the United States. '^ It
was," says Professor Love, '^ a brick structure about 26 feet high, and
contained a transit, an altitude and azimuth instrument, a portable
telescope, an astronomical clock with mercurial pendulum, and other
minor apparatus, all of which he bought in London in 1824 from the
best makers. For want of means and interest, however, the observa-
tory, after Dr. Caldwell's death in 1835, was permitted to go down, and

* Address by Paal C. Cameron in the inangoral proceedings at the Uniyersity of
North Carolina, June 3, 1885, p. 9. AU the material for this sketch of mathematical
teaching at that nniversity has been famished to the writer by Prof. James Lee
Love, associate piofesaor of mathematics at the University of North Carolina. Prof.
Love has not only forwarded pamphlets, but has himselt made careful investigation
into the history of the institation, and kindly commonicated hia resolta to the
writer.

i Fifty Years Slnoe, by William Hooper, 1869, p. 10.

INFLUX OF ENGLISH MATHEMATICS. 79

even the recorcU of observations made there from 1832 to 1835 are not
now known to exist."*

Professor Oaldwell was a man of liberal and progressive views. He
laid wisely the foandatioDS of a great university in library and phllo-
Bophioal apparatnsy as well as in the courses of stndy and in the men he
gathered aronnd him in his fia.calty. In remembrance of his long and
untiring devotion to the institution a monument has been erected to
him, by the alumni, in a grove surrounding the university.

When Caldwell went to Chapel Hill he found the college in a feeble
state, nearly destitute of buildings, library, and apparatus ; the stu-
dents were very rough. We read of *' unpleasant upheavals and vol«
canic eruptions ^ among them. Moreover, the bill of fare with which
the minds of the students were obliged to content themselves was very
meager. For admission in mathematics the elements, of arithmetic
were required from t&e beginning, in 1705, to 1835. In 1800 the require-
ment was ^^ arithmetic as far as the rule of three;" in 1834, ^^ arithmetic
to square root." In our early arithmetics the rule of three was given
for integers before fractions were touched upon, and we imagine that
fractions were not required for admission, nor even any knowledge of
integral arithmetic beyond the merest elements. The mathematical
course offered in 1795 was as follows : (1) Arithmetic in a scientific man-
ner ; (2) algebra^ and the application of algebra to geometry ; (3) Eu-
eUd^s elements ; (4) trigonometry Skud its application to mensuration of
heights and distances, of surfaces and solids, and to surveying and navi-
gation; (5) Vonio sections; (6) the doctrine of the sphere and cylinder ;
(7) the projection of the sphere; (8) spherical trigonometry; (9) the doc-
trine effluxions ; (10) the doctrine of chances amd annuities^ << The first
/btfr courses," says Professor Love, "were to be required for graduation.
The remaining courses were to be taught if requested^ but they were not
requested!^

The text-books used prior to 1868 cannot now be entirely deter-
mined. The first algebra used was probably Thomas Simpson's* It
was certainly studied in 1803 and in 1815, and, perhaps, as late as 1826.
The first geometry studied was Robert Simson's Euclid. On the ap-
plication of trigonometry to mensuration, Ewing's Synopsis was used
flj'St— certainly as early as 1798. About 1810 President Caldwell pre-
pared a course in geometry, based on Simson's Euclid. This was used by
the students in manuscript, copies having been made and handed down
from class to class. Hutton's Geometry was introduced in 1816. In 1822
Dr. Caldwell published his geometry, under the title, <«A Compendious
System of Elementary Geometry." It was used for some years. Bound
with this book in one volume was a treatise on trigonometry. The
plane trigonometry was prepared by himself; the spherical was Robert
Simson's. Ifo record has been found as to the trigonometry used prior
to 1822, though Simson's was probably the one. It does not appear that

PWOTvvwnw

* See alio Mi artiole by Pf of esiior Love in the Nation for Aagoat 16| 1888.

80 TEACHING AND HISTORY OF MATHEMATICS.

the study and use of logarithms was introdaoed until 1811. Natural
philosophy and astronomy were taught from the beginning. Ferguson's
text was the one first used. In natural philosophy Nicholson's was
^ used down to 1809, then Helsham's until 1816.

Dr. Caldwell was professor of mathematics from 1796 to 1817, but his
activity extended in many other directions. He ^^ taught mathematics,
natural and moral philosophy, and did all the preaching." An inter*
esting, though one-sided, picture of him as a teacher of geometry
(about the year 1810) is given by William Hooper, one of the alumni :
«< There being but three teachers in college (president, professor of lan«
guages, and tutor), the Seniors and Juniors had but one recitation per
day. The Juniors had their first taste of geometry, in a little element-
ary treatise, drawn up by Dr. Oaldwell, in manuscript, and not then
finished. Copies were to be had only by transcribing, and in process
of time, they, of course, were swarming with errors. But this was a
decided advantage to the Junior, who stuck to his text, without mind«
iug his diagram. For, if he happened to say the angle at A was equal
to the angle at Bj when in fact the diagram showed no angle at B at
all, but one at C, if Dr. Caldwell corrected him, he had it always in his
power tS say, ^^ Well, that was what I thought myself, but it ain't so
in the book, and I thought you knew better than I." We may well
suppose that the doctor was completely silenced by this unexpected
application of the argumentum ad hominem. • • • The Junior hav*
ing safely got through with his mathematical recitation at eleven o^dock,
was free till the next day at the same hour."* It will be rememberedy
that the blackboard — ^that simple machine which doubles the teaching
power of an instructor in geometry — was then unknown in America.

Fluxions were not taught at that time. William Hooper says in bis
humorous way, <* As for chemistry and dififerential and integral calculus,
and all that, we never heard of such hard things. They had not then
crossed the Boanoke, nor did they appear among us till they were
brought in by the northern barbarians about the year 1818."t These
northern barbarians were Elisha Mitchell and Denison Olmsted. The
latter introduced chemistry, mineralogy, and geology into the univer-
sity. Dr. Mitchell was a New Englander. He graduated at Yale in the
class of 1813 with Olmsted. He began teaching immediately after grad-
uation, and in 1816 was appointed tutor at Yale. At the University of
North. Carolina he held the chair of mathematics from 1817 to 1825, and
performed his duties with energy and success. When Dr. Olmsted was
called to Yale, he assumed the vacant chair of chemistry, which posi-
tion he filled with great credit until his death in 1857. He lost his life
by falling over a precipice, in the darkness, while engaged in the scien-
tific exploration of Mitchell's Peak in western North Carolina.

In 1818, after the arrival of Mitchell, spherical trigonometry, conic
sections, and fluxions were introduced into the course of study leading

'Fifty Years Since, p. 83. t Page 17 of his address.

INFLUX OF ENGLISH MATHEMATICS. 81

to the degree of A. B. The coarse was as follows : Freshman year^ arith-
metic completed, algebra begun ; Sophomore year^ algebra completed,
geometry ; Junior year^ plane trigonometry, logarithms, mensuration,
navigation, spherical trigonometry, conic sections, fluxions ; Senior yea/r^
astronomy, natural philosophy. It wiU be noticed that the course began
now in tbe Freshman instead of the Sophomore year, as formerly. If it
was faithfully carried out, then it must have been very creditable to the
institution at tbat time. It remained nearly unchanged for seventeen
years. As regards the text-books, it is probable that Simpson's Al-
gebra was used by Mitchell ; also Button's, and since 1822, OaldwelPs
Geometry and Trigonometry, and Yince's Oonic Sections. In 1823,
Day's work on mensuration was taught. Ko record has been found as
to what text-books were used when fluxions were first introduced. It
is possible, however, that Yince's and Button's were the ones. In as-
tronomy Kicholson's was used for a long time. Cavallo's Ifatural Phi-
losophy and Wood's Mechanics were used, the latter since 1822.

Mr. Paul O. Oamerob gives an interesting reminiscence of B. F.
Moore, a once prominent lawyer. ^< Often has he entertained me," says
Oameron, ^^ with the lights and shades of his college life; how^grandly
he marched through the recitations in the languages taught in the first
and second years of his college life ; how deep and suddenly he went
und&r when he struck the mathematical course of the Junior year ; how
he wrote to his father and appealed to him to take him home and place
him behind the plow. Bis father refuses^ and tells him to make known
his difficulties to his professor. Be hands his father's letter to Dr.
Mitchell, who invites him to his study and gives him instruction by the
use of his knife and a piece of white pine, cutting for him blocks of
mathematical figures, to be used in the demonstrations of his proposi-
tions. Turning the light on him in this way, he was enabled to con-
tinue his course with satisfaction."

UNIVERSITY OF SOUTH CAEOLINA.*

The South Carolina College threw open its doors for students in Jan-
nary, 1805. The first mathematical teacher at the college was Elisha
Bammond of Massachusetts. Be was a graduate of Dartmouth Col-
lege, and when called to this position, was principal of Mt. Bethel
Academy in Kewberry, S. C. After one year's service he resigned and
returned to the academy. Judge Evans, a student under him, says
" Bis personal appearance and manners were very captivating, and his
popularity for the period of his connection with the college was scarcely
inferior to that of Dr. Maxey." Dr. Maxey was the president.

Eev. Joseph Caldwell, the father of the University of I^orth Carolina,
was then invited to the chair of mathematics, but he declined. Paul
B. Perrault was elected to the place, but in 1811 he was removed for

* For the larger part of oar information respecting this institution, we are indebted
to Professor E. W. Davis, professor of mathematics and astronomy at the university.

881— H*o. 3 6

82 TEACHING AND HISTORY OF MATHEMATICS.

^^ neglect of college daties." He is said to have been ^^ well Bkilled in
mathematics,'' but '^ wanting in that dignity which a Freshman wonld
expert in a learned professor." After his separation from the college
he was attached to the Army as a topographical engineer.

The mathematical professor for the next four years was George Blaek-
bnm; He was a graduate of Trinity Gollege, Dublin. He taught in a
military academy in Philadelphia; afterward he was teacher in Virginia)
and was then called to the chair of mathematics and astronomy in Will-
iam and Mary College. Thence he went to the South Carolina College.
In 1812 he was employed by the State of South Carolina to run the
boundary line between North and South Carolina. An old student
says of him : ^^ He was a man of quick and vigorous understanding, an
able mathematician, and a most excellent instructor." Another : ^^ Pro-
fessor Blackltnrn was a first-rate mathematician ; he taught mathemair
ics as a science, and not as a matter of memory* From him I learned
the demonstration of many difficult problems ; and with his aid I under-
stood much of that abstruse and difficult science as applied to natural
philosophy and astronomy." He made students thiuk. What detracted
somewhat from his power as a teacher was his irritability.

In the better colleges of that day, the curriculum in mathematics em-
braced a short course on fluxions, or calculus. Though the plans of
study included then about all the subjects pursued in the avenge
American college of to-day, these subjects were not taught with the
same thoroughness. Moreover, we are now teaching at^least twice as
much under each branch as was taught at the beginning df this century.
In consequence of this, students of former times began the study of
fluxions when, for lack of prepatory drill in lower branches, they were
far less able to wrestle with the difficulties of the transcendental analysis
than are our students of to-day. Professor Blackburn's teaching of the
calculus, as narrated by M. La Borde, in his History of South Carolina
College (p. 82), presents a picture of a Senior class vainly struggling
with the intricacies of this subject. The class lost interest in the study
and was very remiss in its attendance upon him, and those who did
attend failed so completely in unraveling the mysteries of the transcen-
dental analysis, as to force from the lips of the professor the remark,
^^ that it might be that half of his class were very smart fellows, for he
never saw them ; but the half who attended his recitations were as la-
borious as oxen, but as stupid as asses." It need hardly be said that
this was the cause of a students' rebellion.

After leaving the college. Professor Blackburn made latitude and
longitude observations for the State map, under Governor Allston.
Later he settled in Baltimore, where, with Dr. Jennings, he founded
Asbury College. His last days were spent in Colombia, S. C.

From 1815 to 1820, Christian Hanckel, a Philadelphianand graduate
of the University of Pennsylvania (class of 1810), was professor of
piatb^matics. He took hol;^ orders at St. Michael's, Charleston. His

INFliUX OF ENQU8H MATHEMATICS^ 83

miMO indaoement to aooept the ehair was the obaiace to build Qp tbe
Protestant ISpiacopal Charch in Golambia. On leaving the college^ he
went to St. FaaFs Ohurchy Charleston.

The requirements for admission were, according to oatalogue, at the
beginning, ^< arithmetic^ including proportion," This, most probably,
did not include fractions. In 1836 the terms were <^ arithmetic, includ-
ing fractions and the extraction of roots."

In the earliest course of mathematics at this college, the Freshmen
took up arithmetic } the Sophomores, common and decimal fractions with
extraction of roots } the Juniors, geometry, and theoretical and practi*
cal astronomy ; the Seniors, exercises in higher mathematics as directed
by the faculty. We are not certain that this curriculum embraced alge-
bra. If taught then, it was a Senior study. Fractions were, it seems,*
not only not required for admission, but were not studied before the
Sophomore year.

The course for the year 1811 was considerably stronger. The Fresh-
men were instructed in vulgar and decimal fractions, with extraction
of roots ; the Sophomores had lectures on algebra ; the Juniors studied
Button's course of mathematics ; the Seniors had Icctares by the '^ pro-
fessor of mathematics, mechanical philosophy, and astronomy. " From
the anecdote told of Professor Blackburn, we know that at this time, or
soon after, " calculus " (probably fluxions) was taught in thtf fourth
year.

KENTUCKY UNIVERSITY.

About the year 1785 was opened in Lincoln Oonnty, Kentucky, a
school called the Transylvania Seminary. Four years later it was
mov^to Lexington, Fayette County, where, in 1790, was held << the first
public college commencement in the West of which we have any record.'^
On January 1, 1799, the Transylvania Seminary and a similar school,
called the Kentucky Academy, were united under the name of Transyl*
vania University. The Transylvania University existed under this
name until 1865, when it was merged in Kentucky University, and the
consolidation has since been conducted under the name and charter of
the latter.

Little has been done, in the past, to preserve the history of these in-
stitutions. Some of the records appear to have been lost, and those that
are still extant give but little general information. The data on the
very special subject of mathematical teaching are exceedingly meagre*
The little information we are about to give was kindly furnished us by
President Ohas. Louis Loos, of Kentucky University.

The records of Transylvania University show that on September 16,
1799, ^^ mathematics " was one of the subjects taught. On October 18,
of the same year, the following books are mentioned in the mathemati*
cal course : First yeatj " Geography by Guthrie or Morse ; algebra by
Sannderson, Simson's Euclid, six books; trigonometry and mensura-
tion^ Gibson; Navigation, Patoun or Morse; Simson's conic sections."

84 TEACHING AND HISTORY OP MATHEMATICS.

Second year, " Natural philosophy and astronomy, Ferguson.^ These
data are by no means destitute of interest. They show from what
sources the young mathematician '^ in the West " drew his intellectual
food, in early days. On October 26, 1799, Rev. James Ely the was
elected professor of <' science," which term was made to include mathe-
matics. In 1803 tbe professor of science (J. Blythe) is called professor
of mathematics and natural philosophy, and his duties were to teach
" geography, arithmetic, algebra, geometry, surveying, navigation,
conic sections, natural philosophy, and astronomy." In 1805 the course
was the same as the one just given, except that geography, arithme-
tic, and surveying are not mentioned.

The entry in the records for March, 1816, gives the following course
in mathematics :

Freshmen, first six books of Euclid, plane trigonometry, surveying,
navigation, geography; Juniors, algebra as far as affected equations,
spherical trigonometry, conic sections, natural philosophy, ancient
geography; Seniors, astronomy." In 1817 Webber^s mathematics is
mentioned as a text-book.

THE UNITED STATES MILITABY ACADEMY.

The germinal idea of the United States Military Academy was put
forth by George Washington, who felt, probably more than any one
else, the necessity of having accomplished engineers in time of war.
The Military Academy was established by Congress in 1802. The act
was limited in its provisions and did not raise the academy above a
military post, where the officers of engineers might give or receive in-
struction when not on other duty. The major of engineers was superin-
tendent, the two captains were instructors, and the cadets were pupils.*

The major was Jonathan Williams ; the two captains were William
H. Barron and Jared Mansfield.

Major Williams, in a report to the Government in 1808, gives us some
notion of the early instruction at the academy. He says that ^^ The
major occasionally read lectures on fortifications, gave practical lessons
in the field, and taught the use of instruments generally. The two
captains taught mathematics, the one in the line of geometrical, the
other in that of algebraical demonstrations." Mansfield taught also
natural philosophy. He had previously been teacher of mathematics
and navigation at I^ew Haven, and then at Philadelphia. He had pub-
lished ^'essays" of some originality on mathematics and physics.
They fell under the notice of Thomas Jefferson, and were the means
that led to his appointment by the President as captain of engineers
for the very purpose of becoming teacher at West Point. But after one
year's teaching he was appointed by Jeffersouj in 1805, to establish
meridian lines and base lines in the North- West Territory for the pur-

* The U. S. Military Aoademy at West Point, by Edwaid D. Mansfield, LL. D.

INFLUX OF ENGLISH MATHEMATICS. 85

pose of public surveys. His position remained vacant until his return,
after the War of 1812.

In 1806 Alden Partridge became assistant in mathematics. He was
a native of Vermont, had entered Dartmouth College in 1803, but be-
fore completing his course became cadet at West Point.

Captain Barron was relieved in 1807 by Ferdinand R. Hassler, who
continued ther-e until 1810, when he resigned. The following year he
was called to the United States Coast Survey. Hassler was a Swiss.
It was again the keen eye of President Jefferson that recognized the
. talent and secured the services of this foreigner, who had shortly before
landed on our shores. Hassler's teaching power must have been ham-
pered somewhat by his limited acquaintance with the English lan-
guage. While at West Point he began writing his "Elements of An-
alytic Trigonometry," published by him in 1826. It was written in
French and then translated for publication by Professor Eenwick.
From its preface we take the following : «' It was the desire of intro-
ducing into the course of mathematics at West Point the most usefal
mode of instruction in this branch that led me to the preparation of
this work as early as the year 1807.'^ Hassler was, no doubt, the first
one to teach analytic trigonometry in this country — ^the first one to dis-
card the old "line-system.''

I About the same time Christian Zoeller, also a Swiss, was made in-
structor in drawing. He was "an amiable man of no high attain-
ments."

Down to the year 1812 the academy was in a chaotic condition.
There was no regular corps of instructors, and no regular classes.
There had been no continuous study of any subject except mathe-
matics. Referring to Hassler, Major Williams says in his report of 1808,
" During the last year a citizen of eminent talents as a mathematician
has been employed as principal teacher," and " being the only teacher
designated by the law, he is the only one that, exclusive of the corps
of engineers, can be said to belong to the institution." In conclusion
the major says : "In short, the Military Academy as it now stands is
like a foundling, barely existing among the mountains, nurtured at a
distance out of sight, and almost unknown to its legitimate parents."

In vain did Jefferson in 1808 and Madison in 1810 recommend to
Congress the enlargement of the academy. It was not until the nation
was roused by the shock of war that Congress began to act. In 1812
Congress made liberal appropriations and passed an act reorganizing
the institution. The next five years are the formation period of the
academy. The first reform to be accomplished was the placing of the in-
struction on a higher level. The first academic faculty was constituted
as follows : Col. Jared Mansfield, professsor of natural and experimental
philosophy j Andrew Ellicott, professor of mathematics ; Alden Part-
ridge, professor of engineering j Christian Zoeller, professor of drawing.
We see from this that Mansfield held now the same place as in 1804,
and Partridge was promoted from assistant to the rank of professor.

86 TEACHING AND HISTORY OF MATHEMATICS.

Mansfield and EUicott had long been in the service of the General Oo^-
ernment and of State governments in the capacity of sarveyors ai«d
astronomers, and had established a wide reputation for both their prac-
tical and theoretical knowledge of mathematics. Bat now they were
old men, and their ideas were somewhat old-fashioned. The workings
of this faculty were not altogether harmonious^ Partridge^ being
strong-willed and eccentric, wanted to have everything.his own way.
He was removed from his place. The appointment of Major Sylvanus
Thayer, in 1817, to the superintendency ctf the academy marks a new
era in its history.

Some notion of the instruction in mathematics at West Point between
1812 and 1817 may be obtained from the folloTing extract from the cur-
riculum which, in 1816, received the official approval of the Secretary of
War.

^^Mathematics. — A complete course of mathematics shall embrace the
following branches, namely : The nature and construction of logarithms
and the use of the tables ; algebra, to include the solution of cubic
equations, with all the preceding rules ; geometry, to include plane and
solid geometry, also ratios and proportions, and the construction of
geometrical problems, application of al gebra to geometry, practical geom»
etry on the ground, mensuration of planes and solids ; plane trigonom-
etry, with its application to surveying and the mensuration of heights
and distances ; spherical trigonometry, with its application to the solu-
tion of spherical problems ; the doctrine of infinite series; conic sections,
with their application to projectiles; fluxions, to be taughc at the op-
tion of the professor and student.'^

There was, however, no instruction in fluxions. E. D. Mansfieldi in
his historical sketch of the academy, does not include fluxions in the cur-
riculum for 1816, but he I'emarks that calculus was added to the course a
year or two later. The text- book then in use was Button's Mathematics.

Thus far the cadets were admitted to the academy without entrance
examinations, and poor results were reached. Many cadets were unfit
by prior study for the subjects they had to pursue. Bank and assign-
ment to the various army corps were not made to depend upon merit.*

Self-taught Mathematicians.

The foremost American mathematician of this time, like David Bit-
tenhouse and Thomas Godfrey, had not enjoyed the privileges of a col-
lege education ; like them, he was self-taught. We have reference to
Nathaniel Bowditch.f

* The CoUege Book, edited by Charles F. Bichardsoa and Henry A. Clark, p. 216.

tXhis sketoh is extracted from the Memoirs of Nathaniel Bowditch, by his son,
Kathaoiell. Bowditch (Boston, 1839); from the Discourse on the Life and Character of
Nathaniel Bowditch, by Alexander Young (Boston, 1833) ; from the eulogy by Pro-
fessor Pickering (Boston, 1838), and from the eulogy by Judge Daniel A. White
(Salem, 1838). A full list of Bowditch's mathematical papers may be found in the
Mathematical Monihly, Vol. II.

INFLUX OF ENGLISH MATHEliATIGS. 87

•

It is instractive to study the history of his early life and to ascertsdn
the inflaences under which his mind was formed, fie was bom at
Salem, Mass., in 1773* His parents were poor, and he had often to con-
tent himself with a dinner consisting chiefly of potatoes, and at near
approach of winter to continue wearing the thin garments of summer.
After attending for a short time a dame's school near Salem, he en-
tered Watson's school, which was the best school in Salem. It was
wholly inadequate to furnish the ground-work and elements of a re-
spectable education. He entered the school at the age of seven and
remained there three years.

Bo^ditch early showed a great fondness for mathematics ; but on
account of his extreme youth his master refused to admit him to this
study until he had procured from his father a special request to that
effect. On one occasion he solved a problem in arithmetic which th0
instructor thought must be far above his comprehension. On being
asked who had been doing the sum for him he answered, '^ Nobody — I
did it myself." He was then accused of falsehood and treated with
much severity.

When he was ten years old he left school to work in the shop of his
father, who was a cooper. He received no regular instruction after
leaving school, excepting a few lessons in book-keeping. He became
soon after an apprentice to a ship-chandler, and afterward was clerk
in a large mercantUe establishment. It was during his apprenticeship
that he disclosed that strong bent for mathematical studies. Every
moment that he could snatch from the counter was given to the slate.
When he was only fifteen years old he made an almanac for the year
1790, containing all the usual tables, calculations of the eclipses and
otiier phenomena, and even the customary predictions of the weather.

When he was fourteen years old he one day got &om an elder brother
a vague account of a method of working out problems by letters instead
ot figures. This novelty excited his curiosity; he succeeded in bor-
rowing an algebra, and *^ that night," says he, *' I did not close my
eyes." He read it, and read it again, and mastered its contents ; and
copied it out from beginning to end.

Subsequently he acquired access to an extensive scientific library of
Dr. Bichard Kirwan, an Irish scientist, which had been captured in the
British channel by a privateer and sold to a society of gentlemen at
Salem. This became the ba^is of the present Salem Atheneum. He
found there the Philosophical Transactions of the Boyal Society of Lon-
don, from which he made full and minute abstracts of the mathematical
papers contained in them. At this time he was too poor to buy books,
and this was the only way in which he could manage to have them for con-
venient reference. The title page of one of these manuscript volumes
states that it contains, with the next volume, ^< A complete collection of
all the mathematical papers in the Philosophical Transactions ; extracts
from various encyclopedias; from the Memoirs of the Paris Academy;

88 TEACHING AND HISTORY OF MATHEMATICS.

•

a complete copy of Emerson's Mechanics ; a copy of Hamilton's Conies;
extracts Irom Gravesande's and Martin's Philosophical Treatises ; from
Bemoalli, etc.^ etc." What perseverance^ what energy, what enthusi-
asm is displayed in this laborions work of copying!

Bowditch was very fond of books, bnt having no guide in the selection
of them his reading was at first of the most miscellaneous character.
Thus he read every article in Chambers' EncyclopsBdia from beginning to
end. He secured a copy of Newton's Principia, but as it was published
in Latin he began the study of that language that be might read that
great work. By great perset^erance he learned enough Latin to enable
him to read any work of science in it. He afterward learned Fsencb
for the purpose of having access to the treasures of French mathematical
science, and at a late period of his life he acquired some knowledge of the
German language. When twenty-one years of age he had read the im-
mortal work of Kewton, and there were few in his State who surpassed
bim in mathematical attainments.

Bowditch did not long remain in the situation of a merchant's clerk*
His mathematical talent, in a town distinguished for enterprise, could
not fail of being called into exercise in connection with the art of navi-
gation. He became a practical navigator. Between 1795 and 1804 he
was on five sea voyages, all under the commandof Captain Henry Prince,
of Salem.

The leisure of the long East India voyages, when the ship was lazily
sweeping along under the steady impulse of the trade winds, afforded
him fine opportunities for pursuing his mathematical studies, as well as
for indulging his taste in general literature. The French mathemati-
cian Lacroix acknowledged to a young American that he was indebted
to Mr. Bowditch for communicating many errors in his works, which he
had discovered in these same long India voyages. It was his practice
both when at home and when at sea to rise at a very early hour in the
morning and pursue his studies. He was often seen on deck ^%alking
rapidly and apparently in deep thought, when it was well understood
by all on board that he was not to be disturbed, as we supposed he was
solving some diflicult problem, and when he darted below the conclu-
sion was that he had got the idea ; if he were in the fore part of the ship
when the idea came to him, he would actually run to the cabin, and his
countenance would give the expression that he had found a prize."

"He loved to study himself," says Captain Prince, ^' and he loved to
see others study. He was always fond of teaching others. He would
do anything if any one would show a disposition to learn. Hence," he
adds, " all was harmony on board ; all had a zeal for study ; all were
ambitious to learn." On one occasion two sailors were zealously dis-
puting, in the hearing of the captain and supercargo, respecting sines
and co-sines. As a result of his teaching, the whole crew, yea, even the
negro cook, acquired the knowledge of how to compute a lunar obser*
vation. When the captain once arrived at Manila, he was asked how he

INFLUX OP ENGLISH MATHEMATICS. 89

contrived to find his way, in the face of a northeast monsoon, by mere
dead reckoning. He replied, that " he had a crew of twelve men every
one of whom could take and work a lunar observation as well for all
practical purposes as Sir Isaac Newton himself, were he alive.'' Dur-
ing this conversation Bowditch sat <^ as modest as a maid, saying not
a word, but holding his slate pencil in his mouth;" while another
person remarked, that <^ there was piore knowledge of navigation on
board that ship than there ever was in all the vessels that have floated
in Manila Bay.

At that period the common treatise on navigation was the well known
work of Hamilton Moore, which had occasioned many a shipwreck, but
which Bowditch, like other navigators, was obliged to use. He found
it* abounding with blunders and overrun with typographical errors.
Of these last errors many thousands of more or less importance were
corrected in the early revisions of the work. Bowditch published sev.
eral editions of Moore's works under that author's name, but the whole
book at length underwent so many changes and radical improvements
as to justify him to take it out on his own name. This is the origin
of Bowditch's Practical Navigator, the best book on navigation then
in existence. The following particulars regarding the publication of
this work have been handed down to us :

The first American edition was printed in 1801, but not published until
1802. The publisher, Mr. Blunt, took the work and a copy of Hamilton
Moore, with all the errors marked, to England, called on the publishers
of Hamilton Moore, and sold the printed copy of Bowditch on condi-
tion that the American edition should not be sold until June, 1802, to
give them an opportunity to get theirs into the English market at the
same time. The London edition was revised and newly arranged by
Thomas Kirby, teacher of mathematics and nautical astronomy. It
was recommended as an improvement on Bowditch, but it contained
many errors. This gave occasion to a British writer, Andrew Mackay^
who published a rival work on navigation, to make Dr. Bowditch's sup-
posed inaccuracies a particular object of attack.* This charge was em-
phatically repelled by Bowditch in the new edition of 1807.
. From Harvard College Bowditch received the highest encouragement
to pursue his scientific studies. In July, 1802, when his ship was wind-
bound in Boston, he went to attend the commencement exercises at
Harvard ; and among the honorary degrees conferred, he thought he
heard his own name announced as a master of arts } but it was not till
congratulated by a friend that he became satisfied that his senses had
not deceived him. He always spoke of this as one of the proudest
days of his life, and amid all subsequent distinctions conferred upon
him from foreign countries, he recurred to this with greatest pleasure.

* Memoirs of American Academy of Arts and Science, Vol. II, 1846, Eulogy on
Bowditch, note C.

90 TEACHIKG AND HISTORY OF MATHEMATICS.

On quitting the sea, in 1803, he was appointed president of an in-
surance company in Salem, the duties of which he continued to dis-
charge for twenty years, when he accepted the position of actuary of
the Massachusetts Hospital Life Insurance Company in Boston. For
many years he discharged the duties of this office with the greatest
fidelity and skill.

He was several times solicited to acjsept positions in various literary
institutions. In 1806 he was chosen to fill the Hollis Professorship of
Mathematics at Harvard. He received from Thomas Jefferson the offer
of the professorship of mathematics at the University of Virginia.
Jefferson said in his letter: << We are satisfied we can get from no coun-
try a professor of higher qualifications than yourself for our mathe-
matical department" In 1820 he was asked to permit his name to be
presented to the President of the United States to fill a vacant chair
at the IT. S. Military Academy at West Point. Bowditch could not be
persuaded to accept any of these positions.

The work for which Bowditch was for a long time exclusively known
was his Practical l^avigator. This gave him a wide-spread popularity
among sea faring people everywhere. Bowditch himself did not con-
sider this work as one which would advance his scientific reputation.
What established his celebrity as a man of science was not his Practical
Navigator, but his translation, with a commentary, of the epoch-mak«
ing work of Laplace, called the M^canique G61este.

Later on we shall speak of this translation at length. Bowditch con-
tributed many articles to the American edition of Bees's Oyclopsedia.

The question may be asked, how should Bowditch be ranked as a
mathematician f In answer to this we may say, that he is acknowledged
by all as having stood at the head of scientific men of this country, and
to have contributed more to his country's reputation than any contem-
porary scientist. But a giant in Liliput is not necessarily a giant in
another country. Though a man of great energy and intellectual powetB,
he can not be pronounced a first-class mathematician. He was a man
of learning, but not a man of genius in the sense that Newton^ Leibnitz,
Cthuss, Abel, Pascal, and Archimedes were men of genius. The esti-
mate that Bowditch made of his own capacities and gifts was, in our
opinion, accurate, fair, and just. He did not overrate his talents, nor
did he, with assumed humility, purposely undei rate his powers, fie is
reported as having once said, '^People are very kind and polite, in men-
tioning me in the same breath with Laplace, and bleuding my name
with his. But they mistake both ne and him; we are very different
men. I trust I understand his works, and can supply his deficiencies,
and record the successive advances of the science, and perhaps append
some improvements. But Laplace was a gcuius, a discoverer, an in-
ventor ; and yet I hope I know as much about mathematics as Playfair I "

The career of Bowditch furnishes as with an excellent illustration of
how much may be accomplished through indefatigable energy and per-

INFLUX OF ENGLISH MATHEtlATICS. 91

servance by a mind which, though naturally far above the average
mind, is, nevertheless, lackimg the powers of real genius.

A mathematician of considerable local reputation was Enoch Lewis
(1776-1856). He was a native of Pennsylvania and a Quaker. In 1799
he became teacher of mathematics at the West Town Boarding School,
established by the Society of Friends. He was the author of treatises
on arithmetic, algebra, and trigonometry. ,

Under the tuition of Enoch Lewis, for six months, at the Friends'
Boarding School at West Town, was John Gummere, who was then
about twenty years old. Excepting in reading, writing, and arithmetic,
he had received no instruction whatever up to that time. After teach-
ing elementary schools for six years, he determined in 1814 to open a
boarding school in Burlington. The following story characterizes the
young man.*

He determined to give courses of lectures in natural philosophy and
chemistry, and proposed to his brother, who had joined him in the
school, that he should take the latter. The brother replied that he
had never opened a book on chemistry. '' Neither have I," said John,
" on natural philosophy." It was then objected that they could not
obtain the appropriate instruments and apparatus in this country.
"But we can get them," he said, "from London." It was suggested
that they might fail in so making themselves masters of their respect-
ive subjects as to pursue them advantageously. " But we shall not
fail," said he ; " only determine and the thing is half done." An order
was sent to Londoti for apparatus, both philosophical and chemical, a
better supply of which was provided for his institution (at an expense
of several thousand dollars) than was to be found in any private insti-
tation in this country.

Gummere acquired considerable reputation as a teacher and writer.
He was for over forty years teacher in Friends' schools in Pennsylvania
and New Jersey. He once declined the proffered chair of mathematics
at the University of Pennsylvania. He contributed astronomical papers
to the American Philosophical Society. The most celebrated of his
works was his Surveying (1814), which went through a large number of
editions. It was more extensively known and more highly prized than
any other work on st^rveying. His treatise on theoretical and practical
astronomy was once used as a text-book at West Point and other
leading scientific institutions. In preparing it he had greatly profited
by French models.

Mention should be made here of the mathematical studies of Walter
Folger, a lawyer, of Nantucket. They will throw light upon the kind
of instruction which was then being given at our American ports, in
iught schools for navigators. After attending common schools, Folger
studied land-surveying, in which, without the least assistance, he be-
came exceedingly skillful. " In the winter of 1782-83 he attended an

■A.^^— Illfc I M^lfc^— — JfcXM

* Memorials of the Life and Character of John Gammere, by William J. AlliBOn.

92 TEACHING AND HISTORY OF MATHEMATICS.

eveniDg school in which he stadied* navigatioiiy and readily acquainted
himself with these branches. Nothing of a mathematical character
seemed ever to present any difficulties to his mind. He mastered al-
gebra and fluxions without assistance, and while in his teens he read
Euclid as he would read a narrative, no problem arresting his progress ;
and yet, so little did he know of language^ or of anything appertaining
to it, that he had reached the years of manhood, as he often confessed,
before he knew the definition of the word grammar.^ *

^^His father finally succeeded in obtaining for him a work of naviga-
tion, to which, for the first time, was appended Dr. Maske1yne'<s method
of obtaining the longitude at sea by means of lunar distances. This
delighted him, and at the age of eighteen, [when] prostrated with sick-
ness, he familiarized himself with the problem, and the engagement so
diverted his mind from his infirmities that he speedily regained his
strength. He immediately applied all his influence to the encourage-
ment of the use of this method among his fellow-townsmen, then univer-
sally engaged in the prosecution of whaling voyages. To numbers he
gave personal instruction, and the first American ship-master who de-
termined his longitude by lunar observations is said to have been one
of his pupils." A similar school was held in Philadelphia by Robert
Patterson.

SUBVETlNa OP GOVBENMENT LANDS.

In a new and growing country like ours it was only natural that the
art of surveying should have been early cultivated. But to a surveyor
some knowledge of the rudiments of geometry and trigonometry was
indispensable. As early as 1761 there was published, or reprinted, iu
Philadelphia a work entitled, Subtential Plane Trigonometry, by Thomas
Abel, presumably an English teacher. In 1786 there was reprinted in
Philadelpia an edition of Robert Gibson's Practical Surveying, which
first appeared in London in 1767. This enjoyed an extended circulation.
In 1799 appeared in Wilmington the first popular American treatise on
surveying, by Zachariah Jess, a teacher and practical surveyor, of Dela-
ware. In the preface to Gummere's Treatise on Surveying (1814) we
read : " The works of Gibson and Jess are the only ones at present in
general use. The former, though much the better of the two, is de-
ficient in many respects.'' In 1796 was published in New York, The Art
of Surveying Made Easy, by John Love, and at Litchfield, An Accurate
System of Surveying, by Samuel Moore. In 1806 Eev. Abel Flint pub-
lished his Geometry and Trigonometry, with a Treatise on Surveying.
Flint graduated at Yale in 1786, was tutor at Brown till 1790, after-
ward studied theology, and then became pastor at Hartford, Conn.

The publication of Gibson's Surveying in 1785 was very timely, for
it was in this very year that Congress passed an ordinance specifying

• "A Brief Memoir of the late Walter Folger, of Nantucket," by WiUiam MitcheU,
in the American Joornal of Science and Arti| lecond series, Vol. IZ, No. 27, May, 1850.

INFLUX OF ENGLISH MATHEMATICS. 93

that surveyors, as they were respectively qualified, should proceed to
divide the western territory into townships of 6 miles square by lines run-
ning due north and south, and others crossing these at right angles as
near as may be. Each township should be subdivided into lots of one
mile square. This system was not universally approved, for it tended
to delay the sale of public lands till they could be correctly measured.
In the Madison Papers (Vol II, p. 040) we read that the Eastern States
favored the plan adopted, while the Southern were *' biased in favor of
indiscriminate location." Kentucky and Tennessee adhered to the old
plan of indiscriminate location. This occasioned so much litigation in
those States that it has been said that as much money was annually
expended there in land-title litigation as would defray their taxes for
the support of the severest war. Lands surveyed by the United States,
on the other hand, were comparatively without any legal difficulty. In
fact, one great object of the Government system was the removal of all
temptation to incur the curse pronoanced by Moses on him '^ who re-
moveth his neighbor's landmark." The comers of each section were
carefully located by marked trees, whose species, diameter, distance,
and bearing were entered upon the field-notes. If the marked tree at
any one corner were destroyed, then its location could be determined
from the other corners. Though a great improvement on previous
modes of surveying, it is inaccurate and rude indeed as compared with
the refined triangulation surveys now carried on by the United States
Coast and Geodetic Survey.

Most conspicuous in the execution of the early Government surveys
were Andrew Ellicott ttoid Jared Mansfield. Ellicott was engaged in
a large number of surveys. At various times he was appointed com-
missioner for marking the boundaries of Virginia, Pennsylvania, and
New York J in 1789 he was selected by Washington to survey the land
lying between Pennsylvania and Lake Erie ; in 1790 he was employed,
with his brother Joseph, in surveying and laying out the city of Wash-
ington; in 1792 he was made Surveyor-General of the United States; in
1796 he was appointed United States Commissioner, under the treaty of
San Lorenzo el Eeal, to determine the boundary between the United
States and the Spanish possessions on the south. It is stated that he
sent observations to Delambre, of France, remarking that they were
made by a " self-taught astronomer, and the only practical one now in
the United States.'^ This was after the death of David Kittenhouse.

More prominently connected with the survey of the North-West Terri-
tory than Ellicott was Jared Mansfield. He was a graduate of Yale
College. In 1801 (?) he published Essays, Mathematical and Physical.
From the perusal of his works alone the illustrious Thomas Jefferson
was induced to bring him into public life. In 1803 he was appointed
surveyor-general of the North-West Territory. His first work was* to
determine astronomically certain lines of latitude and the principal
meridians on which the surveys were to proceed. To carry out this

94 TEACHING AND HISTOEY OP MATHEMATICS.

work astronomical instrnments were needed. President Jefferson or«
dered the purchase from London of a transit instrument, a telescope,
an astronomical clock, and a sextant. The first principal meridian be-
gan at the mouth of the Great Miami ; thef second at a point 5 miles
south-west of the confluence of Little Blue Biver with the Ohio; the
third at the confluence of the Ohio and the Mississippi Eiders ; the
fourth at the junction of the Illinois and Mississippi } the fifth at the
mouth of the Arkansas Biver. A large number of other meridians, or
<^ base-lines," have since been established.*

In Tiew of the fact that our Government has had, all in all, nearly
3,000,000 square miles of land to sell or to otherwise dispose of, and
that the sale had always to be preceded by a survey, it must be evident
that there was a demand for surveyors. They could earn a comparao
tively easy subsistence while a student of pure mathematics might have
gone a begging for a living. About 1816 a friend of Gomte in this coun-
try warned that French mathematician and philosopher against the
purely practical spirit that prevailed in this new country, and against
coming here, by saying: ^<I£ Lagrange were to come to the United
States he could only earn his livelihood by turning surveyor."

MATHEMATICAL JOURNALS.

The number of mathematical journals published in this country since
the beginning of this century is much greater than one might suppose.
A full historical sketch of these periodicals has been given by Dr. David
S. Hart in the Analyst (Vol. II, pp, 131-8, 1875), and we shall make free
use of his valuable article.

The oldest mathematical journal in America was the Mathematical
Correspondent. It was established by gentlemen in New York and other
dties, who had long felt the want of a periodical which should do for
America what tbe Ladies' Diary had done for England. George Baron
was editor-in-chief. It was to be issued quarterly. The first number
was issued in New York City on May 1, 1804. ( )nly eight numbers ever
appeared. An essay in this magazine on Diophantine analysis, by
Bobert Adrain, was the first attempt to introduce the study of this sub-
ject in America.

The main cause of the discontiDuance of the journal lies in the prej-
udice which the editors, who were of Hibernian descent, entertained
against American authors. A contributor, who called himself '< A Bab-
bit," was permitted by the editors to sneer at several works written by
American authors, such as Shepherd, Pike, Walsh, and others. The
editors themselves also spoke in the most contemptuous manner of Col.
Jared Mansfield, the superintendent of the Military Academy at West
Point. Baron advertised on the cover of No. 2 of the Correspondent a

* For farther information on the early earveyS) see Niles'n Begisier, VoL XII, pp.
97,406; Vol. XVI> p. 362,

HTFLUX OF ENGLISH MATHEMATICS. 95

leetnre delivered by him in Few York, which contains, as he says, " a
complete refatation of the false and sparions principles ignorantly im-
posed on the public in the new American Practical Navigator, written
by K. Bowditch." The sab-editors endorsed the above. Bat some of
these attacks, especially <^ A Babbit's," seem to have created troablci^
and on p. 154 the editor says : <' ^ A Babbit' will not in any future num-
ber be permitted to propose qnestions concerning the blunders of stupid
Sheph^ds; we had rather soar aloft with the eagle than waddle in the
mud with the goose." For some hidden reason, Baron resigned the
editorship. Many of the subscribers neglected to pay, and the paper
soon died out.

The next periodical was the Analyst, or Mathematical Museum, edited
by Bobert Adrain. The first number was published in Philadelphia in
1808. Five numbers only appeared. We have spoken of this periodical
at some length when we wrote about Bobert Adrain. It contained the
valuable original work of Adrain on the Law of Probability of Errors.
Besides the editor, N. Bowditch, Alexander M. Fisher, and Melatiah
Kafih were among the contributors to the Analyst. ,

In 1818, William Marrat became editor of the Scientific Journal, which
was published at Perth Amboy, N. J., in monthly numbers. limine num-
bers are all that are known to have appeared. The cause of the discon-
tinuance seems to have been the departure of Mr. Marrat for England.

In 1835 Bobert Adrain started in New York a second periodical, the
Mathematical Diary, which was published quarterly daring the first
two years and annually during the last two. The last number contains
an excellent likeness of Lagrange, and a sketch of his life. After the
first year the editorship of the journal passed into the hands of James
Byan, the author of several mathematical works. In the preface to
the first number of the Mathematical Diary, Bobert Adrain said : ^< The
principal object of the present little work is to excite the genius and
industry of those who have a taste for mathematical studies by afford-
ing them an opportunity of laying their speculations before the public
in an advantageous manner. * * * It is well known to mathema-
ticians that nothing contribates more to the development of mathemati-
cal genius than the efforts made by the student to discover the solu-
tions of new and interesting questions." These words may have been
prompted by his own experience. We have already pointed out how
the Analyst, which was edited by him seventeen years previously, was
the medium of publishing the first proofs of the all important Law of
the Facility of Error in Observations.

Nearly all the more prominent mathematicians of America were con-
tributors to the Diary. Among them were Bobert Adrain, N. Bow-
ditch, Theodore Strong, Eugene Nulty, Benjamin Peirce, Benjamin
Hallowell, William Lenhart, M. CKShannessy, Henry J. Anderson, and
others.

In 1832 the publication was suspended on account of an unfortunate

96 TEACHING AND HISTOBY OP MATHEMATICS.

qnarrel among the matliematiciaiiB. Mr. Samuel Ward, a then recent
gradaate of Golambia College, had in part the management of the last
number, in which he inserted a dialogue, written by himself, exhibiting
in a ridiculous light Henry J. Anderson, then professor of mathematics
at Columbia College. High words passed between the parties and their
firiends, which resulted in the complete breaking up of the Matbemati*
cal Diary. Samuel Ward was afterward editor of Young's Algebra.
In later years he followed wholly different pursuits. He became known
in Washington as the '^ king of the lobby,'' and as the giver of the best
dinners of any man in America.

According to Dr. Hart, a journal called the Mathematical Companion,
was started by John D. Williams in 1828, and continued for four years.
The periodical, says Dr. Hart, was evidently gotten up as a rival of the
Mathematical Diary. The writer has never seen a copy of this period!*
caL There is one in the Harvard library.* Mr. Williams had many
opponents, and a bitter contest was carried on between the two parties.
He finally issued his fourteen famous <' challenge problems," directed
against all the mathematicians in America, excepting only Dr. Bow-
ditch, Professor Strong, and Eogene Nulty. Six of these are impossible.
All the others have been solved by several persons.!

The periodicals which we have named were devoted entirely to math,
ematics. In addition to these there were publications which were given

* Dr. Aitemas Martin sends ns the fall title of the Journal, as found in Bolton's
Catalogne of Scientific and Technical Periodicals, 1665 to 1882, published by the
Smithsonian Institution, p. 360 — ''The Mathematical Companion, containing newre-
eearches and improvements in the mathematics, with collections of questions proposed
and resolved by ingenious correspondents. Edited by Williams; 1 vol., 18 mo., New
York, 1828-^31."

tin the Educational Notes and Queries, edited by W. D. Henkle, Vol. II, No. 11,
January, 1876, will be found a copy of a communication to a newspaper made by John
D. Williams in 1832^ containing the '* fourteen challenge problems," and beginning aa
follows:

"Messrs. Editors.— It is this day six months since, under the signature of Diophanfuif
1 proposed through the medium of your paper to the mathematicians of America, a
collection of problems in Diophantine analysis. No correct solution having as yet
been received to the whole of them, I take this opportunity to fulfill my pledge to
furnish such, and inclosed they will come to your hands. I now desire to re-propose
them for the eusning six months; and shall except from my challenge the Hon. Na-
thaniel Bowditch, LL. D.,. etc., of Boston, Mass. ; Mr. Eugene Nulty, of Philadel-
phia; and Prof. Theodore Strong, of Rutgers College, New Brunswick, N. J., only.
The list of gentlemen challenged stands then as follows: Prof. Robert Adrain, Uni-
versity of Pennsylvania ; Henry J. Anderson, Columbia College, N. Y. ; Benjamin
Peirce, Harvard University, Cambridge, Mass. ; Mr. J. Ingersoll Bowditch, Boston,
Mass. ; Marcus Catlin, Hamilton College, Clinton, N. Y. ; M. Floy, jr., New York ; C.
Gill, Sawpitts Academy, N. Y. ; L. L. Inoonnew, Cincinnati, Ohio ; Benjamin Hallo-
well, Alexandria, Va. ; Samuel Ward, 3rd New York— it being presumed that there aie
none in the United States with the exception of the above list would think of attempt-
ing their solution." Then follow the fourteen questions. All problems being in Dio-
phantine analysis would tend to show that thlB subject was then a comparatively
fJAVorite study.

INFLUX OF ENGLISH MATHEMATICS. 97

•

to science, or to nsefal information in general, bat which gave part of
their space to a " mathematical department." Foremost among these
was the Ladies and Gentlemen's Diary, or United States Almanac, eto.,
edited by Melatiah Kash, for the years 1820, '21, '22, It contained mach
valnable information in astronomy and philosophy, enigmas, charades,
qneries, and mathematical problems, to be answered in the sncceeding
numbers. Other almanacs which generally contained mathematical
problems were Thomas's Almanac, pablished at Worcester, Mass.,
which existed for more than one handred years; the Maine Farmer's
Almanac } two publications, each called the ^^ Farmers' Almanac ; " the
Knickerbocker Almanac; the Anti-Masonic Almanac, commenced in
1828 at Bochester, N. Y. Other journals having a mathematical de-
par tpient were the American Monthly Magazine, commenced in New
York in the year 1817 ; the Portico, which was started in Baltimore in
1816 and continued two or three years.

The mathematical journals spoken of were all of the most elementary
kind, and, excepting No. IV of the Analyst, which contained Adrain's
investigations on least squares, added nothing to the stock of mathe-
matical science. These journals had an educational rather than scien-
tific value. The proposal and solution of problems was the main work
done by their contributors. Now, it will certainly be admitted that
solving problems is one of the lowest forms of mathematical work. The
existence of mathematical journals shows that since the beginning of
this century there always were some persons interested in mathematics,
but the number was so small that mathematical journals never were
a financial success. All the early mathematical periodicals had merely
an ephemeral existence*
881— No. 3 7

Ill

THE INFLUX OF FEENOH MATHEMATICS.

Daring the latter part of the eighteenth centary we see the French
people rising with fearful ananimity, destroying their old institntions,
and apon their ruins planting a new order of things* With this period
begins the interest in popular education in France. A new impetus
was given also to higher scientific education, which continued to be &r
in advance of that of the rest of Europe.

In 1794 was opened in Paris the Polytechnic School and in the fol-
lowing year the Schools of Application. The Polytechnic School
gained a world-wide celebrity. The professors at this institution were
men whose names are household words wherever science has a votary.
Lagrange, Lacroix, and Poisson laid the basis to its course in analytic^
mathematics ; Laplace, Ampfere, and others to that of analytical me-
chanics and astronomy. Descriptive geometry and its applications had
for their first teachers the founder of this science, the illustrious Monge
and his celebrated pupils, Hachette and Arago.

The success of the Polytechnic School was phenomenal. It was the
nurse of giants. Among its pupils were Arago, Biot, Bourdon, Oauchyf
Ghasles, Duhamel, Dupin, Oay-Lussac, Le Yerrier, Poncelet, Begnault.
The Polytechnic School is of special interest to those who live in America,
because the (J. S. Military Academy at West Point was a germ from it.

Compared with the French mathematicians who flourished at the
beginning of this century the contemporary American professors were
mere Liliputians. The masterpieces of French scholars were unknown
in America. What little mathematical knowledge existed here came
to us through English channels. For that reason that epoch was called
the period of the influx of English mathematics. As compared with
colonial times, considerable attention was paid to mathematical studies
during that period. But there was still a great dearth in original
thinkers on mathematics among us. The genius of our people was ex-
ercised in different fields, and so the little science we had was borrowed
from others.

But the time came when French writers were at last beginning to

make their influence felt among us. We recognized their superiority

over the English and profited by it. Mathematical studies received a

new impetus. But even then ours was not the glory of the sun, but

INFLUX OP rRENCH MATHBMATldS. 99

only of the moon. The new period produced amon^ us only one mathe-
matician displaying real genius for original research.

It is naturally humiliating to an American when a foreign mathema-
tician like Todhunter, well known for the fairness and candor of his
views, pronounces a judgment on Americans like the following: <^I
have no wish to depreciate their labors ; I know that they possess able
mathematicians, and that in the department of astronomy they have
produced meritorious works ) but I maintain that as against us their
utmost distinction almost vanishes. And yet, with their great popula-
tion, their abundant wealth, their attention to education, their freedom
from civil and religious disabilities, and their success in literature, we
might well expect the most conspicuous eminence in mathematics." ^

No thinking American will pronounce this estimate of American
mathematicians as entirely unsound ; it is, in fact, quite correct. The
reasons for this want of productiveness certainly do not lie in any lack
of power in the American mind. They will be found rather (1) in the
want of interest in and appreciation of abstract scientific work on the
part of the American people, and (2) in the bad methods of mathemat-
ical instruction in our elementary and higher institutions of learning.
There has been no incentive in this country for any large body of men
to direct their life-work, day by day, in the line of mathematical inves-
tigation. In former years our professors in colleges were, with few
exceptions, over- worked in the recitation room; their routine work
absorbed all their energies, thereby rendering their minds unfit for *
original research. ^ Again, every teacher had a stomach ; his wife and
children had stomachs; the human being must be fed; a livelihood
must be earned ; the professor's salary was low; not unfrequently he
had to add to his duties as instructor in college those of a preacher or
private teacher, in order to make his living. Such conditions were not
favorable for the growth of science.

But, in spite of all difficulties, there was much progress. The im- ;
provements in mathematical text-books and reforms in mathematical in-
struction were due to French influences. French authors displaced the
English in many of our best institutions. It is somewhat of a misfor-
tune, however, that we failed to gather in the full fruits of the French
intellect. We followed in the path of French writers whose works had
ceased to be the embodiment of the later results of French science ;
many of the works which we adopted were beginning to be " behind
the times," when introduced in America. We used works of Bezout,
Lacroix, and Bourdon. But Bezout flourished before the French Rev-
olution, and Lacroix wrote most, if not all, of his books before the be-
ginning of this century. In 1821 Oauchy published in Paris his Oours
W Analyse. If thoughtful attention and study had been given by our
American textbook writers to this volume, then many a lax, loose, and
unscientific method of treating mathematical subjects might have been

•The Confiict of Studies and Other Essays^ by I. Todhonter. London, 1873| p. 160*

100 TEACHING AND HISTORY OF MATHEMATICS.

corrected Jit the outset. The wretched treatment of infinite series, as
found in all our text-books, excepting the most recent, might have been
rejected from the very beginning.

In thinking of the influx of French mathematics, we must guard
against the impression that French authors and methods entirely dis-

' placed the English. Euglish books continued to be used in some of
our schools. Many an old English notion has remained with us to the
present day. We still have the English weights and measures. The
old line-system in trigonometry, which we got from the English, but
which they long since rejected, has until very recently been finding
favor among many of our teachers.

There have been improvements in the methods of instruction, but
not so extensive as might be wished. Traditional methods have long
had almost full sway. The mathematical teaching lias been bad. One

' of the most baneful delusions by which the minds, not only of students,
but even of many teachers of mathematics in our classical colleges,

.have been afflicted is, that mathematics can bo mastered by the favored
few, but lies beyond the grasp and power of the ordinary mind. This
chimera has worked an untold amount of mischief in mathematical edu-
cation. The students entered upon their studies with the feeling that
there was no use trying to learn mathematics, and the teacher felt that
there was no use trying to teach it. This humiliating opinion of the
powers of the average human mind is one of the most unfortunate
delusions which have ever misled the minds of American students and
educators. It has prevailed among us from the earliest times. In the
latter part of the last century, the notion was general among us that
girls could not be taught fractions in arithmetic, and that lady teachers
were unfit, for want of mental capacity, to give instruction in arithme-
tic. Warren Burton says that a school-mistress " would as soon have
expected to teach the Arabic language as the numerical science.'' But
this delusion has now vanished. The best instruction in elementary
arithmetic is now given by lady teachers. Among the contributors to
the American Journal of Mathematics there are two ladies. In the
same way the delusion will soon vanish that the average college stu-
dent is not able to grasp the more advanced branches of exact science.
The trouble has been, all along, not so much in the lack of ability in
students, as in the wretched character of the mathematical instruction.
Such is the opinion of Professor Olney, one of the mrjst efficient drill-
masters and teachers of mathematics that this country has produced.
In the preface to his General Geometry and Calculus he says: "Nor is
it impracticable for the majority of students to become intelligent in
these subjects. They do not lie beyond the reach of good com mon minds,
nor require peculiar mental characteristics for their mastery. The dif-
ficulty hitherto has been in the methods of presentation, in the limited
and totally inadequate amount of time assigned them, and more than
all in the preconceived notion of their abstruseness."

INFLUX OP FRENCH MATHEMATICS. , 101

One of the causes of the bad instruction in our colleges has been the
system of tutorships. Fortunately, this relic of scholasticism is now
rapidly disappearing. Young students who needed a skilled teacher of
long experience to guide them and to awaken in them a spirit of free
inquiry were intrusted to inexperienced youths who had just gradu-
ated from college, and who had themselves never folt the glow of the
spirit of independent inquiry. Students did not find their mathematics
interesting, nor did they understand it well. Their hatred of mathe-
matics had its cause in these two facts, which stand in the closest pos-
sible connection with each other. " We might say, either that thestudy
failed of being understood, because it was uninteresting, or that it
awakened no interest, because it was not well understood. Both these
statements were true."* Professor Eddy truly says that very few stu-
dents " do really become in^ny true sense masters of the mathematical
subjects which they study, or indeed have sufficient practice in the
principles which they attempt to learn, to be capable of judging whether
they have been so mastered as to accomplish the ends which should be
sought in mathematical training.'' The great desideratum in our pre-
paratory schools and colleges has been less memorizing, less cramming,
more thorough training in the fundamental branches, more obiect teach-
ing, more drill, more frequent and well-guided original inquiries, greater
freedom from formalism, a stronger spirit of free inquiry.

Says Professor Eddy: "When, as often happens, our collegje grad-
uates go abroad for i^ost-graduate study in departments requiring pre-
vious mathematical training, what do they find their requirements in
this direction to amount to ? I think I may say that a large proportion
of them find themselves almost hopelessly lacking in the essentials of
such training, and not at all fitted to make proper improvement of the
advantages of which they have sought to avail themselves. Our young
men are unequal to the mathematical studies which those of the same
age, but of European academic training, successfully carry. Now,
where does the difficulty lie? Not in any inferior talent for the^e
studies, as I have the best of reasons for believing, but from a lack of
opportunity for obtaining a comprehension of the infinitesimal calculus,
in which they usually find theuiselves almost wholly wanting." Nor
are they always able to manipulate, with any degree of ease, the more
complicated expressions of ordinary algebra. They have been taught
by a " daily lecture instead of a daily drill,'^ a method of teaching which
is like "explaining tactics instead of practicing them." Or, whenever
text-books were used, " the recitations were mere hearings of lessons,
without comment or collateral instruction."!

Professor Eddy's reminiscences of his own study of mathematics in
college are not pleasant. Nor is his experience exceptional. On the con-

•"College Mathematics," by Henry T. Eddy, ia the Proceedings of the Amer-
ican Association for the Advancement of Science, Vol. XXXIII, 1884.
t Harvard Reminiscences, by A. P. Peabody, p. 201.

102 TEACHINa AND HISTOBY OP MATHEMATICS.

trary it has been the rale rather tban the exception in onr classical ool-
leges. In reply to a request made by the writer to give his recollections
of the mathematical teaching at one of our oldest classical colleges, a
now prominent professor of mathematics replied that he did not think
he had << any such recollections " as he <' should care to put in print."
Another one gives his reminiscences^ but marks his letter ''personal and
private."

If our classical colleges had caught something of the spirit that must
have prevailed at the Polytechnic School in Paris in the days of La-
grange, Laplace, Lacroix, Ampere, when it produced such thinkers as
Arago, Oauohy, Le Yerrier, then the list of our prominent mathemati-
cians and astronomers would doubtless have been doubled or tripled.
We got from the French some of their old text-books, but we failed to
catch their love of scientific study and inquiry.

On a previous page it has been stated that Americans had come to
recognize the superiority of French mathematicians over the English.
It should have been added that we did not see this superiority until it
was pointed out to us by the English themselves. The influx of French
mathematics into the United States was preceded by an influx of French
mathematics into England* In Britain there were men who had come
to deplore the very small progress that science was making there, as
compared with its racing progress on the continont. In 1813 the
''Analytical Society" was formed at Cambridge. This was a small
club established by Peacock, John Herschel, Babbage, and a few other
students at Cambridge, to promote, as it was humorously expressed, the
principles of pare " D-ism," that is, of the Leibnitzian notation in the
calculus, against those of " dot-age," or of the Newtonian notation. This
struggle ended in the introduction into Cambridge of the Continental no-
tation {£) to the exclusion of the flnxional notation (j/). This was a

great step in advance, not on account of any great superiority of the
Leibnitzian over the Newtonian notation, but because the adoption of
the former opened up to English students the vast storehouses of Con-
tinental discoveries.

The movement against the flnxional notation began in this country
almost ten years later than it did in England, and proceeded more
quietly. John Parrar, of Harvard, translated from the French the Dif-
ferential and Integral Calculus of Bezout, which employed the Continental
notation, in 1824. Professor Fisher, of Yale, who died in 1822, published
mathematical articles in Silliman's Journal, employing the new nota-
tion. At an earlier date than this there were men connected with West
Point who had been trained in the Continental system. Thus, F. B.
Hassler, educated at the University of Bern, was teacher of mathemat-
ics at West Point from 1803 to 1810. Probably neither calculus nor
fluxions were taught there during that time, for, as late as 1816, we read
in the West Point curriculum that/ttj^oii« were " to be taught at the

INFLUX OF FBENGH MATHEMATICS. 103

option of professor and stadenf In 1817, Orozet, trained at the Poly-
technic School in Paris, became teacher of engineering at the Military
Academy. In this country be, sometimes at least, used the Newtonian
notation. He did so, for instance, in the solution, in French, of a prob-
lem which he published in the Portico, of Baltimore, in 1817. The
Leibnitzian notation must hare been introduced at the Military Acad-
emy very soon after the year 1817.

Bobert Adrain nsed the English notation in his earlier writings. In
the Portico, Yol. Ill, he does so, but in Nash's Ladies and Gentlemen's
Diary, No. II, published in New York in 1820, he employs the notation
dx. We are told that while he was at Oolumbia College, between 1813
and 1826, he wrote a mannscript treatise on the Differential and Integral
Oalculus. We know also that he was a diligent student of the works
of Lagrange and Laplace, which contained the notation of Leibnitz
thronghont. The first article in the Memoirs of the American Acad-
emy of Arts and Sciences, which contains the '<d-istic" notation, was
published in 1818 by F. T. Schubert. It is well known that Bowditch
began the translation of the M6canique Celeste of Laplace as early as
1814, At that time he was, therefore, thoroughly conversant with
pore '^D-ism." He had been converted to the new ^' ism " on the long
sea voyages, from 1795 to 1804, when he studied Lacroix's Calculus.
In general, it may be stated that the change of notation took place in
the United States about the close of the first quarter of this century.

The publication of Bowditch's Laplace, begun in 1829, gave a pow-
erful stimulus to the study of French mathematics and to the general
advancement of mathematical learning in America. Says Edward
Everett: ^' This may be considered as opening a new era in the history
of American science."

This may be a convenient place to consider that work at some length.
As it originally appeared in France, the M6canique G61este was de-
scribed by the Edinburgh Review, one of the li ading scientific journals
in Great Britain, as being of so abstruse and profound a character that
there were scarcely a dozen men in all that country capable of reading
it with any tolerable facility. These remarks created great curiosity
in Bowditch to explore the work. He began translating it in 18I.4t and
pursued it with such ardor and persistence that he accomplished it in
only two years.

In order to state briefly the object of the work of La Place, we quote
from his preface to it as follows :

"Toward the end of the seventeenth century, Newton published his
discovery of universal gravitation. Mathematicians have since that
epoch succeeded in reducing to this great law of nature all the known
phenomena of the system of the world, and have thus given to the theo-
ries of the heavenly bodies and to astronomical tables an unexpected
degree of precision. My object is to present a connected view of these
theories which are now scattered in a great number of works. The
whole of the results of gravitation upon the equilibrium and motions

104 TEACHING AND HISTORY OF MATHEMATICS.

of tlie flaid and solid bodies which compose the solar system and the
similar systems existing in the immensity of space, constitute the object
of Celestial Mechanics^ or the application of the principles of mechan-
ics to the motions and flgnres of the heavenly bodies. Astronomy, con-
sidered in the most general manner, is a great problem of mechanics,
in which the.elements of the motions are the arbitrary constant quanti-
ties. The solution of this problem depends, at the same time, upon the
accuracy of the observations and upon the perfection of the analysis.
It is very important to reject every empirical process, and to complete
the analysis, so that it shall not be necessary to derive from observa-
tions any but indispensable data. The intention of this work is to
obtain, as much as may be in my power, this interesting result."

Though the translation was completed as early as 1817, the publica-
tion did not begin until 1829. In 1817 the income of Bowditch was so
Ismail that he could not afford to have the translation published. The
Amencan Academy of Arts and Sciences offered to publish the work at
their own expense. He was also solicited to publish it by subscription.
But his independence of spirit induced him to decline these proposals.
He was aware that the work would find but few readers, and he did not
wish any one to feel compelled or to be induced to subscribe for it, lest
he should have it in hi8 power to say, *' I patronized Mr. Bowditch by
buying his book, which I can not read." Later on he was able to com-
mence the publication at his own expense.

The objects which Bowditch endeavored to aecomplish by his trans-
lation and commentary, as stated by his biographers, were as follows:

(X) To supply those steps in the demonstration which could not be
discovered without much study, and which had rendered the original
work so difficult. The difficulty arose not mei«ly from the intrinsic com-
plexity of the subject and the medium of proof by the higher branches
of mathematics, but chiefly from the circumstauce that the author,
taking it for granted that the subject would be as plain and easy to
others as to himself, very often omits the intermediate steps and con-
necting links in his demonstrations. He jumps over the interval and
grasps the conclusion by intuition. Bowditch used to say, " I never
come across one of Laplace's < Thtis it plainly appears^ without feeling
sure that I have hours of hard work before me to fill up the chasm and
find out and show how it plainly appears."*

(2) The second great object of the translation was to continue the
original work to the present time, so as to include the many improve-
ments and discoveries in mathematical science that had been made
during the twenty-five years succeeding the first publication. It is
gratifying to know that the most eminent of contemporary mathomati-

* ''The M^canique Celeste is by no means easy reading. Biot, \?ho assiHted La-
place in revising it for the press, says that Laplace himself was frequently uuablo
to recover the details in the chain of reasoaing, and if satisiicd that the conclusions
were correct he was content to insert the constautly recurring ioruiula, ' II est aisd ^
voir.' " W. W. R. Ball's Short Butory of MathemaHoif p. 387.

Jr. J

INFLUX OF FRENCH MATHEMATICS. 105

ciana pronounced his commentary a snccess, and agreed that Bowditch
had attained tlie end he had in view, namely, to bring the work up with
the times. Says Lacroix, July 5, 1836 : *^ I am more and more aston-
ished at a task so laborious and extensive. I perceive that you dd not
confine yourself to the mere text of your author and to the elucidations
which it requires, but you subjoin the parallel passages and subse-
quent remarks of those geometers who have treated of the same sub-
jects ; so that your work will embrace the actual state' of the science at
the time of its publication.'' Legendre, July 2, 1832, says : " Your work
is not merely a translation with a commentary; I regard it as a new
edition, augmented and improved, and such a one as might have come
from the hands of the author himself, if he had consulted his true inter-
est, that is, if he had been solicitously studious of being clear." Mr.
Babbage, of England, August 5, 1832, says : ^' It is a proud circum-
stance for America, that she has preceded her parent country in such
an undertaking ; and we in England must be content that our language
is made the vehicle of the sublimest portion of human knowledge, and
be grateful to you for rendering it more accessible." Similar testimony
was given by Bessel and Encke in Germany ; Puissant in France; Sir
John Herschel, Airy, Francis BaUy in England, and Cacciatore in Italy.

Bowditch once remarked that however flattering the testimony from
foreigners might be, yet the most grateful tribute of commendation he
had ever received was contained in a letter from a backwoodsman pf
the West, who wrote to him to point out an error in his translation of
the M^canique Celeste. "It is an actual error," said he, ''which had
escaped my own observation. The simple fact that my work had reached
the hands of one on the outer verge of civilization who could under-
stand and estimate it was more gratifying to my feelings than the
eulogies of men of science and the commendatory votes of Academies."
In America, many college professors were enabled by means of the trans-
lation and commentary to read and understand the M6canique Celeste,
who would otherwise have looked upon this work as a sealed book.

During the first thirty-five or forty years of this century but little was
accomplished in this country in the line of astronomical observations.
More was done in that respect during the^ays of David Eittenhouse
than in the early part of this century. But, all at once, a great impetus
was given to this kind of scientific work. In 1830 was erected the Yale
College Observatory ; in 1831 the observatory at the University of North
Carolina; in 1836 the Williams College Observatory; in 1838 the Hud-
son Observatory, Ohio ; in 1840 the Philadelphia High School Observ-
atory and the West Point Observatory; in 1842 the National Observa-
tory at Washington. Since then a large number of other observatories
with excellent instruments have been built.

A plan for a National Observatory was submitted to the Government
by Mr. Hassler, in bis project for the survey of the Atlantic coast, as
early as 1807. The proposition met with no favor. For many years

106 TEACHING AND HISTOEY OF MATHEMATICS.

Oongross opposed every such scheme. John Qaincy Adams, in his
annual message of 1825, strongly urged this snbjeot upon the atten-
tion of Congress. In one place he said, ^' It is with no feeling of pride,
as am American, that the remark may be made that, on the compara-
tively small territorial surface of Europe there are existing upward
of one hundred and thirty of these light houses of the skies ; while
throughout the whole American hemisphere there is not one." Presi-
dent Adams's appeal was received with a general torrent of ridicule.
*< The proposition," says Loomis,* ^^ to establish a light-house in the
skies became a common by-word of reproach." It was not till 1842
that an appropriation was passed for an observatory , under the disguised
name of a ^^ Depot of Charts and Instruments."

It need hardly be said that in later years theU. S. Government has
been very liberal in the encouragement of science.

Elementabt Schools.

The beginning of this period is marked by a great revival of element-
ary education. Pestalozzian ideas had gained a foothold in England,
and were now commencing to force their way into the western conti-
nent. In 1806 F. J. N. Neef, once ai^ assistant to Pestalozzi, oame to
this country, and began teaching and disseminating the ideas of the
Swiss reformer. The first fruit of Pestalozzian ideas in the teaching
of arithmetic among us was Warren Oolbum's Intellectual Arithmetic
upon the Inductive Method of Instructiou, known as the << First Les-
sons."

- Warren Golbum worked, while a boy, at the machinist's trade.
He then entered Harvard and graduated in 1820, having ^^ mastered
calculus and read a large part of Laplace." He then taught a select
school in Boston. At this time he began preparing his little book. Of
special interest is the following statement of Mr. Batchelder, of Cam-
bridge, which shows how the First Lessons were prepared : ^^ I remember
once, in conversing with him with respect to his arithmetic, he remarked
that the pupils who were under his tuition made his arithmetic for him ;
that he had only to give attention to the questions they asked and the
proper answers and explanations to be given, in order to anticipate the
doubts and difficulties that would arise in the minds of the pupils." He
had read Pestalozzi, most probably, while in college. A manuscript
*€opy of his First Lessons was furnished by Colburn to his friend George

* B. Emerson for use in a school for girls, and the former received valua-
ble suggestions firom the latter. The success of the book was almost

' immediate. Fo school-book had ever had such sale among us as this.

' Over three and one-half million copies were used in this country, and it
was translated into several European languages.

Oolburn's First Lessons embodied what was then a new idea among us.
Instead of introducing the young pupil to the science of numbers, as did

* Beoent Progreas in Astronomyi especially in the United States, by Blias Tjoomis.
New Tork, 1666, p. 205.

INPIiUX OP FRENCH MATHEMATICS. 107

old Dilworth, by the question, •• What is arithmetic ! " and the answer,
^'Arithmetic is the art or science of computing by numbers, either whole
or in fractions," he was initiated into this science by the following sim-
ple question : ^' How many thumbs have you on your right hand t How
many on your left t How mapy on both together t " .The idea was to
begin with the concrete and known, instead of the abstract and unknown,
and then to proceed gradually and by successive steps to subjects more
difficult. In the publication of this book, the study of arithmetic in the
schools of this country received its best impulse. '^ It led to the adoption
of methods of teaching that have lifted the mind from the slavery of
dull routine to the freedom of independent thought." (Edward Brooks.)

Oolburn's First Lessons was followed in 1826 by his Arithmetic upon
the Inductivjd Method of Instruction, being a Sequel to Intellectual
Arithmetic. This was considered by its author to be superior to the
First Lessons, but it did not meet with so great success. In 1825 he pub-
lished his Algebra upon the Inductive Method of Instruction. Mr. Ool-
bum did not long engage in teaching. Three years after graduation from
college he was appointed superintendent of a manufacturing company
at Waltham, and, soon after, of one at Lowell, Mass. He possessed
£^eat mechanical genius and administrative ability.

Though the First Lessons met with ready appreciation in Kew Eng-
land, it must not be imagined that there was no opposition to it. Old
notions could not be laid aside at once, and even where the new ideas
had gained entrance, new books could not always be had readily.
Now-a days we are apt to forget the difficulty and expense of trans-
portation during the times preceding our railroad era. Says J. Stock-
toui in the preface to his Western Calculator (fourth edition, 1823,
Pittsburg, Pa.), ^< to furnish our numerous schoolsy^ in the western (I)
country, with a plain and practical treatise of arithmetic, compiled and
printed among ourselves, thereby saving a heavy annual expense in
the purchase of such books, east of the mountains, and likewise the car-
riage thereof j have been the motives which induced the compiler to
undertake this work."

In spite of all obstacles Colburn's books gained ground steadily.
Other books were written upon the same idea by different teachers. Old
books underwent revision, so as to embody the new methods in part.
Thus, the celebrated Schoolmasters' Arithmetic of Daniel Adams, first
published in lb 01 , was made to undergo a radical change. The old work
was «* synthetic." " If that be ,a fault of the work," says the author,
<< it is a fault of the times in which it appeared. The analytic or induc-
tive method of Pestalozzi ♦ ♦ • is among the improvements of later'
years. It has been applied to arithmetic with great ingenuity by Mr.
Oolbum in our own country." " Instructors of academies and common
schools have been so long attached to the old synthetic method of in-
struction, that, unhappily, many are ^till (1829) strongly opposed to
the introduction of the valuable works of Coiburn." <'This [Adams's]
work combinei^ the new and the old."

108 TEACHING AND HISTORY OF MATHEMATICS,

The great success of Oolburn's book did not prevent the appearance
of arithmetical works that were quite as worthless as any of earlier
years. There appeared others, on the other hand, which possessed no
little merit and became very popular. As examples of the latter we
would mention the arithmetics of the two brothers, Benjamin D. and
Frederick Emerson, both of whom were well-known teachers in Boston.

The arithmetics of later days are combinations of the old, as found in
our early arithmetics, and the new as found in the works of Colburik
For example : Our old arithmetics generally rejected reasoning, but
gave rules ; Colburn's books reject rules, but encourage reasoning. The
better class of our later arithmetics contain rules, but, at the same time,
give demonstrations and encourage students to think.

About the year 1825 or 1830, the French notation of numbers began
rapidly to displace the English. Large numbers came to be marked off
in periods of three digits instead of six. The earliest book in which we
have noticed the adoption of the French notation is Bobert Patterson's
edition of Dilworth's School-master's Assistant, Philadelphia, 1805; the
latest in which we have seeu^the English notation used is-M. Gibson's
revised edition of Abijah and Josiah Fowler's Youth's Assistant, Jones-
borough, Tepn., 1850. Some of our recent books explain both, but u^e
the French.

It is a rather curious fact that the process of cancellation did not come
to be generally used in our arithmetics before about 1850. In 1840 0,
Tracy published an arithmetic in which cancelling was freely used, a
feature which was then " entirely peculiar to this treatise," and which
distinguished it " from all others." John L. Talbott's Practical Arith-
metic (Cincinnati, 1853) gives the "cancelling system," but only in the
appendix, and re'hiarks in the preface to it,' " In Europe this system
has been very generally adopted in the higher schools, and in this conn-
try it is fast becoming known — and, so far as it is known, it supersedes
the usual modes of operation." Charles Davies takes pains to remark
on the title-page of his University Arithmetic (1857) that " the most
improved methods of analysis and cancellation^ have been employed.

The order in which the various arithmetical subjects have come to be
taught has been generally improved upon. Federal money and com-
pound interest no longer precede common and decimal fractions, but
come after them. Fractions have been moved much further toward the
front part of our books. The placing of fractions toward the end of
arithmetics had been due to the fact that the majority of pupils in olden
times did not pursue mathematics long enough to master fractions, and
were thus put through a course in arithmetic with only integral num-
bers. Those who did study fractions were made to learn the rules of
interest and proportion over again " in vulgar fractions," and then again
" in decimal fractions." Some of the old topics, such as single and
double position, have since been quite generally dropped, but we think
that there is still room for Improvement in that respect. Nothing would

INFLUX OP FRENCH MATHEMATICS, 109

be lost and moch gained if alligation, square and cube root, mensura-
tion, and some of the more difficult applications of percentage should
be dropped from our arithmetics. At least one new subject has been
quite generally and, we think, appropriately introduced into our books —
the metric system.

Before the time of Oolburn, mental arithmetic was quite unknown in
our schools. Since then mental and written arithmetic have not always
been so closely united as they should be. The methods used in the two
were frequently quite diverse. Too often they were taught almost like
distinct sciences, so that a pupU might be quite proficient in the one
without knowing anything of the other.

Grube's method of teaching numbers to children has been in use
among us, especially in the East, but has never been generally adopted.
It is such a refined method that few teachers possess the skill to apply
it readily. The method has a desirable tendency to train ready and
rapid calculators, and has much to commend itself to teachers.*

Since the -beginning of this century arithmetic has come to be re-
garded as the most important, because the most practical, science in
our elementary schools. Every farmer wished his sons to be good cal-
culators; every business man desired to be "quick at figures;" hence
its value was high in the estimation of all. Bookmakers were quick to
profit by this sentiment. They began to multiply the number of text-
books in the course until there were two books in mental arithmetic,
and three in written, in several of the series in general use. As a rule,
the examples in our arithmetics have not been well graded ; difficult
examples have been introduced, so early in the course as. to embarrass
and discourage even the best students. Many examples were regular
puzzles, not only to young boys and girls, but to almost any one not
trained in algebra. There are numerous problems that should never
have found a place in our arithmetics. We could quote from arithme-
tics dozens and dozens of such problems, but we shall give only one.

The 137th problem of the miscellaneous questions in the third part
of Emerson's Korth American Arithmetic, published in 1835, is as fol-
lows :

If 12 oxen eat np 3^ acres of grass in 4 weeks, and 21 oxen eat up 10 acres in 9
weeks, liow many oxen will eat np 24 acres in IS weeks ; the grass being at first
equal on every acre, and growing uniformly.

The idea of placing a problem of such difficulty in a book for boys
and girls ! The history of this problem in this country shows very
plainly that it is beyond the power, not only of pupils, but even of
teachers of arithmetic. Many teachers whose minds had been trained
by the study of algebra and geometry and, perhaps, even higher
branches of mathematics, wrestled with it in vain. There existed so
much uncertainty regarding its true solution that a premium of lifty

* For further information on Grube's method, see Prof. T. H. SaJfford's monograph
on Mathematical Teaching, pp. 19.

110 TEACHING AND HISTORY OP MATHEMATICS.

dollars was offered in June, 1836^ for the most << Incid analytical soln-
tion" of this qnestion, A committee was appointed, with P. Mackin-
tosh as chairman, to examine the solutions presented and award the
prize. The committee reported 112 eolations received, of which only
48 gave the true answer, and awarded the prize to Mr. James Bobin-
son, principal of the department of arithmetic, Bowdoin School, Bos-
ton.*

Think of it! Oat of 112 of, presumably, the best arithmeticians in
the country, ouly 48 got correct results ; and yet this problem was in*
tended to be solved by boys and girls.

But the history of our problem is not yet complete. Nearly twenty-
five years later a revision of Mr. Bobiuson's solution was submitted to
the National Teachers' Association, at Washington, by the Hon. Finley
Bigger, then Begister of the U. S. Treasury ; it was referred to the Math-
ematical Monthly for publication, and was printed in Yol. II, No. 3,
December, 1859, pp. 82-85. Mr. Bigger assumed, ^^for the purpose of
elucidation," that the question was susceptible of two constructions,
and obtained two answers in addition to the true one. The editor of
the Monthly appended an algebraic solution, and showed that there
was only one answer that would satisfy all the conditions of the prob-
lem, and that Mr. Bigger was wrong in his conclusions.

There is no ambiguity in the problem. Twenty-three years later, Dr.
Artemas Martin published several solutions of the problem in the Mathe-
matical Magazine. Dr. Martin does not consider Mr. Bobiuson's solu-
tion very ^Mucid," and pronounces it liable to at least one other objec-
tion — it makes ^^ mincemeat" of the oxen, inasmuch as fractions of oxen
occur throughout the analysis of the question.

There is another curious fact connected with the history of this prob-
lem. Neither Mr. Emerson, nor the committee, nor Mr. Bobinson, nor
Mr. Bigger, nor the National Teachers' Association, nor the Mathe-
matical Monthly, alludes to the fact that the question is taken from the
Arithmetica Universalis of Sir Isaac Newton, published in 1704, which
contains a '' lucid analytical solution." Mr. Emerson's statement of the
problem differs from that of Newton in this, that, owing to a misprint,
the fraction i instead of ^ is given by the former in the number of acres
contained in the first pasture, which mistake produces the absurd result
of 37j^| oxen, instead of 36. The above question goes by the name of
the ^' pasturage problem."

There exists a general feeling among mathematicians and educators
that the teaching of arithmetic has been overdone in our schools.
Parents have desired their older boys to be good mathematicians. But
they failed to perceive the truth that the best review of arithmetic con-
sists in the study of algebra; they looked upon algebra as utterly des-
titute of value. In consequence the boys have been made to waste

* Hendrickft's Analyst, Vol. Ill, p. 75; also the Matbematioal Magazine, edited by
Dr. Artemae MartiOi Vol. I, pp. 17 and 43.

INFLUX OF FRENCH MATHEMATICS. Ill

their time at the study of circulating decimals, difficult problems in
stocks and exchange, in general average, in alligation medial and alliga-
tion alternate, in square and cube root, and in combinations and per-
mutations. From the manner in which these subjects have been treated
in our arithmetics, a student derives very little mental training from
them. The presentation of duodecimals is not only nnphilosophical,
but decidedly absurd..

Protests have been made from time to time against the over-study of
Arithmetic. Thus in 1866 the Superintendent of Public Instruction of
Galifornia said in his Beport (p. 119): <^The crack classes are the
arithmetic classes, and the merits of a whole school not unfrequently
rise or fall with the exploits of the first class in arithmetic on ^ examina-
tion day.' Arithmetic is well enough in its place, but the sky is not a
blackboard,, nor are mountains all made of chalk. Children have facul-
ties other than that of calculation, and they need to be exercised on
appropriate subjects." This doubtless voices the sentiments of many '
thinking teachers. Five years ago the writer heard Prof. Simon STew-
comb, in a lecture at the Johns Hopkins University, protest against ex-
isting practices in the teaching of arithmetic.

Says Prof. T. H. Safford, of Williams College : " The mathematics
have their (disciplinary) value, and a very high one it is; but the lower
mathematics, especially aiithmetic, have been overdone in a certain
direction; I mean that of riddles, puzzles, brsCin-spinning, as the Ger-
mans call it. While our boys and girls are given problems to solve
which quite exceed their thinking powers— I don't suppose I could ever
have gone successfully through Greenleaf s liTational Arithmetic till I
had graduated from college — their minds are quite undeveloped in the
power of observation, and they are often imperfectly trained in the four
ground rules, especially in decimal fractions.''*

A very remarkable and encouraging step toward reform was taken
in 1887 by the Boston School Board. It passed the following orders con-
cerning the study of arithmetic:!

^^ 1. Home lessons in arithmetic should be given out only in excep-
tional cases.

^< 2. The mensuration of the trapezoid and of the trapezium, of the
prism, pyramid, cone, and sphere ; compound interest, cube root and
its applications; equation of payments, exchange, similar surfaces, met-
ric system, compound proportion, and compound partnership, should
not be included in the required course.

" 3. All exercises in fractions, commission, discount, and proportion,
should be confined to small numbers, and to simple subjects and pro-
cesses, the main purpose throughout being to secure thoroughness, ac-
curacy, and a reasonable degree of facility in plain ordinary ciphering.

* The Development of Astronomy in the United States, 1888, p. 27.
i The Academy f January, 1888, article: ''Arithmetic in Boston Schools," by General
Francis A. Walker, President of the Massachnsetts Institute of Technology.

112 TEACHINa AND HISTORY OP MATHEMATICS.

*<4. In < practical problems,' and in examples illustrative of arithmet-
ical principles, all exercises are to be avoided in which a fairly intelli-
gent and attentive child of the age concerned would find any consider-
able difficulty in making the statement which is preliminary to the
performance of the properly arithmetical operations. When arithmet-
ical work is put into the form of practical or illustrative problems, it
must be for the purpose of interesting and aidiug the child in the per-
formance of the arithmetical operations, and with a view to their com-
mon utility.

<<5. In oral arithmetic no racing should be permitted; but the dicta-
tion should be of moderate rapidity.

*< 6. The average time devoted to arithmetic throughout the primary
and grammar school course should be three and a half hours a week ;
and in the third primary grade not more than two hours, and in the
first and second primary grades not more than three and a half hours
each per week.''

The considerations which led the School Board to introduce these
changes are admirably set forth by General Francis A. Walker. The reg-
ulation regarding home lessons in arithmetic may be a good one under
the conditions existing in Boston at the time of its adoption, but can
hardly be recommended for general adoption. It sounds somewhat
arbitrary. The reasons which led to its adoption are, (1) a tendency
among grammar school teachers to unduly magnify the importance of
arithmetic; (2) the injustice done as between pupil and pupil by giving
homo lessons, since the facilities for study at home are so very different
' among pupils ; (3) the absence of the teacher prevents any authorita-
tive interposition to put a stop to excessive, and therefore damaging,
study over problems in the lesson. ^^In the old flogging days of the
Army and Navy," says General Walker, " it was always required that
the snrgeon should stand by, to feel the pulse of the poor wretch under
the lash, to watch the signs of approaching nervous collapse, and, in
his discretion, to forbid the punishment to proceed further. But in the
case of onr young children who are assigned home lessons in arithmetic,
no such humane provision exists. Were the work being done in the
open school room, the severest master would, when he saw that the
child did not understand the problem, could not do the work, and that
it was only becoming more excited and fatigued by repeated attempts,
interpose either to give assistance or to put a stop to the exercise. In the
case of home lessons, however, the ambitious and sensitive child finds
no relief. The work may go on long after the child should have been
in bed until a state is reached where further persistence is not only in
the highest degree injurious, but has no longer any possible relation to
success."

<<£egarding the remaining five orders, considered sis a body," says
General Walker, ^^it may be said that the committee, in framing them*

INFLUX OP FRENCH MATHEMATICS. 113

were actuated by the belief that both loss of time and misdirection of
effort, with even some positively injurious consequences, were involved
in the teaching of arithmetic, as carried on in some of the Boston schools.
And here let me say, to prevent misapprehension, that the committee
at no time intended to reflect on the schools of our own city as compared
with those of neighboring cities and towns. Personally, I believe that
the teaching of arithmetic has been more humane and rational of late
years in the schools of Boston than in those of most New England
> towns and cities. What, then, are the faults complained oft

u First — ^That the amount of time devoted to this study is in excess
of what can fairly be allotted to it, in the face of the demands of other
and equally important branches of study.

" Secondly — That the study of arithmetic is very largely pursued by
methods supposed to conduce to general mental training, which, in a
great degree, sacrifice that facility and accuracy in numerical compu-
tations so essential in the after-life of the pupil, whether as a student
in the higher schools or as a bread-winner.

<' T^irtJZi^— That, as arithmetic is taught in many, perhaps in most
schools, the possible advantages of this branch of study, even as a
means of general mental training and of the development of the reason-
ing powers, are, whether by fault of the text-book or of the individual
teacher or of the standards adopted for examination, largely sacrificed
^ through making the exercises of undue difficulty and complexity, which

^ not only destroys their disciplinary value but becomes a means of posi-

tive injury.^'

The whole paper of General Walker is well worth reading. In one
respect, however, we can not endorse the action of the Board. It seems
to us that the metric system should be retained, even if the tables of
apothecaries' weights and fluid measure, and of the mariner's measure,
had to be omitted to make room for it. The memorizing of the tables
in the metric system is not difficult. Moreover, what problems offer
better opportunities for a good, thorough course in the use of decimal
fractions than those involving meters and decimeters.

But there is still another reason for urging the spread of a knowledge
of the metric system in elementary schools. If the masses have once
acquired sufficient knowledge and familiarity with it as to see its trans-
cending superiority over the old traditional tables of weights and meas-
ures now in use, then we may look forward more hopefully to the early
approach of the tini^ when the French weights and measures will be
^ declared the only legal ones in the United States.

European nations that are generally regarded as being much more
conservative than our own, have introduced them, to the exclusion of
older ones. Even the miniature republic of Switzerland has, within the
last ten years, adopted the metric system. The change was brought
about without serious inconvenience.
881— No. 3—8

114 TEACHINO AND HISTOKt OP MATHEMATICS.

tmiTKD STATBB MILITARY ACADEMY**

In 1817 began a new epoch in the history of the United States Mili-
tary Academy. At this time Maj. Sylvanus Thayer became superin-
tendent, and under him the Academy entered upon a career of unusual
prosperity. Thayer was a native of Massachusetts, graduated at Dart-
mouth College, and then entered the Military Academy as a cadet in
1807. He was appointed lieutenant in the corps of engineers in 1808.
At the close of the War of 1812 he was sent abroad by the Government
to look into the military systems of Europe, particularly of France.
After his return the Academy was reorganized according to French
ideals, but without discarding entirely English teachings. Prof. Oharles
Da vies says that in the construction of the course of study at West
Point,** the beautiful theories of the French were happily combined
with the practical methods of the English systems, and the same has
'since been done, essentially, in the schools of England and France.^
Maj. E. 0. Boynton, in his History of West Point, summarizes the
services of Major Thayer in the following manner : ** The division of
classes into sections, the transfers between the latter, the weekly ren-
dering of class reports, showing the daily progress, the system and
scale of daily marks, the establishment of relative class rank among the
members, the publication of the Annual Begister, the introduction of
the Board of Visitors, the check-book system, the prepondering influ-
ence of the blackboard, and the- essential parts of the regulations for
the Military Academy as they stand to this day, are some of the evi-
dence of the indefatigable efforts of Major Thayer to insure method, or-
der, and prosperity to the institution. It was through the agency of
Major Thayer that Prof. Claude Orozet, the parent of descriptive geome-
try in America, and one of the first successful instructors in higher
, mathematics, permanent fortifications, and topographical curves, be-
came attached to the Academy." Orozet had been a French officer un-
der Kapoleon, and a pupil at the Polytechnic School in Paris.

Thayer was superintendent at West Point from 1817 to 1833. The
great reputation which the Academy obtained was chiefly due to his
efforts. His discipline was very strict The last years of his administra-
tion were years of trial to him. It is said that his discipline was counted
too stern, and that he was not sustained, as he should have been, at the
War Department Difficulties arose between him and the President of
thelTnited States, resulting in hisleavingthe Academy. General Francis
H. Smith says of him : t ** Colonel Thayer held the reins with a firm
hand during his entire administration, and if, at times, he transcended
the limits of legitimate authority, no private pique or personal interest
swayed his judgment. He was animated by the single desire to give

* For Official Begisters of the Military Academy and for valuable loformatlon
garding it, we are indebted to the kindness of W. C. Broirn, First Lieatenant Fisit
Cay airy, Adjutant.

t West Point Fifty Years Ago, New York, 1879, p. 6,

INFLUX OP FKENCH MATHEMATICS. 115

efficiency to kis discipline, and to train every graduate upon the high-
est model of the true soldier.'^

Andrew Bllicott was professor of mathematics from 1813 to 1820.
The following description of him applies to the time preceding the
arrival of Thayer. Says B, D. Mansfield : " There are some who will
recollect Professor Bllicott sitting at his desk at the end of a long
room, in the second story of what was called the Mess Hall, teaching
geometry and algebra, looking and acting precisely like the old-fash-
ioned' school-master, of whom it was written,

" * And still they gazed, and still the wonder grew
That one small head conld carry all be knew.'

^^ In the other end of the room, or in the next room, was his acting
assistant, Stephen H. Long. • • • The text-book used was Hut-
ton's Mathematics, and at that time the best to be had. * * * It
was a good textbook then, for there were no cadets trained to pursue
deeper or more analytical works."

As already stated. Superintendent Thayer caused the classes to be
divided into sections. From the reminiscences given by John EL. B.
Latrobe, who entered the Academy as a cadet in 1818, we see that the
various sections received their mathematical instruction from assist-
ants, and that the professor of mathematici^ occasionally visited the
sections. Mr. Latrobe says : * ^' I do not remember upon what princi-
ple our class of one hundred and seventeen members was divided into
four sections ; I recollect, however, that I was put into the first section.
* * * Our recitation room was next the guard room, on the first
floor of the North Barracks. Here, on a rostrum, between two win-
dows, sat Assistant Professor S. Stanhope Smith, and here, with the first
volume of Hntton's Mathematics in hand, I began my West Point edu-
cation. • • •

^^ I am not sure that we had desks, but rather think that we were
seated on benches against the wall, with a blackboard to supply the
place of pen and ink and slates, although I am not certain about the
slates. Generally we had the section room to ourselves. Sometimes,
however, Mr. Ellicott would pay us a visit and ask a few questions,
ending with giving us a sum in algebra, to explain what was meant by
^ an infinite series,' which was the name he went by in the corps.^

" I have,^ continues Latrobe (p. 29), " no other recollection of him as
an instructor, except once when, while learning surveying, we were
chaining a line from a point in front of his house to an angle of Fort
Olinton, and back again. Our accuracy quite astonished the good old
professor, to whom we did not admit that it was owing to our having
used the same holes that the pins had made in going and returning. ^

Professor Bllicott died at West Point and was baried in the cemetery
there. " My last visit to it as a cadet, " says Latrobe, "was when I

>^ . ■ ■ ^ - ■ ^ - _■ ■ — ... — _ — - _ .. ■■ ■ , ^ _ ^^ ^

*Beport Assooiatioa of Graduates of the U. 8. Military Academy, 1887, p. 8^

116 TEACHING AND HISTORY OP MATHEMATICS.

was on the escort that fired the voUies over the grave of Andrew EUi-
cott, the professor of mathematics who lies buried there. "

Of Button's Mathematics Latrobe says: ''I have often heard those
who have been more recently educated at West Point speak dispar-
agingly of the Huttonian day, as though any one could have graduated
then.'' That this was not the case becomes evident when he says, " that
the first sifting in June, 1819, of my one hundred and seventeen comrades
of the year before, reduced the number to fifty-nine, the next sifting to
forty-eight, and the number that got through the meshes of the sieve was
but forty. Of the others, some resigned, some were ' turned back ' to go
over the year's course a second time, and some were found to be defi-
cient altogether. These last were called, in the parlance of the cadets,
' Uncle Sam's bad bargains.' "

Jared Mansfield, the professor of natural and experimental philosophy,
outlived Ellicott by ten years. Both were veteran surveyors and math-
ematicians. Mansfield retired from his chair in 1828. Mr. Latrobe says
that Colonel Mansfield, " although a most competent instructor, was
very near-sighted, and I am not prepared to say that this defect was
not sometimes taken advantage of. " Professor Church (class of 1828)
says of him : " Professor Mansfield at my time was very old, yet quite
enthusiastic in his branch of study, generally a mere listener to demon-
strations, complimentary to a good one, but coldly silent to a bad one. "

The great impulse to the study of mathematics at West Point was,
however, due to younger men. One of these was Claude Crozet. After
graduating at the Polytechnic School in Paris, he had been artillery
ofiQcer under Kapoleon. From 1816 to 1817 he was assistant professor
of engineering at the Academy, and from 1817 to 1823 full professor.
E. D. MansAcld has given us some interesting recollections of Crozet's
earliest teaching at West Point. The Junior class of 1817-18 was the
first class which commenced thoroughly the severe and complete course
of studies at the Academy. Of Professor Crozet, Mansfield says that
he was to teach engineering, but when he met the class he found that
he would have to teach mathematics first, as not one of tbcm had bad
the necessary preliminary training in pure mathematics for a course in
engineering. "The surjirise of the French engineer, instructed in the
Polytechnique, may well be imagined when he commenced giving his
class certain problems and instructions which not one of them could
comprehend and perform."

Among the preliminary studies we find that descriptive geometry
was included. " We doubt," says E. D. Mansfield, " whether at that
time more than a dozen or two professors of science in this country
knew there was such a thing; certainly they never taught it, and
equally certain there was no text-book in the English language."
This science, founded by Monge, was then scarcely thirty years old.
Crozet meant to begin by teaching this branch, but a new difficulty
arose. Just then he had no text-book on the subject, and geometry

INFLUX OP FRENCH MATHEMATICS, 117

conUl not be tangbt orally. What was to be done f " It was here at
this precise time that Crozet, by aid of the carpenter and painter, in-
troduced the blackboard and chalk. To him, as far as we know, is due
the introduction of this simple machine. He found it in the Polytech-
niqne of France.^' (B. D. Mansfield).

Crozet was, however, not the first one to nse the blackboard in this
country. Of Rev. Samuel J. May, of Boston, it is said that, " to the
work of teaching a public school he then brought one acquisition which
was novel in that day, and which it has taken a half century to intro-
duce into elementary schools, private and public — a knowledge of the
uses of the blackboard, i/cMch he had seen for the first time in 1813 in
the mathematical school Icept by Eev. Francis Xavier Brosius, a Catholic
priest of France, who had one suspended on the wall with lumps of
chalk on a ledge below and cloth hanging on either side." • One thing
is certain : The blackboard was introduced in this country by French-
men. Its importance in the school room can hardly be overestimated.
Simple and inexpensive as it is, its introduction into our colleges was
not instantaneous. For geometrical teaching large tablets with printed
diagrams were nsed in our best colleges long after Crozet had taught
its use at West Point.

Crozet, says E. D. Mansfield, did not more than half understand
English. "With extreme diflBculty he makes himself understood and
with extreme difficulty his class comprehend that two planes at right
T angles with one another are to be understood on the same surface of

the blackboard, on which are represented two different projections of
the same subject.'' The first problems were drawn and demonstrated
on the blackboard by the professor; afterward they were drawn afid
demonstrated by the pupils, and then carefully copied into accurate
drawings.

In 1821 Crozet published his Treatise on Descriptive Geometry, for
the use of cadets of the U. S. Military Academy (New York). The '
first 87 pages were given to the elementary principles, and the next 63 ,
pages to the application of descriptive geometry to spherics and conic
sections. This is, according to our information, the first English work -J
of any importance on descriptive geometry, and the first work pub-
lished in this country which exhibits to the student that gem of geom-
etry — Pascal's Theorem.

Crozet has been called the father of descriptive geometry in this
country. He taught this as preparatory to engineering. It may justly
be said, also, that the course of military science was greatly developed
' by him.

Mr. Latrobe favors us with the following recollections of him : " There
are persons whose appearance is never effaced from the memory. Of
this class was the professor of the art of engineering. Col. Claude
Crozet, a tall, somewhat heavily-built man, of dark complexion, black

• « American Educational Biography/' Barnard's Journal, Vol XVI, p. 141, 1866.

}

118 TEACHING AND HISTOBT OF MATHEMATICS.

hair and eyebrows, deep-set eyes, remarkable for their keen and bright
expression, a firm mouth and square chin, a rapid speech and strong
French accent. I can, even after the lapse of between sixty and seventy
years, fancy that I see the man before me. He had been an engineer
under Napoleon at the battle of Wagram and elsewhere, and the anec-
dotes with which he illustrated his teaching were far more interesting
than the 'Science of War and Fortification,' which was the name of our
textbook at the time« When he left the Academy he became chief
engineer o^ the State of Virginia, which is indebted to him for the sys-
tem that made her mountain roads the best, then, in America. Per-
haps my recollection of Colonel Crozet is strengthened by my having
' seen him long after I ceased to be his pupil."

Ellicott was succeeded in the professorship of mathematics by David
B. Douglass. He held it till 1823, when he was transferred to the de-
partment of engineering, where he taught till 1831. Professor Ohurch
(class of 1828) says of him : ^' Professor Douglass, of engineering, had
the reputation of beiug an able engineer and a fine scholar, yet he
was by no means a clear demonstrator. His style was diffuse and there
was a great want of logical sequence in his language. Most of the
course of engineering was given to the class by him from the black-
board.'' He was afterward the chief engineer of the Groton water-
works.

One of the text-books mentioned by E. D. Mansfield as having been
used was the Mechanics of Dr. Gregory (" Old Greg.^), who wa« pro-
fessor at the Boyal Military Academy at Woolwich. His works are
collections of rules rather than expositions of principles, and are want-
ing in analysis. Gregory is at his best when he descends to the mi-
nutisB of practice. For several years no adequate text-book was found
for civil engineering. In 1823 Major O'Gonner translated a Treatise on
the Science of War by De Vernon, which had been prepared in 1805 by
the order of the French Government and was the text-book in the poly-
technic school. This translation was used at the Academy for several
years. ^^ It was a miserable translation," says General Francis H. ^mith,
^<but it was the best that could be had, and each member of the first
class was required to take a copy, costing some $20."

After being vacated by Douglass, the chair of mathematics was taken
by one whose name became known to nearly every school-boy in our
laud— Charles Davies. He was a native of Connecticut, graduated at
the Academy in 1815, and then was made aasistant professor of ma£he-
matics. He held the full professorship for fourteen years, until 1837.
He earned for himself a wide reputation, not as an original investigator
in mathematics, but as a teacher and as a compiler of popular text-
books. He wafi always described by his pupils as an excellent instructor.
Professor Church (class of 1828) says of him : *^ Professor Davies was
then young, enthusiastic, a clear and logical demonstrator, and an admir-
able teacher. He had at once imbibed the spirit and iuily sympathized in

INFLUX OP FRENCH MATHEMATICS: 119

the desires of the saperiDtendent, and labored earnestly to carry them
oat in baildiog up a logical system of instruction and recitation, Tvbich
required not only a tboroagh understanding of the details of and rea-
sons for everything proposed, but a clear, concise, and complete exam-
ination of it;'

When Church was a cadet, according to his own statement, the meth-
ods of instruction were entirely new, and text-books very imperfect.
The professors and teachers had themselves to learn the true use of the
blackboard, and the strict and detailed manner of demonstration. In\
algebra the best text-book that could be obtained was a poor translation I
of Lacroix. In geometry we had a translation of Legendre ; in trigo-
nometry, a translation of Lacroix ; in descriptive geometry, a small work '
by Crozet, oontaining only the elements without application to the inter- /
section of surfaces or to warped. surfaces. These, with the whole of
shades, shadows, and perspective, stone-cutting, and problems in engi-
neering, were given by lecture to the class. Kotes were taken by the
cadets, the drawings made in our rooms before the next morning, then
presented for examination, and at once recited upon previous to the fol-
lowing lecture. The sections in mathematics, philosophy, and engi-
neering were of twenty cadets each, and were kept in three hours daily.
Blot's work on analytical geometry was used, and Lacroix's calculus.

Those who have toiled over Davies' text-books may enjoy the follow-
ing reminisoences of him : ^' Don't you remember," says General F. H.
Smith (class of 1833), ^^ when muttering out an imperfect answer to one
of his questions, how he would lean forward with one of his significant
smiles and say, < How's that, Mr. Bliss V But I will not now dwell upon
bis long and faithful career in the department of mathematics. The re-
sults of his labors are to be seen in the distinguished career of his pupils
and in his series of mathematical text-books, which are as household
words everywhere in the United States."

When John H. B. Latrobe was a cadet, Davies was as yet only assist-
ant professor. Latrobe speaks of him as follows : <^ My next professor of
mathematics, in my second year's course, was one that I have no difficulty
in describing and whom I can never forget, Oharles Davies. Personally
and mentally he was a remarkable man. Of the middle size, with a
bright, intelligent face, characterized by projecting upper teeth, which
procured for him the name of ^Tush' among the cadets, his whole figure
was the embodiment of nervous energy and unyielding will. His fear-
less activity at a fire which happened in a room in the South Barracks,
in 1819, added the name of 'Bush' to the other. He was a kindly
natured man, too, and tbe patient perseverance that he devoted to the
instruction of his class was not the least remarkable feature of his
character. It was with Professor Davies that I began the study of
descriptive geometry, for which no books in English had then been
published. He had no assistance beyond the blackboard and his own
Ultimate knowledge of the subject and faculty of oral explanation. For-

120 TEACHIKG AND HISTOBT OFv MATHEMATICS.

innately this was exceptionally great, and even then there was no little
amount of actual labor requisite to enable the pupil to understand the
difference betweeen the horizontal and vertical planes and the uses to
be made of them. It is to Professor Davies that I have always at-
tributed in a great measure my subsequent successes at West Point,
and hence this especial notice of him as a tribute to his memory. A
much more enduring tribute is that awarded by the countless benefici-
aries, the colleges, schools, and individuals who have profiled by his
numerous publications in connection with mathematical science."

Professor Davies taught for many years before he conceived the idea
of issuing a series of text-books. Some of his books — as his Legendre
and Bourdon — were adaptations from French works, modified to sup-
ply the wantf of our schools ; others were prepared on his own plan.
While connected with the Academy as professor he published his
Descriptive (Geometry, 1826 (a more extensive work than Grozet's) ;
Brewster's Translation of Legendre, 1828 ; Shades, Shadows, and Per-
spective, 1832 ; Bourdon's Algebra, 1834; Analytical Geometry, 1836;
Differential and Integral Oalculus, 1836 ; a Mental and Practical Arith-
metic. Overwork in the preparation of these textbooks caused bron-
chial affection, which forced him to resign his professorship in 1837.
He visited Europe and soon after his return occupied the professorship
of mathematics at Trinity College, Hartford, Gonn.,but ill-health again
induced him to exchange the position for that of paymaster in the
Army and treasurer at West Point. These offices he resigned in 1846.
In 1848 he became professor of mathematics and philosophy in the
University of New York, but in the following year he retired to Fishkill
Landing, on the Hudson, that he might have leisure to complete his
series of text-books. After teaching in the normal school at Albany, he
was made professor of higher mathematics at Golumbia College, in 1867.

In 1839 appeared his Elementary Algebra; in 1840, his Elementary
Geometry and Trigonometry ; in 1846, his University Arithmetic ; in
1860, his Logic of Mathematics ; in 1862, his Practical Mathematics ; in
1866, jointly with William G. Peck, a Mathematical Dictionary.

Davies' series constituted a connected mathematical course, from
primary arithmetic up to calculus. His books were, as a rule, perspic-
uous, clear, and logically arranged. They were not too difficult for the
ordinary student, and contained elements of great popularity. The
original editions would be found quite inadequate for the wants of
schools of the present day. *<The first translations of Bourdon and
Legendre were imperfect" (Prof. G. W. Sears, class of 1837). Davies
himself greatly modified some of his text-books in later editions. In
his revisions he was greatly aided by his son-in-law, Prof. William G.
Peck. The most recent revisions are those made by Prof. J. Howard
Van Amringe, of Golumbia College.

Brewster's Legendre underwent some changes in the hands of Davies.
In the original work, as also in the translation of Brewster and Farrar,

INFLUX OP FRENCH MATHEMATICS. 121

each proposition was enunciated with reference to and by aid of the par*
ticular diagram used for thei demonstration. But Davies gave the
propositions without reference to particular figures and, to that extent,
returned to the method of Euclid. In later editions Davies did not use
Brewster's translation, but took the original and translated and adapted
it to the courses in American schools. In trigonometry he was wedded
to the line system.

The reasoning sometimes employed by Professor Davies in his books
has been found to be open to objection. This is certainly true of his
treatment of infinite series. In his Legendre the treatment of the
circle is not such as will carry conviction to the young mind. Thus, he
says in one edition, that " the circle is but a regular polygon with an
infinite number of sides." * A trained mathematician w%o feels that he
can give more rigorous proofs by sounder methods, whenever he may
wish to do so, will employ this idea of the circle, and of curves in gen-
eral, with profit and satisfaction. After much study he may even
arrive at the conviction that the method of limits and that of infini-
tesimals are essentially alike. But it is the experience of the majority
of our teachers that the infinitesimal method and the treatment of the
circle as a polygon, appear to beginners as enigmatical and obscure.
Of our more recent geometries, the best and the most popular have
abandoned those methods.

ISov is Davies' explanation of a limit and of the first differential
co-efficient satisfactory. Listen to the testimony of one of his pupils :t
*^ I had not been a teacher of the calculus long * * * before I dis-
covered that I had almost everything to learn respecting it as a rational
system of thought. Difficulties were continually suggested in the
course of my reflection on this subject about which I had been taught
nothing, and consequently knew nothing. I found, in short, that I had
only been taught to work the calculus by certain rules, without know-
ing the real reasons or principles of those rules ; pretty n^nch as an
engineer, who knows nothing about the mechanism or principle of an
engine, is shown how to work it by a few superficial and unexplained
rules."

It is our opinion that under Professors Davies and Church the philos-
ophy of mathematics was neglected at West Point. If this criticism
be true of West Point, which was for several decennia unquestionably
the most influential mathematical school in the United States, how much
more must it be true of the thousands of institutions throughout the
country which came under its influence f If this stricture were not cor-
rect; then such a' book as Bledsoe's Philosophy of Mathematics would
never have been written ; there would have been no occasion for it.

Of Davies' assistants, we shall mention Lieutenant Boss. General F.
H. Smith says : '* There was associated with Professor Davies, * • •

* Davies' Legendre, 1856, Book Y, Scholium to Proposition XII.
tProf. A. T. Bledaoe, Philosophy of Mathematics, 1867, p. 214, note.

l22 / TEACHING AND HISTOBY OP MATHEMATICS.

as his chief assistant in mathematics, having charge of the fourth class,
Lieat. Edward 0. Boss, of the class of 1821. He was the best teacher
of matbeonatics I ever knew, and it is singular, too, that he had no fac-
ulty of demonstration. He gave to our class many extra discussions iu
the difficult points in algebra, particularly on what he called the final
equations, for he was not pleased with Farrar's translation of La Croix,
our tex^book in algebra, and he was preparing his translation of Bour-
don. In putting upon the blackboards these extra demonstrations
every line appeared as if it had been printed, so neat was he in the use
of his chalk pencil. But when he commenced to explain he would twist
and wriggle about from one side of the board to tiie other, pulling his
long whiskers, and spitting out, in inordinate volumes, his tobacco juice.
The class was as ignorant when he closed as when he began. We copied,
word for word, what was written, weU knowing that on. the next day
the first five would be called upon to make the discussion. We read to
him what we had placed on the board. Then commenced his pow^ as
a teacher. In a series of orderly questions he would bring out the
points of the discussion, step by step, sometimes occupying half an hour
with each cadet, and when the three hours of recitation were over we
knew the subject thoroughly. He was an expert in his power of ques-
tioning a class. He did this without note or book, and gave snch esan-
estness and vividness to his examinations that he kept his class up to
the highest pitch of interest all the time.''

Oen^ul Smith gives us a description also of Oourtenay, '* Edward
H. Oonrtenay, who graduated at the bead of Boss's class, was our pro-
fessor of natural and experimental philosophy, fifty years ago. There
never was a clearer minded — a more faithful teacher«-or a more modest
one than Professor Oourt^iay. Well do I remember the hesitating man-
ner with which he would correct the grossest error on the part of a
member oi his section**-/ hordly think $o* He resigned his professorship
in 1834, and after holding many offices of high dignity, as professor and
civil engineer, he was elected professor of mathematics ib the Univer-
sity of Virginia in 1842.''

Oonrtenay was instructor at West Point from 1821 to 1834, excepting
tbe four years from 1824 to 1828. During his first years of teaching he
was assistant professor of natural and experimental philosophy, and
then assistant professor of engineering. After the resignation of Jared
Mansfield he was appointed professor of natural and experimental phil-
osophy, and acted iu that capacity for five years. In 1833 he trans-
lated from the French of M. Boueharlat an elementary treatise on me-
chanics, and made additions and emendations.

The chair oi military and civil engineering, made vacant by the resig-
nation of Prcrfessor Douglass, was filled by the appointment of Lieut
Dennis H. Mahan. Mahan graduated at West Point in 1824, holding
the first place in his class of thirty-one members. ^'After remaining at
the Academy as an instructor for two years, he was ordered to Europe

INFLUX OP FEBNCH MATHEMATICS. 123

to study pnblio engineerin g works and military institutions. By special
favor of the French Ministry of War, Lientenant Mahan was allowed tb
join the Military School of Application for Engineers and Artillerists at
Metz, where he remained for more than a year, under the instruction of
men whose names were then, and are now, widely known in science."*
When, after his return, he entered upon the duties of his department at
West Point, he supplemented the meagre volume of O'Connor wittv ex-
tensive notes. These notes developed into his well-known treatises on
^ Civil Engineering and Field Fortifications.

Such is th^ brief record of the professional career of Professor Ma- \
han ; but it fails to convey any adequate idea of the influence which !
he exerted upon engineering science in this country. To appreciate '-
tibiis, it must be remembered that for many of those forty-one years ^
(during which he was professor) West Point was our only schoolof
mathematical and physical science where the rigid requirements and
high standard now deemed essential were even attempted. Every offi* .
eer of the present corps of engineers who has served long enongh to
win reputation pi the performance of the civil duties assigned to that
corps, and many of the eminent civil engineers of the country as well|
now gratefully remember how, before those old blackboards in that nn-
pretending recitation room at West Point, they learned from Professor
Mahan, with the rudiments of their profession, a high-toned discipline
and the fundamental truth that without precision of ideas, rigid analy-
T sis, and hard work there can be no such thing as success.

'^ But if civil engineering owes mnch to our late colleague, military
engineering and the science of war owe more. For many years, and up
to the day of his death, he was in that branch of the profession con-
fessedly the highest scholastic authority in America."*

Tfa^ death of Mahan was pathetic. In his last years he often had fits
of melsmcholy, and, in an instant of acute insanity, he plunged ftom a
steamer Into the Hudson and drowned.

Professor Davied was succeeded in the chair of mathematics by Pro£
Albert £. Church. Church was a native of Connecticut, graduated at
West I^olnt in 1828, served as assistant professor from 1828 to 1831,
also from 1833 to 1837. He was then acting professor of matliematics
for abont a year, and in 1838 he became full professor, retaining the ^
chair till his death m 1878. He published four works which have been I
used considerably in American colleges. His Differential and Integral
Calculus, 1842, was more extensive than that of Davies. In the new.
edition of 1851 a chapter on the Elements of the Calculus of Variations
^ was inserted. In 1851 appeared his Analytical Geometry, in which he

followed somewhat the work of Biot on this subject. In 1857 his Plane
and Spherical Trigonometry was published. In 1865 appeared his Bl-
ements of Descriptive Geometry, in the preparation of which he was

*' " ■■■■! Ill.ll »!■■ ■>«■■ I . 1,1. II ^ h,, M.— i»— ^

* Biographical Meiuolrs of the National Academy of Sciences, Vol. 11, 1886, p. 32.

124 TEACHINa AND HISTORY OF MATHEMATICS.

aided by the French works of Leroy and Oliver, and by the elaborate
American work of Warren. Later editions give also the application df
the subject to shades, shadows, and perspective. The Descriptive
Geometry met with larger sales than any of his other works.

As a teacher, Professor Charch is. spoken of by General F. H. Smith as
follows : " Prof. Albert E. Church was an assistant professor of math-
ematics when my class entered in 1829. He occasionally heard my sec-
tion in the third class course and exhibited then the clearness and
perspicuity which marked his long career as a professor of mathematics.'*
'/ Prof. Arthur 8. Hardy (class of 1869) gives the following remin-
iscences of the mathematical teaching in his day:*

" The class was divided into sections of from ten to fifteen. The alpha-
betical arrangement, first adopted, became in a few weeks a classifica-
tion by scholarship — transfers up and down being made weekly. The
descent was easy, but it was hard to rise a section. The last section we
called Lbh Immortels (lazy mortals?). In each section each student
recited daily. The sections were taught by army officers detailed at
Professor Church's request. The latter had no section, but generally
visited each daily. Each recitation was one hour and a half long.
Professor Church's visits were dreaded. He usually asked questions.
His questioning was searching. He was a stickler for form — ^it was
not enough to mean right.

"Personally he did not inspire me; he had no magnetism — was dry
as dust, as his text-books are. He delivered one lecture on the calculus.
I never got a glimpse of the philosophy of matjhematics — of its history,
methods of growth. The calculus was a machine, where results were
indisputable, but its mechanism a mystery. I do not think he had a
great mathematical mind.* It was geometrical, rather than an analytic
one. A problem, which he and Professor Bartlett once attacked to-
gether, the latter solved by a few symbols on a piece of paper, while the
former drew a diagram with his cane on the gravel — to Bartlett's dis-
gust, who despised geometry. Church's text-books are French adapta-
tions, minus the luminousness and finish of form of French text-books.

" The only instance of Church's being disconcerted was on being told
by a cadet that the reason for -f- becoming — in passing through zero
was that the cross-piece got knocked off in going through. You can
imagine that the would-be wit was placed in arrest.

" The mathematical recitation at West Point was a drill-room. In my
* judgment its result was a soldier who knew the manoeuvres, but it did
not give an independent, self-reliant grasp of methods of research. In
descriptive geometry, the Academy had a magnificent collection of
models, but they were shown us after the study was finished— in other
words, mental discipline was the object — practical helps and ends were
secondary. Great changes have been made since."

William H. C. Bartlett (class of 1826) was assistant professor for

*Xjetter to the writer, November 18, 1888*

INFLUX OF FRENCH MATHEMATICS. 125

several years at West Point. He was permanently appointed professor
of natural and experimental philosophy in 1836. In 187 L he was re-
tired from military service at his own request, and shortly after he
accepted the place of actuary £or the Mutual Life Tusurance Company
of New York.

The need of an astronomical observatory being felt at West Point,

Professor Bartlett went abroad in 1840 to order instruments and visit

observatories. On his return it was necessary to provide room for the

^ instruments in the new library building of the school, on account of

the great prejudice existing in Congress against a separate observatory ••

Bartlett published treatises on Optics, 1839 ; Acoustics ^ Synthetic
Mechanics, 1850 ; Analytical Mechanics, 1853 ; Spherical Astronomy,
1855. He contributed also to Silliman's Journal. His Analytical Me-
chanics is the first American woik of its kind which starts out with,
and evolves everything from, that precious intellectual acquisition of
the nineteenth century — the laws of the indestructibility of matter and
energy. Dr. E. S. McCulloch (who, by the way, rewrote Bartlett's
Mechanics without allowing, his own name to appear anywhere in the
revised edition) says : t " More than thirty years ago, at West Point,
Professor Bartlett, in. his treatise on Analytical Mechanics, still used
there as a text-book, had deduced the whole science from one single
equation, or formula, well known to every cadet as his equation A ;
^ and he thus expressed and discussed fully what now is generally called

the Law of the Conservation of Energy.''

Bartlett's successor was Prof. Peter S. Michie, the present incumbent
in the chair. Michie graduated in 1863, and has been instructor there
since 1867. He has published Wave Motion, Eelating to Sound and
Light, 1882 ; Hydrostatics ; and Analytical Mechanics, 1886. The first
edition of the last treatise was never published \ the second edition,
1887, differs considerably from the first. It is on the plan of Bartlett^a
book on the same subject, but it is confined to mechanics of solids. It
contains also a good introduction to graphical statics, a subject which,
in recent years, has come to be studied in this country. The first to
place a treatise on graphical statics in the hands of American engi-
gineers was A. Jay Du Bois,J professor at Lehigh Universitj':, Pa.
This subject owes its development chiefly to Oulmann, who, in 1866^
published in Zurich his Oraphische Statilc, In technical schools in
Europe this method has been favorably received. In this country^
original contributions of great value have been made to this subject
* by Prof. Henry T. Eddy, of the University of Cincinnati.§

* Tho Development of Astronomy in the United States, by Prof. T. H. Safford, 1888,
p. 19.

t Papers road before the New Orleans Academy of Sciences, 1886-87, Vol. I., No.
1, p. 120.

t The Elements of Graphical Statics and their Application to Framed Structures,
New York, 1875.

$ Van Nostrand's Engineering Magazine, 1878. Article : ** cV New General Method
in Graphical Statics."

126 TEACHING AND HISTORY OF MATH]?MATICS.

In 1841 Professor Church was aided in his department by five assist^
ants. This number has been increased since, and is now nine. These
assistants have been, we believe, always selected from yoang graduates
of the Academy. The course of study in pure and applied mathematics
ws^s, in 1841, as follows : Fourth Class (first year), Davies' Bourdon,
Legendre, and Descriptive Geometry; Third Olassy Davies' works on
Shades and Shadows, Spherical Projections and Warped Surfaces, Sur-
veying, Analytical Geometry, and Calculus ; Second Olassy Courtenay's
Boucharlat's Traits de M^caniqne, Boget's Electricity, Magnetism,
Electro Magnetism and Electro-Dynamics, Bartlett's Optica, Gummere^s
Astronomy; First ClasSy Mahan's Treatises on Field Fortifications,
Lithographic Notes on Permanent Fortification, Attack and Defence,
Mines and other Accessories, Composition of Armies, Strategy, Coarse
of Civil Engineering, Lithographic Notes on Architecture, Stone Cat-
ting, Mechanics (studied by the first section only).

As Church's and Bartlett's text-books came from the press they were
introduced in place of earlier ones. Thus, Davies' Geometry, Calculus,
Descriptive Geometry, and Trigonometry, Oummere's Astronomy,
and Courtenay's Boucharlat's Mechanics were gradually displaced by
new books. But some of Davies' books have been retained to the
present day. We may here state that the power of selecting text-books
does not lie with each individual professor, but with the Academic
Board.

After the death of Mahan, in 1871, the chair of military and civil
engineering was given to Junius B. Wheeler, of the class of 1855. He
retired in 1884, and was succeeded by James Mercur. Professor Wheeler
gradually substituted books of his own in place of Mahan's treatises.

Professor Church's successor is Pro£ Edgar W. Bass, of the class of
1868. By him more attention is given to the philosophical exposition
of fundamental principles than was given by his predecessors. Davies'
and Church's text-books are still used, but they are much modified by
copious notes by Professor Bass. In calculus the notation of Leibnitz
has always been used, but now the Modem is also given. At present
the calculus is based upon the Newtonian conception of rates, but his
notation is not used. In 1879 determinants and least squares were
introduced into the course of study. Peck's Determinants and Chan*
venet's Least Sqnares being the text-books nsed.

The present mode of instruction in mathematics involves recitations
by cadets at the blackboard, lectures and explanation of the text, namer-
ous applications of each principle, and written recitations by the sta-
dents. The sections number from nine to twelve students, for one and
a half hour's instruction. Three hours are allotted for the study of each
mathematical lesson. Becitations are daily, Sundays excepted.

The course of study for 1888 is as follows : Fourth classj Davies' ele-
ments of Algebra, Legendre's Geometry, Ludlow's Elements of Trigo-
liometry, Davies' Surveying, Church's Analytical Geometry; Third eUus^

INFLUX OF FREKCH MATHEMATICS. 127

GhHfcb's Analytical GteOfinetry^ Descriptive GkKnnetry with its applica-
tiona to Spherieal Projections, Bass's Introdnction to the Differential Oal-
CttlnSy Church's GalcalaSy Oharch's Shades, Shadows, and Perspective^
Ohalivenet^s Treatise on the method of Least Squares ; Second das^^
Hichie's Mechanics, Bartlett's Astronomy, Micbie's Elements of Wave
Motion relating to Soand and Light; First cloftSj Wheeler's Civil Engi-
neering, Field Fortifications, Mercnt's Mahan's Permanent Fortificaitions
(edition of 1887) ; Wheeler's Military Engineering (Siege Operations and
^ Military Mining), Elements of the Art and Science oi>War, and Mahan's

Btereotomy* For reference, is nsed a book called Boyal Engineers'
AideM6moii«, Parti I and IL

It may be stated, in eondasion, that the 17^ S. Military Academy has
oontribated to the educational force of the country no less than thirty*
five presidents of universities or colleges, twenty-seven principals of
academies and schools, eleven regents and chancellors of educational
instttations and one hundred and nineteen professors and teacherst malt-
ing a total of ofie hundred and ninety-two instructors of youth distrib-
uted tiiroughout the country.*

HASYABD GOIXEaS.

In 1807 John Farrar succeeded Samuel Webber in the chair of )
' mathematics and natural philosophy. Farrar was a native of Massa-

'^^ chusetts. After graduating at Harvard he studied theology at Andover, \ «/
but having been appointed tutor of Greek, in 1805, he never entered the -
ministerial office. He retained his chair till 1836, when he resigned in
consequence of a painful illness that finally caused his death. He was
a most amiable, social, and excellent man, and endeared to his friends.
By the students he was familiarly called *' Jack Farrar."

Prof. Andrew P. Peabody gives the following reminiscences of him : t
*< He delivered, when I was in college, a lecture every week to the Junior
class on natural philosophy, and one to the Senior class on astronomy.
His were the only exercises at which there was no need of a roU-calL
Ko student was willingly absent. The professor had no notes, and
commenced his lecture in a conversational tone and manner, very much
as if he were explaining his subject to a single learner. But whatever
the subject, he very soon rose from prosaic details to general l^iws and
principles, which he seemed ever to approach with blended enthusiasm
and reverence, as if he were investigating and expounding divine mys-
teries. Bis face glowed with inspiration of his theme. His voice, which

^ . was unmanageable as he grew warm, broke into a shrill falsetto; and
with the first high treble notes the class began to listen with breathless
stillness, so that a pin-fall could, I doubt not, have been heard through
the room. This high key once reached there was no return to the lower

'Annual Beport of the Board of Visitors to the U. S. Military Academy made to
the Secretary of War, for the year 1886.
t Harvard Keminiscenoes, by Andrew P. Peabody, Boston, 1888, p. 70.

128 TEACHING AND HISTOBT OF MATHEMATICS.

notes, nor any intermission in the outflow and qaickening rash of lofty
thought and profoand feeling, till the bell announced the close of the
hour, and he piled up all the meaning that he could stow into a parting
' sentence, which was at once the climax of the lecture, and the climax
of an ascending scale of vocal utterance higher,! think, than is within
the range of an ordinary soprano singer. I still remember portions of
^ his lectures, and they now seem to me no less impressive than they did
in my boyhood."*

Josiah Qnincy t says in his diary, which he kept while a student at
college, that by the prolixity of Professor Everett in his lectures, ^^ we
gained a miss from Farrar for the fourth time this term. This was
much to the gratification of the class, who in general hate his branch,
though they like him."

Professor Farrar did not distinguish himself by original research in
mathematics, but he was prominent and among the first to introduce
V important reforms in the mathematical teaching in American colleges.

He was the first American to abandon English authors and to place
translations of Continental works on mathematics in the hands of sta-
i dents in the New World.
, In 1S18 appeared Farrar's Introduction to the Elements of Algebra,
, selected from the Algebra of Enler. Notwithstanding the transcend-
ing genius of Euler as a mathematician and the high estimation he was
held on the Continent, his algebra was scarcely to be met with previous
to this time, either in America or England. It was written by the au-
thor after he became blind, and was dictated to a young man entirely
without education, who by this means became an expert algebraist.}
Farrar's Euler was a very elementary book, and was intended for stu-
dents preparing to enter college. It differed from the English works in
, this, that it taught pupils to reason, instead of to memorize without un-
derstanding.

In the same year appeared also Farrar'^ translation of the Algebra of
Lacroix, which was first published in France about twenty years pre-
viously. Lacroix was one of the most celebrated and successful teach-
ers and writers of mathematical text-books in France. Farrar trans-
lated also Lacroix's Arithmetic, but this does not appear to have been

* Professor Poabody continn 68 his reminisoences as follows: "I recall distinctly a
lecture in which he exhibited, in its yarioos aspects, the idea that in mathematical
science, and in it alone, man sees things precisely as Qod sees them, handles the Very
scale and compasses with which the Creator planned and built the universe ; another
in which he represented the law of gravitation as coincident with, and demonstrative
of, the divine omnipresence ; another, in which he made ns almost hear the music of
the spheres, as he described the grand procession, in infinite space and in immeasur-
able orbits, of our own system and the (so called) fixed stars. His lectures were
poems, and hardly poems in prose; for his language was unconsciously rythmicalf
and his utterances wore like a temple chant."

t Pigurea of the Past, p. 23.

% North American RevieW| 1818.

INFLUX OF FRENCH MATHEMATICS. 129

received as favorably in this country as the other works of the Harvard '
professor. 1

In 1819 was published at Cambridge the Geometry of that famous , >
French mathematician, Legendre. A similar translation was made in |
England by David Brewster. Legendre has been the greatest modem \
rival of Euclid. In France, in most schools in America, and in some \
English institutions, the venerable and hoary-headed Euclid was made
to withdraw and make room for Legendre.

If the question be asked, what is the difference between the geome-
tries of Euclid and Legendre, we would answer that the main object of
Legendre was to make geometry easier and more palatable to students.
This he succeeded in doing, but at a sacrifice of scientific rigor. The fol-
lowing are the principal points of difference between Euclid and Legeu^
dre : (1) Legendre treats the theory of parallels differently ; (2) Legen-
dre does not give anything on proportion, but refers the student to
algebra or arithmetic. The objection to this procedure is that in arith-
metic and algebra, the properties of proportion are unfolded with regard
to numbers, but not with regard to magnitude in general. From a
scientific point of view this is a serious objection, especially if we remem-
ber that in geometry incommensurable quantities arise quite as frequently
as commensurable quantities do. Euclid's treatment of proportion dis-
plays wonderful skill and rigor, but is very difficult and abstract for
students beginning the study of geometry; (3) Euclid never supposes
a line to be drawn until he has first demonstrated the possibility and
shown the manner of drawing it. Legendre is not so scrupulous, but
makes use of what are called '' hypothetical constructions.'^ (4) Legen-
dre introduces new matter, especially in solid geometry, changes the
order of propositions, and gives new definitions (as, for instance, his
definition of a straight line).

In 1820 Farrar published his translation of Lacroix's Trigonometry. 1
The original gave the centesimal division of the circle, but in the trans- ;
lation the sexagesimal notation was introduced. This trigonometry '
adopted the ^' line system." Bound together with this book was the
"Application of Algebra to Geometry." This was selected from
the Algebra of Bezout. Eegarding this selection Professor Farrar
says : " It was the intention of the compiler to have made use of the
more improved treatise of Lacroix or that of Biot upon this subject;
but as analytical geometry has hitherto made no part of the mathemat-
ics taught in the public seminaries of the United States, and as only
a small portion of time is allotted to such studies, and this is in many
'instances at an age not sufficiently mature for inquiries of an abstract
nature, it was thought best to make the experiment with a treatise
distinguished for its simplicity and plainness." •

The next book in the " Cambridge Course of Mathematics," as Far-
Tar's works were called, was an Elementary Treatise on the Application

* See advertisement to the Treatiee.
881—No. 3 9

•

130 TEACHING AND HISTOBT OF MATHEICATICS.

of Trigonometry (1828), in the preparation of which were used Gag-
noli's and Bonnycastle's Trigonometries, Delambre's Astronomy, Be*
zont's Navigation, and Pnissant and Malortie's Topography.

In 1824 were published the First Principles of the Differential and In-
tegral Oalculas, '^ taken chiefly from the mathematics of Bezoat." This
is the first textbook published in America on the calculas and employ-
ing the notation of Leibnitz. It is based on the infinitesimal method.
Bezout flourished in France before the Bevolution. His works were,
therefore, at this time, rather old, but his calculus was selected in pref-
erence to others ^^ on account of the plain and perspicuous manner for
which the author is so well known, as also on account of its brevity and
adaptation in other respects to the wants of those who have but little
time to devote to such studies." •

The introduction is taken from Gamofs RSflexions^ and gives the ex-
planalion by the " compensation of errors.'*

The translation of Bezout's calculus is only in part the work of Profes-
sor Farrar. After having begun it, he was obliged to go to the Azores^
on account of the health of his wife, and the translation was completed
by George B. Emerson. He had it printed with his introduction and
notes, so that when Professor Farrar returned he found it ready for use
in the college, t

Farrar's translations and selections from French authors were at once
adopted as text-books in some of our best institutions. Several books
in the series were used at the XJ. S. Military Academy and at the Uni-
versity of Virginia.

The professor of mathematics and natural philosophy was always as-
sisted by tutors. They generally taught the pure mathematics to the
lower classes. In 1825 there were three. One of them, James Hay-
ward, had been tutor for five years, and had striven to reform the teach-
ing of elementary geometry. He was made professor in 1826, but a
year later he severed his connection with the college and engaged in
civil engineering, in which he became a high authority. The original
survey of the Boston and Providence Railway was made by him.
Among the other tutors of note who served during the time of Profes-
sor Farrar were Thomas Sherwin, A. P. Peabody, and Benjamin Peirce.
Dr. Peabody is now Plummer professor of Christian morals, emeritus.
Among Harvard men of Farrar's time are also Charles Henry Davis,
who afterward served on the Coast Survey and established the Amer-
ican Ephemeris, and Sears Cook Walker, who, later, became a noted
astronomer.

We now proceed to inquire into the terms for admission and the courses
of study. Since 1816 the whole arithmetic has been required for ad-
mission to Harvard College. In 1819 a trifling amount of algebra was
added. The catalogue of 1825 specifies the requirements as follows :

»■ I ■ ■ I. I ■■! ■ I I !■■ ■■■II— ■ I

'AdTertisement to the translation.

t Barnard's Jonmal, 1876, " Schools as They Bhoold Be/' by George B. Emeraon.

IHFLUX OF FBENCH HATHEIIATICS. , 131

<< Fandamental rales of arithmetic ; vulgar and decimal fractions ; pro- ^
portion^ simple and compound ; single and double fellowship ; alliga-
tion, medial and alternate; and algebra to the end of simple equa.
tions, comprehending also the doctrine of roots and powers, and arith-
metical and geometrical progression.'' The books used in the exami-
nation were the Cambridge editions of Lacroix's Arithmetic and Enter's
Algebra. In 1841 Enter's Algebra or the First Lessons in Algebra were
required. Ko other changes were made until 1843. The catalogue for
that year mentions for admission Davies' First Lessons in Algebra to
<< Extraction of Square Boot ; " and <' An Introduction to Geometry from ^
the most approved Prussian text-books, to VII. — Of Proportions." No '
other subjects were added until 1866-67, though there were some
changes in the text-books. In 1850 Davies' and Hill's Arithmetics are <
mentioned ; in 1853 Davies' and Chase's Arithmetics ; in 1859 Davies', ,
Chase's, or Eaton's Arithmetics, Euler's Algebra, or Davies' First Les-
sons, or Sherwin's Common School Algebra, and the Introduction ,to
Geometry ; in 1865 Chase's Arithmetic, Sherwin's Algebra, Hill's Sec- \
end Book in Geometry, Parts I and II, or <^ An Introduction to Geom- ^
etty as the Science of Form" as far as p. 130.

In addition to these statements taken from catalogues it will be in-
terestiug to add the following account, given by Prof. William F. Allen,
of the class of 1 851 : * << The requirements for admission were not much
above a common school. That is, I got my arithmetic and algebra in a
country district school (well taught). Geometry I picked up for myself
in a very small quantity. I remember at the entrance examination I
was asked what an angle was. I thought I knew, but I think I con-
vinced the examiner that I didn't ; however, I got in clear."

During the first ten or eleven years of his teaching Professbr Farrar
used the books of Samuel Webber. A second edition of Webber's
Mathematics appeared during Professor Farrar's incumbency. In 1818
the course of study in mathematics was as follows :t Freshmenj alge- '
bra and geometry, during the first and second term and three weeks i
of the first term. Sophomores^ algebra, trigonometry and its appli* ;
cations to heights and distances, and navigation during the third term. \
Juniorsj natural philosophy and astronomy (Enfield's), mensuration of I
superficies and solids, and surveying during the third term. In place /
of Hebrew, on the written request of their parents or guardians, stu-
dents were permitted to attend to mathematics with the private in-
structor, or Greek, or Latin, or French ; Seniors^ conic sections and
spherical geometry during the first term and half of the second. We .
are informed, moreover, that for the attendance on the private iu-
stuctor in mathematics, which was optional, there was a separate charge,
at the rate of $7.50 per quarter.

There was a public examination of each class in the third term and a

• Letter to the writer, November 6, 1888. f North American Review, March, 1818.

132 TEACHINa AND HISTOEY OP MATHEMATICS.

public exIiibitiOQ of performances in composition, elocution, and in the
mathematical sciences, three times a year. Prizes were also given.
The Bowdoin prize dissertation was read in the Ghapel in the third
term. Of tbese prizes, the first premium was given in 1815 to Jared
Sparks, of the Senior class, for a dissertation << On the character of Sir
Isaac Kewton, and the influences and importance of his discoveries."
The title of this essay would show that Sparks had, very probably,
studied fluxions, though this branch was not included in the cirriculum
for 1818, given above. Fluxions never had been a regular study, oblig-
atory upon all the students, though provisions seem to have been made
for those wishing to prosecute it.

During the twenty-nine years that Farrar was professor, from 1807 to
1836, 275 theses were written by students on mathematical subjects at
Harvard, and deposited in the library of the college. Of these abont
one-fourth contain solutions of fluxional problems (or problems involving
the difierential and integral calculus) ; a little less than one-half are
on the calculation and projection of eclipses; the remaining on algebra,
mechanics, surveying, etc. Many of these papers are interesting me-
morials of men since become in different degrees famous. Thus George
Bancroft wrote, in 1817, a thesis, " luvenire Motum Verum Modornm
Lunae ; " George B. Emerson, on '* Fluxional Solutions of Problems in
Hanponicks ^ (1817) ; Warren Colburn, on "Calculation of the Orbit of
the Comet of 1819 ; '' Sears Cook Walker, in 1825, on « The Transit of
Venus in 1883," and "The Effect of Parallax upon the Transit in 1882; »
Benjamin Peirce, in 1828, on " Solutions of Questions • • • from
the Mathematical Diary, etc. ; " Wendell Phillips, in 1831, on " Some
Beautiful Besults to which we are Led by the Differential Calculus in
the Development of Functions.'' •

The catalogue for 1820 shows that Webber's Mathematics and Euclid's
Geometry had been discarded. Farrar's new books came now to be
used. The Freshmen studied Legendre's Geometry and Lacroix's Al-
gebra. Analytic methods began to acquire a foothold. Conic sections
were displaced by analytic geometry, which, with trigonometry, was
begun in the Sophomore and concluded in the Junior year. The Cata-
logues from 1821 to 1824, inclusive, do not give the course of study. In
1824 the Juniors studied, during the second term, differential calculus
from Bezout's work, unless they exercised their privilege of electing mod-
ern languages in place of mathematics. The catalogue of 1830 shows
some slight changes in the course. The Freshmen studied Legendre's
Plane Geometry, algebra, solid geometry ; the Sophomores, trigonome-
try and its applications, topography, and calculus ; the Juniors, natural
philosophy and mechanics in the second term, and electricity and mag-
netism in the third term ; the Seniors, optics and natural philosophy.

The following remarks by Dr. Peabody applying to this period are

* *< Mathematical Theses of Junior and Senior Classes, 1782-1839, by Henry C. Bad-
ger," Bibliographical Contribations of the Library of Harvard College, No. 32.

INFLUX OF FRENCH MATHEMATICS. 133

instrnctive : " The chief labor and the crowning honor of successful
scholarship were in mathematics and the classics. The mathematical
course extended through the entire four years, embracing the differential |
calculus, the mathematical treatment of all departments of physical /
science then studied, and a thoroughly mathematical treatise on as- /
tronomy. (Gum mere's, afterward replaced by Farrar's almost purely i
descriptive treatise."*)

The year 1832 marks an epoch in the history of mathematical teach- 1 /
ing at Harvard. It was then that Benjamin Peirce became professor ^
of mathematics and natural philosophy.

While there had been men in this country who had cultivated mathe-
matics with ardor, they had seldom possessed the talent and aspirations
for original research in this science. We have had many who were
called '* mathematicians," but if this name be used in the highest sense,
and be conferred upon only such persons as have been able to discover
mathematical truths not previously known to m^n, then it can fall
upon very few Americans. The fiiere ability of mastering the contents
of even difficult mathematical books, or of compiling good school-books
in this science, does not make him a mathematician worthy of standing
by the side of Legendre, the Bernoullis, Wallis, Abel, Tartaglia, or
Pythagoras — to say nothing of such master minds as Archimedes, Leib-
nitz, and Kewton. But at last we have come to a name which we may
pronounce with pride as being that of an American mathematician.
We need not hesitate to rank along with the names of Wallis and
John Bernoulli that of Benjamin Peirce. •

It has been said that a young boy detected an error in the solution
given to a problem by Nathaniel Bowditch. *' Bring me the boy who
corrects my mathematics," said Bowditch, and Benjamin Peirce, thirty
years later, dedicated one of his great works " To the cherished and re-
vered memory of my master in science, Kathaniel Bowditch, the father
of American Geometry." The title of " father of American Geometry,"
which Peirce confers upon his beloved master, has been bestowed by
foreign mathematicians upon Peirce himself. Sir William Thomson
referred, in an address before Section A of the British Association, to <
Peirce as, " the founder of high mathematics in America." On a similar
occasion Arthur Oayley spoke of him as the "father of American
mathematics."

Benjamin Peirce was born at Salem in 1809. He entered Harvard ,
College at the age of sixteen, and devoted himself chiefly to mathemat- |
ics, carrying the study far beyond the limits of the college course. .
Thus he attended lectures on higher mathematics by Francis Grund.
While an undergraduate he was a pupil of Nathaniel Bowditch, who . /
perceived the genius of the young man and predicted his future
greatness. Bowditch directed him in the development of his scientific

* Harvard Beminiscences, by A. P. Peabody, p. 203.

134 TEACHINa AND HISTOBT OF 1£ATHEMATICS.

powers, and gave him valaable instraction in geometry and analysis.
When Bowditoh was pablishing his translation and commentary of the
M6caniqne O^Ieste, Peirce helped in reading the proof sheets, and there-
by contribnted gi^atly toward rendering it free £rom errors. This critical
reading of that great work of Laplace must have been an edacation
to him in itself. Indeed, a great part of Peirce's scientific labors was
in the field of analytic mechanics.

Dr. Peabody gives the following reminiscences of Peirce :• "While
Benjamin Peirce the yottnger was still an under-graduate • • • it
was said that in the class-room he not infrequently gave demonstrations
that were not in the text-book, bnt were more direct, summary, or
pnrely scientific than those in the lessons of the day. Oollege classes
were then farther apart than they are now; but even in our Senior
year we listened, not without wo£ner, to the reports that came np to
our elevated platform of this wonderful Freshman, who was going to
carry off the highest mathematical honors of the university. On grad-
uating, he went to Northampton as a teacher in Mr. Bancroft's Bound
Hill School, and returned to Cambridge in 1831 as tutor. The next
year the absence of Professor Farrar in Europe left him at the head of
the mathematical department (which he retained till his death)^ the fol-
lowing year receiving the appointment of professor; while Mr. Farrar
on his return was still unable to take charge of class instruction."

In 1842 Peirce was appointed Perkins professor of astronomy and
mathematics. This position he held until his d^ath, in October, 1880.
Tutor Henry Flint is the oiily person ever connected with the college
for a longer time.

We shall first speak of the mathematical text books written by Peirce,

then of his record as a teacher, and, lastly, of his original researches.!

I As soon as he entered upon his career as teacher of mathematics at

I Harvard he began the preparation of mathematical text-books, tn

/ 1833 appeared his Elementary Treatise on Plane Trigonometry, and in

1836 his Elementary Treatise on Spherical Trigonometry. The two were

, published in a single volume in later editions. In 1836 appeared also

his Elementary Treatise on Sound ; in 1837, his Elementary Treatise on

/ ( Plane and Solid Geometry and his Elementary Treatise on Algebra;

during the period 1841-46 he wrote and published In two volumes his

Elementary Treatise on Curves, Functions, and Forces; in 1855, was

published his Analytical Mechanics.

Bev. Thomas Hill, ex-President of Harvard and an early pupil of
Peirce, speaks bf these books as follows : <^ They were so full of novel-
ties that they never became widely popular, except, perhaps, the Trigo-
nometry; bnt they had a permanent influence upon mathematical teach-
ing in this country ; most of their novelties have now become common-
places in all text-books. The introduction of infinitesimals or of limits

* Harvard ReminisoeDoes, p. 181.

t We shaU draw fieely firom the Memorial Collection, by Moaea King, 1881.

INFLUX OF FBENCH MATHEMATICS. 135

into elementary books ; the recognition of direction as a fundamental
idea ; the nse of Hassler's definition of sine as an arithmetical qaotient|
free from entangling alliance with the size of the triangle; the similar
deliverance of the expression of derivative fanctions and difEerential
co-efficients fi^om the superfluous introduction of infinitesimals; the
fearless and avowed introduction of new axioms, when confinement to
Euclid's made a demonstration long and tedious — ^in one or two of
these points European writers moved simultaneously with Peirce, but
in all he was an independent inventor, and nearly all are now generally
adopted."

The ratio system in trigonometry was used before this by Hassler in
his masterly, but ill-appreciated, work on Analytic Trigonometry, and
also by Charles Bonnycastle in his Inductive Geometry. But this sys-
tem met with no favor among teachers. The most popular works on
trigonometry, such as the works of Davies and Loomis, as also those
of Smyth, Hackley, Bobinson, Brooks, and Olney, adhered to the old
and obsolete " line system,'' and it was not till within comparatively
recent years that the " ratio system " came to be generally adopted.
The old '^line system" was brought to America from England, but the
English discarded it earlier than we did. In 1849 De Morgan wrote
that the old method of defining trigonometric terms' was universal in
England until very lately.

The final victory of the system in this country is due chiefly to the
efforts of Peirce, Chauvenet,* and their followers. It is signiflcant that
Loomis, in a late edition of his trigonometry, has been driven by the
demands of the times to abandon the old system.

The advisability of using infinitesimals and the idea of direction in
elementary text- books will be discussed in another place.

About the beginning of the second quarter of this century consider-
able dissatisfaction came to exist among the public about the college
system as it was then conducted in this country. The people demanded
a change from the old scholastic methods. Then for the first time arose
the now familiar cry against forcing the ancient languages upon all
students entering college. It was demanded that greater prominence
be given to modern languages, tp English literature, to practical me-
chanics, and that the student should have some freedom in the selection
of his studies. Though some few modifications were made here and
there in the college courses, the " New Education " did not secure a
firm hold upon our colleges until the third quarter of the present
century. ^ '

in these reforms Harvard has always taken a prominent part. The
elective system there has been traced back to 1824, when Juniors could
choose a substitute for 38 lessons in Hebrew, and Seniors had the
choice between chemistry and fluxions. Benjamin Peirce was an en-
thusiastic advocate of the elective system.

We now proceed to give the courses in mathematics during the early

\ .^

136 TEACHINa AND HISTORY OF MATHEMATICS.

part of Peirce's counection with the college. His own text- books were
adopted as soou as they came from the press. lu 183G aud 1837 the
Freshmen used Walker's Geometry, Smyth's Algebra, Peirce's Plane and
Spherical Trigonometry ; the Sophomores, Farrar's books on Analytical
Geometry, Calculus, and Natural Philosophy ; the Juniors continued the
Katural Philosophy.

In the catalogue for 1838 we notice important changes. The Freshmen
studied Peirce's Geometry and Algebra ; the Sophomore class was di-
vided into three sections, of which the first pursued practical mathe-
matics, including mensuration, dialing, construction of charts, surrey-
ing, the use of globes and instruments in surveying, during the first
term ; and during the second term the general principles of civil engi-
neering, nautical astronomy, and the use of the quadrant. This section
was evidently intended to meet the demands of the time for practical
knowledge, without having first laid a broad and secure theoretical
foundation. But little could be accomplished in civil engineering with-
out a knowledge of calculus. The second section reviewed arith-
metic, geometry, and algebra; then took up conic sections, fluxions, and
the mathematical theory of mechanics. The third section, intended for
students of mathematical talents and taste, pursued analytic geometry,
theory of numbers and functions, differential and integral calculus, and
mechanics.

But this arrangement did not prove satisfactory. The facts are that
Professor Peirce's text-books were found very difficult, and Peirce him-
self was not a good teacher, except for boys of mathematical genius.
Peirce was anxious to introduce the elective system, so that students
without mathematical ability would not be forced to pursue mathematics
beyond their elements. In May, 1838, a vote was passed, permit-
ting students to discontinue their mathematics at the end of the Fresh-
man year if they chose to. The catalogue for 1839 announced that
^^ every student who has completed during the Freshman year the
studies of geometry, and algebra, plain trigonometry with its applica-
tions to heights and distances, to navigation, to surveying, and that of
spherical trigonometry, and who has passed a satisfactory examination
in each to the acceptance of the mathematical department and a commit-
tee of the overseers — may discontinue the study of mathematics at the
end of the Freshman year, at the written request of his parent."

Beferring to these changes the president said, the following year, that
the liberty to discontinue mathematics at the end of the year had been
found highly acceptable to both students and parents and had, thus far,
been attended by no ill consequences; that elections in the secondary
course had had a tendency to encourage those capable of profiting by
the study of that branch; thatthose possessing mathematical talentwere
stimulated; that of fifty-five, only eight continued mathematics; and
that the head of the department considered the voluntary system
Boperior. The difficulties in the mathematical course for the Sopho-

TSFLUH OF FRENCH HATHEMATICS.

137

mores seemed to be removed. But how aboat the Fresbmeii f Mathe-
matical studies were not popular with them ; they complained of over-
work. In 1839 the committee on studies reported that ^^ the mathe-
matical studies of the Freshman class are so extensive as to encroach
materially upon the time and attention due to other branches," and pro-
posed to remove the time when mathematical studies may be discon-
tinuedy from the end of the Freshman to the first term of the Sopho-
more year.

The catalogue for 1838-39 gives no mathematics for the Junior and
Senior years. The following year Peirce's Treatise on Sound was studied
by the Juniors. In 1841 an extended mathematical course was of-
fered in the Junior and Senior years. The Juniors were to study Peirctfs
Treatise on Sound and the Galculas of Variations and Eesiduals; the
Seniors, Poisson's M^canique Analytique and Celestial Mechanics.
The number of students venturing to enter such difficult but enchanted
fields of study were but few. In 1843 there were only two sections in-
stead of three as before. One was called the course in Practical Mathe-
matics, comprising Peirce's Plane and Spherical Trigonometry ; the
other was called the course in Theoretical Mathematics, in which Peirce's
Algebra was concluded, and his Curves, Functions, and Forces, studied
as far as " Quadratic Loci." These two courses continued through the
Junior and Senior years. The studies offered varied somewhat from
year to year.

In obedience to the practical demands of the times, the Lawrence
Scientific School was opened in 1842 as a branch of Harvard. It
began as a school of chemistry. But by the year 1847 the plan of this
school was broadened so as to embrace other sciences. ^' There shall
be established in the University an advanced school for instruction in
theoretical and practical science and in other usual branches of academic
learning." Instruction was to be given by Professor Horsford in chem-
istry, by Professor Agassiz in zoology and geology, by Professor Lev-
ering in experimental philosophy, by William Bond in practical astron-
omy, and by Professor Peirce in higher mathematics, especially in
analytical and celestial mecli mics. The course offered by Professor
Peirce to students in this school, in 1848, was as follows :

-V

Course in Mathematics and Astronomy.

I.— CURVES AND FUNCTIONS.

Begular course.

Peirce. Carves and F.unctions.

La Croix. Calcal Differential et Integral.

Cauchy. Les AppUcations da Calcul In-
finitesimal k la Gdom^trie.

MoNGE. Application de TAualyse ^ la
G^om^trie.

Parallel course,

Biot. Gdom^trie Analytique.

Cauchy. Cours d'Analyse de P£cole Ro-

yale Polytechniqae.
Hamilton's researches respecting qua

ternions. (Transactioii:^ of the Boyal

Irish Academy, Vol. XXI.}

188

TEACHINa AND HISTOBY OF MATHEMATICS.

11.— ANALYTICAL AND CBLESIIAL MECHANIOS.

Regular course,

Lai*lacb< M^caniqne Celeste, translated^

with a Commentary, by Dr. Bowditoh.

Vol. I.
BOWDITCH. On the Computation of the

Orbits of a Planet or Comet ; Appendix

to Vol. Ill of his translation.
Airy. Figare of the Earth, from the En-

cgolopcBdia Metropolitana,
AlKY. Tides, from the £ncjfoloj3<iedia Metro-
politana.

Parellel couree,

Poissox. M^caniqne Analytiqne.
Lagrange. M^cauiqne Analytiqne*
Hami lton. General Methodin Dynamics,

from the London Philosophical Tran8«

actions for 1834 and 1835.
Qauss. Theoria Motns Corpomm CoBleso

tiam.
Bessel. Untersachnngen.

Lever RiER. D^veloppements snr Pin*
sieurs Points de la Th^orie des Pertur-
bations des Plan^tes.

Leverribr. Les' Variations S^cnlaires
des £l6mens des Orbites, poor lee Sept
Plan^tes Princi pales.

Leverrier. Th^orie des Monvemente de
Merc are.

Leverrier. Hecherches snr les Monve-
ments de la Plan^te Herschel.

Adams. Explanation of the Observed Ir-
regularities in the Motion of Uranus, on
the Hypothesis of Disturbances caused
by a more distant Planet.

m.— MECHANICAL THEORY OP LIGHT.

Regular oauree.

Airy. Mathematical Essays.

MacCdllagb, On the Laws of Crystal-
line Reflection and Refraction. (Trans-
actions of the Royal Irish Academy,Vol.
XVIII.)

Parallel course,

Cauchy. Exercices d' Analyse et dePhys-
ique Matb^niatiqnes.

Neumann. Theoretiscbe Untersnchung
der Gesetze, nach Welchen das Licht
reflectirt und gebrochen wird. (Trans-
actions of Berlin Academy for 1835.)

Sacli a conrse of studies had never before been open to American
students in any American college. Sach a coarse, or any other equally
advanced, was never presented in any other American institation be-
fore the arrival at the Johns Hopkins University of Professor Sylvester.
It must be admitted that the great mass of Harvard students never
studied more mathematics than was absolutely required for their degree,
but now and then Peirce had a pupil who liked mathematics, under-
stood the greatness of his teaching, and appreciated and loved his
character. Peirce was the center of an influence which led to the start-
ing of many a since distinguished scientific career. Prof. T. H. Safford,
one of his favorite pupils, says : ^' Among distinguished scholars of the
years which I remember, were Prof. G. P. Bond, afterward of the Ob-
servatory ; Dr. B. A. Gould, celebrated as an astronomer ; Bev. Thomas
Hill, for a while president of Harvard ; Prof. J. D. Eunkle, of the In-
stitute of Technology, Boston ; Prof. J. E. Oliver, of Gomell Univer-

INFLUX OF FEENCH MATHEMATICS. 189

Bity ; Prof. A. Hall and Prof. S. Newcomb, of the TT. 8. Kavy ; Mr. W.j
P. G. Bartlett, since deceased; Mr. G. W. Hill, of the Nautical Al-|
manac Oflace ; Mr. Ghauncey Wright, known as a philosopher ; Prof. *
James M. Peirce and President Eliot, of Harvard 5 Prof. O. M. Wood-/ ^
ward, of St. Louis ; Rev. G. W. Searle, of St. Paul's (R. 0.) Churchy
New York City ; Prof. W. Watson, formerly of the Institute of Tech-^
nology 5 Professor Byerly, now at Harvard.''*

Of Peirce as a teacher, Dr. A. P. Peabody gives us an interesting
> aocount.t It refers to the fij*st year that Peirce was professor at Har-

vard.

" For the academic year 1832-33, 1, as tutor, divided the mathemat-
ical instruction with Mr. Peirce. • • • He took to himself the in-
dtruotion of the Freshmen. The instruction of the other three classes
we shared, each of us taking two of the four sections into which the
class was divided, and interchanging our sections every fortnight.

• • ♦ In one respect I was Mr. Peirce's superior, solely because I
was so very far his inferior. I am certain I was the better instructor
of the two. The course in the Sophomore and Junior years, embracing
a treatise on the Differential Calculus, with references to the calculus in
the text-books on mechanics and other branches of mixed mathematics,
was hardly within the unaided grasp of some of our best scholars; and,
though no student dared to go to the tutor's room by daylight, it was

J no uncommon thing for one to come furtively in the evening to ask his

teacher's aid in some difficult problem or demonstration. For this pur-
pose resort was had to me more frequently than to my colleague, and
often by students who for the fortnight belonged to one of his sections.
The reason was obdous. No one was more cordially ready than he to
give such help as he could; but his intuition of the whole ground was
so keen and comprehensive that he could not take cognizance of the
slow and tentative processes of mind by which an ordinary learner was
compelled to make his step-by-step progress. In his explanations he
would take giant strides ; and his frequent, ^ you see,' indicated what

he saw clearly, but that of which his pupils could get hardly a glimpse.

• • •

«< Our year's work was on the whole satisfactory, and yet I think that
we were both convinced that the differentiaf calculus ought not to have
been a part of a prescribed course. There was a great deal of faltering
and floundering, even among else good scholars. * * * Our exam-
inations were mva voce^ in the presence of a committee of reputed ex-
^ perts in each several department. We shrank from the verdict of our

special committee in no part of our work except the calculus. As the
day approached for the examination in that branch we were solicitous
that Robert Treat Paine, who was on the committee, should not be
present; for we supposed him to be the only member of the committee
who was conversant with the calculus. He did not come, and we were

•Letter to the writer, Noyember 6, 1888. t Harrard BeminificenceSi 1888, p. 183.

140 TEACHING AND HISTOBY OP MATHEMATICS.

glad. • • • If there were defects and shortcomings, there was cer-
tainly no one present who coald detect them."
Peirce had no saccess in teaching mathematics to students not math*
. ematically inclined. liepoated and loud complaints were made at Har-
Y ^ yard that the mathematical teaching was poor. The majority of students
. disliked the study and dropped it as soon as possible. Says Pro£ Will-
< iam F. Allen (class of 1851) in a letter to the writer :

^^ I am no mathematician, but that I am so little of one is due to the
/ wretched instruction at Harvard. Professor Peirce was admirable for
students with mathematical minds, but had no capacity with others*
He took only elective classes, and of course I didn't elect. Only two
"^ did in our class of about sixty, and I believe they soon dropped it. In
my Freshman year I had very good instruction from Mr. Child, now
the professor of English literature, and editor of ballads. I had alge-
bra and geometry with him, and did fairly well. In the Sophomore
year (trigonometry and analytic geometry) we had a different instructor,
and it was a mere farce. In analytic geometry I was taken up once in
the course of the term, on rectangular co-ordinates in space, and I
knew perfectly well (although I was never so told) that at examination I
should be called up upon rectangular co-ordinates in space. (Written
examinations had never been heard of.) When examination day came
(a committee in attendance) the tutor was sick, and a shudder ran
through the class. But he heroically pulled himself together and held
his examination in person, and I was examined upon rectangular co-
ordinates in space. The sum of my knowledge of analytical geometry
at the present day is that there are such things (or were) as rectangular
co-ordinates in space — and I suppose there must also be some out of
space. • • • Peirce's text-books were used. His geometry I liked
much, also the algebra, only that it was pretty hard. • • •

*^ I graduated in 1851, and I remember when I was in Germany two or
three years later, I met a gentleman who had just returned from Amer-
ica — a young German Oelehrte — and he assured me that there was not
one mathematician in the United States, and only one astronomer,
Peirce. It was not an agreeable thing to be told, for a patriotic young
American as I was then, but I suppose it was not far from the truth."

In January, 1848, Thomas Sherwin, by order of the committee for
examination in mathematics reported that in 1847 there were present
for examination but one Senior in Bowditch's Laplace, and only five
Juniors in Curves and Functions. He went on to say that mathe-
matics could be made attractive, that, hence, arose the inquiry, why
this study was so very decidedly unpopular at the University, and
why so general an opinion prevailed throughout the community, that
the student stood less creditable in this branch than in others. The
answer to this was that the text books were abstract and difficult, that
few could comprehend them without much explanation, that Peirce's
works were symmetrical and elegant, and could be perused with pleas-

INFLUX OF FRENCH MATHEMATICS. 141

are by the adalt mind, but that books for yoang students should be more
simple. The report then says that there are mathematical works of no
small merit, which embraced the same subjects as the text-books
now used, which were much less difficult of comprehension, such as
Bourdon's Algebra, J. E. Young's Treatises, and a recent edition of
Button's Mathematics.

The majority report was followed by a minority report by Thomas
Hill and J. Gill, which differed regarding the text-books at Harvard.
> '* Your minority of the committee believe that these text- books, by their

beauty and compactness of symbols, by their terseness and simplicity of
style, by their vigor and originality of thought, and by their happy
selection of lines of investigation, offer to the student a beautiful model
of mathematical reasoning, and lead him by the most direct route to the
higher regions of the calculus. For those students who intend to go far-
ther than the every-day applications of trigonometry, this series of books
is, in the judgment of the minority, by far the best series now in use."

While the good qualities of Peirce's text-books, as described by the
minority, must be acknowledged, it is nevertheless true, that owing to
their compactness and brevity, which characterize all the writings of
Peirce, the books seemed obscure to beginners.* Still, however, they
continued to be used at Harvard for many years longer.

Professor Peirce said in his report of November 6, 1849, on the teach-
/ ing of higher mathematics in the college and the Lawrence Scientific

School, that he had two pupils. One of these students was a member
of the Lawrence Scientific School, and the other was the child, T. Henry
Safford, who had attracted so much attention for his early development
of mathematical ability. ^^ These two students attended lectures on
analytical mechanics, and young Safford showed himself perfectly com-
petent to master the difficult subject of research, and once or twice sur-
prised his teacher by the readiness with which he anticipated the ob-
ject of some peculiar form of transformation. Up to this time Safford
fully realizes his early promise of extraordinary powers as a geometer,
but his friends cannot free themselves from apprehension, when they
perceive that the growth of his body does not correspond to that of his
intellect." He then states that with the mathematical pupil of the
school the professor read also Lagrange's Micamque Anahjtiqtie and La-
place's TMorie Analytique des Probabilites.

*Iii aDotber place, Hey. Thomas Hill speaks of these hooks as follows (memorial col-
lection by Moses King) : " His text-hooks were also complained of for their condensa-
^ tion, as being therefore obscure ; hut under competent teachers, the brevity was th©

cause of their superior lucidity. lu the Waltham High School his books were used
for many years, and the graduates attained thereby a clear and more useful, applicable
knowledge of mathematics than was given at any other high school in this country ;
nor did they find any difficulty in mastering even the demonstration of Arbogast^s Poly-
nomial Theorem, as presented by Peirce. The latter half of the volume on the inte-
gral oalcnlns, full of the marks of a great .analytical genius, is the only part of lUl
his text-books really too difficult for students of average ability.''

142 TEAGHINQ AND HISTOBY OF MATHEMATICS.

As regards the nnmber of stndents electing mathematics, the com-
mittee of overseers stated, in 1849| that as long as the choice is offered,
the lighter labor will always be preferred, and that this tendency will
probably get stronger. ^^ Hebrew roots and p6lynomial roots will be
neglected in a garden abounding with French boaqaets and Italian
masic; and even now it can not surprise as that, while the Smith pro-
fessor of French and Spanish language and literature, and instructor
in Italian, is surrounded by a gay crowd of utilitarian admirers, the
Perkins professor of mathematics and astronomy is working in his deep
mines for one infant prodigy and one eminent Senior." Some Juniors
studied analytical statics, and gave the best evidence of successful devo-
tion to the subject.

The elective system was abandoned almost completely in 1850. Mathe*
matical studies were elective only in the Junior and Senior years. In
1867 the elective system was again adopted at Harvard, and on the
most liberal scale. Sophomore mathematics were again no longer ^^ re-
quired." Peirce^s books still held their ground. The only invaders
wero J. M. Peirce's Analytical Geometry, and, in 1865, Puokle's Analytic
Geometry. In 1869 the committee of overseers reported that mathe-
matics was a required study only for the Freshmen; that elective mathe-
matics were taken this year by one hundred Sophomores, six Juniors,
and eight Seniors ; that the Sophomores and Juniors could elect either
pure or applied. They also stated that '^ the number electing this de>
partment in the upper classes is never expected to be large, as the
studies are advanced beyond what most students have either aptitude
or occasion for."

We find that, "for the last few years of his life Professor Peirce had
for his pupils only young men who were prepared for profounder
study than ever entered into a required course, or a regularly planned
curriculum ; but he never before taught so efficiently, or with results
so worthy of the mind and heart and soul, which he always put into
his work." •

It will be instructive to listen to what former pupils of Peirce have
to say of him. Prof. Truman Henry Safford, of Williams Oollege, says
in a letter to the writer: <<I was a student at Harvard in the class of
1854. Prof. B. Peirce taught the Sophomores, I believe (I entered the
Junior class), but not very well; he had hardly patience enough, I sup-
pose. To the Juniors and Seniors he lectured on higher algebra, the
calculus, and analytical mechanics. His lectures were substantially
lM)ntained in his text- books — Algebra, Curves and Functions, and Ana-
lytic Mechanics. They were very interesting and inspiring to those
who could follow them. There was but little practice ; the examples
in the book were generally worked out. In my class a number (twelve
or so) took the first year's work ; the second, which included integral
calculus, complex numbers, and analytical mechanics, was taken by

* Harvard Eemimscenoesi by A. P. Peabody, p. 186.

"dttlttx of fkench hathematics. 143

fonr onlj. One of ihem ^m O. K. Lowell, afterwards a cavalry gen-
eral in the Oivil War, a nephew of Professor Lowell ; another, George
Putnam, Esq. ; a third, W. 0. Paine, afterwards a West Point scholar,
where he was first in his class, and a lieutenant of engineers, bat he
resigned as a captain. The fourth was myself. I had heard some of
Professor Peirce's lectures some years before^ while a school-boy, but
could not follow them so well."

For some years following 1838 Prof. Joseph Levering taught classes
in mathematics. Of him and Peirce, Edward E. Hale says : * '< The
classical men made us hate Latin and Oreek ; but the mathematical
men (such men ! Peirce and Levering) made us love mathematics, and
we shall always be grateful to them."

Says Thomas Wentworth Higginson: ^<As to mathematical instruc-
tion, this reform (elective system) was an especial benefit, for Professor
Peirce's genius revelled in the new sensation of having voluntary pupils,
and he gave a few of us his Curves and Fanctions as lectures, with run-
ning elucidations. Nothing could be more stimulating than to see our
ardent instructor, suddenly seized with a new thought and forgetting
our very existence, work away rapidly with the chalk upon a wholly
new series of equations ; and then, when he had forced himself into the
utmost comer of the blackboard and could get no farther, to see him
come back to earth with a sigh and proceed with his lecture. We did
not know whither he was going, but that huddle of new equations
seemed like a sudden outlet from this world, and a ladder to the stars.
He gave a charm to the study of mathematics which for me has never
waned, although the other pursuits of life soon drew me from that early
love. This I have always regretted, and so did Peirce, who fancied that
I had some faculty that way, and had me put, when bat eighteen, on a
committee to examine the mathematical classes of the college. Long
after, when I was indicted for the attempted rescue of a fugitive slave,
and the prison walls seemed impending, I met him in the street and told
him that if I were imprisoned I shoald have time to read Laplace's
MAianiqiie Celeste. ^In that case,' said the professor, who abhorred the
abolitionists, * I sincerely wish you may be.' "

Among the more prominent mathematical tutors of this period may
be mentioned G. W. Eliot, now president of Harvard, and James Mills
Peirce, a son of Benjamin Peirce. The latter graduated in 1853, was
tutor from 1854 to 1858, and from 1860 to 1861, when he was made as-
sistant professor of mathematics. In 1867 he became university profes*
Bor of mathematics.

Benjamin Peirce presided for some years over a mathematical society.
It comprised eight or ten men of some reputation in Boston and Gam*
bridge, who met to discuss mathematical topics. Each member would
present to the society such novelties as his inquiries into some particu-
lar branch had suggested, and ^' in the discussion which followed, it

* How I wfts Edaoatedy Foram, I, April, 1886, p. 61.

144 TEACHING AND HISTORY OF MATHEMATICS. ,

woald almost invariably appear that Feirce had, while the paper was
being read, pushed out the author's methods to far wider results than
the author had dreamed.* His mind moved with great rapidity, and
it was with great difficulty that he brought himself to writing out even
the briefest record of its excursions."

We now proceed to a brief account of Benjamin Peirce^s original re-
searches. Several original articles were contributed by him to the
Mathematical Miscellany and to the Cambridge Miscellany. Peirce had
planned an extended treatise on Physical and Oelestial Mechanics, to
be developed in four systems, of Analytical Mechanics, Celestial Me-
chanics, Potential Physics, and Analytic Morphology. Of these four,
only one appeared, the system of Analytic Mechanics, in 1855. The sub-
stance of this was prepared as a part of a course of lectures for math-
ematical students at Harvard. The publication was undertaken at
the request of some of his pupils, especially of J. D. Bunkle. He
consolidated the latest researches into a consistent and uniform treatise,
and earned <^ back the fundamental principles of the science to a more
profound and central origin." It was very far from being a mere com-
pilation. In his books he supplanted many a traditional method iu
mathematics by concise and axiomatic definitions and demonstrations
of his own invention. As an instance of this we mention his assump-
tion as self evident that a line which is wholly contained upon a limited
surface, but which has neither beginning nor end on that surface, must
be a curve re-entering upon itself. By this new axiom he reduces a
demonstration which would otherwise occupy half a dozen pages to a
few lines, t

Peirce's Analytical Mechanics was generally acknowledged at the time,
even in Germany, to be the best of its kind. I An American student in
^ j Germany asked once an eminent German professor what book he would
recommend on analytical mechanics. The reply was instantaneous,
<< There is nothing fresher and nothing more valuable than your own
Peirce's recent quarto."

Benjamin Peirce was much interested in the comet of 1843, and in a
few lectures he aroused by his great eloquence an interest in astronomy
which led to the foundation of the observatory of Cambridge. His
mathematical ability was first brought into general notice in con-
nection with the discovery of Neptune. Messrs. Adams, of Cambridge,
and Leverrier, of Paris, had calculated, from theory alone, where this
planet ought to appear in the heavens, if visible, and Galle, of Berlin,
discovered on September 23, 1S4G, the planet at the place indicated to
him by Leverrier. Peirce began to study the planet's motion, and came
to the conclusion that its discovery was a happy accident; not that
Leverrier's calculations had not been exact, and wonderfully laborious,

• Nation, October 14, 1880.
/ t Roy. Thomas Hill, in the Memorial Collection, by Moses King.
IKatare. October 28, 1880.

INFLUX OP FRENCH MATHEMATICS. 145

and deserving of the hij^liest honor, but because thfere were, in fact,
two very different solutions of the perturbations of Uranus possible;
Leverrier had correctly calculated one, but the actual planet in the sky
represented the other, and the actual planet and Leverrier's ideal one
lay in the same direction from the earth only in 1846.

Astronomers of to-day would hardly accept Peirce's conclusions.
"His views came, probably, from a misapprehension of Leverrier's
methods. There are two methods by which, in theory, the problem
could be approached, that of general and that of special perturbations.
Leverrier used the latter, while Peirce's criticisms seem directeil against
the former.'' •

On February 2, 1847, Mr. 0. Walker, of Washington, discovered that
a star observed by Lalande in May, 1795, must have been the planet
ISfeptane. This observation afforded the means of an accurate deter-
mination of the orbit. Walker's orbit of Neptune furnished Peirce
with materials for still more thorough investigation of the theory and
re-determination of the perturbations. These perturbations enabled
Walker to get an orbit more correct, which Peirce used again in his
turn. Thus, eighteen months, after the discovery of Neptune its orbit
was calculated by American astronomers so accurately that the con<
formity between the predicted and observed places was far more close
for Neptune than any other planet in the heavens. t

A few years later Peirce published his investigations on Saturn's
rings. The younger Bond had seen the ring divide itself and re-unite,
and had been led by this to deny the solidity of their structure. Peirce
followed with a demonstration, on abstract grounds, of their non-sol-
idity.l The same subject was afterward investigated again iu England
by James Clerk Maxwell.

Admiral 0. H. Davis, a relative of Peirce, succeeded in persuading
Gongress to pay for the calculation of an American almanac for the
sailors, so that we would not bo dependent upon foreigners, which
might be troublesome in case of war. The Nautical Almanac Office
was established at first in Cambridge, under Davis's business manage-
ment and Peirce's scientific control.§ One of the assistants in the
office, appointed in 1849, was J. D. Eunkle, then one of Peirce's pupils
in the Lawrence Scientific School. He helped in the preparation of the
American Ephemeris and Nautical Almanac, iu which he continued to
engage till 1884.

The publications of this office gained scientific recognition from the
first. In 1852 were printed Peirce's Lunar Tables, to be used in making

. ♦ Prof. G. C. Comstook, Washburn Observatory, in a letter to the writer. Prof.
C. A. Tonng claims that the discovery was not an accident ( General Astronomy jj^»
371).

t Proceedings American Association for Advancement of Science, Vol. VIII, 1854,
address by B. A. Gould, jr., p. 18.

t Astronomical Journal (Gould's), Vol. II, p. 5.

J Development of Astronomy in the United States, by T. H. Safford^ 1888, p. 21.

881— No. 3 10^

146 TEACHING AND HISTORY OF MATHEMATICS.

compntations for the Naatical Almanac* They were intended to serve
only a temporary purpose until Hanseii*s long expected tables shoold
make their appearance, bat they continued to be used after that. He
made very laborious and exact calculations of the occultations of the
Pleiades, which furnished means of studying the form both of the earth
and the moon.

From 1852 to 1867 Peiroe had the direction of the longitude observa-
tions for the 17. S. Ooast Survey, and in 1867, after the death of Baohe,
he was appointed Superintendent of the XJ. S. Ooast Survey, which office
he held till 1874.

Beojamin Peirce was more celebrated, in his day, as a mathematical
astronomer than as a cultivator of pure mathematics. His most im-
portant researches in pure mathematics were not placed in reach of the
mathematical public until after his death. In our opinion, Peirce will
be remembered by future generations for his investigations on Linear
Associative Algebra, quite as well as for his other scientific achieve-
ments. He will be remembered as an algebraist as well as an astrono-
mer. His thoughts were turned especially toward the logic of mathe-
matics and the limits and extension of fundamental processes. He read
several papers on algebra before the American Academy for the Advance-
ment of Science. In 1870 one hundred lithographed copies of a memoir
on Linear Associative Algebra, read before the National Academy of
Sciences, were taken, for distribution amoTig his friends. This memoir
was at last published in the American Journal of Mathematics, Vol.
lY, No. 2, with notes and addenda by O. S. Peirce, son of the author.
Bei\jamin Peirce himself considered this memoir the best of his scientific
efforts. The lithographed copies contain the following modest Intro-
ductory remarks by the author, which are omitted in the American
Journal of Mathematics :

" To MY Fbiekds : ^

<^ This work has been the pleasantest mathematical effort of my life.
In no other have I seemed to myself to. have received so full a reward
for my mental labor in the novelty and breadth of the results. I pre-
sume that to the uninitiated the formulas will appear cold and cheer-
less, but let it be remembered that, like other mathematical formolsD,
they find their origin in the divine source of all geometry. Whether I
shall have the satisfaction of taking part in their exposition, or whether
that will remain for some more profound expositor, will be seen in the
fiiture."*

* Peirce distiDgaishes bis algebras from each other by the namber of their fanda-
mental conceptioDS, or of the letters of their alphabet. Thns, an algebra which has
only ODo' letter in the alphabet is a eingle algebra ; one that has two a double algebra,
and so on. His investigation does not usually extend beyond the sextuple algebra.
This classification he calls ''cold and uniustnictive, like the artificial Linnsan sys-
tem of botany.'' ** Bnt it is useful in a preliminary inTCstlgation of algebras until a
sufficient variety is obtained to afford the material for a^ natural classification." He
then begins his researches with iingle algebra, then goes to double algebra, and sooDi

INFLUX OP PEENCH MATHEMATICS. 147

' Peiroe's memoir is a wonderfal volame. It is almcNst entiUed to tank
<* as a Prindpia of the philosophical study of the laws of algebraical
operation."

One of the pall-bearers at the ftineral of the greatest American alge-
braist was prof. J. J, Sylvester.

. Daring the last ten years of his life Benjamin Peirce was relieved
of much of the labor and responsibility falling npon the head of a de-
partment in a university by his son, Prof. James Mills Peirce. Though
not the heir of his father's geninS) Prof. J. M. Peirce is a thorough and
able mathematician. He excels his father in being an excellent teacher.
In 1857 he published an Analytic Gleometry^ which was used for some
years as a text-book at Harvard. He has also published Three and Four
Place Tables of Logarithmic and Trigonometric Functions, 1871 ; Ele-
ments of Logarithms, 1873, and Mathematical Tables, chiefly to Four
Figures, Ist Series, 1879.

Connected with the mathematical department are, since 1870, Prof,
O.J. White; since 1876j Prof. W. E. Byerly; since 1881, Prof. Benja^
mitt O. Peirce, and Mr. George W. Sawin.

Professor Byerly published in 1880 his Elements of the Differential
Calculus, and in 1882 his Elements of the Integral Calculus. Byerly's
Calculus is a scholarly work. In the rigorous treatment and judicious
selection of subjects and adaptability to class use ic is, we believe,
surpassed by no other American work. Professor Byerly uses the
notation, D,^, which was first employed in this country by Benjamin
Peirce. In answer to a letter of inquiry regarding the history of this

notation Professor Byerly says :* ** It was certainly used with some

I - -- 1 I • - ■■ - ■ ■ . ■ III

np to sextuple, making nearly a hnndred algebras, wbioh he shows to be possible.
Of all these, only three algebras had ever been heard of before. Of the two single
algebras we have one — the common algebra, inolading arithmetio. Of the three
doable algebras we have one, the oalculas of Leibnitz and Newton. Of over twenty
quadruple algebras we have the quaternions of Hamilton.

Prof. Arthur Cayley, in his presidential address before the British Association, in
1883, speaking of Pe'iroe's Linear Associative Algebra, 8aid : '' We here consider sym-
bols Af Bf etc., which are linear functions of a determinate number of letters or units,
i, J, kf I, eto«,with co-efficients which are ordinary analytical magnitudes, real or Im-
aginary (viz, the co-efficients are in general of the form x -f- iy, where i is the before-
mentioned imaginary, or */—!). The letters i, j, etc., are such that every binary com-
bination i', ijf jif etc., (the ij being in general not equal to j't) is equal to a linear func-
tion of the letters, but under the restriction of satisfying the associative law, viz, for
each combination of three letters ij.k=ijkf so that there is a determinate and unique
product of three or more letters ; or, what is the same thing, the laws of combination
of the units <, j, h, are defined by a multiplication table giving the values of <*, i/, Ji,
etc. ; the original units may be replaced by linear functions of these units, so as to
give rise, for the units finally adopted, to a multiplication table of the most simple
form ; and it is very remarkable how frequently in these simplified forms we have
nilpotent or Idempotent symbols ({^=0, or i'^i, as the case may be), and symbols
I, J, such that ij=ji=:0 ; and, consequently, how simple are the forms of the multlpU-
oation tables which define the several systems, respectively.'^

* Letter to the writer, December 27, 1888.

148 TEACHING AND HI6T0BT OF MATHEMATICS.

freedom in England and on the Continent in the first half of this centniy.
It is given in Barlow's Mathematical Dictionary, 1814, was used by
Cauchy about 1830, by Tortolini in 1844, by Schlomilch in 1846, and by
Boole and Oarmichael somewhat later, and each of the, authors I have
mentioned uses the symbol as if it were a familiar one and without
reference to its history.?

Prof. Benjamin Osgood Peirce has published Elements of the Theory
of the I^ewtonlan Potential Function, 1886.

Since 1867 great changes have been made in the requirements for
admission to Harvard and in the arrangement of the mathematical
courses. Since that time the elective system has been in operation in
full force. The terms for admission have been much increased.

From the selected sheets of the Harvard University Catalogue for
1888-89 we take the following regarding the requirements for admis-
sion, omitting whatever has no bearing on mathematics :

The examinations for admission embrace two classes of studies, elementary and
advanced. .

The elementary stadies are not supposed to be equivalent to one another ; Greek,
Latin, and mathematics have much greater weight in the examinations than any of
the rest.

The advanced studies are supposed to be equivalent in regard to time spent upon
them at school, and will have the same weight in the examinations. Each of the
advanced stadies is tanght in college in an elective coarse (or two halfocourses) occu-
pying three hours a week for a year ; and the standard required at the entrance ex-
aminations is the same as in the corresponding college courses.

The elementary studies are prescribed for all candidates, except under the condi'
tions named below (Paragraph I) ; and every candidate is farther required to present
himself for examiination in not less than two of the advanced studies.

I. The advanced study numbered 6 together with one of the three numbered 7 (see
below under "advanced studies in mathematics 'Oi 3 (physics), and 9 (chemistry),
may be substituted for either elementary Greek or elementary Latin.
Elementary Studies in MaihematiCB —

(a) Algebra, through quadratic equations. (The requirement) in algebra embraces
the following subjects: Factors, common divisors and multiples, fractions, ratios,
and proportions; negative quantities and the interpretation of negative results;
the doctrine of exponents; radicals and eqaations involving radicals; the binomial
theorem for positive integral exponents and the extraction of roots ; putting ques-
tions into equations, and the reduction of equations ; the ordinary methods of elimi-
nation, and the solution of both numerical and literal equations of the first and seo-
ond degrees, with one or more unknown quantities, and of problems leading to snob
equations.) (b) Plane geometry.

Advanced Studies in Mathemxttics —

6. A£athematics, — (a) Logarithms; plane trigonometry, with its applications to sur-
veying and navigation, (h) Either soUd geometry or the elements of analytic geom-
etry.

7. Maihematios. — (a) Either the elements of analytic geometry or solid geometry.
(h) Either elementary mechanics or advanced algebra.

The following booits will serve to indicate the nature and amount of the require-
ments in logarithms and trigonometry, analytic geometry, and mechanics :

Logarithms and Trigonometry. Wheeler's Logarithms (Cambridge, Sever) or tha
nnbracketed portions of Peiroe's Elements of Logarithms (Boston, Ginn & Co.).

INFLUX OF FRENCH MATHEMATICS. 149

Wheelei's Plane Trigonometry (saiAe pabllshers). Problems in Plane Trigonometry
(Cambridge, Sever). Peirce's Mathematical Tables (Boston, GIdu & Co.).
Analytic Geometry* Briggs's Analytic Geometry (New York, Wiley & Co.).,
Mechanics. GUkkI win's Elementary Statics (Loodon^Bell &, Sons; Cambridge,
Sever).

Advanced Algebra. Went worth's College Algebra (Boston, Ginn & Co.), to arti-
cle 498, omttting Chapters XIX, XX, XXIV, XXV, XXVII, XXVIH. The exami-
nation will be mainly occupied with the portions of algebra, as tbns defined, which
are not included in the elementary requirement in algebra ; but elementary questions
are not necessarily excluded.

All in all, there are nine "advanced'^ studies to choose from. Since
one can enter the college after passing an examination on all the ''ele-
mentary'' studies, and on at least two of the ''' advanced'' studies, it
follows that the least amount of mathematics required for admission, as
a regular student, is that stated above under the heading << Elementary
Studies in Mathematics."

The following are the

Courses of instruction in mathematics.

(1888-89.)

A, Logarithms.— Plane Trigonometry, with its applications to Surveying and Navi-

gation. Half-oourae. T«., Th,^ Sat,, at 11 (first half-year). Professor C. J.
White.

B, Analytic Geometry (elementary course). Half-course, Tu,, Th,, Sat., at 11 (second

half-year). Professor C. J. White.

C, Analytic Geometry (extended course). Man,, Wed,, Fri,, at 2. Professor Bybrlt.
Z>. Algebra. Half-course, Mon, f Wed,, Fri,, at 11 and Z( first half-year), Mr. Sawin.
G, Algebra (extended course). Half-course. Tu,, Th,, 8at,, at 10 (first half-year), Mr.

Sawin.

E, Solid Geometry. Half-course, Tu,, Th,, Sat., at 10 (second half-year), Mr. Sawin.

F. Elementary Mechanics. Half-course. Mon,, Wed,, Fri,, at 12 (second half-year).

Mr. Sawin.

Kot to be given after 1888-89.

Courses A, B, ^,and F correspond to Advanced Mathematical Studies embraced,
as optional studies, in the examination for admission to college.
1.' Practical Applications of Plane Trigonometry. —Spherical Trigonometry. ^Appli-
cations of Spherical Trigonometry to Astronomy and Navigation. Wed,, Fri.,
at 3. Professor C. J. White.

Course 1 is open to Freshmen who have passed the examination in Plane Trig-
onometry.

2. Differential and Integral Calculus (First Course). Mon,, Wed,, Fri., at 11. Pro-

fessor C. J. White.
Course 2 is open to those only who have taken Course B or Course C,

3. Analytic Geometry; higher course. Mon,, Wed,, Fri,, at 10. Professor J. M.

Peirce.
Course 3 is intended for students who have taken Course C; but those who
have taken Course B may elect it, if deemed qualified by the instructor.

4. The Elements of Mechanics. Tu,, Th,, Sat,, at 9. Professor B. O. Peirce.

Course 4 is intended for students who take or have taken Course 2.
Candidates for Second-Tear Honors may take Courses 2 and 3, or 2 and 4.
Other courses may be accepted on special petition.

5. Dlffbiential and Integral Calculus (Second Course;. Mon., Wed., Fri,, at 11. Pro-

fessor Btsblt.

160 . TEACHING AND HISTOBY OF MATHEMATICS.

[6. Quaternions and Theoretical Meobanics. Jfo»., Wed,f Fri., at 12, Frofesaor J.
M. Peibcb.]
Omitted in 18^8-89.
[7. Higher Plane Corves. Profewor J. M. Pbibob.]

Omitted in 1888-89.
6. Analytic Mechanics. Mon,, Wed, ^ Fri., at 10. Professor Bybblt.
9. Qaatemions and Theoretical Mechanics (Second Course). Mon, at 1S« Fri. 11-1.

Professor J. M. Peircb.
10. Trigonometric Series ; Introduction to Spherical Harmonics.—Theory of the Po-
tential. Tu., Th,j at 12, Wed, at 10. Professors Btxrlt and B. O. PxiROB.
[11. Hydromechanics. Professor B. 0. Pbibcb.]

Omitted in 1888-89.
13. The Theory of Functions. Man, at 11, Wed, 11-1. Professsr J. M. Peirob.
20. Special Advanced Study and Research.— The work of the following contaes will
consist in investigations and reading, to be carried on by the students in the
courses, under the guidance of the instructors. Students will be ezpeoted to
present their results from week to week in the form of lectures and theses.
(a) Questions in the Theory of Functions. Wed,, 3.30-5.30. Professor J. M.

Peircb.
(h) Higher Algebra (First Course). Mr. Sawin.

Some few stadies in the college coarse are prescribed, bat all mathe-
matical studies are elective. No mathematics need therefore be studied in
college. A stadent can, if he chooses, get - the degree of bachelor of
arts withoat having had more mathematics than plane geometry and
algebra throagh quadratic equations — ^the minimum requirement for
admission.

We conclude this article by quotations from a letter by Prof. L. M.
Hostuns, of the University of Wisconsin, who, in the year 1884-86, was
honored with a fellowship at Harvard, and studied higher mathematics
there.

<^ There were two courses, in < quaternions and theoretical mechanics,^
given by Prof. J. M. Peirce, each three lectures weekly for the year.
The first course gave the elements of quaternions and the dynamics of
a particle, covering about the ground of Tait and Steele's dynamics of
a particle, but treated by quaternion methods largely. The second
course contin ued with higher applications. * • * A third course, on
analytic mechanics, was offered by Prof. J. M. Peirce, consisting of lect-
ures, following Benjamin Peirce's Analytic Mechanics, for the first half-
year, and for the second Bouth's Bigid Dynamics, the part telating to
moving axes and relative motion, oscillations about equilibrium, oscil-
lations about a given state of motion, motion of a rigid body under any
forces — in short, the first five chapters of Volume II. * * * I at-
tended also a course in " arbitrary functions,'' by Prof. W. B. Byerly.
This covered most of the ground of Biemann's Partielle Differential'
gleichungen* The main subject treated is the methods of solution of
partial differential equations subject to given conditions, a class of prob-
lems constantly arising in physics. The course naturally includes the
proof and discussion of Fourier's Theorem, and the treatment of the dif-
ferent kinds of spherical harmonics, since these are of great use in thft

INFLUX OP FRENCH MATHEMATICiS. 151

solution of certain classes of partial differential eqaations. On the whole,
I found this as attractive a part of pure mathematics as I ever en*
tered; • • •

<^I may remark that the branches of higher mathematics to which
most attention is paid at Harvard seem to be theoretical mechanics and
quaternions. This is doubtless due to the influence of Benjamin Peirce,
whose attainments in the former line are well known, and who was also
among the first to recognize the high value of quaternion methods.
• • • I am able to give both of them (Professors J. M. Peirce and
Byerly) high praise as teachers of mathematics. Both are clear, logical
lecturers, and popular with the students. • • • Of him [Prof. B. O.
Peirce] I have little personal knowledge, but am sure no professor was
held in higher estimation, both as to attainments and ability^ as a
teacher. • • • In 1884-85 the number of graduate students taking
mathematics was five. • ♦ •

<^A mathematical ' seminar' was maintained with a good deal of in-
terest, with weekly meetings throughout the greater part of the year.
These meetings were under the supervision of the mathematical faculty,
and were rather informal in nature, though a formal programme was
usually carried out, usually by volunteer lectures or solutions.'^

TALE COLLEGE*

The successor in the chair of mathematics and natural philosophy to
the lamented Professor Fisher was Matthew R. Button. He was a
graduate of the college and was professor from 1822 to 1825. He
was the author of a work on conic sections and spherical trigonometry.
This work was subsequently revised by D wight, who "laid the students
and teachers of that day under everlasting obligations by his simplifi-
cation and abbreviation of those endless algebraic formulae in Button's
Oonic Sections."

From 1825 to 1836 Denison Olmsted occupied the chair which had
been made vacant by the death of Professor Dutton. He was born in
Hartford, Conn., in 1791, graduated at Yale in 1813, became tutor there
in 1815, was elected professor of chemistry at the University otlTorth
Oarolina in 1815, and finally returned thence to' assume the duties of
the chair of mathematics and physics at Yale. He had made natural
philosophy and chemistry his specialty, and possessed no special fitness
for the teaching of mathematics.

Professor Olmsted was renowned as a teacher rather than an orig-
inal investigator. His teaching power was indeed great, and he exerted
a beneficial influence, not only in college, but also upon the education
in the common schools of Connecticut. In 1831 appeared his Natural
Philosophy, which superseded the antiquated work of Enfield. In the
next year was written his School Philosophy, a more elementary work j
and in 1839 his Astronomy. He wrote also the Eudiments of l^atural
Philosophy and Astronomy, which passed through some fifty editions.

152 fEACHINQ AND HISTORY OP MATHEMATICS.

His I^ataral Philosophy and Astronomy came to be almost universally
used in oar colleges. The Philosophy was later revised by Prof. E. S.
Snell, of Amherst College, and, still later, by Prof. J. Fickliu, of the
University of Missonri. In point of scientific accuracy Olmsted's works
were sometimes rather defective. They were somewhat old-fashioned.

As early as 1830 a good telescope of moderate size was procured at
Yale OoUege. For want of an observatory it was difficult to make ac-
curate observations with it But it served, nevertheless, the excellent
purpose of famishing a means of observing the great November shower
of meteors, which occurred not long afterward. These showers, Hal-
ley's Gomet, and the telescope enabled Professor Olmsted to arouse a
great deal of astronomical enthusiasm at Yale, and for a few years a
number of students turned their attention to astronomy. Of the math-
ematicians and astronomers who graduated in those days are Stanley
and Mason, long since deceased ; and Loomis and Lyman, who are now
aged professors at Yale, The ablest mathematician and astronomer
which Yale has produced is William Ghauvenet. As a teacher he was
never connected with his alma mater ^ though a professorship was of«
fered him twice.

In 1835 the chair of mathematics and natural philosophy was divided
into two separate ones — Olmsted retaining that of natural philosophy,
and Anthony D. Stanley being elected to that of mathematics.
- One of the most prominent of early tutors in mathematics at Yale was
F. A. P. Barnard, of the class of 1828, which was known as a ^^mathe-
matical class," for the mathematical talent it embraced. While tutor at
Yale he prepared an edition of Bridge's Gonic Sections, in which the
work was substantially rewritten and also considerably enlarged.

We proceed to examine the mathematical courses during the time
of Professors Button and Olmsted.

In 1824, Arithmetic was the requirement in mathematics for admis-
sion ; in 1833, '' Barnard's or Adam's Arithmetic; " in 1845, Arithmetic
and Day's Algebra to quadratics.

The mathematical course for 1824 was, for Freshmen, Day's Algebra
during the first two terms, with no mathematics for the third term ; for
Sophomores, six books of Playfair's Euclid during the first and part of
the second term, and Day's Mathematics (including plane trigonometry,
logarithms, mensuration of surfietces and solids, isoperimetry, navigation
and surveying) andDutton's Gonic Sections, during the rest of the year;
for Juniors, Dutton's Spherical Trigonometry during the first t«rm, and
Enfield's Astronomy and Yince's Fluxions during the third term. The
Seniors had no mathematics in their course. In 1825 the study of Euclid
was begun at the close of the Freshman year. Yince's Fluxions still
appeared in that year as a text-book.

The writer has not been able to see catalogues for the years 1826-^2.
In 1833, Olmsted's Natural Philosophy was in use. *^ Fluxions" were
also named, but this meant then, most likely, the differential and in*

INFLUX OF FRENCH MATHEMATICS. 153

' _

tegral calculus. In the Sophomore year, Bridge's Conic Sections (prob-
ably Barnard's edition) was used in place of Button's. Ko changes in
the course were made for several years after.

The teaching of mathematics to the two lower classes in college was
generally intrusted to young and inexperienced tutors, who had, as a
rule, a very meagre acquaintance with the subjects which they were
supposed to teach. It is therefore not surprising that poor results
were generally reached, and that the study of mathematics was very
unpopular. Especially unpopular was the study of conic sections.
No efforts seem to have been made on the part of tutors to make this
study more attractive and to show its usefulness, by pointing out its
application in the study of physics and astronomy. Moreover, the old
books on conic sections were as dry as dust. The dissatisfaction among
students finally culminated, in 1830, in what is known as the '* conic
sections rebellion.'' Bebellions among students were then not unfre-
quent. Some years previously had taken place the '^ bread and butter
rebellion," caused by the poor quality of board that the students were
receiving. I^either their physical nor their intellectual food seems to
have been palatable to them. The ^^ conic sections rebellion "was a
refusal, on the part of the Sophomores, to recite in the manner pre-
scribed by the college rules. They petitioned that the method of rec-
itation required by the college be changed, that they might '^ explain
conic sections from the book, and not demonstrate them from the
figure."* We should judge from this that the practice had hitherto
prevailed of simply asking the student to explain certain parts of the ^
subject, with the book open before him, without requiring him to go to
the blackboard (if blackboards were used) to explain the lesson from
his own figure independently of the book. We have not been able to
ascertain at what time the blackboard was introduced into the mathe-
matical class-room at Yale, but it is not unlikely that the above rebel-
lion arose in the attempt, on the part of the faculty, to introduce such
improved methods as the use of the blackboard would suggest. The
new methods may have called for greater effort on the part of the stu-
dent, and may thus have brought about the " rebellion."

The general impression prevailed at Yale, in those days, that the
mathematical course there was a very difficult and thorough one. '^ This
fancy certainly derived some support from comparison with the class-
ical course, as compared with which the m,athematical was undoubtedly
a good one."f

Mr. Bristed, who entered Yale in 1835, says that in mathematics the
classes studied books rather than subjects, and crammed from one day
to another. " A great deal of the work," says he " of the second and
third year consisted of long calculations of examples worked with loga-
rithms, which consumed a great deal of time without giving any insight

• Tale College : a Skerch of its History, by William L. Kingsley, p, 137.

t Five Years in English Universities, by Charles Astor Bristed^ 3d ed., 1873, p. 456.

164 TEACHlNa AND HISTORY OP MATHEMATICS. ^

into principles, and were eqaaJly distasteful to the good and the bad
mathematicians.'' <^ They (the best mathematicians) complained that
'With the exception of two prizes for problems daring the Freshman and
Sophomore years, and an occasional ^original demonstration' in the
recitation room, they had no chance of showing their superior ability
and acquirements; that much of their time wa6 lost in long arithmetical
and logarithmical computations; that classical men were continually
tempted to ' skin ' (copy) the solutions of these examples, and thus put
themselves unjustly on a level with them." The bad practice of giving
long and tedious examples to work has been quite prevalent in our
colleges until within recent years, especially in trigonometry. For or-
dinary class- work four-place logarithmic tables are sufficient, we should
think. Prof. J. M. Peirce, of Harvard, has done much toward inaugu-
rating a reform in this matter, by his publication of four-place tables.
Such tables are of sufficient accuracy even in connection with the ordi-
nary physical experiments which the student may make in the labora-
tory.

In 1836 Anthony D. Stanley became professor of mathematics. He
held this place until his death in 1853. He was a native of Connecti-
cut and graduated at Yale in 1830. Two years later he was appointed
tutor and afterward professor. We are told that Professor Stanley
took special interest in the theory of numbers, and that he had once an
excellent occasion to show his skill. << In 1835 an anonymous writer in
the Stamford Sentinel challenged the entire faculty of Yale to arrange
^the nine digits in such order that their square root could be extracted
without a remainder. In a few days Mr. Stanley over the signature
<^X'' gave the required solution, and added that the question admitted
of more than one answer, and called upon the proposer to produce
them. To this challenge his opponent made an evasive reply, in which
he stated the number of solutions to be nine, but did not communicate
any solution." * Stanley found twenty-eight different solutions^ but even
a larger number is possible.

It seems that when Stanley was appointed professor he did not im-
mediately enter upon the discharge of the duties of the chair, but- went
to Europe two years and spent most of the time in Paris, where he at-
tended lectures at the Sorbonne and OoUege of France. In 1846 he
published Tables of Logarithms, which were uncommonly accurate. In
1848 appeared his Elementary Treatise on Spherical Oeometery and
Trigonometry. In the preceding year he published an article in the
American Journal of Science, ^* On the Variation of a Differential Co-
efficient of aFunction of any Number of Variables." ^< In this memoir,"
says Professor Loomis, *^ he resolves a problem which had already oc-
cupied the attention of La Grange, Poisson, Ustrogradsky, and Pagani,
the latter of whom was the only one who obtained a correct solution of
it. Professor Stanley here gives a solution of the same problem more

Tale CoUege : a Sketch of its History, by William L. Kingsl^.

INFLUX OF FRENCH MATHEMATICS. 155

simple and concise than Pagani's, an^ which was discovered before re-
ceiving the solution of that mathematician.^* In 1849 he snfLered from
a severe cold, and he sought relief in Italy and Egypt. On his return
he assisted in completing the revision of Day's Algebra, which he had
begun before leaving. He died in 1853.

Somewhat later than Prof. Oharles Davies, Prof. Elias Loomis began
the publication of mathematical text-books, which, like Davies' works,
became extremely popular throughout the United States. Professor
Loomis has been connected with Yale Oollege since 1860, but not as
professor of mathematics. Indeed, his specialty has not been mathe-
matics. His original contributions to science have been in other fields*
At Yale he has been professor of natural philosophy and astronomy.
His chief scientific work has been as d meteorologist and astronomical
observer. In his younger days Professor Loomis was a man of wonder-
ful activity, but now he is nearly four score years old and an invalid.

Professor Loomis was born in Oonnecticut in 1811 ; was graduated at
Yale Oollege in 1830. Af^er graduating he occasionally contributed
solutions to Ryan's Mathematical Diary. He was for a time tutor in Yale.
Together with Professor Twining, of West Point, he made observations
for determining the altitude of shooting-stars. These were, most likely,
the first concerted observations of the kind made in America. He was
the first one in America to discover Halley's Oomet in 1835. The next
year he was chosen professor of mathematics and natural philosophy at
the Western Reserve Oollege, with permission to spend a year in Eu-
roi>e. In Paris he attended lectures of Arago, Biot, Poisson, Dulong,
Ponillet, and others. He returned with astronomical, physical, and
meteorological instruments, and during the next season an astronomical
observatory was erected at the Western Reserve Oollege, in Ohio. Only
three observatories existed in the Uuited States before this, namely, at
. the University of North Oarolina, at Yale, and at Williams Oollege.

In 1844 Professor Loomis became professor in the University of the
Oity of New York. ^« Having here no instruments for observation, he
was induced to undertake the preparation of a text-book on algebra ;>
especially designed for the use of his own classes. This book prepared
the way for a second, and the second was followed by a third, until, nl- ;
timately, his textbooks embraced the whole range of mathematics and .
natural philosophy, astronomy, and meteorology."* His principal;
mathematical and astronomical textbooks are, Planer and Spherical '
Trigonometry, 1848; Analytical Geometry and Oalculus, 1857; Ele-
ments of Algebra, 1851 ; Elements of Geometry and Oonio Sections,
1851; Practical Astronomy, 1855; Elements of Arithmetic, 1863. His
treatise on astronomy, now obsolescent, received in its day high com-
mendation from leading astronomers. Some of his mathematical text-
books were, at first, very thin, but were gradually enlarged in subse-

* Yale College; a Sketch of its History, by William L. Kingsley.

156 TEACHING AND HISTORY OF MATHEMATICS

qaent editions. Thus, his Analytical Geometry and Oalculas were at
V \ first combined in one small volume, while, subsequently, the two sub-
jects were published separately in volumes about as large, each, as the
earlier combined volume.

\ The books of Loomis were written in a clear, simple style, and were
well adapted for use in the class-room. There was nothing in them
which any student of ordinary ability and application did not readily

/ master. These characteristics made Loomis's works very popular. A
student desiring to secure a somewhat extended knowledge of the vari-
ous mathematical subjects would hardly have found Loomis^s works
to answer his purpose ; nor would the works of Davies have given him
better satisfaction. He would have found more of what he wanted in
the books of Peirce and Ohauvenet. Nor were Loomis's works always
up with the times. The treatment of series is bad, both in his Algebra
and in his Calculus. Again, take the following statement, for instance:
<' No general solution of an equation higher than the fourth degree has
yet been discovered."* This piece of historical information is unsatis-
factory ; for, in the first place, M. Hermite has given a transcendental
solution of the quintic and, in the second place, Abel and Wantzel have
proved that an algebraic solution of equations higher than the fourth
degree is impossible. Perhaps the best mathematical work, in point of
accuracy, is his Elementary Geometry. It has been said of American
writers that, while they have given up Euclid, they have modified Le-
gendre's Geometry so as to make it resemble Euclid as much as possi-
ble. This applies to Loomis with greater force, perhaps, than to any
other author. His trigonometry has been wedded to the old ^Mine sys-
tem," and it is only within the last two or three years that a divorce
has been secured.

While Loomis has made no original contributions to pure mathemat-
ics, he has not been idle in other lines of research. He has contributed
one hundred or more papers (chiefly on astronomical, meteorological,
and physical subjects) to the American Philosophical Society, Oonnecti-
cut Academy, Smithsonian Institution, American Journal of Science,
and Gould's Astronomical Journal. Some of his papers have been re-
printed in Europe. His Contributions to Meteorology was translated
into French.

Professor Stanley's successor in the mathematical chair at Yale is
Professor Hubert Anson Newton. He graduated at Yale in 1850, after
which he studied higher mathematics. In 1852 he was made tutor, and
when he entered upon that office in 1853 he was given charge of the
entire mathematical department at once, owing to the illness of Profes-
sor Stanley. In 1855 he was elected full professor, with permission to
spend one year abroad. In 1856 he began the active discharge of the
duties of the chair, which he still holds. Professor Newton's publica-
tions have been restricted almost exclusively to scientific papers, which

* Treatise on Algebra, 1873, p. 334.

INFLUX OP PEENCH MATHEMATICS. J 57

have appeared in the Miemoirs of the I^ational Academy of Sciences and
in the American Journal of Science. He is best known to science for
his observations on shooting-stars and star-showers. He wrote for the
BncyclopsBdia Britannica the article on " Meteorites.'' His work in
pare mathematics includes a paper '^ On the Construction of Certain
Curves by Points,'' published in the Mathematical Monthly, and on
" Certain Transcendental Curves."

' Since 1871 Eugene L. fiichards has been assistant professor of mathe-
matics. .He is the anthor of a Trigonometry.

In 1873 John B. Clark, who had been professor at the University of
Michigan, was chosen professor of mathematics at Yale. Since 1881
Andrew W. Phillips has been assistant professor of mathematics ; also
William Beebe since 1882. Phillips and Beebe have written a novel
and successful treatise on Graphic Algebra.

In 1871 J. WUlard Gibbs was elected professor of mathematical
physics. He graduated at Yale in 1858, and after graduation contiur
ued his mathematical and physical studies. He was tutor from 1863
to 1866. Afterward he went to Europe and spent three years in study
at Paris, Berlin, and Heidelberg. Much of his time has been given to
thermodynamics. He contributed in 1873 to the Connecticut Academy
an article on Graphical Methods in Thermodynamics of Fluids. In the
same year appeared A Method of Geometric Bepresentation of the Ther-
modynamic Properties of Substances by Means of Surfaces.

But Professor Gibbs's studies have been carried on also in the field of
pure mathematics. He has published a treatise on the Elements of
Vector Analysis, which is a triple algebra, as distiuguished from quater-
nions, a quadruple algebra. Vector analysis has been applied by Pro-
fessor Gibbs to about the same kind of problems as quaternions. The
advantage claimed for vector analysis over quaternions is that the
former reaches solutions more simply and directly, and that its prin-
ciples can iSe developed more concisely. In 1886 Professor Gibbs read
an exceedingly interesting paper before the American Association for
the Advancement of Science on Multiple Algebra, which contains an
excellent sketch of the development of this science in the hands of Grass-
man, Hamilton, Hankel, Beujamin Peirce, Sylvester, Cayley, and others*
As to the applications of multiple algebra, Professor Gibbs says:*

^< Maxwell's Treatise on Electricity and Magnetism has done so much
to familiarize students of physics with quaternion notations that it
seems impossible that this subject should ever again be entirely divorced
from the methods of multiple algebra.

^< I wish that I could say as much of astronomy. It is, I think, to be re-
gretted that the oldest of the scientific applications of mathematics, the
most dignified, the most conservative, should keep so far aloof from the
youngest of mathematical methods.''

We now return to the courses of study at Yale College. The catalogue

* Proceedings Amerioan ABSociation for the Adyanoement of Science, 1886, p. 62.

158 TEACHING AND HISTORY OF MATHEMATICS.

of 1845 shows that << Day's Algebra to qaadratios '' was added to
*^ arithmetic '^ as a reqairement for admissioa to college. In 1852 Thorn*
soq's was the arithmetic recommended. In 1855 the requirements were
again increased by the addition of two books in Playfair's Enolid. In
1870 the terms were higher arithmetic, Loomia's Algebra to qoadratics,
and two books of Playfaii's Enclid (or the iirst, third, and fonrth
books of Davies' Legendre, or of Loomis' Elements of Geometry) ; in
1885, arithmetic, algebra as far as logarithms in Loomis, first book in >
Euclid, and the first thirty- three exercises thereon ip Todhnnterfs edition
(or the first four books in another geometry) ; in 1887, higher arithmetic
(including the metric system of weights and measures), algebra (Loomis
as far as logarithms), plane geometry. All candidates for admission
are examined on the same studies, no matter what courses they may
wish to pursue in college. It is also worthy of remark that, since 1885^
the use of Enclid as a text-book in geometry has been discontinued at
Yale, and Princeton is now the only prominent college in the country
which still adheres to Euclid.

We come now to the mathematical course in college. In 1848 it was
as follows :

Freskmenj Day's Algebra, Playfair's Euclid; Bophomores^ Day's iSMh*
ematics. Bridge's Conic Sections, and Stanley's Spherical Geometry
and Trigonometry; Junicrsj Olmsted^s Natural Philosophy, Mechanics^
Hydraulics, Hydrostatics, Olmsted's Astronomy, Analytical Geometxy
or Fluxions (optional).

Fluxions seem to have been optional all the time, though in previous
catalogues they appear as a regular study. Analytical geometry was
also optional for that year. In 1852 Loomis's Analytical Geometry
and Galculus appear in the catalogue as Sophomore studies. In 1864
Bridges Conic Sections or Analytical Geometry appear as part of the
work of the Sophomore year ; and Church's Difierential Calculus in the
Junior year. But analytical geometry aud calculus were efeotive stud-
ies. ^^ Those desirous of pursuing higher mathematics are allowed to
choose analytical geometry in place of regular mathematice in the third
term Sophomore, and calculus in the Junior for Greek and Latin."

In 1858 Loomis's Calculus is given in the Sophomore year, and
Todhunter's in the Junior.

The course was as follows in 1870 : Freshmen^ Loomis's Algebra, Play-
fair's Euclid, Loomis's Conic Sections; SophomareSj Loomis's Trigonom-
etry, Stanley's Spherical Geometry, Davies' Analytical G4H>metry;
Juniors^ Calculus, Loomis's Astronomy. The next year, Ohauvenet's
(Geometry was used with Euclid in the Freshman class.

In 1885 the course was — Freehmen^ Todhunter's Euclid (Books III
and lY ), Chan venet's Geometry, Richard's Plane Trigonometry, Phillips
and Beebe's Graphic Algebra; Sophomoresj Loomi6's Analytical Ge-
ometry (plane and solid), Dana's Mechanics; Juniors^ Loomis's Astron-
omy (required). Calculus, Geodesy, Descriptive Geometry (all three
elective) ; Seniara^ Calculus, Vector Analysis (both elective).

INFLUX OP FRENCH MATHEMATICS. 159

The course for the year 1887-88 is snbstantially the same as that of
1885. We qaote from the datalogae the following account of it :

'* In geometry the exercises consist in recitations from the text-book^
the original demonstration of theorems, *and applications of the prin*
ciples to the solation of numerical problems.

<< After the student has gained facility in the use of' trigonometrical
tables, the principles of plane trigonometry are applied to the problems
of mensuration, surveying, and navigation, and those of spherical trigo-
nometry to the elementary problems relating to the celestial sphere.

^'In algebra the elementary principles of the theory of equations
are illustrated graphically, and the student is exercised in the numerical
solution of equations of the higher degrees and the graphical represen-
tation of the relations of quantities.

^' In analytical geometry the student is carried through the elementary
properties of the lines and surfaces of the second degree, and is intro-
duced to the theory of map projection.
N <^ These are studies of the Freshman and Bophomore years, and, to-
gether with the elements of astronomy which are pursued in Junior year,
are regarded as essential parts of a liberal education.

^^In the Junior and Senior years opportunity is given in the elective
courses to obtain a wider knowledge of analytical geometry and trigo*
nometry with ttieir applications to geodesy and astronomy. A longer
and shorter course are provided in Junior year in differential and integral
calculus. The shorter course is designed for students who desire to be-
come acquainted with the methods of the calculus, but whose principal
studies are not of a mathematical character. The longer course is de-
signed for such as expect to make a serious study of any department of
pure or applied mathematics.

^< In Senior year advanced subjects in the calculus and the elements
of analytical mechanics form one line of study.

^' An elementary and an advanced course are provided in what is
called vector analysis. The object of. these courses is to introduce the
student to the methods of multiple algebra in geometry, mechanics,
and physics. The matter taught is not entirely unlike that usually
given in courses in quaternions, but the method followed is in some re-
spects nearer to Grassmann's than to Hamilton's. The elementary
oourse is confined to the simplest algebraic relations of vectors. The
advanced course includes differentiation with respect to position in
space, and the theory of linear vector functions.

*^ Students who show special aptitude are exercised in the working up
of subjects which require the use of the library and more prolonged
investigation than the daily exercises of the class-room. Such work
begins in Freshman year. There is a considerable collection of models,
which are used to assist the imagination in the various branches of
fltudy.^

In Fovember, 1877, a Mathematical Club was formed at Tale. Pro-

160 TEACHING AND HISTORY OP MATHEMATICS.

fessor Oibbs has been the leading spirit in it. He has, in recent years,
presented papers showing the application of vector analysis to the com-
putation of elliptic orbits. The work of the club has, however, not been
confined to pure mathematics. Professor l^ewton has presented sub-
jects on meteors and astronomy, and Professor Hastings has given re-
sults of experiments made by him on light.

COLLEGE OF NEW JEBSEY.

In 1830 Albert B. Bod became professor of mathematics. He seems
to have been a favorite teacher. His pupils cherish fondly the recollec-
tion of <^his brilliant genius and the interest which he infused into the
study of higher mathematics, as well as the magnetic charm of his man-
ner, as by the wonderful acuteness and perspicuity with which he mas-
tered and explained the most abstruse problems." The same qualities
shone attractively in his lectures on architecture.* He discharged the
duties of his office with signal ability till his death, in 1845. The family
to which he belonged had for several generations been remarkable both
for mathematical taste and talent. His father constructed the engine
of the Savannah, the first steam-boat that crossed the Atlantic.

The scientific and mathematical departments of Princeton were first
made prominent by the labors of Professors Henry and Alexander.
Stephen Alexander was graduated at Union College, JN'ew York, in
1824, at the age of eighteen, with high honors. He then engaged in
teaching. In 1830 and 1831 he was in Albany making numerous astro-
nomical observ^ations and communicating them to the Albany Institute.
He and Joseph Henry were relatives. "Professor Henry was a son
of the elder Alexander's sister, and in 1830 he married his cousin, Miss
Alexander, thus establishing a double relationship, which unquestion-
ably shaped the whole life and fortune of his younger and favorite
cousin and brother-in-law.t In 1832 Professor Henry was elected to the
chair of natural philosophy at the College of New Jersey. Alexander
went with Henry and his family to Princeton. He there entered the
Theological Seminary as a student, but in 1833 he was appointed tutor
in the college. " In 1834 he was made adjunct professor of mathematics,
and in 1840 he was promoted to the full professorship of astronomy,
which he retained until 1876. During the long intervening period the
style and duties of his professorship were several times more or less
modified. For several years after the death of Professor Dod he was
professor of mathematics and astronomy. When Professor Henry went
to Washington he gave up the mathematics and became professor of natu*
ral philosophy and astronomy, but he always held fast to astronomy."

In 1847 John Thomas Daffield became connected with the mathemati-

* The Prioceton Book, 1879.

t Biographical Memoirs of the National Academy of Soienoesi Vol. II, p. 886, '' Bio-
graphical Memoir of Stefthen Alexander," by C. A. Young. Oar remarks on Profeasor
Alexander are drawn chiefly from this sketch.

INFLUX OF FRENCH MATHEMATICS. 161

cal department. He graduated at Princeton College in 1841, afterward
studied theology, and then was appointed tutor in Greek* From 1847
to 1850 he served as adjunct professor of mathematics. During two
years he had charge of a Presbyterian church in connection with his
duties in the college. He published, also, a volume of sermons. He
has been professor of mathematics since 1850. For many years the
mathematical teaching at Princeton was in the hands of Professor Duf-
field and Professor Alexander. The former possessed great power in
teaching young students, while the latter led their way into the more
advanced mathematics and astronomy.

In 1850 the requirements for admission were arithmetic and the ele-
ments of algebra through simple equations. The Freshmen studied
Hackley's Algebra and Playfair's Euclid ; the Sophomores finished Euclid
and then took up plane and spherical trigonometry, navigation, etc. ;
the Juniors studied analytical geometry (Young's), Alexander's Differ-
ential and Integral Calculus, and mechanics.

Princeton is one of the very few colleges in this country which have
retained Euclid as a text-book in geometry to the present day. ^< Euclid
is used as a text-book in geometry because of its historical associations
and its decided superiority for the purpose of mental discipline to any
modern text-book."*

Bev. Dr. E. G. Hinsdale, who was a student at Princeton from 1852
to 1856, gives the following reminiscences of the mathematical teaching
there : " The requirements for admission were geometry — four books
of Euclid, algebra through quadratics. The text-book in algebra dur-
ing the Freshman year was Hackley's. The fact was, that but few who
entered were fully prepared, and therefore we had a rapid review of
the subjects db initio j finishing Hackley the first year. In the Sopho-
more year we finished Euclid's geometry, also surveying and naviga-
tion (elementary). Both subjects were taught in a special way by Prof.
John T. Dufiield, whose syllabus taken down from dictation was a
marvel of clearness. The notes of that syllabus I have with me. It
has never been printed. The Junior class studied Young's Analytical
Geometry and Conic Sections. The first half of the year we were taught
by Professor Duffield, the last half by Professor Alexander, who had
the chair of physics and astronomy.

^' In the Senior class mathematics was taught by Professor Alexander,
a gentleman of marked ability in the higher branches of his depart-
ments. He used no text-book in either department. Both subjects
were taught orally. An elaborate compendium of mathematical physics
was dictated to the class by the professor, accompanied by explanations
of formulse and experimental illustrations. The same way was adopted
by the professor in teaching the mathematics of astronomy. His sylla-
bus in tliat department, however, was Sprinted, not published,' for the
use of the class.

* Catalogne of the College of New Jersey, 1888-89, p. 42.
881— No. 3 ^11

162 TEACHING AND HISTORY OF MATHEMATICS.

^< Professor Alexander had a distingaished repatation among Us
confreres. Professor Peirce, of Cambridge, spoke of him repeatedly
in public lectures a43 the < Kepler of the nineteenth centary,' always in
connection with his theory as to the asteroids, accompanied by mathe-
matical demonstrations that they once formed one wafer-shaped planet
which, ' somewhere, somehow,' was shattered into fragments."

Rev. Horace G. Hinsdale says : • " He [Alexander] pushed his re-
searches into the depths of mathematical and astronomical sciencOi
availing himself of his acquaintance with the principal languages of
Europe. He printed for the use of his students treatises on ratio and
proportion, differential calculus, and astronomy. He was unselfish in
his devotion to the interests of the college and the advancement of
learning. He aroused the admiration of his pupils by the evident ex-
tent of his knowledge and his ardor in imparting it, although it must
be said that he often became so profoundly interested in setting forth
the philosophy of mathematics as to forget that their acquaintance with
the subject was, of necessity, far less than his own, and so to outrun
their ability to follow and comprehend him. The closing lectures in his
course in astronomy, in which he discussed the nebular hypothesis of
Laplace, were characterized by a lofty and poetic eloquence, and drew
to his class-room many others than the students to whom they were
addressed. Even ladies from the village and elsewhere — so far did the
traditional conservatism of Princeton give way before a wholesome pres*
sure — invaded Philosophical Hall.'^

Professor Yoang says : '< He was familiar not only with the ordinary
range of mathematical reading, but with many works of higher order.
He had large portions of the M^canique 061este almost at his finger's
ends, and was well acquainted with the works of Kewton, Eulery and
Lagrange."

As was the case with all college professors in former years, and is
still true with most of them, Professor Alexander's time and strength
were so consumed by the routine duties of the office, that little remained
for anything else. Still he accomplished a great deaL He published
articles in various scientific journals, and presented a large number of
papers, orally, before scientific societies ; and the only record of these
communications which we now have is a mere notice or a brief abstract
of a paper read on such and such a date.

In 1848 he read before the American Academy for the Advancement
of Science a paper on the Fundamental Principles of Mathematics.
Prof. 0. A. Young says of it : ^< It is an interesting, suggestive, and elo-
quent essay. The subject gives the author an opportunity to indulge
his inherited Scotch love for metaphysics and hair-splitting distinctions,
and he finds in it also opportunity for imagination and poetry to an
extent which makes the paper almost unique among mathematical dis-
quisitions."

"Quoted by Piot 0. A. Toimg in his memoir.

INFLUX OF FBSNOH liATHlSHATlOS. 168

Professor Alexander was an astronomet, bat his special fbrto was not
that of the observer* In fact, he had no adeqnate instraments or ob-
servatory. Long did he labor to secure a good observatory for the
college) and, at last, in 1882, a great telescope was pointed toward the
stars. << There was something pathetic in his exclamations of satisfao*
tion and delight, for the great instruments so long dreamed of^ had only
come too late for him to use it."

In 1876 Alexander was made professor emeritns, and Oharles Greene
Bockwood became connected with the mathematical department. Pro-
fessor Bockwood was graduated at Yale in 1864, and, befbre going to
Princeton, was professor of mathematics in Bowdoin and Bntgers stto-
oessively. He has acquired reputation by his studies of earthquakes,
and has contributed articles on vulcanology and seismology to the Be-
ports of the Smithsonian Institution, 1884^6.

From 1878 to 1883 Dn O. B. Halsted was a teacher in mathematics;
until 1881 as tutor, and from that time on as instructor in post-graduate
mathematics.

The present mathematical coirps consists of Proftossors John Thomas
Duffield, 0. 0. Bockwood, H. B. Fine, and Tutor H« D. Thompson. 0.
A% Young is the successor to Alexander as professor of astronomy.

The conservatism of Princeton College is noticeable in some features
of the mathematical instruction. Euclid has been retainediiA a textbook
to the present day. Todhnntei's edition has been used now for many
years. Until recently Loomls's text-books were used largely, though
not exclusively. In the academical under^graduate department the toh
lowing mathematics were taught in 1881 : Freshman year. Bay's Uni^
versity Algebra, Todhuntei's Euclid, and Mensuration ) Baph&mre year^
Loomis's Plane Trigonometry, Navigation, Surveying, Spherical Trigo-
nometry, and Analytical Geometry; Junior and iSav^r years, Analytical
Geometry of two and three dimensions, and Oalculus« Under Professor
Duffield oral instruction is made prominent* It might be more correct
to say that mathematics is taught by him << mainly by lectures-^the
text-books being used by way of reference, and as furnishing examples
for practice." << The students are required to take notes of the lectures
and submit their note^books for examination at the end of each tei^m."
Until q uite recently electl ves were introduced very sparingly. At pres-
ent all studies are prescribed during the first two years ; mathematics
is elective during the last two years.

Modern higher mathematics was first introduced in Princeton Ool-
lege by Dr. G. B. Halsted. His examination papers on quaternions,
determinants, and modem higher algebra, are the first ones that have
ever been set at Princeton. One feature of the mathematical instruc-
tion at this institution that has been in vogue during the last ten years
(perhaps longer) is, we think, to be recommended for more general
adoption. Considerable attention is given to the study of the history
of mathematics. The writer has before him examination paperS| writ-

164 TEACHING AND HISTOBT OF MATHEMATICS.

ten in answer to qaestions set by Halsted in 1831.* From the answers
we infer that qaestions like these have been asked : Who wrote the
iirst algebra that has come down to ns ? What was its natare ? What
part did the Hindoos play in the development of algebra ? Its growth
during the Benaissance? The laws underlying ordinary algebra? etc.

The present mathematical course, according to the catalogue of
188S-89, is as follows :

For admission to the academical department of the college, the
mathematical requirements are : '^ Arithmetic, including the metric
system ; algebra, through quadratic equations involving two unknown
quantities — ^including radicals, and fractional and negative exponents ;
geometry, the first and second books of Euclid, or an equivalent — ^that
is, the propositions in other text-books relating to the straight line and
rectilinear figures, not involving ratio and proportion.''

Studies in the academical department : '* In the Freshman year there
are two exercises a week during the first and second terms, in algel)ra,
and two exercises a week during the third term, in plane trigonometry,
under Professor Fine; in geometry there are. two exercises a week
throughout the year, under Mr. Thompson. The text-book iii algebra
is Wells's University Algebra, to be supplemented by a course on the
theory of equations, by the professor. Loomis's Trigonometry is the
text-book in trigonometry. Euclid is used as the text- book in geometry
because of its historical associations and its decided superiority for the
purpose of mental discipline to any modern text-book. The^.first six
and the eleventh books of Euclid are supplemented by a course in solid
and spherical geometry. Since a thorough knowledge of geometry and
familiarity with its more important propositions can be obtained only
by extended practice in the demonstration of theorems and problems
not contained in the text-book, this exercise occupies a prominent
place in the course of instruction.

<( The Sophomore class has three exercises a week throughout the
year in mathematics, under Professor Duffield. For the first term the
studies are analytical trigonometry, mensuration, and navigation ; for
the second and third terms, surveying, spherical trigonometry, analyt-
ical geometry, and the elements of the differential calculus.

^' In the Junior year mathematics is an elective study. The class
has two exercises a week throughout the year, under Professor Duf-
field. For the first and second terms the studies are analytical geom*
etry and the differential calculus ; for the third term, the integral cal-
culus. Loomis's Trigonometry is the text>book durinc^ the first and sec-
ond terms of the Sophomore year. Bowser's Analytical Geometry and
Calculus during the third term Sophomore and Junior year — supple-
mented largely by oral instruction, and numerous exercises in addition
to the examples for practice of the text-books.

* Odo of these was written by H. B. Fine, now assistant professor of matbematici
at Princeton ; another by A. L. KimbaU, now associate professor of physics at the
Johns Hopkins Uniyersity.

INFLUX OF FRENCH MATHEMATICS. 165

<<The Senior class in mathematics (elective) has two exercises a week
throughout the year, under Professor Pine. The course for the current
year is analyti6al geometry of three dimensions, differential and inte-
gral calculus. Williamson's text-books on the calculus are used, sup-
plemented by lectures on determinants, differentiation and integration
of functions of the complex variable, definite integrals."

In 1873 was founded, as a branch of Princeton OoUege, a scientific
school called the '^ John 0. Green School of Science." Its courses lead
to the degree of bachelor of science. Two years later a course in civil
engineering was organized in this school. The mathematics in the
scientific school is taught by Professor Bockwood. The course is
framed so as to supply the necessary foundation in knowledge and
training for the later studies of physics and mechanics, and especially
finds its natural continuation in the applied mathematics of the course
in civil engineering. Constant blackboard practice is a prominent
feature of the instruction. Euclid is supplanted by Ohauvenet's Geom-
etry. Other text-books used are Wells's Algebra, Bowser's Analytical
Geometry and Calculus. The calculus is begun at the end of the
Sophomore year and then finished in the Junior. With the geometry,
which is illustrated by models, is combined a thorough course in men-
suration and an introduction to the elements of modern geometry.
Thus, a synthetic course in conic sections is made to precede analytical
geometry — an idea highly to be recommended. Calculus is required
of all students in the scientific department. More advanced studies
in pure mathematics are elective.

Descriptive Geometry is taught by Professor Willson from Warren's
treatise. ^

In addition to the college courses, there are at Princeton Univeraity
courses leading to the degrees of master and doctor.

Post-graduate mathematics have been taught since 1881.

"The University courses this year (1888-89) are in differential equa^
tions, in the theory of functions, and in higher algebraic curves and sur-
faces. They are based on the treatises of Forsyth and Boole, Hermite
and Clebsch and Gordan, and Salmon and Olebscb, respectively. Pro-
fessor Fine conducts these courses.

DABTM0T7TH OOLLEaE.

In 1833 Ira Young succeeded Ebenezer Adams as professor of math-
ematics and natural philosophy. His father was a carpenter, which
trade he followed till he attained his majority. He early manifested
much mechanical ingenuity. At twenty-one he began a course pre-
paratory to entering college, and graduated at Dartmouth in 1828. He
served in the college, first as tutor, then as professor, until his death in
1858. He is said to have been an admirable teacher.

From 1838 to 1851 Stephen Chase was a professor of mathematics at
Dartmouth. He was a graduate of this college. While he was pro-

166 TEACHIKO AND HISTORY Ot MATHEMATICS.

fesaor be pnbliabed an algebra. An old alnmnus speaks of bim as a
teacber, ^« tbe ligbt of wbose geniaa, aa it gleams tbrougb one of oqr
tezt^boolca, yet lingers in oar halls,''

The eatalogae tov 18S4 shows that, sinee 1838, a remodeling of the
college oonrse had taken plaoe« There were now four departmental
viz, the classical, matbematioal aod pbysioaly rhetorical^ and the depart*
ment of intelleetnal and moral pbilosopby. The Frefhmen in the math,
ematieal and physical department studied Playfair's Eudid, reviewed
Adams's Arithmetic, and commenced Day's Algebra during tbe first
term ; continued Day's Algebra in tbe second term ; and completed
Euclid in the third.

The Sophomores continued Day's Algebra, devoting their attention to
applications to geometry and logarithms. They then took up plane
trigonometry and its applications. During the second term Bridgets
Oonio Sections and Ourvature, and Playfair's Spherical Geometry and
Trigonometry occupied their attention. They began also Beaoat'a
Oaloulus, which was finished during the third term. The Juniors par-
sued Olmsted's Natural Philosophy, Day's Mathematics (heights and
distances, and navigation), Olmsted's Hydrostatics, and Astronomy*
The Seniors had no mathematics, according to catalogue.

The next year (1839) indicates several changes. Legendre's Geometry
and Bourdon's Algebra displaced old Euclid and Day's Algebra, Da-
vies' Analytical Geometry and Oalculus were also used. The influence
of the Military Academy at West Point was now beginning to be felt at
Dartmouth.

Begarding the mathematical teaching at this time, John M, Ordway,
professor of applied chemistry and biology at the Tulane University of
Louisiana, writes us as follows: ^< When I entered Dartmouth College
in 1840, the Freshmen were instructed in algebra and geometry by a
tutor. We used Daviea' Bourdon's Algebra and Davies' Legendre^s
Geometry. In the Sophomore year we studied Davies' Surveying, and
Plane and Spherical Trigonometry, Davies' Analytical Geometry and
Davies' Calculus. The instruction was given by Professor Stephen
Chase, an excellent scholar, but a somewhat peculiar man. He showed
very little mercy to the duller students, and hence was not very popular.
The analytical geometry and calculus had not been introduced many
years, and it was a sort of traditional idea of the classes that preceded
ours, that these sabjects were very hard. We, however, did not per-
petuate this tradition, for our class as a whole did not find these higher
mathematics so very difficult. We had some field exercises in surveying
and leveling. The professor went out first with half a dozen chosen stu-
dents of the class, and they afterward went out with their respective sec-
tions of the class. Before our time there had been some solemn onrn-
ings of the mathematical text-books at the end of the year, but we had
no such nonsense while I was in college.

<^ Professor Chase also gave the instruction in phyaicS| which was

INFLUX OF FRENCH MATHEMATICS. 167

quite matbematical. He and Professor Yonng, the father of the present
Professor Oharles Young, of Princeton, had planned and partly written
a work on physics, in which the demonstrations were to be made by the
calonlns and analytical geometry ; bat meanwhile Professor Olmsted
published his Natural Philosophy, and as a matter of courtesy they
dropped their own work and introduced the poorer one of 01msted«*
Olmsted used the cbmmon geometry and algebra, and his book was
rather old-fashioned and contained some absurd errors. There was one
question in the book, 'If the pebble that David threw weighed 2 ounces
and Gk)liath weighed 800 pounds, with what velocity must the stone
have moved to prostrate the giant t ' The answer given was (about)
2,800 feet per second, or greater than that of a cannon ball. The .pro-
fessor called me up on this question, in the recitation, and asked me if
I saw any absurdity in the matter. I told him yes, the answer should
have been 2,800, and not 2,800 feet per second. Then the professor
went on to explain that Gtoliath must have had a skpll that would be
penetrated by a stone moving with much less velocity. He had entirely
overlooked the mathematical absurdity of getting a concrete answer out
of mere abstract numbers. I went to him after recitation to explain my
idea more fiilly, and told him that had Mr. Olmsted been a Frenchman
he would have made the answer 2,800 meters per second, and that
would have been just as correct, or 2,800 milea would have done just as
well. This he acknowledged, but seemed never to have thought of it
before, the physiological absurdity having shut out from his perception
the mathematical error.

'^ While Professor Ohase gave the mathematical teaching of physics,
ProfldBSor Young lectured on the subject with the help of a very good
set of apparatus.

<< In the Junior year Professor Young taught astronomy, using Olm-
sted's Astronomy for a text-book. This work was better than the
physics, but it rejected the calculus, which would have made many of
the demonstrations much plainer. Professor Young was an excellent
teacher and was very popular. He could be severe enough, but it was
in a quiet, dry way that was not offensive. He would call up a fellow
who had not studied the lesson well and put several questions, receiv-
ing the wrong answers without any sign of surprise or demur, and finally
say, ' The reverse is true,' aiid call up another man.

<« We had some astronomical instruments, but with the exception of
the telescope very few of the students ever used any of them. Our
examinations in those days were all oral. They were held in the pres*
ence of a committee of old graduates summoned to Hanover for the
purpose, their expenses being paid by the college. These old fellows

* Olmsted's Introdaction to Natural Philosophy was pahlished in 1831 ; Young was
elected professor in 1833 and Chase in 1838. While Young and, we believe, also
Chase, served as tutors before they were appointed professors, it is, nevertheless, not
likely that refeienee can be bad, in the above, to the JIni edition of Olmsted's work
on natural philosophy.

168 TEACHING AND HISTORY OF MATHEMATICS.

were rather rusty sometimes and gave the boys some amnsement by
their occasional old-time questions. The examinations were really a
farce^ though the results were counted in with the rest of the marks.
It was rather funny to see how some fellows who had been rated very
low all the year would be made out by the examiners to be among the
very best.^

In 1849 Chase's Algebra appeared, and began to be used at Dart-
mouth. Three years later Loomis's series was introduced, excepting
his Algebra.

In 1851 Ghase was succeeded by John Smith Woodman, of the class
of 1842. After graduation he taught school in Charleston, S. 0., after-
ward made a tour through Europe on foot, then studied and practiced
laWy and finally was elected professor at his alma mater.

In 1854 James Willis Patterson, of the class of 1848, became professor
of mathematics. He had previously been tutor two years. From 1859
till 1865 he was professor of astronomy and meteorology. He after-
ward entered politics, was elected to the Legislature and finally to the
United States Senate.

About the middle of the present century attempts were made to or-
ganize a system of education based chiefly upon the pure and applied
sciences, modern languages, and mathematics. Of this class were the
scientific schools connected with colleges, such as the Lawrence Scientific
School at Harvard, the Sheffield Scientific School at Yale, .the School
of Mines at Columbia, and the Chandler Scientific School at Dart-
mouth. These schools have done efficient work and supplied a long-
felt want.

The Chandler Scientific School was established in 1851. The instruc-
tion was designed to be ^4n the practical and useful arts of life, com-
prised chiefly in the branches of mechanics and civil engineering." At
first J. W. Patterson is given in the catalogue as Chandler professor of
mathematics, but Professor Woodman was the one wiio labored longest
in this school. He taught in it from its establishment, became profes-
sor of civil engineering in 1856, and was practically at the head of it.
He retained those posts until his death in 1871.

The mathematical course la this school was low at the beginning.
Loomis's books were used, also Puissant's Mathdmatics. Descriptive
geometry, shades, and shadows were also introduced.

In the catalogue of 1865, Bobinson's series, from the Algebra to the
Differential and Integral Calculus, is given. In 1866 Church's Analyt-
ical Geometry and Calculus were studied in the Chandler Scientific
School. Two years later the college dropped Bobinson's series and
returned to Loomis's*

Since 1870 the tex^books used have been, in algebra, Olney, Quimby ;
in geometry and trigonometry, Olney ; in analytical geometry, Loomis,
Church, Olney ; in calculus, Church, Olney ; in analytical mechanics,
Peck| Wood ; in descriptive geometry^ Church ; in quaternionS| Hardy.

INFLUX OF FRENCH IIATHEIIATICS. 169

The terms of admission to the college were, in 1828, arithmetic, alge«
bra through simple equations ; in 1841, the same *, in 1864, the same,
with the addition of two books of (Loomis's) geometry; in 1886 and foif
some years previous, all of plane geometry was required ; in 1888, arith-
metic, including the metric system, algebra to quadratics, and plane
geometry.

The college offers now two courses, one leading to the degree of
bachelor of arts, the other (the Latin-scientific conrse) to the degree of
bachelor of letters. The coarse of study for the year 1888-80 is as
follows :

In the Prescribed Oourtes, lit in each cats an advanced dioition /or ttudents judged to he qua^fi€d to

purtue a more exUnded eourte.

. PB£SCBIBED COURSES.
Fbbshman Tbab.

1. I. Algebra, inoMding Theory of Equations (Qaimby). Sixiy-fiv€ hourt,
II. Algebra. Stxty-five hours,

2. I. Solid, with advanced, Geometry (Olney). Forty-five hours,
II. Solid Geometry. Forty-five hours,

3. I. Plane trigonometry (Olney), Inolnding applications to Sarreying ; Spheri-

cal Trigonometry. Sixty-two exercises (including ten exercises of field work
of three hours each).
II. Same as 3, 1, omitting Spherical Trigonometry.
SoPHOMORB Year.

4. I. Analytic Geometry (Olney). Forty hours,

II. Spherical Trigonometry and Conic Sections. Forty hours,

5. Sarreying with field work and plotting. Eighty-seven hours.

Course 5 is open only to students of the Latin-Scientifio Course*

6. Descriptive Greometry ; Drawing. Sixty hours.

Course 6 is open only to students of the Latin-Soientiiic Course,

ELECTIVE COURSES.

7. a. Differential Calculas. > Applications to Analytic Geometry. Lectures,
b. Integral Calculus. 5 Ninety-four hours.

Course 7, a and h, is elective with French 2 followed by Mathematics 8.

8. Elementary Mechanics (Wood). Fifty hours.

Course 8 preceded by French 2 is elective with Mathematics 7, a and b.
Junior Year.

9. Analytic Mechanics ; Lectures. Sixty hours.

Course 9 is open only to students who have completed Course 7, and is elective with
Latin and Oreek,
10. Descriptive Geometry; Shades, Shadows, and Perspective (Church). Forty-
four hours.
Course 10 is elective ujiih Latin, Greeks German, Physics.

The minimum amonnt of mathematics on which a degree can be ob-
tained, is a coarse ending with spherical trigonometry and conic sec-
tions. Analytic geometry is not necessary.

The course in pure mathematics in the Otandler Scientific School is
much the same as the above. The catalogue mentions in that depart-
ment Olney as the textbook in calculus and Peck as that in analytical
mechanics.

170 TEACHINa AND HI8T0BT OF MATHEHATICS.

In I86O9 John B. Varney was appointed professor of mathematics,
and served for three jears. Daring the next six years John £• Sinclair
filled this position. In 1872 F. A. Sherman, the present professor of
mathematics in the Chandler Sohool of Scienoe, was elected. From
1872 to 1878 0. F. Emerson was connected with the mathematical de-
partment. Since then he has occapied the chair of natural philosophy
and has devoted his energies chiefly to the development of the physical
laboratory. P. H. Pettee has been professor of mathematics since 1877,
and is now teaching mathematics and engineering in the New Hamp-
shire Agricultural Experiment Station, which is a branch of Dartmouth
College. At present T. W, D. Worthen is associate professor of math-
ematics in the college.

Since 1878 Arthur S. Hardy has been the head of the mathematical
department at Dartmouth. He is professor of mathematics and of civil
engineering. Previous to the above date he held the professorship of
civil engineering in the Chandler Scientific School. Professor Hardy
was graduated at West Point in 1869. For three years he was professor
of civil engineering at Iowa College. He then spent one year in study
at the Nicole imp^riale des ponts et chauss^es in Paris, and on his return
went to Dartmouth. In 1881 appeared his Elements of Quaternions, the
first American book on this subject. It is elementary and well adapted
for use of those students in our colleges who may desire to know some-
thing of the wonderful researches of Sir William Bowan Hamilton. A
neat little publication of much interest is Professor Hardy's transla-
tion from the French of Argand's Imaginary Quantities. He published
also New Methods of Topographical Surveying, 1884. Professor Hardy
possesses two qualifications that are rarely combined ; he is a successful
mathematician and also a successful novelist.

BOWDOIN OOLLEaE.*

In 1825 William Smyth became adjunct professor of mathematics,
and in 1828 was given the full chair, which he held until his death in
1868. He was an alumnus of thecoUege. After his graduation, in 1822,
he studied theology at Andover, and then became tutor at his alma
mater. He was led to abandon Greek and take the department of
mathematics as an instructor, from his success in popularising algebra
by means of the blackboard.

The introduction of the blackboard in our colleges must have caused
important changes in the methods of teaching mathematics, especially
geometry. Unfortunately no record of these changes has been pre-
served except at one or two institutions. We are happily able to quote
the following account of its introduction at Bowdoin, taken from the
history of. the college, written by A. S. Packard. He says (p. 91) that
the blackboard was introduced by ^^ Proctor (afterward Professor)

* We are indebted for the information herein contained chiefly toaoommonioalioii
firom Prof. George T. Little, of Bowdoin College.

INFLUX OF FBENCH MATHEMATICS. 171

Smyth in 1824. That novelty, let me here say, made a sensation. When
he had tested the experiment in the Sophomore algebra, and with great
snocess, a eonsiderafale portion of the Janiors requested the privilege of
reviewing the algebra under the new method at an extra hoar — a won-
der in college experienoe ; and that blackboard experiment, I am sare,
led to his appointment as assistant professor of mathematics a year
after. Of this also I am snre, that he had then first detected a math-
ematical element in his mentt^ equipment. His forte had been Greek.''

Professor Packard gives also an interesting account of the modes of
teaching immediately before the blackboard came to boused. <<The
blackboard caused important changes in the manner of teaching gen-
erally, but espedaUy in the mathematical branches. In arithmetic, a
Preshmaii study, and algebra, to which we were introduced at the open-
ing of the Sophomore year, each student had his slate, and when he
finished his work he took the vacant chair next the teacher's and under-
went examination of the process or principle involved. In geometry we
kept a MS., in which we drew the figures and demonstrated from that.
I have been shown a very neat MS. kept at Harvard by the late Dr.
Lincoln^ the father, and bearing the date 1800. • • * It may sur-
prise my heuers that I professed to teach the algebra of the Sopho-
more class in Webber^ft Mathematics-^he first tutor, I believe, to whom
the duty was entrusted. That was the class of 1824. Franklin Pierce^
of the class, in his earlier years of college life, more fond of fhn than
of surds and equations, took his seat by my side for a quia with his
slate and solution of a problem. When asked how he obtained a cex^
tain process; he replied very flrankly, *I got it from Stowe^s slate.'
• • • With blackboard such transfers are less easy. • • • It will
cause more surprise that conic sections in Webber, a Junior branch, fell
under my charge. The manner of reciting was simply to explain the
demonstration in the text-book."

In 1834 the requirements for admission were increased so as to in-
clude ^' six sections of Smyth^s Algebra.'' These six sections indude
nearly the entire Algebra, logarithms and the binomial theorem being
exdoded. In 1867 the requirements were raised so as to read arithme-
tic, the first eight sections of Smyth's New Elementary Algebra (ta
equations of the second degree), and the first and third books of Da-
vies' Legendre. The requisites remaiued practically the same from
18Q7 to 1887, though the tett-books recommended were several times
changed. Since that time all of plane g^metry has been required.

The calculus first appears as a study in the annual catalogue of 1830,
the notation of Leibnitz being then used* Fluxions were probably
never taught at Bowdoin.

Professor Smyth became an exceedingly able teacher and gained
celebrity as a successful writer of mathematical text*books. His pub-
lications were, a work on Plane Trigonometry, followed by his Algebra,
Analytical Qeometi7 (1855), and Calculus (1856). AH of these passed

172 TEACHING AND HI8T0BT OF MATHEMATICS.

through repeated editions and enjoyed an extensive sale. As they came
from the4)ress they took the place of the Gambridge Mathematics at
Bowdoin. In the preparation of his Algebra he followed Bourdon and
Lacroix as models, and it contains many of the excellences and some of
the defects of these works. A remarkable feature is the very late in-
troduction and explanation of negative quantities. They appear on
page 89, after the solution of simultaneous linear equation. In his cal-
culus he uses infinitesimals. <*As a logical basin of the Oalcolas,'' says
he (p. 229), << the method of Newton, and especially that of Lagrange,
has some advantage. In other respects the superiority is immeasura-
bly on the side of the method of Leibnitz." At the end of the book he
very briefly explains the methods of I^ewton and Lagrange. A few
pages are also given to the *^ Method of Variations'' and ** Applications
to Astronomy."

The following account of Smyth and his works is taken from an obitu-
ary address by his colleague, Professor Packard : ^^ As the first fraits,
he issued a small work on Plane Trigonometry, availing himself of the
ingenuity of the late Mr. L. T. Jackson, of this town, in preparing blocks
on a novel plan for striking off the diagrams. The first edition of his
Algebra from the press of Mr. Grifin, of this town, appeared in ISSO,
which first adapted the best French methods to the American mind, re-
ceived warm commendation from Dr. Bowditch, and was adopted as a
text-book at Harvard and other institutions. It passed through several
editions and then gave place to two separate works, the Elementary Al-
gebra and the Treatise on Algebra. Then followed an enlarged edition
of the Trigonometry and its application to Surveying and Navigation,
and treatises on Analytic Geometry and on the Galcnlns, the last being
so clearly and satisfactorily developed and with so much originality as to
receive emphatic approval in high quarters, particularly from the late
Professor Bache.''

^^ In explanation he was precise, simple, and clear. He had great power
of inspiring interest; his own enthusiasm, which often kindled, espe-
cially in certain branches of his department, at the blackboard, being
communicated to bis class. Later classes will carry through life his
setting forth of what he termed the ^ poetry of mathematics,' as exem-
plified in the Oalculus."

Of the graduates of Bowdoin during Smyth's time who distinguished
themselves in the mathematical line, we mention John H. 0. GofSn
(class of 1834), who, soon after graduation, was appointed professor of
mathematics in the U. S. Navy. He was for many years in the Naval
Observatory and, in 1866, took charge of the American Ephemeris and
Nautical Almanac.

Professor Smyth's successor, in 1865, was Charles Greene Bockwood,
who had graduated at Yale in 1864, and in 1866 received the degree of
Ph. D. When he left Bowdoin to^accept a position at Rutgers College,
in 1873, Charles Henry Smith took his place. In 1887, Professor Smith

INFLUX OF FRENCH MATHEMATICS. 173

was sacceeded by Prof. William Alboin Moody, the present incambent
of the chair^of mathematics. Professors Eockwood and Smith left the
college with the reputation of able and skillful teachers. ^^ The latter
was, iu my judgment, remarkably suocessfal," says Professor Little, ^'In
securing good and faithful work from all." The writer has before him
a report on geometry by Professor Smith, presented to the Maine Ped-
agogical Society in 1884, and containing some good recommendations
on the study of it« elements. He strongly recommends a course in em-
pirical geometry of the sort marked out by G. A. HilPs Geometry for
Beginners, Mault's Natural Geometry, and Spencer's Inventional Geom-
etry, to precede the course in demonstrative geometry. '

Mathematics have never been taught at Bowdoin by lectures, though
the instruction has been frequently supplemented by lectures. Since
1880 all mathematics have been elective after the Sophomore year;
since 1886, all afjber the Freshman year. An elective in calculus, not
then a required study, was offered from 1870 to 1880. In the year 1882-
83 the Freshmen studied Loomis's Algebra, and Loomis's Geometry and
Oonio Sections, in two parallel courses during the first two terms; the
third term of the year being given to Plane Trigonometry (Olney). The
Sophomores had Olney's Spherical Trigonometry during the first term ;
during the second and third term they had the choice between analyti-
cal geometry, and Latin and Greek. Calculus was elective for Juniors*
In the Senior year no mathematics were offered. The text-book in
astronomy was Newcomb and Holden. A feature in this mathematical
course to be recommended is that analytical geometry is preceded by a
short course in conic sections (treated synthetically). The course for the
year 1888-89 differs from the preceding in this, that plane geometry is
required for admission ; that Wentworth's Algebra has taken the place
of Loomis's; that differential and integral calculus are studies in the
second and third terms of the Sophomore year ; that an advanced course
in calculus (Williamson's) is offered during the first two terms of the
Junior year, and quaternions during the third term.

GBOBaETOWN OOLLEaB.*

t From 1831 to 1879 Father James Gurley was the head of the mathe-
matical and astronomical department at Georgetown College. He was
bom in Ireland, October 25, 1796. He entered the Society of Jesus
September 29, 1827, and came here in 1830. In 1843 Father Gurley
built the college observatory. Here he calculated, from his observa-
tions, the longitude of Washington. The astronomers at the U. 8.
Naval Observatory had found a longitude differing a little from Father
Ourley's result. When, however, the laying of the Atlantic cable

* For what iqaterial we possess on the teaching; at this college we are iDdebted to
tbe kindness of Prof. J. F. Dawson^ S. J., professor of physios and mechanics at
Georgetown College.

174 TEACHING AND mSTORT OF ICATHEMATICS.

brought Washington into telegraphic oommnnication Trith Oreenwieh,
it was fonnd that Father Garley's calculation was the correct one.
Since 1879 Father Gnrley has not been able to teach ; he is still living
at Georgetown, and is in the fall possession of all his fiiu^alties.

Since 1830 Father Oarley has generally had two assifitants, or asso-
ciates, in mathematics.*

Bev. James Olark was bom October 21, 1809. He entered West
Point at the age of sixteen, and graduated in the class of 1829. He
served in the Army several years. In 1844 he entered the Society of
Jesus, and came to Georgetown in 1845. In 1849 he went to Worcester
College, Massachusetts, then recently established, but remained there
only one year, returning to Georgetown in 18S0. From 1862 to 1867 he
was president of Worcester College. He returned to Georgetown in
1867, but was appointed president of Gonzaga College, Washington, in
1869. This office he held until 1875, when he again took his old posi*
tion in Georgetown. In 1879 Father Clark became unable to teaoh,
and on September &, 1885, he died at Georgetown. For some years he
taught calculus from his own manuscript, and intended to publiah a
texMM>ok but for some reason did not do sa

About the year 1848, political troubles in Europe induced a consider-
able emigration to America of some of the most able members of the
Society of Jesus, and the faculty of Georgetown College was increased
by a considerable accession of learning and talent. We mention as the
most conspicuous. Fathers Sestini and Secchi.

Bev. Benedict Sestini Was born in Italy, March 20, 1816. He entered
the society in 1836. In 1847 he was astronomer of the Soman Observa-
tory. In 1848 he was compelled to leave Italy by the revolutionists,
and came to Georgetown. He taught here until 1857 ; then he taught

* Daring 183d-'42, 1843-'45, 1847-'48, and ld65-'67, Bev. James Ward, S. J., gave ia-
stmction in mathematics; 184W46, Bev. Thomas Jenkins, 8. J ; 1840-'41 and 1842-M3,
Bev. Angnstine Kennedy, S. J. ; 1844>M5, Bev. George Fenwick, S. J. ; 1846-^47 and
1869-^1, Bev. Joseph O'Callaghan, 8. J. ; 1848-'49, Bev. Angelo Secohi, 6. J. ; 1849^68,
Bev. Edward McNerhany, S. J. ; 1852-'54, Bev. Anthony Vanden Henvel, S. J. }
l854-'60, Bev. John Prendergast, 8, J. ; 18dO-'61, 18e8-'63, 1870-71, 1870-'?" , Bev. C.
Bahan; 1861-^63 and 1871-74, Bev. G. Strong, 8. J. ; 1845-'49, 1850-'62,-au/ 1875-79,
Bev. James Clark, B. J. ; 1848-'57, and l863-'69, Bev. B. Sestini ; 1663-'64, Bev. Aloy-
Bins Varsi, 8. J. ; 1864-'65, Bev. James Major, S. J. ; 18G7-'69, Bev. Antonio Cichi, 8.
J. ; 1869-70, Bev. Patrick Forhan, 8. J. ; 1871-72, Bev. Patrick Gallagher, 8. J. ;
1872-73, Bev. Jerome Dangherty, 8 J. ; 1873-^4, Bev. Edmund Tonng, 8. J. ; 1074-
78, Bev. J. Byan, 8. J. ; 1874-78, Bev. M. O'Eane, 8. J. ; 1878-'d3, Bev. J. B. Bioh-
ards, 8. J. ; 1879-'83, Bev. Henry T. Tarr, 8. J. ; 1881-^86, Mr. Thomas MoLonghlia,
8. J. ; 1883-'84, Bev. Timothy Brosnaham, 8. J. ; 1883-'84, Bev. John O'Bourke, 8. J. ;
1884-'85, Bev. Edward Devitt, 8 J. ; 1884-'85, Bev. Thomas Stack, 8. J. ; 1885-^88,
Bev. Samuel H. Frishy, 8. J. ; 1885-'87, Mr. Joseph Gorman, S. J.; 1887-^, Mr.
David Hearn, 8. J. ; 1888-'—, Bev. John Hagen, 8. J., Bey. John Leby, & J., Mr.
James Dawson, 8. J., Mr. J. Gorman, 8. J.

The frequent changes in the corps of instrnctors are due to the custom of the
Society of Jesus. ^'In the society a teacher is liable any year to be sent to another
oollegOy and is rarely left more than four or five years in one plaoot''

INFLUX OF FRENCH MATHEMATICS. 175

fhiee years in Gonzaga GoUege, two years in Worcester College, and
one year in Boston College. In 1863 he returned to Ceorgetown, where
he tanght until 1869 ; he was then removed to Woodstock College* In
1886, advanced in years and broken down in health, he was sent to
Frederick, to the novitiate of the society, where he still remains await-
ing his end.* His books were used several years at Woodstock College
(the scholastioate of the society). At one time they were in rather ex-
tensive use, but at present they have gone out of use almost com<
pletely.

Bev. Benedict Sestini published the following mathematical works :
A Treatise on Analytical Geometry, Washington, 1852 ; A Treatise on
Algebra, Baltimore, 1885 f Elementary Algebra, second edition. 18J5(f } ;
Elementary Geometry and Trigonometry, 1856 ; Manual of Geometrical
and Infinitesimal Analysis, Baltimore, 1871. The method of treatmejut
of the various subjects in these works is not entirely conformable to that
generally in vogue in this country at the time of their publication. The
last named work is a thin volume of 130 pages, making no pretension
of being a complete work on the subject. It was intended primarily for
students in the author's own classes at Woodstock College, in Maryland,
and as an introduction to the study of physical science.

With Father Sestiui came Rev. Angelo Secchi, the astronomer. He
was bom in 1818, and entered the society in 1833. He was compelled to
leave Italy in 1848, on account of the revolution. He remained at
Georgetown very little more than a year. In 1850 he returned to Italy,
and was placed in charge of the Boman Observatory, where he labored
until his death, February 26, 1878.

At present the mathematical course consists of geometry, plane and
spherical trigonometry, analytical geometry, differential and integral
calculus, mechanics, and astronomy. Algebra is taught in the prepar-
atory department. This course has remained practically the same since
1829, except that the time given to mechanics has been increased.

Elective studies have never been offered at the college, nor has the
practice of lecturing ever been in vogue. Since 1829 more time has
been given to mathematics than formerly. About the year 1820 the
Society of Jesus adopted a new ^^ ratio studiorum^^ or plan of studies,
giving to mathematics more attention than had hitherto been accorded
to them. This brought about the change at Georgetown in 1829. The
methods of the Society of Jesus have been strictly adhered to. '< The
professor first explains the lesson, pointing out the important parts,
the proofs, the connection with other parts of the subject, etc., and giv-
ing other proofs if those in the book do not suit him. On the following
day he calls on one of the class for a repetition ; after the repetition

* Professor Dawson has endeavored to find out something aboat the early life and
edacation of Father Sestini, bat with no sucoess. Father Sestini himself can not give
any information on the sabjeot; hXH health has failed very much, and his memory can
not be relied upon.

176 TEACHING AND HISTORY OF MATHEMATICS.

the members of the class bring forward their difficulties, Bnggestions,
etc., after which the following lesson is ezpLained. . Problems are fre-
quently given to test the knowledge and inventive powers of the stu-
dents.'^

Father Sestini's text-books were used several years. They were re-
placed by those of Davies. In 1860 Gnmmere's surveying was intro-
duced, and the Algebra, Geometry, and Trigonometry of Davies ; Sesti-
ni's Analytical Geometry and Calculus were retained. In 1870 Sestini's
Analytical Geometry and Calculus were replaced by Davies' Analytical
Geometry and Ch urch's Calculus. In 1872 and 1873 Loomis's Analytical
Geometry was used. In 1874 Olney's Algebra, Trigonometry, and Calcu-
lus were introduced ; Davies' Geometry and Gummere's Surveying were
retained. In 1878 the Algebra and-Geometry of Loomis were used, and
in 1879 his Trigonometry, Analytical Geometry, and Calculus. Two
years ago Wentworth's series was introduced, with Taylor's Calculus.
Peck's Mechanics was used until 1881, when it was replaced by Dana's.
In calculus the notation of Leibnitz has been employed <^ as far bac^ as
we have any records."

At the college observatory no work has been done for some years;
but in January, 1889, Eev. John Hagen, S. J., was placed in charge of
the observatory and will make regular observations. Father Hagen
was formerly at Prairie du Ohien, Wis. He is a mathematician of con-
siderable ability and has contributed articles to the American Journal
of Mathematics.

COBNELL UNIVEKSITY.^

When Dr. Andrew D. White entered upon the organization of Cornell
XTnivei^sity and the selection of a faculty, tlie first professor appointed
was Evan William Evans. He occupied the chair of mathematics at
Cornell from the time of its opening, in 1868, till 1872, when he resigned
on account of failing health. Professor Evans was a native of Wales,
came to this country with his parents when a child, was graduated at
Yale in 1851, and studied theology for a year. He then became princi-
pal of the Delaware Institute, Franklin^ N. Y.^ was tutor at Yale irom
1855 to 1857, and, later, professor of natural philosophy and astronomy
in Marietta College, Ohio, where he remained until 1864. Before en-
tering upon his work at Cornell University he was occupied for three
years as mining engineer, and spent one year in European travel. He
died not long after resigning his position at Cornell.

In the same year that Professor Evans was selected to the mathemat-
ical chair, Ziba Hazard Potter, a graduate of Hobart College, was ap-
pointed assistant professor of mathematics. This position he held for
fourteen years.

* The writer is indebted to the kindness of Professor OUver for sending annual
reports and giving information on the mathematical courses ■}€ study at Cornell Uai*
yersity.

INFLUX OF FRENCH MATHEMATICS. 177

In 1869 William E. Arnold, major U. S. Volunteers, entered upon the
duties of assistant professor of mathematics and military tactics, and
served seven years in that capacity.

Appointed as assistant professor at the same time as Professor Arnold,
was Henry T. Eddy. He is a native of Massachusetts, was graduated
at Yale in 1867, and then studied engineering at the Sheffield Scientific
School. In 1868 he became instructor of mathematics and Latin at the
University of Tennessee, at Knoxville. At Cornell he received the (de-
grees of C. E. and Ph. D. for advanced studies in pure and applied
mathematics. In 1872 he went to Princeton, where, for one year, he
was associate professor of mathematics. Since 1874 he has held the
chair of mathematics at the University of Cincinnati. The year 1879-80
was spent by him in study abroad.

Professor Eddy has won distinction as an original investigator.
His Eesearches in Graphical Statics (New York, 1878) and his Neue
Otmatructionen in der graphischen Statik (Leipzig, 1880) are contributions
of much value, and, we believe, the first original work on this subject
by an American writer. Professor Eddy is contributing largely to sci-
entific and technical journals. In 1874 appeared his Analytical Geom-
etry. At the meeting of the American Association for the Advancement
of Science, in Philadelphia in 1884, Eddy was Vice-President of Section
A, and delivered an address on " College Mathematics.^ Having been
connected as student or teacher with several higher institutions of learn-
ing, both classical and scientific, he was able to speak from his own obser-
vation and experience of the defects of the mathematical instruction in
the United States. His address contains many valuable suggestions.

In 1870 Lucien Augustus Wait, who had just graduated at Harvard,
was appointed assistant professor. He held this position for about ten
years, when he was made associate professor, which position he still
holds. Some time ago he spent one year in Europe on leave of absence.
Professor Wait is an energetic and excellent teacher of mathematics.

For three years succeeding 1873 William E. Byerly, a graduate of
Harvard, and now professor theee, was assistant professor at Cornell
University. Professor Byerly is a fine teacher, and by his publications
has made his name widely known among American students of the
more advanced mathematics.

Since 1877 George William Jones has been assistant professor. He
is a graduate of Yale, 1859, He ^' is thoroughly logical, and the best
drill-master" in the mathematical faculty at Cornell University.

As has been seen from the above, two of the former assistant pro-
fessors at Cornell have since won distinction elsewhere. The same is
true of some of the instructors in mathematics. Before us lie the names
of the following former instructors in mathematics at Cornell : George
Tayloe Winston (one year, 1873, now at the University of North Caro-
lina), Edmund De Breton Gardiner (one year, 1876), Charles Ambrose
Van Velzer (one year, 1876, now professor pf ipathenpiatics at University
881— No. 3 12

178 TEACHING AND HISTORY OF MATHEMATICS.

of Wisconsin), Madison M. Garver, and Morris E. Conable (each for part
of one year, about 187G).

At present there are four instructors, viz : James McMahon (since
1884), Arthur Stafford Hathaway (since 1885), Duano Studley (since
1887), George Egbert Fisher (since 1887).

Mr. McMahon is a graduate of the University of Dublin, Ireland,
1881. He has a fine mathematical mind, and has obtained gold medals
for his proficiency in mathematics and mathematical physics, and also
an appointment to a scholarship at his alma mater. He has not pub-
lished much, but has assisted in the preparation of text-books on mathe-
matics issued by the Cornell professors.

Mr. Hathaway graduated at Cornell in 1879, and then pursued grad-
uate studies at the Johns Hopkins University, under Sylvester and his
associates till 1884. While in Baltimore he frequently contributed
papers to the mathematical society at the university, which were sub-
sequently published in the Johns Hopkins University Circulars, He
has made the theory of numbers his specialty, and has contributed sev-
eral original articles on the subject to the American Journal of Mathe*
matics. He gives a new theory of determinately-combining ideals.
Mr. Hathaway is not only an able mathematician, but also an expert
stenographet. When Sir William Thomson, of the University of Glas*
gow, delivered a course of lectures on Molecular Dynamics at the Johns
Hopkins University, in October, 1884, Mr, Hathaway exercised his
" power to seize on every passing sound." » These stenographic noted
of Thomson's lectures were printed by the papyrograph process and
published. At Cornell, Hathaway has assisted in the preparation of
textbooks, and is now, with Professor Jones, preparing a Projective
Geometry,

It will be noticed that Harvard University has contributed the largest
share of mathematical talent to the faculty of Cornell. Not only are
Byerly and Wait graduates of Harvard, but also Oliver, the present
occupant of the mathematical chair at Cornell. These three sat at the
feet of that Gamaliel, Benjamin Peir^, and caught the inspiring words
of their great master.

James Edward Oliver was born in Maine,'in 1829, and wAs graduated
at Harvard in 1849. He had then already displayed extraordinary
^lathematical power, and was at once appointed assistant in the office
of the American Nautical Almanac, at that time in Cambridge. In the
Harvard catalogues of 1854 and 1855 we find J. E. Oliver and T, H.
Safford enrolled as mathematical students in the Lawrence Scientific
School, and taking advanced courses of mathematics, such as were
offered at that time by no other institution in the land. In 1871 Oliver
became assistant professor of mathematics at Cornell, and two years
later was given full possession of the chair.

Professor Oliver is an extraordinary man, and it is interesting to lis-
ten to what his former pupils have to say of him. Says Prof. 0. A. Yao

INFLUX OP FRENCH MATHEMATICS. 179

Yelzer : " He is indeed a wonderful man. If Professor Oliver had some
of Sylvester's desire for reputation, he would have been heard from long
ago, and would have been known all over the world.'' Says Mr. A. S.
Hathaway: "Professor Oliver is ^ rare genius, powerful, able, but
without the slightest ambition to publish his results. He works in
mathematics for the love of it. I have seen work of bis done one or
two years ago. Practically the same work appeared in the American
Journal of Mathematics, written by prominent authors, that I had
urged him to publish, and which he had promised to do, but whieh,
with his characteristic dilatoriness and diffidence in this respect, he
failed to do until it was too late. I consider him fully equal in point
of natural ability to Professor Sylvester, and he is better able than
Professor Sylvester, I think, to acquire a knowledge of what others hav0
done. He lacks, however, the energy and ambition of Professor Syl-
vester, and does not concentrate his powers on any one subject. His
work is im-methodical, and leads in whatever direction his mind is bent
at the moment. The result is that he is a far more amiable and con-
genial person to meet than Professor Sylvester. He never obtrudes self
upon you, and wherever you may lead he will follow. Indeed, his sim-
plicity of character and interest in everything that interests anybody
else is one of his greatest charms. There are few subjects in which he
does not know more than most people— you find it out when yon are
talking with him— but he does not seem to know it, at least he never
obtrudes it." ^ ^

Professor Wait is described as a << live energetic business manager,
who was appointed to the position of associate professor to supplement
Professor Oliver's shortcomings, and to take care of the practical man-
agement of the department. A better man could not have been chosen
to associate with Professor Oliver. The latter finds in Professor Wait a
ready promoter of his ideas and plans, and one who is capable of carrying
them out in the smallest detail, and of taking charge of the department
without troubling the chief."

Professor Jones is a good drill-master. The bulk of the work on
mathematical text-books is done by him. His style has been adopted
throughout. Professsor Oliver's style is more classical and polished,
but that of Professor Jones is more suitable for elementary text-books.
In consequence, everything written by any one else, has been re-shaped
more or less by him.

The mathematical faculty of Cornell have published several tex^
books, going by the name of " Oliver, Wait, and Jones's Mathematics."
The works in question are, a Treatise on Trigonometry, a Treatise on
Algebra, and Logarithmic Tables. In preparation are also a Drill-
Book in Algebra, which will be specially adapted to the work of the
preparatory schools, and a Treatise on Projective Geometry.

The Treatise on Trigonometry has been used successfully at Cornell
for eight years, and their Treatise on Algebra for two years. "For th§

180 TEACHING AND HISTOBY OP MATHEMATICS.

regular classes (in algebra) the more difficult parts have been ont oat;
bat every year nearly all that was omitted by them has been taken up
by volunteer classes (all Freshmen) with great satisfaction and profit"
After eight years of use the Trigonometry has been wholly rewritten.

The Treatise on Algebra is not a book intended for beginners, bat
primarily for students entering the Freshman class at Cornell, and who
have bad extensive drill in elementary algebra. Most of our American
colleges would find the book too difficult for use, on account of deficient
preparation on the part of students entering.

If we compare Oliver, Wait, and Jones's Algebra with algebras used
in our colleges ten or fifteen years ago, we discover most radical differ-
ences and evidences of a speedy awakening of mathematical life among
us. A great shaking has taken place among the <^ dry-bones " of Amer-
ican mathematical text-books, and no men <^ shake" more vigorously
than the professors at Cornell. Among the improvements we would
mention a clearer statement of first principles and of the philosophy
of the subject, the introduction of new symbols, a more extended treat-
ment and graphic representation of imaginaries, and a more rigid treat-
ment of infinite series. With some corrections and alterations in a subse*
quent edition, we have little doubt that the book will become the peer of
any algebra in the English language.

At Cornell great efforts are made to teach the logic of mathematicSt
but it is hard to attain the desired standard on account of the way that
preparatory schools train their pupils. T]|e preliminary training in
algebra generally gives students the idea that algebra is merely a mass
of rules, and that students have simply to learn the art of applying them.
In consequence of this, there is a constant rebellion among the average
Freshmen to the logical study of algebra. Formulae and substitutions
are his stand-by.

The attendance of students has been very large at Cornell. Com-
pared with some other departments of the university, the teaching
force in mathematics has been rather small. In 43onsequenoe of this,
the time and energy of the professors have been taxed unusually by
work in the class-room. In the appendix to the Annual Report of the
President of Cornell University for 1886-87, Professor Oliver speaks
of this subject, and also of the general work of the mathematical de-
partment. He says:

^< We are not unmindful of the fact that by publishing more, we could
help to strengthen the university, and that we ought to do so if it were
possible. Indeed, every one of us five is now preparing work for pub-
lication or expects to be doing so this summer, but soch work progresses
very slowly because the more immediate duties of each day leave us so
little of that freshness without which good theoretical work can not be
done.

^' A reprint of our algebra, increased to 412 pages, has, however, ap-
peared this year, and has p^ttracted favorable notice from the press and

INFLUX OF FRENCH MATHEMATICS. 181

from distinguished mathematicians. All Ave of us have in some way
contributed to the work, but much more of it has been done by Profes-
sor Jones than by any one else. The chapters with which we propose
to complete the book deal mainly with special applications, or with
topics peculiar to modern analysis. Meanwhile we have successfully
used the volume in all the Freshman sections this year. • <» •

<< The greatest hindrance to the success of the department, especially
in the higher kinds of work, lies, as we think, in the excessive amount
of teaching required of each teacher ; commonly from seventeen to
twenty or more hours per week. The department teaches more men, if
we take account of the number of hours' instructiou given to each, than
does any other department in the university. Gould each teacher's
necessary work be diminished in quantity, we are confident that the
difference. would be more than made up in quality andincreased attract-
iveness.''

From the Beport for 1887-88, p, 76, we clip the following :

^' Of course one important means toward this end [of securing the
attendance of graduate students] is the publication of treatises for teach-
ing, and of original work. A little in both lines has been done during
the past year, though less than would have been but for the pressure
of other university work, and less than we hope to accomplish next
year. Professor Oliver has sent two or three short articles to the An-
alyst,* and has read, at the National Academy's meeting in Washing-
ton, a preliminary paper oi^he Sun's Eotation, which will appear in the
Astronomical Journal. Professor Jones and Mr. Hathaway have lith-
ographed a little Treatise on Projective Geometry. Mr. McMahon has
sent to the Analyst a note on circular points at infinity, and has also
sent to the Educational Times, London, solutions (with extensions) of
various problems. Other work by members of the department is likely
to appear during the summer, including a new edition of the Treatise
on Trigonometry."

As to the terms for admission to the university, in mathematics, the
requirements in 1869 were arithmetic and algebra to quadratic equa-
tions ; but plane geometry also was required for admission for the course
in arts. ^^ I judge from an old < announcement,' " says Professor Oliver,
*< that in 1868, when the university opened, some students were ad-
mitted with only arithmetic." In recent years the requirements have
been arithmetic, algebra through quadratics^ radicals, theory of expo-
nents, and plain geometry. In the engineering and architectural courses
solid geometry has been added.

In and after 1889, candidates will have two examinations, the ^< pri-
mary " and the " advanced." The " primary " examination will cover
the following subjects in mathematics :

In ArithmetiCy including the metric system of weights and measures; as much as

is contained in the larger text-books.

- - ^^--^~

* The name of the mathematical Jonmal in question is not AnalyBt, but AnnaU tf

182 « TEACHING AND HISTORY OP MATHEMATICS.

In Plane Geometry; as mnoh as is contaiDed in the first five books of Chanvenet's
Treatise on Elementary Geometry, or in the first five books of Wentworth's Elements
of Plane and Solid Geometry, or in the first six books of Newcomb's Elements of Ge-
ometry, or in the first six books of Hamblin Smith's Elements of Geometry.

In Algebra, through quadratic equations, and including radicals and the theory of
exponents; as much as is contained in the corresponding parts of the larger treatises
of Newcomb, Olney, Ray, Robinson, Todhonter, Wells, or Wentworth, or in those
parts of Oliver, Wait, and Jones's Treatise on Algebra that are indicated below, with
the corresponding examples at the ends of the several chapters : Chapters I, II, III ;
Chapter IV, except theorems 4, 5, 6 ; Chapter V, except $$ 3, 5, and notes 3, 4, of
problem 2 ; Chapter YII, $ 11 ; Chapter YIII, $( 1» 2> the first three pages of $ 8 and
i 9 ; Chapter XI, except $ 9, problem 9 of $ 12, and ($ 13, 17, 18.

For admisBioQ to the coarse leading to the degree of bachelor of arts,
no further knowledge of mathematics will be necessary, in any case.

For admission to the courses leading to the degrees of bachelor of
philosophy, bachelor of science, bachelor of letters; to the .coarse in
agriculture ; and (in and after 1890) for all optional students, there will
be required, in addition to the ^< primary ^ examination, an <^ advanced "
eiamination in two advanced subjects, ^' one of which must be French
or German or mathematics." If the applicant chooses mathematics, he
will be examined on* all the Freshman mathematics, namely, solid geom-
etry and elementary conic sections, as much as is contained in New*
comb's Elements of Geometry ; advanced algebra, as much as is con-
tained in those parts of Oliver, Wait, and Jones's Treatise on Algebra
that are read at the university (a list is sent on application to the Beg.
istrar) ; and trigonometry, plane and spherical, as much as is contained
in the unstarred portions of Oliver, Wait, and Jones's Treatise on
Trigonometry.

It was the desire of Mr. Oomell and President White to establish a
university giving broad and general training, in distinction to the nar-
row, old-fashioned college course with a single combination of studies.
The idea was well expressed by Oomell when he said that he trusted
the foundation had been laid to ^' an institution where any person can
find instruction in any study." We shall proceed to give the course of
study in mathematics, and let the reader judge for himself whether or
not the idea of the founder has been carried out in the mathematical
department.

We begin with studies which have been required for graduation. The
mathematical course has always included, for all candidates for bacca-
laureate degrees except (at one time) a few natural history and analytic
chemistry students, one term each of solid geometry, advanced algebra,
and trigonometry (either plane, or plane and spherical). At one time
students in history and political science had onetermof theory of prob-
abilities and statics instead of spherical trigonometry. There have also
always been required in all engineering courses and in architecture,
analytic geometry and calculus ; and, sometimes, analytic geometry in
certain other courses, as those in science and philosophy. At present
the amount required is one term of analytic geometry and one term of

INFLUX OP FRENCH MATHEWTATICS. 183

caJculus, in the course of architecture, and one term of analytic geom-
etry and two terms of calculus in the engineering course. At pres-
ent, the students in mechanical and -electric engineering take also an
extra term in projective geometry and theory of equations in the Fresh-
man year. These are the "required'' mathematics in the different
courses.

In addition to these,i" elective ''mathematics has always been offered
by the university to upper classmen, and also, of late years, to Freshmen
and Sophomores. The number of these elective courses has gradually
increased, till now they are as follows (Register 1888-89) :

ELECTIVE WORK.*

[Any course not desired at the "beginning of the fall term by at least three students,
properly prepared, may not be given.]

11. Problems in Geometry, Algebra and Trigonometry, supplementary to the pre-
scribed work in those subjects, two hours a week. Professor Jones.

12. Advanced work in Algebra, including Determinants and the Theory of Equa-
tions, two hours a week. Professor Wait.

13. Advanced work in Trigonometry, one hour a week. Professor Wait.

[The equivalents of courses 8,^12, and 13 are necessary, and course 11 is useful, as
a preparation for most of the courses that follow. ]

14. Advanced work in Analytic Geometry of two and three Dimensions, viz :—
(a) First year, Lines and Surfaces of First and Second Orders. 3 hours. Professor

Jones.

(6) Second year. General Theory of Algebraic Curves and Surfaces. 3 hours. Pro-
fessor Oliver.

15. Modern Synthetic Geometry, including Projective Geometry. 2 hours. Pro-
fessor Jones.

16. Descriptive and Physical Astronomy. 3 hours. Mr. Studley.

17. The Teaching of Mathematics. Seminary work. 1 hour. Professor Oliver,
and most of the teachers in the Department.

18. (a) Mathematical Essays and Theses: (&) Seminary for discussion of results of
students' investigations. Professoj Oliver.

19. Advanced work in Differential and Integral Calculus. 3 hours. Mr. Fisher.

20. Qnantics, with Applications to Geometry. Requires courses 8, 12, 14 (a), and
preferably also 11, 13, 19. May he simultaneous with 14 (5). 3 hours. Mr. Mc-
Mahon.

21. Differential Equations : to follow course 19. 3 hours. Mr. Hathaway.

22. Theory of Functions. Requires course 19, and preferably 21. (a) First year,
3 hours. (6) Second year, 2 hours. Professor Oliver.

23. Celestial Mechanics. 3 hours. Professor Oliver.

•

25. Finite Differences. 2 hours. Professor Oliver.

27. Rational Dynamics. Professor Wait.

28, Molecular Dynamics ; oVf 29, Theory of Numbers. 3 hours. Mr. XIathaway.

30. (a) Vector Analysis; or, (fc) Hyper-Geometry; or, (o) Matrices and Multiple
Algebra. 2 hours. Professor Oliver.

31. Theory of Probabilities and of Distribution of Errors, including some sociologic
applications. 2 hours. Professor Oliver, or Professor Jones.

41. Mathematical Optics, including Wave Theory and Geometric Optics. 2 hours.

Professor Oliver.

^ —

* Numbers 1 to 10, inclusive, refer in the catalogue to required studies in mathe-
matics.

184 TEACHING AND HISTORY OP MATHEMATICS.

48. Mathematical Theory of Souud. 3 hours. Mr. McMahon^

44. Mathematical Theory of Electricity and Magnetism. Professors Oliysb and

Wait.

In most of the aboTe branches of pure mathematics, an additional year's instmc-

tion, 1 or 2 hoars per week, may he given if desired.

For several years (from 1874 to 1887, we believe) there has been also
a " course in matbematics,'^ with a fixed curricalum, leading to the
degree of "bachelor of science in mathematics/' but it was dropped when
the numerous prescribed curricula and resulting degrees were cfonsoli-
dated into a few ** general courses," of which the work is mainly pre-
scribed in the first two years and mainly elective in the last two, and a
few *" technical courses," whose work is mainly iirescribed thronghont.
That old "course in mathematics" comprisedsome language and caltnre
studies, botany, geology, logic, English literature, descriptive geom-
etry, analytical mechanics, lectures and laboratory work in physics,
while, perhaps, two-fifths of all the student's time was given to pure
mathematics, including analytical geometry, calculus, difierential equa-
tions, finite differences, quaternions, imaginaries, mathematical essays,
seminary-work, etc. The object of this course was to give the best
equipment to students intending to become teachers of mathematics,
professors, and investigators. The students in this course were few,
but earnest, and some of them have since been making their mark as
teachers and investigators.

As to the mathematical text-books which have been used at different
times, we make the following statement :

In Elementary Geometry, Loomis till about 1873; since then, Ghau-
venet.

In Elementary Geometric Conies, Loomis, thenPeck, though the pres-
ent professors " don't much like either."

In Modern Synthetic Geometry, Professor Evans used no book, but
gave lectures. The same has sometimes been done since. At other
times, Cremona's Oeo7n6trie Projective, or the recent English translation,
was used. But now a little lithographed treatise on Projective Geome-
try, written for the purpose by Jones and Hathaway, is being used.
Professor Oliver has taught, also, Casey's Sequel to Euclid, and, once,
Steiner's Conies.

In Algebra, first Loomis, then Davies' Bourdon, Olney, Wells, New-
comb, and now Oliver, Wait, and Jones's, Todhunter's, Bumside and
Panton's Theory of Equations.

In Determinants, Muir, Dostor, Hanus, and lectures.

In Quantics, Salmon's Higher Algebra.

In Trigonometry, first Loomis's (including a little of mensuration,
surveying, and navigation), then Greenleaf's, Chauvenet's, Wheeler's,
and now Oliver, Wait, and Jones's, Todhunter's.

In Analytic Geometry, first Loomis (for two dimensions) and Davies
(for three dimensions), also Church, then Peck, Todhnnter, Aldis, and
now Smith (English work) with the three dimensions by lecture. With

INFLUX OF FBENCH MATHEICATICS. 185

more advanced stadents have been used also Salmon's Oonie Sections,
Higher Plane Curves, and Analytic Geometry of Three Dimensions.

In Differential and Integral Galcalas, first Loomis and Church, then
Peck, Todhunter, Williamson, Taylor, Meunier-Joannet, Homersham
Cox, Woolhouse, Smyth, Byerly ; and now Taylor for the few students
in the one*term course, the abridged Eice & Johnson's Differential and
Bice's Integral Calculus (one term each) for the two-term course for
engineers, and Williamson and Todhunter for advanced work, with
Bertrand for occasional reference and special work.

In Imaginaries, Argand was used, but now preference is given to
Chapter X of Oliver, Wait, and Jones's Algebra.

In Equipollences, Belavitis was once used.

In Quaternions, Kelland, Tait, Hardy, Hamilton's Lectures, Hamil-
ton's Elements.

In Theory of Functions there have been used Laurent^s Fonotions
MliptiqueSj Hermite's Cours cP Analyse ; and now Briot and Bouquet's
ThSorie des Fiynctions Flliptiques and Halphen's Traite des Fonctions
FlUptiqiies,

In Theory of Numbers, Dedekind's edition of Lejeune Dirichlet's
Zahlentheorie has been used recently.

In Least Squares^ Merriman.

In Diffierential Equations, Boole, Forsyth.

In Finite Differences, Boole.

In Descriptive Astronomy, Loomis, Newcomb, and Holden, with
Young's ^ The Son" and Chauvenet (for eclipses) for collateral reading.

In Mechanics, Duhamel^s MScanique Analytiquey and now Minchin's
Analytical Statics and Williamson's Analytical Dynamics.

Quaternions have not been taught now for several years, because the
professors are convinced that the benefit of that study is with most
students better gotten with a mixed course in matrices, vector addition
and subtraction, imaginaries^ and theory of functions.

Among the fundamental ideas of President White, in organizing the
university, was a close union of liberal and practical education. There
have, therefore, from the beginning, existed separate departments of
civil engineering, of mechanic arts, and of physics, each with a sepa-
rate professor at its head. Astronomy is taught partly in the depart-
ment of civil engineering and partly in that of mathematics.

Pupils in mathematics are always encouraged to do original work,
but it is only by older and maturer students that researches are made
which are of sufficient value to merit publication. The writer has
before him two printed theses, written to secure the degree of doctor of
philosophy at Cornell University. One is by C. E. Linthicum, "On the
Bectification of Certain Curves, and on Certain Series Involved" (Balti-
more, 1888) ; the other is by RoMin A. Harris, on "The Theory of Im-
ages in the Bepresentation of Functions" (Annals of Mathemetics,
June, 1888). Both of these are very creditable to the writers and to
the university, and the latter appears to us to fill a gap.

186 TEACHING AND HISTORY OF MAT^EMATICS.

There are always some under-gradaate students who do good work
in the more advanced mathematical electives, but at present it is by
resident graduates of Oornell and other colleges that the best advanced
work is expected to be done. Great efforts have been and are being
made to secure the attendance of graduate students in advanced courses
in mathematics. During the year 1885-86 eleven graduate students
were engaged in the study of the higher mathematics. The number for
the year following is not known to the writer, but the president's re-
port indicates that the attendance on advanced courses in mathematics
was increasing, and that about one-fifth of the graduate students were
taking their chief work in mathematics. In the last report Professor
Oliver says :

<< During 1887-88 eleven graduate students have taken more or less
of their work with us. Allowing for such as were partly in other de-
partments or remained but part of the year, we fi.ud that the mathe-
matical department has had about one-seventh of all the graduate work
in the university. This would seem to be our full share of this desirable
kind of teaching, when it is considered that the higher mathematics is
difScult, abstract, and hard to popularize ; that of course we can not
attract students to it by laboratories and large collections (except of
books), nor by the prospect of lucrative industrial applications ; and
that our department's whole teaching force, composed of only about
one-eleventh of all the active resident professors and instructors in the
university, and including only one-thirteenth of the resident professors,
has to do about one-ninth oi all the teaching in the university."

We are sure that many, perhaps all of our professors of mathematics
will see in the following remarks by Professor Oliver the reflection of
their own experience as teachers : %

<^ We have always had to contend with one other serious difficulty.
There is a wide-spread notion that mathematics is mainly important for
the preliminary training of certain crude powers, and as auxiliary to
certain bread-winning professions, and that only literary studies can
afford that fine culture which the best minds seek for its own sake.
Time, no doubt, will rectify this misapprehension ; but meanwhile it
binders our success."

The methods of teaching mathematics at Cornell are various. The pro-
fessors sometimes lecture, especially when there is no suitable text- book
at hand. This method, when a rather full syllabus is given out before-
hand, and plenty of p]x>blems are assigned to the students for solution,
has sometimes proved very successful. The lecturer perhaps calls upon
the class for suggestions as he proceeds with his topic, and then assigns
to them for home study some problem very much alike in principle to
the one they have just been discussing together.

But oftener it is preferred to base the teaching upon a book that the stu-
dents can study for themselves, supplementing it by lectures and expla-
nations, and holding the class to recitations and examinations upon it
In all the work, and especially in that for advanced classes, the pupils

INFLUX OF FBEKCH MATHEMATICS. 187

are treated by the professor as fellow- stadents, and he avoids assnming
toward them the air of master and dictator. Independent thought is
constantly encouraged, even when this leads the students to criticise
the things they are being taught. Mere memory-work and rote-learn*
ing — still in vogue in many of our schools — ^is discouraged in every way
possible.

Some of the mathematical teachers at Oomell have been accustomed
to test their pupil's mastery of the subject by written examinations^
given in the midst of the term's work without warning, or on weekly
reviews. There is also a written examination at the close of each term ;
but students who have done their term's work with a certain degree of
excellence beyond what would be strictly requisite to ^' pass them up "
in the subject are often exempt from this examination.

Since 1874 mathematical clubs have existed at times. Different
members of it would give in turn the results of their mathematical
studies in lines a little outside of the regular work of the class-room,
and the matter thus presented was then open to discussion by the whole
company present. Professor Oliver has generally presided at these
meetings and taken his turn at presenting topics and work for discus*
sion. The attendance upon these clubs has generally been small, in-
cluding only the professors, instructors, and a few advanced students.
Sometimes the meetings would be kept up for a few months or a year
with a good deal of spirit, and then with change of membership the in-
terest would flag, and the club would be discontinued for a while.
Much of the work presented was the work of immature students and
not worth publishing. But these clubs have helped to keep up an in-
terest in mathematics and to stimulate the spirit of originality.

For the past three or four years the club has been merged into a
^< seminary" for the discussion of aims and methods in teaching math-
ematics. Here the professor proposes such problems as these : ^^Why
do we teach mathematics at all, and what practical rules does this sug-
gest to us in order that our teaching may be most effective and useful
toward the end proposed f" ^' What is the place of memory in math-
ematical teaching!" '* What are the relative advantages of lecturing
and book work, and how are they best combined?" <^ How can we best
teach geometry!" '^What is the nature of axioms in geometry, and
how modified when we consider the possibility of non Euclidian space t"
The professor proposes some such problem, then calls for discussion and
adds his own views. If possible he develops on the blackboard a sylla-
bus or tabular view of the different heads under which the theory must
fall* Then these are discussed in order, either at that or at subsequent
meetings. In the latter case the discussions are often opened by es-
says from members of the seminary. This method of conducting the
seminary is most fruitful of results, espficially if we remember that the
chief object of the graduate department in mathematics is to train
teachers of this science. The coming teacher will acquire possession of
better methods and higher ideals of mathematical teaching.

188 TEACHING AND HISTORY OP MATHEMATICS.

VIEGINIA MILITARY INSTITUTE.

The Yirginia Military Institate at Lexington, Ya., is a State institu-
tion, and was organized in 1839 as a military ^nd scientific school. It
is a footer child of the U. S. Military Academy at West Point. At its
organization General Francis H. Smith was made its superintendent.
This position he has now held for half a century. What the Virginia
Military Institute has been and is, is due chiefly to his long and faithful
service as superintendent.

General Francis Henney Smith is a native of Virginia. He graduated
at West Point in 1833, and was assistant professor of mathematics there
during, the first two years after graduation. He then occupied the
chair of mathematics for two years at Hampden-Sidney College. At
the military institute he added to his duties as superintendent those of
professor of mathematics and moral philosophy.

Smith has published a number of mathematical text-books. Some of
his books have suffered from frequent typographical errors. In 1845
appeared his American Statistical Arithmetic^ in the preparation of
which he was aided by B. T. W. Duke, assistant professor of mathe-
matics at the institute. The book was called << Statistical Arithmetic,"
because the examples were selected as far as practicable from the most
prominent facts connected with the history, geography, and statistics
of our country. This novel idea made that arithmetic the medium for
communicating much important information and a better appreciation
of the greatness and resources of our country.

Other arithmetics appeared by the same author, which enjoyed quite
an e;(tensive circulation. About 1848 was published also a series of
algebras, as a part of the mathematical series of the Virginia Military
Institute.

A valuable contribution to the list of college text-books was the
translation, by Professor Smith, in 1840, of Biot's Analytical G^metry.
The original French work of Biot was for many years the text-book for
the U. S. Military Academy at West Point. When, about ten years
previous, Professor Fartar prepared his Cambridge mathematics, he
chose Bezout's work on the '^ application of algebra to geometry," in
preference to the works of Lacroix and Biot, for the reason that these
works were thought to be too advanced for our American colleges,
which had up to that time paid no attention whatever to analytical
geometry. Bezout's work can hardly be called an analytical geometry.
The only works on this subject which were published in this country
after the Oambridge mathematics and previous to Smith's Biot were
the elementary treatise of J. B. Young (which followed Bourdon as a
model) and the work from the pen of Professor Davies, of West Point
Smith's translation of Biot reached a second edition in 1846. After-
ward the book was revised. An edition of it appeared in 1870.

In 1867 Smith published an edition of Legendre'a Geometry. Bdi*

INFLUX OF FBBKCH MATHEMATICS. 189

tions of this work had appeared in this country by Farrar and Davies.
Smith's translation was from a later French edition, which contained
additions and modifications by M. A. Blanchet, an 6\hve of the £cole
Folytechniqne.

In 1868 appeared from the pen of General Smith a Descriptive Oeoni-
etry. The study of this subject had been introduced in the institute at
a time when it was hardly known by name in other schools and colleges
of Virginia.

The organization of the Virginia Military Institute and the methods
of teaching have been much the same as at the U. S. Military Academy.
Indeed, the institute is frequently called the " Southern West Point.'^
The division of classes into sections and the rigid and extended appli-
cation of the '^ marking system '' have been adopted from West Point.
The marking system seems to have originated in France, and to have
been introduced into this country by West Point.

The relative weight given to the different subjects of instruction
forming the general merit-roll of each class is, according to the Official
Eegister of 1887-88, as follows :

.12. Sarveying 1

13. Moral and political philosophy .. 1

14. Ordnance and gannery..... 1

15. Drawing 1

16. Geography X

17. Infantry tactics 0.5

18. Geology... 0.5

19. Descriptive geometry 1

20. Logic 0.5

21. Rhetoric 0.5

22. Latin 1.5

1. Mathematics (grade) 3

2. Ciyil engineering 3

3. Military engineering 1

4. Chemistry.. 2

5. Mechanics 2

6. French 1

7. German 1.5

8. English 1

9. Physics 1.5

10. Mineralogy 1

11. Astronomy 1

The success of the educational work of the school turns largely upon
the method of dividing classes into sections, whereby the students are
accurately graded according to scholarship, and each secures a propor-
tionately large share of the personal attention of the instructor. Each
section is '< under the command of a ^ section-marcher,' taken from the
first cadet on the section-roll. The sections are formed on parade, at
the appointed hours ; the roll is called by the section-marcher, absen-
tees are reported to the officer of the day, whose duty it is to order all
not properly excused to the class duty. The section-marcher then
marches his section to the class-room, reports the absentees to the pro-
fessor, and then transfers to him the responsibility which he had thus
far borne. The professor examines the section on the appointed lesson,
is responsible for the efficiency of his instruction, and once a week makes
an official report to the superintendent of the progress of his section.
These reports are duly recorded, and constitute an important element
in the standing of each cadet at his semi-annual or general examina-
tions.'' • ^

* The Inner Life of the Virginia Military Institute Cadet, by Francis H. Smith,
LL. D., 1878, p. 22.

190 TEACHING AND HISTORY OP MATHEMATICS.

As at West Point, so at this institution, a candidate for admission 1ia9
been required to know no other mathematical study than arithmetic.
*^ The four ground rules of arithmetic, vulgar and decimal fractions, and
the rule of three" admitted a candidate, as far as mathematics is con-
cerned.

The course of study has been the same as at West Point, but the
books used have not always been the same. The books used at the
beginning were as follows : Bourdon's Algebra, Legendre's Geometry,
Boucharlat's Analytical Geometry (in French), Boucharlat's Differential
and Integral Calculus (in French), Da vies' Descriptive Geometry.
These were, later, displaced by other books, chiefly Smith's own works,
viz, Smith's Algebra, Smith's Descriptive Geometry (after De Fouroy),
Smith's Legendre's Geometry, Smith's Biot's Analytical Geometry,
Gourtenay's Differential and Integral Calculus, Buckingham's Calculus.
As at West Point, so here, there have been no elective studies.

During the first twenty years of its existence the Virginia Military
Institute was flourishing. It '^ had just placed itself before the public
as a general school of applied science for the development of agricultural,
mineral, commercial, manufacturing, and internal improvement interests
of the State and country when the army of General Hunter destroyed its
stately buildings and consigned to the flames its library of ten thousand
volumes, the philosophical apparatus used for ten years by * Stonewall'
Jackson, and all its chemicals. The cadets were then transferred to
Biehmond, and the institution was continued in vigorous operation
until the evacuation of Biehmond on the 3d of April, 1865."*

The War left sad traces on the institution, besides the destruction of
its buildings, library, and apparatus. Three Of its professors had been
slain in battle : Stonewall Jackson, who had been professor of natural
and experimental philosophy since 1850 ; Maj. Gen. B. E. Bodes, a
graduate of the institute, and, in 1860, appointed professor of civil and
military engineering ; Col. S. Crutch field, also a graduate of the insti-
tute, and, since 1858, professor of mathematics. Among the slain were
also two assistant professors and two hundred of its alumni.

Kotwithstanding the impoverishment of the people immediately after
the War, it was decided in 1865 to re-open the institution. Without one
dollar at command to offer by way of salary to the professors, the board
of visitors called back all who survived, and filled the vacancies of those
who had died. Work was begun with earnestness. On the 18th of Oc-
tober, 1865, the day designated for the resumption of academic duties,
sixteen cadets responded. At the end of the academic year the num*
ber of cadets was 65. Such vitality under such discouragements
prompted the legislature to restore the annuity the next winter. It was
not very long before the Virginia 5iilitary Institute was restored to all
its former lustre. In 1870 the buildings of the institute were restored
and equipped with laboratories and instruments.

• Official Begiflter, 1S87-68,

INFLUX OF FRENCH MATHEMATICS. 191

The Official Register for 1887-88 gives 11 9 cadets in the academic school.
The stadies in mathematics for t^at year are as follows : Fourth class —
First year: Smith's Algebra, Davies' Legendre's Geometry and Trigo-
nometry (revised by Van Amringe), Exercises. (Recitations from 8 to
11 daily.) Third class— Second year: Smith's Biot's Analytical Geome-
try, Buckingham's Diiierential and Integral Galcalus. (Recitations f^om
9 to 11 daily.) Second class — Third year : Mahan- Wheeler, Davies' Sur-
veying (Van Amringe), Gillespie's Surveying, field work. (Recitation
from 10 to 11.) First class — Fourth year; Rankine's Applied Mechanics
and Rankine's Civil Engineering, lectures, and field practice.

UNIVERSITY OF VIBaiNIA.

President Jefferson devoted the golden evening of his life to the
founding and building up of the University of Virginia as a nursery for
the youth of his much-loved State. This greatest university of the
South has from its beginning had features peculiar to itself. The entire
abandonment of the class system, and the course arrangement of its
studies, are its most prominent distinguishing features. From the very
beginning the method of instruction has been by lectures and examina-
tions. " Text-books are by no means discarded, but the professor is
expected to enlarge, explain, and supplement the text. Every lecture
is preceded b}" an oral examination of the class on the preceding lecture
and the corresponding text. This method stimulates the professor to
greater efforts, and excites and maintains the interest and attention of
the students a hundred fold."*

The university was opened for students in March, 1825. It then had
eight distinct schools, but at the present time it has nineteen, ^' each
affording an independent course under a professor, who alone is respon-
sible for the system and methods pursued." One of the eight original
schools was that of mathematics, pure and applied. The first profes-
sor of mathematics (from 1825 to 1827) was Thomas Hewett Key, of
England. He was a graduate of Trinity College, Cambridge. Besides
his ability as a mathematician, he possessed great classical and general
attainments. He resigned his position in order to accept the professor-
ship of Latin in the London University.

His successor was Charles Bonnycastle, of England, who, upon Mr.
Key's resignation, was transferred from the chair of natural philosophy
to that of mathematics, which he continued to fill until his death, in
1840. He was the son of John Bonnycastle, who was widely known in
England and America for his mathematical text-books, and was edu-
cated at the Royal Military Academy at Woolwich, where his father
was professor. His father's books exhibit those faults which were com-
mon to English works on mathematics in his day. It is fair to presume,

* Dr. Gessner HaiTison, in Dayokinok's Cy clopsBdia of AmeuQan Literature ; Artiolei
*• University of Virginia."

192 TEACHING AND HIST0B7 OF MATHEHATI08.

however, that Oharles belonged to that coterie of English mathemati-
cians of which Hersohel, Fe&cock, Whe well, •and others were members,
and which introdaced the Leibnitzian notation and also the ratio defini-
tion of the trigonometric functions into Cambridge. At the University
of Virginia he enjoyed the reputation of a man of great ability in math-
ematics and of broad general knowledge. His lighter writings indicate
that he could have shone also in the fields of literature. We are happy
in being able to quote the following, from Dr. James L. Cabell, profes-
sor of physiology and surgery at the University of Virginia:*

<^ Though apparently an earnest and enthusiastic student of the higher
mathematics, it was the constant habit of Professor Bonnycastle to make
extensive and varied excursions into other fields of study, such as his-
tory, metaphysical philosophy, and general literature. I remember to
have seen in his private library after his death several volumes of
works on moral philosophy with copious marginal notes written by him.
I recall in this connection the fact that he used to speak with emphasis
and some indignation on the absurd charge that the study of mathe-
matics tends to render its votaries insensible to the force of probable
evidence, and that when strict mathematical investigation cannot be
had, persons whose mental discipline has been secured by such training
become either obstinately skeptical or wildly credulous. He insisted
that all one-sided training had a natural tendency to narrow the intel-
lect and that this applied to all other branches of learning and all pro-
fessional pursuits as well as to mathematics. The obvious remedy lies
in a liberal and broad culture. It was doubtless with a view to enforce
his precepts by occasional examples that he was in the habit of deliver-
ing at the opening of each session of the university a popular lecture,
the topics of which, having apparently a very remote connection with
mathematical studies, were actually suggested by some recent publica-
tions in the department of general literature. These addresses were
greatly admired by the crowds of young men who attended them, includ-
ing, in addition to his own class, representatives from all the other de-
partments of the university. He was also a contributor to a literary
magazine published by the faculty in 1828-29. Some of his articles
were stories of more than ordinary merit in this class of literary pro-
ductions, and would probably have made his fortune if such magazines
as Harper's, Scribner's, etc., had existed at that day with a competent
development of public taste.

*'The only distinct impression which I can now recall as to Professor
Bonnycastle's method of teaching has reference to his attempts to in-
doctrinate his pupils at every stage of their studies with the philosophy
and essential principles of the subject under consideration. At that
time most, if not all, the usual text- books and all the school teachers
gave only rules which the student was to apply. So far as the students
knew, these rules might be wholly arbitrary. Professor Bonnycastle

* Letter to the writer^ January 4, 1889.

INFLUX OF FRENCH MATHEMATICS. 193

injsisted on the necessity of placing the student in a position to recog-
nize the trne significance of every principle laid down. This was done
by oral lectares characterized by remarkable lucidity of statement and
by a marvellous fertility of striking illustrations. These lectures were
fully appreciated by the better sort of students in the advanced classes,
but were thought by most of us to be thrown away upon the younger
and less ambitious members of the lower clasaes. The general verdict
of all classes of hearers ascribed to Mr. Bonnycastle genius and attain-
ments of the highest and most varied character.'' *

The text-books used by Bonnycastle in pure mathematics, in connec-
tion with his lectures, were the Arith^letic, Algebra, and Hifferential
Calculus of Lacroix, the first two in Farrar's translation. The th,eory
of the integral calculus was taken from YouVig, the examples firom Pea-
cock's Collection. In geometry he used his own work on Inductive
Geometry (1834).

In pure mathematics there were in his time three classes : the ^^ First
Junior," " Second Junior," and *♦ Senior." " Of these the First Junior
begins with arithmetic; but as the student is required to have some
knowledge of this subject When he enters the university, the lectures
of the professor are limited to the theory, showing the method of nam-
ing numbers, the difierent scales of notation, and the derivation of the
rules of arithmetic from the primary notion of addition ; the addition,
namely, of sensible objects one by one. The ideas thus acquired are
appealed to at every subsequent step, and much pains are taken to
exhibit the gradual development from elementary truths of the ex-
tensive science of mathematical analysis." (Catalogue for 1836.) After
a thorough course in arithmetic students were well prepared for alge-
bra. In teaching the rules for adding and subtracting, etc., they were
compared with the corresponding rules in arithmetic, and the agree-
ment and diversity were noticed and explained. The elements of geom-
etry were taught and illustrated by models. The book on Inductive
Geometry was prepared especially for the use of his students. It in-
cludes geometry, trigonometry, and analytical geometry. In the defi-
nition of the trigonometric functions the ratio system is used. ^' The
chief result which the author hoped to secure by the proposed innova-
tion was such an arrangement of the subject as would enable him to

^ ■ ' ™ ■ ^ I !■ iM ■■ mm^^^^^ ■■ MM ■ 11 ■ I I ^^.1 ■ I I II ■ ■ ■■■■■■■■ ^w^— ^1^— ^^ ■, I ^^M^— I I !■ I ■ ■ ^ - I ■ »■■ ^ ■■^—a — ^— — ^— ^i^^w^^^^^M^^^^^i^^— ^^^^^p^i^i^^^

* In another part of his letter Dr. Cabell says: ''I felt bound to tell you tbat
owing to my complete want of mathematical knowledge, even to the extent of igno*
ranee of the terminology of the science, I was utterly incompetent to form a critical
jndgment of Professor Bonnycastle's method of teaching. I can, however, recaU with
some vividness the impression made npon me at the time when he caused me and my
fellow-stndents to understand the significance of processes which we had previously
applied in a purely arbitrary method. It is probable that we exaggerated the merit
of our new professor by contrasting him with the very imperfect and defective stand-
ards of the common schools of Virginia at that day. I believe, however, that these
defects were common to the whole country when Professor Bonnycastle introduced a
reform which in a few years may have become general."

881—No. 3 13

194 TEAOHING AND HISTORT OF MATHEMATICS.

dispense with the distinctions hitherto made between the different
branohes of geometry, and thus permit him to' treat the problems em-
braced auder the heads of synthetic geometry, analytic geometry, and
the two trigonometries, as composing one uniform doctrine, the science
of Quantity and Position.''*

The general plan appears to be a good one, in the main. Bat its eze>
cation is not satisfactory. The work covers 631 crowded pages. The
form in which the subject is presented is bad. Theorems and their
demonstrations are in the same kind of type, and the eye finds nothing
to assist and relieve it in passing over the crowded pages of prolix ex-
planation# 'Sot is the reasoning always good.t

His Inductive Geometry is, we believe, the only mathematical work
whicK he published while he was professor at the University of Vir-
ginia.

Both algebra and geometry were begun in the '< First Junior" class
(catalogue 1836), and then continued in the <' Second Junior '' class. Oal-
cuius was begun in this class and then completed in the ^^ Senior'^ class.

The notation of Leibnitz was used at the University of Virginia from
the very beginning.

In the Virginia Literary Museum, a weekly journal issued in 1829
by the professors of the university, we read of an examination of the

* Preface to Indactiye Oeometrj.

t '^Angles are so evidently portions of space sarroanding their vertex, and this space
80 manifestly the same in all cases, that we are forced to regard it, directly or indi-
rectly, as the standard to which all angles should be referred " (p. 112). The reason-
ing by which the sum of the three angles of a triangle is shown to be two light

•Dglas, is AS follows (p. 123) : '* The lines AB^CD^CE, * * * that enclose a small
triangle at C, are separated by the openings a, (, o, that are nearly equal to the angles
of the triangle ; two of these openings, namely, a and o, are identical with angles of
the triangle, and the third, (, which forms a space indefinitely extended, diflfers 6om
the opening we call the angle C merely by the small space included in the triangle.
" This last, by bringing the triangle nearer to C, may be rendered as small aa we
please ; and thus a triangle can always be assigned whose angles shall differ from a,
\f Of and, consequently, the sum of whose angles shall differ £rom two right angles
by less than any assignable quantity. Some difference between the results appearsy
it is trae, always to remain ; but if we examine more attentively the idea tJiat we
Me able to form of infinite space, we shall find the difference in qnestion merely ap-
parent, and shall perceive the sum of the three angles to be rigidly equal to two i^ht
angles.'' This reasoning is bad. It involveef umecessanlyi the eonsideiation of in*
finite epaces.

OTPLUX OF FRENCH MATHEMATICS, 196

Senior elass in mathematicB, on Thursday, July 16, 1829 : << Tbe mem-
bers of the class were examined in application of algebra to geometry
and the theory of carves, as contained in the lY chapter of Laoroix's
TraiU Au Oalcul JHff^rentiel et du Oalcul InUgraL In the differen-
tial and integral calculus they were examined by examples taken from
the questions on these subjects published by Peacock & HerscbeL
The class have studied the differential calculus chiefly from the treat-
ise of Boucharlat, and the integral from Boucharlat, Laoroix, and the
examples before mentioned. They have proceeded to the integration
of partial differential equations of three or more variables, and the ques-
tions proposed were chosen to this extent." These extracts show that
the course of mathematics taught by Professor Bonnycastle was remark-
ably far advanced, compared with the work done in the ordinary college
or university in this country at that time.

Besides the three classes above given there was from the beginning
a class in mixed mathematics (really a graduate class). Under Bonny-
castle the text-books in this study were Yenturali's Mechanics and the
first book of Laplace's Mieanique CSleste. The principles were applied
to various problems. A separate diploma has been given to students
completing this course of mixed mathematics.

Professor Bonnycastle left a large number of mathematical MSS. in
the keeping of Professor Henry, of the Smithsonian Institution, who a
short time before his death sent them to be deposited in the library of
the University of Virginia.

After the death of Bonnycastle, Pike Powers, now a minister at Bich-
mond, held the chair until J.J. Sylvester was elected professor, in 1841,
Mr. Powers was a young mathema,tician of fine gifts and attainments,
and a pupil of Bonnycastle. Professor Sylvester was then already gener*
ally recognized as a man of brilliant genius and profound mathemati-
cal learning. He resigned in about half a year, and afterward ac-
cepted a professorship in the Boyal Military Academy at Woolwich.
We sh^U have to say more about him in connection with the Johns
Hopkins University. Prof. Pike Powers was again appointed, tempo-
rarily, to teach the mathematics.

The next possessor of the mathematical chair was Edward H. Gourt-
enay, from 1842 to 1853. He was the first regular occupant of this chair
who was educated in this country. He was born in Baltimore, in
1803. After having been examined for admission to the U. S. Military
Academy at West Point, in 1818, the examiner remarked : ^^ A boy from
Baltimore, of spare frame, light complexion, and light hair, would cer-
tainly take the first place in his class." Courtenay completed the four
years' course in three years, and graduated at the head of bis class in
1821. From that time till 1834 he was connected as teacher with tbe
Military Academy, excepting the period from 1824 to 1828. After leav-
ing West Point he was for two years professor of mathematics at the
University of Pennsylvania, then he became division engineer for the

196 TEACHINa AND HISTORY OF MATHEMATICS.

New York and Erie Eailroad. He was employed by the United States
Government as civil engineer in the construction of Fort Independence,
Boston Harbor, from 1837 to 1841. Just before his appointment to the
professorship at the University of Virginia he was chief engineer of
dry-dock, Xavy-Tard, Brooklyn, N", ¥.•

Mr. Gourtenay was a mathematician of noble gifts and a great teacher.
^' His mind was quick, clear, accurate, and discriminating in its appre-
hensions, rapid and certain in its reasoning processes, and far-reaching
and profound in its general views. It was admirably adapted both to
acquire and use knowledge.''! He was modest and unassuming in his
manner, even to diffidence. He would never utter a harsh word to
pupils or disparage their efforts. ^< His pleasant smile and kind voice,
when he would say, ^ Is that answer |>er/5?c% correct 1 ' gave hope to
many minds struggling with the difficulties of science, and have left
the impression of affectionate recollection on many hearts.") '

Eegarding his work at the University of Virginia, Professor Venable
(at one time a pupil of Gourtenay) says that his course in pure mathe-
matics was prepared and written out (or rather printed on white cloth
in large letters) with great care — following Bonnyoastle in the use of
Young in the treatment of the differential and integral calculus. His
course in this branch embraced differential equations and the calculus
of variations. His MSS. on these two subjects for the Senior class fill
nearly one hundred and fifty pages of his printed work. His notes on
the calculus were published in 1857, after his death, and became a
valued text-book in many institutions. *^ In its publication the plan,
language, and even the punctuation have been followed with a fidelity
due to the memory of a friend." The work was more extensive than
any which had yet appeared in this country on the same subject. Gourte-
nay added descriptive geometry to the regular course of pure mathe-
maticB. He prepared extensive notes for his class in mixed mathematics,
which embraced a full course in the applications of the calculus to
mechanics and to the planetary and lunar theories (perturbations).

In 1845 the course in the School of Mathematics was as follows ;
Junior class, theory of arithmetic, algebra, synthetic geometry ; Inter-
mediate class, plane and spherical trigonometry, land surveying, navi-
gation, descriptive geometry and its application to spherical projection,
shadows, perspective ; Senior class, analytical geometry, calculus. The
class in mixed mathematics studied selections from Poisson, Francoeur,
Pontecoulant, and others. This embraced the mathematical investiga-
tions of general laws of equilibrium and motion, both of solids and fluids.
The text-books for that year were, Lacroix's Arithmetic, Davies' Bour-
don, Legendre's Geometry, Davies' Surveying and Descriptive Geome-
try, Davies' Analytical Geometry, Young's Differential and Integral
Galculus.

* Courtenay'fl CaloaloB^ p. iv. t Ibid., p. v. t Ihid., p. viU

INFLUX OF FRENCH MATHEMATICS. 197

After the death of Goartenay the chair of mathematics was filled by
Albert Taylor Bledsoe. He was a native of Kentucky, and graduated
at West Point in 1830. He was one year adjunct professor of mathe-
matics and French at Kenyon OoUege, Ohio ; then one year professor
of mathematics at Miami University, Ohio. Afterward he practiced
law for eight years at Springfield, 111. Before his coming to the Uni-
versity of Virginia he was professor of mathematics and astronomy at
the University of Mississippi. He remained in his new position till
1863, then became assistant secretary of war in the Southern Confed-
eracy. After the War he became principal of a female academy in Bal- '
timore and editor of the Southern Beview. He died in 1877 at Alex*
andria, Ya, '

Prof. Francis H. Smith, of the University of Virginia, who was asso-
ciated with Bledsoe in the faculty of the institution, writes us about
him as follows : ^' He succeeded here an* eminent teacher, Prof. Edward
H. Courtenay ; SjUd, while the two men were most unlike in every respect,
Dr. Bledsoe's evident ability so impressed his class, that the prestige
of the mathematical class sufiered no loss in his hands. From his life-
long addiction to metaphysical studies, he entered with great zeal upon
the philosophy of mathematics, a subject which every infantile mathe-
matician is bound to have an attack of, but which in its widest rela-
tions may very well tax the powers of the most mature and advanced
geometer. In this field I think Dr. Bledsoe won a place by the side of
Bishop Berkeley and Auguste Comte. His treatise on the Philosophy of
Mathematics was put in print and had a considerable circulation. He
established a new course of lectures here in connection with the usual
mathematical curriculum, upon the History and Philosophy of Mathe-
matics. That feature survives to this day. As a manipulator of mathe-
matical formulsB and solver of mathematical problems, Dr. Bledsoe was
not strikingly able. I have known many men of far less strength who
were his superiors in mere algebraic dexterity. Yet, I was convinced
from several incidents which came to my knowledge during his teach-
ing here that had his life, after he left West Point, been devoted to the
science, he would have left the pure mathematics simplified in statement
and improved in form. His originality and force were^ obvious to me,
to whom he freely communicated his difficulties and successes, during
his entire residence here. I learned that while at the Military Academy
these traits were strikingly exhibited by his solving a problem in the
tangencies of circles which had up to that time baffled the geometrical
skill of the academy, and which had been left unsolved by Archimedes
himself. The solution given by Dr. Bledsoe was afterward published
in the Southern Eeview, of which the doctor was editor and proprietor
for a number of years before his death. He had in the latter years of his
life completed a treatise on synthetical geometry, of the Euclidian type,
and, I think, had found a publisher, but whether it evex" got printed I
am not aware. Dr. Bledsoe's greatest work was in the field of meta-
physical theology, constitutional law, and review articles. ^

198 o'EAGHma akd histoby of mathematics.

Hift Philosophy of MathematioS; published in 1867, exhibits brilliant
controversial powers. It initiated a reactionary movement among as
against the nnphilosophical exposition of the calcnlns in the colleges of
oar land. The book is somewhat verbose in its style. The bnlk of it
consists of criticisms of varions text-books. Comparatively little space
is given to what the anthor considers to be the true explanation of the
subject. It seems to us that the criticisms which he makes are generally
good and well founded, but that he fails in proposing a sound substitute
for the explanations which he rejects. The influence of the book has
been beneficial in so far as it has caused many teachers to meditate
upon the philosophy of the calculus.

He gave lectures also on the history of mathematics — a subject which
received little or no attention in our colleges at that time. He prepared,
but never published, a work on analytical geometry, in which, by the
discussion of one equation which contained, wrapped up within itself,
the whole folio of Apollonius on conic sections, he developed the prop-
erties of the circle, ellipse^ hyperbola, and parabola.**"

Bledsoe pursued, In the main, the course in pure mathematics laid
down by his predecessor, except that Oourtenay's Galculus was used in
place of Young's. For the class in mixed mathematics he used (in 1864)
Bartlett's Analytical Mechanics, Newton's Principia, and Pratt's Me-
chanical Philosophy. Pontecoulant^s Systdme du Monde was also used
by him for his class.

Professor Bledsoe was not very strict with students in their daily
work, but on approach of examination day he knew how to prepare a
tough set of questions.

By temporary appointment, Alexander L. Nelson taught mathematics
during part of the session 1853-54 ; Robert T. Massie during part of
the session 1861-62, and Francis H. Smith, of the School of Natural
Philosophy, from 1863 to 1865.

Daring the War the university barely subsisted ; but scarcely was
peace restored ere the institution, amidst perplexing pecuniary embar-
rassments, prepared with resolute energy to enlarge its capacity for
useful work by multiplying its schools. In 1867 the School of Applied
Mathematics with reference to Engineering was established.

In 1865 Oharles S. Yenable was appointed to the chair of mathe-
matics, a position which he still occupies. He is a native of Virginia,
and was born in 1827. After graduating at Hampden-Sidney College
in 1842, he remained one year at the college as a resident graduate^ pur-
suing mathematics under Col. B. L. E well (a West Point graduate, and
afterward president of William and Mary Oollege), and English liter-
ature and history under Maxwell. He then became tutor in mathe-
matics, in which capacity he continued two years, devoting part of his
time to the study of law. In 1845, he went to the University of Vir-
ginia and sperjt one session in the study of law, mathematics^ and Ian-

* Bledsoe'A Philtwophy of MftthematioSi p. 130.

INFLUX OF FRENCH MATHEMATICS. 199

gaages. Here he took the mathematiQal lectures of Professor Oonrte-
nay. He was then elected professor of mathematics at Hampden-Sid*
ney College, to sncceed Ewell. After remaining there one year he
obtained leave of absence, returned to the University of Yirginia, and
studied mixed mathematics and engineering under Professor Oourteuay.
He returned to Hampden-Sidney in 1848, and filled the chair of mathe-
matics till June^ 1852. He then obtained leave of absence again, and
visited Germany for the farther prosecution of studies. In Berlin he
studied astronomy under Encke, and mathematics with Dirichlet and
Borchardt. He then went to Bonn, studying some months under Pro*
fessor Argelander, the director of the observatory of Bonn. While in
Germany astronomy was his chief branch of study. He then travelled
in Southern Europe, studied for some time in Paris, visited England,
and then returned to Hampden-Sidney Oollege, in 1853. In 1856 he
was elected to the chair of natural philosophy and chemistry at the
University of Georgia, to succeed John Le Conte, and in 1857, profes-
sor of mathematics and astronomy in the South Carolina GoUege. In
1858 he published an edition of Bourdon-s Arithmetic. Yenable took
part in the attack upon Fort Sumter, and took active part in the War
until its close.* Since his connection with the University of YirginiCb,
Professor Yenable has issued a series of text-books, consisting of First
Lessons in Numbers, 1866, revised in 1870; Mental Arithmetic, 1866 ;
Practical Arithmetic, 1867, revised in 1871 ; Intermediate Arithmetic,
1872; Elements of Algebra, 1869; Elements of Geometry, 1875; Notes
on (analytical) Solid Geometry.

These rank among the best and most rigorously scientific school*
books published in this country. In his arithmetics, the attempt is
made '^ to render the reasoning of such arithmetics as those of Bourdon,
Briot, DeMorgan, and Wrigley, easily accessible to the young."* His
Elements of Geometry is ^< after Legendre," but it differs from the orig-
inal in the discussion of parallels, in the use of the methods of limits
instead of the method of the reductio ad absurduniy in the fuller treat*
ment of certain parts of the subject, and in giving, at the beginning, a
chapter on the Theory of Proportion (in which the theory of limits is
used for incommensurables) instead of presupposing a knowledge of
proportion, as is done by Legendre. One feature is carried out in this
geometry more extensively than in any other of our books, namely, the
insertion of '^ hints to solutions of exercises." A teacher who does not
make his pupils solve original problems in geometry, is a failure* But
the exercises given in most books are not sufficiently graded, and the
young beginner is very apt to get discouraged. The *' hints" given in
this book serve the excellent purpose of assisting and encouraging the
pupil in his first attempts at original work. In 1887 Professor Yenable
published an Introduction to Modern Geometry, which serves as an

*Oar sketch of the early career of Professor Venable is taken from La Boide's His-
tory of South Carolina CoUege, 1874, p. 474.

200 TEACHING AND HISTOBT OF MATHEMATICS.

appendix to his geometry. The tfeatment of the sabject is metrical
rather than descriptive.

The method of ins traction under Professor Tenable has been essen-
tially the same as that followed by his predecessors. It consists of lect-
nres, prelections on approved text-books, and exercises for testing and
developing the power of the student in original solutions. Great stress
is constantly laid on the solution by the student of original exercises.
In this respect, each meeting of the cl ass is a seminarium. In delivering
their lectures, some professors of the university write condensed notes
on the blackboard, others give syllabuses. The students very soon get
up printed or lithographed notes on the lectures. The practice of
reading the lectures does not prevail at the university.

One might suppose that in an institution where students have the
privilege of attending whatever school they please, the enrollment in
the school of mathematics would be comparatively small. This has,
however, not been the case here. The attendance on this school is, as a
rule, greater than on any other school of the academic department. In
three or four sessions, since the War, the number of students in the
school of Latin has been greater, but by not more than half a dozen
students. The fall attendance is in itself good evidence of the careful
teaching and efficient work in the mathematical department. In order
to present a fuller picture of the services of Professor Venable, we quote
firom a letter of B. H. Jesse, professor of Latin at the Tulane IlDiversity
of Louisiana, and a former student of the University of Virginia. ^<In
my day Colonel Venable was absolutely the most popular among the
students of all the professors in the University of Virginia. At the
same time his control was perfect over all his classes, and indeed over
any and all bodies of students with whom he came in contact. Doubt-
less his experieuce as an officer of rank in the Confederate service, his
long practice in teaching, and his never failing kindness of heart and
sympathy with young men, produced both the popularity and the power
of controL

" Ever since I have known the institution well, now nearly twenty
years, he has been, more than any other man, active and able in pro-
moting her best interests. To him in large degree was due the increase
by the State, in 1875 or 1876 , of her annual contribution from fifteen
thousand dollars to thirty thousand dollars. This increase was accom-
panied with the condition that all Virginia students able to pass the
entrance examinations to the academical schools should be educated in
those schools free of charge. To him chiefly was due the raising of the
endowment fund whereby the McCormick telescope was gained for the
university. To him chiefly has been due the large increase in attend-
ance upon the university in late years. Twice he has been Chairman
[of the faculty] and twice has he laid the office down voluntarily, when
the university, guided safely by his wisdom and energy through some
serious difficulties, had reached excellent condition again. He has had^

INFLUX OF FfiENCH HATHBHATIC8. 201

to my certain knowledge, many flattering calls to other fields, far more
profitable in money, but he has immediately declined them all to stand
fast by his alma mater J^

The high and rigid standard inangurated by Bonnycastle and Oonrte-
nay has been rigoroasly adhered to. The standard of gradaation has
always been high, in fact, y&j high in comparison with the standards
in most other American colleges. The mathematical coarse has been
broadened, as the preparation of students nnder the infiuence of the
university npon the academies and colleges has become broader and
better. " We have many excellent preparatory schools in Virginia,'^
says Professor Yenable, " which prepare students well, far into the
differential and integral calculus in such works as Todhunter's and
Oourtenay's Oalculus."

The course in mathematics, as stated in the catalogue for 1887-88, is
as follows:

I. PuBx Mathematics.

JcNiOR Class.— This class meets three times a week (4^ hoars) and studies theory
of arithmetical notations and operations; algebra, throQjsh the binomial theorem;
geometry, plane and solid ; geometrical analysis, with nnmeroas exercises for original
sblntion ; elementary plane trigonometry, embracing the solaiion of triangles, with
the nse of logarithms, and some applications to problems of ''heights and distances."
The preparation desirable for it is a good knowledge of arithmetic, of algebraic opera-
tions through equations of the second degree, and of the first three books of plane
geometry.

IVsBt&ooib*.— Todhanter's Algebra; Yenable's Legendre's Geometry, with ooUeotion of exeroiBes;
Todhonter's Trigonometry for Beginners.

Intermediate Class. — ^This class meets twice a week (3 hours) and studies geo-
metrical analysis, with exercises for original solution; plane trigonometry, with
applications ; analytical geometry of two dimensions ; spherical trigonometry, with
applications; elements of the theory of equations. The preparation desirable for
this class is a thorough knowledge of algebra through the binomial theorem, and
logarithms; of synthetic geometry, plane and solid, with some training in the solu-
tion of geometrical problems ; and a knowledge of the elements of plane trigonometry,
including the use of logarithmic tables.

T«9^&ooJb«.~SnowbAll'8 Trigonometry, Pbckle's Conio Sections, the Professor's CoUectiou of JBxer-
oises in Plane Geometry.

Senior Class.-— This class meets three times a week (4| hours) and studies analyt-
ical geometry of three dimensions, through the discussion of the conicoids and some
curves in space ; differential and integral calculus, with various applications ; a short
course in the calculus of variations ; the theory of equations, and lectures on the
history of mathematics.

TecOrbookt.—The Professor's Notes on Solid Geometry (Analytical) ; Todhanter's Di£forential Calcu-
lus; William«on's Integral Calcnlus'^; Todhnnter's Theory of Equations.

Candidates for graduation in pure mathematics are required to pursue in the nni*
yersity the studies of both the Intermediate and Senior Classes.

II. Mixed Mathematics.

This course is designed for those students who may desire to prosecute their
studies beyond the limits of pure mathematics. It embraces an extended course of

*In former years Professor Yenable used Courtenay's Integral Calculus, which was
supplemented with notes which ''nearly equalled the text." (Prof. B. H. Jesse.)

202 TEAOHma and histobt of mathematics.

reading under the instraction and gnidanoe of the professor on the applicatloiis of the

differential and integral calcnlns to mechanics, physical astronomy, and selected por-
tions of physics. The class in mixed mathematics meets twice a week (3 hours).
Text-books.— 'Price's Infinitesimal Calcolas, Vols. II and IH j Cheyne's Planetary Theory.

Mathematical physics and spherical astronomy are taught in the
school of natural philosophy, in charge of Prof. F. H. Smith. Norton's
Astronomy is one of the test-books. In this schooli under practical
physics, are studied also the method of least squares. ,

In addition to the undergraduate course in mathematics there is now
a more extended course, occupying a large part of two sessions of nine
months. It is given to graduates who are candidates for the degree of
doctor of philosophy in the mathematical sciences. This course in-
cludes, in addition to the course in mixed mathematics, the study of
modern higher algebra, modern higher geometry (Steiner's or some like
work), a fuller study of the differential and integral calculus (Price and
Hoiiel), determinants (taught at the university for the last fifteen years),
a fuller course in differential equations, probabilities, and other selec-
tions. If the candidate chooses astronomy for his secondary branch, then
he studies Gauss's Tlieoria MotuSj and enters into the practical compu.
tation of orbits. Should he choose physics, then he studies some of the
advanced treatises in the line of mathematical physics.

In order to give a better idea of the course leading to the degree of
doctor of philosophy, we quote from a letter of Dr. S. M. Barton:

^' This doctorate course consisted of graduate studies in pure and
mixed mathematics and mathematical and practical astronomy, and the
text-books read, and on which I was examined, were as follows : Hoilel's
Calcul Infinitesimal, four volumes ; Chasles's TraitS de OSomStrie Sup6-
rieure ; Price's Infinitesimal Calculus, Vol. Ill (Statics and Dynamics of
Material Particles) ; Cheyne's Planetary Theory; Aldis's Bigid Dynam-
ics ; Notes and Examples selected by the Professor.

" The above were required in the mathematical department. In as-
tronomy the textbooks and requirements were : Gauss's Theoria Motus;
Notes on the Computation of Orbits, by Prof. Ormond Stone ; Notes on
Least Squares, Pertuibations, Variations of Constants, etc., by Professor
Stone ; Computation of the Orbit of Barbara (No. 234). This last was
of course a work of several months.

" I was allowed to select my own subject for a thesis, which was ac-
cepted by the faculty and printed before I stood my last examinations.

^< In the preparation for this thesis I was obliged to read, outside of
the studies laid down in the course, the method of equipollences, and
the principles of quaternions, and various articles bearing on the sub-
ject, in which I made use of the following works: Exposition de la
M6thode des Equipollences, by Bellavitis, translated into French by
Lais^nt. La Vraie Thiorie des Quantitis Negatives, etc., by Mourey.
Articles in the Nouvelles Annales de MathAnatiques. Kelland and Tait's
Introduction to Quaternions. Tait's Qaaternious.

INFLUX OF FRENCH MATHEMATICS. 208

<< In pnrsning these doctorate stadies I, of coitrse, made ase of many
books for reference, among which I might mention Salmon's Oonic Sec-
tions and Higher Plane Ourves, and Geometry of Three Dimensions.
Gregory's Examples. Vols. I and II of Price's Oalculns. Some older
works by Peacock and others, as well as some more elementary trea-
tises. • • •

*^ I can not refrain * * * from alluding to one striking feature
of the mathematical teaching at the University of Virginia, namely,
independence in the student; and by independence I mean the spirit of
self-reliance which enables the student to work out and elucidate for
himself.

^^ The student is taught from the start to depend upon himself.

<' This spirit of self-reliance pervades the mathematical department,
and it promotes originality, as well as gives zest to the work.

^^This would seem to be the only true way to teach mathematics, bat
many of our elementary teachers do little or nothing to inculcate this
great principle."

The thesis referred to above is entitled "Bellavltis's Method of Equi-
poUences" (1885). It contains an outline of the calculus of equipoUences
and of its relation to quaternions. It shows that while equipoUences
are more readily mastered, and yield on the whole more expeditions
solutions of plane problems than quaternions, the latter are immeasur-
ably superior in elegance, logical simplicity, and extent of application.

Since Professor Venable has been connected with the University of
Virginia, the department of mathematics has graduated many students
who have become prominent as teachers and scientists in their specialty.
Chief among these are Prof. G. E. Yawter, professor of mathematics in
Emory and Benry College for some years, now in charge of the Miller
Manual Training School ; Prof. G. Lanza, professor of mathematics at
the Massachusetts Institute of Technology ; Prof. W. M. Thornton, of
the school of applied mathematics. University of Virginia ; Professor
Graves, professor of mathematics at the University of If orth Carolina ;
Professor Gore, professor of physics and astronomy at the University
of North Carolina; Professor Bohannan, professor of mathematics at
the University of Ohio (Columbus) ; Prof. H. A. Strode, principal of
Kenmore University High School, Virginia ; Prof. W. H. Echols, pro-
fessor of engineering and president of the school of mines at the Uni-
versity of Missouri ; Prof. W. H. Eichancer, professor of mathematics
at the school of mines, University of Missouri ; Prof. T. U. Taylor, as-
sistant professor of mathematics. University of Texas.

Applied mathematics, t. e., mathematics applied to civil engineering,
was taught in the school of mathematics almost at the beginning of the
university. In 1832 a class in engineering was organized as a separate
department under the professor of mathematics, and was maintained as
an attachment to the school of mathematics until 1850. It was then
left out of the catalogue from the fact, no doubt, that the successful

204 TEACHING AND HISTOBT OP MATHEBIATICS.

working of such a course imposed too heavy a burden upon the mathe-
matical professor. Id 1865 the department of civil engineering was
revived and placed nnder the joint charge of the professors of mathe-
matics, physics and chemistry. In 1867 Prof. Leopold J. Boeck was
made assistant professor and placed in charge of the school of applied
mathematics, comprising courses in civil and mining engineering.
These led to the degrees of civil and mining engineer, respectively.
In 1868 Professor Boeck was promoted to the full professorship of
applied mathematics. He held the chair until 1875, when he resigned,
and was succeeded by Wm. M. Thornton, as assistant professor. Pro-
fessor Thornton was subsequently promoted to the full professorship
of applied mathematics. This school has sent out a large number of
engineers of sound training.

Mention should be made here of the school of practical astronomy,
nnder the direction of Prof. Ormond Stone. He is also director of the
McCormick Observatory, and editor of the Annals of MathematicSb

UNIVERSITY OF NOBTH OABOLINA.*

Professor Mitchell's successor in the chair of mathematics was James
Phillips, from 1826 to 1867. Professor Love speaks of him as follows :
^< He was born in Eugland in 1792. It is not known at what school he
received his early education. The greater portion of his mathematical
education was gotten by private study. He came to America in 1818
and opened an academy in Harlem, K. Y. Here he won reputation as
an instructor, and by contributions to the mathematical publications of
the day. In 1826 he came to North Carolina as professor of mathe-
matics and natural philosophy.

^^ He was a patient student of the masters in mathematics, of Fergu-
son, Newton, Delambre, Laplace, and others. He prepared a text-
book on conic sections which was published and used as an introduction
to analytic geometry. He left in manuscript the greater portion of a
series of text-books on mathematics, including the calculus. These
were most carefully prepared, but for some reason he never published
any of them. Probably the War was the cause of his not publishing.
He left directions when he died that all his MSS. should be burned.
Among them were also many translations from French mathematical
works.

" That Dr. Phillips never published more is very much to be regretted.
He had great mathematical ability, and was an extremely careful and
lucid writer. Like Dr. Mitchell, he divided his time and energy. Both
of them were ministers and spent much time in the preparation of ser-
mons. Dr. Phillips left hundreds of manuscript sermons; and these he
directed to be burned with all his other MSS. He died suddenly of

* For all the information here given on the Unirersity of Korth Carolina, the writer
is indebted to Prof. Jamee L. Love, aesooiate professor of mathematics at the ual'-
varsity.

INFLUX OF FRENCH MATHEMATICS. 205

apoplexy in the college chapel, where he had gone to condact morning
prayers, on the 14th of March, 1867."

The requirements for admission were raised in 1835 so as to inclnde
all of arithmetic. It seems that in the same year a little of algebra —
"Young's Algebra to simple equations''— was also required. The in-
crease in the requisites for entering college were brought on at this time
with excessive haste, and we are not surprised that, after three years'
trial, algebra was withdrawn. It was not again required until 1855,
when candidates were examined on " algebra through equations of the
first degree." No alterations were made till 1868.

As regards the courses. of study, Professor Love says: "In 1835
arithmetic was dropped, algebra was completed in the Freshman year,
and conic sections and analytic geometry begun in the Sophomore ye«r.
In 1839 mechanics was introduced into the Sophomore and Junior years,
civil engineering into the Senior year, and since that date analytic
geometry has been completed in the Sophomore year. Oalcnlus was
begun in the Sophomore year in 1841, and from that date to 1868 it was
sometimes in the Sophomore year and sometimes in the Junior year.
For fifty years, from 1818 to 1868, first fluxions and then differential and
integral calculus were required of all graduates. A three-years' course
in engineering was introduced in 1854. It included in addition to the
regular course required for graduation, descriptive geometry, drawing,
shades and shadows, mechanics, civil engineering, and geodesy. Thiis
course was continued until 1862.

"An attempt was made in 1855 to offer some election of courses in the
Sophomore and Junior years. Two courses were offered, the one analyt-
ical, the other geometrical. The latter embraced geometry, plane and
spherical trigonometry, mensuration, surveying, navigation, natural
philosophy, and astronomy. The analytical course included, in addi-
tion, analytical geometry, differential and integral calculus, statics and
dynamics, acoustics and optics. During the Freshman year the two
courses were identical, but for the Sophomore and Junior years different
text-books were used, even for the same subjects, in the two courses.
After two years' trial, these double courses were given up. From 1857
to 1868 the one mathematical course was as follows : Freshman year^
algebra, geometry; Sophomore year ^ plane and spherical trigonometry
with applications, analytical geometry, differential and integral calculus;
Junior year^ natural philosophy and astronomy."

Our list of books used by Professor Phillips is quite complete ; Eyan's
Algebra was used in 1827 ; Young's Algebra was introduced in 1836;
Peirce's was studied from 1844 to 1868. In geometry, Legendre was
used for a time. About 1843 Peirce's Geometry was introduced, and
not dropped till 1868, except for the years 1865 to 1857, when Perkins
and Loomis were used each one year. From 1857 to 1868, Munroe's
" Geometry and Science of Form" was used in the Freshman class as an
introduction to geometry. The idea of premising a course in demon-

206 TEACHING AND HISTOBY OF. MATHEMATICS.

Btrative geometry by a short one in empirical geometry is yery eom<
mendable. In descriptive geometry, Davies^ was introdnced in 1854;
also his Shades and Shadows. In 1844 Peirce's trigonometry was in-
trodnced ; Perkins's was used from 1855 to 1856 ; Oharles Phillips's
from 1856 to 1860 ; Loomis's from 1860 to 1868. In conic sections James
Phillips's was tanght from 1830 to 1847, when Peirce's book was intro-
dnced. From 1851 to 1868 Loomis's was studied, except from 1853 to
1855, when Ohnrch's and Smith's Biot's were used, each a year.

tn calcnlns the notation of Leibnitz was introdnced in 1830. Hat-
ton's work was supplanted in 1847 by Peirce's Onrves, Fnnctions, and
Forces, which was followed in 1851 by Loomis's, Thi» was used until
1868, except in 1853, when Ohnrch's was taught for one year. In as-
tronomy, Peirce's book was introduced in 1847, Herschel's in 1865, and
Norton's in 1857.

Before the Civil War the university was prosperous and popular.
The courses in mathematics described above were certainly very credit-
able for their day. Hon. Wm. H. Battle spoke of the university as fol-
lows : <^ In the extent and variety of its studies, the number and ability
of its instructors, and the number of its students, it surpassed nearly
all similar institutions in our own section of the country, and was be-
ginning to rival the old, time-honored establishments of Yale and
Harvard. In the year 1858 its catalogue showed a larger number of
under-graduates than that of any other college in the United States,
except Yale. All this success was accomplished in a very short time.
A glance at the rapidly increasing ratio of its graduates will illustrate
the truth of my remark. For the first ten years after the date in which
degrees were conferred by the university, the number of students who
received the baccalaureate was 53 ; for the second decade it was 110 ;
for the third, 259 ; for the fourth, 146 ; for the fifth, 308 ; for the sixth,
448; and for the seventh the annual number was going on at a rate
which would have produced 883, nearly the double of that which im-
mediately preceded it."*

During the Civil War nearly all Southern colleges dosed their
doors, but not so the University of North Carolina. It was the boast
of its president that ** during the four years of war the college bell never
failed in its daily calls', that the faculty was ever in place for duty, and
< that all grew fat on sorghum and com bread ; ' that the institution
was maintained in full working order J* The severest blow to the pros-
perity of the university came after the War. In 1868 the old faculty
was turned out by the ** reconstructed ^ State government, and from
1869 to 1871 a new faculty labored to make the university jiopular
again. But political feeling was too high ; the university was closed
fix)m 1871 to 1875.

'Address delivered before the two literary aocieties of the UolTenity of Noiib
Carolina, June 1, 1865, by Hon. Wm. H. Battle.

IKFLUX OP FRENCH MATHEMATICS. 207

Gharles Phillips became professor of mathematics in 1875. He had
been tntor from 1844 to 1853^ associate professor from 1855 to 1860, and
professor of engineering from 1853 to 1860. In 1879 he was made pro-
fessor emeritas of mathematics, and Balph H. Oraves, jr., who had
been professor of engineering since 1875, became now professor of
mathematics. Professor Graves is a graduate of the University of Vir-
ginia and a former pnpil of Professor Yenable. Since 1885 James Lee
Love has been associate professor of mathematics. He graduated at
the nniversity at the head of his class, and then took a graduate course
in mathematics at the Johns Hopkins University in the year 1884-85.

Under the present able corps of instructors, mathematical teaching
is again flourishing. Since the re-opening, in 1875, the requirements iu
mathematics for admission have been : arithmetic, and algebra to quad-
ratic equations. The course in mathematics has been as follows : Fresh-
men, algebra, geometry ; SophmoreSy plane and spherical trigonometry,
logarithms, plane analytical geometry ; Juniors^ theory of equations,
differential and integral calculus, natural philosophy ; Seniors, mechan-
ics, astronomy. The studies of the flrst and second years have been
required of all graduates. The studies of the third year, except natural
philosophy, have been elective. Mechanics and astronomy were
required in all courses leading to degrees until 1885. Since that time
mechanics is elective in all courses, and astronomy elective in the A. B.
course. Since 1885 post-graduate electives have been offered in solid
analytic geometry (Smith's), determinants, differential equations, mod-
ern algebra, and quaternions. From 1876 to 1879 a three-year course
in engineering was offered. Since 1879 the course has been partially
withdrawn ; and at present (1888) it includes only a one-year course in
surveying, descriptive geometry, and projective drawingw^

Eobinson^s University Algebra was used from 1869 to 1871, and since
1876 Schuyler's, Venable's, NewcomVs, and WelPs — Kewcomb's most.
In gex)metry the books have been, since 1875, those of Venable, Weut-
worth, Newcomb, and J. W. Wilson. In descriptive geometry and pro-
jective drawing Warren's is taught. Davies' Trigonometry was used
from 1869 to 1871, Wheeler's since 1875, and Kewcomb's since 1882.
In calculus the works of Peck, Courtenay, Bowser, Byerly, and Tod-
hunter have been in use. Since 1883 Williamson has been the text-
book. Kewcomb and Deschanel are the books in astronomy and phys-
ics.

In 1883 the Elisha Mitchell Scientific Society was organized. The
professors of mathematics take part in its exercises. Meetings are held
once each month for the presentation of papers on any scientific subject.
The society publishes a Journal, with abstracts of the more important
papers read, and the writer has before him Vol. V, Part I, in which
appear two papers by Professor Graves on geometrical subjects. These
have been published also in the Annals of Mathematics, to which Pro-
fessor Graves is a frequent contributor.

208 TEACHING And history op MATHEMATICa

UNIVERSITY OP SOUTH CAROLINA.*

The successor of Bev. Dr. Hanckel in the chair of mathematics was
James Wallace. He entered apon his daties in 1820, and remained at
the college for fourteen years. Some years previous to his coming to
this institution he had been professor at Oeorgetown College, in West
Washington. He possessed mathematical ability and fine attainments
in his specialty. While at Columbia, S. C, he contributed to the South*
em Review articles on "Geometry and Calculus/' Vol. Ij "Steam
Engiue and Eailroad,'' Vol. VII ; " Canal Navigation," Vol. VHI. In
the first of the above articles a somewhat severe criticism of Hassler's
Trigonometry is given. Wallace upholds the geometrical method and
the line system. He contributed also to Sllliman's Journal, in onenum-
ber, giving an account of a new algebraic series of Stainville in Oer-
gonne's Annals, *but, by mistake, it was not duly accredited, and ap«
peared like Wallace's work. This drew him into a controvorsy wiUi
Nathaniel Bowditch.

Wallace's ability is shown by his treatise on the Use of the Globes
and Practical Astrouomy (New York, 1812), This work was in advance
of any other American treatise on astronomy of its day. The work had
512 pages, was printed closely, with lengthy notes in small type. Some
parts required little or no knowledge of mathematics on the part of the
reader ; others assumed a knowledge of geometry, trigonometry, conic
sections, and algebra, and the last part also of fluxions. The title
page bears the motto, '^ Quid munus Beipublicw majm aut melius afferre
po$8imu8j quam si Inventutem "bene erudiamus t — Cicero."

M. La Borde says in his History that Wallace did not place very
high value upon the above work. '^ He said the MS. of a work to
which he had devoted twenty years of his life was destroyed by fire,
and he thought that but for that accident he would have left something
worthy of remembrance."

As a teacher Wallace was in some respects the opposite of Blackburn.
The latter was somewhat hot-tempered, but Wallace was a patient and
laborious teacher, who loved his art. ^' No obtuseness of perception,
no degree of stolidity could provoke him to ill-temper." Upon leaving
the college he retired to a small farm near Columbia, where he died in
1861.

After the departure of Wallace, Lewis B. Gibbs held a temporary ap-
pointment for one year, or part of one. In 1835 Thomas S. Twiss was
appointed. He occupied the chair for eleven years. He was bom in
Troy, N. Y., graduated at West Point, and, before his election, was
teaching a classical school at Augusta, Ga. He was remarkable for in-
dustry, punctuality, and '^ watching and waiting " to catch students in
mischief. He enjoyed the reputation of arraigning more offenders than

* The material for tliis sketch was kindly furnished as by Prof. S. W. Davis, pro-
fessor of mathematios and astronomy at the unirerdty.

INFLUX OP PBENCH MATHEMATICS. 209

any other two members of the faculty. Fpon leaving the college he be-
came president of some iron works in the Spartanbarg district. From
here he retamed to his old home in Kew York.

The next mathematical professor, Matthew J. Williams, was likewise
a West Point graduate (class of 1821). He was a native of Georgia,
and had an early bent for arithmetic. At the Military Academy he was
one of foar to attain a maximum mark in mathematics. In 1825 he was
stationed at Old Point Comfort, Va., then at Fort Howard, Wis. He
resigned from the Army in 1828, and studied law in St. Louis. He prac*
ticed law in Georgia until 1835, when he received an appointment to the
South Carolina Conference of the Methodist Episcopal Church, at Cokes-
bury, Abbeville County. Thence he was called to the South Carolina
College. He resigned in 1853 on account of severe disease. His health
failed during his last year at West Point, and he seems to have had a
constant struggle with sickness from that time on. As a teacher, he
was ^' zealous, industrious, and thorough." His enthusiasm knew n6
bounds. He was esteemed as a scholar, a man, and a Christian. When
his health began to decline and there was fear he would have to give up
his work, the president of the college wrote in his report to the trustees:
^' I cau not express to you how much I value his services in the depart-
ment which he fills, and I should regard it as a most deplorable calamity
to the college to be deprived of his labors."

His successor, Charles F. McCay, a Pennsylvanian, was, at the time
of his election, a professor at the University of Georgia, and a colleague
of John and Joseph Le Conte. He was elected president of the South
Carolina College in 1855. In an attempt to act as a <^ go-between " in a
disagreefhent between faculty and students, he incurred the displea^re
of both parties. After his resignation he went into business, and tsnow
actuary of an insurance company in Baltimore. From what we can
learn, he was a man of ability and a good teacher.

From 1857 to 1862 Charles S. Yenable filled the chair of mathematics.
Since the War he has been for nearly a quarter of a century profe>sor
at the University of Virginia, and has established for himself a lasting
reputation as a teacher of mathematics. While professor at Columbia,
he was, as yet, a young man, and was not so popular as a teacher.

We proceed to give the courses of study for the period preceding the
War. In 1836 the terms for admission were, " arithmetic, including
fractions and the extraction of roots." In 1848 was added, " algebra to
equations of the first degree." In 1851 Davies' Bourdon was the algebra
used. In 1853 the whole of Bourdon's Algebra was required for en-
trance. This requisition appears to have been excessive, and in 185d it
was reduced to " Bourdon's Algebra to Chapter IX " (thus omitting the
general theory of equations and Sturm's theorem), or ^^ Loomis's Algebra
to Section XVII" (omitting permutations, combinations, series, loga-
rithms, and general theory of equations). The catalogues, from 1857 to
1862, contain this : ^^A thorough knowledge of arithmetic being essential
881— No. 3 14

210 TEACHING AND HISTOBY OF MATHEMATICS.

to Bucoess in all classes of the college, applicants most be prepared for
a fall and searching examination in this study.''

In 1836 the course of study was as follows :

^^Freshman year : Bourdon's Algebra to equations of the third degree,
ratios aud proportions, summation of infinite serieSi nature and con-
struction of logarithms, Legendre's plane geometry. Sophomore year :
Legendre's solid geometry, constructions of determinate geometrical
equations, Davies' mensuration and surveying, including methods of
plotting and calculating surveys, measurement of heights and distances,
and use of instruments in surveying. Junior year : Descriptive geom-
etry and conic sections, principles of perspective, analytic geometry,
fluxions — direct and inverse methods in their application to maxima,
minima, quadrature, cubature, etc. Senior year : Natural philosophy
and astronomy.

<( There shall be daily recitations of each class, one after morning
prayers, one at 11 A. h., one at 4 p. m. On Saturday morning there
shall be one recitation."

In the introduction of descriptive geometry into the course, we no-
tice West Point influences. The ^'fluxions" above mentioned must
mean ^'differential and integral calculus." Mr. Twiss, the professor at
this time, was a graduate of the Military Academy at West Point, and
was not likely to teach fluxions and the I^ewtonian notation.

In 1838 the Freshmen finished the whole, both of algebra and geom-
etry ; the Sophomores had plane and spherical trigonometry in place of
solid geometry.

In 1841 Davies' Oalculus was studied in the Junior year. Three years
lat^r, the Sophomores were taught from Davies' works on Mensuration
and Surveying,AnalyticalGeometry,and Descriptive Geometry. In 1848
(M. J. Williams, professor) Loomis's Conic Sections were studied. De-
scriptive geometry and calculus were taught by lectures. After complet-
ing the calculus, in the Junior year, Olmsted's Mechanical Philosophy
was taken up. The Seniors had courses in astronomy and (Mahan's)
civil engineering. Owing to a rise in the terms for admission Bourdon's
algebra was omitted in the first year, the studies for the other classes
remaining the same. In 1854 descriptive geometry was thrown out
of the course. Professor McCay was not a West Point graduate, and
attached, probably, less importance to this branch. In 1857 spherical
geometry was transferred from the second to the third year.

In 1868 the Freshmen studied geometry (Legendre), reviewed algebra
(applications of algebra to geometry) ; the Sophomores, mensuration,
surveying and leveling, conic sections (Loomis), mechanics (gravity,
laws of motion) ; the Juniors had lectures on calculus, spherical trigo-
nometry, mechanical philosophy (Olmsted) ; the Seniors, astronomy,
civil engineering, natural philosophy (Olmsted).

In 1860 Professor Venable introduced at the end of the first year
^eoretioal arithmetic^ using his own edition of Bourdon. He use4 9\bo

INFLUX OF FBENCR MATHEMATICS. 211

Loomis's Geometry iQ place of Legendre. la 1861 Loomis's Geomelr j is
mentiooedy " with original problema.'^ Algebra was reviewed and ap-
plied to "geometrical problems.'' We judge tbat extra efforts wor^
made by Professor Veuable to improve on the traditional methods of
teaching, by reqairlng the student to do a great deal of original worli:
in the line of solving problems.

In 1863 the buildings of the college ^' were taken possession of by the
Confederate Government, and used as a hospital until the close of the
War.'' Its charter was amended by the Legislature in 1865, and in the
following year it was re-opened as the University of South CaroUua.

The mathematical chair was given to £. P. Alexander, a grad.nate of
West Point and a man of great ability. During the War he was a Oon-
federate brigadier, distinguished himself at Gettysburg, and introdueed
^< signalling " into the Confederate army. As a teacher he was much
liked* He was very practical and to the point in his methods aud illas-
trations. Since leaving the college, he has been connected with rail-
roads, either as president or otherwise.

Prof. T. E. Hart, a graduate of Heidelberg, taught mathematics from
1870 to 1872. He was then and is now in very poor health, sufiiering
&om paralysis. While he was professor his classes b ad often to gO; to his
house for recitation.

From 1873 to 1876 A- ^« Cummings held the mathematical chair. At;
this time the college passed through the darkest period of its history.
These were the unfortunate years of '^ reconstruction." In addition to
the numerous obstacles which American colleges generally have had to
encounter, the colleges in the South have had tocontend with great polit*
ical upheavals. Like the University of I^orth Carolina, the IJniversitty
of South Carolina closed its doors. From 1876 to 1880« the institntiw
was without faculty and without students.

When the institution opened, in 1866, its course of study was remod*
eled. In this reorganization the plan of the University of Virginia
was followed. In the prospectus we read that '^ the university consists
of eight schools ;" that students are allowed to choose the departments
which they wish to pursue, provided they enter at least three schools.
In certain cases, however, students will be allowed to enter less than
three schools."

The prospectus continues, as follows : ^' During the present year there
will be no examinations or other requirements for admission, except
that the applicant must be at least fifteen years of age ] but in order to
ensure uniformity of preparation in certain departments, a preparatory
course has been prescribed, and after this year applicants (under eight-
een years of age) will be required to bring a satisfactory certificate of
proficiency, or to stand an examination. For applicants over eighteen
years of age, no examination or certificate will be required during the
next year."

212 TEACHING AND HISTOBY OF MATHEMATICS.

'< In all the different schools the method of instraction is by means of
lectares and the study of text-books^ accompanied in either case by rigid
daily examinations."

In the <' school of mathematics, and civil and military engineering
and construction," the requirements for admission were : ^'Arithmetio
in all its branches, including the extraction of square and cube roots ;"
*' algebra, through equations of the second degree."

From 1867 to 1872 the terms were as above, together with "a knowl-
edge ot the first four books of geometry," which, "though not indis-
pensable, is very desirable."

In 1872 the management of the university fell under the Beconstmc-
tion administration; negroes were admitted, and a four years' prepara*
tory course was given. The catalogue of 1872-73 says :

" In arithmetic, attention should be paid to all the rules and calcula-
tions usually given in written arithmetic, and too much importance can
not be paid to a thorough preliminary drill in mental arithmetic."

In the "college of literature, science, and the arts," the requirements
are, in addition, for the classical course^ "algebra, as far as equations of
the second degree," and it is " recommended that they also master the
first four books in Davies' Legendre, or the equivalent;" for the scien-
tific course, " algebra, up to radical quantities."

In the catalogue for 1876, the requirements were " the whole of arith«
metic," and " algebra as far as equations of the second degree."

The course of study in mathematics was, in 1866, algebra from eqna-
tions of the second degree to general theory of equations and loga-
rithms, geometry, plane and spherical trigonometry, surveying and the
use of instruments, in the first year ; in the second year, descriptive
geometry, analytical geometry, calculus, mathematical drawing. Text-
books : Loomis's books on algebra and geometry, Davies' Shades, Shad-
ows, and Perspective, Church's Analytical Geometry and Csdcalns.
In the " department of mechanical philosophy and astronomy," Prof.
John Le Conte's Mechanics was taught, also Olmsted's Astronomy,
with Herschel's Outlines and Norton's Astronomy for reference. In
1867 Loomis's Astronomy was used, as well as his series of mathemati-
cal text-books from his Algebra to his Calculus. In 1870 everything
is the same as given above, except that mechanical philosophy and
astronomy were temporarily taught by the professor of mathematics.

In 1872 Robinson's University Algebra and Loomis's Geometry were
studied in the first year; Robinson's Trigonometry, Mensuration, Sur-
veying, and Spherical Trigonometry in the second year ; Robinson's
Analytical Oeometry and Conic Sections the third year. Later on
Ficklin's Algebra was introduced.

** In 1879 the trustees of the university were empowered by act of
the General Assembly to establish a College of Agriculture and Me-
chanics at Columbia, and to use the property and grounds of the col*
lege for this purpose. This was accordingly done in 1880«"

INFLUX OP FBENCH MATHEMATICS. 218

*'Ia 1881 the Legislature granted an annual appropriation for tbe
support of the schools of the university, and in 1882 the South Caro-
lina College was reorganized by the appointment of a full faculty. It
went into active operation the fall of the same year."

From 1882 to 1888 Benjamin Sloan was the professor of m£U:hematics.
At present he is professor of physics and civil engineering. He is a
South Carolinian, graduated at West Point in 1860, served in New
Mexico before the War, and then entered into the Confederate service.
The story goes that when he entered upon the duties of his chair at the
college, he ordered a bookseller to get Courtenay's Calculus. " Calcu-
lus I "replied the bookseller, "what are you going to do with it?"
" Teach it," was the reply. " You can't do that, no South Carolina boy
ever studies calculus." Though this be merely the opinion of a jovial
bookseller, it is, we fear, not without some truth when applied to the ten
years preceding the reorganization and re-opening of the college in 1882.
For four years it was under Beconstmction rule, and for six years its
doors were closed to students.

Professor Sloan is a first-class teacher. He requires a great deal of
original work of students, and inspires considerable enthusiasm. In
his manner he is very quiet and easy. Among the students he is
liked and popular.

In 1888 Dr. E. W. Davis was elected to the mathematical chair. He
graduated at the University of Wisconsin in 1879, and after spending
some time at the Washington Astronomical Observatory, went to the
Johns Hopkins University, where for four years he studied mathe-
matics under Professor Sylvester and his associates. As a subsidiary
study Davis pursued physics under Professor Hastings. At this great
university he soon caught the spirit and enthusiasm which is so con-
tagious there. His mind was chiefly bent toward geometrical studies,
and the papers from his pen, which are published in the Johns Hop-
kins University Circulars and the American Journal of Mathematics
are evidences of his power as an original investigator. Before his ap-
pointment to his present position he was professor of mathematics for
four years at the Florida Agricultural College in Lake City. In his
teaching Professor Davis possesses great power in causing students to
think. He is a bold advocate of greater freedom fi:om formalism in
mathematical instruction.

The terms for admission on the re-opening of the institution were, in
mathematics, arithmetic, and algebra through equations of the first
degree. Sadicals were added in 1883. In 1884 the terms were, arith-
metic, and algebra to equations of the second degree. No additions
have been made since.

The mathematical course in 1882 consisted, in the first year, in the
study of Kewcomb's Algebra, Chauvenet's Geometry (six books) ; in
the second year, in the study of l^ewcomb's Plane and Spherical Trig-
onometry^ Puckle's Conic Sections ; in the third y ear^ in the farther study

214 TEACHINa AND HISTORY OF MATHEMATICS.

of conic sections (Packle, Olney), and oalcnlus (Olney, Todhunter). In
applied mathematics courses were given in the second year on sorvey-
ing (Gillespie) and drawing, Peck's Mechanics, Wood^s Strength and
Resistance of Materials, and Walton's Problems in Elementary Me-
chanics, astronomy (Loomis, Newcomb, and Holden), and Mafaan's Civil
Engineering. In 1884 Warren became the text-book in descriptive
geometry. In 1885 Taylor's Calculus was introduced; in 1886 Watson's
Descriptive Geometry and Merriman's Least Squares ; in 1887 ITew-
comb's Analytic Geometry.

The mathematical text-books for 1888 are, in the first year, Todhun-
ter's Algebta for Beginners, Byerly's Chauvenet's Geometry; in the sec-
ond year, Blaklie's Plane and Spherical Trigonometry, Peiroe's Tables;
in the third year, Taylor's Calculus, Church's Descriptive Geottietry; in
the fourth year, Newcomb and Holden's Shorter Course in Astronomy.

This year (1888-89) a graduate department has been added. In
mathematics it offers the following branches: Algebra (theory of equa-
tions, theory of determinants, etc.), geometry (projective geometiy,
higher plane curves^ etc.), calculus (differential equations and finite
differences), elliptic functionu, astronomy, and quaternions.

UNIVERSITY OF ALABAMA.*

l!he TTniversity of Alabama was opened in 1831, with Gurdon Salton-
stall in charge of the mathematical teaching. Two years later William
W. Hudson became professor of mathematics, and held the position until
1837, when Frederick Augustas Porter Barnard became connected with
the institution, and had charge of tiie mathematical department till
1849. The wonderful activity of this powerful man in the various de-
partments of science gave a great stimulus to higher education in the
State. He had previously been tutor at his alma mater ^ Yale. In 1849
he assumed the duties of the chair of chemistry at the University of
Alabama. While connected with the institution as professor of mathe-
matics and natural philosophy be wrote and published an arithmetic,
which came for a time into pretty general use in Alabama. In 1846 he
was appointed astronomer by the State, to settle a boundary dispute
between Alabama and Florida. He was appointed astronomer for the
State of Florida also, so that he represented both States in the settle-
ment of the dispute. Professor Barnard was always fond of matlie-
matics. He has written a number of valuable articles on mathematical
subjects for Johnson's New Universal Cyclopasdia.

By old students Professor Barnard is always spoken of in most land-
able terms. Says Dr. B. Manly : " To me the study of physics, astronomy,
etc., under Prof. F. A. P. Barnard, • • • and of chemistry and
kindred sciences under Prof. E. H. Bramby, long deceased, were the

* Nearly all the material for this article was bent us by Prof. T. W. Palmer, pio-
ftiaor of mathematics at the anirersity*

INFLUX OF FRENCH MATHEMATICS; 215

most attractive parts of my college course.'' Mr. John A. Foster, now
chancellor of the south-eastern chancery division of Alabama, was a
student and then a tutor of mathematics at the university in the time
that Barnard taught there. He says :

*' I entered the Sophomore class of the University of Alabama at
Tuscaloosa in the autumn of 1844, and received my diploma in August,
1847, in a class of eighteen. During my college course Prof. F. A. P.
Barnard was the professor of mathematics and John G. Barr was the
assistant professor of mathematics. Dr. Barnard afterward became
the president of the University of Mississippi, and in 1861, being a
Union man, resigned and went North, where he was for some time
engaged in the scientific department of the Government, and afterward
was president of Columbia College in the city of New York. A very
short time ago I observed that he has retired from this work.

" Professor Barnard was not less distinguished as a scientist than as
a mathematician. His reputation is world wide. I was a great friend
of his, and up to 1858 I was a constant correspondent with him. I need
hardly say that his instruction was thorough and far in advance of the
methods which prevailed at that time. There has never been a better
teacher of mathematics, and those now living still claim that the country
is but now getting to the methods of teaching practiced by him more
than forty years ago. Withal, he was a warm and generous friend, and
was very popular with those who were his pupils. During the summer
of 1844 or 1845 he went to Europe and spent some time in France, and
on his return to the university he brought with him the newly discovered
Daguerrean process, and took pictures experimentally before his class.
He was hard of hearing and had a deep guttural voice, but no one had
a happier faculty of making himself clearly understood. He married
an English lady while I was his pupil.

" Capt. John G. Barr, the assistant professor, was worthy to occupy
the position as second to this distinguished man. In 1847 he raised a
company and went to the Mexican War, where he served with distinc-
tion until its close. Soon after he was appointed to a diplomatic posi-
tion by the United States Government, and died at sea when on his way
out to assume the duties of his official station. He was an able and
successful teacher of mathematics."

Mr. Foster engaged in educational work till 1869 (being for some
years president of a college in La Grange, Ga.), when he went to the
practice of law.

The mathematical teaching at the university for the three years suc-
ceeding 1849 was in the hands of Prof. Landon Cabell Garland, now
the honored chancellor and professor of natural philosophy and astron-
omy of Vanderbilt University. His successors as instructors of mathe-
matics at the University of Alabama, before the War, were Profs.
George Benagh (1852-60), Robert Kennon Hargrove (1855-57), James
T. Mnrfee (1860-41), and William Jones Vaughn (1863-65).

216 TEACHING AND HISTOBT OF MATHEMATICS.

Prof. B. E. Hargrove, after teaching mathematics for two years,
joined the ministry of the M. E. Ghorch South, and, a few years ago,
was elected bishop by the general conference which met at liTashville,
Tenn.

The terms for admission to the university wer^, 1833-^6, arithmetic;
1857-59, arithmetic, and algebra through equations of the second de-
gree; 1860-62, arithmetic, and algebra to equations of the first degree;
the records for the next three years are lost.

Down to 1852 the professor of mathematics was at the same time pro-
fessor of physics, according to the usual custom in American colleges
at that day. In 1833 the Freshman class completed algebra (Golbumi
Lacroix) and commenced geometry (Farrar'sLegendre) ; the Sophomore
class studied geometry, trigonometry, and conic sections. The Junior
and Senior classes were taught mechanics, statics, heat, light, elec-
tricity, etc. The books used were the Cambridge Mathematics of Pro-
fessor Farrar. This course continued without change until 1842, when
surveying, mensuration, etc., were made an important part of the Sopho-
more work. In 1843 Davies' text-books were adopted. In 1845 Peirce's
Algebra was introduced, but after two years it was displaced by Davies'.
In 1849 the calculus was added to the Junior course. GQie text used
was Church's until 1855, when Loomis's was adopted. From 1860 to
1865 the records are so incomplete that it is impossible to state whether
or not any changes were made during that time.

Before the War, the university was prospering. *'In the Junior and
Senior classes," says Mr. Foster (class of 1847), ''much attention was
given to applied mathematics. Physics, astronomy, surveying, and
navigation were taught. The university was but a college with a fine
corps of professors, and presented advantages offered by very few other
institutions of learning at that time."

The War naturally interfered with the successful working of the uni-
versity. In 1865 the university buildings were destroyed by fire, and
the institution was not opened again until 1869. The condition of the
country at that time was not favorable for the advancement of educa-
tion. In recent years, however, decided and encouraging progress has
been made. A thrill of aspiration and enthusiasm has been running
through Southern colleges.

The first year after the re-opening Prof. K B. Chambliss taught the
mathematics; the next year. Prof. J. D. F. Bichards; and the year fol-
lowing. Prof. Hampton S. Whitfield, and the fourth year Prof. David
L. Peck. In 1872-73 Prof. W. J. Vaughn held the mathematical chair;
and from 1873 to 1878 Prof. H. S. Whitfield again. In 1878 Professor
Vaughn assumed the duties of this chair for the third time, and dis-
charged them for four years. Since 1882 Prof. Thomas Waverly Palmer
has filled the chair and taught with marked success.

Vaughn is now professor of mathematics at Vanderbilt University.
(< Though he has never written text-books," says Professor Palmer, << still

INFLUX OF FSENCH MATHEMATICS. 217

lie is justly regarded as one of the ablest mathematicians in onr Ameri-
can colleges." Prof. J. K. Powers, president of the Alabama State
J!!l'ormal School, who studied at the university from 1871 to 1873, says
that he had completed the course in pure mathematics before going there,
and that he took applied mathematics there. <^Prof. Wm. J. Vaughn
at that time filled the chair of applied mathematics. He was (and is)
an accomplished mathematician, an attractive instructor, a fine general
scholar, and a charming gentleman. At that time the chair of pure
mathematics was filled by Prof. D. L. Peck and Prof. H. S. Whitfield.
I Jenew nothing of their methods, hxitpure mathematics was not popular
in those days. In after years, when Prof. Vaughn assumed control of
that work, no department of the university was more popular."

Of Professor Palmer, Chester Harding (class of '84, now a cadet at the
U. S. Military Academy at West Point) says : " This gentleman, a gradf-
uate of the class of '81 of the university, had so satisfactorily filled the
position of assistant professor during the preceding term, that his elec-
tion was secured, as young as he was, against the claims of other appli-'
cants of extensive experience, reputation, and influence."

From 1869 to 1871 only the elements of arithmetic were required for
admission. During the next two years, algebra to equations of the
second degree was added. In 1873 the requirements were reduced to
arithmetic alone. "So change was made until 1878, when algebra through
equations of the second degree was required. Gradual changes have
been made every year since, and now the whole of algebra and three
books of geometry are required. * '

The catalogue for 1887-88 states that the candidate for admission
<' must pass a satisfactory examination in arithmetic, in algebra through
arithmetical and geometrical progression, and in the first two books of
geometry. The examination in arithmetic will include the whole sub-
ject as embraced in such works as White's, Robinson's, GofPs, Greenleaf s,
or Sandford's higher arithmetic. In algebra, particular stress will be
placed upon the use of parentheses, factoring, highest common factor,
lowest common multiple, simple and complex fractions, simple equa-
tions with one or more unknown quantities, involution, evolution, theory
of exponents, radicals (including rationalization, imaginary quantities,
properties of quadratic surds, square root of a binomial surd, and solu-
tion of equations containing radicals), quadratic equations, equations
of the quadratic form, simultaneous quadratic equations, ratio and pro-
portion, arithmetical and geometrical progression."

In 1874 the calculus was dropped from the university course, but was
introduced again in 1878.

In 1881 there wa« a reorganization of the courses of study. Two
courses of mathematics were arranged, one for classical and scientific
students, and one for engineering students. The course for classical or
scientific students embraced algebra, geometry, plane and spherical
trigonometry, and analytic geometry. These subjects were completed

218 TEACmXG AND mSTOBY OP MATHEMATICS.

in the Sophomore class. Since 1881 no changes have been made in tlie
classical and scientific courses.

The engineering course embraced all subjects that were taught in the
classical and scieatific, but to the Sophomore work was added descrip-
tive geometry, and to the Junior class calculus. This course has been
modified since. At present it consists of higher algebra and geometry
for the Freshmen ; plane and spherical trigonometry, analytical geom-
etry, descriptive geometry, theory of equations, for the Sophomores;
calculus, determinants, and quaternions for the Juniors.

Determinants and quaternions, which are regularly in the course since
1887, have been taught irregularly for several years. Quaternions are^
according to catalogue, now taught in the third term of the Sophomore
year, before the completion of analytic geometry. This is a somewhat
new departure in the arrangement of mathematical studies, and one
which is worthy of respectable and thoughtful consideration.

As to text-books, in 1871 Davies' Algebra and Geometry were used;
also Church's Analytic Geometry and Calculus. In 1872 and 1873 the
books were Bobinson's Algebra and Geometry, and Loomis's Trigo-
nometry and Analytic Geometry. In 1878 Peck's Analytic Geometry
and Galculus were introduced. The books used at present are Well's
Algebra, Wentworth's Geometry, Trigonometry, and Analytic (Geom-
etry, Bowser's Analytic Geometry, Taylor's Calculus, Church's Descrip-
tive Geometry, Peck's Determinants, and Todhunter's Theory of Equa-
tions.

Cadet Chester Harding, who was a student at the TTniversity of
Alabama from 1881 to 1884, gives the following reminiscences of the
mathematical teaching there : *< The training in mathematics was more
extensive in scope and thoroughness in the engineering course than in
the others, including in that course the elementary principles of de-
scriptive geometry and calculus, while in the others the instruction
ceased with the study of the conic sections and surfaces of the second
order in analytical geometry.

<^ I chose the engineering course and began my instructions in the
departmentof mathematics with trigonometry under Prof. W.J. Vaughn,
who now fills the chair of mathematics at Yanderbilt University. Our
text-book was Wheeler's Trigonometry. The trigonometric functions
were taught as ratios, and stress was laid upon the circular system of
measuring angles. • • •

'^ Analytical geometry came next, and our text-book was Professor
Wood's Elements of Co-ordinate Geometry. Of the class of thirty in
this branch, all were beginners but two or three who had been required
to repeat the course because of their deficiency in the preceding year.
Our progress was therefore slow at first, and much time was spent by
the professor in explanations and illustrations. I see the first lessons
still marked in the text-book I have before me now, and some were bat
two and a half o£ the quarto pages. These, however, were expeeted to

INPLTTX OP FRENCH MATHEMATICS. 219

be thoroughly mastered, and many pains were taken to have the prin-
ciples well absorbed by the stadents. Mere exercise of memory was
little sought after in the mathematical department, and any originality
on the part of a student in the deduction or application of a principle
was highly commended.

" The course in analytical geometry closed with the end of the ses-
sion, at which time a satisfactory written examination in the study was
required of every member of the class. In the scientific and classical
courses, mathematics terminated with the Sophomore yeat. In the
Junior year the students of the engineering course^ however, took up
the study of calculus.

*< At the end of my Sophomore year Professor Vaughn resigned his
chair at the University of Alabama to accept a similar position at
Vanderbilt. ♦ ♦ ♦ i

*< In my Junior year the schedule of studies was so arranged that but
three hours a week were devoted by my class to mathematics. This
limited time permitted us to complete but one text-book, Prof. W. G.
Peck's Elements of the Differential and Integral Calculus. From this
text, however, we derived a knowledge of the practical utility of calcu-
lus, and became familiarized with the rules of differentiation and integra-
tion. I can hardly say that we acquired a more thorough knowledge
than this ; and indeed it seemed, from the time assigned to the study,
to be without the purpose of the faculty that more than a groundwork
should^ be acquired, for practical good in the understanding of the ap-
plications of calculus to mechanics and engineering. During my Jun-
ior year we also studied under Prof. E. A. Hardaway, in the depart-
ment of engineering, the elements of descriptive geometry, using as a
textbook Binn's Elements of Orthographic Projection.

*^ With the close of the Junior year the regular course in pure mathe
matics was ended."

As regards the conditions for graduation which have existed at van-
cos times. Professor Palmer says : <^ As a rule mathematics was required
of every student for graduation, from 1831 to 1866. After the reorgani-
zation in 1869, mathematics was also required until 1875, when the elec-
tive system was adopted ; it was entirely optional with the student then
until 1880, when every student was required to take this subject through
analytic geometry."

At present there are no electives, and all the mathematics in each
course is required for a degree in that course.

tJNIVEESITY OF MISSISSIPPI.

The educational record of Mississippi in the early period of her organ-
ised existence is quite honorable. Between 1798 and 1848 there had
been established one hundred and ten institutions, under the various
names of universities, colleges, academies, and schools. This proves
that an entire obliviousness to the educational wants of the people did

220 TEACHING AND HISTOBY OF MATHEMATICS.

not prevail. Oar gratification isubated, however, by the consideration
that these organizations proved inefSicient, and that there was really
but very little beneficial progress.

In 1848 was organized upon a firmer foundation the University of
Mississippi. Considering the many difficulties that were encountered^
the record of the university during its infant years before the War was
honorable. Two names, both well known to the educational public,
devoted their energies to promote its early growth — F. A. P. Barnard,
now president of Columbia College, and A. T. Bledsoe, afterward pro-
fessor at the University of Virginia and, still later, editor of the South-
ern Methodist Eeview.

From the' beginning until 1854, Albert Taylor Bledsoe was professor
of pure and applied mathematics, and astronomy. The mathematical
requirements for admission were, at first, a knowledge of arithmetic.
The catalogue of 1857-58 says : " Arithmetic— especially the subject of
fractions, vulgar and decimal, proportion, and the extrSrCtion of roots ;"
the catalogue for 1859-60 adds to this, '^ algebra as far as simple equa-
tions." In the former catalogue we read also, '^that, hereafter, no
student will be'admitted to any class in the university who shall fail to
pass an entirely satisfactory examination on the subjects or authors re-
quired for admission to the class."

According to the catalogue of 1854, the Freshmen studied Davies'
University Arithmetic, Davies' Bourdon, and Davies' Legendre ; fine
Sophomores continued Davies' Bourdon and Legendre, and then took up
Plane and Spherical Trigonometry and Surveying ; the Jt^nior^ studied
Descriptive Geometry, Shades and Shadows and Perspective, Davies'
Analytical Oeometry, and Descriptive Astronomy; the Seniors j Davies'
Differential and Integral Calculus, and physical astronomy. In the
introduction into the course of descriptive geometry, in the use through-
out of Davies' text-books, and in the apparent thoroughness (for that
time) of the mathematical course, we observe the influence of the U. S.
Military Academy, through Professor Bledsoe, a West Point graduate.

When Bledsoe resigned to accept a professorship at the University
of Virginia, Frederick Augustus Porter Barnard, a young man of re-
markable mathematical talents, took his place. Barnard was a native of
Massachusetts and entered Yale college in 1824. Before admittance to
college he had given no time to mathematical study beyond the ele-
ments of arithmetic, but in college he began to exhibit decided mathe-
matical talent and taste. His tutor, W. H. Holland, later professor of
mathematics in Trinity College, Hartford, said of him : ^< I have never
known any person except the late lamented Professor Fisher, who pos*
sessed so extraordinary natural aptitude." After graduation he was,
for a time, tutor at Yale, then professor at the University of Alabama,
and, in 1854, became Bledsoe's successor at the University of Missis-
sippi. At the meeting of the board of trustees, in July, 1856, the chair
of pure and applied mathematics and astronomy was divided into twoy

INFLUX OF FBBNCH MATHEMATICS. 221

the chair of pare mathematics, and the chair of natural science, civil
engineering, and astronomy. Professor Barnard held the latter, though
he continued to exercise supervision over the former, and was also
elected president of the university. He filled these oilicos until the
suspension of the exercises of the university, in 1861.

From 1856 to 1861 Jordan McOuUogh Phipps was teacher of mathe-
matics—at first adjunct professor, afterward full professor. Daniel B.
Carr was tutor. The department of mathematics, physics, and engi-
neering seems to have been the strongest at the institution. In conse-
quence of frequent complaint that the general statement previously
presented in the annual catalogues of the university had been unsat-
isfactory, a complete account of expenses and of the courses of instruc-
tion was given in the catalogues issued at this time. From the one of
1857-68 we quote the following:

<^ Instruction in pure mathematics commences with the beginning of
the Freshman year, and is continued till the close of the Sophomore.
In order to secure greater ef&ciency of instruction, the class will be di-
vided into sections, which will be met by the instructor separately ;
and all operations in this and every other branch of mathematical sci-
ence will be actually performed by the student in his presence, upon
large wall-slates Or blackboards. The instructor will also avail himself
of the same means of illustrating processes, or principles, and explain-
ing difficulties.

<' The first subject attended to is algebra. It will be the instructor's
endeavor to secure a thorough acquaintance with the elementary prin-
ciples of the science, and a perfect familiarity with its practical opera-
tions. The subject of fractions will be especially dwelt on, after which
will follow the resolution of simple equations, numerical and literal, in-
volving one or more unknown quantities. In taking up, next in order,
quadratic equations, the first object will be to secure on the part of the
student a perfect understanding of the form of the binomial square ;
and this will be afterward applied to the completion of imperfect
squares, in the several cases in which one of the terms of the root is a
number, or a letter, or a numerical or literal fraction. The method be-
ing generalized, will then be applied to the reduction of abstract equa-
tions, and the statement and resolution of problems involving quad-
ratics. Where the equation is denominate, the student* will be re-
quired to interpret the result, to explain the ambiguous sign, and to
distinguish cases in which the conditions of the problem involve an im-
possibility. • • ♦

'< The subject of algebra will be completed by the discussion of the
general theory of equations, their formation, their solution, and their
properties, including in the course the ingenious theorem of Sturm.

" In all parts of this subject, encouragement will be held out to stu-
dents to exercise their ingenuity ih devising various modes of arriving
at the same results ^ and special merit will be attached to the processes

222 TEACHING ANB HISTOJSY OF MATHEIUTIC8.

whieb are the most saceinct or elegant. As a stimiilas to tiua qieeiM
of ing^eimitj, problems not embraced in the text-book maj &om time to
time be proposed by the instmctor ; and varietiea in the mode of state*
ment both of these and of those which oecnr in the regular coarse^ will
be called for from any who may choose to present tiieaou

^^ Geometry, plane, solid, and spherical, will oocapy the latter portion
of the Freshman year. In this branch of science all demonstrations will
be made from figures drawn npon the blackboards, or waU-slates, by
the student reciting, and promptness and aeeoracy in this part of the
basiness will be urgently inculcated and regarded as a merit. The
student will, moreover, be advised to avoid a servile imitation of the
exact forms of the diagrams given in the text-book, and will have his
ingenuity exercised either in forming other figures to iUnstrate the same
propositions, or in demonstrating the propositions from figures c<m*
structed for him. He will also be required to adopt a mode of lettering
his figures different firom that of the book ; or to give the demonstrationa
without the use of letters at aU, by pointing to the parts of the figure
successively referred to in the demonstration.

^^ It will always be regarded as specially meritorious in a student to
present a demonstration of any proposition founded on any legitimate
method differing from that of the author ; and the in^ructor will, him-
self, from time to time, illustrate this practice, by way of awakening the
ingenuity of the stadent. For the purpose of still further eneooraging
originality of investigation, and exciting honorable emulation^ the plan
already described as to be pursued in algebra, will be continued here,
of propounding propositions not contained in the text-book, of which
demonstrations will be subsequently called for, and which will secure
special distinction to such as satisfactorily solve them.''

Equally full is the account of the mathematical work in the Sopho-
more year. The studies for that year were plane and spherical trigo-
nometry, mensuration, surveying, leveling, navigation, and analytieal
geometry. Considerable field-work was done in surveying. The lev-
eling rods employed had the common division to feet and fhiotions, and
also the French metrical division.

The catalogue then proceeds as follows :

'< The course of pure mathematics will conclude with the subject of
the differential and integral calculus, which will be taught at the end
of the Sophomore or the beginning of the Junior year. This will em«
brace the doctrine of functions, algebraic and transcendental, the dif-
ferentiation of functions, successive differentials, theorems of Taylor
and Maclaurin, logarithmic series, the development of a circular arc in
terms of its functions, or of the functions in terms of the arc, partial
differentials, differential equations of curves, principles of maxima and
minima, expressions for tangents and normals, singular and multiple
points, osculating circles, involutes and evolutes, transcendental curveS|
and spirals ; the integration of regularly formed differentials«integra?

INFLUX OP FBENGH MATHEMATICS. ^ 223

Hon by series, integration of rational and irrational fractions, special
methods of integration, th^ rectification of curves, the quadrature of
curves and curved surfaces, the cubature of solids, and the integration
of diflfere'ntials of two or more variables.'^

In natural philosophy great efforts were made to secure a complete
set of apparatus. In the catalogue for 1857*^ we read as follows :
'' It is probable that, with the opening of the ensuing session, the elec-
trical apparatus of the University of Mississippi will be superior to any
similar collection in the United States.'^

In astronomy the celestial motions were beautifully represented by
Barlow's magnificent planetarium, eleven feet in diameter — < 'a piece
of mechanism unrivaled in ingenuity, accuracy, and elegance.'' A port-
able transit instrument was also available for observations of meridian
passages, and a sextant and a prismatic reflecting circle furnished means
of making direct measurements of altitudes and arcs. The catalogue
then says:

<^ The course of civil engineering, distinctly so called, falls entirely
within the Senior year ; but it is in considerable part only a further
development and application of principles embraced in the sciences of
t>ure mathematics and physics previously taught. The course will em-
brace geometrical and topographical drawing, the use of field instru-
ments, such as the engineer's transit, the goniasmometer or pantom^ter,
the leveling instrument, the theodolite, the sextant, the reflecting cir-
cle, and the plane table, descriptive geometry, trigonometrical survey-
ing and geodesy, marine surveys, materials of structures, engineering
statics, carpentry, masonry, bridge construction, surveys for location
and construction of roads and railroads, laying out curves, staking out
cuts and fills, hydraulic engineering, drainage, canals, locks, aqueducts,
dams, sea walls, river improvements, and the dynamics and economy of
transportation. • • •

'* Throughout every part of the course, the student will be constantly
encouraged and stimulated to consult other authorities on the subjects
taught, besides the text-books : and the instructors will often refer them
on special subjects, to such authorities. The following list embraces
the text-books (first in order), and the authors to whom reference will
most frequently be made :

f Algebra : Perkins, Hackley, Peiroe.

Gbohstrt : Perkins, Playfair (Euclid), Peirce.

Tkigonomktry : Perkins, Hackley, Peirce.

Surveying : Gillespie, Davies, Gammere.

Analytical Geometry : Davies, Peirce.

Calculus : Davies, Peirce, Church, Jephson.

Natural Philosophy: Olmsted, Bartlett, Whewell, Brewster (Optics),
Herschel (Light and Sound), Peirce.

Astronomy : Olmsted, Gummere, Bartlett, Loomis.

Civil Engineering : Mahan, Moseley^ Wiesbach, Gillespie, Haupt, Bourne,
Pambour."

224 TEACHING AND HISTOBT OF MATHEBfATICS.

The Qourses for the remaining years before the War were essentially
the same as the one we have described. The constant use of the black-
board is emphasized throughout. The fact that pains are taken to
explain the term as meaning <^ large wall-slates" rather tends to show
that blackboards were then a novelty in Mississippi. As far as we can
judge from the catalogues, the instruction was methodical and of high
efficiency. A serious drawback to high scholarship was found, no
doubt, in the lack of pireliminary culture and training in students enter*
ing the university.

The attendance of students was good. The number of graduates from
the department of arts from 1851 to 1859, inclusive, was 268. During
the last year before the War the number of students in the college was
191, of whom 16 were " irregular" in grade.

Owing to the univereal enlistment of males, even youths, in the Con-
federate States army, the university exercises were suspended in 1861,
until October 1865. In 1865 General Claudius W. Sears, exbrigadier-
general of the Confederate States army, and a graduate of West Point,
was elected professor of mathematics. This position he still holds.

The mathematical requirements for entering were, in 1866, ^< arithme-
tic and algebra, including equations of the first degree." The course
of pure mathematics for the regular under-graduate curriculum was
completed at the end of the Sophomore year, and consisted of Bourdon's
Alge\)ra, Legendre's Geometry, Trigonometry, Mensuration, Surveying,
and Analytical Geometry.

A more extended course than waid required for the degree of bachelor
of arts could be obtained in the department of applied mathematics and
civil engineering, which was in charge of General F. A. Shoup, a grad-
uate of West Point, and now of the University of the South. The
course of instruction in his department formed no necessary part of the
under-graduate course. It was designed to meet the wants of such
students as intended to make- civil engineering or some other of the
mechanic arts a profession. In this course analytical geometry and
calculus were, of course, indispensable, and they could be studied
while students were pursuing their branches in the department proper.
The course could be completed by an ordinary student who came fairly
well prepared in preliminary branches in about two years.

In 1870 the plan of instruction in the university was altered so as to
include (1) a department of preparatory education, (2) a department of
science, literature, and arts (leading, respectively, to the degrees of B.
A., B. S.,B. Ph., C. E.), and (3) a department of professional education
(law).

The terms for admission into the bachelor of arts and bachelor of
science courses were, in mathematics, arithmetic, and Davies^ Ele-
mentary Algebra through equations of the second degree. Oandidatea
for the bachelor of philosophy course and civil engineering were exam-
ined on the whole of Davies' Elementary Algebra. These requirementa

INFLUX OF FBENCH MATHEMATICS. 225

have remained unchanged till the present time. The department of
civil engineering was. discontinued in 1876. In 1872 the first year's
mathematical work in the course leading to the B. A., B» S., and B. Ph.
degrees consisted in the study of Davies' Bourdon's Algebpa, and Le-
gendre's Geometry and Plane Trigonometry. Daring the first half of
the Sophomore year Church's Analytical Geometry and Davies' Land
Surveying (with use of instruments in the field) were studied. This
completed the course in pure mathematics. A. B. students were taught
Smith's Mechanics and Hydrostatics, fiydraulics, and Sound (Bartlett)
in the Junior year, and Bartlett's Optics and Astronomy in the Senior
year. B. S. students had Oummere's Astronomy in the second half of
the third year. (The B. S. and B. Ph. were the only three years'
courses.)

At the. present time (1888) the mathematical course is decidedly
stronger. Van Amringe's edition of Davies' text-books are used, ex-
cept in analytical geometry and calculus, which are studied from the
works of Ohurch. The calculus is now studied during the latter part
of the Sophomore year.

Prof. G. W. Sears has now occupied the mathematical chair for
twenty-three years. One of his old pupils, Prof. Edward Mayes, says
of him, ^Hhat hie ^quizzes' 'like all possessed,' pretends that he does
not know anything about it, and asks 'all sorts of impertinent ques-
tions.'" As Sydney Smith said of Alexander Pope, ''I studied under
him, and have lively recollections."

KENTTJOKY UNIVBESITY.

The records of the Transylvania University for several years follow-
ing 1817 appear to have been lost. In 1825 Thomas J. Matthews, the
father of the late Justice Stanley Matthews, of the Supreme Court of the
United States, is mentioned as being ''professor of mathematics and
natural philosophy." The subjects taught by him were "arithmetic,
geometry, surveying, leveling, natural philosophy, and book-keeping."
The entry for 1829 shows that Pestalozzian ideas had gained a foothold
at the university, inasmuch as Golbum's Algebra is mentioned as the '
mathematical text-book for the Freshmen, The Sophomores studied
Playfair's Geometry and Trigonometry 5 the Juniors, Day's Navigation,
Surveying, Heights and Distances, Leveling^ the Seniors^ Bezout's
Fluxions. Bezout's text-book had been translated from the French by
Professor Farrar, of Harvard. It employed the notation of Leibnitz,
and did not therefore teach "fluxions." The use of this term as a
synonym for "differential and integral calculus" was, we believe, pecu-
liarly American.

In 1832 John Lutz was elected " professor of mathematics and natural
philosophy, " and in 1837 Benjamin Moore. The latter resigned after
one year's service.

881— Ko. 3 15

226 TEACHING AND HISTOBY OF MATHEl^TICS.

The records from 1839 to 1865 can not be foand. From old oatalogOfiB
we glean the following : In 1844 B. T. F. Allen was professor, and the
subjects tanght were, in the Freihman year, Davies' Bourdon and Le>
gendre ; in the Bophomore year, plane and spherical trigonometry,
heights and distaifces, mensaration of superficies and solids (DaTies^),
navigation (Day's), conic sections (Davies' Analytical Geometry), Sur-
veying (Davies'), descriptive geometry (Davies') ; in the Junior year,
differential and integral calculus (Davies') ; in the Senior year, Olm-
sted's Astronomy.

In 1848 James B. Dodd held the chair of mathematics and natural
philosophy. At this time the course was as follows : Freshman year,
arithmetic reviewed, Loomis's Algebra, five books of Legendre ; Bophth
more year, geometry completed, plane and spherical trigonometry and
their applications, analytical geometry (Davies', 6 books) ; Junior year,
Church's Galculus.

In 1850 the mathematics for the Junior and Senior classes consisted
of descriptive geometry, analytical geometry, calculus, and analytical
mechanics ; but they were optional with the student.

Prof. James B. Dodd was the most prominent mathematical teacher
that was connected with Transylvania University. He was a native of
Virginia, and a self-made mathematician. In 1841 he became profiessor
of mathematics at the Gentenary College in Mississippi, and in 1846
was elected professor at the Transylvania (Jniversily. He published
several books, viz., an Elementary and Practical Arithmetic, High School
Arithmetic, Elementary and Practical Algebra, Algebra for High
Schools and Colleges, and Elements of Geometry and Mensuration.
Some of these reached several editions. Professor Dodd contributed
also to the Quarterly Beview of the M. E. Church South. In 1849 he
was appointed president pro tempore of the university.

In 1865 Transylvania University was merged into Kentucky Univer-
sity* The chair of mathematics in Kentucky University has been filled
from 1859 to the present time by Henry H. White. From 1870 to 1876
James G. White acted as a^unct professor. From 1876 to 1878 he was
professor. In mathematics the requirement for admission has been
algebra through equations of the first degree. When Prof. Henry H.
White first became connected with the university as professor, the
course was as follows : Algebra completed, plane and solid geometry,
application of algebra to geometry, plane and spherical trigonometiy,
surveying and navigation, analytical geometry, differential and integral
calculus, mechanics, and astronomy, with original problems and exer-
cises throughout the course when practicable. In 1864 the course was
modified by dropping applications of algebra to geometry; in 1879, by
the addition of conic sections (treated geometrically) ; and in 1884, by
dropping conic sections and navigation.

The tez^books used by Prof. Henry H. White at different times are
as follows : In Algebraj Davies' Bourdon, Towne, Peck ; in Oeometry^

INFLUX OF FBENOH MATSEHATICS. 937

Daviei^ Legendre, Feck; in Trigawmetrpj Dayies, Feck; in Sturt^ifii^
and NaiHgationj Davies, Loomis ; in Analytical Oeometry^ LoamUj Peek ;
in Oaioulu8^ Loomis, Peck; in Meohanie$j Olmsted, BneU'a Olmatod,
Peck ; in AsirMomyj Olmsted, Snell's Olmsted, Peek.

There have been no electives in mathematics ap to this timet except
that the stadent now has the choice between langaages and oaJoulneu

UNIVERSITY OF TENNESSEE.*

<< The foundation of this university Is connected with the earliest
history of Tennessee.

« In X794, by the first General Assembly of the * Territory south of the
Ohio,' was chartered Blount OoUege, named in honor of William Blounty
Governor of the Territory, and afterward one of the two United States
Senators first chosen from the State of Tennessee,

*^ In 1807, under an act of Congress providing for the establishment
of two colleges in Tennessee, East Tennessee College was chartered,
and soon after the franchise and property of Blount College were trans-
ferred to the new institution. • • ♦

<< In 1840 the name of East Tennessee College was changed, by act
of Legislature, to East Tennessee University.

^< In 1869 the Legislature gave in trust to the university the pro*
ceeds of the sale of public lands, donated by act of Congress of July 2,
1862, < tb the several States and Territories which may provide col-
leges for the benefit of agriculture and the mechanic arts.'

" In 1879 the name of East Tennessee University was changed, by an
act of the Legislature, to the University of Tennessee.'' t

It is a source of regret to us that we have not been able to obtain
any information whatever on the mathematical instruction at this in-
stitution during the first eighty years of its existence. Ever since it
took the name of a university, it has been in an almost continual state
of reorganization. These constant upheavals have resulted in the loss
of almost all its records. ^' The requirements for admission and grad-
uation," says Professor Carson, <' have probably been changed, on an
average, every two years." ' The terms for admission were not rigidly
adhered to, and the standard for graduation has not always been high.

The catalogue of 1874-75, the earliest one that we have, gives John
Kerr Payne as professor of mathematics and mechanical philosophy.
The collegiate department comprised at this time three distinct courses
viz., the agricultural course, the mechanical course, and the classical
course. The standard for admission to the first two courses was,
until 1874, lower than to the last course. In 1874-75 the mathematical
studies in the agricultural course were according to catalogue, as fol-

■ ■ I I II I • I I I II . I I ■■ — ^»^^^— — W I I

* The writer i^ indebted to Prof. Wm. W. Carson, professor of mathematios and oIyII
enf^ineerins at the University of Tennessee, for aU the InfonoatiQii htttia pontiahiod*
t Catslogue of the UnlTermty of Tpjmomoe, laSMQ*

228 TBACmNG AND mSTORY OF MATHBB£ATICS.

lows : Freshmetiy Bobisson's TTniversity Algebra, beginning with qnad-
ratic equations, Ohauvenet's Geometry, beginning at tbid third book,
Loomis's Gonio Sections; Sophomores^ Oharch's Descriptive Geometry,
Loomis's Trigonometry and Surveying; Juniors^ Olmsted's Natural
Philosophy and Astronomy. In the mechanical and classical courses,
the schedule was the same in mathematics, except that spherical trigo-
nometry, Loomls's Analytical Geometry and Calculus, and civil engi-
neering, were added.

The biennial report of the trustees for 1881 gives James Dinwiddle as
professor of pure mathematics, and Samuel H. Lockett as professor of
applied mathematics and mechanical philosophy. The report shows that
the university was then organized into distinct schools, like the ITni-
versity of Virginia. These schools have existed, probably, since 1879.
Of the school of pure mathematics, the report says :

^^The subjects taught in the subcollegiate year of this school are ele-
mentary algebra, and four books of geometry. In the first collegiate
year algebra and geometry are finished, and plane trigonometry is
studied. In the second collegiate year are studied spherical trigonom-
etry and analytical geometry of two dimensions, and in the third year
differential and integral calculus."

The work in the school of applied mathematics is described as fol-
lows:

<< Elementary experimental physics is taught in the first college year.
The various subjects of statics and dynamics of solids, liquids, and
gases ; of acoustics, hQat, light, electricity, and magnetism, are treated
without the aid of the mathematics, and are illustrated by numerous
experiments. The apparatus has been specially selected for that par-
pose.

<<In the analytical mechanics, the power of the whole range of the
mathematics is brought to bear upon the investigation of the laws of
forces of nature, and the student is made familiar with the power and
utility of mathematics by the solution of a large number of practical
problems. Astronomy has thus far been taught without instruments,
but the board of trustees has appropriated five hundred dollars for the
purchase of a telescope. Surveying comprehends plane surveying, lev-
eling, topographical surveying, and mining surveying ; the use of the
compass, transit, Y level, plane table, chain, and leveling rod; also
plotting, making profiles and cross-sections, and topographical drawing
with pen ^nd brush. A large share of the student's time is given to
field work and practice.

<^ Descriptive geometry is the foundation of both the science and art
of drawing. It is followed by a course of problems in shades, shadows,
and perspective — mechanical drawing.

^< The course of engineering consists of the subjects treated in Pro-
fessor Gillespie's Beads and Railroads and Professor Wood's revision ot
Mabftn's Oivil Engineering, and of a coarse of lectures by the instraotor

INFLUX OF FBENCH MATHEMATICS. ' 229

on sarface and thoroa^ drainage, on agricaltaral, hydranlio, and ma-
rine engineering, and a brief outline of the science and art of military
engineering. The engineering drawing consists of a course of instrno*
tion in the drawing of plans, sections, elevations, and details of bridges,
tannels, canal locks, etc.

^< For the above engineering conrse students can substitute mechan-
ism, machinery, and machine drawing."

The catalogue for 1883-84 mentions as text-books in the school of
pure mathematics: ^^ White's or Olney's Arithmetic; Davies' Bour-
don, or Olney's Algebra ; Olney's Trigonometry ; Bowser's or Peck's
Analytical Geometry; Bowser's or Peck's Oalculus; Bledsoe's Philoso-
phy of Mathematics.

<< Extra examples, illustrating the different subjects taught, are given
throughout the course."

This is the first time that we find Bledsoe's Philosophy of Mathe-
matics named as one of the text-books in a college course. According to
catalogue, it was used in the third collegiate clas^, which completed
analytic geometry and then took up <' differential and integral calculus,
and the philosophy of mathematics." The idea of teaching the philos-
ophy of mathematics is certainly a good one, but the subject is hardly
presented by Bledsoe in a form suitable for a young student.

In the school of applied mathematics the books given in the cata-
logue for 1883-84 are. Gage's Physics; Loomis's Astronomy; Davies'
S'ew Surveying ; Smith's Topographical Drawing ; Church's Descrip-
tive Geometry ; Wood's, or Bankine's Mechanics ; Mahan's Civil En-
gineering ; Searles's Field Engineering.

In June, 1888, a reorganization and a re-classification of the various
schools took place. The work of the ^' school of mathematics and civil
engineering" for the year 1888-89 is as follows :

I. MATHEMATICS.

Fitii oZa««— (Snb-Freshman) : Algebra (through sards and qaadratios) ; Geometry
(three books).
Second oZa««— (Freshman) : Geometry, Algebra.

Tlhird oIa««— (Sophomore) : Trigonometry ; Graphic Algebra ; Analytical Geometry.
Fourth chua — (Junior) : Calcalas.

Each class is taught in sections small enough to be well handled by
the instructor. Great stress is laid, throughout the course, on the
written solution of original problems— the aim being to induce clear-
ness of thought by precision in expression. Each student is required
to use the level, transit, and compass, from the beginning of his Fresh*
man to the end of his Sophomore year. On entering the Freshman
class the use and adjustments of the level are explained to him. He
then practices with it, at times convenient to himself, until, by running
such lines as may be required of him and submitting profiles and cross-
sections, he shows his ability to handle the ordinary problems of drain-

250 TEAOHXNQ AKD HISTORY OF 2CA.THEHATICS.

iig« and irrigation. The graphical problems in geometry are solved,
ftometimee with drawing instraments on paper^ and sometimes with
engineering instraments on the gronnd. Thns habits of aoonracy are
enforoed early in the coarse by the nse of instraments of precision, and
an elementary knowledge of surveying afforded.

For admission to the first class the applicant is examined in arith-
metic only.

The text-books now in nse are as follows: Hall and Knight^s Algebra
for the Sab-Freshman class, Wentworth's Algebra for the Freshman
dass, Wentworth's (i^metry, Wells's Trigonometry, Packl^s Oonlo
Sections (with leotares), Newcomb's Oalcalos. The Oalcalas is tanght
mainly by lectures, the textbook being used as a guide. As tanght at
present, it is based on the idea of fluxions, demonstrated by limits, and
employs the notation of Leibnitz. In pure mathematics no higher
branches than the calculus have been taught at the university, except
daring the se^ion 1886^7, when a class in quaternions was tanght.
At present agricultural students must finish trigonometry, all others
Mialytioal geometry, while the engineering stadents must finish calculus.

It. CIVIL EN6IKBERING.

1. (Sophomore) t Descriptive Qeometry ; Land, City, sad Mine Sarveying.
3. (Janior): Stone Catting ; Aatronomy.
3. (Janior) : Elementftry Meohantcs ; Analytical Mechanics.

4* (Janior): Sanreys; Sonndings; Maps; Profiles; Cross-eectlons ; Estimates*;
Laying out Work ; Engineering Materials and Methods.

The time of this class is mainly spent in practical work. It makes
barometric reconnaissances ; makes a map of some portiou of the bed of
the Tennessee Biver; does the field and office engineering work for a
line of communications to join two selected points, etc.

5. (Senior) : Analytical Mechanics ; Applied Mechanics.

6. (Senior) : Engineering Strnotares ; Specifications and Contracts.

7. (Post-gradaate) : Economics of Beads ; Sewerage; Water Sapply; Hydraulics;
Arohitectnie.

The department is admirably equipped with the various engineering
instruments. Of the more important (such as levels, transits, sextants,
aneroids, etc.) it has a number of each. It has, with great care and ex-
pense, procured instruments of the finest workmanship and latest at-
tachments, so that its students of engineering may see how much to
expect the instrumentmaker to contribute toward the attainment of
accuracy and speed. Exercises requiring their use are continually re-
quired of every class.

The first six of these classes are required for the degree of bachelor of
science in civil engineering— the seven for the degree of civil engineer.

At present the University of Tennessee is entering upon a career of
remarkable prosperity. Like most of the higher institutions of learning
in the South, it is experiencing a great revival. More thorough work
and a higher standard of scholarship are everywhere perceivable.

IKFLX7X OF FBENCH MATHEMATIC& 231

. ' The present prosperity of the XlDiversity of Tennessee is doe chiefly
to the aggressive leadership of its President, Dr. 0. W. Dabney, a grad-
uate of the University of Virginia, and lat^ of the University of Oiit-
tingen. Be accepted the presidency in Angnst, 1887, under conditions'
giving him great fbeedom to manage the institution according to his
own ideas. In Jane, 1888, the professorships were declared vaeuit,
and were then filled by men selected by the president. Prof. William
W. Oarson, who had been elected to the chair of mathematics in 1885,
was now elected professor of mathematics and civil engineering. Pro-
fessor Oarson, a gradnate of Washington and Lee, was civil engineer
fbr a number of years. Of the other teachers of pure and applied math-
ematics, Prof. T. F. Bargdorff served about a dozen years in the U« 8.
Kavy, and Prof. E. B. Oayle about an equal length of time^in the U« S*
Army. The three other instructors in this school are young men.

TULANX TTNITEBSITY OF LOUISIANA.

TheTulane University came into existence as such in 18i84, when, by
a contract with the State of Louisiana, the administrators of the Tulane
educational fund became the administrators of the University of Lou-
isiana in perpetuity, agreeing to devote their income to its development.

The University of Louisiana had its origin in the Medical Department,
which was establish^ in 1834. This school has numbered among its
professors and alumm the most distinguished medical men of Louisiana
and the South. A law department was organized in 1847 ; and in 1878
the academic department of the University of Louisiana was opened.
It existed under that name till 1884, when it was absorbed into Tu-
lane University. Considering that the academic department of the Uni-
versity of Louisiana received from the State an annuity of only ten thou-
sand dollars, it met with excellent success. A number of very earnest
and well-trained young men were graduated during the six years of its
existence. Its faculty consisted of only seven professors, but they were
men of energy and ability. B. H. Jesse was dean of the fiusulty and
professor of Latin. He was educated at the Univei^ity of Yirginia, and
^ was a man of unusual executive ability. His individuality was strongly
felt in the institution. He organized the department, taking the Uni-
versity of Virginia as his model. There was no curriculum or prescribed
course of study. The parent or guardian had to choose, with the advice
of the faculty, the branches to be pursued by the student. His cast of
mind, as well as his future vocation, could thus receive due weight. In
1883 there were eight " schools.'^ The student was required to attend
at least three, but he was discouraged firom electing more than four, in
order to prevent superficial work.

The school of mathematics was in charge of J. L. Gross, the profes-
sor of mathematics. Professor Gross was, before the War, a student
at the Virginia Military Institute, and a pupil of Prof« Frauds H.
Smith. The school of matiiematics was organized into three zegular

232 TEACHIKG AND HISTOBY OF MATHEMATICS.

classes, the Janior, Intermediate, and Senior. Daring part of the tinre
it was found necessary to establish also an introductory class for stu-
dents deficient in preliminary studies. The requirements for admission
to the Junior class were a knowledge of arithmetic and Loomis's Ele-
ments of Algebra. The Junior class studied Loomis's Treatise on
Algebra, and Loomis's (later Wentworth's) Plane and Solid Geometry.
The Intermediate class was taught Loomis's Plane and Spherical Trigo-
nometry, and Loomis's Analytical (Geometry. The Senior 6lass com-
pleted the course in mathematics by the study of Church's Descriptive
(Geometry, and Loomis's Differential and Integral Oalculus. Professor
Gross is, we believe, the first teacher who ever carried classes in Kew
Orleans through the calculus.

Very efBcient work was done by students in the school of physics.
This was in charge of Prof. Brown Ayres. Professor Ayres received
his general education at the Washington and Lee University, and his
training as a specialist at the Stevens Institute and the Johns Hopkins
University. At the last institution he was honored with a fellowship
in physics. He is a true lover of science, and, with great proficiency
in the theoretical and mathematical parts of his subject, combines great
mechanical ingenuity and skilL In his prelections on text-books he is
extremely clear, and his experiments are always very successful and inter-
esting. His great aim is to awaken in students a genuine love for pure
science. In his school students had frequent opportunities of applying
their knowledge of pure mathematics to physical problems. The theory
of the combination of observations by the method of least squares was
a study in his course. During several years he taught also analytical
mechanics, using the work of De Yolson Wood.

In 1884 the University of Louisiana was absorbed into the Tulane
University of Louisiana. Paul Tulane, who had been in business in
New Orleans for fifty years, donated the greater part of his large
fortune for higher education in New Orleans. Owing to his munificence,
Tulane University has the good fortune of being free from those pecun-
iary embarrassments with which the University of Louisiana had
always to contend. Under the presidency of Col. William Preston
Johnston, an educator of great ability and wide reputation, the courses
of study as they had existed in the University of Louisiana were reor-
ganized.* Not trusting in the ability of immature students, or even of
parents unaccustomed to consider the due proportions and sequence of
studies, to properly formulate their own ideals in education, Tulane
College ofiered a series of six equivalent curricula with prescribed
branches, all leadi ng to the degree of bachelor of arts. These six courses
of study were denominated, respectively, the Classical, Literary, Math-
ematical, Natural Science, Commercial, and Mecb anical Courses. In the

* For farther in formation regarding the plan and workings of Tnlane University,
see President Johnston's address ou <* Edacation in Loaisiana,'' before the National
itdacational Conyention^ Topeka, Kmxl, July 15| 1885.

INFLUX OF FRENCH MATHEMATICS. 233

spring of 1880, the commercial coarse was discontinaed, and the math-
ematical conrse had' its name changed to physical science course.

All the professors of the University of Loaisiana continaed to hold
their respective chairs under the new regime. Several new'professom
were added to the faculty.

The mathematical requirements for admission to Tulane College are
a knowledge of algebra to quadratics and of plane geometry. The
course in mathematics is the same for all Freshmen. After completing
the algebra they take up solid geometry, plane and spherical trigo«
nometry, surveying and leveling, and navigation. In the Sophomore
year, classical and literary students pursue analytical geometry, three
hours per week, before Christmas. This completes the mathematics for
students in those two courses. In the three other courses mathematics
is pursued six hours per week throughout the year, and consists in the
study of analytical geometry and difierential calculus. In the first half
of the Junior year, students in the physical science course and mechani-
cal course pursue the study of integral calculus. These branches are
taught by Professor Cross from Loomis's text-books, excepting that
Wentworth's book is used in geometry.

The mathematical teaching has, thus far, been strictly confined to the
ordinary college branches. No work of university grade, as distin*
guished from college grade, has yet been attempted. *^The end kept
always in view is to impress the principles of mathematical truth clearly
and deeply on the mind, by careful explanations, by daily examinations,
and by constant application of these principles by the students them-
selves to numerous examples taken from the text- books and from other
sources.'^* Professor Cross believes in making a clear presentation to
the student of the principles of mathematics, without applying them
to any great number of special cases. In his opinion, much valuable
time is wasted in the solution of problems. If a student can give, for
instance, the general solution of a quadratic equation, then there is no
need of solving dozens of special exercises under this head. In ge-
ometry careful attentio]i is given to the correct understanding of the
demonstrations given in the book, but little or no effort is made to solve
original exercises. In the class-room Professor Cross preserves strict
discipline and is earnest in the discharge of his duties. When the
routine work of the day is over, his mind finds relaxation and rest in a
good game of chess or checkers.

Students in the mechanical and physical science courses study an-
alytical mechanics under Professor Ayres six hours per week during
the second half of the Junior year. This subject has been exceedingly
well taught. The text-book used heretofore in connection with lectures
has been Wood's Analytical Mechanics. ' This is a good text-book, in-
asmuch as the subject is taken up more or less inductively, and a large

* Catalogue of the Tulane University of LoolBiana, 1888-89, p. 46.

234 TEACHING AND HISTOBT OF MATHEMATICS.

number of special and well-graded problems is given to be worked by
the student Wood makes extensive nse of the calcnlus in his Analyti-
cal Mechanics. The experience has been at this institution, as also at
others, that students who have gone through Loomis's Oalculus are
hardly well enough prepared in that branch to pursue with ease a
course in analytical mechanics. Some important parts of the integral
calculus, particularly definite integrals, receive exceedingly meager
treatment in this book. The course in analytical mechanics serves to
impress more deeply and lastingly the principles of the calculus and
displays to the student its wonderful power in the solution of all sorts
of mechanical problems. This year (1888-^9) Michie's Analytical Me-
chanics will be used as a text-book by Professor Ayres. It contains a
beautiful chapter on graphical statics. In the Senior year students in
the mechanical course take up the subject of applied mechanics. Pro-
fessor Ayres is using, this year, Cotterill's Applied Mechanics, a stand-
ard work of great merit.

In 1883 a very fine collection of physical apparatus was purchased
by the university at a great expense. In optics the collection is excel-
lent. The university is fortunate in having a physicist who knows
how to make use of delicate instruments. Since the above date Pro-
fessor Ayres has devoted much of his time and energy toward building
up a good laboratory. A practical physical laboratory is somewhat of
a novelty in the South. Tulane University o£Bsrs npw as good and effi-
cient courses in experimental physics to students of ooUege grade as any
university in the country.

Since Tulane CTniversity is dependent for its supply of students
chiefly upon its own high school, wise provisions have been made for
more thorough instruction in that department With Professor Ashley
D. Hurt as head-master the high school has been prosperous and
thorough in its work. Both teachers and pupils are working with great
earnestness, and it is gratifying to know tliat the number of students
entering the college after graduating from the high school is decidedly
on the increase.

The JS'ew Orleans Academy of Sciences holds its meetings at the
Tulane University. The professors of the university are its leading
members. There is a general meeting once every month for all mem-
bers of the academy. In addition to this, there are section meetings.
<< Qection A," the mathematical and physical section, meets the second
Tuesday of every month. Professor Ayres has been the leading spirit
in this section, and has contributed many an interesting paper on
physics and mathematics. Two years ago the academy began publish-
ing an annual volume, containing the principal papers read during the
year. The publication for the year 1887-88 contains an article on the
^* History of Infinite Series," and an interesting article by Professor
Ayres on <^ Physics and Psychology.'' During the last two years the

INFLUX OF FREKCnS MATHEBIATICS. 235

aoddemy baa been in a fioarishing condition , and the quality of the
work done has been improving continually.

In the fall of 1887 the H. Sophie I^ewcomb Memorial College for
Younc Women was opened as a branch of Tulane University. It was
^founded on an endowment madr by Mrs. J. L. Newcomb, of New Tork»
This institution is under the able management of President Brandt Y«
B. Dixon, who is also professor of metaphysics and mental science at
the Tulane University. It is the aim to put the Newcomb College on
an equal footing with the Tulane College. Young women will thus
have the same facilities for higher education in JS^ Orleans that
young men have.

The first year (1887'-88) was a year of organization. Many features
of the school were of necessity only tentative. The great obstacle to
high scholarship is the lack of proper preparation on the part of appU-
cants. Eor this reason it has been necessary to establish a preparatory
department The Newcomb College offers four parallel and equivalent
courses of study«««the Classical, Literary, Scientific, and Industrial. In
the two preparatory years, higher arithmetic and algebra are studied.
It is the intention to introduce also a course on inventional geometry*
The first year in college is devoted to geometry, the seoond to the com-
pletion of algebra and to trigonometry. To students taking the soiea*
tifie and industrial courses, analytical geometry is ofiered in the Junior
year, and caloulns and astronomy in the Senior year« During the first
year in the history of the college there were classes in algebra, geome*
txy, and trigonometry. Wentworth's text-books were used. In the pre*
paratory department there were two elasses, one in arithmetic and
algebra, and the other in algebra. The latter class did faithful and
thorough work in Wentworth's Complete Algebra through quadratic
equations. This division did as good work as any class of young men
which the professor has taught. If not always quite as penetrating in
the solution of problems as young men, the young ladies worked more
faithfully and perseveringly. The lowest class of college grade finished
plane geometry and then reviewed algebra as far as logarithms. The
work in geometry was quite satisfb^ctory. A great effort was made to
induce students to solve original exercises. While paralogisms were
very frequent, especially at first, the efforts were not without some sue*
cess. The solving of original exercises in geometry is too much neg-
lected in our schools ; nor are our text-books always satisfactory on this
subject. In the opinion of the writer, the number of exercises should
be greatly increased, and very great care should be taken to either
omit the difficult exercises or give " hints " as to their mode of solution.
Students should not be permitted to get disheartened in this sort of
work. " The inventive power grows best in the sunshine of encourage-
ment*" Wentworth has greatly improved his text-book in his revised
edition of 1888, by the insertion of seven hundred additional exercises.

236 TEAcmNa Ain> history of mathematics.

The professor has found that the interest which popils take in their
studies may be increased if the solution of problems and the cold logio
of geometrical demonstrations are interspersed by historical remarks
and anecdotes. A class in arithmetic will be pleased to hear about the
Hindoos and their invention of the ^< Arabic notation ^^ they will mar-
vel at the thousands of years which elapsed before people Iiad even
thought of introducing into the numeral notation that Columbus egg^
the zero ; they will find it astonnding that it should have taken so long
to invent a notation which they themselves can now learn in a few weeks*
The class will take an interest in the history of decimal firactions and
the various notations that were used once in place of our decimal "pauiL
After the pupils have learned how to bisect a given angle^ surprise them
by telling of the many futile attempts which have been made to solve
by elementary geometry the apparently very simple problem of the tri-
section of an angle. When they know how to construct a square whose
area is double the area of a given square, tell them about the duplica-
tion of the cube— how the wrath of Apollo could be appeased only by
the construction of a cubical altar double the given altar, and how
mathematicians long wrestled with this problem. After the class have
exhausted their energies on the theorem of the right-angled triangle, tell
them something about its discoverer — ^how Pythagoras, jubilant over
his great accomplishment, [is said to have] sacrificed a hecatomb to the
Muses who inspired him. When the value of mathematical training
is called in question, quote the inscription over the entrance into the
academy of Plato, the philosopher : ^^ Let no one who is unacquainted
with geometry enter here." To more advanced students the history of
mathematics becomes instructive and profitable as well as interesting.
It seems to me that students in analytical geometry should know some-
thing of Descartes, who originated this branch of geometry, that, tak-
ing up differential and integral calculus, they should become familiar
with the parts which Newton, Leibnitz, and Lagrange played in creat-
ing the transcendental analysis. No one can claim to have a fair knowl-
edge of this subject who knows not something about the three methods
taught by these great analysts. In his historical talk it is i>OBsible for
the teacher to make it plain to the student that mathematics is not a
dead science in which no new discoveries are or can be made, but that
it is a living science in which racing progress is being made all the
time.

UNITBBSITY OF TEXAS.

The University of Texas opened its doors to students for the first
time in 1882. The first professor of mathematics was Leroy Brown,
who served one year. He was succeeded by G. B. Halsted as professor
of pure and applied mathematics. At the same time with Halsted, A.
Y. Lane was elected assistant instructor in mathematics. He was ad-
vanced to the position of assistant professor of applied mathematics
in 1885.

INFLUX OF FEBNCH MATHEMATICS. 237

Prof. O. B. Halsted was graduated in Princeton in ISTS, and received
the degree of doctor of philosophy at the Johns Hopkins University in
1879, where he had studied for two years under Professor Sylvester, and
had held a fellowship in mathematics. Before taking his degree he
spent some time in Berlin, prosecuting mathematical studies* In 1878
he was appointed tutor in mathematics at Princeton College, and three
years later instructor in post-graduate mathematics.

Dr. Halsted has established a wide reputation as a mathematician
and logician. He has contributed to the American Journal of Mathe-
matics, the Annals of Mathematics, the Mathematical Magazine, the
English Philosophical Magazine, and several other scientific journals.
He has published two books. An Elementary Treatise on Mensuration
(Boston, 1881), and The Elements of Geometry (New York, 1885). His
books and scientific articles have been favorably reviewed in leading
foreign journals. His Metrical Geometry (mensuration) is the best book
of its kind that haa been published in this country. It pontains many
new and interesting features. Of these we would mention his treat-
ment of solid angles (the words ateregon and steradiany now quite gen-
erally adopted, were manufactured by him and first used here) and his
discussion of the prismatoid, deriving a general formula for its volume.
He introduced a distinction between the words sphere and globe (mak-
ing one to mean a surface ^nd the other a solid), which is worthy of
general adoption.

The distinguishing feature of the two works of Halsted is their sci-
entific rigor. Teachers who favor a rigid treatment of geometry will
find it in his Elements. The book rejects the ^^ directional method'' as
wholly unscientific; also the use of the word ^^ distance " as a funda-
mental geometric concept. The word sect, first used in his Mensuration,
is introduced here, meaning " the part of a line between two definite
points.'' Many teachers do not endorse the introduction of this new
technical term in elementary geometry, as they think that there is no
particular call for it. The author is certainly right in protesting against
the use of the word ^^ distance" in two different senses. That there has
really been a want for some of the other new technical terms first in-
troduced by Halsted is evident by the fact that they have been adopted
in standard works, such as the Encyclopaedia Britannica.

Like his Mensuration, his Elements of Geometry possesses many
novelties. In his book on Bectangles he introduces a strictly geometric
algebra, where a and b mean sects, and, by definition, ab means their
rectangle, thus avoiding measurement and the use of numbers. Batio
and proportion are strictly treated, but without limits. The book on
two-dimensional spherics gives a novel method of treating spherics.
His demonstration of the two-term x>rismoidal formula has been trans-
lated into French by the editors of a mathematical journal published in^
Belgium. Halsted is the first writer in this country to preface a geom-
etry by a preliminary chapter on logic. Judged from a scientific point

258 TEACHING AND HISTOEY OF MATHEMATICS.

of view, Tre believe Halsted's Geometry to be the peer of any geometry
pabli»hM in America.

Professor Lane has contribated one article on ^^Bonlettes" to the
Amerioan Journal of Mathematics, and has written a neat little book
on Adjustments of the Gompass, Transit and Level. Professor Lane
tanght ohiefly the applied mathematics, U «., mathematics applied to
engineering, and reached good results in his work. In Jane, 1888, he re-
signed his professorship, and his place was filled by the selection of a
native Texan, T. U. Taylor. Professor Taylor is a graduate of the Uni-
versity of Virginia, and before accepting the present position was pro*
fessor of pare and applied mathematics in the Miller Manual Labor
School of Virginia.

The mathematical requirements for admission have beeu'&om the
beginning the same as they are now, except that Prof. L. Brown ex-
amined students in Wentworth's Geometry instead of Halsted's, As
stated in the catalogue of 1887--88, the terms for admission ai« as fol-
lows: ^^Arithmetic, including proportion, decimals, interest, discount,
and the metric system ; algebra, including theory of exponents, radicals,
simple and quadratic equations ; and the elements of plane geometry
(corresponding to the first six books of Halsted's Geometry).

^'Passing these examinations, a student will be admitted to the
Freshman class in the course of science, or to the Junior class of the
law department.''

Great efforts are being made to cause the high schools in the State to
work in line with the nniversity. High schools desiring the privilege
of sending their graduates to the university without examination are
inspected by committees from the faculty of the nniversity, and if the
work of a school be found satisfactory the school is <^ approved.'' Thus
far the number of irregular students in the academical department of
the university has been large, but as the institution grows older, the
students entering with a view of taking a four-years' course and grad-
uating will doubtless rapidly increase.

During the first year of the university there were, naturally, no
classes formed in the higher mathematics. At thebeginntngof the
second year, in addition to the lower classes, there was a Sophomore
class in analytic geometry, and a Junior class in differential and inte-
* gral calculus. At the beginning of the third year, in addition to these,
there was a Senior class in quaternions, and since then there have
always been Freshman, Sophomore, Junior, and Senior classes in math-
ematics*

At the beginning, Wentworth's Algebra and Geometry were nsed by
Professor Brown. When Professor Halsted entered upon his duties
at this university he " found that the lack of ri^or in Wentworth's Geom-
etry was so exasperating" that he *^ could not continue to use it with
comfort or a clear conscience," and so he put in form for the printer his
own manuscript on geometry. His geometry has been used since its

INFLUX OF FBENCH MA.THSMATICS, 239

issue, sapplemented by Halsted's Mensoratioo. The analytic geometry
used is Packless Conic Sections. Until the present year (1888-39) By-
erly's Calculas has been taught. Po8t:graduate courses in matbematies
are now offered to students.
The present mathematical course is as follows (catalogue 1887-88) :

The Freshman class wiU etndy algebra, solid geometry, qpheries^ mensiirfttioii, plane
and spherical trigonometry, with their applications to snrreyiDg, navigation, etc.

The Sophomere class will study analytical geometry, graphic algebra, and theory of
equations.

The Junior ol<us wiU study analytical geometry of three dimensions, differential
and integral calculus. This course of study will embrace the applications of the oal*
cuius to mechanics and physios.

The Senior cltus wiU study determinants, quaternions, invariants, and qnan«
ties. • * *

In the higher classes will be discussed the history and logical structure of the math-
ematical sciences, and the logical theory of the calculus, the theory of limitB, and the
infinitesimal method.

2Va^(ooX».— Wentworth's Complete Algebra ; Halsted's Geometry (John Wiley di
Sons, New York); Halsted's Mensuration, 3d £d. (Ginn d& Co.); Wentworth'A Trig-
onometry, Surveying, and Navigation ; Graphic Algebra, by Phillips &, Beebe ;
Puckle's Conic Sections. 5th £d. ; Smith's Solid Geometry ; Newcomb's Differential
and Integral Calculus ; Theory of Equations, by Bumside and Panton, 2d Ed. ;
Muir's Determinants ; Scott's Determinants ; Salmon's Modem Higher Algebra, 4t]i
Ed. ; Hardy's Quaternions.

Engineering students are required to take the four*years' course ;
science students, the studies for the first three years ; arts students,
those of the first two years ; and letters students, those of the first year.

Two post-graduate courses are offered :

I. A course preparatory to original investigation in the objective sciences. This
wiU include infinitesimal calculus, the method of least squares, kinematic, linkage,
differential equations, the calculus of finite differences.

IVx/^oo^.*— Williamson's Difierential Calculus, Williamson's Integral Calculus,
Clifford's Kinematic, Forsyth's Differential Equations, Boole's Differential Equations,
Boole's Calculus of Finite Differences, Merriman's Method of Least Squares.

II. A course preparatory to original investigation in the subjective sciences. This
will include projective geometry, the theory of numbers, the algebra of logic, the
theory of probability, non-Euclidian geometry.

Text-books, — Cremona's Projective Geometry ; Lejenne Dirichlet's Zahlentheorief 3d
Ed.; Macfarlane's Algebra of Logic ; Boole's Laws of Thought; Todhunter's History
of the Theory of Probability ; Frischauf s Absolute Geometrie,

The catalogue for 1887-88 gives one student taking postgraduate
studies in mathematics.

The university is open to both sexes. ^^A number of young ladies
still show that they are capable of mastering even the abstruse modem
developments of this oldest of the sciences." (Professor Halsted, June,
1888.)

WASHINGTON UNIVEESITT.

Up to the date of writing we have not been able to secure the infor-
mation desirable for a sketch of the mathematical teaching at this uni-
versity, but an excellent biographical notice of Professor William

240 TEACHIKG AND HISTOBT OF MATHEBIATICS.

Ghaavenet, the first professor of mathematics at Washington IJniver-
sity, has been written for as by his son, Begis Ghaavenety now presi-
dent of the State School of Dlines, at Golden, Golo. Professor William
Ghauvenet ranks among the coryphaei of science in America. He and
BcDJamin Peirce have done more for the advancement of mathematical
and astronomical science, and for the raiding to a higher level of the in-
stmction in these subjects, than any other two Americans. It is oar
wish, on that account, to place before the reader a somewhat full sketch
of the life and works of Professor William Ghauvenet. The biograph-
ical notice above referred to is as follows :

*^ William Marc Ghauvenet, father of the subject of this sketch, was
bom at Narbonne, France, in 1790, and came to the United States in
1816. He was the youngest of four brothers, another of whom also
came to this country but has lefb no descendants. William Marc was
a man of education and culture, versed in several languages, and a con-
stant reader. He came to America, however, in connection with a manu-
facturing enterprise which had its headquarters in Kew York, with a
branch at Boston. The latter department was under Mr. Ghauvenet's
charge, and here he married, in 1819, Miss Mary B. Kerr, of Eoxbury.
This was before a heavy defalcation in the Kew York house, which
broke up the enterprise so badly that all investments in it proved to be
total losses. Mr. Ghauvenet having an idea that rural life would suit
his taste, bought a small farm close to Milford, Pike Gounty, Pa., and
it was here that his only child, William Ghauvenet, was bom, May 24,
1820.

<^ By the advice of friends Mr. Ghauvenet soon gave up his attempt
at farming, and settled in Philadelphia, where his son grew to man-
hood. His rapid progress at school attracted such attention from his
instructors, especially in mathematics, that his father easily yielded to
their advice, and sent him to Yale Gollege, where he graduated in 1840,
^fadle princepa ' in mathematics, and high in standing in all other
branches. The honorary societies, ^ Phi Delta Kappa ' and ^ Ghi Delta
Theta,' denoting respectively the fifteen of highest standing and the fif.
teen best writers of the class, each claimed him as a member.

^< Upon his return to his home be was, after a brief incumbency in a
subordinate position, appointed professor of mathematics in the Navy.
Late in 1841 he married Miss Catherine Hemple, of Philadelphia.
Shortly after this he served a brief term on a United States vessel, as
instructor to midshipmen, but did not go upon a foreign cruise, and was
soon detailed to the * Naval Asylum,' then situated at Philadelphia.
Here midshipmen were sent at that time, to receive instruction and
examinations, principally in mathematics and the theory of navigation.
The young professor was struck with the imperfections in the education
of naval officers, and it was very largely through his efforts, aided by
such influences ns he could bring to bear on the matter, that a commis-
sion was appointed to draft a plan for a fixed ' Naval Academy,' corre-

INFLUX OF FRENCH MATHEMATICS. 241

sponding to the Military Academy at West Point. Six naval officers
constituted this commission. Professor Chanvenet being of the number.
The appointment of so young a man (he was but twenty-four at the
time) on a commission of such importance indicates what must have
been his record, and the impression he made upon his seniors in years
and rank.

^^ The Naval Academy was formally called into existence in the year
1845, being located at Annapolis, Md. Professor Chauvenet was ap-
pointed to the chair of mathematics, and resided at the academy until
his resignation from the Navy in 1859.

<< It was not long after this change of residence that he began to plan
his work on trigonometry, which was published in 1850. Its title, <A
Treatise on Plane and Spherical Trigonometry,' partly indicated that
it was not a students' class* book merely, but that it took up most of the
more advanced applications of the subject. It soon assumed the posi*
tion it still retains as the standard reference work in its line.

'^ Some time before this publication, Professor Ohauvenet had per-
suaded his father to retire from business and accept a position at the
'academy. He came as instructor in the French language, and remained
at his post until his death in 1865.

^^ It having been decided to erect an astronomical observatory at the
academy, Professor Ohauvenet was made professor of astronomy and
put in charge of the observatory. As he became more and more inter-
ested in his work, the idea of his next treatise, ^Spherical and Practical
Astronomy,' grew upon him, and, just previous to his resignation, had
assumed such form that he issued a prospectus for its publication as a
subscription work. This was never canied out.

^< In 1859 he was notified that his application for the professorship of
mathematics at Yale College would be followed by his election to that
position. Almost simultaneously with this came a call to St. Louis,
Mo., where he was offeifd the same chair in the then newly-established
Washington University. After much deliberation he accepted the
latter, and removed with his family (including at that time his mother)
to St. Louis, in the fall of 1859.

<^ Chancellor floyt, who was at the head of the ^ Washington ' at this
time, died early in the ^ sixties,' and Professor Chauvenet was elected
to the vacancy. He still continued his duties as professor of mathematics,
however, and now resumed his work on the ^Astronomy.' The risks
of publication were great, and his means did not enable him to guar-
antee the publishers against loss. The Civil War was in progress, and
the time seemed inopportune for such an undertaking. It was to the
liberality of certain friends, chiefly to the initiative of Mr. (afterward
Judge) Thomas T. Oantt, of the St. Louis bar, that a guarantee fund
was raised, sufficient in the opinion of the publishers to prevent any
loss to them. The work, in two octavo volumes, was published in 1863.

^< Few works of a scientific nature, by American authors, have been
881— No. 3 16

242 TEACHING AND HISTORY OF MATHEKATICS.

received with sach universal favor, by those competent to jndge of its
merits, as was this. Its reputation was qaite as great in Europe as
here, while of coarse it is not (as it was never intended to be) a treatise
much known outside of scientific, and more especially astronomical, cir-
cles. Its scope, and the rigorous methods adopted, are sufficiently
indicated in the author's preface. It retains to-day its standard char-
acter, as fully as when this was first recognized by the soientiAo world
upon its publication.

^^ Professor Ohauvenet^s mother died in St. Louis, not long after the
appearance of the Astronomy, and it was but a few mouths later that
the first symptoms of the disease that proved finally fataljx) him, made
their appearance. Partial recovery and resumption of his duties was
followed by a long period of alternating hopes and fears, during which
time he tried in vain difTerent parts of the United States, from South
Carolina to Minnesota. Daring this illness he worked at his only ele-
mentary publication, the 'Geometry,' which he undertook, partly because
he had long thought that the popular texts of the day were marked by
too strict an adherence to strictly ' Euclidian ' methods, and partly be-
cause he wished to provide an income for his family, by the publication
of a text for which he had reason to suppose there would be a larger
sale than was possible with advanced treatises. The publication of
this work shortly antedated his death, which occarred at Stt Paul)
Minn., December 13, 1870»

<^ Professor Ohauvenet left, so to speak, two distinct impressions be-
hind him. By far the larger circle, in numbers, of those who knew him,
were of those to whom his scientific attainments, though known, were as
traditions merely, since they were in a field whose extent was to them
only a matter of vague conjecture. To these he left the impression of a
man of wide and varied culture, and keen critical taste. Probably few
scientists of distinction were more keenly interested in lines outside of
their own specialties. He was not only a crflic in music, but to his
latest day a pianist of no mean ability, always expressing a preference,
in his own playing, for the works of Beethoven, which he rendered with
an interpretation which never failed to excite the admiration of musi-
cians whose execution surpassed his own. His knowledge of English
literature was extensive, but he read and re-read a few authors, at least
in the latter part of his life, and his great familiarity with many of these
gave point to the old adage, < fear the man of few books,' though perhaps
not in the sense in which these words were originally intended. He was
a ready writer, and contributed at times reviews, partly scientiflo, to
various journals. His style was clear and unaffected, while, in the re-
view of a pretentious or ignorant author, he had the gift of a delicate
saroasm, so light at times as only to be visible to one reading between
the lines. For other pretenders he could drop this mask, and write with
severity ; but only twice in his life, to the knowledge of the present
writer, did he ever do so. In addition to his more important writings,

INFLUX OF FRENCH MATHEMATICS. 243

he was the author of a ^ Lunar Method,' still used in the Kavy, aud in*
vented a device called the * great circle protractor,' by which the naviga-
tor is enabled (knowing his position) to lay down his course on a ^great
circle' of the globe, without further calculation. This invention was
purchased by the United States Government not long after the close of
the Civil War.

<< Professor Ohauvenet's scientific reputation needs littie comment on
the part of the present writer. He was one of a group of scientists in
his own or cognate lines, who were the first to secure recognition abroad,
as well as at home, for the position of the exact sciences in the United
States. Among his more intimate scientific friends were Benjamin
Peiroe and Wolcott Oibbs (Harvard), Dr. B. A. Gould, and many others
whose names are as household words in the history of scientific prog-
ress in this country. At the formation of the National Academy of
Sciences he was one of the prominent members. But while his scientific
reputation will outiast his personal memory, it is doubtful if to those
who knew him, even of his scientific associates, it will ever be as pres-
ent as his strong personal attractiveness, the result at once of an easy
and varied culture, and of a simple dignity of character, which im*
pressed alike his family, his friends, and his pupils. His family, con-
sisting at the time of his death of his wifCj four sons, and a daughter,
are all still living (1889)."

The only mathematical book written by Chauvenet and not mentioned
in the above sketch is a little book entitled Binomial Theorem and
Logarithms, published in 1843 for the use of midshipmen at the I^aval
School, Philadelphia.

As regards the quality of Professor Ohauvenet's books, Prof. T. H.
SafGordf of Williams College, says : <^ This excellent man and lucid
writer was admirably adapted to promote mathematical study in this
country. His father, a Frenchman of much culture, trained him very
thoroughly in the knowledge of the French language, even in its niceties.
They habitually corresponded in that language ; and the son wad en-
abled to study the mathematical writings of his ancestral country in a
way which enabled him to reproduce in English their ease and grace of
style, as well as their matter. In these respects his works are far more
attractive than those of ordinary English writers; his Trigonometry is
much the best work on the subject which I know of in any language ;
his Spherical and Practical Astronomy is frequentiy quoted by eminent
continental astronomers ; and his Geometry has raised the standard of
our ordinary text- books, of which it is by far the best existing."*

Ghauvenet's books, especially his Geometry, have been used in the
beet of our schools. Becentiy a revised edition of his Geometry has been
brought out by Professor Byerly, of Harvard. Among the chief modi-
fications made by him are the following : (1) The ^^ exercises," which

* Mathematical Teaohing, by Prof. T. H. ^afford, iaa7| p. 9.

244 TEACHma Ain> HISTOBY of MATHElfATIOS.

in the original are at the end of the book, are most of them plaoed in
direct connection with the theorems which they serve to illustrate. (2)
The admirable little chapter in the original edition on *^ Modern Geom-
etry" is omitted. (3) The ^^directional method'' is introduced. The
first is, no doubt^ a change for the better ; the second and third are, we
think| to be regretted. It seems to us that the day has come when a
college course should set aside some little time to the study of modem
methods in elementary geometry, and not confine itself to the andent.
The introduction of the '^ directional method^'' in our opinion, robs the
book of some of that admirable rigor for which the original work of
Chauvenet is so justly celebrated.

His Trigonometry and Astronomy are the first American works to
introduce the consideration of the general spherical triangle^ in which the
six parts of the triangle are not subjected to the condition that they
shall each be less than 180^, but may have any values less than 360^.
This feature is mainly due to Gauss. The methods of investigation fol-
lowed in these two books are chiefly those of the German school, of
which Bessel was the head.

UNIVERSITY OF MIOHiaAN.*

The IJniversity of Michigan opened in 1841 . In its organization Prus-
sian ideas predominated. But the regime which existed during the first
ten years in the history of the university did not prove efficient A
re-organization was therefore effected in 1852. The board of regents
were, from that time on, rendered independent of the Legislature by
intrusting their election to the people. The German method of govem-
in£: the faculty by an annual president elected by that body was aban-
doned in 1852, and it was henceforth the duty of the board of regents
to appoint a chancellor for the university.

The first appointment to a professorship at the University of Michigan
was that of George Palmer Williams, in 1841. He was first assigned to
the chair of ancient languages. On the work of this department, how-
ever, he did not enter, but exchaDged it for that of mathematics and
natural philosophy.

Professor Williams was born in Woodstock, Vt., in 1802. After grad-
uating at the University of Vermont he studied theology at Andover,
then became tutor at Kenyon College, and later professor of languages
in the Western University of Pennsylvania. Thence he returned to
Kenyon College, where he remained until 1837, when he entered upon
the services of the board of regents of the University of Michigan, as
principal of the Pontiac Branch.

At the University of Michigan he was professor of mathematics and
natural philosophy until 1854, professor of mathematics from 1854 to

*For part of the informatioD herein contained we are indebted to Prof. W.W.
Beman, of Ann Arbor. The writer is also ander obligation to Charles £. Lowx^yi
Ph. D., for interesting oral communicatioDB.

INFLUX OF FBBNCH MATHEMATICS* 245

18639 and professor of physics firom 1863 to 1875. Williams was a man
of caltare and refinement, and understood well the branches which he at*
'.tempted to teach. As an instructor he lacked thoroughness. < ^ Though
.lie never felt himself called upon to force the reluctant mind into a
bhorongh understanding of that for which it had no liking, he helped
those who desired to study in attaining to the established standard,
jtnd, in a private way, he loved to aid those who desired his help in
transcending that limit. Astronomy, though not nominally in his profes-
.sorship, he taught until the revision of the course in 1854, and a great
enthusiasm was annually awakened among the students as they came
to the calculation of eclipses."*

The mathematical requirements for admission were, in 1847, arith-
metic, and algebra through simple equations. The college course for
that year included algebra, geometry, conic sections, plane and spheri-
oal geometry, and calculus. In 1848 it was the same, save calculus or
analytical geometry, and in 1849 calculus atul analytical geometry. The
text-books were those of Professor Davies, of West Point.

Before its reorganization, in 1852, ^^the institution had flagged some-
what in popular interest ; the number of its students had fallen off ^ a
more vigorous and aggressive leadership was imperatively needed." t
In the year just named. Dr. Henry P. Tappan, of Kew York, was inau-
gurated first chancellor. His connection with the university marks a
new era in its history. During the reconstruction, German ideals were
constantly kept in view. Ho thoroughly understood the workings of
German universities and was a recognized champion among us of uni-
versity education, as distinguished from college education. In the first
catalogue (1852-53) issued by him, we read : <<An institution can not
deserve the name of a university which does not aim, in all the ma-
terial of learning, in the professorships which it establishes, and in the
whole scope of its provisions, to make it possible for every student to
study what he pleases and to any extent he pleases. It is proposed,
therefore, at as early a day as practicable, to open courses of lectures
for those who have graduated at this or other institutions, and for
those who in other ways have made such preparation as may enable
them to attend upon them with advantage. These lectures, in accord-
ance with the educational systems of Germany and France, will form
the proper development of the university, in distinction to the college
or gymnasium now in operation." The university system has been
growing at Ann Arbor, though at first very slowly.

The first fruits of the plan laid down in the catalogue just named was
the appointment to the chair of astronomy, in 1854, of Dr. Francis
Briinnow, of Leipsic, a favorite pupil and assistant of the celebrated
astronomer Encke. BrUnnow remained at the university until 1863,
when he resigned to take charge of the Dudley Observatory. Later, he

* University of Michigan, by Andrew T. Brook, 1875, p. 298.

t The Stndy of History in ▲merioan Colleges, by Herbert B. AdAms, p. 90. •

246 TEACHING AND EtlSTOBY OF MATHEMATICS.

became direotorof the Boyal Observatory in Dablin, Ireland. Under
Mb able management the observatory at the University of Michigan
(called the Detroit Observatory, in recognition of the liberality of citi-
sBens of Detroit who founded it) soon rose to high rank. Besides the
^< Tables of Flora" and the «< Tables of Victoria,'' published at Ann
Arbor, Dr. Bdinnow contributed to science his large work on Spherical
Astronomy and many papers on astronomical subjects. But the influ*
ence of its renowned scholar was felt also in the department of pure
mathematics. It is he who gave the university its start mathematic*
ally. When Professor Olney became a member of the faculty, then the
, university l^ad already made a respectable beginning in the study of
exact science.

The year 1856 marks the earliest dawn of the ^' elective system " at
the University of Michigan. One of the elective studies offered to Sen-
iors in that year was astronomy. Professor Briinnow lectured on this
subject to an elective class of one — James 0. Watson.* With refer*
ence to this class Dr. White happily said, that ^* that was the best
audience that any professor in Michigan University ever had." Briin-
now, with his pupil Watson, reminds us of Gauss, of 06ttingen, who
lectured at that great university to less than half a dozen students,
while Thibant, a mathematician of no scientlflo standing, presented the
elements of mathematics to audiences of hundreds. ^< If I had the
choice," said Hankel, ^< I should prefer being Gauss to Thibaut." If
we had the choice, we should prefer being a Briinnow lecturing to one
or two Watsons, rather than being very ordinary teachers lecturing to
, large classes of easy-going students.

Watson was born in upper Oanada in 1838. He early exhibited ex-
traordinary mental power and activity. When the lad was twelve his
parents were anxiously casting about to secure for him the privileges
of a liberal education. They looked eastward to Toronto and westward
to Michigan. Being in humble circumstances, they chose the latter
place, because education there was free. Young Watson entered at the
Ann Arbor High School, but after an attendance of one day and a half
he was graduated, for it was found that in the sciences he was alto-
gether beyond anything which his teachers had thought of. The pov-
erty of his parents made it necessary for him to partly rely upon his
own support. At this time the future astronomer could be seen going
about sawing wood for boys in college, while his mother took in wash*
ing to support herself and boy. At the university Watson displayed
as much talent for languages as he did for mathematics. The story
goes that he decided between mathematics and Greek, as his specialty,
by throwing a penny. << There slips the penny, for which t '^ A notice-
able exploit in the Junior year was his reading the entire M^canique

* Our remarks on Profeasor Wation are drawn chiefly from an address delivered by
Prof. J. C. Freeman, of tiia U&iVBnitiy of Witoonain, and printed in the J&gis, YoL
J, No. 37| JoiM S4| 1887*

INFLUX OP. FRENCH MATHEMATICS. 247

Celeste of La Place. In the Senior year he took the coarse of lectures
under Briinnow, spoken of above.

While yet very young, Watson contributed numerous astronomical
and mathematical articles to foreign journals. He published in 1867,
at the age of twenty-nine, his great work on Theoretical Astronomy.
Its design appears from these prefatory words: ^< Having carefully read
the works of the great masters, my plan was to prepare a complete
work on the subject, commencing with the fundamental principles of
dynamics and systematically treating, from one. point of view, all the
problems presented.^ The book gives a systematic derivation of the
formulee for calculating the geocentric and heliocentric places, and de-
termining orbits, and for computing special perturbations, including
also the method of least squares, together with a collection of auxiliary
tables, etc. The work was translated into continental languages and
became the text-book in many observatories in Oermany, France, and
England.

When Briinnow left Ann Arbor, in 1863, Watson became his successor.
Watson discovered a considerable number of Asteroids. Twenty -three
times, says Professor Freeman, he knew the joy felt by

** Some watcher of the skies
When a new planet swims into his ken.''

He was led to believe that there existed between Mercury and the
sun a planet hitherto unknown. During his observation of the eclipse
in 1878, at Denver, he caught sight, as he thought, of this new planet.

Watson's genius made the University of Michigan known in scientific
circles throughout the world. His mind was pre-eminently fitted for
his specialty. With a powerful memory and great mechanical genius,
he combined the ability to grasp abstruse problems by a kind of intui-
tion.. He was a man of wonderful activity. Says Professor Freeman :
^^ There was a tireless energy in the man that impressed every beholder.
Some of you recall the foiling you had when Grant or Sherman joined
the army in the field, or when yon saw Sheridan making his last mile
from Winchester to Gedar Greek. Something of the same inspiration
Watson gave his associates.''

During his directorship of the observatory, Watson generally deliv-
ered every year to the student community a course of popular lectures,
but was otherwise relieved from further duties of giving instruction,
excepting to pupils intending to make astronomy their specialty. He
had little patience with the average boy, but his interest in his special
students never flagged. He took great pains to secure for them suita-
ble positions. Old pupils of his may be found holding responsible posi*
tions in the U. S. Navy, Patent Office, and Coast Survey. His two
most favorite pupils were George 0. Oomstock and John Martin Schae-
berly. Watson took the former with him when he left Ann Arbor, in
1879, to take charge of the Washburn Observatory at the University
of Wisconsin. Mr, Scbaeberly remained at the Detroit Observatory until

248 TEAGHma and hibtoby of mathematics.

1888, when he accepted a place at the Lick Observatory. He was sac-
ceeded at Ann Arbor by W. W. Oampbell. After Watson left Ann
Arbor, Prof. Mark W. Harrington became director of the observatory
there.

Daring his first years after graduation, Watson taught, besides
astronomy, mathematics and physics. Thus, from 1859 to 1860 he was
professor of astronomy and instrnctor in mathematics ; from 1860 to
1863, instructor in physics and mathematics. Other young instructors
in mathematics of thijs time were W. P. Trowbridge, 1856 to 1857, a
graduate of the U. S. Military Academy ; and John Emory Clark, 1857
to 1859. Both of them became connected, later, with Yale College, the
former as professor of mechanical engineering, the latter as professor
of mathematics. These young men did much, no doubt, to supply that
thoroughness which was wanting in the teaching of Professor Williams,
the regular professor of mathematics. A beneficial stimulus to the
study of pure mathematics was exerted by the department of engineer-
ing; for good work in that department was impossible without good
preliminary instruction in pure mathematics. Oonnected with the de*
partment of civil engineering, from 1855 to 1857, was W lUiam Guy
Peck, a graduate of West Point He was succeeded by Be Yolson
Wood, who had just graduated at the Bensselaer Polytechnic Institute.
After leaving the University of Michigan, in 1872, Wood became pro-
fessor of mathematics and mechanics in the Stevens Institute of Tech-
nology. He is the author of Besistance of Materials, Boofs and Bridges,
Elementary Mechanics, Analytical Mechanics, revised edition of Ma-
han's Civil Engineering, and Elements of Coordinate Geometry (includ-
ing Cartesian Geometry, Quaternions, and Modem Geometry). Pro-
fessor Wood's text-books contain numerous examples to be worked by
the student. These books possess many good features, and have been
used quite extensively in our colleges and technical schools. Professor
Wood has been a very diligent contributor to a large number of mathe-
matical and scientific periodicals, and has. thereby done much toward
stimulating interest and activity in applied mathematics.

The year 1863 is marked in the history of the University of Michigan
by the departure of Bninnow and the arrival of Olney. Pro! Edward
Olney occupied the chair of mathematics until his death, in 1887. He
was bom in Moreau, Saratoga County, N. Y., in 1827. With slender
opportunities for early education, he achieved through his own energy
distinction as a teacher and scholar. He began his career as a teacher
in elementary schools. Though he had himself never studied Latin, he
began teaching it and he kept ahead of his class, << because he had more
application.'' He thus educated himself in languages as well as iu
mathematics. He acquired great teaching power, and it is to this that
his great success is chiefly due. During the ten years preceding his
appointment at Ann Arbor, he was professor at Kalamazoo College,
Michigan.

INFLUX OF FBEKGH MATHEMATICS. 249

At the University of Miohigan his teaching was marked by great
thoroughness. He was a rather slow man, a(ud took great pains with
the poorer students. He had the happy facnlty of indncing all students
to perform faithful work. It is related that the son of a certain pronv*
inent Congressman once labored under the conceit that his father's repu-
tation would exempt him from the necessity of studying whenever he
felt disinclined to do so. Once, when being called upon to recite, he
answered, '< not prepared." Professor Olney assured him that the lesson
was easy, asked him to rise from his seat, and then proceeded, much to
the amusement of the rest of the class, to develop with him the entire
lesson of the day by asking him questions. In that way was spent the
whole hour. The class was made to assist him in some of the more dif-
ficult points. The Congressman's son concluded, on that occasion, that
it was, after all, more agreeable to his feelings to prepare his mathe*
matics carefully in his own room than to expose his ignorance before
the whole class by being kept reciting for a whole hoar. At times Fro«
fessor Olney enjoyed joking at the expense of those who would not be
injured by it. The result of his teaching was a high average standing
among students. The first important step toward reaching good
results consisted in a strict adherence to the requirements laid down
for admission. If a student failed in his entrance examination, then
Professor Olney took much pains to see that the deficiencies would be
made up under a competent private teacher who was i>ersonally known
to him. The rigid requirements for admission gave the mathematical
department great leverage.

Professor Olney was an active promoter of various humanitarian
enterprises, and was much interested in the educational work of the
Baptist denomination, of which he was a member. He was interested
in the progress of Kalamazoo College (Baptist) quite as much as in
that of Michigan University. His library is now the property of that
college. At the time of his death he was engaged in the revision of
his series of text-books to meet the increased demands of the times.

In 1860, before Olney was connected with the university, the terms
for admission were— -to the classical course, arithmetic, and algebra
through simple equations ; to the scientific course, arithmetic, algebra
through quadratic equations and radicals, and the first and third books
of Bavies' Legendre. In 1864 quadratic equations were added to the
classical course, and to the scientific course the fourth book of Legen-
dre. In 1867 the requirements for the classical course were raised so
as to equal those in the scientific course, but in the tbllowing year quad-
ratic equations were temporarily withdrawn. The fifth book of Legen-
dre was added in the scientific course in 1869. In 1870 all of Legendre
was required, and five books in the classical course. In the' next year
arithmetic, Olney's Complete Algebra, and Parts I and II of Olney's
Geometry (including plane, solid, and spherical geometry), were the
requirements in both courses. No changes have been made since.

250 TEACHING AND HISTORY OP MATHEMATICS.

The college cnrricalam in 1854 was, for both coarses, algebra^ geom-
etry, trigonometry, analytical geometry, and calculus. The next year
calculus was withdrawn from the classical course, but was reinstated
in 1864, and in 1868 was made elective. In 1878 all courses except
those for the degree of B. L. (English) embraced calculus. In 1881 the
1^. L. course included trigonometry. Since then calculus has been
elective in all courses except the scientific. Analytical geometry has
been added to the B. L. course.

During the last eight or ten years the ^' university system" has been
growing rapidly at Ann Arbor. Mathematical studies of university
grade have been offered. Determinants, quaternions, and modern ana-
lytical geometry were first announced in 1878; higher algebra in 1879 ;
synthetic geometry and elliptic functions in 1885; theory of functions in
1886 ; differential equations (advanced) in 1887. The calculus of varia-
tions (probably as much as is contained in Church's or Courtenay's
Calculus) was announced first in 1866.

The text-books which have been used at the University of Michigan,
at different periods, are as follows :

Algebra, — Davies' Bourdon, Ray's— Part II, Olney's Univerfiity Algebra, KewoomVa
College Algebra, Chas. Smith's Treatise on Algebra, Salmon's Higher Algebra, Bora-
side and Panton.

DetermmanU.^-lixjiiTf Scott, Dostor, Peck.

Geometry, — Davies' Legendre, Olney, Ray,

Trigonometry, — Dayies' Legendre, Loomis, Olney.

Synthetio Oeometry. —Reye, Steiner.

Analytic Qeomeiry.—Dayiea, Loomis, Chnrch, Olney, Peirce's Carves, FonctionBi
and Forces, Chas. Smith, Salmon, Frost, Aldis, Wbitworth, Clebsch.

Calculus,— DsmeSf Charch, Loomis, Conrtenay, Olney, Price, Todhanter, William-
son, Jordan.

Differential Equations, —Bodlef Forsyth.

Caloulua of FaHa<ton«. ->Tudhanter, Carll.

9iki«0nticms.— Eelland <& Tait, Hardy, Tait.

Elliptic jPVmottoiitf.— Dur^ge, Bobek, Jordan.

Prof. 6. G. Comstocky of the Washburn Observatory, gives the fol-
lowing reminiscences of the mathematical instruction at Ann Arbor:*

<* I entered the University of Michigan in the fall of 1873, with a
preparation in mathematics consisting of arithmetic, elementary algebra
through quadratic equations and including a very hurried view of
logarithms, and plane, solid, and spherical geometry. The mathemati-
cal course given in the university at that time comprised, in the Fresh-
man year, Olney's University Algebra, inventive geometry (consisting
of an assignment of theorems for which the student was expected to
And demonstrations), and plane and 6X)herical trigonometry. In the
Sophomore year, general geometry and differential and integral calcu-
lus. Des&iptive geometry was required of engineering students, and
was occasionally taught to others.

• Letter to the writer, November 6, 1888.

INFLUX OF FRENCH MATHEMATICS. 251

" The Freshmen were taught by instructora, nsaally yonng men of not
much experience in teaching, but once a week they (the students) went
up to Professor Olney for a review of the week's work, and these occa-
sions were the trials of a Freshman's life. Olney's stern and rigid discipline
had won for him among students the sobriquet " Old Toughy.'' He was
not, howerer, a harsh man, and although the students stood in awo of
him I think that he was generally liked by them. One feature of the
weekly reviews may serve to illustrate his discipline and his power of
enforcing it. He insisted upon the attention of each student being given
to the demonstrations and explanations which the person reciting was
engaged upon, and given so cl(^ely that the latter might be stopped at
any point and any other student required to take up the demonstration
at that point and carry it on without duplicating anjrthing which had
already been given.

*'The University Algebra given the Freshman class contained an
elementary view of infinitesimals, extending to the differentiation of
algebraic functions and the use of Taylor's formula ; and also a presen*
tation of lOci of equations, by which the student became familiar with
the geometrical representation of an equation. The Sophomore thus
came to this study of general'^geometry and calculus with some prelim-
inary notions of these subjects. The study of the calculus was elective,
but every Sophomore was required to take an elementary course in gen-
eral geometry, and to make use here of the principles of the calculus
which he had learned as a Freshman.

<< Professor Olney's tastes were decidedly geometrical in character,
and he constantly sought to translate analytical expressions into their
geometrical equivalents, and much of his success as a teacher is prob-
ably due to this.

*< Professor Beman, on the other hand, is a^ analyst, a * lightning
mathematician ' in the student vernacular, and, in my day, the facility
with which he handled mathematical expressions dazed and discouraged
the student, who usually felt that he did not get much from Professor
Beman.

<< The criticism which I should now make upon the mathematical
teaching which I received, is that little or no attempt was made to point
out the applications of mathematics, and to encourage the student to
apply it to those numerous problems of physical science, of engineering,
and of navigation, which serve as powerful stimulants to the interest.
The student was taught how to solve a spherical triangle, and hQw to
look out logarithms from a table, but was never required to solve such
a triangle and obtain numerical results.

<^ The text-books in use were those written by Professor Olney, none
other being employed even for reference. There were no mathematical
clubs or seminaries, and no facilities offered for the study of mathe*
matios beyond the prescribed oanioalnm.^

252 TEACHIKQ AND HISTOBT OF MATHEMATICS.

Professor Olney is the aathor of a complete set of mathematical
text-books, which have displaced the works of Davies, Loomis, aud
Bobiason in many schools, both in the East and in the West. His
works are quite distinctive in the arrangement of subjects, and mark a
decided advance over the other books just named. In the explanatory
notes added here and there, in the tabular views at the end of chapters,
in the judicious selection of examples, we see the fruits of long experi-
ence in the class-room. His books exhibit him in the light of a great
teacher rather than a great mathematician. He was greatly aided in
his work by Professor Beman, who prepared all the "keys" to the
mathematical books, and did a great deal of critical work. It has been
stated that Professor Olney could never get his publishers to print the
books in the form which seemed the most perfect to him. He consid-
ered the traditional classification of mathematical subjects very defect-
ive, and wished to write a System of Mathematics in which he could
embody his own ideals on this point. He thought, for example, that
a considerable part of algebra should be taught before taking up the ad-
vanced parts of arithmetic, such as percentage and its applications,
and that plane geometry should precede mensaration in arithmetic
By discarding the usual division of mathematics into separate volumes
on arithmetic, algebra, geometry, etc., and by writing a system of math-
ematics he hoped to introduce great improvements. The publishers, on
the other hand, preferred the traditional classification, as the books
would then meet with larger sale. Professor Olney was thus hampered,
to some extent, in the execution of his ideal scheme.

In his published works, the science of geometry is brought under two
great heads, Special or Elementary Geometry, and General Gtoometiy.
The former consists of four parts : The First Part is an empirical geom-
etry, designed as an introduction, in which the fundamental facts are
illustrated but not demonstrated. The Second Part contains the ele-
ments of demonstrative geometry, designed for schools of lower grade.
The Third Part was written to meet the special needs at the University
of Michigan. It was studied in the Freshman class by students who
had mastered the Second Part. The effort is made here to encourage
original research. This part contains also applications of algebra to
geometry, and an introduction to modern geometry. The Fourth Part
consists of plane and spherical trigonometry, treated geometrically.
The old " line-system " is still retained here.

General Geometry was intended to be developed by him in two sep-
arate volumes, but only the first was published. The first treats of
plane loci, the second was intended for loci in space. This first vol-
ume may be very roughly described as covering the field generally oc-
cupied by analytical geometry and calculus. Olney favored the in-
finitesimal method, which he used also in his Elementary Gtoometiy,
where he permits the number of sides of a regular polygon ciroom-
scribed about a circle to beoome ^< infinite,'' and to coincide with the

INFLUX OP FRENCH MATHEMATICS. 253

circle. We are glad that this method is at the present time being more
and more eliminated from elementary text-books. It is worthy of note
that in his calculus Olney gives the elegant method, discovered by
Prof. James 0. Watson, of demonstrating the rule for differentiating a
logarithm without the use of series.

In some courses the subjects have been taught exclusively by lectures,
but the present tendency is to use the best text-book available, and
supplement it with lectures as may be found advisable. Of late years
a good deal of attention has been given to the careful and critical read-
ing of such works as Salmon^s Oonic Sections, Higher Algebra, Geom-
etry of Three Dimensions, Frost's Solid Geometry, Jordan's Cour8
WAnalyse^ Forsyth's Differential Equations, Price's Galculus, Garll's Oal-,
cuius of Yariations, Bumside and Panton's Theory of Equations, Beye's
Oeometrie der Lcbge^ Steiner's Vorlesungen uber synthetiscJie Qeometriej
Olebsch's Vorlesungen iiber Oeometrie der Ebene. It is thought that
better results have been secured in this way than when the student^s
attention is largely given to the taking of notes.

Since the death of Professor Olney, Professor Beman has been filling
the professorship of mathematics. He graduated at the University of
Michigan in 1870. Excepting the first year after graduation (when he
was Instructor in Greek at another institution), he has been teaching
continually at his alma mater — ^from 1871 to 1874 as instructor in math«
ematica, then as assistant professor and as associate professor of math-
ematics, and, finally, as full professor. He has done much toward
introducing the '' university system " in his department, and has been a
contributor to our mathematical journals, particularly to the Analyst
and the Annals of Mathematics.

For several years Oharles K Jones has been professor of applied
mathematics. He has been a very successful teacher of mechanics.
Professor Beman has two or three assistants in the department of pure
mathematics. *

A mathematical club was organized in 1887. It is under the control
of the students, but an active interest is continually shown by the
various instructors. Papers of some length are presented, problems
discussed, etc.

UNITEESITY OF WISCONSIN.

The University of Wisconsin was organized in 1848, and formally
opened in 1850. A preparatory department was established in 1849, and
it was not till 1851 that regular college classes were formed. Like most
other State universities, the University of Wisconsin had a hard strug-
gle for existence during its early years. Our State Legislatures did not
always pursue a wise course toward their higher institutions of learn-
ing. The lands which were granted to the States by the General Gov-
ernment for the support of higher education were disposed of in a man-
ner intended to <^ encourage immigration," rather than to foster a great

254 TEACHIKG AND HISTOEY OF MATHEMATICS.

university. Bat in later years, say since 1875, the jJolicy of the Wis-
consin Legislature has been much more liberal, and the university baa
been advancing with prodigious strides.

The first professor connected with the institution was John W. Ster-
ling. He was the teacher in the preparatory department, which was
started in 1849, in a small building, before the university had any large
buildings of its own. After the college department was organized,
Sterling became professor of mathematics and natural philosophy^ which
position he retained until about 1867, when a separate chair was created
for physics. From that time on until June, 1881, he was professor of
mathematics.

Professor Sterling was bom July 17, 1816, in Wyoming County, Pa.,
and died in March, 1885, at Madison, Wis. He was graduated at the
College of 'Sew Jersey in 1840, and at the Princeton Theological Semi-
nary in 1844. His mathematical and astronomical instruction At Prince-
ton must have been received from Prof. A. B. Dod and ProfL Stephen
Alexander. He went to Wisconsin in 1846, and became professor in
Carroll College, Wankesha: Three years later be entered upon bis long
career as professor at the University of Wisconsin. For one-third of a
century he was connected wiih that institution. Never did man work
more faithfully than did he for its advancement. When the university
was passing through the ^^Sturm und Drang Periode^^ and when it was
without a head, he more than once, as dean of tlie faculty, assumed the
duties of president. He was a man of industry and energy, and was
ready to teach any branch, on an emergency. Among the students he
was popular. He encouraged faint-hearted students, took them to his
table, lent or gave them money when he had little himself. He invaria-
bly treated students like gentlemen of mature judgment and common
sense. The great mass of stndents appreciated this, but occasionally
there were some too young to do so and who should have received
severer treatment and more summary action. In his prime Professor
Sterling was a man of great physical strength. During even bis last
years he walked as erect as a young man of twenty. .

He took a living interest in mathematics even daring the last days of
his life. Though he may not have kept pace with recent advances in
this science, he had a good knowledge of such subjects as were treated
in our ordinary American text-books. He ne^er published any works
of his own. When Professor Watson, the astronomer, came from Ann
Arbor to the University of Wisconsin, in order to take charge of the
magnificent new observatory erected by the munificence of Oovernor
Washburn, an agreement was contemplated or reached between Wat-
son and Sterling to prepare jointly a series of mathematical text-books.
Watson's wonderful mathematical talent and Sterling's long experience
in teaching would, indeed, have made a strong combination, but the
scheme was frustrated by the untimely death, in 1880, of the great
astronomer.

INFLUX OF FRENCH. HATHEMATIC8. 266

In the ckisB-rooih Sterling's discipline was oharaoterized .b3' great
mildness. He would oarefnlly explain to the class the principal parts
of each lesson. Even in the last year of his teaching his prelections were
always very clear, and any student who felt a desire to understand the
subjects which he tanght, could certainly do so by following the exposi-
tion given in the class. ^ While Professor Sterling always explained well,
he was, in his last years at least, not sufficiently exacting ; he would
not compel a boy to study. The consequence was that some got from
hinl a good knowledge of elementary mathematics, while others took
advantage of the professor's leniency. In calculus he taught both the
method of limits and the infinitesimal method. The text-book was
based on the former, but Professor Sterling rather fEivored the latter.
The principles of the calculus were not always unfolded with desired
rigor, and not uufrequently some of the best scholars in the class shook
their heads at the unceremonial rejection of quantities, simply because
they were very, very small.

Among students Professor Sterling went by the familiar name of
<^ Johnnie." In 1881 he was made professor emeritus of mathematics.
Though his active duties in the class-room ceased at that time, he con-
tinued to take a living interest in all matters pertaining to the univer-
sity to the end of his days.

We are not able to give the courses in mathematics during the early
days of the university. During the last years of his teaching Professor
Sterling used Loomis's works throughout. From a communication
received by the writer from Prof. James D. Butler, it would appear that
the works of Loomis were the first ones taught in pure mathematics by
Professor Sterling. In algebra he used Loomis, afterward Davies, and
then again Loomis. In conic sections he used at one time the work of
Coffin. <^ Smith's Analytic Geometry" is also one of the books men-
tioned. This was most likely F. H. Smith's translation of Biot. Other
books me'ntioned are Peck's Mechanics, Bobinson's Astronomy, Snell's
Olmsted's Astronomy, Snell's Olmsted in optics and pneumatics, and
Loomis's Calculus.

In 187&<-77 the Fresnmen studied Loomis's Algebra, beginning with
quadratics, Loomis's Geometry, Loomis's Plane Trigonometry. The
Sophomores were instructed in Loomis's Oonic Sections aud Analytic
Geometry, Practical Surveying (six weeks), and Oalculus, This ended
the course in pure mathematics. The Juniors were offered Peck's Me-
chanics and '^ lectures." The mathematical coarse for engineering stu-
dents embraced also descriptive geometry (Church). All students were
required to pursue mathematics through analytic geometry ; the cal-
culus was elective except to students in civil and mechanical engineer-
ing. Until about the year 1878, William J. L. Nioodemus was professor
of military science and civil and mechanical engineering. He was
spoken of by students as a man of great ability in his liue. On the
death of JSTicodemus one of his pupilsi Allan Darst Oonover, asBomed

256 TEAOHIKG AND HISTORY OF MATHEMATICa

oharge of the department of civil engineering. In mechanical engi-
neering the instruction fell into the hands of Storm Bull, a relative of
the celebrated Ole Ball. Prof. Storm Ball stadied at the Polytechni-
cam in Zurich, Switzerland, and is a thoroagh master of the subjects
which he teaches. Descriptive geometry has been taught by him ever
since his connection with the university.

On Sterling's retirement the management of the mathematical de-
partment was entrusted to Charles A. Van Yelzer, a young man who
for three years had listened to the inspiring words of Professor Sylves-
ter at the Johns Hopkins University. Van Yelzer graduated at Cor-
nell University in 1876. After having been instructor at his alma mater
for one year, he went to the Johns Hopkins University, where he was
honored with a fellowship in mathematics. - His power for original re-
search is exhibited in his contributions to the American Journal of
Mathematics (on ^< Compound Determinants "), the Johns Hopkins Uni-
versity Circulars, the Analyst, and the Mathematical Magazine.

In the fall of 1888 appeared, in two separate volumes, a preliminary
edition of Van Yelzer and Slichter's ^^ Course in Algebra." Slichter is
assistant professor of mathematics at the university. This book is
now used in the Freshman class. The preliminary edition was gotten
up for the purpose of being tested in the classroom. After the test,
such revisions will be made as experience may seem to require. In the
regular edition the two parts will be placed together in one volume.
The work is not intended for beginners, but for students entering the
Freshman class of our colleges, who already possess a tair knowledge
of the elements. It impresses the progressive teacher as being differ;
ent from most other works, and of great excellence. Many an anti-
quated and traditional notion has been thrown overboard, and many
new features have been introduced. They have not been adopted
simply for the sake of producing a book different from others ; on the
contrary, the authors have profited by what seemed good in other alge-
bras.

The first volume of Van Yelzer and Slichter's Algebra embraces, in
addition to the usual subjects, the theory of limits and derivatives.
In the treatment of series the authors not only state, but emphasize the
fact that infinite series must be convergent in order to be used with
safety. Some teachers might doubt the expediency of introducing
Taylor's Formula into a book on algebra on account of the difficulty
encountered in a complete and rigorous proof of it.

The second volume contains chapters on imaginaries, the discussion
of the rational integral function of Xj the solution of numerical equations
of higher degree, graphic representation of equations, and determinants.
These five chapters cover 75 pages. The treatment of these subjects
appears to us admirable. Not more is given on each subject than can
be conveniently taught in any college whose pupils possess a thorough
knowledge of algebra through quadratics before entering the Freshman

INFLUX OP FRENCH MATHEMATICS. 267

class. A pleasant feature of the work is the occasional ^^ historical
notes." This is the first American work on algebra, as far as we know,
which states explicitly that the logarithms invented by Napier are dif-
ferent from the nataral logarithms.

The strongest feature of this algebra is its style. Students who have
been in Professor Van Velzer^s class-room will perceive that his great
power of oral explanation and elucidation has been happily transferred
to the printed page, l^o where is the language of the book above the
comprehension of ordinary students. The objective method of explana«
tion is adopted throughout.

To Professor Van Velzer belongs the credit of introducing the mod-
em higher mathematics into the University of Wisconsin. The writer
knows of a student of great taste for mathematics, who studied Loomis's
Calculus in the year preceding the arrival of Yan Velzer, in Madison,
and who labored under the impression that he had mastered about all
that was to be known in pure mathematics. He was no little surprised
when the new professor, fresh from the Johns Hopkins University, began
to talk about determinants, quaternions, theory of functions, theory of
numbers, and multiple algebra. The student's pride was wounded when
he learned that Loomis's Oalculus could convey only a very mtsagre
knowledge of the transcendental analysis.

In 1883 some alterations were made in the mathematical require-
ments for admission. During the years immediately preceding, the
requirements for all the courses of the university had been, arithmetic,
algebra through quadratic equations, and plane geometry. At the
time named above, solid geometry was added to the requisitions for all
regular courses in the university except the " ancient classical."

The university has established close and friendly relations with the
high schools in the State, and the number of '^ accredited high schools"
is now fifty-six.* Of these only six, however, prepare students for all
courses in the university. This intimate relation with the high schools
has had a wholesome influence upon both the university and the high
schools.

As regards the regular classes of mathematics in the college, we may
say that since the retirement of Professor Sterling, Loomis's Algebra
has been retained until 1888. IS'ow, Van Velzer and Slichter's Algebra
is used. In trigonometry, Wheeler's work was introduced in 1882, Be-
fore that time Loomis's was taught. In solid geometry, Wentworth's
has been used lately, in place of Loomis's. In analytical geometry,
Loomis's work was superseded some years ago by the English work of
Smith. In calculus. Professor Van Velzer has taught Byerly's, but this
year (1888-89) he is using Kewcomb's.

In the Sophomore year, and more especially in the Junior and Senior

years, the elective system has been in operation, with some restrictions.

Since 1881 elective studies in pure mathematics, covering the calculus

and other branches, have been offered every year. There have always

881— No. 3 17

258 TEACHING AND HISTOBT OF MATHEMATICS.

been stadents with a taste for the higher branches of mathematics. In
later years the aitendanoe upon these branches has been on the inereaae.
In the winter term of 1881-*82 determinants were taagbt for the first
time at the University of Wisconsin. In the spring term was organized
a class of five or six students in quaternions. Hardy's text-book was
used. Lately Professor Van Yelzer has preferred the work by Kelland
and Tait. During the year 1882-83 there was an elective class of about
the same number as the preceding, studying Boole's Differential Equa-
tions. Professor Yan Yelzer's constant aim was to induce studenta to
do independent work. He was always glad to listen to such modified
treatment of the lesson in the book as the student might think of. This
method of conducting the recitation gave rise to many interesting and
profitable discussions. Considerable time was given to the subject
of singular solutions. The work of Boole was still used in 1887-88|
but from now on, that of Forsyth will be used, the former being out of
print.

The special courses in pure mathematics during the last two years
have been as follows : Glass of two students in Boole's Differential Equa-
tions in the fall term of 188ft-87, three hours per week ; class of two in
the same text-book, winter term of 1886-87, two hours per week ; class
of six in modern algebra (no text-book), winter term of 188&«87, three
hours per week ; class of six in Boole's Differential Equations, winter
term of 1887-*88, three hours per week ; class of three in Boolc^s Di£Eer-
ential Equations, spring term of 1887-88, three hours per week ; class
of six in Kelland and Tait's Quaternions, spring term of 1887-88, three
hours per week ; class of seven in Smith's Analytical Geometry of Three
Dimensions, fall term of 1887-88, three hours per week ; class of three
in quantics (Salmon), fall term of 1887-88, two hours per week.

It is the practice at the University of Wisconsin to give special honors,
upon the recommendation of the professors iu the several departments,
to the candidates for the bachelor's degree who have done special work
under the direction of the professor of any department and prepared
an acceptable thesis ; but the amount of work required for a special
honor must be at least the equivalent of a full study for one term, and
in case of those branches in which there are longer or shorter elective
courses, the student must have taken the longer course. It has been
specified, farthermore, that candidates for special honors must have a
general average standing of 85, and one of 93 per cent, in the de-
partment of which the application is made.

The number of special-honor students in mathematics in late years
has been quite as great, if not greater, than in any other department,
though the studies of this department are, to say the least, as difficult
as those of any other. In the class of '83 there were three special-honor
men in mathematics. The titles of their theses were as follows : ^ Singu-
lar Solutions of Differential Equations," <^Pole and Polar and Bedpzocal

INFLUX OF FRENCH liATH£MATIOS. 259

Polars in Oarres and Surfaces of the Second Order," and '< Development
and Dissection of Eiemann's Surfaces." Should it be claimed that these
theses are the work of immature students, then we may answer that
for candidates fpr the bachelor's degree they are nevertheless creditable.
The writer of the first thesis (L. M. Hoskins) is now doing excellent work
as instructor in engineering at the university. The writer of the second
thesis (L. S. Hulburt) is now professor of mathematics at the univer-
sity of Dakota. M. UpdegrafF, of the class of '84, wrote a thesis on
<^ Besultants." He holds now a responsible position at the National
Observatory at Oordoba, Argentine Eepublic, South America. Titles
of later " special-honor theses " are, ^'Approximation to the Boots of
ITumerical Equations," ^' Maxima and Minima," '^ On the Equation
sin my. cos ny = sin mx. cos n^?," <' Different Systems of Oo-ordinates."
These theses are certainly indicative of a healthful activity in the
under-graduate mathematical department.

For special studies pursued after graduation and the presentation of
an acceptable thesis, the degree of Master is conferred. The following
are the titles of two theses written to secure the degree of ^* master of
science in mathematics : " ^^ The Hodograph," '< On a Quadratic Form"
(in the theory of numbers).

The present courses in mathematics offered at the University of Wis-
consin (catalogue 1887-88) are as follows :

Sabcourse I, Algebra, Five exercises a week during the fall term. (Professor
Van Velzer and Mr. Siichter.)
Bequired ofFreahmm in all oowBes,

Sabcoarse 11, Theory of EqtMtiona, including the elements of determinants, and
icraphio algebra. Five exercises a week during the winter term. (Professor Van
Yelser and Mr. Siichter.)
Required of Freshmen in the Modern Claeeioal, English, General Soienoe, and Engineering

Couraee.

Saboonrse III, Solid Geometry. Five exercises a week daring the winter term.
(Professor Van Velzer or Mr. Siichter. )
Eequired of Dreehmen in the Ancient Claaaioal Course.

Saboonrse IV, Trigonometry, Five exercises a week dnjing the spring term. (Pro-
fessor Van Velzer and Mr. Siichter.)
Bequired of Freshmen in all courses.

Sabcoarse V, Descriptive Geometry. The topics taaght embrace the projection of
lines, planes, surfaces, and solids, the intersection of each of these with any one of
the others, tangent lines to curves and surfaces and tangent planes to surfaces, prob-
lems in shades and shadows, of lines and surfiEices, linear perspective and isometric
projection. The class-room exercises are accompanied by work in the draughting
room. The text-book used is Church's Descriptive Geometry. Full study during the
spring term, Freshman year, and three-fifths study during the fall term, Sophomore
year. (Professor Ball.)
Required of Freshmen in Civil and Mechanical Engineering. Elective for other students.

SubcoarseVI, Analytic Geometry. Five exercises a week daring the fall term.
(Professor Van Velzer.)
Bequired of engineering Sophomores and scientifio Sophomores who pursue mathenMtical,

physical, or astronomical studies. Elective for other students.

260 TEACHINQ AND HISTORY OF MATHEMATICS.

Sabcoarse VII, Differential Caloulua, Five exercises a week during the winter
term. (Professor Van Velzer.)
Required of engineeting Sophomores and soientifio Sophomoree who pureue maikematieal,

phyeicalf or astronomical studies. Elective for other students,

Sabcoarse YIII, Integral Calculus, Five exercises a week daring the spring term.
(Professor Van Velzer.)
Bequired of engineering Sophomores and seienHjio' Sophomores who pursue mathematioalt

p^ysieal, or astronomical studies. Elective for other students,

Sabcoarse XIX, Method of Least Squares, This is a coarse in the theory of probabil-
ities as applied to the adjustment of errors of observation. It will be first given In
1889. Mast be preceded by subcoorses VI, VII, and Villi three-fifths study daring
the winter term. (Mr. Hoskins.)
Bequired of Seniors in Civil Engineering,

Subcourses IX to Xvill, special advanced eleotives. Courses varying from year to
year are offered in the following subjects : IX, Modem Analyiio fp^eomeiry ; X, Higher
Flane Curves; XI, Geometry of Three Dimensions; XII, Differential [Equations; XIII,
Spherical Sarmonios ; XIV, Elliptic Functions; XY, Theory of Functions; XVI^ Theory
of Numbers; XVII, Quantics; and XVIII, Quaternions,

Very good work has been done, at times, by stadents in the depart-
ment of mathematical physics. Prof. John E. Davies, the professor of
physics, takes a living interest in pare as well as applied mathematics.
His reading in pure mathematics has, indeed, been very extensive.
Mathematical reading is a recreation to him. He wonld not nufte-
qaently take with him some mathematical work — as, for instance, Tait's
Qnatemions — to faculty meetings, that he might pass pleasantly the
otherwise tedious sessions of that august assembly. Many years ago he
made, for his own use, a complete translation of Koenigsberger's work
on Elliptic Functions.

The university offers excellent facilities for the study of astronomy.
The Washburn Observatory has a large equatorial for use in original
work, and also a smaller one for the use of students. After the death of
Professor Watson, Professor Holden became director of the Observatory. .
He held this position until his appointment as director of the Lick
Observatory. Prof. George 0. Gomstock is now professor of astron-
omy and associate director of the Wi^hburn Observatory. Professor
Gomstock is a pupil of Watson, and came to Wisconsin from Ann
Arbor with Watson. Before assuming the duties of his present posi-
tion he was for two or three years professor of mathematics and astron-
omy at the University of Ohio.

The instruction in analytical mechanics is in charge of Mr. L. M.
Hoskins, a young man of very marked mathematical talent. He grad-
uated in 1883 at the head of a class of sixty-five, and was afterward
appointed fellow in mathematics in Harvard University. Through his
influence, the study of analytical mechanics had been made much more
prominent in the engineering courses than it had been formerly. Two
terms are now devoted to it instead of only one. Bowser's Elements
of Analytical Mechanics is the text-book used.

Mr. Hoskins has contributed to the Annals of Mathematics, the
Mathematical MagazinCi and Yau Kostrand's Engineering, Magazine.

INFIilTX OP FRENCH MATHEMATICS, 261

JOHNS HOPKINS TJNIVEBSITY.

President Daniel C. Gilman once said to the trustees of the Johns
Hopkins University, when the question of " How to begin a university ^
was upon their cninds, '^Enlist a great mathematician and a distin-
guished Grecian ; your problem will be solved. Such men can teach ini
a dwelling-house as well as in a palace. Part of the apparatus they
will bring, part we will furnish. Other teachers will follow them."* So
it came to pass that, before there were any buildings for classes, a pro-
fessor of mathematics and a professor of Oreek were secured for the
new university.

When President Gilman was engaged in the all-important work of
selecting men for the above positions, he may h^ve been actuated in his
choice by thoughts similar to those of Prof. G. Chrystal, who, before a
learned body of English scientists, once expressed himself as follows : t
'^ Science can not live among the people, and scientific education can not
be more than a wordy rehearsal of dead text-books, unless we have living
contact with the working minds of living men. It takes the hand of God
to make a great mind, but contact with a great mind will make a little
mind greater. The most valuable instruction in any art or science is to
sit at the feet of a master, and the next best, to have contact of another
who has himself been so instructed."

Is there a student among us who has studied with Sylvester and who
will deny the truth of the above ! Is there a mathematician, who has
sat as a pupil at the feet of Benjamin Peirce, who will deny it ! It is a
fortunate circumstance for the progress of the exact sciences in this
country that, at a time when the " Father of American Mathematics "
was approaching his grave, there came among us another master who
gave the study of mathematics a fresh and powerful impulse. Profes-
sor Sylvester is a mathematical genius, who has no sux>eiior in Eng-
land, except, perhaps. Professor Cayley.

James Joseph Sylvester was born in London in 1814, and was edu-
cated at the University of Cambridge. He came to this country to fill
the professorship at the University of Virginia when he was a very
young man, but his stay among us then was very short. He became a
member of the Royal Society at the age of twenty-five. For some time
he was professor of natural philosophy in University College, London.
In 1855 he became professor. of mathematics in the Eoyal Military
Academy at Woolwich, and in 1876 was elected for the position at the
Johns Hopkins University.

Sylvester's activity has been wonderful. Prior to 1863 he published
112 scientific memoirs, which are recorded in the Royal Society's Index
of Scientific Papers. A most important paper, prilnted in the Philo-

* Annual report of the president of the Johns Hopkins Univeirsity, 1888, p. 29.
iNaturCf September 10, 1885, Section A of Brit. Assocation, opening address hy
Prof. G. Chrystal, president of the section.

262 TEACHING AND HISTORY OF MATHEMATICS.

sophical Transactions of 1864, is Sylvester's Theorem on Kewton's Bole
for discovering the number of real and imaginary roots of an eqaation.
Of this Todhnnter says : • " If we consider the intrinsic beanty of the
theorem, • • • the interest which belongs to the mle associated
with the great name of Kewton, and the long lapscl^of years daring
which the reason and extent of that mle remained nndiscovered by
mathematicians, among whom Maclanrin, Waring, and Euler are ex-
plicitly included, we must regard Professor Sylvester's invetftigations
made to the theory of equations in modern times jastly to be ranked
with those of Fourier^ Sturm, and Gauchy.'' A few of his nnmerons
other investigations, made before coming to Baltimore, are on the Bota-
tion of a Bigid Body ; on the Analytical Development of FresnePs
Optical Theory of Crystals ; on Be version of Series ; on the In volation
of Six Lines in Space, ^' culminating in the result that if these six lines
represent forces in equilibrium they must lie on a ruled cubic surface;"
on a general theorem by which, for instance, the quintic can be ex-
pressed as the sum of three fifth powers. In 1859 he gave a course of
lectures at King's College, London, on the subject of The Partitions of
Numbers and the Solution of Simultaneous Equations in Integers, in
which it fell to his lot ^^ to show how the difficulties might be overcome
which had previously baffled the efforts of mathematicians, and espe-
cially of one bearing no less venerable a name than that of Leonard
Euler," and also laid the basis of a method which has since been carried
out to a much greater extent in his " Constructive Theory of Partitions,"
published in the American Journal "of Mathematics, in writing which
he ^< received much valuable co-operation and material contributions"
from his " pupils in the Johns Hopkins University." t

Pro&ssor Sylvester's most celebrated work has been in modern higher
algebra. A very large portion of the theory of determinants is due to
him, and the epoch-making theory of invariants owes its origin and
early development almost exclusively to his genius and that of Pro-
fessor Oayley.

The Johns Hopkins University offered to Professor Sylvester eiery
facility for original work that could be desired. By the system of
" fellowships " a number of talented young men were drawn to Balti-
more, who were capable not only of understanding the teachings oi
their great master, but, in many cases, also of aiding him in his re-
searches. The univerp.ity, moreover, started the American Journal ol
Mathematics, in which all investigations in mathematics could be pub-
lishcd and thereby bvought before the mathematical public Piofessoi
Sylvester's time waF; not taken up by the usual routine work in school,
but was almost wbolly given to the pursuit of his favorite subjects*
He lectured, perhaps, two or three times per week, but these lectures
generally disclosed some new discovery in algebra.

• Theory of Eqnat'Ions, page 250.

t iDaagural Lector a delivered by Professor Sylvester before the University of Ox-
ford, December 12, 7x885, pabliahed in Nature, January 7, 1886.

INFLUX OF FBEKCH MATHESfATICS. 263

Thoagh he had passed his sixtieth year before he came to the Johns
Hopkins XJuiversity, his mind seemed to be as strong and active as
ever. The group of students he had gathered about him were almost
constantly made to feel the glow of new ideas or of old ones in a new
form. Prom 1877 to 1882, Professor Sylvester contributed thirty arti-
cles and notes to the American Journal of Mathematics ; twenty-two to
the Oomptes Bendus de VAcad4mie des Scienoe$ de VInstitut de France ;
one paper to the Proceedings of the Boyal Society, ^<0n the Limits to
the Order and Degree of the Fundamental Invariants of Binary Quan*
tics ^ (1878) ; four to the Messenger of Mathematics ; four to the Lon-
don, Edinburgh, and Dublin Philosophical Magazine^ six to the JourmU
filr reine and tmgewandte MathematiJcj Berlin.* If this list be complete,
the number of original papers published by him while at the Johns
Hopkins University was sixty-seven. Special mention may be made
here of a proof by Professor Sylvester, printed in the Philosophical
Magazine for 1878, of a theorem on the number of linearly independent
diiferentiants, which had been awaiting proof for over a quarter of a
century. He was led to undertake the investigation of this subject by
a question put to him by one of his students in connection with a foot*
note given at one place in Fad> de Bruno's ThSorie des Formes Binairea.

Since his return to England, Sylvester has been developing a new
subject, which he calls the ^< Method of Beciprocants.'' The lectotes
which he delivered on this subject at the University of Oxford have
been reported by Mr. Hammond and published in the American Jonr*
nal of Mathematics.

Sylvester has manufactured a large number of technical terms in
mathematics. He himself speaks on this point as follows : ^< Perhaps
I may, without immodesty, lay claim to the appellation of the mathe-
matical Adam, as I believe tiiat I have given more names (passed into
general circulation) to the creatures of the mathematical reason than
all the other mathematicians of the age combined.'^t

In his writings, Professor Sylvester is often very eloquent. His style
is peculiarly flowery, and indicative of very powerful imagination.
His articles are frequently interspersed with short pieces of poetry,
either quoted or of his own composition. Thus, in his article in Nature,
January, 1886, is given a short poem, '^ On a Missing Member of a
Family Oroup of Terms in an Algebraical Formula f followed by this
sentence: *< Having now refreshed ourselves and bathed the tips of
our fingers in the Pierian spring, let us turn back for a few brief mo-
ments to a light banquet of the reason."

Since the beginning of the Johns Hopkins University, twenty fel-
lowships have been open annually to competition, each yielding five
hundred dollars and exempting the holder from all charges for tuition.
This system was instituted for the purpose of affording to young men

* U. S. Bureau of Education, Clrciilar of loformation No. 1, 1888, p. 220.
t Nature, Dec. 15, 1687, p. 152, note.

264 TEACHING AND HISTORY OF MATHEMATICS.

of talent an opportunity of continuing tbeir studies in the university,
while looking forward to positions as professors, teachers, and investi*
gators. They have been given to graduate students who showed partio-
ular aptitude for advanced work in their chosen specialty. During
the time when Sylvester was connected with the university there were
nearly always three or four fellowships granted to mathematical stu-
dents, but, in recent years the number has been reduced to two, in con-
sequence of an increase in the number of departments in the university,
among which the fellowships must be divided. Among the first holders
of fellowships in mathematics were Thomas Craig (1876-78), Gteorge B.
Halsted (1876-78), Fabian Franklin (1877-79), W. L Stringham (1878-
80), G. A. Van Yelzer (1878-81), all holding leading and responsible
positions now, as professors of mathematics.

Professor Sylvester's first high class at the new university consisted of
only one student, 6. B. Halsted. who had persisted in urging Sylvester
to lecture on the modem algebra. The attempt to lecture on this sub*
ject led him into new investigations in quantics. In his address on
Commemoration Day at the Johns Hopkins, he ;»poke about this work
as follows :

^^This is the kind of investigation in which I have for the last month
or two been immersed, and which I entertain great hopes of bringing to
a successful issue. Why do I mention it here ? It is to illustrate my
opinion as to the invaluable aid of teaching to the teacher, in throwing
him back upon his own thoughts and leading him to evolve new re-
sults from ideas that would have otherwise remained passive or dormant
in his mind.

^*But for the persistence of a student of this university in urging
upon me his desire to study with me the modern algebra I should never
have been led into this investigation j andcthe new facts and principles
which I have discovered in regard to it (important facts, I believe),
would, so for as I am concerned, have remained still hidden in the womb
of time. In vain I represented to this inquisitive student that he would
do better to take up some other subject lying less off the beaten track
of study, such as the higher parts of the calculus or elliptic functions, or
the theory of substitutions, or I wot not what besides. He stuck with
perfect respectfulness, but with invincible pertinacity, to his point He
would have the new algebra (Heaven knows where he had heard about
it, for it is almost unknown in this continent), that or nothing. I was
obliged to yield, and what was the consequence T In trying to throw
light upon an obscure explanation in our text-book, my brain took fire,
I plunged with re-quickened zeal into a subject which I had for years
abandoned, and found food for thoughts which have engaged my atten-
tion for a considerable time past, and will probably occupy all my
powers of contemplation advantageously for several months to come."

This extract describes the beginning of his scientific activity and pro-
dootiveness in the New World.

INFLUX OP FRENCH MATHEMATICS. 265

It may not be without interest to learn what some of his former pupils
at the Johns Hopkins University have to say about him. Says Dr.
G. B. Halsted :

^< Young Americans could hardly realize that the great Sylvester,
who with Gayley outranks all English-speaking mathematicians, was
actually at work in our land . All youn g men who felt within themselves
the divine longing of creative power hastened to Baltimore, made at once
by this Euclid a new Alexandria. It was this great awakening and
concentration of mathematical promise, and the subsequent facilities
offered for publicafcionof original work, which, rather than any teaching,
made the American renaissance in mathematics. • • •

^' A short, broad man of tremendous vitality, the physical type of
Here ward, the Last of the English, and his brother-in-arms. Winter,
Sylvester's capacious head was ever lost in the highest cloud-lands of
pure mathematics. Often in the dead of night he would get his favor-
ite pupil, that he might communicate the very last product of his cre-
ative thought. Everything he saw suggested to him something new
in the higher algebra. This transmutation of every thing into new math-
ematics was a revelation to those who knew him intimately. They
began to do it themselves. His ease and fertility of invention proved
a constant encouragement, while his contempt for provincial stupidities,
such as the American hieroglyphics for n and e, which have even found
their way into Webster's Dictionary, made each young worker apply to
himself the strictest tests.

<< To know him was to know one of the historic figures of all time,
one of the immortals ; and when he was really moved to speak, his
eloquence equalled his genius. I never saw a more astonished man
than James Bussell Lowell listening to the impassioned oratory of Syl-
vester's address upon the bigotry of Christians.

<^ That the presence of such a man in America was epoch-making is
not to be wondered at. His loss to us was a national misfortune."*

In answer to an inquiry about Sylvester's methods of teaching, Dr.
E. W. Davis (fellow from 1882 to 1884) writes hurriedly as follows :
** Sylvester's methodn ! He had none. ' Three lectures w ill be delivered
on a New Universal Algebra,' he would say; then, ' The course must be
extended to twelve.' It did last all the rest of that year. The following
year the course was to be Substitutions- ThSoHcy by Netto. We all got
the text. He lectured about three times, following the text closely
and stopping sharp at the end of the hour. Then he began to think
about matrices again. * I must give one lecture a week on those,' he
said. He could not confine himself to the hour, nor to the one lecture a
week. Two weeks were passed, and Netto was forgotten entirely and
never mentioned again. Statements like the following were not unfre-
qaentin his lectures: *I haven't proved this, but I am as sure as I can

• Letter to the writer, December 25, 1888.

266 TEACHING AND HISTOBfT OF MATHEMATICS.

be of anything that it mnst be so. From this it will follow, etc.' At the
next lecture it turned out that what he was so sore of was false. Never
mind, he kept on forever guessing and trying, and presently a wonder-
ful discovery followed, then another and another. Afterward he would
go back and work it all over again, and surprise us with all sorts of
side lights. He then made another leap in the dark, more treaaoies
were discovered, and so on forever.'^

Lotus now listen to another of his old pupils, Mr. A. S. Hathaway
(fellow from 1882 to 1884) :

^^ I can see him now, with his white beard and few looks of gray hair,
his forehead wrinkled o'er with thoughts, writing rapidly his figures and
formulsB on the board, sometimes explaining as he wrote, while w4, his
listeners, caught the reflected sounds from the board. But stop, some-
thing is not right, he pauses, his hand goes to his forehead to help Mb
thought, he goes over the work again, emphasizes the leading points, and
finally discovers his difficulty. Perhaps it is some error in bia figorefl,
perhaps an oversight in the roasoning. Sometimes, however, the diffi-
culty is not elucidated, and then there is not much to the rest of the lect-
ure. But at the next lecture we would hear of some new disooveiy that
was the outcome of that difficulty, and of some article for the Journal,
which he had begun. If a text-book had been taken up at the beginning,
with the intention of following it, that text-book was most likely doomed
to oblivion for the rest of the term, or antil the class had been made lis-
teners to every new thought and principle that had sprung fh>m the
laboratory of his mind, in consequence of that first difficulty. Other
difficulties would soon appear, so that no text-book could last more than
half of the term. In this way his class listened to almost all of the
work that subsequently appeared in the Journal. It seemed to be the
quality of his mind that he must adhere to one subject. He would think
about it, talk about it to his class, and finally write about it for tiie
Journal. The merest accident might start him, but once started, every
moment, every thought was given to it, and, as much as possible, he read
what others had done in the same direction ; but this last seemed to be
his weak point ; he could not read without meeting difficulties in the way
of understanding the author. Thus, often his own work reprodooed
what others had done, and he did not find it out until too late.

<<A notable example of this is his theory of cyolotomic funotionSy
which he had reproduced in several foreign journals, only to find that
he had been greatly anticipated by foreign authors. It was manifest,
one of the critics said, that the learned professor had not read Kum-
mer's elementary results in the theory of ide&l primes. Yet Professor
Smith's report on the theory of numbers, which contained a full synopsis
of Rummer's theory, was Professor Sylvester's constant companion.

<< This weakness of Professor Sylvester, in not being able to read what
others had done, is perhaps a concomitant of his peculiar genius. Other
minds could pass over little difficulties and not be troubled by them,

INFLUX OF FEENCH MATHEMATICS. 267

and 80 go on to a final understanding of tbe results of the author. But
not so with him. A difficulty, however small, worried him, and he was
sure to have difficulties until the subject had been worked over in his
own way, to correspond with his own mode of thought. To read the
work of others, meant therefore to him an almost independent de«
velopment of it. Like the man whose pleasure in life is to pioneer the
way for society into the forests, his rugged mind could derive satisfac-
tion only in hewing out its own paths ; and only when his efforts brought
him into the undeared fields of mathematics did he find his place in the
Universe.'^

These reminiscences are extremely interesting, inasmuch as they show
the workings of a great mind. The mathematical reader will surely
enjoy the following reminiscences of " Silly," by one of his favorite pu-
pils, Dr. vW. P. Durfee, professor of mathematics at Hobart Oollege,
Gtoneva, K Y. He was a fellow in mathematics from 1881 to 1883.
Speaking of his recollections of Sylvester, he says :

<< I don't know that I can do better than preface them by an account,
as far as my memory serves me, of the work we did while I was at the
Johns Hopkins University. I say toe^ as I always think of the whole
staff as working together, so thoroughly did Sylvester inspire us all
with the subject which was immediately interesting him. I went to
Baltimore in October, 1881, as a fellow, and, though my previous math-
ematical training had been of the scantiest, I had the courage of igno-
rance and immediately began to attend Sylvester's lectures, while Mr.
Davis and some others thought they would wait for a year and prepare
themselves to profit by them. Sylvester.began to lecture on the Theory
of Numbers, and promised to follow Lejeune Dirichlet's book ; he did
so fbr, perhaps, six or eight lectures, when some discussion which came
up led him off, and he interpolated lectures on the subject of frequency,
and after some weeks interpolated something else in the midst of these.
After some further interpolations he was led to the consideration of his
Universal Algebra, and never finished any of the previous subjects.
This finished the first year, and, although we had not received a sys-
tematic course of lectures on any subject, we had been led to take a liv-
ing interest in several subjects, and, to my mind, were greatly gainers
thereby. The second year, 1882-83, he started off on the subject of
substitutions, but our experience was similar to that of the preceding
year, and I can not now, after the six years which have intervened, dis-
entangle the various topics that engaged his attention. Amongst others
were Turey's series, partitions, and universal algebra. He coald not
lecture on a subject which was not at the same time engaging his atten-
tion. His lectures were generally the result of his thought for the pre-
ceding day or two, and often were suggested by ideas that came to him
while talking. The one great advantage that this method had for his
students was that everything was fresh, and we saw, as it were, the
very genesis of his ideas. One could not help being inspired by such

2(68 TEACHING AND HISTORY OF MATHEMATICS.

teaching, and many of as were led to investigate on lines which he
toached npon. He was always pleased at what one had to suggest, and
generally bore iptermptions with patience. He would often stop to
discuss points that arose, and accepted our opinions as of some worth.
I must qualify these latter statements somewhat, as he was apt to be
partial, and it made all the difference in the world who it was that in-
terrupted him.

<< His manner of lecturing was highly rhetorical and elocutionary.
When about to enunciate an important or remarkable statement he
would draw himself up till he stood on the very tips of his toes, and in
deep tones thunder out his sentences. He preached at us at such
times, and not infrequently he wound up by quoting a few lines of
poetry to impress on us the importance of what he had been declaring.
I remember distinctly an incident that occurred when he was at work
on his Universal Algebra. He had jumped to a conclusion which, he
was unable to prove by logical deduction. . He stated this fact to us
in the lecture, and then went on, ^< Gentlemen ^ [here he raised him-
self on his toes], ^^ I am certain that my conclusion is correct. I will
WAGEB a hundred pounds to one ; yes, I will wageb my life on if
The capitals indicate when he rose on his toes and the italics when he
rocked back on to his heels. In such bursts as these he always held his
hands tightly clenched and close to his side, while his elbows stuck out
in the plane of his body, so that his bended arm made an angle of
about UOo.

^< Personally I had considerable contact with him, as I did work under
his direction that made it necessary for me to see him at his rooms.
On such occasions I always made an engagement with him two or three
days beforehand, and then at his request dropped him a postal, which
reached him an hour or two before I went and reminded him that I was
coming. I always found him interested in my work and full of sugges-
tions.

^< He had one very remarkable peculiarity. He seldom remembered
theorems, propositions, etc., but had always to deduce them when he
wished to use them. In this he was the very antithesis of Oayley, who
was thoroughly conversant with everything that had been done in every
branch of mathematics.

^' I remember once submitting to Sylvester some investigations that
I had been engaged on, and he immediately denied my first statement^
saying that such a proposition had never been heard of, let alone
proved. To his astonishment, I showed him a paper of his own in
which he had proved the proposition ; in fact, I believe the object of
his paper had been the very proof which was so strange to him."

By request of Professor Sylvester, Professor Oayley, the Sadlerian
professor of pure mathematics in Oambridge, England, was associated
in the mathematical work of the Johns Hopkins University from Jan-
uary to June, 1882. Of him Dr. Dorfee says : ^' His subject was Abel-

INFLUX OF FBENCH MATHEMATICS. 269

ian and Theta Eanctions^ and he stack closely to his text. While his
work was of great interest and importance, he did not arouse entha-
siasm and dififnse inspiration, as Sylvester did."

These were proud days for the Johns Hopkins University, when the
two greatest living English mathematicians were lecturing within her
walls.

The first associate appointed in mathematics, when the university
first opened in 1876, was Dr. W, B, Story. He graduated at Harvard
University in 1871, then studied for some years in Germany, receiving
the degree of doctor of philosophy at the University of Leipsic in 1875.
For one year preceding the opening of the Johns Hopkins University
he was tutor of mathematics at Harvard. At the Johns Hopkins Uni-
versity his lectures and his original researches have been chiefly in geom-
etry. He was for several years associate editor of the American
Journal of Mathematics.

Dr. Stx)ry is not only an eminent mathematician, but also a good
teacher. He is ever ready to give private interviews to students and
to explain to them difQcult points, or offer criticisms and suggestions
upon original inquirieis which the student may be engaged in. Dr. Story
is an admirable lecturer, clear, logical, deliberate, proceeding step by
step, so that the student may be sure to follow his reasoning. His work
on the blackboard is written in an elegant hand, and is always scrupu*
lously accurate. In 1884 the university secured a magnificent set of
geometrical models for the study of surfaces. Some of these are often
brought by Dr. Story into the lecture-room to illustrate his subject. In
his lectures Dr. Story generally follows some particular text-book, such
as Glebsch on Oonic Sections, Salmon on Analytic Geometry of Three
Dimensions, or Steiner on Synthetic Geometry, but he often brings in
researches of more recent date, and also inquiries of his own.

Another member of the mathematical staff is Dr. Thomas Oraig. He
graduated with the degree of civil engineer at Lafayette College in 1875^
was one of the first persons elected to a fellowship at the Johns Hop-
kins University, and in 1878 received the degree of doctor of philoso-
phy. He began lecturing at the university when he was a student.
After graduation he was connected for a short period with the U. S.
Coast and Geodetic Survey, for which he prepared in 1879 a Treatise
on the Mathematical Theory of Projections. During his stay in Wash-
ington he studied also Theory of Functions from the work of Konigs-
berger, under the direction of Professor Kewcomb, of the Nautical Al-
manac. Dr. Craig has made the theory of functions and differential
equations his specialty. He has not only kept pace with the most re-
cent rapid advances of these broad and deep subjects, but has added
numerous contributions of his own. Most of them have appeared in the
American Journal of Mathematics, while some have been published in
foreign journals. He is working on subjects which are receiving ex-
tensive development in the hands of Fuchs^ Hermito, Poincar^, Appel,

270 TEACHING AND HISTOBY OF MATHEMATICS.

Darbonxy Picard^ and others. There Beem to be altogether too few
Auericans interested in this line of work and prepared to participate
in its advancement. The mind of Dr. Craig moves with great rapidity.
A quick and brilliant stadent finds his lectures profitable and inspiring.
Some of his courses on differential equations and the theory of func-
tions are very advanced and difficult, and can be followed only by the
maturest of students.

Dr. Oraig associates with the students familiarly. It has been his
practice to invite occasionally students to his house to spend a mathe-
matical evening^ when all sorts of subjects would be discussed in a firee
and easy style.

A somewhat more recent appointment as associate in mathematios is
that of Dr. Fabian Franklin. He graduated at the Oolumbian Univer*
sity in 1869, was fellow in mathematics from 1877 to 1879, and received
the degree of doctor of philosophy in 1880. He was appointed assistant
in mathematics before taking his degree. Franklin always took great
interest in Professor Sylvester's researches while the latter was at the
Johns Hopkins University, and generally was at work on similar linesy
while Dr. Story and Dr. Craig followed more generally lines of inves-
tigation of their own. Some of the articles printed in the American
Journal of Mathematics have appeared under the joint authorship of
Sylvester and Franklin. Professor Sylvester entertained the higheet
opinion of Dr. Franklin.

Dr. Franklin has done more teaching in the under-graduate deparlr
ment than the other members of the mathematical staft^ for the reason
that he excels them all in his power of imparting instruction. His
teaching power is indeed great. It is seldom that a person of so high
mathematical talent is as good an instructor of younger pupils. Dr.
Franklin possesses a remarkably quick eye for short methods. The
student seldom listens to one of his lectures in which proofis are not
given in a shorter, simpler manner than in the book ; seldom is a paper
read in the Mathematical Society which is not followed*, in the ensuing
discussion, by suggestions by Dr. Franklin of a shorter method. His
papers published in the American Journal of Mathematics display the
same power. As a teacher Dr. Franklin is extremely popular among
the students.

In Dr. Story, Dr. Oraig, and Dr. Franklin, Professor Sylvester had
an eminently efficient corps of fellow-laborers. Their mathematical re>
searches have made their names favorably known wherever advanced
mathematics finds a votary.

The instruction for graduates during the time that Professor Sylves-
ter was connected with the university was as follows : *

Courtes of Intiruction, JEtourt per WeeJCf and Attendance, 1876-83.

DeterminaDts and Modem Algebra: Professor Sylvester, 1876-77, 2d half-yeaTj dhn.
(7) ; 1877-78, 2 hrs. (5) ; 1878-79, 2 hrs. (8).

* £leyeQth Annual Beport of the President of the Johns Hopkins Uniyenity, p. 49.

INFLUX OF FEENCH MATHEMATICS. 271

Theory of Kamben : Professor Sylvester; 1879-80, 2 hrs. (8) ; 18d0-81» 2hrs. (6) ; 1881^

82, Ist half-year, 2 hrs. (7).
Theory of Partitions : Professor Sylvester, 1882-83, 2d half-year, 2 hrs. (10).
Algebra of Multiple Quantity : Professor Sylvester, 1881-82, 2d half-year, 2 hrs. (12);

1883-«4, 1st half year, 2 hrs. (6).
Theory of Substitutions : Professor Sylvester, 1882-83, 1st half-year, 2 hrs. (9).
Algebraical G(eometry and Abelian and Theta Functions : Professor Cayley, 1881-82|

2d half-year, 2 hrs. (14).
Quaternions: Dr. Story, 1877-78, 2 hrs. (2) ; 1879-80, 3 hrs. (4); 1881-82, 3 hrs. (7) ;

1882-83, 2d half-year, 3 hn. (4).
Higher Plane Curves : Dr. Story, 1880-81, 2 hrs. (5) ; 1881-82, 1st half year, 3 hrs. (1) ;

1883-84, 2 hrs. (2).
Solid Analytic Qeometry (General Theory of Surfaces and Curves) : Dr. Story, 1881-

82, 2d half-year, 3 hrs. (1) ; 1882-83, Ist half-year, 3 hrs. (6).
Theory of (Geometrical Congruences: Dr. Story, 1882-83, 2d half-year, 2 hrs. (4).
Modem Syntbetio Qeometry : Dr. Franklin, 1877-78, 2 hrs. (2).
Theory of Invariants : Dr. Story, 1882-83, 10 lectures (8) ; 1883-84, 3 hrs. (6).
Determinants : Dr. Franklin, 1880-81, 1st half-year, 2 hrs. (9) ; 1882-83, 20 lecturei

(9).
Modem Algebra : Dr. Franklin, 1880-81, 2d half-year, 2 hrs. (6) i 1881-82, 2d half-
year, 2 hrs. (6).
Elliptio Functions: Dr. Story, 1878-79, 2 hrs. (2); 1879-80 (continuation of the pre-

TiouB year's course), 3 hrs. (4) ; Dr. Craig, 1881-82, 3 hrs. (8) ; 1883-^, 3 hrs. (4).
Elliptio and Theta Functions : Dr. Craig, 1882-83, 3 hrs. (10) ; 1883-M, 3 hrs. (2).
(General Theory of Functions, including Biemann's Theory: Dr. Craig, 187^-80, 30

lectures (2) ^ 1880-81, 1st half-year, 3 hrs. (3).
Spherical Harmonics: Dr. Craig, 1878-79, 10 lectures (6); 1879-80, 20 lectures (6);

1881-82, Ist half-year, 2 hrs. (4) ; 1883-84, 2d half-year, 1 hr. (4).
Cylindric or Bessel's Functions: Dr. Craig, 1879-80, 10 lectures (2).
Partial Differential Equations : Dr. Craig, 1880-81, 2d half-year, 2 bra. (5) ; 1881-82,

2d half-year, 3 hrs. (9) ; 1882-83, 2d half-year, 2 hrs, (2) ; 188^-84, 2d half-year, 2

hrs. (4).
Calculus of Variations: Dr. Craig, 1879-80, 12 lectures (9); 1881-82, 1st half-year, 2

hriL (8) ; 1882-83, 1st half-year, 2 hrs. (6).
Definite Integrals: Br. Craig, 1876-77, Ist half-year, 3 hrs. (5) ; 1882-83, Ist half-year,

2 hrs. (3).

Mathematical Astronomy : Dr. Story, 1877-78, 3 hrs. (2) ; 1882-83, 3 hrs. (2) ; 1883-84,

3 hrs. (2).

Elementary Meohanios : Dr. Craig, 1876-77, 2d half-year (8).

Statics: Dr. Franklin, 1882-83, 2d half year, 3 hrs. (5).

Analytle Meohanios : Dr. Craig, 1877-78, Ist half-year (6) ; Dr. Story, 1880-81, 2d

half-year, 2 hrs. (6) ; Dr. Craig, 1881-82, Ist half-year, 3 hrs. (8) ; 1882-83, 1st

half-year, 3 hrs. (4) ; Dr. Franklin, 1883-84, 3 hrs (6).
Theoretical Dynamics: Dr. Craig, 1878-79, 15 lectures (6) ; 1883-84, 2 hrs (5).
Mathematical Theory of Elasticity: Dr. Story, 1876-77, 2d half-year, 2 hrs. (4);

1877-78, 2 hrs. (2) ; Dr. Craig, 1881-82, 3 hrs. (4) ; 1883-84, 2d half-year, 2 hrs. (3).
Hydrodynamios: Dr. Craig, 1878-79, 24 lectures (7); 1880-81, Ist half-year, 2 hrs.

(6) ; 2d half-year, 4 hrs. (3) ; 1882-83, 2d half-year, 3 hrs. (5).
Mathematical Theory of Sound: Dr. Craig, 1883-84, 3 hrs. (5).

It may be of interest to give a list of the advanced students of math-
ematics daring the seven years that Sylvester was connected with the
Johns Hopkins University, and their present occnpation. Dr. Graig
and Dr. Franklin are, as we have seen, instrnctors in mathematics at
the Johns Hopkins* The list continues as follows : Gt. B. Halsted,

272 TEACHING AND HISTORY OF MATHEMATICS.

professor of matbematics, University of Texas; W. I. Stringham, pro-
fessor of mathematics. University of California ; 0. A. Van Vdzer, pro-
fessor of mathematics, University of Wisconsin ; O. H. Mitchell, pro-
fessor of mathematics, Marietta College, Ohio ; B. W. Prentiss, in the
office of the U. S. Nantical Almanac, Washington ; H. M. Perry, In-
stractor in mathematics, Gascadilla School, Ithaca, !N.Y. ; W. P. Dorfee,
professor of mathematics, Hobart College, 'S. T. ; 6. S. Ely, examiner,
U. S. Patent Office ; E. W. Davis, professor of mathematics. University
of South Carolina; A. S. Hathaway, instructor in mathematics, Oomell
University; G. Bissing, examiner, U. S. Patent Office; A. L, Daniels,
instructor in mathematics, Princeton College, 1883-84.

The success in training students for independent research has been
very great. To convince himself of this, the reader need only ex-
amine the abstracts of papers prepared by students, which have been
published in the Johns Hopkins Circulars and in the American Journal
of Mathematics. Each one of the names given above will be found to
appear repeatedly in those publications, as a contributor.

In December, 1883, Professor Sylvester started for England to enter
upon his new duties as Savilian professor of geometry in the University
of Oxford. The robe of the departing prophet dropped upon the shoul-
ders of Professor Newcomb. No American would have been more
worthy of succeeding Sylvester. As an astronomer his llame has long
shone with a luster which fills with pride every American breast.

Simon ]^ewcomb was born in Wallace, Xova Scotia, in 1835. After
being educated by his father he engaged for some time in teaching. He
came to the United States at the age of eighteen, and was engaged for two
years as teacher in Maryland. There he became acquainted with
Joseph Henry, of the Smithsonian Institution, and Julius E. Hilgard, of
the U. S. Coast Survey. Becognizing his talent for mathematics, they
secured for him, in 1857, a position as computer on the Kautical Alma-
nac, which was then published in Cambridge, Mass. In Cambridge he
came under the influence of Prof. Benjamin Peirce. In the catalogues
of 1856 and 1857 his name appears as a student of mathematics in the
Sheffield Scientific School. He graduated in 1858, ahd continued as a
graduate student for three years thereafter. He was then appointed
professor of mathematics in the U. S. Navy, and stationed at the Kaval
Observatory. He was chief director of a commission created by Con-
gress to observe the transit of Venus in 1874. In that year the Boyal
Society of England awarded him a gold medal for his Tables of Uranns
and Neptune. In 1870 he undertook to investigate the errors of Han-
sen's Lunar Tables as compared with observations prior to 1850. The
results of this onerous task were published in 1878. In the years 1880
to 1882 he and Michelson measured the velocity of light by operations
on such a large scafe and such refined methods as to throw in the shade
all earlier efibrts of a similar kind. For the purpose of this measure-
ment they set up fixed and revolving mirrors on opposite shores of the
Potomac, at a distance of nearly 4 kilometers.

INFLUX OF FRENCH MATHEMATICS. 273

Since 1877 he has been in charge of the office of the American Ephem-
eris and Nautical Almanac. Since 1867 that office has been in Wash-
ington, instead of Cambridge. Professor Newcomb's predecessor in this
office was J. H. 0. Coffin, who in 1877 was placed on the retired list,
having been senior professor of mathematics in the Navy since 1848.

Professor Newcomb has quite a large corps of assistants in Wash-
ington. His researches in astronomy during the last ten or twelve
years have been described in the Nation of September 6, 1888, as fol-
lows:

" The general object of this work is the determination of the form,
size, and position of the orbits of all the large planets of the solar sys-
tem, from the best and most recent observations, and the preparation
of entirely new and uniform tables for predicting the future positions
of these objects. The first of the four sections of the work relates to the
general perturbations of the planets by each other, and the part already
in hand comprises the four inner planets. Mercury, Venus, the Earth,
and Mars, in which fourteen pairs of planets come into play. Twelve
of these were completed some months since, and only the action of Ju-
piter on Yenus and Mars remained undetermined. In the next place,
the older observations of the planets must be recalculated, and thus
problems constantly arise which can not be met by general rules. All
the observations at Oreenwich from 1765 to 1811 have been completely
reduced with modem data. Earlier Oreenwich observations were sim-
ilarly treated by Dr. Auwers, of Berlin, who liberally presented the com-
plete calculations as his contribution to the work of the Nautical
Almanac Office. In a recent report of this work Professor Newcomb
gives further details of his progress in the treatment of other classes
of planetary observations.

<< Following this collation of all the available observations of each
planet, comes the theoretical preparation of their corresponding posi-
tions at the time of observation. This forms the most laborious and
difficult part of the work ; and had Leverrier's tables, the best hitherto,
been used without modification, Professor Newcomb would have found
it impractiable to complete it with the number of computers at his com-
mand ; but he has skillfully avoided the difficulty by a reconstruction
of Leverrier's work in such form that it should be much less laborious
to use, while sufficiently accurate for the purpose required. These theo-
retical positions must next be compared directly with the observations,
one by one, and the differences between the two are then, by suitable
mathematical processes, construed as implying the nature and amount
of certain corrections to the planef s motion in its orbit. More than a
full year must still elapse, says Professor Newcomb, before the work on
the four inner planets will have advanced to the stage where this direct
comparison is ready to be made. There remain the four outer planets,
on the two more important of which, Jupiter and Saturn, Mr. Hill, of
the same office, has been engaged for many years, and his new theory
881— No. 3 18

274 TEACHiNa Am) histobt of mathematics.

of their complicated motion is already in the printer's hands. The two
onter planets, Uranos and Neptune, have not yet been begun."

Of Professor Newcomb's labors Professor Cayley has said : " Professor
NewcomVs writings exhibit, all of them, a combination on the one hand
of mathematipal skill and power, and on the other of good hard work,
devoted to the furtherance of astronomical science."

His book on Popular Astronomy (1877) is well known. It has been
republished in England and translated into (German. The treatise on
Astronomy by Newcomb and Holden, and their <^ Shorter Oourse" on
Astronomy, are works which have been introduced as text-books into
our colleges almost universally.

Professor Newcomb's scientific work has not been confined to astron-
omy. He has carried on investigations on subjects purely mathemati-
caL One of the most important is his article on ^< Elementary Theo-
rems Belating to the Gtoometry of a Space of Three Dimensions and of
Uniform Positive Ourvature in the Fourth Dimension," published in
Borehardt^s Journal, Bd. 83, Berlin, 1877. Full extracts of this very
important contribution to non-Euclidian geometry are given in the
EndyclopaDdia Britannica, article ^^ Measurement." It is gratifying to
know that through Professor Kewcomb America has done something
toward developing the fia>r-reaching generalizations of non-Euclidian
geometry and hyper-space. In Volume I of the American Journal of
Mathematics he has a note <^ On a Glass of Transformations which Bur-
faces may Undergo in Space of more than Three Dimensions," in which
he shows, for instance, that if a fourth dimension were added to space,
a dosed material surface (or shell) could be turned inside out by sim-
ple flexure without either stretching or tearing. Later articles have
been on the theory of errors in observations. In former years he also
contributed to the Mathematical Monthly and the Analyst.

Professor Kewcomb has written a series of college text-booka on
mathematics. In 1881 appeared h|s Algebra for Oolleges and his Ele-
ments of Gtoometry ; in 1882 his Trigonometry and Logarithms, and
School Algebra; in 1884 his Analytical (Geometry and Essentials of
Trigonometry ; in 1887 his Dififerential and Integral Calculus. These
works have been favorably reviewed by the press, and are everywhere
highly respected. Professor Newcomb's fundamental idea has been to
lead up to new and strange conceptions by slow and gradual steps.
^^AU mathematical conceptions require time to become engrafted upon
the mind, and the more time the greater their abstruseness." The stu-
dent is gradually made familiar in these books with the oonceptions of
variables, functions, increments, infinitesimals, and limits, long before
he takes up the calculus, so in the study of the calculus he is not con-
fronted, at the outset and all at once, by a host of new and strange
ideas, but possesses already a considerable degree of familiarity with
them. With the publication of Newcomb's Algebra has begun a con-
siderable <* shaking" of the ^' dry bones" in this science, and we now
possess works on this subject that are of considerable merit.

INFLUX OP FRENCH HATHXMATICS. 275

Professor liTewcomb stadies political eooaomy as a recreatioD, and
every now and then there is a commoticm in the camp of political econ-
omists, caused by a bomb thrown into their midst by Professor New-
comb, in the form of some magazine article or book.

In 1884 Professor Newcomb added to hia duties as superintendent of
the Nautical Almanac that of professor of mathematics and astronomy
at the Johns Hopkins University. He generally delivers at that in-
stitution two lectures per week. The effect of his connection with
the mathematical department has been that the mathematical course
is more thoroughly Bystematized and more oareftilly graded than
formerly, and that the attention of students is drawn also to higher
aatronomy, theoretical and practical. An observatory for instruction
is now provided by the university. Besides a telescope of 9^ inches
aperture there la a meridian circle with collimators, mercury-basin,
and other appliances. Professor Newcomb entered upon his duties
at the Johns Hopkins University in 1884 by giving a course of lect-
ures on celestial mechanics. Among other things it embraced his
own paper on the <^ Development of the Pertnrbative Function and
its Derivative in Sines and Oo*sines of the Eccentric Anomaly and in
Powers of the Bccentricities and Inclinations." The lectures were well
attended by the graduate students. At the blackboard Professor
Newcomb does not manipulate the crayon with so great dexterity as do
his associates, who have been in the lecture-room all their lives, but his
lectures are clear, instructive, original, and popular among the students.

Since the departure of Professor Sylvester the following courses of
lectures have been given to graduate students:

CauraeB of Imtruotionf Sour$ per WmJc, and Attendance, 1884-^88.

AIl|^ytlosl and Celestial Meohanioe : Prot Newcomb, 1884-^86, 2 hn. (11).
Praotioal and Theoretical Aetronomy : Prof. Newcomb, 1885-^, 9 hn. (9) ; 188&-'87,

2 hn. (7).

Theory of Special Perturbations : Profl Newcomb, 1887-'88, Ist half-year, 2 hrs.
History of Astronomy : Prof. Newcomb, 1887-'88, Maroli and April, 2 hrs.
Computation of Orbits : Prof. Newcomb, 1887-'88, May, 2 hrs.
Theory of Nnmben: Dr. Story, 1884-^85, Ut half-year, 2 bn. (9).
Modern Synthetic Geometry: Dr. Story, 1884-^85, Ist half-year, 3 hn. (8).
Introductory Conne for Graduates : Dr. Story, 1884-'85, 5 hn. (10) ; 1885-^86, 5 hra.

(7) ; 1886-^87, 5 hrs. (10) ; ISST-'SS, 5 hn.
Modern Algebra: Dr. Story, 1884-^85, 2d half-year, 2 hn. (9).
Qnaternions: Dr. Story, 1884-^85, 2d half-year; 3 hn. (8); 188^'87. 3 hn. (5);

1887-^88, 3 hn.
Finite Differences and Interpolation : Dr. Story, 188&-'86, Ist half-year, 2 hrs. (5).
Advanced Analytic Geometry : Dr. Story, 1885-^86,3 hrd. (4); 1886-'87, 2 hrs. (8);

1887-'88, 2 hrs.
Theory of Probabilities: Dr. Story, 1885-^86, 2d half-year, 2 hn. (5).
Calculus of Variations: Dr. Craig, 1884-'85, Ist half-year, 2 hrs. (5).
Theory of Functions : Dr. Craig, 1884-^85, 3 hrs. (5) ; 1885-'86, 1st half-year, 3 hrs.

(4); 1886-'87, 3 hn. (6); 1887-'88, Ist half-year, 3 hrs.
Hydrodynamics: Dr. Craig, 1884-^85, Ist half-year, 3 hrs. (6) ; 1885-'86, 1st half-year,

3 hn. (4) J 1886-^87, Ist half-year, 3 hrs, (4) ; 18a7-'88, 1st half-year, 3 hn.

276 TEACHINQ AND HISTOBY OF MATHEMATICS.

Linear Differential Equations: Dr. Craig, 1884-'85, 2d half-year, 3 hrs. (3); 1885-^,

2 hrs. (4) ; 1887-'88, 2d half-year, 2 hrs.
Theoretical Dynamics : Dr. Craig, 1887-'88, 2d half-year, 2 hre.
Differential Equations: Dr. Craic:, 1887-'88, 2 hrs.

Mathematical Theory of Elasticity : Dr. Craig, 18d5-'86, 2d half-year, 3 hrs. (4).
Elliptic and Abelian Fanctions : Dr. Craig, 1885-- 86, 2d half-year, 3 hrs. (4) ; 1886-'87,

Ist half-year, 2 hrs. (6).
Abeliair Fanctions: Dr. Craig, 1887-^88, 2 hrs.
Problems in Mechanics : Dr. Franklin, 1884-^85, 2 hrs. (5) ; 1885-'86, 2 bra. (6) ; .1886-

'87, 2 hrs. (8) ; 1887-^88, 2 hrs.

Sinoe the fall of 1834 Dr. Story has been giving every year an Intro-
ductory Ooarse to graduate students, which consists of short courses of
lectures on the leading branches of higher mathematics. They are in-
tended to give the student a generaLl view of the whole field, which
afterward he is to enter upon and study in its details.

The Johns Hopkins University went into operation primarily as a
University J giving instruction to students who had graduated from col-
lege. A regular college course was, however, organized, and it has been
growing rapidly fh>m year to year. In the college the student has the
choice between several parallel curricula, which are assumed to be equally
honorable, liberal, and difficult, and which therefore lead to the same
degree of bachelor of arts. Seven groups have been arranged. Some
of them embrace no mathematics at all; but, in those courses whereit does
enter, the instruction is very thorough. Take, for instance, Dr. Story's
lectures on conic sections; the method of treatment is entirely modem,
and presupposes a knowledge of determinants. A syllabus has been pre-
pared for the use of the students. The lectures resemble the course
given in the work of Clebsch. The student who may have studied
such books as Loomis's Analytical Geometry, and who may labor with
the conceit that he has mastered analytical geometry and conic sections,
will soon discover that he has learned only the ABO, and that he is
wholly ignorant of the more elegant methods of modem times.

Connected with the mathematical department of the university has
always been a mathematical seminary, which during the time of Syl-
vester constituted in fact the mathematical society of the university.
The meetiugs were held monthly. In it the instructors and more ad-
vanced students would present and discuss their original researches.
Care was taken to eliminate pai)ers of little or no value by immature
students. Professor Sylvester generally presided. " If you were fortu-
nate," says Dr. B. W. Davis, "you had your paper first on the pro-
gram. Short it must be and to the point. Sylvester would be pleased.
Then came his paper, or two of them. After him came the rest, but no
show did theystand ; Sylvester was dreaming of his own higher flights
and where they would yet carry him.''

Since the time of Kewcomb this mathematical seminary has been
called the Mathematical Society. It is carried on in the same way afl
before. Three mathematical seminaries proper have since existed, one

MATHEMATICAL JOURNALS. 277

conducted by Professor Kewoomb, another by Dr. Story, and the third
by Dr. Craig. The meetings are held in the evening, and weekly. Each
instructor selects for his seminary topics from his special studies ^ iN'ew-
comb, astronomical subjects; Story, geometrical subjects or quater-
nions; Craig, theory of functions or differential equations. Professor
NewcomWs seminai;y work is closely connected with his lectures. The
student elaborates some particular points of the lectures or makes prac-
tical, application of the principles involved. In one case the compu-
tation of the orbit of a comet was taken up. Dr. Story, in the year
1885-86, took up the subject of plane curves for his seminary, and dwelt
considerably on quartie and quintic curves, giving matter from Mobius
and Zeuthen, and the result of his own study on quintics. The stu-
dent was expected, if possible, to begin where he had left off and carry
on investigations along lines pointed out by him. Dr. Story's talk on
this subject in this seminary suggested to one of the students a subject
of a thesis for the doctor's degree. In the fall of 1888 Dr. Story began
his seminary work with the seventeenth example, p. 103, in Tait's Qua-
ternions. Dr. Craig's seminary has generally been upon subjects in con-
tinuation and extension of those upon which he is lecturing at the time.
If, for instance, he is lecturing on functions, following the ^^ Oours de
M. Hermitey^ he may in his seminary bring up matter from Briot and
Bouquet. At other times he has introduced work into his seminary
intended to be preparatory to certain advanced courses which he
expected to offer.

Mathematioal Jottbnals. -

• 1

The mathematical journals which we are about to discuss were of a
much higher grade than those of preceding years. First in order of
time is the Mathematical Miscellany ^ » semi-annual publication, edited
by Charles Gill. He was teacher of mathematics at the St. Paul's Col-
legiate Institute at Flushing, Long Island. Eight numbers were pub-
lished ; the first in February, 1836, and the last in November, 1839.
Like many other journals of this kind, it had a Junior and Senior de-
partment — ^the former for young students, the latter for those more
advanced. The first number was entirely the work of the editor, ex-
cepting two or three new problems. Mr. Gill was much interested in
Diophantine analysis. In 1843 he published a little book on the Ap-
plication of Angular Analysis to the Solution of Indeterminate Prob-
lems of the Second Degree, which contains some of his investigations
on this subject.

Another enthusiastic worker in the field of Diophantine analysis, and
a frequent contributor to Gill's journal, was William Lenhart, a favorite
pupU of Robert Adrain. Having been afflicted for twenty-eight years
with a spasmodic affection of the limbs, occasioned by a fall in early
life, which confined him in a measure lo his room, he had devoted a
oonsiderable portion of his time to Diophantine analysis* To him i0

"578 teachhstg xnd history op mathematics.

attributed the solation of the problem, to divide nnity into six parts
such that, if unity be added to each, the sams will be cabes.

The evident defect in Lenhart's processes was their tentative char-
acter. In fact, this criticism applies to all work done in Diophantine
analysis by American computers, down to the present time. It is true
even of old Biophantus himself. To this ancient Alexandrian alge-
braist, who is the author of the earliest treatise on algebra extant, as
well as to his American followers of modern times, general methods
were quite unknown. Each problem has its own distinct method, which
is often useless for the most closely related problems. It has been re-
marked by H. Hankel that, after having studied one hundred solu-
tions of Diophantus, it is difficult to solve the one hundred and first.
It is to be regretted that American students should have wasted so
much time over Diophantine analysis, instead of falling in line with
European workers in the theory of numbers as developed by Gauss and
others. Previous to the publication of the American Journal of Math-
ematics, our journals contained no contributions whatever on the theory
of numbers, excepting the Mathematical Miscellany, which had some
few articles by Benjamin Peirce and Theodore Strong, which involved
Gaussian methods. Among the contributors to the Mathematical Mis-
cellany were Theodore Strong, Benjamin Peirce, Charles Avery, Mar-
cos Gatlin of Hamilton Gollege, George B. Perkins, O. Boot, William
Lenhart, Lyman Abbott, jr., B. Docharty, and others.

The next mathematical periodical was the Cambridge Miscellany of
MaihematicSy Physics^ and Astronomy j edited by Benjamin Peirce and
Joseph Lovering, of Harvard, and published quarterly. The last prob-
lems proposed in GilPs journal were solved here. Four numbers only
were published, the first in 1842. The list of contributors to this Jour*
nal was about tiie same as to the preceding. The most valuable arti-
cles were those written by the editors.

Daring the next fifteen years America was without a mathematical
journal ; bat in 1868, J. D. Bunkle, of the I^autical Almanac offloe in
Boston, started the Mathematical Monthly. He has since held the dis*
tingaished position of professor of mathematics at (and, for a tiiii6|
president of) the Massachusetts Institute of Technology, where he has
been especially interested in developing the department of manual
training. As will be seen presently, the time for beginning the pabli*
cation of a long-lived mathematical journal was not opportune. Three
volumes only appeared. On a fly-leaf the editor announced to his sab*
scribers that over one third of the subscribers to Volume I disoontin-
aed their subscriptions at the close. ^^I have supposed," he says,
*< that those who continued their subscription to the second volume woold
not be BO likely to discontinue it to the third volume, and I have made
my arrangements accordingly. If, however, any considerable number
ahoold discontinue now, it will be subject to veiy serious loss. • • •
I ask as a £avor for all to continue to Yolume III, and notify me doling

MATHEMATICAL JOUBNALS/ , 279

the year if they intend to dtscontinae at its close. I shall then know
whether to begin the fourth volume. I shall not realize a dollar." This
announcement discloses obstacles which all our journals that have been
dependent entirely upon their subscribers for financial support hare
had to encounter, and which none except the more recent could long
resist. Moreover, the Civil War was now at hand. " On account of
the present disturbed state of public affairs, the publication of the
Mathematical Monthly will be discontinued until further notice.'' This
was the end of the Monthly, in 1861.

Gnie salient features in the plan upon which the periodical was con-
ducted, as stated by David S. Hart, • were : " The publication of five
problems in each number, adapted to the capacities of the young stu-
dents, to be answered in the third succeeding number. The insertion
of notes and queries, short discussions and articles of a fragmentary
character, too valuable to be lost ; and, lastly, essays not exceeding
eight pages, on various subjects, in all departments of mathematics.
Besides, there were notices and reviews of the mathematical works
issued, both old and new. Among the most interesting articles are the
account of the comet of Donati, with elegant descriptive plates, written
by the astronomical professor of Harvard University (Vol. I, Nos. 2 and
3); a complete catalogue of the writings of John Herschel (Vol. m,
No. 7) ; articles on indeterminate analysis, by Eev. A. D. Wheeler, of
Brunswick, Me. (Yol. II, Kos. 1, 6, and 12), and the Diophantine analy-
sis (Vol. m, STo. 11). Other articles on the Diophantine analysis by
Mr. Wheeler would have been inserted, if the Mathematical Monthly
had been continued. ^ The Economy and Symmetry of the Honey-bees'
Cells,' by Ohauncey Wright (Vol. II, No. 9). Simon Newcomb gives sev-
eral interesting ^ Notes on Probabilities.' In Vol. II, No. 2, there is an
article containing a complete list of the writings of Nathaniel Bowditch,
accompanied with short sketches of the same, which is extremely inter-
esting. • • • The periodical is embellished by portraits of N. Bow-
ditch, Prof. Benjamin Peirce, and Sir John Herschel, which are finely
executed.'' The Monthly presented a very neat appearance to the eye.
In the mathematical notation employed and in the treatment of mathe-
matical subjects, Benjamin Peirce's influence was clearly perceptible.
From a scientific point of view, the Monthly excelled any of its prede-
cessors.

* Since 1861, we had no mathematical periodical in the United States
for thirteen years. In January, 1874, was published in Des Moines,
Iowa, The Analyst: A Monthly Journal of Pure and Applied Mathemat-
ics^ edited and published by Joel E. Hendricks, a self-taught mathema-
tician. Mr. Hendricks did the printing of the journal himself. It con-
tinued until November, 1883. No previous journal of mathematics in
this countiy had been published regularly for so long a time as this.
Its long life and beneficial infiuence are due to a very great extent to

* Analjst, Yol. II, No. 5, p. 131, Des Moiues, Iowa.

280 TEACHING AND HISTORY OF HATHEBiATICS. j

the untiring energy and self-sacrificing interest of its editor. Its dis-
continuance, after nine years, was not due to want of support, but to
the failing health and strength of Mr. Hendricks. At first it appeared
monthly, afterward bi-monthly. '

The list of contributors included the most prominent teachers of
mathematics in this country. The namSs were no longer those found in
the Mathematical Miscellany or Cambridge Miscellany. A new gener-
ation of workers had come.

As in previous periodicals, so in this, a great part of each number
was devoted to problems. Though the solution of problems is the low-
est form of mathematical research, it is, nevertheless, important, not for
its scientific, but for its educational value. It induced teachers to look
beyond the text-book and to attempt work of their own. The Analyst
bears evidence, moreover, of an approaching departure from antiquated
views and methods, of a tendency among teachers to look into the history
and philosophy of mathematics and to familiarize themselves with the
researches of foreign investigators of this century. Thus, discussions
regarding the fundamental principles of the differential calculus were
carried on. Levi W. Meech gave an << Educational Testimony Concern-
ing the Calculus;'^ W. D. Wilson, of Cornell, gave "A New Method of
Finding Differentials ; ^ Joseph Ficklin, of Missouri, showed how one
might '^ find the differential of a variable quantity without the use of in-
finitesimals or limits \^ C. H. Judson, of South Carolina, gave a valu-
able '^investigation of the mathematical relations of zero and infinity,"
which displayed the wholesome effects of the study of such authors as
De Morgan. Judson dealt powerful blows against the reckless reason-
ing that had been in vogue so long, but, during an occasional unguarded
moment, he was hit by his opponents in return. De Yolson Wood, of
the Stevens Institute, and Simon I^ewcomb, of Washington, discussed
the doctrine of limits.

Another subject considered in the Analyst was the possibility of an
algebraic solution of equations of the fifth degree. A. B. Nelson, pro-
fessor of mathematics in Centre College, Danville, Ky., translated from
the German an article written in 1861 by Adolph Von Der Schulen-
burg, entitled, << Solution of the General Equation of the Fifth De-
gree." The translation and publication of it seem to have been called
forth by a statement of W. D. Henkle in the Educational Kotes and
Queries, to the effect that proofs of the impossibility of such a solution
had been given by Abel and Wantzel. I^elson's paper was followed by
a translation &om Serret's Cout% WAlghhre Supirieure^ by Alexandcor
Evans, of Elkton, Md., of Wantzel's <^ Demonstration of the Impossi-
bility of Besolving Algebraically General Equations of a Degree Higher
than the Fourth.'' Evans also contributed the (non-algebraic) << Solu-
tion of the Equation of the Fifth Degree,'' translated from the Theory
of Elliptic Functions of Briot and Bouquet. W. E. Heal, of Wheeling,
Ind., followed with an article pointing out the error in Schalenburff s

MATHEMATICAL JOUBNALS. 281

solution. One might have supposed that this question had now come to
a rest, but not so. About two years later "T. S. B. Dixon, of Ohicago,
thought he had found a solution, and he published it in the Analyst,
but, in the next number, he stated that he had discovered << the weak
link in the chain'' of its logic.

Of the articles on modern higher mathematics, we mention the "Brief
Account of the Essential Features of Grassmann's Extensive Algebra,''
by W. W. Beman, of Ann Arbor ; " Symmetrical Functions, etc.," and
" Eecent Eesults in the Study of Linkages," by W. W. Johnson, and
an article on determinants by 0. A. Van Velzer, of the University of
Wisconsin.

Among the historical papers is the very complete and interesting
" Historical Sketch of American Mathematical Periodicals," by David
S. Hart, of Stonington, Conn. ; a " History of the Method of Least
Squares," by M. Merriman. Merriman also published Eobert Adrain's
second proof of the principle.

Among other articles of interest are " Multisection of Angles," and
"A Singular Value of ;r," by J. W. Nicholson, of The Louisiana State
University, at Baton Eouge. The latter article was commented upon
by W. W. Johnson, then professor of mathematics in St. John's College,
Annapolis, Md., who was a frequent and most gifted contributor to the
Analyst. Asaph Hall wrote on comets and meteors, George E. Perkins
on indeterminate problems, E. B. Seitz on probability. Other impor-
tant contributors were Daniel Kirkwood, David Trowbridge, Artemas
Martin, and G. W. Hill.

Well known among the mathematical public of America is Artemas
Martin. ^ Before speaking of his two periodicals we shall briefly sketch
his life. This gives us at the same time an opportunity of mentioning
many publications which, though not purely mathematical, contained a
mathematical department. We can think of few American periodicals
of the last thirty years, paying any considerable attention to elementary
mathematics, for which Dr. Martin has not been a contributor. Dr.
Martin was born in 1835. In 1869 he moved to Erie County, Pa., where
he lived on a farm for fifteen years, engaged as a market- gardener. He
is almost wholly self-taught. His leisure moments were devoted to the
study of the "bewitching science." Through the inflaence of the Hon.
W. L. Scott, Member of Congress from Erie, Martin was appointed, in
1885, librarian in the office of the U. S. Coast and Geodetic Survey.
He has a large private library containing a very fine collection of
American books on mathematics. When the writer was in Washington
he enjoyed the great privilege of examining this collection and of seeing
many a quaint and curious volume of great rarity.

Martin began his mathematical career when in his eighteenth year, by
contributing solutions to the Pittsburg Almanac and soon afterward
contributed problems to the " Eiddler Column " of the Philadelphia
Saturday Evening Post, and was one of the leading contributors for

282 TEACHP^G AND HISTOBT OF MATHEMATICS.

twenty years. In 1864 he began contributing problems and solations
to ClarVs School Visitor^ afterward the School-day Magazine, published
in Philadelphia. In Jane, 1870, he took charge of the <^ Stairway De-
partment '' as editor, the mathematical part of which he had conducted
for some years before. In 1875 he was chosen editor of the department
of higher mathematics in the Normal Monthly^ published at Millersville,
Pa., by Edward Brooks. The Monthly was discontinued in 1876. In
this journal he published a series of sixteen articles on Diophantine
analysis. He contributed to the mathematical department of the llli-
noia Teacher (1865-67); the Iowa Instructor y 1865; the National Hdu-
cator^ Kutztown, Pa.; the Tates County Chronicle^ a weekly paper of
Kew York, the mathematical department of which was edited by Samuel
H.Wright; Barneses Udticational Monthly ; the Maine Farmertt^ Almanac;
Educational Notes and Queries, edited and published by W. D. Henkle,
of Ohio. Dr. Martin published thirteen articles on <^ average'' in
Wittenherger, from 1876 to 1880 inclusive. The mathematical depart-
ment of this was edited by William Hoover, afterward professor of
mathematics in the Ohio University at Athens. Martin's name is
&miliar also to the readers of the School Visitor , a journal started in
1880, and edited and published monthly by John S. Boyer in Oettys-
burg, Darke Oounty, Ohio; of the Davenport Monthly, Davenport,
Iowa ; and of The Bizarre, conducted by S. 0. and L. M. Gh)uld, in Man-
chester, K. H. All these journals devoted a portion of their space to
mathematics, and to all these Dr. Martin contributed. The mathe-
matics they contained were of course of an elementary kind. He con-
tributed also to English journals on elementary mathematics. Besides
the above periodicals we mention the Railroad Gazette (New York and
Ghicago), which contained problems in applied mathematics; the
Mathematician, edited by Boyal Cooper, 1877, and utterly worthless ;
and the Wheel, Kew York, 1868, of which only one number ever ap-
peared, in which the question was discussed how many revolutions
upon its own axis a wheel will make in rolling once around a fixed
wheel of the same size.*

In the spring of 1877 Artemas Martin issued the first number, of his
Mathematical Visitor, which he still continues to publish annually.
^' Although he has never served an hour as apprentice in a printing office
to learn the art preservative, he has done all the type-setting for his
publications, except that for the first three numbers of the Visitor, and
has printed all the numbers of the Visitor except the first five on a self-
inking lever press only 6} x 10 inches inside of chase. The numbers of
the Visitor he has printed himself have been pronounced by competent
judges to be as fine specimens of mathematical printing as have ever
been executed. The Magazine he puts in type and gets the preaswork

* For a more complete Uit of Jonmale containing mathematical departaM&l% see
Tk$ BUfarr4, Not€9 and Qtttrim, Volume Y, No. 18^ December, 188&

MATHEMATICAL JOURNALS. 283

done at a printing office, as his press is too small to safely print it,
although he printed one number on it."*

Of the Visitor generally six hundred copies have been printed. The
list of contributors exceeds one Tiundred. In the introduction Dr. Mar-
tin says: "It was stated nearly. three-quarters of a century ago that
the learned Dr. Hutton declared that the Ladies' Diary had produced
more mathematicians in England than all the mathematical authors in
that kingdom.'^ The aim of the Visitor is, if possible, to reach similar
results in this country. It is devoted to the solution of problems.
They deal more particularly in Diophantine analysis, average, and prob-
ability.

In January, 1882, Dr. MaTtin issued the first number of the Mathe-
matical Magazine^ which is published quarterly. It was intentionally
made more elementary than the Analyst of Mr. Hendricks or the An-
nals of Mathematics. It was devoted mainly to arithmetic, algebra,
geometry, and trigonometry. One of the features is the solution and
discussion "of such of the problems found in the various text-books in
use as are of special interest, or present some difficulty." Many of the
articles found in the Magazine and Visitor came from the pen of the
editor himself. Numerous different proofs of the Pythagorean propo-
sition were given in the former, of which we may mention one by James
A. Garfield. It was taken from a magazine of 1876 or 1877, and was
found pasted on a fly-leaf of an old geometry. It resembles somewhat
the old Hindoo proof. Dr. G. B. Halsted contributed several articles on
the prismoidal formula. J. W. Kicholson gave a <' universal demonstra-
tion" of the binomial theorem, without, however, giving a moment's
thought to the question of convergency, whenever the series is infinite.
William Hoover gave an interesting little article on the history of the
algebraic notation. David S. Hart wrote on the history of the theory
of numbers, including the indeterminate and Diophantine analysis. He
also contributed several articles on the subject last mentioned. A
somewhat lengthy discussion was carried on, on the usefulness of log-
arithms, by P. H. Philbrick, professor of engineering at the State
University of Iowa, and H. A. Howe, professor of mathematics at the
dniversity of Denver. The former attempted to show that the use of
logarithms greatly augmented the labor of <^ numerical computation ^
and led to very erroneous results. Some of the calculations in the mag-
azine in which numerical answers are carried to twenty or more decimal
places have no value, either educational or scientific. The names of the
contributors for the magazine were about the same as for the Visitor.

To show the good that elementary journals like this may do, we
give, as an example, the career of E. B. Seitz. He passed his boyhood
on a form, and afterward pursued a mathematical course of two years
at the Ohio Wesleyan University. In 1872 he began contributing
problems proposed in the '' Stairway " depairtment of the School-day

*The Buffalo Express, August 29, 1886. •

284 TEACBIKa AKD HISTOBT OF liATHEMATIC8.

Magazine conducted by Dr. Martin. His energies were stimnlated, and
he became a leading contribator to our periodicals. He astonished his
Mends by his skill in solving difficult problems^ and their admiration
for his talents became almost unbounded. His mathematical mind had
received the first stimulus from our elementary periodicals. Had he
not died in the prime of life, he miglA have done good original work,
provided he had begun to look higher than merely to the solution of
difficult problems in our elementary journals. The solving of problems
is very beneficial at first, but it becomes a waste of time if one confines
himself to that sort of work. The solution of problems is not a high
form of mathematical research, and should serve merely as a ladder to
more ambitious efforts.

Another journal devoted mainly to the solution of problems is the
School Messengevy now called the Mathematical Messenger^ edited and
published bi-monthly by O. H. Harvill, at Ada, La. One of the ablest
contributors to it is J. W. Nicholson, professor of mathematics at the
Louisiana State University. The Messenger commenced February,
1884.

The Annals of Mathematics is the continuation, under a new name and
different form, of the Analyst. It is edited and published at the Uni-
versity of Virginia by Prof. Ormond Stone and Prof. William M. Thorn-
ton. It is of somewhat higher grade than the Analyst, though more
elementary than the American Journal of Mathematics. It contains
more articles on mathematical astronomy and other subjects of applied
mathematics than the American Journal. Our distinguished math-
ematical astronomer, O. W. Hill, contributes several articles in his
specialty. Profis. Asaph Hall, B. S. Woodward, H. A. Howe, and Wil-
liam M. Thornton contribute various articles on applied mathematics.
Professor Oliver, of Oornell, has several papers, one on <^A Projective
Belation among Infinitesimal Elements,'' and another on the '^ General
Linear Differential Equation." Prof. W. W. Johnson writes on
" Olaisher's Factor Tables,'' the " Distribution of Primes," and other sub-
jects. Professor Halsted gives his demonstration of Descarte's theorem
and Enter's theorems. The name of Bohannan, now professor at Ohio
University, appears often« Prof. O. H. Mitchell, of Marietta College,
discusses the equation of the second degree in two variables. Profl B.
H. Graves has geometrical articles ; William E. Heal writes on repetends;
B. T. Moreland, on the momental ellipsoid } J. P. McOullogh, on Bone's
theorem. A little space in each number is devoted to the proposing
and solving of problems. The list of contributors is too large to be
given here in full.

When Professor Sylvester became actively connected with the Johns
Hopkins University, in 1877, the university established the American
Journal of Mathematics^ for the publication of original research in pure
and applied mathematics. It was the design that this should not be a
journal devoted to the pa'blication of solutions to problems, but that it

MATHEMATICAL JOUBNALS. 285

should be of so high a grade as to command a place by the side of the
best Earopean journals of mathematics. It is a source of pride to us
that this great aim has been reached. The American Journal of Mathe-
matica is to-day as well known and as highly respected in Europe as in
America. Among its contributors are found not only the leading scien-
tists of America, but also such foreign investigators as Oayley, Oliflford,
* Orofbon, Fa& de Bruno, Frankland, De Gasparis, Glashan, Hammond,
Hermite, Kempe, Lipschitz, Loudon, Lucas, MacMahon, Muir, Petersen,
Poincar6, Roberts, Weichold, and G. P. Young.

The subject which has received most attention in the American JoW'
nal of Mathematics has been Modern Higher Algebra. The contribu-
tions of Sylvester on this -subject loom large. In Volume I is found
<<a somewhat speculative paper" entitled, <<An Application of the
Kew Atomic Theory to the Graphical Representation of the Invariants
and Oovariants of Binary Quantics,'' followed by appendices and notes
relating to various special points of the theory.* Sylvester contributed
various memoirs on binary and ternary^ quantics^ including papers by
himself, with the aid of Dr. Franklin, containing tables of the numer-
ical generating functions for binary quantics of the first ten orders, and
for simultaneous binary quantics of the first four orders, etc The list
of his articles is too extensive to be mentioned here. Since his return
to England he has contributed to the Journal a series of '^ Lectures
on the Theory of Beciprocants,'' reported by J. Hammond.

The larger number of American contributions are from persons who
were, or still are, connected with the Johns Hopkins University, either
as teachers or students. Dr. W. E. Story, of the Johns Hopkins ITni-
versity, has written on *' If on-Euclidean Trigonometry,'* "Absolute
Olassification of Quadratic Loci, etc.," and other, chiefly geometrical,
subjects. Dr. T. Oraig has contributed numerous papers, mainly on
the theory of functions and differential equations. Dr. F. Franklin has
aided Professor Sylvester in the preparation of papers, and has also
made various independent contributions. After the return to England
of Professor Sylvester, Professor Kewcomb became editor-in-chief. His
valuable articles have been noticed elsewhere. Among the contribu-
tors who were once students at the Johns Hopkins University, but
are now not connected with it, are E. W. Davis, W. P. Durfee, G. S.
Ely, G. B. Halsted, A. S. Hathaway, O. H. Mitchell, W. I. Stringham,
0. A. Van Velzer, A. L. Daniels, 0. Yeneziaui, D. Barcroft, and J. 0.
Fields. The Journal has two lady contributors, Mrs. 0. Ladd Frank-
lin, of Baltimore, and Miss 0. A. Scott, of Bryn Mawr OoUege. The
great memoir on <^ Linear Associativre Algebra," by Benjamin Peirce,
was published in the American Journal of Mathematics; also articles
by his son, 6. 8. Peirce, on the "Algebra of Logic ^ and the " Ghosts in
Diffraction Spectra." Papers on applied mathematics have been written
by Professor Rowland, of the Johns Hopkins University, and George

* F1<2e Professor Cayley's article on Professor Sylvester in Nature, January 3, 1889.

286 TEACHINa AND HISTOB7 OF HATHEMATIGS.

William Hill, of the Nautical Almanac Office Mr. Hill has done ad-
mirable work in mathematical astronomy. For his researches on the
lunar theory, published in the American Journal, and for other astro-
nomical papers, published elsewhere, he was awarded the gold medal of
the Boyal Astronomical Society, in 1887.* Among the writers for the
American Journal is Prot W. W. Johnson, of the XT. S. Kaval Acad-
emy at Annapolis. He is also a frequent contributor to leading Boro-
pean journals, and commands a place among the very foremost of
American mathematicians. In the list of American writers to the
Journal are H. T. Eddy, J. W. Gibbs, E. MeOUntock, A. W. Phillips,
J. Hagen, E. W. Hyde, H. B. Finei and others of no less power and
originality.

The XT. S. Coast Ain> Obodetio Subyet.

In giving the origin of the IT. S. Ooast Survey it is desirable to begin
with a sketch of the preliminary training of its first snperintendent^
Ferdinand B. Hassler. He was bom in the town of Aaran, Switxer-
landy in 1770. He was sent to the University of Bern to study law, bat
he soon drifted into mathematics and became a favorite pnpil of Profl
John G. Tralles.t Tralles and Hassler undertook the topographical
survey of the Oanton of Bern. In 1791 a base-line was measnred| and
a net of triangles established. The instruments on hand being found
insufficient for long distances new ones were ordered from Bamsden,
in London. On the receipt of these, in 1797, the survey was resumed,
but soon discontinued. The conquering armies of the French came
marching into Switzerland. The feeble republic was forced to submit
to the dictatorial orders of the war minister of France, which required,
among other things, that the places then occupied by the Swiss engi*
neers should be vacated and filled by French. A swarm of sixty
French engineers appeared, but soon disappeared without accomplish-
ing anything. The revolutionary tendencies of the times and the un-
settled state of the country induced Hassler to quit Switzerland. His
fatherland seemed to bear no roses for him. Having landed in Phila-
delphia, in October, 1805, he formed the acquaintance of Prof. Bobert
Patterson and Mr. Oamet, of New Brunswick, to whom he showed his
mathematical books and instruments.

About this time Congress was considering the feasibility of a survey
of the coasts and harbors. Professor Patterson sent to President Jef-
ferson a sketch of Hassler's scientific career in Switzerland, and Mr.
Clay, the Bepresentative from Philadelphia, in 1806, asked Hassler
whether he would be willing to undertake the survey, in case that the

* Vide Monthly Kotioes on the Boyal Afitronomical Society, Vot XLYII, No. 4,
February, 1887.

t Translation from the German of Memoirs of Ferdinand Rudolph Hassler, by Emil
Zsohokke, pablished in Aaraa, Switzerland, 1877, with Supplementary Dooament«,
published in Nice, 18i^2«

THE n. S. COAST AKD GEODETIC SUBYET. 287

Ctovenunent should decide upon one. Mr. Hassler was, of coarse, willing.
The law authorizing the survey was passed in February, 1807. Hass-
ler received one of the twelve circulars which were sent to scienti&c
men for plans of the contemplated survey. By the direction of Presi-
dent Jefferson, a commission (formed, it appears, of the very gentlemen
who had proposed plans, excepting Mr. Hassler) examined the various
plans at Professor Patterson's, in Philadelphia. They rejected their
own projects and recommended to the President the one suggested by
Mr. Hassler. The survey proposed by him was of a kind that had never
been previously attempted in this country; it was to be a triangulaUon^
and the sides of the triangles were to be from ten to sixty miles in
length, such as had, at that time, just been executed in France and was
in progress in England. The project was quite in advance of the sci-
ence of our country. It was fortunate for us that a man of Mr. Hass-
ler's learning, ability, and mechanical ingenuity was available to the
Government. He had, meanwhile, been appointed by Jefferson pro-
fessor at West Point. This position he resigned after three years, and
accepted the professorship of mathematics at Union College, Schenect-
ady, N. T. Politics delayed the work of the survey. The first thing
to be done was to procure the necessary instruments. In 1811 Hassler
was sent to England by our Government to direct the manufacture of
suitable instruments. Shortly after his arrival in Great Britain the War
of 1812 broke out, and he was four years in London, in the disagreeable
position of an alien enemy, and half the time left by our Government
without compensation. He returned to this country in- 1814, with a
splendid collection of instruments, which had cost nearly forty thousand
dollars.

In August, 1816, a formal agreement between the Government and
Mr. Hassler was reached, to the effect that he should undertake the
execution of the survey. He immediately entered upon the preliminary
steps of reconnoitering and the numerous collateral experiments neces-
sary for such a survey. Two preliminary baselines were measured :
One in If ew Jersey, in the rear of the Highlands, on North Biver, and
nearly six miles in length ; the other on Long Island, and of about five
miles. Down to the year 1818 eleven stations were .occupied, forming
the elements of 121 triangles.

To a scientific man, familiar with the many preliminary details which
are indispensable to accurate scientific work, but which do not always
appear in the ultimate results, the progress which Hassler was making
would have seemed highly satisfactory. Congress, however, was dis-
pleased. In April, 1818, Mr. Hassler received official notice that he was
suspended, accompanied with the remark that the little progress hitherto
made in the work had caused general dissatisfaction in Gongress, Pos-
sibly the feeling on the part of American engineers against this for-
eigner because he had been preferred to one of them had something to
do with this suspension. To Hassler this was a very severe blow ; his

288 TEACHIKG AITD HISTORY OF MATHEMATICS. ^

brightest hopes seemed dashed into fragments* A year or two later he
prepared a defense of himself. He wrote an account of his plans and
methods and published it in the Philosophical Transactions of Phila-
delphia (Vol. HI, New Series, 1825). By this article' he hoped to vindi-
cate his schemes. It attracted the attention of scientific men every-
where. It was reviewed by leading astronomers in Europe — Bessel,
Btruve, Schumacher, F^russac, FrancoDur, Krusenstem, and others — all
agreeing that Mr. Hassler's plans were good, and testifying to his in-
ventive genius for solving the difficulties of the Ooast Survey, as well
as to the certainty that his plans, if carried out, would lead to success.
Bessel was certainly a competent *judge, for, in addition to his theo-
retical knowledge, he had had experience in geodetic work in Germany.
He had words of only the highest praise for Hassler's scheme.* *

After his suspension from the survey, Hassler engaged in various
occupations. For a while he was a farmer in northern New York. He
afterward went to Jamaica, Long Island, and then to Bichmond, Va.,
giving lessons in mathematics to sons of prominent men. While in
Bichmond he published his Elements of the Geometry of Planes and
Solids, 1828. His Elements of Analytic Trigonometry appeared in 1826.
Subsequently he published an Arithmetic, Astronomy, and Logarithms
and Trigonometric Tables, with introductions in five languages.

After twelve years in rural retreat, Hassler was recsdled to official
activity. He became United States ganger, and then was intrusted,
from 1830 to 1832, with the difficult mission of regulating the standards
of weights and measures throughout the United States, which at that
time were very various.

In 1828 the question of the Goast Survey was again agitated. The
Secretary of the BTavy reported to Congress favorably on Hassler's
work, which had been suspended so suddenly ten years previously.
The Secretary said that ^^ he [Hassler] had accomplished all that was
possible in so short a time." In 1832 Mr. Hassler was reinstated, with
the title of " Superintendent of the United States Goast Survey.''

In the interval from 1818 to 1832 nothing of permanent value had
been accomplished. Attempts had been made to survey portions of the
coast, under the direction of the I^avy Department, but there had been
no general or connected survey. The charts prepared had been expen-
sive and unsafe, and not very creditable to the country.

In 1832 began the most successful and most famous period in Mr.
Hassler's life. Though sixty-two years old, there still glowed in him the
fire of youth. The survey was begun with vigor. He had a traveling
carriage prepared for him, which conveyed him rapidly to all parts of
the survey. In this carriage he could seat himself at a writing table
or dispose himself for sleep. The work was prosecuted according to the
plans first laid out by him. He labored under the great disadvantage
of having no skilled assistants. His corps of workmen had aU to be

• Vide Silliman's Joomal, Vol. IX, p. 226.

THE U, S. COAST AND OEODETIO SUEVBT, 289

trained and edaeated to the refined methods which he was introducing.
Gfhe work of the survey had to be systematized. It continued under his
direction until the time of his death, in 1843. He left the work well
advanced between Karragansett Bay and Cape Henlopen, and the sur-
vey sufficiently organized in all its varied details. Hi9 course was, how-
ever, frequently criticised in Congress, and it was not always easy to
get the necessary appropriations.

Mr. Hassler was very self-confident and independent. . This was one
cause of the occasional opposition to him. Though not conceited, he
was conscious of his superiority over the great mass of men with whom
he came in contact in Washington. The following anecdote is charac-
teristic of him : At one time the cry of ^^retrenchment and reform"
had become popular, and a newly appointed Secretary of the Treasury
thought he could not signalize his administration more aptly than by
reducing the large salary of the superintendent. He sent for Mr. Hass-
ler and said, <<My dear sir, your salary is en£>rmous ; you receive $6,000
per annum — ^an income, do you know, quite as large as that of the Sec-
retary of State.'' '^ True," replied Hassler, '< precisely as much as the
Secretary of State and quite as much as the Ghief of the Treasury ; but
do you know, Mr. Secretary, that the President can make a minister of
State out of anybody ; he can make one even out of you, sir; but if he
can make a Hassler, I will resim my place."

Hassler's successor was Alexander Dallas Bache, a great-grandson of
Franklin and a graduate of West Point. He exercised a very marked
influence over the progress of science among us. He graduated at the
head of his class, and the great expectations that were then entertained
of him have been fully realized. For eight years he devoted himself to
physical science, while professor at the University of Pennsylvania,
and gained a wide reputation. The Ooast Survey made rapid progress
under his management. Congress began to show better appreciation of
this sort of work, and niade more liberal appropriations. This enabled
him to adopt a more comprehensive scheme. Instead of working only
at one locality, as had been done previously, he was able to begin
independent surveys at several places at once, each section employing
its own base. He proposed eight sections, which number was increased
on the annexation of Texas, and again on the annexation of Oalifomia.

Two of the most important, improvements of modem geodesy were
perfected and brought into use at the beginning of Bache's superin-
tendency, namely, Mr. Talbott's method of determining latitudes and
the telegraphic method of determining longitude. Various other re-
finements in every branch of work were introduced. Systematic obser-
vations of the tides, a magnetic survey of the coast, and the extension
of the hydrographic explorations into the Gulf Stream were also insti-
tuted by Bache.

Having extended the scope of the Survey, Bache needed a greater
number of assistants, but the supply was not wanting. Says Prof. T*

881— No. 3 19

290 TEACHIKG AND HIBTOBY OF MATHEMATICS.

H. Bafford,* ^ be found available for its higher geodetic works a number
of West Point officers, of whom T. J. Lee was one, and HamphreySt af-
terward chief engineer of the Army^ another. One of the leaders in
practical astronomy of the topographical engineers was J. D. Graham ;
and tbe wc^k which had been done by that corps npon the nalTonal and
State boandaries had familiarized a good maiay Army officers with fleld
astronomy and geodesy.

^< Bache, who had been ont of the Army nearly twenty years employed
his great organizing and scientific capacity in training the Ooast Surrey
eorps (indnxling detailed Army officers) into practical methods for its
various problems ; and the connection between the West Point officers
and the able yonng civilians, who are now the veterans of the Survey^
was extremely wholesome.

<< Lee prepared a work (Tables and Formulae) which has served an
excellent purpose in bridging the gap between theory and practice;
especially for the last generation of West Point officers. "

Graduates of West Point are now more closely employed in military
and other public duty, and are no longer employed in the Coast Survey.

The work of the Survey was interrupted by the Civil War. Soon
after its close Bache died (1867). Benjamin Peirce, his successor in the
superintendency, said of him : ^^ What the Coast Survey now is, he
made it. It is his true and lasting monument It will never cease to
be the admiration of the scientific world. * * * It is only necessary
conscientiously and faithfully to follow in his foot-steps, imitate his ex-
ample, and develop his plans in the administration of the Survey."

Under Peirce, the survey of the coasts was pushed with vigor, and
it rapidly approached completion. He proposed the plan of connecting
the survey on the Atlantic Coast with that on the Pacific by two chains
of triangles, a northern and a southern one. This project received the
sanction of Congress, and thus the plan of a general geodetic survey
of the whole country was happily inaugurated.

Beigamin Peirce's successor on the Coast Survey was Carlile Pollock
Patterson. He was a graduate of Georgetown College, Kentucky, and
had for many years previous to this appointment, in 1874, been connected
with the Survey as hydrographic inspector. Under him the extension
of the Survey into the interior of our country, as inaugurated by Peirce,
was continued. By the completion of this work this country will con-
tribute its &ir share to the knowledge of the figure of the earth, which
has hitherto been derived entirely from the researches of other nations.
On account of this extension, the name, *^ U. S. Ooast Survey, " was
changed, in 1879, to <^ U. S. Ooast and Geodetic Survey. "

Patterson died in 1881, and Julius Brasmus Hilgard became his suc-
cessor, Hilgard was bom in Zweibriicken, Bavaria, came to this conn-
try at the age of ten, and at the age of twenty was invited by Bache to
become one of his assistants on the Survey. Hilgard soon came to be

*3iathematloaI Teaohings, p. 6.

THE Tf. S. COAST AND GEODETIC SURVEY. 291

recognized for great ability and skill, and rose t6 the position of assist,
ant in charge of the Office in Washington. He held the superintend-
ency from 1881 to 1885, when he resigned. His work consisted chiefiy
of researches and discussions of results in geodesy and terrestrial
physics, and in the perfecting of the methods and instruments em-
ployed. The superintendency was next intrusted to Frank M. Thorn,
who was sncoeeded in July, 1889, by T. 0. Mendenhall, who now fills
the ofSce.

The work of the iT. S. Ooast Survey has been carried on with great
efficiency from its very beginning, and reflects great credit upon Amer-
ica. In makiug the computations for the Survey, the method of least
squares for the a<^ustment of observations has found extended appli-
cation. Valuable papers on this subject by Bache and Schott have
been printed in the reports of the U. S. Coast Survey.* Charles A.
Schott graduated at the Polytechnic School in Carlsruhe, came to this
country in 1848, and has since that time been an efficient worker on the
U. S. Coast Survey. He is now chief of the computing division.

It will be remembered that interesting researches on least squares
had been made quite early in this country by Bobert Adrain. Benjamin
Peirce invented a criterion for the rejection of doubtful observations.!
It proposes a method for determining, by successive approximations,
whether or not a suspected observation may be rejected. Tables are
needed for its application. Objections have been made to its use, be-
cause it ** involves a contradiction of reasoning." f The criterion is
given by Chauvenet in his Method of Least Squares (1864), and has
been used to some extent on the U. S. Coast Survey, but it has found
no acceptance in Europe. Chauvenet gives an approximate criterioti
of his own for the rejection of one doubtful observation, which is de-
rived, he says, " directly from the fundamental formula upon which the
whole theory of the method of least squares is based." But this cri-
terion, too, has been criticised as being ^^ troublesome to use, and as
based on an erroneous principle." Stone, in England, offered still an-
other criterion for the rejection of discordant observations, but Glaisher
pronounces it untrustworthy and wrong. No criterion has yet been
given which has met with general acceptance. Indeed, Professor New-
comb considers it impossible that such a one should ever be invented.
Says he (in his second paper mentioned below) : " We here meet the
difficulty that no positive criterion for determining whether an observa-
tion should or should not be treated as abnormal is possible. Several
attempts have indeed been made to formulate such a criterion, the best
known of which is that of Peirce."

* See reports for the years 1850, '55, '56. '58, '61, '64, '66, '67, '75.

t Qoald's Astronomical Joarnai, Vol. II, pp. 161-3.

t See Prof. Mansfield Merriman's article in the Transactions of the Conneotient
Academy, containing a list of writings relating to the method of least squares and
the theory of the accidental errors of ohseryationi which comprises 408 txtlea by 193
authors.

TEACHIHG AMD HfiBTOBT OP KATHEMATZCS.

Talnable papers on leart squares haTe been eontribntod in tiiis oonn-
try by 6* P. Bond,* of Harvaid ; Simon Newooflib,tG.S. Pieree,| and
Tnunan H. SafEsrd.$ The text-books on this snbjeet generally nsed iu
oor sdiools are tiiose of Ghanvenet, Heniman, and T. W. Wright.

* "On. the 1180 of EqaiTalent Paeton in the Method of Least Sqiuiea," MemoirB Ameri-
can Aeadem J, YoL YI, pp. 179-212.

t^'AXeehflnieal Bepreeentation of a Familiar Problem," Montlilj Kotiee* e£ the
AftroDomieal Society, Louden^ YoL XXXm, pp. 573-'4; ''A Genendised Theory of
the CombinJition of Obeeryations flM> as to Obtain the Best Bteolts," American Jour-
nal of Kathcnmtics, Yol. YIIL

t " On the Theory of Errors of ObserTations," Beport XT. S. Coast Soirey, 1870, pp.
200-224.

f *' On the Method of Least Sqaates," Proceedings American Academy, YoL XL

IV.

THE MATHEMATICAL TEACHING AT THE PRESENT TIME.

The mathematical teaching of the last ten years indicates a ^^rup-
ture" with antiquated traditional methods,,and an <^ alignment with the
march of modern thought." As yet the alignment is by no means recti-
fied. Indeed it has but barely began. The ^' rupture " is evident from
the publication of such works as Newcomb's series of mathematical text-
bookSy recent publications on the calculus, the appearance of such alge-
bras as those of Oliver, Wait, and Jones, Phillips and Beebe, and Yan
Yelzer and Sllchter; of such geometries as Halsted's ^< Elements" and
^< Mensuration;" of such trigonometries as Oliver, Wait, and Jones's;
of CarlPs Calculus of Variations; Hardy's Quaternions; Peck's and
Hanus's Determinants ; W. B. Smith's Co-ordinate G^metry (employ-
ing determinants) ; Craig's Linear Differential Equations.

Determinants and quaternions have thus far generally been offered
as elective studies, and have formed a crowning pinnacle of the mathe-
matical courses in colleges. It is certainly very doubtful whether this
is their proper place in the course. It seems quite plain that the ele-
ments of det^minants should form a part of algebra, and should be
taught early in the course, in order that they may be used in the study of
co-ordinate geometry. What place should be assigned to quaternions is
not quite so plain. Prof. De Yolson Wood introduces their elements in
his work on co-ordinate geometry. The professors of Cornell have not
taught quaternions directly for some years, but are convinced that most
students derive more benefit by a mixed course in matrices, vector ad-
dition and subtraction, imaginaries, and theory of functions. The early
introduction of determinants seems more urgent than that of quaterni-
ons. We think, however, that great caution should be exercised in in-
corporating either of these subjects in the early part of mathematical
courses. Those universities and colleges which are, as yet, not strong
enough to maintain a high and rigid standard of admission, and whose
students enter the Freshman class with only a very meagre and super-
ficial knowledge of the elements of ordinary algebra, would find the in-
troduction of determinants and imaginaries as Freshman studies a
hazardous innovation. One of the very first considerations in mathe-
matical teaching is thoroughness. In the past the lack of thoroughness
has poisoned the minds of the American youth with an utter dislike and
bitter hatred of mathematics. Whenever a subject is not well under-
stood, it is not liked; whenever it is well understood^ it is generally
liked.

294 TEACmKG AND HISTOBY OF MATHEMATICS.

There is almost always some one author whose text-books reach
Tery extended popalarity among the great mass of schools. Such au-
thors were Webber, Day, Davies, and Loomis. If we were called upon
to name the writer whose books have met with more wide-spread circu-
lation during the last decennium than those of any other author we
should answer, Wentworth. Mr. Wentworth was born in Wakefield,
N. H., fitted for college at Phillips Exeter Academy, graduated at Har-
vard College in 1858, and then returned to Phillips Exeter Academy,
where he has been ever since. He had for instructors in mathematics,
at the academy, Prof. Joseph G. Hoyt, afterward chancellor of the
Washington University in St. Louis ; and, in college, Prof. James MiUs
Pelrce. " The characteristics of my books," says Mr. Wentworth, " are
due to what I have found from a long experience is absolutely necessary
in order tl^at a pupil of ordinary ability might master the subject
of his reading. To learn by doing, and to learn one step thoroughly
before the next is attempted, constitute pretty much the whole story."
In point of scientific rigor Wentworth^s books are superior to the popu-
lar works of preceding decades. It seems to us that the book most
liable to criticism is his Elementary Geometry (old edition). He has
been greatly assisted in the writing of his books by leading teachers
from different parts of our country. Some of the books bearing his
name are almost entirely the work of other men.

It is to be hoped that the near fhture will bring reforms in the mathe-
matical teaching in this country. We are in sad need of them. From
nearly all our colleges and universities comes the loud complaint of in-
efficient preparation on the part of students applying for admission;
from the high schools comes the same doleful cry. Errors in mathe-
matical instruction are committed at the very beginning, in the study
of arithmetic. Educators who have studied the work of Prussian
schools declare that our results in elementary instruction are far infe-
rior. Says President 0. K. Adams, of Cornell University : • "In the
lowest grades of schools our inferiority seems to me to be very marked.
The results of the earliest years of the European course, I mean those
devoted to teaching the boy, say from the time he is nine years of age
until he is fourteen, when compared with the fruits of the courses pur-
sued during the corresponding years in the average American school,
are immeasurably superior.'' President Adams institutes a comparison
between Brooklyn and Berlin schools. Speaking of a Brooklyn boy of
fifteen, he remarks : " In the first place it must be said that he has had
forced upon him six hours a week in arithmetic, during the whole of
the seven primary grades. Then on emerging from the primary school,
and coming into the grammar school, he is required to take an average
of four hours a week in the same study, during all the eight grades.
That is to say, during the whole of the boy's career in school, from the

* New England AMooiation of CoUefres and Preparatory Schools ; Addzeaaea and
Fvooeedings at the Annual Meeting, 1888, p. SM.

MATHEMATICAL TEACHING AT THE PBE8EKT TIME. 297

(a)UKIYEIt8ITISB AND COLLXGKS— Coiltilllied.

"Stone of institatloD.

Location..

Name of person
reporting.

Title or position of person
reporting.

Uiiiyenity of Colorado . . .

Boulder, Colo

LM.DeLong

Professor of mathematics.

SUte Sohool of Mines . . . .

Golden, Colo

Paul Meyer

Do.

Stom Af^ricaltaral School

StorrSfConn

W.P.Washbnm..

Professor of chemistry and
mathematics.

1ft

TrinitT College

Hartford, Conn

F.S. Lnther ......

Professor of mathematicfl

and astronomy.

XTniyenity of DakoU

Veimillion, Dak ....

L.S.Halbnrt

Professor of mathematics.

Dakota School of Mines . .

Rapid City, Dak....

L.L.Conant

Do.

Georgetown College

Washington, D.C...

J.F.Dawson

•

Professor of physios and
mechanics.

National Deaf-Mate Col>

....do •••••■•••••■■.•

€ Joseph C. Gtordon
' 1 A. G. Draper

Assistant professor of

lege.

mathematics.

Howard UniTersity

....do

G.W.Cook

Tutor in mathematics.

De Luid IJniTeraity

De Land, Ha

B. Gentry

Seminary West of the Sa-
wannee RiTer.

G.M. Edgar

President and professor of
mathematios and natural
science.

Florida State Agricnltaral
College.

Lake City, Fla

L.H.Orlemaa

Professor of mathematics.

Bowdon College ••••••....

Bowdon. Qa

CO.Stabbs

Do.

North Georgia Agrieult-
nzal College.

Dahloneg%Ga

W.& Wilson

Do.

Cave Spring, Ga

Athens, Ga .........

E.T.Whatley ....
W. Bntherford. . . .

President.

TJniveniity of Georgia. . . .
Bttreka College. ..••«.....

Biireka,Ill

GA. vin«p _, ,

Do.

Illinois State Kormal ITnl-
rersity.

Normal, III

J.W.Cook

Ihstmotor in mathematics.

Lombard University

Galesbnrg,Ill

J.y.N.Standish..

and astronomy.

M'Kendxee CoUege

Lebanon, 111

A.G.Jepson

German>EngUsh College..

Lincoln Uniyenlty

Lake Forest Universi^ ..

GaUNBSblll

Fr.Schaub

President.

Lincoln. ni .....r.T-

Professor of mathematics.

Lake Forest^ 111

M.MoNeiU

and astronomy.

8ft

ITniTerslty of Illinois, ....

r S. W. Shattnok ..
(S.H.Peabody....

Begent (president).

Illinois CoUeire

Jacksonville. Ill ....

J.H. Pratt

Ph.D.

Korth-Westem CoUege . . .

Napersville.111

HF.Kletsing....

Indiana TTnivf^niity

Bloominston. Ind . . .

J.Swain

Do.

Wabash College

Crawfordsville, Ind .

J. Norris ..........

Do.

Earlham College ...•••...

Biohmond. Ind

W. B. Morzan

Do.

Bose Polytechnio Insti*

tate.
De Pan w Uniyeraity

TerreHaate,Ind...

C. A. Waldo

Greencastie, Ind ....

A. Martin

President

Franklin Collese

Franklin. Ind

B. J. Thompson. . .
O.E.Cofian

Profpseor of mathematics.

Indiana Normal College . .

Covington, Ind

Hanoyer Colleire r

Hanov^i*. Ind .....■■

F. L. Morae . ......

Professor of mathematios.

State TJniyersity of Iowa.

Iowa City, Iowa ....

L. G.Weld

Acting professor of malh-
ematics.

University of Des Moines .

Des Moines, Iowa. . .

-T.M.BlakB]ee....

Ph. D., Yale, 1880.

Oskaloosa College

Oskaloosa, Iowa

J. A. Seattle

President.

Upper Iowa Universitiy ..

Fayette^ Iowa

J.W.BreseU

Do.

296

tEACHING AND HIST0B7 OF MATHEMATICS.

8or Safford strongly recommends the parallelism of the two main
mathematical subjects — arithmetic including algebra, and geometry
iAcluding trigonometry and. conic sections. Thereby the study of
algebra and geometry can be extended over a longer period of time. Ac-
cording to his ideal programme of study, primary arithmetic is accom-
panied by notions of form and drawing ; arithmetic through rule of
three, by rudiments of geometry ; universal arithmetic and simple equa-
tions, by one or two books in plane geometry; algebra through quad-
ratics, by plane geometry ; advanced algebra, by solid geometry, conic
sections, plane trigonometry, etc.

^^ Of course this programme is somewhat variable, but the main prin-
ciple, that a.course of arithmetic must run parallel with one of geometry
from the beginning of a school coarse to the end, is one which is laid
down by the best educators since Pestalozzi's time."*

In order to enable the writer to give a view of the present condi-
tion of mathematical teaching, the Bureau of Education sent to various
universities, colleges, normal schools, academies, institutes, and high
schools, a printed letter with a series of questions to be answered. We
give a list of the institutions which sent in replies, and state the le*
suits as fully as our space will permitt

STATISTICS ILLUSTRATING THE PRESENT CONDITION OF MATHEMAT-

ICAL INSTRUCTION IN THE UNITED STATES.

(a) Universities and Colleges.

Name of institatioii.

Location.

Name of peraon
reporting.

Title or position of person
reporting.

UiiiveTsity of Alabama . . .

Toaoaloosa, Ala

T,W. Palmer

Professor of mathematics.

Spriofc Hill College

Central Female College. . .

Mobile, Ala

A. S. Wagner

S.R Foster

Do.

Tascalooea, Ala

President'

AJabama Polytechnio In-
•titate^ AKricnUuraland
Mechanical College.

AabnnifAlA

0. D.Smith

•

Professor pfmatihamatJcs.

HantOTille Female College

Hunt8Tille.A]a

A.B.Jonoa

President

Talladega College

Talladega, Ala

Jesse Bailey

PrincipaL

Alabama Conference Fe-
male College.

Fesk6gie,Ala

JohnMaasey

President

Pierce Christian College. .

College City, Cal...

D.E.Hnghea

Professor of mathematica,
astronomy, and civil en-
gineering.

Stignatiaa College

San FranclBOO, Cal. .

T.C.Leonard

Teacher of higher mathe-
matics.

State Agricnltaral College

Fort Collina, Colo..

7.£.Stolbrand...

Professor of mathematics
and master of literary sci-

TJniyersity of Denyer ....

Denver, Colo

H. A. Howe

Professor of mathematloa
and astronomy.

* Monograph on Mathematical Teaching by T. H. Safford. p. U.

t From this list are omitted some few reports which were sent in too late for insertion, or whioh did
■ot give the aaraa of the institatlon or the peraon reporting, or vhioh were illegible.

MATHEMATICAL TEACHING AT THE PBESESTT TIME. 297
(a) Ukiyebsitiss and Collxgks— Continued.

Name of InttltatloD.

Location..

Name of peraon
reporting.

Title or poaition of peraon
reporting.

Uniyeraity of Colorado . . .

Boulder, Colo

LM.DeLong

Profeaaor of mathematica.

SUt« School of Minea —

Golden, Colo

Paul Meyer

Do.

StoiTs Ajjpricnltaral School

Storra,Conn

W.P.Waahbnm..

Profeaaor of chemiatry and
mathematica.

Trinity CoUeffO

Hartford, Conn

•

F.S. Lnther

and aatronomy.

UniTenity of Dakota

Veimillion, Dak —

L.S.Hnlbnrt

Profeaaor of mathematica.

Dakota School of Hinea ..

Rapid City, Dak....

L.L.Conant

Do.

Georgetown College

Waahington,D.C...

J.F.Dawaon

•

Profeaaor of phyaica and
mechanica.

Kational DeafMnte Col-

....do.. ...... .......

€ Joaeph C. Gtordon
I A. G. Draper

Aaaiatant profeaaor of

lege.

mathematica.

Howard UniTeraity

....do. ....... •••....

G.W.Cook

Tutor in mathemaUoa.

De Land TJniTeraity

Seminary Weat of the Sa-

De Land. Fla .......

B. Gentry

Teacher of mathematica.

G.M.£dgar

Preaident and profeaaor of

wannee Riv^r.

•

acience.

Florida State Agricnltnral
College.

Lake City, Fla

L.H.Orlemaa

Bowdon Coilece •■.»••••.

Bowdon. 6a. ........

C. 0. Stubba

Do.

North Georgia Agiienlt-
nzal College.

Dahloneg^Ga

W.&Wilaon

Do.

Heaxn Inatitate

Caye Spring, Ga —

E.T.Whatley ....

Preaident.

ITniyeraity of Georgia —
KiiirAirik Colleire- ««....

Athena, Ga

"W. Bntherford. . . .

Enreka^Ill

G. A. Killer

Do.

niinoia State Normal ITnl-
versity.

Normal, lU

J.W.Cook

Ihatmotor in mathematioa.

Lombard Univeraity

Gale8barg,Ill

J.y.N.Standiah..

and aatronomy.

M'KendreeCoUege

Lebanon, 111

A.G.Jepaon

Profeaaor of mathematica.

Lincoln Uni yenlty

Lake Foreat TJniyerai^ ..

Gkilena,m

Fr.Schaub

Preaident.

Lincoln, ni T.-T

Lake Foreatk 111

M. McNeill

and aatronomy.

Unlyeraity of Dlinoia, ....

Urbana,Dl

r&W.Shattnck..
(S.H.Peabody....

Begent (preaident).

niinoia CoUeee

Jaokaonyille. Ill ... .

J.H. Pratt

Ph.D.

North-Weatem College . . .

Naperayillclll

HF.Kletaing....

Profeaaor of mathematica.

Bloominston. iDd . . .

J. Swain ...*.^t...-

Do.

Wabaah CoUeire

Cr^wfordfv^ille. Ind

J. Norria..........

Do.

Earlham College

B-iohmond. Ind ......

W. R Monran

Do.

Boae Polytechnio Inati-
tate.
De Pan w Uniyenlty

Terre Hante, Ind . . .

C. A. Waldo

Gi'AAnc«4tle. 7nd . - T -

A.Martin, r..

President.

Franklin Collese

Franklin, Ind

B. J. Thompaon. . .
O.E.Coflan

Indiana Normal College . .

Coyington, Ind

Hanoyer CoUeire..

Hanoyer. Ind .......

F.L. Morae

Profeaaor of mathematica.

State Unlyeraity of Iowa.

Iowa City, Iowa ....

L.G.Weld

Acting profeaaor of math-
ematica.

Uniyenity of Dea Moinea.

Dea Mdnea, Iowa. . .

-T.M.BlakBlee...-

Ph. D., Yale, 1880.

ftS

Oakalooaa College

Oakalooaa, Iowa

J. A. Seattle

President.

Upper Iowa Uniyeiaity ..

Fayette^ Iowa

J.W.BreaeU

Do.

298

TEACHma AKB HISTOBT OF MATHEICATICS.
(a) U111VER8ITIB8 Aia> COLLBOB9— Continned.

Kune of inatitaiion.

Location.

Name of person
reporting.

Title or position of person
reporting.

Oswego College for Youog
Ladies.

Oswego, Kan

S.H. Johnson

PrindpaL

TTnlT^rvity of 'KsilM4 r . . ,

Lawrence, Kan

E. Miller

ProfniMOr of maihemfttlfiL

Ottawa XTnlyenity

Ottawa, Kan

M.L.Ward

Professor of mathematies
and political adenoe.

Bethany CoH^ge and Nor-
mal Institnte.

Lindsborg, Kan

W. A. Granyllle ..

Professor of mftthematloii

Waahborn College

Topeka,Kan

P.MoVIcar

President.

Sff

Xaiisaa State Agrictdtonl
College.

Manhattan, Kan

•

D.KLants

Professor of mathemntioib

West Kentaoky Classical
and JSormal College.

SoQth CarroIlton.Ky

W.CQaynor

President.

Millerslrarg I^Bmale Col-
lege.

MiUersbarg, Ky

C.Pope

Do.

Bereft College

Berea,Ky....'.

P.D.Dodge

Aoting profeoaor of matb-
ematica.

Xmiaenoe College and
Normal SohooL

H. TPorlpg..^,,,,, ,

Teacher of mathematloa,
Latin, and Greek.

OcdeoCollese

Bowling Green, Ky.
North Middletown,

W. A. Ohmdialn . .

Piesident.

Kentaoky Claasioal and

aW.Pearoy

Do.

Business CoUoge.

Ky.

Hamilton Female College

Lezingtott,Ky

J. W. Porter

Profeaaor «f matheiBatiflt
and Latin.

Xeaflhlft Male and TWnalA

KMohie^La .... .x^

O.W.Thigpen....

Professor of mat1iMnstl<w

College.

Mount St Mary's CoUoge.

Bmmitobnrg.Md...

J.A.lfitehell

Professor.

Weaten Maryland CoQege

Weatminster,Md...

W.B.McDaniel..

Prafcssor of malkomatloa.

Baltimore City College ....

Baltimore, Md

W.EUiott

PrinoipaL

Johns Hopkins ITniTersity

.... do •••....••.••.•.

Ot Neweemb ......

Professor of mallMtmatiCB

and astronomy.

IT. S.KaTal Academy

Annapolis, Md

W. yr. Hendriek-
son, J. M. Bice.

.ProfossoriOfmnttwmwriioa.

Maryland Agricnltnral
College.

Agrionltnral College
P.O.,Md.

H. E. AJyord.. . . ..

Presideol

St. John's College

AnnapoliSi Md

Orono^Me

J.W.Cain

J.N. Hart..

Maine State CoUege of Ag.
rienltnre and Meehanio

Arts.

Waterv11l6,Me

L.X. Warren

PmfMUMT Af ni^f|>innatira.

East Maine Conferenoe
Seminary.

A.F.Cha8e

l*riacipaL

Bowdoin College

BrQnswiek,Me

W. A. Moody

Professor of mathemailea^

Bates College

Lewistoii,Me

J.M.Rand

Do.

Agrionltnral College

Amherst, Mass

CD. Warner

aadphyaica.

Wealeyan Academy

Wllbrabam.Maas...

B.&Annls

Instraetorin matheiaattoi

The Society for the CoUe-
giate Instmction of
womeiL

Cambridge^ Mass . . .

A. Oilman

Sectetaiy.

Smith Collece

Northampton, Mass.

E.P.Cnahing

TfA^fiAT Aff mfttlifiinallM.

Tissftll flemtnarr

Anbumdale. Mass ..

L.M. Packard...

TnatmiAtn»ia ma*:ti ■wi p^^nf.

Swain Free Seho^

New Bedfonl, Mass.

A.lngTabam

Master.

Thaver A nadsasT .........

Bratntree, Mm*

0. A. Pitkin

^•af »Af m-li— H- f ^

phyaioa.

MATfiEMATtCAL TEACHING At l^fiE f&CSENT TIME. ^99

(a) IJkitbrsities and CoLLEGBS-^ontinned.

Kame of institation.

location.

Name of person
reporting.

Title or position of person
• reporting.

Amherst Colloeo

Amherst. Mass

W.CBsty

J.D.Bnnkle

Professor of matbAmiktl<iA.

MassachuBettB Institxite

Boston, Mass

Professor of mathematios.

of TeehBoloj^y.

Williston Seminary

Basthampton, Masa.

W.CBoyden

TVilliamB GoUese

Williamstown, Mass
Worcester, Mass . . . .

T. H. Safford

Professor of astronomy.
Profeaaor of higher math»i

Polytechnio lostltate

J.B.Slnelair

BUltiOS.

Mount Holyolce Female
Seminary.

South Hadley, Mass.

B.W.Bardwell...

Direetor of observatory.

Michigan Mining School..

Honghton, Mich . . . .

KM. Edwards ...

Professor of mining ani
engineering.

Battle Creek College

Battle Creek, Mich..

J.H. Haughey

Mathematical department

Adrian Oollese ...........

Adrian. Mich

Hmsdale,Mich

Minneapolis, Minn..

6.B.Mo£]ioy ....

A.E.Hayne8

J.F.Downey

Chairman of the fiMolty.
Professor of malJiematififl

Hillsdale Colleee

Minnesota State Unirer-

sity.

and astronomy. .

Hamline Uniyersity of
Minnesota.

Hamline, Minn

E.F.Mearkle

Washington XTniTersity ..

St. Louis, Mo

CM. Woodward..

Do.

Kansas City Ladies' Col-
lege.

Independenoe, Mo . .

J.M.Chan^

President

MissonriState TTnirersiiy .

Columbia, Mo

W.&Smlth

and astronomy.

School of Mines, tTnlyer*
sityofMissoari. ^

EolIa,Mo

W.H. Echols

Professor of applied math-
ematies.

College of the Chrtstian
Brothers.

St. Louis. Mo.....*...

Brother Panlian ..

Preaidettt.

100

William Jewell College . . .
Drurv Colleire

Liberty, Mo

J.E. Clark

^T F, Amadon .....

Professor of mathematiot.

101

Springfield, Mo

ProfefHior of mAtlMmatloa

and physics.

108

Cooper Normal College . . .

DalevillcMiss......

T.F.MoBeath....

President

103

Agricnlinraland Mechan*

StarkvIUcMiss

H. C.Davis

Acting prof eiflor of mathe-

ieaJ College.

matics.

104

TTniyersity of Mississippi.

University P. O.,
Miss.

C.MSears

Professor of mftthmBatiot.

105

Doane College

Crete. Nebr

A. B. Show... .

Librarian.

106

University of Nebraska. . .

Lincoln, Nebr

H. E. Hitchcock ..

Profeaaor of mAtbonatioa.

107

New Hampshire College
of Agrionltnreand Me-
chanic Arts.

Hanover, N.H

C.H.Pettee

Do.

108

Wake Forest College

WakeFttrest,N.C..

L.RMills

Professor of pure mathe-
matics.

109

TTniyersity of North Caro-
lina.

Chapel HiU,N.C ..

J. L. Love. .....a..

Associate professor of

mathematica.

110

Trinity CoUecre

Randolph County,

N.C.
Burke County, N.C-

J. M. Bandy

B. L. Abcrnethy ..

Professor of maithAmatins.

Hatherford College

President

112

College of the Sacred
Heart.

Vineland,N.J

P. O'Connor

Professor of mathematical

113

Niagara University

Niagara, N.Y

E.A.Antm

Do.

114

TTttion Colleffe .........

Schenectady, N. Y . .

B.H. Bipton

rW. G.Peck

Do.

Columbia College

New York, N.Y... .

{T.S.Fiske

Tutor in mathematioii.

300

TEAOHINa AND HISTOBY OF MATHEMATICS.

(a) UKiYERsmss and CollrOes— Continued.

Name of inRtitatioii.

Location.

NameofpersoJD
reporting.

Title or position of penon
reporting.

Unirenity of Rochester . .

Rochester, N.Y

G.D.Olds

Professor of mathematios.

117

St Lawrence Uniyersity . .

Canton. N. Y. ...... .

H. Priest

Profefwor of mathematiM

and physios.

118

The College of the City of
New York.

New York, N.Y....

AS. Webb

President

119

Syracnae TJnireraity

Syracnae, N.Y

J. & French

Professor of matheniatie&

120

IT. S. Military Academy. . .

West Point, N.Y...

E. W.Bass

Do.

121

Packer Collegiate iDsti-

Brooklyn, N.Y

W.L.C.Steyens..

tute.

•

and physics.

122

Brooklyn Collegiate and
Polytechnic Inatitnte.

..a.CLO. .•*•..........

B. Sheldon

Professor of pnre mathe-
matics.

128

Ohiti CTniversltv

Athens. Ohio •••..• .

William Hooyer . .

Professor of matbematiea.

124

Ohio State Univeraity ....

COlnmbqa. ObiO

Rr D. Bohanniin . r .

Professor of mathematics

and astronomy.

126

Miami Uniyersity

Oxford, Ohio

J.y.CoUina

Do.

126

Caae School of Applied

Cley eland. Ohio

C.Staley

President.

Science.

127

Hiram Collefire.-.T.. •.«....

Hiram. Ohio

C Bancroft .......

Professor of mathematliw

and astronomy.

128

Oberlin Colleee

Oberlin. Ohio

H.C.IUng

Associate professor of
mathematics.

120

Deniaon Uniyersity

GranyiUe, Ohio

J.L.Gilpatriok...

Professor of mathematios.

130

Marietta CoUeee

Marietta. Ohio

O.H. Mitchell

ProffiMior at mathematics

and astronomy.

131

Bachtel Collese

Akron, Ohio ..••.•..

C. S. Howe

Do.

132

Uniyersity of Cincinnati . .

Cincinnati, Ohio .*. . .

H. T.Eddy

Professor of mathemfttloa,
ciyil engineering, and as-
tronomy.

183

Pacific Uniyersity

Forest Groye, Ore-

W.N.Ferrill

184

The State Agricnltnrol

gon.
Coryallis, Oregon . . .

J.D.Letcher......

Professor of mathematiea

College of Oregon.

•

and engineering.

185

Dickinson College

Bryn Mawr College

Carlisle. Pa

F.Dorell

Professor of mathematfca.

188

Bryn Mawr, Pa

Charlotte A. Scott.

Associate piofesaor of

mathematics.

187

Pardee Scientific Depart-
ment of Lafayette Col-

Easton, Pa...

J. G.Foz

Professor of ciyil and topo-
graphical eugineoiing.

lege.

188

Lehigh Uniyersity

Soath Bethlehem, Pa.

C.L.Doolittle

Professor of matbsmatioa.

180

Swarthmore College

Swarthmore, Pa ... .

S. J. Cunningham .
r Isaac Sbarpless..
i Frank Morley . . .

Da
Professor of mathematiiea.

140

Hay erford College

Hayerford, Pa

Instroctor in mathematica.

141

Muhlenberg College

Central Pennsylyania Col-

Allentown. Pa ..■•..

D.Garber.....

Professor of aatronomy.
Professor of mathematica.

142

New Berlin, Pa

H.R. Kelly

lege.

143

Brown Uniyersity

Proyidence, R. I ....

N.F.Dayia

Assistant profiBasoor «f
mathematics.

144

Forman Uniyetatty

Oreenyille, & C

C.H. Jodson

Professor of mathematica
and meohaaioal phUo^
opby.

140

Uniyersity of Sonth Caro*
Una.

Colombia, S.C

E.W.Dayis

Profesaor of matbematlM
and astronomy.

liB

Colombia Pemale College.

....do

J.G.Clinkscalea..

MATHEMATICAL 'TEACmNG AT THE PRESENT TIME.

301

(a) Umiybbsitiss and Colleges— Continued.

Name of institation.

Location.

Name of person
reporting.

Title or position of person
reporting.

147

Fiok Uniyenitv

Nashville. Tenn ,..-

H.H. Wright

and instructor in vocal

music.

148

Univenity of Tennessee. .

Knoxvillo, Tenn ....

Wm. W.Carson...

Professor of mathematics
and civil engineering.

149

Giant Hemoxial UniTer-
sity.

Athens, Tonn

D. A. Bolton

Professor of mathematics.

160

Bethel CoUese •

MclCf nzie. Tenn ....

TV- W. Hi^milton

Do.

151

Chaltanooga University . .

Chattanooga, Tenn. .

E. A. Bobertson'. . .

Do.

152

Tanderbildt University ..

Nashville, Tenn

Wm. J. Vaughn...

Do.

153

Uniytf sity of Texas

Austin, Tex

G.KHalsted

Professor of pure and ap-
plied mathematics.

154

Agiicaltnral and Meohan-
ical College of Texas.

College Station, Tex.

L.L.M*Tnnis

Professor of mathematlca.

155

Bandolph-Maoon College . .

Ashland, Va

B.B.Smithey

Do.

156

Bmoryand Henry College.

Emory, Va

S.M. Barton

Do.

167

Hampden-Sidney College..

Hampden Sidney, Va

J. B. Thorn ton

Do.

158

Uniyersity of Virginia —

Charlottesville, Va. .

C.S.Venal)le

Do.

163

Bethel Military Academy.

Bethel Academy P.
O..Va.

E.S. Smith

Instructor in higher mathe-
matics and modem lan-
guages.

160

Virginia Agrienltaral and
Meohanioal College.

Blacksburg, Va

J. E. Christian....

Professor of mathematiot
and civil engineering.

161

Polytechnic Institute

New Market, Va. . . .

W.H. Smith

President.

162

Vermont Methodist Sem-
inary.

Montpelier, Vt

E. A. Bishop

PrincipaL

163

Konrich University

Northileld,Vt

J*. B. Johnson

Professor of mathematiov.

164

University of Washington .

Seattle, Wash

J. M. Taylor

Do.

165

Wheeling Female College.

Wheeling, W.Va...

H.B.Blaisdell....

President

166

West Virginia College

Flemington, W. Va .

T.E.Peden

Do.

167

Kinon College T.««ar....rn

BiDon. Wis

C H. Chandler ....

PmfAiMnr nf mftfTiAmaflna

•

and physics.

168

BdoitCoUeffe .•..

BeloitWis

T.A. Smith

Do.

Are atudenU entering your inBtitution thorough in tl^e mathemaHos required for admieaion f

"No," "not generally," "by no means," "seldom:" 3,5,6,7,6,9,12,13,14,16,17,
19,20,21,22,23,25,27 (but growing better), 29,31,33,36,39,40,42,43,44,47,49,51,
62, 53, 56, 57, 58, 59, 60, Gl.. 63, 66, 67, 68, 69, 70, 72, 73, 76, 80, 82, 83, 86, 88, 91, 92, 93 (but
marked improvement every year), 94, 96, 97, 99, 102, 104, 105, 109, 110, 111, 113, 117, 160,
161, 162, 163, 164, 165, 166, 167, 168 (and this is one of the evils of our times).

" Fairly so," " reasonably so : " 10, 11, 15, 30, 32. 34, 37, 46, 55, 84, 95.

" Not as thorough as we desire : " 35, 38, 41, 71, 74, 87, 107, 108, 114, 116.

" Yes : " 1 (most of them), 28, 45 (generally), 54 (or fail to enter), 64, 65 (usually), 78
(a fair proportion), 79 (generally), 89, 90 (or fail to enter), 100 (or they are placed in
preparatory department), 101, 112 (generally), 115.

Is the mathematical teaching by texP-hook or by lecture?

This question was answered by all who sent in reports. The following forty- six
answers were "by text-book," without indicating that any lectures whatever were
given: 6,7,11,12,14,17,18,20,21,24,25,26,29,30 (it is impossible to teach mathe-
matios by lecture), 32, 33, 37, 44, 50, 51, 53, 63, 65, 68. 73, 74, 85, 88, 101, 102, 105, 107, 109,
113, 114, 125, 138, 141, 142, 147, 149, 150, 151, 161, 163, 164. ^

802 TEACHING AND HISTORY OF MATHEMATICS.

The following sixty-five answers were " mainly by text-book,'' " text- book prinoi-
pally/' "text-book as a basis/' '* text-book and informal lectures," or some similar
phrase, indicating that the text-book predominated: 1,3,8,9,15,19,^,28,31,34,38,
40, 41, 43, 45, 47, 48, 52, 55, 57, 58, 59, 60, 62, 66, 71, 72, 75, 77, 80, 86, 87, 89, 90, 91, 92, 93, 95,
97, 100, 103, 106, 110, 116, 118, 119, 120, 122, 123. 128, 133, 134, 135, 136, 137, 143, 144, 148,
156, 156, 159, 162, 166, 167, 168.

The following flfty-five answered "by both," or "by text-book and lecture," with-
out saying which predominated : 2, 4, 5, 10, 16, 23, 27, 35, 36, 39, 42, 46, 49, 54, 56, 61, 64,
67,69,70,76.78,79,82,83,84,94,96,98,99,104, 108, 111, 112, 115, 117, 121, 124, 136. 1»7,
129, 130, 131. 132, 139, 140, 145, 146, 152, 153, 154, 157, 158, 160, 165. The answer of num-
ber 13 is ** by lectures, except elementary geometry ; " and of 81, " by lecture."

What mathematioal journaU are taken f

The following answered " none : " 1, 3. 5, 6, 7, 9, 14, 15, 17, 20, 21. 23, 34, 35, 39, 30, 33,
33, 36, 39, 40, 43, 44, 45, 48, 49, 50, 53, 56, 57, 58, 59, 60, 63, 65, 69, 70, 71, 73. 74, 76, 77, 78, 79,
80, 81, 82. 85, 87, 88, 89, 91, 94, 96, 99, 101, 102, 103, 104, 105, 107, 108, 112, 114, 116, 117, 131,
122, 128, 129, 133, 134, 141, 147, 149, 150, 151, 159, 160, 162, 163, 166, 167, 168. Some of
these answers were "none by the college," "none that are pnrely mathematioal,''
"several scientific and engineering journals," but most of them were simply "none."
In addition to this list, numbering eighty-four, we may safely add thirty-three that
did not answer this question in their report, making one hundred and seventeen in-
stitutions out of one hundred and sixty eight that take no mathematical Journal of
any sort devoted to pure mathematics.

The following reported as taking simply the American Journal of Mathematics :
10, 27, 28, 54, 55, 72, 75, 84, 95, 131, 145.

The following, as taking simply the Annals of Mathematics: 4,8,13,16,83,43,64,
90, 100, 110. 148. 157.

University numbered 11 is taking if, il;, Z, marked below;* 12,&,({,«i; 19, ft; 35, 5, d,
f, a ; 37, Ic ; 38, 5, d, 9, n ; 41, (2, m, I ; 46, h, k^ etc. ; 47, 5, d; 51, h, nearly all the foreign
Journals are expected after this year ; 67, all the leading ones ; 68, nearly all mathemat-
ioal journals; 83, h, dtj; 86, &, A, u; 92, iS;, Z, 0, etc. ; 93, &,<2; 97, h, J, etc.; 98, htd,n;
106, a, 5, Jahrbuch d. Vortsch. d. Math. ; 109, <2, j,p ; 111, any we can get ; 115, a, h, e,
d,e,f,g,h,ij,m,nyp,e,t,u; 152,6,<l,/,m,i?,»,t/ 153,&,d[,t; 154,(7,^; 155,df,|>; 158,d,
p, and others ; 161, q ; 164, d, k, I.

Are there any mathematioal eeminariee or eluhe, and haw are they condmted f

All answered in the negative, except the following :

15. No clubs, unless special classes for voluntary and outside reading be so
designated. Such classes are conducted like all other classes.

38. A olub. The meetings of the club occur on alternate Tuesdays. Member-
ship about 35 ; topics are assigned to or chosen by the student at his option ;
assistance is given him as he may need. The work is pedagogical, rather than
original.

41. One. Reading and exposition of the more difficult parts of Williamson's
Calculus.

51. In connection with the Science Club ; by lectures.

67. There is a mathematical society, in which there is free choice of snbjeots
for communication, and there are two or three seminaries for post-graduate
students, conducted by the teacher on special lines.

*(a) Aotft Msthematioa. (b) American Joaisal of ICaUwinatios. (0) AnnaU di HatemAtioa. (d)
Annals of Mathematios. <«) Archir d. Math. n. Phyaik. (/) Bulletin des Solenoes Math, ot Aatraa.
(g) Bulletin de Socl6t6 Math, (h) Comptes Rendns. (i) Journal de Math. (Lionville). (j) Joaraal
£ reine n. aDgew. Math. (CreUe). (ft) Mathematioal Mafratine. (I) Mathematioal Vuitor. (m) Mathe-
matiaohe Annalen. (ti) Meeaenffer of Mathematios. (0) Prooeedinjcs London Math. Society, (p)
KoavaUaa Annalea de Math, (g) School Visitor, (r) School Messenger. («) Quartarly Jotuaal of
Math, (t) Zeitsohrift f. Math. u. Phjs. (n) Zeitsohrift t Yennessongskando.

MATHEMATICAL TEACHING AT THE PBB8EKT TIME. 303

Art iStytm iwf§ na^emoAioal aeminariei or oiudtfi and how uro iktyoonduotedt-^ontinnodL
115. Yes, one. It proposM to disouu the literaiare of matiiematioa, to Bolra

problems given by members, and to make original investigations.
136. No clabs, bat seminaries, thronghpart of regular oonrae, bat not very

formal ; they are intended to afford staddnts opportanlty of working problems

under guidance.
143. There are men in each class studying for honors. The principal part of

their work is the solution of original problems. I meet them freqaently for

discussions and suggestions.

158. Ko clubs; the lectures and recitations regularly extend through an ^iir
and a half, and at each original solutions of problems are given, and next time
are called for. Each meeting of ewih. class is a seminarium.

Are there any acholar$hip$ orfellowehipBfor gradiiate studenta in mathemaiiee t

All who answered this said ** no/' except the following :

67. Yes.

84. None for mathematics exolasively.

86. One is oocasionally given to a man of high promise.

93. Fellows are allowed to choose mathematics.

97. One fellowship ; two scholarships await instantly State appropriation
for support.

115. Yes, an annnal fellowship in science.

136. One Tellowship awarded yearly to a properly qualified graduate of any
college.

145. Yes, there is one, retainable for two years.

159. Oocasionally conferred on deserving students wishing to prosecute their
■ studies at other institutions.

le the percentage of etudenta electing higher mathematica increasing or decreasing f

The following reported ''increasing:'' 1, 5, 6, 8, 9, 12, 15 (among scientific students),
16. 21, 24, 25, 26, 27, 28. 31, 32, 33(f), 35, 37, 38, 39, 40, 42 (f ), 46, 50, 51, 53, 54, 55, 56 (with
gentlemen), 57, 59, 63,73,75,76,78,79,82,88, 90,93,97,98, 99, 101, 102, 106,108,111,
112, 113, 116, 117 (f ), 119, 123, 124, ISfe, 126, 128, 129, 133, 137, 140, 142, 144, 145, 149, 150,
151, 153, 154, 156, 157, 158, 159, 160. 161, 164, 165, 166.

The following reported ''about constant," "neither increasing nor decreasing:"
3, 4, 23,30, 48, 67, 70, 74, 80, 84, 86, 91, 92, 100, 104, 107, 114, 115,118, 122, 127, 130, 131, 141,
147, 148. 162. 163.

The following reports indicated a '' decrease :" 15 (among classic students), 44, 49,
56 (with ladies), 83.

The following reported that none of the mathematioal studies ware elective : 10,
11, 17, 18, 20, 22, 36, 41, 45, 47, 52, 65, 71, 85, 87, 94, 96, 103, 105, 106, 120, 133, 135, 139, 142,
146, 155.

Doea the interest in mathematica increaae aa atudenta advance to higher auhjecia f

"Yes:" 3, .'i, 6, 8. 10, 11, 13, 16, 17, 18, 21, 22, 23, 26, 28, 29, 30, 32, 35, 36, 37, 38, 40, 41,
42, 46, 48, 50, 51, 52, 53 (very much), 54 (generally), 55, 57, 59, 60, 61, 63, 63, 66 (t), 73, 77,
79,82,84,87 (generaUy ), 88. 90. 92, 93, 94, 95, 98. 101, 102, 104 (?), 105 (f), 108, 111, 112,
113, lie, 118, 119, 122, 123, 126, 131, 132, 133, 134, 139. 140, 145 (generally), 146, 147, 150,
151, 153, 154, 156, 157, 158 (emphatically so), 159, 160. 161, 163, 164, 165, 166, 167.

"With the best students only," "with those students whose mental tendencies are
along mathematical lines,'' or some similar remark : 1, 4, 9, 12, 24. 25, 31, 33, 34, 43, 44,
45, 47, 56, 58, 68, 70, 71, 72, 74. 75, 76, 80, 81, 85, 107, 114, 120. 121, 124, 125, 127, 128, 129. 130,
136, 137, 141, 142, 143, 148, 149, 152. 155.

"The interest increases so long as the student sees the bearing of his work upon
practical scientific investigation or can be assured that it has such a bearing." "It
increases as application to practical matters is shown : " 15, 19, 69, 103.

304 TEAOHINa AND HISTbBT OF MATHBMATIOS.

"All who nndentand tlio principles ahow a growing interest," "where proper
preparation on the part of the student has been attended to and the teacher is a live
man, it does" (HO), "The interest is according to the clearness of apprehension of
mathematical truths. Hence, the more eyolyed or abstruse the matter, the greater
the interest to those who succeed " (144) : 37, 91, 97, 110, 115, 144.

"No : " 2, 7, 20, 34 (for poor students), 39, 43 (for the majority), 49, 86, 100, 106, 109,
135, 162, 168.

Are prizes awarded for excellence of daily class-room work, or for success in orginal
research t

" No prizes awarded : » 1, 4, 6, 7. 8, 10, 13, 14, 16, 17, 19, 20, 21, 23, 28, 29, 30, 31, 32, 33,
34, 35, 37, 38, 39, 40, 41, 43, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 63, 64, 65, 68, 69,
70, 71, 72, 73, 76, 78, 79, 80, 81, 84, 85, 87, 88, 89, 90, 91. 93, 94, 95, 98, 100, 101, 103, 104, 105,
106, 107, 108, 111, 112, 114, 115 (except class honors), 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 131, 132, 133, 135. 136, 138, 139, 140, 141, 142, 144, 145, 147, 148, 149, 150, 151.
152, 154, 156, 157, 158, 160, 161, 164, 166. 167, 168. .

"For both:" 2, 9,36,67 (bestowal of scholarships and fellowships is based upon
both the considerations), 82, 99, 134, 155 (but 1 do not believe in prizes).

" For work in the class-room : " 18, 22, 25, 26, 62. 66, 77, 86, 96, 118, 137, 143, 159, 162.

'' For outside work, not generally original : '/ 15, 116.

"For original investigation only : " 97, 102, 117.

" Yes, prizes are awarded : " 3, 5, 11, 12, 24, 27, 54, 57, 74 (scholarship of |300 to befit
Sophomore in mathematics), 92, 97, 109, 110, 113, 165.

What mathematical suhjeets are preferred by students t

The answers given point to no definite conclusion. For want of space, they are
here omitted, except the following: " Their preferences are generally for the particular
eul^eot tohich they have had the best elementary training in " (148).

Are topics assigned to students for special investigation f

1. Yes.

2. Problems are proposed. .

3. Sometimes prize problems are given to students.

5. Yes.

6. Yes.

7. For the higher classes. ,

8. Yes; often.

9. Yes.

10. Not in general.

11. To a small extent.

12. Occasionally.
14. No.

17. Yes.

18. Sometimes.

19. Occasionally.

20. No.

21. Yes.

22. No.

23. Yes.

24. Yes.

25. No.

26. There are.

27. Independent problems given in all the classes for solution, reported oo
paper.

23. Once each term.
89. No.

liATHEMATIGAL TEACHING AT THE PRESENT TIME. 305

Are iopiei assigned to etudenUfor tpeeial lii«e0f<|faf<o»f— Continued.

30. Not mnoh. The man who punnea original inyestigation with the arer-
age Btadent will make a failure*

31. We haye not been acooetomed to do so.

32. Not to any extent that deaerres mention.

33. Praotieallj, no.

34. Not to nnder-gradnates. There are no graduate studente in mathematics
at present.

35. Tes.

36. Tea; bat neceaaarily elemeixtarj.

37. YeBy air.

38. Tea, in connection with the clnb and for graduation theaea.

39. Tea.

40. Sometimea ; bnt our olaaaea are generally ao cloaely occupied by their
varioas atndieai there ia bat little time for extra work.

41. No.

42. Tea.

43. Tea.

44. Frequently.

45. They are, and form a rery eaaential ptrt of the work.

46. Stadenta in the higher olaaaea are aaaigned anch topica.
47« None adTanoed enough,

48. Tea.

49. Sometimea.
51. Tea.

58. Occaaionally.

53. Tea.

54. Tea.
56. No.
56. Tea.

58. Tea.

59. Tea; with good aneecaa*

60. Tea.

63. Tea.

64. No.

65. No.

67. They are in the aeminarlea.

68. No.

69. A few.

70. To a limited extent.

71. No.

72. Not to any great extent.

73. Occasionally.

74. In eleotiye diyisiona.

76. To some extent.

77. Barely.

78. Tes.

79. No.

81. Tes.

82. Tes.
84. No.
65. No.

86. Tes, to post-graduatest

87. Tes.

88. Not often.

881— No. 3 ^20

306 TEACHlKa A9D HI8T0BY OF UATHEUATICS.

Are topici a$8igued to t h i d m U fer tpt^dl iitw9tigmtlon T-^oni^ianiL

89. In iqnplied meehaoiei, yes.

90. Not for origiDal inYeatigatioD, bnt otlierfnse.

91. Occasionally.

92. Te0, for theeia in general geometry and calculus.

93. TeSy especially in elementary geometry, analytical geunielry, andealciilns.

94. Tes.

95. fiare.

96. Only to a limited extent.

97. Sach assignment has hitherto been only exotptiona], berettOer to be made
regular.

98. In applied mathematics tbsses are required on special sabjeots> and origi-
nal inv^estigation encouraged.

99. Tes.

100. No.

101. Yes.

102. Tes, in all the different branches, especially in applied mathematics.

103. The graduating and other theses are on subjects divided among the
departments.

104. No.

105. No. .

. 106. They are.

107. Yes.

108. Yes.

109. No.

• 110. Yes, this is encouraged in all the classes, but la sectind best In the

higher classes.

111. Yes.

112. Occasionally original theorems and problems are given.

114. Occasionally, results submitted in original theses.

115. No, except the work done in the seminary.

116. Yes, in all classes of all departments.

117. No.

118. Yes, to a large extent in geometry.

119. In pure mathematics very seldom; not in applied mathemaiioSk

120. Yes.

121. No.

122. No.

123. No.

124. We have this in view for next term,

125. None as yet.

126. Yes.

127. Yes.

128. Yes.

129. To some extent.

130. Yes.

131. Occasionally.

132. No.

133. To some extent^

134. Occasionally.

135. No.

136. I should consider this exercise profitable only to very advaaeed efcndantif
and have not had occasion to employ it yet.

137. Yes, to some extent.

138. No.

MATHEMATICAL TEACHING AT THE PBESENT TIME. 807

Are topics otHgned to students for special investigation f--Contioued

139. No.

140. Oocasioually. '

141. Not to any extent.

142. Yes, in applied geometry, surveying, and pbysioo.

143. Occasionally to advanced students.

144. Only exercises in theorems and problems.

145. Yes.

146. Yes.

147. No.

148. The solution ol problems related to the recitationa is required. Nothing
else.

149. No.

150. No. %

152. No.

153. Yes.

154. Yes. )

155. In the^gher classes topics are occasionally assigned.
i;56. No.

157. Original exeroisea are given at intervals.

158. To graduate students^ candidates for the degree of Ph. D»

159. Bat few outside of text-book.

160. Yes.

161. No.

162. No,

163. No.

164. Yes.

165. Yes.

166. Yes.

167. To some extent.

168. Occasionally.*

(a) Is any attention given to the history of mathematics f (h) Does it maJce the suljeci
more interesting t

(a) " Yes : " 1, 5, 9, 18. 34, 35, 37, 39, 42. 44, 46. 48, 52, 53. 61, 63. 64, 65, 72, 80, 81, 90, 92,
97, 98, 99, 101, 102, 108, 112, 114, 116, 123, 126, 129, 131, 135, 136, 138, 145, 152, 153 (great),
154.156,157,158,160,164.

(a) " Very little," ''only incidentally," " not much", etc.: 4, 6, 8, 11, 12, 13, 16, 17, 19,
21, 23, 24, 25, 27. 28, 30, 38, 40. 41, 43, 45, 47. 51, 54, 55, 56, 58, 59, 60. 66, 67. 68, 73, 74, 75, 76,
78, 79, 82. 86, 88, 91, 93. 94, 100, 104, 106, 107. Ill, 115, 118, 119, 120, 121, 124, 125, 127, 128,
130. 133, 137, 143, 144, 147, 151, 159. 163, 168.

(rt ) * ' No : " 3, 7, 10, 14, 20, 22, 29, 32, 33, 36, 49, 50, 57, 69, 70, 71 , 77, 83, 85, 87, 89, 95, 96,
103, 105, 109, 110, 117, 122, 132, 134, 139, 140, 142, 146, 148, 149, 150, 161. 162, 166, 167.

(6) ** Yes," "it does," "most decidedly," was the experience of all who had given any
attention, in the class-room, to mathematical history, except the following, who were
in doubt : 11, 15, 16, 47, 56, 76, 104, 120. Even these were inclined to say " yes." M one
answered that it did not make the subject more interesting — a clear case.

How does analytical mathematics compare in disciplinary value mth synthetical t

1. I regard both methods equally important.

4. I think synthetical has much the greater disciplinary value ; analytical
has much the greater value for practical application. Analysis is the princi-
pal tool for investigation and work.

* Widely differont viewa Bedia to be implied la the above anawors aa to what constitutes a " toplo
fox special inyestigatioxu"

808 TEACHINO AND HISTOBY OF MATHElfATICS.

HwB doei onaiiftiMU mafhematiCB compare in disoipUnarsf vdUe toiih <jrRtft«ftMlf— GontPd*

5. Analytioal saperior.

6. The former nsed more largely in the Grammar Department.
8« Analytical mathematics gives the better mental discipline.

9. I think both necessary to fall mental developmenty bnt if I were obliged
to choose I should prefer analytioal.

10. I can not say fairly, for my teaching has been wholly in analytical mathe-
matics. In my studies I prefer that method.

11. I use combination of both and so can not well answer.

12. The development of reason is more regular, rapid, and subetantial in
geometry than in any other branch of the mathematical course. For advanced
students I would count algebraic analysis a superior discipline.

13. It seems to be a question of individuality.

15. I regard analytical mathematics as the more valuable and the more im-
portant.

16. The former is superior.

17. It is superior.

18. Analytical seems to be better.

19. Common geometry, considered as an application of logic, espeoially in
the demonstration of easy ''riders" and in very simple exercises in construc-
tions, is of pre-eminent value to quite young and undisciplined minds. At dif-
ferent stages each has its peculiar and really unmeasurable value.

20. They are of equal value.

81. I have not data enough for an opinion.

22. Superior, if the two are divorced; but the synthetical should be nnited
with it.

23. Favorably.

24. Analytical greater.

25. With the majority of students more satisfactory results are obtained
through the synthetical method of reasoning.

26. Analytical preferred.

27. As a rule, I have found that students stand better in geometry than in
algebra. When analytioal geometry is clearly comprehended, it affords the best
discipline for the mind.

28. Synthesis seems to give better discipline.

30. Analytical preferable.

31. The former is a better test for form and figure, the latter seems to tax
the memory.

32. We have no classes sufficiently advanced to test the relative value ex-
tensively.

33. If the work be the same in both, the syntheticaL

34. My own preference is analytical.

35. Could not get along with either method left out (Professor Shattuek).
Each has its special function ; as well ask whether braces or tie-rods are of
most service in a bridge-truss (Regent Peabody).

36. Synthetical more valuable.

37. Disciplinary value of former is greater.

38. We give the analytical the first consideration after the student is led op
to it.

40. Superior.

41. I think the former the more valuable as an instrument of xeseaxoht tha
latter as a means of discipline.

42. The analytical is more valuable simply as a means of discipline,

43. The synthetioal Is better for younger students ; the analytioal fbr tliosa
nKve mature.

MATHEMATICAL TEACHING AT THE PRESENT TIME. 309

Bom does analytical mathematics compare in disciplinary value with synthetioalt-^onVd,

45. It is evidently far snperior.

46. Each affords excellent discipline.

47. It is superior.

48. For college grade we think the analytical prodaoes the best results.

49. Better.

50. Better for discipline, but we have not ased it as yet.

51. Somewhat superior in value.

52. Superior.

53. Analytical training is more beneficial.

54. Favorably.

55. They are superior.

56. Prefer the analytical.

57. We use both methods, but give preference to former.

58. Can we do without either f I should say both are necessary, but analyt-
ical is less taught.

59. Analyzing the whole into its elements is valuable, but building the whole
from elements is very valuable.

60. Superior for advanced students.

63. The analytical the more valuable.

05. Analytic mathematics is far superior in its disciplinary value.

67. The latter is probably the more valuable discipline in early stages of a
mathematical educatiour; but after the elements of geometry are mastered,
probably the reverse is true.

68. In general we prefer analytical methods.

69. Latter preferred.

70. Doubtful; students prefer synthetical.

73. In my judgment the analytical method is to be preferred.

75. For the average student the synthetical gives better results.

76. I think analytical mathematics better for mental discipline.
79. Its disciplinary value is less than that of the synthetical.
60. The synthetical is more valuable.

81. They interact; but the latter is an indispensable prerequisite for the
former.

82. I should place analytical as greater in disciplinary value.

64. Analytical is inferior to synthetic.

86. Both methods are essential and I am not aware of any difference. Per-
haps I do not understand the question.

87. Superior; yet this depends, perhaps, on the mind of the individual
student.

90. Very favorably, so far as our experience has gone.

91. I prefer the former for advanced students— the latter for beginners, or
students of a low grade.

92. Synthetical seems best for the less advanced students.

93. I do not believe that the two can be separated and compared. I believe
with Sir William Hamilton, '^Analysis and synthesis are only the two necessary
parts of the same method. Each is the relative and correlative of the other.''
Neither without the other would be of much value.

94. The synthetical is absolutely necessary as a foundation of good work ;
after the foundation, tbe former is desirable.

95. Do you mean graphical (or geometrical) by synthetical f I think de-
scriptive geometry has the finest disciplinary value.

97. As commonly taught, unfavorably ; as taught here, with special stress oa
Morphology and by aid of determinants, very favorably.

98. In favor of the former.

310 TEAOHINQ AND HISTORY OP MATHEMATICS.

How does analytical mathematioa compare in diaoiplinary value with eynlKetieal f — Confd.
99. Are in favor of the analytical,

101. Superior. ^

102. Tliat depends npon the peculiar natural bent of the pupil's mind. For
some, analysis, and for others, synthesis is more valuable.

104. Can not institute comparison, we use both in combination.

106. I would answer this by saying, that I consider special geometry better
for mental discipline than analytical geometry, and geometry better than
algebra.

109. Analytical superior.

110. In my judgment the analytical is so far superior to the synthetical that
there is left little room for comparison. Permit me to say that reason wants
lightf not darkness,

111. They are about equal. We use Peck's and Davies' methods.

112. The comparison is in favor of analytical mathematics.

113. The analytical method, in my opinion, produces better results than the
synthetical.

114. Superior.

115. Each has its special value ; both are desirable (Professor Peek). An-
alytical gives the more rigorous training. Each plays its own part (Tutor
Flske).

116. Synthetical better for training in formal logic ; in other respects analyt-
ical is unquestionably superior.

117. Synthetical seems to give better results.

118. For older students the analytical methods ard superior; fov those below
the Sophomore class, this is doubtful.

119. I think the analytical is better.

120. Analytical mathematics predominates here, and therefore has the greater
disciplinary value. If comparing equal times in the two, I should say syn-
thetical.

122. Synthetic best for discipline ; analytic best for use.

123. Former is better.

125. It is hard for me to answer this. Perhaps the latter is superior for doll
or average students, while the former is preferable for the more able.

126. Both necessary for proper discipline.

128. Well.

129. Analytical mathematics is the better.

130. Sometimes seems to me superior ; sometimes seems to me inferior, de-
pending npon the mental character of the student.

132. We teach no synthetical mathematics in the nniversity, except a book
of elementary mechanics, which is good in its place, but analytical mathe-
matics alone develops real mathematical power.

133. I regard the analytic method as much superior in way of developing
habit and power of investigation.

134. I use both and would not willingly part with either. Deem ihem of
about equal value.

135. Equally valuable though in different way.

136. I should be inclined to give preference to analytical ; but where there is
a strong natural mathematical bent, possibly more disoipline is derived firom
synthetical mathematics.

137. Bather unfavorably with the average student.

139. Superior.

140. Better.

143. Both valuable ; both necessary.

144. Analytical is favorable for advanced students ; synthetical, for younger
•tudents.

MATHEMATICAL TEACHIHG AT THE PRESENT TIME. 311

Sow dM8 analytieal maihemnUos oompare in disciplinary value with synthetical t — Cont'd.

145. The value of the diseipline depends upon the oloseness of the etad<»pt's
application rather than upon the methods employed.

146. Superior to it.

147. I consider the analytical far superior to the synthetical.

148. In my opinion, the analytical is far snperior .to the synthetical.

149. I think the analytic is better.

150. The former is of more disciplinary value than the latter.

151. The analytical mathematics in most cases most satlsfiactorily fulfils the
end or object of mathematical study.

152. The former, in my opinion, is preferable in almost every respect.

153. Analytical mathematics is vny fkt iniisrlor to synthetic in disciplinary
value.

155. Analytical mathematics has, I think, a higher disciplinary value than
synthetical.

156. The synthetical is more valnable, I think, but by nomeanfl should either
be adopted to the exclusion of the other.

158. Impossible to make a comparison In so short space.

159. I regard analytical mathematics as possessing higher di8ciplinj>>ryvalne|
when properly taught.

i^. Analytieal mathematics is saperior to synthetical in disciplinary value.
' 161. Favorably, both should be used.

163. I favor analytieal mathema4>ic8«

164. Analysis is snperior in disciplinary valae.

165. Superior.

166. They are about equal.

167. Any true method of study seems to bos to use them bothf with so frequent
changes that comparison is difficult. Moreover, their relaUve value differs
with different pupils.*

(a) What method of treating the calculus do you favor, that of UmitSf the ivifiniteeimal, or
some other t {b) Does the infinitesimal seem rigorouSf and to eaOsfy the mind of the
etudent f

I. (a) Limits. (6) Does not satisfy the student.

3. Calculus is not taught in this college.

4. (a) I favor the method of rates, though I use the method of limits and
infinitesimals— the latter in mechanics, (b) It does not in my experience.

5. We do not teach it.

8. (a) That of limits, (b) At first it does not seem rigorous to the student,
nor to satisfy his mind.

9. (a) We think; with many others, the subject needs both, (h) Not suffi-
ciently S0| and hence the advantage of calling 'limits "-to its aid.

10. (a) Calculus is not taught here. Personally I prefer infinitesimals, (b) I
think so, more than that of limits, which is better for the mathematioian than
the student.

II. (a) I favor none exclusively. I teach "rates"/*' *^fi^*t^*™*^*"» ^^^
" limits." <5) It does not seem rigorous.

12, (a) The method of limits (now made familiar in geometry) seems most
satisfactory, (b) Not to beginners. Later this method f^ould be studied also.

13. (a) Defining /' (a) as co-efficient in development: f(x) =/ {a) +/' (a)

(d^-a) -f- (^) It does not seem rigorous as usually representedi but

could be made so, but I doubt whether for beginners.

15. (a) I favor the method known as that of " rates.*^ (b) I think not.

•w>-

*Wcft oollstend roftdioK en this queatfon see Pmeiient €. V. Xliot's artioU^ " Whttt is a Liberal
Xdaoationf '* in the Century Magazine, Juie, 18M; Beport «f the <SngUs^) Commission in 18S2|
B^port of the French Commissioners in 1870.

312 TB^OHINQ AND HISTORY OF MATHEMATICfiL

(a) TFTiat method of treating the oaloulae do youfavoff that of Umiia, the inJUUteHmal, or
$ome other t (&) Does the infiniieaimal seem rigoroue, and to aatiefjf the mind of the
etudent f-^Continned.

16. (a) A oombination of limits and infinitesimaU. (i) Combined with the
method of limits, it does; alone, no.

17. (a) Infinitesimal ; a little aboat limits, (b) I have never yet had a eta-
dent to whom I ooold not make it perfectly satisfactory.

18. (a) Limits, (i) No.

19. (a) In theory, Buckingham's " direct method of rates; " practically^ the
infinitesimal as set forth by Olney and others, on account of its practical ad*
vantages, (h) The philosophy at' the base of this method seems to involve one
in a maze of absurdities^ but I have had too little experience with pupils in the
calculus to speak positively upon this point.

20. We do not teach the calculus.

81. (a) Doctrine of limits, (h) It does not.

22. (a) That of limits. (5) Not in every case.

23. (a) Limits.

24. (a) Limits.

25. (a) The method of limits, (h) It does not.

26. (a) Limits.

27. (a) I explain and illustrate both limits and infinitesimal analysie. ((>
When properly explained and illustrated, I think it does.

28. (a) The infinitesimal. (() It does, i. e., generally.

30. Method of limitSy not the Newtonian of passing to cero.

31. (a) Have been accustomed to take the limits.

32. We have no classes in calculus.

33. (a) Had experience only with infinitesimals, (h) No ; certainly Olney's
presentation can be improved upon.

34. (a) The infinitesimal, if properly presented. (5) Tee, when the student
can appreciate mathematical reasoning*

35. (a) Teach both methods, do not favor either, (b) Yes; Lagrange's method
of derived fhnctions is considered the best in theory (Professor Shattnok).

36. (a) Method of Umits. (») No.

37. (a) Limits.

38. (a) Method of limits, (h) No.

39. (a) Limits, (i) No.

40. (a) I teach the infinitesimal, prefer it in general. (() Occasionally a
student will not accept its theories ; I then try him on limits and show him
their relation.

41. (4) The latter, with a sprinkling of the former. (() The infinitesimal
method is Just as rigorous, when understood, as the method of limits, but it is
my experience that the latter more quickly removes the logical diffloultiee in
the way of the hegkmer,

42. (a) The method of limits, (h) It does, provided its relation to the method
of limits be shown ; otherwise not.

43. (a) The infinitesimal, {h) Students have generally preferred it to the
method of limits.

45. (a) The infinitesimal, (b) It does when it is known that results do not
vary.

46. (a) I use the method of limits ; the method of infinitesimals is also pre-
sented, (b) One method seems to do as well as the other if properly presented.

47. (a) Method of rates (see Taylor, Bice and Johnson, Buckingham).

48. (a) Genarally by limits, (b) To the first part, yes; to the second part»
genaraUy not very satlsftetory to those going over the suld«ot for the fint tlmft.

49. (a) This infinltestmri. (b) Not always.

MATHEMATICAL TEACHING AT THE PRESENT TIME. 818

(a) What fMthod of treating the caleulue do you favors that of limittf the infinitesimal, or
some other? (b) Does the infinitesimal seem rlgorouSf and to aatisfff the mind of the
student t — Continned.

51. (a) Method of limits ; use both, (h) Not so rigoroos as that of limits.

52. (a) The infinitesimal, (b) So far as I know it does.

53. We have no classes in oalcnlas.

55. (a) The calcnlns is not a part of oar coarse of study ; personally, I prefer
the method of limits.

56. (a) The infinitesimal, (h) Yes.

57. (a) This is elective— no students yet.
68. (a) Have nsed the integral.

59. (a) InfinitesimaL (() YbS| when the stadent is well drilled in what should
precede.

60. (a) So far, the method of limits, (b) Have not found it so, generally.

62. We do not teach anything higher than trigonometry.

63. (a) That of limits. (() No.

64. (a) InfinitesimaL (b) Tes.

65. (a) The infinitesimal, (d) Not entirely ; but the ideas of the calculus axe
olitained better through this than any other method.

66. (a) That of limits, {h) Not altogether so.

67. (a) The two methods named are not essentially distinct ; I regard the
method of infinitesimals, based upon the doctrine of limits, as the best mode of
treating the subject, (b) Not unless it is based upon the doctrine of limits.

68. (a) The method of rates, passing later to the method of limits.

69. No calculus.

70. (a) Infinitesimal, (b) Not entirely.

71. (a) The method of rates, combined with the method of limits, (b) It is
very little used, and only after the others have been taken up.

72. (a) Limits, (b) Students have seemed satisfied when that method has
been used.

73. (a) The infinitesimal, {b) Healthfhl.

74. (a) Infinitesimal method for students taking brief course, (i) Generally,
yes.

75. (a) The infinitesimal, {b) By sufficient explanations.

76. (a) Limiting ratios preliminary to the more direct method ofinfinitesi-
mals. {b) Somewhat, but often fundamental investigations are made more
intelligible to the beginner by this method.

78. (a) Limits. (() It does when properly taught.

79. (a) Limits, (b) No.

80. (a) The method of limits, (b) Tes, if the student persists until it n oon-
quered.

81. (a) The first at the outset. All should be introduced (see Wundt,
Logik).

82. (a) That of limits. (5) Hardly.

83. (a) Limits. (») Yes.

84. (a) Both limits and infinitesimals. (5) Not when the two methods are
presented together.

86. Calculus is not studied with us.

86. (a) Limits, decidedly. (() Not until the student has mastered the method
of limits.

87. (a) Infinitesimal, (b) Ova students seem to understand this best.

88. (a) That of limits.

89. (a) Limits, (b) No.

90. (a) The infinitesimal on account of its simplicity, bat the new method
by Qentral C. F, Buckingham is excellent, (i) Not always.

314 TEACHIBTG AKD HI8T0B7 OF lUTHBIUTICS.

(a) What method of troatin^ the caleulua do youfawfr^ that of UmiiSf the infimitedmal, or
Bomo other f (b) Does the infiniteemal eeem rigorone, and to eatieff the m4nd ef ihe
student f^ Continued.

91. (a) The method of limits, (h) It does not.

92. (a) Infinitesimal, {h) It does.

93. (a) The flnxionary method, (b) Not entirely; it is taught from tezi>
book— flnxionary by leetnres.

94. (a) Infinitesimal. (&) Yes.

95. (a) Infinitesimal. (6) It does.

97. (a) The German method of limits, not the popalar English and French.

98. (a) The rigorous method of limits, (ft) ^o; there is an eTidmt loss of
Dsith at this point for students on fint reading.

99. (a) We use both. (&) Yes.

100. (a) Limits. <() Do not teach it.

101. (a) Newtonian fluxions, (ft) No.

102. (a) Limits, (ft) No ; I &nd few pupils satisfidd with it,

103. (a) Do not teach calculus ; favor limits.

104. (a) We use both; sometimes we prefer one to the other,

106. (a) The infinitesimal forpraoUcal use, but that of rates as a logical basis,
(ft) Not as satisfaotory as the theory of rates as given by Bnokiogham.

107. (a) Limits for general proof and infinitesimals lor doing examples, (ft)
No, not alone.

108. (a) Limits.

109. (a) Limits.

110. (a) The method of limits is the only logical, or rational, way of treating
it ; though the infinitesimal has an advantage in application, (ft) No ; how
a quantity can have another quantity taken from it and not deoreaso the quan-
tity so diminished, is the skeleton that will not down.

112. (a) The infinitesimal, (ft) Yes; both rigorous andsatisfactorj.

113. (a) The method of limits, (ft) It does not.

114. (a) Limits, (ft) Not perfectly.

115. (a) The metliod of infinitesimals, if properly taught, (ft) Perfectly so,
when properly taught (Professor Peck) . (a) Limits. ( ft) Only when axplained
in connection with that of limits (Tutor Flske).

116. (a) Limits, (ft) Not unless based on the theory of liwita.

117. (a) Limits, {ft) I do not find it to.

118. (a) We use infinitesimals. <ft) Yes ; in general.

119. (a) I favor the method ofjluxietis, bnt use the infinitesimal, mainly be*
cause I could not get a suitable text-book in fluxional method until recently,
(ft) Tee ; better than the method of limits ; I have no trouble after a little ex*
planation.

120. («) The method of rates (the Newtonian), using also the prinoiples of
limits in connection therewith, (ft) No, decidedly no; if not established by
the principles of limits. *

121. (a) The infinitesimal for practical use ; limits as a means to an end. (ft)
Tcb; sufficiently so for all practical purposes.

IS^. (a) Infinitesimals for use ; for demonstration, limits, (ft) Yes,

123. (a) Infinitesimal, (ft) Yes.

124. (a) We teach both methods simaltaneoady. Having understood thor-
oughly the rigor of the method of limits, the student has no trouble in hand-
ling infinitesimals, practically, in meciianical problems, (ft) Yes, after he has
once thoroughly understood Taylor's theorem, not as a formula for derdopment
in series merely, but as the means of determining the valae of a fanolion at ona
point from its yalne at anotiier.

125. (a) The first, bnt the infinitesimal should also be given eara£al stata-
mant. (ft) Yes, if properly presented.

MATHEMATICAL TEACHING AT THE PRESENT TIME. 816

(a) JVhat method of treating the calculus do you favor, that of limiiSf the infi/HiteHmalf or
some other? (&) Does the infinitesimal seem rigorous, and to satisfy the mind of the
student f— Continued?

126. (a) Infinitesimal. (6) Yes.

127. (a) The conception of calculus as the *' science of rates, (b) To some
minds it is not satisfactory.

128. (a) Limits, supplemented by the conception of rates, (h) As usually
stated, no.

129. (a) The infinitesimal, (h) Tes.

130. (a) About as giyen in the calculus of J. M. Taylor, (b) Not in the ear-
lier part of his course, but later. Yes.

131. (a) Limits, (b) The students find it easier , and most coUegs students arenot
very critical,

132. (a) We use Todhunter's treatisesi who employs limits in the differential
calculus, and infinitesimals in the integral calculus, and we find it to work
well, (h) Not at first, but later, when calculus is used in analytical mechanics
and mathematical physics, it carries oonrlction and satisfaction.

133. (a) The infinitesimal, {h) It may seem rigorous at first, but I think
ultimately he is better satisfied of its advantages as mental drill.

134. (a) Our course does not include calculus.

135. (a) Infinitesimal in the main, though with many references to the theory
of limits. (&) As so taught, it does.

136. (a) I prefer the infinitesimal method, but I do not hesitate to use such
assistance as can be derived from the method of limits, {b) It appears to me
that it is only familiarity with the proofs that makes either method seem rig-
orous ; but the difficulties seem no greater in r^ard to ono than the other.

137. (a) The infinitesimal for beginners, limits for the advanced, (b) I think
it does if properly presented.

138. (a) Infinitesimals at first, afterward the method of limits may be intro-
duced, (b) With 99-100, yes.

139. (a) Limits is most mathematical, infinitesimals most easily compre-
hended, (b) It is satisfactory to ordinary students.

140. (a) Limits for the theory, infinitesimals for the practice. (&) No.

141. (a) Limits.

142. (a) Infinitesimal.

143. (a) I hardly know. Each has its advantages and disadvantages. I now
use the method of rates as given in Kice and Johnson's treatises. (&) It does
not, as usually presented.

144. (a) Limits for theory, inclusive of rates and of infinitesimals. (() It
sometimes seems to sa4isify the students, but never the professor.

145. (a) The infinitesimal, but in connection with the other methods, {b) It
does, if any method does.

146. We do not teach calculus.

147. (a) The infinitesimal at first ; I use both to some extent, {b) Not al-
together.

148. (a) That of fluxions, with demonstrations by limits, (b) No ; its equa-
tions (with one exception) have never been proved true, and may easily be
shown false. Its results, however, are absolutely true, as experience provea,
and as may be shown by theory.

149. (a) That of limits.

150. (a) Infinitesimal, (b) It does.

151. (a) Limits. (5) It does not.

152. (a) The method of limits. ((} I don*t use the method in teaching.

153. (a) Infinitesimals founded upon limits, (b) The infinitesimal method is
never rigorous unless founded upon limits.

816 TEACHING AND HISTOHT OF MATHEMATICS.

(a) What method of treating the calculus do you favor, that ofHmit3f the inflniteeimal, or
some other t (5) Does the infinitesimal eeem rigorous, and to satisfy the mind of the
student t — Continued. ,

154. (a) Limits* (h) It is the practical method, bnt is not satisfactoryi at
first, to the student.

155. (a) I am decidedly in favor of the method of limits. (&) It does not seem
rigorous, and does not satisfy the mind of the student.

156. (a) The infinitesimal. (() Perhaps not at first.

157. (a) That of limits. <

158. (a) Limits, (b) Yes; can be presented in a perfectly satisfactory man-
ner.

159. (a) The theory of limits, as presented by Todhunter. (5) My experi-
ence with classes has been to the contrary. It does not.

160. (a) Limits, (b) It has not done so with my classes.

161. (a) By limits, (h) It does not.

163. (a) By limits. (() No.

164. (a) The method of rates and fluxionsi as developed by Bice and John-
Bon. (() No.

165. (a) No specialty.

166. (a) Limits, (h) Yes.

167. (a) The method of rates, combined with that of limits, (b) I have not
been able to make it as real to my students as I desire.

168. (a) The infinitesimal with some use of that of limits, (b) Yes; except
with a few students.

(a) Do sdentifie or elassioal stjtdents shino greater aptitude for mathematics t (b) Which

sext

1. (a) No special difference is noticed.

2. Classical.

3. Scientific

5. Classical.

6. (5) Male.

8. (a) No difference ; our CivU Engineering students have shown most apti-
tude. (5) Male sex.

9. (a) I think it depends on talent.

10. (6) Male, generally.

11. (&) Girls for rote work, boys for original work.

12. (a) Classical. ((} Male.

15. (a) 1 observe little or no difference.

16. (a) Classical. (6) Male.

17. {b) Male.

20. (a) Scientific, (b) Male.

21. (a) Classical. (() With us, females.

23. (a) Scientific.

24. (a) Scientific, (b) Equal.

25. (a) Scientific, {b) Male.

26. (a) Scientific. (5) Boys.

27. (a) Scientific, as a rule.

28. (a) Scientific, (fr) Male.

29. (b) I note no material difference in our work.

30. (a) X am inclined to think scientific, (b) Male, but sometimes femato.

31. (a) Have not been able to note great difference, (b) In my nlmiw tho
young ladies have, as a rule, excelled.

82. (a) Soiantifio. {b) The male.

MATHEBiATICAL TEACHING AT THE PRESENT TIME. 317

(a) Do admtifio or olasaioal atudonis show greater aptitude for mathematicB f (b) Which

sex f — Continned.
33. (a) The greater the drill, the greater the aptitude for aDything. There-
fore classical. (5) The young men for persistent, the young ladies for instan-
taneous grasp.

35. (a) Scientific. (&) Male.

36. (a) Good students in either course, (h) Males as a rule.

37. (a) Classical. (5) Males.

38. (a) I would say classical. (() There are more males than females.
40. (a) Difficulfc to answer. (5) The ladies are not inferior.

43. (a) Generally the scientific, (h) 1 see no difference.

43. {a) No difference noted, (b) Males average higher, but a female often
«tands first.

44. (b) A greaternumber of males succeed, but a few females excel.

45. (a) Cannofc see any difference.

46. (a) Classical students not required to study analytical mathematics. No
comparison is possible, as we have no preparatory school, (b) Young men
form the larger part of our higher classes. As far as comparison is possible, the
two sexes show about equtil aptitude for the study.

47 (a) Our best have been classical here, but the reverse has been my experi-
ence elsewhere. (5) Girls in the text-book ; boys outside.
49. (a) No difference. (&) Male.

51. (a) There is no difference, (b) Some of our best students in mathematics
have been young ladies.

52. (a) No difference. (&) Interest about equal.

53. (a) The scientific. (5) Male.

54. (a) Generally scientific. (&) About equally.

55% (a) The former, (b) No perceptible difference as to Bez«
56. (a) Classical. (5) Gentlemen.
67. (a) Generally the former.

59. (b) Equal in applied, but more males iapure,

60. (a) Scientific.

61. (a) Classical.

63. (b) Male.

64. (a) Scientific.

65. (a) Scientific. (5) Male.

66. (a) Classical students.

73. (a) Scientific (b) No difference.

75. (a) Good students in other departments are eqnal in mathemaiies as a
rule, (b) It is rather difficult to answer directly. The ladies average fully as
high as the gentlemen.

76. (a) Scientific, (b) Male.

77. (a) Classical. (&) Male.

79. (a) Classical, thus far. *

80. (a) Classical.

82. (a) Scientific, usually, (b) I find little difference.

83. (a) Classical.

84. (b) The male sex.

65. (a) The classical, as a rule ; our very best mathematical students have
been scientific.

86. (a) No difference.

87. (a) No classical, (b) Yonng men.

88. (a) Classical.

90. (a) Scientific, (b) Males.

91. (a) Scientific. (6) Male for analysis, female for book work.

318 TEAOHINO AND HISTOBT OF HATHEMATIOS.

(•) Do B^ntijU or eU$Heal 9tit^nt9 ehow greaUr apHtudefor mathematkBt (&) WkUk

sex t — Con tinned.

92. (a) Soientifio or philosopbieaL (h) Nearly alike on the text-book work.
The gentlemen seem more saoeessfal in oriffinal inyestigation. May not the
reason for this be found in the fact tha^i it has been assnmed for an indefinite
period that woman is not capable of doing sneh work, and ao ahe has not been
required to do it, thus leading to a dwarfing of this part of her mind f

93. (a) Seem not to divide on this line* (&) See no difference. Few girls
elect calcnlns.

94. (a) No appreciable difference, (b) The females do closer work on lessona
and tasks assigned.

d5. (a) Scientifla.

97. (a) Classical, and students of the exact (not merely scientifie) aoienoes.
(5) Male.

98. (a) Scientific, (h) Male.

99. (a) Scientific. j

100. (h) Male.

101. (a) Scientific, {b) Male.

102 (a) The scientific, (b) Males in quantity^ females in polity,

104. (a) No difference, high classics generally carry high mathematics.

105. (a) No difference. (6; Hard to say. Boys a little better reaaanen.

106. (a) Classical, {b) On the whole, yonng men.
108. (a) Can't say. (b) Males.

113. (a) Scientific.

114. (a) Classical students frequently show a greater aptitude, but scientific
students, after having a practical end in view, more frequently become accom-
plished mathematicians.

115. (a) Scientific (Professor Peck), (a) Mixed, {b) Male (Tutor Fiske).

116. (a) Classical generally.

117. (a) Scientific, (b) Male.

118. (a) We see little difference.

119. (a) Classical, (b) About equal.

121. (a) Scientific.

122. (a) Scientific.

123. (a) Bather the classical, (b) Young men.

124. (a) Scientific. The mathematics in the classical oonrse ends with trig-
onometry, (b) Only a few girls take mathematics. Can't answer aatisfac-
torily. Some girls do excellent work. I doubt whether sex has mnoh to do
with natural mathematical ability.

125. (a) Other oonditions being equal, no difference, (b) No differenoe, fol-
lowing similar preliminary training.

127. (a) Difficult to decide, (b) Not much difference.

128. (a) Our experience not a fair test — students have been so largely clas-
sical, (b) Male, on the whole.

129. (a) Classical.

130. (a) Scientific.

131. (a) About the same, (b) Male.

132. (a) No difference in aptitude, but classical students find no time for ex-
tended courses, (b) Comparatively few females excel, though some are aa
good as any of the males.

133. (a) Very little difference. (&) Males.

135. (a) Scientific students, (h) Male.

136. (a) Classical students. (6) No experience.

137. (a) Scientific or technical.

138. (a) So far as we can test it, olassioaL

]iCA.TQEMATICAL TEACHING AT THE PBESBUrT TIME. 319

(a) Do BtAentifie or eloBtUial aimdmU Aow ffre»t$r ^Utude for maihemaHoo f (>) Wkkh

eesp f---<}(mtinaed .

139. (a) No difforenee in aptitnde. (h) No difTerenoe ia mx«

140. (a) Classioal are more frequently the best.

142. (a) Classical, (h) The male by » large pereentage.

144. (a) Classical.

145. (a) Scientifte.

146. (a) I notice no differeneek

147. (a) The classical are superior, (b) No difference.

148. (a) The scienilfio, bntj for ih^ reason (I think) that those who dislike it
elect a literary course.

149. (a) Classical, I think, (d) Male and female equally is mj observation
here.

150. (a) Classical, (h) Malss.

151. (a) Classical, (h) Male sex.

153. (a) Scientific, (h) Male.

154. (a) We have no elassieal students.

155. (a) Scientific.

156. (a) Sdentifio.

157. (a) Scientific.

159. (a) Scientific.

160. (a) Scientific.

161. (a) The classical, (b) Male.

162. (a) I think classical, (b) Male.
164. (a) Scientific. (5) Male.

166. (a) Classical, (b) I do not see any diffexenee.

167. (a) Generally the scientific, (b) Males as a rule, at least in higher
branches than elementary trigonometry.

168. (a) The classical.

(a) In what other subjeota are good mathematioat etudenis iM8t ettooessful f (b) In what

least suooeeafult

I. (a) Qood students in mathematics generally stand well in all other studies.

4. (a) As a rule none but the better students pursue mathematics more than
two years. Good mathematical students are successful in the scientific branches
taught here* (i) In English.

5. (a) Geography, logic, history, chemistry, and natural philosophy, (b)
Grammar, rhetoric.

8. (a) In logic and in the physical sciences, (b) Ancient languages and Eng-
lish literature*

9. (a) In philosophy, chemistry, analytical mechanics, geodesy, etc. (b) I
have not observed.

10. (a) I can not say. (5) Do not know ; hard to determine.

II. (a) Am not certain ; should say physics and ancient languages, (b) Can't
Bay.

12* (a) I do not know certainly, but I have never noticed any inverse relation
between linguistic and mathematical endowments. Chemistry and mathemat-
ics are less friendly.

15. (a) Scientifio subjects, especially physics. But I observe that good mathe-
maticians usually do well in almost any subject which interests them, (b) Sub-
jects which involve much '* committing to memory.''

16. (a) A good student is successful everywhere. I have found that my best
students in mathematics were, as a rule, ** best students" in other departments.

17. (a) In any subject in which continued reasoning is necessary. (&) I am
unable to specify.

820 TEACHINQ AND HI8T0BY OF MATHEMATICS.

(a) In what other gubfeets are good maXhemaUeal %i%dmU mo9i mmoms/WZ f (() In what

leaat sucoeasful t — Continaed.

18. (a) In natural philosdpliy, in metaphysiosy and generally in Qreek. (b)
literature.

19. (a) They do equally well in Latin, as a rule, (h) English.

20. (a) Mechanics.

HI, (a) A student good in mathematics is apt to be snocessful in all branches.

22. (a) My experience goes to show that a student who is good in mathemat-
ics is capable of coming off with good staQ4&rd in almost any other study, I
have known a £bw apparent exceptions, (h) In languages.

24. (a) Sciences, (h) Classical studies.

25. (a) Physics.

26. (a) Physics, logic, chemistry, (b) Bhetoric.

27. (a) In engineering, physics, astronomy. Mathematical training seems to
make lawyers more successful in the clear statement of their eases, {h) Liter-
ary pursuits.

28. (a) In mental, moral, and natural philosophy, (h) In belles-lettres.
30. (a) In almost eyery other. (5) Perhaps, literature.

32. We can hardly give an intelligent answer to this question with our grade
of work.

33. (a) Mathematics, as we are compelled to teach it, is largely mechanical;
therefore, in subjects not requiring great originality. (() Answered.

35. (a) In engineering. . (b) In languages.

37. (a) Varies with the student.

38. As a rule, our mathematical students are excellent in all their atndies.
Languages are not un&equently hard for good mathematical students.

39. (a) Greek, Latin— often— mainly— various forms of graphics, (b) Scien-
tific research, i, «., natural sciences.

40. (a) The various branches of natural science, metaphysical studies.

41. (a) Usually in all other subjects of our course, (b) Occasionally in lan-
guages.

43. (a) We often have fine work upon topics related to general geometry.

44. (a) Chemistry, physics, languages, (b) History, literature.

45. (a) Oenerally in whatever is undertaken, I believe success in any braneh
is in proportion to application.

46. (a) Whatever they undertake, (b) Whatever they give the least atten-
tion to.

47. (a) Logical, (b) Linguistic.

48. (a) As a rule in all subjects requiring Judgment, reason, discrimination.
(b) In subjects requiring the memory as the chief element of the mind*

49. (a) Sciences. (5) Languages.

50. (a) Languages, {b) History.

51. (a) All others, that is, according to circumstances.

52. (a) Our good mathematical students are good in languages and sciences.

53. (a) Chemistry and physics, (b) Have not noticed.

54. (a) Some in one subject and some in another, according to native aptitnde
and application.

55. (a) In chemistry, physics, and logic. Goodmathematical students rarely
show weakness in any study, (d) Literature (and modem languages f).

56. (a) Natural science, (d) Language.

57. (a) Natural science.

59. (a) Themo/oritjf of good mathematical students are good in everything
else, but sometimes a mathematical mind fails in letters, and vice rerto, as
appreciate only demotutrative reasoning, and some moral,

60. (a) Physics, astronomy, and natural science. (6) Languages.

MATHEMATICAL TEACHING AT THE FBESENT TIME. 321

(a) In what other auhjeoU are good mathematiodl students most sticeessful t (h) In what

least suocessful t — Continued.
63. (a) I find good mathematical stadents saocessful, generally, in all other
BQhjects.
66. (a) In most subjects.

68. (a) Our best students are about equally successful in all mathematical
branches.

69. (a) Natural science, (h) Can't si^.

70. (a) Languages, (h) History and literature.

73. (a) Physics and astronomy, (h) My best mathematicians are best in
other lines.

74. (a) Since I have observed here (four years), the best mathematical
students are usually also among the best in all studies ; otherwise in natural
sciences, English and Greek, history and political science, (fi) Languages.

75. (a) The good mathematicians are those whose general standing is high,
but of course there are exceptions to this ; I should say that they are more likely
to excel in the sciences, logic, and metaphysics.

76. (a) Chemistry, logic, mental science. (6) Language, history, rhetoric,
oratory.

79. (a) Languages, I should say, in general.

80. (a) Latin and science. (&) History and literature.

82. (a) As a rule, I think, in all subjects, although occasionally I find one
^ho is weak in language and literary studies.
84. (a) Physics and mechanics. (&) The languages.

86. (a) No special difference so far as I know.

87. (a) Chemistry, physics, and applied mechanics, (h) Languages.

88. (a) Our records show that good mathematical students are successful in
all other subjects.

89. (a) In all other subjects taught in the school.

90. (a) In historical studies, natural philosophy, and mathemat^ial astron-
omy, (h) Literary, but not always.

91. (a) In the lateral sciences, e, ^., physics, chemistry ; also in logic, (h)
Languages and history.

92. (a) As a rule, I think, in the sciences, and especially in original investiga-
tions in science, (b) So far as my observation goes, in languages, as a rule.

93. (a) With rare exceptions they are good in all the subjects. The converse
is not so general, <. 0., students often excel in one or two departments without
excelling in mathematics.

94. (a) They average well all around, (b) No uniformity.

95. (a) Draughting, physics, chemistry, logic.

96. (a) As a rule, those good in mathematics are good in all others, but es-
pecially in natural sciences, psychology, and logic.

97. (a) In all the more introspective, and such as require prolonged and stren-
uous thought, not mainly observation (like stone — or bug— lore). (5) In these
latter so-called experimental sciences.

98. (a) Applied arts, engineering, physics, etc. (h) Languages, metallurgy,
analytical chemistry.

99. (a) Philosophy. (5) Composition.

101. (a) Philosophical, (h) Linguistic.

102. (a) In natural sciences, history, geography, logic, (h) Languages.
104. (a) AH scientific pursuits, drawing, arts, generally.

106. (a) In such as require concentration of mind, and close reasoning. (5)
If in any, in such as depend upon observation and experiment.

107. (a) Generally in all others, if they are interested.

109. (a) Natural philosophy, chemistry, Greek, Latin. They can generally
do well, wherever they try. (b) English, political science.

881— No. 3 ^21

322

TEACHING AND HISTORY OF MATHEMATICS.

(a) In what other subjects are good mathematical students most suooeesful t {b) In v>hat

, least successful? — Continued.

110. (a) My observation has been that where students were good in mathe-
matios, they were good in all their other studies.

111. (a) Logic and analytical studies.

112. (a) They generally stand high in all subjects.

113. (a) As a rule they are successful in all other studies ; more bo in meta-
physics, theology. (6) The higher study of literature.

114. (a) Very difficult to generalize. Many excellent mathematicians are
''all-around'' men. Others excel in science, and are least suocessfhl in lan-
guages and speculative subjects.

115. (a) A good student in mathematics is generally a good student in all
other branches (Professor Peck).

116. (a) Generally in logic and psychology.

117. (a) Mechanics, physics, chemistry, logic, (h) Classics.

118. (a) They are generally good in all subjects. (() Subjects requiring
memory only.

120. (a) Chemistry (including heat, physiology, eleotrioity, and magnetism),
mineralogy and geology, engineering, ordnance and gunnery, and law. (h) Draw-
ing, Spanish, and French, relatively. Good mathematical students are gener*
ally good in all other branches (Professor Bass, professor of mathematics).

Charles W. Lamed, the professor of drawing at West Point, answers as fol-
lows : -I differ somewhat from the inferences to be drawn from the answer to
this question by the professor of mathematics.

In so far as any influence is to be implied by mathematical proficiency upon
other studies, an examination of the standing of the last five graduating classes
tends to show very positively that law belongs to the category of those studies
in which there exists the greatest discrepancy, and this, notwithstanding that
law is studied two years after mathematics is completed and when habits of
study and ability to master a wider range of subjects is more highly developed
by the study of intermediate synthetic studies.

There is a much greater range of discrepancies also between the group of
studies comprised under the head of chemistry (which includes eleotrioS| min-
eralogy and geology, and heat) and mathematics than would naturally be in-
ferred from the grouping made. Even in natural philosophy the aggregate
discrepancies were greater than I had supposed probable.

The standings of the graduating classes of 1888, 1887, 1886, 1885, and 1884 in
law, chemistry, drawing, English, and French were reduced to the same stand-
ard, and the differences between these and mathematics in each case were ob-
tained, and the aggregate in each subject for these classes is as follows:

Discrepancies as compared with m^thematioi.

1888
1887
1886
1885
1884

Law.

Chemiatry.

Drawing.

French.

856

342

481

418

925

714

1.021

930

1,102

1,096

1,094

1,166

336

260

832

295

195

144

866

854

Snglish.
41S

too

1,I76
Sll
Ml

In regard to drawing it is proper to observe that a marked distinetton exists
between technical graphics and free-hand drawing. The standing giren inclades
ftee-hand drawing, occupying one-fourth of the coarse. In this the posseasion

MATHEMATICAL TEACHING AT THE PRESENT TIME. 323

(a) In what other 9ubjeoU are good maihemaiieal etudenU n^et eueoeeeful t (ft) In wJuU

Uaat raooM«/«Zf— Continued. '
of natural graphical talent exercises a much greater influence in producing
discrepancies in standing. In technical graphics, however, throwing out a
few men, perhaps one-half dozen in each class, with pronounced natural ability,
standing in plane and descriptiye geometry has a decidedly beneficial influence
on standing in drawing. In other respects, intelligence, whether mathematical
or liberal, will tell in the work. Leaying out four or five exceptional men in
each class the discrepancies in drawing, even with £ree«hand included, fall
below those of French, English, and law.
131. (a) Physics and astronomy, (h) Latin, French, etc.

122. (a) First-rate mathematical students generally do well in all other
studies.

123. (a) In mathematics of physics, (h) Possibly the biological sciences and
languages.

124. (a) My experience is that a man who is good in mathematics has mental
ability sufficient to make any subject of an ordinary college course compara-
tively easy. Good in mathematicft*-good eyerywhere.

125. (a) Applied sciences, of course — astronomy, physics, logic, and meta-
physics, (h) Languages and literatare, sometimes. StiU hardly think that is
true, as a rule.

126. (a) Advanced work in physics and engineering. (() Our best students
In mathematics are best everywhere.

129. (a) Good mathematical students are good in all their work, (b) Rarely
unsuccessful in any line of study.

130. (a) Physics, astronomy, logic.

131. (a) Classics, sciences, but there are many exceptions, (h) English,
probably.

133. (a) Think that, on the whole, our best mathematical students are best,
generally, in other studies.

134. (a) Sciences.

136. (a) Greek is often combined with mathematics.

137. (a) They are generally good all around. (5) In languages, but only in
exceptional cases. .

138. (a) In all other subjects, (b) None.

140. (a) Physics and astronomy. The good ones are also usually good in
classics and everything.

141. (a) Generally also in the classical studiea

142. (a) Physics, chemistry, logic, (h) Moral and mental philosophy.

143. (a) All subjects requiring accurate thought. In our college this is es»
peoially noticeable in mental and moral philosophy, (b) Those requiring
mere memory.

144. (a) Usaally in all others.

145. (a) Some in one, some in another. No general rule. (5) No general

rule.

146. (a) In this they differ, though they are possibly better in scientific
studies, (b) Languages.

148. (a) I have not observed that a successful student in mathematics is
more apt to succeed in one subject than he is in another, except where the sub-
ject rests on mathematics.

150. (a) Generally in everything else.

151. (b) Belles-lettres.

152. (a) Naturally in subjects depending upon a knowledge of mathematics,
and generally in whatever else they may study.

153. (a) In logic, physics, engineering, medicine. (() Languages. '

824 TEACHIKG AKD HISTORY OF MATHEMATICS.

(a) In what other aubjeota are good mathematical eiudente moit mooM^uZf In what leati

succea^ult — Continned.

154. (a) In all branches that require aoourate obBerration and olose reason*
ing* W In languages.

155. (a) In branches of natural science, (h) In the languages, I think.

156. (a) In logic and, as a rule, the natural sciences.

157. (a) They are apt to be more successful in all subjects connected with the
sciences than in the study of languages.

158. (a) As a rule, a] 1 students standing very well in mathematics will aohiere
success in any other subject. I have seen but few exceptions.

159. (a) In moral and mental philosophy, logic and civil law. (h) Synthetic
languages.

160. (a) In the mathematical sciences. A good mathematical student is good
at everything he undertakes.

161. (a) Physios and chemistry, (h) Languages and history.

162. (a) Oftener classics.

163. (a) Engineering and physics. (5) Languages.

164. (a) In any subjects requiring reflection, (b) Those requiring perception
and memory only.

166. (a) Logic, chemistry, philosophy, political economy, and astronomy.
(5) Langnage.

167. (a) I believe that a really flrst-oloss mathematical student is generally
successful in nearly all subjects, but those a grade lower are most likely to
excel in the physical sciences than in other lines, (b) Perhaps in bellee-lettree.

168. (a) I don't think I can tell, for there is such diversity ; yet I think that
those who are good in mathematics are good all-around students, as a rule.

What i8 the relative prominence of matTiematica in your oouree ofetudy as shown hy houra

per week and per year t

1. In classical and scientific courses the same number of hours is given to
mathematics as to any other study. In the engiDeering course about twenty-
five per cent. more.

2. About one-fourth part of the class-time is devoted to the study of mathe-
matics.

3. No study has more attention, and some have less.

4. Five hours per week out of sixteen hours for recitation and lectures are
devoted to mathematics for three years ; the last year three hours per week
during the year.

5. More prominence given to mathematics than to any other study.

6. Large.

7. Occupies more time than any other subject.

8. In classical course one-fourth of student's time is devoted to mathe-
matics ; relatively more in scientific and civil eugineering course.

9. The principle studies in our college receive equal attention ; mathematics
one hour and a quarter daily.

10. More stress on mathematics as a whole than upon any other subject, I
think.

11. Mathematics takes one-fourth of the time in the scientific course, odo-
seventh in the literary, and over one-fifth in the classical.

12. It leads Greek, and is on a par with Latin and physics.

13. Considered of fundamental importance aud continued throoghout the
four yeass of study ; twenty hours per week for the four classes (pure mathe-
matics only).

14. During the year, four hours per week out of a total of fifteen honrs,

15. In the classical course about ten per cent, of the work is in mathematics,
and in the scientific course about fifteen per cent. I have counted only the
prescribed work and the pure ma thematic*, so-called.

MATHEMATICAL TEACHING AT THE PRE8FNT TIME. 325

fVkat iB tho relative prominence of mathematioe in your course of 8tv4y as shoum ly hours
, per Moeehand per year f — Continued.

16. It has the same prominence as do the classics.

17. Preparatory, two-thirds of the entire work done is mathematics ; first year,
one-half; second, about two-fifths ; third, one-fifth ; fourth, only applied math-
ematics.

18. One hour of mathematics each day^ i. 6., six hours per week ; about three
and three-fourths of other studies.

19. Our regular course of study practically covers five years, divided into
three terms each ; mathematics occupying one-third of each term for the ten
terms ending with first term of the Junior year (Professor Gordon). In the
early years of the course, equal to any other subject except English (Professor
Draper).

20. Mathematics and classics each occupy five times as many hours as sci-
enoe,

21. Twenty per cent,

22. Daily recitation required of eyeiy student. •.

23. It is of the first prominence.

24. No special prominence observed.

25. Six hours x>er week; about one hundred and eighty per year.

26. One-fifth of time.

527. Mathematics has seventeen hours per week ; the ancient languages, fifteen
hours per week ; English, twelve hours per week.

28. One-fourth of all the time during Freshman and Sophomore years is de-
voted to pure mathematics ; and one-tenth of all the time in the Junior and
Senior years.

29. Mathematics extends througli two-thirds of the course. Takes about one*
fourth of time during that period.

30. The time spent is about the same as in the average college.

31. More time is given than to any other one topic.

32. It ranks with the natural sciences and the ancient and the foreign Ian-*
guages.

33. As prominent as any other chair, if not more so.

' 34. Freshman year, three-fifteenths ; Sophomore year, three-fifteenths ; Junior

year, two-fifteenths (elective) ; Senior year, two-fifteenths (elective).

35. In some courses one-third the time for one year ; in others one-third the
time for' seven terms out of twelve, with applied mathematics for eleven ternu
more. In both, eighteen units out of thirty-six.

36. Nearly one-third in Freshman and Sophomore years.

37. The same as other studies; three and three-fourths hours per week.

38. One year's work is required of all students. Four years are required of
mathematical students. Many elect mathematics for one or more years.

39. The time is about equally distributed between Latin, Greek, modem
languages, and mathematics.

40. Difierent in the various courses. Mathematics is more prominent in the
Boientific course, claiming about one-fourth the student's time (perhaps one-
third).

41. Freshman year twenty-one hours out of sixty-one, 798 per year; Sopho-
more eighteen out of fifty-four, 684 hours per year ; Junior thirteen out of
flixty, 494 per year. In this estimate two hours of preparation are usually
reckoned with each hour of recitation.

42. Sixteen hoars per week.

43. Upon an equality with Latin and Greek.

45. It is taught five hours per week until the end of the Sophomore year.
47. Co-ordinate with Latin and Greek.

326 TEACHING AND HISTORY OF MATHEMATICS. \

What iB the relative prominence of mathematioe in your course of study as shown hy hown

per week and per ^far f—Continned.

48. Aboat one hour oat of eyery four.

49. Five horns a week for two years and one year additionali which is electiTe,
60. About the same.

51. As great as that of any other sabject.

52. Equal with language and science, until Junior year.

53. Has more time than any other study.

55. Fiye out of fifteen per week for three years of the course. Ko mathe-
malics in the last year of the course.

56. It receives more time than any other subject taught.

57. First.

59. About two-fifths of the time in the yarious schools.

60. As about five to four in comparison with language and natural soienoe.

61. First.

62. It ranks with any other study in prominence.

63. It i« desired to make it equally prominent with other subjects.

64. One hour daily in class-room.

65. Four hours per week are devoted to mathematics throughout the coarse.
It receives about equal attention with any other subject.

66. Quite as prominent as classics.

67. Under our ''group'' system, under-graduate students who include in their
•< group" of studies a minor course in mathematics devote one-third of their
time for one year (as measured by hours per week) to mathematics ; those who
take a major course in mathematics devote to it one-third of their time for two
years. The whole time of an nnder-gradnato course is three years. A student
need not include any mathematics in his group. Our entrance requirements
include trigonometry and some analytical geometry.

68. About one-fourth of required time is devoted to pure mathematics.

69. Stands near bottom of the list.

70. Classics, science, and mathematics have equal prominence.

71. During the first year for all students, thirty-three and one-third per cent,
of recitation periods is for mathematics. The time for preparation would be
larger. During the second and third years the engineers gave about twenty-five
per cent, to pure and twenty-five per cent, to applied mathematics. During
the Seinior year about twenty-five per cent, to applied mathematics. Other
students give but little time to mathematics affcer the first year.

73. Four hours out of fifteen per week in Freshman year.

73. Thirty-three and one-third per cent.

74. Freshman year four hours per week, i. e., twenty-five per cent, is required
throughout the year. Four hours per week elective is offered in Sophomore
and Junior years.

75. Five hours per week for the first two years of the course.

76. It stands third in the course.

77. Less prominent than the classics, except in the academy and in the in*
dnctive science courses.

78. It stands on the same level with Latin and Qreek— our courses being (like
those of Harvard College) elective.

79. It is on an equality with Greek.

50. Algebra, plane geometry, plane trigonometry are required, five recitations
per week during Freshman and Sophomore years. Mathematics is elective three
hours per week during rest of course.

82. Five hours per week for thirty-eight weeks per year, or nearly one^third
of the whole work.

84. In first year, one-fourth the time ; second and third, one-fifth ; none in the
last.

MATHEMATICAL TEACHING AT THE PRESENT TIME, 327

What is the relative prominence of mathematioB in your oaurse of study as shown ly h,our$

per umk and per year? — Continned.

85. One-third of the time*

86. Of required workj mathematios has about fifteen per cent, out of the fifty-
two hours weekly.

87. Our students average in three years six and one-third hours per week of
mathematics, to four and one-half of language^ to four of physics and chemistry,
mineralogy and geology altogether.

68. It is different in the different years. For the four years it is about one to
ten.

89. One to six.

90. About par.

91. The hours are about equal, taken as a whole. For the degree of B. S.
they far exceed.

92. A high importance ; fifteen terms (including the preparatory course) ;
five hours per week, forty weeks per year, for five years.

93. Sub-Freshman year, two-sevenths of entire work ; Freshman year two*
ninths ; Ju|iior year(elective) one-fourth ; Senior year (elective) one-sixth. This
does not include mechanics, surveying, and other applied mathematics.

95. Four hours per week ; most other studies (non-professional), three.

96. Daily recitations, one-half hour each.

97. In the scientific courses it is first ; in the classical and literary, second
(Latin and Greek, respectively, English first).

98. Mathematics is the ground work of the institution, preparatory and coin-
oident with the courses in engineering.

99. Two hours daily devoted to mathematics during the session of ten
months.

100. Fonrteen-sixtieths of the four years' course.

101. Twenty-five per cent

103. About one-third of the whole time ; ten hours per week, 480 hours per
year.

103. Each class averages five hours per week, per year.

104. Same, no prominence.

105. In the preparatory course, total hours per week fifteen *, mathematical
average,- three and one-third. In college classes, total hours per week, fifteen^
mathematical average, two and eleven-twelfths.

106. The mathematical course for the majority of our courses is completed the
Freshman year, having five-sixteenths of the time.

107. Leading study.

108. Five twenty-fourths of the whole.

109. Mathematics ranks with Latin and Greek throughout, each getting four^
fifteenths of the time in first two years ; elective in third.

110. Until this year, more than half the time was given to mathematics ; now,
perhaps, one-third.

111. We require two and one-half hours per week.

112. Considering the hours devoted to mathematios, it ranks with any other
rabject. •

113. Mathematics and Latin have each three hours a day.

114. Freshman year, one-third; Sophomore year, four-fifteenths ; Junior and
Senior years, optional.

115. The courses in my department are elective. Question cannot well be
answered. >.

116. Five hours per week out of fifteen in Freshman year ; three hours per
week out of fifteen. Sophomore year for classical ; four out of fifteen for soien-
tiflo.

328 TEACHIKa AND HI8T0RT OF HATHElCATICa

What i$ the relative praminenee ofmatkemaiiee in ycmr eoune of etmdjf a$ 9kowm frjr Jbourt

per week and per year t — Conimned.

117. Freshman year, fire hours per week ont of sixteen ; Sophomore^ fiye out
of seventeen in first term, and three ont of eighteen, second term.

118. In scientific conrse sixteen one-hnndredths of whole time, or twentj-
eight one-hnndredths, inolnding descriptire geometry and mechanics and as-
tronomy. In classical conzse thirteen one-hundiedths and sixteen one-hnn-
dredths.

119. Fonr horns per week reqnired dnring the entire Freshman and Sopho-
more years, and the second term of the Junior year. From two to six hours per
week may be electiye dnring the other terms.

120. Dnring the first year, time devoted to mathematies is to time devoted to
modem languages as fonr and one-half is to three. During the second year,
time devoted to mathematics is about the same as the time devoted to languages
and drawing.

123. It is on an equality with the subjects taught in the other departments.

124. liathematics is one of our most important subjects. Three professon
give Jointly forty-five hours per week to mathematical instruction, for twenty-
six weeks, and thirty-nine hours per week the remaining eleven weeks of the
•ession.

125. Freshmen five-fifteenths, Sophomores three-fifteenths, Junior^ (elective)
two-fifteenths, Seniors (elective) two-fifteenths.

126. It occupies about one -third of the whole course.

127. Full work in all subjects fifteen hours per week. In mathematies five
hours per week whenever any mathematical subject is studied. In the prepar-
atory department algebra is required in aU the courses three terms (one full
year), and in the science courses four terms. Plane geometry is required
through the last term of the Senior preparatory year. Then, in the college
course, we have solid geometry, trigonometry, surveying, analytical geometry,
and calculus, one term each. A second term of analytical geometry and cal-
culus is required of the scientific students ; and in the philosophical conrse,
students elect between a second term of calculus and practical chemistry.

128. All courses are four or five hours a week. In classical course, mathe-
matics have 378 hours, required and elective, out of a total of 4,077 hours.

129. Two hundred and seventy hours distributed through two years with
opportunity of election in addition.

130. Four hours ont of fifteen per week dnring two years for all the class;
then fonr out of fifteen during another year for electives.

131. One^fourth up to second term of Sophomore year ; irom that point all
•nbjects in the course are elective. •

132. We have too many courses to make any general statement.

133. About the same as ancient languages.

134. Greater than others, except English.

135. Freshman year, five hours out of eighteen per week.

136. Mathematics is on an equality with all other courses.

137. A little more than one-fourth of the student's time is given to mathe>
matics in Freshman and Sophomore years ; a little less than one-fourth daring
the remainder of the course.

139. Full course, four hours per week.

140. About twenty-five per cent, of totaL

141. On a par with the classical.

142. First in the course.

143. Pure mathematics thirteen and six-tenths per cent, of required work.
It may be thirty-four per cent, of elective work. It may be one-fifth of the
whola course.

MATHEMATICAL TEACHING AT THE PRESENT TIME. 829

WM %$ the relative prominence of maihemaHos in your course of study as shown hy hour$

per week and per yearf — Continued.

144. Firsfcyear about thirty-three and one^third per cent, of time to mathe-
matios ; second year, fifteen per cent. ; third year, twenty-five per cent. ; fourth
year, fifteen per cent.

145. Out of the required eighteen hours per week, literary students get in
the first year five, second year three, third year nought, fourth year one and one-
half; total, nine and one-half out of seventy-two ; scientific students get in the
first year five, second year four and one-half, third year four and one-half, fourth
year one and one-half; total, fifteen and one-half out of seventy-two.

146. We regard it of greatest importance.

147. In the preparatory course one-third of the study is mathematical, {. e.
185 hours a year out of 555. The same in the Freshman year. In the Sopho-
more year 135 hours out of 555. After that, none.

148. Twenty-five per cent, of the student's time is devoted to mathematics
nntil he completes the Sophomore year. Besides, the students in engineering
devote twelve and one-half per cent, of their time in Junior year to mathematics.

149. About one-fourth the time is devoted to mathematics.

150. Mathematics to science about equal; mathematics to language about
four to one.

151. About the same time is given to mathematics as to other branches, viz»
five recitations per week, except in last year, three times.

152. Mathematics, Latin, and Greek have ^ach four hours per week for
Freshman and Sophomore years ; no other subjects have as much time. After
Sophomore year mathematics is elective.

153. More prominent than any other subject except English and equal to
that.

154. It stands first.

155. It is as prominent as any other branch of study. The Junior class, with
which the college work properly begins, has five recitations per week, each
one hour long. The intermediate class has four per week, the Senior has thr<)e
per week, and in applied mathematics there are three per week.

156. About twenty hours per week, or eight hundred hours per session.^

157. It is given as much time as other subjects, five hours per week in the
Freshman year, five hours per week in the Sophomore year, three hours per
week in the Junior year, and two hours per week in the Senior year.

158. Our system of independent schools and free elective courses enables us
to give a positive statement that, as a rule (with a few exceptional years), the
school of mathematics is the most largely attended school in the academic de-
partment. The number of lecture hours per week for under-graduates is thir-
teen.

159. Mathematics occupies a more prominent position in our schedule than
any other branch.

160. First.

161. It occupies about one-third of the time devoted to the course of instruc-
tion.

162. Five sections per week or nearly one-third of time for first two years.

163. It is probably on about the same footing as the other chief branches of
study.

164. Our course in mathematics is very prominent, requiring one-third the
student's time through the Sophomore year.

165. Three to two.
206. One-half.

167. Classical course : Freshman year, one-third of time required; Sophomore
year, one-third elective; Junior year, one-ninth elective. Scientific course:
Freeman year, one-third required; Sophomore year, two-ninths required.

3S0 TEACHING AND HISTORY OF MATHEMATICS.

What U the relative prominence of maihemaiioB in your oowrse of study oi eihown Ify

per week and per jrearf— Continued,
one- ninth eleotiye; Junior year, one-ninth required; Senior year, one-nhiih
electiye.

168. Almost the least prominent thing in the course, as ours is claaaioal, with
a leaning to natural sciences.

(«) Do you favor memorizing rules in dlgehrat (h) What reforxM are needed in teaching

the same?

1. (a) No.

2. (a) No. (h) It ought not to be taught to such young boys, who contract
the incurable habit of learning it by rote.

3. Principles, but not rules.

4. (a) No. (h) Bules and principles should be deduced from examples; a
more thorough drill in algebraic language, especially in the meaning and nae
of signs, exponents, etc.

5. Do not use text-book too closely.

6. (a) Yes, (fi) More practical application should be giyen,

7. No.

8. No.

9. (a) I prefer formulas. (I) More thoroughness and better understanding
of elementary principles, with reyiews and drilling.

10. (a) No. (() Algebra should be taught just as arithmetic, wholly by the
analytic method.

11. (a) Some of the rules, (h) I do not know.

12. (a) To a limited extent, (h) More of the spirit and reason and less mere
mechanical solution.

13. (a) No. (b) The modem methods, as determinants, etc., should be intro-
duced as soon as possible.

14. (a) Yes.

15. (a) No. (b) A larger number of simple problems ; a less number of diffi-
cult demonstrations, such as those in logarithms, the binomial formula, etc.;
an earlier introduction to the methods and notation of the calculus.

16. (a) No.

17. (a) Very little, (b) Anything which will make it less a collection of dry
bones, and more a living and beautif al science.

18. (a) Yes.

19. (ft) By proper classification the number of propositions could be materi-
ally reduced and the number of important theorems and constructionB fbr origi-
nal work could be materially increased. (Professor Cordon), (a] No. (6)
More attention should be paid to explaining and illustrating the principles in-
volyed in operations, and the embodying of questions to test th^ understanding
of those principles ; e, g., why x — ( = 10 is equal to » = 10 -f 6 ; why does -f-
X — = — t etc. (Professor Draper.)

20. (a) Yes, (() That the sense of the rules shall be known when the mem-
orizing is complete.

21. (a) No. (b) Teaching needs to be less mechanical. The reasons for proo-
esses need to be taught more.

22. No ; the inductive method should be used first.

23. (a) Yes.

24. No ; thorough drill in substituting numerical quantities for literal.

25. (a) I do not. (6) In teaching the elements as few formal demonstrationa
as possible should be used — first a working knowledge, and then philosophice.

26. (a) Do not. (b) Practical examples.

27. No ; more attention to fundamental principles, clear teaching why signa
ftte changed in transposition, etc

MATHEMATICAL TEACHING AT THE PEESENT TIME. 331

(a) Do you favor memorizing rulee in algebra t (h) What rtforme are needed in teaching

the eame f— Continaed.
S6. No ; more prominence to principles and less of method. Students should
do more private work.
69. No; it should be freed from its mechanical character. Algebra should be

30. Not much ; less memorizlDg, more analysis, more thoroughness.

3t. We require the principles iuYolyed, rather than the exact words of a rule.

32. Tes; only the most important rules and theorems should be memorized,
but those thoroughly,

33. Emphatically no; more principle and why ; less toughing, disgusting
gymnastics*

34. No.

35. No; more familiarity with technique ; less mechanism.

36. No.

37. No, sir.

88. (a) Not in general, (b) In general, I should say a more thorough teach-
ing of the principle and reasoning of algebra.

39. (a) Yes. (h) More ought to be taught.

40. (a) Students are urged to state operations and principles clearly and
briefly without much regard to the text, (h) Many problems (original and
otherwise) should be solved mentally.

41. (a) No. (() More careful attention to the interpretation of literal equa-
tions.

42. (a) To some extent.

43. (a) No.

44. (a) I do not. (5) More drill on simple exercises, and fewer difficult
problems.

45. (a) I do not. (() Such as will render the mind able to deal with the
principles in forming rules.

46. (a) Some of them, (b) Too little time seem^ to be given to the study
of algebra.

47. (a) Not verbatim, (b) More rigorous proofs ; more noting of analogies ;
more as a preparation for higher work than the solving of problems as mere
puzzles.

48. (a) Yes.

49. (a) No.

50. (a) I do.

51. (a) No. (b) Methods are learned by practice, and riies evolved there-
f^m.

52. (a) No.

53. (a) Very few. (5) There should be more practical application of its
principles.

54. (a) Not mechanically, (b) The reform of good common sense, and clear,
■imple presentation.

55. (a) No.

66. (a) No. (b) Keep students out of it until they have passed the discus-
sion of arithmetic.

57. (a) Yes. (fr) We need a simpler and at the same time fuller elementary
book.

68. (a) At the beginning, (b) More familiarity with principles.

59. (a) Not generally, (b) The teacher should assist the pupil to make his
own rules.

60. (a) No. (b) A correct reading of algebraic expressions in algebraic lan-
guage, and a clear analysis of work done.

332 TEACHING AND HISTORY OF MATHEMATICS.

(a) Do you favor memorizing mles in algebra f (b) What rtfoi-ma are needed in tea6MM§

the eame t — Continued.

61. (a) I do not. (fi) Require pupils to think and not to be machines or jugs
to be filled.

62. (a) To a certain extent, {b) There is need of impressing the students in
some way with the idea of the practical value of the study and of creating an
interest in it.

63. (a) No. (b) Teachers should wait till their pupils are prepared to begin

the study. It should be thoroughly taught.

64. (a) Yes.

65. (a) Yes. (b) The subject ought to be presented freer from technicalities
than text-books giye it. Unnecessary parts ought to be left out.

66. (a) Yes.

68. (a) No.

69. (a) No. (b) Drill on the principles and raison Witre for formulaa.

70. (a) Yes ; when once thoroughly understood.

71. (a) No. (5) The use and meaning of exponents and of the negative sign
are not made as clear as they should be. More accuracy.

72. (a) No.

73. (a) No. (b) Less rules and more thinking. The less memorizing in
mathematics, the better the results.

74. (a) No.

75. (a) Yes.

76. (a) No. (5) The teacher should lead with the general demonstration of
each subject in form of lectures.

77. (a) No. {b) Pupils should be required more generally to demonstrate
principles and work from them rather than from rules and formulae.

78. (a) We do not teach elementary geometry.

79. (a) Yes. {Jb) The rules should be proved as strictly as any proposition in
geometry.

80. (a) No. ifi) More classification of subjects.

82. (a) No. {b) The chief cause of failure in many cases is not doing enough
miscellaneous examx)les for practice.

83. (a) Yes.

84. (a) Some, (b) None.

85. (a) A more logical arrangement of the different sections of the subject;
more examples, and so given as to form a constant review of the ground already
gone over ; application of business methods to the revision of many rules and
methods.

86. (a) No. {b) More thoroughness, practicality, and solidity of teachings
the German system,

87. (a) To some extent. (&) For a course of study like ours I think more
emphasis should be put on thoroughness than extent of ground covered.

88. (a) Yes.

89. (a) No.

90. (a) No. {b) More attention to reasoning processes.

91. (a) I do not. (&) More independence of books and greater original inves-
tigatiou.

92. (a) No. (b) Less memorizing and more thinking, both on the part of
teacher and student.

93. (a) Yes. {b) Pupils should be taught to state a proposition and follow it
with a general demonstration, as in geometry.

94. (a) Yes — ^No! Teach the pupil to develop the principle, and to formulate
his own rule for it and for his process.

95. (a) Not much. (() Omit attempts to exhaust each subject as it comes up.

96. (a) No.

MATHEMATICAL TEACHING AT THE PBESENT TIME. 333

(a) JJO you favor memorizing rule8 in algebra f (h) What rtforms are needed in teaching

the samef — Continued.

97. (a) No. (h) Blind, unreasoning, meohanioal solution of equations needs
abatement ; the doctrines of forms and series, advancement.

98. (a) No ! ! Battle the hone% of the algebraio ekeleton, as exhilited generally in
this country f and show it in its livingt breathing continuity and beauty of FORM.
Give a conception of the magnificent power of analysis,

99. (a) No. (b) Much desired in text-books, at least many of them.

100. (a) Not verbatim.

101. (a) No.

102. (a) No. (b) More attention to principles and less to problems.

103. (a) F or immature students, yes. {b) The method of teaching must, I
think, vary under different circumstances. The principle idea should be to pre-
vent the student thinking it difficult,

104. (a) No. (6) Knowledge, generally.

105. (a) No.

106. (() I favor thorough mastering of the reasoning used in deducing formuls,
also memorizing for ready use.

107. (a) Not word for word. (J) Digesting subject as a whole, especially on
review.

108. (a) No.

109. (a) No, with few exceptions. (6) A more thorough treatment of a
smaller number of subjects ; use of determinants, less fractions,

110. (a) No. (5) More stress should be laid on factoring, less on the the-
ory — more of the solid work with a broader view of its application.

111. (a) We do not. (6) More mental exercise and less blackboard work.

112. (a) I do not. (5) More attention should be paid to generalization than
it usually receives.

113. (a) No.

114. (a) No.

115. (a) That depends. (J) We need no reforms (Professor Peck), (a) Yes.
(ft) In the preparatory schools more work should be done independently of the
text-bouok, and a more elaborate elucidation of fundamental principles should
be given (Tutor Fiske).

116. (a) No. (5) In general, greater attention to accuracy ; in particular,
more attention to theory of exponents and radicals.

117. (a) Yes ; either those of the text-book or carefully prepared ones. More
" why " needed.

118. (a) Yes ; so far as to secure accuracy of expression and as a mode of fix-
ing methods clearly in the mind.

119. (a) Not to a large extent. (&) I think the student should be taught to
rely upon his logical powers, rather than his memory.

120. (a) No. {b) Methods that develop a clear understanding of each proc-
ess and ability to explain clearly, in place of a knowledge of rules without
nnderstanding.

121. (a) No. (5) More of the inductive method ; and the abolition of much
that may be interesting theoretically, but of little practical use.

122. (a) In very few cases.

123. (a) Yes. {Jb) An improvement in the speed with which the mechanical
processes are done.

124. (a) Yes ; saves time, (b) Get teachers who know more.

125. (a) Hardly, {b) 1. Opposite numbers ought to be given a full treatment,
including all the rules for signs, with illustrations and a considerable number
of examples and problems in their use, before the literal notation is begun, 2. In
the former the reason for the use of 4- &i^d — to mark the series ought to be

834 TEACHING AND HISTOBY OF MATHEMATICS.

(a) Do you favor fMmoriging rules in algebra t (b) What refortM are needed in UaeMug

the eafM t — Continued,
bronght oat Bimply and plainly, and justified. 3. The fact that in element-
ary algebra the letters iJways stand for numbere ought to be reiterated to
avoid obscurity of ideas in the learner's mind. 4. The treatment of the equa-
tion should be analogous to that employed in geometry. The method of writing
references to axioms, etc., at the right of Alq page, familiar to those who have
used Wentworth's Geometry, can be employed in algebra to eyen greater ad-
vantage.

126. (a) No.

127. (a) Some of them, (b) Examinations of students firom many places eon-
vince me that algebra should be taught more thoroughly than it is in most of
the schools.

128. (a) No; (b) Explanation to be really so, and work done at time of ex-
planation as far as possible. Many comparatively simple problems, not poz-
zies. New work in hour. Students to be ranked according to actual work
done in problems. Much board work by entire class.

129. (a) Yes and no. (5) Better elementary text-books, better preparation
on part of teacher ; more rigid demonstrations of the principles of a science.

130. (a) Not in general, (b) More attention should be given tq the axioms
and the fundamental laws and their connection with the subject, and more at-
tention to the theory of simultaneous equations.

131. (a) Yes, most important, but not necessarily in the words of text, (b)
In the larger colleges algebra is mostly taught by tutors, who hold temporary
appointments, and do not expect to make teaching their life work. Algebra
as weU as calculus should be taught by a permanent professor.

133. (a) Yes, but not in rigorous form, (b) Greater facility in their nse with
a more intelligent understanding of them.

134. (a) Yes, but they must also be thoroughly understood.

135. (a) Yes, for average student, (b) Examples given should be made more
modern and practical. The theory of functions should be incorporated, beginning
with simple elements. This will maJce the whole subject of series, eto., easy for Ike
student

136. (a) Very few.

137. (a) Yes. (b) Let us have live, .enthusiastic, and competent teachers—
such as will teach the subject rather than the text-boek.

138. (a) The more important, yes. (b) For the preparatory work, greater
thoroughness is much needed.

139. (a) No. (5) Explain by common sense and not by rule.

140. (a) It does no harm, {b) The current text-books are too arithmeticaL

141. (a) No.

142. (a) No.

143. (a) No. (b) Less mechanical work; more thought* Students shoold
be taught to think I think! I think ! ! !

144. (a) No. (b) With such text-books as Hall and Knight's Elementary
Algebra and same as C, A. Smith's or Todhunter's Higher Algebra ; no rtform
needed,

145. (a) No. (b) Greater attention to mental and inventional algebra, and to
numerical and geometrical applications and illustrations,

146. (a) No.

147. (a) I recommend the memorizing of the rules, unless the pupils furnish
a good working rule of their own (a rare case).

148. (a) No. (b) A more thorough drill in factoring and in firaetiona, and in
putting into words the ideas conveyed by its symbols, equationS| and opeiap
Uons. Also greater precision of expression.

MATHEMATICAL ITEACHING AT THE PRESENT TIME. 335

(a) Do yaufawtr memoriHng rule$ in atgelnra f (h) What reforms are needed in teaching

the fame f— Continued.

149. (a) I do not.

150. (a) No.

151. (a) I do not. (b) The pupils learn to do by doing. Hence, instead of
having pupils waste their time on abstract demonstrations, let them solve nu-
merous problems of every variety. It is only practice that makes perfect.

152. (a) No.

153. (a) No. (b) The founding of all algebra upon the laws of operation.

154. (a) No.

155. ia) Tes.

156. (a) Not at all. (b) The pupil should be taught to think rather than to
worh hy ruU. More thoroughnesa needed.

157. (a) 1 do not. (5) I think that the student should be required to con-
struct his own rules as far as possible.

159, (a) But few. (b) Principles are apt to be lost sight of in the strict and

close adherence to rules.
i 160. (a) No. (b) More thorough drill is needed, especially in the elementary

principles.
. 161. (a) Tes. (b) Broader views of algebraic operations ', more generalizing

and greater exactness of language.

162. (a) Yes.

163. (a) Yes. (fi) The rules should be demonstrated oftener than they are.

164. (a) No. {b) To develop the subject by origiual investigation.

166. (a) Yes.

167. (a) To but very slight extent, (b) Less formality and more "realism f
introduction of principles often held back until higher branches are reached,
e, g., factors of direction, differentials, etc.

168. (a) Only very few. (b) More attention to problems Involviog principles
and less to puzzles.

(a) To what extent are modek used in geometry t (b) To what extent and vHth what success
original exeroises t (o) Do you favor memorizing verbatim the theorems (not the demon-
eirations) in geometry t What reforms we needed in teaching the same t

1. (a) Class-room very poorly supplied, but we use the few we do possess as
much as possible, (b) Such exeroises are given every day and are found to be
very beneficial, (o) No.

2. (a) Moderately, to explain effects of perspective on the black*board. (b)
To a very moderate extent with the great minority of students, to a great extent
with the best.

3. (a) When models have been used it has &oilitated the work, (o) Yes.

4. (a) They are used to a limited extent. I question very much the advan-
tages of using models, except with beginners, or rather with those who are
studying works introductory to regular demonstration, (b) To a limited ex-
tent, (c) Yes; more original work ; more attention to logical processes, clear-
ness and accuracy of statement. I change the figures, i. e,, their relative posi-
tion, so that the demonstration shall be reasoning and not memory.

5. (a) Use them to a great extent, (b) Original elercises with fine success,
(o) No; some.

6. (a) The text-book quite closely followed, (b) Some daily and with good
snecess. (e) Yes.

7. (a) Not used, (b) Used to some extent, (o) No.

8. (a) To a large extent, (b) One-fifth of work in geometry is in original ex-
eroise; the success is good, (o) Yes.

836 TEACHING AND HISTOBT OF MATHEMATICS.

(a) To what extent are modeU used in geomeiry t (h) To what eaetent and wUh what nuieeu
original exercises t (e) Do you favor memorizing verbatim the theorems (not the demon-
atraiions) in geometry f What reforms are needed in teaching the same f — Continaed.

9. (a) To a considerable extent in the lower grades, (b) Very extenslTely
and with very satisfactory results, (o) As a general thing, I am of the opinioa
that too little time is given to the subject to secure the best results,

10. (a) Largely, both in class-room and ont-doors. (b) To no great extent
and with no marked success, as yet.

11. (a) Very little, (b) The exercises in Welsh's Geometry are nsed. S<me
of them seem hard to the students, but on the whole they do fairly with them,
(o) Almost verbatim.

12. (a) Only moderately, (b) They take one-third of the whole time and
make the life of the work, (c) The rigorous requirement of original well-graded
work froiii the very first.

13. (a) In descriptive geometry only, (o) It ought to be taught more from
a comparative point of view.

14. (a) None, (b) To a very limited extent and not with marked Buccess.
(o) No.

15. (a) Models are largely used in geometry in three dimensions, (h) To a
small extent and without marked success, (o) No; better drawing in the
text-books, especially in geometry in three dimensions ; more attention to the
drawing of the students ; less geometry, altogether ; I think the importance of
Euclidian geometry as mental discipline is greatly overestimated.

16. (b) To a very large extent and with excellent success, (o) It is left op-
tional with the student.

17. (a) Very little, (b) Subordinate to a marked degree. I am trying to
change this state of affairs, (o; Yes; more original work, also more compara-
tive, not purely descriptive work.

18. (a) Always used in teaching solid geometry and in teaching conio sec-
tions.

19. (a) Forms are used in solid geometry, etc. , freely, to aid the mental con-
ception of the perfect ideals of mathematics (Professor Gordon ). Bat little used
in plane geometry (Professor Draper), (b) Very simple exercisee, arithmetical
application of geometrical principles, constructions, and problems are fireely
used. Very simple '^ catch '' theorems or '* corollaries'' involving some absurd-
ity are occasionally introduced to be proved ! Students, of ordinary intelligence
generally succeed with exercises graduated to their state of advancement (Pro-
fessor Gordon). About one-fifth of the time is given them. Those who do well
in the text and stand questioning upon it are fairly successful with originals.
(Professor Draper), (c) Tes ; except In a few cases where I think the theorem
itself can be improved. Would begin it in childhood of pupil ; would spend
more time on its elementary principles (Professor Draper).

20. (a) Models are used in solid geometry and spherical trigonometry, (b)
To a limited extent and with good results, (c) Yes. That the sense of the
theorem be known when the memorizing is complete.

21. (a) Very little in plain geometry ; more, but not very extensively, in solid
geometry. (&) Used largely and with unqualified success, (o) Yes. Using
figures just as given in book, using only propositions already proven, and many
other things of a similar kind need reformation.

82. (a) To a limited extent, (b) To limited extent and with good snooess.
(c) Yes, at first ; it promotes accuracy of expression. Greater latitude may be
allowed with advanced students.

33. (a) Models have just been obtained.

24. (a) None, except to illustrate solid bodies. (5) ExtenslTely and ano-
oeBsfully.

MATHEMATICAL TEACHING AT THE PKESENT TIME. 337

(a) To what extent are models used in geometry 9 (b) To what extent and with what euooen
original exercises t (e) Do you favor memorising verbatim the theorems (not the demon*
etrations) in geometry t What rrforms are needed in teaching the same f— Continued.
" 25. (a) From want of fancU the supply is limited to snoh rnde models as
teacher and student can make. (() Original exercises in connection with
every study are used freely and with good results, (c) More original exer-
cises.

26. (o) No. Variation of letters, etc., to represent angles.

27. (a) I have relied upon models to a great extent. I require all studying
solid geometry to construct the five regular polyhedrons with pasteboard, giv-
ing reason why only five can be formed, (h) All classes work at original
propositions. The results have shown the practice to be very important.

28. (a) No models used in plane geometry. The sphere, the cone, and a few
others are used in solid geometry, (b) One-half of all the time for geometry
is devoted to original exercises. Success very satisfactory, (o) Some. More
original work. Demonstrations varying from those of the author should be
encouraged.

29. (a) Not to a great extent, (b) They are freely used with the best results,
(c) Tee, substantially; the gravest error is the memorizing of demonstrations —
an evil that seems unavoidable, if text-books are employed. The ideal method
is oral instruction, in which the mental movements of the pupils are under the
eye of the instructor. It is a pity that a subject that has such possibilities for
pupils should be so taught as to become a mere ** memory grind.''

30. (a) To no great extent, but figures extensively, (b) No great extent.
Human nature is not original. Originality is the exception. The 4iverage
student who spends his time on original exercises will fail of that discipline in
method which he needs, (o) Yes, the student will become benefited in learning
of a formula of words expressing truths. Stick to the Euclidian method ; there
is no *' royal road" to geometry.

31. (a) Such as we are able to improvise.

32. (a) They are not used, (b) We intersperse them throughout the entire
course of geometry, (o) While they should be memorized, the student should
learn to state them also in good language of his own.

33. (a) Practically to no extent, (b) Great success when used, (o) More,
much more original work and simplification of demonstrations.

34. (a) Very little, (b) One-third to one-half of work assigned.^ Great suc-
cess with the better students, (o) No.

35. (a) Used in the study of the geometry of space, of surCaces of the second
order. (5) Geometry is crowded into short time for necessary reasons ; some
original work done, less than would be useful, (o) Tes. Greater familiarity
with definitions and axioms. The constructive method of carrying on demon-
strations (i. 6., omit drawing fig^ure in full, beforehand).

36. (a) We approve of their extensive use. (b) Throughout the course ; sue.
cess indifferent, (o) 7es. More attention to the form of demonstration and ac-
curacy of statement.

37. (a) Not used to any extent. (5) Original exercies are extensively used,
and a greater interest in the study, (o) More attention paid to original exer-
cises well graded.

38. (a) Models are used, (b) All the examples in Wentworth's Geometry are
solved, together with selections outside. We are more successful each
succeeding year, (o) No. In general, less text-book routine and more prob-
lems, not so difficult, but well graded.

39. (a) Very little, (b) Much and with great success, (o) Tes. More demon-
strations should be written out, both in the elements and among original exer-
eises.

881— No. 3 22

838 TEACHING AND HISTOBY OP MATHEMATICS.

(a) To what extent are models used in geometry t (b) To what extent and with what Bueoeee
original exercises t (o) Do you favor memorizing verbatim the theorems {not the demon^
strations) in geometry t JVhat reforms are needed in teaching the same? — Continued.

40. (a) Not very much. I prefer that students should learn as soon as possi-
ble to form mental pictures of the figures and reproduce them ou the
board. (&) Frequent original problems are given and are very yalaable. (c)
Original problems and propositions should be given in connection with the les-
sons from the beginning.

41. (a) They are used in teaching the higher surfaces, especially the warped
surfaces in descriptive geometry. (&) A few original exercises are given with
the text-book work, and with marked success, (o) No. More reliance upon
the imagination for the figures and less upon the blackboard.

42. (a) Limited, (h) Very largely and with great success, (c) Not neces-
sarily.

43. (a) A limited extent, {b) Original demonstrations are required on one
day of each week of second term, (c) Yes.

44. ib) From first to last with good success, (o) I do not. More origixial
work.

45. (a) They are all represented by the blackboard. (&) As much as possi-
ble ; usually daily, (c) I do. Less demonstrations in the book ; more propo-
sitions for the student.

46. (a) Geometry is with us a preparatory study, (b) Constantly, with suc-
cess, (c) No. The geometrical teaching in our public schools seema to be ex-
cellent.

sA7. (a) To a large extent, (b) With good success when in printed form;
otherwise not so. (c) Unless the student can hold himself to an equally clear
form. A union of the old rigor with modern improvements.

48. (a) The usual blocks, etc. (b) About one exorcise out of every ten with
fair success, (c) If the text is given iu definite form and is well worded, yes.

49. (a) None, excepting the elementary models, (o) In part.

50. (a) As far as needed iu all cases, (b) Not much success as yet, but hope-
ful, (c) I do. More use of exercises and original work.

51. (a) Wo have hbout one hundred dollars' worth of models for pure mathe-
matics, (c) No.

52. (a) For solid. (&) From the first and with gratifying success, (c) Yes.
Practical application of principles in concrete problems.

53. (a) Only ordinary models, or those commonly used, (b) Occasionally with
good success, (c) Yes. Less speed and more thoroughness.

54. (a) To a limited extent, (b) Made prominent and with good effect, (e)
- To discourage mere eifort to demonstrate by memory, rather than by intuition

and train of reasouiog.

55. (a) As far as possible, especially in solid geometry. (6) A great many
original exorcises. They are the best measures of the student's ability, (c) No.
Less memorizing of demonstrations and more original work.

56. (b) The representative theorems are all demonstrated by original work
as far as possible, (c) Yes. Any plan that will prevent students irom memoriz-
ing the demonstrations.

57. (o) Not at all. {b) Have not tried this plan yet. (c) No. Not prepared

to suggest.

58. (a) Not much, and mostly in spherical geometry, (b) Have had some
original work with prolit. (r) Yes. More familiarity with relations of parts
to each otlicr, and Iv^a <lep<iKU«nce on tho wording of the demonstrations as
given in the book.

59. (a) Wo usually use diagrams, (h) To considerable extent and with emi-
nent success, (c) Yes. All hail! to tho man who will devise means to prevent
tihe pupil from committing to vianory the demonstrations^

MATHEMATICAL TEACHING AT THE PRESENT TIME. 339

(a) To what extent are models used in geornetry t (5) To tehat extent cmd wUh what euooeee
original exercises? (o) Do you favor memorizing verbatim the thef^reme {not the demon-
strations) in geometry t What reforms are needed in teaching the same f^Continaed.

60. (a) To a ihoderate extent. (i>) Largely, and with great success, (c) Tes.
Gaarding against use of memory too mnoh by students in demonstrations of
propositions. % *

61. (a) In lecturing only, (c) Yes. Thorough undjBistanding of relaiiively
important principles. ^

62. (a) To a very limited extent. (6) We ubc a great many original ertercises
with much success, (o) Tes. Too many allow students to memorize tha demon-
strations and thus miss the great advantage in geometry, a development of the
reasoning faculties.,

63. (a) So far as to illustrate triangles, parallelc^ams, circles, pyramids,
prisms, cones, cylinders, and spheres, (d) Limited, (o) Yes. Dem.onstratioiis
ought not to be memorized. Pupils ought to be shown that the t^rath of eadi
proposition is established by a course of logical reasoning.

64. (a) For illustrating solid geometry, mensuration, conic seotijons. (o) Yes.

65. (a) Not at all. This is due to the school not being provided with models,
and not to the teacher's prefeience. (b) They aro used only orjoasionally, but
with considerable success when used, (o) Yes.

66. (a) Whenever necessary, (o) Yes.

68. (a) Not at all in geometry, to slight extent in descriptl sre geometry, (b)
Cadets are frequently required to submit exercises, (c) No.

69. (a) Largely. (6) Few, but satisfactory, (o) No. Latitude-^o long as
object is clearly stated, and demonstration is concise and co mplete.

70. (a) Very limited, (b) To a limited extent, but with good success, (o)
Yes. More extended use of models.

71. (a) The spherical blackboard and models are used considerably. (&) They
are being introduced with good success, (c) No ; except for those students who
must in order to understand them. Students should learn to depend less on
the printed denonstrations.

72. (a) Definitions are taught by means of models, (fi) The extent varies
with different classes. The success is good with about one-third of the class,
(o) No.

73. (a) Little. (5) To some trifling extent, always with profit, (o) Yes.

74. (a) Where models seem to make principles clearer, or their application
practical, they are used in teaching solid geometry and spherical trigonometry,
(o) Yes.

75. (b) Nearly one-half the time is spent upon original work and with marked
success, (o) Not absolutely.

76. (a) In metrical geometry models are used altogether for illustration.
(b) Our time being limited, we spend little on original exercises, but with
fine success, (o) Yes. The student should be required to carefully write each
demonstration upon the board. r^

77. (a) In solid geometry all the principal figures are thus illustrated, (b)
To a considerable extent in plane geometry and with excellent success, (o)
Yes. More original work and less memorizing of demonstrations.

78. (a) Very little, (b) To a very considerable extent and with marked suc-
cess, (o.) No. More attention should be given to original work.

79. (a) To no great extent, (b) Original exercises are given as optional work
and a few students are very successful in them, (o) Indifferent, provided they
are eiven clearly and conoisely.

bS, (a) I use them very frequently. (Jb) Original exercises form a part of
nearly every lesson. With a few exceptions the results a^ excellent; or at
least satisfactory, (o) Yes. More original work.

340 TEACHING AND HISTOBY OF MATHEMATICS.

(a) To what extent are models need in geometry f (5) To what extent and with what mceem
original exercieee T (c) Do you favor memorizing verhatim the theorems {not the demo*-
straUons) in geometry t What reforms are needed in teachtng the same f — Continned.

82. (a) Somewhat in solid geomet^. (5) To a very large extent in daily
work and with very satisfactory lesnlts. (e) I holi students responsible for a
knowledge of the theorem, but not verbatim.

83. (h) Original exercises are need and with good success, (o) Tes.

84. (a) Kot at aU. (5) To a considerable extent and with much success. («)
Ko. More attention to logical form and precision of statement.

85. (a) In course on '' form." (d) As far as the time allotted wiU allow, and
with great success, (c) Yes. A greater use of objects. A leaving of parts of
the demonstrations to be filled in, thus training for original work.

86. (c) Ko. The adoption of the henristio method.

87. (a) Very little. (5) Original exercises comprise a very large part of the
workv say one-half, in geometry, (o) Yes ; those to be frequently referred to in
subsequent work ; others, no.

88. (h) Much used in geometry, and very successfully, (c) Yes.

89. (.9) None, (h) Numerous practical problems wiUi, I think, good snceesi.
(o) No.

90. (a) To a small extent, (l) A very large extent and excellent results, (e)
No. Cultivation of more originality by means of graded exercises.

91. (a) They are not much used, (h) They are used whenever there is an
opening ; success is good, (o) No.

92. (a) They are used to some extent in solid and general geometry, (b) To
considerable extent, with very satisfactory results, (o) As a rule, yes ; for ths
reason that they are* usually stated much more concisely than the student would
state them. More original demonstrations.

93. (a) Full sets of Schroder's (Darmstadt) models. In solid geometry stu-
dents make models from pasteboard, (b) Such exercises in connection with
nearly every lesson, and with gratifying success, (o) Yes ; number of propo-
sition and book should not be memorized. More problems and practical appli-
cations ; more theorems for original demonstration by pupils.

94. (a) Very sparingly ; find them hurtful rather than helpful. ''Normal
school " methods are a failure in geometry. Have tried both and seen both
tried. (Jb) In connection with nearly every theorem and every lesson. Sneoesi
good, (o) Yes— no, depends on the student and the sort of drill he needs. A
more rigorous insistence on founding everthing on the axioms.

95. {b) Many new exercises with great success, (o) Introduce more exeroises
and require variation in figures.

96. (a) None, (b) Very considerable extent and good success, (c) Not vei^
batim, but clearly and fully in substance.

97. (a) Hitherto but little; henceforth very great (if the appropriation asked
of the State be granted), (fr) If unassisted, or only slightly assist>ed, demonstra-
tion and solution be meant, great and good, (c) No ! Supture with the tradi-
tional Euclidian methods, alignment with the march of modem thought,

98. (a) To a small extent in descriptive geometry (warped surfaces, etc).
(b) To a great extent and as much as possible, and with marked success, (c)
No, only to a slight extent for beginners. More originaJl exercisesi and mere
modem geometry of position.

99. (a) Average. (&) Tested daily, (o) Yes.

100. (a) We use models of the usual geometrical forms for illustration, (b)
Frequent exercises in geometry ; success only moderate, (o) No.

101. (a) In solid geometry, (b) One-third. Good results with fair snoeess.
(e) Yes. Larger per cent, of original work required.

102. (a) None, (b) Extensively used, and results very gratifying, (e) Yes.
More original exercises and a more rigid reference to first principles.

MATHEMATICAL TEACHING AT THE PRESENT TIME. 941

(a) To what extent are modeU used in geometry? (h) To what extent and with wluiteucceea
original exerciaeet (c) Do you favor memorizing verbatim the theorems (not the demons
sfrations) in geometry f VThat reforms are needed in teaching the same f — Continued.

103. (a) None. (&) In geometry, with fair success ; is practical surveying,
leveling, etc. (c) Yes. As a rule, not allowing any lettering on board, etc.

104. (a) None. (&) None, (c) No.

106. (a) To a very limited extent. (&) To the extent which the time will
permit, and with increasing degree of success, (c) I do not. I think reform
needed in regard to grasping the trath, and giving it in good language, of the
pupil's selection.

107. (a) Very little. (&) Slight extent, (o) No. More original work.

108. (a) Small, (o) Yes,

109. (a) Very little. (J) Considerable, with fair success, (o) No. Incorpo-
ration of some treatment of modern geometry.

110. (a) To a considerable extent in teaching solid geometry and spherical
geometry. \Jb) Original exercises constitute half of the work, and with satis-
factory success, (o) Yes. Require more solutions of practical problems , this
tests the ability of the student and teaches him to walk alone.

111. (a) None, (fr) We use few. (o) It is not material with us.

112. (a) Only to a limited extent in illustrating some of the properties of
planes and solids. (6) To a great extent, and with satisfoctory results, (c)
I do. It is necessary that the student should know what he is required to
demonstrate. Theory and practice should go hand in hand.

113. (a) They are used for every demonstration in solid geometry, (o) No.

114. (a) Very little, {h) It has not seemed profitable to spend much time on
original work in geometry, which is a study of the Freshman year, (c) No.

115. (() A great number of original exercises are given with complete suc-
cess, (o) I do not favor memorizing anything ^except such principles as are
needed in after work. For purposes of illustration we have a full set of models
of solid and descriptive geometry. (&) Original exercises are given as regular
and extra work to all classes, (c) Yes. The elementary principles of logio
should be explained in connection with elementary geometry. (Tutor Fiska)

116. (a) None. (5) Increasing number from year to year. Successful with
first third of the class.

117. (a) Students make models in solid geometry. (&) With excellent results
and to a large extent in geometry and trigonometry, (c) Yes. Throw students
more on their own resources.

118. (a) Merely in explaining and illustrating. (5) They are required more
or less throughout the course, especially in geometry, (c) Yes.

119. (a) We use the globe and the usual geometrical solids. (&) Original
exercises are frequently given ; success is very fair, (c) Yes.

120. (a) None are used in plane geometry. We have twenty-six fine models
of warped and single-curved surfaces for use in descriptive geometry, {h) Orig-
inal exercises are given out at each recitation, with great success<<u9 regards
the development of mathematical knowledge, (c) No. With each lesson the
student should have several original exercises involving the principles, to solve
or demonstrate.

121. (a) Always in the teaching of geometry of space, when I find it helpful.
(() Continually given as vofttntar^ text- work, excusing the student from formal
examination in proportion to her success in it. (c) No. The abolition of tft-
direct proof, and the use of symbolic notation, with special attention to form,

122. (a) In solid geometry, (b) To a limited extent, (c) Yes. More fre-
quent direct application to problems in which dimensions are .to be found.

123. (a) To a moderate extent. (&) To a considerable extent and with as
good success as can be expected, (c) Yes. More firequeat tests on original
theoiems and problems.

342 TEACHING AND HISTORY OF MATHEMATICS.

(a) To what extent are nwdela used in geometry f (b) To what extent and with what sueeen
original exercises t (o) Do you favor memorizing verbatim the theorenM(not the demon*
ttrations) in geometry f What reforms are needed in teaching the same f — Continued.

124. (a) Tho stadents make models of the regular polyliedrous. (() Large
numbers giren in each class ; this is one of our chief methods of drill and
training ; t^ is the only way in which fundamental principles can he so thoroughly
ingrained in the mental make-up of a student that he is no longer conscious of an
effort of memory in his hnowledge. (o) Yes, to encourage exact expressiona and
as a tribute to order which is the soul of geometrical reasoning.

125. (a) Very little at present ; hope to use them extensively. ' (b) To as
large an extent as possible. With good success from the majority of students,
(c) I do not. (1) Better trained teachers ; (2) more thinking and less memo-
rizing ; (3) use of thoroughly good text-books, like Byerly's Chanvenet ; (4)
emphasis of logic side ; (5) generalization and summing up of truths proved, etc.

126. (a) No geometry taught excepting descriptive geometry. Students con-
struct their own models. (&) Original exercises in almbst daily use. (c) Tes.
Demonstration of theorems without letters or figures.

127. (a) We have a set of "geometrical solids," which wo use on occasion.
(b) Such exercises are often required, and they are valuable — ^increasing the
interest and testing the student's knowledge.

128. (a) Not largely, but so far as the students seem to need them. (5) Very
largely and with good success, (c) A clean-cut, accurate statement, whether
verbatim or not. Teacher to make sure of actual mastery of prinoiples^no
memory work ; much use of original exercises.

129. (a) So far as is necessary for the pupil to get a clear conception of the
geometrical concept. (&) In the preparatory course for admission to Fresh-
man class, limited. In Freshman used to large extent, (c) Yes.

130. (a) Models of solids are used in solid geometry. (&) A good deal of use
is made of them. Success good with best students, (o) No. More time should
be given to leading the student to discover theorems for himself.

131. (a) Not very much, (b) As much as possible and with gratifying results,
(o) No. More original exercises.

132. (a) Geometry is finished before entrance, except descriptive geometry, in
which we use no models.

133. (a) To a limited extent with sphere and regular polyhedrons, (b) The
original work in Wentworth's Geometry, with fair success, (o) Yes.

134. (a) In solid geometry, {b) Exercises and constructions on eaeh book,
and with good success, (c) No. Something like Wentworth's system of dem-
onstrating propositions.

135. (a) None except sphere and cone, (b) At least one original exercise is
given as a part of each lesson. Great success, (o) No. Stadents should be
taught to master new processes or methods of proof rather than individual
theorems ; so come to look on theorems and proofs as illustrations of processes,
or methods of investigation.

136. (a) Not at aU. (b) Very little, in obligatory mathematics; and in reg-
ular course there is liardly any pure geometry ; but when there is any, such
exercises are helpful, (o) Only in elementary work, and even in that the at-
tendant dangers are great.

137. (a) To a slight extent only, (d) Limited extent, bat with good success.
(o) Yes.

138. (a) To a limited extent, (b) They are much used and with good rssults.
(o) No.

139. (a) Very slight, (b) About fifty original exercises are given and ait
well done, (o) Yes.

140. (a) Slightly, {b) Considerably, with success, (c) No. .

MATHEMATICAL TEACHING AT THE PRESENT TIME. 343

(a) To tvliat extent are models used in geomelrg f (6) To what extentand with what success
original exercises f (c) Do you favor memorizing verhatim the theorems (not the demon-
strations) in geometry f What reforms are needed in teaching the same? — Continned.

141. (a) Small. (&) Considerable extent and "witli commendable success,
(c) Yes. t

142. (&) With good success, as a rule, (c) Yes.

143. (a) Our models are sucli as \re make ourselves. We illustrate, so far as
possible, in solid and descriptive geometry. (6) All that I can Lave time for
and can get the students to solve ; great success with the few, little with the
mass, (c) No ; but I require a clear statement in somebody's words. More origi-
nal work should be given ; the student should be taught to depend upon him-
self more, and less upon book or teacher, to thinlc, to originate, not memorize,
not absorb.

144. (a) Not at all, except for young pupils. (6) Daily use and with good
success, (c) Yes. Young pupils should be drilled in practical exercises, with
nse of ip.strUtnents.

145. (a) Have been used but little. (&) Constantly and successfully, (c) No.
An earlier start, with main aiteniion, at first, to training oiset^vation. Greater
freedom from formalism,

146. (a) Whenever possible. (6) Limited, (c) Yes.

147. (a) I make considerable nse of models, especially in solid geometry, (b)
Original exercises are requited at a few places (two or three lessons). Good
success, (c) Yes ; yet I do not insist on keeping every word, provided the sense
is kept.

148. (a) We do not use models. (&) We lay great stress on original exercises.
When properly selected they are of the utmost service, (c) No. A clearer com-
prehension of the definitions ; a more frequent enumeration of facts already
proved ; a more explicit enumeration of facts to be established in demonstrating
any particular theorem.

149. (a) In a very limited degree, (h) For the past few years I have used
them freely with gratifying success, (c) No.

150. (a) To a limited extent, (c) No.

151. (6) The study of geometry would fall far short of its object if original
work were not required. I devote one recitation hour each week to it, and I
am pleased with the results. I judge of the mental development by the orig-
inal work done by pupils, (c) Yes. The facts of geometry must come first,
concrete object lessons ; can't reason about that concerning which we know
little or nothing.

152. (a) But little, except in descriptive geometry, (h) Much time is given
to solution of problems, both from text-book and from other sources, (c) Yes.

153. (a) Largely, especially in solid geometry. (l>) Continually and copiously
and with great success, (c) No. The rejection of the words ^^ direction^* and

. * * distance " from elemen tary geometry,

154. (a) Largely, especially in conic sections and descriptive geometry. (&)
Weekly exercises and with very satisfactory results, (c) No.

155. (a) To a limited extent. I expect to use models to a greater extent in
the future. (6) Original exercises are greatly used. I value them very highly,
and I am much pleased with the results I have obtained by using them in all
my classes, (c) I do not.

150. (a) None used. (6) Special prominence is given to the use of original ex-
ercises, with encouraging success, (c) Yes. Originality should be encouraged.

157. (a) They are used to a very limited extent, simply because the college
is not supplied with them, (b) Frequent exorcises are given with quite good
success, (c) I do not ; I tliluk that a student should be required always to ex-
press his thoughts in his own language, if for no other reason than to acquire
facility in expression.

|344 TEACHING AND HISTQBT OF HATHEMATIGS.

(a) To what extent are modela need in geometry t (h) To what extent and with what eueeeet
original exercises f (o) Do you favor memorizing verbatim the theorems (not the demon^
strations) in geometry? What reforms are needed in teaching the same f—Conianued.

158. (a) To only a small extent, (b) Largely given, and; I think, with great
snocess in promoting intellectual plack and thoroughness oi^ attainment, (e)
Yes ; and also to learn to state them in one's own words. Subject too larg^ for
space. I will say, however, that the schools should give more exercises for
solution, and train the boys from the beginning in original solution.

159. (a) Models are used in conic sections and for surfaces of revolution in
analytic geometry of three dimensions. (&) Special attention given to original
exercises. A taste for such work is easily developed in every lover of mathe-
matics, (o) No; would prefer that the student thoroughly understand the
truth to be demonstrated and express same in his own language.

160. (a) Very extensively, especially in solid geometry. (&) Largely, and
with decided success, (c) Yes. I would have it made more practical.

161. (a) None. (&) Original exercises are frequent and attended with encour-
aging success, (o) I do. Demonstrations should be less verbose, and expressed
to a greater extent by algebraic symbols.

162. (a) Considerably, (b) With almost every lesson, and with much snocess.
(o) To some extent.

163. (a) Used to illustrate definitions, (b) We devote some time to them now ;
shall devote more ; excellent success, (c) I do. Greater care that the pupil
understand the reasons for every statement.

164. (a) Models used but little in first presenting the subject. (&) Original
exercises given throughout the course, with good success, (o) No. Geometry
should be taught as algebra and arithmetic by original work.

165. (a) Limited, (b) Largely, with great success, (o) Yes.

166. (a) We have none, (b) On each recitation, when there is time, (o) Yes.
Less text-book and more originality.

167. (a) Very little. The attempt is made to lead the student to form his
magnitudes in space, and without even, a drawing, if possible, (b) To a very
considerable extent, especially in test-work ; and with excellent success in
about one-half the oases, (o) No ; but I would insist on concise and aooorate
statement. (1) Less bondage to text-book. (2) Encouraging original demon-
strations. (3) Clearer distinction between leading steps of proof and details.

168. (a) Only slightly, because we are not able to afford them, (b) They are
used as much as time will permit, and with good success, (o) No. Less time
given to theorem-demonstration and more to original exercises, with the proper
change in text books.

Is elementary geometry preceded or aeeampanied by drawing t

<< Preceded'' : 21, 22, 29, 49, 51, 53, 55, 58, 62, 73, 76, 82, 90, 92, 98, 102, 103, 110,
112, 114, 139, 147.

'< Accompanied : '' 8, 10, 12, 13, 14, 20, 24, 25, 27, 28, 30, 33, 39, 41, 42, 43, 44, 48, 54,
56, 57, 59, 60, 61, 66, 70, 71, 84, 94, 96, 97, 99, 101, 105, 106, 122, 127, 134, 135, 137, 138,
140, 142, 150, 153, 154, 457, 159, 164, 165, 166.

" Both preceded and aocompanied " : 4, 9, 11, 69, 85, 87, 88, 107, 111, 119, 124, 146,
160, 161.

''Neither preceded nor accompanied": 1 (except engineering students), 15, 16,
17, 19, 31, 32, 34, 35, 37, 38, 45, 63, 77 (except in industrial course), 81, 91, 93, 100, 108,
109, 115, 116, 117, 120, 125, 128, 129 (except in scientific course), 130,131, 149, 15Q, 155,
156, 167, 168.

''Tee; either preceded or aooompaaied'' : 23, 26, 123, 145, 163.

MATHEMATICAL TEACHING AT THE PRESENT TIME. 345

State HfM of your special preparation for ieaohing matheinaiio8f number of hours you teach

per week, and tchat other subjects you teach,

I. Teach twenty hours per week ; teaoh no other subjects.

3. About ten years before entering the Uniyersity of Alabama, where I spent
five years ; physios and astronomy.

4. I teach twenty-five hours a week, and daring five months in the year
give about six hours a week to special work in surveying ; I teach no other
subjects.

5. Six and a half hours per day in the entire sehool, with one-half hour
recitation for classes at different times.

6. From 1880 to 1886; nine hours; geology, astronomy, elocution.

8. Two years; thirty-five hours per week; book-keeping, six hours per
week.

9. I teach only mathematics, and give five lessons of one and one*fourth hours
each per week.

10. Four years; seventeen hours per week; no other subjects save military,
science and tactics.

II. Five years' study at the Cincinnati Observatory after graduating from
college. At present I teach thirty recitation hours (forty-five minutes each) a
week. Astronomy (popular).

12. It has been my specialty for eight years ; twenty hours per week ; none.

13. Pare mathematics is taught twenty hours per week.

14. Teach mathematics three and four hours per week ; my chief subject is
chemistry, while mathematics is a secondary subject here.

15. I have taught mathematics since graduating from college (1870); I
teach from eight to twelve hours per week. Astronomy is also in my charge.

16. I teaoh from twenty to thirty hours per week, and teaoh no other sub-
ject.

17. Five years; twenty hours per week ; none.

18. The classes of mathematics, except the first class, are taught by the
professors of the regular classical course. Each professor teaches only one
class of mathematics.

19. Ten to fifteen hours per week, mental science and chemistry, etc. (Profes-
sor Grordon). Preparation has nearly all been made since I began and while
teaching ; ten hours ; Latin (Professor Draper).

20. I teach mathematics twenty hours per week. I teaoh no other subjects.

21. Mathematics was my specialty in college two years; teach it fifteen
hours per week. I teach no other subjects.

22. About three hours per diem devoted to teaching. Teaoh, besides, chem-
istry and elocution.

23. I teach twenty hours per week, and also military science and tactics.
25. Three hours per day. I teach nothing but mathematics.

27. I teach only pare mathematics.

28. Thirty; teach no other subject.

29. Fifteen hours a week ; physios five hours a week for six months; astron-
omy five hours a week for four months ; psychology five hours a week for three
months.

30. Sixteen to twenty hours. I teach philosophy, astronomy, logic, moral
philosophy.

32. I give fifteen to twenty hoars of instruction, and teach several natural
sciences, besides some German.

33. One year and a half; twenty hours ; no other subjects.

34. Several graduate courses at Princeton College, and private study. I
teach from eleven to fifteen hours per week. Am also engaged in teadiing
astronomy. ' "

346 TEACHING AND HISTORY OF MATHEMATICS,

8tate time of your special preparation for teaching mathematioSf numher of hours you teaok

per weekf and what other eubjecie you teach — Continued.

35. Have taught and have been a stndent in the higher mathematics the past
twenty years; fifteen hours per week; not any. (Professor Shattuck.) Alge-
bra, five hours per week for two terms ; geometry, same ; natural philoBOphy,
physiology, botany, English, rhetorio, Latin, Greek. (Regent Peabody.)

37. One year ; twelve hours per week, besides teaching some in preparatory
department.

33. I teach mathematics and astronomy about twenty hours per week ; no
other subjects.

39. Fifteen hours ; no other subjects.

40. Teach about twenty hours per week. I teach regularly no other subject
and only the mathematics in the college department.

41. I spend from one to two hours per lesson on mathematical works, directly
• or remotely connected with the recitation. I teach twenty hours per week

and only mathematics.

43. Teach twenty hours per week; teach no other subject except astronomy.

44. Eighteen hours per week ; history and vocal music.

45. Everything pertaining to or suggested by the lesson is prepared. All
lessons are five hours per week with one exception ; geometry has four.

46. I teach mathematics and astronomy fifteen hours per week ; assistant
teaches mathematics fifteen hours per week also.

47. After graduating at Madison, spent four years in post-graduate study at
Tale ; fifteen to twenty hours ; astronomy and political economy.

50. We have four teachers of mathematics, who spend about fifteen honn
A week in their classes.

51. Am at it nearly all my time that can be secured from other work ; first
term, fifteen hours ; second term, twenty hours.

52. Thirty hours per week ; political economy.

53. All afternoon for preparation ; twenty-five a week ; none.

55. Have taught it for eighteen years, six years exclusively. Teach mathe-
matics twenty hours per week ; teach nothing else.

56. I took the two years' collegiate course required and took the post-grada*
ate course, spending three months on special work in mathematics ; fifteen
hours ; natural science.

57. Some class almost every hour in the day, averaging, perhaps, twenty-five
hours per week.

58. Scarcely any two terms the same.

59. About three-fifths of my time is given to mathematics and about two-
fifths to Latin and Qreek.

60. Three years; twenty hours; political science, astronomy, and German.

61. Fifteen hours per week ; mental philosophy and logic.

62. I teach mathematics fifteen hours per week and have some classes in Latin.
My preparation is done each night before the work of the following day.

63. Six to ten hours ; eighteen hours ; Latin, physiology, physical geography,
English literature.

64. One hour daily ; twelve hours per week ; astronomy, natural philosophy,
chemistry, geology, mineralogy, drawing.

65. Four years at college, one in private work, and two at Johns Hopkins
University. Twenty-four houxs per week. Nothing else,

66. Four hours per week.

70. Mathematioal course at Yale College together with four years sabsequent
study. Political economy and English literature.

71. Sinoe completing the course in this college, three years ago, I have spent
in private study a considerable portion of my time ; five to ten hoars per
week ; rhetorio and drawing.

MATHEMATICAL TEACHINa AT THE PRESENT TIME. 347

State time of your special preparation for teaching mathematioaf number of hours you teach

per weeJCf and what other subjects you (eac/^— Continaed.

72. Eif^ht, siX; or four hours per week, according to tern)) whether it be fall,
winter, of spring. Lecture on art.

73. The only special preparation I employ is the light reading of new text-
hooka which come to hand. For over twenty years have had no difficulties in
mathematical instruction. I also teach, as ocoaaion calls, metaphysics, morals,
political economy, history, literature, etc.

74. Teach on an average twelve hours per week.

76. Three years ; about fifteen hours per week ; physics and meteorology.

77. I teach twenty-five hours per week, tye hours of which are devoted to
industrial drawing.

80. Three years in which I did the four years' work in college mathematics,
required and elective, together with outside special work in same subject. I
teach five or six classes per day in mathematics. I have only astronomy, be-
gides mathematics.

82. Harvard College, A. B., with electives in mathematics.

85. A four-years' course in both Btate Normal school and college ; no other
subject. I

86. I teach mathematics about fire to nine hours weekly, astronomy three to
fifteen hours weekly ; and, this term, am teaching algebra. In addition, I have
certain duties connected with the observatory, and a requirement of the founder
of my professorship, viz, I have to contribute to the advancement of astronom-
ical science.

87. Two gentlemen here are occupied in teaching mathematics, exclusive of
analytical mechanics and civil engineering ; occupied in class-room eighteen or
twenty hours.

89. Five hours per week and five hours for assistant ; mining, surveying^ me-
chanics.
90ir Two years ; twenty-two and one-half hours ; no other subjects.

91. Several hours each day ; twenty hours per week. (I only teach the higher
branches.) My assistant teaches all up to and including analytic geometry,
moral science, etc.

92. The mathematics of an ordinary college course and two summet vaca-
tions' study with the late Dr. Edward Olney, of Michigan University ; thirteen
and one-third hours per week ; physics and astronomy.

93. Two years (1871-73) of post-graduate work, and fifteen years since as
specialist ; teach about seventeen hours per week ; part of the work done by
an assistant; astronomy.

94. Twelve and one-half hours; political economy.

95. I have calculus four times per week, mechanics six times, and thermo-
dynamics (Clausius) three times.

97. Two years' private study, three years' study in Europe; fifteen or twenty ;
none.

98. Five hours' class-work per week for five years, full course.

99. Ten per week.

101. Eighteen hours per week ; physics.

103. Preparation, four years' course at the United States Military Academy ;
average time, ten hours per week ; arts apd science of war and tactics.

104. Forty-nine years ; fifteen hours per week ; no other subject.

105. The professor of mathematics has not had special training, but has
special aptitude in this direction. Has usually taught chemistry.

106. My teaching is limited to mathematics ; fifteen hours per week.

107. Four years at Dartmouth College, and two jears at the Thayer School
of Civil Engineering ; about ten hours per week ; mechanics, aabronomyi ma-
teorology^ surveying.

348 TEACHING AND HIST0B7 OF MATHEMATICS.

State time of your special preparation for teaching mathetiuitice, number of houre yon teoA

per weekf and what other eUbjects you t«aoA^m!ontinaed.

108. Twenty hours per week ; no other sabjeot.

109. Graduated Ph. B. at the University of North Carolinaj and spent one
year stndying mathematics at the Johns Hopkins University ; eight hooxs per
week; English four hoars.

110. Time of teaching varies from eighteen to twenty-one hours per week.

112. Teaching hours, sixteen per week ; I teach no other subjects.

113. Generally prepare in one-half hour; teach two hours ; Latin and Greek
one hour each.

114. Five years ; fourteen hours per week ; no other subjects.

115. Our professors teach but one subject.

116. My college course, supplemented by three years' study in Germany ; I
teach ten hours per week ; no other subject.

117. Have been teaching mathematics and science for fifteen years. At pres*
ent most is done here by our instructor.

119. From ten to sixteen hours per week ; I teach nothing but mathematics.

120. I devote eighteen hours per week to teaching ; I do not iastmct in any
other subject.

121. Mathematics and physios, also lectures on general astronomy and some
on physical geography. '

123. Average eighteen ; none. *

124. Graduate of the University of Virginia, with degrees of bachelor of
science, civil engineer, mining engineer ; two years a student of mathematics in
Cambridge University, England, and fourteen months at Gottingen, Germany;
seventeen hours per week (mathematics and astronomy). Assistant Prof. C. £.
Rilboume, graduate of United States Military Academy, teaches ten hours per
week, and Assistant Prof. G. W. MoCoard (Bethany College, W. Ya.) teaches
eighteen hours per week.

125. One and one-half years in university, besides work done privately.
Mathematics, twelve hours per week ; two hours in civil engineering and
astronomy.

127. I teach mathematics from fifteen to twenty hours per week, and French
from five to ten hours.

128. Aside from regular college and private work, a year's partial work at
Harvard. Teach fifteen hours a week. No other subject, except Bible, one
hour a week*

129. From ten to fifteen hoars ; astronomy, surveying, and bridge constmo-
tion.

130. Four years, seventeen to eighteen hours; astronomy and elementary
mechanics are included in the seventeen or eighteen hours.

131. Post-graduate student two years at Massachusetts Agricultural Collegs
and Johns Hopkins University. Teach twelve to sixteen hoars per* week.
Astronomy also.

133. No special training aside from a regular coUego course and private stady ;
ten hours ; mechanics, physics, astronomy.

134. Teach three hours each day ; surveying and mechanics.

135. Oneyear, after graduation from college; eleven hours per week; astron-
omy.

136. Graduated with honors at Girton CoUege, Cambridge, 1880 ; four yean^
subsequent residence and attendance at Professor Cayley's lectures ; ten or
eleven hours per week ; no other subject.

139. Amount of time occupied in teaching is regularly eighteen hoars, often
increased by extra work. I teach no other subject.

t 140. Prof. Isaac Sharpless, L. B. (Harvard); seven hours; none. Prof. Frank
Morley, A. M. and eighth wrangler of Cambridge; foarteen hoars; none.

MATHEMATICAL TEACHIKa AT THE PBESENT TIME. 349

State ftme of your special preparation for teaohing mathetnaiioSf numler ofhoure you teach

per weehf and what other subjeota you teaoh—ConUmied,

142. Usual college course ; twelve hours per week ; Latin.

143. A college course and all the mathematiosj both regular and extra, I could
crowd into it. I teach now thirteen to fifteen hours per week. No other sub-
ject.

144. Algebra, five times; geometry, fiye; trigonometry, three; analytical
geometry, three ; calculus, three ; analytical mechanics, three ; trigonometry
and surveying by an assistant.

145. Five years ; thirteen hours per week ; no other subject.

147. I have never taken a special course in mathematics, but have studied
advanced works to some extent. I teach mathematics four hours a day. I have
in charge vocal music, which takes forty minutes a day.

148. I teach (personally) twenty to thirty hours a week. About one^fourth
of this time is occupied with mathematics, the other three-fourths with mechan-
ics and civil engineering.

. 149. I teach no other subjects. I teach twenty hours per week.
. 150. Twenty-five hours per week; none.

151. No special preparation ; twenty hours per week. I teach no other stud-
ies.

153. I devoted twelve years to my special preparation for teaching mathe-
matics.

154. I am a student of mathematics ; have charge of the department, but
only teach the higher classes.

155. I first took a college course in mathematics. After this I spent several
years in post-graduate work. I teach eighteen hours per week, and I teach no
other subjects.

156. About seventeen hours per week ; no other subject.

157. I have a course in civil engineering. I also have charge of a commercial
course.

158. Number of hours given above one and a half at thirteen per week. But
this does not include office hours of myself and assistant for meeting students
and giving explanation. My assistant does not teach, but simply keeps office
hours for consultation and solution and explanation of difficulties in lectures or
assigned work. I will add that the under-graduate course in pure mathematics
is the most extensive and thorough one given in auy university in the United
States.

159. With assistant, twenty-four hours per week; modem languages (French
and German).

160. Twenty hours per week ; civil engineering.

161. I teach mathematics four hours each day, and also teach physicB and
chemistry.

163. Sixteen hours per week ; eighteen hours ; French, Latin, history.

164. About twenty-five hours per week.

166. We give each' class one hour eacl^ day. Any subject in the course.

165. Twenty years' experience in the mathematical class-room ; about ten
hours per week in mathematics, and five hours per week in physics.

168. The number of hours of teaching varies from five to twenty per week.
College coarse and then three years at Yale; physics and astronomy.

350

TEA.CHma AND HISTORY OF MATHEMATICS.

(h) Normal Schools.

Name of institntion.

Location.

Name of person
reporting.

Title or position of per>
son reporting.

169

State Kornial School

Jacksonville, Ala . ..

C.B. Gibson

President.

170

State Normal School

Florenoe, Ala

if. K. Powers .■>■■ .

Do.

171

Normal School... . ........

Tnskegee, Ala

Pine Bluff, Ark

Maria A. Benson , r

Instrnotor in mathifTnAtira.

172

Branch Normal School of

J.O.CorbiA

PrincipaL

the Arkansafilndostrial

TJniYersity.

173

State Normal School

San Jos6,Cal

B.S.Holway

Teacher in normal aohooL

174

State Normal School

Madison, Dak

William F.Gorrie.

President.

175

Washington Noimal
School

Washington, D. C . . .

£.S. Atkinson....

PilncipaL

176

Southern Illinois Normal
University.

Carbondale, 111

G.Y.Buchanan...

Teacher of mathematics.

177

Tri-State Normal College .

Angola, Ind

L.M. Sniff

President.

178

State Normal School .......

Terre Haute, Ind . . .

Nathan Newby...

Professor of mathematics.

179

State Normal School

Cedar Falls, Iowa. ..

D.B. "Wright

Do.

180

State Norm al School

Emporia, Kans

M.A.Baaey

Do.

181

State Normiil School

Gorham,Me

W.XCorthett

Principal.

182

State Normal School

Baltimore, Md

George L. Smith . .

Professor of matfaematies.

183

State Normal School

Westfield, Mass

J. C. Greenough ..

Principal.

184

State Normal School

Worcester, Mass ....

E.H.BnBsell

Do.

185

Inka Normal Institute

luka. Miss ..•.•>....

B.L. Sherwood ...

Professor of natural sel-

ence and mathematics.

186

State Normal School

Warrensburg, Mo. . .

George H. Howe . .

Professor of mathematics

187

North- Western Normal

Stanberry,Mo

A.Monre ^^t....-

Principal.

SchooL

188

State Normal School

Kirksville, Mo

J.LNelson

Professor of mathematicSb

180

Fremont Normal School ..

Fremont, Nebr

W. H. Clemmons. .

President.

190

Normal College of New
York.

New York, N.T....

J.A.GUbt

Professor of mathematics
and physics.

191

State Normal and Train-
ing School.

Genesee, N. 7

R.A.Waterbury..

Professor of higher math-
ematios and methods in
arithmetic

192

State Normal and Train-
ing School.

Cortland, N.Y

D.E. Smith

Professor of mathematics.

198

State Normal School

Oswego, N.Y..

W.G.Bappleye...

Teacher of mathematics.

194

State Normal and Train-
ing SchooL

NewPaltz,N.Y....

F.S.Capen

Principal.

195

State Normal School

Albany, N.Y

E.P.Waterbury..

Do.

196

State Normal School

Buffalo,N.Y

J.M.Cassedy

Do.

197

State Colored Normal
School.

Plymouth, N.C

H. C.Crosby

Do.

•

198

Normal Training School ..

Cleveland, Ohio

Ellen E.BeTeley..

Do.

199

North- Western Normal
and Collegiate Institute.

Wauseon, Ohio

jr.H.f>iebel

Instructor in mathematics.

200

State Normal School

Ashland, Oregon

J. S. Sweet

President.

201

Drain Academy and State
Normal.

Drain, Oregon

W.C.Hawley

Do.

202

State Normal School

Bloomsburg, Pa

G.E.Wilbur

Professor of higher maths*
matics and history.

203

State Normal School

Clarion, Pa

tf.H. Apple .......

ProfMwor nf vnftthftinatiw.

204

Philadelphia Normal
School.

Philadelphia, Pa....

G.W. Fetter

PrincipaL

295

Cnmberlaod Valley State
Nonnal SchooL

Shippensbnrg, Pa . . .

E.H.Bnghee

Teaoher of matlmnalte

MATHEMATICAL TEACHING AT THE PRESENT TIME. 351

(&) Normal Schools— Continned.

Name of institation.

Location.

Name of person
reporting.

Title or position of per-
son reporting.

206
207

208
200

State Normal School

Central State Normal

SchooL
Bichmond Normal School.

State Normal School

West Chester, Fa...
Lock Haven, Fa ... .

Richmond, Ya

Fairmont, W . Ya. . . .

Farmville, Ya

Johnson, Yt

Oshkoah,Wis

Kivftr Fall A. Wis ....

D.M.Sensenig....
O.W.KitcheU....

S.T. Beach

C. A. Sine

Professor of mathematics.
Instr actor in mathematics.

Principal.
To.

210
211
212

213

Celestia S. Farish.
A. H. Campbell ...
B.F. Webster

Teacher of mathematics.

Principal.

Teacher of mathematics.

State tim3 of your special preparation for teachintj mathematics, number of hours you

teach per week, and what other suljeots you teach,

169. Four years ; twenty hours ; physics, chemistry, and astronomy.

170. Two years ; ten hours ; no other subject.

171. Twenty hours; reading.

172. I personally teach, at present, one class each in algebra, arithmetic, and
geometry, for five days iu the week, one in natural philosophy

173. Twenty hours; no other subject.

174. Algebra five hours; geometry, Latin, zoology, history of education.

176. Regular course in uniyersity ; twenty-five hours ; no other subject.

177. Twenty hours ; no other subject.

179. Twenty hours ; no other subject.

180. Twenty years ; twenty-four hours ; no other subject.

182. Drawing and physics. ^

183. The teacher of mathematics has ten hours ; physiology.

185. Two years ; six hours ; natural science, history, rhetoric, and book-
keeping.

186. Six classes per day, forty minutes each, five days per week ; no other
subject.

188. Five hours per day; astronomy one hour per day.
169. Fifteen hours.

190. Twelve hours ; chemistry, physics.

191. Have taught mathematics almost exclusively for seventeen years, and
principally for twenty-five years ; thirty hours.

192. Election of all mathematics I could get in college course ; eighteen and
three-fourths hours ; class on school law.

193. Graduate of Cornell ; twenty-five hours ; none.

195. Two years; twenty hours; none.

196. Mathematical course at Dartmouth ; fifteen hours ; astronomy.

197. Ten hours ; physiology, history, moral science, and English literature.

198. Teachers of mathematics not specialists.

199. Six hours per day ; no other subject.

200. Twelve hours; book-keeping, philosophy, psychology, art of school
management.

201. Seven hours per week.

202. Four years; twenty-five periods, forty-five minutes each, per week;
civil government.

203. Classical college course ; twenty-five hours ; no other subject.

204. Theory and practice of teaching and school government-

205. Twenty- two hours ; no other subject*

352 TEACHIKG AND HISTOBT OF MATHEMATICS.

State time of your Bpeeial preparation for ieaekimg maihewuitics, nmwiber of Aowv yos
teach per week^ and what other suhjeote you teach — Continiied.

206. Graduate of both elementary and scienttfic conises of MillerBrille State
Normal School, Pennsylvania ; have taaght for twenty years acarooly anything
bntmathematice in three of the normal schools in Pennsylvania. Am anibor
of Nambers Symbolized, an elementary algebra^ and have in preas Kamben
Universalized; thirty hoars.

207. 'Twenty hours ; Latin.

208. Algebra, three hours ; physios, physical geography, rbetorio, Latiiiy efte.

210. Private study at intervals for four years; twenty^five hoais ; no otbor
subject.

211. Usual course in academy, normal school, and college.

212. None, except that spent in my regular course in the normal sohool ; fif-
teen hours ; no other branches.

Are etudente entering your institution thorough in themathematioB required for admieeionf

Of the forty-flve reports received from normal schools three or four give no reply to
this ; all others answer no, excepting the institutions numbered 175^ " generally so ; *
176, " reasonably so ; " 180, " fairly so ; " 190 and 194, "yes."

What are the requirements in mathematies for admieeian f

Number 169 reports, arithmetic and elementary algebra ; 190, arithmetio and a little
geometry ; all others require only arithmetic, generally not the whole of it, but
through fractions and the simpler cases of percentage. Number 197 says, *< funda-
mental rules of arithmetic; " 204 says, ''fractions and percentage." A few institu-
tions admit all who apply, without examination in mathematics.

Is the metric system taught t

AU answered in the affirmative, ex^oepting those bearing the numbers 171, 185, 190,
208.

Which is taught first, algebra or geometry f

All answered '' algebra, " excepting numbers 181, 183, 190, 210, 211.
Numbers 181, 183, 210 take up geometry first.
Numbers 190,211 teach both together.

Sow far do you proceed in the one before taking up the other t j^.

The following carry students first through a full course of elemeniaiy sJgebra:
169, 170, 174, 180, 184, 185, 189. 201, 206, 208, 209.

The following, through quadratics : 173, 178, 197.

The following, to radicals : 193, 196, 212.

The following, to quadratics : 192. 195, 200, 204.

The following, through fractions : 188, 207.

Institution 181 finishes plane geometry before taking up algebra ; 163 gires one
term of geometry before algebra; 210 observes the following order: (1) A course in
form; (2) Rudiments of algebra ; (3) Simple geometric theorems and oonstnictioDs;
(4) More difficult algebra ; (5) More difficult geometry.

Are percentage and its applications taught "before the rudiments of algebra or etftert

All who answered said ''before,'' except the followiDg~192, 194, 210, that said
''after," though some of the simplest parts of percentage were taught before; 181
and 206 said that both were taught together.

The mathematioal course in the normal schools generally embraces a somewhat thor-
ough study of arithmetic, the study of algebra and geometry, and usually a little
trigonometry.

MATHBBtA.TICAL TEACHINQ AT THE PRESENT TIME.

353

(o) AcADBMiES, Institutes, and High Schools.

Name of inBtitntioii.

Location.

Name of person
reporting.

Title or position of per-
son reporting.

Toroli's Tnntltnte for Boys
Fablic High School

Mobilft. Alft.^rr.Txr

A.Toroli

PrincipaL
Superintendent of schoolst

215

Birmingham, Ala . . .

J. H. Phillips

216

TTniversity High School ..

Tuscaloosa, Ala. —

W.H.Verner

PrincipaL

217

Mnrf ^nnft TirntJ tnit« t r

Marianna, Ark

F.A.FutpaU

218

Rogers Academy...

Bogers, Ark

J.W.Soroggs

PrindpaL

219

Hopkins Academy

Oakland, Oal

Geo. C. Edwards..

Teacher of mathematios.

220

St Matthew's HaU

San Mateo. Cal

H.D.BobinsOn...

Tutor of mathematics.

321

Boys' High School

San Francisco, Cal . .

W.N.Bosh

Head-teacher of mathemat-
ical department.

222

Los Angeles High School.

Los Angeles, Cal. . . .

F. A. Dunham ....

Assistant teacher.

223

Girls' High School

San Francisco, Cal..

Fidelia Jewett

Head of department of
mathematics.

224

Oakland High Sbhool

OakhKDd,Cal

S. A. Chambers • . .

Teacher of mathematics.

?!?f>

Hiirh School .......,,.r---

Colorado Springs,
Colo.

Harriet Winfield .

TfiAohai* ctf mfttliAniAtiAA

and science.

22A

School for Bovs ..........

StfAmfoi^. Conn

TT-W-iring

Principal.
Do.

227

Connectlcat Literary In*

Suffleld,Conn

M.H. Smith

sUtation.

•

228

Public High School

New Britain, Conn . .

John H. Peck

Do.

229

Sionx Falls High School ..

Sioux Falls, Dak....

Anna Emerson ...

Assisttnt high school
teacher.

230

Washington High School.

Washington, D.C...

Charlotte Smith ..

Teacher of mathematics.

231

Columbian CoUeze Pre*

••..do •.....•■••■.■■.

H.L.Hodgkins...

paratory SchooL

232

Snarta Academv. ......-•.

SnartabGa

C.B. Little

PrincipaL
Do.

233

Academy of Biohmond

Augusta, Ga

C.H.Withrow....

County.

234

Allen Aoadem7. ..■■■■•...

Chioaflro.Ill

LW.Allen

Preflident.

235

Public High School

Hyde Park, lU

W.H.Beny

PrincipaL

236

North Division High

Chicairo. HI ........ .

0. L Westcott

Do.

School.

237

West Division High
SchooL

....do ...•••.........

G.P.Welles

Do.

238

PftorJa "Rijrb School »

Peoria^ m......

G.E.Knepper

0. Ti. Manchester. .

Do.

239

Joliet High School

South Division High
SchooL

Joliet. HI

Do.

240

Chicaeo.Ill

J. Slocum ........T

241

Jennings Seminary.......

Aurora, Hi >..•••....

J.E. Adams ......

Science and higher mathe-
matics

242

Hiirh School ---_.,..,

Urbana,Ill

Boanoke, Ind

J.W.Hays.......

D. N.Howe

Superintendent of schools.
PrincipaL

243

Boanoke Classical Semi*

nary.

244

Central Grammar High
SchooL

Fort Wayne, Ind . . .

Chester L. Lone ..

Do.

245

Public High School

CrawfordsviUe, Ind .

T.H.Dann

Superintendent of city
schools.

246

Indianapolis High School.

Indianapolis, Ind . . .

W.W.Grant

PrincipaL

247

Union High School

Westfield,Ind

M.E.Cox

Dow

248

Indianapolis Classical
School for Boys.

IndianapoliSp Ind . . .

T.L.Sewall

Do.

249

TndiftnApo1i4 Clasnical

««a*dO •••••«««««••«••

T.L.Sewall,Mary
W.SewaU.

Principals.

School for Girls.

250

New Hope Female Aoad-

Oak Lodge, Choctaw

A. Griffith

SnperintendeDtk

©my.

Nation, Ind. T.

881— No. 3 23

354

TEACHINa AND HISTOBT OF MATHEMATIGEL

(o) AcADEtfiKS, iNSTiTxrTBS, AND HXQH Scbo6L8— Continaed.

251
882
253
264
265
256
257
258
259
260

26t

262
263
264
265
266
267
266

270
271

272

273

274

275
276

277
278

279

281
282
283
284
265
286
287
288
289
290
201
292

Name of inttitation.

High School

High School.

Iowa City Academy

High School

Kew Orleani Seminary . . .
GirlA' High School.:

Madawaaka Training

SchooL
Franklin Female College .

HlghScho<^

High School

Fry eburg Academy

High School

McDonogh School

Washington County Kale

High School.
Centrerille Academy and

High SchooL

Friends' Academy

HaTerhillTraining School
Mount Hermon School —

Cushing Academy.

Prospect High School.

High School .
Baton School

Powder Point School

Admiral Sir Isaac Coffin's
Lancastrian School.

Wheaton Female Semi-
nary.

Lawrence Academy

Smith Academy

Partridge Academy

High School

Hanorer Academy

Lynn High School

Bristol Academy ,

High School

Nichols Academy. ........

High School

Sawin Academy and
Bawse High SchooL

Location.

Dayenport, Iowa. . . .
Des Moines, Iowa.. .
Iowa City, Iowa . . . .
Burlington, Iowa . . .
Davenportb Iowa. . . .

TopekaiKans

Ottawa, Kana .......

Paducah,Ky

New Orleans, La —
...do

Augusta^ Me .

Topsham, Me

Saco, Me

Bath, Me

Fryeburg, Me

Portland, Me

McDonogh, Md

Hagerstown, Md . . . .

CentroTiile, Md.

New Bedford, Mass.

Haverhill, Mass

Mount Hermon,

Mass.
Ashbumham, Mass,

Greenfield, Mass. . . .

New Bedford, Mass
Middleborongh,
Mass.

Dnxbnry, Mate

Nantucket, Mass . . .

Norton, Mass.

Falmouth, Mass . . . .

Hatfield, Mass

Duxbury, Mass

Peabody.Mass

Salem, Mass

Hanorer, Mass

Lynn, Mass

Taunton, Mass

Amherst, Mass

Dudley. Mass

Harerhill, Mass —
Fitohburg, Mass —
Sherbom,Mass

Name of person
reportmg.

T.B.Stratton

J.F.Oowdy

M.R. Tripp

E.Poppe

F.E.6tratton

J. E. Williamson . .

G.L Harvey

A.H.Beals

L. G.Atkinson

M.C.Cttsack

Vetal Cye

D.L. Smith

L.M. Chad wick...

H.B.Cole

M.E.Bnssell

A. E. Chase

D.C.Lyle

G.C.Pearson

A.G.Harley

G.B. Dodge

C. A. Newton

H.E. Sawyer

F.D.Laae

Ida F.Foster

R.G.Hul{ng

A. A. Eaton

F.B.Knapp

E.B.Fox

S.L.Dawea

S. A. Helton

S.L. Cutler

C.F. Jacobs

C.A«Holbrook....

A. L. Groodrich

A.P.Averlll

William Fuller ...
William F.Palmer
S. A. Sherman ...

ILG.Clark

Clarence E.Kelley
H.W.Kittredge..
W.F.Gregory....

Title or position of per-
son reporting.

PrinoipaL
Teacher of

Do.
PrinoipaL

Do.

Do.
SuperinteadcDtb

Do.
President
Department of
xos.

Principal.
Assistant teacher.
PrinoipaL
Assistant.
PrincipaL

PrinoipaL

Assistant teacher.

PrinoipaL

Superintendent.

Instructor in mathemattoi

and German.
Teacher of seienoe

mathematica.
PrincipaL
Do.

Do.
Do.

Teacher of

PrinoipaL

Do.

Dow

Do.
Master.
PrincipaL
Teacher of
PrincipaL

Do.

Dib

HATHEHATICAL TEACHIKa AT THE PRESENT TIME. 355

(0) AOADXMIXS, iNBTmrnSS, AND HtQH SCHOOLS^-Contiliued.

Kame of Xnitltntion.

Location.

Name of person
reporting.

Title OF position of person
reporting.

293

Dmry High School

North Adams, Mass.

EHxabeth H. Tal-

cott
J. 0. Morris

First assistant.

294

Charlestown High School.

Boston, Mass .—....

Head-maeter.

C Adeline L.Sy Ires-
^ ter.

)

295

Girls' Hiffh School

••#•00 ••■••••••••«•»•

f Assistant tM«hers.

(Emerette 0. Patch

296

Pablio Latin School

• • ■ vUO ■•«••■••••■■•«■

G.C.Emery

Teacher of mathematics.

297

Cambridge Latin School..

Oambxidge, Mass . . .

W.F. Bradbury...

Head-master.

298

West BoxbnryHigh School

Boston, Mass

G.C.^ann

Principal.

299

Xngliah and Glaiiioal
High SchooL

Worcester. Mass ....

A.S.Boe

Do.

300

Hifirh School .••••»••■••..

Ann Arbor. Mich-r*

L.D. Wines

Instructor in higher math-
ematics.

301

Michigan liiUtary Acad-

Orchard Lake, Mich .

W.H. Butts

Principal.

302

Hieh School

Menominee, Mich. . .

Jesse Hubbard . . .

Superintendent of city
schools.

808

HiehSohod ....•«..••....

Ypsilanti,Mioh

Grand Bapids, Mich

RrWt Putnam ....

Superintendent of schools.
PrinoipaL

804

Hieh School

W. A. Greeson ....

805

Kiohigan Fomale Semi-

Isabella G.French

Do.

nary.

306

Shattnok School .....

Faribanlt^Minn....
StClondfMirn

Wm. W. Champ-

lin.
C.C.Schmidt

307

FahliC High School

Superintendent.

308

Aagphorg Seminary

Minneapolis, Minn. .

Wilhelm Potter-

Instructor.

809

Hxnneav<di8 Aoadem7 ....

....do ...•■•.........

■en.
E. D.Holmes

Principal.
Superintendent of pnbUo

310

PnbUo High School

Yicksbnrg, Miss. . . .

E. W.Wright

schools.

Smith Academy, Wash-
ington UnlTersity .

St.Lonis,Mo

E.B.Offutt .......

312

St. Joseph High School . . .

St. Joseph, Mo

C.S.Thacht'r

Do.

313

Lincoln High School

Linooln,Nebr

S. P. Barrett

PrinoipaL

314

Bobinson Fettale Semi-

Bxeter, K.H

G.N.Cross

Do.

315

SimondsI^eeHigh School .

Warner, N.H

E. P. Barker

Do.

316

Concord High School

Concord, N.H

J.F.Kent

Do.

817

Breirster Free Academy..

Wolfborongh, N. H.

E.H.Iiord

Do.

318

High School. .k

Portsmouth, N. H . . .

JohnFickard

Do.

819

Pennington Seminary

Pennington, N. J

J.R.Hamlon

Yice-President.

320

Hoboken Academy

Hoboken, N. J

J.Schrenk

PrinoipaL

821

High School

Newark.N.J

H. T.Dawson

Instructor in mathematics.

822

Public High School

Orange,N. J ........

W.W.Cutts

PrinoipaL

823

Newark Technical School.

Newark,N.J

H.T. Dawson

Instructor in mathematioa.

324

Blair Fresbjterial Acad-
emy.
Stevens High School

Blairstown, N. J . . . .

J.H.Shumaker...

PrinoipaL

825

Hoboken, N.J.—...

F. L. Serenoak . . .

Assistant principal and

professor of mathe-

matics.

326

Newark Academy

Newark,N.J

S.A.Fammd

Head-master.

827

Dearborn Horgan School.

Orange, N.J

D. A. Kennedy....

PrinoipaL

328

New Bnmswick High
SohooL

NewBronswlok, N. J

CJakobs.. ••.....

Superintendent of schools.

366

TEACHING AND HISTORY OF MATHEMATICS.

(o) Academies, Institutes, and High Schools— Continued.

Name of institution.

Location.

Name of person
reporting.

Title or position of person
reporting.

820

Fairfield Beminftry

Fairfield, N.Y

J.M.HaU

Teacher of scienMiL

830

College Grammar School .

Brooklyn, N.Y

L.W.Hart

Principal

831

Casenovia Seminary

Casenoria, N. Y ....

A. White!

Ti^Ai*.hAr nf mathemfttifla.

333

TfM Sevnln^rv .-,,.., t>-.>t

Antwerp, N.Y

Troy.Ur.Y

E.M.Wheeler

Principal
TflAchAr of flIiuiiiiML

833

St Mary's Aoademy ......

Adams Collegiate Insti-

tute.
rs-r^niTnar RchAol-.^r.*...,

JohnHogan

L.B. Woodward..

334

AdamSiK.Y

835

Clinton, N,Y

Gh>uTerneur, K. Y. . .

IsaftoO. Best ..^.a.

PrincipaL
Superintendent of sohooia.

33ft

Gonvernenr Seminary ....

J.P.Ferthill

337

TTnion Classical Institute.

Schenectady, N.Y..

E.E.yeeder

Teacher of mathematics.

338

High School

Troy. N.Y.....

J. p. Worden

ProfnHfinr of matbAtniktiflA.

339

Oxford Academy .........

Oxford, N.Y

Brooklyxi, N. Y

F.L.Gamage

C.Harrison

PrincipaL
Headmaster.

840

Brooklyn Latin School

841

The Utica Academy

TTtica,N.Y

G.CSange

M, T. Scndder .

PrincipaL
Do.

842

Borne Free Academy

Rome.N.Y

343

High School

Bqff%]n. N. Y. ...... .

M. T. KarnAA

Do.

344

High School

Foughkeepsle, N. Y.
New York, N.Y... .

TfttnAa WifinA

Do.

345

Buarley School for Girls. .

Jeannette Fine ...

Teacher of mathematics.

348

Friend's Seminary

New York, N.Y....

John M. Child....

PrincipaL

347

Central High School

Binghamton, N. Y ..

Fannie Webster. .

Instructor in mathematiea

848

School for Girls

New YortN.Y

A . Brackett . . . .

PrincipaL

349

Free Academy...... r

ElminkN.Y

E.T. Wilson

350

Delavan Aoademy

Delhi, N.Y

W. D. Graves

PrincipaL
Superintendent.

851

Fort Jervis Academy

Port Jervis, N.Y...

JohnM.Dolph

352

Yonkers High School

Yonkers, N.Y

B. B.Shaw

PrincipaL

353

Hich School

Alhany, N.Y

Syracuse. N. Y ..... .

J. H. Gilbert ....

Professor of m&tliAmfLKAa.

854

High School

O.C. Kinyon

Teacher of physics and
mathematics.

tiS5

Fremont Institute • ........

Fremont, N.O

W.Wills

and Latin.

356

Binshom School ..........

Bingham School
P.O.,N.C.

B. Bingham

Superintendent.

857

High School

Huntersville. N. C

W.W.Orr

President.

358

Green Town Academy

Perrysyille, Ohio...

J.C. Sample

Do.

859

High School

Zanosyllle^ Ohio....

Dayton, Ohio

CleveUnd, Ohio

W. M. Townaend

PrincipaL
Da

8<H)

High School

C.B.8tiyens .. .

361

Mission House College

J.W.Grosshnesch.

Professor.

363

Bishop Scott Academy. . . .

Portland, Oreg

F. E. Patterson . . .

Lieutenant Colonel'-^niatb*
omatics.

863

Dickinson Seminary

Williamsport,Pa ...

G.G.Brower

Teacher of mathematios.

864

Western Pennsylvania
Classical and Scientific
Institute.

Mt Pleasant, Pa....

L. Stephens

President

865

FhiladelphiaSeminary for
Young Ladies.

Philadelphia, Fa....

Carrie A. Bitting..

Librarian.

866

Wyoming Seminary

Kingston, Pa

B.B.Howland

Teacher of'maihematioa.

867

Harry Hillman Aoademy.
William Penn Charter

Wilkes Barre. Pa . . .

B. Scott

PrincipaL

Teacher of mathematios.

868

Philadelphia, Pa....

A. D.Gray

School.

869

Central High School

Hieh School

Chester. Pa.. .......

J.F.Reizart

PrincipaL

870

Titosyille^Pa

CB.Bose ........

partment

871

High School ...••••••••■..

West Chester, Pa...

J. E. Philips

Teacher of mathematlca.

872

High School

Soranton, Pa .......

jr.c.Lange

PrinoipaL

MATHEMATICAL TEACHING AT THE PBESENT TIME. 357

(o) AcADEHiKS, Institutes, and High Schools— Continued.

873
874
375
376
377
878
379
380
381
882

883
384

885
386
387
388
389
390
391
392
893
894

ysune of Institution.

High School

Boys' High School

High School

Fawtacket High School. .

High School

Thetford (Vt. ) Academy
and Boarding SchooL

Brigham Academy

Troy Conference Acad-
emy.

High School

Central Female Institate

High School

Thyne Institute

High School

West Virginia Academy.

]i£sle Academy

Free High School

High School

Location.

Wilkes Barre, Pa . . .

York, Pa

Carbondale, Pa

Harrisburg, Pa

Proridence, K I . . . .

Pawtucket, R I

Charleston, S. C

Chattanooga^ Tenn.

Austin, Tex

Thetford, Vt

Bakersfleld, Vt
Poultney.Vt...

KutlandfVt

Gordonsville, Va. . .

Richmond, Va

Chase City, Va.....
Charleston, W.Va.
Bnckhannon, W. Va
Charlestown, W. Va.

Sheboygan, Wis

Milwaukee, Wis ....
Oshkosh,Wi8

Name of person
reporting.

O.W.Potter

A.Wauner

H. jr. Hoeokenburg

J.H.W«rt

D.W.Hoyt

W.W.Curtis..-.

V.C. Dibble

J.B.Cash

J.B. Bryant

S. F.Morse

F.£.Parlin

C.H.Dunton....

L.B.Folsom

Jas. Dinwiddle. . .

W.F.Fox

jr.H.Veasey

M.R.McOwigaii.

W. Johnson

S.B. Taylor

SL G. Haylett. ...
G.W. Peckham.
B.H.Halsey

Title or position of person
reporting.

Superintendent of sohooli.
PrincipaL

Do.

Dol
Head-master.
Principal.

Do.

Do.
Assistant teacher.

PrincipaL
Do.

Do.

Do.
Superintendent of sehooli.
PrincipaL

Teacher of mathematics.
Principal.

Do.

What re/orma are needed in the teaching of arithmetic t

215. Less adherence to and dependence upon text-books; more tliorongh
primary drill.

318. More easy examples.

223. More mental work, more analytical work, greater qnickness.

225. Increase in nnmber of problems under each principle^ decrease in nam*
bar of ** catch problems'' ; more mental work.

229. There is too much time pat on it in all the lower grades.

232. More attention to rapidity, more every-day soms.

237. Introdnction of qaiok andlabor-saying'methods in all basiness methods.

242. Better use of mathematical language ; arithmetic as a dednctive science.

251. More practice in rapid calculation. Many of the unimportant rules
should be scarcely touched. My pupils waste energy by scattering too much.

255. A more judicious selection of subjects that time be not wasted upon
non-essentials.

257. More mental arithmetic.

262. Something to make it more practical and the student better able to
apply it.

270. Fundamental operations of arithmetic only should be taught before
algebra.

274. Text-books are either so childish as to give np inspiration to work
after the primary grades, or so abstruse and dependent upon logical reason-
ing beyond a child's capacity as to discourage.

275. Insistence upon accuracy in fundamental operations, and alertness of
mind everywhere.

358 TEACHING AND HISTOBT OF MATHEMATICS.

What refartM are needed in ike teaching of aritftmetfe f--Continned.

276. More thoroaghwork in elementary rules and in common and decimal
fraotions.

S77. Scholars are pushed ahead altogether too fast, allowed to work slowly
and incorrectly ; should be drilled in quick addition, etc.

281. More attention to aoouraoy, rapidity, and practical methods.

283. It should be taught as an art rather than as a science.

286. There should be vastly more drill in fundamental processes.

288. Plenty of examples, more oral and ''mental" work.

289. More practice. It seems to me that the agitation for reducing time givea
to arithmetic is a mistake, though greater economy of effort is possible.

294. Fewer subjects, more speed and aoouraoy in computation.

297. The difficulty (especially with female teachers) is too great aubsery-
iency to the text-book — laolc of elaeticity in aecepting metkode.

300. Hire competent teachers only.

304. More mental arithmetic.

307. More mental work, greater accnracy and rapidity. Scope of the subject
reduced.

312. More practical work ; Judicious omission from ordinary text-book; bet-
ter development of principles.

314. More mental, less written work.

317. A diminution in the number of subjects and more independent work by
the pupil.

322. Particular attention to thoroughness, and abundant practice on Ainda-
mental rules and business methods, with the omission of some rules and
methods formerly deemed essential.

331. Keep the key$ out of the way and analyze every problem.

335. To return to the old custom of making the pupil do more thinking.
There are too many helps and too much ''mince* meat.".

338. Many.

340. More philosophy.

341. Return to mental arithmetic, now sadly neglected. More attention to
analysis, less to ingenious devices.

344. Do not permit primary teacher to u»e a figure in presence of children till
they know everything about numbers one to ten.

346. The use of the Grule method with heginnerst of denominate numbers be-
fore abstract, the expansion of the method of analysis in solving problems
usually assigned to proportion.

352. A method that will shorten the time, give the pupils the essentials
thoroughly. This will come, I believe, only through the experiments in indus-
trial education.

353. More simpUcity, less aiming to puzzle, less work that is wholly theo-
retical.

359. Brief methods of calculation should be insisted on, also independence.

370. Lees of it, in much leee time than ie now given to it {Superintendent of
Schools),

382. More attention to mental arithmetic.

386, The use of such books as Colburn's or Yenaole's Mental Arithmetic thor-
oughly at first ; and the rejection of such methods as have recently been in-
jected into the new Colburn's Mental Arithmetic The publio schools are
teaching for show.

389. Books without answers are needed.

392. We should not go too far in seeking to make all divisions in arithmetio
practical. Discipline must be held in mind.

MATHCKATICAL TEAOHING AT THE PRESENT TIME. 359

To what extent are modeU used in teaching geovMtry t

The following reported that models were not used: 216,219,320,326,238,239,246,
253, 256, 257, 263, 266, 267, 277, 278, 288, 299, 300, 302, 316, 334, 352, 370, 378, 384, 393^

The following reported "ocoasionally," "not much," «* very little " : 217,218,222,
231, 233, 236, 240, 241, 242, 243, 244, 245, 251, 255, 258, 265, 269, 276, 282, 293, 294, 295, 296,
307, 309, 318, 324, 326, 332, 335, 337, 338, 339, 341, 348, 350, 351, 354, 356, 3^, 360, 364, 366,
367, 372, 375, 385, 387, 390, 391.

Nearly all the remaining reports stated that models were nsed, specifying, in many
cases, that they were fonnd partlonlarly serrioeable in teaching solid and spherical
geometry.

Those reports which stated that the models were made by the pupils themselves
were classified with the group ** using models.'' To teach plane geometry to very
young students, or solid and spherical geometry to students of any grade, without
the aid of models, is a great mistake.

To what extent and with what euceese original exercieea t

All, except about two dozen, reported that original exercises were frequently used,
with good success. Some said that one-sixth of the time allotted to geometry
was devoted to them, others said one-half of the time ; but the large majority of
those specifying the relative amount of time given to such work answered one-fourth*
Several reporters took occasion to say that the teaching of geometry without iiitr<^
ducing original exercises was necessarily more or less of a failure.

le the metric system taught t

Nearly every report showed that this is taught, though in many schools but little
attention is given to it. We observed only one instance in which it was " dropped,"
after having been taught for some years. How long will it be before this country
will wheel in line with the leading European nations and adopf this system to the
exclusion of the wretched systems now in use among us f

Which is taught firsty algebra or geometry f How far do you proceed in the one hffore
taking up the other t

Excepting a number less than a dozen, all answered that algebra was faught first.
The following complete a course in elementary algebra, before taking up geometry :
215, 218, 225, 226, 230, 235, 237, 238, 239, 245, 246, 247, 260, 263, 266, 267, 272, 273, 278, 282,
283, 284, 289, 290, 292, 294, 298, 299, 301, 307, 314, 316, 318, 321, 323, 324, 326, 327, 328, 335,
337, 338, 344, 345, 353, 354, 355, 359, 360, .369, 370, 372, 376, 377, 380, 382, 383, 385, 387, 389,
394.

The following take up geometry after having carriecL the student through quadratics:
214, 223, 236, 244, 252, 255, 256, 258, 263, 274, 275, 280, 286, 291, 295, 297, 300, 303, 305, 306,
311, 317, 334, 336, 339, 343, 350, 351, 356, 363, 378, 381, 384, 391.

The following, after having carried the student to quadratics : 216, 217, 233, 250,
251, 257, 264, 302, 357, 367, 373, 386.

Through radicals: 228, 243, 262, 322.

Through equations : 224, 268, 330.

To simple equations : 219, 231, 288, 374, 379.

Through factoring : 276, 325.

Through L. C. M. and G. C. D.: 220.

To fractions : 232, 248, 249.

To involution : 229.

Of those who take up geometry before algebra, 222 teaches HilVs Geometry lor 6e«
ginners, 234 teaches the simpler parts of geometry, 242 teaches mathematical draw-
ing, involving about sixty geometric problems (without demonstrations), 315 teaches
geometry one year, 293 observes the following order of studies : (1) Beginning geome-
try; (2) algebra; (3) geometry.

In the two institutions, 269 and 270, algebra and geometry are taught together. Is
this scheme not worthy of more extended trial f

300 TEACHING AND HISTOBT OF HATHEMATtCS.

Are percentage and its applicatione taught he/ore the rudimente of algebra or after t

Nearly all replied that it was taaght ** before."

The following answered that in their institution it was taught ** after" : 217, 224
(arlthmetio being reviewed with aid of algebra), 238, 239, 252, 266 (reYiew),340, 347,
355,360.

In.most, if not all these cases, the demente of percentage had been taaght to the
pupil, before he entered the institution.

In 325 the two subjects are taught " together."

Is it not desirable to introduce the rudiments of algebra earlier than has been the
custom in most of our schools?

Are pupils permitted to use " anetDer-hooha " in arithmetio and algebra t

<<Te8," <' yes, but not encouraged": 214, 216, 217, 218, 219, 226 (with younger claases),
228, 230, 235, 23d, 239 (in arithmetic), 243, 245, 246, 247, 248, 249, 250, 251, 255, 257,
260, 276, 277. 281, 289, 295, 302, 328, 330, 334, 335, 337, 339, 340, 342, 344, 347, 350, 356, 357,
359, 369 (in algebra, but not in arithmetic), 370, 371, 373, 374, 380, 381, 386, 387, 388,
389,391, 394.'

'< No " : 215, 220, 221, 222, 224, 226 (with older classes), 230, 237, 239 (in algebra, but
not in arithmetic), 244, 256 (in algebra), 264, 278, 286, 287, 288, 290, 291, 293, 299, 304,
305, 310, 314, 315, 317, 318, 321, 323, 327, 331, 338, 343, 348, 349, 351, 360, 367, 372 (in alge-
bra), 375, 385, 393.

« Some of the answers:" 214,215,221, 223, 224, 225, 228, 235, 236, 237, 238, 239, 240,
242.244*245.

Are etudente entering your institution thorough in the mathematics required fqr admission t

Some of the institutions, especially academies and institutes, have no require-
ments for admission. In the great majority of reports there was a general complaint
that students were ''not" well prepared or "by no means" well prepared in the
requisites for admission.

The following answered " yes," " fairly so : " 214, 220, 221, 223, 224, 227, 228, 235, 237,
238, 245, 254, 266, 268, 293, 311, 322, 337, 352, 359, 390, 394.

What are the requirements in mathematics for admission to the institution f

'* Practical arithmetic," "common school arithmetic," was the reply given by one
hundred and fourteen institutions.
" Cube root in arithmetio and equations of the second degree in algebra," 217.
Arithmetic and elementary algebra : 222, 230, 273, 288, 303, 357.
Three books in geometry. Brook's Algebra, and arithmetic, 268.
Arithmetic and algebra as far as factoring, 370.
To ratio and proportion in Olney's Practical Arithmetic, 394.
Arithmetic through percentage, 360.
Arithmetic to percentage: 218, 306, 328, 379, 383.
Through fractions in arithmetic : 233, 282, 317.
Fuadamental roles in ftrithmetic : 269, 356, 391*

raSTOBIOAL ESSAYS.

HISTORY OF INFINITE SERIE&fi

The primary aini of this paper is to consider the views on infinite
series held by American mathematicians. Bat the historical treatment
of this or any similar subject would be meagre indeed were we to confine
oar discussion to the views held by mathematicians in this country.
We might as well contemplate the growth of the Euglish language
without considering its history in Great Britain, or study the life-history
of a butterfly without tracing its metamorphic development from the
chrysalis and caterpillar. A satisfactory discussion of infinite series
makes it necessary that the greater part of our space be devoted to the
views held by European mathematicians.

Previous to the seventeenth century infinite series hardly ever oc-
curred in mathematics ; but about the time of Kewton they began to
assume a central position in mathematical analysis.

Wallis and Mercator were then employing the'm in the quadrature of
curves. Kewton made a most important and far-reaching contribution
to this subject by his discovery of the binomial theorem, which is en-
graved upon his tomb in Westminster Abbey. Newton gave no dem-
onstration of his theorem except the verification by multiplication or
actual root extraction. The binomial formula is a finite expression
whenever the exponent of (a +6) is a positive whole number; but it is
a series with an infinite number of terms whenever the exponent is
negative or fractional. Newton appears to have considered his formula
to be universally true for any values of the quantities involved, no
matter whether the number of terms in the series be finite or infinite.

The binomial theorem was the earliest mathematical discovery of
Newton. Further developments on the subject of infinite series were
brought forth by him in later works. He made extensive use of them
in the quadrature of curves. Infinite series came to be looked upon as
a sort of universal mechanism upon which all higher calculations could
be made to depend. Special methods of computation, such as contin-

* This article was read before the New Orleans Academy of Sciences in December,

1887, and printed in the ''Papers" published by that society* VoL I, No. 2. Some

lUffht ohanffes have been introduced here.

861

362 TEACHING AND HISTORY OF MATHEMATICS.

ned fractions, could easily be reduced, it was said, to the general method
of infinite series. It thus appears that series were caltivated by the
early analysts with great zeal. They seem to have placed perfect con-,
fidence in the universality of the method. The mass of mathematicians
never dreamed that the unrestricted use of infinite series was under-
mining mathematical rigor and opening avenues of doubt and error;
they had no idea that in reasoning by means of sel'ies it was necessary
to consider tbeir convergency or divergency. To show what implicit con-
fidence was placed in infinite series I shall quote a passage from the
large, and in many respects excellent, history of miathematics, written
by the celebrated Montucla, who flourished during the latter half of the
eighteenth century.

In Volume III, page 272, he expresses the desirability of having a
more rigid demonstration of the binomial formula than that given by
Newton, so that no rational being might ever entertain the faintest
doubt of its truth. Among the early English mathematicians there was
one who did raise objections to the binomial formula, and of him Mon-
tuclasays: "Thus we have seen a certain Dr. Green, ♦ ♦ • although
professor of physics at the University of Oambridge and a colleague of
Ootes, not only doubt it, but pretend that it was false and say he oould
prove it by examples badly applied; but it does not appear that the En-
glish geometers, not even Cotes, his colleague, deigned to reply to him."
In the light of modem science, this passage ridiculing Green is very in-
structive. Time has turned the tables, and the laugh is no longer ui)0ii
Green, but upon Montucla himself. We now wonder at the reckless-
ness with which infinite series were once used in mathematical reason-
ing. To be sure, talents of the first order, such as Kewton, Leibnits,
Euler, Olairaut, D'Alembert, possessed too much tact and intuitive in-
sight to permit themselves to be dragged to the dangerous extremes
and yawning precipices of error, toward which their own imperfect
theory of infinite series tended to draw them. And yet, some of them
did not escape blunders. The penetrating and teeming mind of Euler^
for instance, is said to have fallen into some glaring mistakes by the
incautious use of infinite series.

Among the mathematicians who, above all others, made the most
unrestricted and reckless use of infinite series, were the Germans.
There flourished in Germany during the latter part of the eighteenth
century a mathematical school which occupied itself principally with
what was termed " combinatorial analysis.'' This analysis was culti-
vated in Germany with singular and perfectly national predilection.
One of the first problems considered by them was the extension of the
binomial formula to polynomials, and the devising of simple rules by
which polynomials could be developed into series. The solution of
this problem was followed by the problem of "reversion of series." In
this connection a quotation from Gerhardt's Oeschichte der Mathematik
in DeutscMand (p. 206) is instructive.

HISTORICAL ESSAYS. • 368

Says be : << The advocates of the combinatorial analysis were of the
opinion that with the complete solution of this problem (of reversion
^series), was given also the general solution of equations. But here
they overlooked an important point — the convergency or divergency
of the series which was obtained for the value of the unknown quantity*
Modern analysis justly demanded an investigation of this point, inas*
much as the usefulness of the results is completely dependent upon it'^
It thus appears that, through the misuse of infinite series, the Germans
were temporarily led to believe that they had reached a result which
mathematicians had so long but vainly striven to attain, namely, the al-
gebraic solution of equations higher than the fourth degree. It will be
observed that their method lacked generality, since it could at best not
yield more than one root of an equation. But in the determination of
this one root the combinatorial school was deceived. The result was a
mere delusion — a mirage produced by the refraction of the rays of
reasoning from their true path while passing through the atmosphere
of divergent series.

We proceed now to the further consideration of the binomial iheo*
rem. After the time of Kewton numerous proofs were given of the
binomial theorem. James Bernoulli demonstrated the case of whole
positive powers by the application of the theory of combinations.

This proof is excellent, and has retained its place in school-books to
the present day. But the general demonstration for the case where the
exponent may be negative or fractional was still wanting, Maclanrin
was among the first to offer a general demonstration. Soon after his
followed a host of proofs, each of which met with objections. It is no
great exaggeration to say that these early demonstrations seemed to
satisfy no one excepting their own authors. Most celebrated is the
proof given by Euler. It is still found in some of our algebras. But
Enter's proof has one fault which is common to nearly all that have
been given of this theorem. It does not consider the convergency of
the series. It seems to me that this fault is fatal. Euler claims to prove
that the binomial formula is generally true, but if this formula is act*
nally taken as being universally true, then it can be made to lead to all
sorts of absurdities. If, for instance, we take, in (a + ft)*, a as 1, 6 ss

— 3, w = — 2, then we get from the formula — = oo.

One might think that absurdities of this kind would have brought
about the immediate rejection of all proofis neglecting the tests of con*
vergency, but this has not been the case.

Another infinite series occupying a central position in analysis is the
one known to students of calculus as Taylor's theorem. It was disoov-
ered by Brook. Taylor and published in London in 1715. One would
have thought that the instant it was proposed, this theorem would have
been hailed as the best and most useful of generalizations. Instead of
this it remained quite unknown for over fifty years, till Lagrange

364 TEACHING AND HISTOBT OF MATHEMATICS.

pointed out its power. In 1772 Lagrange published a memoir in which
he proposed to inake Taylor's theorem the foundation of the differential
calculus. By doing so he hoped to relieve the mind of the difficult con«
ception of a limit upon which the calculus has been built by Newton
and his disciples. The method of limits was then involved in philo- «
sophic difficulties of a serious nature. It was therefore very desirable
that an explanation of the fundamental principles should be given which
should be so clear and rigorous as to command immediate assent. The
illustrious Lagrange attempted to supply such an explanation. He
boldly undertook to prove Taylor's theorem by simple algebra, and then
to deduce the whole differential calculus from Taylor's theorem. In
this way the use of limits or of infinitely small quantities was to be dis-
pensed with entirely. If Taylor's theorem be once absolutely granted,
then undoubtedly all the rest may be made to follow by processes which
are strictly rigorous. But in proving Taylor's theorem by simple alge-
bra without the use of limits or of infinitesimals, Lagrange avoided the
whirlpool of Charybdis only to suffer wreck against the rocks of Scylla.
The principles of algebra employed by him in his proof were those which
he received from the hands of Euler, Maclaurin, and Clairaut. His proof
rested chiefly upon the theory of infinite series. But we have seen
that this very theory was at that time wanting in mathematical rigor.
Oonsequently, all conclusions evolved ftom it possessed the same defect
Though Lagrange's method of treating the calculus was at first greatly
applauded, objections were afterward raised against it, because the
deductions were drawn from infinite series without first ascertaining that
those series were eonvergent This defect was fatal, and to-day La*
grange's <^ method of derivatives," as Ms method was called, has been
generally abandoned even in France.

At the beginning of this century the avidity with which the results
of modem analysis were sought began so far to subside as to allow
mathematicians to examine and discuss the grounds on which the sev-
eral principles were established. The doctrine of infinite series re-
ceived its due share of attention. In building up a tenable theory of
infinite series, the same course became necessary which was followed
some years ago in the erection of the Washington monument in the
District of Columbia ; after the work had proceeded to a certain height,
the old foundation was found to be insecare ; it had to be removed and
to be replaced by another which was broader and deeper. The engineer
to whom more, perhaps, than any other we are indebted for the laying
of a new and firm foundation to infinite series and to analysis in general,
is Gauchy. In the following few but pregnant sentences, taken irom his
Oours cP Analyse (Paris, 1821, p. 2), he states the object he has striven
to attain : ^^ As far as methods are concerned, I have endeavored to
give them all the rigor required in geometry, and never to have
recourse to the reasons drawn from the generalization of algebra.
Seasons of that kind, although they are very generally accepted, espe-

HISTORICAL ESSATS. 365

ciaOy in passing from converging series to diverging series and from
real quantities to imaginary quantities, can be considered, it seems ta
me, only as inductions, fit to give a glimpse of the truth, but which
agree little with the boasted exactness of mathematical science. It is
furthermore to be observed that they tend to give to algebraic formulas
an indefinite extent, while in reality most of these formulae remain true
only under certain conditions and for certain values of the quantities
which they contain." These weighty words of Cauchy became the parole
of a new scientific party. Oauchy himself was eminently successful in
his work. To him we owe the first correct proof of Taylor's theorem.

He took very strong and positive grounds against the use of diver-
gent series. All series that were not convergent, he pronounced faHa-
oiofM. Taylor's theorem he considered as being wrong whenever the
series became divergent. In his Oours cP Analyse no place was given to
those troublesome divergent infinite series that had previously beten the
cause of so much vagueness, uncertainty, and even of error.

But Oauchy was not alone in this protest against the unrestricted
use of the time-honored methods of analysis. A youthful mathematician
from northern Europe, a worthy descendant of mighty Thor, sided with
the French mathematician in tbe contest. This new combatant was the
youthful Abel, who, though he died at the premature age of twenty-
seven, left behind him an imperishable name. As in the times of myth
and fable, Thor, the thunderer, hurled his. huge hammer against the
mountain giants, so Abel, with his massive intellectual hammer, dealt
powerful blows against some of the mathematical methods of his time.
Kotice an extract from a letter written by him in 1826, which expresses
the convictions to which his profound studies had led him. Says Abel :*

<' Divergent series are in general very mischievous affairs, and it is
shameful that any one should have founded a demonstration upon them.
You can demonstrate anything you please by employing them, audit is
they who have caused so much misfortune, and given birth to so many
paradoxes. Can anything be more horrible than to declare that

0=1— 2»+3*— 4»+5»— etc.,

when n is a whole positive number f At last my eyes have been opened
in a most striking manner, for, with the exception of the simplest cases,
as for example the geometric series, there can scarcely be found in the
whole of mathematics a single infinite series whose sum has been rigor-
ously determined ; that is to say, the most important part of mathe-
matics is without foundation. The greater part of the results are correct,
that is true, but that is a most extraordinary circumstance. I am en-
gaged in discovering the reason of this — a most interesting problem.
I do not think that you could propose to me more than a very small
number of problems or theorems containing infinite series, without my
being able to make well-founded objections to their demonstration. Do

* (Euvre$ Comj^UteM deN. H. Jbeh Tome I, Christionia, 1839, p. 264.

366 TEACHING AND HISTORY OF MATHEMATICS.

BO, and I will answer yon. Not even the binomial theorem has yet been
rigorously demonstrated. I have found that

(l + a?)*'=:l + 7W(g + ^(^""^^ a^ + eto,

for all values of w which are less than 1. When xss + l^ the same
formnla holds, but only provided that m is > — 1 ; and when or ss -. 1,
the formula only holds for positive values of m. For all other values
of m the series l + fncD+ etc., is divergent. Taylor's theorem, the foun-
dation of the whole infinitesimal calculus, has no better foundation. I
have only found one single rigorous demonstration of it, and that is the
one given by M. Cauchy in his Abstract of Lectures upon the Infinitesi-
mal Calculus, where he has demonstrated that we have

00»

p {X + CO) x=p (x) + <x>p' {X) + ^ p'' {X) + etc.,

as long as the series is convergent ; but it is usually employed without
ceremony in all cases. • • •

<< The theory of infinite series in general rests upon a very bad foun-
dation. All operations are applied to them as if they wero finite ; but
is this permissible Y I think not.* Where is it demonstrated that the
differential of an infinite series is found by taking the differential of
each term f Nothing is easier than to give examples where this rule is
not correct. • • • The same remark holds for the multiplication
and division of infinite series. I have begun to examine the most im-
portant rules which are (at present) esteemed to hold good in this re-
spect, and to show in what cases they are correct and in what not so.
This work proceeds tolerably well and interests me infinitely."

Such is the unequivocal language of Abel. His early death prevented
him fi*om carrying all his plans into execution. To him we are indebted
for the first rigorous proof of the binomial theorem.t

The views on infinite series held by Oauchy and Abel met with hearty
acceptance by leading mathematicians on the continent. Thus, Poisson
expressed his views in the following language : '' It is taught in the ele-
ments that a divergent series can not serve to calculate the approximate
value of the function from which it results by development, but some-
times it has apparently been thought that such a series can be used in
analytical calculations instead of the function ; and although this error
is far from being general among geometers, nevertheless it is not useless
to point it out, for the results which are obtained by means of divergent
series are always uncertain and most of the time inexacf

The conditions for convergency and divergency of different series be-

* Diriohlet first pointed ont that the most elementary algebraic rale, according to
which every sum is independent of the arrangement and grouping of the terms to be
added, does not necessarily hold trne in infinite series.

t (Euvrea Computet de N, H. AUh Tome J, ChriBtiania, 1839, p. 66.

HI6T0BICAL £SSAYS. 367

gan to be carefoUy investigated. No anif ersal criterion for determin-
ing whether a given series is convergent or divergent was then known j
nor do we possess snch a one even to*day.

A question naturally arising at this point of our inquiry is, whether
the views of Oauchy and Abel and their co-workers met at once with
general acceptance or not. As might almost be expected, they did not,
but encountered firm opposition. The old combinatorial school in Ger-
many would not surrender their orthodox views without a struggle.
They obstinately defended every doctrine of their mathematical creed.
Even such a man as Dr. Martin Ohm, who was really an enemy of the
combinatorial school, and whose achievements in mathematics and
physics place him among the coryphsBi of science^ was not willing to
join Oauchy and Abel in calling divergent series £a»Uacious« In an essay
written by Ohm, entitled, The Spirit of Mathematical Analysis,* he
admits that the great mathematicians of his day^ as Gauss, Dirichlet,
Jacobi, Bessel, Oauchy, do not employ demonstrations conducted with
divergent series, while Poisson speaks decidedly against them. <^But,"
says Ohm, ^< that the series which are used and from which deductions
are drawn ought to be always and necessarily convergent is a circum-
stance of which the author of this essay has not been able at all to con-
vince himself; on the contrary, it is his opinion that series, as long as
they are general, so that we can not speak of their convergency or diver-
gency, must always, when properly treated, necessarily and uncondi-
tionally produce correct results." By a general series Ohm means one In
which the letters represent neither magnitudes nor numberSy but are con-
sidered as perfectly insignificant {inhaltlos). Whenever the letters are
made to represent magnitudes or numbers, then the series is no longer
a general series, but is a <^ numeric " series, and in that case Ohm admits
that an equality can exist between the function and its series only when
the series is convergent. It is very difficult to see exactly what meaning
shall be given to letters upon which algebraic operations are to be per-
formed, when the letters represent neither magnitudes nor numbers.
Nor is it easy to see in what wayformulsB involving these empty, mean-
ingless letters— these '^ ghosts of departed quantities ''—can furnish
rigorous methods in mathematical analysis. In fact, this theory of
general series containing insignificant letters is one of the last shifts to
which the opponents of the new school resorted; one of the last sub-
terfuges before giving up a contest which had become entirely hopeless.

If we pass from Germany to England we meet there with another
mathematician who championed the old cause. I refer to George Pea-
cock, who is well known to matbematicians for his Algebra and his Re-
port, made in 1833 to the British Association, On the Becent Progi^css
and Present State of Certain Branches of Analysis.

Peacock states his views with more clearness than Ohm had stated

his. He bases his argument on what he calls the ^^ principle of the per-

■ — .. , . — - .- — — II I I I 1 1 1 I

*The Spirit of Mathematioal Analysis and its Relation to a Logical System, by Dr.
Martin Ohm; translated by Alexander John Ellis, London, 1843.

368 TEACHINa AND mSTOBY OF MATHEliATICS.

manence of eqaivalent forms^" wUcli be considers to be the real foun-
dation of all rales of symbolic algebra* According to this principle, all
the rales and operations of arithmetic which have been established by
numerical considerations are adopted without reference to relative mag-
nitude ; the symbols of algebra are taken to be perfectly general and
unlimited in value, and the operations to which they are subject are
equally general. To illustrate : In arithmetic we can subtract a smaller
number fh>m a larger, but we cannot subtract a larger from a smaller ;
that is to say, we can subtract 3 from 5, but not 5 from 3. In algebra,
on the other hand, no limitation whatever is placed upon the relative
values of minuend and subtrahend ; there we can subtract 5 from 3
and give the answer a rational interpretation. By the principle of the
permanence of equivalent forms every result obtained from mathemat-
ical operations must always be a correct result, no matter what the
relative values of the quantities be upon which the operations are per-
formed. Peacock applies this principle to the subject of infinite series.
He says (p. 205, Beport for 1833) that «< the series

(1 -f x)^^V ('l + nx + ^(^^^^ a;g + etc.^

indefinitely continued, in which n is a particular value (a whole number),
though general in form, must be true also, in virtue of the principle of
the permanence of equivalent forms, when n is general in value as well
as in form.'' Instead of being always a positive whole number, the ex-
ponent n may, therefore, be negative or fractional, and the above for-
mula still holds true.

Kow, the principle of the permanence of equivalent forms laid down
by Peacock is not self-evident, nor did it become known by intuition ;
on the contrary, it is merely an induction, and can, therefore, hardly be
taken as a reliable basis upon which to settle a disputed question ; for
this very question may be one in which this law established by mere
induction might fail. But even granting the principle of the perma-
nence of equivalent forms to be generally applicable, does it really fol-
low from it that infinite series are true, whether they be convergent or
divergent f In order to discuss this point let us examine a series re-
sulting from the division of the numerator of an algebraical fraction by

its denominator, such as ^^ — .

From arithmetic we get the simple but general statement that the
numerator of a fraction divided by its denominator is equal to the quo-
tient plus the remainder (if there be any remainder). By the principle
of the permanence of eqaivalent forms this must be true of fractions
involving any quantities whatever. Now, if we divide 1 by 1 — a

we get l + a + a* + a^+ l^a ' ^® oberve there is a remainder,
YZTa • ^ ^^ carry the division further there is still a remainder. Is o

HISTORICAL ESSATS, 369

matter how far the division proceeds it will not end, and a remainder

will still exist. We may express this fact by writing :; = 1 + a +

1 — a

a' +..... a* + YZTa* ^^^9 ^ ^ '^^^ * value less than unity the

remainder approaches zero, and we may therefore write ■. ^ = 1 +

a + a^+ etc., ad infinitum. This infinite series is correct whenever
a <1. But, according to Peacock, it would follow flrom the principle of
the permanence of equivalent forms that, if this series is correct for
a <1, it must be true for all values of a. Hence the series is true when
a > 1, in which case the series is divergent. Kow, this conclusion
appears to be inadmissible, because Peacock does not examine the
remainder. When a < 1, the remainder approaches zero, and can there-
fore be neglected ; but if a > 1, then we shall find that the remainder
does not approach zero, imd therefore cannot be neglected.

To neglect it would be to violate the principle of the permanence of
equivalent forms. This principle demands that whenever there is a
remainder it shall always be considered and expressed, no matter how
far the division be continued. If in the above series we take a = 2
and neglect the remainder, then we get

— 1 = 1 + 2 + 2* + 2' + . . . . • ad infinitum^

which is an absurdity. But if the remainder be taken into account,
then we have

-l=l + 2 + 2» + 25+ ...2» + -;^.

This equation is always true, no matter how great n may be; that is to
say, no matter how far the division be continued. From similar con-
siderations in other series it would appear that divergent series are
false and absurd, except when written with the remainder.

And yet not only Peacock, but even De Morgan was not willing to
reject divergent series. Though De Morgan criticised the new school
for the unconditional rejection of divergent series, he cannot be pro-
nounced an enthusiastic supporter of the old school. In an article in
the Transactions of the Cambridge Philosophical Society, Volume VIII,
Part I, he says : ^' I do not pretend to have that confidence in series
which, to judge from elementary writers on algebra, is common among
mathematicians, not even convergent series." His views on this sub-
ject will be more fully elucidated by the following quotation from his
article on " Series ^ in the Penny Cyclopaidia : '' A divergent series is,
arithmetically speaking, infinite; that is, the quantity acquired by
summing its terms may be made greater tha^ any quantity agreed on
at the beginning of a process. • • • Nevertheless, as every alge-
braist knows, such series are frequently used as the repreaentativen of
881— No. 3 ^24

370 TEACHINa AND HISTQEY OF MATHEMATICS.

finite quantities. It was nsaal to admit such series withoat hesitation ;
bat of late years many of the continenta] mathematicians have declared
against divergent series altogether, and have asserted instances in
which the use of them leads to false results. Those of a contrary opin-
ion have replied to the instances, and have argned from general prin-
ciples in favor of retaining divergent series. Our own opinion is that
the instances have arisen from a misunderstanding or misuse of the
series employed, though sufficient to show that divergent series should
be very carefully handled ; bat that, on the other hand, no perfectly
general and indisputable right to the use of these series has been es-
tablished a priori. They always lead to true results when properly
used, but no demonstration has been given that they most always do
so."

About the time when Peacock made his report to the British Asso-
ciation, Oauchy was developing new and valuable results on the subject
of infinite series. With the aid of the integral calculus he was conduct-
ing a careful investigation of the conditions which must be fulfilled in
order that a function be capable of being developed into a convergent
infinite series. He found that four conditions must be satisfied : (1)
The function must admit of a derivative. (2) The function must be
uniform^ that is^ for any particular value for x the function must have
only one value. (3) The function must be finite. (4) The function must
be continuous^ that is, it must change gradually as the variable passes
from one value to another. These results greatly strengthened the posi-
tion held by the new school, and notwithstanding the adroit arguments
brought forth by various mathematicians of the old school in favor of di-
vergent series, the leading mathematicians of to-day have rejected the
old views and adopted those of Oauchy and Abel. In the theory of func-
tions, a branch of mathematics which is now assuming enormous propor-
tions, the convergency of all series employed is carefully and scrupulously
tested. In late years more reliable criteria have been invented for
determining the convergency. Standard treatises on the subject devote
the larger part of their space to the consideration of convergency.
Whenever a series is divergent; then either the remainder is inserted
or the series is unceremoniously rejected. Indeed, divergent series are
now looked upon by our best mathematicians as being nothing more
than exploded chimeras.

Having briefly traced the history of infinite series in Europe, we shall
consider the views on this subject held by American writers. Previous
to the beginning of this century the text-books on algebra used in this
country were all imported from abroad. About the only mathematical
books published in America before 1800 were arithmetics and some few
books on surveying. The earliest imported algebras came from Great
Britain. The most important of them were the algebras of Madaurin,
Saunderson, Oharles Hutton, John Bonnycastle, and Thomas Simpson.
These writers belonged to what we have called the old schooL As

HISTOBICAIi ESSATB. 371

might be expected tbe subject of series was handled by them with the
same looseness and recklessness as by the older school of mathemati-
cians on the continent. Thus, in Hatton's Mathematics, which was a
standard work in its day, considerable attention was paid to series, bat
the terms ^< convergent and << divergent" were not even mentioned.
The earliest American compiler of a coarse of mathematics for colleges
was Samnel Webber. In 1801 he published his ^^ Mathematics," The
algebraical part was necessarily elementary in character, and of coarse
contained no formal criteria for convergency. Whatever defects Web*
ber's Algebra may have, it has also its merits. It is pleasing to observe
that as far as the author had entered apon the subject of infinite series
he was on the right track. Speaking of a certain divergent series he
says that ^^ it is false, and the farther it is continued the farther it will
diverge from the truth " (p. 291). This language possesses the true
ring } it is free from the discords of error, and we regret that American
writers of later date have not imitated it.

In 1814, thirteen years after the publication of Webber's Mathemat-
ics, appealed the algebra of Jeremiah Day, of Yale OoUege. All things
considered, Day's Algebra is superior to Webber's, but on the particu*
lar subject of series it can hardly be said to excel. President Day points
oat, to be sure, that a certain series must converge in order to come
nearer and nearer to the exact value of the fraction from which the
series was derived, but he does not even hint at the insecurity or ab-
surdity of divergent series. He gives no demonstration of the binomial
theorem, but speaks of it as being universally true.

Four years after the publication of Day's Algebra, John Farrar, pro-
fessor of mathematics at Harvard, published An Introduction to the
Elements of Algebra, • • • Selected from the Algebra of Eoler. On
the continent of Earope Baler's writings were at that time justly con-
sidered as the most profound, and as affording the finest models of
analysis. Yet his writings were not faultless. His views on series were
those of the old school. The discussion of series as given in Farrar's
Baler demands oar attention, because subsequent American writers
were doubtless greatly infiaenced by it. On page 76 of this book, the

fraction =-=^ is resolved by division into an infinite series. The foUow-
1— a

ing comments upon it are then made : << There are suf&dent grounds to

maintain that the value of this infinite series is the same as that of the

fraction - — . What we have said may at first seem surprising, but the

consideration of some particular cases will make it easily understood.
• * * If we suppose a=s2, our series becomes =1+2+4+8+16+32+

64, etc., to infinity, and its value must be :J^ that is to say, — ^ s — 1,

which at first sight will appear absurd. But it must be remarked that
if we wish to stop at any term of the above series we can not do so

372 TEACHING AND HISTOBT OF MATHEMATICS.

without joining the firaction which remains." Kow this last sentence is
certainly a true statement. "So fault can be found with it. It simply
means that we must consider the remainder ^ the very thing which the
new school persistently insists upon. But the next statement made by
the author is objectionable. Says he: ^< Were we to continue the series
without intermission, the fraction indeed would be no longer consid-
ered, but then the series would still go on." This really amounts to
saying that when the series becomes infinite, the remainder shall not
be considered. :

Kow, if the remainder is not taken into account, then we can say in
the language of Webber that the further the series is continued, *Hhe
further it will diverge from the truth," hence it must be <^ false."

In addition to this abridgment of Euler's Algebra, Professor Farrar
published a translation from the French of Lacroix's Algebra. La-
croix's works are justly celebrated for their purity and simplicity of
style. Though more cautious in his statements than the majority of ele-
mentary writers, he must still be classed as belonging to the old school.
In his algebra * he speaks of divergent series as leading to consequences
that are ^^ absurd." The binomial theorem is proved by Lacroix for the
case when the exponent is a positive integer, but the proof for the other
cases is omitted. In the light of modem mathematics this was a wise
omission. A correct proof of the general theorem is too difficult for pu-
pils beginning algebra. But the easier proofs are incorrect. Hence it
is preferable to give no proof at all than give a wrong one. But Pro-
fessor Farrart was not satisfied with this omission. In his translation
he tulds a foot-note, with the erroneous statement that the binomial for-
mula is ^< equally applicable to cases in which the exponent is fractional
and negative," and he demonstrates this theorem in the last part of his
Cambridge Course of Mathematics (" On the Differential and Integral
Calculus") without, of course, considering the question of convergency.

Charles Davies, who was appointed professor of mathematics at the
United States Military Academy at West Point in 1823, published ia
1834 an algebra modeled after the large French treatise of Bourdon.
This algebra is familiarly known as ^^ Davies' Bourdon," and, like all
other books of Professor Davies, has had a very extensive circulation
in all parts of the United States. However excellent this treatise may
be in other respects, on the subject of infinite series and the treatment
of the binomial theorem it is very defective.

From what has been said it will be seen that the foreign authors
whom our American writers took for models in compiling their algebras
belonged to the old school. Our early American writers dung faith-
fully to the orthodox opinions of this school. The only dissenting
voice came from Samuel Webber, and it was so feeble that it escaped

■

* Elements of Algebra, by S. F. Laoroiz, translated by John Farrar. Seeond edi-
tion, Cambridge, N. £., 1825, p. 241. (This second American edition was translated
from the eleventh edition, printed in Paris in 1815.)

t Ihid, p. 152,

mSTOBICAL E8BATS. 873

all notice. Bat what, you may ask, were the views held by later Amer-
ican mathematicians Y

In answer to this I need not discuss each author individually. If we
except a few very recent writers, then we may say that on infinite series
the sins of one are quite generally the sins of all. You may consult
the large apd extensive Treatise on Algebra, of Gharles Hackley, or the
Elementary Treatise on Algebra, by James Byan ; the Elementary and
Higher Algebra, by Theodore Strong; the University Algebra, by Ho-
ratio K Bobinson } the Algebra for Colleges and Schools and Private
Students, by Joseph Bay ; the Elements of Algebra, by Msyor D. H.
Hill ; the University Algebra, by Edward Olney ; the Binomial Theorem
and Logarithms, by William Ohauvenet ; the Treatise on Algebra, by
Elias Loomis. YoiLjnay consult these and many others, and you will
find that they are all swayed more or less by the orthodox ideas of the
old school. A few of them give tests for convergency, but npne of them
treat divergent series with that severity which these mischievous ex-
pressions deserve. If divergent series are false, then it ought to be so
stated } the student should be informed of the fact that they are false.
Judging only from American algebras, we might almost conclude that
Oauchy, Abel, Poisson, Dirichlet had never lived, or that their ideas
had been long since expunged from the creed of true science. Of the
algebras which the writer has examined a few of very recent date are
the only ones to which this statement is not applicable. But even
these give demonstrations of the binomial theorem which are deficient
in rigor. The writer has not seen a single proof of this theorem for
negative or fractional exponents in any American algebra which is
not open to well-founded objections. Our writers often begin the gen-
eral proof with an equation in which the sign s expresses always a
numerieal equality^ and, finally, arrive at an equation (the generalized
binomial formula) in which the sign s does not express a numerical
equality^ except under certain limiting conditions. The student is not
informed in what way such a change in the meaning of the sign = has
been brought about, nor is he told by what process of logic this sud-
den metamorphosis is permissible.

It may be argued that the final equation expresses a formal truth.
But is this formal truth anything more than a perforated shell from
which the kernel of useful truth has been removed Y When the equa-
tion expresses merely a formal truth, can it be used for numerical cal-
culations f 1^0. Oan the series be employed in course of analytical dem-
onstrations in place of the function f No, for it leads to uncertainty,
and perhaps even to error. What then is this formal truth good fort

There is an American.algebraist who says that the formula^

(l+a?)'»=l+na?+5^2i:liaj»+ete.,

<*i8 at once true when n is positive or negative, entire or firactional,
real or imaginary, rational or irrationaL" Yet, it was pointed out long

374 TEACmNG AND HISTOBT OF MATHEMATICS.

ago by Abel and others that even when the conditions for oonrergenej
are satisfied there are still other points to be considered before we are
entitled to write the sign of equality between the ftinction and the
series. The expression (1+^)*^ has in general a multiplicity of different
valnes. In fact, the only case in which it has a single valne is when
the exponent n is an integer. Whenever n is a rational fraction, the
expression has more than one value ; whenever n is irrational or im-
aginary, the expression has an infinite number of values.

The series itself, on the other hand^ has always only one value. Now,
if we place the function (1 + x)*^ equal to the series, then the question
arises, which one out of the possibly infinite number of functional valnes
is equal to the one value of the series f

A process in which American books are deficient in rigor is the mul-
tiplication of one infinite series by another. Some of our books exhibit
not the sligl^test hesitation in multiplying by one another any two series
whatever, and placing their product equal to the product of the functions
from which the two series were obtained. The same confidence is placed
in the process of multiplying infinite series as in that of multiplying
finite expressions. But as a matter of fact, when one or both series are
divergent, then their product is an absurd result. It is therefore neces-
sary that both series be convergent. But, strange to say, this neces-
sary condition has not always been found sufficient. There are cases in
which the product of two convergent series may actually be a divergent
series. For instance, Gauchy has shown that the series

i_ 1 + 1 _ 1 + 1

is convergent^ but that its sqnare

-^-(^4)-a+^)+

i& divergent. The investigation of this difficulty has led to the proof
that only the so-called *^ absolutely convergent" series can be multiplied
into each other without liability of error. Thus while our elementary
books teach that all infinite series can be multiplied by one another,
the most recent and most advanced treaties on the subject teach that
only convergent series of a particular kind can be multiplied into each
other so as to lead to trustworthy results.

If we had time we could go on and examine the development of func-
tions into series by the method of indeterminate co-efficients, as taught
in our elementary books. We should meet with several points which
are open to well-founded objections. But we can not enter upon this
subject now.

The writing of a good elementary text-book is one of the most diffi-
cult undertakings. It is hardly advisable to subject to rigorous proof
every rule and every process which ought to go into an elementary
text-book on algebra.

HISTORICAL ESSAYS. 375

Many of tlie proofs would bo either exceedingly difficult to the pupil
or entirely beyond his comprehension. In consequence of this the prob-
lem arises to decide what had best be proved and what might best be
assumed without demonstration. It is easy to see how the opinions of
able and experienced teachers may differ in making this choice. But
there is one point upon which there should be no difference of opinion.
Whatever is placed in an elementary text-book ought to be, as near
as we can make it, the truth and nothing but the truth. If a subject is
BO difficult that it can not be stated in an elementary way without mis*
stating it, then it had best be left out altogether. Whatever reasoning
would be fallacious and wrong when placed in an advanced treatise
must be equally fallacious and wrong when placed in an elementary
book. If divergent series are unreliable, absurd, or false in advanced
articles written by Gayley and Abel, in the Cours cP Analyse by Jordan
and by Oauchy, or in the Calcul JOiffSrentiel by Serret, then divergent
series must be equally unreliable, absurd, or false in the elementary
algebras of Loomis, Davies, or Bobinson. Kow, if divergent series are
actually untrustworthy and fallacious (and the leading mathematicians
of to-day consider them so), would it not be best to make a statement
to that effect in our elementary algebras and to give at least some of the
simplest criteria for determining the convergency. If a correct proof
of the binomial theorem for negative and fractional exponents is too.
long and difficult to find a place in an elementary algebra, why should
it not be entirely omitted from algebra, and inserted afterward in the
differential and integral calculus Y There it can be deduced at once as
an immediate consequence of Taylor's theorem. But in that case we
must be sure that our calculus gives a correct proof of Taylor's theorem.

Unfortunately, m^any of our American works give what may bo called
the old proof of this theorem, which proof is pronounced unsatisfactory
by all standard writers on the calculus. De Morgan does not consider
it a demonstration at all, but treats this old process as <^ nothing more
than rendering it highly probable that

*(a + h) and *(a) + *^(a) h + ^''(a) A*+ etc.,

have relations which are worth inquiring into.^ Todhunter likewise ob-
jects to the old proof, and especially to << the use of an infinite series
without ascertaining that it is convergent."

We regret to say that many of our American books on calculus are
just as reckless and unscrupulous in the treatment of infinite series as
our algebras are. But this assertion can not be applied sweepingly to
all our works on this subject. Take some of the more recent publica-
tions, as, for instance, Byerly's Calculus. On page 118 of Byerly's Dif-
ferential calculus the following statement is made and emphasized by
italics. ^^It is very unsafe to make use of divergent series or to base any
reasoning upon them.^ This doctrine contradicts the doctrine taught in
our algebras. If Byerly's Oalcnlns is correct, then our algebras must

376 TEACHIKa AND HISTOBY OF MATHEMATICS.

be wrong. Imagine the confasion which will arise in the mind of the
Btadent. While he is studying algebra he learns that the binomial
theorem is universally true. When Byerly's Galcalas is placed in his
hands, he discovers that this same theorem is not always true, but
holds good only when certain conditions are satisfied. .The thoughtful
student will become disgusted at such glaring contradictions in the pre-
sentation and explanation of a science which, in the hands of a careful
mathematician, can be made to be the most accurate and consistent of
all sciences. In closing, we give the following summary of the views
presented in this paper :

1. In calculating with or reasoning by means of infinite series, the
question of convergency should always be considered. If a series is
divergent, then the sign of equality should not be placed between that
series and the function from which it was developed. If the sign of
equality he used in that way, then it expresses an absurdity, which is no
less an absurdity when found in an elementary text-book than when
found in a more advanced treatise.

2. Those parts of the subject which are too difficult for correct treat-
ment in algebra, may be assumed temporarily without demonstration,
and may afterward be proved in the differential and integral calculus.
This suggestion applies particularly to the binomial formula for all
.cases in which its exponent is not a positive integer.

ON PARALLEL LINES AND ALLIED SUBJECTS.

There are few subjects in mathematics which have been discussed to
greater extent than that of parallel lines. The various attempts at im-
proving the theory of this subject may be classified under four heads :
I. In which a new definition of parallel lines is suggested. 11. In which
a new axiom, different from Euclid's, is proposed. III. In which efforts
have been made to deduce the theory of parallels from the natore of
the straight line and plane angle. lY. In which, during the present
century, the whole subject of geometrical axioms has been re*inyesti-
gated and searched to the very bottom, and in which the novel and
startling conclusion has been reached that the space defined by Euclid's
axioms is not the only possible non -contradictory space. This gave
birth to what is now termed non-Euclidian geometry.

It is our intention to take up the discussion under the above four
heads, with a view of presenting the ideas advanced by American math-
ematicians or given in text-books used in this country.

I.— NEW DEFINITIONS.

Euclid's definition of parallel lines is as follows: ^Parallel itraight
Unes are such as are in the samepUme^ and which being produced ever so far
both ways do not ineet/^ This definition has been retained by the largei

HISTORICAL E88AYB. 377

nnmber of American writers,^ and seems indeed the most desirable one
to use in elementary geometry.

Parallel lines are lines everywhere equally distant This definition has
been adopted by Hutton,^ Webber,^ T. Walker,* A. Schuyler,' and
probably by other authors whose books have not been examined by the
writer. This definition has never been popular here since the time of
Webber. Ohief among older and foreign authors who used it are Wolf,
Diirer, Boscovich, T. Simpson (in the first edition of his Elements), and
Bonnycastle. Olavius assumed that a line which is everywhere equi-
distant from a straight line is itself straight. This axiom or postulate,
which, by the way, does not hold true in pseudo-spherical space (accord-
ing to the ordinary methods of measurement), lies hidden in disguise in
the above definition. The objections to that definition are that it is an
advanced theorem^ rather than a definition; that it involves a number of
considerationB of great subtlety ; and that it has to be abandoned as a
fundamental definition in the more generalized view which is taken of
this science in what is called non-Euclidian geometry. To be rejected
for similar reasons is the following definition.

TuDO lines that make equal angles with a third linej all being in the same
plancj are parallel. This is given by H. K. Bobinson.' It was used in
France by Yarignon and Bezout, and in England by Gooley.

Parallel lines are straight lines which ha/ve the sanhe direction. This defi-
nition has been growing in favor in this country. The reason of its popu-
larity lies in the fiEust that it appears to contribute to the brevity and sim-
plicity of demonstrations. ^Its validity will be considered further on.
One of the first, perhaps the first, to use it in this country was James Hay-
ward, teacher of mathematics at Harvard GoUege.'^ It was used by
Benjamin Peirce,' K Tillinghast,* Charles W. Hackley,^® Davies and
Peck," Eli T. Tappan," William T. Bradbury," and G. A. Wentworth."

In England the concept of direction was made the basis of a work

1 Of the books examined by the writer, the foUowing employ this definition: Far-
rar, F. H. Smiths and Davies^ in their respective editions of Legendrie; also, Chan-
yenet, Kewcomb, Yenable, Halsted, Loomis, Gmnd, Olney, Hassler, Hnnter, WhitloQk,
and Wentworth (in the revised edition of his geometry, 1888).

* Hutton's Mathematics, edited by Robert Adrain, New York, 1631, YoL I., p. 87o.

* Webber's Mathematics^ Cambridge, N. £., 1806, p. 340.
« Walker's Elements of Geometry, Boston, 1831, p. 30.

» Bchnyler's Elements of Geometry, 1876, p. 33.

^ Elements of Geometry, Plane and Spherical Trigonometry, Cincinnati, 1858, p. 11.

^ Geometry, Cambridge, 1829, p. 7.

" Elementary Treatise on Plane and Solid Geometry, Boston, 1837.

* Plane Geometry for the use of schools. Concord and Boston, 1841.

w Elementary Coarse of Geometry to the use of schools and colleges, KewToifc:,
1847.

u Mathematical Dictionary, Article, « Parallel Lines.''

" Treatise on Plane and SoUd Geometry, *' Ray's Series," Cincinnati, 1864.

u Elementary Geometry and Trigonometry, Boston, 1878.

^ Elements of Plane and Solid Geometry, Boston, 1878. All editions of this moot
popular book, except the rsHssci edition of Jane, 1888, contain the aboTe definition of
parallels.

378 TEACHIKG AKD HISTORY OF ICATHEMATICS.

on geometry by J. M. Wilson, 1868, bat in his new book of 1878 the
whole theory of direction is ignored.

n.— -NEW "AXIOM."

Eadid proves in his Elements (I, 27) that " If a straight line fklling
on two other straight lines make the alternate angles eqnal to one
another, the two straight lines shall be parallel to one another." Bat
before any other step can be made, it is necessary either to prove or
assame that in every other case the two lines are not paralleL

BeiDg nnable to prove this, Enolid assumed it. His assumption con-
sists in what is generally called the twelfth, by some the eleventh
" axiom :" << If a straight line meet two straight lines so as to make the
two interior angles on the same side of it taken together less than two
right-angles, these straight lines, being continnally produced, shall at
length meet on that side on which are the angles which are less than
two right-angles." It has been validly urged against Euclid that this
stateiment is far from being axiomatic. But H. Hankel* has shown
that Euclid himself placed this among the postulates (where it more
properly belongs) and not among the axioms. The mistake of calling
it an axiom was due to later editors. Euclid thus placed the whole
difficulty of parallel lines in an assumption.

It has been objected that this assumption is not sufficiently simple
and obvious. Accordingly, Playfair proposed the following ** axiom":
^^ Two straight lines which intersect one another can not be both par-
allel to the same straight line." This is merely Euclid's *^ axiom " in a
better and more obvious form. It has been adopted by the best Amen-
can works on geometry.t

A large number of our geometries give neither Euclid's nor Playfiftii's
^< axiom," but pretend to prove some ^^ theorem " which states, in sub-
stance, what IS equivalent to Euclid's *^ axiom." This leads us to the

next heading.

in.—" PEooPs."

Since neither Euclid's nor Flayfa.ir's " axiom " is axiomatic, innumer-
able attempts have been made to prove one or the other. Until within
twenty years it was believed by many leading mathematicians that
valid proofs could be deduced from reasonings on the nature of the
straight line. But the researches which led to the development of non-
Euclidian geometry have, at last, made it clear that all such attempts
must necessarily remain fruitless.

We shall call attention to a few so-called proofs found in text-books
used in this country. Hutton| proves the " theorem " that " when a line

*Vorleiungen Uher Complexe Zdklen wnd ikre FwiktianeUf p. 53.

tThe writer has seen it in the geometries of Dayiesi F. H. Smith, Venable, Loomis,
Chaayenet, Hanter, Brooks, Qmnd, Newcomb, Halsted, and Wentworth (in his re-
vised edition, 1888).

t Hntton's Mathematics, edited by Bobert Adrain, New York, 1831, Vol. I, p. S88.

HISTOfilCAL ESSAYS. 379

intersects two parallel lines, it makes the alternate angles equal to each
other '' (which is the equivalent of Euclid's '^ axiom ^) — iin the following
manner : If angles AJEF and IBFD are not equal,
<< one of them must be greater than the other; let it
be HFDj for instance, which is the greater, if pos- A-
sible, and conceive the line FB to be drawn, cut-
ting off the part or angle FFB equal to the angle C-
ABFj cmd meeting the line AB in the paint B. Then,
since the outward angle AEF^ of the triangle BBF^
is greater than the inward opposite angle FFB (th. 8) ; and since these
two angles also are equal (by the construction), it follows that those
angles are both equal and unequal at the same time, which is Impossi*
ble. Therefore, the angle EFD is not unequal to the alternate angle
AFF; that is, they are equal to each other." The error of this proof
lies in the (implied) statement that the line FB must alwayd intersect ^
the line A B, which is, virtually, an assumption of the thing to be proved.
We know that in pseudo-spherical geometry one of the angles (say JS^f!Z>)
is greater than the other, and that the line FB does not cut AB,

This same proof is given in Davies' Elementary Geometry, p. 26. At«
tempts at proving the ^* parallel-axiom ^ were made also by Hassler, by
a writer (James Wallace) in the Southern Iteview (Yol. 1, 1828), and by
A. 0. Twining, in Silliman's Journal (1846, pp. 47 and 89).

Olney* proves Playfair's axiom in this way : " Let AB be the given
line, and O the given point, there can be one and only one perpendic-
ular through O to AB (127). Let this be FF. Now through O one and
onlyoneperpendicularcanbedrawntoJOf. Let this be OD. ThenisOD
parallel to AB by the proposition (just proved in the book), and it is
the only such parallel, since it is the only perpendicular to FF at O.^
The fallacy of this lies in the assumption that every line in the plane,
drawn through the point Q and not cutting the Une AB must neces-
sarily be perpendicular to FF.

An interesting attempt to prove Euclid's axiom is given anonymously
in Orelle's Journal (1834), and translated and published by W. W.
Johnson in the Analyst (Vol. Ill, 1876, p. 103). According to De Mor-
gan, this proof is due to Bertrand. Professor Johnson says that
^Hhis demonstration seems to have been generally overlooked by writ-
ers of geometrical text-books, though apparently exactly what was
needed to put the theory upon a perfectly sound basLs." The error in
the proof seems to lie in the statement that, if lines AB and OD in a
plane lie on the same side of the line AO and are equally inclined to it,
then the infinite space BAOD must always be less than the infinite
space BAFj provided only that angle BAF be not taken less than
angle BAO. That this is not true in spherical geometry, is seen very
readily; nor is it true in pseudo-spherical geometry.

* Treatise on Special or Elementary Geoiiietry, Unlyenity edition, New York, 1872,
p. 70. In later editions this proof is omitted and Playfair's axiom assumed.

380 TEACHING AND HISTORY OF MATHEMATICS.

Those authors who adopt the idea of << direction," and define par-
allels as lines having the same direction, dispose of the whole subject
in a trice. To .them the theory of parallels gives no trouble. The
difficulties of the subject are all hidden from sight by the notion of " di-
rection." The following is the proof given by Hayward* of a '* the-
orem" which says, in substance, the same thing as the ''parallel-axiom."
<^ The straight line has the same direction in every part. An angle is
the inclination of one straight line to another ; that is, the inclination
to each other of these two directions. Two parallel straight lines have
the same direction. Therefore, a straight line (which has but one di-
rection in every part), meeting two straight lines which have but one
direction in all their partS; must have the same inclination to both.
That is, when a strmght line meets two parallel etraight UneSj the angles
which it makes with the one are equal to those which it malces with the other.
Olearer evidence of the truth of this proposition can not be desired."
A little further on we shall consider the question, Is the directional
method scientific I

A mathematician whose attempts to prove the ^^parallel-axiom" were
awarded with the most fruitful results, wasM. Legendre. In the earli-
est editions of his celebrated Elements, he makes a direct appeal to the
senses. In the seventh edition he assumes that a magnitude increases
without limit when perpetual increase is all that is demonstrable. Bat
his early proo& of the ^^ parallel-axiom " did not satisfy even him, and he
temporarily returned to Euclid's mode of treating parallels. Farrar's
second edition of Legendre, brought out in 1825, contains this last pre-
sentation of the subject. Further investigations led Legendre to the
beautiful result that the theory of parallels can be strictly deduced, if
it can previously be shown that the three angles of a triangle are equal
to two right angles. In Farrar's Legendre of 1831 and 1833 is given
Legendre's attempt to prove this theorem without previously assuming
the ^< parallel-axiom." The attempted proof is somewhat long, and in-
troduces an infinite series of triangles. In Volume XII of the Memoirs
of the Institute is a paper by Legendre, containing his last attempt at
a solution of the problem. Assuming space to be infinite, he proved
satisfactorily that it is impossible for the* sum of the three angles of a
triangle to exceed two right angles ; and that if there be any triangle
the sum of whose angles is two right angles, then the same must be
true of all triangles. But in the next step, to show that this sum can
not be less than two right angles, his demonstration failed.

IV.— EEOENT EESULTS.

Some years before Legeudre completed the above investigations, Lo-
batchewsky of Bussia adopted the bold plan of constructing a geom-
etry without assuming the parallel-axiom. He succeeded in this, and

* Elements of Qeometry, p. ix«

HISTOBICAL ESSAYS. 381

•

it opened np the subject of non-Euclidian geometry. His discoveries
were first made public in a discourse at Kasan, February 12, 1826. We
can not here discussthe investigations on this subject that were made
by Lobatchewsky, Gauss, Bolyai, Beltrami, Biemaun, Helmholtz, Klein,
and our own STewcomb.*

We shall only state that the possibility of constructing geometries
upon different assumptions than those made by Euclid has become evi- ^
dent. We know now that, assuming space to be of uniform curvature,
there are really three sorts of geometries possible — ^those of spherical
space, of Euclid's space, and of pseudo-spherical space. Each of these
is consistent in itself. These three do not contradict each otiier, bat
form rather one great system of which each is only a special case.

Much light has been thrown by the above generalizations upon the
vexed subject of geometric ^^ axioms." Do our more recent text-books
profit by these researches t Some of them do. Take, for instance,
Newcomb's Elementary Geometry. On page 14 we read, " We are to
think of the geometric figares as made of perfectly stiff lines which can
be picked np from the paper and moved about without bending or
undergoing any change of form or magnitude." This statement em-
braces a property that is a common characteristic of all three geome-
tries mentioned above, and distinguishes that group from any other
which might be conceived, namely, the property that a figure can
be moved about without undergoing either stretching or tearing.f
<^ Through a given point one straight line can be drawn, and only one,
which shall be parallel to a. given straight line." The assumption,
<^ one straight line can be drawn," divides the Euclidian and pseudo-
spherical geometries from the spherical geometry ; for in the last there
are no (real) straight lines that are parallel to each other. The assump-
tion contained in the words <^ and only one," separates Enclidian geom-
etry from the pseudo-spherical. In the latter more than one line can
be drawn through the same i>oint, none of which intersect a given
straight line. The assumptions thus made completely define the geom-
etry of Euclid from the other two. A good statement of the assump-
tions about Euclid's space is found also in Halsted's Geometry.

This may be a convenient place to inquire into the scientific value of
the term ^^ direction " as a fundamental geometric concept. Professor
Halsted says, in the preface to his geometry : ^^ In America the geome-
tries most in vogue at present are vitiated by the immediate assumption
and misuse of that subtle term, 'direction;' and teachers who know
something of the non-Euclidian, or even the modern synthetic gometries,
are seeing the evils of this superficial < directional' method. • ♦ ♦

* For a bibliograpliy of hyper-space and non-Eaclidian geometry, see a paper by
George Brace Halsted in tbe American Journal of Mathematics, Vol. I, pp. 261-276,
and pp. 384, 385 ; Vol. II, pp. 65-70.

t The property that figures can be moved about '' without bending'' distinguishes
the geometry on an ordinary plane or on a sphere from that on a surface like the cone.

382 TEACHING AND HISTORY OF MATHEMATICS.

The present work^ composed with special reference to use in teaching^
yet strives to present the elements of geometry in a way so absolutely
logical and compact, that they may be ready as rock-foundation for
more advanced study." We quote on this subj eo t also from a promineDt
German, work of Dr. Wilhelm Killing.* <^ The attempts to establish a
natural basis for geometery have, thus far, not been accompanied with
desired success. The reason for that lies, in my opinion^ in this, that
even as geometry has been compelled to abandon the concept of direc-
tion {Begriff der Bichtung) in the senserequired by the paraUel-aziom, so
it will not be able to bold on to the concept of distance {Begriff de9 Alh
$tani69) as a fundamental concept, and musty therefore, pass far beyond
the non-Euclidiau forms of space {Baumformen) in the narrower sense."
ThuS| accouiing to Dr. Killingi geometry has discarded the term direc-
tion as a fundamental concept.

There are several objections which can be urged against the term
<^ direction." When we think of two straight lines as having diOGBrent
directions, we imagine ourselves placed on the point of intersection and
looking along one of the lines, then the other. The term seems clear
as long as we apply it to lines which intersect each other, or which
coincide with one another. In the latter case we say that the two lines
have the same direction. But we, as yet, have no geometric meaning
of the phrase <f the same direction," except when used of lines having
a common point. Simply because lines which intersect each other have
different directions, we can not logically conclude that lines which do
not meet each other have the same direction. This objection was urged
by De Morgan twenty years ago in his review of J. M. Wilson's geome-
try.t Says be, ^^ There is a covert notion of direction,- which, though
only defined with reference to lines which meet, is straightway trans-
ferred to lines which do not. According to the definition, direction is
a relation of lines which do meet, and yet lines which have the same
direction can be lines which never meet. • • * How do you know,
we ask, that lines which have the same direction never meet T Answer-
lines whichmeet have d^^Tisr^^directions, We know they have; but how
do we know that, under the definition given, the relation called direction
has any application at all to lines which never meet T The notion of
limits may give an answer ; but what is a system of geometry which
introduces continuity and limits to the mind as yet untaught to think
of space and of magnitude t "

Benjamin Feirce says, in the preface to his geometry, ^< The term
direction is introduced into this treatise without being defined ; but it
is regarded as a simple idea, and to be as incapable of definition as
lengthy breadth j and thickness.^ But in case of length we have clear and
rigorous means of testing by the method of superposition whether two
lengths are equal or unequal. The same is true of breadth and thick-

*Die Nioht'Euclidischen Baumformen in Analytischer BehandlunQf Leipzig, 1885| p. ir.
t Athenffium, Jaly 16, 1868.

HISTOBICAL SSBAT8. 38S

Z1689. In oaae of directionj on the other hand, comparisons cannot
always be institatedy at least not withont becoming involved in logical
difficulties. We have no satisfactory means of telling whether two non-
intersecting lines in a plane have the same or different directions. We
are not even sure that the relation of direction can be applied to them.
On a pseudo-spherical surface a whole pencil of lines can be drawn
through a given point which do not intersect a given line. The lines in
this pencil do not have the same directions with respect to one another.
The question then arises^ which one, if any, of these lines in the pencil
has the same direction as the given line T If we can not distinguish
between the presence and absence of a quality, then that quality is
useless.

But suppose that, for the sake of argument, we waive the above
objection, and say that parallel lines have the same- direction. After
defining straight line, angle, parallel lines, in accordance with the concept
of ^^ direction," we can reason in the same way as Hay ward does tn the
quotation given under the third heading. But that mode of treating
parallels excludes the possibility of the existence of pseudo«spherical
geometry, inasmuch as it renders absurd the statement that two or
more lines intersecting one another may exist, none of which intersect a
third line, for lines in a plane which have different directions with re-
spect to one another cannot all have the same direction with respect to
a third line. The above use of the term direction involves assumptions
as to the character of space which are too narrow to admit the use of
that term as a fundamental concept. As far as possible, our Euclidian
geometry should be made to rest upon concepts which need not be
abandoned when we take a generalised view of the sdenoe. Our treat-
ment of the elements should be a ^^ rock-foundation for more advanced
study."

One of the many objections to all attempts to found the elements of
geometry on the word ^' direction" is stated by Professor Halsted in the
following manner:* ^^ Direction is a common English word, and in
Webster's Dictionary, our standard, the only definition of it in a sense
at all mathematical is the fourth : ^ The line or course upon which any-
thingis moving * • • ; as, the ship sailed in asoutheasterly direction.'
Direction, to be understood in any strict sense whatever, posits and
presupposes three fundamental geometric ideas, namely, straight line,
angle, parallels. After the theory of parallels founded upon an explicit
assumption has been carefully established, a strict definition of direc-
tion may be based upon these three more simple concepts, and we may
use it as Bowan Hamilton does in his Quaternions. But in American
geometries, for example Wentworth's, the fallacy petitio principii is
three times repeated by defining the three component parts of direction,
each by direction itself."

Professor Halsted objects also to the word ^< distance" as afunda-

* Letter to the writer, November 17| 1888.

884 TEACHINa AND HIBTOBT OF MATHEBfATICS.

mental idea. He says, in the preface to his elements, that the attempt, i
*^ to take away by definition from the familiar word < distance ' its ab-
stract character and connection with length-nnits, only confuses the
ordinary student A reference to the article < Measurement,' in the j
EncydopsBdia Britannica, will show that around the word < distance '
centers the most abstruse advance in pure science and philosophy. An
elementary geometry has no need of the words < direction ' and < dis-
tance.' " This view receives support from Dr. Killing in the above quo-
tation. Professor Halsted has introduced the new word seety meaning
*^ part of a line between two definite points," and corresponding to the !
German word StreoTce. The objections to the word distance are stated
by him in the following words : * <* Distance is also a common English
word, and Webster as its first definition gives, ^ An interval or space
between two objects ; the length of the shortest line which intervenes
between two things that are separate. Every particle attracts every
other with a force • • • in versely proportioned to the square of the
distance. Newton.' Thus, distance posits shortest line and lengthy there-
fore measurement, therefore ratioj never treated before the fifth book in
the Euclidian geometry, and never adequately treated at all in any other
geometry without the use of the whole theory of limits. Yet American
geometries, for example Wentworth's, give in place of the well-known
simple proof of the theorem that any two sides of a triangle are together
greater than the third, the abstruse assumption ^ a straight line is the
shortest distance between any two points,' and that after having ex-
plicitly said that there is only one dietanee between two points."

Before concluding this essay we should like to express our belief that
detailed discussions of the ftindamental geometric concepts should be
avoided with students beginning geometry. Such discussions can be
carried on with more profit when reviewing the subject near the end of
the course, or when beginning the study of non-Euclidian geometry.
In this connection I can not forbear quoting from a letter of Dr.E. W.
Davis, of the University of South Carolina. Says he, '^ This getting
down to the ultimate basis of our assumptions is a long and painM
process, and should not be insisted upon in elementary instruction.
The first beginning in mathematical reasoning should be reasoning that
shows the student facts that are new to him. It disgusts him to have
continually jproi^tfd to him what he has always hnoiDny ox to begin by
asking him to doubt what he can not help but deem true in spite of all
our fine logic. Confidence in logic should be gained by long experience
in predicting by it the unforeseen j before we proceed by it to invalidate
deeply-rooted and universally cherished conceptions." While we fully
indorse these views, we at the same time insist upon a scientific treat-
ment of geometry in our text-books, for the two following reasons : (1)
When the student advances to a more generalized view of the subject,
he will find that his first studies in this line rested upon a rock-founda-
tion, and that the old edifice can be enlarged without being first de-

* Letter to the writer^ November 17, 1888.

HISTORICAL ESSAYS. 385

rss:* molished; (2) A teacher, like an honest preacher, prefers to teach doo-
<uc trine which is, to the best of his knowledge, logically and philosophy
• " ically true.

^'sam:

•►

ON THE FOUNDATION OV ALGEBRA.

From Peacock's Report to the British Association, in 1833, on the Re-
cent Progress and Present State of Certain Branches of Analysis (p. 188)
we quote the following words : ^^Algebra was denominated in the time
of Kewton specious or universal arithmetioj and the view of its principles
which gave rise to its synonym has more or less prevailed in almost
every treatise upon this subject which has appeared since his time. In a
similar manner, algebra has been said to be a science which arises from
that generalization of the processes of arithmetic which results from the
use of symbolical language } but though in the exposition of the prin-
ciples of algebra arithmetic has always been taken for its foundation,
and the names of the fundamental operations in one science have been
transferred to the other without any immediate change in their mean-
ing, yet it has generally been found necessary subsequently to enlarge
this very narrow basis of so very general a science, though the reason
of the necessity of doing so, and the precise point at which, or extent
to which) it was done, has usually been passed over without notice."

From the same Report (p. 284) we quote the following : " In the
early part of the last century the algebra of Maclaurin was almost ex-
clusively used in the public education of this country. It is unduly
compressed in many of its most essential elementary parts, and is un-
duly expanded in others which have reference to his own discoveries.

• # • X^ was, subsequently, in a great measure susperseded, in the
English universities at least, by the large work of Saunderson. It was
swelled out to a very unwieldy size by a vast number of examples
worked out at great length ; and it labored under the very serious defect
of teaching almost exclusively arithmetical algebra, being far behind the
work of Maclaurin in the exposition of general views of the science."

There was indeed, in those days, some opposition at Cambridge (Eng-
land) to the use of negative quantities in algebra. Among Cambridge
algebraists who argued against the use of such quantities were Baron
Francis Maseres (fellow of Clare), author of a dissertation on the nega-
tive sign in algebra (1758), and W. Frend, author of Principles of Alge-
bra (1796-99). Both of these persons set themselves against Saunder-
son, Maclaurin, and the rest of the world ; for they rejected negative
quantities no less than imaglnaries ; and, like Robert Simson, << made
war of extermination on all that distinguished algebra from arithmetic."*

The algebras studied by the early teachers and pupils in this country
were all English works. Maclaurin, Saunderson, Thomas Simpson,

* SoholoB Ac€Ldem%o(B : Some Acconnt of the Studies at EngUsli Uniyersitiea in the

Eighteenth Century, by C. Wordsworth, 1877, p. 68.

881— No. 3 26

886 TEACHING AND HISTORY OF MATHEMATICS.

HattoDy Bonnycastle, and Bridge were authors that coald be foand in
the libraries of onr American professors of mathematics As pointed
oat by Peacock^ these anthers began their treatises with arithmetical
algebra, but gradually and disguisedly introduced negative quantities.

It is to be expected that onr early compilers of algebra and writers
on mathematics should imitate the EDglish. The first publication in
this country of a mathematical work which can, perhaps, lay some little
claim to originality, was the work by Jared Mansfield, entitled, Essays,
Mathematical and Physical, Containing New Theories and Illustrations
of some very Inportant and Difficult Subjects of the Sciences.*

In the first essay, Mansfield says that ^^ affirmative quantities are to
be added, negative ones to be subtracted.." Negative quantities ^^ can
never exist alone or independently ; • • • for to suppose a com-
pound where the elements have been all exhausted by the diminishing
quantities, and something still left, would be very absurd. This, how-
ever, may be the case apparently, and in reality no absurdity follow.
Thus the case above mentioned, 8—12, is absurd in itself, when pore
numbers are considered ; but an algebraist who knows how often the
signs are changed in order to develop the unknown quantity, and that
the quantities are often assumed without knowing on which side the
difiSerence lies, views this expression as nothing else than the difference
of 12 and 8, or as 12—8 ; for those terms which have the sign -f prefixed
to them, have precisely the same effect on those to which the sign — is
prefixed, as those which have the sign— on those which have the sign
+• The signs are totally indifferent, excepting as to the operationS|
and where no operation is to be performed they are to be neglected."

These views suggest an algebra purely arithmetical, which finds it as
impossible to give a clear explanation of negative quantities as it would

of the imaginary V^. In fact, negative quantities are the true <'im-
aginaries" of such an algebra.

Day's Algebra contains a detailed discussion of positive and nega-
tive quantities. "A negative quantity is one which is required to be
subtracted. When several quantities enter into a calculation, it is fre-
quently necessary that some of them should be added together, while
others are subtracted. The former are called affirmative or positive,
and are marked with the sign + ; the latter are termed negative, and
distinguished by the sign — ." But when a negative quantity is greater
than a positive, how can the former be subtracted from the latter!
"The answer to this is, that the greater may be supposed first to exhaust
the less, and then to leave a remainder equal to the difference between
the two." The interpretation of positive and negative quantities is then
given by employing the ideas of gain and loss, ascent and descentj norUk
and south latitude, etc.

* The work contains eight essays. Their titles are as follows : (1) On the Use of
the Negative Sign in Algebra, (2) Goniometrical Properties, (3) Naatioal Astronomy,
(4) Orhicalar Motion, (5) Investigation of the Loci, (6) Fluxionary AnalysiB, (7)
Theory of Gunnery, (8) Theory of the Moon.

HISTORICAL ESSAYS. 387

The treatment of this subject in Day's Algebra ik essentially the same
as that given by all American books, excepting those of recent date.
It is only within the last ten or fifteen years that our writers on alge-
bra (such as Olney, Wentworth, Wells, Thomson and Quimby, Bow-
ser, Ifewcomb, Oliver, Wait, and Jones, Van Velzer and Slichter),
have explicitly eusumed the existence of negative as well as positive
quantities at the very beginning of their text-books, and have clearly
explained that the series of algebraic numbers is assumed as going out
from indefinitely in both directions, a*nd that the signs + and — are
used not only as signs of operation to indicate addition and subtraction,
but also as signs of quality to indicate the nature of the quantities as
positive or negative.

The algebra that is usually found in our school-books, wherein quan-
tities are considered as being one or the other of two diametrically
opposite kinds, has been called single algebra. It differs from pure
arithmetio in assuming the existence not only of positive, but also
of negative quantities. Neither pure arithmetic nor single algebra is
perfect in itself, since each leads to expressions that are meaningless. In
pure arithmetic, a— &, whenever a< &, expresses an impossibility and is
an '^imaginary value.'' In single algebra this ceases to be impossible,
but there we are led to another impossibility, another ^^ imaginary,"

namely, \/^.

By proceeding one step further in our generalization we come to
double algebra J in which the existence of complex quantities (of the form
a ± V~-^ b) is assumed. Geometrically, such a quantity represents a
line of definite length in some one definite direction in a plane. This
algebra is capable of giving meaning to all the expressions to which it
leads and is, therefore, perfect in itself. Some of our recent text-books
(as Wentworth's, Bowser's, NewcomWs, Van Velzer and Slichter's, and
especially Oliver, Wait, and Jones's) give a more or less complete ac-
count of this kind of algebra in their chapters on imaginaries. Triple,
quadruple (quaternions), and other multiple algebras have been in-
vented.

It will be seen that the true foundations of algebra have not been
understood before the present century. The theory of imaginaries in
double algebra has been developed chiefly by Argand, Gauss, and
Oauchy. The philosophy of the first principles of algebra has been
studied by Peacock, De Morgan, Hankel, and others. They established
the three great laws of operations, {. e., the distributive, associative,
and commutative laws. A flood of additional light on this subject was
thrown by the epoch-making researches of Hamilton, Grassman, our
own Peirce, and their followers. They conceived new algebras, whose
laws differ from the laws of ordinary algebra.*

* An excellent historical Hketch of Mnltiple Algebra, by J. W. Gibbs, of Yale, will
be fonnd in the Proceedings of the American Association for the Advancement of
Soience, Vol. ZXXV, 1686,

388 TEACHING AHB HI8TOBT OF lCATHElCATrC&

DIFFERENCE BETWEEN NAPIEK8 AND NATURAL LOGA-

RITHMS^

The term <^ Napierian logarithms " has been used in three diffisrant
flenses : (1) as meaning Napier's logarithms, or the ones invented by him
and published in 1614 in his Mirifiei Logarithmorum Oanonis Deser^
Uo ; (2) as a synonym for ^' natural logarithms f (3) as conveying the
first and second meanings combined, and, thereby, implying that the nat-
ural logarithms are the ones invented by Napier. Though this last use of
the term is inadmissible, because the logarithms invented and published
by Napier are really different from the natural logarithms, it has, never-
theless, been the most prevalent; especially has it been iirevaleDt in
this country.

An examination of the algebras which have been in use in our schocds
will at once coDvince us that this error has been very general. We may
consult the algebras of Bay, Greenleaf, Flcklin, Schuyler, Loomis, Bob-
inson, F. H. Smith, Hackley, Davies, Bowser, Stoddard and HenUe,
Thomson and Quimby, and many others, and we find it stated eitiier
that Lord Napier selected for the base of his system e = 2.718 . • ., or
that he assumed the modulus equal to unity. Either of these two stEtte-
ments is equivalent to saying that the logarithms invented by Napier
are identical with the natural logarithms. Some authors make state-
ments like the following one, taken from the revised edition of Wells's
University Algebra (p. 303): ^^The system of logarithms, which has e
for its base, is called the Napierian system, from Napier, the inventor
of logaritiims."

The objection to statements like this is that they almost invariably
mislead the student. What inference is more natural than that Napie-
rian logarithms were invented by Napier f Some explanation ought
therefore to be made guarding against this error.

But I have seen only two American books doing this, namely, J. M.
Peirce's Mathematical Tables, and Van Velzer and Slichter's Course in
Algebra (of which a preliminary edition has just appeared). In these
two books the truth is conveyed in plain words that Napier's logarithms
differ from the natural. It is the object of this article to explain that
difference.

It is important to note that, in Napier's time, our exponential nota-
tion in algebra had not yet come into use. To be sure, Stifel in Ger^
many and Stevin in Belgium had, previous to this, made some attempts
at denoting powers by indices; but this notation was not immediately
appreciated, nor was it generally known to mathematicians, not even
to the celebrated Harriot, whose algebra appeared long after Napier's
death. It is one of the greatest curiosities in the history of mathe-

* This article has been published in the Mathematical Magazine, Vol. II, No. 1, and
is here reprodnoed with some very slight changes.

HISTORICAL E8SAT8. 389

matics that logarithms should have been oonstraoted before exponents
were used. We know how naturally logarithms flow from the exponen-
tial symbol, but to Napier this symbol was entirely unknown.

The interesting inquiry then arises, What was Napier's treatment of
logarithms t It may be briefly stated as follows:

A o B

\ 1 \

D F

I ,-

Let AB be a line of definite length, DE a line extending from 2> in-
definitely. Imagine two points set in motion at the same time, and
with the same initial velocity; the one point moving from D toward E
with uniform velocity; the other from AtoB with a velocity decreasing
in such a way that when it arrives at any position, 0, its velocity is
proportional to the remaining distance, BG. While the latter point
travels a distance, AO^ suppose the former to move over the space DF.
Napier called DF the logarithm of BO. He first applied this idea to
the calculation of a table of logarithms for the natural sines in trigo-
nometry. In the above figure, AB would represent the sine of 90^ or
the radius, which was taken by him equal to 10,000,000 or lO''. BO would
be the sine of an arc, and DF its logarithm.

^^The logarithm^ therefore, of any sine is a number very nearly ex-
pressing the line which increased equally in the meantime, while the
line of the whole sine decreased proportionally into that sine, both mo-
tions being equal-timed, and the beginning equally swift."*

This treatment of the subject is certainly very unique. Let us now
establish the relation between these Napierian logarithms and our nat-
ural logarithms. Let w=AB, xssDF^ y=BOj then AO=sm^y. The

velocity of the point is \^7^ =ry> ^ being a constant. Integrat-
ing, we have

—Nat log y=:rt+o.

When <=0, then y=»», and c=— Nat log m. The velocity of the point
(7 is rm, when ^=0. Since the two points start with the same velcity,

we have ^=:rm as the uniform velocity of the point F. Hence x=s

rmt Substituting for t and o their values, and remembering that, by
definition, a7=Nap log y, we get

Nap log y=sm Nat log ~

The constant m was taken eqoal to 10^ Sabstitnting we get

Nap log ysrlO^Nat log —

as the equation expressing the relation between Napierian and natural
logarithms. nf i'«»

* Definition 6, p. 3, of Napier's Jfii^ot Logariihmorum Cawmii De9onpHo, do., 1614.

390 TEACHING AND HI8T0B7 OP MATHIJMATICS.

That there is a difference between the two is evident at once. We
easily observe the characteristic property of Napierian logarithms, that
they deorease as the namber itself increases. This property alone should
have been a sufficient guard against declaring the two systems identi-
cal. The Napierian logarithm of 10'^ is equal to zero. The Napierian
logarithms of numbers smaller than it are positive; those of numbers
larger than it are negative, or, in the language of Napier, *^ less than
nothing." In further illustration we give the following:

Nap. log. 1 = 161 180 956.509; Nat. log; 1 =

Nap. log. 2 = 154 249 484.703 ; Nat. log. 2 = 0.6 931 472

Nap. log. 10 =: 138 155 105.578; Nat. log. 10 =r 2.3 025 851

The question may be asked what &a«e Napier selected for his system.
We answer that he did not calculate his logarithms to a base at all. He
never thought nor ever had any idea whatever of a hose in connection
with logarithms. The notion of a base suggested itself to mathema-
ticians later, after the algorithm of powers and exponents, both inte-
gral and fractional, had come to be better understood.

If we inquire what the base to the logarithms in Napier's tables
would have been had he used one, then it will be found that it does not
coincide with the natural base e, but is very nearly equal to its recip-
rocal. In theory, that base is exactly equal to the reciprocal of e, as
will be seen from the following relation,* which is merely another form

of the one given above,

Nap log y

y ^r^\ — w —

The base ~ would not lead accurately to Napier's logarithmic figures,

because the inventor's method of calculation was necessarily some-
what rude and inexact. The modulus of his logarithms is not equal

tol, but nearly equal to — 1. If the base were exactly ~, then the

modulus would be exactly — 1 ; for the modulus of any system of loga-
rithms is the logarithm, in that system, of the Napierian base e.

The first calculation of logarithms to the base of the natural system
was made by John Speidell in his New Logarithms, published in Lon-
don, in 1616, or five years after the first appearance of Napier's loga-
rithmic tables.t

* To make the theory of exponents applicable to Napier's logarithms, it becomes
necessary to divide the namber y by 10^, otherwise the base raised to the zero power
would not be equal to anity. This division really amounts to making the length of
the line AB equal to 1 instead of 10^ If tbis be done, then Nap. log, y must also be
divided by 10^, so as to retain the inventor's conception that the two points on the
lines AB and DE, respectively, move with eqnal initial velocities.

t The error of calling the Napierian and natural logarithms one and the same sys-
tem has been wide-spread. We may pardon the celebrated Montucla, the eldest
prominent writer on the history of mathematics, for making this mistake (MontaolAi

mSTOBICAL ESSATa 391

CIRCLE 8QUAEER8.

It would be strange if America had not produced her crop of " circle-
squarers," just as other countries have done. Our history of them will
be quite incomplete. We have not gone out of our way to seek the ac-
quaintance of this singular race of " mathematicians,'^ nor have we
avoided them. A few individuals have come across our path, and we
proceed to tell about them for the benefit and edification not so much of
mathematicians as of psychologists. The mathematician contemplates
the products of only sound intellects 5 the psychologist studies also the
utterings of minds that are or seem to be diseased.

The history of the quadrature of the circle is not without its sober
lessons to mathematicians. It extends back through centuries almost
to the beginning of geometry as a science.

The student of the history of mathematics is impressed by the fact
that this science, more than any other, has always been a progressive
one. He does not find a period in authentic history during which
mathematics was not cultivated quite successfully by some nation or
other. The earliest contributions were made by the Babylonians and
Egyptians, then came the Oreeks, then the Hindoos, then the Arabs,
and finally the Europeans. Like metaphysics, mathematics has en-
countered fundamental problems apparently of insurmountable difGL-
culty. But it has generally had the good fortune to perceive that for-
tifications can be taken in other ways than by direct attack with open
force ; that, when repulsed from a direct assault, it is well to reconnoitre

Hiatoire dea Maih^maiiques, Tome II, Paris, 1758, p. 21), but there is hardly any
excuse for a modem writer, such as Hoefer (Hiatoire dea Math^atiquea depuia leurs
OrigincB juaqu^au Commencement du Dix-neuvieme Sihle, Paris, 1874, p. 378), for
stumbling over the same stone. The difference between the two systems was pointed
out in Germany by Karsten in 1768, Kaestner in 1774, and Mollweide in 1808, but
no attention was paid to their writings on this subject. A lucid proof of the non-
identity of the two systems was given by Wackerbarth (** Logarithmea Hyperboliquea
et Logarithmea N^pirienaj" Les Mondes, Tome XXVI, p. 626). The French mathema-
tician Biot wrote likewise on this subject {Journal dea Savanta, 1835, p. 259), as did
also De Morgan in England (English Cyclopsedia, Article "Tables''). Still more
recently attention has been called to this matter by J. W. L. Glaisher (Enoyolo-
ptedia Britannica, 9th ed., Article '^ Logarithms''), and by Siegmund Guenther ( Unter^
auchungenzur Geachichte der maihemaiiachen Wiaaenachaftenf Leipzig, 1876, p. 271). The
writings of these scientists do not seem to have received the attention they deserve,
and the erroneous notion of the identity of Napierian and natural logarithms still
continues to be almost universal. 1

Napier's original works on logarithms are very scarce. The Afiryioi Logarithmarum I

Canonia Deacriptio, etc., Edinburgi, 1614, can be found in the Congressional Library
in Washington and in the Bidgway Library in Philadelphia. The latter library has .
also the English edition of the above work, translated by Edward Wright in 1616.
** So rare are these original editions that, of the two greatest historians of logarithms,
Delambre never saw the Latin edition and Montucla never heard of the English,"
(Mark Napier's Biography of Lord Napier, p. 379).

392 TEACHING AND HISTORY OF liATH£\IATICS.

and o6cnpy the sarroandiDg country and dificover the secret paths by
"Which the apparently unconquerable position can be taken.*

From this we can draw the valuable lesson that it is not always best
to " take the bull by the horns.^'

The value of this precept may be seen by giving an instance in which
it has been violated. The history of the quadrature of the circle is in
point. An untold amount of intellectual energy has been expended
upon this problem, yet no conquest has been made by direct assault.
The circle-squarers existed in crowds even before the time of Ar-
chimedes and in all succeeding ages in which geometry was cultivated^
down even to our own. After the invention of the differential calculus
abundant means were introduced to complete the quadrature, if such a
thing were possible. Persons versed in mathematics became convinced
that the problem could not be solved, and dropped it But those who
still continued to make attempts upon this ^^ enchanted casUe," as it
was supposed to be, were completely ignorant of the history of tiie sub-
ject, and generally misunderstood the conditions of the problem. *^ Our
problem,'' says De Morgan,t << is to square the circle with the old oMoto-
ance of means : Euclid's postulates and nothing more. We can not re-
member an instance in which a question to be solved by a definite
method was tried by the best heads and answered at last by that methodj
after thousands of complete failures."

But great advance has been made on this problem by approaching
it from a different direction and by newly discovered paths. Lambert,
an Alsacian mathematician, proved in 1761 that the ratio of the cir-
cumference of a circle to its diameter is incommensurable. Only nine
years ago Lindemann, a German mathematician, demonstrated that
this ratio is also transcendental, and that the quadrature of the circle
by means of the ruler and compass only, or by means of any algebraic
curve, is impossible.X He has thus shown by actual proof that which
keen-minded mathematicians had long suspected, namely, that the
great army of circle-squarers have, for more than two thousand years,
been assaulting a fortification which is as impossible to be torn down
as the firmament of heaven is by the hand of man.

Kow-a days, a person claiming to have solved this problem is ranked
by mathematicians in the same class with inventors of ^< perpetual mo-
tion," and discoverers of the "fountain of perpetual youth." A very
peculiar characteristic of circle-squarers, or quadrators, as Montucla
calls them, is that they cannot be convinced of their errors. The first
American quadrator we shall mention is \Yilliam David Clark Murdock,
who, in a pamphlet of eight pages, bearing no date, gives a Demonstra-
tion of the Quadrature of the Circle.

The next man on our list is John A. Parker, whose work on The Quad-

* H. Hankel, Entwiohelung der Mathemaiik in den Utzten Jahrhunitrienf p. 16.
t English CyclopsBdia; article, "Quadrature of the Circle.''
% MathemaHiohe AnnaleHf Band XX, p. 213.

HISTORICAL ESSAYS. 393

rature of the Circle (1851) was reviewed in the New Bnglander of Feb-
ruary, 1852. The most prominent characteristics of this work, says the
reviewer, are, a contempt for "algebra," and a grudge against "profes-
sors.'* The author proves that all geometers from Euclid to his (Par*
ker's) great forerunner, Seba Smith, have been but blockheads in the
very A B O's of their science. He solves in a twinkling the vexed prob-
lem of the " three bodies." He seems ashamed of his usher, Seba Smith,
and takes him to task for " stealing his thunder.'' Over twenty years of
experience seem to have made him no wiser. In 1874 he republished
his book of over three hundred pages in almost exactly its original
form.

The next publication is the following : The Regulated Area of the
Circle and the Area of the Surface of the Sphere, by Charles P. Bou-
ch6, Citizen of the United States of I^orth America, Cincinnati, 1854.
It covers sixty pages. The author says : " Kotwithstanding the per-
fection at which mathematics may have arrived in rectilineal geometry,
planimetry, and stereometry, yet with regard to the curve line, as well
as the spherical surface, we have remained in great darkness till it
pleased the author of Spheres to afford us some light in this respect,
and from a source little expected, i. e., through the medium of a plain,
but a moderately cultivated, seeker after truth. By the assiduous ap-
plication of mind and the blessing of Ood, I have ultimately succeeded
in correcting some great errors respecting curvilineal geometry P

The next circle-squarer on our list is Lawrence Sinter Benson, the
author of a geometry. In 1879 he published in New York a work called
Mathematics in a Dilemma, in whiQh he also gives an extremely inter-
esting history of his efforts on this subject. He says that after com^
pleting a course of studies at college in 1858, and while residing on his
former place in South Carolina, his mind drifted into geometrical ab-
stra>Btions. He published new modes of demonstrating the quadra-
ture in 1860 and in 1862. He says that he offered << one thousand
dollars to any one who could refute the result which I gave for the cir-
cle, namely, that the perimeter of its equivalent square U exactly equi-
distant heticeen the squares circumscribed and inscribed about the circle ; the
sides of all the squares being respectively parallel. This offer and demon-
stration drew me in many discussions, for mathematicians claim them-
selves able to prove that this intermediate square is just equal to an
inscribed decagon in the circle ; whence they argue that I make the
circle too small. Committees of expert mathematicians— professors in
colleges were selected to decide this issue } but no decision was made.
Therefore, in 1864, while the Civil War was raging, I ran the blockade
and visited Europe, and laid my demonstration before scientific socie-
ties and distinguished mathematicians there." He then says that he
returned to I^ew York and published a simplification of the Elements
of Euclid, with the repudiation of the reductio ad dbsurdum. He says
that these changes met the approval of Professor Docharty of the Col*

\ /

394 TEACHINa AND fflSTOBY OP MATHEMATICS.

lege of the City of New York, and that, in 1873, Charles Davies pub-
lished a book where he also repudiated the (ibsurdum reasoning. <^For
nearly twenty years mathematicians and myself have been at logger-
heads on the issae made by me aboat the circle. I now propose to set
at rest all doubts against the demonstration published by me in 1860
and 1862.'' In more recent years Mr. Benson's efforts to revolutionize
mathematics have been unabated.

Dr. A. Martin tells us of a quadrator who deposited with him a man-
uscript, in 1885, proving that the long sought for ratio is exactly 3^*
Mr. Faber, the writer of it, distinguished himself also in other branches
of mathematical inquiry. In a phamphlet of thirty-four pages, in 1872,
^' Theodore Faber, a citizen of the United States, Few York, " makes
the '^ extraordinary and most significant discovery of a laoMng link in
the demonstration of the world-renowned Pythagorean problem, utterly
disproviDg its absolute truth, although demonstrated as such for twenty-
three centuries. '' In justice to Mr. Benson, it shoidd be remarked
that he, too, is waging war against Euclid, 1, 47. *

* Since writing the above, we have received from Dr. Artemas Martin a copy of
the Notes and Queries, Vol. V, Nos. 6 and 7, June and July, 1888, giving a Bibliogra-
phy of Cyclometry and Quadratures. From this article we see that Theodore Faber
has appeared in print also on the subject of the quadature of the circle. The article
glTes over twenty publications, besides the ones mentioned aboyei by AmerioAii
writers who believe that they have found the true and exact ratio.

APPENDIX.

BIBLIOGRAPHY OF FLUXIONS AND THE CALCULUS.

TSXT-BOOKS PRIKTBD IN THB tJNITED STATBS.

HUTTOK, Cbarlbs. CotcfM of Maihemaiios, in two volumes.

American editions, revised by Hobert Adrain, appeared in 1812, 1816 (f ), and 1832.
Evert Duyckinck brought out an edition in New York in 1828. Another edition ap-
peared in 1831. The second volume contains a short account of the doctrine efflux-
ions) using the Newtonian notation.

ViNCE, Rev. S. The Prindplea of Fluxions, first American edition, corrected and en-
larged. Philadelphia: Eimber&. Conrad. 1812. Pp.256.

Employs the Newtonian notation.
Bezout. First Principles of the Differential and Integral Catirlus, or the Doctrine of
Fluxions, from the Mathematics of Bezout, and translated from the French for use
of students of the University at Cambridge, New England. Boston, 1824. Pp. 195.

This book forms a part of Farrar's Cambridge Mathematics. It is the first work
published in this country employing the notation of Leibnitz and the infinitesimal
method. '^ In the Introduction, taken firom CarnoVs B^flexions sur la MStaphysiquedu
Caloul Infinitisimal, a few examples are given to show the truth of the infinitesimal
method, independently of its technical form." This is done by explaining that there
is a '< compensation of errors.''
Rtak, Jambs. T^e Differential and Integral Calculus, New York, 1828.* Pp. 328.

" The works which I have principally used in preparing this treatise are LacroiX|
Lardner, Boncharlat, Gamier, and Da Bourguet*s Differential and Integral Calculus ;
Lagrange's Caloul des Functions^ Simpson's. Fluxion's, Peacock's Examples on the
Differential and Integral CalouluS; and Hirsoh's Integral Tables" (advertisement).
The first section of the book is given to '' preliminary principles," in which the three
methods of Newton, D'Alembert, and Lagrange are explained. The method adopted
by the author is that of limits, but no formal definition of the term " limit" is given.
The symbol (0), indicating the absence of quantity, is everywhere treated with the
same oourteoy and implicit confidence as though it were actually a quantity. The
inquiry as to whether the laws of analysis are really applicable to this foreign in-
truder into the society of actual magnitudes, or whether it has to be governed by
laws of its own, is nowhere deemed necessary. These remarks apply with equal
force to other works on calculus and to works on algebra.

Young, J. R. Elements of the Differential Calculus, comprehending the general the-
ory of surfaces, and of curves of double curvature. Bevised and corrected by
Michael O'Shannessy. Philadelphia, 1833. Pp. 255.

In the preparation of this American edition, the editor was assisted by Professor
Dod, of Princeton College.

In the explanation of the process of differentiation, he makes h absolutely zero, in
an expression like this :

tjrl:=^Zj^+2xh+h*.

* Byan*a Caloulua is now a rar» book. The copy we have before as was kindly lent to ns by Prof.
•W. Btttherford, of the University of Georgia.

395

S96 TEACHING AND HISTORY OF MATHEMATICS.

'*In both these cases (of whioh that giveu here is one)^ as indeed in e^ery other-
the respective differential co-efficients are only so many particular yalnes of the gen,

eral symbol jr, to whioh ^7"^ always reduces, when fc=0." In the above example

--=3a5'. "The expressions ~ and ^ have, we see, the advantage over the symbol

^i of particularizing the function and the independent variable under consideration

and this, it must be remembered, is all that distinguishes ^ or ^ from ~ for dg.

ax ay

dgf dx, are each absolutely 0." " These differentials, although each = 0, have, never-
theless, as we have already seen a determinate relation to each other (!) ; thus, in
the last example, this relation is such that dy = 25 (a + hx) dx, and, although this
is the same as saying that =2& (a -f 5x) ; yet, as we can always immediately obtain

from this form the true value of ^ or ^. we do not hesitate occasionally to make

dx* - '^

use of it.'^ It will thus be seen that the author has no hesitation whatever in break-
ing up the differential co-efficient.

TOUNO, J. R. Tlie EhnienU of the Integral CalculuSf with its applications to geome-
try and to the summation of infinite series. Revised and corrected by Michael
O'Shannessy. Philadelphia, 1833. Pp. 292.

Davies, Charles. Analytical Geometry and DifferentiaZ and Integral Calculus, 18—.

' Elements of the Differential and Integral Calculus. 1836.

Several editions of Davies' calculus appeared. In the improved edition of 1843

u'—u
(pp. 17 and 18) the author says that flax is the limit toward which the ratio -^

= 2ax + ah approaches in proportion as h diminishes, and hence *' expresses that par-
ticular ratio which is independent of the value of h." Bledsoe objects to this, saying,
" Shall they (teachers) continue to seek and find what no rational beings have ever

found, namely, that particular value of % which does not depend on the value of

At That is to say, that particular value of a fraction which does not depend on its
denominator!'' Davies represents by [dx ''the last value of A, which can not be
diminished, according to the law of change to which ik or x is subjected, without be-
coming 0.'' <'It may be difficult," says the author, " to understand why the value
which h assumes in passing to the limiting ratio is represented hj dx in the first
member and made equal to in the second." To this Bledsoe says : '' Truly this is a
most difficult point to understand, and needs explanation. For if h be made abso-
lutely zero, or nothing on one side of the equation, why should it not also be made
zero on the other side t " ** Why should ' a trace of the letter x ' be preserved in the
first member of the equation and not in the second t The reason is, just because dx is
needed in the first member and not in the second to enable the operator to proceed
with his work."

As regards the conception of the term '' limit," Davies believed that a variable
actually reached its limit. '* The limit of a variable quantity is a quantity toward
which it may be made to approach nearer than any given quantity, and which it
reaches under a particular supposition."*

Davies believed that by the definition of M. Duhamel, according to whioh a varia-
ble never reaches its limit, there seemed to be placed an ** impassable barrier" be-
tween a variable quantity and its limit. " If these two quantities are thus to be
forever separated," says he, *' how can they be brought under the dominion of a com-
mon law, and enter together in the same equation t "t

* Katnre and T7tility of Mathematiot, by CharlM Davies, Kew York, 1878| p. 28L
tlM(l.,p.Bae.

BIBLIOGBAPHT OF FLUXIONS AND THE CALCULUS. 397

Pj&ihob, Benjamin. An Elementary TVeafise on Curvei, FuneiionB, and Foreee, Yol-
ame I, contaming analytic geometry and the differential oalcolns. Boiton and
Cambridge, 1841. Pp. 301. Volnme II, containing oalculus of imaginary quan-
tities, residual oalonlns, and integral oalonlus. Boston, 1846. Pp. 290.

The method followed in these yolnmes is the infinitesimali of which the author was
a great admirer. The differential co-efficients are here denoted by D, 2>', etc. The
second volnme treats of many rather advanced subjects, such as imaginary infinitesi-
mals, imaginary logarithms, imaginary angles, the imaginary angle whose sine ex-
ceeds unity, potential functions, residuals, definite integrals, elliptic integrals, method
of variations, linear differential equations, Biccati's equation, and particular solutions
of differential equations.

Church, Albert £. EUimenU of ihe DifferenUal and Integral Calonlw. New Tork|
1842.
This is in many respects a good work, but the explanation of fundamental princi-
ples therein contained is too brief, and fietils to convey a philosophic knowledge of
them. The difficulties which a student is likely to encounter in a treatise like this
have been well stated by a writer in the Nation of October 18, 1888 : " What vexes
and perplexes him (the student) is that he-seems to himself to comprehend very
clearly what he is doing, and to be doing what all his previous training had taught
him he must not do. It all seems very easy, very simple, and very absurd. He is
told to ' take the limit ' of one side of his equation by striking out a quantity because
it 'is approaching zero,' while on the other side the same quantity- must be careftilly
preserved, because it is one of the terms of the ratio which is the very essence of the
whole process.''

McCartney, WASHlKaTOK. Prine^Ue of iKe DifferenfiaX and Integral CaHouXue^ and

their application to Geometry. Philadelphia, 1844. Pp. 340.

The author makes use of the doctrine of limits, but retains the language of infini-

dff dy

tesimals. " ^ is used as a mere symbol to denote the ultimate ratio, ^ being in reality

X. But inasmuch as the rules for differentiating and the geometrical application of

ultimate ratios are more readily understood by regarding the increments of the ordi-
nate andabqcissa as indefinitely small, we will call these increments in their ultimate
state, indefinitely email qnantitieeJ* ' ' For the sake of convenience," the student is asked
to call dy and dx what he has Just been told that they really are not. Such an expo-
sition of A fundamental principle is quite apt to fail to give satisfiMtion to beginners.

McCartney's Calculus is a book possessing several good features.

LOOHIS, Elias. AnalyOoal Geometry and Caloulue. 1851.

Later the Calculus was published in a separate volume and mnch enlarged. The

unfolding of fhndamenial pij^nciples, as given in the improved edition of 1874, is less

objectionable than that in the preceding works which adopt the method of limits.

The term *' limit of a variable " is here subjected to d^nition, but the student is not

dy
informed whether or not the variable ever reaches its limit. The symbol ^ is made

to represent the limiting value of i' f, . Confusion is apt to arise in the mind of the

student from the fact that dxis'* put for the inor. x in the limiting value " (which
value is zero), and is afterward said to be *' indeterminate" in value, '* either finite
or indefinitely smalL"

Smyth, William. Slemenie of the Differential and Integral Caloulue. 1854.

The author uses the infinitesimal method, but says (p. 229) that '* as a logical basis
of the calculus, the method of Newton and especially that of Lagrange have some ad-
vantage. In other respects the superiority is immeasurably on the side of the method
of Leibnitz."

398 TEACHIKG AND HISTORY OF MATHEMATICS.

CouBTBKAT; EDWARD H. Treatise on the Differential and Integral Caloutue and on
the Calculus of Variations. New York, 1855. Pp. 501.
The exposition of the method of limits, as given in this ill many respects admirable
work, is likewise open to objection, dx is prononnoed to be << indefinitely small " and
equal to h, but when hssO&t the limit, dx continues to remain indefinitely small.

KoBiNSON, Horatio N. Differential and Integral Calculus, 1861.

Some of Robinson's elementary works on mathematics became popular, but not so
his advanced works. His calculus and astronomy met with able but severe criticism
in the Mathematical Monthly. Robinson's work did not appear in a second edition,
but the work of Quinby was added to ** Robinson's Series" in place of it.

DocHABTY, Gbrardus Beekmak. Elements of Analytical Geometry and of the Differ-'
ential and Integral Calculus, New York, 1865. Pp. 306.

The part on the calculus covers 204 pages.

The method of limits is employed and treated in the manner customary with us at
the time the book was written.

Spare, John. The Differential Calculus : with Unusual and Particular Analysis of its
Elementary PrincijpleSf and Copious Illustrations of its Practical Application.
Boston, 1865. Pp. 244.
This work I have never seen. Dr. Artemas Martin, who kindly sends me its title
calls it a unique work, as may be seen from the following, which he quotes from its
preface: "The calculus being algebra, a strictly numerical science, the present
treatise claims to have labored successfully in putting on the true character as such.
No insinuation is allowed to prevail that it is any part whatever of analytical geom-
etry or that it is other than the natural sequel and supplement of common algebra;
useful, indeed, as an appliance, to borrow, in investigation of the fow kinds of
geometrical quantity."

QUXNBT, I. F. A New Treatise on the Elements of the Differential and Integral Calenlus.

New York, 1868. Pp. 472.

Here, as in other works based on the method of limits, the student encounters at

the outset the perplexing statement that ^, where denotes " absolute zero/' is equal

to some particular quantity.

Strong, Theodore. A Treatise on the Diffei'ential and Integral Calculus. New York,
1869. Pp.617. ^

This work was printed, but, we understand, never published. The author died
while the work was in press. Theodore Strong was professor at Rutgers College
from 1827 to 1863, and enjoyed the reputation of being one of the very deepest and
most erudite mathematicians in America, He was a very frequent contributor to
our mathematical periodicals. To students who possessibd taste for mathematical in-
vestigation he was a good teacher, but to those who had no taste he was unintelli-
gible. He had an unconscious tendency to diverge into regions where the ordinary
student could not follow him. This same tendency is exhibited in his Calculus, and
also in his Elementary and Higher Algebra, published in 1859. Both works possess
many original features, but the novelties contained in them are not always improve-
ments. These books are defective in arrangmeent, and not at all suited for use in
the class-room. In his general view of the calculus Strong follows Lagrange, but his
mode of presentation is quite new. He believed that his treatment divested the cal-
culus of all its old metaphysical encumbrances. He attempted to show how the
foundations of this science could be established without the intervention of any of
the antiquated hypotheses. " It is hence clear," says he, ** that the differential and
integral calculus are dedaoible from what has been done, without using infinlteal*
male or limiting ratios " (p. 271).

BIBLIOGRAPHY OF FLUXIONS AND THE CALCULUS, 899

PxcE, William G. Practical Treatise on the Differential and Integral Ca lonluB^ with some
of its applications to mechanics and astronomy. New Yprk and Chicago, 187(V
Pp. 208.
Employs the infinitesimal method.

Sestini, B. Manual of Geometrical and Tnfiniteeimal Analysis. Baltimore, 1871. Pp.
131.*
The infinitesimal method is used.

Olnby, Edward. Genei*al Geometry and Calcuhis, New York, 1871.

The part on the infinitesimal calculus covers 152 pages. The in&nitesimal method
is used. It is the experience of the large majority of teachers in this country that
the infinitesimal method, taken by itself, unaided by any other method, does not seem
rigorous to a student Ifcginning the study ol the calculus, 4ucl does not fully satisfy his
mind.

Bice and Johnson. The Elements cf the Differential Calculus j founded on the mecnod
of rates or fluxions. (Printed for the use of the cadets of the U. S. Naval Acad-
emy.) New York, 1874.
Without abandoning the ordinary notation, the writers return, in this work, to the
method of Newton. Newton's method of rates or fluxions is employed In subsequent
treatises written by the same authors, and also in the works of Buckingham and Tay-
lor. « By these writers much-longed-for improvements in the philosophical exposition
of the fundamental principles of the transcendental analysis have been introduced.

Johnson, W. Woolsey. Integral Calculus*

Rice and Johnson. An Elementary Treatise on the Differential Calculus, founded on
the method of rates or fluxions. New York, 1877. Pp. 469.

Rice and Johnson. Differential Calculus (abridged).

Rice and Johnson. Differential and Integral Calculus (abridged).

Clark, James G. Elements of the Infinitesimal Calculus (in "Ray's Series")* New
York and Cincinnati, 1875. Pp. 441.
The doctrine of limits is made the basis of this work. The author follows mainly
the excellent philosophical treatise of M. Duhamel.

Buckingham, C. P. Elements of the Differential and Integral Calculus. By a new
method, founded on the true system of Sir Isaac Newton, without the use of in-
finitesimals or limits. Chicago, 1875.

Bterly, W. E. Elements of the Differential Calculus, with examples and applications.
Boston, 1880.
The doctrine of limits is used as a foundation of the subject and preliminary to the
adoption of the more convenient infinitesimal method. The notation D^y is em-
ployed.

Byerly, W. E. Elements of the Integral Calculus, with a key to the solution of differ-
ential eqnatious. Boston, 1882.

Bowser, Edward A. An Elementary Treatise on the Differential and Integral Calculus.
New York, 1880.
Adopts infinitesimal method.

Taylor, James M. Elements of the Differential and Integral Calculus, Boston, 1884.
The author employs the conception of rates.

•A copy of IhiB work waa leat to us by Prof. U. y. Pawson, S. J., of Georgetown College. West
Washington.

400 TEACHING AND HISTOBT OF MATHEMATICS.

Newcomb, Simok. ElemenU of the Differmtial and Integral Cdlctilus, New Tark,
% 1837.

The author nseR the method of infinitesimals; based on the doctrine of limits. An
infinitesimal quantity is here defined as one **m the act of approaching zero as a
limit." This definition of an infinitesimal has now been very generally adopted.

It has been said that years ago a cadet at West Point, extremely fond of mathe-
matics, thus estimated the calcnlas : ''The inventors of the differential and integral
oalcalns have claimed that this branch of so-called science belongs to the depart-
ment of mathematics, and, laboring under that delasion, have introduced it into the
course of academical instruction for the torture of students. Such classification is
obyiously incorrect, because the principles of mathematics fall within the scope of
the reasoning iSftculty. The calculus, on the contrary, lies without the boundaries of
reason.''* That such should have been the impression received by the student of
the early works on calculus is not at all strange. Our recent publications on the
subject have, however, made decided progress in the philosophical exposition of the
ftmdamental principles. With a modem book and a competent teacher there is no
reason why the ordinary student should not get a rational understanding of the
calculus.

■ — ■

*Life of (General Nathaniel Lyon, p. SO. The passage is quoted in the Aaal^at, YoL X, 1874» "Edu^

eattonal Testimony Concerning the Calcnlas."

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