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Upper machine

Channel : lower is that of Latham

Monoplanes at Rheims.

that in which Bleriot crossed the English

THE

SCIENCE- HISTORY
OF THE UNIVERSE

FRANCIS ROLT- WHEELER

Managing Editor

IN TEN VOLUMES

THE CURRENT LITERATURE PUBLISHING COMPANY

NEW YORK

1909

INTRODUCTIONS BY

Professor E. E. Barnard, A.M., Sc.D.,
Yerkes Astronomical Observatory.

Professor Charles Baskerville, Ph.D., F.C.S.
Professor of Chemistry, College of the City of New York.

Director William T. Hornaday, Sc.D.,
President of New York Zoological Society.

Professor Frederick Starr, S.B., S.M., Ph.D.,
Professor of Anthropology, Chicago University.

Professor Cassius J. Keyser, B.S., A.M., Ph.D.,
Adrain Professor of Mathematics, Columbia University

Edward J. Wheeler, A.M., Litt.D.,
Editor of 'Current Literature.'

Professor Hugo Munsterberg, A.B., M.D., Ph.D., LL.D.,
Professor of Psychology, Harvard University.

EDITORIAL BOARD

Vol. I — Waldemar Kaempffert,
'Scientific American/

Vol. II — Harold E. Slade, C.E.

Vol. Ill — George Matthew, A.M., .

Vol. Ill — Professor William J. Moore, M.E.,

Assistant Professor of Mechanical Engineering, Brooklyn

Polytechnic Institute.

Vol. IV — William Allen Hamor,

Research Chemist, Chemistry Department, College of the

City of New York.

Vol. V — Caroline E. Stackpole, A.M.,
Tutor in Biology, Teachers' College, Columbia University.

Vol. VI— Wm. D. Matthew, A.B., Ph.B., A.M., Ph.D.,

Assistant Curator, Vertebrate Paleontology, American

Museum of Natural History.

Vol. VI — Marion E. Latham, A.M.,
Tutor in Botany, Barnard College, Columbia University.

Vol. VII— Francis Rolt-Wheeler, S.T.D.

Vol. VII— Theodore H. Allen, A.M., M.D.

Vol. VIII— L. Leland Locke, A.B., A.M.,
Brooklyn Training School for Teachers.

Vol. VIII— Franz Bellinger, A.M., Ph.D.

Vol. IX— S. J. Woolf.

Vol. IX— Francis Rolt-Wheeler, S.T.D.

Vol. X — Professor Charles Gray Shaw, Ph.D.,
Professor of Ethics and Philosophy, New York University.

Leonard Abbott,
Associate Editor 'Current Literature/

THE

SCIENCE - HISTORY
OF THE UNIVERSE

VOLUME VIII

PURE MATHEMATICS

By L. LELAND LOCKE

FOUNDATIONS OF MATHEMATICS

By PROFESSOR CASSIUS J. KEYSER

MATHEMATICAL APPLICATIONS

By DR. FRANZ BELLINGER

INTRODUCTION

By PROFESSOR CASSIUS J. KEYSER

Copyright, 1909, by
CURRENT LITERATURE PUBLISHING COMPANY

CONTENTS

PART I— PURE MATHEMATICS
Introduction by Professor Cassius J. Keyser

CHAPTER PAGE

I Number I

II Calculation 34

III Powers of Numbers 58

IV Algebra 82

V Geometry 107

VI Trigonometry 153

VII Analytic Geometry 164

PART II— FOUNDATIONS OF MATHEMATICS
Special Article by Prof. Cassius J. Keyser . 191

PART III— MATHEMATICAL APPLICATIONS

I Early Non-Mechanical Applications 239

II Chronology and Horology 253

III Surveying and Navigation 273

IV Mechanical Principles 291

V Machines 303

VI Aviation 315

INTRODUCTION

The general reader, for whom this writing is primarily
designed, though he be college-bred, and may thus have
had a mathematical discipline extending possibly through
an elementary course in the calculus, probably entertains
very erroneous or very inadequate notions respecting the
proper character of mathematics, and especially respect-
ing alike its marvelous growth in modern times and the
great range and variety of doctrines that the term has
come to signify. With a view to correcting such errors,
at least in some measure, if they exist, and in order to
enhance the reader's interest and to enlighten his appre-
ciation, it seems worth while to preface the exposition
proper with some general indications — albeit they must
needs be mainly of an exterior kind — of the nature and
the extent of the science whose foundations are to be
subsequently explained.

Let it be understood, then, that, while mathematics is
the most ancient of the sciences, it is not surpassed by
any of them in point of modernity, but is flourishing even
to-day as never before, and at a rate unsurpassed by any
rival. To compare it to a deep-rooted giant tree of
manifold high and far-branching arms is not an adequate
simile. Rather is the science like a mighty forest of such

x INTRODUCTION

oaks. These, however, literally grow into and through
each other, so that by the junction and intercrescence
of limb with limb and root with root and trunk with trunk
the manifold wood becomes a single living, organic, grow-
ing whole. The mathematical achievements of antiquity
were great achievements. The works of Euclid and
Archimedes, of Apollonius and Diophantus, will endure
forever among the most glorious monuments of the human
intellect. And just now, owing to Dr. Heath's superb
English edition of Euclid's 'Elements' — a beautiful trans-
lation of the thirteen books from the definitive text of
Heiberg, with rich bibliography and extensive commen-
tary setting the whole matter in the composite light of
ancient and modern geometric research — one sees even
better than ever before how great, mathematically, was
the age that produced the immortal Alexandrine classic.
Yet the 'Elements' of Euclid is as small a part of Mathe-
matics as the Tliad' is of Literature; as the Tandects' of
Justinian is of human Jurisprudence; or as the sculpture
of Phidias is of the world's total Art.

Not the age of Euclid, but our own, is the golden age
of mathematics. Ours is the age in which no less than
six international congresses of mathematics have been
held in the course of ten years. To-day there exist more
than a dozen mathematical societies, containing a growing
membership of over two thousand men and women rep-
resenting the centers of scientific light throughout the
great culture nations of the world. In our time more
than five hundred scientific journals are each devoted in
part, while more than two score others are devoted ex-
clusively, to the publication of mathematics. It is in

INTRODUCTION xi

our time that the 'Jahrbuch (iber die Fortschritte der
Mathematik' ('Yearbook for the Progress of Mathe-
matics'), tho it admits only condensed abstracts with
titles and does not report upon all the journals, has, never-
theless, grown into nearly forty huge volumes in as many
years. It requires no less than the seven ponderous tomes
of the forthcoming 'Enkyclopadie der Mathematischen
Wissenschaften' ('Encyclopedia of the Mathematical Sci-
ences') to contain, not expositions, not demonstrations,
but merely compact reports and bibliographic notices
sketching developments that have taken place since the
beginning of the nineteenth century. This great work is
being supplemented and translated into the French lan-
guage. Finally, to adduce yet another evidence of like
kind, the three immense volumes of Moritz Cantor's
^Geschichte der Mathematik' ('History of Mathematics'),
tho they do not aspire to the higher forms of" elaborate
exposition, and tho they are far from exhausting the pe-
riod traversed by them, yet conduct the narrative down
only to 1758. (A fourth volume in continuation of Can-
tor's work has recently appeared. It was composed main-
ly by other hands.) That date, however, but marks the
time when mathematics, then schooled for over a hundred
eventful years in the fast unfolding wonders of Analytic
Geometry and the Calculus, and rejoicing in these the
two most powerful instruments of human thought, had
but fairly entered upon her modern career. And so fruit-
ful have been the intervening years, so swift the march
along the myriad tracks of modern Analysis and Geome-
try, so abounding and bold and fertile withal has been
the creative genius of the time, that to record, even

xii INTRODUCTION

briefly, the discoveries and the creations since the closing-
date of Cantor's work would require an addition to his
great volumes of a score of volumes more.

It is little wonder that so vital a spirit as that of Ma-
thesis, increasing in intensity and more and more abound-
ing as the ages have passed — it is small wonder that since
pre-Aristotelian times it has challenged the mathematician
and the philosopher alike to tell what it is — to define
mathematics ; and it is now not surprising that they should
try in vain for many hundreds of years; for, naturally,;
conception of the science has had to grow with the growth
of the science itself.

Cassius J. Keyser.

Editorial Note— Beginning on page J9J will be found a notable
paper on "The Foundations of Mathematics/' written especially
for this series by Professor Keyser.

MATHEMATICS

CHAPTER I

The notion of number is extremely slow to develop, both
in the individual and in the race, yet it has its origin at
such a remote period in the evolution of man that only a
possible reconstruction of its history may be given. Such
an account may built up mainly from three sources, a
study of the knowledge and use of number among peoples
lowest in the scale of civilization at the present time, the
genesis of the number concept in the mind of the child and
a comparison of root words of the various languages, past
and present.

Number is coeval with spoken language, and probably
antedates by a long period any written language or sym-
bolism. Primitive man recorded the results of hunting or
fishing excursions, the number of warriors in the opposing
camp, or the number of days' journey from home by the
use of pebbles, shells, knots in cord, nicks in woods, scores
on stone, and, most important for the present study, by the
fingers and toes.

The mode of recording numbers by knots on cord gave
rise to the term "quipu" reckoning, from the Peruvian lan-
guage, quipu meaning knot. Edward Clodd, in 'The Story
of the Alphabet,' has this reference: "The quipu has a
long history, and is with us in the rosary upon which
prayers are counted, in the knot tied in a handkerchief to

2 MATHEMATICS

help a weak memory, and in the sailor's log-line." Herod-
otus tells that when Darius bade the Ionians remain to
guard the floating bridge which spanned the Ister, he "tied
sixty knots in a thong, saying : 'Men of Ionia, do keep this
thong and do as I shall say : so soon as ye shall have seen
me go forward against the Scythians, from that time begin
and untie a knot each day ; and if within this time I am
not here, and ye find that the days marked by the knots
have passed by, then sail away to your own lands/' "

The quipu reached its more elaborate form among the
ancient Peruvians. It consisted of a main cord, to which

from Peru.

were fastened at given distances thinner cords of different
colors, each cord being knotted in divers ways and each
color having its own significance. Red strands stood for
soldiers, yellow for gold, white for silver, green for corn,
and so forth, while a single knot meant ten, two single
knots meant twenty, double knots one hundred, two double
knots two hundred. Each town had its officer whose
special function was to tie and interpret the quipus. They
were called Ouipucamayocuna, or knot-officers (compare
'harpedonapfce,' or rope-stretchers, in connection with the
Geometry of the Egyptians).

The knot-reckoning is in use among the Puna herdsmen

NUMBER 3

of the Peruvian plateaux. On the first strand of the quipu
they register the bulls, on the second the cows, these again
they divide into milch-cows and those that are dry; the
next strands register the calves, the next the sheep, and so
forth, while other strands record the produce; the differ-
ent colors of the cords and the twisting of the knots giv-
ing the key to the several purposes. The Paloni Indians
of California have a similar practice, concerning whom
Dr. Hoffman reports that each year a certain number are
chosen to visit the settlement at San Gabriel to sell native
blankets. "Every Indian sending goods provided the sales-
man with two cords made of twisted hair or wool, on one
of which was tied a knot for every real received, and on
another a knot for each blanket sold. When the sum
reached ten reals, or one dollar, a double knot was made.
Upon the return of the salesman each person selected from
the lot his own goods, by which he would at once perceive
the amount due, and also the number of blankets for which
the salesman was responsible." Hawaiian tax-gatherers
kept accounts of the assessable property throughout the
island on lines of cordage from four to five hundred fath-
oms long.

A method of keeping the accounts of the British ex-
chequer before the use of writing paper was by means of
tally sticks. These were of willow about 8 or 10 inches
long. Notches were cut, a deep one for a pound, a small
one for a shilling. The stick was then sawed half in two
near one end and split down to this cut, each half bearing
a record of the notches. The shorter piece was given to
the depositor and the bank retained the longer.

A great mass of these sticks was still in the basement
of the Parliament houses when it was decided to burn
them in 1834. Samuel S. Dale describes the bonfire. He
says, "A pile of little notched sticks bearing strange-look-
ing inscriptions in abbreviated Latin and old English
script, the evidence of thrift for a thousand years, tokens
of all the motives that prompt men and women to save,

4 MATHEMATICS

love, hate, greed and sacrifice, hope and fear, frugality
and fraud, the proceeds of honest toil and of crime, held
for ages that the missing pieces carried away by successive
generations might be redeemed, their presence a mute evi-
dence of the blasted hopes of depositors for a thousand
years. They were fed steadily to the flames from early
morning until a few minutes before seven o'clock in the
evening of Thursday, October 16, 1834, when suddenly a
furnace flue, overheated by the unusual fire, started a
blaze in a room above, and in a few hours the House of
Lords and the House of Commons were in ashes, along
with nearly all the old wooden tally sticks and all the
basic standards of weight and measure for the British
Empire." A few of the old tally sticks were saved.

When the savage in his first dim gropings for truth
recognises that two objects are more than one, the first
step is taken toward the formation of the number concept.
That a long pause ensued before the next step was taken
is evidenced by the number of cases, cited by various
writers, of tribes whose only number words are for 'one'
and 'many' or 'one/ 'two' and 'many.' This word for
'many' plays the same role in the language of the savage
as 'infinity' in ordinary parlance, a number inexpressibly
or inconceivably great. The growth of expressibility of
number may be compared with the ever-widening ripples
when a pebble is dropped into still water, the outer ripple
representing the upper bound of conceivable number. All
the region beyond would be, in the language of the savage,
'many.'

The Hindu number system is the first ever devised
which has no outer bound. This fact has led to a more
precise use of the word 'infinity' in modern mathematical
terminology.

The possibilities of the Hindu system are well illustrated
by the answers to the celebrated Archimedean "cattle
problems." These answers, ten in number, were com-
posed of 206,545 figures each. Such a number if printed

NUMBER 5

in small pica type would be nearly a quarter of a mile in
length.

The ability to form a definite conception of a number
grows with intelligence, but in the presence of numbers
of such magnitude it is opportune to ask what relation
exists between the power to conceive the number and the
ability to represent it. There seems to have been a curi-
ous crossing over of the two. The poverty of the aborigi-
nal language should not be taken as evidence of inability
to use larger numbers. It simply means that the verbal
expression paused for a longer time after the number 'two*
than did the number sense. Instances are given of peoples
whose number names do not go beyond ten, but who
reckon as far as one hundred. The number sense grows
along with other mental development, but has not kept
step with the verbal and symbolic expression of large
numbers. It is questionable if the number 10,000 stands
for a distinct conception if it is measured by units. One
obtains an idea of such a number only by grouping it, say,
into a hundred hundreds.

There are several distinct steps in the formation of a
number system: The recognition of increase by adding,
in succession, single objects to a group, counting, attach-
ing a number name to the group counted, as 'three' sticks
(such a number in which the object or unit is named is
called a concrete number), the final separation of the
number notion from the objects counted or abstraction
(one asks how many sticks in the group and the answer is
'three,' an abstract number), the indicating of the number
name by a symbol, the choosing of a method of grouping
and finally the perfection of the system by arrangements
and combinations of the number words and symbols. It
is a long way from the 'mokenam,' 'one' ; 'uruhu,' 'many,'
of the Bococudos to the modern notion of number of the
mathematician, "the class of all similar classes."

Number in its primitive sense answers the question,
"How many ?" It is a pure abstraction which results from

6 MATHEMATICS

counting. Cardinal number tells how many of the group,
as 'seven' trees, while the ordinal number of any one of
the objects indicates the position of the particular object
in the series, as the 'sixth' tree. These two ideas are
equally fundamental, each being derivable from the other.
Counting is simply pairing off, or, in mathematical lan-
guage, establishing a one-to-one correspondence between
the individuals of a group of objects counted, as pebbles,
the fingers, marks or scores, number names or the symbols
for these number names.

In the first stages it would be comparatively easy to in-
vent a word and a symbol for each number, but as the
need for larger numbers grew some method of grouping
became necessary. In 'Problemata,' attributed to Aristotle,
the following discussion takes place: "Why do all men,
barbarians as well as Greeks, numerate up to ten, and not
to any other number, as two, three, four, or five, and then
repeating one and five, two and five, as they do one and
ten, two and ten, not counting beyond the tens, from which
they again begin to repeat? For each of the numbers
which precedes is one or two and then some other, but
they enumerate, however, still making the number ten
their limit. For they manifestly do it not by chance, but
always. The truth is, what men do upon all occasions
and always they do not from chance, but from some law of
nature. Whether is it, because ten is a perfect number?
For it contains all the species of number, the even, the
odd, the square, the cube, the linear, the plane, the prime,
the composite. Or is it because the number ten is a prin-
ciple? For the numbers one, two, three, and four when
added together produce the number ten. Or is it because
the bodies which are in constant motion are nine? Or is
it because of ten numbers in continued proportion, four
cubic numbers are consummated (Euclid viii, 10), out of
which numbers the Pythagoreans say that the universe is
constituted? Or is it because all men from the first have

NUMBER 7

ten fingers ? As therefore men have counters of their own
by nature, by this set, they numerate all other things."

Dr. Conant gives an illustration which typifies the be-
ginnings of this grouping in 'The Number Concept.'
"More than a century ago," he says, "travelers in Mada-
gascar observed a curious but simple mode of ascertaining
the number of soldiers in an army. Each soldier was
made to go through a passage in the presence of the prin-
cipal chiefs ; and as he went through a pebble was dropped
on the ground. This continued until a heap of ten was
obtained, when one was set aside and a new heap begun.
Upon the completion of ten heaps, a pebble was set aside
to indicate one hundred, and so on until the entire army
had been numbered."

That man carries in the fingers the natural counting
machine is shown by the fact that the great majority of
number systems have been based on five, ten or twenty.
A typical case of such a number system is that of the Zuni
scale :

i — topinte taken to start with.

2 — kwilli put down together with.

3 — ha'i the equally dividing finger.

a awjte . \ a^ tne fingers all but one

' done with.

5— opte the notched off.

6— topalik'ya I another brought to add to

( the done with.

7— kwillilik'ya j two brou&ht to an^ held

( up with the rest.

8 hailik'ye . . \ tnree brought to and held

( up with the rest.

o tenal'ik'ya ( all but all are held up with

" ( the rest.

io — astem'thila all the fingers.

ii — astem'thla topaya'thl'- i all the fingers and another

tona l over above held.

And so forth to 20.

8 MATHEMATICS

20 — kwillik'yenastem'thlan.two times all the fingers.

ioo — assiastem'thlak'ya the fingers all the fingers.

l,ooo — assiastem'thlanak'ye- ( the fingers all the fingers
nastem'thla \ times all the fingers.

Arithmetic has been defined as the science of number
and the art of computation. This twofold nature of the
subject is due to the fact that the Greeks divided the sub-
ject into 'Arithmetic' proper, which is the science of num-
bers, a subject for the philosopher, and 'Logistic,' or com-
putation, which was to be taught to the slave.

Notation and numeration are respectively the writing
and reading of numbers. A theory of the building up of
a number system is given by Dean Peacock in his article
on arithmetic in the 'Encyclopedia Metropolitana' : "The
discovery of the mode of breaking up numbers into classes,
the units in each class increasing in decuple proportion,
would lead, very naturally, to the invention of a nomen-
clature for numbers thus resolved, which is simple and
comprehensive. By giving names to the first natural
numbers, or digits — i.e., the first nine numbers, called
digits, from counting on the fingers — and also to the units
of each class in the ascending series by ten, we shall be
enabled, by combining the names of the digits with those
of the units possessing local or representative value, to
express in words any number whatsoever. Thus the num-
ber, resolved by means of counters in the manner indi-
cated by Fig. 2, would be expressed (supposing seven,
six, five, and four denote the numbers of the counters, in
A, B, C, D, and ten, hundred, and thousand, the value
of each unit in B, C, and D) by seven, six tens, five hun-
dreds, four thousands; or inverting the order, and making
slight changes required by the existing form of the lan-
guage, by four thousand, five hundred, and sixty-seven."

The successive columns A, B, C, D are called orders.
The number of ones in any order required to make one of
the next higher order, in this case ten, is called the radix,

NUMBER

9

scale or base of the system. In the above formation when
nine have been put in the column A, the tenth would be
placed in column B and the nine removed from column A,
Such a system is called a decimal or "ten times" system.
One of the earliest devices for reckoning consisted of a
board strewn with sand on which parallel lines were drawn
with the finger. These lines fulfil the same office as the
compartments above marked A, B, C, D. Upon the lines
the counters were laid. This reckoning board was called
an abacus from an old Semitic word abaq, meaning sand.
The development of the abacus from the sand-board to

D c a A

# • •

# • . • «

• » • o

* m ■ • •

* » • *

Fig. 2 — Old Method of Computation with Counters.

the swan pan of the Chinese and the counting frame of
the kindergarten is to be considered in connection with
reckoning.

It was the custom of the Romans to drive a nail in the
temple of Minerva for each year. When, as with count-
ers, the number of marks exceeded the power of the eye
to grasp at a glance, grouping was used.

The simplest method of writing a number is by a mark
or stroke for each unit, or one in the number, as |.| 1 1 1 1 1|
for seven. The stroke was universally used by primitive
peoples as a symbol for one. The drawing of the tomb-
board of Wabojeeg, a celebrated war chief who died on
Lake Superior about 1793, shows this clearly. His totem,

io MATHEMATICS

the reindeer, is reversed. The seven strokes number the
war parties he led, the three upright strokes symbolize

^>W^'<

Fig. 3 — Tomb-board of Wabojeeg.

wounds received in battle. The horned head tells of a
desperate fight with a moose.

The scoring of each fifth one counted may be regarded
as the second step in the development of a satisfactory
number symbolism. Such a method of recording succeed-
ing events is not uncommon to-day. The thresher often
so marks each sack of grain as it leaves the machine, and

NUMBER ii

in loading and unloading vessels it is frequently the mode
used by the tallyman. Thus twenty-two would be written

BMMJtllll

Of the numerous systems of notation which have been
devised, three are distinctive from their mode of forma-
tion, from their logical completion, and from their* ex-
tended use : The Greek, the Roman, and the Hindu, some-
times incorrectly called the Arabic. Consider a number
formed by counters placed in the various compartments A,
B, C, D (Fig. 2). The largest number of counters that
may be put in any one compartment is nine; that is, there
are nine numbers for each compartment. The Greeks
adopted as their number symbols the letters of their alpha-
bet in order, the first nine letters for nine numbers, 1, 2,
3, 4, 5, 6, 7, 8, 9, of column A ; the next nine letters for the
numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 of column B.
As the alphabet consisted of but twenty-four letters, to
fill out column C three obsolete letters were interpolated.
In the accompanying scheme, taken from Gow's 'History
of Greek Mathematics,' the starred letters are those not
belonging to the alphabet.

The limit of the system with letters of the alphabet
alone is 999. When it became necessary to write larger
numbers, a stroke like an inverted prime was put before
and usually somewhat below the letter, as seen in the
number 1,000, to increase the value of the letter one thou-
sandfold. For 10,000 a new letter was used, the M, the
first letter of juvpioi or myriad. The symbols were always
written in descending order from left to right. The
largest number now possible in the Greek notation is
99999999. The use of the alphabet as numerals seems to
date from about 500 B.C. The Greek mode of writing frac-
tions is quite simple, the denominator being written over
the numerator, or the numerator is written with one ac-

12 MATHEMATICS

cent, followed by the denominator twice with two accents,
as lg or itfKaf'Ka". If the numerator is unity it is omitted.
Vs would be written *£' or \pv. Special signs were some-
times used for y2, y$, addition and subtraction.

Archimedes devised a plan by which the Greek number
system might be prolonged indefinitely and which has
been thought by some to contain the germ of the modern

*' p'Y' S'e'C' S'YO't*

1 Z 3 <t S t, 7 9 1 10

k' a' /*' *>' s' o' w *<?«;?

ao Jo 4 ° so ^o 70 ?o <fc

p' <s> t' w f y y' <aj'*~%

too 100 3oo 'ioo so* 6oo 700 loo <f00

/K — t^oo j/3 =2ooo

Mu <ru At &\

/ 0fc 0 O O

Xo.ooo

-Greek Number System.

notion of logarithm. "In a pamphlet entitled 7pa{i/izri?$
(in Latin Arenarius, the sand-reckoner), addressed to
Gelon, king of Syracuse," says Gow, "Archimedes begins
by saying that some people think the sand cannot be
counted, while others maintain that, if it can, still no arith-
metical expression can be found for the number. 'Now I
will endeavor,' he goes on, 'to show you, by geometrical
proofs which you can follow, that the numbers which have
been named by us and are included in my letter addressed

NUMBER 13

to Zeuxippus, are sufficient to exceed not only the number
of a sand-box as large as the whole earth, but of one
which is as large as the universe. You understand, of
course, that most astronomers mean by "the universe" the
sphere of which the center is the center of the earth and
the radius is a line drawn from the center of the earth to
the center of the sun.' Assume the perimeter of the earth
to be 3,000,000 stadia (a stadium was nearly 200 yards),
and in all the following cases take extreme measurements.
The diameter of the earth is larger than that of the moon
and that of the sun is larger than that of the earth. The
diameter of the sun is thirty times that of the moon and is
larger than the side of a chiliagon (a polygon of 1,000
sides) inscribed in a great circle of the sphere of the
universe. It follows from these measurements that the
diameter of the universe is less than 10,000 times that of
the earth and is less than 10,000,000,000 stadia.

"Now suppose that 10,000 grains of sand not < 1 poppy-
seed, and the breadth of a poppy-seed not < 1/v> of a finger-
breadth. Further using the ordinary nomenclature, we
have numbers up to a myriad myriads (100,000,000). Let
these be called the first order and let a myriad myriads be
a unit of the second order and let us reckon units, tens,
etc., of the second order up to a myriad myriads; and let
a myriad myriads of the second order be a unit of the
third order and so on ad lib. If numbers be arranged in
a geometrical series, of which 1 is the first term and 10
is the radix, the first eight terms of such a series will
belong to the first order, the next eight to the second
order and so on. Calling these orders octads and using
these numbers, following the rule that spheres are to one
another in the triplicate ratio of their diameters, Archi-
medes ultimately finds that the number of grains of sand
which the sphere of the universe would hold is less than
a thousand myriads or ten millions of the eighth octad.
This number would be expressed in our notation as 1 with
sixty-three ciphers annexed." There seems to have been

14 MATHEMATICS

no attempt to apply this method further, the ordinary-
system being sufficient for the needs of the time.

The main principle underlying the Roman system was
to provide a symbol for each column or order, the symbol
being repeated for each unit in the order. The following
reconstruction of the Roman process is made for the pur-
pose of comparison with the other two systems and is not
offered as the probable historical course.

For each unit of column A a Roman I was used, it
being the nearest to the primitive stroke or score | ; X
was used for the second order, C for the order of hundreds,
and M for thousands. These are called unit letters. So
far the gap from i to 10 is too great, it being necessary
to write I nine times for 9. A half-way symbol was then
provided for each interval : V for 5, L for 50, and D for
500. These are called half-unit letters. It is altogether
probable that the half-unit letter is a relic of the pause in
finger reckoning when the first hand was completed.
Many of the decimal systems still preserve this trace of a
quinary base.

The half-unit symbol may have arisen in connection
with the use of the reckoning board, placing counters on
the spaces as well as upon the lines as the notes of the
musical staff. Fig. 5 indicates the method of writing
7,868 on the sand-board. It is very probable that the use
of the spaces was derived from the half-unit letter rather
than in the reverse order.

So far the system is built upon an additive basis, the
value, of a symbol of equal or less value written at the
right of a given symbol being added to the value of the
given symbol; thus if 20 is to be written, another X is
written at the right of the X for 10, as XX, while 16 would
be written XVI. At this stage four would be written;
IIII, a form still to be seen on a clock face. A still further
improvement, lessening the number of symbols, was the
adoption of a subtractive principle. This means that a
symbol of lesser value written at the left of a given sym-

NUMBER 15

bol has its value taken from the value of the greater
symbol. In this way 4 would be written IV. Two facts
are here noticeable. The subtractive principle need be
used but twice in each column; in the column A, for ex-
ample, in writing 4 and 9, 3 might be written IIV with
no advantage over III. A half-unit letter is never used
in the subtractive sense; that is, L is used for 50 rather
than LC.

The third and final step was the adoption of the mul-
tiplicative principle (also seen in the Greek notation).
In the Roman scheme it appeared as a dash or vinculum

v o so 00

-A? e — e 2000

D O SOO

-Q OOO S-&-& =

L O SO

-X e , t-a

v o f

-f. 000 £

Total 78£d

Fig. 5 — Roman Units and Half-units.

drawn over a letter to increase its value a thousandfold;
as in Fig. 5, a V with a stroke across the top indi-
cates 5,000. The Roman mind was not of a scientific cast
and one would scarcely expect to find the number system
worked out to logical perfection. In fact, there is a
decided lack of uniformity in the manner of writing num-
bers used by various Roman authors.

The following set of rules compiled by Dr. French
seems to be the logical working out of the system: "Af-
firmative Rules: (1) The value of a unit letter is re-
peated with every repetition of the letter; (2) the value

16 MATHEMATICS

of a letter written at the right of a letter of equal or
greater value is added to that value; (3) the value of a
unit letter written at the left of the next higher unit or
half-unit letter is subtracted from the value of that letter ;
(4) a vinculum placed over a letter increases its value a
thousandfold. Negative Rules : ( 1 ) A half-unit letter
is never repeated; (2) a half-unit letter is never written
before a letter of greater value; (3) a unit letter is never
written before a letter of greater value except the next
higher half-unit and unit letters — i.e., 99 is never written
IC; (4) the vinculum is never placed over I; (5) a let-
ter is not used more than three times in any order."

Little may be said of the origin of the Roman Numer-
als. It is generally supposed that the system was inher-
ited from the Etruscans. Various and interesting have
been the theories advanced to explain the choice of the
symbols. One is that the I is a sort of hieroglyphic form
of the extended finger, V for the hand, and X for the
double hand. Another theory is that decern is related to
decussare, to cut across, and that the cutting across of a
symbol multiplies its value by 10; thus I cut across is X.
C is the initial letter of centum, one hundred.

Traces of the subtractive principle have been found on
brick tablets from the Temple Library of Nippur, recently
deciphered by Professor Hilprecht of the Babylonian Ex-
pedition of the University of Pennsylvania. These bricks
probably date from about the twentieth century B.C.

Each of the wide symbols indicated a ten, the final
straight wedge a one, the twenty and one being combined
in a subtractive sense to give nineteen.

The fundamental principle of assigning a symbol to each
column destined the Roman system of notation to ulti-
mate disuse. By this principle an indefinitely large num-
ber would mean an indefinitely large number of columns,
and hence an indefinitely large number of symbols. No
difference how many symbols were in use, it would be
easy to specify a number which could not be written.

NUMBER 17

Such a system must finally give way to another with no
such limitations.

The Babylonian number system was based on 60, both
for whole numbers and fractions. The possible explana-
tion of this sexigesimal system is that the year was reck-
oned as 360 days, thus dividing the circle into 360 parts,
and they were probably aware of the division of the circle
into 6 parts by stepping off the radius 6 times on the cir-
cumference, and by so doing arriving at 60 parts of the
circle in each part stepped off. 60 proved to be a par-
ticularly favorable base, being divisible by 2, 3, 4, 5, 6,
10, and 12. A large mass of information as to the mathe-
matical accomplishments has recently been revealed by
Professor Hilprecht, who has examined more than 50,000
cuneiform inscriptions from the Temple Library of Nip-
pur.

The Babylonians had a strange custom of deriving their
numbers from a large number which may be called a
basal number. This basal number is 12,960,000 or 604.
This number is, according to the theory of Professor Hil-
precht, the famous "Number of Plato," Republic, Book
VIII. "This number is constructed from 216, the minimal
number of days of gestation in the human kind, and is
called the lord of better and worse births." If the 216 be
interpreted as days, together with 12,960,000, the latter
number gives 36,000 years, the "great Platonic year,"
which was the length of the Babylonian cycle. Thus is
implied that Plato's famous number and the idea of its
influence upon the destiny of man originated in Baby-
lonia.

The Aztec system of numeration had the score for its
basis. There were special signs for the first five numerals ;
for twenty, for its square, four hundred, and for the cube,
eight thousand. Certain combinations of signs symbolized
the other numerals.

The Chinese had, from earliest times, constructed a
system of numerals, similar in many respects to what the

18 MATHEMATICS

Romans probably inherited from their Pelasgic ancestors.
It is only to be observed that the Chinese mode of writing
is the reverse of the Arabic, and that beginning at the top
of the leaf it descends in parallel columns to the bottom,
proceeding, however, from right to left, as practiced by

1 * Yth.

2 ^ Irr.

1,000,000 ylK Chad.

10,000,000 J3 King.
100,000,000 ;JSC Kyau

Fig. 6 — Chinese Number System.

most of the Oriental nations. Instead of the vertical
lines used by the Romans, therefore, horizontal ones are
found in the Chinese notation. Thus 'one' is represented
by a horizontal stroke with a barbed termination, 'two*
by a pair of such strokes. The mark for 'four' has four
strokes with a flourish. Three horizontal strokes and
two vertical ones form the mark for 'five/ and other sym-
bols exhibit the successive strokes abbreviated as far as

NUMBER 19

'nine/ 'Ten' is figured by a horizontal stroke, crossed
with a vertical score, to show that the first rank is com-
pleted, while a hundred has two vertical scores connected
by three short horizontal ones.

The Hindu system was based on the principle of as-
signing a symbol to each of the nine numbers of the first
column, 1 for one, 2 for two, 3 for three, 4 for four, 5 for
five, 6 for six, 7 for seven, 8 for eight, and 9 for nine.
The Hindu notation may be reconstructed as follows: It
is required to write the number pictured in the accom-
panying cut. There are four in the A column, or four
ones, three in the B column, or three tens, five in the C
column, or five hundreds, one in the D column, or one
thousand, and four in the E column, or four ten thou-
sands. Using the symbols above, 4 is written in the A
compartment, 3 in the B compartment, etc. So long as a
box arrangement is used with the compartments named,
the method would be considered complete. In fact, the
above number could be written just as well without the
cells, as 41534, and the order for which any symbol stands
would be determined by its position with reference to the
others. This is called the place-value property, and is
the important feature of the system.

But one thing is lacking: the method fails when any
column is empty. Suppose columns A and C above to be
vacant; there would be then 4 E's, 1 D, 3 B's, and no
A's nor C's. This could be written in cells, but could not
be written without some scheme of labeling the columns.
To avoid this difficulty a new symbol, O, was invented.
It was called cipher from an Arabic word meaning empty.
The above number may now be written 41,030.

In the Hindu notation each symbol has in addition to
its intrinsic value an acquired value resulting from its
position. Thus the 3, standing in the second place, has
the value thirty; 3 being its intrinsic value and the ten
being its acquired or place value. Thus both the multi-
plicative and additive principles are involved in place-

20

MATHEMATICS

value 325 is 3 X I(>0 + 2 X I0 + 5- Writing two symbols,
now called figures, side by side adds them after the left-
hand figure has been multiplied by ten.

It is readily seen that there is no limit to the number
of columns that may be used without increasing the num-
ber of symbols; that is, the Hindu notation begins at
units' column and may be carried indefinitely to the left.
The smallest number that may be written, so far, is Unity,
or one. The two final steps in the perfecting of the sys-
tem, the invention of the decimal point, which permits
of the writing of numbers indefinitely small, striking off

• • •

• • • : •

• • • •

• . • •

£

0

C

B

A

H

1

s

3

H

~*

e

D

c

B A

4

1

3

Fig. 7 — Hindu Notation Arrangement.

the right-hand barrier, and the discovery of the expo-
nential notation and logarithms, which facilitate compu-
tations, will be considered later, together with the long
struggle between the Roman and Hindu systems for su-
premacy.

The origin of the Hindu notation is shrouded in mys-
tery. It is customary for Orientals to attribute any great
discovery or invention to the direct revelation of the gods.
Professor Hilprecht gives an illustration of this trait.
"According to Berosus, a Babylonian priest who lived
some time between 330 and 250 B.C., the origin of all
human knowledge goes back to divine revelation in pri-
meval times. 'In the first year there made its appearance

Fig# 8 — Oannes; Babylontan God of Mathematics and
Learning.

22 MATHEMATICS

from a part of the Erythraean Sea, which bordered upon
Babylonia, a living being endowed with reason, who was
called Oannes. According to this tradition, confirmed by
Apollodorus, the whole body of this creature was like
that of a fish, and it had under a fish's head another or
human head, and feet similar to those of a man subjoined
to the fish's tail, and it also had a human voice; and a
representation of him is preserved even to this day. This
being, it is said, in the day time used to converse with
men, without, however, taking any food; he instructed
men in the knowledge of writing, of sciences and every
kind of art; he taught them how to settle towns, to con-
struct temples, to introduce laws and to apply the princi-
ples of geometrical knowledge, he showed them how to
sow and how to gather fruit; in short, he instructed men
in everything pertaining to the culture of life. From that
time [so universal were his instructions] nothing else has
been added by way of improvement. But when the sun
set, this being Oannes used to plunge again into the sea
and abide all night in the deep ; for he was amphibious.' "

Professor Florian Cajori thus sums up the leading con-
clusions due to Woepcke as to the historical development
of the Hindu numeral system : "The Hindus possessed
the nine numerals, without the zero or cipher, as early
as the second century after Christ. It is known that about
that time a lively commercial intercourse was carried on
between India and Rome, by v/ay of Alexandria. There
arose an interchange of ideas as well as of merchandise.
The Hindus caught glimpses of Greek thought, and the
Alexandrians received ideas on philosophy and science
from the east. The nine numerals, without the zero, thus
found their way to Alexandria, where they may have
attracted the attention of the Neo-Pythagoreans. From
Alexandria they spread to Rome, thence to Spain and the
western part of Africa.

"Between the second and eighth centuries the nine
characters in India underwent changes in shape. A prom-

K © o

•

o

o O O 2

hr <^ \,

o-

<h

t'V'Of o\ c^

IC CO 55

<

<

|o to oo Oo

PC < <^

>

>

< <

3 JS

X

3"

^ Aa vo ^o

p tr'ov

<3.

o
o

*3

=511

-^N

o<

Ko H\^^

1

1

♦O (o p

i.

Q.

// N h d

_ —

-»

—

•V ** «»4 *-1

Sanscrit letters of
the II. Century, a.d.

Apices of Boethius
and of the Middle
Ages.

Gubar-numerals of
the West Arabs.

1

a <

2

as
a

D e van agari - n u m*
erals.

From iheMirrour
of the World, printed
by Caxton, 14S0.

From the Bam-
berg Arithmetic by
Wagner (?), 14S3.

From De Arte
Supputandi by
Tonstall, 1522.

»-1 fo

24 MATHEMATICS

inent Arabic writer, Albiruni (died 1038), who was in
India during many years, remarks that the shape of Hindu
numerals and letters differed in different localities, and
that when (in the eighth century) the Hindu notation
was transmitted to the Arabs the latter selected from
the various forms the most suitable. But before the East
Arabs thus received the notation it had been perfected
by the invention of the zero and the application of the
principle of position.

Fig. 10 — Nana Ghat Inscription, Containing One of the
Earliest Forms of Hindu Numerals.

"Perceiving the great utility of the Columbus-egg, the
zero, the West Arabs borrowed this epoch-making sym-
bol from those in the East, but retained the old forms
of the nine numerals which they had previously received
from Rome. The reason for this retention may have been
a disinclination to unnecessary change, coupled, perhaps,
with a desire to be contrary to their political enemies in
the East. The West Arabs remembered the Hindu origin
of the old forms, the so-called Gubar or "dust" numerals.
After the eighth century the numerals in India underwent
further changes, and assumed the greatly modified forms

NUMBER 25

of the modern Devanagari numerals." Professor Moritz
Cantor recently expressed the opinion that the use of
the zero was probably due to the Babylonians, 1700 b.c.
There are two methods of reading numbers in general
use, in both of which the orders are grouped, beginning
with the. first order, or the order of units. In the French
method each group consists of three orders, such a group
being called a period. The names of the first three orders,
beginning with the lowest, are units, tens and hundreds.
These names are applied also to the three orders in each
period followed by the name of the period. The names
of the first 12 periods follow:

1. Units. 5. Trillions. 9: Septillions.

2. Thousands. 6. Quadrillions. iot Octillions.

3. Millions. 7. Quintillions. 11. Nonillions.

4. Billions. 8, Sextillions. 12. Decillions..

In the English method each period consists of six or-
ders, named units, tens, hundreds, thousands, ten thou-
sands, and hundred thousands. The names of the periods
follow :

1. Units. 3. Billions. 5. Quadrillions.

2. Millions. 4. Trillions. 6. Quintillions.

In both systems the number names are read in descend-
ing order from left to right, and in all cases compounds
are formed in the same way, except in the interval from
10 to 20. Professor Brooks, in 'Philosophy of Arithme-
tic,' gives the following account of number naming:
"A single thing is called 'one'; one and one more are
'two'; two and one are 'three'; and in the same manner
we obtain 'four,' 'five,' 'six,' 'seven,' 'eight,' and 'nine/
and then adding one more and collecting in a group we
have 'ten.' Now regarding the 'ten' as a single thing, and
proceeding according to the principle stated, we have one
and ten, two and ten, three and ten, and so on up to ten
and ten, which we call two tens. When we arrive at ten
tens we call this a new group, a 'hundred.' This is the
actual method by which numbers were originally named;

26 MATHEMATICS

but unfortunately, perhaps, for the learner and for sci-
ence, some of these names have been so modified and
abbreviated by the changes incident to use, that, with
several of the smaller numbers at least, the principle
has been so far disguised as not to be generally perceived.
If, however, the ordinary language of arithmetic be care-
fully examined, it will be seen that the principle has been
preserved, even if disguised so as not always to be im-
mediately apparent. Instead of one and ten we have sub-
stituted 'eleven/ derived from an expression formerly sup-
posed to mean one left after ten, but now believed to be
a contraction of the Saxon 'endlefen/ or Gothic 'ainlif
(ain, one,. and lif, ten) ; and instead of two and ten, we
use the expression meaning, two left after ten, but now
regarded as arising from the Saxon twelif, or Gothic
tvalif (tva, two, and lif, ten). In the numbers following
twelve, the stream of speech 'running day by day' has
worn away a part of the primary form, and left the words
that now exist. Thus, supposing the original expression
to be three and ten, if we drop the conjunction we have
three ten ; changing the ten to teen and the three to thir,
we have thirteen." In a similar manner twenty is a con-
traction of two tens. It is to be noticed that Professor
Brooks has always used the form- two and ten rather than
ten and two. That such use leading to the forms from
10 to 20 is the exception rather than the rule is seen
when it is recalled that from 20 on the larger number
is always read first.

The word million seems to have been used first by
Marco Polo (1254-1324). During the next 300 years it
was used by writers in several senses, and not until the
sixteenth century did it succeed in finally securing its
place in the number system. Billion in the English sys-
tem is equivalent to one thousand French billions, or a
trillion.

An example will suffice to show the two methods of
reading a number. Thus, 436,792,543,896,578, according

NUMBER 27

to the French method, is read four hundred thirty-six
trillion, seven hundred ninety-two billion, five hundred
forty-three million, eight hundred ninety-six thousand,
five hundred seventy-eight; while the English method
would be four hundred thirty-six billion, seven hundred
ninety-two thousand, five hundred forty-three million,
eight hundred ninety-six thousand, five hundred seventy-
eight.

The primitive form of the abacus was a board strewn
with sand, upon which lines were drawn and pebbles were
used as counters. On the Egyptian abacus the lines were
at right angles to the operator, and Herodotus states that
they "calculate with pebbles by moving the hand from
right to left, while the Greeks move it from, left to right,"
thus indicating that the units' column was taken with the
Egyptians on the extreme left. The varying values of
the. counters when changed from one column to another
is referred to in the comparison of Diogenes Laertius,
."A person friendly with tyrants is like the stone in com-
putation, which signifies now much, now little," which
recalls Carlyle's ranking of men with the pieces on a
chessboard. A single example of a Greek abacus is ex-
tant in the form of a marble table discovered on the
island of Solamis in 1846, and now preserved in Athens.
This table is 5 feet long and 2^ feet wide, and the lines,
which are parallel to the operator, are in a good state of
preservation.

Difficulty of calculation with Roman numerals rendered
necessary the use of the abacus, inherited from the
Greeks, and in turn, the ease with which the ordinary
computations were performed with its aid prevented the
perfecting or inventing of a usable system of notation.
Horace (Sat. I, 6, 75) alludes to the practice of boys
marching to school with the abacus and box of pebbles
suspended from the left arm : "Quo puero magnis ex
centurionibus orti, Lasvo suspensi loculos tabulamque
lacerto." In the time of greatest Roman luxury (Juvenal,

28

MATHEMATICS

Sat. II, 131) the counters were of ivory, silver, and
gold.

A more serviceable form was developed under Roman
usage, in which the table or board was replaced by a
thin metal plate with grooves cut entirely through, in

, '

yt .., *

•3 •"
x u

-3

: i

6.
0 *

1 1

h

'

3

Xi.3 I UvJ'Hii.x

Fig.

— Salamis Abacus. The Only Known Early Greek
Specimen.

which were metal buttons which could be slid from one
end of the groove to the other. If at one end, a button
registered one in that groove; if at the other, it was value-
less. In place of a long groove containing 9 buttons, a
shorter groove registered 4, and a still shorter one, imme-

NUMBER 29

diately above, had a value of 5. At the right of units',
column were two short columns in which could be regis-
tered twelfths, the Roman fraction, still preserved in
name, in ounce and inch. Several of these metal abaci
are to be found in museums.

Another form of abacus still in general use in the
Orient is that of a frame across which wires are strung,
upon which are movable beads. This is the 'swanpan' of
the Chinese and the 'tchotu' of the Russians. In 1812 the
abacus was carried from Russia to France, in the form of
the counting frame, as a device for teaching number in
primary work, and is now found in all kindergartens, a
slight evidence of belief in the "culture-epoch" theory
that the training of the child mind should follow the
steps in the mental development of the race.

At the decadence of Rome the Roman notation and
abacus reckoning remained as an inheritance to central
Europe. The Arabs being in possession of the Hindu
numerals carried them to Spain, and they were used in
the commercial towns bordering the eastern end of the
Mediterranean Sea. Some of the more aspiring youths of
England and France journeyed to Spain to acquire the
learning of the Greeks and Hindus which had been pre-
served and cultivated assiduously by the Moors. Others,
merchantmen of Italy, perceived the advantage gained in
the use of these numerals in the Phoenician towns, and
they in turn carried the knowledge home.

Of the former who visited Spain was Gerbert (d. 1003),
afterward Pope Sylvester II. Gerbert's abacus was of
leather, and contained 27 columns. In place of the old
counters new ones of horn were used, upon each of which
one of the first nine numerals was written. Thus the
first step in the use of Hindu numerals was taken. Of
the latter, merchantmen of Italy, was Leonardo of Pisa,
who in 1202 wrote a treatise on mathematics called 'Liber
Abaci/ It begins thus : "The nine figures of the Hindus
are 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine figures and

«

30 MATHEMATICS

this sign, o, which in Arabic is called sifr, any number;
may be written/'

The long struggle of 500 years for supremacy between
the line-reckoning, or abacus, and the Hindu numerals,
began. In one of the cuts is seen a page of line-reckoning

ADDITION.

Matter*
be eaficft tfcap m tbis arrets to abte
but tteo fummes at ones togptber:
%tfto be it, pou tnaf c ab&c moicas J &*! rck
fouanone* tbcrcfoie tbfjeunc pou fypn<*
e&be tteo rummcd,FouSn!i fpj&e fee botonr
one of tbcuMe fouccb not ibbtcbr, Mb tben
bp (i fciato a ipite erode tbc otber IpntSAlnb
aftcrtfcar&c fcreebotntcibeotbcr fumrne > fo
tfjattbat ipnemape
Jebertbenetbcimaa
if pott fcbottlbe abb:

fcMito* &6f$ fo SJ4* > P0S8

im£ muft fee pour fftmcB
aspoufceberc*

&nb tycn if pou
!pft* poamapcaba*
tbc oic to tbc otber in ttiz fame ptae. 02 tl*
pou map afcbf tl) :iu botftc robitbrr in flWflb
place :tobif b ^ap ,bpcaufe w 55 moft pls^ur ft

Fig. 12 — Reckoning on the Line (1558).

from an early English textbook, The Ground of Artes/
by Robert Recorde, 1558. This work, which ran through
at least 28 editions, is in the form of a dialogue between
master and pupil. The following extract concerns the
difficulty the pupil has in multiplying by a fraction as
to why the product should be less than the number mul-

NUMBER 31

tiplied. The master explains the definition of multipli-
cation, but the scholar is not satisfied, and the master says :
"Master. — If I multiply by more than one, the thing
is increased; if I take it but once, it is not changed;
and if I take it less than once, it cannot be as much
as before. Then, seeing that a fraction is less than
one, if I multiply by a fraction, it follows that I do
take it less than once."

"Pupil. — Sir, I do thank you much for this reason;
and I trust that I do perceive the thing."
The use of counters had not disappeared in England
and Germany before the middle of the seventeenth cen-
tury.

Various methods of finger reckoning have been devel^-
oped, and are commonly found in the older arithmetics.
The accompanying cut is from Recorde's 'The Ground of
Artes,' 1558, and gives a general idea of this practice.

According to Pliny the image of Janus or the Sun
was cast with the fingers so bent as to indicate 365 days.
Some have thought that Proverbs iii, 16, "Length of days
in her right hand," alludes to such a form of expressing
numbers.

An interesting illustration is given by Leslie: "The
Chinese have contrived a very neat and simple kind of
digital signs for denoting numbers, greatly superior to
that of the Romans. Since each finger has three joints,
let the thumbnail of the other hand touch these joints in
succession, passing up one side of the finger, down the
middle, and again up the other side, thus giving nine
marks applicable to the decimal notation. On the little
finger these signify units, on the next tens, on the next
hundreds, etc. The merchants of China are accustomed,
it is said, to conclude bargains with each other by help
of these signs, and to conceal the pantomime from the
knowledge of bystanders.

The Korean schoolboy carries to school a bag of count-
ing-bones, each about 5 inches long, and somewhat thin-

^

X1\ 4/(

8 0

J)0

200

pOOC

^

JOOO

2000

3000

4ooo

5000

6000
7000
3000

m

)Q00

Fig. 13 — Finger Reckoning.

NUMBER

33

ner than the ordinary leadpencil. A box of square sticks,
4 inches in length and about y2 inch square, called sangi,
is used in a very ingenious fashion by the Chinese for the
solution of algebraic equations.

The form of reckoning board adopted in the Middle Ages
has left some words and customs. Fitz-Nigel, writing about
the middle of the twelfth century, describes the board as
a table about ten feet long and five feet wide, with a ledge
or border, and was surrounded by a bench, or 'bank,' for
the officers. From this 'bank' comes the modern word
bank as a place of money changing. The table was cov-

Fig. 14 — Chinese Digital Notation.

ered after the term of Easter each year with a new
black cloth divided by a set of white lines about a foot
apart, and across these another set which divided the
table into squares. This table was called "scaccarium,"
which formerly meant chessboard, from which is the term
exchequer, the Court of Revenue.

CHAPTER II

CALCULATION

Under the term 'logistic' the Greeks treated what is
now ordinarily termed computation or calculation, the lat-
ter word coming from a Latin word meaning 'pebble/ in-
asmuch as the reckoning was done with counters or peb-
bles. Calculation is the process of subjecting numbers
to certain operations now to be denned. There are six
fundamental operations in arithmetic, all growing out of
the first. Formerly these were differently classified, some-
times as high as nine being considered, the other three
being special cases or complications of the fundamental
six.

These six operations are divided into two groups, the
direct operations, of which there are three, and the in-
verses, each of which has the effect of undoing one of
the former three.

DIRECT INVERSE

i. Addition. 4. Subtraction.

2. Multiplication. 5. Division.

3. Involution. 6. Evolution.

When one object is put with a group of like objects,
forming thus a new group having one more object than
the original group, the process is said to be that of addi-
tion, and is indicated by +. (This sign appears in a
work by Grammateus in 1514, and in 15 17 in a book by
Gillis vander Hoecke. Thus, 1 apple added to 2 apples
34

CALCULATION 35

gives 3 apples, or with abstract numbers, 2 H- i = 3« The
objects or numbers added are called 'addends* or 'sum-
mands/ and the resulting group or number is the 'sum/
The ending end or -and, so common in mathematical ter-
minology, is Latin present passive participle; in this case

C2M>itfc. tr

ai^ie fctn JU «bbircnt)ieqtiartt«tet emed ita^ flr/tcmR3tlDt oftrrthcti l*c \= Wllfc |« foal)
tneno/a\eVi.mitVi:piimxmttp:ima/(ccur\i)i . J j * ■

m«pfcub8/terti«mittcrtiajc.Tn&fna6b:aui»Hiccrtbept!tt)tqua{»flfntrerti T^ UflcrnaitniM
^etfok^jei^cnal&^iftmc^/rnb— /m:ripija£rtt),,{<ciip;j.1.C:atcuimtnftuacnntbtmti6(t

ro«nneinauantiter&«anbec&cno2fen-f' , , C*"^01'^ 1M-.1M#. .

•ber-fofo!mannfol|eqoantitetflb&itn&rrtCC»f"atIon2!c|Mt«..ct-- ^mcC4m»Bm

$3gefai3tbfie$eycber.-{-ober — icpfcbwn&m rustic/ atotwhttfc j tan& ^

Old 9pM.-j-7tl. <T pit.— 4W. fcn^ » «^V *

ff p».-WCT. « p» — lotft. 4 >

|«IF«/p».-f-'i!ft. >4pji.— 14^1. C?tnnim!ofaftrcciim& J ewft 7«/i& v

CSiccnber Kegel. ^_ , . ,' > • *

3|1in&cr5bernqusntttct-f-^?>fn^ro«®ft"nail0i^trtriini1^ 1 MnftJ— y « ct"*
trrn—/»nb-}-ubmrifft— /fofol^ecnbcr,^ *

flrdfffm/rnju&eniliaaboWtitcnbiP/feRC-' CrnulnpliMnrmorntfcoflnMMcjMlm.

ale <rp:!.-j-<;N: 4p:i.-j-»N. 'STXlOilOimulfipliffrcnuiDrn^fcewtCf i>\ttfh?

i*p;t.-—4N. gptr— <?M. _^ ^-^iitoctfftllcnallcDcnoinmtroUdu.fniDiiiKJfu

' ,8p?i,— I-2N. topii.— an. *i« aloft ttmultiplifcrcn mrt fimpcten ncrr.iiurfce

fTQjcbmr Kegel <rcorftfhDcimonunrrmulftptuc«niiac6ffiudiucvr

folubrrfl&rfMwvontoman^^

o«genfd)jcib.—3f?cofibfr/basbicrn&erqua;(oiDitmu!fip!icccrtmet4eotmtita!ooo;m.
ritrtftbfriri|ftbUfibern/fo5ie^ccmsrciibf««lwa6tmu:t«pli(crHS:»»8mct5foimtt*^«rtifli;
anfrcm/vmbjti bemcHTcn ff pc -j - alo ~

Fig. 15 — First Use of Plus and Minus Signs.
Left Column a Page from Grammateus ; Right, from Gillis vander
Hoecke.

addends is to be translated literally the 'being added'
numbers.

Addition is, in its simplest form, the putting together;
or uniting of two numbers; and all additions of this na-
ture may be broken up into a series of repetitions of the
fundamental process of increasing a number by unity.
Thus, if it be desired to add 3 apples to 5 apples, it may

36 MATHEMATICS

be done at a single step, or at three partial steps, which
may be indicated thus:

5 apples + i apple = 6 apples;

6 apples -J- i apple = 7 apples;

7 apples -J- i apple = 8 apples;
or 5 apples -f- 3 apples = 8 apples;

the three steps resulting the same as the single step given
last, which justifies the statement above that addition
rests upon the fundamental process of increasing a num-
ber by unity.

Like numbers are those in which the same unit is
used; 7 apples and 3 apples are like numbers, as also 7
and 3; 4 trees and 9 stones are unlike numbers, as are
5 ones and 7 tens ; that is, in a number 435, written in the
Hindu notation, 4 in hundreds' order is not like 3 in tens'
order, nor like 5 in units' order. It is fundamental that
only like numbers may be added; before 3 tens is added
to 5 ones, the 3 tens must be changed into 30 ones. This
is a very simple matter, only being, as it were, a shift in
thought, and it accounts in a great measure for the sim-
plicity of the operations with Hindu numerals. In 435,
the 4 may be thought of, in turn, as 4 hundred or as 40
tens or as 400 ones. The place-value feature permits
of numbers being immediately broken up into parts, and
these parts treated one at a time. Thus, in addition, like
orders are written in the same column and the columns
are added separately. This process is illustrated in the
following example:

432 = 4 hundred + 3 tens -|- 2 ones ;
265 = 2 hundred -j- 6 tens -f- 5 ones ;

697 = 6 hundred -f- 9 tens -f- 7 ones.

The sum of the ones, 5 + 2 = 7, is first found, and writ-

CALCULATION 37

ten below the column of ones, and the other orders are
added in succession.

A difficulty arises when the sum of a column is greater
than 9, the largest number that may be written in a col-
umn. An example will make this clear:

387 = 3 hundred + 8 tens + 7 ones;
256 = 2 hundred + 5 tens + 6 ones;

643 = 5 hundred + 13 tens + 13 ones;
or 5 hundred + 14 tens + 3 ones;
or 6 hundred + 4 tens + 3 ones.

The 13 ones is changed to 1 ten and 3 ones; the 3 is
written in ones' column and the 1 ten is added in ("car-
ried to") the tens' column. The 14 tens is treated in
a similar way.

Addition obeys the commutative law; that is, the addi-
tion may be performed in any order. 54-3 = 3 + 5. It
is immaterial whether the 3 is added to the 5 or the 5 is
added to the 3.

The associative law is also valid for addition. If 5
and 7 are to be added to 4, it does not matter whether
the 5 be added and then the 7, or the 5 and 7 first united
and then added to the 4. This is expressed by means of
parentheses. The parentheses mean that the numbers
within are first united: 4 + 5+7 = 4+ (5 + 7). If
two numbers are added, the sum is a number. This state-
ment seems like mere verbiage, but will take on meaning
when considered in the light of the other operations.

Subtraction is the inverse operation of addition. Addi-
tion is putting one number with another to form a third,
and subtraction is taking one number from another to
form a third. If addition has been stated in the form :
given two numbers, to find their sum, subtraction would
be stated: given the sum of two numbers and one of

38 MATHEMATICS

them, to find the other. The sum of two numbers is 8,
and one of them is 5, what is the other? would be solved
by taking 5 from 8, leaving 3. Subtraction is indicated
by — . The number taken away is called the 'subtrahend/
and the number from which the subtrahend is taken is
named 'minuend.' The resulting number is called 're-
mainder,' or 'difference,' depending upon which of the
two phases of subtraction is considered. These two points
of view may be brought out by concrete examples.

If A has $10 and pays out $7, how many dollars has
he remaining? In this example the $7, or subtrahend,
was originally a part of the minuend $10, and is taken
away. The $3 is then called 'remainder.' Again: If A
has $10 and B has $7, how many dollars must B earn to
have as many dollars as A? Here the $10 of A and the
$7 of B are distinct numbers, and the resulting number
is called the 'difference.'

In subtraction, the subtrahend is written before the min-
uend, with like orders in the same column. Each column
is subtracted separately:

476 = 4 hundred + 7 tens + 6 ones ;
263 = 2 hundred -f- 6 tens -f- 3 ones ;

213 = 2 hundred -f- 1 ten + 3 ones.

Two methods are in general use in the case that the num-
ber in an order of the subtrahend is too large to be taken
from the number in the same order of the minuend. Both
methods are inherited from the Hindus, having come
down from the earliest printed textbooks, and seem to be
of about equal difficulty.

The method of Decomposition, or Borrowing, consists
of taking 1 unit from the next higher order, changing
it to the order in question, adding to the number in that
order, which makes the subtraction possible. 7 hundred
+ 2 tens -j- 4 units = 7 hundred + I ten + lA units =
6 hundred + 11 tens + 14 units.

CALCULATION 39

724
— 269

455
6 hundred -}- 11 tens + J4 units
— 2 hundred + 6 tens + 9 units

= 4 hundred + 5 tens + 5 units.

The method of Equal Additions is based on the fact that
the same number may be added to both minuend and sub-
trahend without changing the value of the difference ; that
is, 724 — 269 = (724 + 100 -f 10) — (269 + 100 + 10) •
The 10 in the minuend is thought of as 10 ones, while in
the subtrahend it is necessary to think of it as 1 ten. Sim-
ilarly for the 100. The example used above is worked
by means of equal additions, and will show the transfor-
mations involved:

7^4 is repiaced by 724 + 100 + 10
— 269 — 269 + 100 + 10

hundreds tens ones hundreds tens ones
724+10 tens + 10 ones = 7 + (2+10) + (4+10) = 7 12 14
269+1 hundred+i ten= (2 + 1) + (6+1) + 9 = 3 7 9

In use with the first method it may be said
724
— 269

9 from 14, 5;

6 from 11, 5;

2 from 6, 4.
With the second method,

9 from 14, 5 ;

7 from 12, 5;

3 from 7, 4.

Another mode of thinking of subtraction is called the

40 MATHEMATICS

Austrian method, or the method of "making change."
That the greater portion of subtractions in the business
world is concerned with making change has led to a wide
use of the method in the school-room. It consists in
building to the subtrahend until the minuend is reached.
That it is the natural method is evidenced by the fact)
that it is almost invariably used by those who have never
had the benefit of, or have forgotten, school training :

987
— 236

One says 6 and 1 are 7 ; writes 1 ;
3 and 5 are 8 ; writes 5 ;
2 and 7 are 9; writes 7.

Its introduction as a distinct method is due to Augustus
de Morgan, England's foremost writer on arithmetic.

It is readily seen that subtraction does not obey the
commutative law. One may subtract 5 from 8, but not
8 from 5. This leads to the query, If one number is
subtracted from another, is the result always a number?
The answer is 'yes>' if the minuend is larger than the
subtrahend. Otherwise, that the result is not a number,
such as those heretofore considered. These will be called
natural numbers. If 5 is to be subtracted from 8 no diffi-
culty arises; but if attempt be made to take 8 from 5,
the fact arises that no such operation is possible. Such
a condition brings the arithmetician face to face with one
of the most important considerations in mathematics, one
without which the complete structure, modern mathemat-
ics, would not be possible. It is the principle of con-
tinuity, or principle of no exception, due to Hankel. It
may be stated in this form : There shall be no exception
to the applicability of any operation. If the result is not
found in such numbers as already belong to the system,
call this result a number of a new kind and determine
its properties.

CALCULATION 41

Suppose a man has $50 and spends $40, he has left $10.
This operation is subtraction. Suppose he spends $60 in-
stead of $40. This seems very much the same kind of
an operation. It is agreed to call this subtraction also,
and say that he has a debt of $10, which is a new kind
of number. The natural numbers may be represented by
dots with any chosen interval between them :

1 2 3 4 5 6 7 8 9 10

If one goes 4 dots to the right from the third dot, he is at
dot 7, or 3 + 4 = 7. If one goes 5 dots to the left from
dot 9 he is at dot 4. This going to the left is expressed
by as — or subtraction, 9 — 5 = 4- But if one starts at
dot 5 and attempts to go 8 dots to the left, no dot is found
to mark the stopping point. The fiat of the mathema-
tician says, let there be a dot there. In this manner a
series of dots is obtained extending to the opposite di-
rection,
— 6 — 5—4—3—2—1 01234567

These may be named or marked at pleasure. Call the first
one, at the left of 1, o, the second — 1, the third — 2, etc.
The reason for the choice of these names is apparent?
If a man has $1 and spends $1, he has no dollar remain-
ing, and the symbol for an empty place is o. If he now
spends $1 he is $1 in debt. As this is the opposite of $1
credit, it is appropriate to mark it — 1, giving it a sign —
to distinguish it from 1. If it is desired to mark the 1, a
plus sign, +> is put before it, calling all numbers to the
right of o positive numbers and those to the left nega-
tive numbers. Then 5 — 8 = — 3, while 8 — 5 = -f- 3.
All the numbers, as now represented, are called whole
numbers or 'integers/ If it is agreed always to mark the
ones at the left of o, one may mark the ones at the right,
or not, at will, and no confusion will arise, o is now a
number dividing the positives from the negatives. It is
called zero.

42 MATHEMATICS

The properties of a negative number which are most
important are two: (i) A negative number may be rep-
resented by a dot as far to. the left of o as the correspond-
ing positive number is to the right. (2) A negative
number destroys the effect of, or annuls, a positive num-
ber of the same value when added to it; thus, + 8 -{-
( — 5) = -|- 3, the — 5 destroying -f 5 of the + 8, leav-
ing + 3.

If in an addition example, all the addends are the same,
as in 2 -{- 2 -j- 2 -J- 2 = 8; the form is shortened into
4X2 = 8, tne first number, or the 'multiplier/ indicating
how many addends were taken. The second number,
showing the addend, is called the 'multiplicand/ The St.
Andrew's cross, indicating that the operation of multi-
plication is to be performed, was introduced by William
Oughtred in 1631. Robert Recorde, about 1557, intro-
duced = as the sign of equality, which he says is

"A paire of parallels or Gemowe lines of one
length, thus = becaufe noe 2 thyngs can be moare
equalle."

Multiplication is, then, in essence, repeated addition.

The Commutative Law is seen to be valid in this opera-
tion : 7 rows of 3 dots is the same as 3 rows of 7 dots ; or

3X7 = 7X3-

Multiplication also obeys the associative law; that is,
in a multiplication example where mor^ than two num-
bers, or factors as they are called when used in multipli-
cation, are involved, these factors may be grouped in
any manner.

3 X 7 X 5 = 3 X (7 X 5) = (3 X 5) X 7-
The 3 may be multiplied by the 7, and this result, called
a product, may then be multiplied by 5 ; or the 7 and 5
may first be multiplied and then the 3 used, etc.

A negative number multiplied by a positive gives a nega-
tive product. If in the line of dots one goes 5 dots to the
left, 3 times, one arrives at dot — 15, or — 5X3 = — I5-

CALCULATION 43

But if one attempts to multiply 3 by — 5, no meaning is
attached. One may perform a certain act 3 times, or 1
time, or 0 times (which means that the act is not per-
formed), but to attempt to perform an act — 5 times is
meaningless. In keeping with the Principle of No Ex-
ception, such an operation must be given a meaning, and
it is done by widening the definition of multiplication ;
but in doing so the old multiplication (repeated addition)
must be kept as a special case.

It should be noted that this application of the Principle
of Continuity is a purely arbitrary process. It may be
said since the multiplication by a negative has no mean-
ing, simply reject it and say it cannot be performed.
Such was the usage for a long time, and had it continn
ued so the whole system of mathematics would have been
like an unsymmetrical tree, simply allowed to develop
and branch in any manner. The filling out or com-
pleting the meaningless cases is like a process of graft-
ing which rounds out and gives a symmetrical growth.

One method of procedure here would be as follows:

— 5X3 = — 15, and knowing that with positive num-
bers the commutative law holds, it is agreed to still let
it be valid, from which, — 5X3 = 3X — 5> but — 5X3
= — 15 ; therefore, 3X — 5 = — *5> an<^ tne conclusion
is multiplication by a negative number changes the sign
of the multiplicand and then multiplies it. Another and
better method is to define the operation of multiplication
in such a way that it will be applicable in all cases. Such
a definition is the following: Multiplication is the per-
forming that operation on the multiplicand which, if per-
formed on unity (or one), produces the multiplier. To
multiply 3 by — 5.

The operation upon 1 which produces — 5 is to change
the sign of 1 and repeat it 5 times. Do the same with 3,

— 3, — 3, — 3, — 3, — 3, the sum of which is — 15, as
before. It will be seen that this definition of multiple

44

MATHEMATICS

cation includes repeated addition as a special case. In
the same manner it is seen that — 3 X ( — 5) == ~f~ J5-

Considering the four cases,

+ 3X(+-5)=+i5. +3X (-5) =-iS

. - 3 X (— 5) = + 15, - 3 X (+ 5) = - 15,

it is clear that when the two signs are both -f- or both — ,
that is, alike, the product is +; when they are unlike, the
product is — . The conclusion is, then, in multiplication,
two like signs produce -{-, and two unlike signs produce — .
The sign -f- is read plus, and — is read minus.

One of the commonest forms of the early methods for
multiplication is the Tessellated Multiplication, very much
akin to the usage of to-day.

3 6 ? 2 S

2

f

S

3

8

0

7

3

9

7

0

?

*

7

9

4

0

K

2.

/

f

1

0

2 3 7 7 3' 7 .5- ? Q

Fig. 16 — Tessellated Multiplication.

Another was the Quadrilateral Multiplication. In this
form the partial products do not progress to the left, as
in the Tessellated style, and are added obliquely, as shown
by the arrows. These were not drawn in the work.

Lucas de Burgo called the third form Latticed Mul-
tiplication. The multiplicand is the outside top horizon-
tal row, the multiplier the outside right vertical column.
The product of any figure of the multiplicand by a figure
of the multiplier is found in the square formed by the
intersection of the column and row in which the figures

CALCULATION

45

multiplied are found; thus 9 X 2 is found in the third
column from left, and second row from the bottom. These
products are added in the oblique columns cut out by the

6y2ff

Fig. 17 — Quadrilateral Multiplication.

diagonal lines to the left. Less purely mental work is

performed in this method than in either of the other two.

Napier, the inventor of logarithms, made use of this

3 & ? t s-

3 T £- S" o

Fig. 18 — Latticed Multiplication.

method in a device called Napier's rods, which were usu-
ally of bone, and enabled the operator to perform the mul-
tiplications mechanically.

46

MATHEMATICS

From these methods was evolved the modern form. As
in addition and subtraction, the numbers are broken up
into orders:

437
56

2622
2185

24472

Hundreds. Tens. Ones.

4 3 7

5 6

24
15

18
35

42

or,

42 ones = 4 tens -f- 2 ones ;

18 tens + 4 tens = 22 tens = 2 hundred + 2 tens;

24 hundred -f- 2 hundred = 26 hundred = 2 thousand + 6
hundred.

In the second row of partial products,

35 tens = 3 hundred -f- 5 tens;

15 hundred + 3 hundred = 18 hundred = 1 thousand + 8
hundred ;

20 thousand -f- 1 thousand = 21 thousand = 2 ten thou-
sands + 1 thousand.

The two partial products then appear thus, and are added :
2.622
2185

24472

The product of any two whole numbers is a whole num-
ber. The product of 0 and any whole number is 0.
The inverse operation of multiplication is called di-

CALCULATION

47

vision. In its simplest form it is repeated subtraction. If
it is asked how many 2's in 8? the answer would be de-
termined by subtracting 2 from 8 in succession as many
times as possible, noting the number of times, 4, as the
answer. Division has two phases. One may think of
finding how many times one number is contained in an-
other, which is 'Division,' proper, a species of measure-
ment, or one may wish to divide a number into equal parts,

■pant n*

Oicnne o<

t>am J c <o jparrt <]■>{ r.

1 — Co 1 ] > c — 1

aoslio protht tu inrnid. cbe foiw dm motfw
rndkitHim fmUicbmo^h quad Liff.ro «» l»"*
SmmMMiSo li nrmpli foi fota.iw nee in bun*

Di w«U pe fare lopiediftofcacbtero.joe.j 14-
f,9.5> , 4.e notree f»rlo per U quarro modi come
garaifotto.

9 <4..

9 ?4/«

r*9J*'^

m

JSBKpJf

',;M'4l<

3. 1 6

\6i\it\6T J

vNl t\i4i<

Fig, 19 — Right Column Is a Page from the First Printed
Arithmetic (Treviso) ; Left Column Is a Page from
Calandri, Showing Italian Long Division.

the number of such parts being given. This form is called
'Partition/ With abstract numbers no such distinction
need be made, but with concrete numbers it is important.
The name of the number to be divided is 'Dividend,'
of the dividing number 'Divisor,' and of the resulting
number 'Quotient.' If any part of the Dividend is left
undivided it is called 'Remainder/ There are various

48 MATHEMATICS

signs used to indicate division ; — or 8/2 may be regarded

2
as indicating that 8 is to be divided by 2, as also 8 : 2. The
sign in general use, -+-, was used by Dr. John Pell (1610-
1685), altho this sign had been in use with other meaning
by earlier German writers.

Three methods or algorithms for what is now termed
long division deserve to be mentioned. One of the epoch-
making works on arithmetic was written by Luca Paciuolo,
a Franciscan monk. This book, published in Venice in
1494, gives the first of these methods, the galley or scratch

1

^&7S(7 m-7U7 K§&'M
m m v

Fig. 20 — Galley Division.
I., Completed example; II., III., IV. and V., successive changes

method, a dividing upward. It is a relic of the old method
of reckoning on sand, where the figure is scratched out
as soon as used. The above example of the method is
from Purbach.

CALCULATION 49

Thus to divide 59078 by 74. In the first step, 7 is divided
into 59, and the quotient 7 is written, 7 7's are 49; 49
from 59 is 10, which is written above 59. The dividend is

10/
now /078. 7 4's are 28; 281 from 100 is 72, which is writ-
ten still above the last dividend. The new dividend is

7/
/»/

now /yS, and the division continues, each figure being
scratched out as soon as used.

The first downward division, the present Italian method,
appears in a printed arithmetic by Calandri (1491), altho
it is found occasionally in manuscript form during the
fifteenth century. See Fig. 19.

Consider the example following.

74)59078(798ff
5i8

727
666

618

592

26 Rem.

I shows the completed form of solution, and II the suc-
cessive steps, obtained by separating the number into
orders.

50 MATHEMATICS

II

74)59078(79
49 28
5i 8
5 1 8

727
63 36
66 6

6 6 6

6 1 8

56 32

59 2

5 9 2

2 6 Remainder

The three lines show the partial product in the three
stages of its reduction.

The third, or Austrian, method consists in omitting the
partial products and performing the subtraction at once :

74)59078(79811

618
26

Comparing the three methods as to two points, (1)
beginning on the left to subtract the partial product, (2)

CALCULATION

5i

writing the partial product, the following scheme will show
their relations:

Gallev. Italian. Austrian.

(1) Yes No No

(2) No Yes No

The Galley method is so called on account of the final
form, which resembles a boat under full sail. The Aus-

gAlci jXrXTteHfte 7 <T peji 8

-An Elaborate Form of Galley Division.

trian method, which probably will ultimately replace the
Italian, is constructed from a combination of the best fea-

52 MATHEMATICS

tures of both the older methods (2) of the galley and (1)
of the Italian.

As in the inverse process of subtraction it was found
that the operation did not always result in a natural num-
ber, and it was necessary to create a new kind of num-
ber, the negative, thus widening the number system to
form the class of whole numbers, or integers, it is to be
expected that a like condition exists in the case of division.

If 2 be divided by 1 the quotient is 2; but if one at-
tempts to divide 1 by 2 no corresponding whole number
is found. Considering the second phase of division, the
separating of a number into 2 equal parts, it is agreed
to let this quotient be a number such that it requires two
of them to make I, or unity. This new number is named
one-half, and written by putting the number divided above
a short horizontal line, and the divisor below the line, as
h The class of such numbers is called 'Fractions,' from
the Latin, frangere, to break. The number below the
line is called the 'denominator,' or namer, telling what
the part is; the number above the line tells how many
parts are taken, and is called the 'numerator,' or number-
er. This function of the numerator will be apparent later.

The first widening of the number system, which arose
in the case of the inverse operation, subtraction, created
exactly as many new numbers as there were already in
the system before the new numbers entered. Every com-
bination of two numbers with a minus sign between them
gives a positive or natural number, when the larger num-
ber appears before the sign; and a negative — that is, a
new — number when the smaller number comes first. In
division, the case is the exception rather than the rule
where either order of the numbers results in a whole num-
ber, as f and f , and if one order does so result the other
does not, as f and f. It is apparent, then, that the new
numbers taken in under the name fraction are infinitely
greater in number, when compared with the number al-
ready in.

:

..,.";

^

- =^yyiH tf

>xS ,.«•

Oldest Geometrical Drawings Known — the Ahmes Papyrus

CALCULATION 53

A fraction may be interpreted in any one of three ways :
the fraction f may be thought of as ( i ) 3 units divided
into 2 equal parts, (2) 1 unit divided into 2 equal parts,
and 3 of these parts taken, as, 3 times i; (3) an indi-
cated division not yet performed. The distinction between
(1) and (2) may be seen from a figure, where unity
or 1 is represented by a line 1 centimeter in length.

If the numerator of a fraction is 1, it is called a unit
fraction, as h h h A 'proper' fraction has a numer-
ator less than its denominator, as 7, f, tV All other
fractions are 'improper,' as 8/3, 5/2, 4/2. Such a fraction
can always be changed to either a whole number, as

1 \i

/" I 1 ) ' 1 3

=t2.
2.

V2 = 2, or a whole number and a unit fraction, as */2

= i'A-

The whole numbers were represented by dots arranged
on a line at equal intervals, extending to the right and left
indefinitely from a chosen dot, marked o, or zero.
—6 —5—4—3—2—1 0 1 2 3 4

The creation of the number x/2 introduces a point mid-
way between o and 1, and by combination with each of the
whole numbers in the manner 8/2 = il/2 also places a point
midway in each interval. The fraction % places a point
half way from o to l/2. By continuing this process it is
seen that distance between the dots representing fractions
is made smaller and smaller, as the various fractions
take their places on the line. When all of the fractions
have been represented, if one chooses a particular dot, say

54 MATHEMATICS

s/t, one can always find another dot among those placed
whose distance from the given dot, 3/t, is less than any-
assigned length of line. The proof of this may be put
in the form of conversation between A and B. If the
dot i is i inch from the dot o, A is to show that a dot
may be found in the collection of dots which represent
fractions which shall be nearer to the dot */i than any
fractional part of an inch which B may name.

bcbtSt bififigpr bet fe&m t#yl aim •

T fcfeflc ftgur {fi tm befcefft atrt ft ertel VM atrtQ
ITTX gamsen/af fo mag man and) tin f&nfftaft/ayt
fed>fiail/ain fibwtail o&er jt»ai fed)ftail 2c- ynb a\U
anbet briidj befd)mbeu/2U$ v ' "vl I viT I vj 2a

VI £>$ fcin S c&e ad>tatl/fcae fait fetyetoil bet
VlTT <*d;t aw gam3 mad)cn .
IX £>$ Stgwr besatgt arm rtcwrt ayttffxail fcao fan
XT IX nail/bcx XIa w gams madjett ♦

XX 2>$ 5tg»r bet^atc^ct/jwertijtgt atftwitbr^
XXXI Age tail /t>ae few swensigt tail ♦fcer ain$*
iw&rctjjtgfc awgattsmad?en *

lip 2>$ feut ^wat^iintert tat' l/fcer &'er&tw#
U11C.LX fcertrnOfc^tgUmgaw^ma^ert'

Fig. 22 — Fractions with Roman Numerals.

B says, "Is there a dot nearer to 3/? than 1/w of an
inch?"
A's reply is, "Choose the dot 31/70> whose distance from

s/z is y™."

B then says, "Find me a dot nearer than 1/wo of an
inch."

CALCULATION 55

A's answer is, 'The dot 301/700 is only 1/no of an inch from
*/"', and so forth for any value B may name.

The dots are said to be 'dense/ and it might be thought
that the whole line is filled up, that it has become a con-
tinuous line rather than a collection of discrete dots. But
such is not the case; there are infinitely more dots on
the line that do not represent fractions than there are
dots that do represent them. The third of the inverse
processes, evolution, will reveal the existence" of the miss-
ing dots, and by its aid they, as a new type of number,
will be included in the number system, which will then
be represented by a continuous line.

Fractions are treated in the most ancient mathematical
handbook known, written by an Egyptian scribe, Ahmes,
or Moon-born, some time before 1700 B.C.. This papyrus,
now preserved in the British Museum, is entitled 'Direc-
tions for obtaining the knowledge of all dark things,"
and covers practically the whole extent of Egyptian math-
ematics, no substantial advances being made until the time
of Greek influence. Another papyrus, that found at
Akhmim, written perhaps after 500 a.d., gives the same
treatment of fractions as is found in the work of Ahmes.
Thus Egyptian Mathematics was in its most flourishing
condition when Abram left Ur of the Chaldees, and re-
mained stationary for a thousand years. (See Frontis-
piece.)

The writer gives, in most cases, no general rule for
obtaining results, simply a succession of like problems,
the easy step of generalizing by induction seemingly be-
ing beyond his power. Whole numbers receive no treat-
ment, the work beginning with fractions, which subject
was evidently very difficult, as the author confines his at-
tention solely to unit fractions and fractions with numer-
ator 2. Fractions of the latter type are changed into the
sum of two or more unit fractions. Thus Ahmes changed
a/» into a/e and Vis, and gives a table of such changes of

5^ MATHEMATICS

fractions between yz and 2/99. By the aid of this table any;
fraction of odd denominator could be so broken up.

In this way Ahmes could solve such a problem as "Di-
vide 5 by 21." The 5 is first broken into 2 and 2 and 1 ;
from the table is found 2/2i = y14 and 1/a; 5/2i = */»
and (yM and y«) and (y« and y«) = y21 and (2/M and
7*0 = 7» and y, and ya = y, and 2/2i = y, and x/w and
/*2. The fractions were written side by side, with no sign
for addition between them.

While the Egyptians met the difficulties of fractions by
reducing them to fractions having a constant numerator,
I, the Babylonians avoided the same difficulties by treating
only fractions with a fixed denominator, 60, and the Ro-
mans also used a single denominator, 12.

The usual rule for the division of fractions by inverting
the divisor and then multiplying is not common in the
textbooks of the sixteenth century. It is given as follows
by Thierfeldern (1578) :

"When the denominators are different invert the divisor
(which you are to place at the right) and multiply the
numbers above together and the numbers below together;
then you have the correct result. As to divide % by Yz,
invert thus : ji X V5 = "/» = llA"

The close of the eighth century found the Hindu deci-
mal notation practically perfect as a means of writing
whole numbers. The final perfection of the method by
applying it to fractions, in the form of decimals, did not
occur until the time of Simon Stevin (1 548-1620). In
seven pages of his work, published in 1585, Stevin leaped
what had been an impassable gap for 900 years. The
reason for this pause is not difficult to determine.

In decimal fractions, or decimals, unity, or 1, is divided
into ten equal parts, each part called a 'tenth'; a tenth
is divided into ten equal parts and each part called a 'hun-
dredth'; thus the orders on the right of units' column are
symmetrically named, adding the suffix -th, with those on
the right. As the number of orders on the left is unlim-

CALCULATION 57

ited, so the number of orders on the right is unbounded,
and one is enabled to write numbers of unlimitedly small
value; the smaller the value of the number (less than i),
the larger the number of orders required to express it.
The units' column is marked by placing a period after it
(sometimes the period is midway between the top and bot-
tom lines of the type, as 3-8, but ordinarily it is written
on the base line, as 3.8 for 3 and 8 tenths). In reading
decimals the decimal point is always read 'and.'

In the first grouping of units there was no reason for
putting ten in a group rather than any other number, the
use of ten simply growing out of the use of the hands as
a counting machine. In fact, it would have greatly sim-
plified some applications of the number system if primi-
tive mathematicians had been born with six fingers on
each hand. A duodecimal, or 12 scale, would enable the
writing of such common fractions, Yz,2/z,TAy duodecimally,
in the form .4, .8, .2; whereas decimally they have a con-
tinually recurring set of figures, % = .3333, etc; % =
.1666, etc. Charles XII. of Sweden, a short time before
his death, while lying in the trenches before a Norwe-
gian fortress, seriously debated introducing the duodeci-
mal system of numeration into his dominions.

On the other hand, there is a very decisive predeter-
mining feature in the case of the division of the unit. Ne-
cessity arose for halving or dividing objects into two equal
parts long before separation into ten parts was even
thought of; while the difficulty of dividing into ten equal
parts is apparent. The use of the period (or comma) to
mark the unit order began with Pitiscus, 161 2. With all
the advantages of the decimal notation carried to the
right of the units' column, it was not until the nineteenth
century that decimals came into ordinary arithmetic.

CHAPTER III

POWERS OF NUMBERS

If in the product of several numbers, these numbers or
factors happen to be a repetition of the same number, or,
in other words, if the factors are equal, the product is
called a power of the number which was repeatedly used
to produce it. The process of finding a power of a num-
ber is involution. The term power was used by the early
Greek writers in this sense. The powers are named fol-
lowing the ordinal names of the number of times the fac-
tor is used. If the factor 2 is used 5 times, as 2 X 2 X 2
X 2 X 2, or 32, 32 is said to be the fifth power of 2. The
second power is called the square of the number, as it
was early known that the number of square units in a
square is equal to the second power of the number of
units of length in one side.

If a square is 5 inches on each side, its surface may be
measured, using a small square 1 inch on each side. Such
a unit is called a square unit or square inch, or a unit of
square measure. This square inch may be laid along one
edge 5 times, thus forming 1 row of 5 square inches; 5
such rows may be formed one above the other, completely
using up or covering the original square. The area or
surface of the square is then said to be 5 + 5 square
inches or 25 square inches. The number of square units
in a square is then the second power of the number of
units of length in one side. This fact, which was early
known, led to the naming the second power of a number
58

POWERS OF NUMBERS

59

the square of the number. In a similar manner the
volume of a cube is found by taking for the unit of cubical
measure a cube I inch on each edge.

A cube is a solid figure in which all of the edges, meet-
ing in a corner, are at right angles to each other, and in
which all the edges are equal. In this cube each edge is
5 inches. Its volume is found by taking for a unit of
cubical measure a cube i inch on each edge. This unit

Fig. 22, — Square Measures.

or cubic inch is laid along one edge as many times as
possible, or 5 times, thus forming a row of 5 cubic inches.
On the bottom, 5 such rows may be formed, giving a
layer of 5 X 5 cubic inches. It requires 5 such layers to
fill up the given cube, or 5 X 5 X 5 CUDic inches. This use
of the third power of the number of inches on the edge
gives the name 'cube' of a number to the third power of the
number. Since no solid figure exists with 4 edges at right
angles, this process of naming the powers ceases with the
third, or cube. In the figure, taken from a paper by Miss
Benedict, are shown various symbols which have been
devised for the indication of powers.

6o

MATHEMATICS

Writing the number of the power a little above and to
the right of the number, as f for 7 X 7 X 7> 1S due to
the French mathematician and philosopher, Des Cartes.
The 3, which indicates the number of times the 7 is used
as a factor, is called an 'exponent/ while the 7 is termed
the 'base.' The exponential notation permits the writing
of very large (or very small) numbers much more com-

/

/

/

/

s 1 X
f 1 /

1 /

l/

)

— 1 1 1 •

Fig. 24 — Cubic Measures.

pactly than can be dene without its use. Modern re-
searches in Astronomy and Physics have rendered neces-
sary the use of extremely large numbers (as well as ex-
tremely small), the lower orders of which are either un-
known or of small consequence. The number of vibra-
tions per second of light waves in the visible spectrumi
vary from 3.94 (io14) to 7.63 (io)14. The wave lengths

t *C*£ g M i eg M + °£ £ g i » 1 1 t ri

+4©-!

+ J -f o-ia. 4. 4.

£cLa-+

t- + + + +

J2 u2P(^q^

Q!2,t£2

«£->^

to * * * *

1

0 tit %v a nx

<* u J

c

«*■ 0 0-

0 c£

<3?ty *?<*^* £

o?^0< dxx

- °c

Z^^ 2 Z

v r c ©

u^s

^ ^0 -

£ or 3 U U U

1 tt

ce? or 0? ^ ^ «->>

5>^ £

J!,2^<?*u

>> ^> £ ^

df cEf of c£> >->>-•>

J*

dC

8-2^-S^S^S^ * -5 3 ^ U5^ 5 * *

k. y *- o i <u (J Y) .. -*3 <3 -£ c* o 5 «£» *i « S <u

62 MATHEMATICS

of the spectrum vary from .0000007621 meter to
.0000003933825 meter. In the exponential notation these

1 1

numbers would be written 7621 X — , an(l 3933-825 X — •

io10 io10

If two powers of the same number are to be multiplied,
the exponents are added, as

fXf = f + 5 = f

(7X7X7) X (7X7X7X7X7) =7X7X7X

7X7X7X7X7

If two powers of the same number are to be divided, the

exponent of the divisor is subtracted from the exponent

of the dividend.

f-t-f = 7X7X7X7X7-^-7X7X7 = 7X7-

If a power of a number is itself to be raised to a power,

as in finding the third power of f, the result is obtained

by multiplying the exponent 2 by the 3, exponent of the

power to which "f is to be raised:

(fy = fxfX7' = 7'=?x'-

A corresponding process takes place in extracting a
root of a power :

f-yr- = f72X72X72 = f = 7e + s
As the exponent indicates the number of times the base
is used as a factor, it must be a natural number, since
using a number as a factor — 3 times, or ^ a time, is
meaningless. The Principle of No Exception is here ap-
plied as before, and a meaning is given to exponents of
the form, — 3, 2/$, o, which will be at the same time con-
sistent with the meaning of a whole number used as an
exponent.

If 7s be divided by y5, the quotient is 1 ; but if the expo-
nents be subtracted, as is done when division is performed,
the quotient is 7s — 5 = 70, which should be equal to 1. In a
similar manner it may be shown that any number with an
exponent o is equal to 1. Is this reasonable in the light of
the use of an exponent to tell how many times a factor
appears? In 3 X 5> or I5> 7 1S not used as a factor; or in

POWERS OF NUMBERS 6$

other words, it is used zero times, which may be written

3X5X/ = 3X5Xi=3X5-

Carrying the reasoning a step further,

f _5_ 78 = (7X7X7><7><7) ^ (7X7X7X7X7X7X7X7) = J*

But subtracting exponents,

7 -^7 =7 =7 •
Therefore 7 ~ 3 = -§, which may be stated generally. The

sign of an exponent may be changed by changing the po-
sition of the number from one side of the denominator
line to the other.

The meaning to be attached to y2/3 is determined in a
similar manner. It will be assumed that 2/$, when used as
an exponent, while as yet it has no meaning, will follow
the law above for multiplication ; that is, to multiply 7V3
by itself, the exponents are added,

7% X 7^ = 7* + § = 7*

Repeat the process,

7§ X 7§ X 7§ = 7* + § + * = 7* = 72-

When a number is used as a factor 3 times it is said
to be cubed. The inverse process, of finding the number
when its cube is given, is called finding the cube root.

Since the cube of 78 = y2, 7 must be the cube root of
72; that is, the numerator of a fractional exponent tells the
power that is to be taken, and the denominator tells the
root to be taken. 32% means that 32 is to be squared and
its fifth root found, 322 = 1024. The fifth root of 1024 is
4, since 4X4X4X4X4 = I024> whence 32% = 4;
165 = 1 6* = 4, since 4 X 4 = J6; 71 is, of course, 7.

The use of exponents in computations greatly facilitates
the work. Exponents so used are called logarithms.

It will be agreed that 10 will be used as a base, and that
every number is some power of 10, understanding by
power, 10 with an exponent which is not necessarily a
whole number. io° = 1, 101 = 10, io2 = 100, io3 = 1000,
etc. Since the exponent of 1 or io° is o, and of 10 or io1 is

64 MATHEMATICS

i, any number between i and 10 must have for an ex-
ponent, or logarithm, a number between o and i. In the
same manner, any number lying between 10 and ioo will
have a logarithm whose value is between I and 2. These
facts may be put in a very brief form :

Logarithm of 1 is o.

Logarithm of 10 is 1.

Logarithm of 100 is 2.

Logarithm of 1000 is 3.

Logarithm of 8 is a decimal lying between o and I.
The value of this decimal, found by an elaborate process
of calculation, is .903090 -f- an unending decimal. Tables
have been calculated of these exponents, to every number
an exponent, or logarithm, and to every logarithm a num-
ber. If it be required to multiply one number by another,
the logarithm of each number is found in the table ; these
two logarithms are added, giving, according to the method
of adding exponents, the logarithm or exponent of the
product. Opposite this logarithm is found the number
or product desired. Thus, by the use of logarithmic tables
the operation of multiplication is replaced by the much
easier and shorter operation of addition, and division is
replaced by subtraction. This final step in the perfecting
of the methods of computation was the invention of John
Napier, Baron of Merchiston (1550-1617). It seems to
be an easy consequence of the exponential notation, but,
curiously enough, was discovered by Napier before the
invention of exponents by Des Cartes in 1637, altho the
first steps toward this exponential notation are found in
the works of Simon Stevin (1548-1620).

In October, 1608, Hans Lipperhey invented the tele-
scope. In the summer of 1609 it was perfected by Galileo,
and from this date began the conquest of the heavens.
The next century, terminating with the death of Sir Isaac
Newton, 1727, was the golden age of astronomy, in which
the movements of the celestial bodies were subjected to
mathematical law. It is a striking coincidence that the

POWERS OF NUMBERS 65

invention of the telescope, which so increased the need
for tedious calculation, should occur almost simultaneously
with the invention of logarithms, which to such a degree
shortened these calculations. The greatest of French
mathematicians -and astronomers, La Place, paid this trib-
ute to Napier: "The invention of logarithms, by shorten-
ing the labors, doubled the life of the astronomer."

"It is one of the greatest curiosities of science that
Napier constructed logarithms before exponents were
used," says Cajori, "and the fact that logarithms naturally
flow from the exponential symbol was not observed until
much later, by Euler."

Following is a description of Napier's method: "Let
AE be a definite line (AE is taken to be io7, a proceeding
very similar to the basing of the Babylonian number sys-
tem, or 604), A'D' a line extending from A' indefinitely.

BCD a

^ j-H 1

A' B' C D'

Imagine two points starting at the same moment, the one
moving from A toward E, the other from A' along A'D'.
Let the velocity during the same moment be the same
for both. Let that of the point on line A'D' be uniform ;
but the velocity of the point on AE decreasing in such
a way that when it arrives at any point C its velocity is
proportional to the remaining distance CE. If the first
point moves along a distance AC, while the second one
moves over a distance A'C, then Napier calls A'C the
logarithm of CE." The adaptation to the number 10 was
suggested at a meeting of Napier with Henry Briggs,
who was professor of geometry in Gresham College, Lon-

66 MATHEMATICS

don. Briggs' own words indicated his admiration for the
invention: "Napier, Lord of Markinston, hath set my
head and hands at work with his new and admirable log-
arithms. I hope to see him this summer, if it please God,
for I never saw a book which pleased me better and made
me more wonder." Briggs was delayed in his journey
to meet Napier, who said to a friend: "Ah, John, Mr.
Briggs will not come." Just at that moment Briggs ar-
rived, and it is said that almost one-quarter of an hour
passed by, each beholding the other without speaking a
word. Briggs at last spoke : "My lord, I have under-
taken this long journey purposely to see your person, and
to know by what engine of wit or ingenuity you first came
to think of this most excellent help in astronomy, viz., the
logarithms ; but, my lord, being by you found out, I won-
der nobody found it out before, when now known it is
so easy."

Computations of logarithms to the base io soon fol-
lowed, and are known to-day by the name Briggs loga-
rithms.

In 1647 Gregory St. Vincent discovered that the use of
a base denoted by E = 2.718281828459046 + . . . had
a peculiar relation to the equilateral hyperbola. Such
logarithms are called hyperbolic, or natural, altho occa-
sionally incorrectly termed Naperian, and are of immense
service in Pure Mathematics. Since Napier did not use
exponents, he cannot be said to have used a base in his
system. If, however, his logarithms are expressed as ex-
ponents, the base or number which is raised to the power
would be very nearly 1/e, where 1/e is the base of the natu-
ral system.

The invention of logarithms was designed to simplify
the labor of calculation. An attempt along another line
has been to perform the calculations mechanically. Na-
pier, with the "rods or bones," succeeded in a way with
multiplication. The first successful attempt to perform

Fig. z'j — Leibnitz Calculating Machine.
From Poleai, the Earliest Treatise on Mechanical Calculation.

POWERS OF NUMBERS 69

the first four operations by machinery alone was that of
Blaise Pascal (1623-1662), when a lad of eighteen. The
close application to this work undermined a not over
strong constitution, and he died at the early age of 39.

The Pascal machine, which is here illustrated, was con-
structed on the principle of a wheel upon the circumfer-
ence of which were marked the first 9 numerals. One
turn of this wheel caused the next wheel, similarly marked,
to pass through a tenth of a revolution, and so forth.
Pascal's machine was not built, however, strictly on a deci-
mal scale, as it was designed for monetary work. A simi-
lar attempt was made by Leibnitz, the German mathema-
tician.

The most elaborate calculating engine ever attempted
was designed by Charles Babbage (1791-1871), on which
he expended a private fortune of over $100,000, and to-
ward which the British Government contributed $80,000
and a fireproof building for its construction. While the
machine was never completed, the work on it left an in-
delible stamp on British artizanship. The most success-
ful machine was constructed by George and Edward
Scheutz, who were inspired by the attempt of Babbage.
This machine, which computes and prints logarithmic and
other tables, finally came into the possession of the Dud-
ley Observatory at Albany, N. Y. The last few years
have seen a great advance in the art of constructing com-
puting machines for purely commercial purposes.

The inverse process of involution is evolution, the prob-
lem of which is to determine one of a given number of
equal factors when their product alone is given. The
factors so found are called square root, cube root, fourth
root, etc., depending upon the number of factors involved.
The square root of 4 is 2, the cube root of 2j is 3. The
simplest method of extracting a root is to divide the num-
ber by its lowest prime factor and continue the process.
It may be illustrated in finding the cube root of 216. Since

70

MATHEMATICS

there are three factors 2, and three factors 3, there are
three factors 2X3,or6;or the cube root of 216 is 6.

2 ) 216

2 ) 108

2 ) 54

3) 27

3) 9

3l3

The symbol of evolution is V > an abbreviation, r,
for root, followed by the vinculum; a figure is placed
above the V to indicate the root taken, except in the case
of square root, when it is usually omitted.

The ordinary algorithm or scheme for finding square
root is given in a paraphrase of the work of Theon, of
Smyrna, who flourished about 139 a.d. : "We learn the
process from Euclid, II, 4, where it is stated, 'If a straight

D

£

* F

2 /oo
/o

2.

A B

line be divided by any point, the square on the whole line
is equal to the squares of both parts, together with twice
the oblong which may be found from those segments.' Scs
with a number like 144, we take a lesser square, say 100,
of which the root is 10. We multiply 10 by 2, because
in the remaining gnomon, ABCDEF, there are two ob-

72

MATHEMATICS

longs, and divide 44 by 20. The remaindei, 4, is the
square of AB, or 2."

Cube root is found in a similar manner, based on the
cube instead of the square. Thus the cube on the sum
of two lines, a and b, is equal to the cubes on a and b and
3 flat figures a on two edges and b on the third, together
with 3 oblong figures b on each of 2 edges and a on the
third; this is expressed by a formula: (a + b)3 = a3 +
3a2b + 3ab2 + b8.

Fig. 29 — Illustration of Cube Root.
1566.)

(Trenchant's Arithmetic,

That evolution does not always result in a number of
our system, a fraction, which will now be called a 'rational'
number, is seen if one attempts to find the square root
of 2. This may be done with any degree of approxima-
tion by annexing ciphers on the right of units' column,

resulting in an endless decimal, 1.4142 That

this number cannot be expressed as a fraction is proved
in Euclid's 'Elements of Geometry,' altho the proof
is attributed to some commentator. Suppose W— —
where m and n represent the numerator and denominator
of a fraction, and have no common factor. Then multi-
plying this equation by itself, member by member,
2 = m2/n2, which says that m2 is divisible by n2, which can-

POWERS OF NUMBERS

73

not be, since m and n have no common factor. In a
square, side I, the diagonal is represented by 1/2.

It is proved in Euclid I, 47, that the square of AC is
equal to the sum of the squares on BC and AB. The
square on AB is 1, on BC 1, and the sum of these is 2.
The square on AC is 2, then AC is 1/2. If AC and AB
have a common measure — that is, if a third line exists
which is contained a whole number of times in AB and

AC — y_2 would be represented by the quotient of two

1
whole numbers, as — , which is shown above to be im-

possible. If AB is taken as this third line, it is contained
in itself once, and in AC more than once and not twice;
or, the ratio of these two numbers, ™, is less than 2 and
more than 1. This may be put in the form, t < — < 2.
If Vio of AB is taken, there results 1.4 < ^ < 1.5. If
V10 of this is used, 1.41 <™ < 1.42. Continuing, 1.414
<5i < 1.415, 1.4142 ■<" < 1.4143, and so on indefinitely.
These two lines are said to be incommensurable; that is,
they have no common measure. Euclid does not treat
of incommensurables as such, as his mode of representing*
numbers by lines, which will be spoken of later, and the

74 MATHEMATICS

peculiar device used by him in dealing with ratios, avoid-
ed the difficulty. Theodorus (c. 400 b.c.) showed that the

lines represented by V 3> V 5> / 7> V 8> V *o» V IX» V I2>

y 13, ■/ 14, |/ 15, and j/ 17, are incommensurable with the
unit line.

Going back to the number system following division, it
was found to be representable by a series of dots, between
any two of which existed a third dot, yet the dots do not
form a continuous line. If one chooses as the side of
the above square the distance from dot 0 to dot 1, and then
lays off AC from 0, the end C will give a dot which is
not found in the system of rationals. The final widening
of the number system, so far as arithmetic is concerned,
takes place here when such expressions as V~z^V~^Tt or
the ratio of the circumference to the diameter of a circle
= 3.14159 . . ., e, the base of the Naperian system of
logarithms are called numbers, altho none of them is rep-
resentable fully by any number of orders in the Hindu
notation. Such numbers are called irrationals, and are
divided into two classes : surds, which are expressible by
a combination of root signs, and transcendentals, which
are not, as n and £. A transcendental is sometimes de-
fined as a number which is not the root of any algebraic
equation, with positive integral exponents and rational co-
efficients.

Irrationals were discovered by the Pythagoreans. The
following story is told concerning irrationals: "It is
said that the man who first made the theory of irrationals
public died in a shipwreck because the unspeakable and
invisible should always be kept secret, and that he who
by chance first touched and uncovered this symbol of life
was removed to the origin of things, where the eternal
waves wash around him." Such is the reverence in which
these men held the theory of irrational quantities.

Greek arithmetic, the science of numbers as distin-

POWERS OF NUMBERS

75

guished from logistic, or calculation, has its beginnings
with Pythagoras (circa 569-500 B.C.), who founded a
brotherhood holding common philosophical beliefs, which
were based on mathematics. The Pythagoreans did not
commit their work to writing and held it secret from those
outside their own circle, and the glory of any discovery
was given to Pythagoras himself as the founder of the
school.

The properties of numbers studied by the Pythagoreans
may be classed under four heads which give rise to four
types of numbers : 'Polygonal' numbers, or those num-
bers which if indicated by dots can be arranged in poly-
gons or regular figures; 'factors' of numbers, numbers
forming a 'proportion/ and numbers in 'series/

0

0

0

0

O

O

/

6%

3

0

0

0

0

0

O

0

0

0

0

O

O

&:

0

0

0

0

O

O

b o\
00 o.

$

5^/di

0

0
F

0

0

O

O

Fig

00 ' 0 ' o-
to

31 — Triangular

O O Q

6

Lg- 30

— Gnomon.

Numbers

All numbers (whole) are divided into two classes, even
and odd. The odd numbers, I, 3, 5, 7, . . . are called 'gno-
mons'— that is, an odd number is always the difference
between two square numbers, and can therefore be rep-
resented by the figure which remains when a square is
cut from the corner of a larger square. Thus in the figure
36 is a square number, since it can be arranged in the
form of a square with 6 dots on a side. The lower right-

76 MATHEMATICS

hand square 16 is taken from 36 and there remains the
gnomon 20.

The product of two numbers is said to be plane, and if
the number cannot be represented by a square it is called
oblong. Triangular numbers are those which can be ar-
ranged in the form of a triangle :

In the triangular number 10, one side of the triangle is
4. The following passage from Lucian (given by Ball)
has reference to this fact. A merchant asks Pythagoras
what he can teach him. The following conversation en-
sues :

Pythagoras — I will teach you how to count.
Merchant — I know that already.
Pythagoras — How do you count?

Merchant — One, two, three, four

Pythagoras — Stop ! What you take to be four is
ten, a perfect triangle, and our symbol.
It may be said that the whole treatment of numbers by
the Greeks through the time of Euclid was geometrical.
The ease with which numbers could be represented by
lines led to a habitual linear symbolism such as is used by
Euclid (circa 300 B.C.), where the second, seventh, eighth
and ninth and tenth books either deal with magnitudes,
which include lines as well as numbers, or numbers them-
selves which are represented by lines.

The first proposition of the seventh book of Euclid is
taken from T. L. Heath's 'Euclid,' Vol. II, p. 296, the most
valuable commentary that has appeared in English :

Two unequal numbers being set out, and the less
being continually subtracted in turn from the greater,
if the number which is left never measures the one
before it until an unit is left, the original numbers will
be prime to one another (that is, will contain no com-
mon factor).

For, the less of two unequal numbers AB, CD being
continually subtracted from the greater, let the num-
ber which is left never measure the one before it until

POWERS OF NUMBERS

77

an unit is left. I say that AB, CD are prime to one
another — that is, that an unit alone measures AB, CD.

For, if AB, CD are not prime to one another, some
number will measure them.

Let a number measure them, and let it be E ; let CD,
measuring BF, leave FA less than itself; let AF,
measuring DG, leave GC less than itself, and let GC,
measuring FH, leave an unit HA.

- G

B O

t

Since, then, E measures CD, and CD measures BF,
therefore E measures BF. But it also measures the
whole BA; and therefore it will also measure the re-
mainder AF.

But AF measures DG; and therefore E also meas-
ures DG. But it also measures the whole DC ; there-
fore it will also measure the remainder CG.

But CG measures FH, therefore E also meas-
ures FH.

But it also measures the whole FA ; therefore it
will also measure the remainder, the unit AH, tho
it is a number, which is impossible.

Therefore no number will measure the numbers
AB, CD ; therefore AB, CD are prime to one another.
This theorem leads to the usual method of determining
the largest number which is a common factor of two given
numbers. The smaller is divided into the larger, the re-
mainder from this division into the former divisor. The

78

MATHEMATICS

final remainder which is contained without a remainder is
the largest common divisor; if this last divisor is unity
the numbers are said to be prime to each other.

5^5

Fig. 32 — Albert Durer's Engraving Melancholy, Showing
Magic Squares.

With the Greeks is found much mysticism, imbibed from
the Egyptians. The Pythagoreans sought the origin of
all things in number.

POWERS OF NUMBERS 79

One is the essence of all things ; four is the symbol of
perfection corresponding to the human soul ; five is the
cause of color; six of cold; seven of mind, health and
light; eight of love and friendship. A perfect number is
equal to the sum of its factors : 28 = 1 + 2 + 4 + 7 + 14.
Other numbers are excessive or defective. Amicable num-
bers are those each of which is equal to the sum of the
factors of the other, as 222 = 1 +2 + 4 + 71 + 142 and
284 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 45 + 55
+ no.

To Eratosthenes is due a method of picking out prime
numbers, numbers which have no factors except the num-
ber itself and unity. The even numbers, except 2, contain
no primes. All the others, as far as one wished to go,
were written upon a papyrus. Every third number con-
tains 3 as a factor and was cut out of the papyrus, so with
every fifth, seventh, and so forth. The remaining numbers
on the papyrus are prime. The papyrus with the holes
where the numbers were cut out was called Eratosthenes'
sieve. The last important Greek writer on arithmetic was
Diophantus of Alexandria, who flourished about 150 b.c.
His work will be mentioned in connection with Algebra.

One of the famous theorems in the theory of numbers,
due to Ferrnat, concerns the number of primes contained
in the form Fn = 22n + 1 where n is any number. Fermat
believed that every value of n gives a prime and showed
this, for n = o, 1, 2, 3, 4.

Euler in 1732 found that for n = 5 the number has a
factor, 641. Factors have been found for each of the
following values of n: 6, 7, 9, n, 12, 18, 23, 36, 38.

Fermat asserted without proof that nn + yn= znis un-
solvable except in certain self-evident cases. Mathemati-
cians have not as yet been able to prove or disprove this
statement.

Dedekind's view of the irrational as a "schnitt," or cut,
may be given in his own words, "If all points of the
straight line fall into two classes, such that every point

to

26

6

s

ts

tS

/o

n

$

U

/

'7

/2

2S

S

2d

7

/2

s

/£

,s

22

2

fS

9

//\22

2d

3

/S

7

/4

2/

i2

2S

s

9

J5

20

//

2?

*

22

2.

/#

24

/

'7

26

/

/s

7

/*

2t

3

/*

zs

ft

^7

4

/J

V

2

/o

26

6

6

//

/£

26

3

s6

t6

2/

&

9

td

20

2$

*

'A

Fig. 33 — The Nine Sections of a Magic Cure. (Andrews.)

47

22
II
62

19

26

35
38

16
63
46
21
36
39
18
27

23
48
61
12
25
20

37
34

64
9
24
45
40

33
28
17

49
60

1

8
13

4i
32

2
5

52
57
44
29
16
55

59

5°

7

4

53

14

3i'

42

6
3

"58"
5i
30
43
54
15

Fig. 34 — Closed Knights Tour, Magic Square. (Wenzelides.)

A

BC

D

E

F

G

H

1

K

L

M

N

0

P

a

u

-Jf

11

-3?

17

3/

7?

3Z

Sf

Z2

-23

U

-II

-77

i-

*f

Fig. 35 — Euler's Magic Square.

POWERS OF NUMBERS 81

of the first class lies to the left of every point of the sec-
ond class, then there exists one and only one point which
produces this division of all points into two classes, this
severing of the straight line into two portions." If the
point represents a rational number, well and good; if not,
the 'exists' posits such a point and it is said to represent an
irrational number.

The formation of magic squares which reveal the won-
drous symmetry of numbers has had a fascination for
mathematicians of all lands. The earliest record of a
magic square is found in Chinese literature of about 1125
a.d. ('Chinese Philosophy/ by Dr. Paul Carus, quoted by
W. S. Andrews). The wood-cut by Albert Diirer contains
the first magic square found in the Christian Occident.
Successive numbers, beginning with 1, are to be so placed
in square array that the sum of each column, the sum of
each row and the sum of each diagonal shall be the same.
A curious form of the magic square was worked out by a
Moravian, Wenzelides, in which the numbers, in addition
to having the arrangement of a magic square, follow the
knight on a chessboard, one square forward and one
square diagonally. (Fig. 34.)

Magic cubes have also been constructed, in which the
numbers are arranged in cubical array.

An unsolved problem found among Euler's papers is to
place a number in each of the sixteen squares A, B, C . . .
such that the sum of the squares of the numbers shall
fulfil the conditions of a magic square, and in addition
the products of the numbers taken horizontally two at a
time, and also vertically two at a time, shall be the same.

Euler stated that he had found a general means of solu-
tion which is not given. The particular case here given
was found in the papers he left. (Fig. 35.)

CHAPTER III

There is no hard-and-fast dividing line between Algebra
and Arithmetic. Algebra was called by Sir Isaac Newton
Universal Arithmetic, a generalization of those processes
which have to do with number. It is a generalization in
the application of the processes rather than in the processes
themselves. The most important generalization is in the
notion of number itself. In arithmetic it was represented
by a continuous line, indefinite in extent both to the right
and left. A combined result of the three inverse proc-
esses, subtraction, division and evolution, widens this num-
ber system to cover the entire plane.

Algebra has been defined as the Science of the Equation,
but the equation is also a valuable asset of arithmetic.
When the savage first recognises that 2 is made up of I
and i, setting these ideas over against each other and bal-
ancing them, the equation has become a factor in his
thought, altho it has had no symbolic or verbal expression.
The algebraic use of the equation differs essentially from
the general use to which it is put in arithmetic. In the
latter it was arrived at after a process of thought and
sums up that thought; that is, it becomes a formula in
which are found only known terms. It is seen after an
elaborate course of reason and experiment that the square
described on the sum of two lines a and b is equivalent to
two squares, one on a and another on b, and two rectangles
or oblongs formed by a for one side and b for the other.

ALGEBRA 83

This is put in the shape of a formula, (a + b)2 = a2 + 2
ab + b2, where nothing is found in it except the known
lines a and b. Thus in arithmetic the equation is the
vehicle by which truth already discovered is expressed.

On the other hand, in Algebra the equation is the tool
by which the discovery is made. The unknown number,
or the number to be found, is represented by some symbol
or word, and from the statement of the problem a balance
is set up which the operator manipulates until such un-
known is determined. The equation is the most useful and
powerful tool in the hands of the algebraist, and this par-
ticular distinction just made may be said to be the impor-
tant one. The main purpose of Algebra is to evolve a
mechanism by which the equation may be so manipulated
that it will reduce to a simple equation between the un-
known number on the one hand and a known number on
the other.

If the average school boy were asked for his notion of
algebra his probable reply would be that it has something
to do with x and y. In paging over a recent text-book on
the subject the remark was made that the whole language
seemed to be made up of x's and y's. While the develop-
ment of a comprehensive symbolism is one of the impor-
tant features of the Algebra of to-day it was not always so.
The modern symbolism in Algebra did not reach its pres-
ent perfection until the eighteenth century, and in the past
ten years a new symbolism has sprung up in which words,
which are ambiguous at best, are entirely replaced by sym-
bols in the whole course of the reasoning.

However, Algebra to-day is characterized by a more
general symbolism for number. The use of a single letter
for the unknown number and of other letters for the
known numbers involved greatly facilitates the operations
with these numbers and enables the stating of a general
law in a single step. In the formula cited above a is a line ;
it may be regarded as a number which is found by measur-
ing the line by a unit. Two elements come in which make

84 MATHEMATICS

this a more general number than could be expressed in the
Hindu notation. If the unit is changed the number a is
changed. In this way a may be said to stand for any posi-
tive number whatsoever. Again a and b may be any two
lines at will and the statement is still true. The Principle
of Continuity or of No Exception, invoked in the widening
of the number system, gave new numbers which in general
obey the laws of the old. Thus the above statement, which
originated with a and b as lines, is equally true if a and b
are negative numbers. Summing up this point, it may be
said that in addition to representing numbers by the Hindu
method, Algebra represents numbers by means of letters,
and while such numbers are regarded as known, yet it may
be that no particular value is thought of in the discussion,
and they may be given any value at will.

Again, a number which is in a constant state of flux or
change may be the subject of thought, as the price of wheat
on the exchange or the velocity of a railroad train. It
would be exceedingly difficult to represent such a number
with no more mechanism than arithmetic affords, but Al-
gebra allows of its representation by a letter. The last
letters of the alphabet are usually allotted to these variable
numbers and the first letters to constants or numbers which
do not vary. Another and in some ways parallel distinc-
tion is made in using the last letters for unknowns and the
first letters for knowns. These are simply two phases of
the same convention.

This use of a letter for a general number is found in the
works of Aristotle, where he says in one place: "If A is
the moving force, B that which is moved, G the distance
and D the time," etc.

Still a more general representation of number may be
arrived at through the idea of functionality. A number is
said to be a function of one or more other numbers if it
depends for its value upon the value of the other number or
numbers. Thus the volume of a rectangular solid depends
on the length of base, the width of base and the altitude,

ALGEBRA 85

In some cases it is known exactly what the relation termed
functionality is, but in the great majority of cases such
functionality or dependence cannot be put in any more
definite form.

If a, b are respectively the length and width of the base
and c the altitude and V the volume of the rectangular
solid, functionality is expressed by V = F (a, b, c). This
functionality may be more definitely expressed as V =
a X b X c One says the state of the weather, S, depends
upon temperature, T, humidity H, direction and velocity
of wind, D and V. But no more definite form can be writ-
ten than S = F (T, H, D, V).

If x, y are two variable quantities, with dependence of y
upon x, this is put in the form y = f (x).

The number system of arithmetic was developed from
the simple process of counting and gave rise to an idea
of number (the field of real numbers) which was asso-
ciated with a line, a space notion. Real number may be
thought as arising from sequence in time. 51 is thought
of not as a collection of 51 units, but as an element in a
series after 50 and before 52. In counting one arrives at
51 after 50 and before 52. In this way Algebra may be
conceived of as the science of time series as opposed to
geometry, the science of space, thus treating of the a priori
elements of Kant, time and space. The two, Algebra and
Geometry, have been closely interwoven in their historical
development, especially in the beginnings of each. It has
been seen how the Greeks built their theory of number
upon its line representation, and it is a commonplace that if
a relation can be pictured to the eye by means of a figure
in space the reasoning is greatly assisted.

Such a view is sometimes misleading. If intuition alone
had been trusted to determine whether or no all points had
been used up by fractions, the answer would have been
'yes' and the irrationals would have been omitted.

Environment and racial conditions have been the deter-
mining factors in the growth of Algebra and Geometry.

86 MATHEMATICS

Egypt was an agricultural country, land was of value and
Geometry as the science of measurement began there.
The Arabians were a nomadic people ; land was only valu-
able at the time it was being grazed by the flocks and
herds; the peculiarly clear atmosphere, resplendent with
myriads of stars, nightly turned the Arab's attention to
the celestial bodies as he tended the flock, and he was led
to cultivate those branches of analysis and astronomy
which he received as the product of the subtle, imaginative
mind of the Hindu. Thus Geometry and Algebra, each
arising from the needs and characteristics of a race pe-
culiarly adapted to its cultivation, were developed side by
side, each borrowing something from the other, but pre-
serving its own distinctive qualities until the time of
Descartes (1637), when by his invention of the analytic
geometry the two streams converge again, each becomes
in full the interpretative agency of the other.

Less than fifty years ago it began to be more and more
realized that while Geometry always interpreted Algebra
correctly if it itself were correctly interpreted, yet the
notions of Geometry were only conventionally and approxi-
mately represented by a figure and that intuition guided by
the eye was not always to be trusted. So a new movement
sprang up to completely arithmetize Geometry. Its first
and great apostle was Karl Weierstrass, the "father of
precision/' born at Ostendfelde October 13, 1815.

The investigations of the foundations of mathematics
of the past ten or fifteen years, carried on by a host of
mathematicians in Italy, France, Germany, England and
the United States, has carried this work farther — to base
all geometry upon number. Thus the continuity of the
whole field of mathematics has been established and a com-
plete symmetrical system has been built up or created, be-
ginning with the simple notion of putting one with one,
growing like a great oak from the acorn until to-day it is
impossible for one mind in a lifetime to embrace it in all
its ramifications.

ALGEBRA 87

A simple equation is one in which there is one unknown
quantity and it is involved only in its first power or degree,
as x -f- 7 = 15. It is easily seen that the only value of x
for which this equation can be true is 8 or x -f- 7 = 15 if
x = 8. A simple equation then may be looked upon as a
single condition which is satisfied if a certain value is
given to the unknown.

The Egyptian treatise on mathematics by Ahmes gives,
after his treatment of unit fractions, eleven problems, each
resulting in a simple equation. The equation given is
quoted by Cajori.

Ha' neb-f ma-f ro sefex-f hi-i xePer_f em sa sefex ;

Heap its f , its £, its \, its whole, it gives 37

i.e. «(! +i +} +1) = 37

Another problem reads "Heap, its 2/3, its 1/2, its X/T,
its whole, it gives 33." Which put in modern form, omit-
ting the sign of addition which was not used by Ahmes,

1 v, y2 v. * = 33-

The method of solution is to determine by what I 2/3 7>
1/1 must be multiplied to give 33 and the answer is 14 a/4
V97 7* 7e79 7tt6 7i94 7388- Such was the laborious and awk-
ward solution of a simple equation.

The mathematics of the Hindus from Brahmagupta, born
598 a.d., to Bhaskara, born 11 14, was made known to the
English-speaking world by H. T. Colebrooke (1817).
These treatises are clothed in mystic and obscure language
and are very difficult of translation. The story of the ori-
gin of the work by Bhaskara is given by Brooks : "The
work is named for the author's daughter, Lilavati, who it
appeared was destined to pass her life unmarried and with-
out children. The father, however, having ascertained a

88 MATHEMATICS

lucky hour for contracting her in marriage, left an hour-
cup on a vessel of water, intending that when the cup
should subside the marriage should take place. It hap-
pened that the girl, from a curiosity natural to children,
looked into the cup to see the water coming in at the hole,
when, by chance, a pearl separated from her bridal dress,
fell into the cup, and rolling down to the hole, stopped the
influx of water. When the operation of the cup had been
thus delayed, the father was in consternation, and examin-
ing, he found that the small pearl had stopped the flow of
water, and the long expected hour had passed. Thus dis-
appointed, the father said to the unfortunate daughter, T
will write a book of your name, which shall remain to the
latest times, for a good name is a second life and the
groundwork of eternal existence.' "

The following problem from the Lilavati serves to show
the poetic form in which they are garbed:

"Out of a heap of pure lotus flowers, a third part, a fifth,
a sixth, were offered respectively to the gods Siva, Vishnu,
and the Sun ; a quarter was presented to Bhavani ; the re-
maining six were given to the venerable preceptor. Tell
me, quickly, the whole number of flowers."

"Out of a swarm of bees, one-fifth of them settled on the
blossom of the cadamba and one-third on the flower of
the silind'hri ; three times the difference of these numbers
flew to the bloom of a cutaja. One bee, which remained,
hovered and flew about in the air, allured at the same
moment by the pleasing fragrance of a jasmine and pan-
danus. Tell me, charming woman, the number of the
bees."

The following examples are taken from The Ganita-
Sara-Sangraha, previously quoted, translated by M. Ranga-
charya, of Madras. The source of this material is an
article by Professor David Eugene Smith in 'Bibliotheca
Mathematica' (December, 1908) :

"One-fourth of a herd of camels was seen in the forest;

ALGEBRA 89

twice the square root (of that herd) had gone on to the
mountain slopes ; and three times five camels (were) how-
ever (found) to remain on the bank of a river. What is
the numerical measure of that herd of camels?"

A quadratic equation is one in which appears as the
highest power of the unknown the second power. Thus
the equation x2 — 7 x -f- 12 = o contains the second
power of x and is therefore a quadratic, yielding as the
two values of x, 3 and 4. The question naturally arises,
How can x be at the same time 3 and 4? The quadratic is
the expression of a double condition ; it is satisfied not by
3 and 4 at the same time, but by 3 or by 4.

As is seen by substituting 3 or x, giving 32 — 7X3 +
12 = o, or, 9 — 21 -f- 12 = o, again 4, 42 — 7X4+ I2
= o, or, 16 — 28 + 12 = 0. The equation x2 — 7 x -f-
12 = o is true if x is 3, or if x is 4.

Various devices have been used to solve the quadratic,
which may be written in the general form ax2 -j- bx + c
= o, where a, b, c may have any values whatever except
that a may not be o (if a = o, the second degree term
would vanish and the equation would no longer be quad-
ratic). The simplest mode is by "completing the square."
If the equation to be solved is x2 -j- 6x = 16, it is seen
by comparing with the expression for the square of (a +
b), a2 + 2 ab -f- b2 that the left member of the equation
in order to be a perfect square should have the term 9
added to it. Adding this to the other side also the balance
is preserved.

x2 + 6 x + 9 = 16 + 9 = 25.

Now since both sides are perfect squares, the square
roots may be found. The square root of x2 + 6 x + 9 =
x + 3. And the square root of 25 may be + 5, or — 5,
since (+ 5) X (+ 5) = 25, and (— 5) X (— 5) =25.
This two-fold condition is then expressed by writing
V25 = ± 5. Where as above it is understood that,
either -f- 5, or — 5 is to be taken. Equating the square

90 MATHEMATICS

roots of the two members x + 3 = i 5, and breaking
this up into two conditions,

x + 3 = + 5orx + 3 = — 5

X = 2 x = 8

Bhaskara, who solved such equations, says "the second
value in this case is not to be taken, for it is inadequate,
people do not approve of negative roots."

Such equations as the above were readily solved by the
Hindus. Hankel says of them : "If one understands by Al-
gebra the application of arithmetical operations to complex
magnitudes of all sorts, whether rational or irrational
numbers or space-magnitudes, then the learned Brahmins
of Hindostan are the real inventors of Algebra."

About 150 years after Mohammed's flight from Mecca,
the study of Hindu science was taken up at Bagdad in
the court of Caliph Almansus. In 773 a.d. there appeared
at his court a Hindu astronomer, with astronomical tables
which were translated into Arabic. The first Arabic
treatise now known is that of Muhammed ibn Musa Alch-
warizmi. The work, which was translated probably by
Athelard of Bath, and which is the first work in which
the word Algebra (or in the Arabic aldschebr walmuka-
bala) occurs, begins : "Spoken has Algorithmi. Let us give
deserved praise to God, our leader and defender." The
word Algorithmi is the Latin form of the author's name,
from which comes the word algorithm, signifying a rule
for computation. The two words used as a name for
Algebra mean "restoration and opposition" and have ref-
erence to the transposing of the terms of an equation
and discarding equal terms from both members.

An equation of the form y = 2 x -f- 5 expresses a con-
dition between two unknowns or variables. Such an equa-
tion is said to be indeterminate, since any number of pairs
of values of x and y will satisfy it.

If x = 1, y = 7 ; if x = o, y = 5 ; if x = — 1, y is 3 ;

ALGEBRA 91

if x = — 2, y = 1 ; if x = — 3, y = — 1, and so on
indefinitely. This relation between x and y may be shown
graphically by a method which is the foundation of the
Analytic Geometry, invented by Des Cartes (1637), from
which date, it may safely be said, modern mathematics
takes its rise. The principle upon which it is based is that
a point in a plane may ba located if its distances are known
from two intersecting lines, called axes. These axes are
chosen for convenience at right angles, altho this is im-
material except for simplicity.

The study of indeterminate equations is called Diophan-
tine analysis, from Diophantos of Alexandria, the last
great Greek mathematician, of whose work six books re-
main, which treat of such problems as:

To find a right-angled triangle such that the difference
of its sides is a square, and also the greater alone is a
square, and, thirdly, its area + the less side is a square.
A solution to this problem is to take 1, 2 for the lengths
of the sides. The Fermat equation xn + yn = zn is an
indeterminate equation.

The most famous problem of this type is the "cattle'*
problem, attributed to Archimedes, the most celebrated
problem of antiquity. It is in the form of an epigram,
and has been translated by T. L. Heath as follows:

"Compute, O stranger ! the number of cattle of Helios,
which once grazed on the plains of Sicily, divided accord-
ing to their color, to wit: (1) White Bulls = y2 -\- y3
of the Black Bulls + Yellow Bulls; (2) Black Bulls = %
and y, of the Dappled Bulls + the Yellow; (3) Dappled
Bulls = V. + Vi of the White + Yellow; (4) the White
Cows = Yz and y^ of the Black Herd (Bulls and Cows =
Herd) ; (5) the Black Cows = yA and l/$ of the Dappled
Herd; (6) the Dappled Cows = */„ and V6 of the Yel-
low Herd; (7) the Yellow Cows = */« + */, of the White
Herd.

"He who can answer the above is no novice in numbers,

92 MATHEMATICS

nevertheless he is not yet skilled in wise calculations; but
come, consider all the following numerical relations be-
tween the Oxen of the Sun: (8) If the White Bulls were
combined in one total with the Black Bulls, they would
be in a figure equal in depth and breadth, and the far-
stretching plains of Thrinacia would be covered by the
figure (square) formed by them; (9) should the Yellow
and Dappled Bulls be collected in one place, they would
stand, if they ranged themselves one after another, in the
form of an equilateral triangle. If thou discover the so-
lution of this at the same time; if thou grasp it with thy
brain; and give correctly all the numbers; O stranger!
go and exult as a conqueror ; be assured that thou art by all
means proved to have abundance of knowledge in this
science."

The Hillsboro (111.) Mathematical Club worked on this
problem from 1889 to 1893. The answer given for the
number of white bulls will reveal the magnitude of the
numbers involved: 1,596,510,804,671,144,531,435,526,194,-
370, . . . 385,150,341,800.

Where the . . . indicate the omission of 68,834 periods
of three figures each. Each of the ten answers is com-
posed of 206,545 figures.

Another of these famous puzzles is attributed to Euclid :
"A mule and a donkey were walking along laden with
corn. The mule said to the donkey, Tf you gave me one
measure I should carry twice as much as you ; if I gave
you one, we should both carry equal burdens.' Tell me
their burdens, O most learned master of geometry."

If two equations, x + 2 y = 4, and x — y = 1, are
given, both x and y are determined. Such a system is
called a linear system, and a single pair of values of x and
y may be found which satisfies both conditions. The
statement of the two equations may be thought of as re-
quiring that the position be found in which the generating
point of either line will simultaneously lie on its own line

ALGEBRA

93

and also on the other. The graphical solution indicates
that the point S (x = 2, y = i), or more briefly put,
S (2, i), is the desired point.

y

■••flCi—

■fj^.

rX»

y>

In the study of such systems, Leibnitz (1646-1716) dis-
covered a symmetrical arrangement of the known numbers,
or the coefficients as they are called, which has been of
immense service. This symmetrical array is called a de-
terminant.

The system of three equations

ax + by + cz = d
lx + my -fnz = p
rx + sy + t z = q

may be solved for x by writing a fraction whose numera-
tor is made up of the numbers on the right for a first col-
umn and the coefficients of y and of z for the other two,
and the denominator is the three columns of coefficients
of x and y and z.

94

MATHEMATICS

The following is the arrangement:
d b c
p m n
q s t

The evaluation of this may be shown in the method used
for finding the numerator.

The numbers connected with each arrow to the right
are multiplied and given the + sign, those connected with
arrows pointing to the left are multiplied and given the —
sign. The sum of the six terms is the numerator, or, dmt
+ bnq -\- cps — smq — bpt — dsn: — similarly for the
denominator.

An equation of the form x2 — 4 x — 5 = 0, called a
quadratic, or equation of the second degree, has been
solved by "completing the square." Another method is by
means of a graph x2 — 4 x — 5 is placed equal to y and
the graph drawn by taking particular values for x, and

ALGEBRA

95

from these determining the values of y which goes with
each. A table of these values (taken from Boyd's Alge-
bra) shows the process. It is required to find the values
of x which makes y = o or which satisfy x2 — 4 x — 5 =
y, when y = o. In the figure y = 0 when the curve crosses
the x axis X' X, or the values are — I ~h 5.
x2 — 4 x — 5 = 0, then, when x = — 1, or -j- 5.

For y = xa

— 4x— 5

; x

y

0

- 5

+ 1

- 8

+ 2

- 9

+ 3

- 8

+ 4

- 5

+ 6

0

+ 6

+ 7

etc.

etc.

-1

0

-2

+ 7

-3

+16

etc.

etc.

Another figure taken from the same text shows the
method of solving the simultaneous quadratic system.
x2-|-y2 — 2xy — 4X — 8y — 20 = 0

xy = — 2.

P, Q, R, S are the points of intersection of the two
curves, and the value of x and y for each can be read di-
rectly from the figure.

Solving the equation, x2 — 6x = — 13, by completing
the square, adding 9 to both members, x is found to be
equal to 3 db V — 4, and the question arises, What is the
measuring of V — 4? It is known that (+ 2)2 = + 4,
and that ( — 2 )2= + 4. No number in the system so far
considered will, when squared, give a negative number,

96

MATHEMATICS

and means must be devised by which such a number may
be interpreted.

V — 4 may be factored into V 4, V — I or 2 V — i.

If 2 be multiplied by — i, the result is — 2, or the point

--1-f--f-

-4---J---M--

/fffr-

■T" T 1

■ i | i p t

+ 10; j , i

— j-J-j--

•--4--T-4-"

■-|-f--i-i-

— J-4--J--

I ! j j

+ 5 l j ;

' ! ! !

-j-H

! ! ! !

j { ! D,

r^rjQ j i

|K ! !+5

: J L^[

| N\ j j M

0 I

i i^. 4 —

^fs| !

/L-^-p^i

Tn'TT"'

- 5 : j ! •

j a i * i :

A is changed over to the position B. V — I multiplied
by itself must produce — i, from the notion of square root,
or V- iXV — i =— I-

Then 2XV — i XV — I== — 2- If 2 be multi-
plied twice in succession by V — I, the result is moving

ALGEBRA

97

A to B. Then it is reasonable to suppose that one mul-
tiplication or 2 X V — l should move it halfway.
All that is now necessary is to choose the path. If A
should be moved along the line A B, half the motion
would carry it ,to o, or y — i X 2 — °- But o X 2 — °»
and that would require that V — I = o. But this is
not desirable. The next simplest path is a semicircle.

If two multiplications carry A to B, a single multiplica-
tion should carry it to C. This is found to be a satisfac-
tory definition, for by 3 multiplications A is carried around
toD.

r 2XV-1XV — iXV — i=2X(— OX

V — 1= — 2 V — 1, that is, D is marked with the —
sign of C, which should be so, and a fourth multiplication

98 MATHEMATICS

gives 2, that is, 4 multiplications carries A through a com-
plete revolution. The V — x is indicated by i, which has
the function of a sign, merely indicating that the number
before, which it is placed belongs on the vertical line CD,
while a number without such a sign is on the horizontal
line AB, that is, a real number. A number represented
on AB is called a pure imaginary, the name 'imaginary*
or 'fictitious' number being given to expressions of this
kind which constantly arose in the solution of equations
and to which no meaning had been attached. Bhaskara
says: "The square of a positive as well as of a negative
number is positive, and the square root of a positive num-
ber is double, positive and negative. There can be no
square root of a negative number, for this is no square."

The Italian algebraists called them "impossible num-
bers." It was not until 1797 that Caspar Wessel devised
a method of representation of imaginaries, but it did not
attract particular attention. Again in 1806 Jean Robert
Argand independently arrived at the representation given
above. It is a curious fact that the entire known biog-
raphy of Argand could be written in half a dozen lines,
yet his work is the basis of one of the most extensive
fields in all mathematics.

The number system now consists of real numbers rep-
resented on a horizontal line and pure imaginaries on a
vertical line. The combination of these two classes forms
the class complex numbers, which covers the entire plane.
In the figure 3 + 2 i is found by stepping off 3 units to
the right of O and 2 units up, giving point P.

On the axis of real numbers, O4, the point marked 3
represents the number 3, but it was found to be sometimes
more convenient to think of 3 as represented by the seg-
ment of line beginning with O and ending with 3. With
the number 2 + 3 i it will be thought of as represented by
the point P or by the line segment OP at will.

The angle MOP is called the amplitude of P, and is de-

ALGEBRA

99

noted by <P. The length of OP, which is V 22 + 3* = V *3>
is termed a modulus and indicated by mod P.

Complex numbers obey the laws laid down for real
numbers. They may be subjected to the six operations of
addition, subtraction, multiplication, division, involution
and evolution. The mode of addition is the same as that
employed in adding real numbers.

P3+U

If the real numbers are thought of as line segments, and
2 is to be added to 3, it is done by placing the initial point
O of O 2 on the terminal point 3 of O 3. The point then
occupied by point 2 of O 2 in its new position is 5 and
O 5 is the segment sum of O 3 and O 2. If the two com-
plexes 2 + 31 and 5 + 2 i are to be added they are rep-
resented* as in the figure, the first by OP and the second
by OQ. Starting at P, lay off OL, 5 units to the right

ioo MATHEMATICS

and 2 units up. OR, which is the diagonal of a parallelo-
gram on OP and OQ, is the sum of 2 + 3 i and 5 + 2 i.
The number system now covers the entire plane; to
every point in the plane there is a number and vice versa.
The plane is two-dimensional, that is, by the Cartesian co-
ordinates x y a point is determined by two values, x and
y, or in the Argand diagram by the two real numbers a
and b in the complex a -f: bi. Space is three-dimensional

in points. To locate a point in a room completely it is
necessary to specify its distances respectively from, say, the
floor and each of two intersecting walls, or by 3 numbers.
To take in all points in space, a third line or axis would
be drawn perpendicular to the plane of the paper in the
Argand diagram at point O. Now if a third sign of di-
rection j were used, and the number system extended to
take in space, what would result? The apparent discrep-
ancy between the number system, which is two-dimen-
sional, and space, which is three-dimensional, has been a
source of a great deal of study and involves some of the
most important theorems of algebraic analysis.

ALGEBRA 101

A general equation of the form
a0 xn -f- ax xn - * + a2 xn-2 -f . . . + a^ x + an == o
is said to be of the nth degree, where the exponents are
all whole numbers. It has been proved that if such an
equation is satisfied by a single value of x it is satisfied by
n values, that is, it has n roots. These roots may all be
real or part real and part complex. If there are complex
roots they enter in pairs which are conjugate, that is, if
a + bi is a root so also is a — bi. The condition that "if it
is satisfied by a single root" is very important.

Why should it not be ? It was found that the quadratic
could be easily solved, and very many special equations of
higher degree. The cubic or equation of the third degree
taxed the powers of the algebraists, and it was not until
1545 that a general solution was found. It seems almost
axiomatic that the general equation must have a root, but
such things are not taken for granted. The first proof
that the general equation with whole numbers for ex-
ponents and coefficients, real or complex, was given in
Argand's memoirs. Since that time a number of proofs
have been offered, the principal contributor being Cauchy.
This is called the fundamental theorem of Algebra.

Now since the hypothesis is proved, the conclusion that
there are n roots is easily proved, such proof being famil-
iar to any schoolboy. The next concern is, what is the
nature of the roots? Weierstrass proved that the roots
all are of the form a + bi, that is, complex numbers of the
two-dimensional system.

This at once settles the question raised above, whether
or not it is possible to extend the number system to the
three mensions of space. If the extension is made, such
numbers would not be the roots of algebraic equations; in
other words, such numbers would not be subject to the
ordinary laws of Algebra.

Two diverging lines of thought begin here: if such ex-
tension of the number system be made what formal laws
of Algebra shall be rejected? Having determined the

102 MATHEMATICS

nature of the roots of equations, to devise laws by which
an equation may be solved. The second of the two will
be considered first. It has been seen that the quadratic
is solvable. Equations of higher degree have been solved
in special cases. The general solution of the cubic next
received attention.

The following account of the solution of the cubic is
from Ball's History of Mathematics:

"Nicolo Fontana, generally known as Nicholas Tartaglia
— that is, Nicholas the stammerer — was born at Brescia
in 1500 and died in Venice on December 14, 1557. After
the capture of the town by the French in 15 12, most of the
inhabitants took refuge in the cathedral and were there
massacred by the soldiers. His father, who was a postal
messenger at Brescia, was among the killed. The boy
himself had his skull split through in three places, while
both his jaws and his palate were cut open; he was left
for dead, but his mother got into the cathedral, and finding
him still alive managed to carry him off. Deprived of
all resources, she recollected that dogs when wounded
always licked the injured place, and to that remedy he
attributed his ultimate recovery, but the injury to his
palate produced an impediment in his speech from which
he received his nickname. His mother managed to get
sufficient money to pay for his attendance at school for
fifteen days, and he took advantage of it to steal a copy-
book from which he subsequently taught himself to read
and write ; but so poor were they that he tells us he could
not afford to buy paper, and was obliged to make use of
tombstones as slates on which to work his exercises.

"He commenced his public life by lecturing at Verona,
but he was appointed at some time before 1535 to a chair
of mathematics at Venice, where he was living when he
became famous through his acceptance of a challenge from
a certain Antonio del Fiori. Fiori had learned from
his master, one Scipione Ferreo (who died at Bologna
in 1526), an empirical solution of a cubic equation of the

ALGEBRA 103

form x3 -j- qx = r. This solution was previously un-
known in Europe, and it is probable that Ferreo had
found the result in an Arab work.

"Tartaglia, in answer to a request from Colla in 1530,
stated that he would effect the solution of a numerical
equation of the form x3 + px2 = r. Fiori, believing that
Tartaglia was an impostor, challenged him to a contest.
According to this challenge, each of them was to deposit
a certain stake with a notary, and whoever could solve
the most problems out of a collection of thirty propounded
by the other was to get the stakes, thirty days being al-
lowed for the solution of the questions proposed. Tarta-
glia was aware that his adversary was acquainted with the
solution of a cubic equation of some particular form, and
suspecting that the questions proposed to him would all
depend on the solution of such cubic equations, set himself
the problem to find a general solution, and certainly dis-
covered how to obtain a solution of some if not all cubic
equations. When the contest took place all the questions
proposed to Tartaglia were, as he suspected, reducible to
the solution of a cubic equation, and he succeeded within
two hours in bringing them to particular cases of the
equation x3 -f- qx = r, of which he knew the solution.
His opponent failed to solve any of the problems which
were proposed to him, which as a matter of fact were all
reducible to numerical equations of the form x3 + px2 = r
(notice that in this form the x2 term is present, while in
the other the x term appears). Tartaglia was therefore
the conqueror, and he subsequently composed some verses
commemorative of his victory."

Tartaglia, as was the custom in those days, did not re-
veal his method of solution. He hoped to publish a trea-
tise on Algebra of which the crowning feature would be
the making known to the world this newly discovered solu-
tion of the cubic; but in this he was to be disappointed
through the treachery of Girolamo Cardan, the most
famous astrologer of the time. This Cardan was a most

104 MATHEMATICS

strange admixture of genius and madness, a gambler if
not a murderer, an ardent student of science, solving
problems which had long baffled investigation. The elder
of his two sons was executed for poisoning his wife, while
it is said that Cardan cut off the ears of the younger in
a fit of rage. In 1570 Cardan was imprisoned for heresy
on account of having published the horoscope of Christ.

R-E GVIA.

Deducito tertiam partem numeri rerum ad cubum,cui addes
quadratum dimidrj numeri a:qiiationis,& totiits actipe radicem, fdli
cet quadratam,quam feminabisjunicp dimidium numeri quod iam
in fe duxeras,adrjcies,ab altera dimidium idem minucs,habcbisq$ Bi
nomuim cumfua Apocome, inde detracla r& cubica Aporomaccx rs
cubica fui Binomrj,refiduu quod ex hoc rclinquitur.eft rei eftimatio*
Excmplum.cubus & tf pofitiones, a:quan*
tur 20,ducito 2- , tertiam partem tf , ad cu**
btim,ficS,ducfo dimidium numeri infe,
fit 1 oo,iunge 1 00 & 8,fit 1 o8>accipe radi*
cem quae eft is 1 08, & cam gcminabis,alte |
ti addes 1 o,dimidium numeri,ab altero mi
nues tantundem,habebis Binomiu j* 1 08
p: 1 o,& Apotomen rz i 08 m: 1 o , horum
•actipe R25* cub" & mimic illam que eft Apo ,
tomac,ab ea qua: eft Bfnomrj, habebis rei arftimationem, Rr v: cub: Rr
i 08 p: 1 o m:R2 v: cubica Rt 1 0$ m: 1 o«

Aliud,cubtis pi $ rebus a:quetur io,duc 1, tertiam partem *, ad
cubum,fit f ,duc c.dimidium 1 o,ad quadratum, fit 2C,iunge 2c Sc t ,

H 2 fiunc

Fig- 36 — First Published Solution of the Cubic Equation ;
from Ars Magna. (1545.)

He afterward settled at Rome, where he received a pen-
sion in order to secure his services as astrologer to the
court. Having foretold that he should die on a particular
day, he felt called upon to commit suicide to preserve his
reputation.

In 1545 Cardan completed and published the Ars Magna,
the most advanced treatise on Algebra which had appeared

cub9p:tfreb9a?qlis 20
. % 20

$ 19

108
IS loSpno
rz 1 o$m: 1.0
R?v:cu.R2io8pno
m:Rrv:cu.R> io£m:i£

ALGEBRA 105

up to that time, and in which was given Tartaglia's solu-
tion of the cubic. This method has since been known,
as Cardan's method. Cardan also published the work of
his pupil, Ferrari, on the biquadratic or equation of the
fourth degree. This solution is sometimes known by
Bombelli's name, to whom is due the credit of represent-
ing the three roots to the simplest form in the so-called
irreducible case.

From this time on mathematicians devoted a great
amount of time in attempting the solution of equations of
higher degree. In his 'Reflections on the Resolution of
Algebraic Equations,' Lagrange (1736-1813) gave a
scientific classification of the methods already applied to
the cubic and biquadratic, but was unable to apply them
to the quintic or equation of the fifth degree. In this dis-
cussion the foundation was laid for the study of substi-
tutions, but other matters pressing for attention made
necessary the laying aside of this work. He determined
to take up the subject at some future time, but never
did so.

It was reserved for the brilliant young Norwegian, Neils
Henrik Abel (1802-1829), to give a rigid demonstration
of the impossibility of solving the quintic or higher equa-
tions by means of radicals.

The extension of the number system to three dimensions
was attempted by Argand and resulted in failure. A
corollary of Weierstrass' theorem that the root of an alge-
braic equation must be of the form a -f- bi is that no fur-
ther extension can be made and have the numbers still
conform to the laws of algebra. In the formation of the
complex number there are two units, 1, or the unit along
the axis of reals, and i, the unit along the axis of pure im-
aginaries. If the system is to be extended to space a third
unit is to be chosen — call it j — which will be measured on a
perpendicular to the two axes already used. A number of
this form would be a -f- bi -f- cj. When the negative num-
ber was introduced, it was assumed that in multiplication

106 MATHEMATICS

It should obey the commutative law that 1 X * = i X *•
This was a pure assumption, made in order to give a mean-
ing to multiplication by a negative. It was the subject of
years of meditation with William Rowan Hamilton, as to
what would be necessary in order to extend the system so
as to include the new unit j. At last, on the 16th of
October, 1843, while walking with his wife along the Royal
Canal in Dublin, the discovery flashed upon him that the
commutative law might be rejected, and he engraved with
his knife on a stone in Brougham Bridge the fundamental
formula of the new algebra which is called Quaternions.
This bridge is since known as Quaternion Bridge.

In 1844 appeared a classic work on analysis, the 'Aus-
dehnungslehre' of Hermann Grassmann, in which the
number system is carried to n dimensions. This work
attracted so little notice on account of its "philosophische
allgemeinheit" it is said that after eight years but one
man had read it. In 1862 a new edition was published
which received no more appreciation than the first, and at
the age of fifty-three its author, with a heavy heart, gave
up mathematics for the study of Sanskrit.

The generalization of algebra is carried out by assuming
any number of units i, j, k, 1, etc., forming numbers with
them as a -f- bi + cj 4* dk + el -f- • • • an<3 choosing to re-
ject one or the other of the laws of ordinary algebra (for
at least one must be rejected) and then building up a con-
sistent algebra upon the remaining laws. In 1870 Ben-
jamin Pierce, one of the foremost mathematicians that
America has produced, published his 'Linear Associative
Algebra/ givng the elements of 162 algebras, in which the
numbers are linear functions of the units and obey the
associative law.

CHAPTER IV

GEOMETRY

Geometry is the science of space and is concerned with
relations which exist between its various elements, linear,
superficial and solid. The earliest measurements were
linear, and for the unit was taken some portion of the
human body — for example, finger-breadth, palm, span, foot,
ell, cubit and fathom — but the body does not possess any
convenient unit for the measurement of either surface or
solid. The oldest geometrical work known uses the square
unit for areas and the cubical unit for solids. How and
when the choice of such units was made is difficult to
say. The study of primitive races made possible the re-
construction of the steps in the formation of the number
concept, but such study is silent in regard to the begin-
nings of geometry.

The word geometry, from the Greek, meaning "to meas-
ure the earth," has its origin, as is the case with most
sciences, in the needs of the human being at some particu-
lar time, as is indicated by Herodotus (II, 109) where he
says that Sesostris (c. 1400 B.C.) divided the land of Egypt
into rectangular plots for the purpose of more convenient
taxation; that the annual floods, caused by the rising
of the Nile, often swept away portions of a plot, and that
surveyors were in such cases appointed to assess the neces-
sary reduction in the tax. Hence, in my opinion, arose
geometry and so came into Greece.

Ahmes gives a number of problems concerning the cal-
107

108 MATHEMATICS

culation of the contents of barns, but as the shapes are
unknown it is impossible to interpret them. As with his
work in arithmetic, no rules are given, but a number of
problems solved in a similar manner. The method used
in finding the contents of a barn is to multiply together
two of the dimensions and this by one and one-half the
third. He also finds the area of a square, of an oblong,
of an isosceles triangle and of an isosceles trapezoid, the
latter two being incorrectly found. In the isosceles tri-
angle, a triangle with two equal sides, Ahmes takes half
the product of the base and one of the equal sides and fol-
lows the analogous proceeding with the isosceles trape-
zoid. While the error is slight in the examples given, it
is sufficient to show that the results were only empirical
and that Ahmes was unable to extract the square roots
which are necessary in an exact solution. The area of a
circle is found by deducting from the diameter its
one-ninth and squaring the remainder, which gives
the value of the ratio of the circumference to the
diameter of a circle, usually indicated by #, to be 3.1604,
a value much more nearly correct than those used by
many later writers.

Another glimpse of Egyptian geometry is given by
Democritus (c. 460-370 B.C.) : "In the construction of plane
figures with proof no one has yet surpassed me, not even
the Harpedonaptae of Egypt. To Professor Cantor is due
the credit of making clear the exact meaning of this word,
which is a compound of two words, meaning "rope
stretchers'* or "rope fasteners." Cantor says: There is
no doubt that the Egyptians were very careful about the
exact orientation of their temples and other public build-
ings. But inscriptions seem to show that only the north
and south lines were drawn by actual observation of the
stars. The east and west lines were drawn at right an-
gles to the others. Now it appears, from the practice of
Heron of Alexandria and of the ancient Indian and prob-
ably also the Chinese geometers, that a common method of

GEOMETRY 109

securing a right angle between two very long lines was to
stretch round three pegs a rope measured in three por-
tions, which were to one another in the ratio 3:4:5. The
triangle thus formed is right-angled. Further, the opera-
tion of rope-stretching is mentioned in Egypt, without ex-
planation, at an extremely early time (Amenemhat I).
If this be the correct explanation of it, then the Egyptians
were acquainted 2,000 years B.C. with a particular case of
the proposition now known as the Pythagorean theorem.
Egyptian geometry, as well as the other sciences, was in
the hands of the priestly caste, whose conservatism is il-
lustrated by the fact that Egyptian doctors used only the
recipes of the ancient sacred books, for fear of being ac-
cused of manslaughter if the patient died. That no prog-
ress was made beyond that of Ahmes is borne out by the
Edfu inscriptions of 107-88 B.C., two hundred years after
Euclid, in which the formula given by Ahmes for the
isosceles trapezoid is still given but applied to any four-
sided figure, a proceeding of which Ahmes himself would
not have been guilty.

That the early Greek geometers derived their first
knowledge from the Egyptians is derived from many
sources. Eudemus (c. 330), pupil of Aristotle, wrote a
history of geometry in which occurs this passage : "Geom-
etry is said by many to have been invented among the
Egyptians, its origin being due to the measurement of
plots of land. This was necessary there because of the
rising of the Nile, which obliterated the boundaries apper-
taining to separate owners. Nor is it marvelous that the
discovery of this and other sciences should have arisen
from such an occasion, since everything which moves in
development will advance from the imperfect to the per-
fect. From mere sense-perception to calculation, and
from this to reasoning, is a natural transition." The last
step is the one taken by the Greeks — the Egyptian geom-
etry was concrete, a thing of sense, and to Thales is due
the honor of creating the beginnings of abstract geometry,

no MATHEMATICS

a product of reason, the object of which is to establish
precise relations between the parts of a figure, so that
some of them could be found from others in a purely
rigorous manner.

Thales of Miletus (640-546 B.C.) was a merchantman
when his native city was in its most flourishing condition,
and resided for a long period in Egypt, from whence he re-
turned to his native city in his old age, bringing with him
the knowledge of geometry and astronomy. Tradition in-
forms us that he was one of the first gifted with the
acumen to form a 'trust.' Learning from the stars that
the crop of olives would be abundant during a certain
year, Thales secured control of all of the oil-presses, and
in the following fall made a large profit through his fore-
sightedness. (Aristotle.) He announced beforehand an
eclipse of the sun which happened May 28, 585 B.C., during
a battle between the Medes and Lydians, and to this fact
is attributed his inclusion in the ranks of the Seven Wise
Men, for as Plutarch says, he "apparently was the only
one of these whose wisdom stepped in speculation beyond
the limits of practical utility ; the rest acquired the name of
wisdom in politics." In a conversation concerning Amasis,
King of Egypt, between Niloxenus and Thales, given by
Plutarch, the former says : "Altho he (Amasis) admired
you (Thales) for othe- things, yet he particularly liked
the manner by which you measured the height of the pyra-
mid without any trouble or instrument ; for by merely
placing a staff at the extremity of the shadow which the
pyramid casts, you formed two triangles by the contact of
the sunbeams, and showed that the height of the pyramid
was to the length of the staff in the same ratio as their
respective shadows." From Proclus it is learned that
Thales devised a method of determining the distance of
ships at sea by a theorem which is now known as Euclid
I, 26.

Pythagoras, concerning whose life there is a great
deal of obscurity, was probably induced by Thales to visit

GEOMETRY in

Egypt when a young man, where he lived many years, aft-
erward visiting Crete and Tyre and perhaps Babylon.
Returning to Samos, his home, he found it under the
tyranny of Polycrates, and migrated to Italy, where he
lived and taught for more than twenty years. His broth-
erhood falling under suspicion owing to its secrecy,
Pythagoras fled to Metapontum, where it is supposed he
was murdered in a popular outbreak about 500 b.c.

To Pythagoras, who raised geometry to the rank of a
science, are many of the most important theorems. He
is said to have introduced weights and measures among the
Greeks, to have discovered the numerical relations of the
musical scale, to have proved the theorem of squares on
the sides of a right triangle, to have discovered that the
plane around a point is filled by six equilateral triangles,
four squares or three hexagons, to have found the con-
struction of a figure upon a line which is similar to a
given figure and equivalent to a second given figure. The
word mathematics is due to the Pythagorean school, and
to them is attributed the division of a line into extreme
and mean ratio, called the Golden Section, so that the
whole line is to the greater segment as this segment is
to the lesser, from which construction is derived that of the
inscription in a circle of the regular five and ten sided
polygons.

Proclus says that Pythagoras discovered the "construc-
tion of the cosmic figures," "the five bodies in the sphere,"
concerning one of which Iamblichus says that Hippasus
was drowned for the impiety of claiming its discovery,
whereas the whole was his discovery, for "it is thus
they speak of Pythagoras, and they do not call him by his
name."

The five regular solids were alternately compared by
the Pythagoreans with the five worlds and with the five
senses of man. Kepler, led astray by the speculations of
the philosophers, conjectured that they were in some
way connected with the orbits of the five worlds. He ac-

ii2 MATHEMATICS

cordingly arranged the five solids in order, each inscribed
in a sphere, which in turn was inscribed in the next figure
and with the sun at the center. The surfaces of the
spheres carried the orbits of the planets. He found the

Fig. 37 ■

-Kepler's Analogy of the Five Solids and the Five
Worlds.

ratio of the distances to be remarkably near the ratio of
the actual distances from the sun. He made known his
remarkable pseudo-discovery in the 'Mysterium cosmo-
graphicum' (1596), which had at least one beneficial effect

GEOMETRY 113

in that it brought him to the notice of Galileo and Tycho
Brahe and opened the way for the future true discoveries
which have placed his name in the galaxy of the immor-
tals.

Plutarch, in relating the discovery of the construction
of a figure similar to one and equivalent to another, says
that Pythagoras offered a sacrifice in thanksgiving, think-
ing it finer and more elegant than the other concerning
the squares on the sides of a right triangle. Pythagoras

Fig- 38 — Inner Portion of Kepler's Cosmographicum. (See
Fig- 37-)

thought that the distances of the heavenly bodies from the
earth formed a musical progression, from which comes the
expression "the harmony of the spheres."

The Pythagorean theorem that the square described on
the hypothenuse of a right triangle is equivalent to the
sum of the two squares described on the sides is the most
famous theorem of geometry. It is said that over a thou-
sand distinct proofs have been offered for it. The proof
given by Pythagoras has never been found. He probably
was led to the investigation of the figure from the observa-
tion of the special case which is common in flooring with

H4

MATHEMATICS

square tiles, as in the figure. The Egyptians were fa-
miliar with the right angle property of the particular tri-
angle with sides 3, 4, 5. Within the last few years it has
not only been shown that the Hindus were familiar with

Fig. 39 — Perigal's Dissection.

the Pythagorean theorem in all its generality and the
theory of the irrational long before the time of Pythagoras,
but Bitrk goes so far as to assert that the much-traveled
Pythagoras obtained his knowledge from India. The
proof given in the school text of to-day and is the classic

GEOMETRY

ii5

one given by Euclid, which, notwithstanding the strictures
of Schopenhauer as "a mouse-trap proof and "a proof
walking on stilts, nay, a mean, underhand proof," is one
of the most beautiful ever offered.

One of the most celebrated forms of proof is known as
Perigal's dissection, in which the squares are so cut that
H + P + R-f-L + Ein the figure may be arranged to
form the large square. Another form of dissection is

given in the second figure in the shape of a puzzle, in
which the parts A, B, C, D, E are to be cut out and ar-
ranged so as to exactly cover the large square.

This theorem is the limiting case between two theorems
which may be stated together: The square on the side
opposite an acute (obtuse) angle is equal to the sum of the
squares on the other two sides diminished (increased) by
twice the rectangle of one of those sides and the projec-
tion of the other upon it. The figure of the Pythagorean
theorem was called by the Persians the Princess, and
other two figures were the Sisters of the Princess. The
figure of one of these cases is here given which Gorre-

n6

MATHEMATICS

sponds to the figure given by Euclid for the Pythagorean
theorem. In the accompanying figure if the triangle in
question is ABC, AB is the side opposite the acute an-
gle BCA. CE is the projection of CA upon CB. If CA
is allowed to revolve about point C to the position CA",

Fig. 40 • — One of the 'Sisters of the Princess.'

A"B will have become a side opposite an obtuse (greater
than a right) angle. But in the turning it passes through
the condition of perpendicularity CA', and the right tri-
angle CA'B is the boundary between the two cases. When
this condition occurs the projection CE is zero and the
Pythagorean theorem results. The three cases are stated

GEOMETRY 117

in a single law in trigonometry called the Law of Cosines,
which in turn but one case of a general law in spherical,
plane and pseudospherical geometry.

The third century b.c. produced the three greatest mathe-
maticians of antiquity, Euclid, Archimedes and Apollonius,
of which the earliest was Euclid. Very little is known of
his life. Proclus gives this account of him: "Not much
younger than these (Hermotimus and Philippus) is
Euclid, who put together the Elements, collecting many
of Eudoxus' theorems, perfecting many of Theaetetus',
and also bringing to irrefragable demonstration the
things which were only somewhat loosely proved by
his predecessors. This man lived in the time of the
first Ptolemy. For Archimedes, who came immediately
after the first Ptolemy, makes mention of Euclid : and fur-
ther, they say that Ptolemy once asked him if there was
in geometry any shorter way than that of the elements,
and he answered that there was no royal road to geometry.
He is younger than the pupils of Plato, but older than
Eratosthenes and Archimedes; for the latter were con-
temporary with one another, as Eratosthenes somewhere
says."

That Euclid founded a school at Alexandria is known
from this passage from Pappus: "Apollonius spent a
very long time with the pupils of Euclid at Alexandria,
and it was thus that he acquired such a scientific habit of
thought." Stobaeus relates that "some one who had be-
gun to read geometry with Euclid, when he had learned the
first theorem, asked Euclid, 'But what shall I get by learn-
ing these things?' Euclid called his slave and said, 'Give
him threepence, since he must make gain out of what he
learns.' " The importance of Euclid's elements was rec-
ognised by the Greek philosophers, who posted on the doors
of their schools: "Let no one enter here who is unac-
quainted with Euclid."

The purpose of the elements is to begin with a few
common notions which are statements assumed to be

n8 MATHEMATICS

evident to any reasoning being, and togetker with five
assumptions from these build step by step a complete
chain of theorems. That he succeeded is evidenced by
the following passage from Brill : "Whatever has been
said in praise of mathematics, of the strength perspicuity
and rigor of its presentation, all is especially true of this
work of the great Alexandrian. Definitions, axioms and
conclusions are joined together link by link as into a
chain, firm and inflexible, of binding force, but also cold
and hard, repellent to a productive mind and affording
no room for independent activity. A ripened understand-
ing is needed to appreciate the classic beauties of this
great monument of Greek ingenuity. It is not the arena
for the youth eager for enterprise; to captivate him a
field of action is better suited where he may hope to
discover something new, unexpected."

The work of Euclid was so perfect that it has remained
for 2,000 years the model from which text-books in ele-
mentary geometry have been written. It is safe to say that
it is the greatest work that a single human mind has ever
produced. The Elements was divided into thirteen books,
best known to-day through three translators: Simson,
Heiberg and T. L. Heath; the latter work appeared in
1908, and is of immense value in the realization of the
great geometer's work.

Euclid defines a point as that which has no part, a line
as breadthless length, and a straight line as a line which
lies evenly with the points on itself. Five postulates and
five common notions form the foundation upon which the
superstructure is built. The following are granted:

1. That a straight line may be drawn from any point
to any point.

2. That a finite straight line may be produced continu-
ously in a finite straight line.

3. That a circle may be drawn with any center and any
radius.

4. That all right angles are equal to one another.

GEOMETRY 119

5. That if a straight line falling on two straight lines
make the interior angles on the same side less than two
right angles, the two straight lines, if produced indefi-
nitely, meet on that side on which are the angles less
than the two right angles.

It will be noticed that the plane geometry is built on
three elements, the point, the straight line and the circle.
This may be put otherwise: the three are only the circle
and its two limiting forms, the point being the circle when
its radius has become zero, and the straight line the form
when the radius of the circle has increased to infinity.
These three elements limit Euclidean geometry to two
instruments, the undivided straight-edge and the compass.
Euclid assumes that the circle may be drawn, but the
straight line has been drawn. It is a significant fact that
it was not until 1864 that an instrument was invented by
Peaucillier by which a straight line could be drawn by
mechanical means.

Postulate 2 implies that space is continuous, not dis-
crete, and also assumes its infinitude.

The five common notions are :

1. Things which are equal to the same thing are equal
to each other.

2. If equals be added to equals, the sums are equal.

3. If equals be subtracted from equals, the remainders
are equal.

4. Things which coincide with one another are equal to
one another.

5. The whole is greater than any part.

Common notion 4 implies the free mobility of rigid
bodies in space. Bertrand Russell says what is called mo-
tion in geometry is merely the transference of attention
from one figure to another, and actual superposition nomi-
nally employed by Euclid is not required. Common notion
5 separates the finite from the infinite. The modern defini-
tion of an infinite element is that which is equal to a part
of itself.

120 MATHEMATICS

According to Proclus, every problem and every theorem
which is complete with all its parts perfect purports to con-
tain in itself all of the following elements : enunciation, set-
ting out, definition or specification, construction or ma-
chinery, proof and conclusion. The enunciation states
what is given and what is sought. The setting out marks
off what is given beforehand and adapts it to the investi-
gation. The definition makes clear the particular thing
sought. The construction adds what is needed for the pur-
pose of finding out what is sought. The proof draws the
required inference by reasoning scientifically from ac-
knowledged facts. The conclusion reverts again to the
enunciation, confirming what has been demonstrated.

The fifth proposition of Book I, asserting the equality
of the base angles of an isosceles triangle, has been called
the 'pons asinorum/ or bridge of asses, the inference be-
ing that if the youth had ability to master this theorem,
his future career in geometry was assured. An important
set of theorems in Book I is concerned with the conditions
of equality of triangles, which may be stated as follows:

I. Two triangles are equal if the three sides of one are
respectively equal to the three sides of the other.

II. Two triangles are equal if two sides and the included
angle of one are respectively equal to the corresponding
parts of the other.

IF. Two triangles are equal when a side and the two
adjacent angles are equal respectively to the correspond-
ing parts of the other.

III. Two triangles are equal when two sides and an an-
gle opposite one of them are equal respectively to the cor-
responding parts of the other (containing, however, an
ambiguous case).

III'. Two triangles are equal when two angles and a
side opposite one of them are equal respectively to the
corresponding parts of the other.

It will be noticed that these are arranged in pairs, with
the exception of I, which would be paired with the theorem

GEOMETRY 121

stating the equality of the triangles provided the corre-
sponding angles are equal, which is not necessarily true
in plane geometry.

The primed number of each pair may be gotten from the
unprimed by changing side to angle and vice versa.

A side is determined by the two end points and an angle
by the two including lines, the point and line being the two
limiting cases, one on either side of the circle. Such
a property of certain theorems is called reciprocity or
duality, and enables one to think of such a theorem as a
theorem in points or a theorem in lines as well. This
statement well illustrates duality: two points (lines) de-
termine a line (point). In the triangle theorems the
breaking down of reciprocity in I is due to the fact that the
three angles of a triangle are not independent, as in the
case in sphere geometry. If two are given the third may
be found by subtracting the sum of the two from two
right angles. Three elements (a majority of the five which
may be independent) are required for the determination
of a triangue.

The most important of the remaining theorems of
Book I are those treating of parallels (which will be con-
sidered later) and the Pythagorean theorem. Book III
treats of circles. Book IV of the inscription of regular
polygons in the circle, one of the famous problems of the
ancients, and which leads to the usual method of deter-
mining the approximate value of the ratio of the circum-
ference to the diameter of a circle. The remaining books
through Book IX are mostly concerned with the geometry
of lines — that is, arithmetic treated geometrically. The
last three books are concerned with the geometry of space
and culminate in the regular solid figures which may be in-
scribed in a sphere. While Euclid has been the guiding
star of geometrical text-books for twenty centuries, yet
the tides of darkness have been so dense at many times
that only the faintest gleams of light were discernible.
About 1570 Sir Henry Savile, warden of Merton College,

122 MATHEMATICS

strove to arouse an interest by a course of lectures on
Greek geometry, which were published in 1621. Conclud-
ing, he says: "By the grace of God, gentlemen hearers, I
have performed my promise; I have redeemed my pledge.
I have explained, according to my ability, the definitions,
postulates, axioms and the first eight propositions of the
Elements of Euclid. Here, sinking under the weight of
years, I lay down my art and my instruments." (Cajori.)

Savile says : "In the beautiful structure of geometry
there are two blemishes, two defects ; I know no more."
These were the assumption of the fifth postulate and the
theory of proportion. The non-Euclidean geometry has
vindicated Euclid's position in the first, and it has taken
500 years from the time of Savile to appreciate the theory
of proportion.

The purpose of Euclid was to build up, with a minimum
of assumptions, a logical structure in which reason is the
sole factor. In such a system the figure that is drawn is
simply a guide to the thought and might be entirely dis-
pensed with. Unless it is used with care, it may by subtly
involving intuition ensnare one into error. The following
example of the result of such misleading is well known:
ABCD is a square. AB is bisected perpendicularly at E.
DF is drawn equal to BD. AF is bisected perpendicularly
at G. The two perpendiculars meet at H. CH, DH, AH,
and FH are drawn in the triangles ACH and FDH.
CH = DH, AC = FD, AH = FH. Therefore by the
theorem of equality of two triangles having sides respec-
tively equal, the triangles ACH and FDH are equal and
the corresponding angles ACH and FDH are equals. But
angle ACH = angle BDH, from which angle FDH =
angle BDH; a magnitude equaling a part of itself which
contradicts the fifth common notion, that a whole is greater
than any part of it.

This elimination of observation from the geometry
taught the schoolboy has led to attacks in recent years on
the advisability of the use of Euclid as a school text. J. J.

GEOMETRY

123

Sylvester, one of England's two greatest mathematicians,
in answer to Huxley's statement that "mathematics is that
study which knows nothing of observation, nothing of ex-
periment, nothing of induction, nothing of causation,"

gave voice to the following: "I should rejoice to see
Euclid honorably shelved or buried 'deeper than e'er plum-
met sounded' out of the schoolboy's reach." The Perry
movement, inaugurated in England by John Perry in 1901,

I24 MATHEMATICS

has in a measure resulted in departing from Euclid so as
to make geometry more of a subject of experiment and
observation.

The second great mathematician of this period was
Archimedes, born at Syracuse in 287 B.C., studied at Alex-
andria, returned to Sicily and died in his native city in
212 b.c. Aside from his mathematical contributions, his
mechanical ability was marvelous.

Archimedes was killed during the sack of Syracuse by
the Romans under Marcellus. A soldier found him in the
garden tracing a geometrical figure in the sand as was
customary in those days. Archimedes told him to get off
the figure and not spoil it. The soldier, insulted, thrust
him through with his dagger.

The figure of a sphere, inscribed in a cylinder, was cut
on his tomb in commemoration of his favorite theorem
that the volume of the sphere is two-thirds that of the
cylinder and its surface is four times that of the base of the
cylinder. Cicero rediscovered the tomb in 75 b.c. and
gives a beautiful account of his search in Tusc. Disp.,
V. 23.

"Shall I not, then, prefer the life of Plato and Archytas,
manifestly wise and learned men, to his (Dionysius'), than
which nothing can possibly be more horrid, or miserable,
or detestable?

"I will present you with an humble and obscure mathe-
matician of the same city, called Archimedes, who lived
many years after; whose tomb, overgrown with shrubs
and briers, I in my qusestorship discovered when the Syra-
cusans knew nothing of it, and even denied that there was
any such thing remaining; for I remembered some verses
which I had been informed were engraved on his monu-
ment, there was placed a sphere with a cylinder. When I
had carefully examined all the monuments (for there are
a great many tombs at the gate Achradmse) I observed a
small column standing out a little above the briers, with
the figure of a sphere and cylinder upon it; whereupon I

GEOMETRY 125

immediately said to the Syracusans — for there were some
of their principal men with me there — that I imagined that
was what I was inquiring for. Several men, being sent
with scythes, cleared the way, and made an opening for
us. When we could get at it, and were come near to the
front of the pedestal, I found the inscription, tho the
latter part of all the verses were effaced almost half way.

"Thus one of the noblest cities of Greece, and one which
at one time likewise had been very celebrated for learning,
had known nothing of the monument of its greatest genius,
if it had not been discovered to them by a native of Ar-
pinum."

The work on the Quadrature (or finding the area of a
segment) of the Parabola is one of the most important
works of Archimedes. The proof of the principal theorem
of this work depends upon the "method of exhaustions" in-
vented by Eudoxus, and which is the forerunner of the
modern powerful implement of analysis, the calculus. The
lemma is thus stated by Archimedes: "The excess by
which the greater of two unequal areas exceeds the less
can, if it be continually added to itself, be made to exceed
any finite quantity." The theorem itself asserts that the
area of a segment of the parabola is equal to four-thirds
of a certain triangle inscribed in it.

Another important work, 'The Sphere and the Cylinder,'
containing sixty propositions, was sent to his friends in
Alexandria, in which he purposely misstated some of his
results, "to deceive those vain geometricians who say they
have found everything but never give their proofs, and
sometimes claim they have discovered what is impossible."

The work of Archimedes is of particular interest at the
present time owing to the discovery of a lost work by
Professor Heiberg in Constantinople during the summer
of 1906. The purpose of this work, which is addressed to
Eratosthenes, is well summed up in the following state-
ment and makes clear the method by which Archimedes
arrived at his discoveries: "I have thought it well to

126 MATHEMATICS

analyze and lay down for you in this same book a peculiar
method by means of which it will be possible for you to
derive instruction as to how certain mathematical ques-
tions may be investigated by means of mechanics. And I
am convinced that this is equally profitable in demonstrat-
ing a proposition itself; for much that was made evident
to me through the medium of mechanics was later proved
by means of geometry, because the treatment by the former
method had not yet been established by way of a demon-
stration. For of course it is easier to establish a proof, if
one has in this way previously obtained a conception of
the questions, than for him to seek it without such a pre-
liminary notion. . . . Indeed, I assume that some one
among the investigators of to-day or in the future will
discover by the method here set forth still other proposi-
tions which have not yet occurred to us." Says Professor
Smith: "Perhaps in all the history of mathematics no
such prophetic truth was ever put into words. It would
almost seem as if Archimedes must have seen as in a
vision the methods of Galileo, Cavalieri, Pascal, Newton,
and many of the other great makers of the mathematics
of the Renaissance and the present time."

Very little is known of the life of the third member of
this great trinity, Apollonius of Perga, "the great geome-
ter." It is supposed that he was born about 260 B.C. and
died about 200 b.c. He studied at Alexandria for many
years and probably lectured there. His great work on the
conic sections contains practically all of the theorems of
the text-books of to-day. The work was divided into seven
books, perhaps originally into eight, and while very tedi-
ous, is characterized by strict Euclidean rigor.

A cone is the figure generated by a line passing through
a fixed point and constantly touching the circumference of
a circle. If O is the point and C the circle, the line OC
turns while still passing through O, so that point C
traverses the circle. The complete cone consists of the
symmetrical figure above O as well as the figure below and

GEOMETRY

127

both are extended into space indefinitely. A conic section
is a curve which is formed by passing a plane through the
cone. One of the best methods of quickly constructing
these sections is to immerse a wooden or tin cone in a

vessel of water. The line formed around the cone by the
surface of the water will be the section. There are three
general cases which arise, besides several special ones, as
will be seen by the inspection of the figures, which are

128

MATHEMATICS

vertical cross-sections — that is, the eye is supposed to be
on a level with the surface of the water and sees this sur-
face as a line S.

,&

In Fig. I, where the plane S cuts the two opposite gen-
erators PC and PB, an ellipse is formed. If the plane S

GEOMETRY

129

happens to be at right angles to the axis of the cone as in
la, a circle is the result.

In Fig. II the upper half or nappe of the cone has been
lowered — that is, the cone has been revolved about P until
the axis PB has become parallel with the plane S. The
curve formed is an open curve and is called a parabola.

If the cone be still further turned until both nappes cut
the water as in Fig. Ill, the hyperbola is the resulting

D

Y<

<

d'

E

jp

n

^'_.

E9

i

r 1
1
i

\ >

\

v

X'

-AX

F

0

F'

)a

X

D

f
Y

If

2/

Fig. 41 — Ellipse.

curve. This curve consists of two branches, both of which
are open.

If the plane S passes through the point P during this
investigation, the degenerate conies are formed. I gives
a degenerate circle or ellipse, which is a point where the
radii have become zero; II gives a line which may be
regarded as made up of two coincident lines; in III these
lines become distinct and intersect at P.

It is thus seen that the parabola is the limiting case

130

MATHEMATICS

through which the varying ellipse passes as it merges into
the hyperbola.

These three curves may be defined by a single law of
motion of a point in a plane, and for purposes of study this
is more convenient.

D'

M\

j/*

i

/

/

/

I

7Vr

i

fv

|

1

£

SL

i

Fig. 42 — Parabola.

A point so moves that its distances from a fixed point, F,
called the focus, and from a fixed line, Diy, called the
directrix, are in a given ratio, e, the eccentricity of the
curve. Now the form of the curve and the class to which
it belongs, ellipse, parabola or hyperbola, depends upon
the value given to e. In the figure F is the fixed point, P

GEOMETRY

131

is the moving point on the curve and DD' is the directrix
or fixed line.

In Fig. I e is less than 1 and the curve is an ellipse. It
is seen that it is symmetric to the line YY' and therefore
must have another directrix, DD', on the right and also a
second focus, F".

43 — Hyperbola.

In Fig. II e = 1 and the curve is the parabola. This
curve constantly recedes from the line, yet ever curves to
it. It may be thought of as the left half of an ellipse of
which the right focus has been pulled out to the right an
infinite distance ; it is an open curve — that is, the two arms
of the curve never join again.

132

MATHEMATICS

In Fig. Ill is seen the third case, where e is greater than
i ; the hyperbola with two branches. In the generation of
this curve the point starting at A' recedes indefinitely
downward to the right. It next appears coming back on
the upper half of the left branch, passing along that
branch to an infinite distance and finally coming back

Fig. 44 — Asymptotes of Hyperbola.

along the upper right of the right branch. It is convenient
sometimes to think of the two ends of the curve being
joined by a single infinite point and thus preserve con-
tinuity in the motion of the moving point. The two
branches of the hyperbola constantly approach without

GEOMETRY 133

ever reaching the two intersecting lines OX' and OY' in
the figure; that is, the curves are said to be asymptotic
to these lines, which are called the asymptotes of the
curve.

In the full-page figure is seen the relation which exists
between the foci and directrices of the plane figure and
the cone itself. The plane AB cuts the ellipse from the
cone. If a sphere be dropped in the cone so that it will
be in the cone and just touch the plane the point of touch-
ing or tangency will be a focus. Two such spheres are
possible, the small one above the plane and the large one
below; the foci are F and F'. These spheres touch the
cone in circles. If planes be passed through these circles,
as AC and BC, they will cut the original cutting plane
AB in the lines AM and BN, which are the directrices.

The futility of the argument that it is vain to cultivate
truth for truth's sake is well seen in the case of the
Conies of Apollonius. This monumental work lay dor-
mant and did not reach fruition until seventeen centuries
after, when Kepler found the paths of the planets to be
ellipses and Newton subjected to law the wanderer of the
celestial seas, the comet, whose path is an ellipse if it is
a regular visitor of the solar system. If the path of the
comiet is not an ellipse, it is a parabola, and it comes but
once under the influence of the sun and then forever loses
itself in the vastness of space.

Antiquity has left us three famous problems: The
quadrature of the circle, the duplication of the cube, called
the Delian problem, and the trisection of the angle, or
more generally the problem of the inscription of the regu-
lar polygons in a circle.

The quadrature of the circle, popularly known as squar-
ing the circle, is the problem of finding the side of a square
which has the same area as a given circle. The philoso-
pher Anaxagoras occupied himself with this problem in
his prison. Hippocrates of Chios made one of the most

Fig. 45 — Relation Between Plane and Solid Figure.

GEOMETRY 135

famous attempts at its solution, which resulted in finding
a lune or surface in the shape of a crescent bounded by
two arcs, which was equal in area to a square. Archi-
medes showed that the problem is equivalent to finding
the area of a right-angled triangle whose sides are re-
spectively the perimeter of the circle and its radius, and
further showed that the ratio of these two sides is more
than 3*/7 and less than 2?°/^. This ratio is indicated by the
Greek letter #, introduced by W. Jones in 1706 and crystal-
lized in use by Euler.

Archimedes' method of determining its value was by
inscribing and cricumscribing polygons of 96 sides and by
comparing the ratio of the perimeter of the circumscribed
polygon to the radius determined a value greater than ft,
and by using the inscribed polygon he arrived at a value
less than ft. The present text-book method is to determine
a formula or algorithm by which the perimeter of a poly-
gon of 2n sides may be found from the perimeter of the
polygon of n sides. By carrying this process on indefi-
nitely the ratio may be found to any degree of approxi-
mation.

The ancient Egyptians took the value 256/81> equal to
3, 1605 ; 3 was the value used by the early Babylonians
and also by the Jews (I Kings vii, 23; II Chronicles iv, 2).

A quaint picture is found in the beginning of Halley's
edition of Apollonius and again reproduced in Heath's
volume. The legend below describes Aristiphus, the So-
cratic philosopher, shipwrecked on the island of Rhodes,
where he found the sand of the seashore covered with
geometrical drawings. His exclamation was, "Good cheer.
I see evidences of the Man himself."

Ludolph van Ceulen devoted a considerable portion of
his life to the computation of #• Dying in 1610, he re-
quested that the result to 35 places, which he had obtained,
be cut on his tombstone. Archimedes chose to have his
favorite theorem graven on his tomb, as also James Be-
rnoulli, who, while investigating the properties of the

Fig. 46. — Geometrical "Footprints in the Sand."

GEOMETRY

137

equiangular spiral, discovered the remarkable way in
which curves deduced from it reproduced the original
curve, and he requested that this figure should be carved
on his tomb with the inscription "Eadem numero mutata
resurgo."

Perhaps the limit of perseverance in this direction was
reached by William Shanks, who in 1872 carried the result
to 707 places. Some idea of the accuracy of this value

Eadem numero
mutata resurgo.

Archimedes.
Design on Tombs.

may be inferred from Professor Newcomb's remark that
if the circumference of the earth were a perfect circle,
ten places of decimals would make its circumference
known to a fraction of an inch.

In 1770 Lambert discussed the statement that # is irra-
tional, that it cannot be expressed by a terminating deci-
mal or the ratio of two whole numbers. In 1794 Legendre
proved the irrationality of both it and *\ Hermite in 1873
proved e, the base of the natural logarithms, to be tran-
scendental— that is, it is inexpressible as a root of any
algebraic equation with integral coefficients — and in 1882

138 MATHEMATICS

Lindemann gave a similar proof for the transcendentalism
of #. Euler derived the relation between e and ^expressed
by the following formula, which is one of the most re-
markable in mathematics :

£<■* — — i.

A method of approximating it is by the theory of prob-
ability. On a plane a number of straight lines are drawn
parallel to each other and a units apart. If a stick of
length 1, less than a, is dropped at random on the plane of
these lines, the probability that it will fall across one of

2I
the line is — , from which, by a large number of trials in

which the number of times is recorded that the stick
crosses a line, an approximate value of n is obtained. In
1864 Captain Fox made 1,120 trials and obtained n =
3.1419- (Ball.)

In 1685 Kochausky gave a simple construction by which
the length of a semicircle may be constructed with an ac-
curacy correct to 4 decimal places. At the end point A
of diameter BA draw tangent AF. Take the angle ACE,
equal to 300, and EF, equal to 3 times the radius. Draw
BF and which is the required line? (Halsted.)

The value of ft to 52 places of decimals is tt — 3.141,-
592, 653, 589, 793, 238, 462, 643, 383, 279, 502, 884, 197,-
169, 399. 375. I05» 8.

"Circle squaring" has not entirely died out, but the
mathematical knowledge of the cyclometer of to-day does
not extend much beyond elementary arithmetic. For the
lack of the requisite knowledge to appreciate the problerr
has been substituted a dogged perseverance which should
achieve results if applied in a calling more befitting theii
abilities. Professor de Morgan, whose experience witl,
the several cyclometers certainly puts him in a position to
know their frailties, especially those of James Smith, 0$
Liverpool, says : "The feeling which tempts persons to
this problem is that which, in romance, made it impossible
for a knight to pass a castle which belonged to a giant or

GEOMETRY

139

an enchanter. This rinderpest of geometry cannot be
cured when once it is seated in the system. All that can
be done is to apply what the learned call prophylactics to
those who are yet sound. When once the virus gets into
the brain, the victim goes round the flame like a moth —
first one way and then another, beginning again where it
ended, and ending where he began."

Smith's value for it is 3^, which he attributes to a
French well-sinker, of which De Morgan says: "It does

the well-sinker great honor, being so near the truth, and
he having no means of instruction." Further speaking of
Smith, he says: "He is, beyond a doubt, the ablest head
at unreasoning, and the greatest hand at writing it, of all
who have tried in our day to attach their names to an
error. Common cyclometers sink into puny orthodoxy by
his side. The behavior of this singular character induces
me to pay him the compliment Achilles paid Hector — -to
drag him around the walls again and again." Again : "As
to Mr. James Smith, we can only say this : he is not mad.
Madmen reason rightly upon wrong premises; Mr. Smith
reasons wrongly on no premises at all. His procedures

140 MATHEMATICS

are not caricature of reasoning; they are caricature of
blundering. The o!d way of proving 2 = 1 is solemn
earnest compared with his demonstration."

The origin of the Delian problem, which occupies a
large space in the history of Greek geometry, is given in a
letter from Eratosthenes to King Ptolemy Energetes:

"Eratosthenes to King Ptolemy greeting.

"There is a story that one of the old tragedians repre-
sented Minos as wishing to erect a tomb for Glaucus and
as saying, when he heard that it was a hundred feet every
way:

" 'Too small thy plan to bound a royal tomb.
Let it be double ; yet of its fair form
Fail not, but haste to double every side/ "

But he was clearly in error, for when the sides are
doubled the area becomes four times as great and the
solid content eight times as great. Geometers also con-
tinued to investigate the question in what manner one
might double a given cube while it remained in the same
form. And a problem of this kind was called doubling the
cube, for they started from a cube and sought to double it.
While then for a long time every one was at a loss, Hip-
pocrates of Chios was the first to observe that if between
two straight lines of which the greater is double of the
less it were discovered how to find two mean proportionals
in continued proportion, the cube would be doubled; and
thus he turned the difficulty in the original problem into
another difficulty no less than the former. Afterward,
they say, some Delians attempting, in accordance with an
oracle, to double one of the altars (to rid them of a pesti-
lence) fell into the same difficulty. And they sent and
begged the geometers who were with Plato in the Academy
to find for them the required solution, and while they set
themselves energetically to work and sought to find two
means between two given straight lines, Archytas of
Tarentum is said to have discovered them by means of
half-cylinders and Eudoxus by means of so-called curved

GEOMETRY 141

lines. It is, however, characteristic of them all that they
indeed gave demonstrations, but were unable to make the
actual construction or to reach the point of practical ap-
plication, except to a small extent Mensechmus, and that
with difficulty." Perhaps the most beautiful solution aside
from that of Archytas is by means of the cissoid or "ivy-
like" curve invented by Diodes. This curve is formed
by drawing the horizontal diameter of a circle and draw-
ing pairs of equal half chords perpendicular to this diame-
ter. Through the upper extremity of one of these chords
and the opposite end of the horizontal diameter is drawn
a chord. The point of intersection of this chord with the
other one of the pair of half chords is a point of the cissoid.

In discussing the possibility of a geometrical solution
of a problem it has not always been clear just what is
meant by possibility. Euclid limited his tools to the
straight-edge and compass, so that every geometrical
problem must ultimately reduce to a finite number of con-
structions which are of one or more of the three classes :
finding the intersection of two straight lines, or a straight
line and a circle, or of two circles. At first glance one
would say that the impossibility of a construction by such
methods could never be completely established, that per-
haps some time some one would hit upon the happy combi-
nation necessary for the solution, and so far as geometry
itself is concerned, it has as yet thrown no light on the
subject. It is here that Algebra furnishes the clue. Since
geometry admits of the construction of the square root of
the product of two lines, it may be said that the necessary
and sufficient condition that an analytic expression can be
constructed with the straight-edge and compasses is that
it can be derived from the known quantities by a finite
number of rational operations and square roots. (Klein:
'Famous Problems in Elementary Geometry.')

It is at once seen that the Delian problem reduces to
finding x where x3 = 2 and therefore is unsolvable as a
Euclidean problem.

142 MATHEMATICS

The trisection of an arbitrary angle, while one of the
famous unsolved problems, was not so enshrined in ro-
mance as was the Delian problem. The bisecting or divid-
ing of an angle into two equal parts was very easy of
solution, but not so the trisection. In very special cases,
as that of the right angle, no difficulty is experienced.
The earliest solutions were by means of the hyperbola
and the conchoid of Nicomedes. Since that time many
and various have been the solutions offered, all depending
either on higher plane curves than the circle or upon
mechanical instruments other than the ruler and com-
passes. Speaking of the latter, Plato says: "The good of
geometry is set aside and destroyed, for we again reduce
it to the world of sense, instead of elevating and imbuing
it with the eternal and incorporeal images of thought, even
as it is employed by God, for which reason He always is
God."

It is easily shown that trisection cannot be reduced to
the necessary conditions and therefore it must be classed
as an unsolvable Euclidean problem.

Closely allied with this problem is the other of inscrib-
ing regular polygons in a circle. It has long been known
that polygons may be inscribed if the number of sides is
given by n = 2h, 3, 5 or the product of any two or three
of these numbers. Gauss showed that the operation is
possible for every prime number of the form p = 22n -f- i„
but impossible for all other primes.

Giving n the values o, 1, 2, 3, 4, the primes 3, 5, 17, 257,
65537 result. With n = 5, 6, 7, primes do not result.
Thus is seen that the regular polygon of 7, 9, 11, etc., sides
are not constructable. The polygon of 17 sides has been
constructed by many writers. One construction is given
by Klein in 'Famous Problems.' To the investigation of
the polygon of 65,537 sides Professor Hermes devoted ten
years of his life.

The modification of the instruments used in construc-
tions has been considered successfully by Mascheroni, who

GEOMETRY 143

used compasses alone. All forms which involve rationals
may be dealt with with the straight-edge, while Poncelet
conceived the idea of using the straight-edge and a fixed
circle.

That angle trisectors still exist is attested by the publi-
cation some years ago, with great eclat, that a Western
school girl had succeeded where the mathematicians of
twenty centuries had failed. Verily "fools venture in
where angels fear to tread."

Euclid's definition of parallel lines is straight lines
which, being in the same plane and produced indefinitely
in both directions, do not meet one another in either direc-
tion. Euclid's fifth postulate differs from all the others,
and as Staekel remarks, "It requires a certain courage
to declare such a requirement alongside the other exceed-
ing simple assumptions and postulates," and there is no
better proof of the subtlety and power of the old Greek
geometer than his assumption as undemonstrable that
which required twenty-two centuries to prove as such.
Euclid postpones the use of this postulate until nearly half
of the first book is complete and then assumes it as the
inverse of one already proved, the seventeenth, and uses
it only to prove the inverse of another already proved, the
twenty-seventh.

Proclus demanded a proof as the inverse was demon-
strable and his time on it has been the bone of contention
until the difficulty was cleared up in the nineteenth century
by the most brilliant generalization in the whole field of
mathematics. Playf air's form of this postulate (also
stated by Proclus) is : Through a given point not on a
straight line, one line and but one can be drawn which is
parallel to the given line. Comparing this statement with
the one that one and but one perpendicular can be drawn
from a point to a line, they appear of equal difficulty. On
this slippery ground many good and bad mathematicians
have lost their footing. Lagrange at one time wrote a
paper on parallels in which he hoped he had overcome the

144 MATHEMATICS

difficulty and began to read it before the Academy, but
suddenly stopped and said: "II faut que j'y songe encore"
(I must think it over again). He put the paper in his
pocket and never afterward referred to it.

Legendre showed that this assumption is equivalent to
the statement that the sum of the angles of a triangle is
equal to two right angles, and also proved that if ever a
triangle is found in which the sum of the angles can be
shown to be exactly two right angles, then this is true for
any other triangle. Gerolamo Saccheri in 1733 in a work,
'Euclid Vindicated from All Faults,' obtained the first
glimpse of the modern theory of parallels, and had it not
been for his confidence in the existence of a parallel, would
no doubt have had the credit which belongs now to others.
He presents the curious spectacle of laboring to erect a
structure for the purpose of afterward pulling it down on
top of himself, constructing systems in which he sought
for contradictions in order to prove the hypotheses false.

Wolfgang Bolyai, a Hungarian, was in his college days
a friend of Gauss, the greatest mathematician Germany
has ever produced. He was professor in the Reformed
College of Maros Vasarhely. The son, Johann Bolyai de
Bolyai, is best described by his father when he relates that
the boy in mathematics "sprang before him like a demon."
As soon as he enunciated a problem the child solved it and
asked him to go farther. ' At the age of thirteen he lec-
tured in his father's absence. Writing to his father No-
vember 3, 1823, at the age of 21, he says : "I have not got
my object yet, but I have produced such stupendous things
that I was overwhelmed myself, and it would be an eternal
shame if they were lost. Now I can only say that I have
made a new world out of nothing." And his discovery
was nothing more nor less than to reject the postulate
which had been intuitively accepted since the time of
Euclid, and without this axiom builds up a non-self-con-
tradictory geometry. It was published as an appendix
of twenty-eight pages in a work of his father's. In 1829

GEOMETRY

145

Nicholaus Ivanovitch Lobatchewsky, a brilliant young
Russian, issued his 'New Elements of Geometry with a
Complete Theory of Parallels,' in which the same axiom
is rejected. And so almost simultaneously the new field
was created by two young men, one a Magyar and the
other a Russian, in almost precisely the same manner.

B

Fig. 48 — Lobatchewsky's Parallels.

If P is a point not on the line AB, the lines on the right
of p are divided into two classes, those which cut AB and
those which do not. The line which separates the two
classes is said to be parallel to AB. On the left of p
there is also a parallel. Euclid's axiom would say that
one of these is the prolongation of the other, but such can-
not be proved. Lobatchewsky began his geometry with
the assumption that they are not one and the same line;
in other words, through the point P there are two parallels
to AB, one on either side of p.

In the figure PC and MB make equal angles with PM.
Several cases arise.

I. PC meets AB in the two points, one on the right of
PM and the other on the left (these points may
be distinct or coincident).
II. PC meets AB in one point, and

(a) there exists but the one line PC, which has
this property, or

(b) there exists on the left of PM a second line
having a similar property.

III. PC does not meet AB on either side, however far
produced.

146 MATHEMATICS

In order to give objective reality to these hypotheses,
the geometry of a surface of a sphere will be considered.
It will be necessary to inquire into the meaning of straight
line, since obviously no line may be drawn on a spherical
surface which has the property of straightness, in the com-
mon acceptance of the term. It is not always clear just
what property is meant when the term straight is used.
A very common conception of straightness is that prop-
erty by which if a portion of the line terminated by two
points A and B is placed on any part of the line so that
A and B lie in the line, then the line is said to be straight

if, when this segment is rotated, keeping A and B in the
line, all points between A and B lie evenly in the line.
But this is an unnecessarily complicated statement. An-
other conception which is equally fundamental and much
more fruitful is that the straight line is the minimum line
between the two points. Such a line will be called a geo-
desic. It is the line which the navigator would naturally
choose, other conditions being equal, when sailing between
two points on the surface of the earth, or if a cord is
stretched between two points on the surface of a sphere,
without friction, it will mark a geodesic. It is easily
shown that the only geodesic that may be drawn on the
surface of a sphere is cut out by a plane passing through
the center of the sphere, or the geodesic is a great circle*

GEOMETRY 147

It will be convenient to speak of the spherical surface as
a sphere and the great circle as a straight line or geodesic.
A geodesic on a sphere is determined by two points
(just as the geodesic or straight line in the plane), ex-
cept in the special case of the two points being the ex-
tremities of a diameter. The sum of the angles of a tri-
angle formed by three geodesies is greater than two right
angles. The excess is denoted by E. The area of such
a triangle is proportional to E. In the plane triangle the
sum of the angles is exactly equal to two right angles and
its area is entirely independent of the magnitude of the
angles or their relations one with another. Two triangles
are equal on the sphere if the three angles of one are equal
respectively to the three angles of the other. This was
the case of duality which broke down in the plane. The
surface of the sphere is a two-dimensional manifold of
points; in other words, it has extension in two ways, but
has no thickness. If such a surface could be stripped
from the sphere it could be folded and rolled up by bend-
ing one side inward. If such deformation be performed
without tearing or stretching, it is evident that any theo-
rem concerning lines on the surface would be still valid,
and any figure could at will be moved freely about in the
surface without in any way altering the relations of the
various parts. Likewise the geometry of a portion of a
plane is unaltered if it be rollsd up in the form of a cylin-
der, or cone. Such a property is said to belong to sur-
faces of constant curvature. If R is the radius of a cir-
1 1

cle, — is called the curvature, since as R increases —
R R

decreases and vice versa. If the radius becomes larger
the curvature becomes smaller and the surface flattens out.
Through any point of a surface let all the geodesies be
drawn, and in the plane of any geodesic let that circle be
drawn which most nearly conforms with the geodesic at
the point. The geodesies form a pencil and the curva-

148 MATHEMATICS

ture of each geodesic is the curvature of its particular
circle. Now if all the circles have the same radius, and
this radius is the same for circles at any other point of the
surface, it is said to have constant curvature. This may
be put analytically. A certain expression is taken involv-
ing quantities that are known and which is fully deter-
mined when the line-element of the surface is given. This
expression is an invariant of the surface — that is, it is in-
dependent of the coordinates used to define a point. This
expression is indicated by K and called the Gaussian
measure of curvature. When K is the same for all points
of a surface, the surface is said to have constant curva-
ture.

Suppose the radius of the sphere Rto increase indefinitely ;
1

— , or the curvature, is positive and becomes indefinitely
R

small. The surface flattens out and approaches, as a limit,
the plane with curvature O ; that is, the plane is the limit-
ing case of a spherical surface as curvature or
K approaches zero. Now allow K to pass through

1
zero and become negative; since K = — is nega-

R
tive the radius must be negative or turned in di-
rection. Formerly it was directed inward, and for the
moment it will be convenient to think of it as projecting
outward from the surface. As K passes through O it is
very small and R is very great but negative, or the sur-
face first flattened into a plaae and very slowly curves
the other way, givng a saddle-shaped surface. A surface
of this nature which has constant curvature is generated
by the revolution of the tractrix about the axis to which
it is asymptotic. The tractrix is a curve such that the tan-
gent PT is always a constant. This curve is the projec-
tion on a plane of one of the curves of a skew arch. If
this curve be revolved about the axis OX, it will give a

GEOMETRY

149

saddle-shaped surface called the pseudosphere. On this
pseudosphere a triangle has the appearance of Fig. 50 and
the sum of the angles is less than two right angles. This

Fig. 49 — Curve of Equal Tangents.

deficiency is denoted by D and is proportional to the area
of the triangle.

Going back to the hypotheses, it is seen that the spherical
surface meets the conditions of I. Through a given point

&

Fig. 50 — Pseudo-Spherical Triangle.

P outside a line, no line can be drawn which does not in-
tersect the given line in two distinct points.

150

MATHEMATICS

The geometry of such a surface (of positive curvature)
is called Riemannian or Gaussian.

The plane satisfies hypothesis III if it be assumed that
no other such line may be drawn. The geometry of the
plane is termed Euclidean, and lib is true on the pseudo-
sphere. Through a point outside a given line two parallels
to the line may be drawn. The appearance of the parallels
is indicated by the figure. The geometry of a surface of
constant negative curvature is called Lobatschewskian.

f\

Fig. 51 — Parallels

Pseudo-Spherical Surface.

33

The curvature of a point (regarded as a sphere of zero
radius) is infinite. Starting with a point, let the radius
increase and curvature decrease. As the curvature runs
continuously through the values from -f- 00 down to zero
the surface has a Riemannian geometry of no parallels.
When curvature passes through zero, for an instant the
surface is a plane with the property of one parallel, the
curvature becoming negative. The Lobatschewskian geom-
etry applies and there are two parallels. Continuing the
curvature becomes larger and larger negatively, with ra-
dius becoming smaller until finally the surface closes up
again into a point and the complete course has been run.
Paralleling the case with the conic section, the parabola
was seen to be the boundary between the ellipse and the

GEOMETRY 151

hyperbola. So the Riemannian geometry is said to be
elliptic, the plane parabolic and the pseudosphere hyper-
bolic; these terms come, however, from a different prop-
erty of the spaces.

It is a curious fact that in the simple Riemannian plane
the straight line cuts through the plane without cutting
it in two. This cut cannot well be pictured, but an idea
of its meaning may be got by thinking of the surface of a
ring with a cut extending around the outside of it.

In Lobatchevskian space the unit of measure is a con-
tinuously decreasing length, while in Riemannian space it
is continuously increasing.

Riemann, in his celebrated paper on The Hypotheses
which Lie at the Basis of Geometry/ first advanced the
theory that space might be unbounded without being in-
finite, in these words : In the extension of space-construc-
tion to the infinitely great, one must distinguish between
unboundedness and infinite extent. That space is an un-
bounded three-fold manifoldness is an assumption which
is developed by every conception of the outer world. The
unboundedness of space possesses a greater empirical cer-
tainty than any external appearance. But its infinite ex-
tent by no means follows. On the other hand, if we as-
sume independence of bodies from position, and there-
fore ascribe to space constant curvature, it must neces-
sarily be finite, provided this curvature has ever so small
a positive value. If we prolong all the geodesies from
one point in a surface of constant curvature, this surface
would take the form of a sphere.

The question as to whether the space of experience is
Euclidean, Lobatchevskian or Riemannian is one which
can never be determined. Are there two parallels, one or
none? could only be settled in one of two ways, by reason
or by measurement. A better form for the question is as
to whether the sum of the angles of a triangle is less than,
equal to, or greater than two right angles. As to reason,
the geometry of one hypothesis is just as consistent as

152 MATHEMATICS

that of another. As to measurement, it is conceivable
that an error in the measurement of the three angles of a
triangle which may be drawn on this page would not show
an error which would easily be detected if the triangle
were drawn with sides 10 miles in length.

The largest triangles ever possible to measure have as
a side the diameter of the earth's orbit, the opposite vertex
being a celestial body. That no deviation from two right
angles in the sum for this triangle is found is no evidence
that if it were a million times as great the deviation
would not be appreciable. The most that can be said is
that if space is curved, the curvature is slight.

The study of non-Euclidean spaces enables one better
to appreciate the insight of the old Greek geometer who
2,000 years ago realized that the proof of his fifth postu-
late was beyond his powers.

All measurement in mathematics is concerned either
with that of lines or of angles. Euclid developed a com-
plete theory of measurement of lines, but aside from the
right angle and several of its exact divisors — as ^ of a
right angle, etc. — the only relations which he determined
were those of greater and less; thus, if the sides of a
triangle are 3, 4, 5, it is known by geometry that the angle
opposite the side 5 is a right angle, and further that the
angle opposite the side 4 is greater than that opposite 3,
but exactly how much Euclid gives us no means of de-
termining.

CHAPTER V

TRIGONOMETRY

Trigonometry is the science of the triangle with ref-
erence to the particular problem of finding the value of
the unknown parts when three independent parts are
given, as finding the angles when the three sides are
given, etc. In a right triangle, ABC, lettered as in
Fig. 49, six ratios are involved, which remain the same
so long as the angles are not changed, the size of the
triangle changing at will but preserving its shape. These
six ratios are functions of the angles ; that is, they depend
for their value upon the values of the angles. They are
named in the table below, with the abbreviations usually
assigned to them given last.

— = sine A = sin A — = cos B
c c

b b

— = cosine A = cos A — = sin B

— = tangent A = tan A = tg A — = cot B
b b

b b

— = cotangent A = cot A — ctn A — = tan B
a a

i53

154 MATHEMATICS

c c

— = secant A = sec A — = esc B

b b

c c

- — = cosecant A = cs A — = sec B

a a

In the first column the functions are arranged in pairs,

the second of the pair having its name from the first, with
the prefix co. The origin of this prefix is from the rela-

HB

Fig. 52 — Ratios of a Triangle.

tion which exists between A and B, the sum of which
is 1 right angle. It therefore takes B to fill a right angle
together with A, or B is said to be the complement of
A or co-A. Looking at the second column, one sees that

b
the ratio — is the sin B or sin of co-A or cosine A.

c
These six ratios were originally used in connection with
a right triangle alone. When it became desirable to con-
sider angles greater than 1 right angle, such angles not
being found in a right triangle, the definitions for sine,

TRIGONOMETRY

155

cosine, etc., were so framed as to apply to any angle, posi-
tive or negative. This was done by means of a line rep-
resentation. A circle of radius unity is chosen, and di-
vided into 4 quadrants by means of a horizontal and a
vertical line through the center. It is agreed that the
angle shall begin at OA, and shall be considered positive

if it extends in a counter clockwise direction; directions
of other lines are given by the arrows on the two axes.
Take a point P on the terminal side of the angle and on
the circumference of the circle; since the angle may be
of any magnitude, the point P may be in any one of the
4 arcs AB, BA',A'B', or BA. The construction here given

156 MATHEMATICS

applies to any position of P. It will be supposed that
P is in the arc AB, and the relations between the new
and old definitions of the functions will be apparent.
Draw OP, which will be directed outward from O. Drop
a perpendicular from P to OA, calling the foot of the per-
pendicular M ; then PM/OP = sin AOP, where the vertex
of the angle is O, and OM/OP = cos AOP; but the
circle was a unit circle, and OP=i; whence MP = sin
AOP and OM = cos AOP. From A erect a perpendicu-
lar cutting OP produced in T. Then AT/OA = tan
AOP = AT, sec AOP = OT. From B draw a parallel
to OA cutting OP produced at S. BS = cot AOP and
OS = esc AOP.

If OP, beginning at OA, swings through a complete
revolution about O, all angles from O to 4 right angles
will be passed through.

There are two units employed in measuring angles:
the degree, with its subdivisions minute and second, and
the radian. The degree is V^o of a complete circumfer-
ence, due to the Babylonian year, which was made up of
360 days. The degree, symbolized by °, is divided into 60
equal parts, each called a minute (indicated by a single
prime, '), another Babylonian division; the minute is again
divided by 60, giving the second, ", The unit of radian
measure is the angle which cuts off an arc equal to the
radius of the circle. It is nearly 57.3 degrees. Since
27rr = circumference, 4 right angles = ^n radians, or

27rr, oo° = — , 1800 = n\ 2700 = ^~ The number of radians

2 2

is given by arc/radius.

In the figure of the line functions, if P returns, by mak-
ing a complete revolution, or 2?rr, to A, and continues
turning in the same direction, an angle is formed which is
greater than 27rr', but the functions of this angle are ex-
actly those of the angle formed during the first revolution.
This property of again passing through the same values
with every complete turning is called periodicity. The

TRIGONOMETRY

157

periodicity of the six trigonometrical functions is well
exhibited by a diagram in which distance along the hori-
zontal line represents the magnitude of the angle meas-
ured in radians, and the perpendicular to this line at any
point is the value of the function for the angle indicated
by the point.

y - sin x

Fig. 53 — Curve of Lines.

X

+1

37T

H

2?FX

y = cos x

Fig. 54 — Curve of Cosines.

The simplest relation is that between the sine and cosine
of an angle, which comes directly from the Pythagorean
theorem, sin2 A + cos2 A = 1. One of the most impor-

158 MATHEMATICS

tant properties of these functions is that they have an
addition law; that is, if two angles are added, the sine
of the sum is not the sum of the sines of the two angles,
but it may be expressed through functions of the angles.
This is the most fruitful property. The addition theorem
for sine and cosine follow where A and B are any two
angles :

Fig- 55 — Curve of Secants.

sin (A + B) = sin A cos B + cos A sin B
cos (A + B) = cos A cos B — sin A sin B
In the practical application to the solving of triangles
three laws are used, which may be expressed by formula :

Law of sines:

sin A

sin B

Law of cosines : c2 = a2 -f- b2 — 2 ab cos A
which is the law spoken of as summing up in one state-

TRIGONOMETRY

159

merit the Pythagorean theorem, with the acute and obtuse
cases.

a + b tan ^ (A + B)
Law of tangents: =

tan

(A- B)

Tables have been constructed by which the function of
any angle, and conversely the angle of any function, may
be obtained as accurately as the needs of science demand.

y -sec x
Fig. 56 — a., Nailer's Rules ; b., Polar Triangles.

Spherical trigonometry is the science applied to a tri-
angle on the surface of a sphere. The sides are now also
expressed in angular measure. In the solution of the right
triangle a mnemonic device, found by Napier, the inventor
of logarithms, eliminates the necessity of committing to
memory the relations of the functions. In the figure, C
is a right angle, and before the parts A, c, B are written
co-, which means that in the lines which follow that the

i6o

MATHEMATICS

complement of each part is to be taken rather than the
part.

Napier's Rules of circular parts : Sin of middle part is
equal to the product of the cosines of the opposite parts,
or equal to the product of the tangents of the adjacent
parts.

It is seen that omitting the right angle C, which is
indicated by putting the C within the triangle, that there
are five remaining parts ; now choosing a part, and calling
it a middle part, as a, there are two parts, b, co-B, adja-

Section of a Model of a Cubic Surface. (Blythe.)

cent to a, and two parts, co-A, co-c, which are opposite
to a. Apply the Rules above.

sin a = cos co-A • cos co-c

= sin A sin c
sin a = tan b • tan co-B
= tan b cot B
In this way the ten necessary relations in the right triangle
may be written at will.

There is a very interesting relation in spherical geom-
etry concerning what are called polar triangles. If the
angular points A, B, C of a triangle are used as centers,
and the arc of I right angle is used as a radius, striking
3 arcs which form a triangle, this triangle, indicated by

TRIGONOMETRY 161

A'B'C, is called the polar triangle of ABC. The rela-
tion is reciprocal : ABC is polar of A'B'C. The property
which is to be noted is, that a side of a triangle (or angle)
is the supplement of the opposite angle (or side) of the
polar triangle.

A + a' = 1800
a + A' = 1800

The law of cosines in spherical trigonometry is the
most general case of the universal law which is expressed
in its simplest form by the Pythagorean Theorem :

cos c = cos a cos b -f- sin a sin b cos C
If the radius of the sphere is allowed to become great
without limit — that is, the spherical surface flattens out
and approaches a plane, in the limit — this formula be-
comes the Law of Cosines in plane trigonometry :

c2 = a2 + b2 — 2 ab cos C
If, now, the angle C becomes a right angle, the formula
reduces to

c2 = a2 + b2
or the Pythagorean Theorem.

In the figure used in the definition of the trigometric
functions by lines each function belonged to the angle
AOP. Since the arc AP has the same measure as the
angle, and the sector AOP — i.e., the portion of the circle
bounded by the two radii and the arc — is measured by the
arc AP, it is convenient to say that the six ratios are
functions of the sector as well as of the angle.

The circle was seen to be a particular case, with a fixed
form or shape of the ellipse, which varied as the cone was
turned. The hyperbola varies in shape also with the turn-
ing; there is a position of the cone which gives a form
of the hyperbola analogous to the circle. This form is
called the equilateral hyperbola. Its most familiar use
is in representing the relation between the pressure and
volume of a gas, which is expressed by pv = a constant.

A set of functions belonging to the equilateral hyperbola
has been devised which is distinguished from the set per-

1 62

MATHEMATICS

taining to the circle by calling the first set circular func-
tions and the second hyperbolic functions. In the figure,
the sector of the hyperbola bounded by OA, OP and the
arc OP will be denoted by u. From the foot of the per-

pendicular MP, MT is drawn tangent to the circle. The
sector of the circle AOT will be called v. The hyperbolic
functions of the sector AOP will be denoted by sinh u,
cosh u, etc. v is said to be the gudermanian of u, or
v = gd u. Some of the relations existing between the
functions of u and v are:

cosh u = sec v;

sinh u = tan v;

tanh u = sin v, etc.

The discussion just given is of but a special case of
these functions. The name hyperbolic was not originally
given on account of the properties here stated.

One would expect that the term Elliptic function would
be used for some similar relation in connection with the
ellipse, but such is not the case. The desirable use of
the word would be to denote the more general case of the
circular functions. The term arose in connection with

TRIGONOMETRY 163

some expressions which appeared in the early attempts
to rectify or measure an arc of the ellipse. They may,
however, be regarded as an extension or branch of trig-
onometry, since they have two properties, analogous to two
properties of trigonometric functions, namely : they admit
of an addition theorem and periodicity.

The trigonometrical functions are simply periodic. In
the sine curve let the angle be taken 300. The value of
the sine for 300 is indicated by the perpendicular line MP.
If a point Q be taken 2,-it units from M, the sine line QN
will be the same as MP; Apt will give the same sine. These
points of periodicity are points of a line. The elliptic

functions are doubly periodic. It requires the entire plane
to indicate the values of the independent variable.

Rudiments of trigonometry are found in the Ahmes
papyrus, where the dimensions of square pyramids are
to be found. In these computations appears a word,
'sept/ which has a value of about .75. This is the cosine
of 41 ° 24' 34", which is very nearly the slope of the edges
of the existing pyramids. In Ptolemy's 13 books of the
Great Collection, or the Almagest, spherical trigonometry
is developed and applied to astronomy. The names "min-
ute" and "second" are from the Almagest. Half chords
were first brought into favor by Al Battain, an Arab prince
(c. 850-929), in whose work first appears the Law of
Cosines for the spherical triangle. The greater part of
the plan used in the trigonometry of to-day is the work
of Regiomontanus or Johannes Miiller (1436-1476).

CHAPTER VI

ANALYTIC GEOMETRY

The final union of algebra and geometry by means of
the analytic geometry is usually attributed to Des Cartes.
Algebra has been used at various times in connection with
geometry by Apollonius and Vieta in particular, but in
their works the idea of motion is wanting. Des Cartes,
by introducing variables and constants, was enabled to
represent curves by algebraic equations. A point in a
plane is determined by its distances from two intersecting
lines, which, for convenience, may be taken as perpen-
dicular to each other. By allowing these two distances
to vary, the point moves and generates a curve. By ex-
pressing the relation between these two variable distances
in the form of an equation, the curve becomes subject to
investigation following the laws of algebra. This Is the
great contribution by Des Cartes, and by it "the entire
conic sections of Apollonius is wrapped up and contained
in a single equation of the second degree." (Cajori.)

The plotting of an equation of the first degree which
results in a straight line was spoken of in connection with
algebra, as was also an equation of the second degree.
The general equation of the second degree is written in
the form

Ax2 + Bxy + Cy2 + Dx + Ey + F = o.
Two processes are applied to change the form of an equa-
tion, which evidently depends upon the axes chosen. One
of these is to translate (or move parallel to themselves)
164

ANALYTIC GEOMETRY 165

the axes, and the other is to rotate them about the point
of intersection, which is called the origin. If the general
constants, A, B, C, D, E, F, are such that the equation
can be reduced by one or both of these operations to the
form b2x2 + a2y2 = a2b2, the curve is an ellipse ; if to the
form x2 -f- y2 = r2, the circle, b2x2 — a2y2 = a2b2, is the
equation of the hyperbola, and y2 = 2 px is the parabola.
If the left member of the equation can be factored, it is
a degenerate conic. The equation of the third degree
gives a curve which is called the cubic. Newton gave a
classification of the cubic curves, the general form of
which is a closed loop and an open branch. The curves
of higher degree comprise some of the historic curves.

In addition to the algebraic curves there is a great
class of curves called transcendentals. To this class be-
long the curves of the trigonometric functions given in
p. 157. The most famous of the transcendentals is the
cycloid, the path of a point on the rim of a carriage
wheel as the wheel rolls on the ground. If the wheel
rolls on the circumference of a circle, instead of on a
line, the curve generated is called an epicyloid, and is
one of the curves used in laying out gear wheels.

Some idea of the number of curves that have been
investigated may be gathered from the fact that an Italian
writer listed these curves, with a short description of each,
filling a large book of about 700 pages.

The method of Des Cartes is easily carried to three
variables. An equation of this fofm might be z == f (xy).
The plane determined by the two perpendicular lines OY
and OX is the old XY plane; perpendicular to it the new
Z-axis, OZ. Since x and y are independent of each other,
any value, as OM, may be laid off for x on the X-axis;
perpendicular to this axis a value of y, say MN, is plotted.
Putting these values in the equation, z is determined,
which is laid off at right angles to the plane XOY, or
NP; that is, P is one point of the surface represented by

i66

MATHEMATICS

the equation. If a corresponding point is found for every
point in the XY plane, the entire surface will be plotted.

An equation of the second degree in three variables, x,
y, and z, represents one of what are called quadric sur-
faces. Such surfaces are of two classes; on a surface of

the first class, such as the ellipsoid, no straight lines may
be drawn and the geodesies are all curved lines. The
ellipsoid is generated by a variable ellipse moving parallel
to itself. In the second class of surfaces, called the ruled
surfaces, the geodesies are straight lines. The hyperbo-
loid of one sheet may be generated by a line moving
parallel to itself while constantly touching two circles in

ANALYTIC GEOMETRY

167

parallel planes, the planes being oblique to the moving line-
Such a surface has two sets of line generators, one set
inclined to the right, and the other to the left.

The cubic surface, or surface of the third degree, con-
tains 27 straight lines, a fact discovered by Dr. Cayley in
1849. In tne drawing of the section of one of these sur-
faces some of these lines are seen. The blackened por-
tion indicates where the solid model is cut, only a part of
the surface being shown.

The principal advances in analytic geometry have been
along three lines :

1. Changes in the system of coordinates.

2. Changes in the element used.

3. The introduction of the imaginary element.

In 1857 President Hill, of Harvard, gave a list of 22
systems of coordinates then in use, and since that time
many more have been added. One of the most useful
systems is known by the term polar coordinates, in which
a point P is located by the distance r = OP from the
origin and the angle 0 between OP and the initial line
through O.

This system greatly simplifies some of the equations o£

1 68 MATHEMATICS

the Cartesian system; for example, r = a constant is
the equation of a circle in polar coordinates. The general
equation of the straight line in Cartesian coordinates is
Ax -f- By + C = o. This equation is seen to lack homo-
geneity, or likeness, two of the terms containing variables
and the third term being a constant. This unlikeness is
removed if, in place of choosing as determining coordi-
nates the distances from two intersecting lines, three lines
are taken which intersect in pairs; that is, do not pass
through the same point. Instead of using the three dis-
tances the three ratios of these distances are taken as the
trilinear coordinates of a point.

.P

In Euclid's choice of elements, the primary element is
the point, with the circle and line as secondary, each of
these being an aggregate of points. A point in motion
generates a line or curve; the curve in motion, not along
itself, generates a surface, which if moved outside of itself
gives a solid. And the whole geometry is a point geom-
etry, made up problems in which a certain point is to be
found, the intersection of two lines, a line and a circle,
or of two circles.

Looking at these elements from another viewpoint,
they are but the circle which Euclid could draw and its
two limiting cases, as the radius becomes indefinitely small,
and becomes indefinitely great. The latter Euclid could
not draw, whence he assumes straight-edge as one of his
instruments. The symmetry of the three suggests that

ANALYTIC GEOMETRY 169

the line might just as well be taken as the point. A line
is made up of an infinite number of points arranged in a
certain way, and a point is made up of an infinite number
of lines arranged in a definite manner.

A theorem which is thought of as a relation between
points, it is evident, may be by simply interchanging the
words 'point' and 'line' becomes the expression of a rela-
tion between lines. This principle of Duality was first
worked out in its entirety by Jean Victor Poncelet, a
brilliant young French lieutenant of engineers, who was
made prisoner in the French retreat from Moscow in
1812. Fnding himself in prison, without books or any
means of enjoyment, he occupied himself with investi-
gations in geometry, and wrote his classic work on 'The
Projective Properties of Figures,' in which the principle
of Duality is completely worked out.

The analytical or algebraic investigations of geometry
very often result in giving values which involve the imag-
inary element i. Every equation of the second degree
represents a conic, and if two such equations are solved
simultaneously for the points of intersection, four such
points result. If the equations are those of circles, it
is seen that two circles at most intersect in two real
points. The other solutions result in imaginary solu-
tions. The coordinates of these two points are conjugate
imaginaries; one is of the form a + ib and the other
of the form a — ib. These two points are indicated by
I and J and are called the two circular points at infinity,
for it is found that every two circles, besides intersecting
in two real or two imaginary points in the finite region
of the plane, also intersect in I and J. Again, it requires
five points to determine or pick out a conic section, and
it is known that three points determine a circle. What
about the two missing points? They are I and J, which
lie on every circle in the plane. In this conception, a
circle is the aggregate of all of the points in its circum-
ference and the two points I and J.

170 MATHEMATICS

If a circle has its radius indefinitely diminished it ap-
proaches as a limit a point, a degenerate conic which was
its center. The equation of a circle with the center at the
origin of coordinates is x2 + y2 = r2. If r be made zero
the equation is x2 -f- y2 = o, which may be factored, giving
x = iy and x = — iy. These are the equations of two
imaginary lines called isotropic lines, which have some in-
teresting properties.

Through every point of the plane pass two isotropic
lines.

These isotropic lines make the same angle with every
real line through the point.

The distance between any two points on an isotropic
line is zero, from which property they are called minimal
lines.

The isotropic lines join the real point through which
they pass with I and J respectively.

Perpendicularity between two real lines through the
real point is a relation between the two lines and the two
isotropic lines through the point.

The algebraic treatment of geometry permits the in-
vestigation of imaginary elements with exactly the same
rigor as that of the real elements, and the only distinc-
tion between real and imaginary elements is not one of
existence but of adaptability to the picturing processes of
the mind. The term imaginary originally implied non-
existence, but the development of algebraic processes has
entirely swept away that meaning. The whole question
of existence with the geometer is not one of material exist-
ence; points, lines and planes are but creations of thought
without materiality. That which exists is that which is
consistent in thought, coherent and non-contradictory.
A real element is one which may be represented, as a line
by a mark or string, a surface by a sheet of paper, and
the imaginary is one of which no such picture or image
may be formed.

The disposition to seek decision upon matters which

PROBABILITY 171

do not come within the domain of present knowl-
edge, that intuitive desire of mankind to rely upon the
doctrine of chance, seems to be a universal trait with
humanity. That such an instinct should arise and be
cultivated in every branch of the human race is but a
corollary of the fact that the future is hidden. Proba-
bility is more or less a factor in the life of every indi-
vidual. It may be said that in no contingency which
arises is there more than probable evidence upon which to
proceed. Voltaire puts the case more strongly. "All
life," says he, "rests on probability." As a moral guide
it is said that the following theory was taught by 159
authors of the Church before 1667: 'If each of two oppo-
site opinions in matters of moral conduct be supported by
a solid probability, in which one is admittedly stronger
than the other, we may follow our natural liberty of
choice by acting upon the less probable.'

This gaming instinct has left as a heritage a number of
games of great antiquity, varying from those in which
skill and mental acuteness is the predominant factor
down to those in which no element enters except that of
pure chance. The best type of the first class is the game
of chess, while perhaps midway comes cards and finally
dice. Games akin to chess and checkers are represented
in Egyptian drawings as early as 2000 B.C.

Professor Forbes puts the origin of chess "between
three and four thousand years before the sixth century of
our era." Altho this antiquity is to be doubted, it must
be considered as extremely old. The game of chatu-
ranga is said to have been invented by the wife of Ravana,
King of Ceylon, when his capital, Lanka, was besieged
by Rama. That the game was in some way connected
with war seems evident. The Chinese name for chess
is literally "the play of the science of war." The word
chaturanga means the four divisions of the army, ele-
phants, horses, chariots and foot soldiers.

The intricacies of the game are seen when it is known

172

MATHEMATICS

that there are as many as 197,299 ways of playing the first
four moves, and nearly 72,000 different positions at the
end of these moves.

The move of the knight is one move forward and one
diagonally, and from this has been framed a famous prob-
lem : So to move the knight that it occupies but once each
of the 64 squares of the board. This problem gives rise
to some very odd geometrical designs on the board, if a

34

49

22

11

36

39

24

1

21

10

35

50

23

12

37

40

48

33

62

57

38

25

2

13

9

20

61

54

63

60

41

26

32

47

58

61

56

53

14

3

19

8

55

52

59

64

27

42

46

31

6

17

44

29

4

15

7

18

45

30

5

16

43

28

Fig- 55 — Knight's Move in Magic Square

straight line is drawn between each two successive posi-
tions. The solution here given is that of De Moivre. The
number of possible solutions has been shown to be over
31,054,144.

The origin of cards is as uncertain as that of chess.
They appeared in Europe about 1200. If one seeks to go
back from this, one trail leads through Spain to Africa
and Egypt, another over the Caucasus to Persia and
India, and perhaps another is picked up in China. In
the Chinese dictionary (1678) it is said that cards were
invented in the reign of Seun-ho, 1120 a.dv for the amuse-

PROBABILITY 173

merit of his various concubines. Tradition says that
cards have existed in India from time immemorial and
that they were invented by the Brahmans.

Fig. 56 — Knight's Tour on Single and Double Chess Boards.
(Falkener.)

One form of cards, the Tarot card, was brought into
Europe from the East by gipsies, who used them for

174 MATHEMATICS

divination purposes. They undoubtedly have been con-
nected with witchery from the very beginning.

A number of famous problems have been devised with
cards. The first to be spoken of is Gergonne's, or the
three-pile problem. In this trick 27 cards are dealt face
upward in three piles, dealing from the top of the pack,
one card at a time to each pile. A spectator is requested
to note a card and remember in which pile it is. Taking
this pile between the other two the operation is repeated,
and the third time is noted the middle card of each pack.
Ask now for the pile and it is the card noted in this pile.
Now if the three piles are taken up face down in the same
order and dealt from the top it is the fourteenth card.
Gergonne generalized the problem to a pack containing-
mm cards.

The mouse-trap is another noted game with cards. A
set of cards marked with consecutive numbers from 1 to
n are dealt in any order face upward in the form of a
circle. The player begins with any card and counts
round the circle. If the kth card has the number k on
it, a hit is scored and the player takes up the card and
begins afresh. The player wins if he takes up all the
cards. If he counts up to n without taking up a card,
the cards win. In Tartaglia's work occurs a similar
problem : A ship, carrying as passengers fifteen Turks and
fifteen Christians, encounters a storm; and the pilot de-
clares that in order to save the ship and crew one-half
of the passengers must be thrown into the sea. To choose
the victims, the passengers are arranged in a circle, and
it is agreed to throw overboard every ninth man, reckon-
ing from a certain point. In what manner must they be
arranged that the lot will fall exclusively upon the Turks ?

The number of combinations possible in various card
games is enormous. With the whist deal this number is
53,644,737,765,488,792,839,237,440,000.

Dice and dolasses go back in history at least 3,000 years.
Apollo taught their use to Hermes. These Greek gods

PROBABILITY 175

probably got their knowledge from Egypt, where dice, and
it is even said loaded ones, have been found in the tombs.
Gaming with dice was common with the Romans, who
had two forms, one like those of the present and the other
oblong and numbered on but four sides. On these the
deuce and the five were omitted. The convulsion of na-
ture which overwhelmed Pompeii found a party of gentle-
men at the gaming table, and they have been uncovered
two thousand years after, with the dice firmly clenched in
their fists. Seneca brings the gambling Emperor Claudius
finally to Hades, where he is compelled to play constantly
with a bottomless dice-box.

The two theories of choice and chance are very closely

Fig- 57 — Roman Dice.

bound up together. Choice is made up of two branches,
those problems which deal with arrangements and those
with combinations alone. A problem of the first type is
to find the number of ways in which 10 men may be
seated at a round table. The first man has manifestly
no choice; he may be seated anywhere; after he is seated
the second man has 9 choices, the third 8 and so on until
the tenth man, who has but 1 choice. It is a principle
that if a thing may be done in a ways and another in b
ways, the two together may be done in a X b ways.
Therefore the 10 men may be seated in9X8X7X6X
5X4X3 X2X ! ways, which is denoted by 9 ! or 9
factorial. The general expression for n things taken r
at a time is n ! / (n — r) !

176 MATHEMATICS

If there is no distinction between the objects — that is,
the order is immaterial — a choice is called a combination;
as to find in how many ways a committee of 4 men may
be chosen from 25 men. The mode of solution is to find
in how many ways 25 men may be arranged if chosen 4
at a time, and divide by the number of arrangements pos-
sible with the 4 men.

If an event happens a times and fails b times, the prob-

a

ability of the event happening is and the prob-

a + b
b a

ability of it failing is . — are the odds in favor and

a-fb b
b

— are the odds against the event happening. This may
a

be illustrated in finding the probability of throwing at
least 4 with 2 dice. The number of favorable cases is the
number of cases in which 4, 5, 6, 7, 8, 9, 10, 11, 12 may
be thrown. The number of unfavorable cases is the
number of ways in which 2 and 3 can be thrown. 2
can be thrown in one way by throwing 1 and I. 3
can be thrown in two ways, 2 and 1 and 1 and 2. The
number of unfavorable cases is 3. The total number of
cases is 6 X ^ or 36. The number of favorable cases is
then 36 — 3 or 33, and the probability of throwing at least

33 11
4 is — or — .
36 12
If 52 cards be dealt to 4 players, the probability that a

11
particular player will hold 4 aces is .

4165
An application of the theory of probability may be
given in determining the expectancy of a player in the
ordinary "crap" game: A and B play with two dice, A

PROBABILITY 177

throwing, and B being the "banker." If A throws 7 or
11 he wins; if he throws 3, or 2 aces, or 2 sixes, B wins.
But if he throws 4, 5, 6, 8, 9 or 10 he continues throwing
to duplicate his first throw, in which event he wins; if
in throwing a 7 comes up, B wins. To determine the
chances of the two players.

2
The chance of throwing 7 or n is — ; of 2, 3 or 12 is

9
1 2

— ; of 4, 5, 6, 8, 9 or 10 is — . If A throws 4 his chance

9 3

1 2

of winning the second throw is — . — ; of the third throw is
12 3
T\ of f of [1 - (J2 + *)] or rL of f of f.
A's chance of winning on 4 is

I + A of f [1 +jt + (I) 2 '+ (I) 3 + • • • 1 = ♦■
A's chance of winning on 5 is

t + toff[i + H+ (H) ■ + (H) 3 + ..-] = If

A's chance of winning on 6 is

I + 35<r of f [1 + ff + (If) 2 + (M) 3 + ...] = If-
A's chance of winning on 8, 9 or 10 is the same as for
6, 5, or 4.

A's chance is then HI + It + If) = tWs-
722 763

B's chance is 1 - — = .

1485 1485

763

The odds in favor of B are . (Zerr's solution.)

722
One very important application of probability is to
determine the probable error in a number of observa-
tions. In 1805 Legendre gave his Law of Least Squares,
which may be simply stated as follows : The most prob-
able value of a measured quantity is that in which the
sum of the squares of the differences between this quan-

178 MATHEMATICS

tity and the observed values, provided they are equally
good, is a minimum.

Probability finds its greatest function, however, in de-
termining the probable death-rate upon which are based
insurance premiums. When it is recalled that at the
present time the greatest amount of money that is in-
volved in any single business is that in insurance, the
words of Augustus de Morgan, penned in 1838, seem more
than prophetic:

"The theory of insurance, with its kindred science of
annuities, deserves the attention of the academic bodies.
Stripped of its technical terms and its commercial asso-
ciations, it may be presented from a point of view which
will give it strong moral claims to notice. Tho based on
self-interest, yet it is the most enlightened and benevolent
form which the projects of self-interest ever took. It is,
in fact, in a limited sense and a practicable method, the
agreement of a community to consider the goods of its
individual members as common. It is an agreement that
those whose fortune it shall be to have more than the
average success shall resign the overplus in favor of
those who have less. And tho, as yet, it has only been
applied to the reparation of the evils arising from storm,
fire, premature death, disease, and old age, yet there is
no placing a limit to the extensions which its applica-
tion might receive, if the public were fully aware of its
principles and of the safety with which they may be put
in practice."

The science of probability had its origin in a problem
proposed in 1654 to Blaise Pascal by Chevalier de Mere,
a professional gambler. It is now known as the problem
of points. Two players want each a given number of
points in order to win: if they separate, how should the
stakes be divided? Pascal's solution is as follows: Two
players play a game of 3 points and each player has
staked 32 pistoles.

Suppose that the first player has gained 2 points and

PROBABILITY 179

the second player 1 point; they have now to play for a
point on this condition, that if the first player wins he
takes all the money at stake, namely, 64 pistoles, and if
the second player gains each player has 2 points, so that
if they leave off playing each ought to take 32 pistoles.
Thus, if the first player gains 64 pistoles belong to him,
and if he loses 32 pistoles belong to him. If, then, the
players do not wish to play this game, the first player
would say to the second: "I am certain of 32 pistoles if
I lose this game, and as for the 32 pistoles, perhaps I
shall have them and perhaps you will have them: the
chances are equal. Let us then divide these pistoles
equally and give me also the 32 pistoles of which I am
certain." Then the first player would have 48 pistoles
and the second 16 pistoles.

Next, suppose that the first player has gained two
points and the second player none, and that they are about
to play for a point; the condition then is that if the first
player wins this point he secures the game and the 64
pistoles, and if the second player gains this point they will
be in the position just examined, in which the first player
is entitled to 48 pistoles and the second to 16 pistoles.
Thus, if they do not wish to play the first player would
say to the second, "If I gain the point I gain 64 pistoles;
if I lose I am entitled to 48 pistoles. Give me the 48
pistoles of which I am certain, and divide the other 16
equally, since our chances of gaining the point are equal."
Thus the first player gets 56 pistoles and the second 8
pistoles.

Finally, suppose that the first player has gained one
point and the second player none. If they proceed to
play for a point the condition is that if the first player
gains it the players will be in the position first examined,
in which the first player is entitled to 56 pistoles; if the
first player loses the point each player is then entitled
to 32 pistoles. Thus if they do not wish to play, the first
player would say to the second, "Give me the ^2 pistoles

i8o MATHEMATICS

of which I am certain and divide the remainder of the 56
pistoles equally — that is, divide 24 pistoles equally." Thus
the first player will have the sum of 32 and 12 pistoles —
that is, 44 pistoles — and consequently the second player will
have 20 pistoles.

Thus the science which underlies the greatest business
of the twentieth century had its origin at the gaming
table. Pascal corresponded with his friend Fermat re-
garding the problem, and the subject continued to be de-
veloped to such an extent that Professor Todhunter's
'History of Probability,' from which the above problem is
taken, covers 624 pages.

The theorem at the base of Probability is thus stated
by James Bernoulli : "If a sufficiently large number of
trials is made, the ratio of the favorable to the unfavorable
events will not differ from the ratio of their respective
probabilities beyond a certain limit in excess or defect,
and the probability of keeping within these limits, how-
ever small, can be made as near certainty as we please by
taking a sufficiently large number of trials." The in-
verse problem of reasoning from known events to prob-
able causes is much more complicated. De Morgan thus
states the principle of the inverse probability: "When an
event has happened and may have happened in two or
three different ways, that way which is most likely to
bring about the event is most likely to have been the
cause."

Another principle, due to Bayes, is thus stated : Knowing
the probability of a compound event and that of one of
its components, we find the probability of the other by.
dividing the first by the second.

Michell more than a century ago gave a classic at-
tempt to apply the inverse theorem when he strove to
find the probability that there is some cause for the fact
that the stars are not uniformly distributed over the
heavens.

The following witty dictum is from Poinsot:

THE CALCULUS i8r

"After having calculated the probability of an error,
it is necessary to calculate the probability of an error in
the calculations."

One thus gets in an endless regression by in turn cal-
culating the probability of the correctness of the next
preceding calculation.

Poincare closed his lectures on the calculus of prob-
abilities with this skeptical statement: The calculus of
probabilities offers a contradiction in the terms itself
which serve to designate it, and if I would not fear to
recall here a word too often repeated, I would say that it
teaches us chiefly one thing — i.e., to know that we know
nothing.

An idea floating about in the minds of mathematicians
for centuries, most nearly approached in the method of
exhaustions used by Archimedes and in the method of
indivisibles of Cavaleri, pupil of Galileo, was, by aid of
the introduction of the notion of variable into geometry,
finally evolved almost simultaneously and independently
by the two greatest mathematicians of the period, Sir
Isaac Newton and Gottfried Wilhelm Leibnitz, and has
become the mighty engine of analysis, the first and only
mathematical subject to be dignified by the article "The,"
The Calculus.

This subject is based upon two fundamental and com-
paratively easily understood operations: the direct opera-
tion, Differentiation, and its inverse, Integration. A few
preliminary ideas are necessary.

A variable quantity is said to have a limit when it ap-
proaches a constant quantity in such a way that the dif-
ference between the variable and the constant quantity
can be made to become and remain less than any pre-
viously assigned value. The constant quantity is called
the limit of the variable. The condition is very often
added that the variable never actually reaches its limit, but
this is not necessary and very much narrows the applica-
tion of the notion. Starting with the number i, add to

382 MATHEMATICS

it its one-half, and continue the process indefinitely, each
time adding one-half of the next preceding addition, thus:

i + 72 + y4 + 1/8 + y1e+ • . •

It is evident that this sum never reaches 2, but may, by
proceeding far enough, be made to differ from 2 by as
small a number as we please.

Inscribe in a circle a regular polygon; take the mid-
point of each arc and join it with straight lines to the
two adjacent vertices of the polygon. A new polygon is
formed with double the number of sides of the original.
Continuing indefinitely, a polygon may be formed which
in area and perimeter differs from the circle by as little
as we please ; but the circle is never actually reached.

A quantity which aproaches zero as a limit is called an
infinitesimal. An infinitesimal is not necessarily an ex-
ceedingly small quantity; the smallness is not the im-
portant matter, but the fact that it can be made small.

Zeno's paradox of Achilles and the tortoise rested upon
the consideration of infinitesimals. Achilles was a cer-
tain distance behind the tortoise and attempting to over-
take it. Zeno argues that he can never do so, for, says
he, while Achilles travels half the distance between them
the tortoise has traveled a certain distance ; while Achilles
is traveling half the remaining distance the tortoise has
moved forward, etc. If these half distances were traveled
in finite intervals of time Zeno's argument would be cor-
rect. But the intervals of time are approaching zero
as well as the distances.

The differential calculus is based on finding the limit
of the ratio of two infinitesimals. Suppose a train travels
without stop from A to B, a distance of ioo miles, in ioo
minutes, and it is required to find its speed. One says a
mile a minute, but the train started from rest at A and
comes to rest at B, whence there are points at which the
speed is less than that given and at other points greater,
so that the speed assigned is not the speed at every point,
but what might be called an average speed. Suppose

THE CALCULUS 183

it is required to find the speed at a particular point, C; one
would proceed in this manner: Measure a distance of say
1,000 feet along the track of which C is the middle point;
time the train over this distance. The ratio of the dis-
tance to the time is the speed or rate, but it cannot be said
that this is the rate at C; it is an average rate over the
1,000 feet. Take a shorter distance, say 500 feet ; the ratio
of this shorter distance to the shorter time is more nearly
the rate at C than the former. Continue this process, and
the ratio of the distance to the time as each becomes in-
definitely small comes nearer and nearer to the exact rate
at C. If the motion of the train was subjected to a law by
which the limit of this ratio could be found, that limit
would be the rate at C. Differentiation is this process of
finding the limit of the ratio of two infinitesimals that are
mutually dependent.

A geometric example will be given.

It is required to find the direction in which the point
moves which generates the curve in the figure as it passes
through the particular position, P. This direction will be
along a tangent line, PR, since if the point were to con-
tinue in the direction in which it is moving at P, it would
move in a straight line tangent to the curve at P. Take a
second point, P', on the curve and pass a line through P
and P'. Now if P' moves along the curve toward P this
line swings around toward the limiting position PR. The
direction of PP' is fixed by the angle MPP', of which the
tangent is MP'/PM. As P' approaches P, both MP' and
PM approach zero, but they have a limiting ratio which is
equal to MN/PM, or the tangent of the angle MPN.

The mode of applying this operation algebraically is
quite simple. The coordinates of P are given, say (xx yx).
A second point, P', is chosen with coordinates (xx -J- h, yx
-f- k) and subjected to the condition that P' lies on the
curve. This is done by finding the relation of h and k
by substituting xx -\- h and y1 -j- k in the equation of
the curve in place of xx and yr The limit of the ratio

184 MATHEMATICS

of h = PM and k = MP' is then found as h and k ap-
proach zero.

In the parabola y2 = 8 x, find the direction or slope of
the tangent at point P whose coordinates are 2 and 4.
Take P/ (2 + h, 4 -|- k), a point on the curve. Since it
lies on the curve these coordinates must satisfy the equa-

tion of the curve. Putting 2 '-f- h for x and 4 + k for y
one has

(4 '+ k)2 = 8 (2 + h)
or 16 + 8 k '+ k2 = 16 + 8 h
or 8 k + k2 = 8 h
k k 8

Solving for — one has — = . As k and h ap-

h h 8 + k

proach zero together their ratio becomes 1, or the tangent

THE CALCULUS 185

of the angle which the line of direction makes with the
X-axis is 1, from which the angle may be found, by con-
sulting a table of tangents, to be 450, or the line which is
tangent to the parabola at the point (2, 4) makes an angle
with the X-axis of 450.

d

The sign of the operation of differentiation is — .

dx

The inverse operation, or integration, may be looked at
from two viewpoints. If one chooses to consider it as
simply the inverse operation, in order to perform it it
would only be necessary to take cognizance of the steps
in the direct process and reverse them. This would seem
to be a very simple matter, but in practice frequently be-
comes extremely diificult or impossible. The second phase
of integration is that of a summation of infinitesimals,
y = f (x) is the equation of a curve; if y is differentiated
with respect to x, the result is a new function of x, say

d dy

X. Then — = y or — = X, from which dy = X dx. This

dx dx

X being a function of x if plotted gives a curve as in the
figure.

The y of any point of the curve is found by putting the
corresponding value of x in the equation y = X (x), as
x gives y, x2 gives y2, etc.

In dy = X dx, take dxx = the interval (xx x2) and let
(Xj x2) = (x2 x3) = (x3 x4), etc.

Then for x = xv dyx = yt X dxi = y2 X (xi xa) = area
of rectangle xt R.

For x = x^ dy2 = y2 X dx = Y* X (x2 xs) = area °f
rectangle x2 S.

For x = x3, dy3 = y3 X dx = y3 X (x3 XJ = area of
rectangle x3 T.

Now if dx and dy each be made to approach zero and
the sum of the dy's be taken to find y, this sum will be
equal to the sum of the areas of these rectangles, as each

i86

MATHEMATICS

rectangle has its base diminished toward zero. When this
occurs the small shaded triangles approach zero and the
sum of the rectangles approaches the area bounded by the
curve, the X-axis, yi, and y*.

This is written y = S Xdx = area APQB.

Where S means the sum of all terms of the form x,
«dx as dx approaches zero.

If X be placed equal to Y and the curve plotted as

above, and also y = f (x), the relations of the two curves
is that the ordinate of any point of the second curve indi-
cates the area under the first curve from a chosen point
on the curve to the point for which the ordinate is taken.

When integration is regarded as above as a summation
the sign ^ is sometimes used, altho it is customary to
write the usual sign of integration S.

With the invention of the Analytic Geometry and the
Calculus, modern mathematics begins. Speaking of its
development from the date 1758, which closes the period

THE CALCULUS 187

covered by the third volume of Moritz Cantor's 'Geschichte
der Mathematik,' Professor Keyser says : "That date, how-
ever, but marks the time when mathematics, then schooled
for over a hundred eventful years in the unfolding won-
ders of Analytic Geometry and the Calculus and rejoic-
ing in the possession of these the two most powerful
among instruments of human thought, had but fairly en-
tered upon her modern career. And so fruitful have been
the intervening years, so swift the march along the myriad
tracks of modern analysis and geometry, so abounding and
bold and fertile withal has been the creative genius of the
time, that to record even briefly the discoveries and the
creations since the closing date of Cantor's work would
require an addition to his great volumes of a score of vol-
umes more."

And throughout all this wonderful growth nothing is.
lost or wasted, the achievements of the old Greek geome-
ters are as admirable now as in their own days, and they
remain the eternal heritage of man.

THE FOUNDATIONS OF

MATHEMATICS

PROFESSOR CASSIUS J. KEYSER

ADRAIN PROFESSOR OF MATHEMATICS,
COLUMBIA UNIVERSITY

THE FOUNDATIONS OF
MATHEMATICS

A traditional conception, still current everywhere ex-
cept in critical circles, has held mathematics to be the
science of quantity or magnitude, where magnitude in-
cluding multitude (with its correlate of number) as a
special kind, signified whatever was "capable of increase
and decrease and measurement." Measurability was the
essential thing. That conception of the science was a
very natural one, for magnitude did appear to be a sin-
gularly fundamental notion, not only inviting, but de-
manding, consideration at every stage and turn of life.
The necessity of finding out how many and how much
was the mother of counting and measurement, and Math-
ematics, first from necessity and then from pure curiosity
and joy, so occupied itself with these things that they
very plausibly came to seem its whole enjoyment.

Nevertheless, numerous great events of a hundred
years have been absolutely decisive against that view.
For one thing, the notion of continuum — the "Grand
Continuum," as Sylvester called it — that great central
supporting pillar of modern Analysis, has been con-
structed by Weierstrass, Dedekind, Georg Cantor, and
others, without any reference whatever to quantity, so
that number and magnitude are seen to be more than
independent — they are radically disparate. When the at-
191

192 FOUNDATIONS OF MATHEMATICS

tempt is made to correlate the two, the ordinary concept
of measurement as the repeated application of a constant
finite unit, undergoes such refinement and generalization
through the notion of Limit, or its equivalent, that count-
ing no longer avails, and measurement retains scarcely
a vestige of its original meaning. And when we add
the further consideration that non-Euclidean geometry —
primate among the emancipators of the human intellect —
employs a scale in which the unit of angle and distance,
tho it is a constant unit, nevertheless appears, from the
Euclidean point of view, to suffer lawful change from
step to step of its application, it is seen that to retain
the old words, and call mathematics the science of quan-
tity or magnitude, and measurement, cannot be accepted
as telling either what the science has actually become
or what its spirit is bent upon.

Moreover, the most striking measurements, as of the
volume of a planet, the weight of a sunbeam, the growth
of cells, the valency of atoms, rates of chemical change,
the penetrative power of radium emanations, are none of
them done by direct repeated application of a unit or by
any direct method whatever. They are all of them ac-
complished by one form or another of indirection. It
was perception of this fact that led the famous philoso-
pher and respectable mathematician, Auguste Comte, to
define mathematics as "the science of indirect measure-
ment." But the thought is not yet sufficiently deep or
comprehensive. For there is an immense range of admit-
tedly mathematical activity that is not in the least con-
cerned with measurement, whether direct or indirect.
Consider, for example, that splendid creation of the nine-
teenth century, known as Projective Geometry. Here is
a boundless domain of countless fields, where reals and
imaginaries, finites and infinites, enter on equal terms,
where the spirit delights in the artistic balance and sym-
metric interplay of a kind of conceptual and logical coun-
terpoint— an enchanted realm where thought is double

FOUNDATIONS OF MATHEMATICS 193

and flows throughout in curiously winding but parallel
streams. In this domain there is no concern with num-
ber or quantity or magnitude, and metric considerations
are entirely absent or completely subordinate. The fact,
to take a simplest example, that two points determine
a line uniquely, or that the intersection of a plane and
a sphere is a circle, or that any configuration whatever
— the reference is here to ordinary space — presents two
reciprocal aspects according as it is viewed as an ensemble
of points or as a manifold of planes, is not a metric fact
at all: it is not a fact about size or quantity or magni-
tude of any kind. In this region of thought it was posi-
tion, rather than size, that seemed to some the central
matter, and so it was proposed to call mathematics the
science of measurement and position.

The conception, thus mightily expanded, yet excludes
many a mathematical realm of vast extent. Consider,
for example, that limitless class of things known as opera-
tions— limitless alike in number and in kind. Now it so
happens that there are many systems of operations such
that any two operations of a given system, if they be
thought as following one another, together thus produce
the same effect as some single operation of the system.
Such systems are infinitely numerous, and present them-
selves on every hand. For a simple illustration, think of
the totality of possible straight motions in space. The
operation of going from point A to point B, followed,
by the operation of going from B to point C, is equivalent
to the single operation of going straight from A to C.
Thus it is seen that the system of such operations is a
closed system: that is, combination of any two of the
operations yields a third one, not without, but within the
system. The great notion of Group, thus simply exem-
plified, tho it had barely emerged into consciousness a
hundred years ago, has meanwhile become a concept of
fundamental importance and prodigious fertility. It not
only affords the basis of an imposing mathematical doc-

194 FOUNDATIONS OF MATHEMATICS

trine — the Theory of Groups — but therewith serves also
as a bond of union, a kind of connective tissue, uniting
together a large number of widely dissimilar doctrines
as organs of a single body. But — and this is the point
to be noted here — the abstract operations of a group of
operations, tho they are very real things, are neither mag-
nitudes nor positions.

This way of trying to come at an adequate conception
of what mathematics is, namely, by attempting to charac-
terize in sucession its distinct domains, or its varieties of
subject matter, or its modes of activity, in the hope of
finding a common definitive mark, is not likely to prove
successful. For it demands an exhaustive enumeration,
not only of the fields now occupied by the science, but
also of those destined to be conquered by it in the future,,
and such an achievement would require a prevision that
none may claim.

Fortunately, there are other paths of approach that
seem more promising. Every one has observed that math-
ematics, whatever it may be, possesses a certain mark,
namely, a degree of certainty not found elsewhere. So
it is, proverbially, the exact science par excellence. Ex-
act, no doubt, but in what sense? An excellent answer
is found in a definition given about one generation ago
by a distinguished American mathematician, Professor
Benjamin Peirce: "Mathematics is the science which
draws necessary conclusions." This formulation is of
like significance with the following, yet finer, mot, by that
scholar of Leibnizian attainment and brilliance, Professor
William Benjamin Smith: "Mathematics is the universal
art apodictic." These statements, tho neither of them
is adequate or final, are both of them telling approxima-
tions, wondrously penetrating insights, at once foreshad-
owing and neatly summarizing for popular use, the epoch-
making thesis established mainly by the creators of mod-
ern logistic, namely, that mathematics is included in, and
in a profound sense may be said to be identical with,

FOUNDATIONS OF MATHEMATICS 195

Symbolic Logic. Observe that the emphasis falls on the
quality of being "necessary"; that is, correct logically.,
or valid formally.

But why are mathematical conclusions correct? Is it
that the mathematician has a reasoning faculty different
' in kind from that of other men ? By no means. What,
then, is the secret? Reflect that conclusion implies pre-
mises, that premises involve terms, that terms stand for
ideas, concepts or notions, and these latter are the ulti*
mate material with which the spiritual architect, called
the Reason, designs and builds. Here, then, one may
expect to find some light. The apodictic quality of mathe-
matical thought is not due to any special kind of faculty
peculiar to the mathematician, nor to any peculiar mode of
ratiocination, but is rather due to the character of the
concepts with which he deals. What is that distinctive
characteristic? The answer is: precision and complete-
ness of determination. But how comes the mathema-
tician by such precision and completeness? There is
no mystery or trick involved: some concepts admit of
such precision and completeness, others do not — at least
not yet; the mathematician is one who deals with those
that do. The matter, however, is not so simple as it
may now seem, and the attentive consideration of the
reader is invited to what is yet to follow.

The Two Movements of Logico-mathematical Thought.
— The foregoing thesis, which will be more narrowly
examined in the latter part of this article, is the
joint result of two modern movements of thought, which
have had separate origins, have followed separate paths,
and, having been carried on by two distinct and even
alien groups of investigators, have recently converged, to
the astonishment of both groups, upon the thesis in ques-
tion. One of these movements originated at the very
center of mathematics itself, and may be appropriately
designated as the critico-mathematical movement. The
other, which may be called the logistical movement, took

196 FOUNDATIONS OF MATHEMATICS

its rise in other interests and in what seemed to logicians
and mathematicians alike to be a very different and
even a scientifically alien field — the interests and the field
of what has come to be known as Logistic or Symbolic
Logic.

The Critico-mathematical Movement. — For more than
a century after the inventions (i.e., the discoveries) of
Analytical Geometry by Descartes and Fermat, and the
Infinitesimal Calculus by Leibniz and Newton, mathema-
ticians devoted themselves almost riotously to application
of these powerful instruments to problems of physics,
mechanics and geometry, without much concerning them-
selves about the nicer questions of fundamental princi-
ple, logical cogency and precision of concept and argu-
mentation. In the latter part of the eighteenth century
the efforts of "the incomparable Euler," of Lacroix, and
others, to systematize results, served to reveal in a start-
ling way the necessity of improving foundations. Con-
structive work was not, indeed, arrested by that dis-
closure. On the contrary, new doctrines continued to
rise and old ones to expand and flourish. But a new
spirit had begun to manifest itself. The science became
increasingly critical as its towering edifices more and
more challenged attention to their foundations. Mani-
fest already in the work of Gauss and Lagrange, the
new tendency, under the powerful impulse and leadership
of Cauchy, rapidly developed into a momentous move-
ment. The Calculus, while its instrumental efficacy was
meanwhile marvelously improved, was itself advanced
from the level of a tool to the rank and dignity of a sci-
ence. The doctrines of the real and of the complex,
variable were grounded with infinite patience and care,
so that, owing chiefly to the critical constructive genius
of Weierstrass and his school, that stateliest of all the
pure creations of the human intellect — the Modern The-
ory of Functions, with its manifold branches — came to
rest on a basis not less certain and not less enduring

FOUNDATIONS OF MATHEMATICS 197

than the very integers with which we count and tell the
number of coins in the coffer or cattle in the field. The
movement still sweeps on, not only extending to all the
cardinal divisions of Analysis, but, through the agency
of such as Lobachevski and Bolyai, Grassmann and Rie-
mann, Cayley and Klein, Hilbert and Lie, Peano, Pieri
and Pasch, recasting the foundations of Geometry also.

In the light of all this criticism of mathematics by
mathematicians themselves, the science assumed the ap-
pearance of a great ensemble of theories, competent no
doubt, interpenetrating each other in a wondrous way, yet
all of them distinct, each built up by logical processes on
its own appropriate basis of pure hypotheses, or assump-
tions, or postulates. As all the theories were thus seen
to rest equally on hypothetical foundations, all were seen
to be equally legitimate; and doctrines like those of Qua-
ternions, non-Euclidean Geometry and Hyperspace, for
a time suspected because based on postulates not all
of them traditional, speedily overcame their heretical
reputations and were admitted to the circle of the lawful
and orthodox.

The Logistical Movement. — It is one thing, however,
to deal with the principal divisions of mathematics sever-
ally, underpinning each with a foundation of its own;
as, for example, the theory of the cardinal numbers (the
positive integers) was assumed as the basis for the up-
building of function theory. That, broadly speaking, was
the plan and the effect of the critical movement above
sketched. But it is a very different and a profounder
thing to underlay all the divisions at once — both those
that are and those that are yet to be — with a simple foun-
dation, with a foundation that shall be such, not merely
for the divisions but for something else, distinct from
each and from the sum of all, namely, for the organic
whole, the science itself, which they constitute. It is
nothing less than that achievement — the founding, not
of mathematical branches, but of mathematics — which,

198 FOUNDATIONS OF MATHEMATICS

unconsciously at first, consciously at last, has been the
aim and destined goal of the logistical movement —
research in symbolic logic.

The advantage of employing symbols in the investi-
gation and exposition of the formal laws of thought is
not a recent discovery. As every one knows, symbols
were thus employed to a small extent by the Stagirite
himself. The advantage, however, was not pursued; be-
cause for two thousand years the eyes of logicians were
blinded by the blazing genius of the ''master of those
that know." With the single exception of the reign of
Euclid, the annals of science afford no match for the
tyranny that has been exercised by the logic of Aristotle.
Even the important logical researches of Leibniz and
Lambert, and their daring use of symbolical methods, were
powerless to break the spell. It was not till 1854, when
George Boole, having invented an algebra to trace and
illuminate the subtle ways of reason, published his sym-
bolical 'Investigation of the Laws of Thought,' that the
yet advancing revolution in logic really began. Altho
it was neglected for a time by logicians and mathema-
ticians, it was this work of Boole, who was both logician
and mathematician, that inspired and inaugurated the
scientific movement now known and honored throughout
the world under the name of Symbolic Logic. Under the
leadership of C. S. Peirce in America, of Bertrand Rus-
sell in England, of Schroder in Germany, of Couturat in
France, and of Peano and his disciples and peers in Italy
— supreme histologist of the human intellect — the deeps
of logical reality have been explored in the present gen-
eration as never before in the history of the world. Not
only have the foundations of the Aristotelian logic — the
Calculus of Classes — been recast, but side by side with
that everlasting monument of Greek genius there rise
to-day two other structures, fit companions of the ancient
edifice, namely, the Logic of Relations and the Logic of
Propositions.

FOUNDATIONS OF MATHEMATICS 199

And now the base of this triune organon — the Calculus
of Classes, the Calculus of Propositions, the Calculus of
Relations — is surprising in its seeming meagerness, for
it consists of a score or so of primitive propositions —
the principles of logic — and less than a dozen primitive
notions called logical constants. Yet more surprising,
however, is the fact — justly described as "one of the
greatest discoveries of our age" — that this foundation of
logic is the foundation of mathematics also. So one may-
say: Symbolic logic is mathematics, mathematics is sym-
bolic logic; the twain are one.

The Thesis. — The thesis, accordingly, which it is the
purpose of the following paragraphs to explain with some
detail, is this: All mathematical notions are definable
directly or indirectly in terms of a few undefined or prim-
itive notions (called logical constants), and in mathemati-
cal argumentation there enter as fundamental not more
than about twenty undemonstrated or primitive proposi-
tions (called principles of logic).

What these primitives are will be seen presently. It
is to be at once mentioned and to be constantly borne in
mind that, if nothing be assumed, nothing can be deduced.
Accordingly, in mathematics, as in any other science, the
ideas that occur fall into two classes, the undefined and
the defined; and the propositions fall into two classes, the
undemonstrated and the demonstrated. In any case, the
primitives — the undefined and the undemonstrated — are,
to some extent, a matter of arbitrary choice and con-
venience. Simplicity is desirable, but not essential. What
is necessary is that the set of notions chosen for primi-
tives must be such that all other ideas of the science in
question must be definable in terms of them; and what-
ever system of propositions be chosen for primitives must
be such that all the other propositions of the science are
demonstrable in terms of them. The set of primitive,
propositions must be compatible among themselves, and
it is desirable, though not necessary, that the system

200 FOUNDATIONS OF MATHEMATICS

should be non-tautological or irreducible; that is, that
none of them be logically deducible from the others. The
primitives contemplated in the foregoing thesis consti-
tute the foundation of modern logic. It is to be shown
that no new primitives are required in mathematics. This
done, it follows that mathematics, instead of being a sci-
ence that merely uses logic, is really a prolongation of it
— a proper part, and, indeed, the principal part, of the
superstructure of logic. If, then, an edifice includes both
the basal masonry and that which is built upon it — and
such appears to be the better use of the term — the pro-
priety of identifying mathematics and logic is sufficiently
evident.

It remains a moot question which of the three above-
mentioned branches of modern logic, if any one of them
is entitled to the distinction, is logically prior to the oth-
ers. As, however, discourse would seem to be quite im-
possible without propositions, in the following sketch we
adopt the obvious recommendation of common sense, and
begin with the calculus or logic of propositions. The
set of primitive notions and propositions here presented
is that which at present seems most likely to be finally
adopted with least modification. Tho it is the result of
the thought of numerous investigators, it may be called
the Peano-Russell system, as suggesting the two men who
have done most to produce it.

The Logic or Calculus of Propositions. — In this logic,
besides the notion of truth, which remains undefined and
constantly employed, the primitive ideas are two: (i)
material implication and (2) formal implication. The
notion of implication is not defined. We know, however,
that it is a relation that, when it is found, is found to
subsist between propositions. The idea of proposition is,
however, defined. It is, namely, any thing that is true
or fake, or any thing that implies any thing. It is im-
portant to distinguish between a genuine proposition, as
Xerxes was a soldier, from what has merely the form

FOUNDATIONS OF MATHEMATICS 201

of a proposition, as y was a soldier, this last being, as it
stands, neither true nor false. Such forms as the last,
containing a variable (y or some other), are known as
prepositional functions, the notion being one of the primi-
tives of the logic of classes. The distinction between
material and formal implication is to be acquired very
much as a child learns to distinguish cats from dogs.
And the very young logician often confounds them. For
one thing, material implication subsists only between gen-
uine propositions, while formal implication is the kind
that holds between propositional functions. Thus, 'Xerxes
was a soldier implies Xerxes was a man' is an example
of material implication, but 'A was a soldier implies A
was a man' is an example of formal implication. If, in
the last, one replaces the variable A by some constant,
as Columbus or Cesar, the function is replaced by a
proposition, and formal by material implication. In ac-
tual discourse, as it runs in the world, the distinction in
question is often disguised. If p and q are propositions,
then the proposition, p implies q, asserts a material im-
plication, and means that either q is true or p is false.
The proposition — if 2 is 4, 5 is 10 — states a material
implication, but the implication in the statement — if x
is twice x, any multiple of x is twice that multiple — 1
is formal. To borrow Mr. Russell's illustration: "The
fifth proposition of Euclid follows from the fourth : if the
fourth is true, so is the fifth; while if the fifth is false,
so is the fourth. This is a case of material implication,
for both propositions are absolute constants. . . . But each
of them states a formal implication. The fourth states
that if x and y be triangles fulfilling certain conditions,
then x and y are triangles fulfilling certain other condi-
tions: and the fifth states that if x is an isosceles triangle,
x has the angles at the base equal." [Cf. Russell's 'Prin-
ciples of Mathematics,' Vol. I, p. 14.]

The primitive propositions of propositional logic are ten
in number, and are as follows :

202 FOUNDATIONS OF MATHEMATICS

(i) 'p implies q' implies 'p implies q';

(2) 'p implies q' implies 'p implies p' ;

(3) 'p implies q implies 'q implies q';

(4) If p implies q and if p is true, then p may be
dropped and q asserted;

(5) 'p implies p and q implies q' implies 'pq implies p' —
the expression "p and q" being denoted by the symbol pq ;

(6) 'p implies q and q implies r' implies 'p implies r';

(7) 'q implies q and r implies r and p implies (q im-
plies r)' implies 'pq implies r';

(8) 'p implies p and q implies q and pq implies t' im-
plies *p implies (q implies r)';

(9) 'p implies q and p implies r' implies 'p implies qr' ; and

(10) 'p implies p and q implies q* implies *('p implies q'
implies p) implies p'.

Experience has shown that it is in various ways advan-
tageous without compensating disadvantages to reduce all
matter, whenever it is possible, to symbolic form. In case
of the foregoing primitives such reduction is readily ac-
complished by employing the symbol 0 (an inverted c)
to denote the word "implies," and by using periods or
dots in place of the word "and," as well as to indicate
the relative ranks of the various copulas o of a same
proposition. A very little practice suffices to enable one
both to translate into symbolic forms and to interpret
them. Thus, in symbolic form the primitives in question
stand as follows :

(1) p 0 q .0 . p d q;

(2) p 0 q . D . p d p;

(3) poq.o.qoq;

(4) If p 0 q, and if p be true, p may be dropped and q asserted ;

(5) p Dp . q D q . D . pq 3 p;

(6) pDq.qor.D.por;

(7) qDq.ror.pD.qDr-.D.pqDr;

(8) p o p . q d r . pq 3 r . o : po.qor;

(9) p d q . p d r . d . p o qr;

(10) p3p.q3q.3:.p3q.3p:3p.

FOUNDATIONS OF MATHEMATICS 203

Of these, (1) means that poq is a proposition; (2)
means that what implies something is a proposition; (3)
means that what is implied is a proposition; (4) is pe-
culiarly interesting as illustrating the limits of formalism
— it does not admit of symbolic statement, a fact not to
surprise or mystify since it is a priori obvious that dis-
course is essentially prior to symbolism and is necessary
to tell the meaning of it. The meaning of (4) may be
made clear by a familiar example. If Socrates is a man,
and if all men are mortal, then Socrates is mortal. Let
the two premises be granted true, how justify the asser-
tion of the conclusion as a true proposition — to be hence-
forth so taken? The answer is (4) — an exceedingly
subtle principle introduced into logic by Peano. Cou-
turat calls it the principle of deduction. (5) means that
the joint assertion of two propositions, p and q, implies
the assertion of the former; (6) is evidently the familiar
principle of the syllogism; (7) states that, if a certain
proposition implies that a second one implies a third, then
the third is implied in the joint assertion of the other
two. Thus, the proposition — Socrates lived in Athens — -
implies that, if Athens was then a city of Greece, the
population of Greece once contained a philosopher. Now
all this, says (7), implies that the proposition — the popu-
lation and so on — is implied in the joint assertion of the
two propositions: Socrates lived in Athens; Athens was
then a city of Greece. (8) is the converse of seven; (9)
means that, if a proposition implies each of two proposi-
tions, it implies both of them; that is, that the assertion
of the first carries implicitly the joint assertion of the
other two. The reader can easily illustrate. Finally,
(10) tells us that if p is implied by the proposition, p
implies q, then p is implied by the proposition that 'p im-
plies q' implies p.

The reader will have observed that the foregoing prin-
ciples differ in respect to simplicity and obviousness. He
must be reminded that they were not selected because

204 FOUNDATIONS OF MATHEMATICS

they were simple or obvious, but because they were found
to be expedient. It is their serviceability that recom-
mends them. They shine in their agency and use.

That the primitive propositions are true propositions
the reader may convince himself by means of a test now
to be explained. The proposition, p implies q, means that
q is true or p is false and nothing more. This consid-
eration readily serves to justify the remarkable state-
ment: in respect to material implication, every false
proposition implies all propositions, and every true propo-
sition is implied by every proposition. Let us now apply
this proposition as a criterion to test the truth of one
of the primitives, say (8). Suppose, first, that p, q, r are
all true. Then qor is true, hence po . qor is true, and
hence (8) is true. Next suppose p is false, and q and r
true. ThenpD . qor is true, and hence (8) is true. Again,
suppose p true, q false, and r true. Then po . qor is true,
and hence (8) is true. Once more, suppose p and q true,
and r false. Then pop is true, qcr is false, and hence the
joint assertion preceding the third dot is false; hence (8)
is true. A like result follows under all the other possible
suppositions respecting the elements p, q and r. And in
like maner for the remaining nine primitives.

The conception of a science in a state of perfection*
requires that all other notions entering the structure of
the propositional calculus be defined in terms of implica-
tion (and truth), and that all the other propositions of
that calculus be demonstrated as theorems by means of
the above-given primitive propositions. Among such
superstructural notions and theorems are the following
cardinal ones:

The logical product of two propositions, p and q, is
their joint assertion, and is symbolized by p^q or simply
by pq. In terms of implication and truth, the definition is:
if p implies p and if q implies q — i.e., if p and q are propo-
sitions— pq signifies that r is true if p implies that q im-
plies r.

FOUNDATIONS OF MATHEMATICS 205

The logical sum of p and q is denoted by writing pwq.
It is a proposition s implied by p and by q and implying
every proposition that is implied both by p and by q.
The sum of p and q is the same as the disjunction or
alternation, p or q.

The negative, — p, of a proposition p is defined to be
such a proposition that, if r be any proposition whatever,
— p implies that p implies r.

Two propositions are said to be equivalent when and
only when each of them implies the other; that is, if
poq and qop, then p = q, and conversely.

The fact that the product of a proposition by the same
proposition is equivalent to the proposition — pp = p — is
called the law of tautology for propositional multiplica-
tion. And for addition it is the fact that PWP~P-

Cardinal among the theorems of the propositional cal-
culus are the following:

The product, p~ — p, of a proposition and its negative is
false — the law of contradiction.

The sum, pw — p, of a proposition and its negative is
true — the law of excluded middle or third.

The negative of the negative of a proposition is equiva-
lent to the proposition; that is, — ( — p)=p. Such is
the law of double negation.

Logical multiplication of propositions is commutative,
associative and distributive; that is, P~q = q~P, p~(q~r)
== (p~q)~r, and p~(q~r) = (p~q)~(p~r).

The same three laws hold for logical summation of
propositions.

The Calculus or Logic of Classes. — This logic is char-
acterized by three primitive or undefined ideas or no-
tions and by two primitive or undemonstrated proposi-
tions. The primitive ideas are: (1) Propositional func-
tion; (2) the relation of an individual to a class contain-
ing it; (3) and the notion expressed by the phrase such
that, or its equivalent in the same or another language.
The notion (1) is denoted by such symbols as <£(x), ^(x),

206 FOUNDATIONS OF MATHEMATICS

f (y), etc. It is familiar to everybody. Of it Mr. Russell
('Principles of Mathematics,' Vol. I, p. 19) says: "We
may explain (but not define) this notion as follows: <£(x)
is a propositional function if, for every value of x, <£(x)
is a proposition, determinate when x is given." Thus
x 4- 2 = o is a propositional function, for it yields a prop-
osition, true or false, on replacing the variable x by any
constant, as 1, 5, — 2, Socrates, Wednesday, or love. Again,
tan 450 = 1, tan 6o° = 4, are propositions, while tan x = 1,
tan x = x, are propositional functions. Once more, x is a
triangle, is a propositional function, but 'J°nn Jones is a
triangle' is a proposition. Primitive (2) is denoted by
the letter ; thus, to say that the individual k belongs to
the class a, we write ksa. The important distinction be-
tween the relation denoted by and that of part to whole
was first pointed out by Peano. To say that a class a is
a part of or is included in a class b, we write: aob, the
symbol 0 being that which in the logic of propositions
denotes "implies." Thus the syllogism, aob . xea . d . xfb,
means: the class a belongs (as a part) ,to the class b, the
individual x belongs to the class a, therefore the indi-
vidual x belongs to the class b. But if a, b, and x are
all classes, the syllogism is: aob . xoa . o . xob The third
primitive, such that, is denoted by the symbol 3 (inverse
of £). Thus, to say the ensemble of x-values that
render the function 0(x) a true proposition, or verify or
satisfy it, we write: X2[0(x)], which may be read "the
x's such that <£(x) is true."

The two primitive propositions of this calculus are as
follows :

(1) (p(x) is true when and only when x belongs to the
ensemble of terms satisfying <P(x).

(2) If <P(x) and ^"(x) are equivalent propositions for
all values of x, then the class of x's such that 0(x) is true
is identical with the class of x's such that ^"(x) is true.

FOUNDATIONS OF MATHEMATICS 207

These primitives may be stated symbolically as follows :

(1) Ks[x3(p(x)]ocp{k)]

(2)4>(x)=W(x) . 0: X3<p(-x) . =x?!Pr(x).

The chief among defined ideas and proved propositions
of class-logic are the following:

A class of terms is composed of the constants that sat-
isfy a propositional function.

A propositional function that is false for every value of
the variable in it defines a null-class.

An individual x is identical with an individual y if and
onlv if y belongs to every class that contains x; otherwise
x and y are diverse.

The class a is said to be included in the class b, and
then we write aob, when and only when every proposi-
tion of the form xga implies, for the same x, that x^b.

The classes a and b are said to be identical if each
includes the other.

A class a is said to exist when and only when the logical
sum of all propositions of the form x^a is true.

The logical product of two classes a and b is the class
of terms x such that the logical product of the two propo-
sitions, x£a, xeb, is true.

The logical sum of two classes a and b is the class of
terms x such that the logical sum of the two proposi-
tions, x«a, xeb, is true.

The logical product of a class c of classes is the class
of terms x such that ucc implies xau.

The logical sum of a class c of classes is the class of
terms x such that, if u«c implies u«k for all u's, then xfk
for all k's.

When, as often happens, it is necessary to distinguish
formally between a singular class (one having but one
term) and its term, it is customary to place the Greek
letter z before the symbol for the class. Thus, if a be a
singular class, za is its term. Also the inverse 1 of the
Greek letter, if placed before the symbol for a term, gives
a symbol for the singular class having that term for sole

208 FOUNDATIONS OF MATHEMATICS

term. Thus, if x be a term, ?x denotes the corresponding
singular class.

The laws of tautology for class multiplication and
summation are the facts that the logical product of a
class by the class is the same class, and the sum of a
class and the class is again just the class.

If we write x — ea for the negative of x« a, then the
negative, — a, of the class a, is defined by — a . = . x £
(x — £ a) ; that is the class, — a, is the class of terms x
such that x is an a is false.

The negative of the negative of a class is this class:!
— ( — a) = a, the law of double negation for classes.

The laws of commutation, association and distribution
are valid for the logical multiplication and addition of
classes; thus, if a, b, c be classes, then ab = ba, a-f-b =
b + a, a(bc) = (ab)c, a + (b + c) = (a + b) + c,
a(b + c)=ab + ac, a -f- (b + c) = (a + b) + (a + c).

The foregoing class-logic definitions in terms of ideas
in the propositional logic serve to exhibit the close con-
nection and interpenetration of the two logics. Their
parallelism, too, is striking. Thus to the propositional syl-
logism, 'p implies q and q implies r' implies 'p implies r/
corresponds the class syllogism; if a is included in b, and
b in c, then a is included in c. The parallelism is not,
however, thoro-going, and may not be incautiously em-
ployed. For example, if p, q and r denote propositions
and if a, b, c denote classes, then we have pqor = per woqor,
but not aboc = aoc wboc.

The Calculus or Logic of Relations. — We come now to
the latest (in point of development), the subtlest, the
profoundest, and, for mathematics, the most significant
division of modern logistic. Founded by Charles S.
Peirce upon the extensional view of relations — the view
that a relation consists of the class of couples between
which it subsists — elaborated and expounded on the same
view by Schroder ('Algebra der Logik'), the Calculus of
Relations was then refounded by Bertrand Russell, in

FOUNDATIONS OF MATHEMATICS 209

1900, upon the intensional view of relations, and by him
dressed in the garb of a slightly modified Peano symbol-
ism. It is this last theory, mainly due to Mr. Russell,
of which the following account is a sketch :

This logic is characterized by two primitive ideas and
eleven primitive propositions.

The primitive ideas are: (1) the notion of relation —
symbolized by R and written rel; (2) the notion of iden-
tity— denoted by the symbol |\

The primitive propositions are as follows:

(1) R being a relation, xRy means for all x's and
y's that x has the relation R to y.

(2) Given any R, there is a relation R' — called the
converse of R — such that xR'y is equivalent to yRx.

(3) If x and y be any two definite terms, there is a
relation that x has to y and that does not subsist between
any other couple of terms.

(4) If K be a class of relations, the logical sum of
the relations of K is a relation, where by logical sum is
meant the class of relations R such that, if an R relates
an x to a y, there is in K a relation R' relating that x to
that y and that, if an R' of K relates an x to a y, that x
is related to that y by an R.

(5) If K be a class of relations, the logical product of
the relations of the class is a relation, where by this
product is meant the class of relations R such that if an
R relate an x to a y, then each relation R' of K relates
that x to that y and that, if an x be related to a y by each
R' of K, that x is related to that y by one of the R's.

(6) If any term x is related to a term y by a relation
Ri, and if y is related to z by R2, x is related to z by a
relation R1R2, called the relative product of Ri and R2.

(7) The negative, — R, of a relation R is a relation,
where — R means that the proposition, x — Ry, is equiva-
lent to the proposition x is not related to y by R.

(8) The symbol (as employed in the class-logic) is,
or expresses, a relation.

210 FOUNDATIONS OF MATHEMATICS

(9) Identity (the primitive notion) is a relation,

(10) Any term x is identical with that term x.

(11) Identity implies identity.

If we denote the assertion that a thing exists by writ-
ing before its symbol the symbol 3 (inverse of the letter
E), denote the logical sum and product of a class K
of relations respectively by the symbols, W'K and ^'K,
and denote by $ the class of terms that may stand before
an R — i.e., its domain — and by g the codomain or class of
terms that may come after R, then the foregcing primi-
tive propositions may be written in symbollic form as
follows :

(1) R e rel . 0 : xRy . = . x has the relation R to y ;

(2) R e rel . d . 3 rel ~ R/ 3 (xR'y . = . yRx) ;

(3) 31 rel ~R 3 (§ = ix . § = [iy) ;

(4) W'K e rel ;

(5) ~'Ke rel;

(6) R2R2 erel;

(7) — Re rel;

(8) e e rel ;

(9) 1' e rel;

(10) x e rel;

(11) I'D 1'.

It will be observed that a relation has sense; that is,
xRy means to assert that R relates the antecedent x to
the consequent y, and not y to x.

The class of the antecedents is the domain of the rela-
tion; that of the consequents is the co-domain; and the
logical sum of the domain and the co-domain is the field of
the relation.

Relations admit of important classifications. Thus a
relation R is uniform if each of its antecedents has the
relation to one, and but one, of the consequents. A rela-
tion R is biuniform if R is uniform and its converse R
is also uniform. R is symmetric if xRy implies yRx; it is
non-symmetric if xRy and yRx are both true for some
but not all pairs of values of x and y; and asymmetric if,

FOUNDATIONS OF MATHEMATICS 211

when xRy is true, yRx is false. R is transitive if the
logical product of xRy and yRz implies xRz ; non-transi-
tive if the three statements are true for some but not all
triplets of values of x, y, z ; and intransitive if xRz is false
when xRy and yRz are both true. Thus the relation of
equality is both symmetric and transitive; the relations,
greater than and less than, are transitive but asymmetric;
the relation, implies, is non-symmetric but transitive; and
the relation £ is asymmetric and non-transitive.

A relation R is reflexive if, like equivalence, for ex-
ample, it holds between an x and that x.

The relation R is included in the relation R' if xRy
implies xR'y for all values of x and y; and R and R' are
equivalent if each includes the other.

Among the theorems that enter the logic of relations
the two following ones, which are converses of one an-
other, are specially noteworthy :

(1) The relative product of a relation and the converse
relation is a symmetric and transitive relation ;

(2) Every relation that is symmetric and transitive is
equivalent to the relative product of a uniform relation
and the converse relation.

The last states the principle of the so-called definition
"by abstraction."

The Thesis Justified. — A sketch of modern logic having
been premised, the above-stated thesis regarding the con-
nection of mathematics with symbolic logic remains now
to be justified by taking up serially the ideas upon which
the chief divisions of mathematics have been built up, and
presenting them in terms of the primitives (above given)
of logic. Conceiving mathematics as falling into Analysis
and Geometry, we may begin with the former, tho in this
connection some ideas, as that of order, belong as well
to geometry as to analysis. The reader should note that
all definitions are given directly or indirectly in terms
of the above-given logical ideas.

The Cardinal Theory of Cardinals. — The cardinal num-

212 FOUNDATIONS OF MATHEMATICS

bers may be defined either with or without use of the
notion of order, giving rise to two theories of the cardi-
nals, namely, the cardinal and the ordinal. It will be
instructive to present the cardinal theory first.

Two classes, a and b, are said to have the same cardi-
nal number when there is a biuniform relation whose
domain includes a and such that the class of consequents
of the terms of a is identical with b. It follows that
two null-classes have the same number. This is called
zero, and denoted by the symbol o. Plainly, too, two sin-
gular classes have the same number. It is called one, and
denoted by the symbol I. It is to be noted that we have
defined sameness of number of two classes, but have not
yet defined number of a (given) class. Two classes hav-
ing the same number are said to be equivalent. Now
equivalence is a reflexive, transitive and symmetric rela-
tion, so that, a class a being given, there is a class of
classes each equivalent to a and to any other class in the
class of classes. The number of the class a is defined to
be the class of classes each equivalent to a. Two classes
without a common term are said to be disjoint. If a and b
are two disjoint classes, and if oc and (5 are their cardinal
numbers, then the arithmetic sum of <x and (5 is the car-
dinal number y of the logical sum of a and b. The com-
mutativity (a + fi = ft + a) of arithmetic addition is evi-
dent in the fact that the notion of order does not enter the
definition of such addition. Arithmetic multiplication (of
cardinals) is definable as follows: Let k be a class of
disjoint classes of which none is a null class. The class
of classes formed by taking (to compose a class) one,
and but one, term of each of the classes k, is named the
multiplicative class of the classes k. The cardinal num-
ber of this multiplicative class is named product of the
cardinal numbers of the classes k. The notion of order
being absent, the validity of the commutative law ( <*/?
= /3a) is obvious. And the laws of distribution and asso-
ciation are readily shown to be valid.

FOUNDATIONS OF MATHEMATICS 213

It is noteworthy that in the foregoing there enters no
distinction of finite and infinite class or number, and that
the theory is applicable, therefore, alike to finite and to
infinite cardinals. A class is said to be infinite or finite
according as it contains or does not contain a part, or
sub-class, such that a biuniform relation (a one-to-one
correspondence) subsists or does not subsist between the
terms of the class (the whole) and the sub-class (the
part). And the number of a class is said to be infinite
or finite according as the class is infinite or finite.

The Ordinal Theory of Cardinals. — This begins by ad-
joining to the foregoing definitions of zero (o) and one
(1), the two definitions: (1) The successor of a cardinal
n is the cardinal n -f- 1, the arithmetic sum (already de-
fined in logical terms) of n and 1 ; (2) N is the class of
cardinals that belong to every class c that contains both
zero and the successor of every cardinal that it contains.
This last definition states the principle of mathematical
induction. It readily admits of proof that N is an infinite
class, but that all the cardinals in N are finite, so that,
unlike the cardinal theory, the ordinal theory of cardinals
applies only to finite cardinals. It is not difficult to estab-
lish the propositions that zero is in N ; that, if a is in N,
the successor of a is in N ; that, if a is in N, the successor
of a is not zero ; that, if the successor of a is identical with
the successor of b, a and b are themselves identical; and,
without using other than logical primitives, to erect the
entire arithmetic of the finite integers.

The Notion of Order. — The definition of this exceed-
ingly important notion is a notable achievement of recent
investigation. Whatever order is, it was noticed that it
might -be linear, any two terms of the ordered class being
the one before, the other after, with or without a term
between, the class so ordered being called an open series ;
or it might be circular, of which a term cannot be said
to be before or after another, but of which we are en
abled to say merely that a pair of terms, a, b, is separated

2i4 FOUNDATIONS OF MATHEMATICS

by a pair, c, d, if the four terms are arranged thus:
acbda . . . or adbca . . ., a class thus ordered being
described as a closed series. The sense of the disposition
ab is disregarded, so that ab and ba are the same; ac-
cordingly, a triplet of terms is essential to linear order;
thus abc (or cba) differs from acb (or bca), and enables
us to say that one of the terms is between the other two.
Similarly, disregarding sense, three terms cannot be in
circular order, for abca is then the same as acba. Hence
four terms are the element in case of circular order.

What order is has been ascertained by inductive study
of the various relations that generate order. These, which
reduce apparently to six distinct varieties, cannot be here
presented. It is found, however, that any order, no matter
by what relation it is generated, is generable by a transitive
asymmetric relation. That is to say, if we have any ordered
class of terms, the order, whatever it may be, is regardable
as being set up by some asymmetrical transitive relation
R, such that, x and y being any two terms of the class
xRy, or else yRx is true but one of them is false; that,
R being transitive, the logical product of xRy and yRz
implies xRz ; that the converse of R is also transitive and
asymmetric; and that, given any term x of the class the
remaining terms fall into two classes y and z such that
xRy and zRx; and thus, of any three terms, x, y, z of the
class, one of them, as y, is between the other two — i.e.,
xRy and yRz, or zRy and yRx. A simple example is that
of the class N of finite cardinals ordered by the relation
greater than. Another example is that of the class oc
points of a line of unit length extending from o to I, the
points o and I being both included; the points being
taken in their so-called natural order of increasing dis-
tance from o ; the order may be regarded as established by
the asymmetric transitive relation, farther from o.

Ordinal Numbers. — We are now prepared to define or-
dinal numbers, or types of order, which must not be con-
founded with the terms of the familiar series, first, sec-

FOUNDATIONS OF MATHEMATICS 215

ond, third, and so on* Two series, u and v, are said to
be like when there is between them a biuniform rela-
tion such that for every pair of terms ai, bi of u and
their correspondents a2, b2 of v, if ai precedes bi, aa pre-
cedes ba; or, the likeness may be affirmed of the two
relations by which the series u and v are generated. It is
noteworthy that likeness is to series or their generating
relations analogous to equivalence in case of classes. Like
equivalence, the relation of likeness is reflexive, sym-
metric and transitive.

The ordinal number, or order-type, of a series u is
the class of series each like to u. If a series be a
finite class, its ordinal number is uniquely determined by
its cardinal number, the two numbers obey the same laws
of operation and are (owing to the failure of man to
distinguish between them) denoted by the same symbol.
Thus the cardinal three and the ordinal three are both
denoted by 3; yet they are radically different things;
for the cardinal three contains, for example, the class
composed of the individuals a, b, c, but not the series
a, b, c as such ; while the ordinal three contains that series
and the distinct series b, a, c, among others. In the field
of infinites, the difference between the concept of ordinal
number and that of cardinal not only may, but must, be
observed. For the laws of operation are then no longer
the same for the two kinds of numbers. For cardinals,
whether finite or not, the commutative law of addition
holds without exception; not so, however, for ordinals.
For example, denote by <x the infinite ordinal number of
the endless series ai, a2, a3, . . . , and by 3 the ordinal
number of the series bi, b2, bs; then the ordinal number Of
the series bi, b2, bs, ai, a2, a3, . . . is naturally 3 -f- a ; that
of the series ai, a2, a3, . . . bi, b2, bs, is « + 3J ^mt: the last
two series are not similar, so that 3+a is not the same
number as a -f- 3 ; hence not all ordinals obey the commu-
tative law of addition. And so for other laws of opera-
tions. The calculus of infinite cardinals and the distinct

216 FOUNDATIONS OF MATHEMATICS

calculus of infinite ordinals are among the most beautiful
and inspiring creations of mathematics. Philosophers and
theologians have yet to learn to appreciate the significance
of these doctrines, both of which are due mainly to the
subtle creative genius of Georg Cantor, tho others have
made important contributions to their development and
refinement.

Rational Numbers. — Rational numbers, or fractions, are
defined to be certain relations between the integers or
cardinal numbers. This may be made clear as follows:
Let the small letters, a, b, c, d, e, . . . denote integers.
Suppose that ab = c, db = e, . . . It is obvious that to
b there corresponds a relation, conveniently denoted by
B, which consists in the fact that ab = c, db = e, ... .
Similarly, to any other integer, as m, there corresponds
a relation M such that pMq means that pm = q. Now
suppose that ab = cd ; then we may write ab = p, cd = p,
whence aBp and dCp. From the last follows pCd, while
from this and aBp follows aBCd. The compound relation
BC, uniquely determined by the integers b and c, is named
fraction, and denoted by the familiar symbol 6/c. All such
relations together constitute the class of fractions or so-
called rational numbers. Rational numbers having the
cardinal i for denominator are usually denoted by the
symbol for the numerator, and are thus made to appear
as cardinals. Cardinals, however, they are not, as is
evident by comparing definitions: a cardinal is a class;
a rational is a relation. Upon this relational basis the
entire theory of rationals is easily built up.

Positive and Negative. — It is to be noted and kept in
mind that cardinal numbers and rational numbers are
neither positive nor negative. Each of them is signless.
Numbers having sign (-j-or — ) are defined as follows:
If two integers are consecutive, there is a relation between
them, the same for every pair of consecutives, by virtue of
which one of them precedes and the other follows. De-
note this relation by R. Then, a and b being integers, the

FOUNDATIONS OF MATHEMATICS 217

proposition aRb means that a + I =b. The relation R is
asymmetric but intransitive. If aRb and bRc, then aRRc
or aR2c, and so on. The powers of R are also asymmetric
relations. The converse of Rp is ftp, that is (R)p; s
that aRps = sftpa, the left-hand member signifying simply
that a + P = s, and the right-hand member that s — p = a.
The relations Rp and Rp are defined to be respectively
the positive and negative integers, commonly denoted by
+ p and — p. Next let a, b, c, . . . denote rational num-
bers or fractions. Suppose that the sum of a and b is c,
then corresponding to b there is a relation B such that
aBc means that a-f-b = c, that mBn means m + b==n,
and so on. This relation B is defined to be a positive frac-
tion, and is denoted by -f- b. The converse relation B
is named negative fraction, is denoted by — b, and is such
that mBn means n — b = m. The reader should not fail
to discriminate the integer a and the positive integer -f- a ;
the former *is a class, the latter a relation. Similarly, the
fraction a and the positive fraction +-a are distinct : both
are relations, but the relations are by no means the same.
Real Numbers. — Consider the ensemble Ei of all the
rational numbers less than the rational number 1, and the
ensemble E2 of all rationals whose squares are less than
the rational 2. Each of the ensembles possesses the prop-
erties: it does not contain all the rational numbers; it
contains every rational number that is less than any ra-
tional whatever (any variable rational) contained by it;
that is, if it contains the rational x, it contains every
rational less than x; it contains no number greater than
all the other numbers in it. Any class of rational num-
bers that has the three properties stated is named segment
(of rationals). Given any segment, s, the class composed
of all other rationals may be conveniently denominated
cosegment of s (complement of s). A segment of ra-
tional numbers is called a real number, which is thus a
class. The real number Ei is named one, and denoted
by 1. The real number E2 is named square root of 2, and

218 FOUNDATIONS OF MATHEMATICS

denoted by the usual symbol. Segments fall into two
classes, according as their cosegments contain or do not
contain a minimum number, one that is smaller than every
other number in the cosegment. The segments, or reals,
of the latter kind are called irrationals. Those of the
former kind are commonly called rational numbers, tho
they are obviously very different from the rationals, mere-
ly corresponding to them. Thus the symbol 2, for ex-
ample, denotes the cardinal two, the positive integer two,
the rational two, and the real number two, all different
ideas manageable by the same laws of operation. The
theory of real numbers, as thus defined, turns out to be
identical with that arising from the usual definition of
reals, and has the advantage of not having to assume a
limit where there is none, as, for example, in case of the
foregoing segment E2. The notion of limit, -lot yet em-
ployed, will be defined in the following section.

The Concept of Continuum. — This most important con-
cept is definable in terms of order and without use of
metric or magnitudinal considerations. The process is
due to that primate among subtile thinkers, Georg Cantor.
Denote by V the order-type represented by the ensemble
of rational numbers taken in order of magnitude. Any
series of this type has the following three properties:
(1) It is denumerable; (2) it has neither a first nor a
last term; (3) it is compact; that is, between any two of
its terms there is another term of it. By calling it de-
numerable it is meant that a biuniform relation subsists
between its terms and the terms of the series 1, 2, 3, 4,
. . . That it is denumerable may be shown easily. Ar-
range the rationals in a series by beginning with 1/i, fol-
lowing this with those having 3 for sum of numerator and
denominator, these with the fractions having 4 for sum of
terms, and so on, omitting any fraction that is equal to a
predecessor in the series. The series is: 1/h %/s, 2/i, l/»,
*/i, V*, 2/3» "A V1' • • • > the fractions having same number
for sum of terms being arranged according to increasing

FOUNDATIONS OF MATHEMATICS 219

magnitude. It is now plain that we can correlate the first
term of the series with 1, the second with 2, the third with
3, and so on, so that each term gets paired with an integer,
and conversely; hence the series of rationals or any
other series of type rj is denitmerable.

A series of the type of the series 1, 2, 3, . . . is named
progression. A progression all of whose terms are terms
of a series V is called a fundamental progression of ?/;
an ascending progression of its terms follow in the same
order or sense as those of V, but descending if in the con-
trary sense. 'A class of terms belonging to a series is
said to have a limit x when and only when x immediately
follows (or precedes) the class but does not immediately
folow (or precede) any one term of the class. A funda-
mental progression of a series y has a limit x if x be in rj
and immediately follows (or precedes) all the terms of
the progression. Again, a series is said to be perfect when
and only when all of its fundamental progressions have
limits and all of its terms are limits of fundamental pro-
gressions.

These definitions premised, we are now prepared to
define continuum. A series is said to be continuous if it
is perfect and contains a series of type V. It admits of
proof that an ensemble that belongs to a perfect series,
is denumerable and has a term between every pair of
terms of the containing series, is of type V. Hence we
may say that a series S is continuous if it is perfect and
if it contains a denumerable class having a term between
every two terms of S. A standard example of a con-
tinuum is the class of the real numbers equal to or greater
than zero and equal to or less than 1. This continuum is
commonly represented by the class of points of a line seg-
ment of unit length, it being assumed that the series of
such points and the mentioned series of real numbers are
like. '

Multiple Series and Geometry. — The remainder of this
article, which aims at merely sketching modern thought

220 FOUNDATIONS OF MATHEMATICS

on the foundations of mathematics, will be devoted to
Geometry. For many centuries, indeed down to the early
part of the last century, the term geometry meant Euclid-
ean geometry, and the propositions constituting it — the
axioms and postulates, together with the theorems de-
duced therefrom — were regarded, not merely as a set of
assumptions and deductions from them, thus constituting
a coherent body of doctrine suspended in the intellectual
air, but as true statements about actual space. And so
geometry has often been said to be the science of space,
where "space" was used to denote actual or sensuous
space, and not, as in recent years, merely the ensemble
of elements, whether existent or not, about which geome-
try discourses. One of the Euclidean premises, however,
namely, the so-called parallel axiom, seemed to critical
.minds to be not sufficiently "self-evident," and yet baf-
fled all attempts (of which there is a vast literature, and
still increasing by occasional contributions of the ill in-
formed) to deduce it as a theorem from the other Euclid-
ean axioms. At length appeared the geometries of Lo-
bachevski and Bolyai, in which the axiom in question*
was denied. The fact that these geometries contradict
the Euclidean at many points (for example, regarding
the sum of the angles of a triangle) and are at the same
time both free from interior contradiction and from con-
tradictability by experimental measurement or other ex-
perience, lead first to the suspicion and then, through the
discovered possibility of manifold geometries each con-
sistent with itself but inconsistent with the others, to the
conviction that the attempt to describe space results in
an experimental science like physics or biology, that the
so-called geometry thus arising is but a branch of what
is commonly denominated applied mathematics (tho there
is, strictly speaking, no such thing as applied mathemat-
ics), and that geometry, regarded as a branch of mathe-
matics, is to be regarded and justified, not as a description
of actual space, but, like every other branch of mathe-

FOUNDATIONS OF MATHEMATICS 221

matics, as a hypothetico-deductive system. A given geom-
etry consists of certain assumptions A and certain theo-
rems T deducible from A. The truth of the geometry
resides in the implication of the theorems T by the as-
sumptions A, and not in their practical usability in the
business of the work-a-day world — not in any applications
to the concrete facts of the universe.

In recent years numerous memoirs on the foundations
of geometry have been produced by European and Ameri-
can mathematicians. A striking result of such many-sided
investigation is that the subject matter of what is called
geometry is multiple series ; that is, series of two or more
dimensions. These terms may be explained as follows:
A series si generated by an asymmetrical transitive rela-
tion R is said to be simple, no matter what the nature of
the terms of Si. Suppose, now, that each term of Si is
itself a simple series or an asymmetric transitive relation
(for the relation, and not the terms, is the essence of a
series). The class of all the terms in all the fields of the
terms of Si is said to be a series of two dimensions. Call
it sz. For an image, the reader may think of Si, as the
series of the lines of a plane that are parallel to a given
line. Each line (term of Si) is a simple series (asym-.
metric relation) of points. The plane s2 is the field of all
the points of all the lines of Si. Next suppose the terms
of S2 to be each of them an asymmetric transitive relation.
Thus arises a three-dimensional series Ss, the field of the
fields of the fields of the terms of Si. The process here
indicated, or its reverse, will, if continued, lead to the
concept of a series of n dimensions. It is noteworthy
that the ordinary complex numbers of the type x -f- iy,
where x and y are real numbers and i is the square root
of — i, constitute a double series, and that the result of
assigning to y the value zero is, contrary to customary
speech, not a real number.

Projective Geometry. — The study of such multiple se-
ries, or of the relations generating them, has yielded three

222 FOUNDATIONS OF MATHEMATICS

grand types of geometry: Projective, Descriptive and
Metric. These agree in the fact that they are concerned
with multiple series of what are called points. But the
terms of the series might as well be called roints, "slithy
toves," "wabes," or any other names, for, as will be seen,
"point" is to be merely the name of a class-concept, no
matter what, whose individuals satisfy certain relations
prescribed by the hypotheses or assumptions or postulates
or so-called axioms (all the terms are in use) that are
chosen for undemonstrated propositions of whatever ge-
ometry is being built up. In what respects the three grand
divisions differ fundamentally will appear in the sequel.
For each of the varieties in question there have been
found various systems of basal hypotheses, so that an
(undemonstrated) proposition of one system may be a
theorem, a proposition demonstrated on the basis of an-
other system serving for a basis of the same geometry.

The following system of basal assumptions for projec-
tive geometry is due to Pieri : T principii della Geometria
di posizione composti in sistema logico deduttivo' ['Memo-
rie della R. Accad. delle Scienze di Torino,' second series,
vol. XLVIII, 1898]. An analysis of the system is found
in Russell's 'Principles of Mathematics,' and also in Cou-
turat's 'Les Principes des Mathematiques.' The basis
upon which Pieri erects the beautiful edifice of projective
geometry consists of the following assumed (undemon-
strated) propositions:

I. Points form a class.

II. There is at least one point.

III. If a is a point, there exists a point other than a.

IV. If a and b are two different points, the straight
line ab is a class.

V. Each term of this class is a point.

VI. If a and b are two different points, the straight
line ab is contained in the straight line ba.

VII. If a and b are different points, a belongs to the
straight line ab.

FOUNDATIONS OF MATHEMATICS 223

VIII. If a and b are distinct points, the straight line ab
contains at least one point distinct from a and from b.

IX. If a and b are distinct points, and if c, a point of
the straight line ab, is distinct from a, then b is a point
of the straight line ac.

X. Under the hypothesis of IX, the straight line ac is
contained in the straight line ab.

XL If a and b are distinct points, there exists at least
one point not belonging to the straight line ab.

XII. If a, b and c are three non-collinear points, and
if a' is a point of be other than b and c, and b' a point of
ac other than a and c, then the straight lines aa' and bb'
have a point in common.

XIII. If a, b and c are non-collinear points, there exists
at least one point that does not belong to the plane abc.

XIV. If a, b, c are collinear points, their fourth har-
monic does not coincide with c.

XV. If a, b, c are three distinct points of a straight
line, then if d, a point of the line, be distinct from a and
from c, and does not belong to the segment abc, it belongs
to the segment bca.

XVI. If a, b, c are three distinct collinear points, then
if the point d belongs to both of the segments bca and
cab, it cannot belong to the segment abc.

XVII. If a, b, c are distinct collinear points, and if d
belongs to the segment abc, and e to the segment adc,
the point e belongs to the segment abc.

XVIII. If the segment abc is divided into parts X and
Y such that each of them contains at least one point and
that every point x of X precedes every point y of Y in
the order abc, there exists at least one point z of the seg-
ment abc such that every point of abc that precedes it
belongs to X and every point of abc that succeeds it
belongs to Y.

Some of these propositions plainly presuppose certain
definitions. These are now to be given, along with some
commentaries designed to indicate the spirit and course

224 FOUNDATIONS OF MATHEMATICS

of the author's thought. Certain diagrams, which the
reader may readily construct, tho they are not essential,
will serve to make clear. Such propositions as II and
III show that no more points are to be assumed than are
indispensable. The existence of others is to be proved.
Thus, in the matter of fundamental assumptions, William
of Occam's famous dictum is regulative: 'Entia non sunt
multiplicanda praeter necessitatem.' The meaning of IV
and V is that two points a and b determine a class of
points, named straight line, and denoted by ab, where by
"determine" is meant that, given any pair of points, there
is a certain definite relation R that holds between the
pair and a corresponding unique class of points. The
offices of a and b being indistinguishable, it follows from
VII that b, too, belongs to ab. From X it readily follows
that a straight line is completely determined by any two
of its points. Number XI, with preceding postulates, im-
plies the existence of at least several straight lines. Num-
ber XII, which is not valid in either the Euclidean or
the Lobachevskian (called by Klein hyperbolic) geometry,
leads to the conception of the (projective) plane. The
class of points on the straight lines containing a, and each
of them a point of be, is named plane, and denoted by
abc. It is then proved that the planes abc, acb, bac, bca,
cab, and cba, are one and the same ; also that a plane is de-
termined by any three of its non-collinear points, whence
it follows that a plane containing two points of a straight
line contains the entire line. The term fourth harmonic
of XIV is defined as follows: The fourth harmonic of
three collinear points a, b, c, or (as it is often called)
the harmonic conjugate of c with respect to a and b, is
a point x of ab such that there exist two distinct points
u and v collinear with c, but not on ab, and such that x
is collinear with the intersections of au with bv and av
with bu. The point x is constructed by means of a figure
(indicated in the foregoing definition) known as the
von Staudt Quadrilateral. It is noteworthy that the defi-

FOUNDATIONS OF MATHEMATICS 225

nition implies neither the existence nor the unicity of x.
The former is readily demonstrable by means of the first
twelve postulates, but the latter requires XIII ; for the
unicity depends upon the theorem of homologous trian-
gles (found in every book of projective geometry), and
it is a most rotable fact that this plane theorem does not
admit of proof except by the help of points outside the
plane — a most suggestive fact. What is true in a given
domain of experience may, nevertheless, not be provable
within that domain.

The straight line has been introduced as a whole, as an
orderless class. Pieri endows it with order, thus giving
it the character of a series of points, as follows: Given
a, b, c, three collinear points. Let y be any other point
of the line, and z the harmonic conjugate of y with re-
spect to a and c. Let x be the harmonic conjugate of
b with respect to y and z. By taking a new y, and hence a
new z, a new x is obtained. The class of x's thus obtain-
able is named segment abc. It is shown that b belongs
to the segment, that its extremities a and c do not, and
that the segment abc is the same as the segment cba.
The segment has the property : if a, b, c, d be four points
of a straight line, and if a', b', c', d' be four points so situ-
ated on another straight line that the lines aa', bb', cc", dd'
have a point in common, then d' belongs to the segment
aW when and only when d belongs to the segment abc.
If d does not belong to the segment abc, and is distinct
from a and c, then, the four points being collinear, the
points a and c are said to separate the points b and d.
It is proved that the relation of separation is symmetric;
that is, that the points a and c are also separated by
b and d; furthermore, that the statement is valid if in it
we exchange the points of either couple. The ordering of
the points of a line is then completed by means of the
postulates XV, XVI and XVII. Continuity is introduced
by number XVIII. The effect of the postulate XIX,
namely, if a, b, c, d are four non-complanar points, and e

226 FOUNDATIONS OF MATHEMATICS

a point in none of the planes, abc, abd, acd, bed, then
there exists a point common to the line ae and the plane
bed, is to restrict the geometry to a space of three dimen-
sions. This restriction is essential to the duality of ordi-
nary projective geometry in virtue of which the notions
point and plane may be interchanged. If we wish to pass
to projective geometry of hyperspace, postulate XIX must
be omitted and other suitable postulates added. One such,
for example, would be: if a, b, c and d be four points not
belonging to a same plane, there exists at least one point
not in the hyperplane abed, where by hyperplane is meant
the class of points on the lines determined by the points
of a plane and a point not in the plane.

If, now, a definition ©f projective geometry (of three
dimensions) be required, the answer is: it is the theory
consisting of the foregoing nineteen postulates (or an
equivalent set), together with the propositions logically
deducible from them. And, similarly, projective space (of
three dimensions) is any class of things (for convenience
called points) that are related as prescribed by the fore-
going or an equivalent set of postulates.

The one undefined notion in projective geometry, as
above founded, is that of straight line. In order that
the doctrine shall be quite expressible in terms of logical
constants, it is necessary and sufficient that the straight
line be defined in such terms explicitly. Such a defini-
tion is: A projective straight line ab is a relation R
between the points a and b, R being symmetric, aliorela-
tive (not subsisting between a point and that point) and
transitive, in so far as transitivity is not restricted by,
aliorelativity.

Descriptive Geometry. — The doctrine of which some
account is to be rendered here is not the descriptive ge-
ometry commonly so called, created by Gaspard Monge,
and in elementary form presented to technological stu-
dents as the semi-practical art of graphically representing
space configurations by means of their projections on a

FOUNDATIONS OF MATHEMATICS 227

plane. This last is about identical with projective geome-
try, or the geometry of position, as popularly understood.
The descriptive geometry to be dealt with here is a new
theory, having been created by Pasch ('Vorlesungen iiber
neuere Geometric/ 1882) and formulated in the symbols
of modern logic by Peano (T principii di Geometria logi-
camenta exposti/ 1889, and 'Sui fondamenti della Geo-
metria in Rivista di Matematica,' Vol. IV, 1894). How it
differs from projective geometry in procedure and funda-
mentals will appear in the light of the following postulates
(as given by Peano) and commentaries upon them. For
fuller analyses of the postulates, the reader may consult
the above-cited works of Russell and Couturat. The
Peano postulates (undemonstrated propositions) of de-
scriptive geometry are as follows. The meaning of some
of them will be clear only by aid of definitions to follow.

I. There is at least one point.

II. Given a point a, there is a point x distinct from a.

III. Between two coincident or identical points there
is no point.

IV. Between two distinct points there is a point.

V. The segment ab is contained in the segment ba.

VI. The point a is not between a and b.

VII. If a and b are two distinct points, there are points
that belong to a'b.

VIII. If c is a point of the segment ab, and if d is a
point of the segment ac, d is also a point of the seg-
ment ab.

IX. If c and d belong to a segment ab, they coincide,
or d is between a and c or is between c and b.

X. If c and d belong to the ray a'b, they coincide, or
d is between b and c or c is between b and d.

XI. If b is between a and c, and c between b and d,
c is between a and d.

XII. If r is a straight line, there exists at least one
point outside of r.

XIII. If a, b, c are non-collinear points, and if d is

22& FOUNDATIONS OF MATHEMATICS

between b and c, and e between a and d, then there is a
point common to ac and the prolongation of be.

XIV. If a, b, c are three non-collinear points, and if
d is between b and c, and f between a and c, the segments
ad and bf have a common point.

XV. Given any plane, there exists at least one point
outside the plane.

XVII. If p is a plane, a a point outside the plane, and b
a point on the prolongation of one of the segments join-
ing a to points of p, then, if x is any point, it belongs to
p, or else p and the segment ax or else the segment bx
have a common point.

XVIII. Let k be a class of points in the segment ab;
there exists a point x of the segment ab, or coinciding
with b, such that no point of k is between x and b, and
that, y being any point taken between a and x, there exist
points of k between y and b.

Such are the basal assumptions of descriptive geome-
try. A few explanatory words will make their meaning
clear and will serve to show the concept of descriptive
space and the corresponding geometry in the process of
gradually coming into being.

By segment ab is meant the class of points between
the points a and b. In this geometry the notion of seg-
ment is central like that of straight line in projective
geometry. By III the segment aa or xx is a null-segment,
one void of points, an empty class. By IV a segment ab
is null if its extremities a and b are identical (coincident).
V shows that segments ab and ba are one and the same:
to be between a and b is the same as to be between b and
a; a segment is without direction, or sense. By VI the ex-
tremities of a segment are not points of it. By the sym-
bol a'b (in number VII), called the prolongation of ab
beyond b, is meant the class of points x such that b is
between a and x. VII postulates the existence of such
prolongation. The existence of ab' is a consequence, as is
also the fact that a'b = ba' and that ab' = b'a. Such pro-

FOUNDATIONS OF MATHEMATICS 229

longations, which are not segments, are called rays. Num-
ber VIII enables us to prove that segment ab contains the
segments ac, be and cd; that the ray a'c contains the rav
a'b; that the logical product of the propositions, b is
between a and c, c is between a and b, is false ; and that,
consequently, the segment ab and the rays a'b and ab'
have no common point. By help of IX it is demonstra-|
ble that the segment ab is the logical sum of the segments
ac and cb and the point c; that, if c is between a and b
and d between c and b, then c is between a and d; that,
if c is between a and b, d between a and c, and e between
c and b, then c is between d and e; that, under the same
hypothesis, the segments ac and cb have no common point ;
and that, if c and d belong to the segment ab, the segment
cd is contained in the segment ab. Such are properties
of segments. Those of rays are found by means of X
and XI to be that, under the hypothesis of X, the ray
a'b is the logical sum of the segment be, the point c and
the ray a'c; under the same hypothesis, the segment cd is
contained in the ray a'b; and by XI, if b belongs to the
segment ac or to the ray ac', the rays a'c and b'c coincide.
The straight line ab (a term occurring in XII) is de-
fined to be the logical sum of the points a and b, the seg-
ment ab and the rays a'b and b'a. The first eleven
postulates suffice to show that the straight lines ab and
ba are identical; that, if c is different from a and belongs
to the straight line ab, the straight lines ab and ac are
identical; and that, if c and d are distinct points of the
straight line ab, the straight lines ab and cd are one and
the same; or, what is equivalent, that a straight line is
determined by any two distinct points of it. Postulates
XII and XIII provide for the concept of plane, as will
presently be seen. If h and k be two classes of points,
the symbol hk will denote the class of all the points on the
segments joining the points of h to those of k; h'k the
class of points on the prolongations of the segments each
beyond its k point, whence the meaning of hk' is also

23o FOUNDATIONS OF MATHEMATICS

clear, and that, too, of such symbols as a (be), a'(bc), etc.
From XIII follows that a(bc)=b(ac). This figure or
class of points is named triangle and denoted by triangle
abc. The plane abc is defined to be the class composed
of the (non-collinear) points a, b and c, the segments ab,
be, ca, the prolongations ab', ba', be', cb', ca', ac', the
triangle abc, and the figures a' (be), b'(ca), c'(ab),
c(a'b'), a(b'c'), b(c'a'). Postulate XIV is essential to
prove that a plane is uniquely determined by any three
non-collinear points of it. And numbers XV and XVII
are respectively necessary that space shall have three
dimensions and that it shall be continuous.

Obvious among the notable differences of projective
geometry and descriptive geometry are the following.
In the former the straight line is a closed series of points
(like the circumference of a circle) ; in the latter the
straight line is an open series of points. Two projective
straight lines of a (projective) plane, or a projective line
and plane, always have a point in common ; but a descrip-
tive plane contains many pairs of non-intersecting
straight lines and a descriptive line and a descriptive plane
may or may not have a common point. One point of a
descriptive line divides it into two parts, and a pair of
points divide it into three parts one of which is a seg-
ment determined by the two points. It requires three
points to determine a segment of a projective straight
line, two points separate the line into two portions, and
•one does not divide it into parts. Two projective planes
have a line in common but two descriptive planes may
or may not have a common line, tho they have a com-
mon line or no common point.

It is an interesting and instructive fact that upon the
foregoing descriptive postulates it is possible by suitable
choice of elements to build up a projective space and
geometry. This may be done as follows, and the process
further reveals the differences and relationships of the
two. varieties of space. Let a and b be any two given lines

FOUNDATIONS OF MATHEMATICS 231

of a descriptive plane #., and let P be any given point
of descriptive space. The two planes determined by P
and a, and P and b, have a common line L. The class
of lines L thus determined by allowing P to take all posi-
tions in descriptive space is named sheaf of lines. These
will have a common point (called the vertex of the sheaf)
or not according as a and b have a common point or not.
Again, if Si and S2 be two sheaves and P a point (not
on the common line of the sheaves if they have one) . P,
Si and S2 determine a plane #, namely, that containing
those lines of Si and S2 that contain P. The class of
planes it thus obtainable by varying P, is named pencil
of planes. The planes of the pencil will have a common
line (called the axis of the pencil) or not according as
Si and S2 have a common line or not. Finally, let Si, S2,
and S3 be any three sheaves whose lines are not all in
the planes of a same pencil, and let S* be a sheaf such
that there is a sheaf S whose lines are common to the
pencils S1S.3 and S2S4. The class of sheaves Si that fulfil
the condition will be named hyperpencil of sheaves. If
now we denote the new entities, sheaves, pencils and
hyperpencils, respectively by the names, points, lines and
planes, it can be shown that these points, lines and planes
constitute a projective space, altho as seen the new
elements are defined in terms of descriptive space.

Metric Geometry. — In recent years various investiga-
tors, American and European, have proposed various log-
ically equivalent systems of postulates for this the most
ancient form of geometry. Of such systems, that found
in Hilbert's 'Grundlagen der Geometrie' (also in English
and French) is the most famous. We prefer, however,
to present here that of Pieri as being more interesting
and not less profound. In this system there are two un-
defined terms, namely, point and movement. It will be seen
that point is merely a name for the element of any system
of elements (if such there be) that satisfy the postulates.
And movement does not mean ordinary motion, but only a

232 FOUNDATIONS OF MATHEMATICS

transformation, or change of attention from one thing
to another. Even so the process is disregarded, only
the initial and the final stages and not any passage are
regarded. The postulates are as follows. Subsequent
explanations will make them clear.

I. Point and movement are genuine concepts or classes.

II. There exists at least one point.

III. If p is a point, there exists a point different from p.

IV. Every movement is a biuniform correlation between
two figures.

V. Whatever be the movement M which makes the
point y, for example, correspond to the point x, there is
a movement u that makes x correspond to y.

VI. Two movements, M and v, effected successively the
one on the result of the other, are equivalent to a single
movement.

VII. For each pair of distinct points there is an ef-
fective movement that leaves them fixed.

VIII. If a, b and c are three distinct points, and if
there exists an effective movement that leaves them fixed,
every other movement that leaves a and b fixed leaves c
fixed.

IX. If a, b and c are three collinear points, and if d
is a point of (the line) be other than b, the plane abd
is contained in the plane abc.

X. If a and b are distinct points there exists a move-
ment that leaves a fixed and transforms b into another
point of the straight line ab.

XI. If a and b are distinct points, and if two move-
ments that leave a fixed transform b into another point
of the straight line ab, this point is the same in both
movements.

XII. If a and b are distinct points, there is a move-
ment that transforms a into b and that leaves one point
of the straight line ab fixed.

XIII. If a, b and c are three non-collinear points, there

FOUNDATIONS OF MATHEMATICS 233

is a movement that leaves a and b fixed and transforms c
into another point of the plane abc.

XIV. If a, b and c are three non-collinear points, and if
d and e are points of the plane abc common to the
spheres ca and cb, and different from c, then d and e
coincide.

XV. If a, b and c are distinct non-collinear points,
there exists at least one point outside the plane abc.

XVI. If a, b, c and d are four non-complanar points,
there exists a movement that leaves a and b fixed and
transforms d into a point of the plane abc.

XVII. If a, b, c and d are four distinct collinear points,
the point d cannot be upon only one of the segments ab,
ac, be.

XVIII. If a, b and c are three collinear points, and
if c is between a and b, no point can be at once between
a and c and between b and c.

XIX. If a, b and c are three non-collinear points, every
straight line of the plane abc that has a point in the seg-
ment ab has a point in the segment ac or in the segment
be, or it contains one of the points a, b, c.

XX. If k is a class of points in the segment ab, there
exists in the segment, or coincides with b, a point x, such
that no point of k is between x and b, and that for every
point y between a and x there is a point k between y and
x or coincident with x.

Two figures (classes of points) coincide when and only
when they are composed of the same points. IV means
that a movement is a one-to-one relation between two
figures. The movements V and u (V) are each the other's
converse; they are mutually converse biuniform rela-
tions. By VI the relative product of the movements M
and is a movement. The relative product Mu leaves
every point fixed, or, as we say, transforms all points each
into itself. In contradistinction from such movements,
others are described as effective. VII provides for rota-
tion of a figure about two of its points. A straight line

234 FOUNDATIONS OF MATHEMATICS

ab is defined to be the class of all points that remain fixed
in case of every movement leaving a and b fixed. It is
a matter of proof that a straight line is determined by
any two distinct points of it. VIII is not valid in space
of four or more dimensions, and hence no special postu-
late restricting our geometry to three dimensions is neces-
sary. It is readily proved that any movement whatever
transforms any and every triplet of collinear points into
such a triplet; in other words, a movement is a collinea-
tion. By plane abc is meant the figure composed of the
points of the lines joining a to points of be, or b to points
of ac, or c to points of ab, it being assumed that a, b and
c are non-collinear points. It is a theorem that every
movement converts a plane into a plane. Postulate IX
is necessary to prove that a plane is determined by any
three non-collinear points of it.

By the sphere b» is meant the class of points such that
for each of them there is a movement transforming it
into b while leaving a fixed. The point a is the center
of the sphere. It is demonstrable that every movement
transforms spheres into spheres ; that any movement that
leaves the center of a sphere fixed transforms the sphere
into itself; and that, if two spheres have but one common
point, that point is collinear with the centers of the
spheres. X, XI, and XII provide for transforming a
line into itself; and XIII and XIV make the like pro-
vision for the plane. A circle is the logical product of a
sphere and a plane containing its center. The center of
the circle is that of the sphere. The notion of perpen-
dicularity is introduced by the definition: the pair (a, c) of
points is said to be perpendicular to the pair (a, b) when
and only when there is a movement that leaves a and b
fixed and transforms c into another point of the straight
line ac. The notion is readily extensible to straight lines.
XV provides for a plurality of planes, and XVI for the
transformation of one plane upon another. The notion
of equidistance is introduced by the definition: a point

FOUNDATIONS OF MATHEMATICS 235

a is equidistant from two points b and c when and only
when it is the center of a sphere containing b and c.
It is demonstrable that, in a plane containing the dis-
tinct points a and b, the class of points equidistant from
a and b is the straight line perpendicular to the straight
line ab and containing the mid-point of the segment ab;
that a straight line perpendicular to two straight lines
ab and ac is perpendicular to every straight line that:
contains a and is contained in the plane abc; and other
theorems respecting perpendicularity are readily proved.
A point is interior to a sphere if it is the mid-point of
two distinct points of the sphere. If not, it is exterior,
or else is a point of the sphere. A point of a plane con-
taining a circle is interior or exterior to the circle accord-
ing as it is interior or exterior to the sphere having the
same center as the circle and containing the circle. A
sphere having for center the mid-point of two points a
and b, and containing them, is called the polar sphere of
the points a and b. The notion between is introduced by
the definition : a point x is between points a and b if it
is contained in the straight line ab and is interior to the
polar sphere of a and b. The class of points between
two points a and b is named segment ab. The segment ab
is less than the segment cd when and only when there
exists a movement that transforms a into c and b into a
point between c and d. Two segments (or other figures)
are congruent if there exists a movement transforming
one of them into the other. It is demonstrable that if
two segments are not congruent, one of them is less than
the other. The notion angle is defined and to it are
extended the ideas of less than and congruence. If a,
b and c are non-collinear points, the triangle abc is the
figure composed of the points of the segments, each join-
ing a and a point of the segment be. The three theorems
regarding congruence are proved; and so on and on. By
XX, which provides for continuity, is deduced the Archi-

236 FOUNDATIONS OF MATHEMATICS

median "axiom" as a theorem. Thence follows the idea
of measurability of segments.

General Remarks. — No geometry involves ideas not
found in logic or definable in terms of logical constants,
and no geometry contains other undemonstrated proposi-
tions than the primitive propositions of logic. The name
point is merely that of a class of things (if there be such
things) that satisfy a certain set of postulates, but geome-
try does not assert the actual existence of any such class
and does not assert the truth of the postulates. What
it does assert is that, if such a class exists, then such and
such a body of theorems are valid regarding the class.
Geometry is thus a body of implications. It says merely
"if so and so, then so and so." This important fact is
somewhat disguised by the categorical form in which pos-
tulates are often stated.

Bibliography. — Instead of giving a list of the works
constituting the vast and rapidly growing modern litera-
ture dealing with the foundations of mathematics in gen-
eral, with the foundations of special branches, and with
modern logic, it will be sufficient to refer the reader to
Russell's 'Principles of Mathematics,' Vol. I (Cambridge,
University Press) and to Couturat's 'Les Principes des
Mathematiques' (Paris, Felix Alcan) and 'Traite de Lo-
gistique' (Alcan), wherein nearly all the important works
are cited in connections showing the bearings of them.
Most of the works are too technical for the general
reader, who will naturally begin with the mentioned trea-
tises of Russell and Couturat, extending his reading grad-
ually according to increasing ability and interest.
Cassius J. Keyser,
Adrain Professor of Mathematics.

Columbia University.

MATHEMATICAL APPLICATIONS

Dr. FRANZ BELLINGER

MATHEMATICAL APPLICATIONS

CHAPTER I

EARLY NON-MECHANICAL APPLICATIONS

Modern life, as a whole, lies under a debt to Mathe-
matics far beyond calculation. Science has shown many
underlying principles which govern matter, life and mind
in their several environments and in their relation each
to each, but it has required the mathematical faculty and
the mathematical knowledge to transpose those principles
into productive value. Mathematics may be termed the
spirit, practical application the flesh, of a single and indi-
visible entity. Hence the term "applied mathematics" is
to be used with caution, since it is inherent' in the nature
of mathematics that it shall not be divorced from any of
its subsidiary uses, but remain as a vigorously vital and
governing law.

Mechanical principles, for example, are mainly mathe-
matical deductions from principles enunciated by 'Pure
Science/ even as that same Pure Science finds itself de-
pendent upon mathematical expression for the enunciation
of those principles. The very words that are spoken or
written bear a definite relation each with the other, and
no more mathematical concept than 'Relation' could well
be thought of. From the abstruse and remote questions
of the affirmation of a stellar parallax in Astronomy to the
'multiplication' of yeast cells in making a loaf of bread,
from the lofty flights into the regions of the mathemati-
239

24o MATHEMATICAL APPLICATIONS

cally infinite to the counting of change over a counter,
Mathematics is applied and practical. It does not always
appear mechanical, because it has not always been trans-
literated into such forms, and these non-mechanical appli-
cations existed in antiquity as they do now. Applied
Mathematics, in that sense, is as old as Applied Thought,
and Applied Thought is coeval with Man.

"To think aright," says Prof. Cassius Keyser in an
illuminative recent lecture on 'Mathematics,' "is no char-
acteristic striving of a class of men; it is a common as-
piration; and Mechanics, Mathematical Physics, Mathe-
matical Astronomy, and the other chief 'Anwendungsge-
biete' (spheres of application) of mathematics, as Geodesy,
Geophysics, and Engineering in its various branches,
are all of them but so many witnesses to the truth of
Riemann's saying that 'Natural science is the attempt to
comprehend nature by means of exact concepts.' A gas
molecule regarded as a minute sphere or other geometric
form, however complicate; stars and planets conceived
as ellipsoids or as points, and their orbits as loci ; time and
space, mass and motion and impenetrability; velocity, ac-
celeration and energy ; the concepts of norm and average —
what are these but mathematical notions? And the won-
drous garment woven of them in the loom of logic — what
is that but mathematics?

"Indeed, every branch of so-called applied mathematics
is a mixed doctrine, being thoroly analyzable into two
disparate parts: one of these consists of determinate con-
cepts formally combined in accordance with the canons
of logic — i.e., it is mathematics and not natural science
viewed as matter of observation and experiment — the
other is such matter, and is natural science in that con-
ception of it, and not mathematics. No fiber of either
component is a filament of the other.

"It is a fundamental error to regard the term Mathe-
maticization of thought as the importation of a tool into
a foreign workshop. It does not signify the transition of

EGYPT AND GREECE 241

mathematics conceived as a thing accomplished over into
some outlying domain like physics, for example. Its sig-
nificance is different radically, far deeper and far wider.
It means the growth of mathematics itself, its extension
and development from within; it signifies the continuous
revelation, the endlessly progressive coming into view, of
the static universe of logic; or, to put it dynamically, it
means the evolution of intellect, the upward striving and
aspiration of thought everywhere, to the level of cogency,
precision and exactitude.

"It is the aggregate of things thinkable logically that
constitutes the mathematician's universe, and it is incon-
ceivably richer in mathetic content than can be any outer
world of sense, such as the physical universe according
to which we chance to have our physical being."

The term 'practical,' in its common acceptation, often
denotes shorter methods of obtaining results than are in-
dicated by science. It implies a substitution of natural
sagacity and mother wit for the results of hard study
and laborious effort. It implies the use of knowledge
before it is acquired — the substitution of the results of
mere experiment for the deductions of science, and the
placing of empiricism above philosophy. But if to "prac-
tical" be given its true and right signification, then it
becomes a word of real import and definite value. In its
right sense it denotes the best means of making the true
ideal the actual; that is, of applying the principles of
science in all the practical business of life and of bodying
forth in material form the conception of taste and genius.

Beyond the obvious application of simple and known
principles, the whole problem of the practical lies in the
measurement, modification and best uses of the forces
of nature. The uses and applications of these must be
fashioned according to certain forms indicated by sci-
entific formulae. These formulae are constructed from the
laws which regulate the cohesion of the particles of the
substance employed — the nature of the force to be applied

242 MATHEMATICAL APPLICATIONS

— the amount of that force and the ultimate end to be
attained. All these fixed laws of force — all their combina-
tions— and all the forms of the material employed in using
them for practical purposes can only be reached through
the processes and language of mathematics.

The language of Geometry and Number furnished the
architect with all the signs and instruments of thought
necessary to a perfect ideal of his work before he took
the first step in its execution. It also enabled him, by
drawings and figures, so to direct the hand of labor as
to form the actual after its pattern — the ideal. The vari-
ous parts may be constructed by different mechanics, at
different places, but the law of science is so certain that
every part will have its right dimensions, and when all
are put together they form a perfect whole.

The influence of mathematical investigations on physical
theories is not restricted to any single stage, but makes
itself apparent throughout the whole course of their evo-
lution. Numbers form the connecting link between theory
and verification, and they always imply mathematical for-
mulae, however simple these may be.

There seems to be historical evidence that a practical
acquaintance with certain rules of number and form was
acquired by ancient peoples, especially by the Egyptians,
before there was any knowledge of mathematics as a
pure science. In Babylonia geometrical figures were
used in augury. Herodotus, Plato and Strabo ascribe the
origin of geometry to the changes which annually took
place from the inundation of the Nile, and to the conse-
quent necessity of settling disputes as to the extent of
property, and of determining the tax due to the govern-
ment. There was a well-developed system of mensura-
tion in the time of the traditional biblical Joseph; and
besides the extraordinary mechanical ability of the Egyp-
tians in handling stone, they were able to construct ac-
curately leveled canals, to ascertain the various eleva-

EGYPT AND GREECE 243

tions of the country, and, tradition says, to deflect the
course of the Nile.

At the time of King Menes, who is supposed to have
performed this extraordinary feat, dykes had been built
and sluices invented, with all the mechanism pertaining
to them. The water supply into plains of various levels
was regulated, and a report was made of the exact quan-
tity of land irrigated, the depth of the water, and the
time it remained upon the surface. All this required much
mathematical skill, and it was not likely to be carelessly
carried on, since the amount of taxes and the price of
provisions for the ensuing year were ascertained at the
time of the inundation. Nilometers — instruments for
measuring the gradual rise or fall of the river — were in
use in various parts of Egypt as early as the twelfth
dynasty.

"The employment of squared granite block and the
beauty of the masonry of the interior of the Pyramids,"
says Geo. Rawlinson, "which has not been surpassed, if
even equaled, at any subsequent age, also prove the de-
gree of skill the Egyptians had reached at a time long
anterior to the rudest attempts at masonry in Italy or
Greece. We may well conclude that the principles of
construction were known to them, as well as the engineer-
ing skill required for changing the course of the Nile,
even before the reign of Menes."

The immense weight of the blocks of stone used in
building shows that the Egyptians were well acquainted
with mechanical powers and a method of applying force
with wonderful success. The largest obelisk in Egypt is
calculated to weigh about 297 tons, is more than 70 feet
in height, and was carried 138 miles from the quarry.

The Egyptians could not only move immense weights;
they could erect obelisks, lift large stones to a consider-
able height and adjust them with the utmost precision;
and this sometimes in spaces that would not admit the
introduction of the inclined plane.

244 MATHEMATICAL APPLICATIONS

Pliny mentions that one obelisk, built by Rameses, was
99 feet in height. He adds : "And, fearing lest the engi-
neer should not take sufficient care to proportion the
power of the machinery to the weight he had to raise, he
ordered his son to be bound to the apex, more effectually
to guarantee the safety of the monument."

Of the science of arithmetic the Egyptians early were
in need, both in their domestic economy and in the appli-
cation of geometrical theorems; but its greatest utility
was in the cultivation of astronomical studies. Indeed,
mathematics was the handmaid of astronomy among the
Assyrians, Babylonians and Egyptians. An ancient writer
says : "The orders and motions of the stars are observed
at least as industriously by the Egyptians as by any peo-
ple whatever; and they keep a record of the motions of
each for an incredible number of years, the study of this
science having been, from remotest times, an object of na-
tional ambition with them."

There is record in Egypt of the solid contents of barns
before the calculation of areas. In the papyrus of Ahmes,
reaching back to about 2500 B.C., there are problems re-
lating to the pyramids which disclose some knowledge
not only of geometrical figures but the principles of pro-
portion, and possibly trigonometry. Cantor is of the
opinion that the Egyptians were familiar with the proper-
ties of the right triangle in case of sides with the ratio
3:4:5 as early as 2000 b.c. This opinion is based on the
orientation of the temples and early records of the "rope-
stretching" method of laying out the land.

The Arabs developed the notion of "specific gravity,"
and gave experimental methods for its determination.
Al Biruni used for this purpose a vessel with a spout
slanting downward. It was filled with water up to the
spout, then the solid was immersed, and the weight of
the overflow determined. This, together with the weight
of the solid in air, yielded the specific gravity. Al Khazini,
in his Book of the Balance of Wisdom, written 1137 B.C.,

246 MATHEMATICAL APPLICATIONS

describes a curious beam balance, with five pans, for
weighing in air and in water. One pan was movable
along the graduated beam. He points out that air, too,
must exert a buoyant force, causing bodies to weigh less.

Thales, in his pyramid and ship measurements, was prob-
ably the first to apply theoretical geometry to practical
uses. He was able to predict an eclipse of the sun in 585
B.C., and several practical applications of geometry are at-
tributed to him. But the illustrious name among the
Greeks, in respect to both mathematical and mechanical
science, is that of Archimedes. The most important ser-
vices of Archimedes were rendered in the science of pure
mathematics, but his popular fame rests chiefly on his
application of mathematical theory tc mechanics.

Heron of Alexandria, called Heron the Elder, was a
mathematician and also a practical surveyor who lived in
the second century b.c. His teacher, Ctesibus, was cele-
brated for his mechanical inventions, such as the water-
clock, the hydraulic organ and catapult. Heron himself
was the inventor of the seolipile, which contains the germ
of the steam-engine, and a curious mechanism known as
"Heron's Fountain."

It is, however, in architecture that the Greeks and
Romans made the most marked advance upon the achieve-
ments of the Egyptians, mechanically as well as artisti-
cally. The three principles of the beam, the arch and the
truss were known to the Greeks and Romans ; indeed, it is
the opinion of H. W. Desmond that they possessed all the
technical knowledge of the medieval builders. It is evident
that they adopted from the Egyptians whatever they
needed. The construction of the arch dates from an early
period. Mathematical skill is a great factor in the develop-
ment of architecture ; the very term implies tools and force
at command and instruments for supplementing the labor
of the hands. The draftsman, in designing a structure,
should be conversant not only with the nature of his
material, but also with the forces to which it is to be sub-

EGYPT AND GREECE 247

jected, their magnitude, direction, points of application
and their effects. The ancient Romans not only con-
structed arches, but the largest domes of brick now in
existence. These structures rest on all sides of the space
to be covered, but there is also the simple or wagon-
head vault, which rests on only two sides of the covered
rectangle, leaving the other two free from all pressure.
Further than this, the Romans invented that highly
ingenious contrivance, the cross-vault, which exerts its
whole pressure solely on the angles of the apartment, leav-
ing all the sides free.

The origin of this construction is simply the crossing of
two vaulted passages lying at right angles to each other
and each corridor required to be left perfectly free. The
crossway is covered by a ceiling that rests solely on the
four angles or corners; the elliptic lines that form the in-
ternal ridges, called groins, can support not only them-
selves but the whole of the upper ceiling. The beauty and
advantages of this kind of vaulting led the Romans to use
it not only over crossways, but over corridors and long
apartments with a boldness of construction that has never
been equaled.

With the decline of Roman power this art of vaulting
was lost, and for centuries the basilicas of Italy and the
churches of all Roman Christendom remained with nothing
but timber roofs. The Byzantine Greeks, however, re-
tained or else reinvented another mode of vaulting possess-
ing many of the advantages of groining, but not all of
them. This system depended on two simple geometrical
principles : First, that every section of a sphere by a plane
is circular, and, second, that every intersection of two
spheres is a plane curve and therefore circular.

The Greek vaulting then consists wholly of spherical
surfaces. A hemispherical dome may be supposed whose
base circumscribes the plane of any apartment or com-
partment, square, rectangular, triangular or polygonal.

248

MATHEMATICAL APPLICATIONS

Imagine the sides of this plane continued upward, as verti-
cal planes, till they meet the hemispheric surface. This
meeting line must in every case be a semi-circle and may
therefore be made an open arch, and the portions of the
dome thus cut off from every side of its base may be omit-
ted altogether, provided their office as buttresses to the

Fig. 2 — Groin Device to Support Weight of Building.

remaining portion above be replaced by the pressure of
some other vault, which may be of any kind, if it be ap-
plied against the semi-circular arch. Hence no walls are
required on the sides of the supposed compartment, all the
weight of the pendentive dome, as it is called, being thrown
on the angles of its plane. Thus this dome serves for
covering an open crossway and is so applied at the mosque
of St. Sophia, at Constantinople. The covered crossway,

EGYPT AND GREECE 249

a 115-foot square, might well be esteemed, in the barbarous
age of its erection, a wonder of the world.

The same idea repeated without end — the same sprouting
of domes out of domes, continues to characterize the Byzan-
tine style, both in Greek churches and Turkish mosques,
down to the present day. Hope describes them as a con-
geries of globes of various sizes growing one out of an-
other. This system of vaulting has been adopted by two
great modern architects, Sir Christopher Wren at St.
Paul's, in London, and by Soufflot at Ste. Genevieve, Paris ;
by the former with great success and in both made to
harmonize well with the Roman style.

There is no more striking and beautiful example of the
application of mathematical principles to practical affairs
than in the history of architecture. The close reasoning of
the mathematician has been behind and above the work of
the draftsman and artisan ; his imagination has reached
out boldly to the projection of new designs, restrained al-
ways by the immutable laws of science; his achievement
it is to unite strength and durability with beauty and
geometric truth with grandeur.

"In architecture," says Fergusson, "there is still to be
taken into consideration not only that subtler and com-
plexer force, the personal genius of the architect, but also
the native genius of the people in which he is a sharer,
that spirituality or temper of mind which is obvious
enough in its stronger manifestations." Thus the nations
that showed a talent for mathematics were building na-
tions, since here was a science which could be definitely
and immediately applied to practical use.

It is necessary to discharge from the mind many uncon-
sciously implied conditions before an exact picture of the
'pre-mechanical age' can be gained. All the raw material
of mechanical science was at hand, as much before as after
the magical words of Newton or of Helmholtz, but mathe-
matical genius had not yet touched the spring which dissi-
pated the inertia of established habit.

250 MATHEMATICAL APPLICATIONS

But before the civilized world could be transformed
from a world utilizing, as one might say, only the more
obvious natural forces to a world rilled with devices for
multiplying hands and feet, for increasing the value of eye
and ear, a news-gathering world where oceans are neigh-
borly high roads and warfare a contest of scientific equip-
ment— before this transformation could happen the mathe-
matician had need to direct his analytic and speculative
powers to the natural phenomena of the universe.

Concerning this stage of the development of mathe-
matics, Cassius J. Keyser writes: "A traditional concep-
tion, still current everywhere except in critical circles, has
held mathematics to be the science of quantity or magni-
tude, where magnitude, including multitude (with its cor-
relate of number) as a special kind, signified whatever was
'capable of increase and decrease and measurement.'
Measurability was the essential thing. That definition of
the science was a very natural one, for magnitude did
appear to be a singularly fundamental notion, not only in-
viting but demanding consideration at every stage and
turn of life. The necessity of finding out how many and
how much was the mother of counting and measurement ;
and mathematics, first from necessity and then from pure
curiosity and joy, so occupied itself with these things that
they came to seem its whole employment.

"Indeed, for direct beholding, for immediate discerning
of the things of mathematics there is none other light but
one — namely, psychic illumination — but mediately and in-
directly they are often revealed or at all events hinted by
their sensuous counterparts, by indications within the radi-
ance of day, and it is a great mistake to suppose that the
mathetic spirit elects as its agents those who, having eyes,
yet see not the things that disclose themselves in solar
light. To facilitate eyeless observation of his sense-
transcending world the mathematician invokes the aid of
physical diagrams and physical symbols in endless variety
and combination; the logos is thus drawn into a kind of

EGYPT AND GREECE 251

diagrammatic and symbolical incarnation, gets itself ex-
ternalized, made flesh, so to speak; and it is by attentive
physical observation of this embodiment, by scrutinizing
the physical frame and make-up of his diagrams, equations
and formulae, by experimental substitutions in and trans-
formations of them, by noting what emerges as essential
and what as accidental, the things that vanish and those
that do not, the things that vary and the things that abide
unchanged as the transformations proceed and trains of
algebraic evolution unfold themselves to view — it is thus,
by the laboratory method, by trial and by watching that
often the mathematician gains his best insight into the
constitution of the invisible world thus depicted by visible
symbols.

"Indeed, the time is at hand when at least the academic
mind should discharge its traditional fallacies regarding
the nature of mathematics and thus in a measure promote
the emancipation of criticism from inherited delusions
respecting the kind of activity in which the life of the
science consists. Mathematics is no more the art of reck-
oning and computation than architecture is the art of
making bricks or hewing wood, no more than painting is
the art of mixing colors on a palette, no more than the
science of geology is the art of breaking rocks or the
science of anatomy the art of butchering.

"Pernicious, because deeply embedded and persistent, is
the fallacy that the mathematician's mind is but a syllogis-
tic mill and that his life resolves itself into a weary repe-
tion of A is B, B is C, therefore A is C, and Q.E.D.
That fallacy is the 'Carthago delenda' of regnant methodol-
ogy. Reasoning, indeed, in the sense of compounding
propositions into formal arguments, is of great importance
at every stage and turn, as in the deduction of conse-
quences, in the testing of hypotheses, in the detection of
error, in purging out the dross from crude material, in
chastening the deliverances of intuition, and especially in
the final stages of a growing doctrine in welding together

252 MATHEMATICAL APPLICATIONS

and concatenating the various parts into a compact and
coherent whole. But, indispensable in all such ways as
syllogistic undoubtedly is, it is of minor importance and
minor difficulty compared with the supreme matters of
Invention and Construction.

"When the late Sophus Lie, great comparative anatomist
of geometric theories, creator of the doctrines of Contact
Transformations and Infinite Continuous Groups and revo-
lutionizer of the Theory of Differential Equations, was
asked to name the characteristic endowment of the mathe-
matician, his answer was the following quaternion : 'Phan-
tasie,' 'Energie,' 'Selbstvertrauen,' 'Selbstkritik.' Not a
word, you observe, about ratiocination. Phantasie, not
merely the fine frenzied fancy that gives to airy nothings
a local habitation and a name, but the creative imagination
that conceives ordered realms and lawful worlds in which
our own universe is as but a point of light in a shining
sky; Energie, not merely endurance and doggedness, not
persistence merely, but mental vis viva, the kinetic, plung-
ing, penetrating power of intellect; Selbstvertrauen and
Selbstkritik, self-confidence aware of its ground, deepened
by achievement and reinforced until in men like Richard
Dedekind, Bernhard Bolzano and especially Georg Cantor
it attains to a spiritual boldness that dares leap from the
island shore of the Finite over into the all-surrounding
boundless ocean of Infinitude itself, and thence brings back
the gladdening news that the shoreless vast of Transfinite
Being, differs in its logical structure from that of our island
home only in owning the reign of more generic law."

CHAPTER II

CHRONOLOGY AND HOROLOGY

Altho the ancients gave so much of their attention to
understanding and recording the facts of astronomy, yet
there was very little systematic attention given to the
computation of time or to the chronological aspect of his-
tory. Chronology is comparatively a modern science, yet
a highly important one. Accurate chronology is essential
to all reasoning from historical facts; the mutual depend-
ence and relations of events cannot be traced without it;
with great propriety it has been called one of the eyes of
history, while geography with equal propriety has been
said to be the other.

Present acquaintance with the truths of astronomy
would have been as deep had Eastern philosophers never
turned their eyes to the realms of space or watched the
harmonious movements of the worlds in the firmament
above. "The moment," says Sir John Herschel, "as-
tronomy became a branch of mechanics, a science essen-
tially experimental — that is to say, one in which any prin-
ciple laid down can be subjected to immediate and decisive
trial and where the experience does not require to be
waited for — its progress acquired a tenfold acceleration;
nay, to such a degree that were the results of all the obser-
vations from the earliest ages annihilated, leaving only
those made in Greenwich Observatory during the single
lifetime of Maskelyne, the whole of this most perfect of
sciences might, from those data and as to the objects in-
253

254 MATHEMATICAL APPLICATIONS

eluded in them, be at once reconstructed and appear pre-
cisely as it stood at their conclusion. The operation, in-
deed, of Arabian knowledge of astronomy in the early
ages was perhaps principally to lend a plausibility to
astrology; the observers of stars, like Columbus pre-
dicting the eclipse, had the power of astonishing when
they prepared to delude."

Fig. 3 — Arabic Astrolabe.

CHRONOLOGY AND HOROLOGY 255

The most obvious measures and divisions of time are
those suggested to all men by the revolutions of the heav-
enly bodies. These are three — days, months and years;
the day from the revolution of the earth on her axis or the
apparent revolution of the sun around the earth; the
month from the periodical changes in the moon; the year
from the annual motion of the earth in her orbit round the
sun. These three divisions are not commensurate, and
this has caused the chief embarrassment in the science of
chronology; it has, in point of fact, been difficult so to
adjust them with each other in a system of measurement
as to have the computed time and the actual time per-
fectly in agreement or coincidence.

The day was undoubtedly the earliest division and origi-
nally was distinguished, it is likely, from the night and
extended from sunrise to sunset. It was afterward con-
sidered as including also the night and was marked as the
time from sunrise to sunrise. But the beginning of the
day has been reckoned differently by different nations for
civil purposes; at sunrise by the Babylonians, Persians,
Syrians and inhabitants of India; at sunset by the Jews,
Athenians, ancient Gauls and Chinese ; at midnight by the
Egyptians, Romans and moderns generally. Astronomers
in their calculations consider the day as beginning at noon,
after the manner of the Arabians.

There have also been various modes of subdividing the
day. The division of time into hours is very ancient, the
oldest hour being the twelfth part of a day. Herodotus
observes that the Greeks learned from the Egyptians,
among other things, the method of dividing the day into
twelve parts, and the "astronomers of Cathaya" still re-
tained that method at the time of Herodotus. The division
of the day into twenty-four hours was not known to the
Romans before the Punic War.

The Greeks, in the time of Homer, seem not to have
used the division into hours ; his poems present the more
obvious parts of the day, morning, noon and evening. But

256 MATHEMATICAL APPLICATIONS

before the time of Herodotus they were accustomed to the
division of the day and of the night also into twelve parts.
They were acquainted also with the division of the day
and night into four parts each, according to the Jewish and
Roman custom.

The Romans subdivided the day and night each into four
parts, which were called vigils or watches. They also
considered the day and night as each divided into twelve
hours; three hours, of course, were included in a vigil.
The day vigils were designated simply by the numerals
first, second, third, fourth, but as the second vigil com-
menced with the third hour, the third vigil with the sixth
hour and the fourth with the ninth hour, the terms first,
third, sixth and ninth are also used to signify the four vigils
of the day. The night vigils were designated by the names
vesper, evening, midnight and cockcrow. The first hour of
the day began with sunrise and the twelfth ended at sun-
set; the first hour of the night began at sunset and the
twelfth ended at sunrise. Of course, therefore, the hours
of the day in summer were longer than those of the night
and in the winter they were shorter.

The division of time into months, without much doubt,
had its origin in the various phases or changes of the
moon. It included the time of the moon's revolution round
the earth, or between two new moons, or two successive
conjunctions of the sun and moon. The mean period is
29 days, 12 hours, 44 minutes. It was considered to be
29^2 days, and the ancients commonly reckoned the month
as consisting alternately of 29 and 30 days.

The Greeks thus reckoned their months. Twelve luna-
tions so computed formed the year, but it fell short of the
true solar year by about nj4 days, making in four years
about 45 days. To reconcile this and bring the computa-
tion by months and years to coincide more exactly another
month was intercalated every two years; in the first two
years a month of 22 days and in the next two a month of
23 days. Thus, after a period of four years, the lunar and

CHRONOLOGY AND HOROLOGY 257

solar years would begin together. But the effect of this
system was to change the place of the months relatively
to the seasons, and another system was adopted. This was
based on the supposition that the solar year was 365^
days, while the lunar was 354, which would in a period of
eight years give a difference of 90 days. The adjust-
ment was made by intercalating, in the course of the
period, three months of 30 days each. Its invention
was attributed to Cleostratus of Tenedos; it was univer-
sally adopted and was followed in civil matters even after
the more perfect cycle of Meton was known.

With the Romans the case was somewhat different.
Under Romulus they are said to have had only ten months,
but Numa introduced the division into twelve, according to
that of the Greeks. But, as has been seen, this formed
only a lunar year, a little more than eleven days short of
the solar year; therefore an "extraordinary month" was
to be inserted every other year. The intercalating of this
and the whole charge of dividing the year was intrusted to
the Pontifices, and they managed, by inserting more or
fewer days, to make the current year longer or shorter, as
they for any reason might choose. This finally caused the
months to be transposed from their stated seasons, so that
the winter months were carried back into autumn and the
autumnal into summer. Julius Caesar put an end to this
disorder by abolishing the intercalation of months and by
adopting a system which was available by the more ac-
curate division of the year.

A consideration of the division of the year takes the
historian back into the twilight of history. It is well
known that the Babylonians had a system of notation called
the sexagesimal, which reveals a high degree of mathe-
matical insight. It was used chiefly in the construction of
a system of weights and measures and reveals some knowl-
edge of geometrical progressions, but the indications are
that it was in the possession of few and was used but

258 MATHEMATICAL APPLICATIONS

little. The base of this system was the number 60. The
Babylonians reckoned the year at 360 days.

The Grecian year, however, which was established by
Solon and continued to the time of Meton and even after,
consisted of 365^4 days. This division was probably not
formed until considerable advance had been made in
astronomical science, and it was long after its first adop-
tion before it attained to anything like an accurate form.

The Roman year seems to have consisted of 365 days
until the time of Julius Caesar, who attempted to remedy
the confusion resulting from the method employed by the
Romans to adjust their computations by lunar months to
the solar year. Caesar instituted a year of 365 days and 6
hours. To remove the error of 80 days, which computed
time had gained of actual time, he ordered one year of 445
days, which was called the Year of Confusion. To secure
a proper allowance for the six hours which had been dis-
regarded, but which would amount in four years to a day,
he directed that one additional day should be intercalated
in the reckoning of every fourth year. Thus each fourth
year should have 366 days, the others 365. This is called
the Julian year and begins to show some of the familiar
landmarks of modern chronology. But even in the plan
of the great Julius there was still a fault, owing to an
error in computed time. The extra day was intercalated
too soon — that is, computed time, instead of gaining six
hours a year as was supposed, gained only 5 hours, 48
minutes and 57 seconds, so that a whole day was not gained
in four years. The intercalated day was inserted too soon
by 44 minutes and 12 seconds, and of course computed
time by this plan lost 44 minutes and 12 seconds every four
years or 11 minutes and 3 seconds every year. In 131
years this makes a loss of computed time of one day or
computed time would be one day behind actual time. In
1582 a.d. this loss had amounted to ten days, and Pope
Gregory XIII. attempted to remedy the evil by a new
expedient. This was to drop the intercalary day every

CHRONOLOGY AND HOROLOGY 259

hundredth year except the four hundredth. The Gregorian
year was immediately adopted in Spain, Portugal and
Italy and during the same year in France, in Catholic
Germany in 1583, in Protestant Germany and Denmark in
1700, in Sweden in 1753. In England it was adopted in
1752 by act of Parliament directing the 3d of September to
be styled the 14th, as computed time had lost eleven days.
This was called the change from Old to New Style. The
Julian calendar, or Old Style, is still retained in Russia
and Greece, whose dates consequently are now 12 days in
arrear of those of other countries of the western hemi-
sphere. It is also retained in the Greek and Armenian
churches.

Different nations have begun the year at different sea-
sons or months. The Romans at one time considered it as
beginning in March, but afterward in January. The
Greeks placed its commencement at the summer solstice.
The Christian clergy used to begin it at the 25th of March,
and this style was practiced in England and in the Amer-
ican colonies until 1752 a.d., on the change from Old Style
to New, when the 1st of January was adopted.

In adjusting the different methods of computing time, or
the division of time into days, months and years, great
advantage is derived from the invention of cycles. These
are periods of time so denominated from the Greek word
meaning a circle, because in their compass a certain revolu-
tion is completed. Under the term cycle may be included
the Grecian Olympiad, a period of four years; the Octae-
teris, or period of eight years; the Roman Lustrum, a
period of five years, and also the Julian year, or period of
four years. The period of 400 years, comprehended in the
system of Gregory, may justly be termed the Cycle of
Gregory. Besides these, there are the Lunar Cycle, the
Solar Cycle, the Cycle of Indiction and the Julian Period.

The Lunar Cycle is a period of 19 years. Its object is
to accommodate the computation of time by the moon to
the computation by the sun, or to adjust the solar and lunar

260 MATHEMATICAL APPLICATIONS

years. The nearest division of the year by months is into
twelve, but twelve lunations fall short of the solar year by
about eleven days. Of course, every change in the moon
in any year will occur eleven days earlier than it did in
the preceding year, but at the expiration of nineteen years
they occur again nearly at the same time. This cycle was
invented by Meton, an Athenian astronomer who lived
about 430 b.c. The improvement was at the time received
with universal approbation, but not being perfectly ac-
curate it was afterward corrected by Eudoxus and subse-
quently by Calippus. The Cycle of Meton was employed
by the Greeks to settle the time of their festivals, and the
use of it was discontinued when these festivals ceased to be
celebrated. The Council of Nice, however, wishing to
establish some method for adjusting the new and full
moons to the course of the sun, with a view to determining
the time for Easter, adopted again the Meton Cycle, and
from its great utility they caused the numbers of it to be
written on the calendar in golden letters, which has ob-
tained for it the name of the Golden Number. This name
is still applied to the current year of the Lunar Cycle and
is always given in the almanacs.

The Solar Cycle is a period of 28 years. Its use is to
adjust the days of the week to the days' of the month and
the year. As the year consists of 52 weeks and 1 day, it
is plain that it must begin and end on the same day, and
if 52 weeks and 1 day were the exact year, or if there were
no leap year, the year would, after seven years, begin again
on the same day. But the leap year, consisting of 52 weeks
and 2 days, interrupts the regular succession every fourth
year, and the return to the same day of the week is not
effected until four times seven, or twenty-eight years.
This cycle is employed particularly to furnish a rule for
finding Sunday or to ascertain the Dominical Letter.
Chronologers employ the first seven letters of the alphabet
to designate the seven days of the week, and the Dominical
Letter for any year is the letter which represents Sunday

CHRONOLOGY AND HOROLOGY 261

for that year. Tables are given for the purpose of finding
it in chronological and astronomical books.

The Cycle of Indiction is a period of 15 years. The
origin and primary use of this has been the subject of vari-
ous conjectures and discussions. It seems to have been
established by Constantine the Great, in the fourth cen-
tury, as a period at the end of which a certain tribute
should be pi i by the different provinces of the empire.
Public acts of the emperors were afterward dated by the
years of this cycle.

The Paschal Cycle is a period of 532 years, after which
Easter falls on the same day of the year.

The cycle which has been perhaps most celebrated is that
termed the Julian Period and was invented by Joseph
Scaliger. Its object was to furnish a common language
for chronologers by forming a series of years, some term
of which should be fixed, and to which the various modes
of reckoning might be easily applied. To accomplish this
he combined three cycles of the moon, sun and indiction,
multiplying 19, 28 and 15 into one another, which produces
7,980, after which all three cycles will return in the same
order, every year taking again the same number of each
cycle as before. This invention would be of great impor-
tance if there was no universally acknowledged epoch, or
fixed year, from which to compute, but its use is almost
entirely superseded by the general adoption of the Chris-
tian era as a fixed standard.

It is essential to correct and exact chronology that there
should be some fixed epoch to which all events may be
referred and be measured by their distance from it. It is of
comparatively little consequence what the epoch is, pro-
vided it is fixed and acknowledged, as it is perfectly easy to
compute in a retrograde manner the time before it, as well
as in a direct manner the time after it. The Greeks for a
long time had no fixed epoch, but afterward they reckoned
by Olympiads, periods of four years. These began yy6
B.C. The Romans often reckoned by lustrums, often by the

262 MATHEMATICAL APPLICATIONS

year of the consul or emperor. The building of the city
was their grand epoch, which began 753 b.c. The present
era began to be used about 360 a.d., according to some
writers, but others state that it was invented by Dionysius,
a monk, about 527 a.d.

The Mohammedan Era, or Hegira, was founded on the
flight of Mohammed from Mecca to Medina, 622 a.d. One
of the interesting vagaries of chronological history is
found in the Era of the French Republic, which the revo-
lutionists attempted to establish. This was introduced in
J793> with a formal rejection of the Sabbath and of the
hebdominal week and a novel arrangement and pedantic
nomenclature of the months. The 22d of September was
fixed as the beginning of the year. The year consisted of
twelve months of thirty days each, which were divided,
not by weeks, but into three decades or periods of ten
days. As this would comprise but 360 days, five were
added at the close of the last month of the year, called
complementary days, and at the close of every fourth year
a sixth day was added, called the Day of the Republic.
The cycle of the four years was termed the Franciade.
This calendar was used about twelve years. The Gre-
gorian calendar was restored on January 1, 1806.

The mechanical instruments that have been made for
the measurement of time present in themselves an inter-
esting pictorial commentary upon the more abstract sci-
ence of chronology. Horology, the art of measuring the
hours or any definite small portions of time, began when
man first marked the shadow of any upright object and
noted its movements in relation to the apparent movement
of the sun. The next step came when he noted that a staff
placed in the ground and pointed toward the north will
always at a particular hour of the day throw a shadow in
the same direction. This fact, undoubtedly observed by
the Babylonians in the most ancient times, suggested the
idea of the sun-dial. This instrument consists of two parts,
the "gnomon," or upright staff or "style," usually a piece

CHRONOLOGY AND HOROLOGY 263

of metal, always placed parallel to the earth's axis and
therefore pointing to the north star, and the dial, another
plate of metal or stone, usually horizontal, on which are
marked the directions of the shadow for the several hours,
their halves and quarters and sometimes smaller divisions.

Fig. 4 — Simple Sun-dial Face, Measured for Latitude of
New York.

Sun-dials were generally known in ancient times. It is
suggested that the circular rows of stones built by the
Druids were used to mark the sun's path and to indicate the
times and seasons. Obelisks are also supposed by some
writers to have been used for measuring sun shadows.
The Greeks were perfectly acquainted with the method of
making sun-dials with inclined styles. Small portable sun-

264

MATHEMATICAL APPLICATIONS

dials were much prized before the introduction of watches,
and were provided with compasses by which they could
be turned round so that the style pointed to the north.
Sun-dials have been found in the ruins of ancient cities

Fig. 5 — Dial Set Up with South Exposure.

of Greece, in Rome, in the excavations of Pompeii and
Herculaneum, and many medieval specimens are well
known.

The objections to a sun-dial are that the shadow of the
style is not sufficiently well defined to give very accurate

CHRONOLOGY AND HOROLOGY

265

results and that refraction, which always makes the sun
appear a little too high, throws the shadow a trifle toward
noon at all times. That is, the time is a little too fast in
the morning and too slow in the afternoon. More than
that, a correction is always necessary in order to find civil,
or clock, time.

The simplest form of sun-dial is the best, and as a regu-
lator of clocks the dial is good within one or two minutes.
The "noon mark" is simply a north-and-south line marked
on a horizontal plane and the style is any object fixed to

en

P3l

Fig. 6 — Early Currency of the United States, Showing
Sun-dial.

the dial and slanted so as to point to the north pole. On
four days of the year the sun is right with mean time and
the shadow mark may be set on those days, or on other
days the noon mark may be set by consulting the table in
the almanac which shows the variation of the sun from
civil time in even minutes. Thus on October 10, 1909, the
noon mark could be made by the shadow of the style at
11.47 by tne clock and it would be right for all time to
come.

A device less dependent upon the climatic conditions
was the water-clock, or clepsydra. It is said that this
instrument was in use among the Chaldeans and an-
cient Hindus. Sextus Empiricus says that the Chaldees

266 MATHEMATICAL APPLICATIONS

used such a vessel for finding their astrological data, but
remarks that the unequal flowing of the water and the
alterations of atmospheric temperature rendered their cal-
culations inaccurate.

In this instrument the water, which falls drop by drop
from the orifice of one vessel into another, floats a light
body that marks the height of the water as it rises against
a graduated scale and thus denotes the time that has
elapsed. As a measure of hours of the day in countries
such as Egypt, where the hours were always equal and
thus where the longer days contained more hours, the
water-clock was very suitable, but in Greece and Rome,
where the day, whatever its length, was always divided
into twelve hours, the simple water-clock was as unsuit-
able as a modern clock would be, for it always divided the
hours equally and took no account of the fact that by such
a system the hours in summer were longer than in winter.

In order, therefore, to make the water-clock available in
Greece and Italy it became necessary to make the hours
unequal and to arrange them to correspond with unequal
hours in the Greek day. This plan was accomplished by
placing a float upon the water in the vessel that measured
the hours, and on the float stood a figure made of thin
copper, with a wand in its hand. This wand pointed to
an unequally divided scale. A separate scale was provided
for every day in the year, and these scales were mounted
on a drum which revolved so as to turn round once in the
year. Thus as the figure rose each day by means of a cog-
wheel it moved the drum round one division or one-three-
hundred-and-sixty-fifth part of a revolution. By this
means the scale corresponding to any particular day of
winter or summer was brought opposite the wand of the
figure, and thus the scale of hours was kept true. In fact,
the water-clock, which kept true time, was made by arti-
ficial means to keep untrue time, in order to correspond
with the unequal hours of the Greek days. One of the
more complicated forms of the water-clock was probably

CHRONOLOGY AND HOROLOGY

267

invented by Ctesibus of Alexandria. In the Athenian
courts a speaker was allowed a certain number of amphorae
of water for his speech, the quantity dependent on the im-
portance of his suit. Both the simple and more elaborate
forms of clepsydrae were introduced into Rome in the
second century b.c.

Chinese Water Clock.

second century b.c. A Chinese water-clock, reputed to be
over 3,000 years old, consisted of four copper jars, on
ascending steps, with small openings and filled every morn-
ing. The purpose of the series was to obviate the irregu-
larity in dropping which would be caused by the greater
weight in the first jar at the beginning of the day.

The running of fine sand from one vessel into another
was found to afford a still more certain measure of time,
so the hour-glass came into being. This instrument con-
sists of two bulbs of glass united by a narrow neck; one

268 MATHEMATICAL APPLICATIONS

of the bulbs is nearly filled with dry sand, fine enough to
run freely through the orifice in the neck, and the quantity
of sand is just as much as can run through the orifice in an
hour, if the instrument is to be really an hour-glass; in a
minute, if a minute-glass. It is said that King Alfred
observed the lapse of time by noting the gradual shorten-
ing of a lighted candle.

The pendulum is the mechanical basis of modern
clocks and was first scientifically investigated by Galileo
in the latter half of the sixteenth century. The story runs
that while he was praying one day in the cathedral at Pisa
his attention was arrested by the motion of the great lamp
which, after being lighted, had been left swinging. Galileo
proceeded to time its oscillations by the only watch in his
possession — namely, his own pulse. He found the times,
as near as he could tell, to remain the same, even after the
motion had greatly diminished. Thus was discovered the
isochronism of the pendulum. Later experiments carried
out by Galileo showed that the time of oscillation was inde-
pendent of the mass and material of the pendulum and
varied as the square root of its length.

Galileo's invention did not become generally known at
that time, and fifteen years later, in 1656, Christian Huy-
gens independently invented a pendulum clock which met
with general and rapid appreciation. The honor of this
invention belongs, therefore, to both Galileo and Huygens.

Wheel-work had been known long before the time of
Galileo and had been skilfully applied by Archimedes.
When therefore some sort of wheel mechanism was needed
to keep the pendulum oscillating, the mechanical means
were at hand. Galileo saw that if the pendulum could be
kept swinging, a timepiece could be constructed which
would be mathematically perfect. There must be some
reservoir of force such that when a pendulum comes back
and touches it the touch shall allow some pent-up power to
escape and to drive the pendulum forward. An arrange-
ment of this kind was contrived by Galileo. He provided

CHRONOLOGY AND HOROLOGY

269

a wheel with a number of pins around it. The pendulum
had an arm attached to it and there was a ratchet with a
projecting arm which engaged with the pins. This ar-
rangement is called an escapement.

The type of escapement invented by Galileo was, for
practical purposes, full of imperfections, and it was left
for later inventors to modify his ideas and to improve on

Fig. 8 — Escapement Principle. Fig. 9 — Mechanism of

'Grandfather's Clock.'

them until an accurate timepiece was achieved. The
balance-wheel was invented, which does the work of the
pendulum, and various escapements, such as the crown or
verge escapement, the anchor-and-crutch escapement, the
dead-beat escapement and the gravity escapement, have
all taken their place in the development of the timepiece.
The prime requisite of a good escapement is that the im-
pulse communicated to the pendulum be invariable, not-
withstanding any irregularity or foulness in the train of
wheels. The compensating balance-wheel is a balance-

2;o MATHEMATICAL APPLICATIONS

wheel whose rim is formed of two metals of different ex-
pansive powers, so arranged that the change of size of the
wheel, as the temperature rises or falls, is compensated for
by the change in position of the parts of the rim.

The anchor escapement was employed in that popular
and excellent timepiece used throughout the eighteenth
and in the early part of the nineteenth century and now
known as the Grandfather Clock. In this clock the pen-
dulum is hung from a strip of thin steel spring, which
allows it to oscillate and supports it without friction. This
manner of supporting pendulums is now very much in use.

The watch differs from the original clock in that it has a
vibrating wheel instead of a vibrating pendulum. As in a
clock gravity is always pulling the pendulum down to the
bottom of its arc, but does not fix it there, because the mo-
mentum acquired during its fall from one side carries it up
to an equal height on the other, so in a watch a spring, gen-
erally spiral, surrounding the axis of the balance-wheel, is
always pulling this toward a middle position of rest, but does
not fix it there, because the momentum acquired during its
approach to the middle position from either side carries it
just as far past on the other side, and the spring has to
begin its work again. The balance-wheel at each vibra-
tion allows one tooth of the adjoining wheel to pass, as
the pendulum does in a clock, and the record of the beats
is preserved by the wheel which follows. A main spring
is used to keep up the motion of the watch, instead of the
weight used in a clock, and as a spring acts equally well
whatever be its position, the watch keeps time altho car-
ried in the pocket or in a moving ship. In winding up
a watch one turn of the axle on which the watch is fixed
is rendered equivalent by the train of wheels to about 400
turns or beats of the balance-wheel, and thus the exertion
during a few seconds of the hand which winds up gives
motion for twenty-four or thirty hours.

The laws of the mechanism of the clock can easily be
understood. The experiments with the pendulum and

CHRONOLOGY AND HOROLOGY 271

with springs revealed certain principles which were early
reduced to six and can be stated thus :

(1) A harmonic motion is one in which the accelerating-
force increases with the distance of the body from some
fixed point.

(2) Bodies moving harmonically make their swings
about this point in equal times.

(3) A spring of any sort or shape always has a resti-
tutional force proportional to the displacement,

(4) And therefore masses attached to springs vibrate
in equal times, however large the vibration may be.

(5) The bob of a pendulum, oscillating backward and
forward, acts like a weight under the influence of a spring
and is therefore isochronous.

(6) The time of vibration of a pendulum is uninfluenced
by changes in the weight of the bob, but is influenced by
changes in the length of the pendulum rod. The time of
vibration of a mass attached to a spring is influenced by
changes in the mass.

Early attempts were made to use a pendulum clock at
sea, suspending it so as to avoid disturbance to its motion
by the rocking of the ship. These proved vain. It there-
fore became desirable that a watch with a balance-wheel
should be contrived to go with a degree of accuracy in
some respects comparable with the accuracy of a pendu-
lum clock. To encourage inventors an Act of Parliament
was passed in the thirteenth year of Queen Anne's reign
promising a large reward to any one who would invent a
method of finding the longitude at sea true to half a de-
gree— that is, true to thirty geographical miles. If the
finding of the longitude were to be accomplished by the
invention of an accurate watch, then this involved the use
of a watch that should not, in several months' going, have
an error of more than two minutes, or the time the earth
takes to turn through half a degree of longitude.

This was the problem which John Harrison, a carpenter
of Yorkshire, made it his life business to solve. His

272 MATHEMATICAL APPLICATIONS

efforts lasted over forty years, but at the end he succeeded
in winning the prize. His instruments have been much
improved by subsequent inventors and have resulted in
the construction of the modern ship's chronometer, a
large watch about six inches in diameter, mounted on
axles, in a mahogany box. The marine chronometer dif-
fers from the ordinary watch in the principle of its escape-
ment, which is so constructed that the balance is free from
the wheels during the greater part of its vibration, and
also in being fitted with a compensation adjustment similar
to that in the balance-wheels of the finer clocks and
watches. The balance-spring of the chronometer is heli-
coidal, that of the watch spiral.

One of the inventions of modern times is the pneumatic
clock, which is one of a series of clocks governed by pul-
sations of air sent at regular intervals to them through
tubes by a central clock or regulator. The movement of
the central clock compresses the air in the tube and causes
a bellows to expand on each dial, thus moving the hands.

Another recent invention is a clock without wheels or
pendulum. It consists solely of two inclined plates with
zigzag tracks and the clock framework supporting them.
A perforated disk connected with the shaft which journals
in the frame and two ball weights suspended in each tower
and connected by means of a cord to the shaft successfully
furnish the motive power. These weights are raised daily.

So the ingenuity of man goes on measuring this earthly
element of time. Laplace said that "Time is to us the
impression left on the memory by a series of events,"
and that motion, and motion only, can be used in meas-
uring it. Thus it is motion, whether of the shadow on
the grass, the dropping of water or the continuous oscilla-
tions of a swinging body, which is the necessary and
unvarying element in all the measurements of time.

CHAPTER III

SURVEYING AND NAVIGATION

One of the earliest necessities of civilization was a sys-
tem of ascertaining by measurement the shape and size of
any portion of the earth's surface and representing the
results on a reduced scale on maps. This is the surveyor's
art and is supposed to have originated in Egypt, where
property boundaries were annually obliterated by the in-
undations of the Nile. In Rome surveying was considered
one of the liberal arts, and the measurement of lands was
entrusted to public officers, who enjoyed certain privileges.

Julius Caesar conceived the idea of a complete survey of
the whole empire. For this purpose three geometers were
employed : Theodotus, entrusted with the survey of the
northern provinces ; Zenodoxus, with the survey of the
eastern, and Polycletus, of the southern. It is stated that
a partial survey was finished 19 b.c. and the whole com-
pleted in 6 a.d. The materials collected were lodged in the
public archives, receiving from time to time marks and
notes to designate the various changes in the provinces. It
was consulted by Pliny. The numerous changes at length
required the construction of another chart with corrected
measurements, which was effected about 230 a.d. under
Alexander Severus. Of this chart the celebrated docu-
ment Tabula Peutingerianse is supposed by some modern
critics to be an imperfect copy.

The mathematicians of the Alexandrian school made a
distinct contribution to the art of surveying. Most authori-
273

274 MATHEMATICAL APPLICATIONS

ties believe Heron of Alexandria to be the author of
"Dioptra," tho some writers have attributed it to another
mathematician of a later date by the name of Heron.
"Dioptra," says Venturi, "were instruments resembling
the modern theodolites. The instrument consisted of a
rod, four yards long, with little plates at the end for aim-
ing. This rested upon a circular disk. The rod could be
moved horizontally and also vertically. By turning the rod
around until stopped by two suitably located pins on the
circular disk, the surveyor could work off a line perpen-
dicular to a given direction. The level and plumb line
were also used." Heron explains, with the aid of these
instruments and of geometry, a large number of surveying
problems, such as to find the distance between two points,
only one of which is accessible, or between two points
which are visible but both inaccessible; from a given
point to run a perpendicular to a line which cannot be
approached; to find the difference of level between two
points, and to measure the area of a field without entering
it. The "Dioptra" discloses considerable mathematical
ability, but it gives rules and directions without proof.

The higher development of the art of surveying, like so
many other mechanical arts depending on mathematics, is
of comparatively recent date. The enormous areas of new
land opened for habitation in the New World, the con-
struction of railroads, bridges and water works have em-
ployed the keenest practical minds in solving large survey-
ing and engineering problems, of which the Government
does a large part.

Surveys may be divided into three classes: First, those
made for general purposes, or information surveys, which
may be exploratory, geodetic, geographic, topographic or
geologic; second, those made for jurisdictional purposes,
or cadastral surveys, which define political boundaries and
those of private property and determine the enclosed areas ;
third, there are surveys made for construction purposes, or
engineering surveys, on which are based estimates of the

SURVEYING AND NAVIGATION 275

cost of public and private works, such as canals, railways,
water supplies and the like and their construction and im-
provement.

The topographic survey, one of those in the first class,
is made for military, industrial and scientific purposes.
The topographic map, made directly from nature by meas-
urements and sketches on the ground, is the mother map
from which all others are derived. It shows with accuracy
all the drainage, relief and cultural features which it is
practicable to represent on the scale chosen. These fea-
tures are numerous and important, if the government maps
of the advanced modern nations are taken as a model. On
the topographical maps issued by the United States Geo-
logical Survey are exhibited hydrography, or water fea-
tures, such as ponds, streams, lakes and swamps; hypsog-
raphy, or relief of surface, as hills, valleys and plains, and
the features constructed by man, as cities, roads and vil-
lages, with the names and boundaries.

The uses of topographic maps are many. For the pur-
poses of a national government or a State they are invalu-
able, as they furnish data from which may be determined
the value of projects for highway improvement, for rail-
ways, for city water supply and sewerage and for the sub-
division into counties, townships and the like. They serve
the military department in locating encampment grounds,
in planning practice or actual operations in the field and,
during war, in indicating the precise situations of ravines,
ditches, buildings, hills and streams. The Post-office De-
partment utilizes them in considering all problems con-
nected with the changing of mail routes, star routes, and
especially in connection with contracts and assignments of
rural free-delivery routes. In the future wooded areas
are to be indicated on the United States Government maps,
so that foresters will find them useful, as well as those peo-
ple who are investigating mineral resources, water power
and land reclamation.

The operations involved in surveying are the measure-

276

MATHEMATICAL APPLICATIONS

ment of distances, level, horizontal, vertical and inclined,
and of angles, horizontal, vertical and inclined, and the
necessary drawing and computing to represent properly on
paper the information obtained by the field work. If the
tract to be surveyed is so large that the curvature of the
earth's surface must be taken into account, it is a geodetic
survey.

The practical basis of surveying is the mathematical
theory of the triangle and the solution of the various prob-
lems of the triangle by means of geometrical formula? and

l-l

-2»m

B

*B jj W ]E

Fig. 10 — Examples of Triangulation.

logarithms. If two angles and one side of a triangle are
known, the third angle and the length of the other two
sides can be computed by easy geometrical rules. The use
of logarithms, which are artific'al numbers so devised that
they shorten the processes of multiplication and division,
reduces the work of computing the long tables of angles
and measurements which often falls to the work of the
surveyor.

Now an actual measurement of a portion of the earth's
surface can be made by any one by means of a rope, a tape
or a chain, thus insuring actual knowledge of the length
of one side, called the base line, of the future triangle. By
means of a telescope and a level, together with other in-

SURVEYING AND NAVIGATION 277

genious devices, placed at the end of the base line, two
objects in a given area are sighted, as, for instance, a
church steeple in one direction and a signal placed at the
other end of the base line. The three points are the apexes
of the triangle, formed by connecting lines. The angles can
be measured by the instruments at the surveyor's hand,
the length of the base line is known ; therefore the length
of the other two sides can be computed.

This principle of triangulation has many variations, and
in actual practice there are many complicating elements.
The topography of an area of any size hangs, not on one,
but on a system of triangles. In the preliminary work an
arbitrary line, or meridian, is established, from which to
compute the measurements. But if the actual position is
required — that is, the location on the earth's surface ac-
cording to latitude and longitude — observations of the sun
or of the fixed stars must be made and the measurements
recorded. The elevation of the pole measures the distance
of the observer from the equator, and this distance is the
latitude of a place, north or south, the pole lying midway
between the highest and lowest positions of the pole star.
In practice other means, not quite so accurate, but useful,
may be used for determining the latitude. One of the
common methods, exact enough for ordinary geographical
reconnaissances, is to measure the angular altitude of the
sun when on the meridian, and from this altitude, with the
aid of the declination taken from the Nautical Almanac,
and with correction for refraction, the latitude is obtained.
This method on land requires the use of an artificial hori-
zon in place of the natural.

But to fix the position of any place on the globe it is
necessary to know at what point on the circle of latitude
it lies, or its longitude. This is a more difficult matter
and one that requires for its determination, astronomically,
the introduction of the element of time. Strictly speaking,
longitude is the angle at the pole contained between two
meridians, one of which, called the prime meridian, passes

2;8 MATHEMATICAL APPLICATIONS

through some conventional point from which the angle
is measured. The longitude of the conventional point is
zero, and longitudes are reckoned east and west from it to
180 degrees in arc and to 12 hours in time, 15 degrees
being equal to one hour. In Great Britain universally and
in the United States generally geographers reckon from
the meridian of the transit circle at the Royal Observatory
of Greenwich in England; the meridian of Washington is
also used occasionally in the United States. On shore the
most accurate method is to compare the time of the two
places by means of the electric telegraph; while at sea,
the local time being determined by observation of some
celestial object, it is compared with Greenwich time, as
shown by a chronometer carefully set and regulated before
sailing.

The instruments used in surveying are numerous, but
the more important are the measuring chain, the vernier,
the level, the barometer and compass, the transit, the sex-
tant and theodolite.

The instruments commonly used in the measurement of
angles are the compass, which determines directions and,
indirectly, angles, and the transit, which determines angles
and, indirectly, directions. The sextant is an angle-meas-
uring instrument, the use of which is confined to certain
particular operations, such as the location of soundings
taken offshore and angular measurements at sea.

The compass consists of a line of sight attached to a
graduated circular box, in the center of which is hung, on
a pivot, a magnetic needle. At any place on the earth's
surface the needle, if allowed to swing freely, will assume
a position in what is called the magnetic meridian of the
place. If the direction of any line is required, the compass
may be placed at one end of the line and the line of sight
may be made to coincide with the line. The needle lying
in the magnetic meridian and the zero of the graduations
of the circular needle-box being in the line of sight, the

SURVEYING AND NAVIGATION 279

angle that the line on the ground makes with the magnetic
meridian is read on the graduated circle.

At a very few places on the earth's surface the needle
points to the true north. When it does not point thus, the
angle that the magnetic meridional plane makes at any
point with the true meridional plane is called the magnetic
declination. This declination is subject at every place to
changes, regular and irregular, so that the magnetic bear-
ings of lines run with the compass are required to be re-
duced to the true bearings.

The sextant is an important instrument in surveying
and navigation, used for measuring the angular distance
of two stars or other objects, or the altitude of a star
above the horizon, the two images being brought into coin-
cidence by reflection from the transmitting horizon-glass.
In the hands of a competent observer, the work of the
sextant is extremely accurate. The first inventor of the
sextant (quadrant) was Newton. A description of this
instrument was found among his papers after his death,
not, however, until after its reinvention by Thomas
Godfrey, of Philadelphia, in 1730. This is the instrument
used by seamen for observations for finding latitude and
longitude.

The transit is used for measuring horizontal angles,
and resembles a theodolite, but is not intended for very
precise measurements. The theodolite has appeared in a
variety of forms. Its purpose is to measure horizontal,
and sometimes vertical, angles. It consists essentially of
a telescope which has a motion about a horizontal axis
which rests in two pillars which are perpendicular to the
axis of rotation of the telescope. These pillars are fixed
at right angles to a plate, which turns upon a vertical
axis and to which is attached a vernier. Around this is
a second plate, graduated, and concentric with the first.

It may also be provided with a vertical circle, and if
this is not very much smaller than the horizontal circle
the instrument is called an altazimuth. If it is provided

280

MATHEMATICAL APPLICATIONS

with a delicate striding level and is in every way con-
venient for astronomical work, it is called a universal
instrument. A small altazimuth with a concentric mag-

Fig, ii — Essential Parts of Theodolite.
A., telescope ; B., eye tube ; C, Ratchet and pinion for moving eye
tube ; D., Screw for adjustment of cross wires ; E., axis of
rotation ; F., pillars supporting axis ; G., compass ;H., upper
plate carrying vernier; I., lower (graduated) plate; J., clamp
and tangent screws for upper plate ; K., levels ; M., ball and
socket joint with four leveling screws; N., spindle axis of
rotation of azimuth plate ; T., tripod.

netic compas is called a surveyor's transit. A theodolite
in which the whole instrument, except the feet and their

SURVEYING AND NAVIGATION 281

connections, turns relatively to the latter, and can be
clamped in different positions, is called a repeating circle.

A hydrographic survey is one that has to do with any
body of water, and may be undertaken for any one of a
number of purposes. One of the most important uses of
hydrographic surveying is to supply maps of the bed of
the sea, or harbor, or bay, or river for the information
of seamen. In this case it is necessary to locate the chan-
nels, dangerous rocks and shoals. In many cases the work
of the hydrographic surveyor goes much further than this,
and determines the cross-sections of streams, their veloci-
ties, their discharge, the direction of their currents, and
the character of their beds.

The topography of the bed of a body of water is de-
termined by sounding — that is, measuring the depth of
the water. If many points are observed a contour map
of the bottom may be drawn, the water surface being the
plane of reference. For depths less than 15 or 20 feet a
pole is used. Soundings made in moderately deep water
are made with a weight, known as a lead, attached to a
suitable line. There is a deep-sea sounding machine,
by the aid of which soundings may be made to great
depths, with a close approach to accuracy. This result,
has been attained by a combination of improvements in
which great ingenuity has been displayed and in which
the inventive genius of Sir William Thomson has been
particularly conspicuous. The principal features of the
most perfect sounding-machine are: (1) The sinker,
which is a cannon-ball through which passes a cylinder
provided with a valve to collect and retain a specimen of
the bottom, the cylinder being, by an ingenious mechani-
cal arrangement, detached from the shot, which remains
at the bottom; (2) the line, made of steel wire, weighing
about 14^ pounds to the nautical mile; (3) machinery
for regulating the lowering of the sinker and for reeling
in the wire with the cylinder attached, in such a manner
that the irregular strain due to the motion of the ship

Fig. 12 — The Solar Transit.

SURVEYING AND NAVIGATION 283

may be guarded against and the danger of breakage thus
reduced to a minimum. In the deepest accurate sounding
yet made the bottom was reached at the depth of 4,655
fathoms.

The determination of the coast line is accomplished
by a general scheme of triangulation, just as the topo-
graphical map of land areas is determined by it; but the
necessity of taking observations from a ship makes the
practice somewhat different. A map of a section of coast
is the double product of the measurements of angles and
base lines and the soundings taken to determine the depth

Fig. 13 — Survey of Coast-line.

of the water. The survey is made by two parties, one on
shore and one in a boat sailing along the coast.

If the reckoning of a ship could be accurately kept as
she runs along a coast, a very good chart could be made
simply by taking exact bearings of various points on the
shore line and noting the time. The track of the ship
would be a base line, and the intersections of the bear-
ings would fix the positions of the shore line. The lati-
tude and longitude would be determined accurately at
intervals of forty or sixty miles, and the intervening"
points could be plotted by plane surveying methods. The
bearing of any terrestrial object can be determined from
a ship by astronomical methods, but owing to currents, lee-
way, and difficulties in steering, the accuracy of the track

284 MATHEMATICAL APPLICATIONS

base cannot be depended upon. Therefore the astronomi-
cal observations are made on shore with the transit and
zenith telescope.

The ship and shore parties proceed along the coast by
carefully determined stages, each party taking angular
measurements from three points and soundings. Both
parties take angular measurements from some fixed ob-
ject farther inshore, and by comparing observations, de-
termining the exact position of the ship at certain in-
tervals, and establishing a system of triangles not only
with the shore party, but with new fixed objects at each
stage, the data for coast line are obtained. The work can
be plotted on a polyconic chart to include the coast, the
scale depending on its extent.

The art of the land surveyor is closely allied to that
of the seaman, who is obliged to find his course, in any
extended voyage, by angular observations of the heavenly
bodies and the mathematical solution of the problems thus
offered. The mariner has more than an academic inter-
est in determining his position — it is a matter of life and
death to him, and navigation depends mainly upon the
acquisition of that knowledge.

Navigation is the art or science of directing the course
of vessels as they sail from one part of the world to an-
other. The management of the sails, or as it may be of
machinery, the holding of the assigned course by proper
steering, and the working of the ship generally pertain
rather to seamanship. The two fundamental problems of
navigation are the determination of the ship's position at
a given moment and the decision of the most advanta-
geous course to be steered in order to reach a given point.
The methods of solving the first are, in general, four:
(i) By reference to one or more known and visible land-
marks; (2) by ascertaining through soundings the depth
and character of the bottom; (3) by calculating the di-
rection and distance sailed from a previously determined
position; and (4) by ascertaining the latitude and longi-

SURVEYING AND NAVIGATION 285

tude by observations of the heavenly bodies. The places
of the sun, moon, planets and fixed stars are deduced
from observation and calculation, and are published in
nautical almanacs, the use of which, together with loga-
rithmic and other tables computed for the purpose, is
necessary in reducing observations taken to determine
latitude, longitude, and the error of the compass.

The calculation of a ship's place at sea, independently
of observations of the heavenly bodies, is called dead-
reckoning. The ship's position is calculated simply from
the distance she has run by the log and the courses steered
by the compass, this being rectified by due allowances
for drift and leeway. In very early times dead-reckoning
was an important branch of knowledge, in which the in-
struments for measuring time, such as the sand-glass,
played a considerable part. The sand-glass is still found
on many sailing ships using the old-fashioned 'log/ The
earliest mode of measuring the speed of a vessel at sea
was by throwing overboard a heavy piece of wood, so
shaped that it resisted being dragged through the water,
and with a line tied to it. The block of wood was the
log, and the string had knots in it, so arranged that when
one knot ran through a sailor's fingers in half a minute,
measured by the sand-glass, the vessel was going at the
speed of one nautical mile an hour, ten knots on the line
ten miles, and so forth. The nautical mile is of such a
length that 60 of them constitute one degree on a great
circle of the earth; therefore the knots are 50 feet and
7 inches apart.

Patent logs are generally used now at sea, those most
commonly found on vessels being either the harpoon or
the taffrail log. The harpoon log is shaped like a torpedo,
and has at one end a metal loop to which the log-line is
fastened, and at the other fans which cause the machine
to spin round as it is drawn through the water. The
spinning of the instrument sets a clockwork machinery
in motion, which records the speed of the vessel upon

286 MATHEMATICAL APPLICATIONS

dials, the rotation of the instrument being, of course, de-
pendent upon the rate at which it is dragged through the
water. In the taffrail log the recording machinery is
secured to the taffrail, and the fan is towed astern at the
end of a long line.

If the sea were a smooth plane surface, without cur-
rents or tides, it would be a simple matter to fix ac-
curately the position of a vessel, and to take her from one
place to another on the earth's surface by dead-reckon-
ing only ; but as it is in constant motion, influenced by
irregular currents and tides and the drift of the waves,
it becomes necessary to have some more accurate method
to insure safe navigation, and this is to be found in the
system of observation of the heavenly bodies, or, in other
words, in the science of nautical astronomy.

Thus the angular measurement of the sun and the fixed
stars, by means of the sextant, becomes a necessity, and
also the solution of the triangle problems by means of
logarithms and trigonometrical formulae. Since the sailor
always has the horizon and zenith with him, he can find
his latitude at any time by taking the meridian altitude of
the sun and correcting that by the declination found in
his nautical almanac. His longitude will be found by the
aid of the sun and a chronometer. The apparent time
at sea he will find by observing the sun's hour-angle ; ap-
parent time must be turned into mean time by applying
the equation of time; and mean time at ship must be
compared with mean time at Greenwich, as ascertained
by the chronometer. The difference between these two is
the ship's longitude.

Nautical almanacs are published by the governments of
Great Britain, the United States, and most other maritime
powers. These are almanacs for the use of navigators
and astronomers, in which are given the ephemerides of
all the bodies of the solar system, places of the fixed
stars, predictions of astronomical phenomena, and the

SURVEYING AND NAVIGATION 287

angular distances of the moon from the sun, planets, and
fixed stars.

The laws of the tides and of storms must also be
studied by the seaman, especially the "law of storms" in
a navigational sense. This expression generally means
the law of circular storms or cyclones, and should be un-
derstood by all who are responsible for the safe conduct
of foreign-going ships. Owing to the nature of the
cyclone, very fair general rules can be made which assist
the mariner in steering a course away from the storm
center. A good many generalizations have been made
in regard to winds in a wide sense. Airey found that the
wind never blows steadily for any period of time ex-
cept from eight points of the compass. When in any
other quarter it is merely shifting round to one of these
points. It never blows at all directly from the south.
The two most prevalent winds are south-southwest and
west-southwest. The first serious study of the circula-
tion of winds on the earth's surface was instituted at the
beginning of the second quarter of this century by W. H.
Dove, William C. Redfield, and James P. Espy, followed
by researches of W. Reid, Piddington, and Elias Loomis.
But the deepest insight into the wonderful correlations
that exist among the varied motions of the atmosphere
was obtained by William Ferrel (1817-1891). He was
born in Fulton County, Pa., and brought up on a farm.
In 1885 appeared his Recent Advances in Meteorology.
In the opinion of a leading European meteorologist, Julius
Hann, of Vienna, Ferrel has "contributed more to the ad-
vance of the physics of the atmosphere than any other liv-
ing physicist or meteorologist."

Ferrel teaches that the air flows in great spirals toward
the poles, both in the upper strata of the atmosphere and
on the earth's surface beyond the 30th degree of latitude ;
while the return current blows at nearly right angles
to the above spirals, in the middle strata as well as on
the earth's surface, in a zone comprized between the par-

288 MATHEMATICAL APPLICATIONS

allels 300 N. and 300 S. The idea of three superposed
currents blowing spirals was first advanced by James
Thomson, but was published in very meager abstract.

Another theory of the general circulation of the atmos-
phere was propounded by Werner Siemens, of Berlin,
in which an attempt is made to apply thermodynamics to
aerial currents. Important new points of view have been
introduced recently by Helmholtz, who concludes that
when two air currents blow one above the other in differ-
ent directions, a system of air waves must arise in the
same way as waves are formed on the sea. He and A.
Oberbeck showed that when the waves on the sea attain
lengths of from 16 to 33 feet, the air waves must attain
lengths of from 10 to 20 miles and proportional depths.
Superposed strata would thus mix more thoroly and
their energy would be partly dissipated. From hydro-
dynamical equations of rotation Helmholtz established the
reason why the observed velocity from equatorial regions
is much less in a latitude of, say, 200 or 300 than it
would be were the movements unchecked.

Another science bearing directly on navigation is the
construction of vessels, both in its architectural aspect
and in its relation to magnetism. The earth being a mag-
net, it induces magnetism in all things on its surface.
When an iron ship is being built, the hammering which
she undergoes causes magnetism of a more or less per-
manent character to be induced in her. This is known as
sub-permanent magnetism, because tho a ship rarely
loses it altogether, it alters very much after the vessel is
launched, through change of position, through being
knocked about in a heavy sea, and from other causes.

In the case of a ship built head south in northern lati-
tudes her blue polarity will be in her bow, and the north
point of her compass needle will be attracted to it. This
will cause westerly deviations as the ship's head passes
through the western half of the compass and easterly
when through the eastern. If her head is north when

290 MATHEMATICAL APPLICATIONS

building her stern will have blue polarity, and she will
have easterly deviation with her head in the western
semicircle of the compass and westerly deviation with
her head in the eastern semicircle. With her head east
when building she will have more blue polarity in her
starboard side than in her port, and with her head west
when building there will be easterly deviation on southerly
courses and westerly deviation on northerly.

A ship, like everything else, has its center of gravity,
tho this center is not a fixed point. It varies with
every change in the position and quantity of the weights
in her. A ship has also her center of buoyancy. This is
the geometrical center of her immersed portion, and its po-
sition can be arrived at with great certainty. Thus, a ves-
sel floating upright and at rest will fulfil certain condi-
tions. First, she will displace a weight of water equal
to her own weight; secondly, her center of gravity will
lie in one and the same vertical line with the center of
gravity of the volume of water displaced, and in that line
is the center of buoyancy.

If weights are moved in a vessel laterally the position
of her center of gravity is changed laterally, too; but
when she is heeled by wind or sea no change occurs in it.
The buoyancy, acting upward through the center of
buoyancy, shifting as it does from side to side as a ship
is heeled over or rolls through the action of wind or
sea, is the upward righting force mainly to be relied upon
to keep a vessel from capsizing.

The knowledge of mathematical laws and principles is
necessary to good seamanship, but perhaps in no art is the
practical and actual handling of apparatus more useful
than in that of the mariner. Theory can but lead the
learner to the edges of the subject; science and practice
must go hand in hand before any substantial acquirements
can be gained.

CHAPTER IV

MECEIANICAL PRINCIPLES

It is the privilege of the modern to make the most of
an environment of mechanism, a development consequent
upon the growing complexity of society. This, while
it adds greatly to the luxury of the whole, reduces the
sphere of the individual, making it no longer possible
to be well versed in many lines; the day of the Jack-at-
all-trades is past and the day of the expert has come.

Numbers form the connecting link between theory and
the application of theory to practical arts. In every me-
chanical principle mathematical formulae are implied,
tho they may be extremely simple. It is for the mathe-
matician to find out how far experimental confirmation
of a theory can be pushed and where a new hypothesis is
necessary. Facts apparently unconnected are found to
have their origin in a common source, and often only
the mathematician can trace their connection. More than
this, the mathematician is able to draw corollaries and
secondary truths from a given principle which the ex-
perimentalist alone does not discover.

"Mechanical science,'' said William J. M. Rankine, "en-
ables its possessor to plan a structure or machine for a
given purpose without the necessity of copying some ex-
istent example; to compute the theoretical limit of the
strength and stability of a structure or the efficiency of a
machine of a particular kind; to ascertain how far an
291

292 MATHEMATICAL APPLICATIONS

actual structure or a machine fails to attain that limit, and
to discover the cause and remedy of such shortcoming; to
determine to what extent, in laying down principles for
practical use, it is advantageous for the sake of simplicity
to deviate from the exactness required by pure science;
and to judge how far an existing practical rule is founded
on reason, how far on custom, and how far on error.''
A signal illustration of the truth of these words is
offered in the famous instance of falling bodies. Aristotle
proved to his own satisfaction, it seemed, and told the

Fig. is — Arm as a Lever.

Weight is raised by shortening of muscle, m., muscle ; w., weight ;
x., point of application ; y., fulcrum.

world at large, that heavy bodies fall to earth faster
than lighter ones; and it was left for Galileo, more than
a thousand years later, to disprove a statement whose
truth or falsity, it would seem, might have been estab-
lished by any one. It required mathematical science to
confute experimental error.

Not only has mechanical nomenclature been largely
taken from animals, but many of the principal mechanical

MECHANICAL PRINCIPLES 293

devices have preexisted in them. Examples of levers of
all three orders are to be found in the bodies of animals.
The human foot contains instances of the first and sec-
ond and the fore arm of the third order of lever. The
knee cap is practically a part of a pulley. There are sev-
eral hinges and some ball-and-socket joints, with per-
fect lubricating arrangements. Lungs are bellows, and
the vocal organs comprize every requisite of a perfect
musical instrument. The heart is a combination of four
force pumps acting harmoniously together. The wrist,
ankle and spinal vertebrae form universal joints. The
eyes may be regarded as double-lens cameras, with power
to adjust local length, and able, by their stereoscopic ac-
tion, to gage size, solidity and distance. The nerves
form a complete telegraph system, with separate up-and-
down lines and a central exchange. The circulation of
the blood is a double-line system of canals, in which the
canal liquid and canal boats move together, making the
complete circuit twice a minute, distributing supplies
wherever needed, and taking up return loads wherever
ready without stopping. It is also a heat-distributing ap-
paratus, carrying heat from wherever it is generated, or
in excess, to wherever it is deficient, and establishing a
general average.

Archimedes was almost the only philosopher among the
ancients, so far as is known, who formed clear and cor-
rect notions concerning the simple machines. He acquired
firm possession of the idea of pressure, which lies at the
root of mechanical science, and of equilibrium. The proof
of the properties of the lever given in Archimedes' "Equi-
ponderance of Planes" holds its place in text-books to
this day. His estimate of the efficiency of the lever is
expressed in the saying attributed to him, "Give me a
fulcrum on which to stand, and I will move the earth."
The "Equiponderance" treats of solids, while the book
on "Floating Bodies" treats of hydrostatics, or the equi-
librium of fluids.

294 MATHEMATICAL APPLICATIONS

It was long a common practice for mechanicians to
recognise six simple machines, or six devices represent-
ing the first principles of mechanics. These are the pul-
ley, the lever, the wedge, the screw, the inclined plane
and the wheel and axle. In the latter part of the eight-
eenth century, however, La Grange simplified the mechani-
cal principles, including them all under two, the principle
of the lever and the principle of the inclined plane. Every
machine that exists, from the egg-beater to the escalator,
is constructed by the application of these principles or
a combination of them.

The lever consists of a bar or rigid piece of any shape,
acted upon at different points by two forces, which sev-
erally tend to rotate it in opposite directions about a
fixed axis. It was beautifully demonstrated by Archi-
medes that the power at one end and the weight, or re-
sistance, at the other are in equilibrium under certain
conditions, the simplest being the case in which the load
is ten times as great as the power, but the power is ten
times as far from the fulcrum as the load is from the
fulcrum; or, stated otherwise, the two forces are in equi-
librium when they are inversely as the length of their
respective arms. There are three different kinds of levers,
according to the relative positions of the three points,
the fulcrum, the point of application of power, and the
point of application of the load. The handle of a com-
mon pump is a lever of the first class, in which the ful-
crum is between the other two points. The piston and
the water are the weight, the hand of the worker is the
power, while the pivot on which the handle turns is the
fulcrum. The ordinary steelyard is another example of
a lever of this class.

The second class Is formed by levers in which the
weight is between the fulcrum and the power, as is illus-
trated by the wheelbarrow. The axle of the wheel is the
fulcrum in this case, the load in the barrow is the weight,

MECHANICAL PRINCIPLES 295

and the handles of the barrow are the levers. The boat
with its oars is another example of this class of levers.

In the third class of levers the point of application of
the power lies between the fulcrum and the load, and is
illustrated by the lifting of a ladder when one end is rest-
ing on the ground. These distinctions are of slight im-
portance, however, since they become confused as the
machines to which they are applied become more com-
plicated. The Archimedean laws, however, which apply
to levers are extremely simple, and illustrate the beauty
with which physical or mechanical phenomena, of ap-
parently diverse types, may often be reduced to law.
First, the two extreme forces must always act in the
same direction; secondly, the middle one must act in
the opposite direction and be equal to the sum of the
other two; and thirdly, the magnitude of the extreme
forces is inversely proportional to their distance from
the middle one.

Probably of all devices of man none is more fre-
quently in evidence than the rope tackle used in hoisting,
and known as the pulley. This is a contrivance for bal-
ancing a great force against a small one, or for lifting
a big load with a small power. Its sole use is to produce
equilibrium. It does not save work, unless indirectly in
some unmechanical way. The pulley is a lever with equal
arms; but when it turns, the attachments of the forces
are moved.

The wheel and axle, also one of the simple machines,
works indirectly on the principle of the lever. In its
primary form it consists of a cylindrical axle on which a
wheel, concentric with the axle, is firmly fastened. A
rope is usually attached to the wheel, and the axle is
turned by means of a lever; the rope acts as in the pul-
ley; that is, upon the principle of the lever, which ex-
plains all the possible phenomena exhibited by the pulley
and the wheel and axle, just as the principle of the in-

296 MATHEMATICAL APPLICATIONS

clined plane explains all the phenomena of the wedge and
the screw.

The inclined plane in mechanics is a plane inclined
to the horizon, or forming with a horizontal plane any
angle whatever except a right angle. It is one of the two
fundamental machines, the other being the lever. The
power necessary to sustain any weight on an inclined
plane is to the weight as the height of the plane to its

-Tower Moved by Windlass and Pulleys. (From a
sixteenth-century print.)

length. This was first proved by Stevin in the sixteenth
century. If the inclined plane, with its horizontal plane
as a base, and the line connecting the two planes be con-
sidered as a right-angled triangle, the weights propor-
tional to the hypotenuse and height of the triangle balance.
The screw and the wedge, both called simple machines,
are special applications of this principle. The wedge con-

MECHANICAL PRINCIPLES 297

sists of a very acute-angled triangular prism of some hard
material, which is driven in between objects to be sep-
arated, or into anything to be split. It is, of course, one
of the commonest of implements, as is also the screw;
but in the apparently simple action of these two devices
lie the germs of some of the most effective instruments
for increasing man's "natural" power. It is necessary
to understand the exact function of each part of this ap-
parently innocuous machine, the screw, in order to follow
its development in the more complicated inventions.

The screw is a cylinder of wood or metal having a
spiral ridge, the thread, running round it, usually turning
in a hollow cylinder in which a spiral channel is cut cor-
responding to the ridge. The convex and concave spirals,
with their supports, are often called the screw and nut,
and also the external or male screw and the internal or
female screw, respectively. The screw is virtually a
spiral inclined plane; only the inclined plane is commonly
used to overcome gravity, while the screw is more often
used to overcome some other resistance. Screws are right
and left, according to the direction of the spiral.

Screws have a variety of uses, the most important of
which are two. First, they are used for balancing forces,
as the jack screw against gravity, the propeller screw
against the resistance of water, and the screw-press
against elasticity. Secondly, they are used for magnify-
ing a motion and rendering it easily manageable and
measurable, as in the screw-feet of instruments, microm-
eter screws, and the like. Hunter's screw is a double
screw consisting of a principal male screw that turns in
a nut, but in the cylinder of which, concentric with its
axis, is formed a female screw of different pitch that turns
on a secondary but fixed male screw. The device fur-
nishes an instrument of slow but enormous lifting power
without the necessity of finely cut and consequently frail
threads. Everything else being equal, the lifting power
of this screw increases exactly as the difference between

298

MATHEMATICAL APPLICATIONS

the pitches of the principal male screw and the female
screw diminishes, in accordance with the principle of vir*
tual velocities.

Archimedes himself made several experimental applica-
tions of his screw, among which were a railway and a
machine for lifting water. In the railway a continuous
shaft rotates on pillars between two lines of rails, and
propels the car by means of a screw which engages in a
pedestal attached to the car. The instrument for lifting
water, technically called the "Archimedean screw," is

Fig. 17 — The Archimedean Screw.

made by forming a spiral tube within or by winding a
flexible tube spirally without a cylinder. When the cylin-
der is placed in an inclined position and the lower end
is immersed in water, its revolution will cause the water
to move upward through the spiral chambers.

The mechanical powers, as the six simple machines
have long been called, are often in evidence in modern
inventions almost in their original simplicity. The screw
propeller, for instance, consists of a continuous spiral
vane on a hollow core running lengthwise of a vessel.
This is but an extension and amplification of the screw
and was also devised by Archimedes. The modern screw
propeller is attached to the exterior end of a shaft pro-

MECHANICAL PRINCIPLES 299

truding through the hull of a vessel at the stern. It con-
sists of a number of spiral metal blades either cast
together in one piece or bolted to a hub. In some special
cases, as in ferry-boats, there are two screws, one at
each end of the vessel. In some war-vessels transverse
shafts with small propellers have been useot to assist in
turning quickly. An arrangement of screws now common
is the twin-screw system, in which two screws are ar-
ranged at the stern, each on one of two parallel shafts,
which are driven by power independently one of the
other. By stopping or slowing up one shaft while the
other maintains its velocity, very rapid turning can be
effected by twin screws, which have, moreover, the ad-
vantage that, one being disabled, the vessel can still make
headway with the other. Some vessels designed to attain
high speed have been constructed with three screws. A
very great variety of forms have been proposed for screw-
propeller blades; but the principle of the original true
screw is still in use. Variations in pitch and modifica-
tions of the form of the blades have been adopted with
success by individual constructors.

The actual area of the screw propeller is measured on
a plane perpendicular to the direction in which the ship
moves. The outline of the screw projected on that plane
is the actual area, but the effective area is, in good ex-
amples, from 0.2 to 0.4 greater than this; and it is the
effective area and the mean velocity with which the water
is thrown astern that determine the mass thrown back-
ward. The mass thrown backward and the velocity with
which it is so projected determine the propelling power.
A kind of feathering propeller has also been used, but has
not been generally approved.

The mechanism of nature has offered suggestions for
many inventions, one of which provides an illustration
of many others. The pedrail, for instance, which is a
rail moving on feet, is constructed on the principle of
the horse. A horse has practically two wheels, its front

300 MATHEMATICAL APPLICATIONS

legs one, its back legs the other. The shoulder and hip
joints form the axes and the legs the spokes. So the ped-
rail has wheels the spokes of which, to any number, are
connected at their outer ends by flat plates. As each angle
of the plates is passed, the wheel falls plumb on to the
next plate. The greater the number of spokes, the less
will be each successive jar, or step; and consequently the
perfect wheel is theoretically one in which the sides have
been so much multiplied as to be infintely short.

With the exception of Archimedes and a few mathema-
ticians of the Alexandrian school, the ancients and the gen-
erations of the Middle Ages slept, so far as mechanical sci-
ence was concerned, in an untroubled peace. Not until the
seventeenth century were some of the Aristotelian myths of
science banished, when Galileo aroused the mechanical
and scientific genius of the age.

Among the curious vagaries of imagination which have
deluded the human mind, none is more interesting than
the idea of perpetual motion, which has been followed for
centuries with fatuous hope. Perpetual motion, in a me-
chanical sense, is a motion that is preserved and continu-
ously renewed of itself without the aid of any external
cause. It is, however, one of the chimeras of the brain
which has its aspects of plausibility for the tyro.

Many historic machines purporting to display the power
of perpetual motion have brought their inventors to
poverty, if not to despair. One authoritative writer says:
"In order to produce a perpetual motion, we have only
to remove all the obstacles which oppose that motion ; and
it is obvious that if we could do this, any motion what-
ever would be a perpetual motion. But how are we to get
rid of these obstacles? Can the friction between two
touching bodies be entirely annihilated? Or has any sub-
stance yet been found that is void of friction? Can we
totally remove all the resistance of the air, which is a
force continually varying? And does the air at all times
retain its impeding force? These things cannot be re-

MECHANICAL PRINCIPLES

301

Fig. 18 — Ferguson's Machine to Show the Fallacy of Per-
petual Motion Schemes.

The axle is placed horizontally and the spokes turn in a vertical
position. The spokes are jointed, as shown, and to each of
them is fixed a frame in whi~h a weight, D, moves. When
any spoke is in a horizontal position, the weight, D, in it
falls down, and pulls the weighted arm, A, of the then verti-
cal spoke straight out, by means of a cord, C, going over the
pulley, B, to the weight, D. But when the spokes come about
to the left hand, their weights fall back and cease pulling, so
that the spokes then bend at their joints and the balls at
their ends come nearer the center on the left side as the
balls or weights at the right-hand side are farther from the
center than they are on the left. It might be supposed that
this machine would turn round perpetually, but it is a mere
balance.

302 MATHEMATICAL APPLICATIONS

moved so long as the present laws of nature continue to
exist.

"Every attempt to produce a self-moving machine has
been in open defiance to the coordinated relations of
force and motion; and any man who comprehends this
law of velocity will no sooner attempt to solve the prob-
lem of perpetual motion than to climb upon his own shoul-
ders as a higher point of observation.

"But in the search for an impossibility so many val-
uable and practical certainties have been demonstrated
that perhaps no time has been absolutely thrown away.
As alchemy fostered and developed chemistry, so the
search after perpetual motion has taught scientists how to
apply force through complicated machinery, and has
given rise to many new devices."

In treating of perpetual motion — "that grand secret
for the discovery of which those dictators of philosophy,
Democritus, Pythagoras, Plato, did travel unto the Gym-
nosophists and Indian priests" — its history would be a
fascinating but tragic tale. Every contrivance hitherto
planned or experimented upon has been proved fallible.
Paracelsus built a "little world," Cornelius Dreble in-
vented a planetarium for King James, and Peregrinus
suggested the "magnetical globe of Terella," which he
thought might be kept in motion by pieces of steel and
loadstones; and Bishop Wilkins himself made an applica-
tion of Archimedes' screw, but all were alike "found in-
adequate to the grand end for which they were designed."

CHAPTER V

MACHINES

In a general mechanical sense a machine is any instru-
ment which converts motion, or rather force, into motion,
as, for instance, a machine designed to convert rapid mo-
tion into slow motion, as a windlass, or, vice versa, as the
connection of a large wheel to a small increases the ve-
locity of the latter. The ordinary tools consisting of a
single device, such as the hammer, or a simple combination
of moving parts, such as shears or tongs, are machines in
the strict technical definition of the term. Many writers
have used the word in a sense other than the strictly tech-
nical one, as Huxley when he says: "The human body,
like all living bodies, is a machine, all the operations of
which will, sooner or later, be explained on physical prin-
ciples."

Among the most ancient machines were those that em-
ployed wind or flowing water as a motor power for turning
wheels. In medieval times even bellows were adapted to
this purpose. The windmill is a familiar device for raising
water from a well or spring for grinding and other pur-
poses. There are two types of these wind-motors, the
vertical being the most common. The vertical motor con-
sists essentially of a horizontal wind-shaft, with a combi-
nation of sails or vanes fixed at the end of the shaft and
suitable gearing for conveying the motion of the wind-
shaft to the pump or to the other machinery.

The typical Dutch windmill was provided with four
303

304

MATHEMATICAL APPLICATIONS

vanes or sail-frames, called whips, covered with canvas
and provided with arrangements for reefing the sails in a
high wind. To present the vanes to the wind, the whole

Fig. 19

-Bellows for Raising Water.
century print.)

(From a fifteenth-

structure or tower was at first turned round by means of a
long lever. Later the top of the tower, or cap, was made
movable. Windmills are now made with many wooden
vanes forming a disk exposed to the winds and fitted with

MACHINES 305

automatic feathering and steering machinery, governors
for regulating the speed and apparatus for closing the
vanes in storms. These improved windmills are chiefly an
American invention and are used for pumping water.

Water power is perhaps, after wind power, the most
natural and the most truly economic source of energy.
The term "water power" is not exact, since the real agent
in water machines is gravity, the fluid itself being only the
medium through which the action of gravity is transmitted
to the prime motor. In order that water may be available
for doing work, it must be in such a position that it can
fall from a higher to a lower level or must be under pres-
sure produced by some external force, such as that of a
weight or spring acting on the surface of the fluid through
a piston or plunger. Under the former condition its
utmost capacity for doing work is the product of the
height through which it can fall into the weight of the
water falling.

For practical purposes there are three ways by which
water power can be applied to the performance of work:
through the velocity of the fluid itself, by weight or by
pressure. Each of these three methods requires a differ-
ent type of motor for its application. An illustration of
the first is the turbine, which is moved by the force of
projected water; the second, the water-wheel, which is
moved by the weight of the falling water; the third, the
hydraulic pressure engine, which operates by the applica-
tion of the hydraulic law of equal pressure.

The old-fashioned mill for grinding flour or corn, which
was once the center of nearly every New England village,
was run by water flowing over the upper wings of a clumsy
wooden wheel. These overshot wheels are now nearly
obsolete, but have been constructed in the past on rather
gigantic plans. The water falls from a sluice or pen
trough near the top and moves the wheel by falling into
floats or shallow buckets. It is regulated by a gate and
falls into the third or fourth bucket from the summit, thus

306

MATHEMATICAL APPLICATIONS

utilizing as much as possible of the gravitational force.
The undershot wheel turns by having the force of the
stream of water act at its lowest point instead of its
summit.

But the numerous disadvantages of the water-wheels
described have caused them to be almost entirely super-

Fig. 20 — Overshot Wheel.

seded by the turbine. This is a water-wheel driven by the
impact or reaction of a stream of water flowing against a
series of radial buckets or by impact and reaction com-
bined. Turbines are usually horizontally rotating wheels
on vertical shafts. They are of various constructions and

MACHINES 307

may be classed as reaction turbines, whose buckets move
in a direction opposite to that of the flow ; impulse turbines,
whose buckets move with the flow, and the combined re-
action and impulse wheels, which include the best modern
types of turbines. In these a very high percentage of the
potential energy of water is converted into work while
passing through the wheels.

Impulse wheels, constructed as large as 18^ feet in
diameter, have been employed to work air compressors
in mines. A wheel of this size weighs 10,000 pounds and
runs at no revolutions per minute; it has energy equal to
300 horse-power. The wheel is made of iron plates riveted
together, which are held concentric with the shaft by radial
spokes. There is a variable nozzle operated by an auto-
matic hydraulic regulator, through which the water is
applied to the wheel. It will run at uniform speed with
varying loads. Turbines are now made from 6 to 80 feet
in diameter and are so cheap, durable and highly effective
that they are fast superseding other types of wheels.

Two other important applications of water power are
found in the hydraulic press and the hydraulic ram. The
hydraulic press is operated by the pressure of a liquid,
under the action either of gravity or of some mechanical
device such as a force pump. It depends on the law of
hydrostatics that any pressure upon a body of water is
distributed equally in all directions throughout the whole
mass, whatever its shape. In the more common forms of
hydraulic press the pressure of a piston upon a body of
water in a cylinder of small area is distributed through
pipes or openings to a piston or a larger area. The statical
force is thus multiplied in the direct ratio of the areas of
the pistons. Therefore if the diameter of a small piston
is one inch and of a larger piston in the cylinder is one
foot, the area of the larger piston will be 144 times the
area of the smaller, and if a load of one ton is applied to
the smaller, the larger will exert an upward statical force
of 144 tons.

308 MATHEMATICAL APPLICATIONS

This interesting machine is used as the basis of a great
number of inventions, such as the hydraulic block, jack,
crane, hoist, lift and others, and for the pressing of paper
and other materials.

The hydraulic ram is a self-contained and automatic
pump, operated partly by the pressure of a column of
water in a pipe and partly by the living force acquired by
the intermittent motion of the column. This machine can
be used to raise water to a height many times greater than
the available head, and it is also adapted to draw water
from a source independent of that which supplies the
power for operating it.

Hydraulic machines are very wonderful to people who
observe their action for the first time. With a common
hydraulic press a laborer, without any other help, can
raise a load of a hundred tons, which is the weight of a
long railway train. At large ship docks any boy can, by
the manipulation of a few handles, lift heavy weights rap-
idly from a ship and place them on the dock.

No single invention in the history of the world has had
so deep and wide an influence as the steam engine. This
truism is one which deserves consideration, even in days
when there is all too much exploitation of the mechanical
inventiveness of the age. If the possibility of travel which
the locomotive has brought within the reach of nearly
every one be considered, apart from any other uses of the
steam engine, its extraordinary influence on the life of the
century is startlingly apparent. Until within fifty years
travel and acquaintance with foreign peoples, historic
monuments and all the artistic accumulations of other
nations and other generations were the privilege of very
few; in these days traveling is the universal epidemic.
More than that, with better acquaintance nation has re-
acted upon nation, so that political and military problems
have taken on a wholly different aspect.

The germ of the steam engine existed in Heron's eolipile,
invented in the second century B.C. This illustrated per-

MACHINES 309

fectly the expansive force of steam generated in a closed
vessel and escaping by a narrow aperture. It consisted of
a hollow ball containing water and two arms bent in oppo-
site directions, from the narrow apertures of which steam
issued with such force that the air, reacting on it, caused a

Fig. 21 — Heron's Eolipile.

circular or rotary motion of the ball. Several attempts
have been made to apply the principle of the eolipile to
rotating machinery.

In 1705 there was invented the first important device
for the practical application of steam power. For about

310 MATHEMATICAL APPLICATIONS

1,500 years after Heron's eolipile no progress had been
made. During the seventeenth century steam fountains
were designed, but they were merely modifications of
Heron's engine, and were probably applied only for orna-
mental purposes. Some effort was also made by Morland,
Papin and Savery to construct practical machines for the
raising of water or driving of mill-works. The first suc-
cessful attempt to combine the principles and forms of
mechanism then known into an economical and convenient
machine was made by Thomas Newcomen, a blacksmith of
Dartmouth, England. It is probable that he knew of
Savery's engine, as Savery lived only fifteen miles away.

Assisted by John Calley, Newcomen constructed an en-
gine— an "atmospheric steam-engine," for which a patent
was secured in 1705. In 171 1 such a machine was set up
at Wolverhampton for the raising of water. Steam pass-
ing from the boiler into the cylinder held the piston up
against the external atmospheric pressure until the passage
between the cylinder and boiler was closed by a cock.
Then the steam in the cylinder was condensed by a jet of
water. A partial vacuum was formed and the air above
pressed the piston down. This piston was suspended from
one end of an overhead beam, the other end of the beam
carrying the pump-rod. Desaguliers tells the story that a
boy, Humphrey Potter, who was charged with the duty of
opening and closing the stopcock between the boiler and
cylinder for every stroke, contrived by catches and strings
an automatic motion of the cock. The fly-wheel was intro-
duced in 1736 by Jonathan Hulls. The next great im-
provements were introduced in Scotland by James Watt in
the latter half of the eighteenth century. Watt was edu-
cated as a maker of mathematical instruments, and in 1760
he opened a shop in Glasgow. Becoming interested in the
steam-engine and its history, he began to experiment in a
scientific manner. He took up chemistry and was assisted
in his studies by Dr. Black, the discoverer of "latent heat."
Observing the great loss of heat in the Newcomen engine,

MACHINES 311

due to the cooling of the cylinder by the jet of water at
every stroke, he began to think of means to keep the
cylinder "always as hot as the steam that entered it." He
has told us how, finally, the happy thought securing this
end occurred to him : "I had gone to take a walk on a fine
Sabbath afternoon. I had entered the Green by the gate
at the foot of Charlotte Street, and had passed the old
washing-house. I was thinking upon the engine at the
time, and had gone as far as the herd's house when the
idea came into my mind that as steam was an elastic body
it would rush into a vacuum, and if a communication were
made between the cylinder and an exhausted vessel, it
would rush into it, and might be there condensed without
cooling the cylinder." Through this invention the piston
was now moved by the expansion of steam, not by air
pressure, as in Newcomen's engine. Watt introduced a
separate condenser, a steam-jacket and other improve-
ments. He deservedly commands a preeminent place
among those who took part in the development of the
steam-engine. The expiration of Watt's vital patent oc-
curred in 1800, and he himself then retired from the active
supervision of his engineering business, having virtually
finished his life's work on the last year of the century.

One of the first and most obvious uses of the steam-
engine was to apply its power to locomotion, both on sea
and land. Before steam lent its power to the propulsion
of ships, navigation was, like the windmill, subject to the
intermittent character of the winds or limited to the man-
power of rowers. The method of moving vessels by pad-
dle-wheels was adopted by the Romans, probably borrowed
from the Egyptians ; but the wheels were turned by handles
within the vessels, operated by men. There are several
obscure references in annals of the seventeenth century
to what is supposed to be the propulsion of paddle-wheels '
by steam. Among the rest there is a description of a steam
propeller, invented by one Genevois, a pastor at Berne,
which was formed like the foot of a duck. This was made

312 MATHEMATICAL APPLICATIONS

to expand and present a large surface to the water when
moved against it and to close up into a small compass
when moved in the opposite direction. In 1774 there is a
tradition of a boat which, when tried upon the Seine near
Paris, moved against the stream, tho slowly, "the engine
being of insufficient power." The construction of this
engine is attributed to the Count D'Auxiron, a French
nobleman.

Many attempts to apply the force of the steam-engine to
the propulsion of paddle-wheels were made in the latter
part of the eighteenth century, and it is said that William
Symington, an English inventor, accomplished a certain
form of steam navigation. But it was left for Robert
Fulton, an American artist as well as inventor, to bring the
trials to a successful issue. In 1809 Fulton's steam vessel,
the Clermont, made her first voyage from New York to
Albany, a distance of about 140 miles, at the rate of five
miles an hour. To those who viewed this spectacle this
first steamer "had a most terrific appearance." She used
dry pine wood for fuel and sent forth a column of ignited
vapor for a distance of many feet above the flue, and
whenever the fire was stirred showers of sparks flew off
into the air. One of the chroniclers states : "Notwithstand-
ing the wind and tide were adverse to its approach, they
saw with astonishment that the vessel was rapidly coming
toward them, and when it came so near that the noise of
the machinery and paddles was heard the crews, in some
instances, shrank beneath their decks from the terrific
sight and left their vessels to go ashore, while others
prostrated themselves and besought Providence to protect
them from the approach of the horrible monster, which
was marching on the tide and lighting its path by the fire
which it vomited."

The Clermont was of 160 tons burden, the paddle-wheels
were 15 feet in diameter and dipped 2 feet in the water.
She was impelled by a machine of four-foot stroke and a
two-foot cylinder. Within a few weeks after the appear-

MACHINES 313

ance of the Clermont, Stevens, of Hoboken, launched a
steam vessel. She could not ply on the waters of the Hud-
son, in consequence of the exclusive patent of Fulton and
Livingston, so she was taken to the Delaware. This was
the first steamer that ever sailed the ocean. From that
time steamboats have multiplied till every water in the
civilized portion of the earth was marked with these agents
of rapid intercourse.

For the purpose of comparison the Cunard steamer
Lusitania, launched in 1907, may be placed beside that of
the Clermont. The Lusitania is 790 feet long and 88 feet
broad. She has a displacement of 45,000 tons and is pro-
pelled by four screws rotated by turbine engines of 68,000
horse-power. Placed in perspective, her length would out-
reach the angular height of the Great Pyramid.

The history of locomotion on land presents a parallel
tale of simple beginnings and extraordinarily rapid prog-
ress. The "Stourbridge Lion" was the first locomotive
brought to America and was tried on the road at Hones-
dale, Pa., on the 8th of August, 1829. Its boiler was i6>4
feet long, the two cylinders were three-foot stroke and its
weight was 7 tons. It was operated around a curve and
up the road for about two miles and then was returned to
the place of starting. The experiment demonstrated that
the track was not substantial enough for so heavy an en-
gine, and it was housed beside the track, where it remained
for fifteen years. It was then removed to Carbondale,
where the boiler was used for stationary purposes and the
remainder was sold for old iron.

This ignominious end to the first attempt to utilize the
steam-engine for locomotion on land in America did not
discourage other people from making other trials. Peter
Cooper, having an interest in the Baltimore and Ohio
road, in 1829 built an engine known as the "Tom Thumb,"
to demonstrate that a locomotive could be built that would
run round short curves. This engine had an upright boiler
20 inches in diameter by 5 feet high, fitted with gun bar-

314 MATHEMATICAL APPLICATIONS

rels for flues. The engine drove a large gear which fitted
into a smaller gear on the axle. The fire was urged by a
fan driven by a belt. The driving-wheels were 2^2 feet in
diameter. In August, 1830, the first railroad car in Amer-
ica propelled by a locomotive was tested on the Baltimore
and Ohio road. The wheels were "coned," which was the
first use of this principle as applied to car-wheels. Cooper's
engine was coupled to a car in front of it containing a
load of 4^ tons, including 24 passengers. The trip of 13
miles was made in 1 hour and 15 minutes and the return
trip in 57 minutes. This was the first locomotive built in
America.

In the locomotive engines used at the present time it is
not unusual to see engines for passenger service which
have a total weight of about 185,000 pounds, cylinders 22
inches in diameter and a piston stroke of 30 inches. The
locomotive will now at least double the speed of the race
horse and will carry not only itself, but three or four times
its own weight in addition, and will go a hundred miles
without stopping, if only the road ahead be clear.

The fastest mechanism of any size, which has ever
cut its way through the water for any considerable
distance is the torpedo boat Ariete, made by a London
firm in 1887. This little craft has a displacement of no
tons and machinery capable of exerting 1,290 effective
horse-power. The speed accomplished at the trial tests
was 30 miles per hour, this being the average of six one-
mile tests.

CHAPTER VI

For more»than two centuries man has been trying to in-
vent a means whereby he might navigate the air, but it is
only since the beginning of the twentieth century that any
degree of success has been attained.

The apparatus used in aviation divides, roughly, into
two classes, dirigible balloons and the so-called "gasless,"
or heavier-than-air machines, represented by the biplane,
the ornithopter, or beating-wing machine, and the heli-
copter, or direct lift machine.

The dirigible balloon has already, relatively speaking,
arrived at some degree of perfection, insomuch as the
serious difficulties connected with this type of aerial loco-
motive have been largely overcome. The gas-bag, with
the volume of gas employed, has been brought to its
smallest practicable size, and the weight of subsidiary
material and machinery has, it is believed, been brought
to its lowest limit of safety. With the inventions of Count
Zeppelin Germany has been in the lead, so far as actual
progress in the making of dirigible balloons is concerned,
but France is a close second. As long ago as 1907 the
Zeppelin dirigible, 413 feet in length, attained a speed of
34 miles an hour and covered more than 200 miles in one
ascent which lasted eight hours. "La Patrie," a dirigible
owned by the French Government, traveled without rest
from Paris to Verdun, 142.8 miles, at a mean speed of
more than 20 miles an hour.
3i5

3i6 MATHEMATICAL APPLICATIONS

Great Britain, Italy, Spain and the United States have
also produced dirigibles, but no essential advance in the
principle has been made. The American Baldwin dirigible
has a gas-bag of 84 feet in length, with a capacity of
18,000 cubic feet. The frame is 66 feet long; the 12- foot
propeller, placed on the forward end of the frame, has a
speed of 450 revolutions a minute. The ship is kept on an
even keel and is lowered or raised by a number of box-
like planes near the forward end, operated by the aviator.
It is driven by a 20-horse-power Curtiss engine. The
frame is almost as long as the gas-bag and is attached to
it by means of a fine strong netting, while the operators
are carried in two cars. The Baldwin is distinctly an
American machine, but bears a general resemblance to
the enormous German dirigibles.

Germany, represented by Count Zeppelin, has made
significant contributions to aeronautics. August, 1909,
was commemorated by a recording-breaking flight of the
dirigible Zeppelin III. from Friedrichshafen to Berlin.
It was a triumph of Count Zeppelin's scientific skill and
his patient courage and perseverance. At the end of the
remarkable journey the roofs, streets and parks of the
German capital swarmed with people, singing and cheer-
ing, as the airship sailed round the palace and cathedral
and landed in the Tempelhof parade ground, where the
Emperor, Empress and many leading officials were wait-
ing to receive the aged Count.

The dirigible, as at present designed, consists of a huge
skeleton framework of aluminium alloy, over which is
stretched continental rubberized fabric. The ship is six-
teen-sided, with long, latticework girders springing out
from the solid central prow, giving the ship the required
shape. It is something more than 440 feet long and has
seventeen separate gas envelopes. It can be used over
water, owing to its floating cars; it can mount duplicate
engines of considerable horse-power, and it has a far
wider range of action and utility than any other aerial

AVIATION 317

vessel. Already it holds every record in distance, altitude
and duration in the air.

The helicopter is a machine with an upright shaft
and revolving blades, which can rise nearly vertically or
at a steep angle and has other points of advantage over
the aeroplane, tho it has not yet been perfected for prac-
tical use. It is said that the helicopter was first suggested
four hundred years ago by the artist, Leonardo da Vinci,
"as a practical, comparatively simple and inexpensive
flying device." One of the most successful helicopters
has two superposed propellers in horizontal parallel planes,
mounted on concentric hollow shafts, revolving in oppo-
site directions and driven by an eight-cylinder 40-horse-
power air-cooled Curtiss motor. The propellers are 17
feet in diameter and the platform is 16 feet square. The
machine possesses in a marked degree the desiderata of
initial stability and flexibility of movement. It has at-
tained a speed of thirty miles an hour.

The aeroplane, it is evident, has not nearly attained its
possible limit of perfection. The great originator of the
flying machine was Lilienthal,, who, aiter exhaustive
study and experimentation with specially designed appa-
ratus modeled after the wings of birds, was the first man
to glide with large wing-like surfaces through the air.
Lilienthal was compelled to use his machine merely as an
aerial coaster, as there was no light motor then in exist-
ence.

Several distinguished aerial engineers have emulated
Lilienthal's zeal, among whom are Herring, the Wright
brothers and Glen H. Curtiss in America, Henri Bleriot
in France, Henry Farman and Latham in England. Her-
ring improved on Lilienthal's machine, changing his de-
sign and providing the glider with a wonderful mechanism
which performed most of Lilienthal's acrobatic feats auto-
matically. To one of these machines he later applied
stored power in the shape of compressed air. Applying
this to two large wooden screw propellers, he was able to

318 MATHEMATICAL APPLICATIONS

fly horizontally, instead of coasting downward for the
short time his power would last. Since then Curtiss has
invented a light motor of great ingenuity, which has suc-
cessfully been applied to the aeroplane and the helicopter.

The actual methods by which practical progress is made
in the equipment and operation of these machines is more
or less shrouded in mystery so far as the public is con-
cerned, but results are evident. The Wright brothers
began work on the Lilienthal basis, as did Herring. They
also worked out their own methods of controlling the
glider by mechanical means. The chief feature of the
Wright aeroplane lies in the application of the petrol
motor to the propelling blades. It is the lightness of this
motor that has made progress possible in this direction.
The propellers force the machine through the air and the
two planes, from which the machine gets its name — bi-
plane— support it. The two planes are rigid at their tips,
which can be twisted in order to prevent too much tilting
when turning. It is guided by a horizontal rudder in
front and another ordinary rudder at the rear. The length
of the planes had become difficult to handle, therefore it
was cut in two and one plane placed above the other. The
whole mechanism is handled by a single operator, who is
seated in the center of the lower plane.

The Aerial Experiment Association operating at Ham-
mondsport, New York, has contributed interesting chap-
ters to the history of aviation. The June Bug, a very effi-
cient type of aeroplane, was constructed by this body. In
winning the trophy on July 4, 1908, the machine rose
rapidly to a height of 20 feet and sped on, traversing a dis-
tance of one mile in 1 minute and 42 seconds, correspond-
ing to an average speed of 35V10 miles per hour. The first
trans-oceanic flight was that of Bleriot, the French ex-
perimenter, who performed in August, 1909, the feat of
crossing the English Channel in a monoplane.

During the last week of August, 1909, the first interna-
tional aviation race-meet held anywhere in the world took

AVIATION 319

place near the city of Rheims, France. It was there that
the best achievements of the heavier-than-air machines
were exhibited and practically every contribution to the
science of aviation by motor placed before the public. The
exhibitions of aerial skill were such as to make the week
a memorable one in the history of aviation. New records
were made and broken every day and the safety of the
flying machines was as remarkable as their efficiency.
Flights were made during rain and when the wind was
blowing twenty-five miles an hour.

Altogether there were thirty-eight aeroplanes entered
in the various contests and races, for which $40,000 in
cash prizes was offered. The machines which made
flights were divided about equally between the monoplane
and the biplane types, altho the latter type was rather
more in favor. Of the machines of this kind five were
Wright biplanes, five were biplanes of the Voisin cellular
type with a tail and three of the Farman type with a tail,
but without vertical partitions between the main planes.
The Curtiss biplane, which is modeled closely after the
pattern used by the Wright brothers, represented America.

These machines were entered in contests for speed in
long-distance flights, for "sprints," for passenger-carrying
power and for duration of flight. Flights of half an hour,
an hour, an hour and a half became common early in the
meeting, and on Tuesday M. Paulhan, driving a Voisin
biplane, broke the record made by Wilbur Wright at Le
Mans, France, in 1908, by flying for 2 hours and 43 min-
utes. In that time he covered 83 miles and only descended
when his fuel was exhausted. The next day his record,
in point of distance, was promptly superseded by M.
Latham, the French aerialist, who made the first, tho
unsuccessful, attempt to fly across the English Channel.
In an Antoinette monoplane M. Latham circled the course
fifteen times, covering a distance of 96 miles in 2 hours
and 18 minutes. This is about the same time that Mr.
Wright remained in the air on his record flight in 1908,

320 MATHEMATICAL APPLICATIONS

but during that time he covered only yy miles. On
the 28th (Friday) Mr. Farman, an Englishman, flying
in a biplane of his own design, once more set the mark at
a higher point. He flew about 118 miles, remaining in the
air more than three hours, breaking the records made
both by M. Latham and M. Paulhan. His performance
won for him the Champagne Grand Prize. Bleriot made
the best time for a single round of the course during the
first part of the week, covering the distance of 6l/5 miles
in almost exactly 8 minutes and 4 seconds.

In the middle of the week the International Aviation
Trophy was contested. France was represented by two
monoplanes, a Bleriot and an Antoinette and a Wright
biplane, while America was represented by one tiny bi-
plane with an eight-cylinder motor, designed and operated
by Glen H. Curtiss. The real race was between Bleriot
and Curtiss, the champions of the biplane and monoplane
types of flying machines, respectively. The morning of
the contest, August 28, was mild, calm and hazy at Rheims.
Curtiss, after a preliminary round of the course, circled
round once in front of the grand stand and crossed the
line at full speed. The aeroplane pitched perceptibly and
the turns were at first rather wide. Nevertheless he made
the two rounds in record time, the second being 4x/5 sec-
onds faster than the first. The total time of the rounds
was 15 minutes 503/,. seconds, corresponding to an average
speed of 47.04 miles an hour.

Bleriot was unable to better this record, tho his mono-
palne flew splendidly, without any rolling or pitching.
His time was 53/5 seconds more than that of Curtiss. The
third place in the competition was secured by Latham, who
flew at a height of about 150 feet and covered the course
in 17 minutes 32 seconds. Lefebvre, the third French
representative, with a Wright biplane fitted with a 40-
horse-power motor, was fourth, making the course in 20
minutes 47 seconds.

The passenger-carrying competition was won by Henry

Wright Biplane Machine on Ground, with Glenn Curtiss
Flying Overhead. (Taken at Rheims.)

AVIATION 321

Farman, who, after making a round with one passenger
in 9 minutes 53*/5 seconds, carried two people around the
course at a speed of 34.96 miles an hour. The total live
weight lifted by his machine was in the neighborhood of
450 pounds. Farman's biplane was the only machine that
succeeded in carrying three people. Bleriot's "No. 12"
monoplane, however, was the first aeroplane to accomplish
this feat, which it did at Douai in June, 1909. At that
time a total weight of 1,234 pounds was carried at about
30 miles an hour with a 30-horse-power motor.

The chief event of the meet at Rheims, however, was
the contest for the James Gordon Bennett cup for the
fastest flight of 30 kilometers. Early in the week it was
evident that Bleriot and Curtiss were the two serious can-
didates for this prize, and the excitement over the two
contestants was intense. Bleriot started on his journey,
crossed the line and made the first turn at a rapid rate,
flying at a low elevation. He disappeared from sight, how-
ever, at the far end of the long course, and presently it
was found that his machine had suddenly dived to the
ground, caught fire and was rapidly being consumed.
This unfortunate accident eliminated serious rivalry to
the American machine, which had already proved its re-
markable powers. Curtiss made the three rounds of the
course in his 60-horse-power biplane in 23 minutes 29
seconds, or at a speed corresponding to 47.6 miles an hour.
The second lap of the course was made at a speed of 47.73
miles an hour. Latham; with the Antoinette monoplane,
was second in this contest and the Wright biplane third.
Thus the Prix de la Vitesse also fell to Curtiss, bringing
to America the lion's share of the honors of the meeting.

The Curtiss biplane carries an eight-cylinder water-
cooled motor, weighing 200 pounds. All valves are me-
chanically operated and the ignition is by magnet. The
weight of the aeroplane' loaded is 700 pounds ; the total
surface is 225 square feet. The thrust developed by the
propellers is 280 pounds and its greatest speed is 47.73

322 MATHEMATICAL APPLICATIONS

miles an hour. The machine is, in comparison to the
other types of biplane, compact and small, weighing less
than half as much as those of his competitors.

The contest seems to have settled many of the moot
questions concerning stability, landing and manipulation
of the machines. The most important factor appears to be
the reliability of the motor. The spectacle during the
week's contests was an unprecedented one, for at times six
machines were in the air at once.

The last few years have seen the revolutionary triumph
of the flying machine over gravity; the coming years will
see its evolutionary subjugation of the treacherous ele-
ment into which it has launched itself.

"Flight is a new mental and physical experience," says
Thos. S. Baldwin, the inventor of the U. S. military
dirigible balloon, in a recent article. "It transposes one to
a world of action and emotion in direct contrast to much
of what one feels and lives on the hard surface of the
globe. It tends to exhilarate and exalt the mind; it
changes the registry and the workings of a number of the
human senses ; and it breathes into the body an overflow-
ing measure of health, endurance and power. The elimi-
nation of the force of gravity affects the habits of gravity.
The mind's freedom is denoted by an enormous increase
of energy and power of action. The gravity of every
square inch of the plane on which one stands or sits, and
of every ounce of one's body, have been neutralized by a
buoyancy of a gas lighter than air or by mechanical force
and pressure upon the air.

The aeronaut brings a measure of this power from the
heavens down to the earth with him as he alights from his
ship. After a long voyage one touches the ground with
the feeling that he can step over tall buildings, leap broad
rivers and fly from place to place. His tread upon the
ground is like walking upon bags of wool. This fact ex-
plains why so small a percentage of persons who fall in
flight are killed. This apparent lightness and buoyancy

AVIATION 323

remains in the very bones for many hours after one has
made a protracted aerial voyage, and lures one back to the
height of the air. It is a sensation of pleasure that the
great majority of humanity have yet to know.

"First we shall fly a step in a crude machine — we have
begun to do that — then in time we shall sail the air in
great ships, and in some remote day man will pass through
the air in his own body solely. No one who has keenly
felt the joy and triumph of flight in his own person can
fail to believe in this last prediction."

But it would be doing Mathematics a grievous injustice
to level its applicative value to mechanical inventiveness,
for if there is one thing that is more sure than another it
is that the development of machinery, marvelous tho it
has been, is but one — and a small — part of the heritage that
Modern Mathematics has given. The scope of logistics
is immeasurable, and there are not wanting evidences that
abstruse subjects supposed to be inherently psychologic
may come under the magic spell of number.

Whether imagination itself shall ever be reduced to a
fourth dimension in space, man cannot yet know; but re-
garding that spiritual essence of man, the mathematician
has always his fixed idea. Cassius J. Keyser couples the
science with what was once known as "the queen of all
sciences," and makes mathematics the key to a vaster
realm than it has hitherto conquered.

"I do not believe," he says, "that the present declined
state of Theology is destined to be permanent. The pres-
ent is but an interregnum in her reign and her fallen days
will have an end. She has been deposed mainly because
she has not seen fit to avail herself promptly and fully of
the dispensations of advancing knowledge. The aims,
however, of the ancient mistress are as high as ever, and
when she shall have made good her present lack of mod-
ern education and learned to extend a generous and eager
hospitality to modern light, she will reascend and will
occupy with dignity as of yore an exalted place in the

324 MATHEMATICAL APPLICATIONS

ascending scale of human interests and the esteem of en-
lightened men. And mathematics, by the character of her
inmost being, is especially qualified, I believe, to assist in
the restoration.

"It was but little more than a generation ago that the
mathematician, philosopher and theologian, Bernhard Bol-
zano, dispelled the clouds that throughout all the foregone
centuries had developed the notion of Infinitude in dark-
ness, completely sheared the great term of its vagueness
without shearing it of its strength, and thus rendered it
forever available for the purposes of logical discourse.
Whereas, too, in former times the Infinite betrayed its
presence not indeed to the faculties of Logic but only to
the spiritual Imagination and Sensibility, mathematics has
shown, even during the life of the elder men here present
— and the achievement marks an epoch in the history of
man — that the structure of Transfinite Being is open to ex-
ploration by the organon of Thought.

"Again, it is in the mathematical doctrine of Invariance,
the realm wherein are sought and found configurations and
types of being that, amid the swirl and stress of countless
hosts of transformations, remain immutable, and the spirit
dwells in contemplation of the serene and eternal reign of
the subtile law of Form, it is there that Theology may
find, if she will, the clearest conceptions, the noblest sym-
bols, the most inspiring intimations, the most illuminating
illustrations and the surest guarantees of the object of her
teaching and her quest, an Eternal Being, unchanging in
the midst of the universal flux."

HI. SLOOS. OSQ. iOhSZ