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Open Court Classics of Science and Philosophy,

iOMETRICAL LECTURES

ISAAC BARROW

J. M, CHILD

GEOMETRICAL LECTURES

ISAAC BARROW

Court Series of Classics of Science and

Thilo sophy, 3\(o. 3

THE
GEOMETRICAL LECTURES

ISAAC BARROW

. HI

TRANSLATED, WITH NOTES AND PROOFS, AND A

DISCUSSION ON THE ADVANCE MADE THEREIN

ON THE WORK OF HIS PREDECESSORS IN THE

INFINITESIMAL CALCULUS

B.A. (CANTAB.), B.Sc. (LOND.)

546664

CHICAGO AND LONDON.
THE OPEN COURT PUBLISHING COMPANY

1916

LECTI ONES

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fi>1/Si'» *'Ao &.'««> n$uw, OH 6)f «tr;t TO ^urt^i «oT»i a'uT&i/yi'jrf
**ir. Plato de Repub. VII.

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Note the absence of the usual words " Habitae Cantabrigioe," which on
the title-pages of his other works indicate that the latter were delivered as
Lucasian Lectures. — J. M. C.

PREFACE

ISAAC BARROW was Ike first inventor of the Infinitesimal
Calculus ; Newton got the main idea of it from Barrow by
personal communication ; and Leibniz also was in some
measure indebted to Barrow's work, obtaining confirmation
of his oivn original ideas, and suggestions for their further
development, from the copy of JJ arrow's book (hat he purchased
in 1673.

The above is the ultimate conclusion that I have arrived
at, as the result of six months' close study of a single book,
my first essay in historical research. By the "Infinitesimal
Calculus," I intend "a complete set of standard forms for
both the differential and integral sections of the subject,
together with rules for their combination, such as for a
product, a quotient, or a power of a function ; and also a
recognition and demonstration of the fact that differentiation
and integration are inverse operations."

The case of Newton is to my mind clear enough. Barrow
was familiar with the paraboliforms, and tangents and areas
connected with them, in from 1655 to 1660 at the very
latest; hence he could at this time differentiate and inte-
grate by his own method any rational positive power of a
variable, and thus also a sum of such powers. He further
developed it in the years 1662-3-4, and in the latter year
probably had it fairly complete. In this year he com-
municated to Newton the great secret of his geometrical
constructions, as far as it is humanly possible to judge from
a collection of tiny scraps of circumstantial evidence ; and
it was probably this that set Newton to work on an attempt
to express everything as a sum of powers of the variable.
During the next year Newton began to "reflect on his
method of fluxions," and actually did produce his Analysis
per sfLquationes. This, though composed in 1666, was not
published until 1711.

viii BARROW'S GEOMETRICAL

The case of Leibniz wants more argument that I am in a
position at present to give, nor is this the place to give it. 1
hope to be able to submit this in another place at some future
time. The striking points to my mind are that Leibniz
bought a copy of Barrow's work in 1673, anc^ was a^^e "to
communicate a candid account of his calculus to Newton "
in 1677. In this connection, in the face of Leibniz' per-
sistent denial that he received any assistance whatever from
Barrow's book, we must bear well in mind Leibniz' twofold
idea of the " calculus " : —

(i) the freeing of the matter from geometry,
(ii) the adoption of a convenient notation.
Hence, be his denial a mere quibble or a candid statement
without any thought of the idea of what the "calculus"
really is, it is perfectly certain that on these two points at
any rate he derived not the slightest assistance from
Barrow's work ; for the first of them would be dead against
Barrow's practice and instinct, and of the second Barrow
had no knowledge whatever. These points have made the
calculus the powerful instrument that it is, and for this the
world has to thank Leibniz; but their inception does not
mean the invention of the infinitesimal calculus. This, the
epitome of the work of his predecessors, and its completion
by his own discoveries until it formed a perfected method
of dealing with the problems of tangents and areas for
any curve in general, i.e. in modern phraseology, the
differentiation and integration of any function whatever
(such as were known in Barrow's time), must be ascribed
to Barrow.

Lest the matter that follows may be considered rambling,
and marred by repetitions and other defects, I give first
some account of the circumstances that gave rise to this
volume. First of all, I was asked by Mr P. E. B. Jourdain
to write a short account of Barrow for the Monist ; the
request being accompanied by a first edition copy of
Barrow's Lectiones Opticce. et Geometries. At this time, I
do not mind confessing, my only knowledge of Barrow's
claim to fame was that he had been "Newton's tutor": a
notoriety as unenviable as being known as " Mrs So-and-So's
husband." For this article I read, as if for a review, the
book that had been sent to me. My attention was arrested

PREFACE ix

by a theorem in which Barrow had rectified the cycloid, which
I happened to know has usually been ascribed to Sir C. Wren.
My interest thus aroused impelled me to make a laborious
(for I am no classical scholar) translation of the whole of
the geometrical lectures, to see what else I could find. The
conclusions I arrived at were sent to the Monist for publica-
tion ; but those who will read the article and this volume
will find that in the article I had by no means reached the
stage represented by this volume. Later, as I began to still
further appreciate what these lectures really meant, I con-
ceived the idea of publishing a full translation of the lectures
together with a summary of the work of Barrow's more
immediate predecessors, written in the same way from a
personal translation of the originals, or at least of all those
that I could obtain. On applying to the University Press,
Cambridge, through my friend, the Rev. J. B. Lock, I was
referred by Professor Hobson to the recent work of Professor
Zeuthen. On communicating with Mr Jourdain, I was
invited to elaborate my article for the Monist into a
2oo-page volume for the Open Court Series of Classics.

I can lay no claim to any great perspicacity in this dis-
covery of mine, if I may call it so ; all that follows is due
rather to the lack of it, and to the lucky accident that made
me (when I could not follow the demonstration) turn one
of Barrow's theorems into algebraical geometry. What I
found induced me to treat a number of the theorems in the
same way. As a result I came to the conclusion that
Barrow had got the calculus; but I queried even then
whether Barrow himself recognized the fact. Only on com-
pleting my annotation of the last chapter of this volume,
Lect. XII, App. Ill, did I come to the conclusion that is
given as the opening sentence of this Preface ; for I then
found that a batch of theorems (which I had on first reading
noted as very interesting, but not of much service), on careful
revision, turned out to be the few missing standard forms,
necessary for completing the set for integration ; and that one
of his problems was a practical rule for finding the area
under any curve, such as would not yield to the theoretical
rules he had given, under the guise of an "inverse-tangent"
problem.

The reader will then understand that the conclusion is

x HARROW'S GEOMETRICAL LECTURES

the effect of a gradual accumulation of evidence (much a^
a detective picks up clues) on a mind previously blank as
regards this matter, and therefore perfectly unbiased. This
he will see reflected in the gradual transformation from
tentative and imaginative suggestions in the Introduction
to direct statements in the notes, which are inset in the
text of the latter part of the translation. I have purposely
refrained from altering the Introduction, which preserves the
form of my article in the Monist, to accord with my final
ideas, because I feel that with the gradual developrhent
thus indicated I shall have a greater chance of carrying my
readers with me to my own ultimate conclusion.

The order of writing has been (after the first full trans-
lation had been made) :— Introduction, Sections I to VIII,
excepting III; then the text with notes; then Sections III
and IX of the Introduction ; and lastly some slight altera-
tions in the whole and Section X.

In Section I, I have given a wholly inadequate account
of the work of Barrow's immediate predecessors ; but I felt
that this could be enlarged at any reader's pleasure, by
reference to the standard historical authorities ; and that it
was hardly any of my business, so long as I slightly expanded
my Monist article to a sufficiency for the purpose of showing
that the time was now ripe for the work of Barrow, Newton,
and Leibniz. This, and the next section, have both been
taken from the pages of the Encyclopedia Britannica ( Times
edition).

The remainder of my argument simply expresses my own,
as I have said, gradually formed opinion. I have purposely
refrained from consulting any authorities other than the
work cited above, the Bibliotheca Britannica (for the dates in
Section III), and the Dictionary of National Biography (for
Canon Overton's life of Barrow) ; but I must acknowledge
the service rendered me by the dates and notes in Sotheran's
Price Current of Literature. The translation too is entirely
my own — without any help from the translation by Stone
or other assistance — from a first edition of Barrow's work
dated 1670.

As regards the text, with my translation beside me, I
have to all intents rewritten Barrow's book; although
throughout I have adhered fairly closely to Barrow's own

PREFACE xi

words. I have only retained those parts which seemed to
me to be absolutely essential for the purpose in hand.
Thus the reader will find the first few chapters very much
abbreviated, not only in the matter of abridgment, but also
in respect of proofs omitted, explanations cut down, and
figures left out, whenever this was possible without breaking
the continuity. This was necessary in order that room
might be found for the critical notes on the theorems, the
inclusion of proofs omitted by Barrow, which when given
in Barrow's style, and afterwards translated into analysis,
had an important bearing on the point as to how he found
out the more difficult of his constructions ; and lastly for
deductions therefrom that point steadily, one after the
other, to the fact that Barrow was writing a calculus and
knew that he was inventing a great thing. I can make no
claim to any classical attainments, but I hope the transla-
tion will be found correct in almost every particular. In
the wording I have adhered to the order in which the
original runs, because thereby the old-time flavour is not
lost ; the most I have done is to alter a passage from the
active to the passive or vice versa, and occasionally to
change the punctuation.

I have used three distinct kinds of type : the most widely
spaced type has been used for Barrow's own words ; only
very occasionally have I inserted anything of my own in
this, and then it will be found enclosed in heavy square
brackets, that the reader will have no chance of confusing
my explanations with the text ; the whole of the Introduc-
tion, including Barrow's Prefaces, is in the closer type;
this type is also used for my critical notes, which are
generally given at the end of a lecture, but also sometimes
occur at the end of other natural divisions of the work,
when it was thought inadvisable to put off the explanation
until the end of the lecture. It must be borne in mind
that Barrow makes use of parentheses very frequently, so
that the reader must understand that only remarks in heavy
square brackets are mine, those in ordinary round brackets
are Barrow's. The small type is used for footnotes only.
In the notes I have not hesitated to use the Leibniz
notation, because it will probably convey my meaning
better ; but there was really no absolute necessity for this,

xii BARROW'S GEOMETRICAL LECTURES

Barrow's a and e, or its modern equivalent, /* and /&, would
have done quite as well.

I cannot close this Preface without an acknowledgment
of my great indebtedness to Mr Jourdain for frequent
advice and help; I have had an unlimited call on his wide
reading and great historical knowledge ; in fact, as Barrow
says of Collins, I am hardly doing him justice in calling him
my " Mersenne." All the same, I accept full responsibility
for any opinions that may seem to be heretical or otherwise
out of order. My thanks are also due to Mr Abbott, of
Jesus College, Cambridge, for his kind assistance in looking
up references that were inaccessible to me.

J. M. CHILD.
DKKIIY, ENGLAND,
Xmas, 1915.

P.S. — Since this volume has been ready for press, I have
consulted several authorities, and, through the kindness of
Mr Walter Stott, I have had the opportunity of reading
Stone's translation. The result I have set in an appendix
at the end of the book. The reader will also find there a
solution, by Barrow's methods, of a test question suggested
by Mr Jourdain ; after examining this I doubt whether any
reader will have room for doubt concerning the correctness
of my main conclusion. I have also given two specimen
pages of Barrow's text and a specimen of his folding plates
of diagrams. Also, I have given an example of Barrow's
graphical integration of a function ; for this I have chosen
a function which he could not have integrated theoretically,
namely, i/«/(i -^4), between the limits o and x ; when the
upper limit has its maximum value, T, it is well known that
the integral can be expressed in Gamma functions; this
was used as a check on the accuracy of the method.

J. M. C.

TABLE OF CONTENTS

PAGE

INTRODUCTION —

The work of Barrow's great predecessors .... I

Life of Barrow, and his connection with Newton . 6

The works of Barrow ...... .8

Estimate of Barrow's genius ... 9

The sources of Barrow's ideas .... . 12-

Mutual influence of Newton and Barrow .... i6-

Description of the book from which the translation has been

made . 20

The prefaces. ....... .25

How Barrow made his constructions ... . 28"

Analytical equivalents of Barrow's chief theorems ... 30

TRANSLATION—

LECTURE I. — Generation of magnitudes. Modesof motion and
the quantity of the motive force. Time as the independent
variable. Time, as an aggregation of instants, compared
with a line, as an aggregation of points • • • • 35

LECTURE II. — Generation of magnitudes by "local move-
ments." The simple motions of translation and rotation . 42

LECTURE III. — Composite and concurrent motions. Com-
position of rectilinear and parallel motions . . 47

LECTURE IV. — Properties of curves arising from composition
of motions. The gradient of the tangent. Generalization
of a problem of Galileo. Case of infinite velocity . . 53

LECTURE V. — Further properties of curves. Tangents. Curves

like the cycloid. Normals. Maximum and minimum lines 60

LECTURE VI. — Lemmas ; determination of certain curves con-
structed according to given conditions ; mostly hyperbolas . 69

LECTURE VII. — Similar or analogous curves. Exponents
or Indices. Arithmetical and Geometrical Progressions.
Theorem analogous to the approximation to the Binomial
Theorem for a Fractional Index. Asymptotes , . , 77

xiv BARROW'S GEOMETRICAL LECTURES

LECTURE VIII. — Construction oftangentsby means of auxiliary
curves of which the tangents are known. Differentiation
of a sum or a difference. Analytical equivalents . . 90

LECTURE IX. —Tangents to curves formed by arithmetical and
geometrical means. Paraboliforms. Curves of hyperbolic
and elliptic form. Differentiation of a fractional power,
products and quotients . . . . . . . 101

LECTURE X. — Rigorous determination of ds/rfjc. Differentia-
tion as the inverse of integration. Explanation of the
"Differential Triangle" method; with examples. Differ-
entiation of a trigonometrical function . . . .113

LECTURE XI. — Change of the independent variable in inte-
gration. Integration the inverse of differentiation. Differ-
entiation of a quotient. Area and centre of gravity of a
paraboliform. Limits for the arc of a circle and a hyperbola.
Estimation of IT . . . . . . . . 125

LECTURE XII. — General theorems on rectification. Standard
forms for integration of circular functions by reduction to
the quadrature of the hyperbola. Method of circumscribed
and inscribed figures. Measurement of conical surfaces.
Quadrature of the hyperbola. Differentiation and Integra-
tion of a Logarithm and an Exponential. Further standard
forms .1 55

LECTURE XIII. — These theorems have not been inserted . 196

POSTSCRIPT —

Extracts from Standard Authorities . . . . .198

APPENDIX —

I. Solution of a test question by Barrow's method . . 207

II Graphical integration by Barrow's method . . . 211

III. Reduced facsimiles of Barrow's pages and figures . . 212

INDEX , ..... 216

INTRODUCTION

THE WORK OF BARROW'S GREAT
PREDECESSORS

THE beginnings of the Infinitesimal Calculus, in its two
main divisions, arose from determinations of areas and
volumes, and the finding of tangents to plane curves. The
ancients attacked the problems in a strictly geometrical
manner, making use of the " method of exhaustions." In
modern phraseology, they found "upper and lower limits,"
as closely equal as possible, between which the quantity
to be determined must lie ; or, more strictly perhaps, they
showed that, if the quantity could be approached from two
" sides," on the one side it was always greater than a certain
thing, and on the other it was always less ; hence it must be
finally equal to this thing. This was the method by means
of which Archimedes proved most of his discoveries. But
there seems to have been some distrust of the method, for
we find' in many cases that the discoveries are proved by a
reductio ad absurdum, such as one is familiar with in Euclid.
To Apollonius we are indebted for a great many of the pro-
perties, and to Archimedes for the measurement, of the conic
sections and the solids formed from them by their rotation
about an axis.

The first great advance, after the ancients, came in the
beginning of the seventeenth century. Galileo (1564-1642)
would appear to have led the way, by the introduction of
the theory of composition of motions into mechanics ; * he
also was one of the first to use infinitesimals in geometry,
and from the fact that he uses what is equivalent to "virtual
velocities " it is to be inferred that the idea of time as the
independent variable is due to him. Kepler (1571-1630)
was the first to introduce the idea of infinity into geometry

* See Mach's Science of Mechanics for fuller details.

2 BARROW'S GEOMETRICAL LECTURES

and to note that the increment of a variable was evanescent
for values of the variable in the immediate neighbourhood
of a maximum or minimum ; in 1613, an abundant vintage
drew his attention to the defective methods in use for
estimating the cubical contents of vessels, and his essay
on the subject (Nova Stereometria Dolioruni) entitles him
to rank amongst those who made the discovery of the in-
finitesimal calculus possible. In 1635, Cavalieri published
a theory of "indivisibles," in which he considered a line as
made up of an infinite number of points, a superficies as
composed of a succession of lines, and a solid as a succession
of superficies; thus laying the foundation for the "aggre-
gations " of Barrow. Roberval seems to have been the first,
or at the least an independent, inventor of the method ; but
he lost credit for it, because he did not publish it, preferring
to keep the method to himself for his own use ; this seems
to have been quite a usual thing amongst learned men of
that time, due perhaps to a certain professional jealousy.
The method was severely criticized by contemporaries,
especially by Guldin, but Pascal (1623-1662) showed that
the method of indivisibles was as rigorous as the method
of exhaustions, in fact that they were practically identical.
In all probability the progress of mathematical thought is
much indebted to this defence by Pascal. Since this method
is exactly analogous to the ordinary method of integration,
Cavalieri and Roberval have more than a little claim to be
regarded as the inventors of at least the one branch of the
calculus ; if it were not for the fact that they only applied it
to special cases, and seem to have been unable to generalize
it owing to cumbrous algebraical notation, or to have failed
to perceive the inner meaning of the method when concealed
under a geometrical form. Pascal himself applied the
method with great success, but also to special cases only ;
such as his work on the cycloid. The next step was of a
more analytical nature; by the method of indivisibles,
Wallis (1616-1703) reduced the determination of many
areas and volumes to the calculation of the value of the
series (o'"+ im+ 2™+ . . . «"')/(« + i)nm, i.e. the ratio of the
mean of all the terms to the last term, for integral values of n ;
and later he extended his method, by a theory of interpola-
tion, to fractional values of n. Thus the idea of the Integral

INTRODUCTION 3

Calculus was in a fairly advanced stage in the days immedi-
ately antecedent to Barrow.

What Cavalieri and Roberval did for the integral calculus, ^
Descartes (1596-1650) accomplished for the differential
branch by his work on the application of algebra to geometry.
Cartesian coordinates made possible the extension of in-
vestigations on the drawing of tangents to special curves to
the more general problem for curves of any kind. To this
must be added the fact that he habitually^ used the index
notation ; for this had a very great deal to do with the
possibility of Newton's discovery of the general binomial
expansion and of many other infinite series. Descartes
failed, however, to make any very great progress on his own
account in the matter of the drawing of tangents, owing to
what I cannot help but call an unfortunate choice of a
definition for a tangent to a curve in general. Euclid's
circle-tangent definition being more or less hopeless in the
general case, Descartes had the choice of three : —

(1) a secant, of which the points of intersection with

the curve became coincident ;

(2) a prolongation of an element of the curve, which

was to be considered as composed of an infinite
succession of infinitesimal straight lines ;

(3) the direction of the resultant motion, by which the

curve might have been described.

Descartes chose the first ; I have called this choice unfor-
tunate, because I cannot see that it would have been possible
for a Descartes to miss the differential triangle, and all its
consequences, if he had chosen the second definition. His
choice leads him to the following method of drawing; a
tangent to a_curve in general. _ Describe a circle, whose
centre is on the axis of x, to cut the curve ; keeping the
centre fixed, diminish the radius until the points of section
coincide ; thus, by the aid of the equation of the curve, the
problem is reduced to finding the condition for equal roots
of an equation.

For instance, \etyz = 4ax be the equation to a parabola,
and (x - p)* +yz = r& the equation of the circle. Then we
have (x - pf + 4ax — r*. If this is a perfect square,
x=p - 20, ; i.e. the subtangent is equal to 2a.

4 BARROW'S GEOMETRICAL LECTURES

The method, however, is only applicable to a small
number of simple cases, owing to algebraical difficulties.
In the face of this disability, it is hard to conjecture why
Descartes did not make another choice of definition and use
the second one given above ; for in his rule for the tangents
to roulettes, he considers a curve as the ultimate form of a
polygon. The third definition, if not originally due to
Galileo, was a direct consequence of his conception of the
composition of motions ; this definition was used by
Roberval (1602-1675) and applied successfully to a dozen
or so of the well-known curves; in it we have the germ of
the method of "fluxions." Thus it is seen that Roberval
occupies an almost unique position, in that he took a great
part in the work preparatory to the invention Qiboth branches
of the infinitesimal calculus ; a fact that seems to have
escaped remark. Fermat (1590-1663) adopted Kepler's
notion of the increment of the variable becoming evanes-
cent near a maximum or minimum value, and upon it
based his method of drawing tangents. Fermat's method
of finding the maximum or minimum value of a function in-
volved the differentiation of any explicit algebraic function,
in the form that appears in any beginner's text-book of to-
day (though Fermat does not seem to have the "function "
idea) ; that is, the maximum or minimum values of f(x) are
the roots of f'(x) = o, -where f(x) is the limiting value of
[f(x + K) -f(x)\lh ; only Fermat uses the letter e or E instead
of h. Now, if YYY is any curve, wholly con-
cave (or convex) to a straight line AD, TZYZ
a tangent to it at the point Y whose ordinate
is NY, and the tangent meets AD in T;
also, if ordinates NYZ are drawn on either
side of NY, cutting the curve in Y and the
tangent in Z; then it is plain that the
ratio YN : NT is a maximum (or a mini-
mum) when Y is the point of contact of
the tangent.

Here then we have all the essentials for the calculus ;
but only for explicit integral algebraic functions, needing
the binomial expansion of Newton, or a general method of
rationalization which did not impose too great algebraic
difficulties, for their further development; also, on the

INTRODUCTION 5

authority of Poisson, Fermat is placed out of court, in that he
also only applied his method to certain special cases. Follow-
ing the lead of Roberval, Newton subsequently used the
third definition of a tangent, and the idea of time as the '
independent variable, although this was only to insure that
one at least of his working variables should increase uni-
formly. This uniform increase of the independent variable
would seem to have been usual for mathematicians of the
period and to have persisted for some time ; for later we find
with Leibniz and the Bernoullis that d(dy\dx) = (cPyldx^dx.
Barrow also used time as the independent variable in order
that, like Newton, he might insure that one of his variables,
a moving point or line or superficies, should proceed uni-
formly; it is to be noted, however, that this is only in the
lectures that were added as an afterthought to the strictly
geometrical lectures, and that later this idea becomes
altogether subsidiary. Barrow, however, chose his own
definition of a tangent, the second of those given above ; <
and to this choice is due in great measure his advance over
his predecessors. For his areas and volumes he followed
the idea of Cavalieri and Roberval.

Thus we see that in the time of Barrow, Newton, and
Leibniz the ground had been surveyed, and in many direc-
tions levelled; all the material was at hand, and it only
wanted the master mind to " finish the job." This was f
possible in two directions, by geometry or by analysis;
each method wanted a master mind of a totally different
type, and the men were forthcoming. For geometry,
Barrow: for analysis, Newton, and Leibniz with his in-
spiration in the matter of the application of the simple and
convenient notation of his calculus of finite differences to
infinitesimals and to geometry. With all due honour to
these three mathematical giants, however, I venture to assert
that their discoveries would have been well-nigh impossible
to them if they had lived a hundred years earlier; with the
possible exception of Barrow, who, being a geometer, was
more dependent on the ancients and less on the moderns
of his time than were the two analysts, they would have
been sadly hampered but for the preliminary work of
Descartes and the others I have mentioned (and some I
have not— such as Oughtred), but especially Descartes.

6 BARROW'S GEOMETRICAL LECTURES

LIFE OF BARROW, AND HIS CONNECTION
WITH NEWTON

Isaac Barrow was born in 1630, the son of a linen-draper
in London. He was first sent to the Charterhouse School,
where inattention and a predilection for fighting created a
bad impression ; his father was overheard to say (pray,
according to one account) that " if it pleased God to take
one of his children, he could best spare Isaac." Later, he
seems to have turned over a new leaf, and in 1643 we nnd
him entered at St Peter's College, Cambridge, and afterwards
at Trinity. Having now become exceedingly studious, he
made considerable progress in literature, natural philosophy,
anatomy, botany, and chemistry — the three last with a
view to medicine as a profession, — and later in chronology,
geometry, and astronomy. He then proceeded on a sort
of " Grand Tour " through France, Italy, to Smyrna, Con-
stantinople, back to Venice, and then home through Germany
and Holland. His stay in Constantinople had a great
influence on his after life ; for he here studied the works of
Chrysostom, and thus had his thoughts turned to divinity.
But for this his great advance on the work of his pre-
decessors in the matter of the infinitesimal calculus might
have been developed to such an extent that the name of
Barrow would have been inscribed on the roll of fame as
at least the equal of his mighty pupil Newton.

Immediately on his return to England he was ordained,
and a year later, at the age of thirty, he was appointed to
the Greek professorship at Cambridge ; his inaugural lectures
were on the subject of the Rhetoric of Aristotle, and this
choice had also a distinct effect on his later mathematical
work. In 1662, two years later, he was appointed Professor
of Geometry in Gresham College ; and in the following year
he was elected to the Lucasian Chair of Mathematics, just
founded at Cambridge. This professorship he held for five
years, and his office created the occasion for his Lectiones
Mathematics, which were delivered in the years 1664-5-6
(Habitce Cantabrigice). These lectures were published,
according to Prof. Benjamin Williamson (Encyc. Brit.

INTRODUCTION 7

(Times edition), Art. on Infinitesimal Calculus) in 1670;
this, however, is wrong : they were not published until
1683, under the title of Lectiones Mathematics. What was
published in 1670 was the Lectiones Optics et Geometries;
the Lectiones Mathematics were philosophical lectures on
the fundamentals of mathematics and did not have much
bearing on the infinitesimal calculus. They were followed
by the Lectiones Optics and lectures on the works of
Archimedes, Apollonius, and Theodosius; in what order
these were delivered in the schools of the University I have
been unable to find out ; but the former were published in
1669, "Imprimatur" having been granted in March 1668,
so that it was probable that they were the professorial
lectures for 1667 ; thus the latter would have been delivered
in 1668, though they were not published until 1675, and then
probably by Collins. The great work, Lectiones Geometries,
did not appear as a separate publication at first : as stated
above, it was issued bound up with the second edition of
the Lecliones Optics; and, judging from the fact that there
does not, according to the above dates, appear to have been
any time for their public delivery as Lucasian Lectures,
since Imprimatur was granted for the combined edition in
1669; also from the fact that Barrow's Preface speaks of
six out of the thirteen lectures as " matters left over from
the Optics," which he was induced to complete to form a
separate work ; also from the most conclusive fact of all,
that on the title-page of the Lectiones Geometries there is no
mention at all of the usual notice " Habitse Cantabrigiae " ; —
judging from these facts, I do not believe that the '•''Lectiones
Geometries" were delivered as Lucasian Lectures. Should this
be so, it would clear up a good many difficulties ; it would
corroborate my suggestion that they were for a great part
evolved during his professorship at Gresham College ; also
it would make it almost certain that they would have been
given as internal college lectures, and that Newton would
have heard them in 1663-1664.

Now, it was in 1664 that Barrow first came into close
personal contact with Newton ; for in that year, he
examined Newton in Euclid, as one of the subjects for
a mathematical scholarship at Trinity College, of which
Newton had been a subsizar for three years ; and it was due

to Barrow's report that Newton was led to study the
Elements more carefully and to form a better estimate of
their value. The connection once started must have
developed at a great pace, for not only does Barrow secure
the succession of Newton to the Lucasian chair, when he
relinquished it in 1669, but he commits the publication
of his Lectiones Optica to the foster care of Newton and
Collins. He himself had now determined to devote the
rest of his life to divinity entirely ; in 1670 he was created
a Doctor of Divinity, in 1672 he succeeded Dr Pearson as
Master of Trinity, in 1675 ne was chosen Vice-Chancellor
of the University; and in 1677 he died, and was buried
in Westminster Abbey, where a monument, surmounted by
his bust, was soon afterwards erected to his memory by
his friends and admirers.

Ill
THE WORKS OF BARROW

Barrow was a very voluminous writer. On inquiring of
the Librarian of the Cambridge University Library whether
he could supply me with a complete list of the works of
Barrow in order of publication, I was informed that the
complete list occupied four columns in the British Museum
Catalogue \ This of course would include his theological
works, the several different editions, and the translations of
his Latin works. The following list of his mathematical
works, such as are important for the matter in hand only,
is taken from the Bibliotheca Britannica (by Robert Watt,
Edinburgh, 1824) : —

1. Euclid's Elements, Camb. 1655.

2. Euclid's Data, Camb. 1657.

3. Lectiones Opticortim Phenomenon, Lond. 1669.

4. Lectiones Optic&et Geometricce, Lond. 1670

(in 2 vols., 1674'; trans, Edmond Stone, 1735)-

5. Lectiones Mathematics, Lond. 1683.
This list makes it absolutely certain that Williamson is

wrong in stating that the lectures in geometry were published
under the title of "Mathematical Lectures." This, how-
ever, is not of much consequence; the important point in

INTR OD UCTION 9

the list, assuming it to be perfectly correct as it stands, is
that the lectures on Optics were first published separately
in 1669. In ^e following year they were reissued in a
revised form with the addition of the lectures on geometry.

The above books were all in Latin and have been
translated by different people at one time or another.

IV
ESTIMATE OF BARROWS GENIUS

The writer of the article on " Barrow, Isaac," in the ninth
(Times) edition of the Encyclopedia Britannica, from which
most of the details in Section II have been taken, remarks : —

''By his English contemporaries Barrow was considered
a mathematician second only to Newton. Continental
writers do not place him so high, and their judgment is prob-
ably the more correct one."

Founding my opinion on the Lectiones Geometric^ alone, I
fail to see the reasonableness of the remark I have italicized.
Of course, it was only natural that contemporary continental
mathematicians should belittle Barrow, since they claimed
for Fermat and Leibniz the invention of the infinitesimal
calculus before Newton, and did not wish to have to con-
sider in Barrow an even prior claimant. We see that his
own countrymen placed him on a very high level ; and
surely the only way to obtain a really adequate opinion of a
scientist's worth is to accept the unbiased opinion that has
been expressed by his contemporaries, who were aware of
all the facts and conditions of the case ; or, failing that, to
try to form an unbiased opinion for ourselves, in the position
of his contemporaries. An obvious deduction may be drawn
from the controversy between Newton and Hooke • the
opinion of Barrow's own countrymen would not be likely to
err on the side of over-appreciation, unless his genius was
great enough to outweigh the more or less natural jealousy
that ever did and ever will exist amongst great men occupied
on the same investigations. Most modern criticism of
ancient writers is apt to fail, because it is in the hands of
the experts ; perhaps to some degree this must be so, yet
you would hardly allow a K.C- to be a fitting man for a jury.

io BARROW'S GEOMETRICAL LECTURES

Criticism by experts, unless they are themselves giants like
unto the men whose works they criticize, compares, perhaps
unconsciously, their discoveries with facts that are now
common knowledge, instead of considering only and solely
the advance made upon what was then common knowledge.
Thus the skilled designers of the wonderful electric engines
of to-day are but as pigmies compared with such giants as
a Faraday.

Further, in the case of Barrow, there are several other
things to be taken into account. We must consider his dis-
position, his training, his changes of intention with regard
to a career, the accident of "his connection with such a man
as Newton, the circumstances brought about by the work of
his immediate predecessors, and the ripeness of the time
for his discoveries.

His disposition was pugnacious, though not without a
touch of humour ; there are many indications in the Lectiones
Geometries alone of an inclination to what I may call, for
lack of a better term, a certain contributory laziness ; in this
way he was somewhat like Fermat, with his usual " I have
got a very beautiful proof of . . . : if you wish, I will send
it to you ; but I dare say you will be able to find it for your-
self"; many of Barrow's most ingenious theorems, one or
two of his most far-reaching ones, are left without proof,
though he states that they are easily deduced from what has
gone before. He evidently knows the importance of his
discoveries ; in one place he remarks that a certain set of
theorems are a " mine of information, in which should any-
one investigate and explore, he will find very many things
of this kind. Let him do so who must, or if it pleases him."
He omits the proof of a certain theorem, which he states
has been very useful to him repeatedly ; and no wonder it
has, for it turns out to be the equivalent to the differentia-
tion of a quotient ; and yet he says, " It is sufficient for me
to mention this, and indeed I intend to stop here for a while."
It is not at all strange that the work of such a man should
come to be underrated.

His pugnacity is shown in the main object pervading the
whole of the Lectiones Geometricce • he sets out with the one
express intention of simplifying and generalizing the existing
methods of drawing tangents to curves of all kinds and of

INTRODUCTION 11

finding areas and volumes ; there is distinct humour in his
glee at " wiping the eye " of some other geometer, ancient
or modern, whose solution of some particular problem he
has not only generalized but simplified.

"Gregory St Vincent gave this, but (if I remember
rightly) proved with wearisome prolixity."

" Hence it follows immediately that all curves of this
kind are touched at any one point by one straight line
only. . . . Euclid proved this as a special case for the
circle, Apollonius for the conic sections, and other persons
in the case of other curves."

His early training was promiscuous, and could have had
no other effect than to have fostered an inclination to leave
others to finish what he had begun. His Greek professor-
ship and his study of Aristotle would tend to make him a
confirmed geometrician, revelling in the "elegant solution"
and more or less despising Cartesian analysis because of
its then (frequently) cumbersome work, and using it only
with certain qualms of doubt as to its absolute rigour.
For instance, he almost apologizes for inserting, at the
very end of Lecture X, which ends the part of the work
devoted to the equivalent of the differential calculus, his
" a and & " method — the prototype of the " h and k " method
of the ordinary text-books of to-day.

Another light is thrown on the matter of Cartesian
geometry, or rather its applications, by Lecture VI ; in this,
for the purpose of establishing lemmas to be used later,
Barrow gives fairly lengthy proofs that

(i) my±xy = mx-/fi, (2) ±yx+gx — my = mx'2/r

represent hyperbolas, instead of merely stating the fact on
account of the factorizing of mx^jb + xy, »;x^jr±xy. The
lengthiness of these proofs is to a great extent due to the
fact that, although the appearance of the work is algebraical,
the reasoning is almost purely geometrical. It is also to be
noted that the index notation is rarely used, at least not till
very late in the book in places where he could do nothing else,
although Wallis had used even fractional indices a dozen
years before. In a later lecture we have the truly terrifying
equation (rrkk - rrff-\- 2fmpa)lkk = (rrmm + ^fn^pa)|kk.
Again we must note the fact that all Barrow's work,

12 BARROW'S GEOMETRICAL LECTURES

without exception, was geometry, although it is fairly evident
that he used algebra for his own purposes.

From the above, it is quite easy to see a reason why
Barrow should not have turned his work to greater account ;
but in estimating his genius one must make allowance for
this disability in, or dislike for, algebraic geometry, read
into his work what could have been got out of it (what I
am certain both Newton and Leibniz got out of it), and
not stop short at just what was actually published. It
must chiefly be remembered that these old geometers could
use their geometrical facts far more readily and surely than
many mathematicians of the present day can use their
analysis. As a justification of the extremely high estimate
I have formed, from the Lectiones Geometries alone, of
Barrow's genius, I call the attention of the reader to the
list of analytical equivalent? of Barrow's theorems given on
page 30, if he has not the patience to wade through the
running commentary which stands instead of a full trans-
lation of this book.

V
THE SOURCES OF BARROW'S IDEAS

There is too strong a resemblance between the methods
to leave room for doubt that Barrow owed much of his idea
of integration to Galileo and Cavalieri (or Roberval, if you
will). On the question as to the sources from which he
derived his notions on differentiation there is considerably
more difficulty in deciding ; and the comparatively narrow
range of my reading makes me diffident in writing anything
that may be considered dogmatic on this point ; and yet if
I do not do so, I shall be in danger of not getting a fair
hearing. The following remarks must therefore be con-
sidered in the nature of the plea of a " counsel for the
defence," who believes absolutely in his client's case ; or as
suggestions that possibly, even if not probably, come very
near to the truth.

The general opinion would seem to be that Barrow was a
mere improver on Fermat. But Barrow was conscientious to
a fault ; and if we are to believe in his honesty, the source
of his ideas could not have been Fermat. For Barrow re-

INTRODUCTION 13

ligiously gives references to the ancient and contemporary
mathematicians whose work he quotes. These references
include Cartesius, Hugenius, Galilseus, Gregorius a St Vin-
centio, Gregorius Aberdonensis, Wallis, and many others,
with Euclides, Aristoteles, Archimedes, Apollonius, among
the ancients ; but, as far as I can find, no mention is made
of Fermat in any place ; nor does Barrow use Fermat's idea
of determining the tangent algebraically by consideration of
a maximum or minimum ; these points entirely contradict
the notion that he was a mere improver on Fermat, which
seems to have arisen because Barrow uses the same letter,'
e, for his increment of x, and only adds another, a, to signify
the increment of y. I suggest that this was only a coinci-
dence ; that both adopted the letter e (Fermat seems to have
used the capital E) as being the initial letter of the word excess,
whilst Barrow in addition used the letter a, the initial letter
of the word additional ; if he was a mere improver on Fermat,
the improvement was a huge one, for it enabled Barrow to
handle, without the algebraical difficulties of Fermat, im-
plicit functions as well as explicit functions. On the other
hand, Barrow may have got the notion of using arithmetic
and geometric means, with which he performs some wonders,
from Fermat, who apparently was the first to use them, though
by Barrow's time they were fairly common property, being
the basis of all systems of logarithms ; and Barrow's con-
dition of tangency was so similar to the method of Fermat
that, while he could not very well use any other condition
with his choice of the definition of a tangent, Barrow may
have deliberately omitted any reference to Fermat, for fear
that thereby he might, by the reference alone, provoke ac-
cusations of plagiarism. As I have already remarked, there
is a distinct admiration for the work of Galileo, and the
idea of time as the independent variable obsesses the first
few lectures ; however, he simply intended this as a criterion
by means of which he could be sure that one of his variables
increased uniformly, or in certain of his theorems in the
later parts that he might consider hisj as a function of a
function ; but in most of the later lectures the idea of time
becomes quite insignificant. This is, of course, explained by
the fact that the original draft of the geometrical lectures
consisted only of the lectures numbered VI to XII (includ-

i4 BARROW'S GEOMETRICAL LECTURES

ing the appendices, with the possible exception of Appendix
3 to Lect. XII); for we read in the Preface that Barrow
" falling in with his wishes (I will not say very willingly) added
the first five lectures." The word " his " refers to Librarius,
which — for lack of a better word, or of editor, which I do
not like — I have translated as the publisher; but I think it
refers to Collins, for, in Barrow's words, " John Collins looked
after the publication."

The opinion I have formed is that the idea of the differ-
ential triangle, upon which all attention seems (quite
wrongly) to be focussed, when considering the work of
Barrow, was altogether his own original concept ; and to call
it a mere improvement on Fermat's method, in that he uses
two increments instead of one, is absurd. The discovery
was the outcome of Barrow's definition of a tangent, wholly
and solely ; and the method of Fermat did not consider this.

The mental picture that I form of Barrow, and of the
events that led to this discovery, amongst others far more
important, is that of the Professor of Geometry at Gresham
College, who has to deliver lectures on his subject ;
he reads up all that he can lay his hands on, decides
that it is all very decent stuff of a sort, yet pugnaciously
determines that he can and will "go one better." In the

Fig. A.

Fig. B.

course of his researches, he is led from one thing to
another until he comes to the paraboliform construction of
Lect. IX, § 4, perceives its usefulness and inner meaning,
and immediately conceives the idea of the differential
triangle. I think if any reader compares the two figures
above, Fig. A used for his construction of the paraboliforms,
Fig. B for the differential triangle, he will no longer inquire

INTR OD UCTION 1 5

for the source of Barrow's idea, unless perhaps he may ,
prefer to refer it to Lect. X, § 1 1.

Personally, I have no doubt that it was a flash of inspira-
tion, suggested by the first figure ; and that it was Barrow's
luck to have first of all had occasion to draw that figure, and
secondly to have had the genius to note its significance and
be able to follow up the clue thus afforded. As further
corroborative evidence that Barrow's ideas were in the main
his own creations, we have the facts that he was alone in
using habitttally the idea of a curve being a succession of
an infinite number of infinitely short straight lines, the pro--
longation of any one representing the tangent at the point
on the curve for which the straight line, or either end of
of it, stood ; also that he could not see any difference
between indefinitely narrow rectangles and straight lines as
the constituents of an area. If his methods had required it,
which they did not, he would no doubt have proved rigor-
ously that the error could be made as small as he pleased
by making the number of parts, into which he had divided
his area, large enough; this was indeed the substance
of Pascal's defence of Cavalieri's method of " indivisibles,"^
and the idea is used in Lect. XII, App. II, § 6.

I have remarked that, in considering the work of Barrow,
all attention seems to be quite wrongly focussed on the differ-
ential triangle. I hope to convince readers of this volume
that the differential triangle was only an important side-
issue in the Lectiones Geometricce; certainly Barrow only
considered it as such. Barrow really had, concealed under
the geometrical form that was his method, a complete treatise
on the elements of the calculus.

The question may then be asked why, if all this is true,
did Barrow not finish the work he had begun ; and the
answer, I take it, is inseparably bound up with the peculiar
disposition of Barrow, his growing desire to forsake
mathematics for divinity, and the accident of having first
as his pupil and afterwards as his co-worker, and one in
close personal contact with him, a man like Newton, whose
analytical mind was so peculiarly adapted to the task of
carrying to a successful conclusion those matters which
Barrow saw could not be developed to anything like the
extent by his own geometrical method. One writer has

1 6 BARROW'S GEOMETRICAL LECTURES

stated that the great genius of Barrow must be admitted,
if only for the fact that he recognized in the early days of
Newton's career the genius of the man, his pupii, that was
afterwards to overshadow him. Also, if I fail 10 make
out my contention that Barrow's ideas were in the main
original, the same remark can with justice be applied to
him that William Wallace in similar circumstances applied
to Descartes, that if it were true that he borrowed his ideas
on algebra from others, this fact, " would only illustrate
the genius of the man who could pick out from other works
all that was productive, and state it with a lucidity that makes
it look his own discovery"; for the lucidity is there all right
in this work of Barrow, only it wants translating into analy-
tical language before it can be readily grasped by anyone
but a geometer.

MUTUAL INFLUENCE OF NEWTON AND
BARROW

I can image that Barrow's interest, as a confirmed
geometer, would have been first aroused by >oung Newton's
poor show in his scholarship paper on Euclid. This was in
April 1664, the year of the delivery of Barrow's first lectures
as Lucasian Professor, and, according to Newton's own
words, just about the time that he, Newton, discovered his
method of infinite series, led thereto by his reading of the
works of Descartes and Wallis. Newton no doubt attended
these lectures of Barrow, and the probability is that he
would have shown Barrow his work on infinite series ; for
this would seem to have been the etiquette or custom of
the time; for we know that in 1669 Newton communicated
to Collins through Barrow a compendium of his work on
fluxions (note that this is the year of the preparation for
press of the Lectiones Opticce et Geometric^}. Barrow could
not help being struck by the incongruity (to him) of a man
of Newton's calibre not appreciating Euclid to the full : at
the same time the one great mind would be drawn to the
other, and the connection thus started would have ripened
inevitably. I suggest as a consequence that Barrow would
show Newton his own geometry, Newton would naturally

INTRODUCTION 17

ask Barrow to explain how he had got the idea for some of
his more difficult constructions, and Barrow would let him
into the secret. " I find out the constructions by this little
list of rules, and methods for combining them." " But, my
dear sir, the rules are far more valuable than the mere find-
ing of the tangents or the areas." " All right, my boy, if you
think so, you are welcome to them, to make what you like of,
or what you can ; only do not say you got them from me, I'll
stick to my geometry."* This was probably the occasion when
Newton persuaded Barrow that the differential triangle was
more general than all his other theorems put together ; also
later when the Geometry was being got ready for press,
Newton probably asked Barrow to produce from his stock
of theorems others necessary to complete his, Barrow's,
Calculus, the result being the appendices to Lect. XII.

The rest of the argument is a matter of dates. Barrow
was Professor of Greek from 1660 to 1662, then Professor
of Geometry at Gresham College from 1662 to 1664, and
Lucasian Professor from 1664 to 1669; Newton was a
member of Trinity College from 1661, and was in residence
until he was forced from Cambridge by the plague in the
summer of 1665 ; from manuscript notes in Newton's hand-
writing, it was probably during, and owing to, this enforced
absence from Cambridge (and, I suggest, away from the
geometrical influence of Barrow) that he began to develop
the method of fluxions (probably in accordance with some
such permission from Barrow as that suggested in the purely v
imaginative interview above).

The similarity of the two methods of Barrow and Newton
is far too close to admit of them being anything else but
the outcome of one single idea ; and I argue from the dates
given above that Barrow had developed most of his geometry
from the researches begun for the necessities of lectures at
Gresham College. We know that Barrow's work on the
difficult theorems and problems of Archimedes was largely
a suggestion of a kind of analysis by which they were reduced
to their simple component problems. What is then more
likely than that this is an intentional or unintentional crypto-
grammatic key to Barrow's own method? I suggest that it

* Of course this is imaginative retrospective prophecy ; I beg that no
one will take the inverted commas to signify quotations.

i8 BARROWS GEOMETRICAL LECTURES

is more than likely, — IT is. As I said, the similarity of the
two methods of Newton and Barrow is very striking.
For the fluxional method the procedure is as follows : —

(1) Substitute x + xo for x and y+yo fory in the given
equation containing the fluents x and y.

(2) Subtract the original equation, and divide through by o.

(3) Regard o as an evanescent quantity, and neglect o
and its powers.

Barrow's rules, in altered order to correspond, are : —

(2) After the equation has been formed (Newton's rule i),
reject all terms consisting of letters denoting constant or
determined quantities, or terms which do not contain a or e
(which are equivalent to Newton's yo and xo respectively) ;
for these terms brought over to one side of the equation will
always be equal to zero (Newton's rule 2, first part).

(i) In the calculation, omit all terms containing a power
of a or e, or products of these letters ; for these are of no
value (Newton's rule 2, second part, and rule 3).

(3) Now substitute m (the ordinate) for a, and / (the sub-
tangent) for e. (This corresponds with Newton's next step,
the obtaining of the ratio x : y, which is exactly the same as
Barrow's e : a. )

The only difference is that Barrow's way is the more suited
to his geometrical purpose of finding the " quantity of the
subtangent," and Newton's method is peculiarly adapted to
analytical work, especially in problems on motion. Barrow
left his method as it stood, though probably using it freely
(mark the word usitatum on page 119, which is & frequentative
derivative of utor, I use) to obtain hints for his tangent
problems, but not thinking much of it as a method compared
with a strictly geometrical method ; yet admitting it into
his work, on the advice of a friend, on account of its
generality. On the other hand, Newton perceived at once
the immense possibilities of the analytical methods intro-
duced by Descartes, and developed the idea on his own
lines, to suit his own purposes.

There is still another possibility. In the Preface to the
Optics, we read that " as delicate mothers are wont, I com-
mitted to the foster care of friends, not unwillingly, my dis-
carded child." . . . These two friends Barrow mentions by
name: "Isaac Newton ... (a man of exceptional ability

INTR OD UCTION 1 9

and remarkable skill) has revised the copy, warning me of
many things to be corrected, and adding some things from
his own work." . . . Newton's additions were probably con-
fined to a great extent to the Optics only ; but the geometrical
lectures (seven of them at least) were originally designed as
supplementary to the Optics, and would be also looked over
by Newton when the combined publication was being pre-
pared. . . . " John Collins has attended to the publication."
Hence, it is just possible that Newton showed Barrow his
method of fluxions first, and Barrow inserted it in his own
way ; this supposition would provide an easy explanation
of the treatment accorded to the batch of theorems that
form the third appendix to Lect. XII ; they seem to be
hastily scrambled together, compared with the orderly treat-
ment of the rest of the book, and are without demonstration ;
and this, although they form a necessary complement for
the completion of the standard forms and rules of procedure.
I say that this is possible, but I do not think it is at all
probable; for it is to be noted that Barrow's description
of the method is in the first person singular (although, when
giving the reason for its introduction, he says "frequently
used by us"); and remembering the authentic accounts of
Barrow's conscientious honesty, and also judging by the
later work of Newton, I think that the only alternative to
be considered is that first given. Also, if that is accepted,
we" have a natural explanation of the lack of what I call the
true appreciation of Barrow's genius. Barrow could see the
limitations imposed by his own geometrical methods (none
so well as he, naturally, being probably helped to this con-
clusion by his discussions with Newton) ; he felt that the
correct development of his idea was on purely analytical
lines, he recognised his own disability in that direction and
the peculiar aptness of Newton's genius for the task, and,
. lastly, the growing desire to forsake mathematics for divinity
made him only too willing to hand over to the foster care
of Newton and Collins his discarded child " to be led out
and set forth as might seem good to them." " Carte blanche "
of such a sweeping character very often has exactly the
opposite effect to that which is intended ; and so probably
Newton and Collins forbore to make any serious alterations
or additions, out of respect for Barrow; for although the

20 BARROWS GEOMETRICAL LECTURES

allusion to the revision properly applies only to the Optics,
it may fairly be assumed that it would be extended to the
Geometry as well ; and if not then, at any rate later, for,
quoting a quotation by Canon Overton in the Dictionary
of National Biography (source of the quotation not stated),
which refers to Barrow's pique at the poor reception that
was accorded to the geometrical lectures — and does not this
show the high opinion that Barrow had of them himself, and
lend colour to my suggestion that they were never delivered as
Lucasian Lectures?; also note his remark in the Preface, given
later, " The other seven, as I said, I expose more freely to your
view, hoping that tJiere is nothing in them that it will displease
the erudite to see" — " When they had been some time in the
world, having heard of a very few who had read and considered
them thoroughly, the little relish that such things met with
helped to loose him more from those speculations and heighten
his attention to the studies of morality and divinity." Does
not this read like the disgust at people forsaking the legiti-
mate methods of geometry for " such unsatisfactory stuff (as I
have suggested that Barrow would consider it) as analysis " ?
Who can say the form these lectures might have taken
if there had been no Newton; or if Barrow had taken
kindly to Cartesian geometry; or what a second edition,
"revised and enlarged," might have contained, if Barrow
on his return to Cambridge as Master of Trinity and Vice-
Chancellor had had the energy or the inclination to have
made one ; or if Newton had made a treatise of it, instead
of a reprint of " Scholastic Lectures," as Barrow warns his
readers that it is, and such as the edition of 1674 in two vol-
umes probably was ? But Barrow died only a few years later,
Newton was far too occupied with other matters, and Collins
seems to have passed out of the picture, even if he had been the
equal of the other two.

VII

DESCRIPTION OF THE BOOK FROM WHICH
THE TRANSLATION HAS BEEN MADE

The running commentary which follows is a precis of
a full translation of a book in the Cambridge Library. In
one volume, bound in strong yellow calf, are the two works,

INTRODUCTION 21

the Lectiones Opticce et Geometricce ; the title-page of the
first bears the date MDCLXIX, that of the second the
date MDCLXX, whilst "Imprimatur" was granted on
22nd March 1669; this points to its being one of the
original combined editions, No. 4 of the list in Section III
of this Introduction. On the title-page of the Optics there
is a line which reads, "To which are annexed a few
geometrical lectures," agreeing with the remark in the
preface to the geometrical section that originally there were
only seven geometrical lectures that were intended to be
published as supplements of the Optics, instead of the
thirteen of which the section is composed. For in all
probability this title-page is that of the first edition of the
Optics, but the Librarius, whoever he may have been,
persuaded Barrow to leave the seven lectures out, enlarge
them to form a separate work, and to publish them as
such in combination with the Optics, as we see, in 1670;
and by an oversight the title-page remained uncorrected.

Of prefaces there are three, one being more properly an
introduction, explaining the plan and scope of the originally
designed " XVIII Lectures on Optics " and the supple-
mentary seven geometrical lectures ; this is in the same type
as, and immediately in front of, the Optics. The other two
are true prefaces or " Letters to the reader " ; they are in
italics : a full translation of both is given later.

On a fly-leaf in front of the Optics is a list of symbols
of abbreviation as used by Barrow ; as these cover the two v
sections and are not repeated in front of the geometrical
section, they furnish additional evidence that the book I
have used is one of the first combined editions. The
similarity of the symbols used by Barrow to those used
at the present day, to stand for quite different things, does
not simplify the task of a modern reader. This is especially
the case with the signs for " greater than" and "less than,"
where the " openings " of the signs face the reverse way to
that which is now usual ; another point which might lead
to error by a casual reader who had not happened to
notice the list of abbreviations, is the use of the plus sign
between two ratios to stand for the ratio compounded from
them, i.e. for multiplication ; the minus sign does not,
however, stand for the ratio of two ratios, i.e. for division,

22 BARROW'S GEOMETRICAL LECTURES

the ease with which the argument may be followed is also not
by any means increased by Barrow's plan of running his work
on in one continuous stream (paper was dear in those days),
with intermediate steps in brackets; and this is made still worse
by the use of the " full stop " as a sign of a ratio (division)
instead of as a sign of a rectangle (multiplication) ; thus
DH . HO : : (DL. LN : : DL - DH . LN - HO : : LH . LB : : )LH . HK
stands, in modern symbols, for the extended statement

DH:HO = DL:LN = (DL- DH) : (LN - HO) = LH:HB

.-. DH:HO = LH:LB = LH-.HK;

whereas DL x LK-LH x HK = KO x LH - HK x LH, on the
contrary, means, as is usual at present, DL. HK - LH . HK =
KO . LH - HK . LH, the minus sign thus being a weaker bond
than that of multiplication, but a stronger bond than that
of ratio or division. Barrow's list of symbols, in full, is : —

"For the sake of brevity certain signs are used, the
meaning of which is here subjoined.
A + B that is, A and B taken together.

A - B A, B being taken away.

A - : B The difference of A and B.

A x B A multiplied by, or led into, B.

A divided by B, or applied to B.
B

A = B A is equal to B.

Acr-B A is greater than B.

A ~^B A is less than B.

A. B : : C-D A bears to B the same ratio as C to D.

A, B, C, D -T-T A, B, C, D are in continued proportion.

A . B c-C . D A to B is greater than C to D.

A.B— nC.D A to B is less than C to D.

The ratios j' j equal to J
.B + C.D^>M.N AtoB, C to D-| are Water than -M to N.

compounded ( ) less than j

The square on A.

The side, or square root of, A.

The cube of A.

The side of the square made up of the

square of A and the square of B.

Other abbreviations, if there are any, the reader will re-
cognise, by easy conjecture, especially as I have used very
little analysis."

INTRODUCTION

The style of the text, as one would expect from a Barrow,
is " classical " ; that is, full of long involved sentences,
phrases such as "through all of a straight line points,"
general inversion of order to enable the sense to run on,
use of the relative instead of the demonstrative, and so on ;
all agreeing with what is but an indistinct memory (thank
goodness !) of my trials and troubles as a boy over Cicero,
De Senectute, De Amicitia, and such-like, studied (?), by the
way, in Newton's old school at Grantham in Lincolnshire.

In this way there is a striking difference between the
style of Barrow and the straightforward Latin of Newton's
Principia, as it stands in my Latin edition of 1822, by Le
Seur and Jacquier. My classical attainments are, however,
so slight that, in looking for possible additions by Newton,
I have preferred to rely on my proof-reading experience in
the matter of punctuation. The strong point in Barrow's
somewhat awful punctuation is the use of the semicolon,
combined with the long involved sentence, and the frequent
interpolation of arguments, sometimes running to a dozen
lines, in parentheses ; Newton makes use of the short con-
cise sentence, and rarely uses the semicolon, nor indeed
does he use the colon to any great extent. Of course I do
not know how much the printer had to do with the punctua-
tion in those days, but imagine this distinction was a very
great matter of the author. Comparing two analogous
passages, from each author, of about 200-250 words, we
get the following table : —

Barrow.

Newton.

commas

semicolons

None

colons

full-stops

None

parentheses

This contrast is striking enough for all practical purposes ;
in addition, Barrow starts three of his five sentences with a
relative, whilst Newton does not do this once in his ten.

Using this idea, I failed to find anything that could, with
any probability, be ascribed to Newton.

24 BARROWS GEOMETRICAL LECTURES

Lastly, one strong feature in the book is the continued use
of the paraboliforms as auxiliary curves ; this corroborates
my contention that Barrow fully appreciated the importance
and inner meaning of his theorem, or rather construction
(see note to Lect. IX, § 4) ; that is, he uses it in precisely
the same way as the analytical mathematician uses its equi-
valent, the approximation to the binomial and the differentia-
tion of a fractional power of a variable, as a foundation of
all his work.

Although there are two fairly long lists of errata, most
probably due to Newton, there are still a great number of
misprints ; the diagrams are, however, uniformly good, there
being no omissions of important letters and only one or two
slips in the whole set of 200, one of these evidently being
the fault of the engraver; nevertheless they might have
been much clearer if Barrow had not been in the habit
of using one diagram for a whole batch of allied theorems,
thereby having to make the diagram rather complicated
in order to get all the curves and lines necessary for the
whole batch of theorems on the one figure, whilst only
using some of them for each separate theorem. In the
text which follows this introduction, only those figures have
been retained that were absolutely essential.

There is a book-plate bearing a medallion of George I
and the words "MUNIFICENTIA REGIA 1715" which points
out that the book 1 have was one of the 30,000 volumes of
books and manuscripts comprised in the library of Bishop
Moore of Norwich, which was presented to the Cambridge
University Library in 1715 by George I, as an acknowledg-
ment of a loyal address sent up by the University to the king
on his accession. It may have come into his possession as
a personal gift from Barrow ; at any rate, there is an in-
scription on the first fly-leaf, "A gift from the author." I
am unable to ascertain whether Moore was a student at
Cambridge at the date of the publication of these lectures,
but the date of his birth (1646) would have made him twenty-
four years of age at the time, and this supposition would ex-
plain the presence of a four-line Latin verse (Barrow had a
weakness for turning things into Latin or Greek verse) on
the back of the title-page of the geometrical section, which
reads : —

INTRODUCTION 25

To a young man at the University*

Humble work of thy brother, pronounced or to be,
Noiv rightly appears, devoted to thee ;
Should' st learn from it aught, both happy and sure
In thy patronly favour permit it endure,

and is in the same handwriting as the inscription.

VIII SQ

THE PREFACES

In the following translation of the Prefaces, ordinary type
is used instead of Barrow's italics, in order that I may call
attention to points already made, or points that will be
possibly referred to later, in the notes on the text, by means
of italics.

The first Preface, which precedes the Optics : —

" Communication to the reader.

" Worthy reader,

" That this, of whatever humble service it
may be, was not designed for you, you will soon understand
from many indications, if you will only deign to examine it ;
nor, that you might yet demand it as your due, were other
authorities absent. To these at least, truly quaking in mind
and after great hesitation, I yielded ; chiefly because thereby
I should set as an example to my successors the production
of a literary work as a duty, such as I myself was the first
to discharge ; if less by the execution thereof, at any rate
by the endeavour at advance, not unseemly, nor did it seem
to be an ostentation foreign, to my office. There was in
addition some slight hope that there might be therein some-
thing of the nature of good fruit, such as in some measure
might profit you, and not altogether be displeasing to you.
Also, remember, I warn those of you, who are more ad-
vanced in the subject of my book, what manner of writing
you are handling ; not elaborated in any way for you alone ;

* With apologies for doggerel ; but the translation is fairly close, line for
line.

26 BARROW'S GEOMETRICAL LECTURES

not produced on my own initiative; nor by long medita-
tion, exhibiting the ordered concepts of leisurely thought ;
but Scholastic Lectures ; first extracted from me by the
necessities of my office ; then from time to time expanded
over-hastily to complete my task within the allotted time ;
lastly, prepared for the instruction of a promiscuous literary
public, for whom it was important not to leave out many
lighter matters (as they will appear to you). In this way
you will not be looking in vain (and it is necessary to warn
you of this, lest by expecting too much you may harm both
yourself and me) for anything elaborated, skilfully arranged,
or neatly set in order. For indeed I know that, to make
the matter satisfactory to you, it would be expedient to cut
out many things, to substitute many things, to transpose
many things, and to 'recall all to the anvil and file.' For
this, however, I had neither the stomach nor the leisure to
take the pains ; nor indeed had I the capability to carry the
matter through. And so I chose rather to send them forth
' in Nature's garb,' as they say, and just as they were born ;
rather than, by laboriously licking them into another shape,
to fashion them to please. However, after that I had
entered on the intention of publication, either seized with
disgust, or avoiding the trouble to be undergone in making
the necessary alterations, in order that I should not indeed
put off the rewriting of the greater part of these things, as
delicate mothers are wont, I committed to the foster care of
friends, not unwillingly, my discarded child, to be led out and
set forth as it might seem good to them. Of which, for I
think it right that you should know them by name, Isaac
Newton, a fellow of our college (a man of exceptional ability
and remarkable skill) has revised the copy, warning me of
many things to be corrected, and adding some things from
his own work, which you will see annexed with praise here
and there. The other (whom not undeservedly I will call
the Mersenne of our race, born to carry through such essays
as this, both of his own work and that of others) John
Collins has attended to the publication, at much trouble
to himself.

" I could now place other obstacles to your expectation, or
show further causes for your indulgence (such as my meagre
ability, a lack of experiments, other cares intervening) if

INTRODUCTION 27

I were not afraid that that bit of wit of the elder Cato would
be hurled at me : —

'"Truly you publish abroad these things as if bound by
a decree of the Amphictyones.'

"At least fairness demanded a prologue of this kind, and
in some degree a certain parental affection for one's own
offspring enticed it forth, in order that it might stand forth
the more excusable, and more defended from censure.

" But if you are severe, and will not admit these excuses
into a propitious ear, according to your inclination (I do
not mind) you may reprove as much and as vigorously as
you please."

The second Preface, which refers to the Geometry : —

"My dear reader,

" Of these lectures (which you will now

receive in a certain measure late-born), seven (one being ex-
cepted) I intended as the final accompaniments and as it were
the things left over from the Optical lectures, which stand forth
lately published; otherwise, I imagine, I shall be thought
little of for bringing out sweepings of this kind. However,
when the publisher [or editor — Librarius — ? Collins] thought,
for reasons of his own, that these matters should be prepared,
separately removed from the others ; and moreover he de-
sired something else to be furnished that should give the
work a distinct quality of its own (so that indeed it might
surpass the size of a supplementary pamphlet); falling in
with his wishes (/ will not say very willingly) I added the
first five lectures, cognate in matter with those following and
as it were coherent ; which indeed / had devised some years
ago, but, as with no idea of publishing, so without that care
which such an intention calls for. For they are clumsily
and confusedly written; nor do they contain anything
firmly, or anything lying beyond the use or the compre-
hension of the beginners for which they are adapted ; where-
fore I warn those experienced in this subject to keep their
eyes turned away from these sections, or at least to give
'them indulgence a little liberally.

" The other seven, that I spoke of, I expose more freely to
your view, hoping that there is nothing in them that it ivill
displease the more erudite to see.

28 BARROW'S GEOMETRICAL LECTURES

" The last lecture of all a friend (truly an excellent "man,
one of the very best, but in a business of this sort an in-
satiable dun *) extorted from me ; or, more correctly,
claimed its insertion as a right that was deserved.

" For the rest, what these lectures bring forth, or to what
they may lead, you may easily learn by tasting the beginnings
of each.

" Since there is now no reason why I should longer
detain or delay you,

" FAREWELL."

HOW BARROW MADE HIS CONSTRUCTIONS

In hazarding a guess as to how Barrow came by his con-
structions, one has, to a great extent, to be guided by his
other works, together with any hint that may be obtained
from the order of his theorems in the text. Taking the
latter first, I will state the effect the reading of the text had
on me. The only thing noticeable, to begin with, was the
pairing of the propositions, rectangular and polar ; the rest
seemed more or less a haphazard grouping, in which one
proposition did occasionally lead to another; but certain
of the more difficult constructions were apparently without
any hint from the preceding propositions. Once, however,
it began to dawn on me that Barrow was trying to write a
complete elementary treatise on the calculus, the matter was
set in a new light. First, the preparation for the idea of a
small part of the tangent being substituted for a small part
of the arc, and vice versa (Lect. V, § 6), this, of course,
having been added later, probably, I suggest, to put the
differential triangle on a sound basis ; then the lemmas on
hyperbolas, for the equivalent of a first approximation in
the form of y = (ax + b)j(cx + d) for any equation in the form
giving y as an explicit function of x ; this first gave the
clue pointing to his constructions having been found out
analytically ; then the work on arithmetical and geometrical
means leading to the approximation to the binomial raised

* Flagitator improbus ; a specimen of Harrovian humour.

INTRODUCTION 29

to a fractional power ; lastly, a few tentative standard forms ;
and then Lect. IX, with the differentiation of a fractional
power, and the whole design is clear as day. Barrow knows
the calculus algebraically and is setting it in geometrical
form to furnish a rigorous demonstration. From this point
onwards, truly with many a sidestep as something especially
pretty strikes him as he goes, he proceeds methodically to
accumulate the usual collection of standard forms and
standard rules for their completion as a calculus. If one
judges from this alone, there is no other possible explana-
tion of the plan of the work.

I then looked round for some hint that might corroborate
this opinion, and I found it, to me as clear as daylight, in
his lectures on the explanation of the method of Archimedes.
In these I am convinced Barrow is telling the story of his
own method, as well as stating the source from which he has
derived the idea of such a procedure. With this compare
Newton's anagram and Fermat's discreet statement of the
manner in which he proved that any prime of the form
4« + i was the sum of two squares. Any reader, who has
been led, by reading this statement, into trying to produce
a proof of this theorem for himself, will agree with me that
Fermat was not giving very much of his method away. And
so it was with all these mathematicians, and other scientists
as well ; they stated their results freely enough, and some-
times gave proofs, but generally in a form that did not reveal
their own particular methods of arriving at them. For
instance, take the construction of Lect. IX, § 10; to my
mind there cannot possibly be any doubt that he arrived
at it analytically ; and the analytical equivalent of it as it
stands is

If y=f(x\ and Mz
then \Adzldx = \\dyjdx + ( M - N)

given the capacity for doing this bit of differentiation, the
construction given would be easily found by Barrow. This
construction is all the more remarkable be,cause the proof
given is unsound, not to say wrong; and I suggest that this
fact is a very strong piece of evidence thaV the construction
was not arrived at geometrically. Mary other examples
might be cited, but this one should be sufficient.

30 BARROW'S GEOMETRICAL LECTURES

ANALYTICAL EQUIVALENTS TO BARROW'S
CHIEF THEOREMS

Fundamental Theorem

If n is any positive rational number, integral or fractional,
''then (i + •#)"<! +n.x, according as «< i ; and this inequality
tends to become an equality when x tends to zero.
[Proved without convergence in Lect. VII, §§ 13-16.]

Standard Forms for Differentiation

1. Ify is any function of x, and z = A/j,

thmdz/dx = - (k/y*) . dy/dx . . ' . Lect. VIII, 9

2. If y is a function of x, and z —y + C,

then dz/dx^dyldx Lect. VIII, n

3. Ify is any function of x, and z2 =jy2 - a2,
then z.dz/dx=y.dy/dx; or, in another form,

if z = V(j2 - a2), then dz/dx = [y/J(y* - a^]dy/dx

Lect. VIII, 13

4. If z2 =jy2 + a2, then 2 . <&/</# =y . dy/dx,

OTdz/dx = [y/J(y2 + a*)]dyldx . ' . . Lect. VIII, 14

5. Ifz2 = a2-jy2, then z.dz/dx = -y.dyjdx,

or dzjdx = - [j/v/(a2 -y*}}dyjdx . . Lect. VIII, 1 5

6. Ify is any function of x, and z = a + fry,

then dzjdx = b . dyfdx ..... Lect. IX, i

7. If/ is any function of x, and zn = an~r.yr,

then (i/s) • dz/dx = (n/r) .(i/y)- dyjdx . . Lect. IX, 3

8. Special case : d(xn}jdx = n . x"'1 or n . (y/x),
where « is a positive rational . . . Lect. IX, 4

9. The case when n is negative is to be deduced from
the combination of Forms i and 8.

10. If y = tanx, then dy/dx = sec2x, proved as Ex. 5 on
the " differential triangle " at the end of Lect. X.

n. It is to be noted that the same two figures, as used
for tan x, can be used to obtain the differential coefficients
of the other circular functions.

INTRODUCTION 31

Laws for Differentiation

LAW i. Sum of Two Functions. — lfw=y + z,
then dwjdx = dy/dx + dz/dx . . . . Lect. VIII, 5

LAW 2. Product of Two Functions. — Ifw=yz,
then ( i /w) . dw/dx = ( i /y) . dyjdx + (i/z). dz/dx . Lect. IX, 1 2

LAW 3. Quotient of Two Functions.— If w=yjz,
then, ifv=i/z, (i/v).dv/dx = -(i/z) .dz/dx, as has already
been obtained in Lect. VIII, 9 ; hence by the above —

(i/w) . dw/dx = (i/y) . dy/dx - (i/z) . dz/dx.

N.B. — Note the logarithmic form of these two results,
corresponding with the subtangents used by Barrow.

The remaining standard forms Barrow is unable ap-
parently to obtain directly; and the same remark applies
to the rest of the laws. So he proceeds to show that

Differentiation and Integration are inverse operations. •£
(i.) If R.z = [ydx, then R. dz/dx =y . . Lect.(L^, n
(ii.) If R.dz/dx=y, then R.z = fydx . . Lect. XI, 19
Hence the standard forms for integration are to be
obtained immediately from those already found for differ-
entiation. Barrow, however, proves the integration formula
for an integral power independently, in the course of certain
theorems in Lect. XL He also gives a separate proof of the
quotient law in the form of an integration, in Lect. XI, 27.

Further Standard Forms for Integration

Lect. XII, App. 3,

B. \Qa*dx = k(ax- i) ) Prob. 3, 4

C. Ptan0d0=bg(cos0) . . Lect. XII, App. I, 2

D. ^ecOde = ^l0g{(i+sme}l(i-sme}} „ „ 5

E. \dx/(a2- x2) = {log (a + x)/(a-x)}/2a (see Form D)

F. \cos 0 d(tan 0} dO = \tan 0 d(cos 6) dd - tan 6 cos 0,
both being equal to ^secOdO, the only example of
"integration by parts" I have noticed . Lect. XII, App. I, 8

G. |^/N/(^ + «2) = M{^ + v/(^ + «2)}/«] „ „ 9

32 BARROW'S GEOMETRICAL LECTURES

Graphical Integration of any Function

For any function, f(x), that cannot be integrated by the
foregoing rules, Barrow gives a graphical method for
^f(x}dx as a logarithm of the quotient of two radii vectores
of the curve r=f(6), and for \dxjf(x} as a difference of
their reciprocals . . . Lect. XII, App. Ill, 5-8

Fundamental Theorem in Rectification

He proves that (ds/dx)- = i + (dyldxf . . Lect. X, 5
He rectifies the cycloid (thus apparently anticipating
Wren), the logarithmic spiral, and the three-cusped hypo-
cycloid (as special cases of one of his general theorems), and
reduces the rectification of the parabola to the quadrature
of the rectangular hyberbola, from which the rectification
follows at once.

(XII, App. Ill, i, Ex. 2; XI, 26; XII, 20, Ex. 3.)

In addition to the foregoing theorems in the Infinitesimal
Calculus (for if it is not a treatise on the elements of the
Calculus, what is it?), Barrow gives the following interesting
theorems in the appendix to Lect. XI. : —

Maxima and Minima

He obtains the maximum value of xr(c-x)s, giving the
condition that x/r = (c - x)/s ; also he shows that this is the
condition for the minimum value of xrj(x—c}s.

Trigonometrical Approximations

Barrow proves that the circular measure of an angle a
lies between 3 sin a/(2 + cos a) and sin a(2 + cos a)/(i + 2 cos a),
the former being a lower limit, and equivalent to the formula
of Snellius; each of these approximations has an error of
the order of a5.

THE
GEOMETRICAL LECTURES

ABRIDGED TRANSLATION

WITH NOTES, DEDUCTIONS, PROOFS OMITTED

BY BARROW, AND FURTHER EXAMPLES OF

HIS METHOD

LECTURE I

Generation of magnitudes. Modes of motion and the
quantity of the motive force. Time as the independent vari-
able. Time, as an aggregate of instants, compared with a
line, as the aggregate of points. Deductions.

[In this lecture, Barrow starts his subject with what he
calls the generation of magnitudes.]

Every magnitude can be either supposed to be produced,
or in reality can be produced, in innumerable ways. The
most important method is that of "local movements." In
motion, the matters chiefly to be considered are the mode
of motion and the quantity of the motive force. Since
quantity of motion cannot be discerned without Time, it
is necessary first to discuss Time. Time denotes not an
actual existence, but a certain capacity or possibility for a
continuity of existence; just as space denotes a capacity
for intervening length. Time does not imply motion, as
far as its absolute and intrinsic nature is concerned ; not
any more than it implies rest ; whether things move or are
still, whether we sleep or wake, Time pursues the even
tenor of its way. Time implies motion to be measurable ;
without motion we could not perceive the passage of Time.

" On Time, as Time, 'tis yet confessed
From moving things distinct, or tranquil rest,
No thought can be"

is not a bad saying of Lucretius. Also Aristotle says: —
" When we, of ourselves, in no way alter the train of our
thought, or indeed if we fail to notice things that are affecting
it, time does not seem to us to have passed" And indeed it
does not appear that any, nor is it apparent how much, time
has elapsed, when we awake from sleep. But from this, it
is not right to conclude that: — "// is plain that Time does
not exist without motion and change of position" " We do
not perceive it, therefore it does not exist," is a fallacious
inference ; and sleep is deceptive, in that it made us connect
two widely separated instants of time. However, it is very
true that: — " Whatever the amount of the motion was, so
much time seems to have passed" ; nor, when we speak of so
much time, do we mean anything else than that so much
motion could have gone on in between, and we imagine
the continuity of things to have coextended with its con-
tinuously successive extension.

We evidently must regard Time as passing with a steady
flow ; therefore it must be compared with some handy
steady motion, such as the motion of the stars, and
especially of the Sun and the Moon ; such a comparison
is generally accepted, and was born adapted for the pur-
pose by the Divine design of God (Genesis i, 14). But how,
you say, do we know that the Sun is carried by an equal
motion, and that one day, for example, or one year, is
exactly equal to another, or of equal duration ? I reply

LECTURE I 37

that, if the sun-dial is found to agree with motions of any
kind of time-measuring instrument, designed to be moved
uniformly by successive repetitions of its own peculiar
motion, under suitable conditions, for whole periods or for
proportional parts of them ; then it is right to say that it
registers an equable motion. It seems to follow that strictly
speaking the celestial bodies are not the first and original
measures of Time; but rather those motions, which are
observed round about us by the senses and which underlie
our experiments, since we judge the regularity of the
celestial motions by the help of these. On the other hand,
Time may be used as a measure of motion; just as we
measure space from some magnitude, and then use this
space to estimate other magnitudes commensurable with
the first ; i.e. we compare motions with one another by the
use of time as an intermediary.

Time has many analogies with a line, either straight or
circular, and therefore may be conveniently represented by
it ; for time has length alone, is similar in all its parts, and
can be looked upon as constituted from a simple addition
of successive instants or as from a continuous flow of one
instant ; either a straight or a circular line has length alone,
is similar in all its parts, and can be looked upon as being
made up of an infinite number of points or as the trace of
a moving point.

Quantity of the motive force can similarly be thought of
as aggregated from indefinitely small parts, and similarly
represented by a straight line or a circular line ; when Time
is represented by a distance the motive force is the same

38 BARROWS GEOMETRICAL LECTURES

as the velocity. Quantity of velocity cannot be found from
the quantity of the space traversed only, nor from the lime
taken only, but from both of these brought into reckoning
together; and quantity of time elapsed is not determined
without known quantities of space and velocity; nor is
quantity of space (so far as it may be found by this
method) dependent on a definite quantity of velocity
alone, nor on so much given time alone, but on the joint
ratio of both.

To every instant of time, or indefinitely small particle of
time, (I say instant or indefinite particle, for it makes no
difference whether we suppose a line to be composed of
points or of indefinitely small linelets ; and so in the same
manner, whether we suppose time to be made up of instants

i or indefinitely minute timelets) ; to every instant of time, I
say, there corresponds some degree of velocity, which the

• moving body is considered to possess at the instant ; to this
degree of velocity there corresponds some length of space
described (for here the moving body is a point, and so we
consider the space as merely long). But since, as far as
this matter is concerned, instants of time in nowise depend
on one another, it is possible to suppose that the moving
body in the next instant admits of another degree of velo-
city (either equal to the first or differing from it in some
proportion), to which therefore will correspond another
length of space, bearing the same ratio to the former as the
latter velocity bears to the preceding ; for we cannot but
suppose that our instants are exactly equal to one another.
Hence, if to every instant of time there is assigned a suit-

LECTURE I 39

able degree of velocity, there will be aggregated out of these
a certain quantity, to any parts of which respective parts
of space traversed will be truly proportionate; and thus
a magnitude representing a quantity composed of these
degrees can also represent the space described. Hence,
if through all points of a line representing time are drawn
straight lines so disposed that no one coincides with another
(i.e. parallel lines), the plane surface that results as the aggre-
gate of the parallel straight lines, when each represents the
degree of velocity corresponding to the point through which
it is drawn, exactly corresponds to the aggregate of the
degrees of velocity, and thus most conveniently can be
adapted to represent the space traversed also. Indeed
this surface, for the sake of brevity, will in future be called
the aggregate of the velocity or the representative of the
space. It may be contended that rightly to represent each
separate degree of velocity retained during any timelet, a
very narrow rectangle ought to be substituted for the right
line and applied to the given interval of time. Quite so,
but it comes to the same thing whichever way you take it; but
as our method seems to be simpler and clearer, we will in
future adhere to it.

[Barrow then ends the lecture with examples, from which
he obtains the properties of uniform and uniformly accel-
erated motion.]

(i) If the velocity is always the same, it is quite evident
from what has been said that the aggregate of the velocity
attained in any definite time is correctly represented by a
parallelogram, such as AZZE, where the side AE stands for

40 BARROW'S GEOMETRICAL LECTURES

a definite time, the other AZ, and all the parallels to it,

BZ, CZ, DZ, EZ, separate degrees of

A B r. n r

velocity corresponding to the separate

instants of time, and in this case
plainly equal to one another. Also
the parallelograms AZZB, AZZC, AZZD,
AZZE, conveniently represent, as has

z z z z z

Fig. i.

been said, the spaces described in the z
respective times. AB, AC, AD, AE.

(2) If the velocity increase uniformly from rest, then the
aggregate of the velocities is represented by the triangle
AEY. Also if the velocity increases uniformly from some
definite velocity to another definite velocity represented
respectively by CY, EY, then the space is represented by a
trapezium, such as CYYE.

(3) If the velocity increase according to a progression of
square numbers, the space described to represent the aggre-
gate of the velocity is the complement of a semi-parabola.
[For which Barrow gives a figure.]

[From (i) and (2) all the properties of uniform motion
and of uniformly accelerated motion are simply deduced,
and the lecture concludes with the remark : — ]

These things, being necessary for the understanding of
things to be said later, and theories of motion that are, I
think, not on the whole quite useless, it has seemed to be
advantageous to explain clearly as a preliminary. Having
finished this task, I direct my steps forward.

LECTURE I 41

NOTE

There is not much in this lecture calling for remark.
The matter, as Barrow says in his Preface, is intended for
beginners. There is, however, the point, to which attention
is called by the italics on page 39, that Barrow fails to see
any difference between the use of lines and narrow rect-
angles as constituent parts of an area. This is Cavalieri's
method of " indivisibles," which Pascal showed incontro-
yertibly was the same as the method of "exhaustions,"
as used by the ancients. There is evidence in later lectures
that Barrow recognized this; for he alludes to the possi-
bility of an alternative indirect argument (discursus apo-
gogicus) to one of his theorems, and later still shows his
meaning to be the method of obtaining an upper and a
lower limit. There is also a suggestion that he personally
used the general modern method of the text-books, that of
proving that the error is less than a rectangle of which one
side represented an instant and the other the difference
between the initial and final velocities ; and that it could
be made evanescent by taking the number of parts, into
which the whole time was divided, large enough. Also
the attention of those who still fight shy of graphical
proofs for the laws of uniformly accelerated motion, if any
such there be to-day, is called to the fact that these proofs
were given by such a stickler for rigour as Barrow, with the
remark that they are evident, at a glance, from the diagrams
he draws.

LECTURE II

Generation of magnitudes by '•''local movements" The
simple motions of translation and rotation.

Mathematicians are not limited to the actual manner in
which a magnitude has been produced ; they assume any
method of generation that may be best suited to their
purpose.* Magnitudes may be generated either by simple
motions, or by composition of motions, or by concurrence
of motions.

[Examples of the difference of meaning that Barrow
attaches to the two latter phrases are given by him in a
later lecture. The simple motions are considered in this
lecture.]

There are two kinds of simple motions, translation and
rotation, i.e. progressive motion and motion in a circle.
For these motions, mathematicians assume that (i) a
point can progress straightforwardly from any fixed
terminus, and describe a straight line of any length ;
(2) a straight line can proceed with one extremity moving

* As an example, take the case of finding the volume of a right circular
cone by integration ; here, by definition, the method of generation is the
rotation of a right-angled triangle about one of the rectangular sides ;
but it is supposed to be generated, for the purpose of modern integration,
by the motion of a circle, that constantly increases in size, and moves
parallel to itself with its centre on the axis of the cone.

LECTURE II 43

along any other line, keeping parallel to itself; the former
is called the genetrix, and is said to be applied to the latter
which is called the directrix ; by these are described paral-
lelogrammatic surfaces, when the genetrix and the directrix
are both in the same plane, and prismatic and cylindrical
surfaces otherwise. In general, the genetrix may, if neces-
sary, be taken as a curve, which is intended to include
polygons, and the genetrix and the directrix may usually
be interchanged. The same kinds of assumptions are
made for simple motions of rotation; and by these are
described circles and rings and sectors or parts of these,
when a straight line rotates in its own plane about a point
in itself or in the line produced ; if the directrix is a curve
(in the wider sense given above), and does not lie in the
plane of the genetrix, of which one point is supposed to
be fixed, the surfaces generated are pyramidal or conical.
From this kind of generation is deduced the similarity of
parallel sections of such surfaces ; and thus it is evident
that the surfaces can also be generated by taking the
genetrix of the first method as the directrix, and the former
directrix as the genetrix so long as it is supposed to shrink
proportionately as it proceeds parallel to itself towards what
was the vertex or fixed point in the first method.

For producing solids the chief method is a simple
rotation, about some fixed line as axis, of another line
lying in the same plane with it. In addition, there is the
method of " indivisibles," which in most cases is perhaps
the most expeditious of all, and not the least certain and
infallible of the whole set.

44 BARROW'S GEOMETRICAL LECTURES

The learned A. Tacquetus * more than once objects to
this method in his clever little book on " Cylindrical and
Annular Solids," and therein thinks that he has falsified
it, because the things found by means of it concerning the
surfaces of cones and spheres (I mean quantities of these)
do not agree in measurement with the truths discovered
and handed down by Archimedes.

Fig. io.t

Take, for example, a right cone DVY, whose axis is VK;
through every point of this suppose that there pass straight
lines ZA, ZB, ZC, ZD (or KD), etc.; from these indeed
according to the Atomic theory the right-angled triangle
VDK is made up; and from the circles described with these
as radii the cone itself is made. " Therefore" he argues,
"from the peripheries of these circles is composed the conical
surface ; now this is found to be contrary to the truth ; hence
the method is fallacious,"

I reply that the calculation is wrongly made in this
manner ; and in the computation of the peripheries of which

* Andreas Tacquet, a Jesuit of Antwerp : published a book on the
cylinder (1651), Elementa Geometrice (1654) and a book on Arithmetic
(1664) ; mentioned by Wallis.

f The numbering of the diagrams is Barrow's and is, in consequence of
abridgment, not consecutive.

LECTURE II 45

such a surface is composed, a reasoning has to be adopted
different from that used when computing the lines from
which plane surfaces are made up, or the planes from which
solids are formed. In fact, it must be considered that the
multitude of peripheries forming the curved surface are
produced, through the rotation of the line YD, from the
multitude of points in the genetrix YD itself; by observing
this distinction all error will be obviated, as I will now
demonstrate.*

At every point of the line YD, instead of the line VK,
suppose that right lines are applied perpendicular to the
line YD, and equal to the peripheries, taken in order,
that make up the curved surface. From these parallel
straight lines is generated the plane VOX, which is equal to
the said curved surface.

Further, if instead of the peripheries we apply the corre-
sponding radii, the space produced will bear to the curved
surface a ratio equal to that of the radius of any circle to
its circumference. In the particular case chosen, the two

* This is the first example we come across of the superiority of Barrow's
insight into what is really the method of integration. In effect, Barrow
points out that if a periphery is thought of as a solid
ring of very minute section, then in this case the V j\^
section is a trapezium, as shown in the annexed | v
diagram, of which the parallel sides are perpendicular i ^
to the axis oi the cone, and the non-parallel sides x \

both pass, if produced, through the vertex. Tacquet x^\

uses the surface generated by the top parallel sides, I \s
PS as if he were finding the area of a circle, by I v\

means of concentric rings (? or he uses the perpendi- _[

cular distance from S on QR) ; Barrow points out that
he should use the surface generated by the slant
side SR.

In modern phraseology, Barrow shows that Tacquet
has made the error of integrating along a radius of the
base (? or along the axis), instead of along a slant side.

46 BARROW'S GEOMETRICAL LECTURES

plane surfaces are triangles and the area of the curved
surface is thus easily found.

There are other methods which may be used conveniently
in certain cases ; but enough has been said for the present
concerning the construction of magnitudes by simple
motions.

NOTE

It would be interesting to see how Barrow would get
over the difficulty raised by Tacquet, if Tacquet's example
had been the case of the oblique circular cone. It seems
to me to be fortunate for Barrow that this was not so.
Barrow also states, be it noted, that the method is general
for any solid of revolution, if the generating line is supposed
to be straightened before the peripheries are applied ; in
which case, the area can be found for the curved surface
only when the plane surface aggregated from the applied
peripheries turns out to be one whose dimensions can be
found.

Thus, if the ordinate varies as the square of the arc
measured from the vertex, the plane equivalent is a semi-
parabola, and the area is z-n-sr/^, where s is length of the
rotating arc, and r is the maximum or end ordinate.

LECTURE III

Composite and concurrent motions. Composition of
rectilinear and parallel motions,

In generation by composite motions, if the remaining
motions are unaltered, then, according as the velocity of
one, or more, is altered, we usually obtain magnitudes
differing not only in kind but also in quantity, or at least
differing in position every time.

Thus, suppose the straight line AB
_is_carried along the straight line AC
by a uniform parallel motion, and at _
the same time a point M descends
uniformly in AB; or suppose that,
while AC descends with a uniform
parallel motion, it cuts AB also
moving uniformly and to the right. A
From motions of this kind, com-
posite in the former case, and con-
current in the latter, the straight line AM may be
produced.

Again, if in the previous example, whilst the motion of
the straight line AB remains the same as before with respect
to its velocity, but the uniform motion of the point M, or

48 BARROW'S GEOMETRICAL LECTURES

of the straight line AC, is altered in velocity, so that indeed
the point M now comes to the point //,, or AC cuts AB in
//., there is described by this motion another straight line
A/*, in a different position from the first.

Further, if once more, while the motion of AB remains
the same, instead of the uniform motion of the point M,
or of the straight line AC, we substitute a motion that is
called uniformly accelerated ; from such composite or con-
current motion is produced the parabolic line AMX, or in
another case the line A/*Y, according as the accelerated
motion is supposed to be one thing or another in degree.

In these examples, it is seen that composite and con-
current motions come to the same thing in the end ; but
frequently the generation of magnitudes is not so easily
to be exhibited by one of these methods as by the other.
Thus suppose that a straight line AB is uniformly rotated
round A, and at the same time the point M, starting from A.
is carried along AB by a continuous and uniform motion ;
from this composite motion is produced a certain line, namely
the Spiral of Archimedes, which cannot be explained satis-
factorily by any concurrence of motions. On the other hand,
if a straight line BA is rotated with uniform motion about a
centre B, and at the same time a straight line AC is moved
in a parallelwise manner uniformly along AB, the continuous
intersections of BA, AC, so moving, form a certain line,
usually called the Quadratrix ; and the generation of this
line is not so clearly shown or explained by any strictly
so-called composition of motion.

Magnitudes can be compounded and also decomposed in

LECTURE III 49

innumerable ways ; but it is impossible to take account of
all of these, so we shall only discuss some important cases,
such as are considered to be of most service and the more
easily explained. Such especially are those. that are com-
pounded of rectilinear and parallel motions, or rectilinear and
rotary motions, or of several rotary motions; preference
being given to those in which the constituent simple motions
are all, or at least some of them, uniform. Moreover, there
is not any magnitude that cannot be considered to have been
generated by rectilinear motions alone. For every line that
lies in a plane can be generated by the motion of a straight
line parallel to itself, and the motion of a point along it ;
every surface by the motion of a plane parallel to itself and
the motion of a line in it (that is, any line on a curved surface
can be generated by rectilinear motions) ; in the same way
solids, which are generated by surfaces, can be made to
depend on rectilinear motions.

I will only consider the generation of lines lying in one
plane by rectilinear and parallel motions ; for indeed there
is not one that cannot be produced by the parallel motion
of a straight line together with that of a point carried along
it ; * but the motions must be combined together as the
special nature of the line demands.

For instance, suppose that a straight line ZA is always
moved along the straight line AY parallelwise, by any
motion, uniform or variable, increasing or decreasing

* In other words, Barrow states that every plane curve has a Cartesian
equation, referred to either oblique or rectangular coordinates ; yet it is
doubtful whether he fully recognizes that all the properties of the curve can
be obtained from the equation.

50 BARROWS GEOMETRICAL LECTURES

or alternating in velocity, according to any imaginable
ratio ; and that any point M in it is moved in such a
way that the motion of the point is proportional to the
motion of the straight line, throughout any the same
intervals of time; then there will be certainly a straight
line generated by these motions.

B C

2
Fig. 17.

For, since we always have

AB:AC = BM:fy, or AB : MX = AM : X/*,
(MX being drawn parallel to AC), it follows that the points
A, M, fj- are in one straight line.

But if, instead, these motions are comparable with one
another in such fashion that, given any line D, then the
rectangle, contained by the difference between the line D
and BM the distance traversed and BM itself, always bears
the same ratio to the square on AB (the distance traversed
by the line AZ in the same time) ; then a circle or an ellipse
is described ; a circle, if the supposed ratio is one of equality
and the angle ZAY is a right angle, and an ellipse otherwise .
and in these there will be one diameter, situated in the line
AZ in its first position, and drawn from A in the direction
of Z, and this diameter will be equal to D.

LECTURE III 51

If, however, the motions are such that the rectangle con-
tained by the sum of the lines D and BM and BM itself bears
always the same ratio to the square on the line AB, a hyper-
bola will be produced ; a rectangular or equilateral hyperbola,
if the assigned ratio is one of equality and the angle ZAY is
a right angle; if otherwise, of another kind, according to
the quality of the assigned ratio; of these the transverse
diameter will be equal to D, being situated in ZA when
occupying its first position, and being measured in a
direction opposite to Z; and the parameter is given by
the given ratio.

But if the rectangle contained by D and BM bears always
the same ratio to the square on AB, it is evident that a
parabolic line is produced, of which the parameter is easily
found from the given straight line D and the quantity of the
assigned ratio.

Also in the first case of these, if the transverse motion
along AY is supposed to be uniform, the descending motion
along AZ will also be uniform ; in the second and third
cases, if the motion along AY is uniform, the descending
motion along AZ will be continually increasing ; and the
same thing being supposed in the last case, in which the
parabola is produced, the point M has its velocity increased
uniformly.

In a similar manner, any other line can be conceived to
be produced by such a composition of motion. But we
shall come across these some time or other as we go along ;
let us see whether anything useful in mathematics can be
obtained from a supposed generation of lines in this way.

52 BARROWS GEOMETRICAL LECTURES

But for the sake of simplicity and clearness let us suppose
that one of the two motions, say that of the line preserving
parallelism, is always uniform ; and let us strive to make
out what general properties of the generated lines arise from
the general differences with regard to the velocity of the
other ; let us try, I say, but in the next lecture.

NOTE

We here see the reason for Barrow considering time as
the independent variable; he states, indeed, that the con-
structions can be effected, no matter what is the motion of
the line preserving parallelism ; but for the sake of simplicity
and clearness he decides to take this motion as uniform ; for
this the consideration of time is necessary. At the same
time it is to be noted that Barrow, except for the first case
of the straight line, is unable to explicitly describe the
velocity of the point M, and uses a geometrical condition
as the law of the locus ; in other words, he gives the pure
geometry equivalent of the Cartesian equation. In later
lectures, we shall find that he still further neglects the use
of time as the independent variable. This, as has been
explained already, is due to the fact that the first five
lectures were added afterwards. In Barrow's original de-
sign, the independent variable is a length along one of his
axes. This length is, it is true, divided into equal parts ;
but that is the only way, a subsidiary one, in which time
enters his investigations ; and even so, the modern idea of
" steps " along a line, used in teaching beginners, is a better
analogue to Barrow's method that that which is given by a
comparison with fluxions.

LECTURE IV

Properties of curves arising from composition of motions.
The gradient of the tangent. Generalization of a problem of
Galileo. Case of infinite velocity.

Hereafter, for the sake of brevity, I shall call a parallel
motion of the straight line AZ along AY a "transverse
motion," and the motion of a point moving from A in the
line AZ a " descent " or a " descending motion," regard
being had of course to the given figure. Also I shall take
the motion along AY and parallels to it as uniform, hence
this motion and parts of it can represent the time and
parts of the time. Now I come to the properties of lines
produced by a uniform transverse motion and a continually
increasing descent.

1. The line produced is curved in all its parts.

2. The velocity of the uniform descending motion, by
which a curve is described (i.e. if there is a common uniform
transverse motion for the chord and its arc) is less than the
velocity, which the increasing descending motion has at N,
the common end of both.

3. Of a curve of this sort, any chord, as MO, falls entirely

54 BARROW'S GEOMETRICAL LECTURES

within the arc, and if produced, falls entirely without the
curve.

This property was separately proved for the circle by
Euclid, for the conic sections by Apollonius, for cylinders
by Serenus.

4. Curves of this sort are convex or concave to the
same parts throughout.

This is the same as saying that a straight line only cuts
the curve in two points ; nor does it differ from the definition
of " hollow," as given by Archimedes at the beginning of
his book on the sphere and the cylinder.

5. All straight lines parallel to the genetrix cut the curve ;
and any one cuts the curve in one point only.

This was proved, separately, for the parabola and the
hyperbola by Apollonius, and for the sections of the
concoids by Archimedes.

6. Similarly all parallels to the directrix cut the curve,
and in one point only.

Apollonius proved this for the sections of the cone.

7. All chords of the curve meet the genetrix and all
parallels to it, produced if necessary.

Apollonius thought it worth while to prove this property
separately for the parabola and the hyperbola.

8. Similarly, all straight lines touching the curve, with
one exception (see § 1 8), meet the same parallels.

This also Apollonius showed for the conic sections in
separate theorems.

LECTURE IV 55

9. Also any straight lines cutting the genetrix will also
cut the curve.

Apollonius went to a very great deal of trouble to prove
a property of this kind in the case of the conic sections.

10. Straight lines applied to the directrix, i.e. parallels
to the genetrix, have a ratio to one another (when the less
is the antecedent) which is less than the ratio of the corre-
sponding spaces traversed by the moving straight line ; i.e.
the ratio of the versed sines of the curve, the less to the
greater, is less than the ratio of the sines.

This property of circles and other curves, it will be found,
is everywhere proved separately for each kind.

NOTE

All the preceding properties are deduced in a very simple
manner from one diagram ; and Barrow's continual claim
that his method not only simplifies but generalizes the
work of the early geometers is substantiated.

The full proof of the next property is given, to illustrate
Barrow's way of using one of the variants of the method of
exhaustions.

11. Let us suppose that a straight line TMS touches a
given curve at a point M (i.e. it

does not cut the curve); and

let the tangent meet AZ in T, A

and through M let PMG be drawn

parallel to AY. I may say that '

the velocity of the descending \4K

point, describing the curve by its O\^

motion, which it has at the point Fig. 20.

56 BARROWS GEOMETRICAL LECTURES

of contact M, is equal to the velocity by which the
straight line TP would be described uniformly, in the
same time as the straight line
AZ is carried along AC or PM (or,
what comes to the same thing, I
say that the velocity of the de-
scending point at M has the same

ratio to the velocity with which ^\

the straight line AZ is moving as °\S

the straight line TP has to the straight line PM).

For, let us take anywhere in the tangent TM any point
K, and through it draw the straight line KG, meeting the
curve in 0 and the parallels AY, PG in D, G. Then, since
the tangent TM is supposed to be described by a twofold
uniform motion, partly of the straight line TZ carried
parallelwise along AC or PM, and partly of the point T
descending from T along TZ ; and since, of these motions,
the one along AC or PM is common with, or the same as,
that by which the curve is described ; and since, when TZ
is in the position KG, AZ will be in the same position as
well ; therefore, when the point descending from T is at K,
the point descending from A will be at 0, the intersection
of the curve with KG (for the straight line KG cannot cut
the curve in any other point, as has already been shown).
Also the point 0 is below K, because the tangent lies en-
tirely outside the curve. Now, if the point 0 is supposed
to be above the point of contact, towards T, since in that
case OG is less than GK, it is clear that the velocity of the
descending point, by which the curve is described, at the

LECTURE IV 57

point 0 is less than the velocity of the uniformly descend-
ing motion, by which the tangent is produced; since the
former, always increasing, in the same time (namely that
represented by GM) traverses a smaller space than the latter
which does not increase at all ; and as this goes on continu-
ally, the former describes the straight line OG whilst the
latter describes the straight line KG. On the other hand,
if the point K is below the point of contact towards the end
S, since OG is then greater than KG, it is clear that the
velocity of the descending point, by which the curve is pro-
duced, at the point 0, in the same way as before, is greater
than the velocity of the uniformly descending motion, by
which the tangent is described j for the former motion,
continually decreasing during the time represented by GM,
traverses a greater space than the latter, which does not
decrease at all, but keeping constant, describes indeed the
space KG. Hence, since the velocity of the point describ-
ing the curve, at any point of the curve above the point of
contact towards A, is less than the velocity of the motion
for TP ; and at any point of the curve below the point of
contact is greater than it ; it follows that it is exactly equal
to it at the point M.

12. The converse of the preceding theorem is also true.

13. From these two theorems, it follows at once that
curves of this kind are touched by any one straight line in
one point only.

This, separately, Euclid proved for the circle, Apollonius
for the conic sections, and others for other curves.

58 BARROWS GEOMETRICAL LECTURES

From this method, then, there comes out an advantage
not to be despised, that by the one piece of work proposi-
tions are proved concerning tangents in several cases.

14. The velocities of the descending point at any two
assigned points of a curve have to one another the ratio
reciprocally compounded from the ratios of the lines applied
to the straight line AZ from these points (i.e. parallels to
AY) and the intercepts by the tangents at these points
measured from the said applied lines. In other words, the
ratio of the velocities is equal to the ratio of the intercepts
divided by the ratio of the applied lines.

15. Incidentally, I here give a general solution by my
method, and one quite easy to follow, of that problem
which Galileo made much of, and on which he spent much
trouble, about which Torricelli said that he found it most
skilful and ingenious. Torricelli thus enunciates it (for the
enunciation of Galileo is not at hand) : —

" Given any parabola with vertex A, it is required to find
some point above it, from which if a heavy body falls to A,
and from A, with the velocity thus attained, is turned hori-
zontally, then the body will describe the parabola."

NOTE

Barrow gives a very easy construction for the point, and
a short simple proof; further his construction is perfectly
general for any curve of the form y = xn, where « is a posi-
tive integer.

1 6. 17. These are two ingenious methods for determining
the ratio of the abscissa to the subtangent.

Barrow remarks that the theorems will be proved more
geometrically later, so that they need not be given here.

LECTURE IV 59

1 8. A circle, an ellipse, or any " returning" curve of this
kind, being supposed to be generated by this method, then
the point describing any one of them must have an infinite
velocity at the point of return.

For instance, let a quadrant ARM be so generated; then
since the tangent, TM, is parallel to the diameter AZ, and
only meets it at an infinite distance, therefore the velocity
at M is to the velocity of the uniform motion of AZ parallel
to itself as an infinite straight line is to PM ; hence, the
velocity at M must certainly be infinite. And indeed it
will be so for all curves of this kind ; but for others which
are gradually continued to infinity (such as the parabola
or the hyperbola) the velocity of the descending point at
any point on the curve is finite.

Leaving this, let us go on to those other properties of the
given curves which have to be expounded.

NOTE

It is to be observed that, although Barrow usually draws
his figures with the applied lines at right angles to his
directrix AZ, his proofs equally serve if the applied lines are
oblique, in all cases when not otherwise stated. That is,
analytically, his axes may be oblique or rectangular. Having
mentioned this point, since my purpose is largely with
Barrow's work on the gradient of the tangent, I shall always
draw the applied lines at right angles, as Barrow does ; ex-
cept in the few isolated cases where Barrow has intentionally
drawn them oblique.

LECTURE V

Further properties of curves. Tangents. Curves like Uie
Cycloid. Normals. Maximum and minimum lines.

i. The angles made with the applied lines by the tangents
at different points of a curve are unequal ; and those are
less which are nearer to the point A, the vertex.

Fig. 26.

2. Hence it may be taken as a general theorem that
tangents cut one another between the applied lines drawn
at right angles to AZ through the points of contact.

3. The angle PTM is greater than the angle XQN.

4. Applied lines nearer to the vertex (and therefore also
any straight lines parallel to other directions) cut the curve
at a greater angle than those more remote.

LECTURE V 61

5. If the angle made by an applied line is a right angle
or obtuse, I say that the arc MN of the curve is greater
than the straight line MN, but less than the straight
line ME.

This is a most useful theorem for service in proving
properties of tangents. For, it follows from it that, if the
arc MN is assumed to be indefinitely small, we may safely
substitute instead of it the small bit of the tangent, i.e. either
MEorNH.

NOTE

We have here the statement of the fundamental idea of
Barrow's method, to which all the preceding matter has led.
This is a fine illustration of Barrow's careful treatment ; and
it is to be observed that this idea is not quite the same thing
as the idea of the differential triangle as one is accustomed
to consider it nowadays, i.e. as a triangle of which the hypo-
tenuse is an infinitely small arc of the curve that may be
considered to be a straight line. It will be found later that
Barrow uses the idea here given in preference to the other,
because by this means the similarity of the infinitesimal
triangle with the triangle TPM is far more clearly shown on
his diagrams ; and many matters in Barrow are made sub-
servient to this endeavour to attain clearness in his diagrams.
For instance, when he divides a line into an infinite number
of parts, he generally uses four parts on his figure, and gives
the demonstration with the warning "on account of the
infinite division " as a preliminary statement.

As an example of the use to which the above theorems
may be put, Barrow finds the tangent to the Cycloid, his
construction being applicable to all curves drawn by the
same method. Note that this is not the general case of the
roulette discussed by Descartes. Barrow's construction and
proof are given in full to bring out the similarity of his
criterion of tangency to Fermat's idea, as mentioned in
the Introduction.

62 BARROW'S GEOMETRICAL LECTURES

6. A straight line AY, moving parallel to itself, traverses
any curve, either concave or convex to the same parts,
with uniform motion (that is to say, it passes over equal
parts of the curves in equal times), and simultaneously
any point is carried, also uniformly, along AY from A ;
by the point moving in this manner there is generated
a curve AMZ, of which it is required to find the tangent
at any point M.

Fig. 27.

To do this, draw MP parallel to AY to cut the curve APX
in P; through P draw the straight line PE touching the
curve APX; through M draw MH parallel to PE; take any
point R in MH, and draw RS parallel to PM ; mark off RS
so that MR : RS = arc AP : PM (i.e. as the one uniform
motion is to the other); join MS. Then MS will touch
the curve AMZ.

For, if any point Z be taken in this curve, and through
it ZK be drawn parallel to MP, cutting the curve APX in X,
the tangent at P in E, MH the parallel to it in H, and MS
in S; then,

(i), if the point Z is above M towards A, PE < arc PX ;
.-. arc PA: PE > arc PA : arc PX.

LECTURE V 63

But arc PA : arc PX = PM : PM - XZ = PM : EH - XZ

arc PA : arc PX = PM : ZH - EX > PM : ZH ;
hence, arc PA : PE > PM : ZH or arc PA : PM > PE : ZH.

But arc PA : PM = MR : RS = MH : KH - PE : KH ;
PE:KH > PE:ZH, and KH < ZH.

Now, since EZ < XZ < PM or EH, the point H is outside
the curve AZM ; hence K is outside the curve AZM.

Similarly [Barrow gives it in full], (ii), if the point Z is
below the point M, 'K will be outside the curve ; therefore
the whole straight line KMKS lies outside the curve, and
thus touches it at M.

After this digression we will return to other properties of
the curve.

7. Any parallel to the tangent TM, through a point E
directly below T, will meet the curve. [Fig. 26.]

8. If E lies between the point T and the vertex A, the
parallel to the tangent will cut the curve twice.

Apollonius was hard put to it to prove these two theorems
for the conic sections.

9. If any twt> lines are equally inclined to the curve,
these straight lines diverge outwardly, i.e. they will meet
one another when produced towards the parts to which
the curve is concave.

10. If a straight line is perpendicular to a curve, and
along it a definite length HM is taken, then HM is the
shortest of all straight lines that can be drawn to the curve
from the point H.

•64 BARROWS GEOMETRICAL LECTURES

11. It follows that the circle, with centre H, drawn
through M, touches the curve.

12. Conversely, if HM is the shortest of all straight lines
that can be drawn from H to the curve, then HM will be
perpendicular to the curve.

13. If HM is the shortest of all straight lines that can be
drawn from H, and if the straight line TM is perpendicular
to it, then TM touches the curve.

14. Further, a line which is nearer to HM is shorter than
one which is more remote.

15. Hence it follows that any circle described with centre
H meets the curve in one point only on either side of M ;
that is, it does not cut the curve in more than two points
altogether.

1 6. If two straight lines are parallel to a perpendicular,
the nearer of these will fall more nearly at right angles to
the curve than the one more remote.

17. If from any point in the perpendicular HM, two
straight lines are drawn to the curve, the nearer will fall
more nearly at right angles to the curve than the one more
remote.

1 8. Hence it is evident that by moving away from
the perpendicular, the obliquity of the incident lines
with the curve increases, until that which touches the
curve is reached ; this, the tangent, is the most oblique
of all.

19. If the point H is taken within the curve, and if, of

LECTURE V 65

all lines drawn from it to meet the curve, HM is the least;
then HM will be perpendicular to the curve or the tangent
MT.

20. Also, if HM is the greatest of all straight •lines drawn
to meet the curve, then HM will be perpendicular to the
curve.

21. Hence, if MT is perpendicular to HM, whether the
latter is a maximum or a minimum, it will touch the curve.

22. It follows that, if a straight line is not perpendicular
to the curve, no greatest or least can be taken in it.

23. If HM is the least of the lines drawn to the curve, and
any point I is taken in it; then IM will be a minimum.

24. If HM is the greatest of the lines drawn to meet the
curve, and any point I is taken on MH produced; then IM
will be a maximum.

For the rest, the more detailed determination of the.
greatest and least lines to a curve depends on the special
nature of the curve in question.

[Barrow concludes these preliminary five lectures with
the remark: — ]

"But I must say that it seems to me to be wrong, and
not in complete accord with the rules of logic, to ascribe
things which are applicable to a whole class, and which
come from a common origin, to certain particular cases, or
to derive them from a more limited source."

NOTE

The next lecture is the first of the seven, as originally

66 BARROW'S GEOMETRICAL LECTURES

designed, that were to form a supplement to the Optics.
Barrow begins thus : —

" I have previously proved a number of general properties
of curves of' continuous curvature, deducing them from a
certain mode of construction common to all; and especially
those properties, as I mentioned, that had been proved by
the Ancient Geometers for the special curves which they
investigated. Now it seems that I shall not be displeasing,
if I shall add to them several others (more abstruse indeed,
but not altogether uninteresting or useless) ; these will be,
as usual, demonstrated as concisely as possible, yet by the
same reasoning as before ; this method seems to be in the
highest degree scientific, for it not only brings out the truth
of the conclusions, but opens the springs from which they
arise.

The matters we are going to consider are chiefly con-
cerned with

(i) An investigation of tangents, freed from the loathsome
burden of calculation, adapted alike for investigation and
proof (by deducing the more complex and less easily seen
from the more simple and well known) ;

(ii) The ready determination of the dimensions of many
magnitudes by the help of tangents which have been
drawn.

These matters seem not only to be somewhat difficult
compared with other parts of Geometry, but also they have
not been as yet wholly taken up and exhaustively treated
(as the other parts have) ; at the least they have not as yet
been considered according to this method that 2 know. So we

LECTURE VI 67

will straightway tackle the subject, proving as a preliminary
certain lemmas, which we shall see will be of considerable
use in demonstrating more clearly and briefly what follows."

The original opening paragraph to the "seven" lectures
probably started with the words, " The matters we are going
to consider, etc.," the first paragraph being afterwards
added to connect up the first five lectures.

Barrow indicates that his subject is going to be the con-
sideration of tangents in distinction to the other parts of
geometry, which had been already fairly thoroughly treated ;
he probably alludes to the work on areas and volumes by
the method of exhaustions and the method of indivisibles,
of which some account has been given in the Introduction ;
when he treats of areas and volumes himself, he intends
to use the work which, by that time, he has done on
the properties of tangents. From this we see the reason
why the necessity arose for his two theorems on the inverse
nature of differentiation and integration.

That Barrow himself knew the importance of what he
was about to do is perfectly evident from the next para-
graph. He distinctly says that Tangents had been investi-
gated neither thoroughly nor in general; also he claims
distinctly that, to the best of his knowledge and belief, his
method is quite original. He further suggests that it will be
found a distinct improvement on anything that had been
done before. In other words, he himself claims that he is
inventing a new thing, and prepares to write a short text-
book on the Infinitesimal Calculus. And he succeeds, no
matter whether the style is not one that commended itself
to his contemporaries, or whether the work of Descartes
had revolutionized mathematical thought; he succeeds in
his task. In exactly the same way as the man who put the
eye of a needle in its point invented the sewing-machine.

Barrow sets out with being able to draw a tangent to a
circle and to a hyperbola whose asymptotes are either given
or can be easily found, and the fact that a straight line is
everywhere its own tangent. Whenever a construction is
not immediately forthcoming from the method of description

68 BARROW'S GEOMETRICAL LECTURES

of the curve in hand, he usually has some means of drawing
a hyperbola to touch the curve at any given point ; he finds
the asymptotes of the hyperbola, and thus draws the tangent
to it ; this is also a tangent to the curve required. Analyti-
cally, for any curve whose equation is y = f(x\ he uses as a
first approximation the hyperbola y = (ax + b}l(cx + d).

He then gives a construction for the tangent to the
general paraboliform, and makes use of these curves as
auxiliary curves. As will be found later, he proves that
i + nx is an approximation to ( I + x)n, leading to the
theorem that if y = xn, then dyjdx = n .yjx. Thus he founds
the whole of his work on exactly the same principles as
those on which the calculus always is founded, namely, on
the approximation to the binomial theorem ; and he does
it in a way that does not call for any discussion of the con-
vergence of the binomial or any other series.

For the benefit of those who are beginners in mathe-
matical history, it may not be out of place if I here reiterate
the warning of the Preface (for Prefaces are so often left
unread) that Barrow knew nothing of the Calculus notation oj
Leibniz. Barrow's work is geometrical, as far as his published
lectures go ; the nearest approach to the calculus of to-day
is given in the "a and e " method at the end of Lecture X.

Again, with regard to the differentiation of the com-
plicated function, given as a specimen at the end of this
volume, I do not say that Barrow ever tackled such a thing.
What I do urge, however, is that Barrow could have done
so, if he had come across such a function in his own work.
My argument, absolutely conclusive I think, is that I have
been able to do so, using nothing but Barrow's theorems
and methods.

LECTURE VI

Lemmas ; determination of curves constructed according to
given conditions ; mostly hyperbolas.

1. [The opening paragraph, as quoted in the note at the
end of the preceding lecture.]

2. Let ABC be a given angle and D a given point ; also
let the line ODO be such that, if any straight line DN is
drawn through D, the length MN, intercepted between the
arms of the angle, is equal to the length DO, intercepted
between the point D and the line ODO; then the line ODO
will be a hyperbola.*

Moreover, if MN is supposed to bear always the same
ratio to DO (say a given ratio R : S), the line ODO will be a
hyperbola in this case also.

3. Here I note, in passing, that it is easy to solve the
problem by which the solutions of the problems of Archi-
medes and of Vieta were reduced to conic sections by
the aid of a previously constructed conchoid.

Thus " to draw through a given point D a straight line, so

* There is a very short proof given to this theorem, as an alternative.
It is hard to see why the comparatively clumsy first proof is retained, unless
the alternative proof was added in revise (? by Newton). There is also
a reference to easy alternative proofs for §§ 4, 9. These alternative proofs
depend on an entirely different property of the curve.

70 BARROW'S GEOMETRICAL LECTURES

that the part of the straight line so drawn, intercepted
between the arms of a given angle ABC, may be equal to a
given straight line T."

For, if the hyperbola (of the preceding article) is first
described, and if with centre D, and a radius equal to the
given straight line T, a circle POQ is described, cutting the
hyperbola in 0, and DO is produced to cut the arms of
the angle in M and N ; then it follows that MN = DO = T.

4. Let ABC be a given angle and D a given point ; and
let the line OBO be such that, if through D any straight
line DN is drawn, the length MN intercepted between the
arms of the angle bears always the same ratio (say X : Y)
to the length MO intercepted between the arm BC and the
curve OBO ; then OBO will be a hyperbola.

5. If MO is taken on the other side of the straight line
BC, the method of proof is the same.

6. INFERENCE. — If a straight line BQ divides the angle
ABC, and through the point D are drawn, in any manner,
two straight lines MN, XY, cutting the straight line BQ in
the points 0, P, of which 0 is the nearer to B; then

MN:MO < XY:XP.

7. Moreover, if several straight lines BQ, BG . . .
divide the angle ABC, and if from the point D the straight
lines DN, DY are drawn, cutting BC, BQ, BG, BA in M, 0,
E, N and X, V, F, Y, DN being the nearer to B; then
NE:MO< YF:VX.

8. From what has gone before, it is also evident that
through B (in one of two directions) a straight line can be

LECTURE VI 71

so drawn that the segments intercepted on lines drawn
through D between the constructed line and BC shall
have to the segments intercepted between BA and BC a
ratio that is less than a given ratio.

9. Again, suppose a given angle ABC and a given point
D ; also let the line 000 be such that, if through D any
straight line DO is drawn, cutting the arms of the angle
in M, N, then DM always bears to NO a given ratio (X : Y say);
then the line 000 will be a hyperbola.

10. A straight line ID being given in position, and a
point D fixed in it, let DNN be a curve such that, if any
point G is taken in ID, and a straight line GN is drawn
parallel to a straight line IK given in position, and if two
straight lines whose lengths are m and b are taken, and if
we put DG = x, and G N = y, there is the constant relation
my + xy = mx^jb; then DNN will be a hyperbola.

n. If the equation is my — xy = mx^jb, the same hyper-
bola is obtained, only G must be taken in DM instead of
DO. If, however, the equation is xy-my = mx^jb, then
G must be beyond M and the hyperbola conjugate to the
former is obtained.

12. If BDF is a given triangle and the line DNN is such
that, if any straight line RN is drawn parallel to BD, cutting
the lines BF, DF, DNN in the points R, G, N, and DN is
joined; and if DN is then always a mean proportional
between RN and NG ; then the line DNN is a hyperbola.

13. If ID is a straight line given in position; and DNN is
a, curve such that, if any point G is taken in ID, and the

72 BARROW'S GEOMETRICAL LECTURES

straight line GN is drawn parallel to IK, a straight line given
in position, and if straight lines whose lengths are g, m, r,
are taken ; and if we put DG = x, and GN =_y, then there is
a constant relation xy+gx- my = mxz/r; then the line
DNN will be a hyperbola.

If the equation is -yx+gx + my = mxz/b, then the same
hyperbola is obtained, but the points G must then be taken
between B and M (B being the point where the curve cuts
the straight line ID); and if the points G are assigned to
other positions, the signs of the equation vary. But it is
not opportune to go into them at present.

14. Two straight lines DB, DA, are given in position, and
along the line DB a straight line CX is carried parallel to
BA ; also, by turning round the point D as a centre, a straight
line DY moves so that, if it cuts BA in X, there is always
the same ratio between the lines BE and CD (equal to the
ratio of some assigned length R to DB, say); then, if DE
cuts CX in N, the line DNN is a parabola.

Gregory St Vincent gave this, but demonstrated with
laborious prolixity, if I remember rightly.

We add the following: —

15. If, other things remaining the same, CX and DY are
moved in such a way that now BE and BC are always
in the same ratio (BD : R, say) ; their intersections will
give a parabola also.

1 6. If, with other things remaining the same, the straight
line CX is not now carried parallel to BA, but to some
other straight line DH, given in position ; and if the ratio

LECTURE VI 73

of BE to DC is always equal to the ratio of DB to R; then
the intersections N will lie on a hyperbola.

17. Moreover, other things remaining the same as in
the preceding, if CX now moves in such a way that BE
always bears the same ratio to BC (BD : R, say) ; the inter-
sections in this case will also lie on a hyperbola.

1 8. Let two straight lines DB, DA be given in position,
and a point D fixed in DB; and let the line DNN be such
that, if any straight line GN is drawn parallel to BA, and
two straight lines whose lengths are g, r are taken, and DG,
GN are called x,y; and if ry - xy = gx\ then the line DNN
will be a hyperbola.

If, however, the equation is xy - ry = gx, we must take
DE = r, and BO = g (measured below the line DB); the
proof is the same as before.

19. Let two straight lines DB, BA be given in position;
and let the straight line FX move parallel to DB, and let DY
pass through the fixed point D, so that the ratio of BE to
BF is always equal to an assigned ratio, say DB to R; then
the intersections of the straight lines DY, GN lie on a
straight line.

20. But if, other things remaining the same, some other
point 0 is taken in AB, which we take as the origin of reckon-
ing, so that the ratio BE to OF is always equal to the ratio
DB to R; then the intersections will lie on a hyperbola.

21. Moreover, other things remaining the same, let the
straight line FX now move not parallel to DB, but to another
straight line DH, so that, a fixed point 0 being taken in

74 BARROW'S GEOMETRICAL LECTURES

BA, the ratio BE to OF is always equal to an assigned ratio
(say DB to ni) ; then the intersections will again lie on a
hyperbola.

22. Let ADB be a triangle and DYY a line such that, if
any straight line PM is drawn parallel to DB, meeting AB
in M, PY is always equal to ^/(PM2 - DB2) ; then the line
DYY is a hyperbola.

COR. If Y8 is the tangent to the hyperbola DYY, then
PM2 : PY2 = PA : PS.

23. If, other things remaining the same, we have now
PY = ^/(PM'2 + DB2) ; then the line DYY is again a hyperbola.

COR. If Y8 is the tangent to the hyperbola, then
PM2 : PY2 = PA : PS.

24. If ADB is a triangle, having the angle ADB a right
angle, and the curve CGD is such that, if any straight line
PEG is drawn parallel to DB, cutting the sides of the triangle
in F, E, and the curve in G, the rectangle contained by EF
and EG is equal to the square on DB; then the curve CGD
is an ellipse, of which the semi-axes are AD, AC.

COR. Let GT be a tangent to the ellipse, then
EF2:EG2 = AE:AT.

25. If DTH is any rectilineal angle, and A is a fixed point
in TD, one of its arms ; if also the curve VGG is such that,
when any straight line EFG is drawn perpendicular to TD,
cutting the lines TD, TH, VGG in the points E, F, G, and
AF is joined, EG is always equal to AF; then the line VGG
will be a hyperbola.

LECTURE VI 75

NOTE. If a straight line FQ is drawn perpendicular
to TH, and QR is taken equal to AE (along TD), and GR
is joined; then GR will be perpendicular to the hyperbola
VGG.

Take this on trust from me, if you will, or work it out
for yourself;* I will waste no words over it.

26. Let two straight lines, AC, BD, intersecting in X, be
given in position ; then if, when any straight line PKL is
drawn parallel to BA, cutting AC, BD, in the points P and
K, PL is always equal to BK ; then the line ALL will be a
straight line.

27. Let a straight line AX be given in position and a fixed
point D; also let the line DNN be such that, if through D
any straight line MN is drawn, cutting AX in M, and the line
DNN in N, the rectangle contained by DM and DN is equal
to a given square, say the square on Z; then the line DNN
will be circular.

Thus you will see that not only a straight line and a
hyperbola, but also a straight line and a circle, each in its
own way, are reciprocal lines the one of the other.

But here, although we have not yet finished our preliminary
theorems, we will pause for a while.

NOTE

It has already been noted in the Introduction that the
proofs which Barrow gives for these theorems, even in the
case where he uses an algebraical equation, are more or less

* "Ad Calculum exige." — I hardly think that Barrow intends "by
analysis," but he may.

;6 BARROW'S GEOMETRICAL LECTURES

of a strictly geometrical character ; the terms of his equa-
tions are kept in the second degree, and translated into
rectangles to finish the proofs. In this connection, note the
remark on page 197 to the effect that I cannot imagine
Barrow ever using a geometrical relation, in which the ex-
pressions are of the fourth degree. The asymptotes of the
hyperbolae are in every case found; and this points to his
intention of using these curves as auxiliary curves for draw-
ing tangents ; cases of this use will be noted as we come
across them ; but the fact that the number of cases is small
suggests that the paraboliforms, which he uses more
frequently, were to some extent the outcome of his re-
searches rather than a first intention.

The great point to notice, however, in this the first of the
originally designed seven lectures, is that the idea of Time
as the independent variable, i.e. the kinematical nature of
his hypotheses, is neglected in favour of either a geometrical
or an algebraical relation, as the law of the locus. The dis-
tinction is also fairly sharply defined.

Barrow, throughout the theorems of this lecture, gives
figures for the particular cases that correspond to rectangular,
but his proofs apply to oblique axes as well (in that he does
not make any use of the right angle). Both the figures and
the proofs have been omitted in order to save space ; all
the more so, as this lecture has hardly any direct bearing
on the infinitesimal calculus.

The proofs tend to show that Barrow had not advanced
very far in Cartesian analysis ; at least he had not reached
the point of diagnosing a hyperbola by the fact that the
terms of the second degree in its equation have real factors ;
or perhaps he does not think his readers will be acquainted
with this method of obtaining the asympotes.

LECTURE VII

Similar or analogous curves. Exponents or Indices.
Arithmetical and Geometrical Progressions. Theorem ana-
logous to the approximation to the Binomial Theorem for a
Fractional Index. Asymptotes.

Barrow opens this lecture with the words, " Hitherto we
have loitered on the threshold, nor have we done aught
but light skirmishing." The theorems which follow, as he
states at the end of the preceding lecture, are still of the
nature of preliminary lemmas ; but one of them especially,
as we shall see later, is of extraordinary interest.

For the rest, it is necessary to give some explanation of
Barrow's unusual interpretation of certain words and phrases,
i.e. an interpretation that is different from that common at the
present time. A series of quantities in continued proportion i
form a Geometrical Progression ; thus, if we have A, B, C, D in i
continued proportion, then these are in Geometrical Pro-
gression, and A:B = B:C = C:D. Barrow speaks of these
as being "four proportionals geometrically," and this
accords with the usual idea. But he also speaks of " four
proportionals arithmetically " to signify the four quantities
A, B, C, D, which are in Arithmetical Progression ; that is,
A-B = B - Cj = C — D; and further his proofs, in most cases,
only demand that A - B = C - D. If A, B, C, D, E, F, . . . N
and a, b, f, d, e, f, . . . n are two sets of proportionals,
he speaks of corresponding terms of the two sets as being
"of the same order" ; thus, B, b ; C, c; . . . are "mean
proportionals of the first, second, . . . order between A and
N, a and n respectively ; and this applies whether the x
quantities are in Arithmetical or Geometrical Progression.

78 BARROW'S GEOMETRICAL LECTURES

An index or exponent is also defined thus : — If the number
of terms from the first term, A say, to any other term, F
say, is N (excluding the first term in the count), then N
is the index or exponent of the term F. Later, another
meaning is attached to the word exponent-, thus, if A, B,
C are the general ordinates (or the radii vectores) of
three curves, so related that B is always a mean of the
same order, say the third out of six means, between A and
C ; so that the indices or exponents of B and C are 3 and
7 respectively ; then 3/7 is called the exponent of the
curve BBB. There is no difficulty in recognizing which
meaning is intended, as Barrow uses n/m for the latter
case, instead of N/M. The connection with the ordinary
idea of indices will appear in the note to § 16 of this lecture
and that to Lect. IX, § 4.

^ i. Let A, B be two quantities, of which A is the greater ; let
some third quantity X be taken ; then A + X:B + X<A:B.
For, since X:A<X:B, X + A:A<X + B:B; hence, etc.

2. Let three points, L, M, N be taken in a straight line

Y L £ M F N G z

Fig. 61.

YZ ; and between the points L, M let any point E be taken,
and another point G outside LN (towards Z) ; let EG be cut
in F so that GE : EF = NL : LM ; then F will fall between
M and Z.

For NE:ME > NL:ML( = GE:EF) > NE:FE,
.-. FE > ME.

3. Let BA, DC be parallel straight lines, and also BD, GP ;
through the point B draw two straight lines BT, BS,

LECTURE VII

cutting GP in L and K; then I assert
that DS : DT = KG : LG

For the ratio KG: LG
is compounded of KG : GB and GB : LG,
that is, of PK : PS and PT : PL,
that is, of DB : DS and DT : DB,
and hence is equal to the ratio DT : DS. ^

4. Let BDT be a triangle, and let any two straight lines
BS, BR, drawn through B, meet any straight line GP drawn
parallel to the base BD in the points L, K ; then I say that
D + KL . RD : KG .TD = RD : SD.

Fig. 62.

[Barrow's proof by drawing parallels through L is rather
long and complicated ; the following short proof, using a
different subsidiary construction, is therefore substituted.

Construction. — Draw GXYZ, as in the figures, parallel
to TD.

Proof. — Since LQ : BZ = GX : XZ = TS : SD, *

and KG : BZ = GY : YZ = TR : RD; '
hence RD : SD = LG . TR : KG . TS /

= LG . TR + RD . KG : KG . TS + KG . SD
= LG.TD+KL.RD : KG . TD.]

5. But, if the points JR, S are not situated on the same
side of the point D,* then by a similar argument,
LG.TD-KL.RD:KG.TD= RD : SD.

* This is a mistake : D should be P, and LG . TD - KL . RD should be
ambiguous in sign.

8o BARROW'S GEOMETRICAL LECTURES

6. Let there be four equinumerable series of quantities
in continued proportion (such as you see written below),
of which both the first antecedents and the last conse-
quents are proportional to one another (i.e. A : a = M : /A,
and F : <f> = 8:0-); then, the four in any the same column
being taken, they will also be proportional to one another
(say, for instance, D : 8 = P : TT).

A, B, C, D, E, F

a, /?, y, 8, e, <f>

M, N, 0, P, R, 8

fa v, o, JT, p, <r

For Aw, Bv, Co, DTT, Ep, Fo- i

on r, > are in continued proportion,
and aM, )3N, yO, SP, eR, <£S J

Therefore, since A/A = Ma and Fo- = «£S, it is plain that
DTT = SP, and hence that D : S = P : IT.

The conclusion applies equally to either Arithmetical or
Geometrical proportionality.*

7. Let AB, CD be parallel lines;
and let a straight line BD, given
in position, cut these. Now let
the curves EBE, FBF be so related
that, if any straight line PG is
drawn parallel to BD, PF is always
a mean proportional of the same

Fig. 65.

If the series are " arithmetical proportionals," then

and the condition for the first antecedents and the last consequents must be
A-i-^-a+M and F + <r = 4> + S; in this case D-fn-= 6 + P, or D-S = P-w.
This is not of very great importance, as, in the following theorems,
Barrow apparently only considers geometrical proportionals. For arith-
metical proportionality, the lines AGB, HEL must be parallel curves, i.e.
EG must be constant, and then the curves .KEK and FBF are parallel
curves, i.e. RK is constant.

LECTURE VII 8 1

41 ^

order between PG and PE; then, through any point E of
the given line EBE, let HE be drawn parallel to AB and CD ;
and let another curve KEK be such that, if any straight line
QL is drawn also parallel to BD, cutting EBE in I, HE in L,
FBF in R, and CD in 8, then QK is always a mean propor-/
tional of the same order between QL and Ql (of that order,
I say, of which PF was a mean proportional between PG and
PE) ; then I assert that the lines FBF, KEK are similar ; that
is, the ordinates such as QR, QK bear a constant ratio to
one another, the ratio which PF bears to PE.

This follows from the lemma just proved, as will be clear
by considering the argument below.

Since QS, QR, Ql "I are proportionals
QL, iQK, Ql I such that

PG, PF, PEJ QS:QL= PG : PE,

PE, PE, PEJ Ql :QI = PE:PE;

hence, QR : QK = PF : PE

NOTE. Instead of the straight lines AB, HE, CD, we can
substitute any parallels we please, even curved lines.

8. Again, let AQPB, ASGD be two straight lines meeting in
A, and let BD be a straight line given in position; also let
EBE, FBF be two curves so related that, if any straight line
PG is drawn parallel to BD, PF is always a mean propor-
tional of the same order between PG and PE; then, having
joined AE, let another curve KEK be such that, if any
straight line QLI is drawn parallel to BD, cutting AE in L,
EBI in I, and FBF in R, QK is always a mean proportional

82 BARROW'S GEOMETRICAL LECTURES

between QL and Ql of the same order as PF was between
PG and PE; then the line FBF is similar to the line KEK;
that is, QR : QK = PF : PE, in every position.

NOTE. For the three straight lines AB, AH, AD we can
substitute any three analogous lines.

9. Also, if AGB is a circle whose centre is D ; and EBE,
FBF are two other curves such that, if any straight line DG
is drawn through D, DF is always a mean proportional of
the same order between DG and DE; then, through E, let a
circle H E, with centre D, be drawn ; and let another curve
KEK be drawn such that, if any straight line DL is drawn
through D to meet the circle HE in L, and EBE in I, DK is
always a mean proportional between DL and Dl, of the same
order as DF was between DG and DE ; then the curves
FBF, KEK will be similar, i.e. DR : DK = DF : DE, in every
position. [DL meets FBF in R.]

NOTE. In this case also, instead of the circles, we may
substitute any two parallel or two analogous lines.

10. Lastly, let AGBG, EBE be any two lines; and let FBF
be another line so related to them that, if any straight line
DG be drawn in any manner through a fixed point D, DF is
always a mean proportional of the same order between DG
and DE ; then let H EL be a line analogous to AGB (i.e. such
that, if through D a straight line DSL is drawn in any
manner, DS and DL are always in the same proportion) ;
lastly, let the line KEK be such that, if DL be drawn in any
manner, cutting EBE in I, DK is always a mean proportional
between DL and Dl, of the same order as DF was between

LECTURE VII 83

DG and DE ; then, in this case also, FBF is analogous to the
line KEK.

n. Let A, B, C, D, E, F be a series of quantities in Arith-
metical Progression ; and, two terms D, F being taken in it,
let the number of terms from A to D (excluding A) be N,
and the number of terms from A to F (excluding A) be M ;
thenA~D:A~F = N : M.

For, suppose the common difference to be X ; then

D = A±N.X and F = A ± M . X ;
therefore A~D : A~F = N . X : M . X = N : M.

12. Hence, if there are two series of this kind, and in
each a pair of terms, corresponding to one another in order,
are taken (say D, F in the first, and P, R in the second) ;
then A~D:A~F = M~P:M~R,

where the series are

A, B, C, D, E, F and M, N, 0, P, Q, R.
For each of these ratios is equal to that which the
numbers, N, M, as found in the preceding article, bear to
one another.

These numbers N, M, in any series of proportionals, we
shall usually call the exponents or indices of the terms to
which they apply ; and where we use these letters in what
follows, we shall always understand them to have this
meaning.

13. Let any quantities A, B, C, D, E, F be a series in
Arithmetical Progression; and let there be another set,
equal in number, in Geometrical Progression, starting

84 BARROW'S GEOMETRICAL LECTURES

with the same term A ; thus, [suppose the two series

are]

A, B, C, D, E, F.

A, M, N, 0, P, Q.

Also let the second term B of the Arithmetical Progression
be not greater than M the second term of the Geometrical
Progression ; then any term in the Geometrical Progression
is greater than the term in the Arithmetical Progression
that corresponds to it.*

For, A + N > 2M > 2Bor A + C, .'. N > C;
hence, M + N > B + CorA + D; butA + 0 > M + N;

.-. A + 0 > A + D, i.e. 0 > D;
hence, M+0> B + DorA + E; butA + P> M+0;

.-. A + P > A + E, i.e. P > E;
and so on, as far as we please.

14. Hence, if A, B, C, D, E, F are in Arithmetical Pro-
gression, and A, M, N, 0, P, Q are in Geometrical Progression,
and the last term F is not less than the last term Q (the
number of terms in the two series being equal) ; then B is
greater than M.

For, if we say that B is not greater than M, then F must
be less than Q ; which is contrary to the hypothesis.

15. Also, with the same data, the penultimate E is greater
than the penultimate P.

* Algebraically: — If the series are a, a + d, a + zd, a + %d, . . . and
a, ar, at*, ar3 ..... , we have, whether r ^ i, the fact that (i-r)(i — r),
(i-r)(i — r*), (i — r^i-r3), . . . are all positive ; hence it follows that

a4-at*>zar, a + ar3 > ar+ar4, a + ar4 > ar+ar3, . . .
Hence, since ar is not less than a + d, it follows that

and so on, in exact equivalence with Barrow's proof.

LECTURE VII 85

1 6. Moreover, with the same da/a, any term in the Arith-
metical series is greater than any term in the Geometrical
series ; for instance, C > N.

For E > P, and hence D > 0, and so on, .'. C > N.*

1 7. Hence it may be proved that : — If there are any four
lines HBH, GBG, FBF, EBE, cutting one another at B, and
these are so related that, if any straight line DH is drawn in
any manner parallel to a straight line BD, given in position,
or if through a given point D any straight line DH is drawn,
DG is always an arithmetical mean of the same order be-
tween DH and DE, and DF is the geometrical mean of the
same order ; then the lines GBG, FBF will touch one another.
For it is evident from the preceding that the line GBG will
lie wholly outside the line FBF.

1 8. Hence also (I mention it briefly in passing), the
asymptotes, straight lines applying to many different
kinds of hyperbolae, and curves of hyperbolic form, may
be denned. t

Thus, let two straight lines VD, BD be given in position,
and let AGB, VEI be two other straight lines; now, any
straight line PG being drawn parallel to DB, let P<£ be
always an arithmetical mean of the same order between
PG and PE, and let PF be the geometrical mean of the
same order. Now, since the straight lines EG, E<£ are
always in the same ratio, the line <£<£<£ is a straight line ;
but the line VFF is a hyperbola or some curve of hyperbolic

* See note at the end of this lecture ; the italics are mine,
f See note at the end of the lecture.

86 BARROW'S GEOMETRICAL LECTURES

form (the hyperbola of Apollonius indeed, if PF is the
simple geometrical mean between PG and PE, but some
curve of hyperbolic form of a different kind, if PF is a mean
of some other kind) ; and it is plain from the last theorem
that the line <t><t><f> is an asymptote to the line VFF, corre
spending to the points of the same kind of mean.

I do not know whether this is of very much use, but
indeed it was an incidental corollary for us to have obtained
it here.

19. Let three straight lines BA, BC, BQ be drawn through
a given fixed point B to a straight line AC, fixed in position ;
then, in QC produced let some point D be taken as a fixed
point. Then it is possible to draw through B a straight
line (BR say), on either side of BQ, such that, if any straight
line is drawn through D, as DN, the part intercepted between
BQ and BR is less than the part intercepted between BA
and BC.

20. Let D, E, F be three points in a straight line DZ, and
let F be the vertex of a rectilineal angle BFC, of which the
arms are cut by a straight line DBC ; let a straight line EG
be drawn through E; then it is possible to draw through
E a straight line EH such that, on any straight line DK
drawn through the point D, the intercept between the
lines EG, EH is less than the intercept between the lines
FC, FB.

21. Let a straight line BO touch a curve BA in B; and
let the length BO of the straight line be equal to the arc
BA of the curve ; then, if any point K is taken in the arc

LECTURE VII 87

BA and KO is joined, the straight line KO is greater than
the arc KA.

22. Hence, if any two points K, L, on the same side of
the point of contact, are taken, one in the curve and the
other in the tangent, and KL is joined; then KL+LO >
arc KA.

NOTE

From § 1 6 we have the following geometrical theorem.

Suppose a line AB is divided into two parts at C, and
that the part CB is divided at D, E, F, G, H in the figure on
the left, and at D', E', F', G', H' in the figure on the right,
so that AC, AD, AE, AF, AG, AH, AB are in Arithmetical Pro-
gression, and AC, AD', AE', AF', AG', AH', AB are in Geo-
metrical Progression; then AD > AD', . . ., AH > AH'.

Expressing this theorem algebraically, we see that, if
AC = a and CB = ax, and the number of points of section
between C and B is n— i, and F is the rth arithmetical,
and F' the rlh geometrical "mean point" between C and
B, then the relation AF > AF' becomes

a 4- r . ax/n > a.[?J{(a + ax)/a}]r;
i.e. i + (r/n)x > (i + x)r/n, where r < n.

Also, as CB becomes smaller and smaller, the difference FF'
becomes smaller and smaller, since it is clearly less than
CB ; that is, the ratio FF'/AC can be made less than any
assigned number by taking the ratio CB/AC small enough.
Hence the algebraical inequality tends to an equality, when
x is taken smaller and smaller.

Again, if we put rx/n = y, we have x = nylr, and then

i +y > ( i + ny/r)r/n or I + (n/r)y < (i +y)n/r,
where n > r; and again the inequality tends to become an
equality if y is taken small enough.

88 BARROW'S GEOMETRICAL LECTURES

Naturally, a man who uses the notation xx for xz does
not state such a theorem about fractional indices. But the
approximation to the binomial expansion is there just the
same, though concealed under a geometrical form. We
may as well say that the ancient geometers did not know
the expansion for sin (A + B), when they used it in the form
of Ptolemy's theorem, as say that Barrow was unaware of
this. Moreover, if further corroborative evidence is needed,
we have it in § 18. Here Barrow states that a line <f><t><j> is
an asymptote to a curve VFF, the distance between the curve
and its asymptote, measured along a line parallel to a fixed
direction, being the equivalent of our FF' in the work above.
Now the condition for an asymptote is that this distance
should continually decrease and finally become evanescent
as we proceed to " infinity." Let us try to reason out the
manner in which Barrow came to the conclusion that his
line was an asymptote to his curve.

The figure on the left is the one used by Barrow for § 18;
as P moves away from V, PE and PG both increase without
limit, but it can readily be seen that the ratio of EG to PE
steadily decreases. This is all that can be gathered from
the figure ; and, as far as I can see, it must have been from
this that Barrow argued that the distance F<£ decreased
without limit and ultimately became evanescent. In other
words, he appreciated the fact that the inequality tended to
become an equality when x was taken small enough. Assum-
ing that my suggestion is correct, the very fact that he has
recognized this important truth leads him into a trap ; for
the line <£<£<£ is not an asymptote to the curve VFF, i.e. as we
understand an asymptote at the present day. Taking the

LECTURE VII 89

simplest case, as mentioned by Barrow, of the ordinary
hyperbola, it is readily seen that the other branch of the
curve passes through the common point of the straight lines
AGB, VEI, and therefore the line </><£<£ cannot be an asymptote,
for it also passes through this common point and touches
the curve there.

This is easily seen analytically, taking the figure on the
right. For, if the equations of VEI and AGB, referred to
YD and a line parallel to DB through the middle point of
AV as axes, are_y = n(x + a) and y = m(x - a), then the equa-
tion to the hyperbola is y2- = mn(xi - a2) ; that of theasymptote,
with which Barrow confuses the line <£<£<£, isy = *J(mri).x;
and that of the line </><£<£ is zy = (m + n}x + (m- n)a ; and
the two lines are not the same unless m = n, i.e. unless VEI
and AGB are parallel. The argument is the same, if DB is
not taken at right angles to YD, or for different kinds of
" means."

The true source of the error is, of course, that it is not
true that F</> decreases without limit, but that it is F<£ : PE
which decreases without limit, whilst PE increases without
limit. This kind of difficulty is exactly on a par with the
difficulties arising from considerations of convergence of
infinite series. Barrow certainly has in his theorem the
equivalent of the binomial approximation as far as it is
necessary for differentiation of fractional powers in the
ordinary method ; it is very likely that he may have found
difficulties with other theorems of the kind discussed above ;
but, as will be seen in the note to Lect. IX, § 4, he is quite
independent of considerations of this sort, i.e. of infinite
series with all their difficulties ; for all that he requires is
the bare inequality, as given in his theorem. By means of
this, at the very least, he was \hefirst man to give a rigorous
demonstration of a method for differentiating a fractional
power of the variable.

As an example of the use that a geometer could make of
his geometrical facts, it may be pointed out that the theorem
of § 1 7 is equivalent to the analytical theorem : —

The curves whose equations are

y = [{« - r] ./(*) + r . F(x}]ln and y = n^[{f(x)}^ . {/(*)}•]
touch one another at all the points common to the two
curves whose equations are y = f(x) and y = f(x).

LECTURE VIII

Construction of tangents by means of auxiliary curves of
which the tangents are known. Differentiation of a sum or a
difference. Analytical equivalents.

Truly I seem to myself (and perhaps also to you) to have
done what that wise man, the Scoffer,* ridiculed, namely,
to have built very large gates to a very small city. For up
to the present, we have done nothing else but struggle
towards the real thing, just a little nearer. Now let us get
to it.

1. We assume the following : —

If two lines OMO, TMT touch one another, the angles
between them (OMT) are less than any rectilineal angle;
and conversely, if two lines contain angles which are less
than any rectilineal angle, they touch one another (or at
least, they will be equivalent to lines that touch).

The reason for this statement has already been discussed,
unless I am mistaken.

2. Hence, if any third line PMP touch two lines OMO,
TMT, the lines OMO, TMT will also touch one another.

* Socrates, the Athenian philosopher : Zeno called him " Scurra Atticus."
the Attic Scoffer.

LECTURE VIII 91

3. Let a straight line FA touch a curve FX in F ; and let
FE be a straight line given in position ; also let EY, EZ be
two curves such that, if any straight line I L is drawn parallel
to EF, cutting FA in G and the curves FX, EY, EZ in I, K, L
respectively, the intercept KL is always equal to the intercept
IG ; then the curves EY, EZ touch one another.

4. Again, let a straight line AF touch a curve AX, and let
EY, EZ be two curves such that, if through a fixed point D
any straight line DL is drawn, cutting the given lines as in
the preceding theorem, KL is always equal to IG ; then the
curves EY, EZ will touch one another.

The two foregoing conclusions are also true, and can
be shown to be true by a like reasoning, if it is given
that IG, KL always bear to orue another any the same
ratio.

5. Let TEI be a straight line, and let two curves YFN, ZGO
be so related that, if any straight line EFG is drawn parallel
to AB, a straight line given in position,

the 'intercepts EG, EF always bear to
one another the same ratio ; also let
the straight line TG touch ZGO, one of
the curves in G and meet IE in T ;
then TF, being joined, will touch the
curve YFN.

For, let a straight line IL, parallel to AB, be drawn, cutting
the given lines as shown in the figure. Then

IL : IN > 10 : IN > EG : EF > IL : IK, and .'. IN < IK ;

92 BARROW'S GEOMETRICAL LECTURES

hence the line TF falls altogether without the curve YFN.*

Otherwise. It can be shown that IL:IK = OL:NK; hence,
by § 4 above, since GL, GO touch, FN, FK also touch.

6. Moreover, if three curves XEM, YFN, ZGO are so related
that, if any straight line EFG is drawn parallel to a line given
in position, EG and EF are always in the same ratio ; also
let the tangents ET, GT to the curves XEM, ZGO meet in T;
then TF, being joined, will touch the curve YFN.t

7. Let D be a given point, and let XEM, YFN be two
curves so related that, if through D any straight line DEF is
drawn, the straight lines DE, DF always bear to one another
the same ratio; and let the straight line FS touch YFN, one
of the curves, and let ER be parallel to FS; then ER will
touch the curve XEM.J

8. Let XEM, YFN, ZGO be three curves such that, if any
straight line DEFG is drawn through a given point D, the

* The reasoning for these theorems given by Barrow is not conclusive ;
it depends too much on the accident of the figure drawn. Although he
states in a note after § 6 that he always chooses the simplest cases, it is
desirable that these simple cases should be capable of being generalized
without altering the argument. In addition, his proof of § 4 is long and
complicated, and necessitates as a preliminary lemma the theorem of
Lect. VII, § 20 ; this is also proved in a far from simple manner, although
there is a very simple proof of it. Still Barrow must have had some good
reason for proving these two theorems by the method of ' ' the vanishing
angle" of § i, for he states that " these theorems are set forth, so that none
of the following theorems maybe hampered with doubts." He seems to
doubt the rigour of the method used in § 5, of which I have given the full
proof for the sake of exemplification ; together with the alternative proof
by § 4. The proof of the latter follows easily thus : — Since FA lies wholly
on one side of the curve FX, EZ lies wholly on one side of EY.

f If_y =f(x), y = F(x), y = <f>(x) are three curves such that there is a con-
stant relation A./+B. F + C.<f> = o, where A + B + C = o, the tangents at
points having equal abscissae are concurrent.

J Homothetic curves have parallel tangents ; this theorem and the next
are the polar equivalents of those of §§ 5, 6.

LECTURE VIII 93

intercepts EG, EF always bear the same ratio to one another
(say as R is to 8) ; and let the straight lines ET, GT touch
two of the curves (say XEM, YFN) in E and G ; it is required
to draw the tangent at F to the curve YFN.

Imagine a curve TFV such that, if a straight line is drawn
in any manner through D, cutting the straight lines TE, TG
in the points I, L and the curve in K, the intercepts IL, IK
bear to one another the same given ratio, R to 8. Then
IK > IN, and therefore the curve TFK touches the curve YKN.
But, by Lect. VI, § 4, the curve TFK is a hyperbola ; * let
FS be the tangent to it. Then SF will touch the curve
YFN also.

Since mention is here made for the first time of a tangent
to a hyperbola, we will determine the tangent to this curve,
together with the tangents of all other curves constructed
by a similar method, or of reciprocal lines.

9. Let YD be a straight line, and XEM, YFN two curves
so related that, if any straight line EOF is drawn parallel to

'P s

Fig. 84.

a line given in position, the rectangle contained by DE, DF
is always equal to any the same area ; also the straight line

* Note the use of the auxiliary hyperbola.

94 BARROW'S GEOMETRICAL LECTURES

ET touches the curve XEM at E, and cuts YD in T ; then,- if
DS is taken equal to DT and F8 is joined, FS will touch the
curve YFN at F.*

Let any straight line IN be drawn parallel to EF, cutting
the given lines as shown ; then

TP : PM > TP : PI > TD : DE ; also SP : PK = DS : DF ;
TP . SP : PM . PK > TD . 3D : DE . DF > TD . SD : PM . PN.
But, since D is the middle point of TS, .-. TD . SD > TP . SP
hence all the more, TD . SD : PM . PK > TD . SD : PM . PN,

.-. PM.PK < PM.PN or PK < PN.
Therefore the whole line FS lies outside the curve YFN.

NOTE. If the line XEM is a straight line, and so coinci-
dent with TEI, the curve YFN is the ordinary hyperbola, of
which the centre is T and the asymptotes are TS and a line
through T that is parallel to EF.

10. Again, let D be a point, and XEM, YFN two curves so
related that, if any straight line EF is drawn through D, the
rectangle contained by DE, DF is always equal to a certain

* The analytical equivalent of this is : —

If y is a function of x, and z = A/j, then (1/2). dz/dx = -(i/_y). dyjdx.
Also the special case gives d(i/x)/dx = — i/x2. It is thus that Barrow starts
his real work on the differential calculus.

LECTURE VIII 95

square (say the square on Z) ; and let a straight line ER
touch one curve XEM ; then the tangent to the other is
found thus : —

Draw DP perpendicular to ER and, having made DP : Z =
Z : DB, bisect DB at C ; join CF and draw FS at right angles
to CF; then FS will touch the curve YFN.

n. Let XEM and YFN be two curves such that, if any
straight line FE is drawn parallel to a straight line given in
position, it is always equal to a given length ; also let a
straight line FS touch the curve YFN ; then RE, being drawn
parallel to FS, will touch the curve XEM.*

12. Let XEM be any curve, which a straight line ER
touches at E ; also let YFN be another curve so related to
the former that, if a straight line DEF is drawn in any manner
through a given point D, the intercept EF is always equal
to some given length Z ; then the tangent to this curve is
drawn thus : —

Take DH = Z (along DEF), and through H draw AH
perpendicular to DH, meeting ER in B ; through F draw FG
parallel to AB; take GL = GB; then LFS, being drawn,
will touch the curve YFN.t

NOTE. If XEM is supposed to be a straight line, and so
coincide with ER, then YFN is the ordinary true Conchoid,
or the Conchoid of Nicomedes ; hence the tangent to this
curve has been determined by a certain general reasoning.

* The analytical equivalent is: — If y is a function of x, and z ~y-±c,
where c is a constant, then dzfdx = dyjdx.

f For the proof of this theorem, Barrow again uses an. auxiliary curve,
namely the hyperbola determined in Lect. VI, § 9.

96 BARROW'S GEOMETRICAL LECTURES

13. Let VA be a straight line, and BEI any curve; and let
DFG be another line such that, if any straight line PFE is
drawn parallel to a line given in position, the square on
PE is equal to the square on PF with the square on a given
straight line Z ; also let the straight line TE touch the curve
DEI; let PE2:PF2 = PT:PS; then FS, being joined, will
touch the curve DFG.*

[This is proved by the use of Lect. VI, § 22, and
corollary.]

14. Let other things be supposed the same, but now let
the square on PE together with the square on Z be equal
to the square on PF; also let PE2 : PF2 = PT : PS ; then FS
will touch the curve GFG.f

[For this, Barrow uses Lect. VI, § 23, and its corollary.]

15. Let AFB, CGD be two curves having a common axis
AD, so related to one another that, if any straight line EEG
is drawn perpendicular to AD, cutting the lines drawn as
shown, the sum of the squares on EF and EG is equal to
the square on a given straight line Z ; also let the straight
line FR touch AFB, one of the curves; and let EF2 : EG2

= ER : ET ; then GT, being joined, will also touch the curve
CGD. t

[For this, Barrow uses Lect. VI, § 24, and its corollary.]

* The analytical equivalent is : — If y is any function of x, and «2= _y2 - a2,
where a is some constant, then z.dzjdx= y.dyjdx ; or in a different form,
if *= </(v2 - a2), then d*/dx—y . (dy/dx)/(y2 - a2). The particular case, when
y=x, is the equivalent of Lect. VI, § 22.

t A similar result for the case of V(v2 + a2) or V(x2 + az).

£The case of z=V(a2-y2) or V(a2-*2). Since T, R are to be taken
on opposite sides of FG.

LECTURE VIII 97

16. Let AFB be any curve, of which AD is the axis and
DB is applied to AD; also let VGC be another curve so
related that, if any straight line ZF is drawn through some
fixed point Z in the axis AD, and through F a straight line
EFG is drawn parallel to DBG, EG is equal to ZF; also let
FQ be at right angles to the curve AFB; along AD, in the
direction ZE, take QR = ZE; then RG, being joined, will be
perpendicular to the curve VGC.

[For this, Barrow makes use of the hyperbola of Lect. VI,
§ 25, as the auxiliary curve ; he did not give a proof of the
theorem of that article, but left it "to the reader."]

17. Let DP be a straight line, and DRS, DYX two curves
so related that, if any straight line REY is drawn parallel to
a straight line DB, given in position, cutting DP in E and
the curves DRS, DYX in R, Y, the ratio RY: DY is always
equal to the ratio DY:EY; also let the straight line RF
touch the curve DRS at R. It is required to draw the
tangent to the curve DYX at Y.

Suppose the line DYO is such that, if any straight line
GO is drawn parallel to DB, cutting the lines FR, FP, DYO
in the points G, P, 0, and DO is joined, GO : DO = DO : PO ;
then the curve DYO touches the curve DYX at Y.

But, in Lect. VI, §12, it has already been shown that
the curve DYO is a hyperbola; let YS touch the hyperbola;
then YS also touches the curve DYX.

NOTE. If the curve DRS is a circle, and the angle GDB
is a right angle, the curve DYX is the ordinary Cissoid ; and
thus the tangent to it (together with many other curves
similarly produced) is determined.

98 BARROW'S GEOMETRICAL LECTURES

18. Let DB, VK be two lines given in position, and let
the curve DYX be such that, if from the point D any straight
line DYH is drawn, cutting the straight line BK in H and the
curve DYX in Y, the chord DY is always equal to the straight
line BH ; it is required to draw the straight line touching
the curve DYX in Y.

With centre D and radius DB, describe the circle BBS ;
let YER, drawn parallel to KB, meet the circle in R ; join
DR. Then RY : YD = YD:DE; hence, the straight line
touching the curve DYX can be found by the preceding
proposition.*

19. Let DB, BK be two straight lines given in position;
also let BXX be a curve such that, if from a point D any
straight line is drawn, cutting BK in H and the curve BXX
in X, HX is always equal to BH ; it is required to draw the
tangent to the curve BXX at X.

Suppose that DYY is a curve such that DY is always equal
to BH (such as we considered in the previous proposition),
and let YT touch this curve in Y, and cut BK in R ; then let
the hyperbola NXN be described, with asymptotes RB, RT,
to pass through X ; then the hyperbola NXN touches the
curve BXX at X. Thus, if the tangent to the hyperbola,
XS is drawn ; XS will also touch the curve BXX.

However, we seem to have trifled with this succession of
theorems quite long enough for one time ; we will leave off
for a while.

* In § 17, it is not essential that the curve RS should pass through D ;
hence this statement is justifiable.

LECTURE VIII

NOTE

In the footnote to § 9, 1 state that it is in this theorem that
Barrow starts his real work on the infinitesimal calculus.
Certainly he has given theorems on tangents before this
point, which have had analytical equivalents ; but these
have been special cases. Here for the first time he gives
theorems that are equivalent to the differentiation of general
functions, not only of the variable simply, but of any other
function that is itself a function of the variable. Thus, the
theorem of Lect. VI, § 22 is indeed equivalent to the differ-
entiation of J(x~ - a2) with regard to x ; but it is in the
theorem of Lect. VIII, § 13 that he gives the equivalent
to the differentiation of *](jp - a2) with regard to x, where
y is any function of x whose gradient is known. Thus
Barrow substantiates the last words of the paragraph with
which he opens the lecture : "Now let us get to it."

He however omits a theorem, which would seem to fall
.naturally into place in this lecture, as a generalization of
the theorem of § 1 1 .

If XEM, YFN, ZGO are three curves and PD is any straight
line such that, if any straight line PEFG is drawn parallel
to a straight line given in position, the intercept PE is
always equal to FG ; also let El, FK touch two of the curves
XEM, YFN ; draw the straight line GL such that, if any
straight line HO is drawn parallel to DEFG, cutting the
given lines as shown, KL = HI; then LG will touch the
curve LGO.

For, if the two curves XEM, YFN are
both convex to the line VP,
since HM = NO, and HI = KL,
KO>NO>HM>HI>KL;
hence the curve lies altogether above the
lineGL.

If both curves are concave to VP, the
argument is similar, but now ZGO falls
altogether below the line GL

If one of the curves is concave and V
the other convex to VP, say XEM, jF«, draw the curve
YFN so that the intercept KN is always equal to the inter-
cept «K ; then the two curves YFN,jrF« touch and have a

ioo BARROWS GEOMETRICAL LECTURES

common tangent KF. Let now the third curve be sG<?;
then, since LO = IM + KN, and \j> = IM - KN, therefore 00
is always equal to 2KN; hence, by § 3 above, the curves
zGo, ZGO also touch, and LG is the common tangent.
Therefore the construction holds in this case also.

I believe the omission of the theorem was intentional ;
and I argue from it that Barrow himself was not completely
satisfied with the theorems of §§ 3, 4, thus corroborating
my footnote. This theorem is of course equivalent to the
differential of a sum. Barrow may have thought it evident,
or he may have considered it to be an immediate con-
sequence of his differential triangle ; but I prefer to think
that he considered it as a corollary of the theorem of § 5.
For this may be given analytically as : —

If niv — ty + (n- r)z, then n , dwjdx = r . dyjdx + (n — r).
dzjdx. If we take one curve a straight line, and this straight
line as the axis, we have d(ky)jdx = k . dyjdx, or the sub-
tangents of all "multiple" curves have the same subtangent
as the original curve. Hence the constructions for the
tangents to "sum" and "difference" curves follow
immediately : —

Let A A A, BBB be any two curves, of which EF is taken
as a common axis ; let NAB be any straight line applied
perpendicular to EF; let the tangents AS, BR, cut EF in
8, R; take Aa, B£ equal to NA, NB respectively, and also let
NC= NA+NB, andm = NA-NB.

Join Sa, R<£ intersecting in T, and draw TV perpendicular
tott.

Then TC will touch the "sum" curve CCC, and YD will
touch the "difference" curve ODD.

It seems rather strange, considering Barrow's usual custom,
that he fails to point out that, in § 12, if the curve XEM is a
circle passing through D, the curve YFN is the Cardioid or
one of the other Limagons.

The final words of the lecture seem to indicate that
Barrow now intends to proceed to what he considers to
be the really important part of his work ; and, in truth,
this is what the next lecture will be found to be.

LECTURE IX

Tangents to curves formed by arithmetical and geometrical
means. Paraboliforms. Curves of hyperbolic and elliptic
form. Differentiation of a fractional power, products and
quotients.

We will now proceed along the path upon which we
started.

i. Let the straight lines AB, YD be parallel to one another ;
and let DB cut them in a given position; also let the lines
EBE, FBF pass through B, being so related that, if any
straight line PG is drawn parallel to DB, PF is always an
arithmetical mean of the same order between PG and PE ;

Fig. 94.

and let the straight line BS touch the curve. Required to
draw the tangent at B to the curve FBF.

p-£

102 BARROW'S GEOMETRICAL LECTURES

Let the numbers N, M (as explained in Lect. VII, § 12)
be the exponents of the proportionals PF, PE ; take DT,
such that N : M = DS : DT, and join TB ; then TB touches
the line FBF.

n x

For, in whatever position the line PG is drawn, cutting
the given lines as shown in the figure, we have
FG:EG = N:M = D8:DT= LG : KG.

Hence, since by hypothesis KG < EG, .'. LG < KG ; and
thus it has been shown that the straight line TB falls wholly
without the curve FBF.*

2. All other things remaining the same, let now PF be
a geometrical mean between PG and PE (namely, the mean
of the same order as in the former case of the arithmetical
mean) ; then the same straight line touches the curve FBF.

For the lines constructed in this way from arithmetical
and geometrical means touch one another ; hence, since BT
touches the one curve, it will also touch the other, t

Example. — Suppose PF is the third of six means between
PG and PE, then M = 7, and N - 3 ; and DS : DT = 3:7.

* Note that in this case, FG : EQ = LG : KG ; and thus this is a par-
ticular case of the curves in Lect. VIII, § 5; the analytical equivalent is
ii(a + 6y) = b.dy\dx.

f Analytical equivalent : — Ifjyis any function of x, and sfn = an~r .y, then
dzfdx = (r/n) . dy\dx, when 2 = y = a.

LECTURE IX

103

3. Again, the preceding hypothesis being made in all
other respects, let any point F be taken in the curve FBF;

Fig. 95-

then a straight line touching the curve may be drawn by a
similar method. Thus, let the straight line PG be drawn
through F parallel to DB, cutting the curve EBE in E,
and let EX touch the curve EBE at E ; take PY, such that
N : M = PX : PY, and join FY.
Then the straight line FY touches the curve FBF.*
For, if through E the straight line CEI is drawn parallel
to AB or YD, and it is supposed that a curve HEH passing
through E is such that, if any straight line Ql is drawn
parallel to DB, cutting the curves EBE, HEH in L, H, and
the straight lines CE, VP in I, Q, QH is a mean between
Ql and QL of the same order as PF was between PG and
PEj then it follows from the preceding proposition that,
if YE is joined, it will touch the curve HEH.

But the curves HEH, FBF are analogous curves (Lect. VII,
§ 7) ; therefore YF touches the curve FBF (Lect. VIII, § 5).

* This is a generalization of the last theorem ; the equivalent is that,
in general, if s* = a"-* .yr, then (1/2) . dz\dx = (n/r) . (i/_y) . dyldx. The
ana'logy of the curves occurs in the case of the arithmetical means, for then
IH:HL = GF:FE.

104 BARROWS GEOMETRICAL LECTURES

4. Note that, if the line EBE is supposed to be straight,
then the line FBF is one of the parabolas or curves of the
form of a parabola (" paraboliforms "). Therefore, that which
is usually held to be "known" concerning these curves
(deduced by calculation and verified by a sort of induction,
but not, as far as I am aware, proved geometrically) flows
from an immensely more fruitful source, one which covers
innumerable curves of other kinds.*

5. Hence the following deductions are evident : —

If TD is a straight line and EEE, FFF are two curves so
related that, when straight lines PEF are drawn parallel to
BD, a straight line given in position, the ordinates PE vary
as the squares on the ordinates PF; and if ES, FT, straight
lines drawn from the ends of the same common ordinate,
touch these curves ; then TP = 2SP. But, if the ordinate
PE varies as the cube of PF, then TP = 3SP; if PE varies
as the fourth power of PF, then TP = 4SP ; and so on in
the same manner to infinity.!

6. Again, let AGB be a circle, with centre D and radius
DB, and let EBE, FBF be two lines passing through B, so
related to one another that, when any straight line DG is
drawn through D, DF is always an arithmetical mean of
the same order between DG and DE ; also let the straight

* See note at the end of this lecture ; where it is shown that this theorem
is equivalent to a rigorous demonstration of the method for differentiating
a fractional power of the variable.

f This is a special case of the preceding theorem ; for PF is the simple
geometrical mean between PE and a definite length PQ ; or the second of
two, the third of three, etc. , geometrical means between PE and PQ ; thus
PF2 = PE . PG, PF3 = PE, PG2, etc. This enables Barrow to differentiate
any power or root of /"(*•), when he can differentiate/^*) itself.

LECTURE IX 105

line BO touch the curve EBE at B; required to draw the
tangent at B to the curve FBF.

This (demonstrated generally, to a certain extent,* in
Lect. VIII, § 8) will here be specially shown to follow more
clearly and completely from the method above. Thus : —

Let DQ be perpendicular to DB, cutting BO in 8 ; and
let N : M = DS : DT j join BT. Then BT touches the curve
FBF.t

7. Hence, other things remaining the same as before,
if the straight line DF is always taken as a geometrical
mean (of the same order as before) between DG and DE,
the same straight line BT will touch the curve FBF also.

For the lines formed from arithmetical means and from
geometrical means of the same order touch one another,
and have a common tangent.

8. Further, other things remaining the same as in the
preceding proposition, let any point P be taken in the curve
FBF; then the straight line that touches the curve at this
point can be determined by a similar plan.

Let the straight line DF be drawn, cutting the curve EBE
in E ; also draw DQ perpendicular to DG cutting EO the
tangent to EBE in X ; make DX : DY = N : M ; join EY, and
draw FZ parallel to EY. Then FZ touches the curve FBF.

Hence, not only the tangents to innumerable spirals,
but also those to a boundless number of others of different
kinds, can be determined quite readily.

* The actual construction for the asymptotes or tangent to the auxiliary
hyperbola is not given.

t Barrow proves his construction by the use of an auxiliary hyperbola
using Lect. VI, § 4, and VIII, § 9.

106 BARROW'S GEOMETRICAL LECTURES

9. Hence, it is clear that, if two lines EEE, FFF are so
related that, when any straight line DEF is drawn from a
fixed point D, DE varies as the square on DF ; and if E8,
FT are the tangents to the curves at the ends E, F, meeting
the line perpendicular to DEF in S, T; then DT = 2D8.
But, if DE varies as the cube of DF, DT = 3D8 ; and so on.*

10. Let YD, TB be two straight lines meeting in T, and
let a straight line BD, given in position, fall across them ;

T
V

Fig. loo.

also let the lines EBE, FBF pass through B and be such
that, if any straight line PG is drawn parallel to BD, PF is
always an arithmetical mean of the same order between PG
and PE; also let BR touch the curve EBE. Required to
draw the tangent at B to the curve FBF.

Taking numbers N, M to represent the exponents of PF,
PE, make N . TD + (M - N) . RD : M . TD = RD:SD, and join
BS; then BS touches the curve FBF.t

* As the theorems of §§ 6, 7, 8, 9 are only the polar equivalents of §§ i,
2, 3, 5, the figures and proofs are not given ; their inclusion by Barrow-
suggests that he was aware of the fact that, with the usual modern notation,
tan <j> = r. dQldr.

f The form of the equation suggests logarithmic differentiation : see note
at end of this lecture.

LECTURE IX 107

For, if any straight line PG is drawn, cutting the given
lines as in the figure, we have EG : FG = M : N ;
therefore FG . TD : EG . TD = N . TD : M . TD ;
also EF.RD:EG.TD = (M - N) . RD : M . TD.

Hence, adding the antecedents, we have
FG.TD + EF.RD:EG.TD = N.TD + (M - N).RD:M.TD

= RD:SD.

Now, LG.TD+EF.RD:EG.TD = RD:SD;VII, §4,
therefore

FG.TD+EF.RD:EG.TD = LG .TD + KL. RD: KG.TD

Hence, since EG > KG,

.-. FG .TD + EF. RD > LG .TD + KL. RD*
.'. ratio compounded of FG/EF and TD/RD > than

that compounded of LG/KL and TD/RD
or, removing the common ratio RD/TD, .'. EG/EF > LG/KL;
hence, by componendo EG/EF > KG/KL > EG/EL (by Lect.
VII, § i); therefore EF < EL, or the point L is situated
on the far side of the curve FBF; and thus the problem
is solved.

u. Moreover, all other things remaining the same, if
PF is supposed to be a geometrical mean of the same order
(plainly as in the cases just preceding) the same straight line
BS will touch the curve FBF.

* This is either a very bad slip on Barrow's part, or else he is making the
unjustifiable assumption that near B the ratio of LK to FE is one of equality.
In either case the proof cannot be accepted. The demonstration can, how-
ever, be completed rigorously as follows from the line

FG . TD + EF . RD : EG . TD = LG . TD + KL . RD : KG . TD.
Hence EG/EF : KG/KL = FG/EF + RD/TD : LG/KL + RD/TD

= EF/EF - RD/TD : KL/KL - RD/TD (dividendo) ;

therefore EG/EF = KG/KL, or EG/FG = KG/GL ; hence, since EG > KG,
it follows that FG > LG, i.e. L falls without the curve.

io8 BARROWS GEOMETRICAL LECTURES

Example. — If PF is a third of six means, or M = 7, N = 3 ;
then
3TD + 4RD:7TD = RD:SD, i.e. SD = 7TD.RD/(3TD + 4RD).

12. It is evident that, if any point F whatever is taken on
the line FBF, the tangent at F can be drawn in a similar
manner. Thus, through F draw the straight line PG par-
allel to DB, cutting the curve EBE at E, and through E let
ER be drawn touching the curve EBE at E ; then make

N.TP + (M-N).RP:M.TP = RP:SP,
and join SF. Then SF will touch the curve FBF.

13. Note that, if EBE is a straight line (i.e. coinciding
with the straight line BR), the line FBF is one of an infinite
number of hyperbolas or curves of hyperbolic form ; and
we have therefore included in the one theorem a method of
drawing tangents to these, together with innumerable others
of different kinds.

14. If, however, the points T, R do not fall on the same
side of D (or P), the tangent BS to the curve EBF is drawn
by making N . RD - (M - N) . TD : M . TD = RD : 3D.

15. Hence also the tangents to not only all elliptic curves
(in the case when EBE is supposed to be a straight line
coinciding with BR), but to an innumerable number of
other curves of different kinds, can be determined by the
one method.

Example. — If PF is the fourth of four means, i.e. M = 5,
and N = 4; then SD = 5TD. RD/(4RD -TD).

NOTE. If it happens that N . RD = (M - N) . TD, then DS

LECTURE IX 109

is infinite ; or BS is parallel to VD. Other points may be
noticed, but I leave them.

16. Amongst innumerable other curves, the Cissoid and
the whole class of cissoidal curves may be grouped together
by this method. For, let DSB be a semi-right angle ; and
let 8GB, SEE be two curves so related that, if any straight
line GE is drawn parallel to BD, cutting the given lines
BS,DS in F, P, PG, PF, PE are in continued proportion;
also let the straight line GT touch the curve SGB atG ; then
the line touching the curve SEE is found by making

2TP-SP:TP = SP:RP;
and, in every case, if RE is joined, RE touches SEE.

The proof is easy from what has gone before.

Now, if the curve SGB is a circle, and the angle of appli-
cation, SPG, is a right angle, then the curve SEE is the
ordinary Cissoid or the Cissoid of Diocles ; otherwise it
will be a cissoidal curve of some other kind. But I only
mention this in passing, and will not now detain you longer
over it.

NOTE

This lecture is remarkable for the important note of § 4.
In it, Barrow calls his readers' attention to the fact that he
has given a method for drawing tangents to any of the
parabolas or paraboliforms ; and apparently he refers in
more or less depreciative words to the work of Wallis,
whilst claiming that his own work is a geometrical demon-
stration, and therefore rigorous. If we take a line parallel
to PG, and DV, as the coordinate axes, and suppose them
rectangular or oblique, then PFM = PGM~N.PEN gives
xta =. aw-N>jj,Nj or y — k.x™!**, as the general equation to
the curve FBF.

no BARROW'S GEOMETRICAL LECTURES

Also, dy\dx = PT/PF = (PT/P8) . (PS/PF) = (M/N) .yfx ;
or, if the axes are interchanged, the equation to the curve
is y = c.x"lM, and then dyldx = PF/PT = (N/M). v/.v.
Note particularly that the form suggests logarithmic dif-
ferentiation.

The theorem of § 6 is a particular case of this, in which
N -- i, i.e. PF is the first of any number of means between
PG and PE, and the equations of the curves arey = k.x2,
6. xs, k. x*t etc. (the " parabolas " as distinguished from
the " paraboliforms ").

It seems strange, unless perhaps it is to be ascribed to
Barrow's dislike for even positive integral indices, that he
does not make a second note to the effect that if the curve
EBE is a hyperbola whose asymptotes are YD and a line
parallel to PG, then the curves FBF are the hyperboliforms.
For, from this particular case, in a manner similar to the
foregoing, it follows that if y = c . x~r, where r is any
positive rational, either greater or less than unity, then
dy/dx = - r(yjx). But Barrow probably intends the recip-
rocal theorem of Lect. VIII, § 9, to be used thus : — If
y = c.x~r, let z — i/y — k.xr ; then from Lect. VIII, § 9,
we- have (i/z) . dz\dx = -(i/y) . dyfdx ; also from the above,
dzjdx = r . z/x ; hence dy/dx = (-r) .y/x. I suggest that
Barrow found out these constructions by analysis, using
letters such as a and e instead of dy and dx, and that the
form of the results suggests very strongly that he first
expressed his equation logarithmically.

Anyway, Barrow was the first to give a rigorous demon-
stration of the form of the differential coefficient of xr,
where r is any rational whatever. As far as I am aware,
it is the only proof that has ever been given, that does not
involve the consideration of convergence of infinite series,
or of limiting values, in some form or other. Moreover,
he gives it in a form, which yields, as a converse theorem,
the solution of the differential equation dyldx = r . dz/dx,
although, of course, this is not noted by Barrow, simply
because he had not the notation.

Again, considering § 8, which is only § 3 with the
constant distance between the parallels, PG, replaced by
the constant radius, DG, we see that, if DB is the initial
line, and the angle BDG is 6, and the angles between the

LECTURE IX in

vector DG and the tangents at E and F are <£ and x>
DF = R, DE = r, and DG = a, then/a« <£ : tan^ = N : M, and
RM = aM - N . rn . hence (dO/<tr)/(<t6/JR) = d&\dr = (N/M). (R/r)
and /a« <£ : /a« x = r ^/^ : R • dOjdR. ; and I suggest that it
was thus that Barrow obtained the construction for this
theorem. I go further. Although it is a consequence of
a consideration of the whole work, the present place is the
most convenient one for me to state my firm conviction
that Barrow's drawing of tangents was a result of his
knowledge of the fundamental principles of a calculus
of infinitesimals in an algebraic form, which may have
been so cumbrous that it was only intelligible to himself
when expressed in geometrical form. I fail to see how else
he could possibly have arrived at some of his constructions,
or elaborated so many of them in the comparatively short
time that he had to spend upon them ; unless indeed he
was a far greater genius than even I am trying to make
him out to be. If he had stumbled on the idea in his
young days, as might be possible, one could better under-
stand these theorems as being gradually evolved ; but we
have his own words against this : " The lectures were
elicited by my office." Thus I suggest that whilst his
geometrical theorems perhaps took definite shape whilst
he was Professor of Geometry at Gresham, his knowledge
of the elements of the calculus dated from before this
time.

Last, but by no means least, the theorems of §§ 10, n,
12 are modifications of §§ i, 2, 3, in which a pair of
inclined lines are substituted for the pair of parallels.
Referring to Fig. 100 on p. 106, take the angle BDT a
right angle, and DT as the axis of x, then the relation
given is a relation between subtangents solely. Further,
instead of BT we can take a fixed curve touching BT at
B ; and we have : —

If PF"1 = PGM - N . PEN, then N/RD + (M - N)/TD = M/SD.

Also, if we take Z1*-1 . PH = PFM, we have by § 5, if WD
is the subtangent to the locus of H, 1/WD = M/SD.

This affords a complete rule for products, and combining
the result with the reciprocal theorem of Lect. VIII, 9,
for quotients also.

H2 BARROW'S GEOMETRICAL LECTURES

Thus, putting N = i, and M = 2, we have for the general
theorem of § 1 2 the remarkably simple results : —

^GGG, EEE are tivo curves, and PEG is a straight line
applied perpendicular to an axis PRT, and GT, ER are the
tangents to GGG and EEE, then

(i) If HHH is another curve, so related to the other two
that 1. PH = PE. PG ; then, if HW is the tangent to HHH,
meeting the axis in W, 1/PW = 1/PR + l/PT; /.*. PW is a
fourth proportional to PR + PT, PR, and PT.

(it) If KKK is another curve so related to GGG and EEE
that PK:Z = PE:PG, then, if KV is the tangent to KKK,
meeting the axis in V, 1/PV = 1/PR-l/PT; or PV is a
fourth proportional to PT- PR, PR, and PT.

The elegance of the geometrical results probably accounts
for the fact that Barrow adheres to the subtangent, as used
by Descartes, Fermat, and others ; and this would tend to
keep from him the further discoveries and development that
awaited the man who considered, instead of the subtangent,
the much more fertile idea of the gradient, as represented
by Leibniz' later development, dyfdx; the germ'of the idea
of the gradient is of course contained in the " a and e "
method, but it is neglected.

Note the disappearance of the constant Z; hence the
curves may be drawn to any convenient scale, which need
not be the same, for all or any, in the direction parallel to
PEG. The analytical equivalents are : —

(i) If w = yz, then (ilw^dwjdx = (i/y)dy/dx + (i/z)dz/dx;
(ii) if v = y/z, then (i/v)dv/dx = (i/y)dy/dx -(i/z)dz/dx.

The first of these results is generally given in modern text-
books on the calculus, but I do not remember seeing the
second in any book. Thus, for products and quotients we
may state the one rule : —

Tr uv dy uvV\ du , i dv i dw i dz~\

1 t -i; := -S I _ . J_ ^ __ — _ I

wz dx wz\_u dx v dx w ' dx z dx\
where u, v, w, z, and y are all functions of x.

LECTURE X

Rigorous determination of dsjdx. Differentiation as the
inverse of integration. Explanation of the " Differential
Triangle " method ; with examples. Differentiation of a
trigonometrical function.

/ i. Let AEG be any curve whatever, and API another
curve so related to it that, if any straight line EF is drawn
parallel to a* straight line given in position (which cuts AEG
in E and API in F), EF is always equal to the arc AE of the
curve AEG, measured from A; also let the straight line ET
touch the curve AEG at E, and let ET be equal to the
arc AE; join TF; then TF touches the curve AFI.

/ 2. Moreover, if the straight line EF always bears any the
same ratio to the arc AE, in just the same way FT can be
shown to touch the curve AFI.*

v/3. Let AGE be any curve, D a fixed point, and AIF be
another curve such that, if any straight line DEF is drawn
through D, the intercept EF is always equal to the arc AE ;
and let the straight line ET touch the curve AGE; make

* Since the arc is a function of the ordinate, this is a special case of
the differentiation of a sum, Lect. IX, § 12 ; it is equivalent to J(as+y)/dx=
a . dsfdx + dyldx ; see note to § 5.

114 BARROW'S GEOMETRICAL LECTURES

TE equal to the arc AE * ; let TKF be a curve such that, if
any straight line DHK is drawn through D, cutting the curve
TKF in K and the straight line TE in H, HK = HT; then let
F8 be drawn f to touch TKF at F; F8 will touch the curve
AlFalso.

4. Moreover, if the straight line EF is given to bear any
the same ratio to the arc AE, the tangent to it can easily be
found from the above and Lect. VIII, § 8.

5. Let a straight line AP and two curves AEG, AFI be so
related that, if any straight line DEF is drawn (parallel to

K.I HG

Fig. 106.

AB, a straight line given in position), cutting AP, AEG, AFI,
in the points D, E, F respectively, DF is always equal to the
arc AE; also let ET touch the curve AEG at E; take TE
equal to the arc AE, and draw TR parallel to AB to cut
AP in R; then, if RF is joined, RF touches the curve AFI.

For, assume that LFL is a curve such that, if any straight
line PL is drawn parallel to AB, cutting AEG in G, TE in H,
and LFL in L, the straight line PL is always equal to TH
and HG taken together. Then PL > arc AEG > PI ; and

* TE, AE are drawn in the same sense,
t By Lect. VIII, § 19.

LECTURE X 115

therefore the curve LFL touches the curve API. Again,
by Lect VI, § 26, PK = TH (or KL = GH) ; hence the curve
LFL touches the line RFK (by Lect. VII, § 3); therefore
the line RFK touches the curve AFI.*

6. Also, if DF always bears any the same ratio to the
arc AE, RF will still touch the curve AFI ; as is easily shown
from the above and Lect. VIII, § 6.

7. Let a point D and two curves AGE, DFI be so related
that, if any straight line DFE is drawn through D, the straight
line DF is always equal to the arc AE ; also let the straight
line ET touch the curve AG E at E ; make ET equal to the
arc AE; and assume that DKK is a curve such that, if any
straight line DH is drawn through D, cutting DKK in K and
TE in H, the straight line DK is always equal to TH. Then,
if FS is drawn (by Lect. VIII, § 16) to touch the curve
DKK at F, FS touches the curve DIF also.

8. Moreover, if DF always bears any the same ratio to the
arc AE, the straight line touching the curve DIF can likewise
be drawn ; and in every case the tangent is parallel to FS.

9. By this method can be drawn not only the tangent
to the Circular Spiral, but also the tangents to innumerable
other curves produced in a similar manner.

10. Let AEH be a given curve, AD any given straight line

* The proof of this theorem is given in full, since not only is it a fine
example of Barrow's method, but also it is a rigorous demonstration of the
principle of fluxions, that the motion along the path is the resultant of the
two rectilinear motions producing it. Otherwise, for rectangular axes,
(dsjdxY = i + (dyldx)*\ for ds\dx = DF/DR = ET/DR = Cosec DET and
ctyldx=CotDEi.

ii6 BARROW'S GEOMETRICAL LECTURES

in which there is a fixed point D, and DH a straight line given
in position ; also let AGB be a curve such that, if any point
G is taken in it, and through G and D a straight line is
drawn to cut the curve A EH in E, and GF is drawn parallel
to DH to cut AD in F, the arc AE bears to AF a given ratio,
X to Y say ; also let ET touch the curve A EH ; along ET take
EV equal to the arc AE ; let OGO be a curve such that, if any
straight line DOL is drawn, cutting the curve OGO in 0 and
ET in L, and if OQ is drawn parallel to GF, meeting AD in
Q, LV : AQ = X : Y. Then the curve OGO is a hyperbola (as
has been shown).* Then, if GS touches this curve, GS will
touch the curve AG B also.

If the curve AEH is a quadrant of a circle, whose centre
is D, the curve AGB will be the ordinary Quadratrix. Hence
the tangent to this curve (together with tangents to all
curves produced in a similar way) can be drawn by this
method.

I meant to insert here several instances of this kind ;
but really I think these are sufficient to indicate the
method, by which, without the labour of calculation, one
can find tangents to curves and at the same time prove the
constructions. Nevertheless, I add one or two theorems,
which it will be seen are of great generality, and not lightly
to be passed over.

J .11. Let ZGE be any curve of which the axis is AD; and
let ordinates applied to this axis, AZ, PG, DE, continually

* Only proved for a special case in Lect. VI, § 17 ; but the method can
be generalized without difficulty.

LECTURE X

117

increase from the initial ordinate AZ; also let AIF be a line
such that, if any straight line EOF is drawn perpendicular
to AD, cutting the curves in the points E, F, and AD in D, the
rectangle contained by DF and a given length R is equal
to the intercepted space ADEZ; also let DE : DF = R : DT,
and join DT. Then TF will touch the curve AIF.

Fig. 109.

For, if any point I is taken in the line AIF (first on the
side of F towards A), and if through it IG is drawn parallel
to AZ, and KL is parallel to AD, cutting the given lines as
shown in the figure; then LF : LK = DF : DT = DE : R, or
R . LF = LK . DE.

But, from the stated nature of the lines DF, PK, we have
R . LF = area PDEG ; therefore LK . DE - area PDEG < DP . DE ;
hence LK < DP < LI.

Again, if the point I is taken on the other side of F, and
the same construction is made as before, plainly it can be
easily shown that LK > DP > LI.

From which it is quite clear that the whole of the line
TKFK lies within or below the curve AIFI.

Other things remaining the same, if the ordinates, AZ,
PG, DE, continually decrease, the same conclusion is

n8 BARROW'S GEOMETRICAL LECTURES

attained by similar argument ; only one distinction occurs,
namely, in this case, contrary to the other, the curve AIFI
is concave to the axis AD.

COR. It should be noted that DE . DT = R . DF = area
ADEZ.*

12. From the preceding we can deduce the following
theorem.

Let ZGE, AKF be any two lines so related that, if any
straight line EOF is applied to a common axis AD, the
square on DF is always equal to twice the space ADEZ;
also take DQ, along AD produced, equal to DE, and join
FQ; then FQ is perpendicular to the curve AKF.

I will also add the following kindred theorems.

13. Let AGEZ be any curve, and D a certain fixed point
such that the radii, DA, DG, DE, drawn from D, decrease
continually from the initial radius DA; then let DKE be
another curve intersecting the first in E and such that,
if any straight line DKG is drawn through D, cutting the
curve AEZ in G and the curve DKE in K, the rectangle
contained by DK and a given length R is equal to the area
ADG ; also let DT be drawn perpendicular to DE, so that
DT = 2R ; join TE. Then TE touches the curve DKE.

Moreover, if any point, K say, is taken in the curve DKE,
and through it DKG is drawn, and DG : DK = R : P ; then, if
DT is taken equal to 2P and TG is joined, and also KS is
drawn parallel to GT; KS will touch the curve DKE.

* See note at end of this lecture.

LECTURE X 119

Observe that Sq. on DG : Sq. on DK = 2R : DS.

Now, the above theorem is true, and can be proved in
a similar way, even if the radii drawn from D, DA, DG, DE,
are equal (in which case the curve AGEZ is a circle and
the curve DKE is the Spiral of Archimedes), or if they con-
tinually increase from A.

14. From this we may easily deduce the following
theorem.

Let AGE, DKE be two curves so related that, if straight
lines DA, DG are drawn from some fixed point D in the
curve DKE (of which the latter cuts the curve DKE in K),
the square on DK is equal to four times the area ADG ; draw
DH perpendicular to DG, and make DK : DG = DG : DH ; join
HK ; then HK is perpendicular to the curve DKE.

We have now finished in some fashion the first part, as
we declared, of our subject. Supplementary to this we
add, in the form of appendices, a method for finding
tangents by calculation frequently used by us (a nobis
ttsitatum). Although I hardly know, after so many well-
known and well-worn methods of the kind above, whether
there is any advantage in doing so. Yet I do so on the
advice of a friend ; and all the more willingly, because it
se^ms to be more profitable and general than those which
I lave discussed.*

* See note at the end of this lecture.

120 BARROWS GEOMETRICAL LECTURES

Let AP, PM be two straight lines given in position, of
which PM cuts a given curve in M, and let MT be supposed
to touch the curve at M, and to cut the straight line at T.

In order to find the quantity of the straight line PT,*
I set off an indefinitely small arc, MN, of /

the curve; then I draw NQ, NR parallel to /flvl

MP, AP; I call MP = m, PT = /, MR =

NR = <?, and other straight lines, determined

by the special nature of the curve, useful A T Q p

for the matter in hand, I also designate Fig. 115.
by name; also I compare MR, NR (and through them,
MP, PT) with one another by means of an equation obtained
by calculation ; meantime observing the following rules.

RULE i. In the calculation, I omit all terms containing
a power of a or e, or products of these (for these terms
have no value).

RULE 2. After the equation has been formed, I reject
all terms consisting of letters denoting known or deter-
mined quantities, or terms which do not contain a or e
(for these terms, brought over to one side of the equation,
will always be equal to zero).

RULE 3. I substitute m (or MP) for a, and / (or PT) for
e. Hence at length the quantity of PT is found.

Moreover, if any indefinitely small arc of the curve enters
the calculation, an indefinitely small part of the tangei 1S
or of any straight line equivalent to it (on account of tl

* See note at the end of this lecture.

LECTURE X 121

indefinitely small size of the arc) is substituted for the arc.
But these points will be made clearer by the following
examples.

NOTE

Barrow gives five examples of this, the "differential
triangle " method. As might be expected, two of these are
well-known curves, namely the Folium of Descartes, called
by Barrow La Galandc, and the Quadratrix ; a third is the
general case of the quasi-circular curves xn+yn = an; the
fourth and fifth are the allied curves r = a . tan 6 and
y = a . fan x. It is noteworthy, in connection with my sug-
gestion that Barrow used calculus methods to obtain his
geometrical constructions, that he has already given a purely
geometrical construction for the curve r = a . tan 0 in Lect.
VIII, § 1 8, if the given lines are supposed to be at right angles.
I believe that Barrow, by including this example, intends to
give a hint as to how he made out his geometrical con-
struction : thus : —

The equation of the curve is x* + xzyz = a2y2 ; the
gradient, as he shows is x(2x2+y2)/y(a2 - x2) ; using the
general letters x and y instead of his p and m. Descartes
has shown that a hyperbola is a curve having an equation
of the second degree, hence Barrow knows that its gradient
is the quotient of two linear expressions, and finds (? by
equating coefficients) the hyperbola whose gradient is
x0(2xx0 +jyy0)/>'0(a2 - XXQ) > tne feasibility of this is greatly
enhanced by the fact that Barrow would have written the
two gradients as

m : t = x(2xx +yy) :y(aa - xx) and

These two gradients are the same at the point x0, _y0; hence

he can find such a hyperbola, it will touch the curve ; he

n draw its tangent, and this will also be a tangent to

I Is curve. The curve does turn out to be a hyperbola ;

— >r its equation is X^X^Q + x$>0 . xy = a2yy0 or x2+y2 =

[y + (yJxo) ' (d-x}\, where d — a2/x. This latter form is

122 BARROWS GEOMETRICAL LECTURES

easily seen to be equivalent to the construction, in Lect. VIII,
§ 1 7, for the curve DYO, when the axes are rectangular ; for
the equation gives DY2 = YE . YR. It also suggests that the
construction for the original curve is transformable into
that of § 17, as is proved by Barrow in § 18, and in order
that Barrow may draw the tangent, §§ 10, 1 1, 1 2 of Lect. VI
are necessary to prove that the auxiliary is a hyperbola of
which the asymptotes can be determined by a fairly easy
geometrical construction. Barrow then generalizes his
theorems for oblique axes. I contend that this suggestion
is a very probable one for three reasons : (i) it is quite
feasible, even if it is considered to be far-fetched, (ii) we
know that mathematicians of this time were jealous of their
methods, and gave cryptogrammatic hints only in their work
(cf. Newton's anagram), and (iii) it is to my mind the only
reason why this particular theorem should have been selected
(especially as Barrow makes it his Example i), for there is
no great intrinsic worth in it.

The fifth example, the case of the curve y — a. tan 0, I
have selected for giving in full, for several reasons. It is
the clearest and least tedious example of the method, it
is illustrated by two diagrams, one being derived from the
other, and therefore the demonstration is less confused, it
is connected with the one discussed above and suggests
that Barrow was aware of the analogy of the differential
form of the polar subtangent with the Cartesian subtangent,
and that in this is to be found the reason why Barrow
gives, as a rule, the polar forms of all his Cartesian theorems ;
and lastly, and more particularly, for its own intrinsic
merits, as stated below. Barrow's enunciation and proof
are as follows : —

EXAMPLE 5. Let DEB be a quadrant of a circle, to
which BX is a tangent ; then let the line AMO be such
that, if in the straight line AV any part AP is taken equal
to the arc BE, and PM is erected perpendicular to AV, then
PM is equal to BG the tangent of the arc BE.

LECTURE X

123

Take the arc BF equal to AQ and draw CFH ; drop EK,
FL perpendicular to CB. Let CB = r, CK =/ KE = g.

A T Q P \

Fig. 121.

- 2/ge/r) = r:m-a,

K L

Fig. 1 20.

Then, since CE : EK - arc EF : LK = QP : LK ; therefore
r\g = e : LK, or LK = gejr, and CL = f-tfgelr'} hence also
LF -

But CL : LF = CB : BH,or/+^/r :
and squaring, we have

f2 + zfselr '• S2 ~ 2/Sefr = rZ '• m* ~
Hence, omitting the proper terms, we obtain the equation

rfma = gr*e +gm*e ; .
and, on substituting m, t for a, e, we get

rfm* = gr*t+gm*t, or rfm*l(gr*+gm'i) = t.
Hence, since m = rglf, we obtain
t=m. ri\(r* + m^ = BG . CB2/CG2 = BG . CK2/CE2.

In other words, this theorem states that, \{-y = tan x,
where x is the circular measure of an "angle" or an "arc,"
then dy\dx = m\t = CE2/CK2 = sefi x.

Moreover, although Barrow does not mention the fact,
he must have known (for it is so self-evident) that the same
two diagrams can be used for any of the trigonometrical
ratios. Therefore Barrow must be credited with the differ-
entiation of the circular functions. (See Note to § 15 of
App. 2 of Lect. XII.)

124 BARROW'S GEOMETRICAL LECTURES

As regards this lecture, it only remains to remark on the
fact that the theorem of § 1 1 is a rigorous proof that
differentiation and integration are inverse operations, where
integration is defined as a summation. Barrow not only,
as is well known, was the first to recognise this ; but also,
judging from the fact that he gives a very careful and full
proof (he also gave a second figure for the case in which
the ordinates continually decrease), and in addition, as will
be seen in Lect. XI, § 19, he takes the trouble to prove the
theorem conversely, — judging from these facts, I say, — he
must have recognised the importance of the theorem also.
It does not seem, however, to have been remarked that he
ever made any use of this theorem. He, however, does
use it to prove formulae for the centre of gravity and the
area of a paraboliform, which formulae he only quotes with
the remark, "of which the proofs may be deduced in
various ways from what has already been shown, without
much difficulty " (see note to Lect. XI, § 2).

The "differential triangle" method has already been
referred to in the Introduction ; it only remains to point
out the significance of certain words and phrases. Barrow,
whilst he acknowledges that the method "seems to be
more profitable and more general than those which I have
discussed," yet is in some doubt as to the advantage of
including it, and almost apologizes for its insertion ;
probably, as I suggested, because, although he has found
it a most useful tool for hinting at possible geometrical
constructions, yet he compares it unfavourably as a method
with the methods of pure geometry. It is also to be
observed that his axes are not necessarily rectangular,
although in the case of oblique axes, PT can hardly be
accepted as the subtangent ; hence he finds it convenient
to tacitly assume that his axes are at right angles. The
last point is that Barrow distinctly states that his method
is expressly "in order to find the quantity of the sub-
tangent," and I consider that this is almost tantamount to
a direct assertion that he has used it frequently to get his
first hint for a construction in one of his problems. The
final significance of the method is that by it he can readily
handle implicit functions.

LECTURE XI

Change of the independent variable in integration. Integra-
tion the inverse of differentiation. Differentiation of a quotient.
Area and centre of gravity of a paraboliform. Limits for the
arc of a circle and a hyperbola. Estimation of TT.

NOTE

. In the following theorems, Barrow uses his variation of
the usual method of summation for the determination of an
area. If ABKJ is the area under the curve AJ, he divides
BK into an infinite number of equal parts and erects
ordinates. In his figures he generally makes four parts
do duty for the infinite number.

He then uses the notation already
mentioned, namely, that the area ABKJ
is equal to the sum of the ordinates AB,
CD, EF, GH, JK.

The same idea is involved when he
speaks of the sum of the rectangles
CD . DB, EF . FD, GH . FH, JK . GH ;
for this sum, where commas are used
between the quantities instead of a plus B D F H K
sign, does not stand for the area ABKJ, but for R . A'BKJ',
where an ordinate HG' is such that R . HG' = HG . FH,
and R is some given length ; in other words, ordinates
proportional to each of the rectangles are applied to points
of the line BK, and their aggregate or sum is found ;
hence this sum is of three dimensions. On the contrary,
he uses the same phrase, with plus signs instead of commas,
to stand for a simple summation.

126 BARROW'S GEOMETRICAL LECTURES

Thus, in § 3 of this lecture, the sum of AZ . AE2, BZ . BF2,
CZ . CG2, etc., is the area aggregated from ordinates
proportional to AZ . AE2, BZ . BF2, CZ . CG2, etc., applied
to the line YD; and it is of the fourth dimension.
Whereas, in § 3, the sum HL. H02 + LK . LY2 + Kl . KY2 +
etc., is aggregated from ordinates equal to HO2, LY2, KY2,
etc., applied to the line HD; and it is the same as the
sum of HO2, LY2, KY2, etc.

i. If VH is a curve whose axis is VD, and HD is an
ordinate perpendicular to VD, and <J>Zi[/ is a line such that,
if from any point chosen at random on the curve, E say,
a straight line EP is drawn normal to the curve, and a
straight line EAZ perpendicular to the axis, AZ is equal to
the intercept AP; then the area VDi/^ will be equal to half
the square on the line DH.

For if the angle HDO is half a right angle, and the
straight line VD is divided into an infinite number of
equal parts at A, B, C, and if through these points straight
lines EAZ, FBZ, GCZ, are drawn
parallel to HD, meeting the curve
in E, F, G ; and if from these
points are drawn straight lines EIY,
FKY, GLY, parallel to VD or HO;
and if also EP, FP, GP, HP are
normals to the curve, the lines
intersecting as in the figure; then
the triangle HLG is similar to the Fig.. 122.

triangle PDH (for, on account of the infinite section, the
small arc HG can be considered as a straight line).

LECTURE XI 127

Hence, HL : LG - PD : DH, or HL . DH = LG . PD,
i.e. HL.HO = DC.D^.

By similar reasoning it may be shown that, since the
triangle GMF is similar to the triangle PCG, LK.LY = CB.CZ;
and in the same way, Kl . KY = BA . BZ, ID . IY = AV . AZ.

Hence it follows that the triangle DHO (which differs in
the slightest degree only from the sum of the rectangles
HL.HO + LK. LY + KI.KY + ID.IY) is equal to the space
VDi/*x£ (which similarly differs in the least degree only from
the sum of the rectangles DC . Di^ + CB . CZ+ BA . BZ

+ AV.AZ);

i.e. DH2/2 = space VD</^.

A lengthier indirect argument may be used; but what
advantage is there ?

2. With the same data and construction as before, the
sum of the rectangles AZ . AE, BZ . BF, CZ . CG, etc., is equal
to one-third of the cube on the base DH.

For, since HL : LG = PD : DH = PD . DH : DH2; therefore
HL.DH2 = LG.PD.DH or LH . H02=DC . D<// . DH ; and,
similarly LK . LY2 = CB . CZ . CG, Kl . KY2 = BA . BZ . BF, etc.

But the sum HL. H0'2+ LK. LY2+ Kl . KY2 + etc. = DH3/3;*
and the proposition follows at once.

3. By similar reasoning, it follows that

the sum of AZ AE2, BZ . BF2, CZ . CG2, etc. = DH4/* ;
the sum of AZ . AE3, BZ . BF3, CZ . CG3, etc. = DH5/5 ;
and so on.f

* See the critical note immediately following.

f The analytical equivalents of the theorems given above are comprised
in the general formula (with their proofs),

fyr(dyldx} . dx=fyr . <ly = yr+ll(r+ i).

128 BARROW'S GEOMETRICAL LECTURES

NOTE
On the assumptions in the proofs of §§ 2, 3.

The summation used in § 2 has already been given by
Barrow in Lect. IV ; he states that it has been established
" in another place " (? by Wallis or others), and that it at
least "is sufficiently known among geometers."

It is easy, however, to give a demonstration according
to Barrow's methods of the general case; and, since in
several cases Barrow is content with saying that the proof
may easily be obtained by his method, and sometimes he
adds "in several different ways," I feel sure that he had
made out a proof for these summations in the general case.

The method given below follows the idea of Lect. IV,
by finding a curve convenient for the summation, without
proving that this curve is the only one that will do. Other
methods will be given later, thus substantiating Barrow's
statement that the matter may be proved in several ways ;
see notes following Lect. XI, § 27, and Lect. XI, App., § 2.

Let AH, KHO be two straight lines
at right angles, and let AH = HO = R
and KH = 8. Let AEO be a curve such
that (in the figure) UE is the first of
»-i geometrical means between UW
and UV,

i.e. UE" = UW-MJY,

or DYn = AD" = R-1. DE.

Let AFK be a curve such that PF is
the first of n geometrical means between
PQ and PL, or PF"+1 = PQn . PL, i.e. 8 . AD"+1 = R"+1 . DF.

Then the curve AFK is a curve that is fitted for the
determination of the area under the curve AED, providing
a suitable value of 8/R is chosen.

For FD/DT= PS/AD = («+ i) . DF'/AD = («+ i) . DE . S/R2 ;
and if S is taken equal to R/(« + i), FD : DT = DE : R ; and
therefore by Lect. X, § n,
the sum DY". DD' + . . . = the sum R"'1 . DE . DD'
= R—1.area ADE
= R". DF = S. AD"+1/R = DY"+1/(« + i ).

LECTURE XI 129

Hence we may deduce the following important
theorems : —

4. Let VDif/4> be any space of which the axis VD is
equally divided (as in fig. 122); then if we imagine
that each of the spaces VAZ<£, VBZ<£, VCZ<£, etc., is
multiplied by its own ordinate AZ, BZ, CZ, etc., respectively,
the sum which is produced will be equal to half the square
of the space

5. If, however, each of the square roots of the spaces is
multiplied by its own ordinate, an aggregate is produced
equal to two-thirds of the square root of the cube of

6. Example. — Let VDt^ be a quadrant of a circle, of
which the radius is R and the perimeter is P ; then the
segments VAZ, VBZ, VCZ, . . ., each multiplied by its
own sine, AZ, BZ, CZ, . . . , respectively, will together
make R2P2/8.

Also the sum AZ . V(VAZ) + BZ . V(VBZ) + etc.

Again, if VDi//- is a segment of a parabola, the sum made
from the products into the ordinates will be equal to 2/9
of VD2 . Dip, and that from the products of the square roots
of the segments into the ordinates will be equal to 2/3 of
^(8/27 of VD3 . DiA3) or ^(VD3 . D^3 . 32/243).

* The equivalents of §§ 4, 5, are respectively fy(fy • dx) . dx = (fy . dx)*
&i\&fy>J(fy. dx}. dx = (fy . dx}W. 2/3 ; or in a more recognizable form,
putting z forfy . dx, they a.refz(dz/dx) . dx = z2/2, and so on.

130 BARROW'S GEOMETRICAL LECTURES

Other similar things concerning the sums made from the
products of other powers and roots of the segments into the
ordinates or sines can be obtained.

7. Further, it follows from what has gone before that, in
every case, if the lines VP intercepted between the vertex
and the perpendiculars are supposed to be applied through
the respective points A, B, C, . . . , say that AY, BY, CY, . . .
are equal to the respective lines VP; then will the space
VD£0, constituted by these applied lines, be equal to half
the square on the subtense VH.

8. Moreover, if with the same data, RXXS is a curve
such that IX = AP, KX = BP, LX = CP, . . . ; then the solid
formed by the rotation of the space VD</^ about VD as an
axis is half the solid formed by the rotation of the space
DRSH about the same axis VD.

9. All the foregoing theorems are true, and for similar
reasons, even if the curve VEH is convex to the line VD.

From these theorems, the dimensions of a truly bound-
less number of magnitudes (proceeding directly from their
construction) may be observed, and
easily verified by trial.

10. Again, if VH is a curve, whose
axis is VD and base DH, and DZZ is
a curve such that, if any point such
as E is taken on the curve VH and
ET is drawn to touch the curve, and

a straight line EIZ is drawn parallel to Fig. 125.

LECTURE XI 131

the axis, then IZ is always equal to AT; in that case,
I say, the space DHO is equal to the space VHD.

This extremely useful theorem is due to that most learned
man, Gregory of Aberdeen : * we will add some deductions
from it.

11. With the same data, the solid formed by the rotation
of the space DHO about the axis VD is twice the solid
formed by the rotation of the space VDH about the same
axis.

For HL:LG = DH : DT = DH : HO = DH2:DH.HO;

HL.DH.HO = LG. DH2 - CD.DH2.
Similarly, LK. DL. LZ = BC.CG2, KI.DK.KZ = AB.BF2,
and ID.DI.IZ = VA.AE2.

But it is well known that t
the sum CD . DH2 + BC . CG2 + AB . BF2 + VA . AE2

= twice the sum of Dl. IE, DK . KF, DL.LG, etc.,
and therefore the solid formed by the space DHO rotated
about the axis VD is double of the solid formed by the
space VDH rotated round VD.

12. Hence the sum of Dl . IZ, DK. KZ, DL . LZ, etc. (ap-
lied to HD) = the sum of the squares on the ordinates to VD,

= the sum of AE2, BF, CG2, etc. (applied to VD).
The same even tenor of conclusions is observable for the
other powers.

* The member of a remarkable family of mathematicians and scientists
that is here referred to is James Gregory (1638-1675), who published at
Padua, in 1668, Geometric Pars Universalis. He also gave a method for
infiriitely converging series for the areas of the circle and hyperbola in 1667.

t For a discussion of this and §§ 12, 13, 14, see the critical note on
page 133.

132 BARROW'S GEOMETRICAL LECTURES

13. By similar reasoning, it follows that

the sum of Dl2 . IZ, DK2 . KZ, DL2 . LZ, etc. (applied to HD)
= three times the sum of the cubes on all the ordi-
nates AE, BF, CG, . . ., applied to VD.

14. With the same data, if DXH is a curve such that any
ordinate to DH, as IX, is a mean proportional between the
ordinates IE, IZ congruent to it; then the solid formed by
the space VDH rotated about the axis DH is double the
solid formed from the space DXH rotated about the same
axis.

15. If, however (in fig. 125), the curve DXH is supposed
to be such that any ordinate, CX say, is a bimedian *
between the congruent ordinates IE, IZ ; then the sum of
the cubes of IX, KX, LX, etc., is one-third of the sum of the
cubes of DV, IE, KF, etc. But if IX is a trimedian*; then
the sum of IX4, KX4, LX4, etc., is equal to one-fourth of the
sum of DV4, IE4, KF4, etc.; and so on for all the other
powers.

* NOTE. I call by the name bimedian the first of two
mean proportionals, by trimedian the first of three, and
so on.

These results are deduced and proved by similar reason-
ing to that of the previous propositions ; but repetition is
annoying.!

1 6. Again, if VYQ is a line such that the ordinate AY is
equal to AT, BY to BT, and so on ; then the sum of IZ-, KZ2,
LZ2, etc., that is, the sum of the squares of the ordinates

f Literally, " it irks me to cry cuckoo."

LECTURE XI 133

of the curve DZO applied to the line DH, is equal to the
sum VA . AE . AY + AB . BF . BY + etc., that is, the figure VDH
"multiplied" by the figure VDQ.*

17. Also the sum of IZ3, KZ3, LZ3, etc.

= the sum VA.AE.AY2 + AB.BF.BY2 + etc.;*
that is, the figure VDH "multiplied" by the figure VDQ
" squared."

These you can easily prove by the pattern of the proofs
given above.

1 8. The same things are true and are proved in an
exactly similar manner, even if the curve VH is convex to
the straight line VD.

NOTE

The equality, which in § 1 1 is said by Barrow to be well
known, namely, the sum CD . DH2 + BC . CG2 + etc.

= twice the sum of Dl . IE, DK . KF, etc.,
is really an equality between two expressions for the volume
of the solid formed by the rotation of VDH about VD; and
the analytical equivalent is Jy2 dx = 2\xydy, with the inter-
mediate step i[[ydydx. Thus these theorems of Barrow
are equivalent to the equality of the results obtained from
a double integral, when the two first integrals are obtained
by integrating with regard to each of the two variables in
turn. He says indeed that the first result, that in § 1 1 , is
a matter of common knowledge, but he remarks that the
others that he uses in the following sections can be obtained
by similar reasoning. From this, and from indications in
Lect. IV, § 1 6, Lect. X, § 1 1, and § 19 of this lecture, I feel

* The equivalents of these theorems are :

16. Jpiyl(dyldx)t.<fy =fy . [yKdyldx)}. dx.
17- Jlyl(dytdx)?.dy =fy [yKdyldx)}* . dx.

134 BARROW'S GEOMETRICAL LECTURES

certain that Barrow had obtained these theorems in the
course of his researches, but, as in many other cases, he
omits the proofs and leaves them to the reader. All the
more, because the proofs follow very easily by his methods.
Take, for the sake of example, the case of the equivalent of
JJ>'3 dy dx, for which I imagine Barrow's method would have
been somewhat as follows.

In order to find the aggregate of all the points of a given
space VDH, each multiplied by the cube of its distance from
the axis VD, we may proceed in two different ways.

Method i. — By applying lines PX, proportional to the
cube of BP, to every point P of the line BF, find the

"><

aggregate of the products for the line BF; then find the sum
of these aggregates for all the lines applied to VD. Here,
since PX is proportional to BP3, the curve BXX is a cubical
parabola, and the space BFX is one-fourth of the fourth
power of BF ; and Barrow would write the result of the
summation as the sum of AE4/4, BF4/4, CG4/4, etc. (applied
to VD).

Method 2. — Find the aggregates along all the parallels
to VD, and then the sum of these aggregates applied to DH.
Here the first aggregates are represented by rectangles whose
bases are IE, KF, etc., and whose heights are equal to Dl3,
DK3, etc., respectively ; and Barrow would write this as the
sum of Dl3 . IE, DK3 . KF, etc. (applied to DH).

Lastly, in fig. 125, ID. Dl3. IZ = VA.AE3.DI, etc.; hence
all the results follow immediately; i.e. fy^dxdy is equal
to either of the integrals jj'4/4 «&> ^xy3 dy; and Barrow
proves that these are equal to one-fourth of ^y*yl(dyldx) dy.

LECTURE XI

135

19. Again, let AM B be a curve of which the axis is AD
and let BD be perpendicular to AD; also let KZL be another
line such that, when any point M is taken in the curve AB,
and through it are drawn MT a tangent to the curve AB,
and MFZ parallel to DB, cutting KZ in Z and AD in F, and
R is a line of given length, TF : FM = R : FZ. Then the
space ADLK is equal to the rectangle contained by R
and DB.*

For, if DH = R and the rect-
angle BDHI is completed, and
MN is taken to be an indefinitely
small arc of the curve AB, and
MEX, NOS are drawn parallel to
AD ; then we have

NO: MO = TF:FM = R:FZ;

.-. NO.FZ = MO.R, and FG.FZ=ES.EX.
Hence, since the sum of such rectangles as FG . FZ differs
only in the least degree from the space ADLK, and the
rectangles E8 . EX form the rectangle DHIB, the theorem is
quite obvious.

20. With the same data, if the curve PYQ is such that
the ordinate EY along any line MX is equal to the corre-
sponding FZ ; then the sum of the squares on FZ (applied
to the line AD) is equal to the product of R and the
space DPQB.

21. Similarly, the sum of the cubes of FZ, applied to AD,
is equal to the product of R and the sum of the squares

T A

«.-- R --.»

D K/

S J

M N^

E/Y

B Q

Fig. 127.

This is the converse of Lect. X, § n.

136 BARROW'S GEOMETRICAL LECTURES

of EY, applied to BD; and so on in similar fashion for the
other powers.

22. Let DOK be any curve, D a fixed point in it, and DKE
a chord ; also let AFE be a curve such that, when any straight
line DMF is drawn cutting the curves in M and F, D8 is
drawn perpendicular to DM, MS is the tangent to the curve
DOK, cutting DS in 8, and R is any given straight line, then
DS : 2R = DM2 : DF2. Then the space ADE will be equal to
the rectangle contained by R and DK.*

23. The data and the construction being otherwise the
same, let KH and Ml be drawn perpendicular to the tangents
KT and MS, meeting DT, DS in H and I respectively; and
let AE be a curve such that DE = V(DK . DH), and also
DF = V(DM . Dl), and so on. Then the space ADE is
equal to one-fourth of the square on DK.

24. If DOK is any curve, D a given point on it, and DK
any chord : also if DZI is a curve such that, when any point
M is taken in the curve DOK, DM is joined, DS is drawn
perpendicular to DM, MS is a tangent to the curve, DP is
taken along DK equal to DM, and PZ is drawn perpendicular
to DK, then PZ is equal to DS; in this case the space DZI
is equal to twice the space DKOD.t

25. The data and the construction being in other respects
the same, let the ordinates PZ now be supposed to be equal

* The analytical equivalents are :

22. yVj2. dQ = 2R ./r2/(r2 . dO/dr) . dO = zftr.

23. frftz.dQ =fr/2.(dr/d6) . dQ = r2/4-

24. /r2 . dO = />2 . (de/dr) . dr.
f See note on page 138.

LECTURE XI 137

to the respective tangents MS ; take any straight line xk,
and distances along it equal to the arcs DOK, DOM, DON,
etc., and draw the ordinates kd, md, nd, etc., equal to the
chords KD, MD, ND, etc. ; then the space xkd will be equal
to the space OKI.

26. Moreover if, other things remaining the same, any
straight line kg is taken, the rectangle xkgh is completed, and
the curve DZI is supposed to be such that MD : DS = kg : PZ ;
then the rectangle xkgh will be equal to the space OKI.

Hence, if the space OKI is known, the quantity of the
curve DOK may be found.

Should anyone explore and investigate this mine, he will
find very many things of this kind. Let him do so who
must, or if it pleases him.

Perhaps at some time or other the following theorem,
too, deduced from what has gone before, will be of service ;
it has been so to me repeatedly.

27. Let VEH be any curve, whose axis is YD and base
DH, and let any straight line ET touch it; draw EA parallel
to HD. Also let GZZ be another curve such that, when any
straight line EZ is drawn from E parallel to VD, cutting the
base HD in I and the curve GZZ in Z, and a straight line of
given length R is taken, then at all times DA2 : R2= DT : IZ.
Then DA : AE = R2 : space DIZG ; (or, if DA : R is made equal
to R : DP, and PQ is drawn parallel to DH, then the rect-
angle DPQI is equal to the space DGZI). [Fig. 131, p. 139.]

The following theorem is also added for future use.

28. Let AMB be any curve whose axis is AD; also let the

138 BARROWS GEOMETRICAL LECTURES

line KZL be such that, if any point M is taken in AMB, and
from it are drawn a straight line MP perpendicular to AB
cutting AD in P, and a straight line MG perpendicular to AD
cutting the curve KZL in Z, at all times GM : PM is equal to
arc AM : GZ; then the space ADKL will be equal to half the
square on the arc AM.

These theorems, I say, may be obtained from what has
gone before without much difficulty ; indeed, it is sufficient
to mention them ; and, in fact, I intend here to stop for
a while.

NOTE

The theorems of §§ 24-28 deserve a little special notice.
The first of these was probably devised by Barrow for the
quadrature of the Spiral of Archimedes ; it included, as was
usual with him, "innumerable spirals of other kinds," thus
representing both, as Barrow would consider it, an improve-
ment and a generalization of Wallis' theorems on this spiral
in the " Arithmetic of Infinites."

It is readily seen that if DZI is a straight line, the curve
AOK is the first branch or turn of the Logarithmic or
Equiangular Spiral; if DZI is a parabola, the curve DOK is
the Circular Spiral or Spiral of Archimedes ; and if the
curve is any paraboliform, the curve DOK is a spiral whose
equation may be rm = A0". In short, Barrow has given a
general theorem to find the polar area of any curve whose
equation is 0 — \f(r)/r2. dr, for all cases in which he can
find the area under the curve y — f(x).

The theorem of § 26 is indeed remarkable, in that it is a
general theorem on rectification. It is stated * that Wallis
had shown, in 1659, that certain curves were capable of
rectification, that William Neil, in 1660, had rectified the
semi-cubical parabola, using • Wallis' method, that the second
curve to be rectified was the cycloid, and that this was

* Ency. Brit. (Times edition), Art. on Infinitesimal Calculus.
(Williamson). These dates are wrong, however, according to other
authorities, such as Rouse Ball.

LECTURE XI

139

effected by Sir C. Wren in 1673. Barrow's general theorem
includes as a special case, when the line DZI is a straight
line, whose equation is y= SJ^ . x, the curve DOK with the
relation ds\dr= ^2 . r, that is the triangle DMS is always
a right-angled isosceles triangle, and therefore the curve is
the Logarithmic or Equiangular Spiral, which may thus be
considered to be the real second curve that was rectified.
Even if not so, we shall find later that Barrow has anticipated
Wren in rectifying the cycloid, as a particular case of
another general theorem ; and in this case, he distinctly
remarks on the fact that he has done so. In general,
Barrow's theorem rectifies any curve whose equation is
6 = I v/(R2-r2)/r2. dr, where R=f(r), so long as he can

find the area under the curve y =f (x).

The theorem of § 27 is even more remarkable, not only
for the value of its equivalent, which is the differentiation
of a quotient, but also because it is a noteworthy example
of what I call Barrow's contributory negligence; for
"although he recognizes its value, and indeed states that
it has been of service to him " repeatedly " (and no wonder),
yet he thinks that " it is enough to mention it," and
omits the proof, which "may be obtained from what
has gone before without much difficulty." Even the figure
he gives is the worst possible to show the connection, as it
involves the consideration that the gradient is negative
when the angle of slope is obtuse. Of the figures below,
the one on the right-hand side is that given by Barrow ;

Fig. 1 31 A.

Fig. 131.

the proof, which Barrow omits, may be given as follows,
reference being made to the figure on the left-hand side.

HO BARROW'S GEOMETRICAL LECTURES

Let the curve VXY be such that, if EA produced meets it
in Y, then always EA : AD = AY : R. Divide the arc EV into
an infinite number of parts at F, L, . . . , and draw FBX,
LCX, . . . , parallel to HD, meeting VD in B, C, . . . , and the
curve VXY in the points X ; also draw FJZ, . . . , parallel to
YD, meeting HD in J, . . . , and the curve GZZ in the
points Z.

— P

Then AY . AD . BD = R . EA . BD = R . (EA . AD - EA . AB),
and BX . AD . BD = R . FB . AD = R . (EA . AD - IJ . AD) ;
hence, if XW is drawn parallel to VD, cutting AY in W, then
WY . AD2 = WY . AD . BD = R . (IJ . AD - EA . AB).

But EA:AT=IJ:AB, or EA . AB = IJ . AT ;
^WY.AD2 = R.(IJ.AD-IJ.AT) = R.IJ.DT.

Now DA2-.R2 = DT:IZ - IJ.DT:IJ.IZ;

R2:IJ.IZ = AD2:IJ.DT = R:WY.

Hence, since the sum of the rectangles IJ . IZ only differs
in the least degree from the space DGZI, and the sum of
the lengths WY is AY ; it follows immediately that
R2 : space DGZI = R : AY = DA : AE.

Now if DT and DH are taken as the co-ordinate axes, then
WY is the differential of AY or Ry/x, and DT = x-y. dx\dy ;
therefore the analytical equivalent of WY . AD2 = R . DT . IJ
is R . d(y\x) . x2 =- R.(x-y . dx\dy) . dy, or d(yjx) = (x . dy
- y . dx)jxz.

Barrow states it as a theorem in integration; but, if I
have correctly suggested his method of proof, he obtains
his theorem by the differentiation of yjx (see pages 94, 1 12).

APPENDIX

i. When many years ago I examined the Cydomctrica of
that illustrious man, Christianus Hugenius,* and studied it
closely, I observed that two methods of attack were more
especially used by him. In one of these, he showed that
the segment of a circle was a mean between two parabolic
segments, one inscribed and the other circumscribed, and
in this way he found limits to the magnitude of the former.
In the other, he showed that the centre of gravity of a cir-
cular segment was situated between the centres of gravity
of a parabolic segment and a parallelogram of equal altitude,
and hence found limits for this point. It occurred to me
that in place of the parabola in the first method, and of
the parallelogram in the second, some paraboliform curve
circumscribed to the circular segment could be substituted,
so that the matter might be considered somewhat more
closely. On examining it, I soon found that this was

* The work of Christiaan Huygens (1629-1695), the great Dutch mathe-
matician, astronomer, mechanician, and physicist, that is referred to may
be the essay Exetasis quadratures circuit (Leyden, 1651), but more
probably is the complete treatise De circuit magnitudine inventa, that
was published three years later. Putting the date of Barrow's study of
Huygens' work at not later than 1656 (note the words in the first line above
that I have set in italics — many years ago, — and remembering that this was
printed in 1670), it follows from Barrow's mention of the paraboliform
curve as something well known to him, and from a remark that the proofs
of the theorems of § 2 " may be deduced in various ways from what has
already been shown, without much difficulty," that Barrow was in
possession of his knowledge of the properties of his beloved paraboliforms
even before this date. Is it not therefore probable, nay almost certain,
that Barrow, in 1655 at the -very latest, had knowledge of his theorem
equivalent to the differentiation of a fractional power"}

142 BARROW'S GEOMETRICAL LECTURES

correct ; moreover, I easily found that like methods could
be used for the magnitude of a hyperbolic segment. As
the proofs for these theorems — better perhaps than others
that might be invented — are short, and clear (because they
follow from or depend on what has been shown above),
I thought good to set them forth in this place. I think,
too, that they are in other respects not without interest.

2. Let us assume the following as known theorems ; of
which the proofs may be deduced in various ways from
what has already been shown, without much difficulty.

If BAE is a paraboliform curve, whose
axis is AD and base or ordinate is BDE,
BT a tangent to it, and K the centre
of gravity ; then, if its exponent is n/m,
we have*

Area of BAE = m/(n + m) of AD . BE,

TD = m/n of AD,
and KD = m/(n + zm) of AD.

Fig. 133-

* The definition of Lect. VII, § 12, uses N/M ; the value of TD/AD is
found in Lect. IX, § 4 ; where also the definition of the paraboliforms is
given.

Now it is clear from the adjoining figure that
if AHLE is a paraboliform, whose exponent is
r/s( = i/a say), then LK/HK = a . LM/AM ; and
conversely.

Let AIFB be a curve such that

FM/R = LK/HK = a . LM/AM ;
then, by Lect. XI, § 19, area AFBD = R . DE.
But, in this case, we have

IG : FM = LM/AM - HN/AN : LM/AM

= AM . LK - LM . HK : LM . AN
= (a-i). LM . HK : LM . AN ;
FG/GI = i/(«-i)of AM/FM.

Hence AIFB is a paraboliform, whose vertex is A, axis AD, exponent
— i.

LECTURE XI— APPENDIX 143

3. Let AEB, AFB be any two curves, having the axis AD
and the ordinate BD common, so related that, if any straight
line EFG is drawn parallel to BD, cutting the given lines in
the points E, F, G, and E8, FT touch the curves AEB, AFB
respectively, TG is always greater than SG ; then I say that
no part of the curve AFB can fall within the curve AEB.

4. Let BAE be any curve, of which AD is the axis, and let
the base ADE be an ordinate to it; also let the point H be
the centre of gravity of the segment BAE, and RS a straight
line through it parallel to BE. Further let another curve
(or any line you please) MRASN pass through the points
R, S, and have the same axis AD; let it cut the first curve
BAE in such a manner that the upper part RKAPS falls
within the curve BAE, but the lower remaining parts, RM,
SN, fall outside it. Then the centre of gravity of the seg-
ment MRASN will be below the point H, that is, towards the
base MM.

5. Let the two straight lines BT, E8 touch the circle AEB,
whose centre is C, and meet the diameter CA in the points
T and S ; also let the straight lines BD, EP be perpendicular
to CA. Then, if AD > AP, TD : AD > SP : AP.

Conversely, if AIFB has an exponent n/m( = a- i), the integral curve is a
paraboiiform, exponent i/a or m/(n-t-m).

Hence, since DB/R = <z.DE/AD, area AIFBD = R. DE = ;«/(« + ;«) of AD.DB.

Similarly, area ALED = AD . DE -(« 4- #*)/(« + 2>«) of AD. DE,

= m/(n + zm) of AD . DE ;
R . a . area ALED : AD . area AIBD = n + m : n + zm.

Now, since FM/R = a . LM/AM, . '. FM . AM . MN = R . a . LM . HK ;
hence, summing, AK . area AFBD = R . a . area ALED ;
therefore AK : AD = n + m: n-\-2m, or KD = m\(n + 2tn) of AD.

In a similar way the centre of inertia could be found.

The proof could have been deduced from the note on § 2 of Lect. XI,
or by drawing a subsidiary curve as in the note to § 27 of Lect. XI.

144 BARROW'S GEOMETRICAL LECTURES

6. Let the two straight lines BT, ES now touch a hyper-
bola AEB, whose centre is C; and let other things be the
same as in the theorem just before ; then TD : AD > SP : AP.

7. Let the axis AD and the base BD be common to the
circle AEB whose centre is C, and the paraboliform AFB ;
also let the exponent of the paraboliform be nfm, where

AD = (m - 2n)l(m - n) of CA,
or m - n : m - zn = CA : AD.

Moreover, let the straight line BT touch the circle ; then BT
will touch the paraboliform also.

8. It should be noted in this connection that, conversely,
if the ratio of AD to CA is given, the paraboliform which
touches the circle AEB at B is thereby determined.

For instance, if AD/CA = s\tt then (t-s)f(2t-s) will be
the exponent of the required paraboliform.

9. With the same hypothesis as in § 7, the paraboliform
AFB will fall altogether outside the circle AEB.

10. Again with the same hypothesis, if with a base GE
(any parallel to BD) and axis AD another paraboliform is
supposed to be drawn, of the same kind as AFB (or having
the same exponent n/m) ; then this curve also, for the part
AE above GE, will fall altogether outside the circle.

11. Also it may be shown that the said paraboliform (of
like kind to AFB and constructed on the base GE), when
produced below GE to DB, will fall altogether within the
circle as regards this part.

12. Further, let AD be the axis and DB the base common

LECTURE XI— APPENDIX 145

to the hyperbola AEB whose centre is C, and the para-
boliform AFB, whose exponent is n/m; also let AD =
(zn - m)l(m - n) of CA ; and let BT touch the hyperbola.
Then BT touches the paraboliform also.

13. Hence again, if the ratio of AD to CA is given,
the paraboliform touching the hyperbola at the point B
is thereby determined. For instance, if AD/CA = s/t,
n\m = (f + s}/(2t + s).

14. With the same hypothesis as in § 12, the paraboliform
AFB will lie altogether within the hyperbola AEB.

15. Also, with the same hypothesis, if you imagine a
paraboliform of the same kind to be constructed with the
base GE and axis AG ; it will fall within the hyperbola on
the upper side of G E.

1 6. Moreover, if this second like paraboliform, constructed
on the base EG, is supposed to be produced to DB; then
the part of it intercepted by EG and BD will fall altogether
outside the hyperbola.

17. Let the circle AEB and the parabola AFB have a
common axis AD and base BD; then the parabola will fall
within the circle on the side above BD, and without the
circle below BD.

If an ellipse is substituted for the circle, the same result
holds and is proved in like manner.

1 8. Let the hyperbola whose axis is AZ and parameter
AH, and the parabola AFB have the same axis AD and base
BD ; then the parabola will fall altogether outside the

H6 BARROWS GEOMETRICAL LECTURES

hyperbola above BD, but within it when produced
below BD.

19. From what has been said, the following rules for the
mensuration of the circle may be obtained.

Let BAE be a part of a circle, of which the axis is
AD, and the base BE; let C be the centre of the circle, and
EH equal to the right sine of the arc BAE ; also let
AD : CA = s : t.

Then (i) (it- s)l($t - zs) of AD . BE > segment BAE;

(2) EH + (4/ - 2j)/(3/ - 2j) of BH > arc BAE ;

(3) 2/3 of AD . BE < segment BAE ;

(4) EH + 4/3 of BH < arc BAE.

20. Similarly, the following rules for the mensuration of
the hyperbola may be deduced.

Let ADB be a segment of a hyperbola [Barrow's figure is
really half a segment], whose centre is C, axis AD, and base
DB ; and let AD : CA = s \ /.

Then (i) (z/ + j)/(3/+zj) of AD . DB < segment ADB;
(2) 2/3 of AD . DB > segment ADB.

NOTE

The results of §§ 19, 20, for which Barrow omits any hint
as to proof, are thus obtained.

§ 19 (i) A paraboliform whose exponent is (t-s)J(2t-s)
can be drawn, touching the circle BAE at B, A, and E, and
lying completely outside it ; the area of it cut off by the
chord BE is, by § u, equal to (zt - 2s)/(^-2s) of AD . BE.
(3) A parabola is a paraboliform whose exponent is 1/2, and
the area of the segment is 2/3 of AD . BE. (2) and (4)
follow from (i) and (3) by using obvious relations for the
circle, and are not obtained independently. This explains

LECTURE XI— APPENDIX 147

why there are only two formulas given for the hyperbola, and
these are formulae for the segment; for there are no
corresponding simple relations for the hyperbola that
connect the sector or segment with the arc.

§ 20. In a similar way, the two limits for the hyperbolic
segment are obtained from a paraboliform whose exponent
is (/ + j)/(2/ + .y), and a parabola.

The formulae of (i) and (2) for the circle reduce to the
trigonometrical equivalent a < sin a . (2 + cos a)/(i + 2 cos a)
in which the error is approximately o,5/45 ; the formulae
of (3) and (4) reduce to the much less exact equivalent
a > sin a. (2 -\-cos a)/3, where a is the half-angle. Thus
Barrow's formula is a slightly more exact approximation
than that of Snellius, namely, 3 sin 20/2(2 +cos za), where
the error is approximately 4a5/45, and is in defect ; Barrow,
in § 29, obtains Snellius' formula in the more approxi-
mate form $sinal(z+cosa). Hence Barrow's formula and
Snellius' formulae give together good upper and lower
limits to the value of the circular measure of an angle.
The equivalent to the first formula for the hyperbola is
sin~l (tan a) > 3 tan a/( i + 2 cos a) ; the error being again of
the order a5.

21. Further, let BAE be the segment of a circle whose
centre is C, axis AD, centre of gravity K ; also let AD : CA =
s : /, and HD : AD= zt-s : $f- zs ; then HD will be greater
than KD.*

22. Let the point L be the centre of gravity of the
parabola (such as was discussed in § 18); then L will be
below K ; i.e. KD is greater than two-fifths of AD.*

23. Let BAE be a segment of a hyperbola whose centre
is C, axis AD, base BE, and centre of gravity K; also let
AD: CA = J:/, and HD : AD = 2t + s: $t+zs • then HD is less
than KD.*

* See note at foot of next page.

148 BARROW'S GEOMETRICAL LECTURES

24. The centre of gravity of the parabola, L say, lies
above K ; i.e. KD is less than two-fifths of AD.*

25. Lest the present method of research, owing to the
great number of methods of this kind for measuring the
circle, may seem to be of little account, we will add one or
two riders (if only for the sake of these, the few theorems
given would deserve employment) ; from which indeed
Maxima and Minima of things of a kind may be determined
in a great number of cases.

Let ABZ be a semicircle whose centre is C ; also let ADB
be a segment; and to this let a paraboliform AFB be
adscribed, whose exponent is n/tn, where AD : CA =
m - 2« : /;/ - n.

If the parameter of the paraboliform t (that is a straight
line such that some power of it multiplied by a power of
the axis of the segment, AD say, produces a power of the
ordinate, DB say) is / ; then / will be a maximum of its
kind.

For, if any straight line GE is drawn parallel to DB, and
a paraboliform of like kind to AFB is supposed to be
applied to GE, of which the parameter is called q; then,
since the paraboliform AFB touches the circle externally,
GF > GE. /. GFm > GE"1, or /»-" . AG" > qm~n . AG",
.-.p>g.

It shouW be observed that /""-"> = ZDm . AD"'-2" and

* These limits are not remarkable for close approximation unless the
segment is very shallow. Thus if the arc is one-third of the circumference,
the limits for the circle are only 2AD/5 and 3AD/5-

f It is to be observed that Barrow here indicates that the equation to the
paraboliform is, in general y = axm/n.

LECTURE XI— APPENDIX 149

^(«-n> = Z0m _ AGro-2n, hence ZD"1 . AD"'-*1 > ZGm . AGm-2n ;
that is ZDm . ADm~2n is a maximum.*

Example i. Let n = i, and w = 3; then/4 = ZD3 . AD =
ZD2 . BD2, or pz = ZD . BD; and AD = CA/2.

Example 2. Let n = 3, and w = 10; then/14 = ZD10 . AD4,
or ? = ZD5 . AD2 = ZD3 . BD4, and AD = 4CA/7.

26. Again, let AEB be an equilateral hyperbola whose
centre is C, and axis ZA ; and to it let a paraboliform AFB,
whose exponent is n/»i and parameter /, be adscribed
(with a base DB) ; also suppose that AD : CA = 2n - m :m-n;
then p will be a minimum of its kind.

It is to be noted that /2(m-"» = ZD'" . AD2"~™, and also
(as in § 25) ?2('"-n) = ZG'" . AG"'-m ; hence ZDm . AD2"-m is
a minimum.t

As in the preceding I have touched upon the mensura-
tion of the circle, what if I add incidentally a few theorems
bearing upon it. which I have by me ? The following general
theorem must, however, be given as a preliminary.

27. Let AGB be any curve whose axis is AD, and let the
straight lines BD, GE be ordinates to it. Then the arc AB
will bear a greater ratio to the arc AG than the straight line
BD to the straight line GE.

* This is equivalent to the algebraical theorem that, if x+y = a constant
then X? . ys is a maximum when xjr = y/s.

•f This is equivalent to the algebraical theorem that, if x—y = a constant,
then xrlys is a minimum when xjr = y/s.

The proof of this theorem is generally ascribed to Ricci, who proved if
algebraically in 1666, and used it to draw the tangent to the general
paraboliform ; thus we see that Barrow's proof was independent of Ricci,
even if Barrow had not discovered it before Ricci ; cf. " many years ago."

ISO BARROW'S GEOMETRICAL LECTURES

28. Let AMB be a circle, of which the radius is CA, and
let DBE be a straight line perpendicular to CA ; also let ANE
be a curve such that, when any straight line PMN is drawn
parallel to DE, cutting the circle in M and the curve in N,
the straight line PN is equal to the arc AM. Then the para-
bola described with axis AD and base DE will fall altogether
outside the curve ANE.

29. From the preceding, and from what is commonly
known about the dimensions of the spaces ADB, ADE,* the
following formula may be easily obtained :

3CA.DB/(2CA + CD) < arc AB.

Further, if the arc AB is supposed to be one of 30
degrees, and 2CA = 113, then the whole circumference,
calculated by this formula, will prove to be greater than
355 less a fraction of unity.

30. Hence also, being given the arc AB, let arc AB = /,
CA = r, and DB = e ; then the following equation may be
used to find the right sine DB :

or, substituting k for $r^/(c)r2 +/2), we have

kp = 4&? -e2 ; or zk - ^/(4/fc2 - kp} = e.

3 i. Let AMB be a circle whose radius is CA, and let the
straight line DBE be perpendicular to CA ; let also the curve
ANE be a part of the cycloid pertaining to the circle AMB;
and lastly let a parabola AOE be drawn with axis AD and
base DE. Then the parabola will fall altogether within the
cycloid.

* See note at the end of this lecture.

LECTURE XI— APPENDIX 151

32. From the preceding, and from what is generally
known about the dimensions of circles and cycloids,* the
following formula may be obtained;

(2CA . DB + CD . DB)/(CA + 2CD) > arc AB.

Further, if the arc AB is one of 30 degrees, and 2CA =113,
it may be shown by this formula that the whole cir-
cumference is less than 355 plus a fraction.

You see then that, from the two formulae stated, there
results immediately the proportion of the diameter to the
circumference as given by Metius.

33. Since in this straying from the track, the cycloid
has brought itself under notice, I will add the following
theorem ; — I am not aware that it has been anywhere
observed by those who have written so profusely on the
cycloid.

If the space ADEG is completed (in § 31), the space AEG
will be equal to the circular segment ADB.

The proof I shall leave out, nor shall I wander further
from my subject.

34. Let two circles AIMG, AKNH touch one another at A,
and have a common diameter AHG ; and let any straight
line DNM be drawn perpendicular to AHG. Then the
segment AIMD will bear to the segment AKNH a less ratio
than the straight line DM to the straight line DN.

35. Let YFZT be an ellipse, of which YZ and HT are the
conjugate axes ; and let the straight line DC be parallel to

* See note at the end of this lecture.

152 BARROW'S GEOMETRICAL LECTURES

the major axis YZ, and let the circle DFCV, whose centre is
K a point on the minor axis FT, pass through the points
D, F, C; then I say that the part DOFPC of the circle will
lie within the part DMFNC of the ellipse.

36. Let DEC be a segment of a circle whose centre is L;
and, any point F being taken in its axis GE, let DMFC be a
curve such that, when any straight line RMS is drawn
parallel to GE, RS:RM = GE:GF; then DMFC is an ellipse
thus determined :— Find H, such that EG : FG = GL:GH;
through H draw YHZ parallel to DC, and let HY equal LE ;
then HY, HF are the semi-axes of the ellipse.

This is held to have been proved by Gregory St Vincent,
Book IV, Prop. 154.

COROLLARY. — Hence, segment DEC : segment DMFC
= EG : FG.

37. Let DEC, DOFC be portions of two circles having a
common chord DC and axis GFE; then the greater portion
DEC will bear to the portion DOFC a greater ratio than that
which the axis GE bears to the axis GF.

•

NOTE

In § 29, Barrow gives no indication of the source of the
" known dimensions," and there is also probably a misprint ;
for the "spaces ADB, ADE," we should read the "spaces
AN ED, AOED," unless Barrow intended ADE to stand for
both the latter spaces. If so, we have from § 2, area
AOED = 2/3 of AD . DE, and the area of AN ED can thus be
found by Barrow's methods : —

Complete the rectangle EDCF, and draw QRS parallel and
indefinitely near to PMN; draw SVZ parallel to AC, cutting

LECTURE XI— APPENDIX

153

/ TQ^^s3*""""

2Jjj_

R \

=• Z Y

Fig. 151.

PN, CF in V, Z, and RT parallel to AC, cutting PN, CF in
T, Y; then we have CP : CM = MT : MR = MT : NV,

.-. CP.NV = CM. MT. A

Hence area AN ED + CD . DE

= the sum of VN . CP+ . . .

= the sum of CM . MT+ . . .

= CM .(the sum of MT+ . . .)

= CM.BD;

Area AN ED = CA . BD - CD . DE.

.-. CA . BD - CD . DE < area AOED

< 2/3 of AD. DE,
.-. 3CA. BD< (3CD + 2AD). DE

or (2CA + CD).arc AB.

It should be observed that the equivalent of the expres-
sion for the area A NED is

Ja cos~l x/a dx = x cos~l x/a - a . <J(a2 - xz).

. The formula finally obtained by Barrow, if we put 2<£ for
the angle subtended at the centre of the circle by the arc
AB, reduces to 2</> > 3 sin 2</>/(2 +cos 2$), which is the
formula of Snellius ; this, as I have already noted, has an
error of the order <£5 ; a handier result is obtained by
taking a = z</>, when it becomes a > 3 sin a/(2 +cos a).

For § 32, since MN (of this theorem) = arc AM = PN (in
fig. 151); hence
area of cycloid = area of AM BD + area of AN ED (fig. 151)

= (CA . arc AB - CD . BD)/2 + CA . BD - CD . arc AB.
The remark made by Barrow in § 33 indicates with
almost certainty that the above was his method for the
cycloid.

Now, since the area of the cycloid is less than the area
of the corresponding parabola, which is 2/3 of AD . DE or
2/3 of AD . (DB + arc AB) ; hence we obtain

arc AB < (2CA . DB + CD . DB)/(CA + 2CD).

This is equivalent to a < sin 0(2 +cos a)/(i + 2 cos a), a
limit obtained before in § 19. Thus Barrow has here two
very close limits, one in excess and the other in defect,
each having an error of the order a5.

154 BARROWS GEOMETRICAL LECTURES

The results obtained, by the use of these approximate
formulae, with the convenient angle of 30 degrees are in
fact 355 '6 and 354'8. The formula obtained by "Adrian,
the son of Anthony, a native of Metz (1527), and father of
the better known Adrian Metius of Alkmaar " is one of the
most remarkable " lucky shots " in mathematics. By con-
sidering polygons of 96 sides, Metius obtained the limits
SrW an^ 3TiiV» and then added numerators and de-
nominators to obtain his result 3^\2g- or ZTI^ ' • •

Barrow seems to be content, as usual, with giving the
geometrical proof of the formula obtained by Metius ; which
must have appeared atrocious to him as regards the method
by means of which the final result was obtained from the
two limits. If only Barrow had not had such a distaste for
long calculations, such as that by which Briggs found the
logarithm of 2 (he extracted the square root of 1-024 forty-
seven times successively and worked with over thirty-five
places of decimals), it would seem to be impossible that
Barrow should not have had his name mentioned with that
of Vieta and Van Ceulen and others as one of the great
computers of TT, For he here gives both an upper and a
lower limit, and therefore he is oply barred by the size of
the angle for which he can determine the chord. Now, he
would certainly know the work of Vieta ; and this would
suggest to him that a suitable angle for his formulae would
be 7r/2n, where n was taken sufficiently large. Por Vieta's
work would at once lead him to the formulae

2 COS 7T/2" = (V/[2+N/{2+\/(2+ )}]>

2 Sin 7T/2" = ^[2 - N/{2 +^(2 + )}],

where there are n — i root extractions in each case.

If, then, he took n to be 48, his angle would be less than
i/24G, and the error in his values would be less than i/2234;
this is about io~45; hence Barrow has practically in his
hands the calculation of IT to as many decimal places as the
number of square root extractions he has the patience to
perform and the number of decimal places that he is willing
to use.

LECTURE XII

General theorems on Rectification.

GENERAL FOREWORD. — We will now proceed with the
matter in hand ; and in order that we may as far as
possible save time and words, it is to be observed every-
where in what now follows that AB is some curved line,
such as we shall draw, of which the axis is AD ; to this

Q C

Fig. 156.

Fig- 157.

axis all the straight lines BD, CA, MF, NG are applied
perpendicular; the arc MN is indefinitely small; the
straight line a/3 = arc AB, the straight line ap. = arc AM, and
pv = arc MN ; also lines applied to a/3 are perpendicular
to it. On this understanding,

i. Let MP be perpendicular to the curve AB, and the

156 BARROW'S GEOMETRICAL LECTURES

lines KZL, a<f>S such that FZ = MP and /x<£ = MF. Then the
spaces a/38, ADLK are equal.*

2. Hence, if the curve AMB is rotated about the axis AD,
the ratio of the surface produced to the space ADLK is that
of a circumference of a circle to its diameter; whence, if
the space ADLK is known, the said surface is known.

Some time ago we assigned the reason why this was so.

N/"

^^^X. H

"">

F Y \v

D XV

S \

: 1L *

P-

. — - —
<r

Fig. 156.

3. Hence the surfaces of the sphere, both the spheroids,
and the conoids receive measurement.! For, if AD is the
axis of the conic section from which these figures arise,
there always exists some one line of the conies, KZL, that
can be found without much difficulty. I merely state this,
for it is now considered as common knowledge.

4. With the same hypothesis, let AYI be a curve such
that the ordinate FY is a mean proportional between the
corresponding FM, FZ. Then the solid formed by the
rotation of the space a8fi about the axis a/3 will be equal

* The equivalent is yds =y. (dsldx)dx.

f For the circle, the figure ADLK is a rectangle ; and the area of a zone
is immediately deducible ; and so on.

LECTURE XII 157

to the solid formed by the rotation of the space ADI about
the axis AD.

5. By similar reasoning, it may be deduced that, if FY
is supposed to be a bimedian between FM and FZ, the sum
of the cubes of the applied lines, such as /*</>, from the
curve a</>8, to the straight line aft is equal to the sum of
the cubes on the lines applied to the straight line AD from
the curve AY I. Similarly, the theorem holds for other
powers.

6. Further, with the same hypothesis, let the curve VXO
be such that EX = MP ; and let the curve -nfy be such that
/^ = PF. Then the space amp ft = the space DVOB.

7. Observe also that, if the curve AB is a parabola,
whose axis is AD and parameter R ; then the curve VXO
will be a hyperbola, whose centre is D, semi-axis DV, and
the parameter of this axis equal to R. Also the space
a(3(j/ir will be a rectangle. Hence it follows that, being
given the hyperbolic space DVOB, is to be given the curve
A MB, and vice-versa. All this is remarked incidentally.*

8. It should also be possible to observe that all the
squares on the lines applied to the straight line a/>, taken
together, from the curve TT|^, are equal to all the rectangles
such as PF. EX, applied to the line DB (or calculated) ; the
cubes on /A£ are equal to the sum of PF2 . EX, etc. : and
so on.

* Yet it has an important significance ; for it is the first indication that
Barrow is seeking the connection between the problem of the rectification
of the parabola and that of the quadrature of the hyperbola. He is not
quite satisfied with this result, but finally succeeds in § 20, Ex. 3.

158 BARROW'S GEOMETRICAL LECTURES

9. Also it may be noted that, PMQ being produced, if FZ
is supposed to be equal to PQ, and /*<£ to AQ ; then the
space a/38 is equal to the space ADLK.

10. Further, let the straight line MT touch the curve AB,
and let the curves DXO, a<£8 be such that EX = MT and
p.(f> = MF. Then the space a/3S is equal to the space DXOB.

Fig. 158. Fig. 159.

11. Hence again, the surface of the solid formed by the
rotation of the space ABD about the axis AD bears to the
space A DOB the ratio of the circumference of a circle to
its radius; therefore, if one is known, the other becomes
known at the same time.

Hence again one may measure the surfaces of spheroids
and conoids.

12. If the line DYI is such that EY2 = EX. MP; then the
solid formed by the rotation of the space a/38 about the
axis a/3 is equal to the solid formed by the rotation of the
space DBI about the axis IB.

13. By similar reasoning, one may compare the sums of
the cubes and other powers of the ordinates with spaces
computed to the straight line DB.

LECTURE XII

159

14. Moreover, let the lines AZK, a^ be such that
FZ = MT and //.£ = TF ; then the space a/ty will be equal
to ADK.

15. Also the sum of the squares on the applied lines
//.£ will be equal to the sum of the rectangles TF . FZ ; the
sum of the cubes on ft£ to the sum of TQ . FZ, . . . (con-
sidering them to be computed to the straight line AD); and
so on for the other powers.

1 6. Again let the straight line QMP be perpendicular to
the curve A MB; and let /38=BD; complete the rectangle
u/3S£; then let the curve KZL be such that FZ = QP. Then
the rectangle a/38£ is equal to the space ADLK.

Therefore, if the space AKLD is known, the quantity of
the curve AMB is also known.

Fig. 1 60.

17. Also, let the straight line TMY be supposed to touch
the curve AMB, and let fty be made equal to BC, and the
rectangle aftyij/ be completed ; let then the curve OXX be
such that FX = TY. Then the space ADOXX . . .—in-
definitely continued — will be equal to the rectangle a/?yi^.

Hence, again, if the space ADOXX . . . has been ascer-
tained, then the curve AMB becomes known.

160 BARROW'S GEOMETRICAL LECTURES

1 8. Moreover, if any determinate length R is taken, and
(38 is taken equal to R ; and if the curve OXX is such that
MF:MP = R to FX; then the rectangle a/3S£ will be equal
to the space ADOXX. Also, if this space is found, the curve
is forthwith known.

Many other theorems like this could be set down ; but
I fear that these may already appear more than sufficient.

19. It should be observed, however, that all these
theorems are equally true, and can be proved in exactly
the same way if the curve AMB is convex to the straight
line AD.

20. Also, from what has been shown, an easy method of
drawing curves (theoretically) is obtained, such as admit
of measurement of some sort; in fact, you may proceed
thus: —

Take as you may any right-angled trapezial area (of
which you have sufficient knowledge), bounded by two
parallel straight lines AK, DL, a straight line AD, and any
line KL whatever; to this let another such area A DEC be
so related that, when any straight line FH is drawn parallel

E L

Fig. 162.

to DL, cutting the lines AD, CE, KL in the points F, G, H,
and some determinate straight line Z is taken, then the

LECTURE XII 161

square on FH is equal to the squares on FG and Z. More-
over, let the curve AIB be such that, if the straight line GFI
is produced to meet it, the rectangle contained by Z and
Fl is equal to the space AFGC ; then the rectangle contained
by Z and the curve AB is equal to the space ADLK. The
method is just the same, even if the straight line AK is
supposed to be infinite.

Example i. Let KL be a straight line, then the curve
CGE is a hyperbola. (Fig. 162.)

Example 2. Let the line KL be the arc of a circle whose
centre is D, and let AK = Z; then the curve AGE will
be a circle; and the arc AB = AD/2 + (DL/2AK) . arc KL
(Fig. 163.)

Example 3. Let the line KL be an
equilateral hyperbola, of which the ,

centre is A, and the axis AK = Z:

B D EL

then CGE will be a straight line, and F< ,

the curve AB a parabola.

Example 4. Let the line KL be a parabola, of which the
axis is AD; then the line CGE will also be a parabola, and
the curve AB one of the paraboliforms.

Example 5. Let the curve KL be an inverse or infinite
paraboliform (for instance, such that FH2 = Z3/AF); then
the curve AB will be a cycloid, pertaining to the circle
whose diameter is equal to Z. (See figure on page 164.)

Per,haps, if you consider, you may think of some examples

that are neater than these.

162 BARROW'S GEOMETRICAL LECTURES

NOTE

The chief interest in the foregoing theorems lies in the
last of all. The others are mainly theorems on the change
of the variable in integration (or rather that the equality
(Sz/S#) . 8y = (SyjBx) . 8z holds true in the limiting form for
the purposes of integration, although of course Barrow
does not use Leibniz' symbols) ; and secondly, the appli-
cation of this principle to obtain general theorems on the
rectification of curves, by a conversion to a quadrature.
It must be borne in mind that Barrow's sole aim, expressly
stated, was to obtain general theorems ; and that he merely
introduced the cases of the well-known curves as examples
of his theorems ; and to obtain the gradient of the tangent
of a curve in general is the foundation of the differential
calculus.

In 1659, Wallis showed that certain curves were capable
of "rectification " ; the first-fruits of this was the rectification
of the semi-cubical parabola by William Neil in 1660, by
the use of Wallis' method. Almost simultaneously this
curve was also rectified by Van Huraet (see Williamson's
Int. Cal., p. 249) by the use of the geometrical theorem : —
" Produce each ordinate of the curve to be rectified until
the whole length is in a constant ratio to the corresponding
normal divided by the old ordinate, then the locus of the
extremity of the ordinate so produced is a curve whose
area is in a constant ratio to the length of the given curve."
Now this theorem is identical with the theorem of § 18;
hence, remembering that the semi-cubical parabola, whose
equation is R .jv2 = x3, is one of those paraboliforms of which
Barrow is so fond, and for which, as we have seen, he could
find both the tangent at any point and the area under the
curve between any two ordinates, noting also the examples
given to the theorem of § 20, it is beyond all doubt that
Barrow must have perceived that for this particular para-
boliform his curve OXX . • • (fig. 160) was the parabola
4j'2 = R. (9^ + 4R). Why then did not Barrow give the
result? The answer, I think, is given in his own remark
before Problem IX in Appendix III, "/ do not like to put
my sickle into another mans harvest" where he refers to the
work of James Gregory on involute and evolute figures.

LECTURE XII 163

Moreover, this supposition may set a date to this section,
namely not before 1659, and not very much later than
1 66 1. For from his opening remarks to the Appendix to
Lect. XI, we can gather that it was Barrow's habit to read
the work of his contemporaries as soon as he could get
them, and try to "go one better," and there are indications
enough in this section to show that Barrow was trying to
follow up the line given by § 7, to obtain the reduction of
the problem of the rectification of the parabola (and prob-
ably all the paraboliforms in general as well) to a quadra-
ture of some other curve; we see, for instance, that he
obtains the connection between a parabolic ' arc and a
hyperbolic area in § 7, and this connection is obtained
in several other places by different methods. He also
seeks general theorems in which the quadrature belongs to
one of the paraboliforms or the hyperboliforms (curves that
can be obtained from a rectangular hyperbola in the same
way as the paraboliforms are obtained from a straight line
in Lect. IX, § 4) ; and the result of using these curves,
whose general equation is ymxn = Rm+", is seen in § 20,
Ex. 5, where he takes m = 2, and «=i, and the derived
curve is the Cycloid. He does not state that thus he has
rectified the cycloid, apparently because in Prob. i, Ex. 2
of App. Ill, he has obtained it in a much simpler manner
as a special case of another general theorem. (See critical
note that follows this problem.)

The great interest, however, of this section centres in the
question of the manner in which Barrow obtained the
construction for § 20. There is nothing leading to it in any
theorem that has gone before it in the section; the only
case in which he has used the construction of a subsidiary
curve, such that the difference of the squares on the
ordinates of the two curves is constant, is in Lect. VI,
§§ 22, 23, and then his original curve is a straight line. The
only conclusion that I can come to is that he uses his
general theorem on rectification (Lect. X, § 5) analytically
thus : —

If Z . (dS/dx) — y, where 8 is the arc of the curve to be
rectified, and Y its ordinate, we must have Z . (dYJdx) equal
to -J(y* - Z2), and therefore Z . Y = J^/(jF2 - 22)dx. The
given construction is an immediate consequence.

164 BARROWS GEOMETRICAL LECTURES

Of course Barrow knew nothing about the notation
or JVO^-Zfyfoj his work would have dealt with
small finite arcs and lines ; but the pervading idea is better
represented for argument's sake by the use of Leibniz'
notation. I suggest that Barrow's proof would have run in
something like the following form : —

Draw JPQR parallel and very
near to IFGH, cutting the curves
as shown in the adjoining diagram,
and draw JT perpendicular to IH ;
then

E L

Z . IT = area PFGQ = PF . FG ;

.'. Z2.IJ2 = Z2.IT2 + Z2.TJ2 = PF2.FG2 + Z2.PF2;

.-. Z.IJ = PF.FH.
Hence, summing, Z . arc AB = area ADLK.

That Barrow had, in § 20, Ex. 5, really rectified the
cycloid is easily seen from the adjoining diagram. Barrow
starts^ I suppose, with the property of the cycloid that, if
IT, IM are the tangent and normal at I, then TM is per-
pendicular to BD. Let AD = Z,
then since Z . IT/TN = FH,* we have
FH2 = Z2. IT2/TN2 = Z2 . TM . TN/TN2
= Z2.TM/TN = Z3/AF.

The area under the curve KHL is
given as proportional to the ordi- B M
nate of what I may call its integral
curve (see note to Lect. XI, § 2), and is easily shown to
be 2AF . FH.

Hence arc Al = areaAFHK/Z = 2AF . FH/Z = 2IT; that is,
equal to twice the chord of the circle parallel to Tl, which
is also equal to it.

* This follows at once from the figure at the top of the page ; for,
Z . IJ = PF . FH, and IT : TN (in the lower figure) is equal to IJ : JT (in the
upper figure) ; and this is equal to IJ : PF or FH : Z ; hence, in lower figure
IT : TN = FH : Z.

APPENDIX I

Standard forms for integration of circular functions by
reduction to the quadrature of a hyperbola.

Here, although it is beyond the original intention to touch
on particular theorems in this work ; * and indeed to build
up these general theorems with such corollaries would tend
to swell the volume beyond measure; yet, to please a friend
who thinks them worth the trouble* I add a few observa-
tions on tangents and secants of a circle, most of which
follow from what has already been set forth.

s
I/

LX^

H- /

Fig. 167.

Fig. 1 66.

GENERAL FOREWORD. — Let ACB be a quadrant of a
circle, and let AH, BG be tangents to it; in HA, AC pro-
duced take AK, CE each equal to the radius CA ; let the

* Observe the words of the opening paragraph which I have italicized.

1 66 BARROWS GEOMETRTCAL LECTURES

hyperbola KZZ be described through K, with asymptotes
AC, CZ ; and let the hyberbola LEO be described through
E, with asymptotes BC, BG.

Also let an arbitrary point M be taken in the arc AB,
and through it draw CMS cutting the tangent AH in 8, MT
touching the circle, MFZ parallel to BC, and MPL parallel
to AC. Lastly, let a/3 = arc AB, ap. = arc AM ; let the
straight lines ay, £/wn/r be perpendicular to aft; and let
ay = AC, tf = AB, fuff = CS, and par = MP.

1. The straight line CS is equal to FZ; thus the sum of
the secants belonging to the arc AM, applied to the line AC,

s equal to the hyperbolic space AFZK.

2. The space a/if, that is, the sum of the tangents to the
arc AM, applied to the line a/t, is equal to the hyperbolic
space AFZK.

3. Let the curve AXX be such that PX is equal to the
secant CS or CT; then the space ACPX, that is the sum of
the secants belonging to the arc AM, applied to the line CB,
is twice the sector ACM.

LECTURE XII— APPENDIX I 167

4. Let CVV be a curve such that PV is equal to the
tangent AS; then the space CVP, that is, the sum of the
tangents belonging to the arc AM, applied to the straight
line CB, is equal to half the square on the chord AM.*

5. Let CQ be taken equal to CP, and QO be drawn
parallel to CE, meeting the hyperbola LEO in 0; then the
hyperbolic space PLOQ multiplied by the radius CB (or the
cylinder on the base PLOQ of height CB) is double the sum
of the squares on the straight lines CS or PX, belonging to
the arc AM, and applied to the straight line CB.*

6. Hence the space ay^/t, that is, the sum of the secants
of the arc AM applied to the line a(3, is equal to half the
hyperbolic space PLOQ.*

7. All the squares on the straight lines p.\{/, applied to
O.JJL, are equal to CA . CP . PX, that is, equal to the parallele-
piped on the rectangular base A PCD whose altitude is CS.

8. Let the curve AYY be such that FY = AS; then, if
a straight line Yl is drawn parallel to AC, the space ACIYYA
(that is, the sum of the tangents belonging to the arc AM,
applied to the straight line AC, together with the rectangle
FCIY) is equal to half the hyperbolic space PLOQ.*

9. Let ERK be an equilateral hyperbola (that is, one
having equal axes), and let the axes be CED, Cl ; also let
Kl, KD be ordinates to these; let EVY be a curve such that,

* These theorems are not at first sight of any great interest ; they
appear only to be a record of Barrow's attempts to connect the quadrature
of the hyperbola in some way with the circle. But later, when we find
that Barrow has the area under the hyperbola, their importance becomes
obvious. (See critical note following App. Ill, Probs. 3, 4.)

1 68 BARROW'S GEOMETRICAL LECTURES

when any point R is taken at random on the hyperbola,
and a straight line RVS is drawn parallel to DC, then SR,
CE, SV are in continued proportion ; join CK ; then the
space CEYI will be double the hyperbolic sector DOE.

10. Returning now to the circular quadrant ACB, let
CE = CA ; and with axis AE, and parameter also equal to
AE, let the hyperbola EKK be described ; now let the curve
AYY be supposed to be such that, when any ordinate MFY
is drawn, FY is equal to the tangent AS ; draw YIK, cutting
CZ in I and the hyperbola in K, and join CK, then the space
ACIYA is double the hyperbolic sector ECK.

1 1. COROLLARY. — Hence, if with pole E, a chord CB, and
a sagitta CA, a conchoid AW is described ; and if YFM
produced meets it in V ; then MV = FY ; and thus the
space AMV is equal to the space AFY.

12. Whence the dimensions of conchoidal spaces of this
kind become known.

13. Let AE be a straight line perpendicular to RS (cutting
it in C); and let CE = CA ; let AZZ, EYY be two conchoids,
conjugate to one another, described with the same pole E
and a common chord RS; from E draw any straight line
EYZ, cutting EYY, AZZ, RS in the points Y, Z, I ; also let
EKK be an equilateral hyperbola, with centre C and semi-
axis CE ; draw IK parallel to AE and join CK.

Then the four-sided space, bounded by AE, YZ, and the
conchoidal arcs EY, AZ is equal to four times the hyper-
bolic sector ECK.

LECTURE XII— APPENDIX I 169

14. We will also add to these the following well-known
measurement of cissoidal space.

Let AMB be a semicircle whose centre is C, and let the
straight line AH touch it ; and let AZZ be the cissoid that
is congruent to it, having this property, that, if any point M
is taken in the circumference AMB, and through it the
straight line BMS is drawn (cutting AH in 8), and also a
straight line MFZ, cutting the cissoid AZZ in Z, MZ = AS;
then in a straight line aft take ap. equal to the arc AM, and to
a/j. let straight lines p.£ be applied perpendicular, and equal
to the versines AF of the arc AM. Then the trilinear space
MAZ is double the space a/*£. Hence, since the dimensions
of the space a/j.^ are generally known, and indeed can be
easily deduced from the preceding theorems, therefore the
dimension of the cissoidal space MAZ is obtained. Anyone
may make the calculation who wishes to do so.

The following rider will close this appendix.

15. Let ACB be a quadrant of a circle, and let AH, BG
touch the circle; also let the curves KZZ, LEO be hyper-
bolas, the same as those that have been used above ; let
the arc AM be taken, and let it be supposed to be divided
into parts at an infinite number of points N ; through these
draw radii ON, and let the straight lines NX (drawn parallel
to AH) meet them in the points X. Then the sum of the
straight lines NX (taken along the radii) will equal to the
space AFZK/(radius), and the sum of the straight lines NX
(taken along parallels to AH) will be equal to the space
PLQO/(s . radius).

APPENDIX II

Method of Exhaustions. Measurement of conical surfaces.

For the sake of brevity combined with clearness, and
especially for the latter, the proofs of the preceding
theorems have been given by the direct method ; by
which not only is the truth firmly established, but also
its origin appears more clearly. But for fear anyone, less
accustomed to arguments of this nature, should hesitate
to use them, we will add a few examples by which such
arguments may be made sure, and by the help of which
indirect proofs of the propositions may be worked out.

1. Let the ratios A to X, B to Y, C to Z, be any ratios,
each greater than some given ratio R to 8 ; then will the
ratio of all the antecedents taken together to all the con-
sequents taken together be greater than the ratio R to S.

2. Hence it is evident that, if any number of ratios are
each of them greater than any ratio that can be assigned,
then the sum of the antecedents bears a greater ratio to
the sum of all the consequents than any ratio that can be
assigned.

LECTURE XII— APPENDIX II 171

3. Let ADB be any curve, of which the axis is AD, and
to this the straight line BD is applied; also let the straight
line BT touch the curve, and let BP be an indefinitely small
part of the line BD; draw PO parallel to DT, cutting the
curve in N. Then I say that PN will bear to NO a ratio
greater than any assignable ratio, R to S, say.

4. Hence, if the base BD is divided into an infinite
number of equal parts at the points Z, and through these
points are drawn straight lines parallel to DA, cutting the
curve in E, F, G ; and through the latter are drawn the
tangents BQ, ER, FS, GT, meeting the parallels ZE, ZF, ZG,
DA in the points Q, R, 8, T; then the straight line AD will
bear to all the intercepts EQ, FR, GS, AT taken together a
ratio greater than any assignable ratio.

5. Among the results of this we have : —

All the lines EQ, FR, GS, AT taken together are equal
to- zero.

The lines ZE, ZQ; ZF, ZR; etc., are equal to one
another respectively.

Also the small parts of the tangents BQ, ER, etc., are
equal to the corresponding small parts of the curve, BE, EF,
etc. ; and they can be considered as coincident with one
another.

Moreover, one may safely assume anything which
evidently is consistent with these.

6. Again, let AB be any curve, of which the axis is AD,
and let DB be applied to it ; also let DB be divided into
an indefinite number of equal parts at the points Z;

172 BARROW'S GEOMETRICAL LECTURES

through these points draw straight lines parallel to AD,
cutting the curve in the points X, and let these be met
by straight lines ME, NF, OG, PH, drawn through the points

X
G

Fig. 176.

X parallel to BD; also let the figure ADBMXNXOXPXRA,
circumscribed to the segment ADB (contained by the
straight lines AD, DB and the curve AB), be greater than
any space S; then I say that the segment ADB is not less
than the space 8.

7. Also if it is supposed that the inscribed figure
HXGXFXEXZDH is less than any space 8; then I say
that the segment ADB is not greater than S.

8. Hence, if there is any space, 8 say, the figure circum-
scribed to which is equal to the figure ADBMNOPRA, and
also the figure inscribed to it is equal to the figure
HGFEZDH ; then the space 8 will be equal to the segment
ADB. For, as has just been shown, it cannot be greater
than it, nor can it be less.

Also these things can be altered to suit other modes of
circumscription and inscription ; it should be sufficient to
have just made mention of this.

LECTURE XIl^APPENDIX II 173

NOTE. — In § 6, Barrow uses the usual present-day method
of translating the error for each rectangle across the diagram
to sum them up on the last rectangle; another point of
interest is the striking similarity between the figure used
by Barrow and the figure used by Newton in Lemma II of
Book I of the Principia, especially as Newton uses the
four-part division of his base, which is usual with Barrow,
whereas in this place Barrow, strangely for him, uses a
five-part division of the base.

METHOD OF MEASURING THE SURFACE OF CONES

Let AMB be any curve, whose axis is AD, and C a given
point in it, BD a straight line at right angles to it. Any
point M in the curve being taken, draw ME touching the
curve, and from C draw CG perpendicular to ME; also let
CV be a straight line of given length, perpendicular to the
plane ADB ; join VG. Then VG will be perpendicular to
MG. Also let RS be a line such that, if a straight line MIX
is dr.awn parallel to AD, cutting the ordinate BD in I, and
the line RS in X, then MP : ME - VG : IX ; or, if the line
AL is such that, when MPY is drawn parallel to BD, cutting
the axis AD in P, and the line AL in Y, then PE : ME
= YG : PY; then will either of the spaces BR8D or ADL
be double the surface of the cone formed by straight lines
through V that move along the curve AMB.

Example. — Let the curve AMB be an equilateral hyper-
bola, of which the centre is C, and let CV = CA = r, and
CP = x (for it helps matters in most cases to use a calcula-
tion of this kind); join MC ; then the rectangle BRSD is
double the area AM BY of the cone.

This elegant example was furnished by that most excellent

174 BARROW'S GEOMETRICAL LECTURES

man, of outstanding ability and knowledge, Sir Francis
Jessop, Kt., an Honorable ornament of our college, of
which he was once a Fellow-commoner ; I shall venture to
adorn my book, as with a jewel, not indeed at his request,
nor yet I hope against his wish, by means of his cleverly
written work on this matter, kindly communicated to me.

PROPOSITION i

If from a point E in the axis km of a right cone ABC/, a
straight line of unlimited length, EC, passes through the sur-
face of the cone, and if with the end E kept at rest, the line
EC is carried round until it returns to the place from which
it started, so that always some part of it cuts the surface
of the cone (say, through the hyperbola CFD and the
straight lines DA, AC situated in the surface of the cone),
the solid contained by the surface or surfaces generated by
the straight line EC so moved and by the portion of the
surface of the cone bounded by the line or lines CFD, DA,
AC, which the straight line EC describes in the surface as
it is carried round, will be equal to the pyramid of which
the altitude is equal to E«, the perpendicular drawn from
the point E to the side of the cone, and base equal to that
part of the conical surface bounded by the line or lines
CFD, AD, AC, generated by the motion of the line EC.

PROPOSITION 2

Let ABC/ be a right cone ; let it be cut by the plane
CFD parallel to its axis km; let the straight lines AC, AD
be drawn from the vertex of the cone to the hyperbolic
line CFD ; and upon the triangle AGO let the pyramid EACD

LECTURE XII— APPENDIX II 175

be erected, having its vertex E in the axis of the cone ; and
let ES be perpendicular to the plane ACD and En to the
side of the cone. Then I say that the conical surface
bounded by the hyperbolic line CFD and the straight lines
DA, AC is to the pyramid EACD on the base ACD as the
altitude of the pyramid ES is to the perpendicular E«.

PROPOSITION 3

Let ABC/ be a given right cone; let it be cut by a plane
(say, in the triangle qrf) and let this plane cut the axis of
the cone produced beyond the vertex in the point q\ also
let the common intersection of it and the surface of the
cone be the hyperbolic line rS/, and let straight lines
Ar, A/ be drawn from A the vertex of the cone, from the
point q a perpendicular ^X to the side A/ of the cone
produced, and from the point A a perpendicular AZ to the
plane qrt. Then I say that the conical surface, bounded
by the hyperbolic line rst and the straight lines rA, M, is
to the hollow hyperbolic figure qr§tq as the perpendicular
AZ is to the perpendicular ^X.

PROPOSITION 4

Let AB/^g" be a given right cone ; and let it be cut by a
plane HFEG passing through the axis below the vertex;
from the point H, where the plane cuts the axis of the cone,
let HK be drawn perpendicular to any side of the cone, and
from the vertex A a perpendicular AL to the plane HFEG.
Then I say that the conical surface, bounded by the lines
FEG, GA, AF is to the plane HGEF as the perpendicular AL
is to the perpendicular HK.

APPENDIX III

Quadrature of the hyperbola. Differentiation and Inte-
gration of a logarithm and exponential. Further standard
forms.

On looking over the preceding, there seems to me to be
some things left out which it might be useful to add.
Anyone can easily deduce the proofs from has already been
given, and will obtain more profit from them thereby.

PROBLEM i

Let KEG be any curve of which the axis is AD, and let
A be a given point in AD ; find a curve, LMB say, such that,
when any straight line PEM perpendicular to the axis AD
cuts the curve KEG in E and the curve LMB in M, and AE
is joined, and TM is a tangent to the curve LMB, then TM
shall be parallel to AE.

The construction is made as follows : — Through any
point R, taken in the axis AD, draw a straight line RZ
perpendicular to AD; let EA produced meet it in 8, and
in the straight line EP take PY equal to RS; in this way
the nature of the curve OYY is determined; then let the
rectangle contained by AR and PM be equal to the space
AYYP (or PM is equal to the space AYYP/AR). Then the
curve AYYP shall have the proposed property.

It should also be easily seen that, other things remaining

LECTURE XII— APPENDIX III 17?

the same, if the curve QXX is such that, if EP cuts it in X,
PX = AS ; then the space AXXP is equal to the rectangle con-
tained by AR and the arc LM, or space AXXP/AR = arc LM.

Example i. — Let ADG be a quadrant of a circle; if EP
is any straight line perpendicular to AD, join DE. It is
required to draw the curve AMB such that, if EPM produced
meets it in M, and MT touches the curve, then MT shall be
parallel to DE.

The construction is as follows : — Draw AZ parallel to
DG, and let DE produced meet it in 8; let the curve AYY
be such that, if PE produced meets it in Y, PY = AS; then
take PM = space AYP/AD ; and the construction is effected.

NOTE.— If the curve QXX is such that PX = DS (or if
AQ = AD and QXX is a hyperbola bounded by the angle
ADG), then arc AM . AD = space AQXP.

Example 2. — Let AEG be any curve whose axis is AD
such that, when through any point E taken in it a straight
line EP is drawn perpendicular to AD, and AE is joined, then
AE is a given mean proportional between AR and AP of the
order whose exponent is n/m. It is required to find the
curve AMB, of which the tangent TM is parallel to AE.

Observe about the curve AM that n : m = AE : arc AM.

Now, if njm = 1/2 (or AE is the simple geometrical mean
between AR and AP), then AEG will be a circle, and AMB
the ordinary cycloid. Hence the measurement of the latter
comes out from a general rule.

These also follow from the more general theorem added
below.

178 BARROW'S GEOMETRICAL LECTURES

NOTE

At first sight the foregoing proposition, stated in the
form of a problem, but (by implication in the note above)
referred to by Barrow as a theorem, would appear to be an
attempt at an inverse-tangent problem. But " the sting is
in the tail " ; this, and most of those which follow, are really
further attempts to rectify the parabola and other curves,
by obtaining a quadrature for the hyperbola. That this is
so is fairly evident from the note to Ex. i above ; and it
becomes a moral certainty when we come to Problem IV,
where Barrow is at last successful.

The first sentence of the opening remark to this
Appendix, which I have put in italics, makes it certain that
these were Barrow's own work. The reference to Wallis at
the end of Problem IV almost " shouts " the fact that it was
through reading Wallis' work that Barrow began to accumu-
late, as was his invariable practice, a collection of general
theorems connecting an arc with an area ; it is also probable
that it was only just before publication that he was able to
complete his collection with the proof that the area under
a hyperbola was a logarithm.

The proof, as Barrow states, for the construction given
in Problem i is very easily made out, by drawing another
ordinate NFQY parallel and near to MEPY and MW parallel
to PQ to cut NQ in W. For we have then
PE/PA = RS/AR = PY/AR = MW/NW = MP/PT, .-. AE//MT.

Example i is not truly an example of the problem ; if we
allow for Barrow's inversion of the figure (a bad habit of
his that probably caused trouble to his readers), to render
this a true example of the method of the problem, AD . PM
should be made equal to the space DYP instead of the
space AYP; this, however, would have made the curve lie
on the same side of AD as the quadrant, at an infinite
distance; so Barrow subtracts the infinite constant, equal
to the area QADY, and thus gets a curve lying on the other
side of the line AD, fulfilling the required conditions.
Example 2, however, is a true example of the problem ;
and it is particularly noteworthy on account of the fact
that it rectifies the cycloid, a result previously attained in

LECTURE XII— APPENDIX III 179

Lect. XII, § 20, Ex. 5 ; but, as has been noted, the matter
is not so clearly put in that as it is here; for, in this
Example 2, since AE2 = AP. AR, the curve AEG is evidently
a circle, and it follows from the property that the tangent
at M is parallel to AE, that the curve AMB is the cycloid;
the theorem states that the arc AM is equal to twice the
chord AE; and thus Barrow has undoubtedly rectified the
cycloid, and thus anticipated Sir C. Wren, who published
his work in the Phil. Trans, for 1673. Moreover, and
Barrow seems to be prouder of this fact than anything else,
Barrow's theorem is a general theorem for the rectification
of all curves of the form given by

X = 2ct cosm!n 0, Y = zamln . |V«2 0 cos(m-n}'n 6 dO.

If m/n = 2, the curve, as Barrow remarks, is a cycloid ;
this is also evident analytically if the equations above are
worked out. If, however, m/n is equal to any odd integer,
the curve AEG has a polar equation r = a cosls 0, and the
curve AMB is one of the form given by the equations

X = a cos* + 10, Y = a(zs + i ) J%*«2 6 cos 2l ~ * 6 dO ;

and this, in the particular case when ^ = i, is the three-
cusped hypocycloid, X2/3 + Y2/3 = a2'3, and the arc of this
curve, is given as 3AE/2 (for my n is Barrow's m - n), or
3 al/3x213 ; and thus the theorem also rectifies the three-
cusped hypocycloid; though, of course, Barrow does not
mention this curve, nor can I see a simple theorem by which
Barrow could have performed the integration, denoted by
\sin- 0 cos 6 dO, by a geometrical construction.

PROBLEM 2

To draw a curve, AMB say, of which the axis is AD, such
that, any point M being taken in it, if MP is drawn perpen-
dicular to AD and MT is supposed to be a tangent to the
curve, then TP : PM shall be an assigned ratio.

Let any straight line R be taken ; find PY, such that
TP : PM (which ratio the assigned relation will give) is equal
to the ratio R : PY (and this is to be taken along the line

1 8o BARROWS GEOMETRICAL LECTURES

PM and at right angles to the axis AD) ; and through the
points Y obtained in this way let the curve YYK be drawn ;
then, if PM is made equal to the space APY/R, the nature
of the curve AMB will be established.

Example i. — Let ADG be a quadrant of a circle, of which
the radius is equal to the assigned length R; let it be
desired that the ratio of TP to PM shall be equal to that of
R to arc AE ; then, since as prescribed, R : arc AE = R : PY ;
PY = arc AE; and hence PM = APY/R.

Example 2. — Let ADG be a quadrant of a circle, and
suppose that the ratio TP : PM has to be equal to that of
PE : R ; then PY will be equal to the tangent of the arc
GE; and the space APYY is equal to R . arc AE. Then
PM = arc AE.

PROBLEM 3

Being given any figure AMBD whose axis is AD and base
DB, it is required to find a curve KZL such that, when any
straight line ZPM is drawn parallel to DB, cutting AD in P,
and it is supposed that ZT touches the curve KZL, then
TP = PM.

The construction is as follows : —

Let OYY be a curve such that, any finite straight line R
being taken, and PMY produced, PM : R = R:PY; then,
taking any point L in BD produced, draw LE at right angles
to DL,* so that DL : R = R:LE; then, with asymptotes
DL, DG,* describe the hyperbola EXX passing through E ;
let the space LEXH be equal to the space DOYP, and pro-

AD is produced to G and LE is in the same sense as DG.

LECTURE XII— APPENDIX III 18 1

duce XH and YP to meet in Z. Then will Z be a point in
the required curve, and if ZT is a tangent to it, TP = PM.

It is to be noted that, if the given figure is a rectangle
ADBC, the curve KZL has the following property. DH is a
geometric mean between DL and DO of the same order as
DP is an arithmetic mean between DA and 0 (or zero).

Now, if any curve KZL is described with this property,
and the tangent ZT is found practically, then the hyberbolic
space LEXH will be found, and this in all cases is equal to
the rectangle contained by TP and AP.*

It can also be seen that

(i) the space ADLK = R(DL-DO);

(ii) the sum of ZP2, etc. = R(DL2 - D02)/2, and the sum
of ZP3, etc. = R(DL3 - D03)/3, and so on ; f

(iii) if it is supposed that </> is the centre of gravity of the
figure ADLK, and <f>\f/ is drawn perpendicular to AD and <££
to DL, then ty = (DL+ DO)/4, and <j>£ = R - AD. DO/LO.

PROBLEM 4

Let BDH be a right angle, and BF parallel to DH ; with
DB, DH as asymptotes, let a hyperbola FXG be described
to pass through F ; with centre D describe the circle KZL ;
lastly let AMB be a curve such that, if any point M is taken
in it, and through M the straight line DMZ is drawn, and
it is also assumed that Dl = DM and IX is drawn parallel

* Here Barrow seeks the curve whose subtangent is constant and obtains
it ; he, however, does not at first seem to perceive the exponential character
of it. For, although he states the property of the geometric and arithmetic
means, it is not till in connection with the next problem that he states that
this has anything to do with logarithms.

f As usual, these quantities have to be applied to AD.

182 BARROW'S GEOMETRICAL LECTURES

to BF, then the hyperbolic space BFXI is equal to twice
the circular sector ZDK. It is required to draw the tangent
at M to the curve AMB.

Draw DS perpendicular to DM, and let DB . BF = R2;
then make DK : R = R : P, and then DK : P = DM : DT ;
join TM ; then TM will touch the curve AMB.

It is to be observed that the curve has the following
property. Dl is a geometric mean between DB and DO
(or DA) of the same order as the arc KZ is an arithmetic
mean between 0 (or zero) and the arc KL That is, if Dl
is a number in the geometric series beginning with DB and
ending with DA, and 0, KL are the logarithms of DB, DA,
then KZ will be the logarithm of Dl. Or, working the
other way (the way in which ordinary logarithms go), if Dl
is a number in the geometric series starting with DO and
ending with DB, and 0 is the logarithm of DO, and the arc
LK that of DB, then the arc LZ will be the logarithm
of Dl.

Now, h the curve is completely drawn and the tangent
to it determined practically, it is evident that the circular
equivalent of the hyperbolic space is given, or the hyper-
bolic equivalent of the circular sector.

That most eminent man, Wallis, * worked out most clearly
the nature and measurement of this Spiral (as well as of
the space BDA) in his book on the cycloid ; and so I will
say no more about it.

* Wallis' chief works connected with the problems of Infinitesimal
Calculus are in course of preparation, and will be issued shortly ; so that
it has not been thought necessary to give here anything further than this
reference.

LECTURE XII— APPENDIX III 183

NOTE

The two foregoing propositions are particularly interest-
ing in their historical associations. Logarithms had been
invented at the beginning of the seventeenth century, and the
method of Briggs (Arithmetica Logarithmica, 1624) was still
fresh. Logarithms were devised as numbers which increased
in arithmetical progression as other numbers related to
them increased in geometrical progression. We know
that Wallis had evaluated the integral of a positive integral
power of the variable, and later had extended his work to
other powers ; Cavalieri had also obtained the same results
working in another way ; also Fermat had used the method
of arithmetic and geometric means as the basis of his
work on integration, and he specially remarks that it is
a logarithmic method; but it was left to Gregory St
Vincent to perform the one remaining integration of a
power when the index was - i. This he did by the
method of exhaustions, working with a rectangular hyper-
bola referred to its asymptotes; he stated (in 1647) that,
if areas from a fixed ordinate increased in arithmetic
progression, the other bounding ordinates decreased in
geometric progression.* This is practically identical with
the special type that Barrow takes as an example to
Problem 3 ; but it was left to Barrow to give the result
in a definite form. At the same time we see that, if
Barrow owed anything at all to Fermat, we must credit
Fermat's remark with being the source of Barrow's ideas
on the application of these arithmetic and geometric
means. As usual with Barrow, he gives a pair of
theorems, perfectly general in form, one for polar and
the other for rectangular coordinates. He proves that
the area under the hyperbola referred to its asymptotes,
included between two ordinates whose abscissae are a, b, is
log (^/a), though he is unaware apparently of the value of
the base of the logarithm. I say apparently, because I
will now show that it is quite within the bounds of
probability that Barrow had found it by calculation ;

* 'Brouncker used the same idea in 1668 to obtain an infinite series for
the area under a hyperbola.

1 84 BARROW'S GEOMETRICAL LECTURES

supposing my suggestion is true, however, Barrow would
at that time have been unable to have proved his
calculation geometrically, or indeed in any other theoretical
manner, and so would not have mentioned the matter ; as
we see, he leaves the constant to be determined practically
(Mechanice), this way being just as good in his eyes as any
other that was not geometrical.

Let AFB be a paraboliform such that PF ^

is the first of m - i means between PG and
PE. Also let VKD be another curve such that

space AVDP = R. PF.

Then, taking AC = CB, to avoid a con-
stant, the equation to AFB is

DTm AD Pfim— 1

— t\ r • r \A ,

and the equation of VDK is

(m. PD)m. AP"1-1 = Rm. PG"1-1.

Now area LKDP = R.(PF-HL)

/. LP . PD = R . PG1-1/"1 . (AP1/™ - AL1/m).

Hence, if we put x for AP, we obtain ^dxjxl~llm = the
sum of m . LP . PD/R . PG1-1/"1 = m . (AP1/"1 - AL'/m).

But if m is indefinitely increased, and R is taken equal
to m, the curve VKD tends to become a rectangular hyper-
bola ; and in Problems 3, 4, Barrow has shown that the
area is proportional to log AP/AL. Hence log AP/AL is the
limiting value of ;;/(AP1/?" - AL1/m), when m is indefinitely
increased, or log x is the limiting value of (xn - i)/«x when
» is indefinitely small.

Now remembering that Briggs in his Arithmetica Logarith-
mica had given the value of 10 to the power of i/254 as
I'oooooooooooooooi278i9i4932oo3235, it would not have
taken five minutes to work out log 10 = 2*3058509 . . .;
hence, calling this number p, Barrow has

jdx/x = p . loglQ x.

Considering Barrow's fondness for the paraboliforms, it
would seem almost to be impossible that he should not
have carried out this investigation; although, if only for
his usual disinclination to "put his sickle into another

LECTURE XII— APPENDIX III 185

man's harvest," as he remarks at the head of Problem 9,
he does not publish it ; he in fact refers to Wallis' work
on the Logarithmic Spiral as a reason why he should say
no more about it. It is to be noted that, in Problem 4,
Barrow constructs the Equiangular Spiral, and then proves
it to be identical with the Logarithmic Spiral. Hence,

if r— = C, r = a° and conversely ; thus d(ax}\dx = K«z

and \a*dx = max, where K, m have to be determined.

If we do not allow that Barrow had found out a value
for the base of the logarithm, yet assuming log x to stand
for a logarithm to an unknown base, Barrow has rectified
the parabola, effected the integration
of tan 6, and the areas of many other
spaces that he has reduced to the quad-
rature of the hyperbola. For instance, /B
in Lect. XII, § 20, Ex. 3, he shows that /
Z.arc AB = area ADLK.

Now ATLK = Z2 . log (J'2 . AT/Z)/2 + Z2/4

.-. ADLK = |Z2.%(AD + DL)/Z
In modern notation, since Z . DL = AD2/2,

'(a;2 + az)dx = ^a? . log [{x + >/(#2 + a'

where the base of the logarithm has to be determined.

Similarly, in Lect. XII, App. I, § 2, he states that the
sum of the tangents belonging to the arc AM applied to the
line a/* is equal to the hyperbolic space AFZK ; that is,

i° tan Od6 = /^-AF/AC = log cos 0.

The theorem of § i is the same thing in another form.

Again, in Lect. XII, App. I, § 4, we have the equivalent
of the integral of sin 6 ; since Barrow's integrals are all

re
definite, we find it in the form I sin 6 d9 = 2CoszOJ2.

re ra

From § 5, we obtain I sec2 9 d (sin 6} = I dOlcos' 6 or
Jo Jo

1 86 BARROW'S GEOMETRICAL LECTURES

•0

sec 6 d& given as £ log {(i + sin 0)/(i - j/« 0)}, which of

course can be reduced immediately to the more usual
form log tan (0/2 +7T/4) ; the same result is obtained from
§§6, 7; or they can be exhibited in the form ^dx/(az - x2)
= (log (a + x)f(a - x)}/2a.

The theorem of § 8 is a variant of the preceding and
proves that J cos 0 d(tan 6} is equal to J tan 0 d(cos 0) - tan
6 . cos 0, both being equal to J sec 6 dO.

The theorem of § 9 reduces immediately to

Thus Barrow completes the usual standard forms for the
integration of the circular functions.

There is one other point worth remarking in this con-
nection, as it may account for the rushing into print of this
rather undigested Appendix ; I have already noted that,
from Barrow's own words, this Appendix was added only
just before the publication of the book. I imagine this
was due to Barrow's inability to complete the quadrature
of the hyperbola to his rattier fastidious taste. But, in
1668, Nikolaus Kaufmann (Latine Mercator) published his
Logarithmotechnia, in which he gave a method of finding
true hyperbolic logarithms (not Napierian logarithms) ; of
this publication Prof. Cajori says: — "Starting with the
grand property of the rectangular hyperbola . . ., he
obtained a logarithmic series, which Wallis had attempted
but failed to obtain." (Rouse Ball attributes the series to
Gregory St Vincent.) This may have settled any qualms
that Barrow had concerning the unknown base of his
logarithms, and decided him to include this batch of
theorems, depending solely on the quadrature of the
hyperbola, and merely requiring a definite solution of the
latter problem to enable Barrow to complete his standard
forms. Kaufmann obtained his series by shifting one axis
of his hyperbola, so that the equation became y = i/(i+x),
expanded by simple division, and integrated the infinite
series term by term, thus obtaining the area measured from
the ordinate whose length was unity, and avoiding the
infinite area close to the asymptote.

LECTURE XII— APPENDIX III 187

PROBLEM 5

Let EDG be any space bounded by the straight lines
DE, DG and the curve ENG, and R any straight line of
given length ; it is required to find a curve AMB such that,
when any straight line DNM is drawn from D, and DT is
perpendicular to it, and MT touches the curve AMB, then
shall DT : DM = R : DN.*

Let KZL be a curve such that DZ2 = R . DN, and, the
straight line DB being drawn, of arbitrary length, let
DB : R = R : BF, where BF, and also DH, is at right angles
to DB. Then through F, within the angle BDH, draw the
hyperbola FXX, and let the space BFXI (where IX is supposed
to be parallel to BF) be equal to double the space ZDL;
lastly, let DM = Dl. Then M will be a point on a curve
such as is required ; and if a straight line MT touches the
curve at any point M, then will TD : TM = R : DN.

PROBLEM 6

Again, let EDG be a given space (as in the preceding) ;
it is required to find a curve AMB such that, if any straight
line DNM is drawn, and DT is perpendicular to it, and MT
touches the curve, then DT shall be equal to DN.

Take any straight line of length R, and let DZ2 = R3/DN ;
also having taken DB (to which DH, and BF (= R3/DB2),
are perpendiculars) assume that through F is drawn, between
the asymptotes DB, DH, a hyperboliform of the second kind
(that is, one in which the ordinates, as BF or IX, are fourth

* The next four problems constitute Barrow's conclusion of his work on
Integration. Probs. 5, 6 give graphical constructions for integration, and
7, 8 find graphically the bounding ordinate or radius vector for a figure
of given area, i.e. graphical differentiation of a kind.

188 BARROW'S GEOMETRICAL LECTURES

proportionals in the ratio DB to R,* or Dl to R). Then
take the space BIXF equal to double the space ZDL; and
let DM = Dl ; then M will be a point on the required curve ;
and if MT touches it, DT = DN.

PROBLEM 7

Let ADB be any figure, of which the axis is AD and the
base is DB, and, any straight line PM being drawn parallel
to DB, let the space APM be given (or expressed in some
way) ; it is required from this to draw the ordinate PM, or
to give some expression for it.

Take any straight line R, and let R . PZ = space APM ;
in this way let the line AZZK be produced; find ZO the
perpendicular to it; then PZ : PO = R : PM.

Otherwise. Take PZ = ^APM) • let ZO be perpendicular
to the curve AZK ; then PM = PO.

PROBLEM 8

Let ADB be any figure, bounded by the straight lines
DA, DB and the curve AMB, and through D let any straight
line DM be drawn ; given the space ADM, it is required to
find the straight line DM.

Take any straight line R, and let DZ = 2ADM/R ; draw
ZO perpendicular to the curve AZK ; let DH, the perpen-
dicular to DM, meet it; then DM2 = R . DO.

Otherwise. Let DZ = V/(4ADM) ; and draw ZO per-
pendicular to the curve AZK ; let DH, the perpendicular to
DZ, meet it ; then DM2 = DZ . DO.

* If DB : R = R : P = P : BF, DB2 : R2 = R : BF, or BF = R3/DB2.

LECTURE XII— APPENDIX III 189

NOTE

These four problems are generally referred to by the
authorities as " inverse-tangent " problems. I do not think
this was Barrow's intention. They are simply the com-
pletion of his work on integration, giving as they do a
method of integrating any function, which he is unable to
do by means of his rules, by drawing and calculation.
Thus, the problem of § 5 reduces to : — " Given any function,
f(x) say, construct the curve whose polar equation is
r =/(#), perfoim the given construction, and the value
of Jo/(*X* is equal to R log DB/DI or R log DB/DM."
Similarly, in Problem 6, the .value of ^dx/f(x) is given as
i/DM - i/DB. The construction as given demands the
next two problems, or one of those which follow, called
by Barrow "evolute and involute" constructions. As an
alternative, Barrow gives an envelope method by means of
the sides of the polar tangent triangle. It is rather re-
markable that as Barrow had gone so far, he did not give
the mechanical construction of derivative and integral
curves in the form usual in up-to-date text-books on
practical mathematics, which depend solely on the property
that differentiation is the inverse of integration.

With regard to the propositions that follow under the
name of problems on "evolutes and involutes," it must be
noted that, although at first sight Barrow has made a
mistake, since the involute of a circle is a spiral and cannot
under any circumstances be a semicircle ; yet this is not
a mistake, for Barrow's definition of an involute (whether
he got it from James Gregory's work or whether he has
misunderstood Gregory) is not the usual
one, but stands for a polar figure equivalent
in area to a given figure in rectangular coor-
dinates, and vice versa. In a sense somewhat
similar to this Wallis proves that the circular
spiral is the involute of a parabola.

Hence, these problems give alternative
methods for use in the given constructions for Problems 5, 6.
Thus in the adjoining diagram, it is very easily shown that
area D^B = £ area DBMP.

190 BARROW'S GEOMETRICAL LECTURES

That brilliant geometer, Gregory of Aberdeen, has set
on foot a beautiful investigation concerning involute and
evolute figures. I do not like to put my sickle into another
man's harvest, but it is permissible to interweave amongst
these propositions one or two little observations pertaining
in a way to such curves, which have obtruded themselves
upon my notice whilst I have been working at something else.

PROBLEM 9

Let ADB be any given figure, of which the axis is AD
and the base is DB ; it is required to draw the evolute
corresponding to it.

With centre C, and any radius CL, let a circle LXX be
described ; also let KZZ be a curve such that, when any
line MPZ you please is drawn parallel to BD, the rectangle
contained by PM and PZ is equal to the square on CL
(or PZ is equal to CL2/PM). Then let the arc LX = space
DKZP/CL (or sector LCX = half the space DKZP) and in
CX take C/* = PM ; then the line B/A/X is the involute of
BMA, or the space C//./3 of the space ADB.

For instance, if ADB is a quadrant of a circle, the line /fy.C
is a semicircle.

COR. i. It is to be observed that if the two figures ADB,
ADG are analogous; and the involutes are C/i/3, Cvy ; and
if C/A : Cv = DB : DG ; then, reciprocally,
L /3f> : L {3Cv = DG : DB.

COR. 2. The converse of this is also true.

LECTURE XII— APPENDIX HI 191

COR. 3. If Cvy, CS/3 are analogous suo modo, that is if,
when any straight line CvS is drawn through C, Cv to CS is
always in the same ratio ; then these will be the involutes
of similar lines.

PROBLEM 10

Given any figure (3G<f>, bounded by the straight lines C/3,
C</>, and another line /&£ ; it is required to draw the evolute.

With centre C, describe any circular arc LE (making with
the straight lines C/3, C<£ the sector LCE) ; then, CK being
drawn perpendicular to LC, let the curve /3YH be so related
to the straight line CK that, when any straight line C/x,Z is
drawn, and CO is taken equal to the arc LZ, and OY is drawn
perpendicular to CK, OY = C//.. Also, let the curve BMP
be so related to the straight line DA that, when DP is equal
to space C/3YO/CL, and PM is drawn perpendicular to DA,
then PM = G/J. also. Then the space DBFA is the evolute

Example. — Let LZE be the arc of a circle described with
centre C, and /fyiC a spiral of such a kind that, if the straight
line C/xZ is drawn in any manner, the arc EZ always bears
to the straight line C/* some assigned ratio (say, R : S). It
is plain that the line /2YH is straight, for we have always
EZ (or KO) : C/* (or OY) = R : 8. Hence, the evolute BMP
is a parabola, since the parts AP, AD of the axis are in the
same ratio as the spaces KOY, KC/3, that is, as the squares
on OY, C/2, or the squares on PM, DB.

192 BARROW'S GEOMETRICAL LECTURES

Corollaries

Theorem i. If on the figure /3C<£ is erected a cylinder
having its altitude equal to the whole circumference of the
circle whose radius is CL; then the cylinder will be equal
to the solid produced by rotating the figure C/3HK about CK.

Theorem 2, Let AMB be any curve of which the axis is
AD and the base is DB, and AZL a curve such that, when
any straight line ZPM is drawn, PZ = V/(2APM) ; and let
OYY be another curve such that, when the straight line
ZPMY is produced to meet it, ZP2 : R2 = PM : PY. Lastly,
let DL : R = R : LE, and through E, within the angle LOG,
describe the hyperbola EXX ; let the straight line ZHX,
drawn parallel to AD, meet it in X. Then the space PDOY
will be equal to the hyperbolic space LXHE.

Hence, the sum of all such as PM/APM = 2LEXH/R2.

Theorem 3. Let AMB be any curve whose axis is AD
and base is DB ; and let the curve KZL be such that, if any
straight line R is taken, and an arbitrary line ZPM is drawn
parallel to BD, ^/APM : PM = R : PZ; then the space ADLK
is equal to the rectangle contained by R and 2^/ADB, or

ADLK/2R =

Example. — Let ADB be a quadrant of a circle; then the
sum of all such as PM/ ^APM = V(2DA . arc AB).

Theorem 4. Let AMB be any curve of which the axis is
AD and the base is DB, and let EXK, GYL be two lines so

* The theorems equivalent to Theorems 2 and 3 are clear enough, even
without the final line in the first of the pair ; Barrow intends them as
standard forms in integration.

LECTURE XII— APPENDIX III 193

related that, any point M being taken in the curve, and the
straight lines MPX, MQY being drawn respectively parallel
to BD and AD, and it being supposed that MT touches the
curve A MB, then TP : PM = QY : PX. Then will the figures
ADKE, DBLG be equal to one another.

NOTE. Of all the propositions so far, this theorem is the
most fruitful ; since many of the preceding are either con-
tained in it or can be easily deduced from it. For, suppose
the line AMB is by nature indeterminate, then if one or
other of the curves EXK, GYL is determined to be anything
you please, there will result from the supposition some
theorem of the kind of which we have given a considerable
number of examples already. If, for instance, the line GYL
is supposed to be a straight line making with BD an angle
equal to half a right angle (in which case the points D, G
are taken to be coincident), then we get the theorem of
Lect. XI, § i.

If GYL is a line parallel to DB, we have Lect. XI, § n.

Again, if PM = PX (or the lines AMB, EKX are exactly
the same), hence follows Lect. XI, § 10.

Further it is plain from the theorem that for any given
space an infinite number of equal spaces of a different kind
can be easily drawn ; thus, if the space DGLB is supposed to
be a quadrant of a circle, centre D, and AMB is a parabola
whose axis is AD, we get this property of the curve EKX (by
putting DB = r> AP = x, PX = y, and k for the semi-para-
meter of the parabola or DB2/2AB), that r-k\z =

194 BARROW'S GEOMETRICAL LECTURES

If, however, AMB is supposed to be a hyperbola, there
will be produced a curve EXK of another kind.

Moreover, on consideration, I blame my lack of fore-
sight, in that I did not give this theorem in the first place
(it and those that follow, of which the reasoning is similar
and the use almost equal); and then from it (and the others
that are added directly below), as I see can be done, have
deduced the whole lot of the others. Nevertheless, I think
that this sort of Phrygian wisdom is not unknown either to
me or to others who may read this volume.

NOTE

When I first considered the title-pages of the volume
from which I have made the translation, I was struck by
the fact that the Lectiones Opticce had directly beneath the
main title the words " Cantabrigia in Scholis publids habits "
(delivered in the public Schools of Cambridge), whereas
no such notification appears on the separate title-page of
the Lectiones Geometries. When later I found that the
title-page of the Lectiones Mathematics also bore this noti-
fication, I became suspicious that at any rate there was no
direct evidence that these lectures on Geometry had ever
been delivered as professorial lectures, though they might
have been given to his students by Barrow in his capacity of
college fellow and lecturer. As I considered the Preface,
I was confirmed in this opinion ; and the above note would
seem to corroborate this suggestion. For surely if these
matters had been given in University Lectures in the Schools,
it would not have been necessary to wait till they were
ready for press before Barrow should find out that his
Theorem 4 was more fruitful and general than all the
others. His own words contradict the supposition that he
initially did not know this theorem, for he blames his
"want of foresight." This raises the point as to the exact
date when Newton was shown these theorems ; this has
been discussed in the Introduction-

LECTURE XII— APPENDIX III 195

Theorem 5. Let ADB be any space, bounded by the
straight lines DAE, DQBK and the curve A MB, also let EXK,
GYL be two curves so related that, if any point M is taken
in the curve AMB, and DMX is drawn, and DQ = DM, and
QYBL, DG are drawn perpendicular to DB, and.DT is per-
pendicular to DM, and the straight line MT touches the
curve AMB ; if, I say, when these things are so, TD : DM =
DM . QY : DX2; then shall the space DGLB be double the
space EDK.

Theorem 6. Again, let AMB be any curve of which the
axis is AD and the base is DB; and let EXK, HZO be two
curves so related to one another and the axes. AD, a/3 so
related to one another that, if a point M is taken anywhere
on the curve AMB, and MPX is drawn perpendicular to AD,
and afj. is taken equal to AM, and //.Z is drawn perpendicular
to a/3, and it is supposed that MT touches the curve AMB,
and cuts DA in T, then TP : PM = /xZ : PX. Then the
spaces ADKE, a/30H shall be equal to one another.

Theorem 7. Let ADB be any space, bounded by the
straight lines DAE, DBK and the curve AMB; also let EXK,
HZO be two curves so related that, if any point M is taken in
the curve AMB, and the straight line DMX is drawn, and a/A
is taken equal to the arc AM, and /xZ is drawn perpendicular
to the straight line aft, and DT is perpendicular to DM,
and the straight line MT touches the curve AMB, then
DT : DM = DM . /xZ : DX2. Then shall the space a00H be
double the space EDK.

But here is the end of these matters.

LECTURE XIII

[The subject of this lecture is a discussion of the roots
of certain series of connected equations. These are very
ingeniously treated and are exceedingly interesting, but
have no bearing on the matter in hand ; accordingly, as
my space is limited, I have omitted them altogether.]

Laus DKO Optimo Maximo
FINIS

In the second edition, published in 1674, there were
added the three problems given below, together with a set
of theorems on Maxima and Minima. Problem II is very
interesting on account of the difficulty in seeing how Barrow
arrived at the construction, unless he did so algebraically.

PROBLEM I. Let any line AMB be given (of which the
axis is AD, and the base DB), it is required to draw a curve
ANE, such that if any straight line MNG is drawn parallel
to BD, cutting ANE in N, then the curve AN shall be equal
to GM.

The curve ANE is such that if MT touches the curve AMB,
and N8 the curve ANE, then SG : GN = TG :

EXTRACTS FROM SECOND EDITION 197

PROBLEM II. With the rest of the hypothesis and con-
struction remaining the same, let now the curve ANE be
required to be such that the arc AN shall be always equal
to the intercept MN.

Let the curve ANE be such that SG :GN = 2TG.GM :
(GM2 -TG2), then ANE will be the required curve.

PROBLEM III. Let any curve DXX be given, whose axis
is DA; it is required to find a curve AMB with the property
that, if any straight line GXM is drawn perpendicular to AD,
and it is given that SMT is the tangent to the curve AM,
then MS = GX.

Clearly the ratio TG : TM (that is, the ratio of GD to MS
or GX) is given ; and thus the ratio TG : GM is also given.

Barrow does not give proofs of these problems. The
only geometrical proof of the second I can make out is as
follows : — .

Draw PQR parallel to GNM, cutting the curves ANE, AMB
in Q, R respectively, and draw MW, NV parallel to AD to
meet PQR in W, R. Then we have NQ = RW - QV from the
supposed nature of the curve ; also from the several differen-
tial triangles, we have RW/GP = MG/GT, QV/GP - NG/GS,
and NQ/GP = NS/SP ; and therefore

NS.GT = MG.GS-GT.NG.

Squaring,

(NG2 + GS2).GT2 = MG2.GS2-2MG.GS.GT.GN
hence, GS . (GM2 - GT2) = 2MG . GT . NG,

or GS:GN = 2MG.GT:(GM2-GT2).

But I can hardly imagine Barrow performing the opera-
tion of squaring, unless he was working with algebraic
symbols ; in this case he would be using his theorem that

(dsldx? - i + (dyldxf. (Lect. X, § 5.)

POSTSCRIPT

Extracts from Standard Authorities

Since this volume has been ready for press, I have con-
sulted the following authorities for verification or contra-
diction of my suggestions and statements.

ROUSE BALL (A Short Account of t/ie History
of Mathematics)

(i). " It seems probable, from Newton's remarks in the
fluxional controversy, that Newton's additions were confined
to the parts " (of the Lectiones Optics et Geometric^) " which
dealt with the Optics."

(ii). "The lectures for 1667 . . . suggest the analysis
by which Archimedes was led to his chief results."

(iii). " Wallis, in a tract on the cycloid, incidentally gave
the rectification of the semi-cubical parabola in 1659; the
problem having been solved by Neil, his pupil, in 1657;
the logarithmic spiral had been rectified by Torricelli" (i.e.
before 1647). "The next curve to be rectified was the
cycloid; this was done by Wren in 1658."

(This contradicts Williamson entirely ; I suggest that, of
the two, Ball is probably the more correct, if only for the
fact that this would explain why Barrow did not remark on
the fact that he had rectified both the cycloid and the
logarithmic spiral.)

POSTSCRIPT 199

(iv). The only thing in Barrow's work that is given any
special notice is the differential triangle ; since Ball states
that his great authority for the time antecedent to 1758
is M. Cantor's monumental work Vorlesungen iiber die
Geschichte der Mathematik, it would appear that Cantor
also does not give Barrow the credit that he deserves.

(v). Fermat had the approximation to the binomial
theorem; for he was able to state that the limit of
<?/{i - (i -e)513}, when e is evanescent, is 3/5. Since we
know that Fermat had occupied himself with arithmetic
and geometric means, it would seem probable that Barrow's
equivalent theorem was deduced from this work of Fermat ;
however, Ball states that these theorems of Fermat were not
published until after his death in 1665, whereas Barrow's
theorem was, I have endeavoured to show, considerably
anterior to this.

(vi). With reference to the Newton controversy we have : —
" It is said by those who question Leibniz' good faith, that
to a man of his ability the manuscript (Newton's De Analyst
per Aequationes), especially if supplemented by the letter of
Dec. 10, 1672, would supply sufficient hints to give him a
clue to the methods of the calculus, though as the fluxional
notation is not employed in it, anyone who used it would
have to invent a notation."

(How much more true is this of Barrow's Lectures, which
contained a complete set of standard forms and rules, and
was much more like Leibniz' method, in that it did not use
series but gave rules that would work through substitutions \
See under Gerhardt.)

" Essentially it is Leibniz' word against a number of
suspicious details pointing against him."

(I hold that the dates are almost conclusive, as they are
given in the fourth paragraph of the preface ; and in this
I do not by any means suggest that Leibniz lied, as will be
seen under Gerhardt. A mathematician, having Leibniz'
object and point of view, would more probably consider
that Barrow's work and influence was a hindrance rather
than a help, after he had absorbed the fundamental ideas.)

200 BARROW'S GEOMETRICAL LECTURES

Professor LOVE (Encyc. lirit. Xlth. ed., Art.
" Infinitesimal Calculus")

(i). "Gregory St Vincent was the first to show the
connection between the area under the hyperbola and
logarithms, though he did not express it analytically.
Mercator used the connection to calculate natural
logarithms."

(ii). " Fermat, to differentiate irrational expressions, first
of all rationalized them ; and although in other works he
used the idea of substitution, he did not do so in this case."

(iii). " The Lectiones Optiae et Geometries, were apparently
written in 1663-4."

(iv). " Barrow used a method of tangents in which he
compounded two velocities in the direction of the axes of
x andy to obtain a resultant along the tangent to a curve."

" In an appendix to this book he gives another method
which differs from Fermat's in the introduction of a second
differential."

(Both these statements are rather misleading.)

(v). "Newton knew to start with in 1664 all that Barrow
knew, and that was practically all that was known about the
subject at that time."

(vi). "Leibniz was the first to differentiate a logarithm
and an exponential in 1695."
(Barrow has them both in Lect. XII, App. Ill, Prob. 4.)

(vii). " Roger Cotes was the first, in 1722, to differentiate
a trigonometrical function."

(It has already been pointed out that Barrow explicitly
differentiates the tangent, and the figures used are applic-
able to the other ratios ; he also integrates those of them
which are not thus obtainable by his inversion theorem
from the differentiations. Also in one case he integrates
an inverse cosine, though he hardly sees it as such. With
regard to the date 1722, as Professor Love kindly informed

POSTSCRIPT 201

me on my writing to him, this is the date of the posthumous
publication of Cotes' work ; Professor Love referred me to
the passages in Cantor from which the information was
obtained.)

(viii). "The integrating curve is sometimes referred to
as the Quadratrix."

(This is Leibniz' use of the term, and not Barrow's.
With Barrow, the Quadratrix is the particular curve whose
equation is

v = (r - x) tan Trxjzr. )

There are a host of other things both in agreement with
and in contradiction of my statements to be found in this
erudite article ; nobody who is at all interested in the
subject should miss reading it. But I have only room for
the few things that I have here quoted.

Dr GERHARDT (Editions of Leibnizian Manuscripts, etc.}

(i). In a letter to the Marquis d'Hopital, Leibniz writes : —
" I recognize that Barrow has gone very far, but I assure
you, Sir, that I have not got any help from his methods.
As I have recognized publicly those things for which I am
indebted to Huygens and, with regard to infinite series, to
Newton, I should have done the same with regard to
Barrow."

(In this connection it is to be remembered, as stated in
the Preface, that Leibniz' great idea of the calculus was the
freeing of the work from a geometrical figure and the con-
venient notation of his calculus of differences. Thus he
might truly have received no help from Barrow in his
estimation, and yet might, as James Bernoulli stated in
the Act a Erudilorum for January 1691, have got all his
fundamental ideas from Barrow. Later Bernoulli (Acta
Eruditorum, June 1691) admitted that Leibniz was far in
advance of Barrow, though both views were alike in some
ways.)

(ii). Leibniz (Historia et Origo Calculi Differentialis)
states that he obtained his " characteristic triangle " from

202 BARROWS GEOMETRICAL LECTURES

some work of Pascal (alias Uettonville), and not from
Barrow. This may very probably be the case, if he has
not given a wrong date for his reading of Barrow, which
he states to have been 1675; this would not seem to be
an altogether unprecedented proceeding on the part of
Leibniz, according to Cantor. It is difficult to imagine
that Leibniz, after purchasing a copy of Barrow on the
advice of Oldenburg, especially as in a letter to Oldenburg
of April 1673 ne mentions the fact that he has done so,
should have put it by for two whole years ; unless his
geometrical powers were not at the time equal to the task
of finding the hidden meaning in Barrow's work.

(iii). Gerhardt states that he has seen the copy of Barrow
referred to in the Royal Library at Hanover. He mentions
the fact that there are in the margins notes written in
Leibniz' own notation, including the sign of integration.
He also lays stress on the fact that opposite the Appendix to
Lect. XI there are the Latin words for " knew this before."
This tells against Leibniz, and not for him, for this Appendix
refers to the work of Huygens, which of course Leibniz
" knew before," and Gerhardt does not state that thei>e
words occur in any other connection ; hence we may argue
that this particular section was the only one that Leibniz
" knew before." The sign of integration, though I cannot
find any mention of it before 1675, means nothing, for it
might be added on a second reference, after Leibniz had
found out the value of Barrow's book. A striking "coin-
cidence" exists in the fact that the two examples that
Leibniz gives of the application of his calculus to geometry
are both given in Barrow. In the first, the figure (on the
assumption that it was taken from Barrow) has been altered
in every conceivable way ; for the second, a theorem of
Gregory's quoted by Barrow, Leibniz gives no figure, and
/'/ was only after reference to B arrow* s figure that I could
complete Leibnitf construction from the verbal directions
he gave. This looks as if Leibniz wrote with a figure
beside him that was already drawn, possibly in a copy of
Gregory's work, or, as I think, from Barrow's figure. I
have been unable to ascertain the date of publication of
this theorem by Gregory, or whether there was any chance
of its getting into the hands of Leibniz in the original.

POSTSCRIPT 203

Professor ZEUTHEN ( Geschichte der Mathematik im

XVI. und XVII. Jahrhundert; Deutsche Ausgabe

von Raphael Mayer).

(i). Oxford and Cambridge seem to be mixed up in the
historical section, for it is stated that Barrow was Professor
of Greek at Oxford and Wallis was the Professor of
Mathematics at Cambridge, as the context suggests that he
was Barrow's tutor.

(ii). "... he produced his important work, the Lectiones
Mathematics, a continuation of the Lectiones Optic<z; this
was published, with the assistance of Collins, the first
edition in 1669-70, the second edition in 1674."

(Thus Williamson's error is repeated ; it would be inter-
esting to know whether Zeuthen and Williamson obtained
this from a common source, and also what that source was.)

(iii). "He" (Leibniz) "utilized his stay" (in London in
I^7S) "to procure the Lectiones of Barrow, which Oldenburg
had brought to his notice." (See under Gerhardt.)

(iv). Zeuthen, most properly, directs far more attention
to the inverse nature of differentiation and integration, as
proved by Barrow, than to the differential triangle. But,
by his repeated reference to the problem of Galileo, he
does not seem to have perceived the fact that the first five
lectures were added as supplementary lectures. Yet he
notes the fact that Barrow does not adhere to the kine-
matical idea in the later geometrical constructions. He
also calls attention to the generality of Barrow's proofs.

(v). He mentions the differentiation of a quotient, as
given in the integration form in Lect. XI, but appears to
have missed the fact that the rules for both a product and
a quotient have been given implicitly in an earlier lecture.

I have not room for further extracts ; each reader of this
volume should also read Zeuthen, pp. 345-362, if he has
not already done so. What he finds there will induce him
to read carefully the whole of this excellent history of the
two centuries considered.

204 HARROW'S GEOMETRICAL LECTURES

EDMUND STONE (Translation of Harrow's Geometrical
Lectures, pub.

This translation is more or less useless for my purpose.
First of all, it is a mere translation, without commentary of
any sort, and without even a preface by Stone.

The title-page given states that the translation is "from
the Latin edition revised, corrected and amended by the
late Sir Isaac Newton." If this refers to the edition of
1670, Stone is in error. But, since at the end of the book,
there is an " Addenda," in which are given several theorems
that appeared in the second edition, it must be concluded
that Stone used the 1674 edition. It is to be remarked
that these theorems are on maxima and minima, and,
according to the set given by Whewell, only form a part of
those that were in the second edition of Barrow ; some two
or three very interesting geometrical theorems being omitted ;
one of these is extremely hard to prove by Barrow's methods,
and one wonders how Barrow got his theorem ; but the proof
"drops out" by the use of dyjdx, which may account for
Barrow having it, but not for Stone omitting it. This seems
to give a clue as well to an altogether unjustifiable omission,
by either Newton or Stone (I do not see how it could have
been Newton, however) at the end of the Appendix to
Lect. XI. Two theorems have been omitted ; their in-
clusion was only necessary to prove a third and final
theorem of the Appendix as it stood in the first edition ;
namely, that if CED, CFD are two circular segments having
a common chord CGD, and an axis GFE, then the ratio of
the seg. CED to the seg. CFD is greater than the ratio of GE
to GF. In Stone the two lemmas are omitted and the
theorem is directly contradicted. The proof given in Stone
depends on unsound reasoning equivalent to : —

Ifa>£, then c + a:d + b > c:d,

without any reference to the value of the ratio of c to d, as
compared with that of a to b. Finally the theorem as
originally given is correct, as can be verified by drawing
and measurement, analytically, or geometrically.

In addition to this alteration, in Stone there is added a
passage that does not appear in the first edition, nor is it

POSTSCRIPT 205

given in Whewell's edition. " But I seem to hear you
crying out ' aXXrjv ?>pvv fiaXavi^e, Treat of something else.' "
In a table of errata the last four words are altered to " Give
us something else." The Greek (there should be no
aspirate on the first word) literally means "Shake acorns
from another oak." If this alteration was made by Stone,
the addition of the passage, after the manner of Barrow, is
an impertinence. The point is not, however, very important
in itself, but taken with other things, points out the com-
parative uselessness of Stone's translation as a clue to
important matters.

The whole thing seems to have been done carelessly and
hastily ; there hardly seems to have been any attempt to
render the Latin of Barrow into the best contemporary
English ; and frequently I do not agree with Stone's render-
ing, a remark which may unfortunately cut either way.

Of course the passage may, though it is hardly likely,
have been added by Barrow ; such an unimportant state-
ment would hardly have been added in those days of dear
books ; it is also to be noted that Whewell does not give it.
The point could only be settled on sight of the edition from
which Stone made his translation. Barrow, however, makes
a somewhat similar mistake with ratios in Lect. IX, § 10,
and Stone passes this and even renders it wrongly. This
error has been noted on page 107; the wrong render-
ing is as follows: — Barrow has FG : EF + TD : RD, by
which, according to his list of abbreviations, he means
(FG/EF). (TD/RD); and not, as Stone renders it, FG/EF +
TD/RD, without noticing that this does not make sense of
the proof.

Perhaps one sample of the carelessness with which the
book has been revised will suffice : he has

A x B = A dividend (sic) by B
A

= A multiplied or drawn into B ;

in any case want of space forbids further examples.

It is this untrustworthiness that make it impossible to
take Stone's statement on the title-page as incontrovertible ;
nor another statement that these lectures on geometry were

206 BARROW'S GEOMETRICAL LECTURES

delivered as Lucasian Lectures ; it is also to be noted that
he gives as Barrow's Preface the one already referred to in
the Introduction as the Preface to the Optics and omits the
Preface to the Geometry.

WHEWELL ( The Mathematical Works of Isaac Barrow,
Camb. Univ. Press, 1860)

(i). Stress is only laid on two points ; one of course is the
differential triangle ; the other is the " mode of finding the
areas of curves by comparing them with the sum of the in-
scribed and circumscribed parallelograms, leading the way
to Newton's method of doing the same, given in the first
section of the Principia."

(ii). " It is a matter of difficulty for a reader in these days
to follow out the complex constructions and reasonings of
a mathematician of Barrow's time; and I do not pretend
that I have in all cases gone through them to my satis-
faction." (This is proof positive that Whewell did not
grasp the inner meaning of Barrow's work ; that being done,
there is, I think, no difficulty at all.)

(iii). The title-page of the Lectiones Mathematics states
that these lectures were the lectures delivered as the
Lucasian Lectures in 1664, 1665, 1666; and Lect. XVI. is
headed

MATHEMATICI PROFESSORIS LECTIONES
(A.D. MDCLXVI).

(iv). Lect. XXIV starts the work on the method of
Archimedes, which would thus appear to be the lectures for
1667, as guessed by me, and as stated by Ball.

(v). Whewell gives the additions that appeared in the
second edition of 1674. These consist of four theorems
and a group of propositions on Maxima and Minima. One
theorem is noteworthy, as its proof depends on the addition
rule for differentiation and the fact that

APPENDIX

1. Solution of a Test Question on Differentiation
by Barrow's Method

II. Graphical Integration by Barrow's Method
III. Specimen Pages and Plate

I. Test Problem suggested by Mr Jourdain

Given any four functions ; represented by the curves </></>, 60,
££» ££> and given their ordinatcs and subtangents for any one
abscissa, it is required to draw the tangent for this abscissa
to the curve whose ordinate is the sum (or difference) of the
square root of the product of the ordinates of the first two
curves and the cube root of the quotient of the ordinates of
the other two curves.

In other words, differentiate

The figures on the following page have been drawn for

~lj{stan xfxz}.

(i). Let N#</> be the ordinate for the given abscissa, </>F,
OT the given subtangents ; let TTTT be a curve such that
R. NTT = N<£. HO; find NP, a fourth proportional to
NF+NT, NF, NT; then PTT will touch the curve THT. [See
note on page 112, rule (i).]

208 BARROW'S GEOMETRICAL LECTURES

T F P N

APPENDIX

209

(ii). Let N££ be the ordinate for the given abscissa, £X,
£Z the given subtangents ; let xx be a curve such that
Nx:R = N£:N£; find NQ, a fourth proportional to
NZ-NX, NZ, NX; then QY will touch the curve xx- [See
note on page 112, rule (ii).J

(iii). Let pp be a curve whose ordinate varies as the square
root of the ordinate of TTTT; then its subtangent NR = 2NP
(page 104).

(iv). Let KK be a curve whose ordinate varies as the cube
root of the ordinate of xx> tnen ^ts subtangent NO = 3NQ
(page 104).

(v). Let o-o- be a curve such that its ordinate is the sum
of the ordinates of the curves KK, pp ; take Hf, Hr double of
NK, Hp respectively; then (k, Rr are the tangents to the
curves whose ordinates are double those of the curves KK,
pp; let these tangents meet in s ; then so- will touch the
curve crcr. (See note on page 100.)

If sd is drawn perpendicular to RC to meet it in d, then
dS will touch the curve 88, whose ordinate is the difference
between the ordinates of the curves KK, pp. (See note on
page 100.)

Geometrical
Relations

Analytical Equivalents

i _ i i

NP "" NF + NT

If u =

0.6, £ . ?!* = i

u dx $

rf* . i

S tr

I _ i I
NQ ~~ NX NL

If v =

£IY i dv i
v dx £

'Zr ~ f

' 5*

NR = 2NP

If U =

V«, U/<g = 2«

1 du

NC = 3NQ

IfV =

V«, V/^ = 3*

/*.

Hence —

. 6(jr)}± V{

_ U rfw i _V _ dv
2/( dx yv dx

_
"

210 BARROW'S GEOMETRICAL LECTURES

APPENDIX 211

II. The Area under any Curve
(Lect. XII, App. Ill, Prob. 5)

In the diagram on the opposite page, the curve CFD is
a given curve, or a curve plotted to the rectangular axes
BD, BC, which Barrow would be unable to integrate by any
of the methods he has given, or, in fact, could give. The
curve that I have chosen is one having the equation
y = ^/(i -x4), and the problem is to draw a curve that
shall exhibit graphically the integral J dxfy for all values
of the limits, subject to the condition that these limits must
be positive numbers and not greater than unity. The first
step is to construct the curve GNE, which is such that
r = ^/(r-04), for which the method of construction is
obvious from the diagram. Comparing this with the
enunciation of Barrow's Prob. 5, the curve shown in the
diagram is Barrow's curve turned through a right angle;
thus, the point N is also the point T in Barrow's enuncia-
tion. Then a curve has to be constructed such that the
several lines DN or DT are the respective subtangents. The
curve produced is BMA, the method of construction being
clearly shown in the figure ; starting with B, each point is
successively joined to its corresponding point on the curve
GNE (so that MT is the tangent at M) and to the next point
on GNE, and the point midway between the two points in
which these cut the next ray from D is taken as the point
on the curve BMA.

With this figure the area represented in Leibniz' nota-
tion by

fi/^i - x^dx or J*i/v/(i - #4)<tO = R/DM - R/DB ;
for, if r =/(0), since DN = r =/((9), from the curve GNE,
and DT = p2 . dQjdp, from the curve BMA, it follows that
\d&lf(&) = jdp/p2 = i/p, for all limits.

The value between the limits o and i works out as
i/DA-i/DB, which is found from the diagram to be
1/4-8 -i, taking DB = i, that is 1-304; the true value

is {r(i/2).r(i/4)}/{4-r(3/4)} = 1-31 about

It is only suggested that this was the purpose of Barrow's
problem, not that he drew such a figure as I have given.

212 BARROWS GEOMETRICAL LECTURES

III. Specimen Pages and Plate

Two specimen pages and a specimen of one of Barrow's
plates here follow. The pages show the signs used by
Barrow and the difficulty introduced by inconvenient or
unusual notation, and by the method of " running on "
the argument in one long string, with interpolations.
The second page shows Barrow's algebraic symbolism.
Especially note

z. I'M r ix

k . m : : r . — = EK
k

which stands for

since k: m = r: EK, .'. EK = — r-.

The specimen plate shows the quality of Barrow's dia-
grams. The most noticeable figure is Fig. 176, to be
considered in connection with Newton's method as given
in the Principia.

APPENDIX 213

L E C T. I X. n

D E F coflcurrentes punftis S, T . erit Temper D T = i D S. Quod

fi D E funt ut cubi ipfarum D F, erit Temper D T = 3 D S , ac fi- F'g- S9»

mili deiRceps modo.

X. Sine reftae V D, T B concurrentes in T, quas deeuflet pofiiio- .
ne&tsitftaDB, tranfeant etiam per B linez EBE, FBF tales, FlS- I0°-
ut dufta quacunque P G ad D.B parallcla, fit perpetuo P F eodcm or-
dine media Arithmctice inter PC, P E ; tangat autem B R, curvam
E B E j oportet linez FBF tangentera ad Bdetermmare.

Sumptis N M ( ordinura in qoibus font P F, P E exponentibus)

fiatN xTD

tur B S ; hzc curvam FBF continget.

Nam utcunque dufta fit P G, diftas lineas fecansut vides. Edque
EG.FG::(4;M.N.ergoFGxTD. EGxTD::NxTD.
MxTD. Item EFxRD.EGxTD :: M — NxRD. MX
T D. Quapropter ( antecedentes conjungendo ) erit FG x T D -»- w«/; Left'
EFxRD.EGxTD::NxTD-t-M— NxRD.MxTD.
Choc eft) ':: f/.)RD.SD.(fj Eft antemLGxTD-1-KLxRD. .... a
KGxTD::RD.SrO. quare FG x TD + FF x R D . EG x V. Uft'
TD::LGxTD-i--KLxRD.KG xTD. hinc, cumfitEG(^) Vr.
t^KG;ericFGxTD + E F x RDc~LG x TD-f-KL x RD; W^f-
velFG.EF4-TD.RDc-LG.KL-l-TD.RD; feu(dem-
pta communi ratione ) FG.EFc~LG.KL. rel componendo
EG.EFc-KG.KL («) c-EG.EL. unde eft E F *n E L . CO '• kS-
itaque punftum L extra curvara FBF fitum eft ; adcoque liquet V1I>
Propofitum.

XI. Quinetiam, reliquis ftantibus iifdem, fi P F fupponatur ejuf-
dem ordinis Geometrice media liquet (plane ficut in modo pratceden-
tibus) eandem B S curvam FBF contingere.

Extmplum. Si P F fit e fex racdiis tertia, feu M = 7 } & N = j ;

XII. Patet etiam, accepto qaolibet in curva F B F pun&o (ceuFJ ^-g
reftam ad hoc tangentem eonfimili pafto defignari. -Nempe per F, '*' '
ducatur refta P G ad D B parallela , fecans curvam E B E ad E . &

petEducatur E Rcumm E B Etangens;fiatque NxTPJ^ M ? x RP.

L MK

214 BARROWS GEOMETRICAL LECTURES

LECT. X. 8J

4 — 3 m m * — 3 ffe — ime^ fuD/tiiuendogue If m —

Exemp. IV.

Sit Qt*4r*trix C M V (ad circulum C E B pertincns cui centrum
A , ) cujus axis V A ; ordinatae C A . M P ad V A perpendkula-
res.

Protraais retfis A M E, A N F, duclifque rcdis E K, F L ad A B
perpcndicularibus , dicantur arcus C B — p , radius A C = r . refta
=*. Eftque jam C A arc. C B : : N R . arc. F E.

hoc eft, r\ p -. : * . ^ = arc. FE &AM.MP;:AE.EK;hoc
eft, k..m::r -^^EKi item A E . E K : = arc. F E . L K . hoc
cftr.^::-^ ,^ = LK. Verum AM . AE :: AP. AK;
hoc eft i.r:sf.^=AK. ergo /— ^= A L.Et^-

( abjeftk lupcrfluis ) = A L q . adedque L F q —
rrf/-t- ifmf* _ rrmnt^i fmpa.

~ "

Eft autem A CLq . dN q : : A L q . L F q ; hoc eft Qj/— f .
Q^ m~\-a;: ALq.LFq. hoc eft// — *ft.mi»-*-zma::
rrff — ifmft.rrm m-^ifmf*. Unde ( fublatis ex nor-
raa rcjeftaneis ) emerget tqmtleiffpt-i-mmpa — rrfa—rrme- feu
k&f*— rrfA—rrmt }vel fubftituendo juxta fr*fcriptum-tkkf*i _ \rfrn

=rrmt . vel — - — f — t. Hinc colligitur effe redam A T —

. hoc eft (quoniam, utnotUmeft,A V =~-) eat A T —
feu, A V. AM:: AM. AT.

M 2 Exemp.

APPENDIX

215

INDEX

Added constant, diff. of
Analogous curves
APOLLONIUS i,

FACE

. 81

7, ii, 13,
54, 57, 63
• 43
. 146

Applied lines

Arc, of circle

approximations . . 147
infinitesimal = tangent . 61
length, see Rectification.

ARCHIMEDES i, 7, 13, 54, etc.

ARISTOTLE . . 6, n, 13

Arithmetical mean greater
than geometrical
mean ... 85
proportionals . . -77

Asymptotes ... 85

BALL, W. W. R. . .198
BARROW'S mathematical

works ... 8

symbols . . 22

BERNOULLI . . . 201

Bimedian . . . .152

Binomial approximation . 87

BRIGGS

183, 184

CANTOR . . . -199
Cardioid . . . .100
CAVALIERI ... 2
Circular functions, diff. of . 123
Spiral . . . .115
Cissoid of Diocles . 97, 109
COLLINS . 7, 14, 19, 26, 27

Composite motions
Conchoid of Nicomedes
Concurrent motions .
Conical surfaces .
COTES
Cycloid

area of .

rectification

PAGE
47
95

. 47
. 173
. 200
62, 198

• '53
161, 164, 177

DESCARTES ... 3
Descending motion or de-
scent 53
DETTONVILLE . . . 202
D'HOPITAL . . .201
Difference curve, tangent to 100

diff. of, see Laws.
Differential Triangle 13, 14, 120
compared with fluxions 17, 18
Differentiation the inverse

of integration . 31, 117
Directrix .... 43
Double integral, equivalent

of . . . .133

Equiangular Spiral . 139

EUCLID . . .13, 54, 57
Exhaustions, method of .170
Exponent of a mean proper- •

tional . . 78, 83
of a paraboliform . . 142

FERMAT
Fluxions

4, 9, 13, 183, 199
• 4> 16

INDEX

217

PAGE

Fluxions, proof of principle. 115

Fractional indices . . 1 1

powers, diff. of . . 104

integ. of . .128

GALILEO . i, 4, 13, 58, 203
Generation of magnitudes . 35
Genetrix .... 43
Geometrical proportionals . 77
GERHARDT . . . 201
Graphical integration . . 32
GREGORY, James (of Aber-
deen) . . 13, 131
involutes and evolutes . 190

GULDIN .... 2

HUYGENS . . 13, 141, 201

Hyperbola, approximation

to curve ... 68
determination of an

asymptote . 69, 73

Index notation . . . 3, 1 1
of a mean proportional 78, 83
Indivisibles ... 2
Infinite velocity, case of . 59
Integration, method of

Cavalieri . . . 125
inverse of differentiation 31, 135

JESSOP

KAUFMANN
KEPLER

174

1 86
i

Laws for differentiation of
a product, quotient,
and sum ... 31
LEIBNIZ . . 5, 9, 200, 202
Logarithmic differentiation . 1 06
Spiral ... . 139, 198

PAGE

LOVE .... 200
Lucasian Lectures 6, 7, 194, 206

Maximum and minimum 2, 32,

63, H9

Mean proportionals . . 77
MERCATOR . . .186
METIUS' ratio for ir 150,151,154

NEIL. . . . 138, 198
NEWTON 3, 9, 16-20, 26, 199, 200
Normals or perpendiculars . 63

Order of mean proportional 77
OUGHTRED ... 5

OVERTON .... 20

TT, limits for . 150, 151, 154
Paraboliforms, centre of

gravity . . .142

tangent construction 14, 104
PASCAL ... 2, 202
Polar subtangent . . 1 1 1
Power, differentiation of . 104
Preface to the Geometry . 27

to the Optics ... 25
Product curve, tangent to . 112

diff. of, see Laws.
Properties of continuous

curves . . 60-65

Quadratrix. 48, 118, 201, 214
Quadrature of the hyper-
bola . . 180, 186

theorems depending on 165, 185
Quotient curve, tangent to . 112

diff. of, see Laws.

Reciprocal, diff. of -94

Rectification, fundamental

theorem . . 32, 115
general theorems . . 155

218 BARROW'S GEOMETRICAL LECTURES

RICCI

ROBBRVAL

Root, diff. of

Rotation, mode of motion

PAGE

149

IO4
42

ST VINCENT . n, 13, 72, 200
Secants, integration theor-
ems . . 165, 167
Second edition, additional

theorems . . . 196
Segment of circle and hyper-
bola . . . 146
Semi-cubical parabola 162, 198
Spiral of Archimedes 48, 115, 119
Standard forms, see pp. 30-32.
STONE .... 204
Subtangent . . . 106
Sum curve, tangent to . 100

diff. of, see Laws.
Symbols, Barrow's list of . 22

TACQUET .... 4

I'AGK

Tangency, criterion of . 90

Tangents, definitions of . 3
integration theorems on . 166

THEODOSIUS ... 7

Time, see Lecture I.

TORRICELLI'S Problem . 58

Translation, a mode of

motion ... 42

Trigonometrical approxima-
tions ... 32
ratios, diff. of . . .122

Trimedian . . . -152

VAN HURAET . . . 162
Velocity, laws of . . 40
WALLIS 2, 11, 13, 138, 162, 198
WHEWELL . . . 206
WILLIAMSON ... 6
WREN . . 139, 179, 198

ZRUTHEN .

203

PRINTED IN GREAT BRITAIN BY NEILI. AND CO., LTD., EDINBURGH-

^A Barrow, Isaac

33 The geometrical lectures of

B253 Isaac Barrow

Applied Sci.

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Full text of "The geometrical lectures of Isaac Barrow, translated, with notes and proofs, and a discussion on the advance made therein on the work of his predecessors in the infinitesimal calculus"

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