(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

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"

Open Court Classics of Science and Philosophy, 



iOMETRICAL LECTURES 



OP 



ISAAC BARROW 



J. M, CHILD 



CO 



GEOMETRICAL LECTURES 



ISAAC BARROW 





Court Series of Classics of Science and 



Thilo sophy, 3\(o. 3 

THE 
GEOMETRICAL LECTURES 

OF 

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 



BY 



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



546664 



CHICAGO AND LONDON. 
THE OPEN COURT PUBLISHING COMPANY 

1916 



Copyright in Great Britain under the Ad of 191 1 



33 



LECTI ONES 

Geometric^; 

In quibus (praffertim) 

GEHERALJA Cttr varum L iacaruw STMPTOUATA 
D E . C L A 7^ A T^ T II !(, 



Audore IsAACoBARRow Collegii 

S S. Trhiitatn in Acad. Cant ah. SociO, & Socictatif lie- 
ti< Sodale. 



Oi <fvftt Aoj/r/ito; H< iruiv]<t TO. nxSHfufla, at tr& wVi^C) o%*( 
r>riu- OITTI GfetJfft , a* i rttrv -mufAttli it, yu pi <iow mt , 
fi>1/Si' *'Ao &.'> n$uw, O H 6)f tr;t TO ^urt^i oTi a'uT&i/yi'jrf 
**ir. Plato de Repub. VII. 




L ON D IN I, 

Typis Cttlielmi Godkd , & proftant venales apud 

Jh-tnitf>t Tt-wmsre, & OR.ivi.innm Pullejn Juniorcm. 

UW. D C L X X. 



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. 

I 



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'"+ i m + 2+ . . . "')/( + i)n m , 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, \ety z = 4ax be the equation to a parabola, 
and (x - p)* +y z = 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 



II 

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 nn d 
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, an d 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. I n ^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) myxy = 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^jrxy. 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- 



i 4 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. 

VI 

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. 

2 



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 

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. 




II 


IS 


commas 


10 


5 


semicolons 


None 


3 


colons 


5 


10 


full-stops 


4 


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 S Q 

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." 



IX 

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 



X 

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 z 2 =jy 2 - a 2 , 
then z.dz/dx=y.dy/dx; or, in another form, 

if z = V(j 2 - a 2 ), then dz/dx = [y/J(y* - a^]dy/dx 

Lect. VIII, 13 

4. If z 2 =jy 2 + a 2 , then 2 . <&/</# =y . dy/dx, 

OTdz/dx = [y/J(y 2 + a*)]dyldx . ' . . Lect. VIII, 14 

5. Ifz 2 = a 2 -jy 2 , then z.dz/dx = -y.dyjdx, 

or dzjdx = - [j/v/(a 2 -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 z n = a n ~ r .y r , 

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

8. Special case : d(x n }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 = sec 2 x, 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 

A. 



Lect. XII, App. 3, 

B. \ Q a*dx = k(a x - i) ) Prob. 3, 4 

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

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

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

F. \cos d(tan 0} dO = \tan 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 x r (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 x r j(xc} 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 a 5 . 



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 





y 


\ 


^r 


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. 




oK 



o 




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 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 is below K, because the tangent lies en- 
tirely outside the curve. Now, if the point 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 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 = x n , 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 

5 



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 = x n , 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 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 = mx z /r; then the line 
DNN will be a hyperbola. 

If the equation is -yx+gx + my = mx z /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 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 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 ^/(PM 2 - DB 2 ) ; then the line 
DYY is a hyperbola. 

COR. If Y8 is the tangent to the hyperbola DYY, then 
PM 2 : PY 2 = PA : PS. 

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

COR. If Y8 is the tangent to the hyperbola, then 
PM 2 : PY 2 = 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 
EF 2 :EG 2 = 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 



79 




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 

6 



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 = AN.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 + > M + N; 

.-. A + > A + D, i.e. > 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*, ar 3 ..... , we have, whether r ^ i, the fact that (i-r)(i r), 
(i-r)(i r*), (i r^i-r 3 ), . . . are all positive ; hence it follows that 

a4-at*>zar, a + ar 3 > ar+ar 4 , a + ar 4 > ar+ar 3 , . . . 
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 x z 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 y 2 - = mn(x i - a 2 ) ; 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 

/M 

r 

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/x 2 . 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 PE 2 :PF 2 = 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 PE 2 : PF 2 = 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 EF 2 : EG 2 

= 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 = _y 2 - a 2 , 
where a is some constant, then z.dzjdx= y.dyjdx ; or in a different form, 
if *= </(v 2 - a 2 ), then d*/dxy . (dy/dx)/(y 2 - a 2 ). The particular case, when 
y=x, is the equivalent of Lect. VI, 22. 

t A similar result for the case of V(v 2 + a 2 ) or V(x 2 + a z ). 

The case of z=V(a 2 -y 2 ) or V(a 2 -* 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. 

7 



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 



99 



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~ - a 2 ) with regard to x ; but it is in the 
theorem of Lect. VIII, 13 that he gives the equivalent 
to the differentiation of *](jp - a 2 ) 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 sf n = a n ~ 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; 

Y 




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 
PF 2 = PE . PG, PF 3 = PE, PG 2 , 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 

S 




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 PF M = PG M ~ N .PE N gives 
x ta =. a w-N > jj,N j 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.x 2 , 
6. x s , 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.x r ; 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 x r , 
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 . r n . h ence (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 = PG M - N . PEN, then N/RD + (M - N)/TD = M/SD. 

Also, if we take Z 1 *- 1 . PH = PF M , 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. 

8 



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 




B 

T" 

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 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 



R 



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 x n +y n = a n ; 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 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* + x z y z = a 2 y 2 ; the 
gradient, as he shows is x(2x 2 +y 2 )/y(a 2 - x 2 ) ; 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 
x (2xx +jyy )/>' (a 2 - 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 x , _y ; 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$> . xy = a 2 yy or x 2 +y 2 = 

[y + (yJ x o) ' (d-x}\, where d a 2 /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 DY 2 = 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. 




M 


t 

R 


// 


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 

f 2 + z fs e l r ' S 2 ~ 2 /S e f r = 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 . CB 2 /CG 2 = BG . CK 2 /CE 2 . 



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 = CE 2 /CK 2 = 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 . AE 2 , BZ . BF 2 , 
CZ . CG 2 , etc., is the area aggregated from ordinates 
proportional to AZ . AE 2 , BZ . BF 2 , CZ . CG 2 , etc., applied 
to the line YD; and it is of the fourth dimension. 
Whereas, in 3, the sum HL. H0 2 + LK . LY 2 + Kl . KY 2 + 
etc., is aggregated from ordinates equal to HO 2 , LY 2 , KY 2 , 
etc., applied to the line HD; and it is the same as the 
sum of HO 2 , LY 2 , KY 2 , 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. DH 2 /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 : DH 2 ; therefore 
HL.DH 2 = LG.PD.DH or LH . H0 2 =DC . D<// . DH ; and, 
similarly LK . LY 2 = CB . CZ . CG, Kl . KY 2 = BA . BZ . BF, etc. 

But the sum HL. H0' 2 + LK. LY 2 + Kl . KY 2 + etc. = DH 3 /3;* 
and the proposition follows at once. 

3. By similar reasoning, it follows that 

the sum of AZ AE 2 , BZ . BF 2 , CZ . CG 2 , etc. = DH 4 /* ; 
the sum of AZ . AE 3 , BZ . BF 3 , CZ . CG 3 , etc. = DH 5 /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), 

fy r (dyldx} . dx=fy r . <ly = y r + l l(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 DY n = 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 = PQ n . 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/R 2 ; 
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 R 2 P 2 /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 VD 2 . 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 VD 3 . DiA 3 ) or ^(VD 3 . 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 = z 2 /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 
VD0, 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 = DH 2 :DH.HO; 

HL.DH.HO = LG. DH 2 - CD.DH 2 . 
Similarly, LK. DL. LZ = BC.CG 2 , KI.DK.KZ = AB.BF 2 , 
and ID.DI.IZ = VA.AE 2 . 

But it is well known that t 
the sum CD . DH 2 + BC . CG 2 + AB . BF 2 + VA . AE 2 

= 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 AE 2 , BF, CG 2 , 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 Dl 2 . IZ, DK 2 . KZ, DL 2 . 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 IX 4 , KX 4 , LX 4 , etc., is equal to one-fourth of the 
sum of DV 4 , IE 4 , KF 4 , 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-, KZ 2 , 
LZ 2 , 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 IZ 3 , KZ 3 , LZ 3 , etc. 

= the sum VA.AE.AY 2 + AB.BF.BY 2 + 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 . DH 2 + BC . CG 2 + 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 Jy 2 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 



">< 


X 


F 






/ 




\ 






F 


> 




^ 












^ 



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 BP 3 , 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 AE 4 /4, BF 4 /4, CG 4 /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 Dl 3 , 
DK 3 , etc., respectively ; and Barrow would write this as the 
sum of Dl 3 . IE, DK 3 . KF, etc. (applied to DH). 

Lastly, in fig. 125, ID. Dl 3 . IZ = VA.AE 3 .DI, etc.; hence 
all the results follow immediately; i.e. fy^dxdy is equal 
to either of the integrals jj' 4 /4 &> ^xy 3 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 



K 


\, 








\ 


Z 


L 


T A 




G 


F 


.-- R --. 


\ 








D K/ 


\ 


^ 


N 





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 = DM 2 : DF 2 . 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. yVj 2 . dQ = 2 R ./r 2 /(r 2 . dO/dr) . dO = zftr. 

23. frftz.dQ =fr/2.(dr/d6) . dQ = r 2 /4- 

24. /r 2 . 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 DA 2 : R 2 = DT : IZ. 
Then DA : AE = R 2 : 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 r m = A0". In short, Barrow has given a 
general theorem to find the polar area of any curve whose 
equation is \f(r)/r 2 . 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= S J^ . 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 /(R 2 -r 2 )/r 2 . 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 ; 



w/ 





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 . AD 2 = WY . AD . BD = R . (IJ . AD - EA . AB). 

But EA:AT=IJ:AB, or EA . AB = IJ . AT ; 
^WY.AD 2 = R.(IJ.AD-IJ.AT) = R.IJ.DT. 

Now DA 2 -.R 2 = DT:IZ - IJ.DT:IJ.IZ; 

R 2 :IJ.IZ = AD 2 :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 
R 2 : 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 . AD 2 = R . DT . IJ 
is R . d(y\x) . x 2 =- R.(x-y . dx\dy) . dy, or d(yjx) = (x . dy 
- y . dx)jx z . 

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. 



N 



M 



B 



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\t t 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 

10 



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 4a 5 /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 a 5 . 

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. /. GF m > GE" 1 , or /-" . AG" > q m ~ n . AG", 
.-.p>g. 

It shouW be observed that /""-"> = ZD m . 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 = ax m / n . 



LECTURE XI APPENDIX 149 

^ ( -n> = Z m _ AG ro - 2n , hence ZD" 1 . AD"'-* 1 > ZG m . AG m - 2n ; 
that is ZD m . AD m ~ 2n is a maximum.* 

Example i. Let n = i, and w = 3; then/ 4 = ZD 3 . AD = 
ZD 2 . BD 2 , or p z = ZD . BD; and AD = CA/2. 

Example 2. Let n = 3, and w = 10; then/ 14 = ZD 10 . AD 4 , 
or ? = ZD 5 . AD 2 = ZD 3 . BD 4 , 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'" . AD 2 "~, and also 
(as in 25) ? 2( '"- n) = ZG'" . AG"'- m ; hence ZD m . AD 2 "- 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? . y s is a maximum when xjr = y/s. 

f This is equivalent to the algebraical theorem that, if xy = a constant, 
then x r ly s 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)r 2 +/ 2 ), we have 

kp = 4&? -e 2 ; or zk - ^/(4/fc 2 - 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 



1 


/ TQ^^s 3 *"""" 


^ 


2Jjj_ 


f 


Y 


R \ 


E 




\ 


= 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(a 2 - x z ). 

. 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 a 5 . 



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 an d then added numerators and de- 
nominators to obtain his result 3^\ 2 g- 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/2 n , 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/2 4G , and the error in his values would be less than i/2 234 ; 
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. 



s 


N/" 


^^^X. H 


""> 




J 




M 


/ 






$ 


^. 




\ 


R 


F Y \v 




/ 




\ 


D XV 








B 


E 




S \ 


: 1L * 










P- 


V 





/ 


f 




+ 


^\ 





. - 
<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 PF 2 . 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 EY 2 = 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 FH 2 = Z 3 /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. 

II 



H 



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 .jv 2 = x 3 , 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, an d 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 y m x n = R m+ ", 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* - Z 2 ), and therefore Z . Y = J^/(jF 2 - 2 2 )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 




B 



E L 



CK 






Z . IT = area PFGQ = PF . FG ; 

.'. Z 2 .IJ 2 = Z 2 .IT 2 + Z 2 .TJ 2 = PF 2 .FG 2 + Z 2 .PF 2 ; 

.-. 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 
FH 2 = Z 2 . IT 2 /TN 2 = Z 2 . TM . TN/TN 2 
= Z 2 .TM/TN = Z 3 /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/ 


r 


. 

LX^ 


/ 




V 




H- / 


<d 




TT 


f 


V 


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 



AX 



H 

Z 


X 
G 

Z 


s 




o 


N 

f 


X 

F 

Z 


\ 


X 

E 

Z 


s 


X 

Z 


\ 



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 qrtq 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. 

12 



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 AE 2 = 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 cos m!n 0, Y = zamln . |V 2 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 cos ls 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, X 2/3 + Y 2/3 = a 2 ' 3 , and the arc of this 
curve, is given as 3AE/2 (for my n is Barrow's m - n), or 
3 a l/3 x 213 ; 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- 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 (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 ZP 2 , etc. = R(DL 2 - D0 2 )/2, and the sum 
of ZP 3 , etc. = R(DL 3 - D0 3 )/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 = R 2 ; 
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 (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 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 = R m . PG" 1 - 1 . 

Now area LKDP = R.(PF-HL) 

/. LP . PD = R . PG 1 - 1 /" 1 . (AP 1 / - AL 1/m ). 

Hence, if we put x for AP, we obtain ^dxjx l ~ llm = the 
sum of m . LP . PD/R . PG 1 - 1 /" 1 = m . (AP 1 /" 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 ;;/(AP 1/? " - AL 1/m ), when m is indefinitely 
increased, or log x is the limiting value of (x n - 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/2 54 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 . log lQ 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(a x }\dx = K z 

and \a*dx = ma x , 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 = Z 2 . log (J'2 . AT/Z)/2 + Z 2 /4 

.-. ADLK = |Z 2 .%(AD + DL)/Z 
In modern notation, since Z . DL = AD 2 /2, 

'(a; 2 + a z )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. 

Jo 

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 = 2Cos z OJ2. 

Jo 

re ra 

From 5, we obtain I sec 2 9 d (sin 6} = I dOlcos' 6 or 
Jo Jo 



1 86 BARROW'S GEOMETRICAL LECTURES 





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/(a z - x 2 ) 
= (log (a + x)f(a - x)}/2a. 

The theorem of 8 is a variant of the preceding and 
proves that J cos d(tan 6} is equal to J tan 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 DZ 2 = 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 DZ 2 = R 3 /DN ; 
also having taken DB (to which DH, and BF (= R 3 /DB 2 ), 
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) l e t 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 DM 2 = 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 DM 2 = DZ . DO. 

* If DB : R = R : P = P : BF, DB 2 : R 2 = R : BF, or BF = R3/DB 2 . 



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 CL 2 /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 /( 2 APM) ; and let 
OYY be another curve such that, when the straight line 
ZPMY is produced to meet it, ZP 2 : R 2 = 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/R 2 . 

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 DB 2 /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 : DX 2 ; 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 : DX 2 . 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 : 
(GM 2 -TG 2 ), 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, 

(NG 2 + GS 2 ).GT 2 = MG 2 .GS 2 -2MG.GS.GT.GN 
hence, GS . (GM 2 - GT 2 ) = 2MG . GT . NG, 

or GS:GN = 2MG.GT:(GM 2 -GT 2 ). 



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 ; 

D 

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 xfx z }. 



(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 


S 


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 



_ 
" 



14 



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 -x 4 ), 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-0 4 ), 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 = p 2 . dQjdp, from the curve BMA, it follows that 
\d&lf(&) = jdp/p 2 = 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-. 

k 

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, Fl S- 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 i me ^ 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 rrfarrme- feu 
k&f* rrfArrmt } vel fubftituendo juxta fr*fcriptum- t kkf*i _ \rfrn 

=rrmt . vel - f t. Hinc colligitur effe redam A T 



. hoc eft (quoniam, utnot U meft,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 

95 

. 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 

2 

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. 



PLEASE DO NOT REMOVE 
CARDS OR SLIPS FROM THIS POCKET 

UNIVERSITY OF TORONTO LIBRARY