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THE GEOMETRY

OF

RENÉ DESCARTES

TRANSLATED FROM THE FRENCH AND LATIN

BY

DAVID EUGENE SMITH

AND

MARCIA L. LATHAM

WITH A FACSIMILE OF THE FIRST EDITION, 1637

CHICAGO • LONDON THE OPEN COURT PUBLISHING COMPANY

1925 C

D6

Copyright by

THE OPEN COURT PUBLISHING COMPANY

1925

PRINTED IN THE UNITED STATES OF AMERICA

THE GEOMETRY OF RENE DESCARTES

ZA-5Q.Z

Preface

If a mathematician were asked to name the great epoch-making works in his science, he might well hesitate in his decision concerning the product of the nineteenth century ; he might even hesitate with respect to the eighteenth century ; but as to the product of the sixteenth and seventeenth centuries, and particularly as to the works of the Greeks in classical times, he would probably have very definite views. He would certainly include the works of Euclid, Archimedes, and Apollonius among the products of the Greek civilization, while among those which contributed to the great renaissance of mathematics in the seventeenth century he would as certainly include La Gcomcfrie of Descartes and the Principia of Newton.

But it is one of the curious facts in the study of historical material that although we have long had the works of Euclid, Archimedes, Apollonius, and Newton in English, the epoch-making treatise of Des- cartes has never been printed in our language, or, if so, only in some obscure and long-since-forgotten edition. Written originally in French, it was soon after translated into Latin by Van Schooten, and this was long held to be sufficient for any scholars who might care to follow the work of Descartes in the first printed treatise that ever appeared on analytic geometry. At present it is doubtful if many mathemati- cians read the work in Latin ; indeed, it is doubtful if many except the French scholars consult it very often in the original' language in which it appeared. But certainly a work of this kind ought to be easily access- ible to American and British students of the history of mathematics, and in a language with which they are entirely familiar.

On this account, The Open Court Publishing Company has agreed with the translators that the work should appear in English, and with such notes as may add to the ease with which it will be read. To this organization the translators are indebted for the publication of the book, a labor of love on its part as well as on theirs.

As to the translation itself, an attempt has been made to give the meaning of the original in simple English rather than to add to the dif- ficulty of the reader by making it a verbatim reproduction. It is believed that the student will welcome this policy, being content to go to the original in case a stricter translation is needed. One of the translators having used chiefly the Latin edition of Van Schooten, and the other the original French edition, it is believed that the meaning which Descartes had in mind has been adequately preserved.

Table of Contents

BOOK I

Problems the Construction of which Requires Only Straight

Lines and Circles

How the calculations of arithmetic are related to the operations of geometry. . 297 How multiplication, division, and the extraction of square root are performed

geometrically 293

How we use arithmetic symbols in geometry 299

How we use equations in solving problems 300

Plane problems and their solution 302

Example from Pappus 304

Solution of the problem of Pappus 307

How we should choose the terms in arriving at the equation in this case 310

How we find that this problem is plane when not more than five lines are given 313

1 It should be recalled that the first edition of this work appeared as a kind of appendix to the Discours de la Méthode, and hence began on page 297. For con- venience of reference, the original paging has been retained in the facsimile. A new folio number, appropriate to the present edition, will also be found at the foot of each page. For convenience of reference to the original, this table of contents follows the paging of the 1637 edition.

VI

TABLE

'Des matières de U

GEOMETRIE.

L'mre Tremier,

DES PROBLESMES QJCJ'ON PEUT

conflruire fans y employer que des cercles & des lignes droites.

O M M E N T /^ calcul d' Ay'ithmeticjtie p rapporte auxopC" rations de (geometric. 2çj

Comment fê fint (jcometricjuement la Multiplication , U _ Dmifion^Cr lextra^ion de laracine c^Harree, 29S

Comment on pent vfer de chiffres en Géométrie, j.çç

Comment il jkut venir aux Equations qui f entent a re foudre les pro- blefmes^ ^00

^uels font les probief mes plans -^ Et comment tlsfe refoluent. ^02

Exemple tiré de Pappus. j 04

Tiejpon/ê a la cjueflion de Pappus. ^o/

Cornent on doitpofer les termes pottr venir a C Equation en cet exeple.^i 0

K k k Corn

Vlï

BOOK II On the Nature of Curn'ed Lines

What curved lines are admitted in geometry 315

The method of distinguishing all curved lines of certain classes, and of know- ing the ratios connecting their points on certain straight lines 319

There follows the explanation of the problem of Pappus mentioned in the pre- ceding book 323

Solution of this problem for the case of only three or four lines 324

Demonstration of this solution : 332

Plane and solid loci and the method of finding them 334

The first and simplest of all the curves needed in solving the ancient problem

for the case of five lines 335

Geometric curves that can be described by finding a number of their points... 340

Those which can be described with a string 340

To find the properties of curves it is necessary to know the relation of their points to points on certain straight lines, and the method of drawing

other lines which cut them in all these points at right angles 341

General method for finding straight lines which cut given curves and make

right angles with them 342

Example of this operation in the case of an ellipse and of a parabola of the

second class 343

Another example in the case of an oval of the second class 344

Example of the construction of this problem in the case of the conchoid 351

Explanation of four new classes of ovals which enter into optics 352

The properties of these ovals relating to reflection and refraction 357

Demonstration of these properties 360

Table.

Comment on trciéue cjue ceprohhfwe cflplan lorpja'tl n' eft point propofé en plm de s lignes. ^ , ,

Di [cours Second.

DE LA NATURE DES LIGNES COURBES.

Q V clic i font tes lignes conrbes <!jt4 on pent receuoiren Cjeometne. ^\ ; La façon de dtjlwgPicY tentes ces lignes courbes en certains aenres:

ht de connoiflre le rapport qti ont toHS leurs poins a ceux des lianes

droites. j i ç

Suite de l' expliCAtion de la c^uejlion de Pappu4 wife au hure preeedenr..

3^3- Sobttion de ceteqmjïion quand elle nejl proposée' cjh en j ou ^Ifrnes. ' 3^4. Demonflration de ccte folution. ^^2

^els font les lieux plans & fohdes & la façon de les trouucr tous. ^^4. ^elle efi la premiere & la plu^ fimple de toutes les lignes courbes cjni

feruent a la cjuejîion des anciens cjuandelle ef propofé e en cinq lignes,

33S' Celles font les lignes courbes qu'on defcnt en trouuant plufeurs de leurs

poins qui peuuent eflre receucs en Géométrie. ^4.0

Celles font au fjt celles qu on defcnt aueç vne chorde,qui peuuent y eflre

receues, 2 ^o

^epour troHuer toutes les proprietez^des lignes courber , il fufflt defca-

uoir le rapport quoht tous leurs poins a ceux des lignes droites ; cr U

façon de tirer a autres lignes qui les coupent en tous ces poms a angles

droits. j^;

Façon générale pour trouuer des lignes droites qui couppent les courbes

données yOU leurs contingentes a angles droits. Z4.z

Exemple de ce te operation en vne Ellipfe : Et en vne parabole du fecoiid

geure, ^^^

tAutre exemple en vne ouale du fc-condgeure. 3 44

Exemple de la conflruRion de ce probief me en la conchoide. 5 j r,

Explication de 4, nouueaux geures d*Ouales qm feruent a l'Optique, ^sï Les propriete'^de ces Ouales touchant Icsreflextons cr les réfractons.

357 DemonjlrAtion de ces proprie tez., ^60

IX

TABLE OF CONTENTS

How it is possible to make a lens as convex or concave as we wish, in one of its surfaces, which shall cause to converge in a given point all the rays which proceed from another given point 363

How it is possible to make a lens which operates like the preceding and such that the convexity of one of its surfaces shall have a given ratio to the convexity or concavity of the other 366

How it is possible to apply what has been said here concerning curved lines described on a plane surface to those which are described in a space of three dimensions, or on a curved surface 368

BOOK III

On the Construction of Solid or Supersolid Problems

On those curves which can be used in the construction of every problem 369

Example relating to the finding of several mean proportionals 370

On the nature of equations 371

How many roots each equation can have 372

What are false roots Z12

How it is possible to lower the degree of an equation when one of the roots

is known 2>12

How to determine if any given quantity is a root ZTh

How many true roots an equation may have yii

How the false roots may become true, and the true roots false 373

How to increase or decrease the roots of an equation 374

That by increasing the true roots we decrease the false ones, and vice versa. . 375

How to remove the second term of an equation 376

How to make the false loots true without making the true ones false 2)11

How to fill all the places of an equation 378

How to multiply or divide the roots of an equation 379

How to eliminate the fractions in an equation 379

How to make the known quantity of any term of an equation equal to any

given quantity 380

De La Géométrie.

Comment 0» peut faire vn verre autant connexe ou concatig en l*vne de fes fuperficieStCju on voudra, ^uirajfemble a vn point donné tout les rayons cjiti vienent d*vn autre point donné. ^ 6^

(Comment on en peut fkire vn t^ut fhce le mefme , 6r cjue la conaexite de i'vne de fs ftperfictes ait la proportion donnée ausc la conuexité ou conçauité de II autre. ^.6 6,

Comment on peut rapporter tout ce quia ejlé dit des lignes courbes dé- faites ^r vne fuperficte_plate,a celles <jui fe defcriuent dans vn ejpaee (jui a s dimenJtonSf oubien fur vne ftperficie courbe» }6i

Liure Troijtefme

DE LA CONSTRUCTION DES

problefmes roIides,ou plufque folides.

DE cfuelles lignes courbes on peut fe jèruir en la conJlruBion de chaf- cjue probUfme. 5 6ç

Exemple touchant l' muent ion deplufiems moyenes proportionelles, 57 e De la nature des Ecjuations. ^71

Combien il peut y auoir de racines en chafque EcjHation, S7Z

.Celtes font les fnuffes racines. ^yZ

Comment on peut diminuer le nombre des dimenfions dtvne Equation,

lorfquon connoifhcjuelcju'vne defes racines, 37 z

Comment on peut examiner fi quelque quantité donnée efi la valeur

d' vne racine, ^7 s

Combien il peut y auoir de vrajes racines en chafque Equation. 57^ Comment omfkit que les fnujfes racines deuienent vrayes , & les vrajes

fautes, . ^'7^

-Comment on peut augmenter ou diminuer les racines d'vneSquation.^74. j:£^V/2 augmentant aw files vrayes racines on diminue lesfhuffes , ou au

contraire, 375

Comment on peut ofler le pcond terme dvne Equation, 376

Comment on fan que les fauffes racines deuienent vrajes fins que les

vrayes deuienent faujfes, S77

(Comment on fait que toutes les places d'vneEquationfoient remplies ^78 (Comment on peut multiplier ou diuifer les racines êH vne Equation, 3 jç (Comment on ofle les nombres rompus d'vne Equation, 379

(fomment on rend la quaiîtité connue de l'vn des t<rmes d'vne Equation

efgale a telle autre qu'on veut. J ^ "

Kkk z ^^c

TABLE OF CONTENTS

That both the true and the false roots may be real or imaginary 380

The reduction of cubic equations when the problem is plane 380

The method of dividing an equation by a binomial which contains a root 381

Problems which are solid when the equation is cubic 383

The reduction of equations of the fourth degree when the problem is plane.

Solid problems 383

Example showing the use of these reductions 387

General rule for reducing equations above the fourth degree 389

General method for constructing all solid problems which reduce to an equa- tion of the third or the fourth degree 389

The finding of two mean proportionals 395

The trisection of an angle 396

That all solid problems can be reduced to these two constructions 397

The method of expressing all the roots of cubic equations and hence of all

equations extending to the fourth degree 400

Why solid problems cannot be constructed without conic sections, nor those problems which are more complex without other lines that are also more

complex 401

General method for constructing all problems which require equations of de- gree not higher than the sixth 402

The finding of four mean proportionals 411

Table. De LA Géométrie.

^^e les racines tant vrayes que fknjfes peunent eftre réelles ou imaginai- res, ^so La rediiEîion des Equations cubiques lorfque le problefme efl plan, ^So La façon de diuifer vne Equation par ijn binôme qui contient [à racine.

^j4els problefmes font jéhdes lorfque l'Equation efl cubique. ^S^

La redutlion des Equations qui ont quatre di wen fions lorfqne le problef- me efl plan. Et quels [ont ceux qui. font folides. 5 S^ exemple de L^vftge de ces reduBions. ^ s 7 '^gle gêner aie pour réduire toutes les Equations quipaffentle quarré de quarré. ^ g ^ Façon générale pour confîruire tous les problefmes jôltdes réduits a vne Equation de trois ou quatre dimenfions, 2Sç Vinuenticn de deux moyenes proportionelles. 2Çf La diuifion de l'angle en trois , ^o^ ^e tous les problefmes folides fe peutient réduire a ces deux confiru- 61 ions. ^çy^ La façon d! exprimer la valeur de toutes les racines des Equations cubi- ques: Et en fuite de toutes cell&s qui ne montent que lufques au quar- ré de quarrè. ^00 T^ourquov les problefmes folides ne peuuent eflre conflruits (ans les fe- rlions coniques y ny ceux qui font plus compofés [ans quelques autres lignes plus co?npfeés. ^ot Façon générale pour confirutre tous les problefmes réduits a vne Eq [■ra- tion qui n'a point plus de fx dimenftons. ^02 L'inuention de quatre moyenes proportionelles.

411

F I N.

Les

BOOK FIRST

The Geometry of Rene Descartes

BOOK I

Problems the Construction of Which Requires Only Straight

Lines and Circles

ANY problem in geometry can easily be reduced to sucb terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.''' Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract other lines ; or else, taking one line which I shall call unity in order to relate it as closely as possible to numbers/"' and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication) ; or, again, to find a fourth line which is to one of the given lines as unity is to the other (which is equivalent to division) ; or. finally, to find one, two, or several mean proportionals between unity and some other line (which is the same

''' Large collections of problems of this nature are contained in the following works: Vincenzo Riccati and Girolamo Saladino, Institutioncs AnaIyticae,'Bo\ogna, 1765; Maria Gaetana Agnesi, Istltusioni Analitkhc, Milan. 1748; Claude Rabuel, Commentaires sur la Géométrie de M. Descartes, Lyons, 1730 (hereafter referred to as Rabuel) ; and other books of the same period or earlier.

'"'Van Schooten, in his Latin edition of 1683, has this note: "Per unitatem intellige lineam quandam determinatam, qua ad quamvis reliquarum linearum talem relationem habeat, qualem unitas ad certum aliquem numerum." Geotnetria a Renato Des Cartes, una cum notis Flori)nondi de Beanne, opera aiqne studio Francisci à Schooten, Amsterdam, 1683, p. 165 (hereafter referred to as Van Schooten).

In general, the translation runs page for page with the facing original. On account of figures and footnotes, however, this plan is occasionally varied, but not in such a way as to cause tlie reader any serious inconvenience.

{ pri'iUrrvf^^M-tJ^ayTl^

297

L A

GEOMETRIE.

LIVRE PREMIER.

^es problefmes qu'on peut conftruire [ans y employer que des cercles 0^ des lignes droites.

^<S^^^ O u s les Problefmes de Géométrie fè peuucnt facilement réduire a tels termes, % qu'il n'eft befoin par après que de connoi- ftre la longeur de quelques lignes droites, 'pour les conftruire. Et comme toute l'Arithmétique n'eft compofée, que Commcc de quatre ou cinq operations, qui font l'Addition, la|p, "j*=^^ Souftradion, la Multiplication , la Diuifîon , & l'Extra- thJeti- •<Stion des racines , qu'on peut prendre pour vne efpece '^^^ ^^ de Diuifion : Ainfî n'at'on autre chofe a faire en Geo- auxope- metrie touchant les lignes qu'on cherche , pour les pre- ^"0 "' ^^ parer a eftre connues, que leur en adioufter d'autres , ou t"e. en ofter, Oubicn en ayant vne, que le nommeray l'vnite' pour la rapporter d'autant mieux aux nombres , & qui peut ordinairement eftre pnfe a dircretion,puis en ayant encore deux autres, en trouuer vne quatriefme , qui foit à r vne de ces deux, comme l'autre eft a IVnitc, ce qui eft le mefme que la Multiplication i oubien en trouuer vne quatriefme, qui foit al' vne de ces deux, comme rvnite'

Pp eft

3

«eome-

LaMulti- plicatioD.

29% La Géométrie.

eft a l'autre, ce qui eft le mefme que la Diuifiorij ou enfin trouuer vne,ou deux ,ou plufieurs moyennes proportion- nelles entre l'vnité, & quelque autre ligne j ce qui eft le mefme que tirer la racine quarrée^ on cubiqu Cj&c. Et ie ne craindray pas d'introduire ces termes d'Arithméti- que en la Géométrie , afEn de me rendre plus intel-

ligibile.

Soit pai* exemple ABlVnite', & qu'il fail- le multiplier B D par C B G, ie n ay qu'a ioindre

les poins A & C, puis ti- rer D E parallèle a C A, &, B E eft le produit de cete Multiplication. Oubiens'il faut diuifer BE par BD, ayant ioint les poins E & D , ie tire A C parallèle a D E, & B G eft le produit de cete diuifîon.

Ou s'il faut tirer la racine quarree de G H , ie luy ad- ioufte en ligne droite F G, qui eft rvnite'^o.: diuifànt F H H en deux parties efgales au point K, du centre K ie tire le cercle F I H, puis eiîeuant du point G vne ligne droite iufquesà I,à angles droits fur FH, c'eft GI la racine cherchée. le ne dis rien icy de la racine cubique, ny des autres, à caufe que l'en parleray plus commodément cy après. ^^^'peut^ Mais fouuent on n'a pas befoin de tracer ainfî ces li- gne

La Divi-

flOQ.

TExtra- éliondela racine quarrcc.

FIRST BOOK

as extracting the square root, cube root, etc., of the given hne.'" And I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater clearness.

For example, let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA ; then BE is the product of BD and BC.

If it be required to divide BE by BD, I join E and D, and draw AC parallel to DE ; then BC is the result of the division.

If the square root of GH is desired, I add, along the same straight line, EG equal to unity ; then, bisecting EH at K, I describe the circle EIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root. I do not speak here of cube root, or other roots, since I shall speak more conveniently of them later.

Often it is not necessary thus to draw the lines on paper, but it is

sufficient to designate each by a single letter. Thus, to add the lines

BD and GH, I call one a and the other b, and write a + b. Then a — b

will indicate that b is subtracted from a; ab that a is multiplied by b;

a

^ that a is divided hy b ; aa or a- that a is multiplied by itself ; a^ that

this result is multiplied by a, and so on, indefinitely.''' Again, if I wish

to extract the square root of ar^b-, I write ^Ja--\-b"; if I wish to

extract the cube root of a^ — b^-\-ab~, I write ^a^ — b^-^ah'^, and sim- ilarly for other roots. '^' Here it must be observed that by a", b^, and similar expressions, I ordinarily mean only simple lines, which, how- ever, I name squares, cubes, etc., so that I may make use of the terms employed in algebra.'*'

''' While in arithmetic the only exact roots obtainable are those of perfect powers, in geometry a length can be found which will represent exactly the square root of a given line, even though this line be not commensurable with unity. Of other roots, Descartes speaks later.

'■*' Descartes uses a", a*, œ', a'"', and so on. to represent the respective powers of a, but he uses both aa and a- without distinction. For example, he often has

aabb, but he also uses -rr^. 4b-

'°^ Descartes writes : ^JC.à^' — d'^-j-abd. See original, page 299, line 9.

'*' At the time this was written, a- was commonly considered to mean the sur- face of a square whose side is a, and b'^ to mean the volume of a cube whose side is b; while b*, b'', . . . were unintelligible as geometric forms. Descartes here says that a~ does not have this meaning, but means the line obtained by constructing a third proportional to 1 and a, and so on.

GEOMETRY

It should also be noted that all parts of a single line should always be expressed by the same number of dimensions, provided unity is not determined by the conditions of the problem. Thus, a^ contains as many dimensions as ab' or b^, these being the component parts of the

line which I have called ^a^ — b^-\-ab-. It is not, however, the same thing when unity is determined, because unity can always be under- stood, even where there are too many or too few dimensions ; thus, if it be required to extract the cube root of a-b- — b. we must consider the quantity a^b" divided once by unity, and the quantity b multiplied twice by unity. ^''

Finally, so that we may be sure to remember the names of these lines, a separate list should always be made as often as names are assigned or changed. For example, we may write, AB=1, that is AB is equal to 1 ;'" GH = a, BD = 6. and so on.

If, then, we wish to solve any problem, we first suppose the solution already effected.'^' and give names to all the lines that seem needful for its construction, — to those that are unknown as well as to those that are known. ''"^ Then, making no distinction between known and unknown lines, we must unravel the difficulty in any way that shows most natur-

'"' Descartes seems to say that each term must be of the third degree, and that therefore we must conceive of both a-b- and b as reduced to the proper dimension.

'*' Van Schooten adds "seu unitati," p. 3. Descartes writes, AB 00 1. He seems to have been the first to use this symbol. Among the few writers who fol- lowed him, was Hudde (1633-1704). It is very commonly supposed that 00 is a ligature representing the first two letters (or diphthong) of "aequare."' See. for example, M. Aubry's note in W. W. R. Ball's Recreations Mathématiques et Prob- lèmes des Temps Anciens et Modernes, French edition, Paris, 1909, Part III, p. 164.

'" This plan, as is well known, goes back to Plato. It appears in the work of Pappus as follows: "In analysis we suppose that which is required to be already obtained, and consider its connections and antecedents, going back until we reach either something already known (given in the hypothesis), or else some fundamen- tal principle (axiom or postulate) of mathematics." Pappi Ale.yandrini Collectiones quae supcrsimt e Hbris manu scripfis edidit Latina interpcllatione ct commentariis instni.vit Frcdericus Hulisch. Berlin, 1876-1878; vol. II, p. 635 (hereafter referred to as Pappus). See also Commandinus, Pappi Alexandrini Mathcmaticae Collec- tiones, Bologna, 1588, with later editions.

Pappus of Alexandria was a Greek mathematician who lived about 300 A.D. His most important work is a mathematical treatise in eight books, of which the first and part of the second are lost. This was made known to modern scholars by Commandinus. The work exerted a happy influence on the revival of geometry in the seventeenth century. Pappus was not himself a mathematician of the first rank, but he preserved for the world many extracts or analyses of lost works, and by his commentaries added to their interest.

''"' Rabuel calls attention to the use of a, b, c, ... for known, and x, y, z, . . . for unknown quantities (p. 20).

Livre Premier. 299

gnes fur le papier, & il fuffift de les defigner par quelques ^ç^^ ^^ lettres, chafcune par vue feule. Comme pour adioufter clnfFresea la ligne B D a G H, ie nomme Tvne a & l'autre b,&c Qfcris tde!"^^' a~h b-^Eta— ^,pour fouftraire b d' a-^ Et a ^,pour les mul- tiplier IVne par l'autre; Et ^,pourdiuifer^zpar^j-Ec a a,

1 5

ou a, pour multipliera par foymefmc; Et/^, pour le multiplier encore vne fois par a , &:ainfl a rinfini ^ Et

'il z z

^ ^-j- b y pour tirer la racine quarrce d' a -h b -^Et

* Ca-'b-i^abbj pour tirer la racine cubique d' a—b -h abb, & ainfi des autres.

Où il cil a remarquer que par a ou b ou femblables, ie ne conçoy ordinairement que des lignes toutes fîm-- pies, encore que pour me feruir des noms vfités en l'Al- gèbre, ie les nomme des quarre's ou des cubes, ôcc,

Ileltaufly a remarquer que toutes les parties dVne mefmeligne,fedoiuent ordinairement exprimer par au* tant de dimenfions l'vne que l'autre, lorfque IVnite'n'eil:

point déterminée en la queftion, comme icy a en con»-

tientautantqu'^^^ ou b dont fecompofe la ligne que

Tay nommée ^C. a- b -i- abb: mais que ce n'eft pas de mefine lorfque Tynite eft déterminée, a caufo- qu'elle peut eftre foulèntendue par tout ou il y a trop ou trop peu de dimenfions : comme s'il faut tirer la racine cubique de aabb — b j il faut penfer que la quantité aabbcd diuifee vne fois par l'vnite', & que l'autre quan- tité b eft multipliée deux fois par la mefme,

P p a Au

^^^ La Géométrie.

Au refte affin de ne pas manquer a fe fauuenir des noms de ces lignes, il en faut toufîours faire vn regiftrc fèpare'' , à mefure qu'on les pofe ou qu'on les change, cfcriuant par exemple .

A B 30 I , c'eft a dire, A B efgal à t. GH 30 ^ BD 00 b, ''zc, Cemmct Ainfî voulautrefoudre quelque problefînc, on doit d'à- nir^rux^^ bord le confiderer comme delîa fair, & donner des noms Equatiôs a toutcs les lignes, qui femblent necefTaires pour le con- uent are- ûruifc^ auffy bien a celles qui font inconnues , qu'aux foudre les autres. Puis fans confiderer aucune difference entre ces mes. lignes connu es, & mconnues , on doit par counr la diffi- culté, felon l'ordre qui monftre le plus naturellement de tous en qu'elle forte elles dependent mutuellement. les vnes des autres, iufques a ce qu'on ait trouue moyen 'd'exprimer vne mefme quantite^'en deux façons : ce qui le nomme vneEquationj car les terme s de l'vnc de ces deux façons font efgaux a ceux de l'autre. Et on doit trouuer autant de telles Equations,qu'ona fuppofc de li- gnes, qui eftoient inconnuë:t. Oubien s'il ne s'en trouue pas tant, & que nonobflant on n'omette rien de ce qui ell defiré en la queftion,cela tefmoigne qu'elle n*eft pas en- tièrement déterminée. Et lors on peut prendre a difcre- tion des lignes connues, pour toutes les inconnues auf. qu'elles ne correfpond aucune Equation. Après cela s'il enrefte encore plufieurs , il fe faut feruir par ordre de chafcune des Equations qui refteut aufly , foit en la con- fiderant toute feul^,foit en la comparant auec lés autres, pour expliquer chafcune de ces lignes inconnues; & faire

ainfî

FIRST BOOK

ally the relations between these lines, until we find it possible to express a single quantity in two ways.'"^ This will constitute an equation, since the terms of one of these two expressions aie together equal to the terms of the other.

We must find as many such equations as there are supposed to be unknown lines ;''"' but if, after considering everything involved, so many cannot be found, it is evident that the question is not entirely deter- mined. In such a case we may choose arbitrarily lines of known length for each unknown line to which there corresponds no equation."''

If there are several equations, we must use each in order, either con- sidering it alone or comparing it with the others, so as to obtain a value for each of the unknown lines ; and so we must combine them until there remains a single unknown line"*' which is equal to some known line, or whose square, cube, fourth power, fifth power, sixth power, etc., is equal to the sum or difference of two or more quantities, "°' one of which is known, while the others consist of mean proportionals between unity and this square, or cube, or fourth power, etc., multiplied by other known lines. I may express this as follows :

or s-= — aa-\-b-,

or c^= a::.- -\-b-jj — c'^

or ::*=-ac^ — ■c^.c-\-d'^, etc.

That is, 2, which I take for the unknown quantity, is equal to b; or, the square of ^ is equal to the square of b diminished by a multiplied by 2; or, the cube of a is equal to a multiplied by the square of s, plus the square of b multiplied by ^. diminished by the cube of c ; and sim- ilarly for the others.

'"^ That is, we must solve the resulting simultaneous equations.

'^"' Van Schooten (p. 149) gives two problems to illustrate this statement. Of these, the first is as follows : Given a line segment AB containing any point C, required to produce AB to D so that the rectangle AD.DB shall be equal to the square on CD. He lets AC = a, CB = b, and BD = x. Then AD = a + b+x,

and CD =zb 4- x, whence ax -\- bx + x- ^b~-'r 2b x + x- and x = 7- .

a — b

^"' Rabuel adds this note : "We may say that every indeterminate problem is an infinity of determinate problems, or that every problem is determined either by itself or by him who constructs it" (p. 21).

'"' That is, a line represented by x, x-, x^, x*, ....

'"^ In the older French, "le quarré. ou le cube, ou le quarré de quarré, ou le sur- solide, ou le quarré de cube &c.," as seen on page 11 (original page 302).

GEOMETRY

Thus; all the unknown quantities can be expressed in terms of a sin- gle quantity/"' whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.'^''

But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by any one at all familiar with ordinary geometry and with algebra, who will con- sider carefully all that is set forth in this treatise.''^'

'"' See line 20 on the opposite page.

^"' Literally, "Only one or two degrees greater."

'^^' In the Introduction to the 1637 edition of La Geometric, Descartes made the following remark : "In my previous writings I have tried to make my mean- ing clear to everybody; but I doubt if this treatise will be read by anyone not familiar with the books on geometry, and so I have thought it superfluous to repeat demonstrations contained in them." See Oeuvres de Descartes, edited by Charles Adam and Paul Tannery, Paris, 1897-1910, vol. VI, p. 368. In a letter written to Mersenne in 1637 Descartes says: "I do not enjoy speaking in praise of myself, but since few people can understand my geometry, . and since you wish me to give you my opinion of it, I think it well to sav that it is all I could hope for, and that in La Dwptriquc and Les Météores, I have only tried to persuade people that my method is better than the ordinary one. I have proved this in my geom- etry, for in the beginning I have solved a question which, according to Pappus, could not be solved by any of the ancient geometers.

"Moreover, what I have given in the second book on the nature and properties of curved lines, and the method of examining them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond .the a, b, c of children. . . .

"As to the suggestion that what I have written could easily have been gotten from Vieta, the very fact that my treatise is hard to understand is due to my attempt to put nothing in it that I believed to be known either by him or by any one else. ... I begin the rules of my algebra with what Vieta wrote at the very end of his book. De eincndatioiic acquationutn. . . . Thus, I begin where he left off." Oeuvres de Descartes, publiées par llctor Cousin, Paris, 1824, Vol. VI, p. 294 (hereafter referred to as Cousin).

In another letter to Mersenne, written April 20, 1646, Descartes writes as follows: "I have omitted a number of things that might have made it (the geom- etry) clearer, but I did this intentionally, and would not have it otherwise. The only suggestions that have been made concerning changes in it are in regard to rendering it clearer to readers, but most of these are so malicious that I am com- pletely disgusted with them." Cousin, Vol. IX, p. 553.

In a letter to the Princess Elizabeth, Descartes says : "In the solution of a geometrical problem I take care, as far as possible, to use as lines of reference parallel lines or lines at right angles ; and I use no theorems e.xcept those which assert that the sides of similar triangles are i)roportional, and that m a right triangle the square of the hypotenuse is equal to the sum of the squares of the sides. I do not hesitate to introduce several unknown quantities, so as to reduce the question to such terms that it shall depend only on these two theorems." Cousin, Vol. IX, p. 143.

10

Livre Premier. 5oi

ain{îenlesdemefjant, qu'il n'en demeure quVne feule, efgale a quelque autre, qui foit connue , oubiea dont le quarré, oulecube,oulequarredequarré', ouïe furfbli- de, ouïe quarre''de cube, &c. foit efgal a ce, qui fe pro- duift par l'addition, ou fouflradtion de deux ou plufieurs autres quantités ^dontlVne foit connue , & les autres foient compofe'es de quelques moyennes proportion» Belles entre rvnite', & ce quarré, ou cube , ou quarre de quarre',&c. multipliées par d'autres connues. Ce que i'e- fcris en cete forte. ;{_ 30 ^. ou

i.

^30 — a ^-^bb. ou

s^ 00 'i-a ^-^bb^s^-'C, ou

4 } î 4

^ 30 ^J5 î^ " c :^-H d. &c. C'eftadire, ^ que ieprens pour la quantité* inconnue, eftefgaléa^, ou le quarré de ^ eft efgâl au quarre de b moins « multiplié par ^. ou le cube de ^ eft efggl à a multipliépar le quarre de i^plus le quarre' de ^ multiplie par ;^moins le cube de c, & ainfi des autres.

Et on peut toufîours réduire ainfi toutes les quantités inconnues à vne feule, lorfque le Problefme fe peut con- ftruire par des cercles & des ligues droites, ou aufîy par des fedtions coniques,ou mefme par quelque autre ligne qui ce foit que d'vn ou deux degrés plus compofce. Mais^ ie ne m'areft^e point a expliquer cecy plus en detail ,^a caufe que ie vous ofterois le plaifir de l'apprendre de vous mefme, & l'vtilité de cultiuer voftrc efpric en vous y exerceant, qui eft a mon auis la principale, qu'on puifle

Pp 3 tirer

11

Quels

fondes

problef-

3°^ La Géométrie.

tirer de cetefcience. Aufîy que ien y remarque rien de Il difficile, que ceux qui feront vn peu verfé's en la Géo- métrie commune, & en l'Algèbre, & qui prendront gar- de a tout ce qui eil en ce traite, ne puifTent trouuer.

C'eftpourquoyieme contenteray icy de vous auer- tir, que pourvu qu'en demcflant ces Equations on ne manque point a feferuir de toutes les diuifîons, qui fe- ront poffibles , on aura infalliblemcnt les plus fimples termes, aufquels la queftion puifTe eftre réduite.

Et que 11 elle peut eftre refolue par la Géométrie ordi- naire, c eft a dire, en ne fe feruant que de lignes droites mes plans ^ circulaires tracées furvnefuperficie plate , lorfque la dernière Equation aura efté entièrement déo]eflee,iln y reftera tout au plus qu'vn quatre inconnu, efgal a ce qui fe produift de l'Addition , ou fouftradtion de fa racine multipliée par quelque quantité connue , & de quelque autre quantité' auiTy connue

Et lors cete racine, ou ligne inconnue fetrouue ayfe-

ment. Car (î i*ay par exemple

1.

.,. loo a :{-i'bb

iefais le triangle re(5tan- gle N L M, dont le co- fte'L M eft efgal à b ra- cine quarrée de la quan- tité connue bb, 8c l'au- j^ trcLNeft ^ ^, la moi- tié de l'autre quantité' connue, qui eftoit multipliée par ^que ie fuppofe eftre la ligne inconnue, puis prolongeant M N la baze de ce tri- angle,

Com- ment ils fe refol- uenc.

12

FIRST BOOK

I shall therefore content myself with the statement that if the stu- dent, in solving these equations, does not fail to make use of division wherever possible, he will surely reach the simplest terms to which the problem can be reduced.

And if it can be solved by ordinary geometry, that is, by the use of straight lines and circles traced on a plane surface,''"' when the last equation shall have been entirely solved there will remain at most only the square of an unknown quantity, equal to the product of its root by some known quantity, increased or diminished by some other quan- tity also known. '^°' Then this root or unknown line can easily be found. For example, if I have 2- = a3 -{- &-/"' I construct a right triangle NLM with one side LM, equal to b, the square root of the known quan- tity b-, and the other side, LN, equal to ^ a, that is, to half the other known quantity which was multiplied by a, which I supposed to be the unknown line. Then prolonging MN, the hypotenuse'"' of this triangle, to O, so that NO is equal to NL, the whole line OM is the required line z. This is expressed in the following way:'^'

But if I have y' = — ay-\-b-, where y is the quantity whose value is desired, I construct the same right triangle NLM, and on the hypote-

''"' For a discussion of the possibility of constructions by the compasses and straight edge, see Jacob Steiner, Die gcometrischen Constructionen ausgefiihrt fnittelst dcr gcradcn Linic und cincs fcstcn Krciscs, Berlin, 1833. For briefer treatments, consult Enriques, Fragcn dcr Elemcntar-Gcomctric, Leipzig, 1907 ; Klein, Problems in Elementary Geometry, trans, by Beman and Smith, Boston, 1897; Weber und Wellstein, Ëncyklopddie der Elementarcn Géométrie, Leipzig, 1907. The work by Mascheroni, La gcometria del compasso, Pavia, 1797, is inter- esting and well known.

'^^ That is, an expression of the form z- ^= a::± b. "Esgal a ce qui se produit de l'Addition, ou soustraction de sa racine multiplée par quelque quantité connue, & de quelque autre quantité aussy connue," as it appears in line 14, opposite page.

'"^' Descartes proposes to show how a quadratic may be solved geometrically.

'^' Descartes says "prolongeant MN la baze de ce triangle," because the hypote- nuse was commonly taken as the base in earlier times.

i^^'From the figure OM.PM = ^M^ If OM = .3, PM = s — a, and since

LM := t, we have .Î (.? — o) ^ fc- or r- 3= ar-f-b-. Again, MN = \/~o- + fc-j whence

OM = 3=ON-(-MN = -a-f\/ja=-|-6-. Descartes ignores the second root, which

is negative.

13

GEOMETRY

mise MN lay off NP equal to NL. and the remainder PM is y, the desired root. Thus I have

■'= -9'' + \h''' + ^'-

In the same way, if I had

.v* = — ax"" -\- i>% PM would he x- and I should have

and so for other cases.

Finally, if I have ;:- = as—h~, I make NL equal to ^ a and LM equal to b as before ; then, instead of joining the points M and N, 1 draw MOR parallel to LN, and with N as a center describe a circle through L cutting MQR in the points Q and R ; then .c, the line sought, is either MQ or MR, for in this case it can be expressed in two ways, namely :'^^'

^ = r + \/^^-'^^

and

' = i"-Vr'-*=-

^-" Since MR.MQ^zLM". then if R = -, we have \iQ = a — s, and so

s {a — a)=: b- or r- := «r — b-.

If, instead of this, MQ = .3, then MR = a — ^, and again, .s- = a^ — b'-. Further- more, letting O be the mid-point of QR,

1

MQ = OM - OQ = - « - Jl a-.-_ ^2,

and

MR

= MO + OR= j'^+yjl a^--b^-

Descartes here gives both roots, since both are positive. If MR is tangent to the circle, that is, if è = — a, the roots will be equal; while if t» > — a, the line MR

will not meet the circle and both roots will be imaginary. Also, since RM.OM:=LM', c.^2., = b^,andRM + QU = s^ + 3^ = a.

14

Livre Premier. 3^3

angle , iufques a O , en forte qu'N O foit efgale a N L, la toute O M eft :^ la ligne cherchée Et elle s'exprime en cete forte

;^ x> ^ « -h- t^~ aa -{- bb.

Que fi i^jyy :xi — a y H- bbjSc qu'y foit la quantité qu'il faut trouuer , ie fais le mefme triangle rectangle NLM, &defabazeMNi'ofteNPefgalea NL, &Ie refte P M eft ^ la racine cherchée. De façon que iay

b b. Et tout de mefme fî i'a-

30 - ^ ^

V'jaa

uois X :x> — a X H- b. P M feroit x . & i'aurois

X :ù ^ - ^

4

^^: &ainfî des autres. Enfin il i'ay

2^ CO a^'- bb: ie fais N L efgale à | ^, & L M efgale à b corne deuât, pusis,au lieu de ioindre les poins M N , ie tire M QJl parallèle a L N. & du cen- tre N par L ayant defcrit vn cer- cle qui la couppe aux poins Q 8c R, la ligne cherchée ;{ eft M Q? oubië M R, car en ce cas elle s'ex- prime en deux façons, a fçauoir \:x:)'^a»r-V ^aa-bb^

&c ^ 7G~a— x/'^aa-'bb.

Et fi le cercle, qui ayant fon centre au point N , pafîe

par le point L, ne couppe ny ne touche la Hgne droite

MQ^, il n'y a aucune racine en l'Equation, de fa^n

qu'on peut affurer que la conftru^tion du problefms

propofé eft impoffible .

Au

15

304 La GEOMETRIE.

Au refle ces mefmes racines fe peuuent trouuer par vne infinité d'autres moyens , & i'ay feulement veulu mettre ceux cy, comme fort fimples, aîHn défaire voir qu'on peut conftruire tous les Problefmes de la Géomé- trie ordinaire, fans faire autre chofe que le peu qui efl compris dans les quatre figures que i'ay expliquées. Ce queienecroy pas que les anciens ayent remarqué, car autrement ils n'eufTent pas prisJa peine d'en efcrire tant de gros liures, ou le fèul ordre de leurs propofîtions nous fait connoiftre qu'ils n'ont point eu lavraye méthode pourles trouuer toutes,mais qu'ils ont feulement ramaf^ fe celles qu'ils ont rencontrées. exemple Et on le peut voir aufTy fort clairement de ce que Pap- Pappus. pus amis au commencement de fonfeptiefme liure, ou après s'eftre arefte'' quelque tems a dénombrer tout ce qui auoit efté efcrit en Géométrie par ceux qui l'auoient precede', il parle enfin d vne queftion , qu'il dit que ny Euclide,ny Apollonius, ny aucun autre n'auoient fceu entièrement refoudre. & voycy fes mots. Je cite Quem autem àicit [Apollonius) in tertio lihro locum ad Jcrfionh- i^^^i ^ quatuor Uneas ah Eucliâe perfeBum non ejje , ne que tine que le 2pJ"e perficere poterat , neque aliqui; alius'-: fed neque fau- affin que lulum quidaddere iî5 , quœ Euclides {cripfityper ea tantum chafcun çQ^jQii ^ qj^^ ufquc ad Eudidù t empara prtvmonjirata

plu4 ayfe- Juntj^C.

ment. £j. ^^^ ^^ aprc^s il explique ainfi qu'elle eft cete que-

Hion.

At locus ad très ^ ^ quatuor linens , in quo (Apolloîiius) magnifiée fe iaBat i & oftentat^nulla habita gratia ei , qui prius fcripferat , cflbujufmodi. Sipofitione datùtnbus

reïlis

16

FIRST BOOK

And if tlie circle described about N and passing through L neither cuts nor touches the Hne MOR, the equation has no root, so that we may say that the construction of the problem is impossible.

These same roots can be found by many other methods ,'''^ I have given these very simple ones to show that it is possible to construct all the problems of ordinary geometry by doing no more than the little covered in the four figures that I have explained.'""' This is one thing which T believe the ancient mathematicians did not observe, for other- wise they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all,'""' but rather gathered together those propositions on which they had happened by accident.

This is also evident from what Pappus has done in the beginning of his seventh book,'"'' where, after devoting considerable space to an enumeration of the books on geometry written by his predecessors,'""' he finally refers to a question which he says that neither Euclid nor Apollonius nor any one else had been able to solve completely ;''" and these are his words :

"Quern autem dicit (Apollonius) in tertio libro locum ad très, & quatuor tineas ah Euclide perfectum non esse, neque ipse perficere poterat, neque aliquis alius; sed neque paululum quid addere its, quœ

'"^^ For interesting contraction, see Rabuel, p. 23, et seq.

'-"' It will be seen that Descartes considers only three types of the quadratic equation in s, nan^ely, S' + as — b~ = 0, z- — as — b- =^ 0, and s- — o5 + &- = 0. It thus appears that he has not been able to free himself from the old traditions to the extent of generalizing the meaning of the coefficients, — as negative and fractional as well as positive. He does not consider the type z- + as + b- = 0, because it has no positive roots.

'^^ "Qu'ils n'ont point eu la vraye méthode pour les trouuer toutes."

'='1 See Note [9].

1='^ See Pappus, Vol. II, p. 637. Pappus here gives a list of books that treat of analysis, in the following words : "Illorum librorum, quibus de loco, 'ava\v6^ei>os sive resoluto agitur, ordo hie est. Euclidis datorum liber unus, Apollonii de pro- portionis sectione libri duo, de spatii sectione duo, de sectione determinata duo, de tactionibus duo, Euclidis porismatum libri très, Apollonii inclinationum libri duo, eiusdem locorum planorum duo, conicorum octo, Aristaci locorum solidorum libri duo." See also the Commandinus edition of Pappus, 1660 edition, pp. 240-252.

'^"^ For the history of this problem, see Zeuthen : Die Lchrc von den Kegel- schnitten im AUerthum, Copenhagen, 1886. Also, Adam and Tannery, Oeuvres de Descartes, vol. 6, p. 723.

17

GEOMETRY

Enclides scripsit, per ea tantum conica, qucc usque ad Eiiclidis tcmpora prœmonstrata sunt, arc." '"'

A little farther on, he states the question as follows : "At locus ad très, & quatuor lincas, in quo {Apollonius) niagnifice se jactat, & ostentat, nulla habita gratia ei, qui prins scripserat, est hujusmodi.^^"^ Si positione dotis tribus redis lineis ab tino & eodem piincto, ad très lineas in datis angulis rectœ lincœ ducantur, & data sit proportio rcctanguli contcnti duabiis dnctis ad quadratiim reliqiiœ: piinctnm contingit positione datum solidum locum, hoc est unam ex tribus conicis sectionibus. Et si ad quatuor rectas lincas positione datas in datis angulis linecc ducantur; & rectanguli duabns ductis contenti ad contcntum duabns reliqitis proportio data sit; similiter punctum datum coni sectioncm positione continget. Si quidem igitur ad duas tantum locus planus ostensus est. Quod si ad plures quam quatuor, punctum continget locos non adhuc cognitos, sed lincas tantum dictas; quales autem sint, vel quam habcant proprietatem, non constat; earum unam, neque primam, & qucc manifestissima videtur, composucrant osten- dentes utilem, esse. Propositiones autem ipsarum hce sunt.

"Si ab aliqiio puncto ad positione datas rectas lineas quinque ducantur rectœ linecc in datis angulis, & data sit proportio solidi parallèle pip edi rectanguli, quod tribus dnctis lineis continctur ad solidum parallelepipe- dum rectangulum, quod continctur rcUquis duabus, & data quapiam tinea, punctum positione datani lincaui continget. Si auteui ad sex, & data sit proportio solidi tribus lineis contcnti ad solidum, quod tribus reliquis continctur; rursus puncturn continget positione datant lineam. Quod si ad plures quam sex, non adhuc habent diccre, an data sit pro- portio cnjiispiam contenti quatuor lineis ad id quod reliquis continctur,

'^'' Pappus, Vol. II, pp. 677, et seq., Commandimis edition of 1660, p. 251. Literally, "Moreover, he (Apollonius) says that the problem of the locus related to three or four lines was not entirely solved hy Euclid, and that neither he him- self, nor any one else has been able to solve it completely, nor were they able to add anything at all to those things which Euclid had written, by means of the conic sections only which had been demonstrated before Euclid." Descartes arrived at the solution of this problem four years before the publication of his geometry, after spending five or six weeks on it. See his letters, Cousin, Vol. VI, p. 294, and Vol. VI, p. 224.

''^' Given as follows in the edition of Pappus by Hultsch, previously quoted: "Sed hie ad très et quatuor lineas locus quo magnopere gloriatur simul addens ei qui conscripserit gratiam habendam esse, sic se habct."

18

Livre Premiek. Boi*

reBis lineis ah uno & eodem'punBe, ad très lineas in àatis art" gulis reU^ Uneœ ducantur , (3 data fit proportio reUanguli contenti duahu^ duBis ad quadrutum reliquœ : punUum con-* tingîtpofitione datum folidum locum , hçc efl unam ex tribus conicisfeBionihus. Et fi ad quatuor reBas lineas pojîtione datas in datis angulis linece ducantur i ^ reBanguli duabus duBis contenti ad content um duabus reliquis proportio data fit: fi militer punBum, datum coni feBionem pofitione cwitin- get. Si quidem igituradduas tantum locus planus ojlenfus cfl, ^luodfi adplures quam quatuor, punBum continget /«- cos non adhuc cognitos^ fed lineas tantum diBas s quales au-* temfntj velquam habeant proprietatem, non confiât: earum unam, nequeprimam^ & quœmanifefiijfimavidetur, compO' jueruntofi}endentes utilemefe. propoftiones autemipfarum hce funt.

Si ab aliquo punBo adpoftione datas reBds lineas quin- que ducantur reBce linete in datis angulis , ^ data fit propor^ tio falidiparallelepipêdi reBanguli-, quod tribus duBis lineis continetur ad folidum par allelepipedum reBangulum , quod continetur reliquis duabus j (3 dataquapiamlinea^ punBum p opt ion e datam Une am continget . Si autem adfex , S? data fit propo rtio folidi tribus lineis contenti ad folidum, quod tribus reliquis continetur i rurfus punBum continget pofitione datam lineam. ^hiodfiadplures quamfex, non adhuchabent dicere^an data fit proportio cuiufpiâ contenti quatuor lineis ad id quod reliquis continetur, quoniam non efi aliquid con* tentum pluribus quam tribus dimenfionibus.

Ou ie vous prie de remarquer en paffant, que lefcru- pulcj que faifoient les anciens dV fer. des termes del'A- rithmetiqueen la Géométrie, qui ne pouuoit procéder,

O q que

19

306 La Géométrie.

que de ce qu'ils ne voyoient pas afTes clairement leur rapport, caufoit beaucoup dobfcuritc, & d'embaras, es la façon dont ils s'expliquoient. car Pappus pourfuit en

ce te forte..

jicquiefcuntaufem his , quipaulo ante talia interf retail fimt. 7ieque unum ali quo pact 0 comprehenfibîlefigniJïca?itcs quodhîs co7itinetur.Licehit aute per coniunïïas prop orti ones hc£C, (3 âiceret ^ demonflrare univerfe m diPcis proport ion?- biis, atque his in hune modum. Si ah aliquo pwiBo adpojl- tione datas-reBas Iméas ducanturrecl(Ç lineœ in datis angu- lis, ^ data fît proportio cotiiunUa ex e<i, quam habet una du' Rarum adunam, (3 altera adalteram^^ alia adaliam^^ te* liqua ad datant lineam, fifint feptemj fivero oFio, ^ r cliqua a d reliquam: pun&um continget pofitione datas lineas. Et fimiliter quotcumque fmt impares vel pares multitudine, €um hœcy ut dixi, loco ad quatuor lineas refpondeant^ nullum igiturpofuerwntita utlinea not a fit » ^c,

La queftion donc qui auoit elle commencée a rofou* dreparEucIide, &:pourfuiuieparApolloDius, fans auoir eftèacheuéeparperfonoe , eftoit telle. Ayant trois oa quatre ou plus grand nombre de lignes droites données par pofîtioHj premièrement on demande vn point, du- quel on puifle tirerautant d'autres lignes droites, vne fur chafcune des données, qui façent auec elles des angles donnes, & que le redangle contenu en deux de celles, qui feront ainfi tirées d'vn mefme point., ait la propor- tion^ donnée auec le quarré de la troifiefme , s'il n'y en a que trois; oubien auec le redangle des deux autres, s'il y en a quatreiOubien,s'il y en a cinq, que le parallélépipède compofede trois ait la proportion donnée auec le parais

lelepipede

20

FIRST BOOK

quoniam non est aliquid contcntnm plurihns quavi tribus dimensioni- hus." '"'

Here I beg you to observe in passing that the considerations that forced ancient writers to use arithmetical terms in geometry, thus mak- ing it impossible for them to proceed beyond a point where they could see clearly the relation between the two subjects, caused much obscur- ity and embarrassment, in their attempts at explanation.

Pappus proceeds as follows :

"Acqiiiescimt aiitem his, qui paulo ante talia interpretati sunt ; neque nnum aliquo pacto comprehensibile significantes quod his continetur. Licebit autem per conjiinctas proportiones hœc, & dicere & demonstrare universe in dictis proportionibus, atque his in hunc modum. Si ab aliquo puncto ad positione datas rectas lineas ducantur rectœ lineœ in datis angulis, & data sit proportio conjuncta ex ea, quam habet nna ductarum ad iinam, & altera ad alteram, & alia ad aliani, & rcliqua ad datam lineam, si sint septcm; si vcro octo, & reliqua ad reliquam: punctum continget positione datas lineas. Et similiter quotcumque sint

'"' This may be somewhat freely translated as follows : "The problem of the locus related to three or four lines, about which he (Apollonius) boasts so proudly, giving no credit to the writer who has preceded him, is of this nature: If three straight lines are given in position, and if straight lines be drawn from one and the same point, making given angles with the three given lines; and if there be given the ratio of the rectangle contained by two of the lines so drawn to the square of the other, the point lies on a solid locus given in position, namely, one of the three conic sections.

"Again, if lines be drawn making given angles with four straight lines given in position, and if the rectangle of two of the lines so drawn bears a given ratio to the rectangle of the other two; then, in like manner, the point lies on a conic section given in position. It has been shown that to only two lines there corre- sponds a plane locus. But if there be given more than four lines, the point gen- erates loci not known up to the present time (that is, impossible to determine by common methods), but merely called 'lines'. It is not clear what they are, or what their properties. One of them, not the first but the most manifest, has been examined, and this has proved to be helpful. (Paul Tannery, in the Oeuvres de Descartes, differs with Descartes in his translation of Pappus. He translates as follows : Et on n'a fait la synthèse d' aucune de ces lignes, ni montré qu'elle servit pour ces lieux, pas même pour celle qui semblerait la première et la plus indiquée.) These, however, are the propositions concerning them.

"If from any point straight lines be drawn making given angles with five straight lines given in position, and if the solid rectangular parallelepiped contained by three of the lines so drawn bears a given ratio to the solid rectangular paral- lelepiped contained by the other two and any given line whatever, the point lies on a 'line' given in position. Again, if there be six lines, and if the solid con- tained by three of the lines bears a given ratio to the solid contained by the other three lines, the point also lies on a 'line' given in position. But if there be more than six lines, we cannot say whether a ratio of something contained by four lines is given to that which is contained by the rest, since there is no figure of more than three dimensions."

21

GEOMETRY

impares vel pares mult itn dine, cum hœc, ut dixi, loco ad quatuor lineas respondeant, nullum igitur posuerunt ita ut linea nota sit, &c}^*^

The question, then, the solution of which was begun by Euclid and carried farther by Apollonius, but was completed by no one, is this :

Having three, four or more lines given in position, it is first required to find a point from which as many other lines may be drawn, each making a given angle with one of the given lines, so that the rectangle of two of the lines so drawn shall bear a given ratio to the square of the third (if there be only three) ; or to the rectangle of the other two (if there be four), or again, that the parallelepiped''"^ constructed upon three shall bear a given ratio to that upon the other two and any given line (if there be five), or to the parallelepiped upon the other three (if there be six) ; or (if there be seven) that the product obtained by mul- tiplying four of them together shall bear a given ratio to the product of the other three, or (if there be eight) that the product of four of them shall bear a given ratio to the product of the other four. Thus the question admits of extension to any number of lines.

Then, since there is always an infinite number of different points satisfying these requirements, it is also required to discover and trace the curve containing all such points.'^"' Pappus says that when there are only three or four lines given, this line is one of the three conic sections, but he does not undertake to determine, describe, or explain the nature of the line required'"' when the question involves a greater number of lines. He only adds that the ancients recognized one of them which they had shown to be useful, and which seemed the sim-

'^^' This rather obscure passage may be translated as follows : "For in this are agreed those who formerly interpreted these things (that the dimensions of a figure cannot exceed three) in that they maintain that a figure that is contained by these lines is not comprehensible in any way. This is permissible, however, both to say and to demonstrate generally by this kind of proportion, and in this man- ner : If from any point straight lines be drawn making given angles with straight lines given in position; and if there be given a ratio compounded of them, that is the ratio that one of the lines drawn has to one, the second has to a second, the third to a third, and so on to the given line if there be seven lines, or, if there be eight lines, of the last to a last, the point lies on the lines that are given in position. And similarly, whatever may be the odd or even number, since these, as I have said, correspond in position to the four lines ; therefore they have not set forth any method so that a line may be known." The meaning of the passage appears from that which follows in the text.

'^^^ That is, continued product.

'^^^ It is here that the essential feature of the work of Descartes may be said to begin.

'^'^ See lino 19 on the opposite page.

22

Livre Premier. 3^7

felepipedecoinpofedes deux qufreftcntj&dVne antre ligncdonnée. Ou s'il y en a fîx, que le parallélépipède côpofédetroisaitla proportion donnée auec le parafle- lepipcde des trois autres. Ou s'il y en a fept^que ce qui fe produid lorfqu'on en multiplie quatre Tvne par l'autre, aitlaraifon donnée auec ce qui feproduift par [a multi- plication des trois autres, & encore d'vne autre ligne donnec; Ou s'il y en a huit, que le produit de la multi- plication de quatre ait la proportion donne'e auec le-pro- duit des quatre autres. Et ainfi cete queftiou fe peuc eftendre a tout autre nombre de lignes. Puis a caufe qu'il y Tw toufiours vneinfînite'dediuerspoins qui peuucnt fa- tisfaireacequi eft icy demande, il eft aufly requis de connoiftre, & de tracer la ligne,dans laquelle ils doiuent tousfe trouuer. & Pappus dit que lorfqu'il n'y a que trois ou quatre lignes droites données , c'eft en vne des trois feétions coniques, mais il n'entreprend point de la determiiier, nyde la defcrire. non plus que d'expli- quer celles ou tous ces poins fe doiuent trouuer, lorfquc laqueftioneftpropofeeenvnplus grand nombre de li- gnes. Seulement ilaioufte que les anciens en auoient imagine vne qu'ils monftroient y eftrevtile , mais qui fembloit la plus manifefte, & qui n'eftoit pas toutefois la premiere. Ce qui m'a donne' occafion d'effayer fî par la méthode dont ie me ièrs on peut aller aulTy loin qu'ils ont efte'.

Et premièrement i'ay connu que cete queftion n'eftant Rcfponfc propofee qu'en trois, ou quatre,ou cinq lignes , on peutl'^^T

r I 11/ ^ '"on de

toufiours trouuer les poms cherches par la Géométrie Pappus. fimplci c*eft a dire en ne fe feruant que de la reigle & du

Q,q 2 compas.

23

3©^ JLa Géométrie,

compas, uj ne fmfàm auti;echofe, t|ue ce qui a défia efte. dit; excepteTeuIement lorfqu'il y a cinq lignes données, fi elles font toutes parallèles. Auquel cas, comme aufly lorfquela queftion eft propofee en fix, ou 7, ou 8, ou 9 lignes, on peuttoufiourstrouuer les poins cherchés par la Géométrie des folides j c'eft a dire en y employar«t quelqu^vne des trois fediions coniques. Excepte' feule- ment lorfqu'il y a neuf lignes données, fi elles font toutes parallèles. Auquel cas derechef, 8c encore en 10,11,12, ou 13 hgnes on peut trouuer les poins cherchés par le moyen d'vne hgne courbe qui foit d'vn degré plus cora- pofée que les fed;ions coniques. Excepte'' en treize fi el- les font toutes parallèles , auqueîcas , & en quatorze, i y, 16, & 17 il y faudra employer vne ligne courbe encore d'va degré' plus compofca que la précédente. & ainfî a l'infini.

Puisiay trouuc'auffy, que lorfqu'il ny a que trois ou quatre hgnes données, les poins cherchés fe rencontrent tous , non feulement en l'vnedes trois fedions coni- ques , mais quelquefois aulTy en la circonférence d'vu cercle , ou en vne Hgne droite. Et que lorfqu'il y en a cinq, ou fix, ou fept, ou huit, tous ces poins fe rencon- trent en quelque vne des lignes, qui font dVn degré plus corapofées que les fecStions coniques , & il eft impofîîble d'en imaginer aucune qui ne foit vtile a cete queftioU; mais ils peuuent aufTy derecheffe rencontrer en vne fe- (Stion conique, ou en vn cercle, ou en vne ligne droite. Et s'il y en a neuf, ou i o, ou n , ou 1 2,, ces poins fe ren- contrent en vne hgne, qui ne peut eftrc que d'vn degr^ plus compofée que les précédentes 3 mais toutes celles

qui

24

FIRST BOOK

plest, and yet was not the most important. '''' This led me to try to find out whether, by my own method, I could go as far as they had gone.'''"'

First, I discovered that if the question be proposed for only three, four, or five lines, the required points can be found by elementary geometry, that is, by the use of the ruler and compasses only, and the application of those principles that I have already explained, except in the case of five parallel lines. In this case, and in the cases where there are six, seven, eight, or nine given lines, the required points can always be found by means of the geometry of solid loci,''"' that is, by using some one of the three conic sections. Here, again, there is an exception in the case of nine parallel lines. For this and the cases of ten, eleven, twelve, or thirteen given lines, the required points may be found by means of a curve of degree next higher than that of the conic sections. Again, the case of thirteen parallel lines must be excluded, for which, as well as for the cases of fourteen, fifteen, sixteen, and seventeen lines, a curve of degree next higher than the preceding must be used ; and so on indefinitely.

Next, I have found that when only three or four lines are given, the required points lie not only all on one of the conic sections but some- times on the circumference of a circle or even on a straight line.'"'

When there are five, six, seven, or eight lines, the required points lie on a curve of degree next higher than the conic sections, and it is impossible to imagine such a curve that may not satisfy the conditions of the problem ; but the required points may possibly lie on a conic section, a circle, or a straight line. If there are nine, ten, eleven, or twelve lines, the required curve is only one degree higher than the pre- ceding, but any such curve may meet the requirements, and so on to infinity.

'"*' See lines 5-10 from the foot of page 23.

'^^' Descartes gives here a brief summary of his solution, which he amplifies later.

[40] -pj^jg term was commonly applied by mathematicians of the seventeenth cen- tury to the three conic sections, while the straight line and circle were called plane loci, and other curves linear loci. See Fermât, Isagoge ad Locos Pianos et Solidos, Toulouse, 1679.

'"' Degenerate or limiting forms of the conic sections.

25

GEOMETRY

Finally, the first and simplest curve after the conic sections is the one generated by the intersection of a parabola with a straight line in a way to be described presently.

I believe that I have in this way completely accomplished what Pappus tells us the ancients sought to do, and I will try to give the demonstration in a few words, for I am already wearied by so much writing.

Let AB, AD, EF, GH, ... be any number of straight lines given in position,''"' and let it be required to find a point C, from which straight lines CB, CD, CF, CH, . . . can be drawn, making given angles CBA, CDA, CFE, CHG, . . . respectively, with the given lines, and

'*'' It should be noted that these lines are given in position but not in length. They thus become lines of reference or coordinate axes, and accordingly they play a very important part in the development of analytic geometry. In this con- nection we may quote as follows: "Among the predecessors of Descartes we reckon, besides Apollonius, especially Vieta, Oresme, Cavalieri, Roberval, and Fermât, the last the most distinguished in this field; but nowhere, even by Fermât, had anv attempt been made to refer several curves of difïerent orders simultane- ously to one system of coordinates, which at most possessed special significance for one of the curves. It is exactly this thing which Descartes systematically accomplished." Karl Fink, A Brief History of Mathematics, trans, by Beman and Smith, Chicago, 1903, p. 229.

Heath calls attention to the fact that "the essential difference between the Greek and the modern method is that the Greeks did not direct their efforts to making the fixed lines of a figure as few as possible, but rather to expressing their equations between areas in as short and simple a form as possible." For fur- ther discussion see D. E. Smith, History of Mathematics, Boston, 1923-25, Vol. II, pp. 316-331 (hereafter referred to as Smith).

26

Livre Premier. ^^^

qui font dVn degré plus compofees y peuuentferuir, & ainfî a l'infini.

Au refle la premiere, & la plus fimple de toutes après les fednons coniques , eft celle qu'on peut defcrirepar i'interfeétiond'vne Parabole, &dVne ligne droite, en la façon qui fera tantoft explique'e. En forte que ie penfè auoir entièrement fatisfait a ceque Pappus nous dit auoir efte'chetché'en cecy par les anciens. & ic tafcheray d en mettre la demonftration en peu de mots.car il m'ennuie défia d'en tant efcrire.

Soient A B, A D, E F, G H, &c. plufieurs lignes don- nées par pofition, &: qu'il faille trouuer vn point, comme C, duquel ayant tire'd'autres lignes droites fur les don- nées, comme C B, C D, C F, & C H , en forte que les anglesCBA,CDA,CFE,CHG,&c.foientdonnds,

Qq 3 &

27

3^^ La Géométrie.

&que ce qui eft produit par la multiplication d' vue par- tic de ces lignes, foit efgal a ce qui eft produit par la mul- tiplication des autres, oubien qu'ils ayent quelque autre proportion donnée, car cela ne rend point la queftion pius difficile. Commet Premièrement ie fuppofe la chofe comme defîa faite^ ^'ofedes ^-pour me demeller de la côfulion de toutes ces- lignes, termes ie confidcre l'vne des donne'es, & IVne de celles qu'il «n VE- fauttrouuer, parexemple A B, & C B , comme lesprin- quation cipalcs, & aufquclles ie tafche de rapporter ainfi toutes exemple. Ics autrcs. Quc le fegment de la ligne A B, qui eft entre les poins A & B, foit nommé x. & que B C foit nomme' y, & que toutes les autres lignes données foient prolon- gées, iufques a ce qu'elles couppent ces deux, aufly pro- longées s'il eft befoin, ôcCi elles ne leur font point paral- lèles, comme vous voy es icy qu'elles couppent la ligne A B aux poins A, E, G, & B C aux poins R,S,T. Puis a caufequetousles angles du triangle A RB font donne''s3 la proportion, qui eft entre les coftés A B, & B R, eft auf- fy donnée, & ie la pofe comme de ^ à ^, de façon qu' A B

eftant x, R B fer i *:' &: la toute C R fera y -+- ~ ' à caufe que le poin t B tombe entre C & R^ car fi R tomboit en- tre C & B,C R feroit ;/---{'& fi C tomboit entre B & R,

CR feroit — ^-i-"7* Tout de mefme les trois angles

du triangle D R C font donnel; , & par confequent aufiy la proportion qui eft entre les cofte's C R, & C D , que ie

pofe comme de ;^à r; de façon que C R eftant y -^- -*

CD

28

FIRST BOOK

such that the product of certain of them is equal to the product of the rest, or at least such that these two products shall have a çiven ratio, for this condition does not make the problem any more difficult.

First, I suppose the thing done, and since so many lines are confus- ing, I may simplify matters by considering one of "the given lines and one of those to be drawn (as, for example, AB and BC) as the prin- cipal lines, to which I shall try to refer all the others. Call the segment of the line AB between A and B, x^ and call BC, y. Produce all the other given lines to meet these two (also produced if necessary) pro- vided none is parallel to either of the principal lines. Thus, in the figure, the given lines cut AB in the points A, E, G, and cut BC in the points R, S, T.

Now, since all the angles of the triangle ARB are known,'"' the ratio between the sides AB and BR is known.'"' If we let AB :BR = r :b,

since AB = x, we have RB = — ; and since B lies between C and R '"',

z

/>x we have CR^v + -— • (When R lies between C and B, CR is equal

to y — —, and when C lies between B and R, CR is equal to — y + — )

Again, the three angles of the triangle DRC are known,'*"' and there- fore the ratio between the sides CR and CD is determined. Calling this

ratio z : c, smce CR = y -{--;:> we have CD = " -f- ^:;^- i hen, smce

'"' Since BC cuts AB and AD under given angles.

'^' Since the ratio of the sines of the opposite angles is known.

'"' In this particular figure, of course.

'*"' Since CB and CD cut AD under given angles.

29

GEOMETRY

the lines AB, AD, and EF are given in position, the distance from A to E is known. If we call this distance k, then EB = A- -f- x ; although EB = fe — X when B lies between E and A, and E=^- — k -{- x when E lies between A and B. Now the angles of the triangle ESB being given, the ratio of BE to BS is known. We may call this ratio a : d.

Then BS = '^^^ + ^^' and CS = ^-L+^''^i±_^^'.i-] ^j^^^ g y^^^ between B

G 2

and C we have CS = , and when C lies between B and S

z

we have CS = ~ — — — . The angles of the triangle ESC are

known, and hence, also the ratio of CS to CF, or s : e. Therefore,

ezy -^ de/; -\- i/fx t -i • \ r- i • • ^ T^r^ i

LP = -^ — . Likewise, AG or / is given, and B(j = / — x.

Also, in triangle BGT, the ratio of BG to BT, or ,z : f, is known. There- fore, BT =-^^ ~-^'^" and CT = ^-'' "^-^'^~^\ In triangle TCH, the ratio

z z

of TC to CH, or z : g, is known,'''' whence CH ^ '^^^ ^ ^^- .

i^'i We have

, dk-\-dx

= y + ~

^y-\-dk^dx

and similarly for the other cases considered below.

The translation covers the first eight lines on the original page 312 (page 32 of this edition.

'"' It should be noted that each ratio assumed has ^ as antecedent.

30

5^x

CD fera t^

hex

-. Apres cela pourceque les lignes A F,

A D, &: E F font données par pofition, la diftance qui eft entre les poins A & E eft au fTy donnée, & fi onlanom- me K, on aura E B efgal a k^ -{- x-^ mais ce feroit /^— x , fi le point B tomboit entre E & A;& -- >^-f- .r^fi E tomboit entre A &B. Et pourceque les angles du triangle ESB font tous donnés, la proportion de BE a BS eftaufly donnée, & ie la pofe comme :^à^ , fibienque BS eft

dk>i< dx „ , ^ ^ f^ zy 'i* dk <i>d x

& la toute C S eft

mais ce feroic

\y •- dk -- dx

file point s tomboit entre B &C5& ce feroic

■ - z.y >i* d k 'i* dx

K.

, fi C tomboit entre B^ & S. De plus les trois angles du triangle F S Cfont donne's, 6c en fuite îa

pro-

31

^^* La Géométrie.

proportion de C S à C F, qui foie comme de ^kc, 5c1â

toute C F fera ^^ . En meime taçon AG

que ie nomme /eft donnée, &B G eft /-- x\ & acaufe dutriangleBGTlaproportion de BG la BTefraufîy

fl'-fx

donnée, quifoit comme de :^ à /! &B Tfera — ^ ,&

C T co ^•'^'^{"^ . Puis derechef la proportion de T C a C H eft donnée , acaufe du triangle T C H , & lapofant

comme de^agy on aura C H 30 — .

EtainfivousvoyeX qu'en tel nombre de lignes don- nées par pofition qu'on puifîeauoir, toutes les lignes ti- rées defTus du point C a angles donne's fuiuant la teneur delaqueftion ,fepeuuent toujours exprimer chafcune par trois termes j dont l'vn eft compofe'de la quantité in- connue j', multipliée , ou diuifce par quelque autre connue^ & l'autre de la quantité' inconnue x, aufly mul- tiplie'e ou diuifce par quelque autre connue , & le trolîel^ me d'vne quantité toute connue. Excepte feulement lî elles fontparalleles joubien a la ligne AB, auquel cas le terme compofe de la quantité AT fera nul ; oubien a la li- gne C B, auquel cas celuy qui eft compofe'de la quantité" y fera nulj ainfi qu'il eft trop manifeftc pour que ie m are- fte a l'expliquer. Et pour les fignes 4-, &: -, qui fe ioi- gnent à ces termes, ilspeuuent eftre changes en toutes les façons imaginables.

Puis vous voyés aufly, que multipliant plufîeurs de ces lignes l'vne par l'autre, les quantités x3cy, qui fe trouuent dans le produit, n'y peuuentauoir que chafcu- ne autant de dimenfions, qu'il y a eu deligues, al'expli-

cation

32

FIRST BOOK

And thus you see that, no matter how many Hues are given in posi- tion, the length of any such hne through C making given angles with these lines can always be expressed by three terms, one of which coh- sists of the unknown quantity y multiplied or divided by some known quantity ; another consisting of the unknown quantity .r multiplied or divided by some other known quantity ; and the third consisting of a known quantity.''"' An exception must be made in the case where the given lines are parallel either to AB (when the term containing .r van- ishes), or to CB (when the term containing 3' vanishes). This case is too simple to require further explanation. '°"' The signs of the terms may be either + or — in every conceivable combination.''''

You also see that in the product of any number of these lines the degree of any term containing x or y will not be greater than the num- ber of lines (expressed by means of .r and y) whose product is found. Thus, no term will be of degree higher than the second if two lines be multiplied together, nor of degree higher than the third, if there be three lines, and so on to infinity.

'^"^ That is, an expression of the form ax + by + c, where a, b, c, are any real positive or negative quantities, integral or fractional (not zero, since this exception is considered later).

[50] Yj-jg following problem will serve as a very simple illustration : Given three parallel lines AB, CD, EF, so placed that AB is distant 4 units from CD, and CD is distant 3 units from EF ; required to find a point P such that if PL, PM, PN

be drawn through P, making angles of 90°, 45°, 30°, respectively, with the parallels. Then PM-= PL.PN.

Let PR = y, then PN = 2y, PM = V2 ( v + 3) , PL = j + 7. If PM " = PN . PL,

we have V^^ i' + >^) | = 2v ( J + 7) , whence :v = 9. Therefore, the point P lies on

the line XY parallel to EF and at a distance of 9 units from it. Cf. Rabuel, p. 79. '°'' Depending, of course, upon the relative positions of the given lines.

2>l

GEOMETRY

Furthermore, to determine the point C, but one condition is needed, namely, that the product of a certain number of hues shall be equal to, or (what is quite as simple), shall bear a given ratio to the product of certain other lines. Since this condition can be expressed by a single equation in two unknown quantities,'"'' we may give any value we please to either .v or y and find the value of the other from this equation. It is obvious that when not more than five lines are given, the quantity x, which is not used to express the first of the lines can never be of degree higher than the second.'"^'

Assigning a value to 3', we have x- =-^ ± a.v ±: h-, and therefore x can be found with ruler and compasses, by a method already explained.'"' If then we should take successively an infinite number of different values for the line y, we should obtain an infinite number of values for the line .r, and therefore an infinity of different points, such as C, by means of which the required curve could be drawn.

This method can be used when the problem concerns six or more lines, if some of them are parallel to either AB or BC, in which case

'""' That is, an indeterminate equation. "De plus, à cause que pour determiner le point C, il n'y a qu'une seule condition qui soit requise, à sçavoir que ce qui est produit par la multiplication d'un certain nombre de ces lignes soit égal, ou (ce qui n'est de rien plus mal-aisé) ait la proportion donnée, à ce qui est produit par la multiplication des autres ; on peut prendre à discretion l'une des deux quantitez inconnues x ou y, & chercher l'autre par cette Equation." Such variations in the texts of different editions are of no moment, but are occasionally introduced as matters of interest.

''^^' Since the product of three lines bears a given ratio to the product of two others and a given line, no term can be of higher degree than the third, and there- fore, than the second in x.

'^^' See pages 13, et seq.

34

Livre Premier. 5^5 .

cation defquelles elles feruent , qui ont elle'' ainfî multi- pliées: enforce qu'elles n'auront iaraais plus de deux dî- menfious, en ce qui ne fera produit que par la multipli- cation de deux lignes; ny plus de trois , en ce qui ne fera produit que par la multiplication de trois , & ainfi a l'in- fini .

De plus, a caufe que pour determiner le point C, il o^^^ou^c n'ya qu'vne feule condition qui foitrequife , à fçauoir que ce que ce qui eft produit par la multiplication d'vn certain ^^°^^ ' nombre de ces lignes foit efgal , ou Ccequi n eft de rien plan lorji plus malayfe] ait la proportion donnée , à ce qui eft pro- "^l-'^^ ^ duit par la multiplication des autres; on peut prendre api^opofé

*"^ , 1 1 . , . - en plus de

difcretion T vne des deux quantités mconnues x ou y , & j lignes. chercher l'autre par cete, Equation, en laquelle il eft eui- dent que lorfque la queftion n eft point propofee en plus decinqlignes, la quantité a: qui ne ferc point a Icxpref- (îon de la premiere peut toufîours n'y auoir que deux di- menfious. de façon que prenant vne quantité connue pourjy, il ne reftera que xxyi-hou-- ax-{- ou — bb, &c ainfî on pourra trouuer la quantité x auec la reigle &le compas, en la façon tantoft explique'e. Mefme prenant faccelîîuement infinies diuerfes grandeurs pour la ligne y y onentrouneraauffyiniSnies pourlahgne Ar,&ain{ion auravncinfiniteMediuerspoins , tels que celuy qui eft marqué C , par le moyen defquels on defcrira la ligne courbe demandée.

11 fe peut faire aufTy, la queftion eftant propofe^e en fîx, ou plus grand nombre de lignes^ s'il y en a entre les don- nées, qui foient parallèles a B A, ou B C , quel'vne des deux quantités x ou y n'ait que deux dimenfîons en

Rr TEqua-

35

^14 i-A GEOMETRIE^

TEquation, Se ainfî qu'on puifTe trouuuer le point C aaec lareigle &: le compas. Mais au contraire fi elles font tou- tes parallèles , encore que la queftion ne foit propofee qu'en cinq lignes, ce point C ne pourra ainfi eftre trou- ue', a caufe que la quantité x ne fe trouuant point en tou- te rEquation,il ne fera plus permis de prendre vne quan- tité connue pour celle qui eft nommeej' , mais ce fera elle qu'il faudra chercher. Et pource quelle aura trois di- menfions,on nelapourra trouuer qu'en tirant la racine dVn€ Equation cubique, cequi ne fe peut généralement faire fans qu'on y employe pour le moins vne fedion co- nique. Et encore qu'il y ait iufques a neuf lignes don- nées,pourvûqu'elles ne foient point toutes parallèles, oiî peut toufiours faire que l'Equation ne monte que iufques auquarrédequarré. au moyen dequoy on lapeutauffy toufiours refoudre par les fedtions coniques, en la façon que i'expliqueraycy après. Et encore qu'il y en ait iuf^ ques a treize , on peut toufiours faire qu'elle ne nionte que iufques au quarré de cube, en fuite de quoy on la peut refoudre par le moyen d'vne ligne , qui n'eft que d'vn degré' plus compofée que les feétions coniques, en la façon que i'exphquerayauflycy après. Et cecy eft la premiere partie de cequei'auoisicyademonftrer^ mais auant que ie pafi^e a la féconde il eft befoin que ie- die quelque chofe en general delà nature des lignes cour- bes.

LA

36

FIRST BOOK

either x or y will be of only the second degree in the equation, so that the point C can be found with ruler and compasses.

On the other hand, if the given lines are all parallel even though a question should be proposed involving only five lines, the point C can- not be found in this way. For, since the quantity x does not occur at all in the equation, it is no longer allowable to give a knowni value to y. It is then necessary to find the value of 3'.'^'^ And since the term in y will now be of the third degree, its value can be found only by finding the root of a cubic equation, which cannot in general be done without the use of one of the conic sections.'^"'

And furthermore, if not more than nine lines are given, not all of them being parallel, the equation can always be so expressed as to be of degree not higher than the fourth. Such equations can always be solved by means of the conic sections in a way that I shall presently explain.'"'

Again, if there are not more than thirteen lines, an equation of degree not higher than the sixth can be employed, which admits of solution by means of a curve just one degree higher than the conic sections by a method to be explained presently.'^*'

This completes the first part of what I have to demonstrate here, but it is necessary, before passing to the second part, to make some general statements concerning the nature of curved lines.

''"''' That is, to solve the equation for y.

''"' See page 84.

i="i See page 107.

^^^ This line of reasoning may be extended indefinitely. Briefly, it means that for every two lines introduced the equation becomes one degree higher and the curve becomes correspondingly more complex.

37

BOOK SECOND

Geometry

BOOK II

On the Nature of Curved Lines

THE ancients were familiar with the fact that the problems of geom- etry may be divided into three classes, namely, plane, solid, and linear problems.'^"' This is equivalent to saying that some problems require only circles and straight lines for their construction, while others require a conic section and still others require more complex curves.'*'' I am surprised, however, that they did not go further, and distinguish between different degrees of these more complex curves, nor do I see why they called the latter mechanical, rather than geometrical.'"' If we say that they are called mechanical because some sort of instru- ment'"*' has to be used to describe them, then we must, to be consistent,

[59] (-£ Pappus, Vol. I, p. 55, Proposition 5, Book Til : "The ancients consid- ered three classes of geometric problems, which they called plane, solid, and linear. Those which can be solved by means of straight lines and circumferences of circles are called plane problems, since the lines or curves by which they are solved have their origin in a plane. But problems whose solutions are obtained by the use of one or more of the conic sections are called solid problems, for the surfaces of solid figures (conical surfaces) have to be used. There remains a third class which is called linear because other 'lines' than those I have just described, having diverse and more involved origins, are required for their construction. Such lines are the spirals, the quadratrix, the conchoid, and the cissoid, all of which have many impor- tant properties." See also Pappus, Vol. I, p. 271.

'""^ Rabuel (p. 92) suggests dividing problems into classes, the first class to include all problems that can be constructed by means of straight lines, that is, curves whose equations are of the first degree ; the second, those that require curves whose equations are of the second degree, namely, the circle and the conic sec- tions, and so on.

'"^ Cf. Encyclopedic on Dictionnaire Raisonne des Sciences, des Arts et des Metiers, par une Société de gens de lettres, mis en ordre et publiées par M . Diderot, et quant à la Partie Mathématique par M. d'Alcmbert, Lausanne and Berne," 1780. In substance as follows : "Mechanical is a mathematical term designating a con- struction not geometric, that is, that cannot be accomplished by geometric curves. Such are constructions depending upon the quadrature of the circle.

The term, mechanical curve, was used by Descartes to designate a curve that cannot be expressed by an algebraic equation." Leibniz and others call them transcendental.

1"'^ "Machine."

40

Geome- tric.

Livre Secokd. Sif

GEOMETRIE.

LIVRE SECOND.

^e la nature des lignes courhes,

T E s anciens ont fore bien remarque , qu'entre les -■— 'Problefmes de Géométrie, les vns font plans , les au- Quelles tresfolidesj&lesautreslineaircs, c'eil adire^queles vns ["J^^^f peuuenteftreconflruits, eu ne traçant que des lignes courbes droites, &:descerclesjau lieu que les autres ne le peu- peuTV uent eftre, qu'on n'y employe pour le moins quelque fe- ^^uoir en d:ion conique, ni enfin les autres , qu on n'y employe '^""^" quelque autre ligne plus compofee. Mais ie m'eftonne de ce qu'ils n'ont point outre cela difliugué diuers de- grees entre ces lignes plus compofées, & ie ne fçaurois comprendre pourquoy ils les ont nommées mecl^ni- ques, plutoft que Géométriques. Carde dire que c'ait efte'', a caufe qu'il efV befoin de fe fèruir de quelque ma- chine pour les defcrire, il faudroit reietter par melrne raifon les cercles & les lignes droitesjvû qu'on ne les de- fcrit fur le papier qu'auec vn compas, & vne reigle, qu'on peut auffy nommer des machines. Ce n'eft pas non plus, a caufe que les inftrumens, quiferuent a les tracer^eftanc plus compofe's que la reigle & le compas , ne peuueut eftre fî iuftes; car il Eiudroit pour cete raifon les reietter des Mechaniques, où la iultelTe des ouurages qui fortent delamaineftdefirec; plutoft que de la Géométrie , ou c'cft feulement la iufteile du raifonnemct qu'on recher-

Rr 2 che,

41

3'<^ La Géométrie.

che, & qui peut fans doute eftre^ufly parfaite touchant CCS lignes , que touchant les autres. le ne diray pas aufly, que ce foit a caufe qu'ils n*ont pas voulu augmenter le nombre de leurs demandes , & qu'ils fe fontcontentés qu'on leur accordaft , qu*ils puflent ioindre deux poins donnés par vne ligne droite , & defcrire vn cercle d'wn centre donne, qui pafîaft par vn point donne.carils n'ont point fait de fcrupule de fuppofer outjr e ceIa,pour traiter des fedîions coniques , qu*on puft coupper tout cône donnd'parvn plan donne. &iln*eft befoin de rien fup- pofer pour tracer toutes les lignes courbes , que ie pre- tens icy d'introduire; finon que deux ou plulîeurs lignes pniflent eftre meues IVne par l'autre , & que leurs inter- férions éo marquent d'autres ^; ce qui ne me paroift en rien plus difficile. Il eft vray qu'ils n ont pas aufly entiè- rement receu les fed:ions coniques en leur Géométrie, & ie ne veux pas entreprendre de changer les noms qui ont efte^approaue's par Ivfàge; mais il eft, ce me fèmble, très clair, que prenant comme on fait pour Géométri- que ce qui eft precis & exad: , & pour Mechanique ce qui ne Teft pas ; & confiderant la Géométrie comme vne fcience, qui enfeigne généralement a connoiftre les mefures de tous les cors, on n'en doit pas plutoft exclure les lignes les plus com pofees que les plus limples, pourvu qu'on les puiflc imaginer eftre defcrites par vn mouue- ment continu, ou par plufieurs qui s'entrcfuiuent & dont les derniers foient entièrement règles par ceu:: qui les precedent, car par ce moyen on peut toufîotirs auoir vue connoiftance exaéte de leur mefure. Mais peuteftre que ce qui a empefche' les anciens Géomètres de reçe-

uou:

42

SECOND BOOK

reject circles and straight lines, since these cannot be described on paper without the use of compasses and a ruler, which may also be termed instruments. It is not because the other instruments, being more complicated than the ruler and compasses, are therefore less accurate, for if this were so they would have to be excluded from mechanics, in which accuracy of construction is even more important than in geometry. In the latter, exactness of reasoning alone'""' is sought, and this can surely be as thorough with reference to such lines as to simpler ones.'"' I cannot believe, either, that it was because they did not wish to make more than two postulates, namely, (1) a straight line can be drawn between any two points, and (2) about a given center a circle can be described passing through a given point. In their treat- ment of the conic sections they did not hesitate to introduce the assump- tion that any given cone can be cut by a given plane. Now to treat all the curves which I mean to introduce here, only one additional assump- tion is necessary, namely, two or more lines can be moved, one upon the other, determining by their intersection other curves. This seems to me in no way more difficult. '°^'

It is true that the conic sections were never freely received into ancient geometry, '°°' and I do not care to undertake to change names confirmed by usage ; nevertheless, it seems very clear to me that if we make the usual assumption that geometry is precise and exact, while mechanics is not f^ and if we think of geometry as the science which furnishes a general knowledge of the measurement of all bodies, then we have no more right to exclude the more complex curves than the simpler ones, provided they can be conceived of as described by a con- tinuous motion or by several successive motions, each motion being completely determined by those which precede ; for in this way an exact knowledge of the magnitude of each is always obtainable.

'"'^ An interesting question of modern education is here raised, namely, to what extent we should insist upon accuracy of construction even in elementary geometry.

'"' Not only ancient writers but later ones, up to the time of Descartes, made the same distinction ; for example, Vieta. Descartes's view has been universally accepted since his time.

'"^' That is, in no way less obvious than the other postulates.

'*°' Because the ancients did not believe that the so-called constructions of the conic sections on a plane surface could be exact.

'"' Since it is not possible to construct an ideal line, plane, and so on.

43

GEOMETRY

Probably the real explanation of the refusal of ancient geometers to accept curves more complex than the conic sections lies in the fact that the first curves to which their attention was attracted happened to be the spiral, '""' the quadratrix,'"*' and similar curves, which really do belong only to mechanics, and are not among those curves that I think should be included here, since they must be conceived of as described by two separate movements whose relation does not admit of exact determination. Yet they afterwards examined the conchoid/"" the cissoid/''' and a few others which should be accepted; but not knowing much about their properties they took no more account of these than of the others. Again, it may have been that, knowing as they did only a little about the conic sections,'"' and being still ignorant of many of the possibilities of the ruler and compasses, they dared not yet attack a matter of still greater difficulty. I hope that hereafter those who are clever enough to make use of the geometric methods herein suggested will find no great difficulty in applying them to plane or solid problems. I therefore think it proper to suggest to such a more extended line of investigation which will furnish abundant opportunities for practice.

Consider the lines AB. AD, AF, and so forth (page 46), which we may suppose to be described by means of the instrument YZ. This instrument consists of several rulers hinged together in such a way that YZ being placed along the line AN the angle XYZ can be increased or decreased in size, and when its sides are together the points B, C, D, E, F, G, H, all coincide with A ; but as the size of the angle is increased,

'"^' See Heath, History of Greek Mathematics (hereafter referred to as Heath), Cambridge, 2 vols., 1921. Also Cantor, Vorlesungen ilber Geschichte der Mathe- niatik, Leipzig-, Vol. I (2), o. 263, and Vol. H (1), pp. 765 and 781 (hereafter referred to as Cantor).

'«°i See Heath, I, 225 ; Smith, Vol. H, pp. 300, 305.

'™i See Heath, I, 235, 238 ; Smith, Vol. H, p. 298.

"'' See Heath, I, 264; Smith, Vol. U, p. 314.

'"^ They really knew much more than would be inferred from this statement. In this connection, see Taylor, Ancient and Modern Geometry of Conies, Cam- bridge, 1881.

44

Li vre Second. ^^^

uoir celles qui eftoient plus compofees que lesfedions coniques, c eft que les premieres qu'ils ont confiderees, ayant par hafard efte la Spirale, la Quadratrice , & fein- blables , qui n'appartienent véritablement qu'aux Me- chaniquesj&nefont point du nombre de celles que ie penfe deuoir icy eftre receues, a caufe qu'on les imagine defcrites par deux mouuemens fepares, & qui n*ont en- tre eux aucun raport qu'on puifTe raefurer exadtement, bienqu'ils ayent après examiné la Conchoide , la Ciflbi- de, & quelque peu d'autres qui en font, toutefois a cau- fe qu'ils n'ont peuteftre pas afles remarqué leurs pro- priete's , ils n'en ont pas fait plus d'eftat que des premie- res. Oubien c'eft que voyant , qu'ils ne connoiffoient encore , que peu de chofes touchant les ferions coni- ques, &qu 'illeur enreftoitmefme beaucoup, touchant ce qui fe peut faire auec la reigle & le compas , qu'ils ignoroient, ils ont creu ne deuoir point entamer de ma- tière plus difficile. Mais pourceque i'efpere que d'orena- uant ceux qui auront Tadreffe de fe feruir du calculGeo- metriqueicy propofe'', netrouueront pas aire's dequoy s'arefter touchant les problefmes plans, ou folidesj ie croy qu'il eft a propos que ie lesinuite a d'autres re- cherches , où ils ne manqueront iaraais d'exercice.

Voyesleslignes AB,A D, A F, & ferablables queie fuppofe anoir efté defcrites par l'ayde de l'inftrumenc Y Z, qui eft compofé de plufîeurs reigles tellement ioin- tes, que celle qui eft marquee YZ eftant areftée fur la ligne A N,on peut ouurir & fermer l'angle X Y Z; & que lorfqu'ileft tout fermé , les poins B, C, D, F, G, H font tous aflemblés au point A ; mais qu'a mefure qu'on

Rr 5 l'oaure.

45

5IS

La Geometrte.

Tomire, la reigle B C, qui eft iointe a angles droits auec XYau point B, poufTevers Z la reigle CD, qui coule fiirY Zenfaifant toufiours des angles droits auec elle, 82 C D poufle D H, qui coule tout de mefme fur Y X en de- meurant parallèle a B Q D E poufTe EF,E F poufTe F G, cellecy poufTe G H. & on en peut conceuoir vne infinite d'autres*, qui fe pouflent confequutiuement en mefme façon, & dont les vnesfacent toufiours les mefmes an- gles auec Y X, & les autres auec Y Z. Or pendant qu'on ouureainfi l'angle XYZ,le point B dcfcritlaligne AB, qui eft vn cercle, &les autres poins D^F, H, ou fe font les interfe(5tions des autres reigles , defcriuent d'autres lignes courbes AD, A F, A H, dont les dernières font par ordre plus copofc'es que la premiere, & cellecy plus que le cercle, mais ie ne voy pas ce qui peut empefcher, qu'on ne concoiueauffy nettement j & auflTy diftindte- ment la defcripcion de cete premiere^que du cercle , ou

du

46

SECOND BOOK

the ruler BC, fastened at right angles to XY at the point B, pushes toward Z the ruler CD which slides along YZ always at right angles. In like manner, CD pushes DE which slides along YX always parallel to BC ; DE pushes EF ; EF pushes EG ; EG pushes GH, and so on. Thus we may imagine an infinity of rulers, each pushing another, half of them making equal angles with YX and the rest with YZ.

Now as the angle XYZ is increased the point B describes the curve AB, which is a circle ; while the intersections of the other rulers, namely, the points D, E, H describe other curves, AD, AE, AH, of which the latter are more complex than the first and this more complex than the circle. Nevertheless I see no reason why the description of the first'"^ cannot be conceived as clearly and distinctly as that of the circle, or at least as that of the conic sections ; or why that of the sec- ond, third,'''' or any other that can be thus described, cannot be as clearly conceived of as the first; and therefore I see no reason why they should not be used in the same way in the solution of geometric problems.™

'"' That is, AD.

"*i That is, AF and AH.

'^^' The equations of these curves may be obtained as follows: (1) Let

YA = YB = a, YC = .r, CD —y, YD = ^; then z : x = x : a, whence s = — • Also s-==x- + y-; therefore the equation of AD is x* = a"(x- + y-). (2) Let YA = YB = a, YE = x, EF = v, YF = :r. Then z : x = x : YD, whence

YD = ^. Also

.r : YD = YD : YC, whence YC == '— -^ x = — . •

z- z .

But YD : YC = YC : a, and therefore

4i

Also, z' = x- + y^. Thus we get, as the equation of AF,

'd! = X- + y-, or x^ = a- (x- + y- ) \

(3) In the same way, it can be shown that the equation of AH is

.r'- = a"(x- + y-)^. See Rabuel, p. 107.

47

GEOMETRY

I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one,™ but I think the best way to group together all such curves and then classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric." that is, those which admit of precise and exact measurement, must bear a definite relation'"' to all points of a straight line, and that this relation must be expressed by means of a single equation.''"' If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class,''"' which contains only the circle, the parabola, the hyperbola, and the ellipse ; but when the equation contains one or more terms of the third or fourth degree'*"' in one or both of the two unknown quantities'"'' (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class ; and if the equation con- tains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely.

[78] "Qui seroient de plus en plus composées par degrez à l'infini." The French quotations in the footnotes show a few variants in style in different editions.

'"' That is, a relation exactly known, as, for example, that between two straight lines in distinction to that between a straight line and a curve, unless the length of the curve is known.

'™' It will be recognized at once that this statement contains the fundamental concept of analytic geometry.

''"' "Du premier & plus simple genre," an expression not now recognized. As now understood, the order or degree of a plane curve is the greatest number of points in which it can be cut by any arbitrary line, while the class is the greatest number of tangents that can be drawn to it from any arbitrary point in the plane.

'*"' Grouped together because an equation of the fourth degree can always be transformed into one of the third degree.

'"' Thus Descartes includes such terms as .r-^', .v-.v-, . . as well as x^^, y*

48

Livre Second. 519

du moms que des fedtions coniques- ny ce qui peut em- pefcher, qu'on ne concoiue la féconde , & la troifiefme, & toutes les autres, qu'on peut defcrire, aufTy bien que lapremi&re; ny par consequent qu'on ne les recoiue toutes en mefme façon, pour feruir aux fpeculations de Géométrie.

le pourrois mettre icy plufieurs autres moyens pour La ùcoa tracer &conçeuoir des liraes courbes, qui feroient <Je "^^ "^'^^"^" plus en plus compolées par degrés a 1 infini, mais pour tes les li- comprendreenfèmble toutes celles, qui font en la natu- ^""'^^^"'^' re , & les diftiuguer par ordre en certains genres j ie ne certains fçache rien de meilleur que de dire que tous les poins, de ^Tcol'. "^ celles qu'on peut nommer Géométriques, c'eft a dirc"°'^^^ ^^ qui tombent fous quelque meflire precife ôc exad:e, ont wont necefTairement quelque rapport a tous les poins dVne-^°"^ '^""^

1- j- • n • / polos a

hgne droite, qui peut eirre exprime par quelque equa-^euxdes tion, en tous par vnemefme. Et que lorfque ceteequa^ Jj^SJ'.^^g tion ne monte que iufques au recftangle de deux quanti- tés indéterminées, oubien au quarréd'vnemefine, la li- gne courbe eft du premier & plus fîmpie genre, dans le- quel il ny a que lé cercle, la parabole, l'hyperbole , & TEllipfe qui foient comprifes. mais que lorfque l'équa- tion monte^iufques a la trois ou quatriefme dimenfion des deux, ou de Tvne des deux quantite^s indéterminées, car il en faut deux pour expliquer icy le rapport d\n point a vn autre, elle eft du fecondrSc que lorfque l'équa- tion monte iufques a la y ou fixiefme dimenfion, elle- eft du troifiefme; & ainli des autres a l'infini.

Comme fi ie veux fçauoir de quel genre eft la ligne E C;, que l'imagine eftre defcrite par i'interfedion de la-

reigîe-

49

320

La GEOMETRIE.

reigle G L, & du plan rediligne G N K L, dont le cofté K N eft indefiniement prolongé vers G , & qui eftant meu fur le plan de deflbus en ligne droite , c'eft a dire en telle forte que fon diamètre. KL fe trouue toufîours ap- pliqueTur quelque endroit de la ligne B A prolongée; de part & d'autre, fait mouuoir circulairement cete reigle G L autour du point G, a caufe quelle luy eft tellement iointe quelle pafle toufîours par le point L. le choiiîs vne ligne droite, comme A B,pour rapporter a fes diuers poinstousceuxdecetelignecourbeEG, &en cete li- gne A B ie choifis vn point, comme A, pour commencer par luy ce calcul. le dis que ie choifis &rvn& l'autre, a caufe qu'il eft libre de les prendre tels qu'on veult. car encore qu il y ait beaucoup de choix pour rendre l'équa- tion plus courte, &: plus ayfécj toutefois en quelle façon qu'ouïes prene, on peut toufîours faire que la ligne pa- roiflè de meûne genre, ainfî qu'il eft ayfe^ a demonftrer.

Apres

50

SECOND BOOK

Suppose the curve EC to be described by the intersection of the ruler GL and the rectihnear plane figure CNKL, whose side KN is produced indefinitely in the direction of C, and which, being moved in the same plane in such a w^ay that its side'^'' KL always coin- cides with some part of the line BA (produced in both directions), imparts to the ruler GL a rotary motion about G (the ruler being hinged to the figure CNKL at L)."" If I wish to find out to what class this curve belongs, I choose a straight line, as AB, to which to refer all its points, and in AB I choose a point A at which to begin the investigation.'"' I say "choose this and that," because we are free to choose what we will, for, while it is necessary to use care in the choice in order to make the equation as short and simple as possible, yet no matter what line I should take instead of AB the curve would always prove to be of the same class, a fact easily demonstrated.''"'

^^^ "Diamètre."

^*^^ The instrument thus consists of three parts, (1) a ruler AK of indefinite length, fixed in a plane ; (2) a ruler GL, also of indefinite length, fastened to a pivot, G, in the same plane, but not on AK; and (3) a rectilinear figure BKC, the side KG being indefinitely long, to which the. ruler GL is hinged at L, and which is made to slide along the ruler GL.

'*^^ That is, Descartes uses the point A as origin, and the line AB as axis of abscissas. He uses parallel ordinates, but does not draw the axis of ordinates.

'*^' That is, the nature of a curve is not affected by a transformation of coordinates.

51

GEOMETRY

Then I take on the curve an arbitrary point, as C, at which we will suppose the instrument applied to describe the curve. Then I draw through C the line CB parallel to GA. Since CB and BA are unknown and indeterminate quantities, I shall call one of them y and the other x. To the relation between these quantities I must consider also the known quantities which determine the description of the curve, as GA, which I shall call a ; KL, which I shall call h ; and NL parallel to GA, which I shall call c. Then I say that as NL is to LK, or as c is to h, so CB, or

y, is to BK, which is therefore equal to - y. Then BL is equal to

- y — h, and AL is equal to x -\- -y — h. Moreover, as CB 13 to LB,

b . l> , .

that is, as -v is to - T — h, so AG or a is to LA or x -\- - y — h. Multi-

ah plying the second by the third, we get — y — ah equal to

b , ,

xy^- y — by,

which is obtained by multiplying the first by the last. Therefore, the

required equation is

ex

y '= cy 7- 3' + ^v — <3^.

52

Livre Second.

321

A près cela prenant vn point a difcretion dans la courbe, comme C, fur lequel ie fuppofe que l'inflrument qui ferc a la defcrire eft applique', ie tire de ce point C- la ligne C B parallèle a G A, &:pourceque C B & B A font deux quantités indéterminées & inconnues , ie les nomme Tvne^ & l'autre a;, maisaffin de trouucr le rapport de IVneàrautrcjieconfidere aufTy les quantités connues qui déterminent la defcription de ccte ligne courbe, comme G A que ie nomme ^, K L que ie nomme b , & N L parallele'a G A que ie nofnme c. puis ie dis^ comme NLeftàLK,oucà/^,ainriCB,ou;^, eftàBK, qui eft

^ b b

parconfequent-;;': ôcBLeft— y-b, &c A Left a: -H

b h

~y — b, de plus comme C B eft à L B, ou j/ à -jy-b, ainfî

a^ ou G A, eft a L A, ou a: -^ -^y -b, de façon que mul-

S f tipliant

S3

J^ La Géométrie.

tip liant la féconde par la troifrefme on produit 77 - ai^

qui eft efgale à xy-h^^yy - by qui fe produit en multi- pliant la premiere par la dernière. & âinfî Tequation qu'il faUoittrouuereft .

y y 30 cy- ^y -h ay - ae. de laquelle onconnoift que la ligne EC eft da premier genre , comme en effedl elle n eft autre qu vne Hy- perbole.

Que fî en Tinftrument qui fèrt a la defcrire on fait qu'au lieu de la ligne droite C N K, ce fdit cete Hyper- bole, ou quelque autre ligne courbe du premier genre, qui termine le plan C NKL; Tinterfedtion de cetc ligne & de la reigle G L defcrira, au lieu de l'Hyperbole E C, vne. autre ligne courbe, qui fera du fécond genre. Com^ me fî C N K eft vu cercle, dont L fôit le centre , on de- fcrira la premiere Conchoidedes anciens j &fî ceft vne Parabole dont le diamètre foit K B , oii defcrira la ligne courbe, que i'ay tantoft diteftre la premiere, & Ia*plus fîmplè pourla^eftion dePappus,lorfqu'il n'y a que cinq lignes droites données par pofîtion. Mais lî au lieu d vne de ces lignes courbes du premier genre , c'en eft vue du fécond, qui termine le plan C N K L, on en defcrira par fon moyen vne du troifîefme, ou fi c'en eff vne du troifi- cfme, onen defcrira vne du quatriefme, & ainfi a l'infini, comme il eft fort ayfea connoiftr^ par le calcul. Et en quelque autre façon, qu'on imagine la defcriptiou d'vne ligne courbe , pourvûqu'elle foit du nombre de celles qucictiomme Géométriques , on pourra toufiourstrou-

uer

54

SECOND BOOK

From this equation we see that the curve EC belongs to the first class, it being, in fact, a hyperbola.'""

If in the instrument used to describe the curve we substitute for the rectilinear figure CNK this hyperbola or some other curve of the first class lying in the plane CNKL, the intersection of this curve with the ruler GL will describe, instead of the hyperbola EC, another curve, which will be of the second class.

Thus, if CNK be a circle having its center at L, we shall describe the first conchoid of the ancients, '^^ while if we use a parabola having KB as axis we shall describe the curve which, as I have already said, is the first and simplest of the curves required in the problem of Pappus, that is, the one which furnishes the solution when five lines are given in position.'"*'

^^^ Ci. Briot and Bouquet, Elements of Analytical Geometry of Two Dimen- sions, trans, by J. H. Boyd, New York, 1896, p. 143.

The two branches of the curve are determined by the position of the triangle CNKL with respect to the directrix AB. See Rabuel, p. 119.

Van Schooten, p. 171, gives the following construction and proof: Produce AG to D, making DG =: EA. Since E is a point of the curve obtained when GL coincides with GA, L with A, and C with N. then EA = NL. Draw DP parallel to KG. Now let GCE be a hyperbola through E whose asymptotes are DP and PA. To prove that this hyperbola is the curve given by the instru- ment described above, produce BC to cut DP in I, and draw DH parallel to AF

meeting BC in H. Then KL : LN = DH : HL But DH = AB = x, so we may

write b : c = x : HI, whence HI = ^, IB — a + c ^, IC = o + c — -7 y.

000

But in any hyperbola IC.BC = DE.EA, whence we have (a + c i- —y)y^ac,

cxy or y^ ^ cy -r' + ay — ac. But this is the equation obtained above, which is

therefore the equation of a hyperbola whose asymptotes are AP and PD.

Van Schooten, p. 172, describes another similar instrument : Given a ruler AB pivoted at A, and another BD hinged to AB at B. Let AB rotate about A so that D moves along LK ; then the curve generated by any point E of BE will be an ellipse whose semi-major axis is AB + BE and whose semi-mmor axis is AB — BE.

'"^ See notes 59 and 70.

'**' Por a discussion of the elliptic, parabolic, and hyperbolic conchoids see Rabuel, pp. 123, 124.

55

GEOMETRY

If, instead of one of these curves of the first class, there be used a curve of the second class lying in the plane CNKL, a curve of the third class will be described ; while if one of the third class be used, one of the fourth class will be obtained, and so on to infinity.'""' These state- ments are easily proved by actual calculation.

Thus, no matter how we conceive a curve to be described, provided it be one of those which I have called geometric, it is always possible to find in this manner an equation determining all its points. Now I shall place curves whose equations are of the fourth degree in the same class with those whose equations are of the third degree ; and those whose equations are of the sixth degree'""' in the same class with those whose equations are of the fifth degree""' and similarly for the rest. This classification is based upon the fact that there is a general rule for reducing to a cubic any equation of the fourth degree, and to an equa- tion of the fifth degree'"'' any equation of the sixth degree, so that the latter in each case need not be considered any more complex than the former.

It should be observed, however, with regard to the curves of any one class, that while many of them are equally complex so that they may be employed to determine the same points and construct the same problems, yet there are certain simpler ones whose usefulness is more limited. Thus, among the curves of the first class, besides the ellipse, the hyperbola, and the parabola, which are equally complex, there is also found the circle, which is evidently a simpler curve ; while among those of the second class We find the common conchoid, which is described by means of the circle, and some others which, though less

'*°^ Rabuel (p. 125), illustrates this, substituting for the curve CNKL the semi- cubical parabola, and showing that the resulting equation is of the fifth degree, and therefore, according to Descartes, of the third class. Rabuel also gives (p. 119), a general method for finding the curve, no matter what figure is used for CNKL. Let GA = a, KL=b, AB = .r, CB = y and KB = r; then LB = s—b, and AL = x + c—b. Now GA:AL = CB:BL, or a : x + s — b — y : :: — b,

, xy — by-^-ab

whence r = * .

a — y

This value of .:: is independent of the nature of the figure CNKL. But given any figure CNKL it is possible to obtain a second value for :: from the nature of the curve. Equating these values of z we get the equation of the curve.

[90] "ÇgUes dont l'équation monte au quarré de cube."

'"' "Celles dont elle ne monte qu'au sursolide."

""' "Au sursolide."

56

Livre Second. Î^J

uer vne equation pour déterminer tous Tes poins en cere forte.

Au refteie mecs les lignes courbes qui font monter cete equation iufques au quarre de quatre , au mefme genre que celles qui ne la font monter que iufques au cube. & celles dont Tequation monte au quarrédecu- be,au mefme genre que celles dont elle ne monte qu'au furfolide. &ain(î des autres. Dontlaraifoneft, qu'iîy a reigle générale pour réduire au cube toutes lesdifScul- te's qui vont au quarre'de quarre , &au furfolide toutes celles qui vont au quarre de cube , de façon qu'on ne les doit point eftiraer plus compofees.

Mais il eft a remarquer qu'entre les lignes de chafque genre, encore que la plus part foient efgalement compo- sées , en forte qu'elles peuuentferuir a déterminer les mefmes poins. Su conftruire lesmefmes problefmes ,il y eoa toutefois aufly quelques vues , qui font plus fimplcs, &qui n'ont pas tant d'eftendue en leur puilfance. cora- mcentre celles du premier genre outre l'Ellipfe l'Hyper- bole & la Parabole qui font efgalement compofees ,Ic cercle y eft aufiy compris , qui mauifeftement eft plus fimplcr & entre celles du fécond genre il y a la Conchoi- de vulgaire, qui afon origine du cercle^ &il y en a en- core quelques autres, qui bien qu'elles n ayentpas tant d'eftendue que la plus part de celles du mefme genre, nepeuuenr toutefois eftre mifes dans le premier.

Or après auoirainfî réduit toutes les lignes courbes a J^,"!"], J^ certains genres , il m eft ayfe'de pourfuiure en la de- ^ion delà

ppus

monftrationdelarefponfe,qiiei'ay tanroftfaite alaque- Tzf,,^ ftion de Pappus. Car preaierement ayant fait voir cy ^'^^ 'ii

Ol Z dcliuS, ccJep-

57

3*4 La GEOMETRIE.

delTus , que lorfqu'il n'y a que trois ou 4 lignes droites données, l'équation qui fert a determiner les poins cher- chés, ne monte quciufqucs au qnarréj il efVeuidcntjque la ligne courbe ou fetrouuent ces poins, eft neceflaire- ment quelquVpe de ceîles du premier genre:a eaufe que cete mefme equation explique le rapport , qu'ont tous les poins deshgnes du premier genre a ceux d'vne ligne droite^ Et que lorfqu'il n'y a point plus de 8 lignesdroi- tes données , cete equation ne monte que iufques au quarredequarré'tputauplus, 5c que par confequent la hgne cherchée ne peut eftre que du fécond genre , ou au deffous.Et que lorfqu'il n'y a point plus de 1 2 lignes don- nées , l'équation ne monte que iufques au quarre'de cu- be, & que par confequent la hgne cherchée n'cft que du troifîefmegenre, ouaudeffous. &ainfi des autres. Et mefme a caufe que la pofition deslignes droites données peut varier en toutes fortes, & par confequent faire châ- ger tant les quantités connues, que les fîgnes H- & -- de l'équation, eu toutes les façons imaginables j il eft eui- dentqn*iln'ya aucune ligne courbe du premier genre, qui ne (bit vtilea cete queftion, quand elle eftpropofeh en4hgnesdroitesjnyaucunedufecondqui nyfoit vti- le, quand elle eft propofee en huit; ny du troifîefme, quand elle eft propofee en douze: ôc ainfi des autres. En forte qu'il n'y a pas vne Hgne courbe qui tombe fous le Solution calcul&puifleeftre recede en Géométrie , quin'yfoit ^^ ^ftioti ^^^^ P°^^ quelque nombre de hgnes. quandeiie Maisil faut icy plus particuHeremeut queiedetermi- pofée^^° ne, & donne la façon de trouuer la ligne cherchée * qui qu'en î fçf i; eu chafque cas, lorfqu'il ny a que 3 ou 4 lignes droi-

58

SECOND BOOK

complicated''"'' than many curves of the same class, cannot be placed in the first class. '"^

Having now made a general classification of curves, it is easy for me to demonstrate the solution which I have already given of the prob- lem of Pappus. For, first, I have shown that when there are only three or four lines the equation which serves to determine the required points'*^' is of the second degree. It follows that the curve containing these points must belong to the first class, since such an equation expresses the relation between all points of curves of Class I and all points of a fixed straight line. When there are not more than eight given lines the equation is at most a biquadratic, and therefore the resulting curve belongs to Class II or Class I. When there are not more than twelve given lines, the equation is of the sixth degree or lower, and therefore the required curve belongs to Class III or a lower class, and so on for other cases.

Now, since each of the given lines may have any conceivable posi- tion, and since any change in the position of a line produces a corre- sponding change in the values of the known quantities as well as in the signs + and — of the equation, it is clear that there is no curve of Class I that may not furnish a solution of this problem when it relates to four lines, and that there is no curve of Class II that may not furnish a solution when the problem relates to eight lines, none of Class III when it relates to twelve lines, etc. It follows that there is* no geometric curve whose equation can be obtained that may not be used for some number of lines."*'

It is now necessary to determine more particularly and to give the method of finding the curve required in each case, for only three or

'^'' "Pas tant d'étendue." Cf. Rabuel, p. 113. "Pas tant d'étendue en leur puissance."

^"^^ Various methods of tracing curves were used by writers of the seventeenth century. Among these there were not only the usual method of plotting a curve from its equation and that of using strings, pegs, etc., as in the popular construc- tion of the elHpse, but also the method of using jointed rulers and that of using one curve from which to derive another, as for example the usual method of describing the cissoid. Cf. Rabuel, p. 138.

'*^^ That is, the equation of the required locus.

[96] «-gj^ sorte qu'il n'y a pas une ligne courbe qui tombe sous le calcul & puisse être receuë en Géométrie, qui n'y soit utile pour quelque nombre de lignes."

59

GEOMETRY

four given lines. This investigation will show that Class I contains only the circle and the three conic sections.

Consider again the four lines AB, AD, EF, and GH, given before, and let it be required to find the locus generated by a point C, such that, if four lines CB, CD, CF, and CH be drawn through it making given angles with the given lines, the product of CB and CF

is equal to the product of CD and CH. This is equivalent to saying

that if

CB = y,

„„ ezy + dek -\- dex

z^ '

and ç^^^g"^y-\-f9\-fg--_

z^

then the equation is

, {cfglz — dcks^)y — (dez^ -\- cfgz — hcgz)xy -\- hcfglx — bcfgx^

ez^ — cgz'

60

Livre Second. ^*^

res données; & enverra par mefme moyen que le pre- mier genre des lignes courbes n'en contient aucunes au- tres, queles trois fecStions coniques, (Se le cercle.

Reprenons les 4 ligues AB, AD, EF,&GH don- nées cy deflus, & qu'il failletrouuer vne autre ligne , ea laquelleilfe rencontre vne infinite de poins tels que C, duquel ayant tireles 4 lignes CB,CD,CF, & CH,a «igles donnes, fur les données, CE multipliée parCF, produift une fomme efgale a C D , multiplie'e par C H.

c z. y >i< b c X,

c'eft a dire ayant fait C B so j , C D oo —

GF^ ^- ,, &CH3Q ^ ,: lequatioeft

-dekzz, "^t "dezzx ^ >i>bifglx

^ i -- W t ^ <, A

i- ^fê^^ j ^ -cfgz^x U ..bcfgxx >i' hcgzx J

}

Sf î

au

61

ja^ La GeometriEo

au moins en fuppofant e i^plus grand que f ^.car s'il eftoit moindre, il faudroit changer tous les fîgnes H- & — . Et il la quantité j' fe trouuoit nulle, ou moindre que rien en ceteequationjlorrqa'onafupporé'Ie point C en l'angle D AG, il faudroitle fuppofer au jGTy en l'angle D A E, on E A R, ou R A G, en changeant les lignes 4- & — felon qu'il feroit requis a cet effect. Et (i en toutes ces 4 po- fitions la valeur d'j/ fe trouuoit nulle , la queftion feroit impoffible au cas propofé. Mais fuppofons la icy eftrc poffible, 5c pour en abréger les termes, au lieu des quan-

titcs ^ elcriuons ±m , ôc au heu de

ez,-- cgzz dezz^i* cfgz--bc^7 . tn ^

efcnuons — ; & ainli nous au-

î z

ez-cgzl^

rons

^ ^" ... 'i^bcfgîx-.bcfgxx jont la raci- yy^zmy- 7- xy -,

€ Z— CgZZ

ne. eft

nx •//" imnx nnxx*^ bcfglx -■ bcfgxx,

y^m- --t- mrïï ^ h-^~^ 7:~TZ^

abréger, au lieu de efcriuonso,&:àu lieu de-

ez-- CgZZ

ô^ derechef pour abréger, au lieu de

tmn bcfgl - . , . I- , nn -bcfn

ez-cgzz e.-cgzz

efcriuons ^. car ces quantite's eftant toutes données, nous les pouuons nommer comme il nous plaift, 6r ainfi nous auons

y TOm-'-X'^-'^ mm-^- oa:-- -.vAr,quidoit cftrela

longeur delà ligne B C, en laiffaut A B, ou .v indeter-

raince.

62

SECOND BOOK

It is here assumed that cz is greater than eg ; otherwise the signs + and — must all be changed.""' If y is zero or less than nothing in this equation/"*' the point C being supposed to lie within the angle DAG, then C must be supposed to lie within one of the angles DAE, EAR, or RAG, and the signs must be changed to produce this result. If for each of these four positions y is equal to zero, then the problem admits of no solution in the case proposed.

Let us suppose the solution possible, and to shorten the work let us

write 2w instead of — ^- s—, and — mstead of ~ ^r^-

ez^ — cgz^ 2 ez^ — cgz^

Then we have

^ 2« hcfqlx — hcfqx^

^ ^ z ' ' ez^ — cgz-

of which the root"*' is

"•^ , / , 2mnx n-x- hcfqlx — hcfqx' 2 \ z z^ ez^ — cgz-

A • r , 1 r 1 • 2w« hcfql , , ,

Again, for the sake of brevity, put + -^ — ^ equal to o, and

«^ bcfg . p

-ly — — 1. -v equal to—; for these quantities being given, we can

z ez — cgz"- f'l

represent them in any way we please.''""' Then we have

y = m — - X + Lfi2 I o.r + - x^.

This must give the length of the line BC, leaving AB or x undeter-

[""J When cs is greater than eg, then ez^' — eg a- is positive and its square root is therefore real.

'**' Descartes uses "moindre que rien" for "negative."

'*®^ Descartes mentions here only one root ; of course the other root would fur- nish a second locus.

'""'In a letter to Mersenne (Cousin, Vol. VII, p. 157), Descartes says: "In regard to the problem of Pappus, I have given only the construction and demon- stration without putting in all the analysis ; ... in other words, I have given the construction as architects build structures, giving the specifications and leaving the actual manual labor to carpenters and masons."

63

GEOMETRY

mined: Since the problem relates to only three or four lines, it is obvi- ous that we shall always have such terms, although some of them may vanish and the signs may all vary.'""'

After this, I make KI equal and parallel to BA, and cutting off on BC a segment BK ecjual to m (since the expression for BC contains -|- m; if this were — m, I should have drawn IK on the other side of AB,"°'' while if m were zero, I would not have drawn IK at all). Then I draw IL so that IK : KL =- ^ : n; that is, so that if IK is equal to x,

KL is equal to ~x. In the same way I know the ratio of KL to IL,

which I may call n : a, so that if KL is equal to - x, IL is equal to

a

-X. I take the ponit K between L and C, since the equation contains

z

— -.V ; if this were -1 — .r, I should take L between K and C ;'""'' while if z z

- X were equal to zero, I should not draw IL. This being done, there remains the expression

LC= x/;n.- + o.r + -A-2,

from which to construct LC. It is clear that if this were zero the point

'^"'^ Having obtained the value of BC algebraically, Descartes now proceeds to construct the length BC geometrically, term by term. He considers QC equal to BK+KL + LC, which is equal to BK — LK + LC which in turn is equal to

~ -^' +\/ Mî2 + OX + —

\ m

1'"=' That is, take I on CB produced.

'"'^ That is, on KB produced. C is not yet determined.

64

Livre Second.

3*7

îBinée. Et il eft euident que la queftion n'eftantpro- pofce qu'en trois ou quatre lignes , on peut toufîours auoirdetels termes, excepfe que quelques vns d'eux peuuenteftrenuls, & que les figues t1- Ôc -- peuuent di- uerfement eftrechangés.

Après celaie fais Kl efgalc & parallèle aB A, en forte qu'elle couppe de B C la partie B K efgale à /w , à caufe qu'il y a icy -f- m; & ielauroisadioufteeentirantcete ligne I K de l'autre code, s'il y ^uoit QU — m; & ie ne Tau- rois point du tout tirée, fi la quantité" ttî euftefte'' nulle. Puis ie tire aufiy I L , en forte que la ligne I K efi: à K L, comme Z eft a «. ceft adiré que IK efiantA:, KL eft

-.V. Et par raefme moyen ieconnois au fly la proportion

qui

65

52$» I^A GEOMETRIE.

qui ell: entre KL, & I L, que ie pofe comme entre n Se a: fibienque K L eftant -x, I L eft - x; Et ie fais que Ie

point K foit entre L &: C , a caufe qu'il y a icy — - x-,

au lieu que i'aurois mis LentrcK & Cjfi i'eulTe en ^- - .r,-

& ien'eufTe pointtiré'ceteIigneIL,fi^A;euft efte'nulle. OrceIafait,iInemereftepluspourlaligne LC, que

ces termes, LCoo m'm'^r ox "-^^^. doùievoy

<5ue s'ils eftoient nuls, ce point C fe trouueroit en la li- gne droite I L3& que s'ils eftoient tels que la racine s'en

ft pufttirer,c'eftadirequew2/»&;^A; :v eftant marqués

dVn mefme figne 4- ou — , 00 fuftergalà4^;7?,oubien

queIestermes/ww&oA:,ouoA; &- xx fuflent nuls, ce point C fe trouuerpit en vne autre ligne droite qui ne fe- roit pas pins malayfee a trouuer qu' I L. Mais lorfque cela n'eft pas, ce point C eft toufiours en l'une des trois fedions coniques , ou en vn cercle , dont l'vn des dia- mètres eft en la ligne I L,&: la ligne L C eft l'vne de cel- les qui s'appliquent par ordre à ce diamètre j ou au con- traire L C eft parallèle au diamètre , auquel celle qui efc «n la ligne I L eft appliquée par ordre. A fçavoir fi le ter«

me ^xx, eft nul cete fe£tion.conique efi vne Parabole-

& s'il eft marqué du fîgne -f- , c'eft vne Hyperbole ; & enfin's'il eft marque du fîgne — c'eft vne Ellipfe. Excepte" feulement fi la quantité' aam eft efgale à pw & que l'an- gle ILC foit droit ; auquel cas on à vn cercle au lieu

d'vne

66

SECOND BOOK

C would lie on the straight line IL ;'""' that if it were a perfect square,

P that is if «r and — x- were both -1-'"^' and o- was equal to Apm, or if m

m' and ox, or ox and -- x-, were zero, then the point C would lie on

another straight line, whose position could be determined as easily as that of IL.'^"*'

If none of these exceptional cases occur,'""^ the point C always lies on one of the three conic sections, or on a circle having its diameter in the line IL and having LC a line applied in order to this diameter,^'"*' or, on the other hand, having LC parallel to a diameter and ÎL applied in order.

In particular, if the term — x- is zero, the conic section is a parabola ;

if it is preceded by a plus sign, it is a hyperbola; and, finally, if it is preceded by a minus sign, it is an ellipse. '^°*^ An exception occurs when

[104] -pj^g equation of IL is j) := m — ~x.

tio6] -phere jg considerable diversity in the treatment of this sentence in differ-

ent editions. The Latin edition of 1683 has "Hoc est, ut, mm & — xx signo +

p notalis." The French edition, Paris, 1705, has "C'est à dire que mm et —xx étant

-m

marquez d'un môme signe + ou ■ — ." Rabuel gives "C'est a dire que mm and

^ XX k.\.2,x\\ marquez d'un même signe +." He adds the follov^ring note: "Il y a

dans les Editions Francoises de Leyde, 1637, et de Paris, 1705, 'un même signe + ou — ', ce qui est une faute d'impression." The French edition, Paris, 1886, has "Etant marqués d'un même signe + ou — ."

[i°8] Note the difficulty in generalization experienced even by Descartes. Cf. Briot and Bouquet, p. 72.

'""' "Mais lorsque cela n'est pas." In each case the equation giving the value of ;y is linear in x and y, and therefore represents a straight line. If the quantity

under the radical sign and x are both zero, the line is parallel to AB. If the

quantity under the radical sign and m are both zero, C lies in AL.

[los] «^j^ ordinate." The equivalent of "ordinition application" was used in the 16th century translation of Apollonius. Hutton's Mathematical Dictionary, 1796, gives "applicate." "Ordinate applicate," was also used.

''•^J Cf. Briot and Bouquet, p. 143.

Gl

GEOMETRY

a'm is equal to p2^ and the angle ILC is a right angle/""' in which case we get a circle instead of an ellipse. ''"'

If the conic section is a parabola, its latus rectum is equal to — and

a

its axis always lies along the line IL.'"'' To find its vertex, N, make IN equal to — ;^, so that the point I lies between L and N if m^ is posi- tive and ox is positive; and L lies between I and N if wr is posi- tive and o.v negative ; and N lies between I and L if in- is negative and ox positive. It is impossible that nr should be negative when the terms are arranged as above. Finally, if m- is equal to zero, the points N and I must coincide. It is thus easy to determine this parabola, according to the first problem of the first book of Apollonius'"".

If, however, the required locus is a circle, an ellipse, or a hyper- bola,'"'' the point M, the center of the figure, must first be found. This

'""' Rabuel (p. 167) adds "If a-m^^pz- or if m=^p the hyperbola is equi- lateral."

'"'' In this case the triangle ILK is a right triangle, whence IK^ = LK'^ -|- K?; but by hypothesis IL : IK : KL = a : s : 7i; then a'-\-n'^ = s-. Now the equa- tion of the curve is

:>' = '«-? + '^\m^ + 02-^ x\ ^ \ nt

and therefore the term in x"^ is

and if a^m=^ pz-, then — = -r;, and this term in x- becomes

^'+"' .,2 2

Therefore, the coefficients of x- and ■v" are unity and the locus is a circle. iu2] "pi^is ffjay ijg ggçj^ 2s follows : From the figure, and by the nature of the

parabola LC^= LN./) and LN = IL-(-IN. Let IN — 4>; then since IL = -x, we

Û 71 ft {I

have LN = - .r -|- 0 and LC = j' — in+—x; whence (;y — ni-\- — x)- ^ (-x-\-<P) p. But {y — m-\- —x)~ =^ m- -\- ox from the equation of the parabola; therefore

- .r/i -|- 0/> =: m^ -(- o.r. Equating coefficients, we have -pr=o; p ^ ~^; <pp=:m^;

02 „ , a»r

a 02

'"'' ApoUonii Pcrgaeii Quae Graece exstant edidit I. L. Heiberg, Leipzig, 189L Vol. I, p. 159. Liber I, Prop. LII. Hereafter referred to as Apollonius. This may be freely translated as follows : To describe in a plane a parabola, having given the parameter, the vertex, and the angle between an ordinate and the corre- sponding abscissa.

'^"' Central conies are thus grouped together by Descartes, the circle being treated as a special form of the ellipse, but being mentioned separately in all cases.

68

Livre Second.

329

d'y ne Ellipfe. Que fi cete fedion eft vne Parabole , fon

colle droit eft efgal à -^, & fon diamètre eft toujours en

la ligne IL. &: pour trouuer le point N, qui en eft le

fommet, il faut faire I N efgale a 7^,- & que le point I

fait entre!. & N,fî les termes font -j-mm-^ox; oubien que le point L foit entre I & N, s'ils font -^ mm — ox; oubien il faudroit qu' N fuft entré I & L , s'il y auoit " m m -^ 0 X , Mais il ne peut iamais y auoir — m m, en la façoaque les termes ont icy cfte' pofe^s. Et enfin le point N feroit le mefme que le point I (î la quan- tité w;7ze(xoit nulle. Au moyen dequoy il dt ayfé de trouucrcereParaboleparlei^^^Problefrae du i^r. jiure

d'Apollonius.

Tt QLie

69

J5o La GEOMETRIE.

Que (î la ligne demâdee efc vn cercIe,ou vne eIlipfe,ou vnc Hyperbole, il faut premièrement chercher le point M, qui en eft le centre , & qui eft toufiours en la ligne

ao m

droite IL, ou on le trouue en prenant ~ pour IM. en

forte que fi la quantité o eft nulle, ce centre eft iuftement au point I. Et fi la ligne cherchée eft vn cercle, ou vne ElHpfej on doit prendre lé point Mdumefme*'cofté que lepointL, aurefpedi du point I, lorfqu'on a -H oatj & lorfqu'on à — o a; , on le doit prendre de l'autre. Mais tout au contraire en l'Hyperbole, fi on a — ox, ce centre MdoiteftreversLj&fîona-^-oAT, il doit eftrede l'au- tre cofte. Après cela le cofte' droit de la figure doit eftre

—jj- H 7^ lorfqu'on a H- w wî , &: que la ligne

cherchée eft vn cercle, ou vne EUipfè ; oubien lorfqu'on a— mm, & que c'eft vne Hyperbole. & il doit eftre

't/'ûozz, A^P^^'p^ I- T. 1 » n f

~~r. Jr~"la hgne cherchée eftant vn cercle,

ou vneElîipfe,ortà->7;2 77?;DTibien fi eftant tne Hyper- bole & la quantité'o o eftant plus grande que 4 mp, on à -f- m m. Qiie fi la quantitcTW m eft nulle, ce cofte droit

eft"^, & fi (? :c eft nulle ,il eft: f^^.^^^. Puis pour le cofté

a, a a, ^

travcrfant, il fauttrouuer vne ligne, qui foita ce cofte' droit, corne «<îw2 eft à^ :^:^,àfçauoir fi ce cofte droit t^t

%

'U' 0 0 zz 4 w P^^> t r f^ 'i/ a a.0 omm ^ aam

""7^"'^" — — — letrauerianteit -— — — ■ -r-— ■

Et en tous ces cas le diamètre de la fedion ek en la ligne I M, & L Ceft l'vnede celles qui luy cft appliquée par ordre; Sibienque £iifant M N efgale a la moitié du cofte

trauer*

70

SECOND BOOK

will always lie on the line IL and may be found by taking I M equal to

-^ — .'"^^ If 0 is equal to zero M coincides with I. If the required locus

is a circle or an ellipse, M and L must lie on the same side of I when the term ox is positive and on opposite sides when ox is negative. On the other hand, in the case of the hyperbola, M and L lie on the same side of I when ox is negative and on opposite sides when ox is positive. The latus rectum of the figure must be

4

if m^ is positive and the locus is a circle or an ellipse, or if m^ is nega- tive and the locus is a hyperbola. It must be

if the required locus is a circle or an ellipse and m^ is negative, or if it is an hyperbola and o^ is greater than 4mp, mr being positive.

oz

But if m' is equal to zero, the latus rectum is — ; and if o^ is equal to ;ro'"''^ it is

4

lAmpz^ For the corresponding diameter a line must be found which bears

the ratio —-5- to the latus rectum: that is, if the latus rectum is

4

o'^s- Anips-

the diameter is

4

a^o^m^ 4a^m^

+

p-z"" ^ pz^

In every case, the diameter of the section lies along IM, and LC is one of its lines applied in order. '"'^ It is thus evident that, by making MN equal to half the diameter and taking N and L on the same side of M,

'"'^ Cf. Briot and Bouquet, p. 156.

'"*' Some editions give, incorrectly, ox for oc.

["'J See note 108.

71

GEOMETRY

the point N will be the vertex of this diameter.''"' It is then a simple matter to determine the curve, according to the second and third prob- lems of the first book of Apollonius.'"*'

When the locus is a hyperbola'^^' and in- is positive, if o- is equal to zero or less than 4pm we must draw the line MOP from the center M parallel to LC, and draw CP parallel to LM, and take MO equal to

4/ '

while if o.v is equal to zero, MO must be taken equal to m. Then con- sidering O as the vertex of this hyperbola, the diameter being OP and the line applied in order being CP, its latus rectum is

and its diameter'"'' is

4w2

''^*'If the equation contains — m" and +nx, then n^ must be ;?reater than 4mp, otherwise the problem is impossible.

'""' Cf. Apollonius, Vol. I, p. 173, Lib. I, Prop. LV : To describe a hyperbola, given the axis, the vertex, the parameter, and the angle between the axes. Also see Prop. LVI : To describe an ellipse, etc.

'"*' Cf. Letters of Descartes, Cousin, Vol. VIH, p. 142.

[ini "Qf^^Q traversant."

72

Livre Second. 35i

traucrfant 6c le prenant du piefme coCté du point M, qu efc le point L, on a le point N pour le fommet de ce diamètre .en fuite dequoy il eCt ayfeMe trouuer la fedtion par le fécond ôc 3 prob. du i", liu. d'Apollonius-

Mais quand cote fedion eftant vne Hyperbole , on à •4- m W5 & que la quantité 0 0 eft nulle ou plus petite que 4;? m, on doit tirer du centre M la ligne MOP parallèle a L C ,' & C P parallèle à L M; & faire M O efgale a

^ ww--^^.oubien la faire efgale à m fila quantite'orc eft nulle. Puis confiderer le point O, corne le fommet de cete HyperbolCi dont le diamètre eft O P , & C P la

Tt 2 lign^^

73

332- La- Géométrie.

ligne qui Iqy eft appliquée par ordres fori coftedroireft

-— ; — 77^:;^ & Ion coite trauersat elc *^ ^mjn-

Excepte'quand o x eft nulle.car, alors le cofte droit db — ^77~. ^letrauerfanteft iw. &ainfî il çft ay/c de la trouuer par le 3 prob.du i^^, ijy^ d'Apollonius. UraTimi Et Ics demonftrations de tout cecy font euidentes.car detoutcccompofant vn efpace des quantités que iay afîign ces ^^cft^'e^^'^pourlecoftedroit, & Je trauerfant, ôcpourlefegment cipiiquc. dudiametreNL,ouOP,fuiuâtlateneurderii,du ii,& d:u 13 theorefraes du i", liure d'Apollonius, on trouuera tous les mefmes termes dont eft compofé lé quarrè de îaligne C P,ou C L,qui eit appliquée par ordre a ce dia- mètre. Gomme en cet ex'emple oftantlM , qui eft

TTT, de N M, qui eft -— 0 0 -I- 4 mpy iay I N, a laquel-

le aiouftant IL, qui eft ~^, lay N L ,^qui eft - X' — -^ — -

•H JT^"^ 0 0 -h 4 ;» /> , ôd cecy eftant multiplie^ par

;^<^ 0-1- 4 »2/?, qui eft le cofte droit de la figure, il vient

rvy 0 0-^4 j?z^ "' ,"; ^ oo-j- ^mp -h ~7 -h z m ?n.

pour le rectangle, duquel il Faut oftet vn efpace qui foi t au quatre de N L comme le cofté'droit eft au trauerfant.

& ce quarré de N L eft ^^f^:- -— -.^- <l « o * 7» rt nam. _ ^ L

74

SECOND BOOK

An exception must be made when ox is equal to zero, in which case the

latus rectum is , ^ and the diameter is 2;;;. From these data the p2-

curve can be determined in accordance with the third problem of the

first book of Apollonius/'^^

The demonstrations of the above statements are all very simple, for,

forming the product'^^^ of the quantities given above as latus rectum,

diameter, and segment of the diameter NL or OP, by the methods of

Theorems 11, 12, and 13 of the first book of Apollonius, the result will

contain exactly the terms which express the square of the line CP or

CL, which is an ordinate of this diameter.

In this case take IM or -^^—- from NM or from its equal

am

Yfz

9.„ Vo'H-4w/).

To the remainder IN add IL or— jt, and we have

2

a aom am

z Ipz Ipz ' '^

Multiplying this by

the latus rectum of the curve, we get

for the rectangle, from which is to be subtracted a rectangle which is to the square of NL as the latus rectum is to the diameter. The square of NL is

^'-"-1 See note 113.

1123] "Composant un espace."

GEOMETRY

Divide this by a-m and multiply the quotient by pc-, since these terms express the ratio between the diameter and the latus rectum. The result is

P 1 1-^1 i 1 ^^"^ ^^ l-n 1 9

— x'^ — ÛX -I- X -yJo^ -I- 4Mp 4- -— — — — — V^ + -if/ip -I- Tfr. m ^ ^ ^ ^ 2/ 2/ ^ ^ ^

This quantity being subtracted from the rectangle previously obtained, we get

CL, =tn^ Jr-ox — — x'^. m

It follows that CL is an ordinate of an ellipse or circle applied to NL, the segment of the axis.

Suppose all the given quantities expressed numerically, as EA=3,

AG = 5, AB = BR, BS= |- BE, GB = BT, CD= |cR, CF-2CS, CH =

— CT, the angle ABR=60° ; and let CB . CF=CD . CH. All these quan-

ties must be known if the problem is to be entirely determined. Now let AB^,r, and €6=3». By the method given above we shall obtain

3;^==2y — xy-\-^x — ,r^ ;

whence BK must be equal to 1. and KL must be equal to one-half KI ; and since the angle IKL = angle ABR ^ 60° and angle KIL (which is one-half angle KIB or one-half angle IKL) is 30°, the angle ILK is a

right angle. Since IK = AB = ;»:, KL = -.v-, IL = ;f a/-, and the quantity

/3 3 represented by z above is 1 , we have a = \\-, ?fi = l, c? = 4, / = -, whence \ 4 4

IM = a/ ~, NM = a/ — -; and since a^w (which is .) is equal to ps^ , and

76

Livre Secon^d. ?33

i/^M~ ^ (?o H- 4;»/? qu'il fautdiuiferpar^tf^ôc

multiplier par;j^^,acaufe que ces termes expliquent la proportion qui eft entre le cofté trauerfant & le droit, &

0 0 in

il s\^\A-xx--oX'\'xV 00 -^ ± mp -.

tn ■' i />

«-^-^ "/ oo-^-A-mp -fr m;«.cequ'il faut ofler du red:anele precedent, ôcontrouue ?w;»-Hoa; — - ATArpourlequar- redeCL, qui par confequent eft vne ligne appliquée p^r ordre dans vne Ellipfe,oudans vn cercle,au lègment du diamètre NL.

Et Convent expliquer toutes les quantite's données par nombres, en faifant par exemple EAa)^, A God y, AB:»BR,BSfX)iBE,GB30 BT, CDco ^CR,CF 002CS, CHx>f CT, & quel'angle ABR foit de 60 degrésj & enfin que le redtangle des deux C B , & C F, foit efgai au re&ngle des deux autres C D ôrC Hj car il faut auoir toutes ces chofesaffin que la queftion foit en- tièrement déterminée. & auec cela fappofànt A B do .v, & G B 30^, on trouue par la façon cy deflus expliquée y y 30 2 j " X y -^ ^ X " X X Sc y CO j .. L.x -h"

/'i-f-4A;'-|^':v: fi bienqueB Kdoit eftre i,& KL doit eftre la moitié de Kl, & pourceqae Tangle I Kli ou A BR eft de ($0 degrés, &îKILquieftla moitic'de K I B ou I K L, de 30, 1 L K eft droit. Et pourceque I K ou ABeftuomme:c,KLeft^A;, Ôc IL- eft a:^|, &lâ quantité qui eftoit tantoft nomm^ ^ eft i , celle qui eftoit a cft î^^ |, celle qui eftoit m eft r, celle qui eftoit 0 eft 4, & celle qui eftoit p eft |,de façon qu'on à / '|

Tt i powr.

77

3M

La GEOMETRIE.

Quels font les lieux plans, & lblides:& la façon de les Uouuer.

pour I M, Se V ^^ pour N M, & pourceque aam qui eft I eft icy efgâl à ps^::^ & que Tangle I L C eft droit , oa trouue que la ligne courbe N C eft vn cercle. Et on peut facilement examiner tousles autres cas en mcfme forte .

Aurefte acaufe que les equations, qui ne montent que iufques au quarre^, font toutes comprifes en ce que ie viens d*expliquer ; non feulement le problefine des an- ciens en 5 & 4 lignes eft icy entièrement acheue'j mais aufly tout ce qui appartient à ce qu'ils nommoient la compolîtion des lieux folides- Ôcparconfèquent auffya celle des lieux plans» a caufe qu'ils font compris dans les folides. Car ces lieux ne font autre chofe, fînon que lors qu'il eft queftion de trouuer quelque point auquel il

manque

78

SECOND BOOK

the ang-Ie ILC is a right angle, it follows that the curve NC is a circle. A similar treatment of any of the other cases ofïers no difficulty.

Since all equations of degree not higher than the second are included in the discussion just given, not only is the problem of the ancients relating to three or four lines completely solved, but also the whole problem of what they called the composition of solid loci, and conse- quently that of plane loci, since they are included under solid loci.'^"' For the solution of any one of these problems of loci is nothing more than the finding of a point for whose complete determination one con-

'^' Since plane loci are degenerate cases of solid loci. The case in which neither x^ nor y- but only xy occurs, and the case in which a constant term occurs, are omitted by Descartes. The various kinds of solid loci represented by the equa- tion y=i±ni±—x±: — ± \ ± m- ± ox ± —x may be summarized as follows :

(1) If all the terms of the right member are zero except -7, the equation repre-

sents an hyperbola referred to its asymptotes. (2) If — is not present, there are several cases, as follows: (a) If the quantity under the radical sign is zero or a perfect square, the equation represents a straight line; (b) If this quantity is not a perfect square and if — .r- = 0, the equation represents a parabola; (c) If it is

not a perfect square and if — x^ is negative, the equation represents a drcle or an

ellipse; (d) If — x~ is positive, the equation represents a hyperbola. Rabuel, p. 248.

79

GEOMETRY

ditioii is wanting, the other conditions being such that (as in this exam- ple) all the points of a single line will satisfy them. If the line is straight or circular, it is said to be a plane locus ; but if it is a parabola, a hyperbola, or an ellipse, it is called a solid locus. In every such case an equation can be obtained containing two unknown quantities and entirely analogous to those found above. If the curve upon which the required point lies is of higher degree than the conic sections, it may be called in the same way a supersolid locus, ''"^' and so on for other cases. If two conditions for the determination of the point are lacking, the locus of the point is a surface, which may be plane, spherical, or more complex. The ancients attempted nothing beyond the composition of solid loci, and it would appear that the sole aim of Apollonius in his treatise on the conic sections was the solution of problems of solid loci. I have shown, further, that what I have termed the first class of curves contains no others besides the circle, the parabola, the hyperbola, and the ellipse. This is what I undertook to prove. I12EJ u^j^ jjgy sursolide."

80

Livre Second. 33/

manquevne condition poureflre entieretncnt determi- ne, ainfî qu'il arritie en cete exemple, tous les poins d'Vne mefme ligne peuuent eftre pris pour celuy qui efl de- mande'. Et fî cete ligne eft droite, ou circulaire , on la nomm^vn lieu plan. Mais fi c'eftvne parabole, ouvne hyperbole, ou vne cUipfè, on la nomme vn lieu folide. Et toutefois & quantes que cela eft, on peut venir a vne E- quationqui contient deux quantite's inconnues, & eft pareille a quelqu'vne de celles que ie viens de refoudre. Que fi la ligne qui determine ainfi lè point cherché , eft d'vndegre'pluscompofeequeles fciflions coniques, on la peut nommer, en mefme façon , vn heu furfohde , & ainfi des autres. Et s'il manque deux conditions a la de- termination de ce point, le heu ou il fè trouue eft vne fu- perficie, laquelle peut eftre tout de mefme ou plate, ou fpherique, ou plus compofee. Mais le plus haut but qu'ayent eu les anciens en cete matière a efte deparue- niralacompofîtiondes lieux folides: Et il femble que tout ce qu'Apollonius a efcrit des fedlions coniques n'a efte'qu'àdefleinde la chercher. ^ u n.

^ Quellcclt

De plus on voit icy que ceque iay pris pour le premier '^ prcmie- genredeshgnes courbes,n en peut comprendre aucunes pîu? fim- autres que le cercle, la parabole, l'hyperbole, &rellipfe.P^'''^*= ,

. /• -, . . , ^ toutes les

qui eit tout ce quel auois entrepris de prouuer. lignes

Que fi la queftion des anciens eftpropofee en cinq li- '°"^^^" gnes, qui foîent toutes parallèles ; ilefteuidentque le uent^en la

point chercheTeratoufîours en vne ligne droite. Maisfi ]lf^Z" elle eftpropofee en cinq lignes, dont ilyenait quatre ciens qui foient parallèles, Sequela cinquiefme les couppe a S pro- angles droits, & mefme que toutes les limes tirées duP°f^^"*

. cinqli-

pOintgncs.

81

53<^ La Géométrie.

point cherche les rencontrent aufîy a angles droits, & enftn que le parallélépipède compofè de trois, des lignes ainfî tirées fur trois de celles qui font paralleles/oit efgal au parallélépipède compofé des deux hgnes tirées Tvne fur Ja quatriefme de celles qui font parallèles & l'autre fur celle qui les couppe a angles droits, & dVne troifîcf. me ligne donnée, ce qui eft ce femble le plus ûm- pic cas qu'on puiflb imaginer après le precedent j le point cherche fera en ja ligne courbe , qui eft defcnte parle raouuementd'vne parabole en la façon cy deffus expliquée.

Soient

82

SECOND BOOK

If the problem of the ancients be proposed concerning five hnes, all parallel, the required point will evidently always lie on a straight line. Suppose it be proposed concerning five lines with the following condi- tions :

(1) Four of these lines parallel and the fifth perpendicular to each of the others ,

(2) The lines drawn from the required point to meet the given lines at right angles ;

(3) The parallelepiped"""' composed of the three lines drawn to meet three of the parallel lines must be equal to that composed of three lines, namely, the one drawn to meet the fourth parallel, the one drawn to meet the perpendicular, and a certain given line.

This is, with the exception of the preceding one, the simplest pos- sible case. The point required will lie on a curve generated by the motion of a parabola in the following way:

[120] Yhat is, the product of the numerical measures of these lines.

83

GEOMETRY

Let the required lines be AB, IH, ED, GF, and GA. and let it be required to find the point C, such that if CB, CF, CD, CH, and CM be drawn perpendicular respectively to the given lines, the paral- lelepiped of the three lines CF, CD, and CH shall be equal to that of the other two, CB and CM, and a third line AI. Let CB=3;, CM=jr. AI or AE or GE=a; whence if C lies between AB and DE, we have CF=2a— V, CD==a— 3;, and CH=v-fa. Multiplying these three to- gether we get y^~2ay-—a-y^2a'' equal to the product of the other three, namely to axy.

I shall consider next the curve CEG, which I imagine to be described by the intersection of the parabola CKN (which is made to move so that its axis KL always lies along the straight line AB) with the ruler GL (which rotates about the point G in such a way that it constantly lies in the plane of the parabola and passes through the point L). I take KL equal to a and let the principal parameter, that is, the par- ameter corresponding to the axis of the given parabola, be also equal to a, and let GA=2a, CB or MA=y, CM or AB=.r. Since the triangles GMC and CBL are similar, GM (or 2a— y) is to MC (or x) as CB

(.ovy) is to BL, which is therefore equal to ^ - - . Since KL is a, BK

2a— y

^y 2a — ay — xy

IS a — - or . Finally, since this same BK is a segment

2a— y 2a— y

of the axis of the parabola, BK is to BC (its ordinate) as BC is to a (the latus rectum), whence we get y^—2ay-—a-y-^2a"^=axy, and there- fore C is the required point.

84

Livre Sicokb. 337

Soient par exemple les lignes cherchées A B,I H,E D, G F, & G A. & qu'on demande le point C, en forte que tirant C B, C F, C D, C H, & C M a angles droits fur les données, le parallélépipède des trois CF, CD, & CH foit efgal a celuy des 2 autres C B, & C M, & d'vne troi- fiefme qui foit A I. le pofè C B y3y. C M O) x\ A I, ou A E, ou G E 00 ^,de façon que le point C eflant entre les lignes A B, &DE, iayCFooa^ —y, C D :» ^ — ^. & C H 30^ H- ^. & multipliant ces trois l'y ne par l'autre,

lay y —layy-- a ay -^ ia efgal au produit des trois autres quieft^ATj/. Après cela icconfidere ta ligne cour- be C E G, que i'imaginc eftre defcrite par l'interfedion, de la Parabole C K N, qu'on fait mouuoir en telle forte que fon diamètre KL eft toufiours fur la ligne droite A B, & de la reigle G L qui tourne cependant autour du point G en telle forte quelle pafle toufiours dans le plan de cete Parabole par le point L. EticfaisKLoo «, &le coftd'droit principal, c'eft adiré celuy qui fè rapporte a l'aiflieudeceteparabole^auflyefgalà^, &GA30 2^7, & CB ou M A 30 j^, & C M ou A B 30 AT. Puis a cau/è des triangles femblables GM C & C B L,G M qui eft 2 ^ -y, eft à M C qui eft ^, ,comme C B qui efty, eft à B L qui eft

X y

par confequent -^^. ^ pourceque L K eft ^, B K eft ^

- xy laa -• ay - xy

— -,oubien — ^^^ — . Et enfin pourceque ce mef- mcB Keftant vn fegment du diamètre de la Parabole eft à B C quiluy eft appliquée par ordre , comme cel- iecyeft au cofté droit qui eft a, le calcul monilre que

y "Zayy —aay -h z-a, eft efgal à a xy. &par confè-»

V V quenc

85

33»

La Géométrie.

quent que le point C eft celuy qui eftoit demande. Et il peut eftre pris en tel endroir de la ligne C E G qu'on ve- uille choifîr, ou aufTy en Ton adiointe ^ E G ^ qui fe de- fcri t en mefme façon, excepté que le fommet de laPara- bol e eft tourne vers l'autre cofté , ou enfin en leurs con- trepofe'es Nlo,nl 0,qui font defcrites par l'interfeétion que fait la ligne G L en l'autre cofté de la Parabole

KN.

Or encore que les parallèles donné'cs A B , 1 H, E D, & G F ne fuficnt point efgalement distantes, & que G A ne les couppaft point a angles droits, ny aafly les lignes

tirées

86

SECOND BOOK

The point C can be taken on any part of the curve CEG or of its adjunct cEGc, which is described in the same way as the former, except that the vertex of the parabola is turned in the opposite direction ; or it may He on their counterparts""'' NIo and «lO, which are generated by the intersection of the hue GL with the other branch of the para- bola KN.

Again, suppose that the given parallel lines AB, III, ED, and GF are not equally distant from one another and are not perpendicular to GA, and that the lines through C are oblique to the given lines. Tn this case the point C will not always lie on a curve of just the same nature. This may even occur when no two of the given lines are parallel.

[i2'] "£j^ leurs contreposées."

87'

GEOMETRY

Next, suppose that we have four parallel lines, and a fifth line cutting them, such that the parallelepiped of three lines drawn through the point C (one to the cutting line and two to two of the parallel lines) is equal to the parallelepiped of two lines drawn through C to meet the other two parallels respectively and another given line. In this case the required point lies on a curve of different nature/^^*^ namely, a curve such that, all the ordinates to its axis being equal to the ordinates of a conic section, the segments of the axis between the vertex and the ordinates'^^"' bear the same ratio to a certain given line that this line bears to the segments of the axis of the conic section having equal ordinates. ''^°'

I cannot say that this curve is less simple than the preceding ; indeed, I have always thought the former should be considered first, since its description and the determination of its equation are somewhat easier.

I shall not stop to consider in detail the curves corresponding to the other cases, for I have not undertaken to give a complete discussion of the subject ; and having explained the method of determining an infinite number of points lying on any curve, I think I have furnished a way to describe them.

It is worthy of note that there is a great difference between this method'"^^ in which the curve is traced by finding several points upon

I12S] Yi^e general equation of this curve is axy — xy~ -\-2a-x ^ a-y — ay-. Rabuel, p. 270.

112»] That is, the abscissas of points on the curve.

[ISO] -pi^g thought, expressed in modern phraseology, is as follows : The curve is of such nature that the abscissa of any point on it is a third proportional to the abscissa of a point on a conic section whose ordinate is the same as that of the given point, and a given line. Cf. Rabuel, pp. 270, et seq.

'"'' That is, the method of analytic geometry.

88

Livre Second, 33?

tirées du point C vers elles, ce point (j ne IaiÏÏ*eroit pas de fe trouuer toufiours en vne ligne courbe, qui feroit de cete mefme nature. Et il s'y peut aufly trouuer quel- quefois, encore qu'aucune des lignes données uefoienc parallèles. Maisfî lorfqu'ilyena 4 ainfî parallèle s, & vne ciuquiefme qui les trauerlê: 6c que le parallélépipède de trois des lignes tire'cs du point cherche, l'vne fur cete cinquiefme, &: lès 1 autres fiir 2 de celles qui font paral- lèles; foitefgal a celuy, des deux tirées fur les deux au- tres parallèles , Ôcd'vne autre Hgne donnée. Ce point cherchcf'eften vne ligne courbe d'vue autre nature, â fçauoir en vne qui eft telle, que toutes les lignes droites appliquas parordre a fon diamètre eftant efgales a cel- les dVne fe<SÎ:ion conique, les fegmens de ce diamètre, quifoDteptrelefommet&ces lignes , ont mefme pro- portion a vne certaine ligne donnée, que cete ligne don- née a aux fegmens du diamètre de la fêd:ion conique, aufquels les pareilles lignes font appliquas par ordre. Et ie ne fçaurois véritablement dire que cete ligne foit moins fîmple que la précédente, laquelle iay creu toute- fois deuoir prendre pour la premiere, acaufêquela de- fcription , & le calcul en font en quelque façon plus faciles.

Pour les lignes qui feruent aux autres cas, ienc mare- fteray point aies diftinguer par efpeces. car ie n'aypas entrepris de dire tout ; &: ayant explique la faconde trouuer vne infinite de poins par ou elles paffectjie pçnfç âuoir aflcs donné le moyen de les defcrire.

Mefm€ ileft a propos de remarquer, qu'il y a grande diflference entre cete façon de trouuer plufieurs poins

Vv 2 pour

89

340 La Géométrie.

font les pour tracer vue ligne courbe, & celle dont on le lert pour l^g"es j.^ fpirale, & fes femblablés. car par cete dernière on ne

courbes ^ t

qu'on de- trouue pas indiffère ment tous les poins dé la ligne qu'on trouu" cherche, maisfèulernent ceux qui peuuent eftre dcter- piuficurs mines par quelque mefurephisfimple, que celle qui eft poin7,qyirequifepourlacomporer, & ainfî a proprement parler peuucnc on ne trouue pasjvude {ç,% poins. c'eft a dire pas vn de ceuL^eû ceux qui luy font tellement propres, qu'ils ne puifîcnt Gcoine- eftre trouuc's que par elle: Au lieu qu'il ny a aucun point dans.les lignes qurferuent a la queftion propofé'e , qui ne fe puifTe rencontrer entre ceux qui fe déterminent par la façon tahtoft expliquée. Et pourceque cete façon de tracer une Hgne courbe, en trouuant indifferêment plu- iîeurs de fês poins , ne s'eftend qu'a celles qui peuuent aufly eftre defcrites par vnmouuement régulier & con- tinu, on ne la doit pas entièrement reietter de la Géo- métrie. Sft^ufly Et on n'en doit pas reietter non plus, celle ou on fe celles fert d'vn fil, ou d'vne chorde repliée, pour determiner ?crit auec ^^g^^^î^ OU là difference de deux ou plufieurs lignes vnechor- droitcs quipeuugnt eftre tirées de chafque point de la pc'ui?e"nc courbe qu'on cherche, a certains autres poins ^ ou fur y eftre Certaines autrcs lignes a certains aneles. ainfî que nous auons fait en la Dioptrique pour expliquer rEllipie &: THyperbole. car encore <]u'on n'y puiiTe reçeuoir au- cunes lignes qui femblent a dès chordes , c'eft a dire qu] deuienent tantoft droites &: tantoft courbes, a cauie que la proportion, qui eft entre les droites &■ les courbes, n'eftant pas connue, & mefme ie croy ne le pouuant eftre par les hommes, on ne pourroit rien conclure de là qui-

fuft

90

.Tcceucs.

SECOND BOOK

it, and that nsed for the spiral and similar curves.'"'' In the latter not any point of the required curve may be found at pleasure, but only such points as can be determined by a process simpler than that required for the composition of the curve. Therefore, strictly speaking, we do not find any one of its points, that is, not any one of those which are so peculiarly points of this curve that they cannot be found except by means of it. On the other hand, there is no point on these curves which supplies a solution for the proposed problem that cannot be determined by the method I have given.

But the fact that this method of tracing a curve by determining a number of its points taken at random applies only to curves that can be generated by a regular and continuous motion does not justify its exclusion from geometry. Nor should we reject the method"^" in which a string or loop of thread is used to determine the equality or difference of two or more straight lines drawn from each point of the required curve to certain other points.''"' or making fixed angles with certain other lines. We have used this method in "La Dioptrique" '"'' in the discussion of the ellipse and the hyperbola.

On the other hand, geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds,'""' and therefore no con- clusion based upon such ratios can be accepted as rigorous and exact.

'^"' That is, transcendental curves, called by Descartes "mechanical" curves.

I133J ç-£ j.j^g familiar "mechanical descriptions" of the conic sections.

'"^' As for example, the foci, in the description of the ellipse.

'"'' This work was published at Leyden in 1637, together with Descartcs's Discours de la Méthode.

1136] Yhis is of course concerned with the problem of the rectification of curves. See Cantor, Vol. II (1), pp. 794 and 807, and especially p. 778. This statement, "ne pouvant être par les hommes" is a very noteworthy one, coming as it does from a philosopher like Descartes. On the philosophical question involved, consult such writers as Bertrand Russell.

91

GEOMETRY

Nevertheless, since strings can be used in these constructions only to determine lines whose lengths arc known, they need not be wholly excluded.

When the relation between all points of a curve and all points of a straight line is known. '"'^ in the way I have already explained, it is easy to find the relation between the points of the curve and all other given points and lines ; and from these relations to find its diameters, axes, center and other lines'"**^ or points which have especial significance for this curve, and thence to conceive various ways of describing the curve, and to choose the easiest.

By this method alone it is then possible to find out all that can be determined about the magnitude of their areas,"'""' and there is no need for further explanation from me.

''^'^ Expressed by means of the equation of the curve. [138] Pqj. example, the equations of tangents, normals, etc.

I"»] Por the history of the quadrature of curves, consult Cantor, Vol. II (1), pp. 758, et seq.. Smith, History, Vol. II, p. 302.

92

Livre Se CONI5. 3fi

fufirexad&afTuré. Toutefois a caufe qu'orrnefe ferr de chordcs en ces conftrud:ions , que pour détermine^ des lignes droites, dont on connoift parfaitement la lon^ geur, cela ne doit point faire qu'on les reîette.

Orde cela feul qu'on fçait le rapport, qu'ont tousles Q^e pont, poins d'vne ligne courbe a tous ceux d'vne ligne droite, J^'^'J^j^'iç en la façon queiay expliqueej il eft ayfé de trouuer auffy proprié- té rapport qu'ils ont a tous les autres poins, & lignes don- ^^^^ nées: & en fuite de connoiftreles diamètres , les aiffieux, couibcs, le^ centres, &: autres lignes , ou poins ^ a qui cliaique ii- ddcaudr gne courbe aura quelque rapport plus particulier , ou^erapporc plus fimple, qu'aux autres: & ainfî d'imaginer diuers toutîeuis moyens pour les defcnre,& d'en choilîr les plus faciles. P°''^^ Et mefme on peut aufTy par cela feul trouuer quafï tout lignes cequipeut^ftre déterminé' touchant la grandeur de Te- «^'^J''^"» fpace quelles comprenent, fans qu'ilfoit befbin- que i-en de cirer donne plus d'ouuerture. Et enfin pour cequi eH detou-j!^"^"" tes les autres propriete's qu'on peut attribuer aux lignes qui les courbes, elles ne dependent que de la grand,eur des an- ^^"JJj"^ gles qu'elles font auec quelques autres figues. Mais lorA "s poins qu on peut tirer des lignes droites qui les couppent a an- droifs. gles droits, aux poins ou elles fpnt rencontrées par cel- lésauec qui elles font les angles qu'on veut mefurer, oiî, cequeie prensicy pour le mefme, qui couppent leurs contingentes- la grandeur de ces- angles ireftpas plus malayfée a trouuer, que s'ils eftoient compris entre deux lignes droites. C'eftpourquoy ie croyray auoir miS' iey tout ce qui ell requis pour les elemens des lignes cour- bes, lorfque i*auray généralement donne' la façon de ti- rer des lignes droites, qui tombent a angles droits fur

Vr 5 tels

9>i

Façon

générale

pour

trouuer

des lignes

droites»

qui coup-

pent les

courbes

données,

ou leurs

coBtia-

ger^tcs>a

angles

droits.

^^^ La Géométrie.

tels déleurs poins qu'on voudra choifîr. Et i'ofe dire que c'eft cccy le problefme le plus vtilc , & le plus gene- ral non feulement que iefçache, mais rnefme que l'aye iamais defîré de fçauoir en Géométrie.

Soit G E la ligne courbe, & qu'il faille ti- rer vne ligne droite par le point C, qui fa- ce auec elle des angles droits. le fùppofc la chofe defîa faite, & que la ligne cherchée eft C P , laquelle ie pro- longe iufques au point P, ou elle rencontre la ligne droi- te G A, que ie fuppoiè eftre celle aux poins de laquelle on rapporte tous ceux de la hgne C E : en forte que fai- fant M A ou C B 30^^, & G M, ou B A X) at, iay quelque equation, qui explique le rapport, qui eft entre x ôç^y* PuisiefaisPCoo/, &PA»r^ouP M y> v -y, &c a caufe du triangle redtangle P M C iay//, qui eft h quar- re de la baze efgal à xx'hvv-'ivy-hyy , qui font les quarrés des deux coftes . c'eft a dire iay x oa

f^sx'-vv-h ivy-^yy^ oubien ^ ao t/ -H V ss — xx,8c parie moyen de cete equation, i'ofte de l'autre equa- tion qui m'explique le rapport qu'ont tous les poins de la courbe C E a ceux de la droite G A,rvue des deux quan- tités indéterminés X ou y. ce qui eft ayfé a faire en mettant partout V ss — vv-i^ ivy-- yy au lieu d'.r , Se le quatre de cete fomme au lieu d^xx^ &fon cube au heu

d'x, &ainudesautres,ficeft;cqueie veuille oûerj ou-

bien

94

SECOND BOOK

Finally, all other properties of curves depend only on the angles which these curves make with other lines. r>ut the angle formed by two intersecting curves can be as easily measured as the angle between two straight lines, provided that a straight line can be drawn making right angles with one of these curves at its point of intersection with the other. '"°^ This is my reason for believing that I shall have given here a sufficient introduction to the study of curves when I have given a general method of drawing a straight line making right angles with a curve at an arbitrarily chosen point upon it. And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know.

Let CE be the given curve, and let it be required to draw through C a straight line making right angles with CE. Suppose the problem solved, and let the required Hne be CP. Produce CP to meet the straight line GA, to whose points the points of CE are to be related.'"'' Then, let MA=CB=y ; and CM=BA=.r. An equation must be found expressing the relation between .r and y.''''' I let PC=i', PA=7', whence FM^v—y. Since PMC is a right triangle, we see that s", the square of the hypotenuse, is equal to s--\-v-—2vy-\-y-, the sum

of the squares of the two sides. That is to say, x= ^s-—v'^-{-2z>y—y- or y= V + '^s^ —X' . By means of these last two equations, I can elimi- nate one of the two quantities x and 3' from the equation expressing the relation between the points of the curve CE and those of the straight line G A. If .r is to be eliminated, this may easily be done by replacing

.r wherever it occurs by ^s' — v^ -\-2vy — yr , x' by the square of this ex- pression, x^ by its cube, etc., while if y is to be eliminated, y must be

replaced by v -\- V/— .^-'^ and y',y^, ... by the square of this expres-

'^*"' That is, the angle between two curves is defined as the angle between the normals to the curve at the point of intersection.

'""' That is, the line GA is taken as one of the coordinate axes.

''^-' This will be the equation of the curve. See also the figure on page 97.

95

SECOND BOOK

sion, its cube, and so on. The result will be an equation in only one unknown quantity, .i' or 3'.

For example, if CE is an ellipse, MA the segment of its axis of which CM is an ordinate, r its latus rectum, and q its trans- verse axis,'""' then by Theorem 13, Book I, of Apollonius,'"'' we have

x^ = ry — -y' . Eliminating x' the resulting equation is

2 Ç" ' "" - ■ g-r

Î 2 , o 2 ^ 2 ^^ 2 I qyy-2qvy + gv'-gs' s —V +ivy—y =ry — - y , or y -\ = 0.

In this case it is better to consider the whole as constituting a single expression than as consisting of two equal parts.'""'

If CE be the curve generated by the motion of a parabola (see pages 47, et seq.) already discussed, and if we represent GA by b, KL by c, and the parameter of the axis KL of the parabola by d, the equation

u"] "Le traversant."

'"'^Apollonius, p. 49: "Si conus per axem piano secatur autem alio quoque piano, quod cum utroque latere trianguli per axem posita concurrit, sed neque basi coni parallelum ducitur neque e contrario et si planum, in quo est basis coni, planumque secans concurrunt in recta perpendicular! aut ad basim trianguli per axem positi aut ad earn productam quselibet recta, quae a sectione coni communi sectioni planorum parallela ducitur ad diametrum sectiones sumpta quadrata aequalis erit spatio adplicato rectje cuidam, ad quam diametrus sectionis rationem habet, quam habet quadratum rectse a vertice coni diametro sectionis parallels ducts usque ad basim trianguli ad rectangulum comprehensum rectis ab ea ad latera trianguli abscissis, latitudinem rectam ab ea e diametro ad verticem sectionis abscissam et figura deficiens simili similiterque posita rectangulo a diametro parametroque com- prehenso; vocetur autem talis sectio ellipsis." Cf. Apollonius of Perga, edited by Sir T. L. Heath, Cambridge, 1896, p. 11.

■'"' That is, to transpose all the terms to the left member.

96

Livre Second.

?<f3

bien fîc'cft^, en mettant en fon lieu j/^- i^ss-xx , 6c le quarré, ou le cube,&c. de cete (bmme, au lieu dyy,o\x

y &c. De façon qu'il rcfte toufîours après cela vne equa- tion, en laquelle il ny a plus quVne feule quantité" indé- terminée, a;, ou^.

Comme fi C E eft vne Ellipfe , 6c que M A foit le fegment de fon diamètre, auquel G M foit appliquée par ordre, & qui ait r pour fon cofté droit , & ^ pour le tra-

uerfantjonàparle 15 th.

du I liu. d'Apollonius.

S6XX>ry"^y y , d*oa oftant XX, il refte fS"-

.r

- vv-b-zvy-yy X) ry--yy, oubien, y y ^ ^^"^V^,^ ^^"^efgala rien, car il cft mieux eu

cet endroit de confîderer ainfî enfemble toute la fbm- me y que d'en faire vne partie efgale a l'autre.

Tout de mcûne fî C E eft la ligne courbe defcrite par le mou- uement d'vne Parabole en la façon cy deiTuj expliquc'e, ôc qu'on ait pofë^pourGA, c^oax KL, & ^ pour le cofte droit du diamètre KL e n JUparabole : l'equatio qui explique le rApport qui

97

344-

La Géométrie.

r-^

- "1 C.d-K "Y ■-• i,h h c d-\

i?yi>i*hb\^^Ahcd K y^ ccdd( ^tidJ ^ - ^ddv-y - ddssC

gui éft entr-e oc Uy, c^y — hyy — c dy H- b c d ^ d xy x> o*

d'où oltant x , on a j — byy — ^ây-hbcd-^Ay V ss—vp-^z.vy—yy, & remetrant en ordre ces termes parle moyen de la multiplication, il vient

- i^b b c d-\

yy -- zb c cddy >ii bb ccddxio

<i(d.d Tj V

Et ainfi des autres. Mefme encore que les poins de la ligne courbe ne fê rapportafTentpasenlafaçonqueiay ditte a ceux d'vne ligne droite, mais en tCKite autre qu'on fçauroit imagi- j]er, on ne laifle pas de pouuoir toufîour s auoir vne telle equation- ^ Comme fi Ç E eft vne ligne , qui ait tel rap- port aux trois poins F, G, &: A, que les lignes droites ti- rées de chafcun de fes poins comme C^iufques au point F, furpafTent la ligne F A d'vne quantité, qui ait certaine

proportiôdon- Ql^^^-s?^^ nce a vne autre

^ quantité' dont

GA furpafleles lignes tire'es des mcfmes poins iufques à G. Faifons GAoo^, AFoor, & prenant àdifcretionlepoint C dans la courbe, que la quantité dont CF furpaflfe FA. foit à celle dont G A furpaffe GC, commè^à^, en ibrteque fi cete quantité qui eft

indéterminée fe nomme .^iFC eftcH-:{,&GCeft^ — ^:{.

PuispofantMAcoy, G -Aedb-y, ScFM eft^^-;', & iicaufe du triangle rWlmgle CM G, oftant le quarré

de

98

SECOND BOOK

expressing the relation between x and v is y^ — by^ — cdy-^bcd-\-d.Y\=0. Eliminating x, we have

y^—l7y-—cdy + [h'd+ dy \s-—v'^-\-2vy—y-=0.

Arranging the terms according to the powers of y by squaring/'"' this becomes

y<^-2hy''-^{h--2cd-\-d-)y*^{Ahcd—2d-v)y'^

-{-(c"d-—d~s--ird-v-—2b-cd)y-—2bc-d-y+b-c-d-=0,

and so for the other cases. If the points of the curve are not related to those of a straight line in the way explained, but are related in some other way,''^'' such an equation can always be found.

Let CE be a curve which is so related to the points F, G, and A, that a straight line drawn from any point on it, as C. to F exceeds the line FA by a quantity which bears a given ratio to the excess of GA over the line drawn from the point C to G.''**' Let GA=&, AF=c, and taking an arbitrary point C on the curve let the quantity by which CF exceeds FA be to the quantity by which GA exceeds GC as d is to e. Then if we let c represent the undetermined quantity, FC=c+:: and

GC = l>--,z. Let MA=;', GM = ô-y, and FM = r+j'. Since CMG is a d

right triangle, taking the square of GM from the square of GC we have

i"«i "j7n remettant en ordre ces termes par moyen de la multiplication."

'"'' "Mais en toute autre qu'on saurait imaginer."

^''" That is the ratio of CF — FA to GA — CG is a constant.

99

GEOMETRY

r' 2be left the square of CM, or --^z^ —j- z-\-2by—y'^. Again, taking the

square of FM from the square of FC we have the square of CM expressed in another way, namely : z--\-2cz — 2cy — y-. These two expres- sions being equal they will yield the value of y or MA, which is

2bd^-\-2cd'

Substituting this value for y in the expression for the square of CM, we have

——2 bd^z--\-ce^z--\-2bcd-z—2bcdez

^^ = b¥+7d-' y-

If now we suppose the line PC to meet the curve at right angles at C, and let PC=j and FA^î' as before, PM is equal to v—y\ and since PCM is a right triangle, we have s^-—z>--\-2vy—y- for the square of CM. Substituting for y its value, and equating the values of the square of CM, we have

2 2bcd'z-2bcdez-2c(Pvz-2bdevz-bd'^s'' + bd''i?-cd''s'^cd'^v^ ^ ^ bd'^-^ce'+e'v-d'v

for the required equation.

Such an equation having been found'""' it is to be used, not to deter- mine X, y, or z, which are known, since the point C is given, but to find V or s, which determine the required point P. With this in view, observe that if the point P fulfills the required conditions, the circle about P as center and passing through the point C will touch but not cut the curve CE ; but if this point P be ever so little nearer to or far- ther from A than it should be, this circle must cut the curve not only

[119] 'pj^ree such equations have been found by Descartes, namely those for the ellipse, the parabolic conchoid, and the curve just described.

100

Livre Second. 345"

de G M du quarre de G C, on a le quarre de C M, qui eft

'' ^..L!o^_l-2 3y--j/j. puis oftant le quarre' de F M

du quarre'de F C, on a encore le .quarre de C M en d'au- tres termes, a fçauoir:^:^ 4-2 <: :^— 2 fj'— y j', & ces ter- mes eftantefgaux auxprecedens, ils font connoiftrej,

ouMA,quicfl;— --TT^j^rr^ -&fubftituantce-

te forame au lieu d)' dans le quarfede C M , ontrouue qu'il s'exprime en ces termes.

bddz.z. »^ ceez.z <^ i bcddz.-- i bcdcz.

bdd ^ cdd ^ " "jy*

Puis fuppofant que la ligne droite PC rencontre la courbe à angles droits au point C, Scfaifant PC 30x, & V k-Xiv comme deuant, PMeftr-y j & a caufe du trîangle redangle P C M,on à //- vv -I- 2 vy-yy pour le quarre de C M, ou derechef ayant au lieu d)' fubftitue la fomme qui luy eft efgale, il vient

►f 1 bcddz. -- 1 bcdez.— i cdd-vz. -- i bdevz. — bddss ►{« bddw- x{, ' bdd >¥ cee ee v --^df

-- cddss^cddvv. 00 opourTequation que nous cherchions.

Orapre's qu'on à trouuevne telle equation , auliea des'enferuirpourconnoiftrelcsquantite's .v,ou7, ou ^, qui font défia donne'es, puifque le point C eft donne, on la doit employer a trouuert;, ou / , qui déterminent le point P, qui eft demande'. Et a cet effed il faut confide- rer,que fi ce point P eft telqu'on le defire, le cercle dont il fera le centre, &: qui paflera par le point C, y touchera la ligne courbe C E, fans la coupper: mais que fi ce point P, eft tant foit peu plus proche, ou plus efloigné du point

Xx A, qu'il

101 ■

^^^ La Géométrie.

A, qu'il ne doit, ce cercle couppera la courbe , non feu- lement au point C, mais aufîy neeefTairement en quel- que autre. Puis il faut aufïyconfîderer, que lorfque ce cercle couppe la ligne courbe C E, l'équation par laquel- le on cherche la quantité' :v, ou 7, ou quelque autre fem- blable, en fuppofant P A & P C eftre connues, contient neceffairement deux racines, qui font inefgales. Car par exemple fi ce cercle couppe la courbe aux poins C & H, ayant tire E Qjparallele a CM, les noms des quantités indéterminées x 5f^, conuiendront aufly bieii aux lignes EQ^&:QA,quaCM, &MAj puis PEeft efgale a PC,.acaufe du cercle, fi bien que cherchant les hgnes

EQ & QA, parPE & P A qu'on fuppofe com- me données , on aura la mefme equation , que fi on cherchoic C M & M A par PC,PA. d'où il fuit euidcmment,que la valeur d'AT, ou d'/, ou de telle autre quantité qu'on aura fuppofee , fera double en cete equation, cell a dire qu'il y aura deux racines ineL gales entre elles; ocdontl'vue feraCM, l'autre EQ, fi c'eft X qu'on cherche- oubien l'vne fera M A , & l'autre Q Ajfic'efty. &ainfi des autres. Il eft vray que fi le point Ene fe trouue pas du mefinecofte de la courbe que le point Cj il ny aura que l'vne de ces deux racines qui fait vraye, & l'autre fera renuerfec, ou moindre que rien: mais plus ces deux poins, C, & E, font proches l'vn de l'autre, moins il y a de difference entre ces deux raci- nes;

p M

QjS

102

SECOND BOOK

at C but also in another point. Now if this circle cuts CE, the equation involving x and y as unknown quantities (supposing PA and PC known) must have two unequal roots. Suppose, for example, that the circle cuts the curve in the points C and E. Draw EQ paral- lel to CM. Then x and 3' may be used to represent EQ and QA respec- tively in just the same way as they were used to represent CM and MA; since PE is equal to PC (being radii of the same circle), if we seek EQ and QA (supposing PE and PA given) we shall get the same equation that we should obtain by seeking CM and MA (suppos- ing PC and PA given). It follows that the value of x, or y, or any other such quantity, will be two-fold in this equation, that is, the equa- tion will have two unequal roots. If the value of x be required, one of these roots will be CM and the other EQ ; while if y be required, one root will be MA and the other QA. It is true that if E is not on the same side of the curve as C, only one of these will be a true root, the other being drawn in the opposite direction, or less than nothing.''""^ The nearer together the points C and E are taken however, the less differ-

ii^o] "j7^ l'autre sera renversée ou moindre que rien."

103

GEOMETRY

ence there is between the roots ; and when the points coincide, the roots are exactly equal, that is to say, the circle through C will touch the curve CE at the point C without cutting it.

Furthermore, it is to be observed that when an equation has two equal roots, its left-hand member must be similar in form to the expres- sion obtained by multiplying by itself the difiference between the unknown quantity and a known quantity equal to it ;^'"^ and then, if the resulting expression is not of as high a degree as the original equation, multiplying it by another expression which will make it of the same degree. This last step makes the two expressions correspond term by term.

For example, I say that the first equation found in the present dis- cussion,'"^' namely

a , çn' — "^çvy + q'v^ — qs^ y + ,

q-r

must be of the same form as the expression obtained by making ^=y

and multiplying y — e by itself, that is, as 'f- — 2ey-\-e'. We may then

compare the two expressions term by term, thus : Since the first term,

nyv '2,p'vv

•f , is the same in each, the second term,'"^' ^-^ ^-^, of the first is

q—r

equal to —2ey, the second term of the second ; whence, solving for v,

r 1 or PA, we have v = e—~e-\-~r, or, since we have assumed e equal to;', q 2

r 1

v=y — -y-\-~ r. In the same way, we can find ^ from the third term, q I

"^^'^ That is, the left-hand member will be the square of the binomial x — a when ;ir = a.

'^'^'■'^ See page 96. The original has "first equation," not "first member of the equation."

[163] That is, the second term in ;y.

104

Livre Secokd. 347

nesj &: enfin elles font entièrement efgales, s'ils font tous denxioins en vn^ c*eft adiré fi le cercle, qui palTe par C, y touche la courbe CE fans la coupper.

De plus il faut confiderer, que lorfqu'ily a deux raci- nes efgales en vue equation, elle a neceflairement la mefme forme,que fi on multiplie par foy mcfme la quan- tité" qu'on y fuppofe eftre inconnue moins la quantité connue qui luy^ft cfgale, & qu'après cela fi cetc dernière fommen'apas tant de dimenfions que la précédente, on la multiplie par vne autre fomme qui en ait autant qu'il luy en manque^ afiîn qu'il puiffe y auoir feparement equation entre chafcun des termes de l'vne , & chafcun des termes de l'autre.

Comme par exemple ic dis que la premiere equation trouuee cy deflus, afçauoir

y y — ; — aoitauoirlamefine forme que

celle qui feproduift en faifànt^ efgala/, & multipliant ye par (by mefiiie,d'où il vient ^y — zey-^-e e, en forte qu'on peut comparer fèparement chafcun de leurs ter- mes, & dire que puifque le premier qui eft; ; eft tout le mefme en Tvne qu'en l'autre, le fécond qui eftenlVnc

qr y - -z (i v y,

—TTr — ^ft €%^^ ^" fecôd de l'autre qui eft - 2 ey ,d'où cherchant la quantité' v qui eft la ligne P A , on à

v'Xie — ~^-H ï?*, oubie

a caule que nous auons fuppofe' e efgal a; , oti a

Xx a ainfi

105

^4& l'A GEOMETRIE.

ainlî on pourroit trouuer s par le troifîefine reime ee co^^^^^^^^^T^'maispourceque la quantité t/ determine affés le point P,qiiî eft le feul que nous cherchions,on n'a pas befoin de pafTer outre.

Tout de mefme la féconde equation trouuée cy dç(- fus, a fçauoif,

i^i dd-^ - idd-uJ '- d d ssC >itd d V -v^

doit auoir mefme forme , que la fomme qui fe produifir lorfqu'on multiplie ^^ '-^ei -A- ee par

4 î 5 4

y -^fj '-^ggn^^^y-^ -i, qui eft

- "^^^ >hee,-' ^eef Ç ^eeggJ ^ e e t?ij

de façon que de ces deux equations i'en tire fix autres, qui feruent a connoiftre les fix quantite^s /^ g, h, \, v, & j : D'où il eft fort ayfe' a entendre, que de quelque genre, qucpuiffe eftrela ligne courbe propofee, il vient tou- fiours par cete façon de procéder autant d'équations, qu'on cft obligé de fuppofer de quantités , qui font in- connues. Mais pour demeller par ordre ces equations, & trouuer enfin la quantité z^, qui eft la feule dont on a befoin, & à l'occafion de laquelle on cherche les autres: Il faut premièrement par le fécond terme chercher/, la premiere des. quantités inconnues de la dernière fom- me, & on trouue/:» ze— ib.

Vu\s par le dernier il faut chercher /^1a dernière des quantite's inconnues de la mefme fomme, ôc on trouuc

bbccdd.

/•^30—

^ ee

Puis

106

SECOND BOOK

2 Of' — qs'

e — ; but since v completely determines P, which is all that is

q—r

required, it is not necessary to go further.''"''

In the same "way, the second equation found above, '''^' namely,

4- (rV^ - 2/)-r./+ d'-i- - d's' )/ - 2âr'dy + /; W' , must have the same form as the expression obtained by multiplying

_v-— 2^3'+^- by y^-\-fy'''+g-y--\-lry-\-k*, that is, as y'-^(f-2e)y'-\-(cf--2ef^c~)y*Jr(Ji'-^eg"-+e-f)y'

-\-(k'—2eJr-\-e-g-)y--{-(e-h"-2ek')y^e'kK

From these two equations, six others may be obtained, which serve to determine the six quantities /, g, h, k, v, and s. It is easily seen that to whatever class the given curve may belong, this method will always furnish just as many equations as we necessarily have unknown quan- tities. In order to solve these equations, and ultimately to find v, which is the only value really wanted (the others being used only as means of finding îO. we first determine /. the first unknown in the above expression, from the second term. Thus, f=2e — 2b. Then in the last terms we can find k, the last unknown in the same expression, from

'"'' That is, to construct PC we may lay off AP = 7' and join P and C. If instead we use the value of e, taking C as center and a radius CP = r, we con- struct an arc cutting AG in P, and join P and C. Rabuel, p. 309. To apply Descartes's method to the circle, for example, it is only necessary to observe that all parameters and diameters are equal, that is, q^r; and therefore the equation

7' = y v-|- — ;- becomes z'= _, ^ = — diameter. That is, the normal passes

through the center and is a radius of the circle. Rabuel, p. 313.

''■''^' See page 99. As before, Descartes uses "second equation" for "first mem- ber of the second equation."

107

GEOMETRY

which fe*^ — ^ — . From the third term we get the second quantity

g--=Ze-—Ahe—2cd^h-^d-.

From the next to the last term we get h, the next to the last quantity, which is'"°'

2^VV2 2^rV2

h' =

ê '

In the same way we should proceed in this order, until the last quantity is found.

Then from the corresponding term (here the fourth) we may find V, and we have

le" T^be^ b'^e 2ce 2bc b^ l^V\

a add dee

or putting y for its equal <f, we get

2y^ ^by"" b'^y 2cy 2bc b^ bh^

for the length of AP. ""1 Found from.

108

L I V R E s E C O N D. 34P

Puis par le troifiefme rerme il faut chercher a la féconde quantité, &ona^^30 ^ ee — ^^be — z cd'r' bb-i-dd. Puis par le pcnukiefnie il faut chercher /j la penultiefîne

quantité, qui eft Z» ' oo

ib b c cdd 1 bccdd .

ei

Etaiiiiî il fau-

droit continuer fuiuant ce mefme ordre iufques a la der- nière, s'il y en auoit d'auantage en cete fomme • car c'eft chofe qu'on peut toufîours faire en mefme façon.

Puis par le terme qui fuit en ce mefme ordre, qui eft icy le quatriefrae, il faut chercher la quantité' v, & On a

vX>-

h b e 1 ce i bc

bec hh c c^

? bee dd ~'~dd" '' dd~~ d ' " ' d -■ ee

©u mettant/ au lieu d'^ qui luy cft efgal on a

-y t ^^yy ^^y -^y* ^^^ bec bbcc.

.... ~~^ -

f/30

d

7

dd d4i ' dd

pour la ligne A P,

Etainfila troifiefme equation; qui eft

Xx 3

yy

y'

K.^'

109

iSO La GEOMETRIE.

tft zbcddz'- xbcdex.--z cddvz, — ibdevK •• bddss ifi b ddvv-

K\-

bdd i^t6t^ eev'

■ ' cdds s >î< c ddvv ,

a la mefme forme que

^^'-if^-^ff, en fuppofant/efgal a ;^, fi bienque il y a derechef equation entre— 2/, ou — 2 :{, &

>i* 1 b c dd -' 1 hc d e — î. cddv --1 hdcTJ .

' Tdd>i<cee>i.eev..ddv d OÙ OU COmioift qUC

« . / /1 bcdd-bcde>i* bddz. ^ ceez

ia quantité v eft -7di:^JJ7..ee^^dd^

C'eftpourquoy composant la ligne A P , de cete fbmme ef^ gale à V dont toutes les quan- tite's font connues, ôc tirant du point Painfî trouue", vne ligne droite vers C, elle y couppe la courbe CE a an- gles droits, qui eft ce qu'il falloit faire. Et ie ne voy rien qui empefche, qu'on n'eftende ce problefme en mefme façon a toutes les lignes courbes, qui tombentfous quel- que calcul Géométrique.

Mefme il eft a remarquer touchant la dernière fom- me, qu'on prent a difcretion , pour remplir le nombre des dimenlîons de l'autre fomme , lorfqu 'il y en man- que , comme nous auons pris tantoft

y ''^ fy ' "^Zg, y y -h /^ '^ -+- >^^ 5 que les lignes -^ & — ypeuuenteftrefuppofestels, qu'on veut, fans que la \U gne Vf ou A P, fe trouue diuerfè pour cela , comme vous pourresayfement voir par experience, car s'il falloit que icm'areftalTeademonftrertous les theorefmes dont ie

fais

110

SECOND BOOK

Again, the third''"' equation, namely,

Ibcd^'z - 2bcdez - 2cdh'2 - 2bdevz - bd^-s' + bd^-v'—cd's^ + cd'h^

2' + -

bd^'+ce^+e'v-d'^v

is of the same form as zr—2fc-\-f- where /=r, so that —2/ or —2z must be equal to

2bcd'^ - 2bcdc - 2cd'^v - Zbdev bd^+ce'' + é\'-d\'

whence

bcd"^ - bcde -\-bd'^z+ ce^z ^'~ cd''-^bde-€''z\d''z '

Therefore, if we take AP equal to the above value of v, all the terms of which are known, and join the point 1' thus determined to C, this line will cut the curve CE at right angles, which was required. I see no reason why this solution should not apply to every curve to which the methods of geometry are applicable.''"'

It should be observed regarding the expression taken arbitrarily to raise the original product to the required degree, as we just now took

that the signs + and — may be chosen at will without producing dif- ferent values of V or AP.'''°' This is easily found to be the case, but if I should stop to demonstrate every theorem I use, it would require a

'"'' First member of the tliird equation.

'"*' Let us apply this method to the problem of constructing a normal to a para- bola at a given point. As before, s^ — x- -^ v- — 2vy ^ y- . If we take as the eciuation of the parabola .r- = ry, and suljstitute, we have

j= =: rv 4- e'= — 2tt + J- or v^ + (r — 2zO.V + ^'- — ^" = 0-

Comparing this with y- — 2cy^ c- — '^, we have r — 2v = — 2c\ v~ — s- = e- ;

t;=J + f. Since e = y, v^^- + y. Let AM = r. and 7' = AP ; then

AM — AP = MP = one-half the parameter. Rabuel, p. 314.

['"^ It will be observed that Descartes did not consider a coefficient, as a, in the general sense of a positive or a negative quantity, but that he alwavs wrote the sign intended. In this sentence, however, he suggests some generalization.

Ill

GEOMETRY

much larger volume than I wish to write. I desire rather to tell you in passing that this method, of which you have here an example, of sup- posing two equations to be of the same form in order to compare them term by term and so to obtain several equations from one, will apply to an infinity of other problems and is not the least important feature of my general method.'""^

I shall not give the constructions for the required tangents and nor- mals in connection with the method just explained, since it is always easy to find them, although it often requires some ingenuity to get short and simple methods of construction.

[160] Yhe method may be used to draw a normal to a curve from a given point, to draw a tangent to a curve from a point without, and to discover points of inflexion, maxima, and minima. Compare Descartes's Letters, Cousin, Vol. VI, p. 421. As an illustration, let it be required to find a point of inflexion on the first cubical parabola. Its equation is y" = a-x. Assume that D is a point of inflexion, and let CD = y, AC = x, PA ^ s, and AE =: r. Since triangle PAE is

similar to triangle PCD we have -^. — =-, whence .v = " . Substituting in

A' + j 5 r

the equation of the curve, we have \'^ — — ^+a-j^O. But if D is a point of

r

inflexion this equation must have three equal roots, since at a point of inflexion there are three coincident ixjints of section. Compare the equation with

y^ — Zey- + Zc-y — e^ = 0.

Then Ze"^ = 0 and e ^0. But c ^ y, and therefore y ^^ 0. Therefore the point of inflexion is (0, 0). Rabuel, p. 321.

It will be of interest to compare the method of drawing tangents given by Fermât in Methodus ad disquircndam maxiniam et minimam, Toulouse, 1679, which is as follows : It is required to draw a tangent to the parabola BD from a

point O without. From the nature of the parabola > -, since O is without the

DI tj i^

curve. But by similar triangles 5£. = ^l^. Therefore —>£^. Let CE = a,

CI = e, and CD = ^; then DI = d — e, and -; — — >7 ^- : whence

a — c (a — e)^

de- — 2ade > — a-e.

Dividing by e, we have dc — 2ad > — a-. Now if the line BO becomes tangent to the curve, the point B and O coincide, de — 2ad = — a-, and e vanishes ; then 2ad — a- and a — 2d in length. That is CE = 2CD.

112

Livre Secokd. $fx

fais quelque mention, ie ferois contraint d'efcrire vn vo- lume beaucoup plus gros que ie ne defîre. Mais