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

Full text of "The geometry of Renâe Descartes"

/ 



"l 



^oofe ^'o.S.4-bfoK 








F 



LIBRARY RULES 

This book may be kepL.-CiO.^ weeks. 

A fine of two cents will be charged for each day books or 
magazines are kept overtime. 

Two books may be borrowed from the Library at one time. 

Any book injured or lost shall be paid for by the person to 
whom it is charged. 

No member shall transfer his right to use the Library to any 
other person. 




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 

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 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^bcfg îx -.bcf gxx 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 -|- 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 -^ 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' 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 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 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 -— -I- 4 mpy iay I N, a laquel- 

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

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

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

rvy 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, & 



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 
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-2 qvy + 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"^ = 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 ie veux 
bien en paflant vous auertir que l'inuention defuppofcr 
deux equations de mefme forme, pour comparer fepa- 
rement tous les termes de l'vne a ceux de l'autre , & ainfî 
en faire naiftre plufieurs d'vne feule , dont vous aues vu 
icy vn exemple, peut fcruir a vne infinité d'autres Pro- 
blefmes, & n'eft pas l'vne des moindres- de la méthode 
dont ie me fers. 

len'adioufte pomt les conftrudtions, par lefquelles on 
peut defcrire les contingentes ou les perpendiculaires 
cherchées, en fuite du calcul que ie viens d'expHquer , a 
caufe qu'il eft toufîours ayfe'de les trouuer: Bienque fbu- 
uenton aicbefoin dVn peu d'adrefle, pour les rendre 
courtes &fîmples, 

Comnje par exemple, lîD Ceft lapremiçre conchoi- E„mpie 

de des anciens^ Je la con- 

dont A foit le po- de"« p°o. 

le, & BH la rede: blefme.ea 
- "la con- 

cn lorte que tou- choidc. 
tes les lignes droi* 
tes qui regardent 
vers A , & font 
coraprifes entre la 
courbe CD, &Ia 
droite B H , com- 
me DB & C E, foient efgales : Et qu'on veuille trouuer 
1^ Hgne C G qui la couppe au point C a angles droits. 
On pourroit en cherchant, dans la ligne B H, le point 
par où cete Hgne C G doitpafler , felon la méthode icy 

expli* 



cL-^ — 


[ 


) 


\\ \f 

i \ \"E. 


B 


h\ 


\ 


A 



113 



Explica- 
tion de 4 
nouuc- 
aux gen- 
res d'O- 
uales, qui 
feruent a 
I'Opti- 
aue. 



3Ji La Géométrie. 

expliquée, s'engager dans vn calcul autant ou plus long 
qu'aucun des precedens: Et toutefois la conftruélion, qui 
deuroitaprc^'sen eftre déduite, eft fort fîmple. Car il ne 
faut que prendre C F en la ligne droite C A , & la faire 
efgale à C H qui eft perpendiculaire fur H B : puis du 
point F tirer F G, parallèle à BA, & efgale à EA: au 
moyen de quoy on a le point G , par lequel doit pafter 
C G la ligne cherchée. 

Au refte affin que vous fçachiees que la confideration 
des lignes courbes icy propofée n'cft pas fans vfage, & 
qu'elles ont diuerfes propriétés, qui ne cedent en rien a 
celles des fêd:ions coniques, ie veux encore adioufter icy 
l'exphcationde certaines Ouales, que vous verres eftre 
très vtiles pour la Théorie de la Catoptrique , &dela 
Dioptrique. Voycy la façon dont ie les defcris. 




Premièrement ayant tire" les lignes droites FA, & 
A R, qui s'entrecouppent au point A, fans qu'il importe 
a quels angles, ieprens en l'vne le point F a difcretion, 
c'eftadireplus ou moins efloigne''du point A félon que 

ie 



114 



SECOND BOOK 

Given, for example, CD, the first conchoid of the ancients (see page 
113). Let A be its pole and BH the ruler, so that the segments of all 
straight lines, as CE and DB, converging toward A and included 
between the curve CD and the straight line BH are equal. Let it be 
required to find a line CG normal to the curve at the point C. In try- 
ing to find the point on BH through which CG must pass (according 
to the method just explained), we would involve ourselves in a calcula- 
tion as long as, or longer than any of those just given, and yet the 
resulting construction would be very simple. For we need only take 
CF on CA equal to CH, the perpendicular to BH ; then through F 
draw FG parallel to BA and equal to EA, thus determining the point 
G, through which the required line CG must pass. 

To show that a consideration of these curves is not without its use, 
and that they have diverse properties of no less importance than those 
of the conic sections I shall add a discussion of certain ovals which you 
will find very useful in the theory of catoptrics and dioptrics. They 



115 



GEOMETRY 

may bè described in tbe following way : Drawing the two straight lines 
FA and AR (p. 114) intersecting at A under any angle, I choose arbi- 
trarily a point F on one of them (more or less distant from A accord- 
ing as the oval is to be large or small). With F as center I describe a 
circle cutting FA at a point a little beyond A, as at the point 5. I then 
draw the straight line 56""" cutting AR at 6, so that A6 is less than Ab, 
and so that A6 is to A5 in any given ratio, as, for example, that which 
measures the refraction,'"'^ if the oval is to be used for dioptrics. This 
being done, I take an arbitrary point G in the line FA on the same side 
as the point 5, so that AF is to G A in any given ratio. Next, along the 
line A6 I lay off RA equal to GA, and with G as center and a radius 
equal to R6 I describe a circle. This circle will cut the first one in two 
points 1, 1,'"'^ through which the first of the required ovals must pass. 
Next, with F as center I describe a circle which cuts FA as little 
nearer to or farther from A than the point 5, as, for example, at the 
point 7. I then draw 78 parallel to 56 and with G as center and a radius 
equal to R8 I describe another circle. This circle w^ill cut the one 
through 7 in the points 1, 1''"^ which are points of the same oval. We 
can thus find as many points as may be desired, by drawing lines paral- 
lel to 78 and describing circles with F and G as centers. 

''°^' The confusion resulting from the use of Arabic figures to designate points 
is here apparent. 

''°'' That is, the ratio corresponding to the index of refraction. 
'^•'1 "Au point 1." 
'^"'^ "Au point 1.". 



116 



Livre Second. 3T3 

ie veux faire ces Ouales plus ou moins grandes, fedece 
point F comme centre ie defcris vn cercle , quipaflfe 
quelquepeu au delà du point A, comme par le point y, 
puis de ce point 5" ie tire la ligne droite s6y qui couppe 
lautre au pomt 6, en forte qu' A 6 foit moindre qu' A y, 
felon telle proportion donnée qu'on veut, a fçauoir fe- 
lon celle qui mefure les Refracftions fî on s'en veut fer- 
uir pour la Dioptrique. Après cela ieprcns auffy le point 
G, en la ligne F A, du cofte'où eft le point f , a difcrction, 
c'eft a dire enfaifant que les lignes AF&GA ont entre 
elles telle proportion donnée qu'on veut. Puis ie fais 
R A efgale à G A en la ligne Ad. & du centre G dcfcri- 
iiantvn cercle, dont le rayon foit efgal à R5,il couppe 
l'autre cercle de part & d'autre au point i , qui eft Tvn de 
ceux par où doit pafTer la premiere des Ouales cher- 
che'es. Puis derechef du centre F ie defcris vn cercle, 
qui paffe vn peu au deçà, ou au delà du point f , comme 
par le point 7, & ayant tire" la ligne droiteyg parallèle a 
S d, du centre G ie defcris vn autre cercle, dont le rayon 
eft efgal a la ligne R8. & ce cercle couppe celuy qui 
pafl€ par le point 7 au point i , qui eft encore iVn de ceux 
delamefme Ouale. Et ainli on en peut trouuer au- 
tant d'autres qu'on voudra , en tirant derechef d'au- 
tres lignes parallèles à 7 8, 5c d'autres cercles des centres 
F,&G. 

Pour la féconde Ouale il n'y a point de difference , fi- 
non qu'au lieu d' A R il faut de l'autre cofte' du pomt A 
prendre A S efgal à AG, & que le rayon du cercle de- 
fcrit du centre G, pour coupper celuy qui cft defcrit du 
centre F & qui paffe par le point y , foit efgal a la 

Yy ligne 



117 



3r4 



La Géométrie. 




ligne S 6; ou qu'il foit cfgal à S 8 , fî c'eft pour coupper 
eeluyqui paiïepar le point 7. & ainfî des autres, au 
moyen dequoy ces cercles s'entrecouppent aux poins 
marqués 2,1, qui font ceux de cete féconde Oualc 
A 2 X. 

Pourlatroifîefme, &laquatriefrne,au lieu de la ligne 
A G il faut prendre A H de l'autre cofté du point A, à 
fçauoirdu mefme qu'eft lepoint F. Et il y a icy de plus 
a obferuer que cete ligne A H doit eftre plus grande que 
A F: laquelle peut mefme eftre nulle, en forte que le 
point F fe rencontre où efl le point A, en ladefcriptioa 
de toutes ces ouales. Apres cela les lignes A R , & A S 
eftant efgales à A FI , pour defcrire la troifiefme ouale 
A 3 Y, ie fais vn cercbe du centre H, dont îe rayon eft 

efgai 

118 



SECOND BOOK 

In the construction of the second oval the only difference is 
that instead of AR we must take AS on the other side of A, equal 
lo AG, and that the radius of the circle about G cutting the circle about 
F and passin_s: through 5 must be equal to the line S6; or if it is to cut 
the circle through 7 it must be equal to S8, and so on. In this way the 
circles intersect in the points 2, 2, which are points of this second oval 
A2X. 

To construct the third and fourth ovals (see page 121), instead of 
AG I take AH on the other side of A, that is. on the same side as F. 
It should be observed that this line AH must be greater than AF, which 
in any of these ovals may even be zero, in which case F and A coincide. 
Then, taking AR and AS each equal to AH, to describe the third oval, 



119 



GEOMETRY 

A3Y, I draw a circle about H as center with a radius equal to S6 and 
cutting in the point 3 the circle about F passing through 5, and another 
with a radius equal to S8 cutting the circle through 7 in the point also 
marked 3, and so on. 

Finally, for the fourth oval, I draw circles about H as center with 
radii equal to R6, R8, and so on, and cutting the other circles in the 
points marked 4.''"^' 

'^°^' In all four ovals AF and AR or AF and AS intersect at A under any 
angle. F may coincide with A, and otherwise its distance from A determines the 
size of the oval. The ratio AS : A6 is determined by the index of refraction of 
the material used. In the first two ovals, if A does not coincide with F it lies 
between F and G, and the ratio AF : AG is arbitrary. In the last two, if F does 
not coincide with A it lies between A and H, and the ratio AF : AH is arbitrary. 
In the first oval AR = AG and the points R, 6, 8 are on the same side of A. In 
the second oval AS =; AG and S is on the opposite side of A from 6, 8. In the 
third oval AS = AH and S is on the opposite side of A from 6, 8. In the fourth 
oval AR =AH and R, 6, 8 are on the same side of A. Rabuel, p. 342. 



120 



Livre Second. 



3SS 




efgal a S 6, qui couppe au point 3 celuy du centre F, qui 
palTe par le point j- & vn autre dont le rayon eil efgal a 
S 8, qui couppe celuy qui pafle par le point 7, au point 
aully marque' 3} Sc ainfî des autres. Enfin pour la dernière 




Yy z 



ouale 



121 



iX^ La Géométrie. 

oualeie fais des cercles du centre H , dont les rayons 
font efgaux aux lignes R ^, R 8, & femblables , qui coup- 
pent les autres cercles aux poins marque's 4. 

On pourroit encore trouuer vne infinité d'autres 
moyens pourdefcrire ces mefmes ouales. comme par 
exemple, on peut tracer la premiere AV, lorfqu'on fup- 
pofe les lignes F A & A G eftre efgales , fi on diuife la 
toute F G au point L, en forte que F L foit a L G , com- 



me A yà A 6^ c'ejflà dire qu'elles ayent la proportion, 
qui mefure les refractions. Puis ayant diuife A L en deux 
parties efgales au point K, qu'on face tourner vne reigle, 
comme F E, autour du point F, en preffant da doigt C, 
la chorde E C, qui eftant attachée au bout de cete reigle 
vers E, fe replie de C vers K, puis de K derechef vers G, 
& de C vers G, ou fon autre bout foit attache' , en forte 
que la longeur de cete chorde foit compofée de celle 
des hgnes G A plus AL plus FE moins AF. &: ce fera 
lemouuementdu point C, qui defcrira cete ouale , a 
l'imitation de cequi a cfte dit en la Dioptriq; de l'ElIipfe^ 

& 



122 



SECOND BOOK 



There are many other ways of describing these same ovals. For 
example, the first one, AV (provided we assume FA and AG 
equal) might be traced as follows : Divide the line FG at L so that 
FL : LG=A5 : A6, that is, in the ratio corresponding to the index 
of refraction. Then bisecting AL at K, turn a ruler FE about the 
point F, pressing with the finger at C the cord EC, which, being 
attached at E to the end of the ruler, passes from C to K and then 
back to C and from C to G, where its other end is fastened. Thus the 
entire length of the cord is composed of GA-|-AL-|-FE — AF, and the 
point C will describe the first oval in a way similar to that in which the 



123 



GEOMETRY 

ellipse and hyperbola are described in La Dioptriqne.^'^"^ But I cannot 
give any further attention to this subject. 

Athou^h these ovals seem to be of almost the same nature, they 
nevertheless belong to four different classes, each containing an infinity 
of sub-classes, each of which in turn contains as many different kinds 
as does the class of ellipses or of hyperbolas ; the sub-classes depend- 
ing upon the value of the ratio of A5 to A6. Then, as the ratio of AF 
to AG, or of AF to AH changes, the ovals of each sub-class change in 
kind, and the length of AG or AH determines the size of the oval.'""' 

If A5 is equal to A6, the ovals of the first and third classes become 
straight lines ; while among those of the second class we have all pos- 
sible hyperbolas, and among those of the fourth all possible ellipses.'"*' 

In the case of each oval it is necessary further to consider two por- 
tions having different properties. In the first oval the portion toward 
A (see page 114) causes rays passing through the air from F to con- 
verge towards G upon meeting the convex surface lAl of a lens 
whose index of refraction, according to dioptrics, determines such 
ratios as that of A5 to A6, by means of which the oval is described. 

^'""1 See the notes on pages 10, 55. 112. 

'^®'' Compare the changes in the ellipse and hyperbola as the ratio of the length 
of the transverse axis to the distance between the foci changes. 

[168] "pi^ggg theorems may be proved as followrs : (1) Given the first oval, with 
AS = A6 ; then RA = GA ; FP = F5 ; GP = R6 = AR — R6 = GA — AS = G5. 
Therefore FP-FGP = FS + GS. That is, the point P lies on the straight line FG. 
(2) Given the second oval, with A5 = A6; then F2 = FS=FA + AS; 
G2=S6=SA + A6= SA + AS ; G2 — F2 = SA — FA = GA — FA = C. There- 
fore 2 lies on a hyperbola whose foci are F and G, and whose transverse axis is 
GA — FA. The proof for the third oval is analogous to (1) and that for the 
fourth to (2). 

It may be noted that the first oval is the same curve as that described on 
page 98. For FP = FS, whence FP — AF = AS, and AR = AG ; GP = R6 ; 
AG — GP = A6. If then A5 : A6 = d : c we have, as before, 
FP — AF : AG— GP = d : c. 



124 



Livre Second. jj'/ 

& de l'Hyperbole, mais ie ne veux point m'arefter plus 
longtems fur ce fuiet. 

Or encore que toutes ces oualesfemblent eftre quafi 
demefmcnature,elles font néanmoins de 4 diuers gen- 
res, chafcun defquels contient fous foy vne infinite d'au- 
tres genres, quiderechefcontienent chafcun autant de 
diuerfèsefpeces, que fait le genre des Ellipfes , ou celuy 
des Hyperboles. Car felon que la proportion , qui eft en- 
tre les lignes A y, A ^, ou femblables, eft différente ,. le 
genre fubalterne de cesouales cft different. Puis félon 
que la proportion, qui eft entre les lignes A F, & A G,ou 
A H, eft change'e, les ouales de chafque genre fubalter* 
ne changent d'efpece. Et felon qu' A G, ou A H eft plus 
ou moins grande, elles font diuerfes en grandeur. Et fî 
les lignes A 5 & A6 fontefgales, an lieu des ouales du 
premier genreoudutroifîefme, on ne defcrit que des 
lignes droites; mais au lieu de celles du fécond on a tou- 
tes les Hyperboles poflîblesj ôc au lieu de celles du der- 
nier toutes les EUipfes^ 

Outre cela en chafcune de ces oualès il faut coufiderer Les pro- 
deux parties, qui ont diuerfes propriétés ; a fçauoirenla ^''"^^"^if^ 
premiere, la partie qui eft vers A, fait que les rayons, qui touchant 
eftant dans l'air vienent du point F, fe retouruent tous iTonf & 
vers le point G, lorfqu'ils rencontrent la fuperficie con- '« refra- 
uexedVn verre, dont la fuperficie eft i A i, &i dans le- 
quel les refraction s fe font telles, que fuiuant ce qui a 
eftéditenlaDioptrique, elles peuuent toutes eftreme- 
furees par la proportion , qui eft entre les lignes A y & 
A fîjou femblables, par l'ayde defquelles on a defcrit cete 
ouale. 

Yy 3 Mais 

125 



55S 



La Géométrie. 




Mais la partie, qui eft vers V, fait que les rayons qui 
vienent du point G fe reflefchiroient tous vers F , s'ils y 
rencontroient la fuperficie concaue dVn miroir , dont la 
figure fuft I V I , & qui fuft de telle matière qu'il di- 
minuaft la force de ces rayons,felon la proportion qui eft 
entre les lignes A 5 & A <5 : Car de ce qui a efté demon- 
ftre en la Dioptrique, il eft euident que cela pofé, les an- 
gles de la reflexion feroient inefgaus, aufTy bien que font 
ceux de la refraction , & pourroient eftre mefures en 
mefme forte. 

En la féconde ouale la partie 2 A ifert encore pour les 
reflexions dont on fuppofe les angles eftre inefgaux. car 
eftantenla fuperficie d'vn miroir compofé de mefme 
matière que le precedent,elle feroit tellement reflefchir 
tous les rayons, qui viendroientdu point G, qu'ils fem- 
bleroient après eftre reflefchis venir du point F. Et il 
eft a remarquer , qu'ayant fait la ligne A G beaucoup 

plus 



126 



SECOND BOOK 



But the portion toward V causes all rays coming from G to converge 
toward F when they strike the concave surface of a mirror of the 
shape of 1\^1 and of such material that it diminishes the velocity of 
these rays in the ratio of A5 to A6, for it is proved in dioptrics that in 
this case the angles of reflection will be unequal as well as the angles 
of refraction, and can be measured in the same way. 

Now consider the second oval. Here, too, the portion 2A2 (see 
page 118) serves for reflections of which the angles may be assumed 
unequal. For if the surface of a mirror of the same material as in the 
case of the first oval be of this form, it will reflect all rays from G, 
making them seem to come from F. Observe, too, that if the line AG 



127 



GEOMETRY 

is considerably greater than AF, such a mirror will be convex in the 
center (toward A) and concave at each end; for such a curve would 
be heart-shaped rather than oval. The other part, X2, is useful for 
refracting lenses ; rays which pass through the air toward F are re- 
fracted by a lens whose surface has this form. 

The third oval is of use only for refraction, and causes rays travel- 
ing through the air toward F (page 121) to move through the glass 
toward H, after they have passed through the surface whose form is 
A3Y3, which is convex throughout except toward A, where it is slightly 
concave, so that this curve is also heart-shaped. The difference between 
the two parts of this oval is that the one part is nearer F and farther 
from H, while the other is nearer H and farther from F. 

Similarly, the last of these ovals is useful only in the case of reflec- 
tion. Its effect is to make all rays coming from H (see the second 
figure on page 121) and meeting the concave surface of a mirror of 
the same material as those previously discussed, and of the form 
A4Z4, converge towards F after reflection. 

The points F, G and H may be called the "burning points" '"°^ of 
these ovals, to correspond to those of the ellipse and hyperbola, and 
they are so named in dioptrics. 

I have not mentioned several other kinds of reflection and refraction 
that are effected'™' by these ovals ; for being merely reverse or opposite 
effects they are easily deduced. 

'""" That is, the foci, from the Latin focus, "hearth." The word focus was 
first used in the geometric sense by Kepler, Ad ViteUioncm Paralipomena, Frank- 
fort, 1604. Chap. 4, Sect. 4. 

'^'"i "Réglées." 



128 



L I V R E s E C O N I>. 3r^ 

plus grande que A F, ce miroir fcroit conucxe an milieu, 
vers A, & concaue aux extrémitez: car telle ei\ la figure 
decetc ligne, qui en cela reprefente plutoftvn coeur 
qu'vneouale. 

Mais fon autre partie X 2 fertpourIesrefraâ;ious,& 
fait que les rayons, qui eftant dans l'air tendent vers F,fe 
détournent vers G, en trauerfant la fuperficie d\'n ver- 
re, qui enait la figure. 

La troificfme ouale fert toute aux refradions , & fait 
que les rayons, qui eftant dans Taif' tendent vers F, fe 
vont rendre vers H dans le verre, après qu'ils ont trauer- 
fô fa fuperficie, dont la figure ell A 3 Y 3, qui eftconue- 
xe par tout,excepté vers A où qUq eft vn peu concaue en 
forte qu'elle a la figure d*vn coeur aufiy bien que la pré- 
cédente. Et la difference qui eft entre les deux parties 
deceteouale, confifteencequelepoinc F cft plus pro- 
che de l'vne , que n'eft le point H- &: qu'il eft plus 
eftoigneMe Vautre, que ce mefme point H. 

En mefme façon la dernière oua.le fert toute aux re- 
flexions, & fait que fi les rayons, qui vienent du point H, 
rencontroient la fuperficie concaue d'vn miroir de mef- 
me matière que les precedens, & dont la figure fuft A4 
Z4, ilsfereflefchiroicnt tous vers F. 

De façon qu'on peut nommer les poins F, & G , ou tî 
lespoinsbruflans de ces ouales, a l'exemple de ceux des 
Ellipfes, &des Hyperboles, qui ont efte ainfi nommés 
enlaDioptrique. 

l'omets quantité" d'autres refradions, & reflexions, 
qui font reiglces par ces mefmes ouales : car n'eftanc 
que les conuerfes, ou les contraires de celles cy, elles en 

peuuent 



129 



DcmoH- 
ft rat ion 
des pro- 
priétés de 
ccsoualcs 
touchant 
les refle- 
xions & 
refra- 
^oas. 




3<^ La Géométrie. 

peuuent facilement eftre déduites. Mais il ne faut pas 
que i omette la demonftration de ceque iay dit. & a cet 
cffecSt, prenons par exemple le pointe a difcretionenla 
premiere partie de la premiere de ces ouales ; puis tirons 

la ligne droite 



CP, quicoup- 
pe la courbe au 
point C à an- 
gles droits, ce- 
qui eft facile 
par le problefme precedent ; Car prenant i pour A G , r 
pour A F, c^^ pour F C j & fuppofant que la propor- 
tion qui eu entre dà^e , que le prendray icy toufîours 
pour celle qui mefure les refracîlions du verre propofc', 
defigneaulTy. celle qui eft entre les lignes A 5, & A 5, ou 
femblâbles, qui ont ferui pour defcrire cetc ouaIe,cc qui 

donned —-^j^ pour G C: on trouue que la ligne A P e/l 

bcdd - bcde ►!< bidz. ^f ceez. . «^ ,.* y, y n. ^ J /T 

bde >i. cdd ^ dd^ .-. .e^ ^^"fi q" ^^ ^ gft^ Q^Q^ft^^ ^y ^^^"s. 

De plus du point Payant tiré'PQ.a angles droits fur la 
droite F G, & P N aufly a angles droits fur G C,confidc- 
ronsquefîPQLeftàPN, comme^eft àr, c'eft à dire, 
comme les lignes qui mefurent les refrad^ons du verre 
connexe A C, le rayon qui vient du point F au point C, 
doit tellement s'y courber en entrant dans ce verre, qu'il 
s'aille rendre après vers G; ainfi qu'il eft très euident de 
cequiaeftéditenlaDioptrique. Puis enfin voyons par 
le calcul, s'iieftvray, que PQfoit à PN; commet eft 
i e, ies triangles red:angIesP Q F, & C M F font fem- 

blables: 



130 



SECOND BOOK 

I must not, however, fail to prove the statements already made. For 
this purpose, take any point C on the first part of the first oval, and 
draw the straight line CP normal to the curve at C. This can be done 
by the method given above,'"'' as follows : 

Let AG=&, AF=c, FC=c-|-5'. Suppose the ratio of d to e, which 
I always take here to measure the refractive power of the lens under 
:onsideration, to represent the ratio of A5 to A6 or similar lines used 
to describe the oval. Then 

e 

a 

whence 

bed} — bcde + bd'^'z + ce^z 



AP = ' 



bde-^-cd'^^d'^z-e'^z 



From P draw PQ perpendicular to FC, and PN perpendicular to GC.''"' 
Now if PQ : PN=c/ : e, that is, if PQ : PN is equal to the same 
ratio as that between the lines which measure the refraction of the 
convex glass AC, then a ray passing from F to C must be refracted 
toward G upon entering the glass. This follows at once from dioptrics. 

''■'' See page 115. 

''"' Here PQ is the sine of the angle of incidence and PN is the sine of the 
angle of refraction. The ray FC is reflected along CG. 



131 



GEOMETRY 

Now let US determine by calculation if it be true that PQ : PN=(i : e. 

The right triangles PQF and CM F are similar, whence it follows that 

FP CM 
CF : CM = FP: PQ, and ^^ =PQ- Again, the right triangles 

Cr 

CP CM 

PNG and CMG are similar, and therefore — -^ — =PN. Now since 

CCr 

the multiplication or division of two terms of a ratio by the same num- 

u 1 . u .1 .• -rFP.CM GP.CM , . -. ... 

ber does not alter the ratio, if ^ : — -z^^ — —a: r, then, dividing 

each term of the first ratio by CM and multiplying each by both CF 
and CG, we have FP . CG : GP . CF=^ : e. Now by construction, 



bcd^ — bcde-\-bd-a-\-cc^c 
F P = r + -^^oq: bdc-e'a+d'-s ' 



or ^^^hcd^-+c"'d"--^bd'::+cd'- 



cd~-\-bde—e-s-\-d-c ' 

and ^^ , <? 

a 

Then 

b^cd--{-bc-d^-\-b-d-c-\-bcd-a—bcdca—c-de::—bdes-—cdes- 

^^■^^= cd-+bde-e^-s+d^-s " 

Then 

bcd- — bcdc-\-bd-c-\-ce-;:: 

^^ = ^- ~7d-~^-bd^^-c+d -7~ ' 
or 

b-de-\-bcde — be~c — cc-c 
~ cd--\-bde—e-!s-\^d-s ' 

and CF=c4--- So that 

b'cde-\-bc-de-\-b^des-\-bcde:s — bce-s — c^e-z—be^z- — ce-z"^ 

GP.CF = — ~ ,7, , , ^^-VZP' • 

cd~-^bde—e-zArd-z 



132 



Livre s ECO NDr 5<^J 

blableSi d'où il fuit que C F eft à C M , comme F P eft a 
P Q j «Scparconfequenc que FP , eftant multipliée par 
C M, & diuifee par C F, eft*efgale a P Q^ Tout de mejp. 
me les triangles re(5taDgIes PNG, & C M G font fem- 
blables; d'où il fuit que G P, multipliée par C M, & diui- 
fee pa'r C G, eft efgale a P N. Puisa cauffe que les mul- 
tiplications, ou diuifions, qui fe font de deux quantité^? 
par vne raefme, ne changent point la proportion qui eft 
entre elIes; fi F P multipliée par C M; & diuifee par C F, 
eft à G P multipliée aufly par C M & diuife^e par C G; 
comme <^eft à e, en diuifant IVne ôcrrautre de ces deux 
fbmmes par CM*, puis les multipliant toutes deux par 
C F, &: derechef par C G,il refte F P multipliée par C G, 
qui doit eftre à G P multiphee par C F, comme d eftà<?« 

OrparlaconftrudionFPeft^^ i^,^ad.i,ddz,..eez. — 
oubien F P oo ^<^^^ '^ ^^^^ >î^ ^ddi. ^ cdd z,. ^ ^ n. 

bdeyi^ cdd ^dd{-eex. ^ ^ ^ ^*^ 

^ " ~d ^-fîbienque multipliant F P par C G il vient 

bbcdd*itbccddiitbbddz.^bcddz,— bcde/c — ccdex, — bdez.^ ^- edez'{^. 
Fde * cdd>i'dd{-'eez ~^ 

. — bcdd<hbcde--bddz-- cee!, . . 

Puis G P eft ^ --^^ J^ ^ ^^^ .. ^^^ • oubicn 

bbde >^ btde--beez --ceez,', _„ « 

G P 30 w«1.7^^ ^rf^t-"^^ & C F eft f H- ^; 
fibienque multipHant G P par C F , il vient 

bbc de 4« bccde — bceex, - - cc eez "i* bbde\ > i> bcde\— beezz, — ceeza. 
bde <i< cdd >i* ddz — eez, 

Et pourcequela premiere de ces (bmmes diuifee par ^, 
eft la mefrae que la féconde diuifee par ^, il eft mauifefte, 
que F P multipliée par C G eft a G P multiplie'e par C F; 

Zz c'eft 

133 



$^2 La Géométrie. 

c*€ft a dire que P Qjft à P N, comme ^ eft à e, qui eft 
tout ce qu'il falloit demonftrer. 

Etfçaches, quecete mefme demonftration s'eftend 
a tout ccqui a efte dit des autres refrad:ions ou refle- 
xions, quifefontdanslesoualespropofces- fans qu'il y 
faille changer aucune clîofe, que les fignes ■+- &•— du 
calcul, c'eftpourquoy chafcunles peut ayfement exa- 
miner de foymefme, fans qu'il foit befbin que ie my 
arefte. 

Mais il faut maintenentjqueiefatisface a ce queiay 
omis en la Dioptrique,lorfqu après auoir remarque^'^qu'it 
peutyauoir des verres de plufîeursdinerfes figures, qui 
facent aufTy bien l'vn que l'autre, que les rayons venans 
d'vn mefme point de l'obiet, s alïemblent tous en vn au- 
tre point après les auoir trauerfes. & qu'entre ces verres» 
ceux qui font fort connexes d'un cofte , & concaues de 
l'autre, ont plus de force pour brufler, que ceux qui fbnc 
efgalement connexes des deux cofte's. au lieu que tout 
au contraire ces derniers font les meilleurs pour les lune- 
tes. ie me fuis contente d'expliquer ceux , que i'ay crû 
eftre les meilleurs pour la prattiquc, en fuppofànt la diffi- 
culté que les artifans peuuent auoir a les tailler. C'efc 
pourquoy,affin qu'il ne refte rien a fouhaiter touchant la 
théorie de cete fcience,ie doy expliquer encore icy la fi- 
gure des verres, qui ayant l'vne de leurs fuperficies au- 
tant connexe, ou concaue, qu'on voudra, nelaiffentpas 
de faire que tousles rayons , qui vienent vers eux d'vn 
mefme point 5 ou parallèles, s'aflfemblent après en vn 
mefme point • & celle des verres qui font le femblable, 
cftantelgalementconuexes des deux coftc's , oflbienla 

conue- 



134 



SECOND BOOK 

The first of these products divided by d is equal to the second divided 
by e, whence it follows that PO : PN=FP . CG : GP . CF=d : e, 
which was to be proved. This proof may be made to hold for the 
reflecting- and refracting properties of any one of these ovals, by proper 
changes of the signs plus and minus ; and as each can be investigated 
by the reader, there is no need for further discussion here.'™' 

It now becomes necessary for me to supplement the statements made 
in my Dioptrique'"'' to the effect that lenses of various forms serve 
equally well to cause rays coming from the same point and passing 
through them to converge to another point ; and that among such lenses 
those which are convex on one side and concave on the other are more 
powerful burning-glasses than those which are convex on both sides ; 
while, on the other hand, the latter make the better telescopes.'"^' I 
shall describe and explain only those which I believe to have the great- 
est practical value, taking into consideration the difficulties of cutting. 
To complete the theory of the subject, I shall now have to describe 

'"'^ To obtain the equation of the first oval we may proceed as follows : Let 

AF = c; AG^b; FC= c+"; GC=:é-4~- Let CM=,r, AM=y. FM^c-\-y; 

a 

GM = & — y. Draw PC normal to the curve at any point C. Let AP = f. Then 
CF^=CM^+FMI Also, c- + 2cc + c- = x- + c- + 2cy + y^, whence 



c + ]/ ;f ' + c^ + 2cy + y^. 



Also, CG"-^= CM"^+ GM^ whence 

b' — 2—z -]- — ::- = .v- + b- — 2bv + y-. 
a a- 

Substituting in this equation the value of ^ obtained above, squaring, and simplify- 
ing, we obtain : 

^(d^--e^)x'+(d--cny--2(e-c + bd-)y-2ec(ec-bd)'Y 

= 4c-(bd + cc)-(x^- + c"--\-2cy-\-y^-). Rabuel, p. 348. 

'"^'Descartes: La Dioptriquc, published with Discours de la Méthode, Leyden, 
1637. See also Cousin, vol. Ill, p. 401. 

'"^' "Lunetes." The laws of reflection were familiar to the geometers of the 
Platonic school, and burning-glasses, in the form of spherical glass shells filled with 
water, or balls of rock crystal are discussed bv Pliny, Hist. Nat. xxxvi, 67 (25) 
and xxxvii, 10. Ptolemy, in his treatise on Optics, discussed reflection, refraction, 
and plane and concave mirrors. 

135 



GEOMETRY 

again the form of lens which has one side of any desired degree of con- 
vexity or concavity, and which makes all the rays that are parallel or 
that come from a single point converge after passing through it ; and 
also the form of lens having the same effect but being equally convex 
on both sides, or such that the convexity of one of its surfaces bears a 
given ratio to that of the other. 

In the first place, let G, Y, C, and F be given points, such 
that rays coming from G or parallel to G A converge at F after 
passing through a concave lens. Let Y be the center of the inner sur- 
face of this lens and C its edge, and let the chord CMC be given, and 
also the altitude of the arc CYC. First we must determine which of 
these ovals can be used for a lens that will cause rays passing through 
it in the direction of H (a point as yet undetermined) to converge 
toward F after leaving it. 

There is no change in the direction of rays by means of reflection or 
refraction which cannot be efitected by at least one of these ovals ; and 
it is easily seen that this particular result can be obtained by using either 
part of the third oval, marked 3 A3 or 3Y3 (see page 121), or 
the part of the second oval marked 2X2 (see page 118). Since 
the same method applied to each of these, we may in each case take Y 



136 



Livre Second. ^'^^ 

conuexite de l'vne de leurs fuperficies ayant la propor- 
tion donnée à celle de l'autre. 



Pofons pour le premier cas, que les poins G, Y, C, & F ^^ peut 
eftant donnes, les rayons qui vienent du point G, oubien ^^'^'^^J^^_ 
qui font parallèles à GAfe doiuent afTembler au point ^YnTcoQ- 
F, après auoirtrauerfevn verre ficoncaue, qu' Y eftant uexe^ou^ 
lemilieudefafuperficie intérieure, l'extrémité' en foie ^"''"^.c 
au point Cenforte que la chorde CMC, S^l^Aeche demies fu-^ 
Y M de Tare C Y C, font données. La queftion va là, ^^^on 
que premièrement il faut coufiderer , de laquelle desvoudra,_ 
oualesexplique'es, lafuperficie du verre Y C , doitauoir fembie a 
la figure, pour faire que tous ks rayons, qui eftant àe-Zl^V^^ 
dans tendent vers vnmefîtie point, comme vers H, qui tous les 

^ . rayons 

n'eft pas encore connu, S aillent rendre vers vn autre, a^Jj^j^. 
fçauoirversF, apr^sen eftrefortis. Carilny a aucun ncnt^d'vi 
effed: touchant le rapport des rayons changé par refle- poi^t 
xion, ou refradiond'vn point a vn autre , qui ne puiffe ^10"°^. 

eftre caufe par qu^lqu'vne de ces ouales. & on voit 
ayfementque ceiTuycy le peut eftre par la partie de la 
troifiefmeOuale,quiatantoft efté marquee 3 A 5 , ou 
par celle de lamefme, qui a efté marquee 5 Y 3 , ou enfin 
parlapartiedelafecondequiaeftémarquée 2X2. Et 
pourceque ces trois tombent icy fous mefme calcul, on- 
doittant pour l'vne, que pour l'autre prendre Y pour 

Zz 2 leur 



137 



î^4 La Géométrie. 

leur fommet, C pour Tvn des poins de leur circonféren- 
ce, & F pour l'vn de leurs poins bruflansj apre's quoy il 
iierefte plus a chercher que le point H, qui doit eftre 
l'autre point brullant. Et on le trouue en confiderant, 
que la difference, qui eft entre les lignes F Y & F C,doit 
eftre a celle, qui eft entre les lignes H Y & H C, comme 
^/eft à Ci c'cft a dire,comme la plus grande des lignes qui 
mefurent les refrad:ions du verre propofe'' eft à la moin- 
dre- ainfi qu'on peut voir manifeftement de la defcri- 
ption de ces ouales. Et pourceque les lignes F Y & F C 
Ibnt données, leur difference i'eft aufî}^ , & en fuite celle 
qui eft entre H Y &: H C ; pourceque la proportion qui 
eft entre ces deux differences eft donnée. Et de plus a 
caufe qde Y M eft donnée , la difference qui ^{t entre 
M H,ô£ H C, I'eft auffy;& enfin pourceque C M eft don- 
née, il ne refte plus qu'à trouucr M H le cofte du triangle 



rectangle C M H, dont on a l'autre cofte CM, & on a 
auffy la difference qui eft entre C H la baze , & M H le 
cofle demande^, d'où il eft ayfe dele trouuer. car fi on 
prent ;^pour l'excès de C H fur M H, & « pour la longeur 

de la ligne C M, on aura j;^-- \ k^ pour M H. Et apre's 

auoir ainfî le point H>s'il fe troune plus loin du point Y> 

que 



138 



SECOND BOOK 

(see pages 137 and 138), as the vertex, C as a point on the curve/''"' 
and F as one of the foci. It then remains to determine H, the other 
focus. This may be found by considering that the difference between 
FY and FC is to the différence between HY and HC as rf is to t' ; that 
is, as the longer of the Hues measuring the refractive power of the lens 
is to the shorter, as is evident from the manner of describing the ovals. 

Since the lines FY and FC are given we know their diff'erence ; and 
then, since the ratio of the two differences is known, we know the dif- 
ference between HY and HC. 

Again, since YM is known, we know the difference between MH 

and HC, and therefore CM. It remains to find MH, the side of the 

right triangle CMH. The other side of this triangle, CM, is known, 

and also the difference between the hypotenuse, CH and the required 

side, MH. We can therefore easily determine MH as follows: 

n- 1 
Let X' = CH-MH and ;/-CM; then —-- -y^ = MH, which deter- 

mines the position of the point H. 

[1701 "Circonférence." 



139 



GEOMETRY 

If HY Is greater than HF, the curve CY must be the first part of 
the third class of oval, which has already been designated by 3A3. 

But suppose that HY is less than FY. This includes two cases: 
In the first, HY exceeds HF by such an amount that the ratio 
of their difference to the whole line FY is greater than the ratio of e, 
the smaller of the two lines that represent the refractive power, to d, 
the larger; that is, if HF^c, and HY=c-|-/î, then dh is greater than 
2ce-\-eh. In this case CY must be the second part 3Y3 of the same 
oval of the third class. 

In the second case dJi is less than or equal to 2ce-\-eh, and CY is the 
second part 2X2 of the oval of the second class. 

Finally, if the points H and F coincide, FY = FC and the curve 
YC is a circle. 

It is also necessary to determine CAC, the other surface of the lens. 
If we suppose the rays falling on it to be parallel, this will be an ellipse 
having H as one of its foci, and the form is easily determined. If, 
however, w^e suppose the rays to come from the point G, the lens must 
have the form of the first part of an oval of the first class, the two foci 
of which are G and H and which passes through the point C. The 
point A is seen to be its vertex from the fact that the excess of GC 
over GA is to the excess of HA over HC as d is to c. For if k repre- 
sents the difference between CH and HM, and x represents AM, then 
x—k will represent the difference between AH and CH ; and if g repre- 
sents the difference between GC and GM, which are given. g-\-x 



140 



Livre Second. 36 j^ 

que n'en eft le point F, la ligne C Y doit eftre la premie- 
re partie de l'ouale du troifiefme genre^qui a tantoft efté 
nommée 3 A 3: Mais ii H Y eft moindre que F Y, oubien 
ellefurpafTe H F de tant, que leur difference eft plus 
grande a raifon de la toute F Y, que n'eft e la moindre 
des lignes qui mefureni! Tes refradionscoipparée auec d 
la plus grande, c'eft a dire que faifant H F 30 c, &: 
•HYoof •^h^dht^'çXws, grande que ^ce-\-eh y &c lors 
C Y doit eftre la féconde partie de la mefme ouale du 
troiiîefrae genre, qui a tantoft efte'nomee 3 Y'3jOubien 
z:^ /; eft efgale , ou moindre que 2 ce-i-eb:* Ôc lors CY 
doit eftre la féconde partie de Touale du fécond genre 
qui a cydeftlisefte nommée 2X2. Et enfin fî le point H 
eftie mefme que le point F,-ce qui n'arriue quelorfque 
F Y & F C font efgales ccte ligne Y C eft vn cercle. 

Après cela il faut chercher C A G l'autre fuperficie de 
ce verre, qui doit eftre vne Ellipfe , dont H foit le point 
bruflantjfî on fuppofe que les rayons qui tombent deffus 
foiët parallèles; & lors il eft ^yfé de la trouuer. Mais fi on 
fuppofe qu'ils vienêt du poinrG.çe doit eftre la premiere 
partie d'vne ouale du premier genre,dortt les deux poins 
bruftans foiët G & H, & qui pafle par le point Cid'où on 
trouue le point A pour le fommet de cete ouale,en confî- 
derâE,que G Cdoit eftre plus grade que GA,d'vne quan- 
tité'', qui foit a celle dont H A furpafle H C,comme dà.e. 
car ayant pris J^pour la difFerence,qui eft entre C H,& H 
MjfîonfuppofeArpour AM,ou auraj^;-- j^, pour la diffe- 
rence qui eft entre A H, & C H; puis fi on prent g pour 
celle, quieftentreG C, &GM, qui font données, on 
aura^H-ATpour celle, qui eft entre GC, & GA; & 

Zz 3 pour- 



141 



CommÔt 
ou peut 
faire vn 
veii"e,qui 
air le mef- 
me efFed 
que le 
précéder, 
& que la 
conuexi- 
ré del'vne 
de fes fu- 
perficies 
aula pro- 
portion 
<lonnée 
aueccelle 
dellautre. 



}^^ La GEOMETRIE. 

pourcequecetc dernière^ -f- a; eft à l'autre .V— ^, com- 
me ^eft à ^ . onkge'+-excodx '- d\t oubien ^__^ 

pour la ligne a:, ou A M , par laquelle on determine le 
point A qui eftoit cherché. 

Pofonsmaintenent pour l'autre cas 5 qu'on ne donnç 
que les poins G C, &: F, auec la proportion qui eft entre 
les lignes AM, & Y M, & qu'il faille trouuer la figure du 
verre AC Y, qui face que tous les rayons, qui vienenc 
du point G s'aflTemblent au point F. 

On peut derechef icy fe feruir de deux ouales dont 
IVne, A C, ait G ôc Hpour fes poins bruflansi & l'autre^ 




C Y,ait F& H pour les fîens.Et pour les trouuer,premic- 
rement fuppofant le point H qui eft commun a toutes 
deuxeftre connu, ie cherche A M par les trois poins 
G,C,H,en la façon tout mainteneut expliquecjafçauoir 
prenant >^our la difference, qui eft entre C H , & H M; 
&:^pour celle qui eft entre G C, &GM: &ACcftant 
la premiere partie de l'Ouale du premier genre , iay 

—jZTT P^^r A M: puis ie cherche au jQfy M Y par les trois 

poinsF, C, H,enforte que CY foit la premiere partie 
dVne ouate du troiûefme genre^ éprenant y pour M Y, 



142 



SECOND BOOK 

will represent the difference between GC and GA ; and since 

g-j-x : x~k=d : e, we have ge^cx^dx—dk, or KM=x= ^^'^ 

d—e ' 

which enables us to determine the required point A. 

Again, suppose that only the points G, C, and F are given, together 
with the ratio of AM to YM ; and let it be rec^uired to determine the 
form of the lens ACY which causes all the rays coming from the point 
G to converge to F, 

In this case, we can use two ovals, AC and CY, wnth foci G and H, 
and F and H respectively. To determine these, let us suppose first 
that H, the focus common to both, is known. Then AM is determined 
by the three points G, C, and H in the way just now explained; that is 
if k represents the dift'erence between CH and HM, and g the differ- 
ence between GC and GM, and if AC be the first part of the oval of the 

first class, we have AM= -, . 

d—c 

We may then find MY by means of the three points F, C, and H. 

If CY is the first part of an oval of the third class and we take y for 

MY and f for the difference between CF and FM, we have the dif- 



143 



GEOMETRY 

ference between CF and FY equal to /+3' ; then let the difference 
between CH and HM equal k, and the difference between CH and HY 
equal k-\-y. Now k-\-y : f-\-y=^e : d, since the oval is of the third class, 

whence MY = "-'- . Therefore, AM + MY = AY = '^'^, '\ whence it 

a—e d—e 

follows that on whichever side the point H may lie, the ratio of the 
line AY to the excess of GC-j-CF over GF is always equal to the ratio 
of e, the smaller of the two lines representing the refractive power of 
the glass, to d — r, the difference of these two lines, which gives a very 
interesting theorem.''"' 

The line AY being found, it must be divided in the proper ratio into 
AM and MY, and since M is known the points A and Y, and finally 
the point H, may be found by the preceding problem. We must first 
find whether the line AM thus found is greater than, equal to, or less 

than -J — . If it is greater, AC must be the first part of one of the 
a—e 

third class, as they have been considered here. If it is smaller, CY 
must be the first part of an oval of the first class and AC the first part 
[177] "Qyj Ç5(- yj^ assez beau théorème." 



144 



Livre Second. 3^7 

&/pourla difference, qui eft entre C F , & F M , i'ay 
f-h-yt pour celle qui eft entre C F, & F Y: puis ayant dé- 
fia /^pour celle qui eft entre C H, &: H M,iay j^ -h y pour 
celIequieftentreCH, &:H Y,que ie fcay deuoir eftre 
àf'T'y comme e eft à â', a caufe de i'Ouale du troifiefmc 

genre, d ouïe trouue que j ou MY elt -7777 puis roi- 
gnant enfemble les deux quantités trouue'es pour A M, & 

MYjietrouue*^— ^"Tjpourlatoute A YjD'où il fuit que 

de quelque cofte'quefoitfuppofe'le point H, cete ligne 
A Y eft toufiours compofée d'vne quantité', qui eft a cel- 
le dont les deux enfemble G C , & C F furpallent la tou- 
te G F, Comme ^,la moindre des deux lignes qui feruent 
a mefurer les refradlions du verre propofe', eft à d— e , la 
difference qui eftentre ces deux lignes; cequi eft vn af- 
fés beau theorefme. Or ayant ainfî la toute A Y, il la 
faut couper felon la proportion que doiuent auoir {ts 
parties A M & M Y- au moyen de quoy pource qu'on a 
défia le point M, on trouue aufly les poins A & Y,- &en 
fuite le point H, par le problefme precedent. Mais au- 
parauantilfautregarder,filalignc A M ainfi trouueeeft 

plus grande que jtt;^^ P^"^ petite, ouefgale. Car fi elle 
eft plus grande, on apprent de là que la courbe A C doit 
eftre la premiere partie d'vne ouale du premier genre^ 6c 
C Y la premiere d'vne du troifiefme, ainfi qu'elles ont 
efté icy fuppofees: au lieu que fî elle ç,{z plus petit© , cela 
monftre que c'eft C Y, qui doit eftre la premiere partie 
d'vne ouale du premier genre J & que AC doit eftre la 
premieredVne du troifiefme : Enfinfi AM eft efgale à 

i-e 

145 



S^8 La Géométrie. 

^ 7^ les deux courbes A C & C Y doiuenc eftre deux hy- 
perboles. 

On pourroit eftendre ces deux problefmes a vne infi- 
nite d'autres cas, que ie ne m'arefre pas a deduire,à caufe 
qu'ils n'ont eu aucun vfage en la Diopcrique. 

On pourroit aufTypafler outre, & dire j Ibrfque iVne 
des fuperficies du verre eft donnée, pouruû qu'elle ne 
ibit quetouteplatejoucompofeede fecflions coniques, 
ou de cercles; comment on doit faire fon autre fuperfî- 
cie, affin qu'il tranfmet te tous les rayons d'vn point don- 
ne', a vn autre point aufTy donné, car ce n'eft rien de plus 
difficile que cequeie viens d'expliquer ; ou plutoftc'eft 
chofe beaucoup plus facile^ à caufex^ue le chemin en efc 
ouuert. Mais iayme mieux, que d'autres le cherchent, 
affinques'ils ont encore vn peu de peine à le trouuer, ce- 
la leur face d'autant plus eftimer l'inuention âQS chofes 
qui font icy demonft rces. 
Au refte ie n'ay parlé en tout cccy, que des lignes cour- 
onpeuc bes, qu'on peut defcrire fur vne fuperficie plate ; mais il 
appliquer eft ayfé de rapporter cequc i'en ay dit, à toutes celles 
eftédic qu'on fçauroit imaginer eftre formées , par le mouue- 
\^ntT mentreguHerdespoinsdê quelque cors, dansvnefpace 
courbes qui a trois dimenlîons. A fçatioir en tirant deux perpen- 
tt7nT diculaires,de chafcun des poins de la ligne courbe qu'on 
fuperficie y^^^ confîderer,flir deux plans qui s'cntrecouppent a an- 
cdies qui gles droits, l'vne fur l'vn, & l'autre fur l'autre, car les ex- 
^^'^i"-''n tremitesde ces perpendiculaires defcriuent deux autres 
"fpace qu" lignes courbcs, vneTur chafcun de ces plans , defquelles 

rrr\lCrll. -/^ 1 /V* t^ ._ _^^ J ^ *. ^ ^.—^ I«>^^->m ^'^ffC 

les 

146 



c 
a 

xnenfioDS. 



^ "°'' ^'' on peut,en la façon cy de{fusexpliquee,determiner tous 



SECOND BOOK 

of one of the third class. Finally, if AM is equal to '^^ , the curves 

d—e 
AC and CY must both be hyperbolas. 

These two problems can be extended to an infinity of other cases 
which I will not stop to deduce, since they have no practical value in 
dioptrics. 

I might go farther and show how, if one surface of a lens is given 
and is neither entirely plane nor composed of conic sections or circles, 
the other surface can be so determined as to transmit all the rays from 
a given point to another point, also given. This is no more difficult 
than the problems I have just explained; indeed, it is much easier since 
the way is now open ; I prefer, however, to leave this for others to 
work out, to the end that they may appreciate the more highly the dis- 
covery of those things here demonstrated, through having themselves 
to meet some difficulties. 

In all this discussion I have considered only curves that can be 
described upon a plane surface, but my remarks can easily be' made to 
apply to all those curves which can be conceived of as generated by the 
regular movement of the points of a body in three-dimensional space.'"*' 
This can be done by dropping perpendiculars from each point of the 
curve under consideration upon two planes intersecting at right angles, 
for the ends of these perpendiculars will describe two other curves, one 
in each of the tw'o planes, all points of which may be determined in the 
way already explained, and all of which may be related to those of a 
straight line common to the two planes ; and by means of these the 
points of the three-dimensional curve will be entirely determined. 

■'""' This is the hint which Descartes gives of the possibility of the extension of 
his theory to solid gcomctrv. This extension was effected largely by Parent (1666- 
1716), CÎairaut (1713-1765), and Van Schooten (d. 1661). 



147 



GEOMETRY 

We can even draw a straight line at right angles to this curve at a 
given point, simply by drav^^ing a straight line in each plane normal to 
the curve lying in that plane at the foot of the perpendicular drawn 
from the given point of the three-dimensional curve to that plane and 
then drawing two other planes, each passing through one of the straight 
lines and perpendicular to the plane containing it ; the intersection of 
these two planes will be the required normal. 

And so I think I have omitted nothing essential to an understanding 
of curved lines. 



148 



Livre Second. }^9 

les poins, & les rapporter a ceux de la ligne droite , qui 
eft commune a ces deux plans, au moyen dequoy ceux 
de la courbe, qui a trois dimenfions, fout entièrement 
determines. Mefme fi on veut tirer vne ligne droite,qui 
couppe cete courbe au point donne a angles droits • il 
taut feulement tirer deux autres lignes droites dans les 
deux plans, vne en chafcun, qui couppent a angles droits 
les deux lignes courbes, qui y font, aux deux poins , où 
tombent les perpendiculaires qui vienent de ce point 
donne', car ayant efleue deux autres plans , vn fur chaf- 
cune de ces lignes droites, qui couppea angles droits le 
plan où elle eît, ou aura Tinterfedlion de ces deux plans 
pour la ligne droite cherchée. Et ainfî ie penfe n'auoir 
rien omis des elemens, qui font neceflaires pour la con- 
noiflance des lignes courbes. 



149 



BOOK THIRD 



Geometry 

BOOK III 

On the Construction of Solid and Supersolid Problems 

WHILE it is true that every curve which can be described by a con- 
tinuous motion should be recognized in geometry, this does not 
mean that we should use at random the first one that we meet in 
the construction of a given problem. We should always choose with 



152 



L A 

GEOMETRIE. 

LIVRE TROISIESME. 

^e la conflruBion des T^rohlefmes , qui 
font Solides^ ou plu/que Solides, 

De quel- 

TJ N c o RE que toutes les lignes courbes, qui peuuent ^" ^'g"" 
-■^eftre defcrites par quelque mouuement régulier, on peut 
doiuent eftre receuës en la Géométrie , ce n'eft pas a ai- ^^ ^""''■» 
re qu'il foit permis de fe feruir indifféremment de la pre- firudion 
miere qui le rencontre, pour la conftruétion de chafque ^^, fî"l!^* 

Aaa pro- me. 



153 



37° La Geo metr ie. 

problefme: mais ilfautauoir foin de choifir toiifiours la 
pins fîmple, par laquelle il foit pofTible de le refondre. 
Et mefme il eft a remarquer, que par les plus fimples on 
ne doit pas feulement entendre celles, qui pcuuent le 
plus ayfement eftre defcrites , ny celles qui rendent la 
conftrudion, ou la demonftration du Problefme propo- 
fé plus facile, mais principalement celles, qui font du 
plus fimple genre,qui puiiïe feruir a determiner la quan- 
tité qui eft cherchée. 




Exemple 

touchant 

l'inuentiô 

de plu- 

fieurs 

moytaes 

propro- 

tioncUcs. 



Comme par exemple ie ne croy pas, qu'il y ait aucu- 
ne façon plus facile, pour trouuer autant de moyennes 
proportionnelles, qu'on veut, nydpnt la demonftration 
foit plus euidcnte, que d'y employer les hgnes courbes, 
qui fe defcriuent par l'inftrument X Y Z cy defTus expli- 
qué! Car voulant trouuer deux moyennes proportion- 
nelles entre Y A & Y E, il ne faut que defcrire vn cercle, 
dont le diamètre foit Y E; &: pource que ce cercle coup- 

pe 



154 



THIRD BOOK 



care the simplest curve that can be used in the sokition of a problem, 
but it should be noted that the simplest means not merely the one most 
easily described, nor the one that leads to the easiest demonstration or 
construction of the problem, but rather the one of the simplest class 
that can be used to determine the required quantity. 

For example, there is, I believe, no easier method of finding any num- 
ber of mean proportionals,"™^ nor one whose demonstration is clearer, 
than the one which employs the curves described by the instrument 
XYZ, previously explained.''*"^ Thus, if two mean proportionals 
between YA and YE be required, it is only necessary to describe 

II"»] Por the history of this problem, see Heath, History, Vol. I, p. 244, et seq. 
'^^1 See page 46. 



155 



GEOMETRY 

a circle upon YE as diameter cutting the curve AD in D, and YD is 
then one of the required mean proportionals. The demonstration 
becomes obvious as soon as the instrument is appHed to YD, since YA 
(or YB) is to YC as YC is to YD as YD is to YE. 

Similarly, to find four mean proportionals between YA and YG, or 
six between YA and YN, it is only necessary to draw the circle YEG, 
which determines by its intersection with AE the line YE, one of the 
four mean proportionals ; or the circle YHN, which determines by its 
intersection with AH the line YH, one of the six mean proportionals, 
and so on. 

But the curve AD is of the second class, while it is possible to find 
two mean proportionals by the use of the conic sections, which are 
curves of the first class. "^'' Again, four or six mean proportionals can 
be found by curves of lower classes than AF and AH respectively. It 
w^ould therefore be a geometric error to use these curves. On the other 
hand, it would be a blunder to try vainly to construct a problem by 
means of a class of lines simpler than its nature allows. ^^"' 

Before giving the rules for the avoidance of both these errors, some 
general statements must be made concerning the nature of equations. 
An equation consists of several terms, some known and some unknown, 
some of which are together equal to the rest ; or rather, all of which 
taken together are equal to nothing ; for this is often the best form to 
consider.'''"^ 

''*'' If we let .V and y represent the two mean proportionals between a and b we 
have a : X = X : y ^= y : b, whence a- = ay ; y- = bx, and xy = ab. Therefore 
X and y may be found by determining the intersections of two parabolas or of a 
parabola and a hyperbola. 

""^ Cf. Pappus, Book IV, Prop. 31, Vol. I, p. 273. See also Guisnée. Applica- 
tion dc l'Algèbre a la Géométrie, Paris, 1733, p. 28, and L'Hospital, Traité Analy- 
tique des Sections Coniques, Paris, 1707, p. 400. 

[183] Yj^g advantage of this arrangement had been recognized by several writer* 
before Descartes. 



156 



LïvRE Troisiesme. 57' 

pela courbe A D au point D, Y D eft IVne des moyennes 
proportionnelles cherché^es. Dont la demonftration fe 
voit a l'œil par la feule application de cet inftrument fur 
h ligne Y D. car comme Y A, ou YB, quiluy eftefgale 
cftaYCjainfiYCeftaYDi&YDa Y£, 

Toutdemefme pour trouuer quatre moyennes pro- 
portionelles entre Y A & Y G; ou pour en trouuer fix en- 
tre Y A & Y N, il ne faut que tracer le cercle Y F G, qui 
couppant A F au point F, determine la ligne droite Y F, 
qui eft iVne de ces quatre proportionnelles j ou Y H N, 
qui couppant A H au point H, determine Y H l'vHe des 
fix, & ainfi des autres. 

Maispourceque la ligne courbe A D eft du fécond 
genre, & qu'on peut trouuer deux moyenes proportio- 
nelles par les fedions coniques,qui font du premier • de 
auflypourcequ'on peut trouuer quatre ou fix moyenes 
proportionellcs, par des lignes qui ne font pas de genres 
fi compofés, que font A F, & A H, ce feroit vne. faute en 
Géométrie que de les y employer. Et c'eft vne faute 
aufiy d'autre cofté de fetrauailler inutilement a vouloir 
conftruire quelque problefme par vn genre de lignes 
plus fimple, que fa nature ne permet. 

Or affin que ie puifie icy donner quelques reigles, Dc h lu. 
poureuiterrvne& l'autre de ces deux fautes, il faut que ^^'^ '^.l*' 
ie die quelque chofe en general de la nature des Equa- '^"^""^* 
tions-c'eft a dire des fommes compofces de plufieurs ter- 
mes partie connus, & partie inconnus , dont les vns font 
efgaux aux autres, ouplutoft qui confideres tous enfem- 
blc font efgaux a rien, car ce fera fouuent le meilleur de 
les confiderer en cete forte, 

Aaa 2 Scaches 

157 



37^*- La g eometrie. 

iipeut'y" Scachés donc qu'en chafque Equation, autant que 
auoir de |^ quantité inconnue a de dimenfions , autant peut il y 
ea chafqi auoir de diuetfes racines, c cft a dire de valeurs de cete 
Equatiô, quantité, car par exemple (î on fuppofe x efgale a 2; ou- 
bien x— i efgal a rien • & derechef a; 30 3 j nubien 
X — 3 33 0; en multipliant ces deux equations .V -- 2000, 
OCX" s x>(?,rvne par l'autre, on aura xx— s x-h6'X>o, 
ovihicnxx20 fx— 5, qui eft vne Equation en laquelle la 
quantité a: vaut 2 Sctoutenfemble vaut $. Que fî dere- 
chef on fait AT — 4 30 0, & qu'on multiplie cete fomrae par 
xX'-s^'^^^o, on aura x^ — ^ x x -h 26 x — z^.'X) 0, 
qui eft vne autre Equation en laquelle x ayant trois di- 
menfions a aufly trois valeurs,qui font 2, 5, &4. 
Quelles j, ais fouuent il arriue, que quelques vnes de ces raci- 
fauflesra- nes font fauffes , ou moindres que rien, comme fi on 
cines. fuppofe quc X defigne aufiTy le défaut d'vne quantité, 
quifoity ,onaAr-f-y00(? , qui eftant multipliée par 
X ^ " 9 X X -h 26 X " 2^00 fait 

X^"4X^ •"19XX'+- 106 x— 120 oo<? 
pour vne equation en laquelle il y a quatre racines , a 
fçauoir trois vrayes qui font 2, 3, 4, &vne faufle qui 
cft f. 
cômcût £j jj Qjj. euidemment de cecy, que la fbmme d'vne 

on peut •' ' J 

diminuer equation, qui Contient plufieuts raciucs , peut toufiours 
^dcT^- ^^ ^^^^ diuifée par vn binôme compofe' de la quantité in- 
mcnfions connuë,moins la valeur de Tvne des vrayes racines, la- 
qimion quelle quc cc foltj ou plus la valeur de l'vne des fauffés. 
lorfqu'on ^q moycn de quoy on diminue d'autant ùs dimeu- 

connoift >, 
qucK ilOnS. 

ou' vne de Et recipioqucment que fi la fômme dVne équation 

158 



THIRD BOOK 

Every equation can have''^'' as many distinct roots (values of the 
unknown quantity) as the number of dimensions of the unknown 
quantity in the equation. ''^°' Suppose, for example, .v = 2 or x — 2 = 0, 
and again, x = 3, or x — 3 = 0. Multiplying together the two equa- 
tions X — 2 = and x — 3 = 0, we have x- — 5.1--1-6 ^= 0, or x- = Sx — 6. 
This is an equation in w^hich .r has the value 2 and at the same time''^"' 
X has the value 3. If we next make a*— 4 := and multiply this by 
X- — Sx-\-6 = 0, we have x^'—9x--\-26.v — 24 = another equation, in 
which X, having three dimensions, has also three values, namely. 2, 3, 
and 4. 

It often happens, however, that some of the roots are false''*'' or less 
than nothing. Thus, if we suppose x to represent the defect''*"' of a quan- 
tity 5, we have .^--]-5 = which, multiplied by x^—9x--\-26x — 24 = 0, 
yields .a-*— 4.r^ — 19.t---|-106.r— 120 = 0, an equation having four roots, 
namely three true roots, 2, 3, and 4, and one false root, 5.''*^' 

It is evident from the above that the sum'^""' of an equation having 
several roots is always divisible by a binomial consisting of the unknown 
quantity diminished by the value of one of the true roots, or plus the 
value of one of the false roots. In this way,''"'' the degree of an equa- 
tion can be lowered. 

On the other hand, if tlie sum of the terms of an equation'""' is not 
divisible by a binomial consisting of the unknown quantity plus or 

'^'^^ It is worthy of note that Descartes writes "can have" ("peut-il y avoir"), 
not "must have," since he is considering only real positive roots. 

[185] -phat is as the number denoting the degree of the equation. 

[1S6] '"Poyj; ensemble," — not quite the modern idea. 

[187] "j^acines fausses," a term formerly used for "negative roots." Fibonacci, 
for example, does not admit negative quantities as roots of an equation. Scntti de 
Leonardo Pisano, published by Boncompagni, Rome, 1857. Cardan recognizes 
them, but calls them "sestimationes falsas" or "fictje," and attaches no special sig- 
nificance to them. See Cardan, Ars Magna, Nurnberg, 1545, p. 2. Stifel called 
them "Numeri absurdi," as also in Rudolff's Coss, 1545. 

[18S] <ij^g défaut." If X = — 5, — 5 is the "defect" of 5, that is, the remainder 
when 5 is subtracted from zero. 

[189] -pj^jj^. jg^ three positive roots, 2, 3, and 4, and one negative root, — 5. 

''°°^ "Somme," the left meml)er when the right member is zero; that is, what 
we represent by /(.r) in the equation /(.r)=0. 

[191] 'Yhat is. by performing the division. 

''""' "Si la somme d'un equation." 



159 



THIRD BOOK 

minus some other quantity, then this latter quantity is not a root of the 
equation. Thus the"™' above equation a-*— 4.r" — iar- + 106.r— 120 = 
is divisible by x—2, .r-3, .r— 4 and .r+5,""" but is not divisible by .v 
plus or minus any other quantity. Therefore the equation can have 
only the four roots, 2, 3, 4, and 5.'""' We can determine also the num- 
ber of true and false roots that any equation can have, as follows i'""' 
An equation can have as many true roots as it contains changes of sign, 
from -f to — or from — to + ; and as many false roots as the num- 
ber of times two + signs or two — signs are found in succession. 

Thus, in the last equation, since -f .r* is followed by — 4.^-^ giving a 
change of sign from + to — , and — 19.r- is followed by +106.r and 
-f 106.r by —120, giving two more changes, we know there are three 
true roots ; and since —Ax^ is followed by —\9x- there is one false root. 

It is also easy to transform an equation so that all the roots that 
were false shall become true roots, and all those that were true shall 
become false. This is done by changing the signs of the second, fourth, 

'^'"'' First member of the equation. Descartes always speaks of dividing the 
equation. 

'^"■•^ Incorrectly given as x — 5 in some editions. 

ti^B) Where 5 would now be written — 5. Descartes neither states nor explicitly 
assumes the fundamental theorem of algebra, namely, that every equation has at 
least one root. 

[190] Yj^jg jg ^Y^ç. -^yeii I.jnown "Descartes's Rule of Signs." It was known how- 
ever, before his time, for Harriot had given it in his Artis analyticac praxis, Lon- 
don, 1631. Cantor says Descartes may have learned it from Cardan's writings, 
but was the first to state it as a general rule. See Cantor, Vol. 11(1) pp. 496 
and 725. 



160 



racines ea 
e 



Livre Troisiesme. ^^^ 

nepeuteftrcdiuifeeparvn biuômecompofédclaquau- on'^eu* 
titeinconnue-r- ou — quelque autre quantité, cela tef^ examiner 

•/.Al 1 1. Il quelque 

moigne que cete autre quantité n eft la valeur d aucune Quantité 
de fes racines. Comme cete dernière donnée 

elilava- 
X'^'-^X^-'l^XX'i' lOÔX—llOOOo leurd'vnc 

peut bien eftrediuifée, par x — 2, & par x— 3,&:par """'^' 
a: — 4, & par at 4- 5 ; mais non point par a; 4- ou - - aucu- 
ne autre quantité', cequi monftre qu'elle ne peut auoir 
que les quatre racines 2.,3,4,ôc y. 

On connoiftaufly de cecy combien il peut y auoir de Combien 
vrayes racines, &: combien de faufles en chafque Equa- luoir de 
tion. A fçauoirily en peut auoir autant de vrayes, que ^"7" 
les lignes -H & — s'y trouuent de fois eft te changes ,• & "hafq" 
autant de faufles qu*il s'y trouue de fois deux lignes 4-, ^4"^"° 
ou deux lignes — qui s'entrefuiuent. Comme en la der- 
nière, a caufe qu'après -i- x'^Hya-' /\.x ^qui eft vn chan- 
gement du ligne H- en-, & après- 19 :v a: il y a -H 105 a:, 
&apres-f-lo6 Arilya— izoqui font encore deux autres 
changemens, onconnoift qu'il y a trois vrayes racines;& 
vue fauire,a caufe que les deux lignes — ,de 4.x\ôci^xx, 
s'entrefuiuent. 

De plus il eft ayfc de faire en vne mefme Equation, q 
que toutes les racines qui eftoicnt fauiTes deuienent onùit 
V rayes, & par mefme moyen que toutes celles qui eftoiêt ^^"Ves 
vrayes deuienent faufles : a fçanoir en changeant tous ^f^"" 
les lignes -h ou - qui font en la féconde , en la nJTt^on" 
quatriefme , en la fixiefme , ou autres places qui le f J^^'^^^ 
defignent par les nombres pairs , fans changer ceux Us vrayes 
de la premiere , de la troifiefme, de la cinquiefme ^*"^"' 
& femblabics qui fe defigiient par les nombres 

Aaa 3 impairs. 

161 



.ornent 



574 La Géométrie. 

impairs. Comme fi au lieu de 

-h X'*-' ^X^ — l^ XX-h- lo6 X — I20 x> a 

on efcric 

-^ X ^ -^ 4.X' — ï^xx — 106 X-' 120C00 
on a vne Equation en laquelle il n'y a quVne vraye ra- 
cine, qui eft j, & trois faufTes qui font 2,5, &4. 
Comcnc Quefifansconnoiftre la valeur des racmes d'vne E- 

on peut . 

augmen- quation,onla veut augmenter, ou diminuer de quelque 
^^■°"^'^*' quantite'connue, il ne faut qu'au lieu du terme inconnu 
lesracines enfuppofcr vn autre, qui fbftplus ou moins grand de ce- 
quatL^, te me fine quantité", &le fubftituer par tout en la place 
fans les du premier. 

connoi- Comme fi on veut augmenter de 5 la racine de cete 
Equation 

X"^-^ 4^X^'^lSXX^'l06 X-' TlOlO 

il faut prendrey au lieu d'x , &: penfer que cctQ quantité' 
y eft plus grande qux de 3, en forte que ^ — 5 eft efgal 
SLXjScâuVieud' X Xj ilfautmettrelequarré'd'y — .3 qui 
eftyj/— 6 y-i- 9 8c ânlieu d' X ^ il faut mettre fon cube 
qui eft^ ' •- 9yy~^ 27 y — 27, & enfin au lieu d' at + il faut 
mettre fonquarrédequarré'qui eft y'*— ity ^-}- f4.yy 
— io8^-f-8r. Et ainiî defcriuant la fbmme précédente 
€Q fubftituant par touty au lieu d'x on a 
y^-^-liy^-h^^yy— lo^y-hSl 
4-47Î-- ^6yy -f- io8y~.ioS 

— ^s>yy -^ ii4y — 171 

— io5y-f-3i8 
— 120 

y^^^Zyi^'iyy ^Sy* :X)0 

oubien 



162 



THIRD BOOK 

sixth, and all even terms, leaving unchanged the signs of the first, third, 
fifth, and other odd terms. Thus, if instead of 

4-.r*-4.r'-19.v--+106.i--120 = 
we vv^rite 

_|_,t-4^4^-n_i9_^.2_io6.r-120 = 

we get an equation having one true root, 5, and three false roots, 2, 3, 
and 4/^"^' 

If the roots of an equation are unknown and it be desired to increase 
or diminish each of these roots by some known number, we must sub- 
stitute for the unknown quantity throughout the equation, another 
quantity greater or less by the given number. Thus, if it be desired 
to increase by 3 the value of each root of the equation 

.i-^_^4.r-'-19.i--106.r-120 = 

put y in the place of x, and let y exceed x by 3, so that y — 3 = x. Then 
for .r- put the square of y — 3, or y- — 6y-\-9; for x^' put its cube, 
y^ — 9y'+27y — 27; and for .i-* put its fourth power,'"'**-' or 

y*- 12_v-'+543'-- 1083'+81. 

Substituting these values in the above equation, and combining, we have 

y* - I2y'^ + 54\- - 108y + 81 

+ 4_v' - 363- + 108y - 108 

- 19v= + 114y- 171 

- 106y + 318 

- 120 



y_ 8\'-''- y--\- 8y = 0,'"^' 

or 3,^_8y_3,+8 = 0, 

'^"'^ In absolute value. 

[19S] "5qj^ quarré de quarré," that is, its fourth power. 

''"^^ Descartes wrote this y* — S^)-- — 3'- + 8y * 00 0, indicating by a star the 
absence of a term in a complete polynomial. 



163 



GEOMETRY 

whose true root is now 8 instead of 5, since it has been increased by 3. 
If, on the other hand, it is desired to diminish by 3 the roots of the 
same equation, we must put 3'+3 = x andv"+63;+9 = x-, and so on. 
so that instead of .v* + 4x'' — 19.r- — 106.r — 120 = 0, we have 

y* ^ 123;^ 4- 54/ + IO83; + 81 

+ 4y' + 363;- + IO83; + 108 

-I9y--- 114v- 171 

— 1063; - 318 

-120 



3,4 -I- 16/' + 713»= — 43; - 420 = 0. 



It should be observed that increasing the true roots of an equation 
diminishes"""' the false roots by the same amount ; and on the contrary 
diminishing- the true roots increases the false roots ; while diminishing 
either a true or a false root by a quantity equal to it makes the root 
zero ; and diminishing it by a quantity greater than the root renders 
a true root false or a false root true."*"' Thus by increasing the true 
root 5 by 3, we diminish each of the false roots, so that the root pre- 
viously 4 is now only 1, the root previously 3 is zero, and the root 
previously 2 is now a true root, equal to 1, since — 2+3 = -f-1. This 
explains why the equation 3'^— 83-- — v-)-8 = has only three roots. 

'"""^ In absolute value. 

■'"'' For example, the false root S diminished by 7 means — (5 — 7)= +2. 



164 



Livre Troisiesme. V^ 

oubien^ ^ — 8^^ •- 1 ^ ^- 8 oo <?. 

oil la vraye racine qui eftoit j eft maintenant 8 , acaufe 
du nombre trois qui luy eft aioufté. 

Que fi on veut au contraire diminuer de trois laraci- 
ne de cete mefme Equation , il faut faire ^ -H 3 ooa; 
&^^-f- 6 y -h ç'x> :«x, & ainfî des autres de façon 
qu'au lieu de 

a:*4-4;v'- i^xx — lOS^x — 12000a 
on met 

-19 yy - 1147 - 171 

— io6y — 318 

— 120 

y^-i- i6y i-\~yiyy-' 4^.-420300. 

Et il eft a remarquer qu'en au2mentant les vrayes ra- Qb'^^» 
cines d'vne Equation, on diminue les fauffes de la mef- tanc les 
me quantité; ou au contraire en diminuant les vrayes,on J/iJJj"^" 
augmente les faufles. Et que fî on diminue foit les vnes diminue 
foit les autres, d*vne quantité qui leur foit efgale, elles f",^^au 
deuienent nulles, &c[ue fi c'eft d'vne quantitéqui les fur- contraire 
pafle, de vrayes elles deuienept faufles, ou de faufles 
vrayes. Comme icy en augmentant de 3 la vrayc racine 
qui eftoit y, on a diminué de 3 chafcune des faufles , en 
forte que celle qui eftoit 4 fi'eft plus qu'i, & celle qui 
eftoit 3 eft nulle, & celle qui eltoit 2 eft deuenueviaye 
& eft I, a caufe que — 2 -+- 3 fait -h i. c'eft pourquoy 
en cete Equation^ * - Syy — i^ -h S so il ny a plus que 
3 racines, entre lefquellcs il y en a deux qui font vrayes, 

I.& 



165 



'' La Géométrie. 

I, ôc 8, &: vne faufTe qui eft auffy i. & en cete autre 

y *-H i6^ ' -^y^yy --4 y -- 410 30« 
il n'y en a qu'vne vraye qui eft 2, a caufe que -H y — 5 fait 
-f- 2, Octrois faufTes qui font j",(5, &7. 
Comcoc Or par cete façon de changer la valeur des racines 
ofter^ïc ^^ns les connoiftre, on peut faire deux chofes, qui auront 
fécond Qy aprés quelque vfage: la premiere eft qu'on peut tou- 
d'vnc E- fîours ofter le fécond terme de l'Equation qu'on exami- 
quatioD. j^g^ ^ fçauoir en diminuant les vrayes racines, de la quan- 
tité connue de ce fécond terme diuifee par le nombre 
desdimenfions du premier, (îl'vn de ces deux termes 
eftantmarque'du figne-t-,rautreeft marqué du ligne— ; 
oubien en l'augmentant de la mefme quantité , s'ils ont 
tous deux le fîgne "f-, ou tous deux le fi gnc—. Comme 
pour ofter le fecon^ terme de la dernière Equatiô qui eft 

y^-^ 16 y ' -h 7^yy — 4 y — 4 lo do d 
ayantdiuiféidpar4, acau(èdcs4 dimenfions du terme 
y 4, il vient derechef 4, c'eftpourquoy icfais ;^ — 4 ooy^ 
& i'efcris 

•^16 ^^-'içz^^HryôS îç_— I024 

— 420 



ou la vraye racine qui eftoit 2, eft 6, a caufe qu'elle eft 

augmentée de 4^ & les faufles qui eftoient y, 6, & 7, ne 

fontplusque 1,2, 6c 3, a caufe qu'elles font diminuc'es 

chafcunede4. 

Tout 



166 



THIRD BOOK 

two of them, 1 and 8, being true roots, and the third, also 1, being false ; 
while the other equation y* — 16y^'-\-7ly- — 4y — 420 = has only one 
true root, 2, since -|-5— 3 = +2, and three false roots. 5, 6, and 7. 

Now this method of transforming the roots of an equation without 
determining their values yields two results which will prove useful : 
First, we can always remove the second term of an equation by dimin- 
ishing its true roots by the known quantity of the second term divided 
by the number of dimensions of the first term, if these two terms have 
opposite signs ; or, if they have like signs, by increasing the roots by 
the same quantity.'""'^ Thus, to remove the second term of the equation 
'\'*+16y+7l3r— 43;— 420^0 I divide 16 by 4 (the exponent of y in 
3;*), the quotient being 4. I then make s — 4 ^ y and write 

2* - I6r' + 96^= - 256^ + 256 

4- 16^' - 192^= + 768^ - 1024 

+ 71x;=- 568^ +1136 

— 4:r + 16 

— 420 



— 25^- — 60^ — 36 = 0. 



The true root of this equation which was 2 is now 6, since it has been 
increased by 4, and the false roots, 5, 6, and 7. are only 1, 2, and 3, 

[202] -pj^^j. jg^ ^y diminishing the roots by a quantity equal to the coefficient of 
the second term divided by the exponent of the highest power of x, with the oppo- 
site sign. 



167 



GEOMETRY 



since each has been diminished by 4. Similarly, to remove the second 
terms of .r*— 2a.r^-)-(2a' — r=).t-=— 2a\v+a* = ; since 2a-^4 = -^we 

1 

must put ^+-rt' = A-and write 



z' + 2a^+la'z^ + la'z + 


y 


-2a^-3a'2^-^a'z- 


h^ 


-\-2a-2- + 2n'z + 


V 






- c'z' - ac'z - 


4^' 


-2a'z- 


a' 


+ 


a' 



2 +{:^a —c jz - (a' -\-ar)z-\- a' --a-c- = 0. 

Having found the value of ,:;, that of x is found by adding -^?. Second, 

by increasing the roots by a quantity greater than any of the false 
roots'""^ we make all the roots true. When this is done, there will be 
no two consecutive + or — terms ; and further, the known quantity 
of the third term will be greater than the square of half that of the 
second term. This can be done even when the false roots are unknown, 
since approximate values can always be obtained for them and the roots 
can then be increased by a quantity as large as or larger than is 
required. Thus, given, 

'^^^ In absolute value. 



168 



LivKE Troisiesme. 377 

Tout de mefme fi on veut ofter le fécond terme de 
x^ — iaxi '^^^^^ xx.. la^x -ha^ooo , 

pourcequetJiuifant i a par 4 il vient ^ ^; il faut faire 
î^-+-^aï)Ar.&^fcnre 



"tai^ -laaii -^a^ ^ 



I./r.4 



4- 



a 



* c c — ace "~aacc 
-+-^4 



— ce —atc — -^aacc 
&{îontrouueapres la valeur de :^, enluyadiouflant ~ a 
on aura celle de at. r ;^,«,n^ 

Cornent 

La féconde chofe, qui aura cvapre's quelque vfâee o" p^uc 

/v» r If, fairequc 

eft, qu on peut touiiours en augmentant la valeur des toutes 
vrayes racines, dVne quantité quifoit plus grande que '"faufTes 
n'eft celle d'aucune des faufles, faire qu'elles deuienent dvnc 
toutes vraves,en forte qu'il n'y ait point deux lignes -f~, ^S^.^^o*^ 

t n • » r f ' deuiencc 

ou.deux lignes -- quiientreluiuent, & outre cela que la vrayes, 
quantité' connue du troifiefme terme foit plus grande, [ç^^yj^"^ 
quelequarré'delamoitiede celle du fécond. Car en- deuienct 
core que cela fe face , iorfque ces faufles racines font ^" "* 
inconnues, il eflayfe néanmoins de iuger a peu pre''- de 
leur grandeur, &de prendre vne quantité, quilesfur- 
pafle d'autant, ou de plus, qu'il n*efl requis a cet effedt. 
Comme fi on a 

Bbb 



16') 



^7^ La Géométrie. 

en faifânt^ - 6 » so -v, on trouuera 

y*- }i»'\y5 HE» î40«/*^ y ♦--45ia »'yy' 4« 19440« ■♦'^ yy"46(;)^«s'^ y ^4ô6j6»« 

>i<» r — jo»»^ HhS^o^'C --zi6o»<j »î<648o»M --777<»»< 

6»»-' ►Î4i44»jr .-li9^«4l ►î-fi84«'l--777<^«< 

►j. 5i »}-' .. 648 «♦? 4« jS88»»^ -- iTiin* 

J tfi iZ<f6 ^ 'j -- 777^ « < 
y"--jy»y'»î'504»» y* - 3780 «^ y»»i- ij'^-o «^ y*-- 172.16 «'y * 30 0. 

OuiKeft manifefte, que yo4 ««, qui eft la quantité' 
connue du troifîefme terme eft plus grande, que le quar- 
rede !*• », qui eft la moitic'de celle du fécond. Et il n'y 
â point de cas, pour lequella quantité, dont on augmen- 
te les vrayes racines, aitbefoina cet efFedt, d'eftre plus 
grande, a proportion de celles qui font données , que 
pour cetuy cy. 
Cômcne ^^j^ ^ caufe que le dernier terme s'y trouue nul, fi on 

on taw * - - ■' 

que cou- ne defire pas que cela foit, il faut encore augmenter tant 

^ bces ^'^^^ P^" ^^ valeur des racines j Et ce ne Tçauroit eftre de 

d'vne E- fi peu, quc cc ne foit afles pour cet effedt. Non plus que 

?o"iea°° lorfqu'on veutaccroiftre le nombre des dimenfious de 

remplies, quelque Equation, ôt faire que toutes les places de Tes 

termes foient remplies. Comme fiaulieude x ' **'•'* 

-«5 30 0, on veut auoir vne Equation, en laquelle la 

quantité'mconnue ait fix dimenfions, & dont aucun des 

termes ne foit nul, il faut premièrement pour 

.^. * * » "*■ — ^30 efcrire 

x' =* * * =*.-^Ar ''■SOO 

puis ayant fait ^—^ 30 A^J on aura 

-- b y y^ a b 

Quileftmanifeftequetantpetitequela quantité' a foit 

fuppof^e 

170 



THIRD BOOK 

x''-^nx''—67i-x'+36n".\-—2l6n\v--\-U96if'.v—7776n'' = 0, 
make v—6n ^ ,^- and we have, 

y^-26n] v-'+540m==1 T*-4320n-'l v"+19440w*l :^--46656n^l v+46656n'' 

' j^ n\' - ZOn-\ + 360nH - 2160m* | + 6480mM - 7776«" 

- 6n-] + 144«^ f - \296n'\ + 5184nM — 7776w'"' 

4- 36n-''J - 648m* I + 3888n^r - 7776«« 

- 216n*J + 2592n-'| — 7776n'' 

+ 1296n^^J — 7776n''' 

- 7776«« 

y^—ZSny^ +504«=^/ —3780«^ +15120«V — 27216»'^3; =0. 
Now it is evident that 504h-, the known quantity""*' of the third term, 

/35 y 

is larger than I ;H ; that is, than the square of half that of the sec- 
ond term ; and there is no case for which the true roots need be in- 
creased by a quantity larger in proportion to those given than for this 
one. 

If it is undesirable to have the last term zero, as in this case, the 
roots must be increased just a Httle more, yet not too Httle, for the pur- 
pose. Similarly if it is desired to raise the degree of an equation, and 
also to have all its terms present, as if instead of x'' — b = 0, we wish 
an equation of the sixth degree with no term zero, first, for .r" — b = 
write x''' — bx = 0, and letting _v — a^^ -v we have 

3,6_6a/+15ay-20fl"r + 15ay-(6a'^+5)y+a''+a& = 0. 

It is evident that, however small the quantity a, every term of this equa- 
tion must be present. 
'^^' I. e., the coefficient. 



171 



GEOMETRY 



We can also multiply or divide all the roots of an equation by a 
given quantity, without first determining their values. To do this, sup- 
pose the unknown quantity when multiplied or divided by the given 
number to be equal to a second unknown quantity. Then multiply or 
divide the known quantity of the second term by the given quantity, 
that in the third term by the square of the given quantity, that in the 
fourth term by its cube, and so on, to the end. 

This device is useful in changing fractional terms of an equation to 
whole numbers, and often''"^' in rationalizing the terms. Thus, given 

po O 

x^— M 3 .r'^-f T^x — ^ = 0, let there be required another equation 

27 27 \T 

in which all the terms are expressed in rational numbers. Let j'= \'^ 
and multiply the second term by ^J^, the third by 3, and the last by 

3 V3. The resulting- equation is y^ — 3y- -{-'^y— q =0- Next let it be 
required to replace this equation by another in which the known quanti- 
ties are expressed only by whole numbers. Let r=3y. Multiplying 

26 8 

3 by 3, -r- by 9, and by 27, we have 

;:^-9s-+26r-24 = 0. 

The roots of this equation are 2, 3, and 4; and hence the roots of the 
'"""' But not always. Compare the case mentioned on page 175. 



172 



Livre Troisiesme. ^^^ 

fuppofee toutes les places de l'Equation ne laiflent pas 
d*eftre remplies. 

De plus on peut, fans connoiftre la valeur des vrayes Commet 

1 . 1 . • 1- ^■ -r on peut 

racines dvne Equation, les multiplier, ou diuiier tou- muUi- 
rcs, par telle quantité connue qu on veut. Cequi fe fait 5|'"re°dcs 
en fuppofant que la quantité' inconnue eftant multipliée, racines 
oudiuifce, par celle qui doit multiplier, ou diuifer les [^JJj^J^[, 
racines, eft efgale a quel<jue autre. Puis multipliant, ou ftr«- 
diuifant la quantité connue du fécond terme, par cete 
mefrae qui doit multiplier, ou diuifer les racines j &par 
fon quarré, celle du troifiefmcj &: par fon cube , celle du 
quatricfmej & ainfi iufques au dernier. Ce qui peut fer- ^ °rTdu?ft 
uir pour réduire a des nombres entiers &rationaux, lesi" °om- 
fradtions, ou lôuuent aufTy les nombres fours , qui fe puVdVnê 
trouHcnt dans les termes des Equations. Comme fi on a Çq^^^'on 

Xs-'Yl XX-^-^X'^^yjZOOy tiers. 

& qu'on veuille en auoir vne autre en fâ place^ dont tous 
lestermcs s'expriment par des nombres ratiouaux; il faut 
fuppofer y 30a; T^ 3 , 5c multiplier par V^ la quantité 
connue dufccondtcrme, qui eft auffy /^3 , & par fon 
quarré qui eft 3 cefle du troifiefme qui eft || , & par fon 
cube qui cft 3 /"5 celle du dernier, qui eft ^1^, , ce qui 
fait 

Puis fi on en veut auoir encore vne autre en la place de 
celle cy, dont les quantités connues ne s expriment que 
par dts nombres entiers^ il faut fuppofer ^^ 30 3 ^ , & mul- 
tipliant 5par5, |<î par 9, & |pari7ontrouue 

V - 9^=^"^ 26 :^»- 24 30 0, OÙ les racines eftant 2,3, 
& 4, on connoift de là que celles de l'autre d'auparauant 

Bbb 2 eftoient 

173 



5^0 La Géométrie. 

eftoient y, I , & |, ôcquc celles de la premiere eftoietrt 

Cômeni Cctc Operation peut aulTf ferurr pour rendre la quan- 
quintitc tïtc confluë dc quclqu'uu des termes de l'Equatiô efgale 
connue ^ quelque autre donnée, comme fi ayant 

des ter- -^* ' * —Ùbx-j^C' 30a 

mesd'vne Qn vcut auoir en faplace vneautrc Equation, en laqueL- 

Equatioii ^ -^ ^-^ 

efcaie a le la quantité' connue, du terme qui occupe là troifiefme 
qu'on''"^ place, a fçauoir celle qui eft icy ^^,roit 5 fl^,il faut fuppo. 

veut •i/~ xcm ^^'c' X» 

ferj^ 30 a: ♦'^ — ,-puisefcrire^ ' * - "i^accyA — ^ V' J 30<7. 
Que les Aurefte tant les vrayes racines que les faufles ne font 
taa'tm'. pas toufîours reclleSj mais quelquefois feulement imagi- 
yes que naiteSj c'cft a dire qu'on peut bien toufiours en imaginer 
pcuucnr autant que lay dit en chafque Equation^ mais qu il.n y a 
cftrcieci- querquefois aucune quantité', qui correfponde a celles 

les ou ^ . ^ ■*• , rr ■ • 

imaginai- qu OU imagine, comme encore qu on en puilie imagi- 
^"' nertroisenccllecy, 'V - 6a:.v^- 15 .V— io30<7, il ny 
en a toutefois quVne réelle, qui eft 2, & pour les deux 
autres, quoy qu'on les augmente, ou diminue, ou multi- 
plie en la façon que ie viens d'expliquer, on nefçauroit 
Tes rendre autres qu'imaginaires . 
lat^du- Or quand pour trouuer la conftrudtion de quelque 
E^°"t!?s pJ*oblefmc,on vient a vne Equation, en laquelle la quan- 
cubiqucs tité inconnue a trois dimenfions ; premièrement fi les 
^^'^^gj^^J'^ quantités connues , quiyfont , contienent quelques 
me eft noinbrcs rompus,illes faut réduire a d'autres entiers, par 
P'^°' la multiplication tantoft expliquée • Et s'ils encontie- 
nentdefburs , il faut auffy les réduire a d'autres ratio- 
naux, autant qu il fera poffible,tant par cete mefme mul- 
tiplication, 

174 



THIRD BOOK 
2 4 

preceding equation are —, 1 and —, and those of the first equation are 

2 / — 1 — 4 ; — 

g V3,y\'3,and— \'3. 

This method can also be used to make the known quantity of any 
term equal to a given quantity. Thus, given the equation 

x'—b-x-\-r = 0, 
let it be required to write an equation in which the coefficient of the 
third term.'""'' namely b-, shall be replaced by 3a-. Let 

Isa' 

and we have 



^=-^'V7^ 



/_3aV + ^V3=0. 

Neither the true nor the false roots are always real ; sometimes 
they are imaginary ;'""'' that is, while we can always conceive of as many 
roots for each equation as I have already assigned,'""*' yet there is not 
always a definite quantity corresponding to each root so conceived of. 
Thus, while w'e may conceive of the equation x" — 6x--\-l3x — 10^0 
as having three roots, yet there is only one real root, 2, while the other 
two, however we may increase, diminish, or multiply them in accord- 
ance with the rules just laid down, remain always imaginary. 

When the construction of a problem involves the solution of an 
equation in which the unknown quantity has three dimensions,'"""' the 
following steps must be taken : 

First, if the equation contains some fractional coefficients,'"'"' change 
them to whole numbers by the method explained above ;'""' if it con- 

[206] Descartes wrote this equation .r * — /)^.r + c^' 30 0, the star showing, as 
explained on page 163, that a term is missing. Hence, he speaks of — b'-x as the 
third term. 

[^'J "Mais quelquefois seulement imaginaires." This is a rather interesting 
classification, signifying that we may have positive and negative roots that are 
imaginary. The use of the word "imaginary" in this sense begins here. 

[208] This seems to indicate that Descartes realized the fact that an equation of 
the nth .degree has exactly n roots. Cf. Cantor, \^:)1. 11(1), p. 724. 

[209] That is, a cubic equation. 

[210] "Nombres rompues," the "numeri fracti" of the medieval Latin writers and 
"numeri rotti" of the Italians. The expression "broken numbers" was often used 
by early English writers. 

'""' That is, transform the equation into one having integral coefficients. 

175 



GEOMETRY 



tains surds, change them as far as possible into rational numbers, either 
by multiplication or by one of several other methods easy enough to 
find. Second, by examining in order all the factors of the last term, 
determine whether the left member of the equation is divisible'^^^' by a 
binomial consisting of the unknown quantity plus or minus any one of 
these factors. If it is, the problem is plane, that is, it can be constructed 
by means of the ruler and compasses ; for either the known quantity 
of the binomial is the required root'""' or else, having divided the left 
member of the equation by the binomial, the quotient is of the second 
degree, and from this quotient the root can be found as explained in 
the first book.'"'' 

Given, for example, y"— 8y' — 124a''— 64 = 0."'" The last term, 64, 
is divisible by 1, 2, 4, 8, 16, 32, and 64; therefore we must find whether 
the left member is divisible by y- — 1, y' + l- 3'" — 2, 3;--)-2, y- — 4, and 
so on. We shall find that it is divisible by y- — 16 as follows : 



+ ^,6 _ 8v* - 124y- - 64 = 
_ye_ 8y^- 4/ 


= 


_ 16/ - 128y2 




- 16 - 16 




+ /+ 8/+ 4 = 


= 



Beginning with the last term, I divide —64 by — 16 which gives +4; 
write this in the quotient ; multiply -|-4 by -|-y- which gives -\-4y" and 

[212] "Qyj divise toute la somme." 

[213] ^\^^^ jg^ ^]^ç ^QQ^ [jijj^ satisfies the conditions of the problem. 

f'"J See page 13. 

'""^ Descartes considers this equation as a function of y-. 



176 



Livre Troisiseme. î^i 

tipli cation, que par diuers autres moyens, qui font afTés 
faciles a trouuer. Ihiis examinant par ordre routes les 
quantite's , qui peuuentdiuifer fans fradion le dernier 
terme, il faut voir, fî quelqu'vne d'elles, iointe auec la 
quantite'inconnue par le ligne -^ ou — , peutcompofer 
vn binôme , qui diuife toute lafommej ôc lî cela eft le 
Problefme eft plan , c'eft a dire il peut eftre conftruit 
auec la reigle & de compas j Car oubien la quantité 
connuëdecebinofmeeftia racine cherche'e • oubien 
l'Equation eftant diuifce par luy , fe reduift a deux di- 
menfions, en forte qu'on en peut trouuer après la racine, 
par ce qui a eft e' dit au premier liure. 

Par exemple fi on a 

y^"Sy^— iî4^' — (5*490 (^. 

le dernier terme , qui eft ^4, peut eftre diuifé fans fra» 

aionpari,2,4, 8,1^, 32, & ^4; C'eft pourquoy il faut 

examiner par ordre fi cete Equation ne peut point 

eftre diuife'e par quelquVn des binômes , yy — i ou 

y y ■+* ^>yy" ^ ^^yy -^^^yy -4 &c.&on troùue qu'el- 
le peut Teftre par y y - 1 5, en cete forte. 

-h y^"9y^'-i24.yy-'64 ooa 

-t^'^-Sy^- ^yy ... 



Vôy^ — izSyy 
16 \6 



- I* 



--H ^ 4 _^ %yy ~H^4 300. 

le commcnceparledernièrterme,& diuife- (^4 parj"' j'°? 
.-!<?, ce qui fait -f- 4, que i'efcris dans le quotient, pnisvocEqua' 
ic multiplie ^4 par H- jr^,ce qui fait -F 4;/^. c'eft pour- "°bi^^^^^^ 
quoy i'efcris - à, y y en la fomme, qu'il faut diuifcr.car il y "^"^ '^^^ 



B b b 5 faut raan " ^ 

177 



3^2 La Géométrie» 

tauttoufiours efcrire le ligne H- ou— tout contraire a 
celuy que produift la multiplication. & ioignant— ii^yy 
auec — 4^j, iay— 128^^, que iediuife derechef par— 16 ^ 
& iay ■+- 8 jj, pour mettre dans le quotient & en le mul- 
tipliant paryy^iay -- Zy ^,pour ioindre auec le terme qu'il 
faut diuifer, qui eft aufTy —8^4, Se ces deux enfemble 
font— 1(5^ % que ie diuile par —16, ce qui fait -T-iy"^ 
pour Te quotient, & — i y <; pour ioindre auec -f-i^^^ ce- 
quifaita, &raonftre que la diuifîon eft achcuee. Mais 
s'il eftoit refte^quelque quantité, oubien qu'on n'cuft pu 
diuifer fans fraétion quelquVn des termes precedens, on 
euft par la reconuu,quelle ne pouuoit eftre faite. 

Tout de mefme fî on ay ^ *"*>' K'**yy-'^^ ^^^ 00 0. 

le dernier terme fe peut diuifer (ans fradion par 
a, aa, aa -+- rr, a * -f- acs^ & femblables. Mais il n'y en a 
que deux qu'on ait befbin de confîderer, afçauoir aa Se 
aa -\- <:^j car les autres donnant plus ou moins de dimen- 
iions dans le quotient, qu'il n'y en a en la quantité con- 
nue du penultiefme terme^ cmpefcheroient que la diui- 
fion ne s'y pli ft faire. Et notés, que ie ne conte icy \qs 
dimenfionsd*^^, que pour trois, acaufequ'il ny a point 
à! y % ny d'j' \ ny d*^ en toute la fomme. Or en exami- 
nant le binôme j/j — aa ^-cc 00 o,on trouue que la diui/îon 
fc peut faire par luy en cete forte. 



-- aac * 
y. »i<ce —aacc ..an—ce 

-- CUiCC -.(M-- ce 



-^fvzyyv^a ^0- Ce. 



178 



THIRD BOOK 

write in the dividend (for the opposite sign from that obtained by the 
multiplication must always be used). Adding — 124y- and — 4^;" I 
have — 128v". Dividing this by —16 I have +83;- in the quotient, and 
multiplying by y~ I have — 8y* to be added to the corresponding term, 
— S'y*, in the dividend. This gives — 163;* which divided by — 16 yields 
-(-y* in the quotient and — y^ to be added to -f-y*"' which gives zero, and 
shows that the division is finished. 

If, however, there is a remainder, or if any modified term is not 
exactly divisible by 16, then it is clear that the binomial is not a 
divisor.'"'*' 

Similarly, given ' 

y^ + o-)y — a*ly- — a" ] 

— a-c'j 

the last term is divisible by a, a-, a'-\-c-, a^-^ac'-, and so on, but only 
two of these need be considered, namely a- and a--\-c-. The others give 
a term in the quotient of lower or higher degree than the known quan- 
tity of the next to the last term, and thus render the division impos- 
sible. '"'' Note that I am here considering y*' as of the third degree, 
since there are no terms in y^, y^, or y. Trying the binomial 

y- — a- — c- = 

we find that the division can be performed as follows : 

+ /+ aM 4 -«4 » 2- «' 1 



- 2^2 j 4 _ rt" I 2 - ^^^* 



^2 cy' ^2.2 çy^ 



_ ^2_^2 



[218] -pi^js is evidently a modified form of our modern "synthetic division," the 
basis of our "Remainder Theorem," and of Horner's Method of solving numerical 
equations, a method known to the Chinese in the thirteenth century. See Cantor, 
Vol. 11(1), pp. 279 and 287. See also Smith and Mikami, History of Japanese 
Mathematics, Chicago, 1914; Smith, I, 273. 

[21"] This is not a general rule. 

179 



GEOMETRY 

This shows that a--\-c- is the required root, which can easily be proved 
by multipHcation. 

But when no binomial divisor of the proposed equation can be found, 
it is certain that the problem depending upon it is solid,'"'*' and it is then 
as great a mistake to try to construct it by using only circles and straight 
lines as it is to use the conic sections to construct a problem requiring 
only circles ; for any evidence of ignorance may be termed a mistake. 

Again, given an equation in which the unknown quantity has four 
dimensions.'""*' After removing any surds or fractions, see if a binomial 
having one term a factor of the last term of the expressioh will divide 
the left member. If such a binomial can be found, either the known 
quantity of the binomial is the required root, or,'^'"' after the division is 
performed, the resulting equation, which is of only three dimensions, 
must be treated in the same way. If no such binomial can be found, 
we must increase or diminish the roots so as to remove the second term, 
in the way already explained, and then reduce it to another of the third 
degree, in the following manner : Instead of 

x^ ± px- ±: qx ± r -=0 
write 

3,6 ± 2py' 4- ip- ± Ar)y- - q- = 0.''^' 

'^''' That is, that it involves a conic or some higher curve. 
'^^"^ A biquadratic equation. 

[220] "Either, or," as in the original. It is like saying that the root of x- — o-=0 
is either x =z a or x = — a. 

[221] Descartes wrote sul:)stantially "Instead of 

+ x^* .pxx .qx .r x 
v^rrite 

+ y*'>.2py'^+ {pp.Ar)yy — qq x 0." 

The symbolism is characteristic of Descartes. 



180 



Livre Tkoisies ME. 3^3 

Ce qui mooftref que la racine cherchée eUaa-hcc. 
Et la preuue en eft ayfée a faire par la multiplication. 

Mais lorfquonne trouuc aucun binôme, qui puifle Qh^^s 
ainfidiuifertoutclafomme de I'Kquation propofee, il mes font 
eft certain que le Problefme qui en depend eft folide. Et f°'v^"' 

. , * '■ lotlquc 

ce n'eft pas vne moindre faute après cela, de tafcher a le l'Equa- 
conftruire fans y employer que des cercles 6c des lignes ''"°. ^^^ 
droites, que ce feroitd employer des fedtions coniques 
aconftruireceuxaufquclsonn'abefoin que de cercles, 
car enfin tout ce qui tefmoigne quelque ignorance s'ap- 
pelé faute. 

Que fi on a vne Equation dont la quantité' inconnue ^^ ^^j^_ 
ait quatre dimenfions, il faut en mefme façon, après en aio» des 
auoir ofte^'les nombres fours, & rompus, s'il y en a, voir fi jjJIJs'qui 
on pourra trouuer quelque binôme, qui diuife toute la ont qua- 
fomme, en le compofantdelVnc des quantités , qui di- m^^^oas, 
uifent fans frajftion le dernier terme. Et fi onentrouue^°^^l"=^^ 
vn, oubieniaquantite'connuëde ce binôme eft la racine m^ cft 
cherchée; on du moins après cete diuifion, il ne refte en P^^°- .^^ 

• \ n r- 1 quels lonr 

l'Equation, que trois dimenfions , en fuite dequoy il ceux qui 
faut derechefTexaminer en la mefme forte. Mais lorf- ^°"/^ ^°''' 
qu'il ne fetrouue point de tel binôme , il faut en au- 
gmentant, ou diminuant la valeur de la racine, ofterle 
fécond terme delà fomme , en la façon tantoft expli- 
qué»'. Etapréslareduire a vneaurre , qui ne contie- 
ne quç trois dimenfions . Cequi fe fait en cete forte. 
AuHeudeH-;c+ * .pxx , qx .r oo o, 

il fau t efcrire -h y^2py ^^Iryy — qq ^o. 

Et pour les figncs H- ou — que iay omis, s'il y a 

eu 

181 



3^4 La Géométrie. 

eu H-/7 en la précédente Equation, il faut mettre en ccl- 
IecyH-2/^,ous'ilyaeu -/>, il faut mettre— 2. p. & au 
contraire s'il y a eu -H r, il faut mettre --4. r, ou s'il y a eu 

— r, il faut mettre -H 4 r. ôcfoit qu'il y ait eu 4- ^, ou 
--^, il faut toufiours mettre — ^//,& -H pp. au moins fî 
onfuppofe que ,v*, Sic y ^ font marquées du fio-nes -h, 
car ce feroit tout le contraire fi ou y fuppofoit le fî- 
gne -. 

Par exemple fi on a -4- a: ♦ * — 4 a; a; - 8 a: -H 5 y 30 
ilfautefcrirecnfbnlieuy^ — 8^^ — i247^--(54 30(7. car 
la quantitc''que iay, nommée/; ellant — 4 , il faut mettre 

— 8^^pour2/jy*. Scelle, que iay nomme'ereftant if^ 
îlfaut mettre */^^Q^, ceft a dire — it^yy y au lieu de 
*^^^^. & enfin q eftant 8, il faut mettre — ^4, pour - qq, 
Toutderacfme au lieu de -h jc '^ * — 17 xx — 20 x— 6'x>o. 
il faut efcrire -H^ ^ — 34^ "^ "+~ 3 1 ^yy - 4co oo 0, 
Car ^4 eft double de 17, & 313 en eft le quarré ioint au 
quadruple de 6, & 400 eft le quarré de 20. 

Tout de mefme aufly au lieu de 

Il faut efcrire 

-«< 

Car;? cH-h^aa - ce, &ipp, eft | ^ ^ - aacc -+- ^ ^ , &! 4 r 
eft— ^a'-'^aacCySlcnEn-'qqçiï'-a^ .'2a^cc -^âac \ 
Apres que l'Equation eft ainfî réduite a trois dimen- 
fions, il faut chercher la valeur à'yy par la méthode défia 
exphquefe. Et iî celle ne peut eftre trouuee , on n'a point 

befoin 



182 



THIRD BOOK 

For the ambig^iious''"^ sig^n put -[-2p in the second expression if -{-p 
occurs in the first ; but if — p occurs in the first, write — 2p in the sec- 
ond ; and on the contrary, put — 4r if +r, and -\-Ar if — r occurs ; but 
whether the first expression contains -\-q or -~q we always write — q- 
and -{-p- in the second, provided that .r* and a'*' have the sign -\- ; other- 
wise, we write -\-q- and — p-. For example, given 

.r* — Ax- — 8.r 4- 35 = 
replace it by 

3,« _ 8y* — 124y- — 64 = 0. 

For since /> = — 4, we replace 2/7y* by — 8y* ; and since r^35, we 
replace (/>-— 4r)y- hy (16— 140)y- or — 1243;-; and since g = 8, we 
replace — g- by —64. 
Similarly, instead of 

.r* - \7x- — 20.r — 6 = 
we must write 

3,6 _ 34^4 ^ 313^^,2 _ 400 = 0, 

for 34 is twice 17, and 313 is the square of 17 increased by four times 6, 
and 400 is the square of 20. 
In the same way, instead of 

we must write 

y'+{a^- 2^2)/ + (^4 _ ^4)_^2 _ ^6 _ 2^4^2 _ ^2^4 ^ Q; 

for 



^^-c\ p^^^a^- aV + ^4^ 4,- ^ _ ^ ^4 + ^2^2_ 



And, finally, 

— q- = — a^ — 2a*c- — a-c*. 

When the equation has been reduced to three dimensions, the value 
of y- is found by the method already explained. If this cannot be 
'""' Descartes wrote "pour les signes + ou — que j'ai omis." 



183 



<1 

2y 


= 


2y' 


= 0. 



GEOMETRY 

done it is useless to pursue the question further, for it follows inevit- 
ably that the problem is solid. If, however, the value of y- can be 
found, we can by means of it separate the preceding equation into two 
others, each of the second degree, whose roots will be the same as 
those of the original equation. Instead of + x^ d= px- ± ç.r dz r ^ 0, 
write the two equations 

-> loi 

-\- x^~ yx-\- —y^±. —p 
and , 1 P , 1 , 

+ x^ + yx -\- Y y — y/ - 

For the ambiguous signs write + —p in each new ecjuation, when p 

has a positive sign, and — ^p when/» has a negative sign, but write 

Ç q 

4- TT when we have — y.r, and — -^ when we have + yx, provided q has 
2y 2y 

a positive sign, and the opposite when q has a negative sign. It is then 
easy to determine all the roots of the proposed equation, and conse- 
quently to construct the problem of which it contains the solution, by 
the exclusive use of circles and straight lines. For example, writing 
3,0 _ 34^4 _^ 313^2 _ 400 = instead of .r* — I7x- — 20a' — 6 = we 
find that y- = 16; then, instead of the original equation 

-f x' — 17 X- — 20.r — 6 = 
write the two equations + x- — 4x — 3=0 and -f~-t"'+'^''*^ -[-2 = 0. 

For, J/ = 4, -- j2 = 8, / = 17, Ç = 20, and therefore 

-^Y' 2^''2y=~^ 

^"^ +^y'-^P + ^=+2. 

^ 2 -^ 2^^ 2y ^ 



184 



Livre Troisîesme. 2^S 

befoin de pafler outre; car il fuit de là înfalliblement:, 
que le problefme eft folide. Mais fi on la trouue , on 
peut diuifer par fon moyen la précédente Equation en 
deux antres, en chafcune defquelles la quantité' incon- 
nuênaura que deux dimenfions, Se dont les racines fe- 
ront le$ mefmes que les lignes. Afçauoir^aulieu de 

il faut efcrire ces deux autres 
-hxx'-yx-i-iyy.'jp. ^^ coo,& 

Et pour leî> fignes H- &— queiay omis, s'ilya4- pen, 
l'Equation précédente, il faut mettre -f- ^ /? en chafcune 
de celles cy; (5c -^/>, s'il y a en l'autre - p. A ai s il faut 

mettre H — -en celle où il y a—y ^;&--—, en celle où il 
ya-i-j'AT, lorfqu'ily a -+- ^ en la premiere. Et au con- 
traire sM y a — <7, il faut mettre — - , en celle, où il y a 

-_yA;;& -h ~*encelleoùilya-f-^Ar. En-fuitc dequoy 

il eft ayfé de connoiftrc toutes les racines de l'Equation 
propofée, & par confequent deconftruirele problefme, 
dont elle contient la folution, fans y employer que des 
cercles, & des lignes droites. 

Par exemple a caufe que faifant 

y^' — S'^y^'^sisyy- 400 30 <?, pour 
x^* — ly XX-' 2QX-' 6 03 0, on trouue que^^ eft i5,on- 
doii au lieu de cete Equation 

-^x^ * "iyxx.— zoX"iQx " 6 70 0, efcrire ces deux 

C c c autres 



185 



l86 La*Geometrie. 

autres H- xx'-- 4 at— 5 so 0. Et •+- .va: H- 4 ^ "^ 2 30 e?. 

car;' eft 4,^;';' eft 8,/^ eft 17, & ^ eft 20, de façonque 

tirant les racines de ces deux Equations, on trouue tou- 
tes les mefmes , que fi ou les tiroit de celle oii eft a; -^ , a 
fçauoir on en trouue vne vraye", qui eft f/ 7 H- 1,6c trois 
fauftes, qui font / 7 -- 2, z -4- |/ 2, & 2 - V'z. 
AiniiayantA;'^— 4 ata:- 8 x~h- ^yt^^^jpourceque la racine 
dej' ^ - ty ^ ~ 1 14-yy ^' " ^4 ^ ^, eft derechef 16 , il faut 
efcrire 
;ca; — 4.'^ -I- j 00 <7, Scxx-h- 4 ,r -J- 7 :x) ^. 

Caricy-Hi_y;'..^/;..f^fait5',&-Hi;7-|: /; 4-t 
fait7. Et pourcequ'on ne trouue aucune racine , ny 
vraye,nyfauflc, en ces deux dernières Equations ^ on 
connoiftdelà que les quatre de l'Equation dont, elles 
procèdent font imaginaires^ & que le Problefme , pour 
lequel on l'a trou uée, eft plan de fâ nature j mais qu'il 
ne fçauroit en aucune façon effreconftruit,acaufe que 
les quantités données ne peuuent fè roindre. 

Tout de mefme ayant 

pourcequ*on trouue aa H- ce pourj'j', il faut efcrire 
jy^ — y aa-)rcc \~h~ach — \^a V aa-i- ce OOo, 6c 
^^-^- V aa-^ ce ^-h-^aa'-i-^a Vaa-^cCXto, 

Car y eft Vaa-^-cc, & -H \yy -^- i,p ciïlaa. Se fy 

cfti;^ -^'aa-hcc» D'oùonconnoift que la valeur de t^ 

eft 



186 



THIRD BOOK 

Obtaining the roots of these two equations, we get the same results as 
if we had obtained the roots of the equation containing x*, namely, one 

true root, V 7 + 2, and three false ones, V 7 - 2, 2 + V 2 , and 2 - V 2. 
Again, given .r*— 4.r-— 8.r+35 ^0. we have 3;"— 8y— 124y-— 64 = 0, 
and since the root of the latter equation is 16, we must write 
x^—4x-\-5 = and x--{-4x~^7 = 0. For in this case, 

and , 1 2 1 >. , ^ 7 

Now these two equations have no roots either true or false,'"'' whence 
we know that the four roots of the original equation are imaginary; 
and that the problem whose solution depends upon this equation is 
plane, but that its construction is impossible, because the given quanti- 
ties cannot be united. '^^^' 
Similarly, given 

^4+ (1.^2 _ ^2\^2_ (^3 + ^,2) ^ + A^4_ A^2^2 ^ Q, 

since we have found 3'" = a- -\- c-, we must write 
and 

[223] -pj^g^ jg^ ^]j j^g roots are imaginary. 

ts24] 'p]^^^ jg ^Yit given quantities cannot be taken together in the same problem. 



187 



GEOMETRY 



For y= ^Ja^ + c^ and + \y'+ \p= ^a^ and 2~ = y « ^Ja^ + c^, then 
we have 



^ = Y ^ ''' + '- +\/ - \a^+^c^+ I a a'^2-+72 



or 



Now we already have z + - a = x, and therefore x, the quantity in 
the search for which we have performed all these operations, is 



To emphasize the value of this rule, I shall apply it to a problem. 
Given the square AD and the line BN, to prolong the side AC to E, so 
that EF, laid off from E on EB. shall be equal to NB. 

Pappus showed that if BD is produced to G, so that DG = DN, and 
a circle is described on BG as diameter, the required point E will be 
the intersection of the straight line AC (produced) with the circum- 
ference of this circle.''"'' 

Those not familiar with this construction would not be likely to dis- 
cover it, and if they applied the method suggested here they would 
never think of taking DG for the unknown quantity rather than CF 
or ED, since either of these would much more easily lead to an equa- 

''''' Pappus Lib. VII, Prop. 72, Vol. II, p. 783. The following is in substance 
the proof given by Pappus. He first gives an elaborate proof of the following 
lemma: Given a square ABCD, and E a point in AC produced, EG perpendicular 
to BE at E, meeting BD produced in G, and F the point of intersection of BE and 
CD. Then CD^ -f FE"^ = DG.^ Then he proceeds as follows: By the construe 
tion given in the problem, I5N'^=BD'-f- BN'^ By the lemma, DG^=CDVfE^. 
By construction, BD = CD and DG = DN. Therefore, FE = BN. 



188 



Livre Troisiesme. 



3h 



oubieii ^ f'aa -h 7c- "^- iaa-h^ ce -{- '-^aV aa-^ ce. 
Et pourceque nous anions fait cy deflus :^H- I^ooat, 
nous apprenons que laquantite.v, pourlaconnoifTance 
de laquelle nous auons fait toutes ces operations, eft 



h -i^-H V^aa -hlcc-^^ ^cc-- \aa -h'^a V'aa -H 



ce. 




Maisaffin qu'on puiffe mieux connoiftre l'vtilite de ^^J.^^jP^^"^ 
cetereiele il faut que ie rapplique a quelqj Problefme.de cesfe- 

Si le quarré A D, & la ligne B N eftant donnes , il faut 
prolonger le cofte A C iufques a E, en forte qu E F,tirec 
d'EversB, foit efgale a NB. On apprent de Pappus, 
qu ayant premièrement prolonge' BD iufques à G , en 
forte que D G foit efgale à D N, & ayant defcrit vn cer- 
cle dont le diamètre foit B G , fi on prolonge la ligne 
droite AC,ellerencontreralacirconference de ce cer- 
cle au point E, qu'on demandoit. Mais pour ceux qui ne 
fçauroiet point cete côflrucStion elle feroit affés dilficile 
à rencotrer,& en la cherchât par la méthode icy propo- 
fée, ils ne s'auiferoiêt iamais de prêdrc D G pour la quâ- 
tité inconnue, maisplutoft C F , ou F D , a caufe que ce 

Ccc 2 font 



189 



i88 La Géométrie. 

font elles qui conduifent le plus ayfement a l'Equatiô.' & 
lors ils en trouueroiêt vne qui ne feroic pas facile a deme- 
fler, fans la reigle que ie viens d'expliquer. Carpofant^ 
pour B D ou C D, & ^ pour E F , & .v pou r D F, on a C F 
00 a -AT, & corne C F ou ^ —^•,eft àF E ou f,ainfî F D ou .v, 

efl a B F, qui^ar confequent eft ^— . Puis acaufe du tri- 
angle redtangle B D F, dont les coftés font l'vn a: & l'au- 
tre a y leurs quarres,'qui font xx-\- a. a^ font efgaux a ce- 

luy de labaze; qui eft ^.^..TJ'x^ c^c, >^^ ^^Ç°" 4"^ multi- 
pliant le tout par xX'-zax-^-aayOW trouue que l'E- 
quation ç,ÇiX^ "xax^ "-^ ^ci(txX'-xa'> X -)r a^'Xi ce xXy 

oubien :v* — 2 /ï :w ^ ^_^1^ x x —2^5 x-V- a * 33 a. Et on 
connoift par les reigles précédentes, que fa racine, qui 
eftîaîongeurdelaligneDF,eft \a H- V'^aa-h^cc 

,^V ^cc — ~ aa-^^aV aa-i- ce. 

Que 11 on pofoit B F, ou C E , ou B E pour la quantité 
mconnuë,. on vieiîdroît derechef à vue Equation, en la- 
quelle il y auroit 4 dimenfîons, mais qui feroit plus ayfée 
a démeficr, 5c on y viendroit affes ayfement ; au lieu que 
fî c'eftoit D G qu'on fuppofaft , on viendroit bea'ucoup 
plus difficilement a l'Equation, mais aufTy elle feroit très 
fimple, Cequeie mets icy pour vous auertir, que lorf. 
que le Problefme propofe'n'eft point folide, fi en le cher- 
chant par vn chemin on vient a vne Equation fort corn. 
pofce,onpeut ordinairement venir a vne plus liraple, en 
le cherchant par vn autre. 

le pourrois encore aioufter dioerfes reigles pour dé- 
melîer les Equations, q^ui vont.au Cube , ou au Quarre 

de 



190 



THIRD BOOK 

tion. They would thus get an equation which could not easily be solved 
without the rule which I have just explained. 

For, putting a for BD or CD, c for EF and x for DF, we have 
CF = a~x, and, since CF is to FE as FD is to BE, we have 

a—x: c =x: BE, 

whence BF^ — ^ — . Now, in the right triangle BDE whose sides are 
a — x 

X and a, X'-\-a-, the sum of their squares, is equal to the square of the 

C'X- 

hypotenuse, which is — — ^ — .. o Multiplying both sides by 

,1 i— c/ , I j CI 

x-—2ox-\-a- 
we get the equation, 

X* —2ax^-]-2a'x- —2a^x-\-a'^ = c~x~, 
or 

x'—2ax"-\-(2a-—c-)x--2a"x+a* = 0, 

and by the preceding rule we know that its root, which is the length of 
the line DF, is 



^«+ \rr«^+ -T<^'^ ~- \nr^^- ^«^+ ~9" V«"+^'- 



If, on the other hand, we consider BE, CE, or BE as the unknown 
quantity, we obtain an equation of the fourth degree, but much easier 
to solve, and quite simply obtained.'""' 

Again, if DG were used, the equation would be much more difficult 
to obtain, but its solution would be very simple. I state this simply to 
warn you that, when the proposed problem is not solid, if one method 
of attack yields a very complicated equation a much simpler one can 
usually be found by some other method. 

''^*^ Taking BF as the unknown quantity, the resulting equation is 
X* + 2<:.r- + {c- — 2a-). v- — 2a- ex — a-c- = 0. 
Rabuel, p. 487. 



191 



GEOMETRY 

I might add several different rules for the solution of cubic and 
biquadratic equations but they would be superfluous, since the con- 
struction of any plane problem can be found by means of those already 
given. 

I could also add rules for equations of the fifth, sixth, and higher 
degrees, but I prefer to consider them all together and to state the 
following general rule : 

First, try to put the given equation into the form of an equation 
of the same degree obtained by multiplying together two others, each 
of a lower degree. If, after all possible ways of doing this have been 
tried, none has been sucessful, then it is certain that the given equation 
cannot be reduced to a simpler one ; and, consequently, if it is of the 
third or fourth degree, the problem depending upon it is solid ; if of 
the fifth or sixth, the problem is one degree more complex, and so 
on. I have also omitted here the demonstration of most of my state- 
ments, because they seem to me so easy that if you take the trouble 
to examine them systematically the demonstrations will present them- 
selves to you and it will be of much more value to you to learn them 
in that way than by reading them. 



192 



Livre Troisiesme. 389 

de qnarre, mais elles fcroient fliperfîucs ; car îorfque les 
Problefmes fout plans ,on en peut toufiours trouuer la 
Gonftrutftion par celles cy, 

le pGurrois aiifTy en adiouflrer d autres pour les Equa^ Regie 
tions qui montent iufques au furfblide, ou au Qnarré de ^0""^ ! 
cube, ou au delà, mais i'ayme mieux les comprendre dukeies 
toutes en vne, & dire en general, que /orfqu on àtafchc'qu^paj: ^ 
de les réduire a mefme forme, que celles d antant de ai- ^•^"'^ '^ 
menfîons,quivieuent delà multiplrcation de denx au- quarré. ^ 
très qui en ont moins, & qu'ayant dénombré tous les 
moyens, par lefquels cete multiplicatioaeft pofTible , la 
chofe n'a pu fucceder par aucun, on doits'aflurer qu'el- 
les ne fçauroient eftre réduites a de plus fîmples. En for- 
te que fi la quantité inconnue a 3 on 4 dimenfions,Ie Pro- 
blefme pour lequel on la cherche eft folide^- & fi elle en a 
5,oni?,ileftd'vndegrépluscompofèi &ainfi des autres. 

Au rcfte i'ay omis icy les demonftrations de la plus 
part de ce que iay dit a caufe qu'elles m'ont femblé fi fa- 
ciles, que pourvûque vous preniesla peine d'examiner 
méthodiquement fi iay failly, elles fè prefenteront a vous 
d'elles mefme: & il fera plus vtile de les apprendre en ce- 
te façon, qu'en les lifant. 

Or quand on eft afllire, que le Problefme propofe eft „^^3°"^^" 
folide, foit que l'Equation par laquelle on le cherche pourcon- 
monte au quatre de quarrê, foit qu elle ne monte que tJus'^ks 
iufquesaucube, onpeut toufiours en trouuer la racine problei^ 
par l'vne des trois fed;ions coniques , laquelle que ce foie allrb- 
ou mefmepar quelque partie del'vnc d'elles, tant petite ^"'^ ^" 
qu'elle puiffe eftre- en ne fe feruât au refte que de lignes quadôde 
droites, ôd de cercles. Mais ieme contenteray icy de "o'sou 

"' j quatre di- 

CCC 5 donner men{Ions. 

193 



390 La Géométrie. 

donner vne reigle generalepourles trouuer tontes par le 
moyen d'vne Parabole, a caufe qu'elle efl en quelque fa- 
çon la plus fimple. 

Premièrement il faut ofler le fécond terme de l'Equa- 
tion propofee, s'il n'eft défia nul, & ainfi la réduire à tel- 
le forme, ^^30*.ap2^,a aq, fi la quantité' inconnue n'a 
que trois dimenfionsj oubienàtelle, ^^^o"*. ap^'{. aaq^, 
a 5 rfi elle en a quatre^oubien en prenant a pour IVnité, 
à telle, ^ ' 30 *. /? ;^. ^, & à telle 



an 



T 




Aprci 



194 



THIRD BOOK 



Now. when it is clear that the proposed problem is solid, whether 
the equation upon which its solution depends is of the fourth degree or 
only of the third, its roots can always be found by any one of the three 
conic sections, or even by some part of one of them, however small, 
together with only circles and straight lines. I shall content myself 
with giving here a general rule for finding them all by means of a para- 
bola, since that is in some respects the simplest of these curves. 

First, remove the second term of the proposed equation, if this is not 
already zero, thus reducing it to the form z^ = -±aps±à'q. if the given 
equation is of the third degree, or z^ = ±apz^±a-qz±a'r, if it is of the 
fourth degree. By choosing a as the unit, the former may be written 



195 



GEOMETRY 



z^ = ±pz±q and the latter z^ = ±p2^±qz±r. Suppose that the para- 
bola FAG (pages 194-198) is already described; let ACDKL be 
its axis, a, or 1 which equals 2AC, its latus rectum (C being within the 
parabola), and A its vertex. Lay off CD equal to |/> so that the points 
D and A lie on the same side of C if the equation contains -\-p and on 
opposite sides if it contains — p. Then at the point D (or, if p =0. at 
C^ erect DE perpendicular to CD. so that DE is equal to -h q, 
and about E as center with AE as radius describe the circle EG, if the 
given equation is a cubic, that is, if r is zero. 



196 



Livre TroIsiesme. ÎPr 

Apres cela fnppofant que la Parabole F AG eft défia 
defcrite, Se que fon aifîieu efc A G D K L, & que fon co- 
fte droit eft «, ou i , dont A C eft la moitié', & enfin que 
k point C eft au dedans de cete Parabole, & que A en efc 
lefommet; Il faut faire C Dsoi/;, & la prendre du mef- 
me cofcé, iju'eft le point A au regard du point C , s'il y a 
"h pen l'Equation . mais s'il y a - /? il faut la prendre de 
l'autre cofte. Et du point D, oubien , fi la quantité 




p eftoitnulle.du point C il faut eflcuer vne ligne a an^ 
gles droits iufques a E, en forte quelle foit efgale n\q. 
Et enfin du centre E il faut defcrire le cercle FG, donc 



197 



^9^ 




La Géométrie. 

ledemidiametre foie 
A E , fi l'Equation 
n'efc que cubique, en 
forte que la quanti- 
tér foit nulle. Mais 
quand il y a -H r il 
faut dans cete ligne 
A E prolonge'e, pren- 
dre d'vn cofte A R 
efgale à r, & de l'autre 
AS efgale au cofté 
droit de la Parabole 
quiefc i, &: ayant de- 

fcrit vn cercle dont le diamètre foit R S, il faut faire A H 

perpêdiculaire fur 
A E , laquelle A H 
rencontre ce cer- 
cle R H S au point 
H,quie{tceluypar 
où l'autre cercle 
F H G doit pafler. 
Et quand il y a — r 
il faut âpres auoir 
ainfî trouuc la ligne 
A H , infcrirc A I, 
qui luy Ibit efgale, 
dans vn autre cer- 
cle , dont A E (bit 
le diamètre, & lors 
c'eftparle point I, 
que 




198 



THIRD BOOK 

If the equation contains -\- r, on one side of AE produced, lay ofif 
AR equal to r, and on the other side lay off AS equal to the latus 
rectum of the parabola, that is, to 1, and describe a circle on RS as 
diameter. Then if AH is drawn perpendicular to AE it will meet the 
circle RHS in the point H, through which the other circle FHG must 
pass. 

If the equation contains — r, construct a circle upon AE as 
diameter and in it inscribe AI, a line equal to AH f"^^ then the first 
circle must pass through the point I. 

'"^' That is, draw a chord equal to AH. 



199 



GEOMETRY 

Now the circle FG can cut or touch the parabola in 1, 2, 3, or 4 
points ; and if perpendiculars are drawn from these points upon the 
axis they will represent all the roots of the equation, both true and 
false. If the quantity q is positive the true roots will be those perpen- 
diculars, such as FL, on the same side of the parabola, as E,'"*' the 
center of the circle ; while the others, as GK, will be the false roots. 
On the other hand, if q is negative, the true roots will be those on the 
opposite side, and the false or negative roots'"'"' will be those on the 
same side as E, the center of the circle. If the circle neither cuts noi 
touches the parabola at any point, it is an indication that the equation 
has neither a true nor a false root, but that all the roots are imagi- 
nary.'""' 

This rule is evidently as general and complete as could possibly be 
desired. Its demonstration is also very easy. If the line GK thus con- 
structed be represented by r, then AK is s-, since by the nature of the 
parabola, GK is the mean proportional between AK and the latus rec- 
tum, which is 1. Then if AC or ^, and CD or ^p, be subtracted from 
AK, the remainder is DK or EM, which is equal to z' — \p — J of which 
the square is 

And since DE = KM = -^ q, the whole line GM = z-\--^- g, and the square 
of GM equals z'^-\-gz+ ^ 'f- Adding these two squares we have 

z^-Pz^^qz^ \g'+ -}/+ \p^\ 

IMS] 'y\y^\_ is, on the same side of the axis of the parabola. 

[229] «Leg fausses ou moindres que rien." This is the first time Descartes has 
directly used this synonym. 

'*^"' It may be noted that Descartes considers the cubic as a quartic having zero 
as one of its roots. Therefore, the circle always cuts the parabola at the vertex. 
It must then cut it in another point, since the cubic must have one real root. It 
may or may not cut it in two other points. It may cut it in two coincident points 
at the vertex, in which case the equation reduces to a quadratic. 



200 



Livre Troisiesme. 393 

(^uc doit pafler F I G le premier cercle cherche. Or ce 
cercle FGpeuccoupper, ou toucher la Parabole en i, 
ou 2, ou 3, ou 4 poins, defquels tirant des perpendiculai- 
res fur laifîieu, on a toutes les racines de l'Equation tant 
vrayes, que faufles. A fçauoir fî la quantité'// eft marquee 
du ligne -H, les vrayes racines feront celles de ces per- 
pendiculaires, qui fe trouueront du mefmecofte delà 
parabole, que E le centre du cercle, comme F L ; & les 
autresj comme G K, feront faufTe^ : Mais au contraire fî 
cete quantité' ^efl marquée du fîgne — les vrayes feront 
celles de Tantrecofté; ôc les fauiïes, ou moindres que 
rien feront du cofte^'ôu eft E le centre du cercle. Et en- 
fin fi ce cercle ne CGuppe,ny ne touche la Parabole en au- 
cun point, cela tefmoigne qu'il n'y a aucune racine ny 
vraye ny faufTe en l'Equation , & qu'elles font toutes 
imaginaires. En forte que cete reigle eft la plus généra- 
le , &: la plus accomplie qu'il foit pofîîble de fou- 
haiter. 

Etlademonftration en eft fort ayfi'e. Cai* fî la ligne 
GKjtrouuéeparceteconftrudtion, fe nomme î^, AK 
fera ^^ a caufe de la Parabole , en laquelle GK doit 
eftre moyene proportiouelIe,entre A K, & le cofte droit 
qui eft i.pui s (î de AKi'ofte AC, qui eft ^ , & C D qui 
eft ~p, il refce D K, ou E M, qui eft ^^— \p- | , dont le 
quarre eft 

X'-pV^'-'^^-^ÏPP'^k^^l- &: a caufe que DE, ou 
KMeft^<7,latouteGMeft:(-+-^/7, dont le quatre' eft 

^^'^^^■^"ï'77»*^^^^^"'^^'^"^^^^ deuxquarrés, on a 

Ddd pour 



201 



394 



La Géométrie. 



d>r 



r 




ponrîequarredelaligneG E, acaufe qu'elle eft la baze 
du triangle re<îtangle E M G. 

Maisacaufe que cete mefme ligne G E eft le demi- 
diametre du cercle F G, elle fe peue encore expliquer en 
d'autres rermes^afçauoirE D eftant^*^, &: AD eftant 

ip -f- ^,E A eft î/ ^ fq-^iPp-^' ip^i^ caufe de Tan- 
gle droit A D E, puis H A eftant moyene proportionelle 
entre A S qui eft i & A R qui eft r,elle efc Vr- & à cau- 
fe de Tangle droit E A H, le quarré deH E , oa E G eft 

-qq'^\PP'^\P '^ ï '^ ^ • fibienque il y a Equation 

entre 



202 



THIRD BOOK 



for the square of GE, since GE is the hypotenuse of the right triangle 
EMG. 

But GE is the radius of the circle FG and can therefore be expressed 
in another way. For since ED = i g, and AD = i /,-!_ i, and ADE is 
a right angle, we have 



Then, since HA is the mean proportional between AS or 1 and AR or r, 
HA= V ;-; and since EAH is a right angle, the square of HE or of EG is 

and we can form an equation from this expression and the one already 



203 



GEOMETRY 

obtained. This equation will be of the form s^ = ps-—qs-\-r, and there- 
fore the line GK, or r, is the root of this equation, which was to be 
proved. If you will apply this method in all the other cases, with the 
proper changes of sign, you will be convinced of its usefulness, without 
my writing anything further about it. 

Let us apply it to the problem of finding two mean proportionals 

between the lines a and q. It is evident that if we represent one of the 

2 23 

z z z 
mean proportionals by a, then a:2=z: = : ,. Thus we have an 

equation between ç and 2> namely, z'^ = a^q. 

Describe the parabola FAG with its axis along AC, and with 
AC equal to ^ a, that is, to half the latus rcclum. Then erect CE 
equal \.o \q and perpendicular to AC at C, and describe the circle AF 



204 



Livre Troisiesme. ^^-^ 

entre cete fbmme & la précédente, cequiefi: lemefine 
que ^ * 30 *p^X" q ^-f- r. & par confequent la ligne trou- 
vée GK qui a efcé nommée ;^efc la racine de cete Equa- 
tion, ainfî qu'il falloic demonftrer. Et fi vous appliqués 
ce mefme calcul a tous les autres cas de cete reigle, en 
changeant les fignes -H & — felon Toccafion , vous y 
trouuerés voftre conte en mefme lbrte,fans qu'il foit be- 
tfoin que ie m'y arefte. 




Si on veut donc fuiuant cete reigle trouuer denxmo» 
yennesproportionelles entre les lignes a & ^; chafcun 
fçaitquepofant ^ pourlVne, comme <ï eft à ;^ , ainfi 



;^à-^, & 7 à ^jdefaçonqu'ily a Equation entre q Se L'inucn- 

rionde 

^j, c'efladire,;^' 0)=* *^^^.EtIaParaboleF AGeilant f^HTô- 

portio- 
2 



X 



Ddd 



j porno- 
de- ndles. 



205 



^^^ La Géométrie. 

defcrite, auec la partie de fon aifïîeu A C, qui eft ^« la 
moitie'du cofte droit ; il faut du point C efleuer la per- 
pendiculaire C Eefgaleà^^j&ducentre E,parA, de- 
fcriuantleccrcle AF, ontrouue FL, &:LA, pour les 
deux moyennes cherchées. 




;n trois. 



Tout de mefme fî on veut diuifer l'angle NOP, ou- 
de^fu^fe" bienlarc, ou portion de cercle N QJL' P, en trois par- 
vn angle ^^q^ ef^ales • faifant N O 30 i , pour le rayon du cercle, & 
NP 30^, pour la fubtendue de lare donne, &NQoo:^, 
pour la fubtendue du tiers de cet arc j l'Equation 
vient, 

i^i3o'''3;^--</. Car ayant tiré les lignes Nd, OQ, 
OT;& faifant QS parallèle a TO, on voit que comme 
NOeftaNQ^ainfiNCLaqRjôcQRaRSj enforte 

que 

206 



THIRD BOOK 

about E as center, passing through A. Then FL and LA are the 
required mean proportionals. '""'' 

Again, let it be required to divide the angle NOP, or rather, 
the circular arc NOTP, into three equal parts. Let NO = 1 be 
the radius of the circle, NP = q he the chord subtending the given arc, 
and NQ^^r be the chord subtending one-third of that arc; then the 
equation is 2^ =3s — q. For, drawing NQ, OO and OT, and drawing 
QS parallel to TO, it is obvious that NO is to NO as NO is to QR as 
QR is to RS. Since NO = 1 and NO = ^, then OR = 3- and RS = 2^ ; 
and since NP or q lacks only RS or 2^ of being three times NO or 2, we 
have q = 32 — 2^ or 2^ = 32 — g.''^"' 

Describe the parabola FAG so that CA, one-half its latus rectum, 

13 1 

shall be equal to -^,-; take CD= ^^and the perpendicular DE= ^ Ç'' 

then describe the circle FA^G about E as center, passing through A. 
This circle cuts the parabola in three points, F, g, and G, besides the 
vertex, A. This shows that the given equation has three roots, namely, 
the two true roots, GK and gk, and one false root, FL.'''"" The smaller 

'•"^ This may be shown as follows: Draw FM ± to EC; let FL=2. From 

the nature of the parabola, FL^=a . AL; AL= — ; ËC^-fCÀ^=ÈA^ ËM^FM' 

a 

=ËF^ EA'=|^-f ''^; Ëm' = (EC - FL)^= /^ y-.-V; FM'=CÏ?= (AL-AC)^ 

= ("-4^5 EF'=Ç'-9^ + ^-+4 — ■s'+'t- ButEF=EA. 
\a 2 I 4^ a-' 4 

4 4 4 a^ 4 

whence s^ = a^q. 

[232] ^ NOQ is measured by arc NQ ; 

ZQNS is measured by è arc QP or arc NQ ; 
ZSQR=ZQOT is measured by arc QT or NQ ; 
.••ZOQN=ZNQR=ZQSR. 
.•-NO : NQ= NQ : QR = QR : RS. 
QR = .s- ; RS = s-\ Let OT cut NP at M. 
NP = 2NR -h MR = 2NQ -}- MR 
= 2NQ + MS — RS 
= 2NQ-1-QT— RS 
= 3NQ — RS. 
Or g = 3.C: — s^. 
Rabuel, p. 534. 

'^^^ G and g being on the opposite side of the axis from E, and F being on the 
same side. 

207 



GEOMETRY 

of the two roots, gk, must be taken as the length of the required Hne 
NO, for the other root, GK, is equal to NV, the chord subtended by 
one-third the arc VNP,'™' which, together with the arc NOP consti- 
tutes the circle ; and the false root, FL, is equal to the sum of ON and 
NV, as may easily be shown. '^^^ 

It is unnecessary for me to give other examples here, for all prob- 
lems that are only solid can be reduced to such forms as not to require 
this rule for their construction except when they involve the finding 
of two mean proportionals or the trisection of an angle. This will be 
obvious if it is noted that the most difficult of these problems can be 

[234] Pqj. pj.QQf^ ggg Rabuel, page 535. 




1^=1 Let AB = &; EB :^ MR = m/fe = NL := c; KK = t;Kk = s;KL = r; 
KG=3); kg=s, FL=z;. Then GM.=y + c, gm=s+c, FN—v—c, GK^=a.AK, 



ai=y\ t 



y — 2 

^ ^ — — ,srk =a.Ak, as = z-,s 



a 



ME = AB-AK = (^ 



mE= b - 



EN= 



-b 



E G^ = EM^ + MG'^ 



E a''^ = A b'^ -f BE^ 
Ëg'= b^— 2^1'' 4- y' +y + 2cy + â 



lab 



y^-\-2a-c-\- aP'y 

y 



2ab 



\z^+2a~c-\-a-z 



y^ + 2a-c-\-à^y z^-\- 2a -c + a-2 

y ^ 



2a^c =^ 2-y-\- 2y~ 
Similarly, 

2a-c = v-y — ■ vy- 
z-y -j- zy- = v-y — vy" v- — z- ^=vy'{- zy 

v — z — y v — y^rz FL = KG-t-/î-^ 

Rabuel, p. 540. 



208 



Livre Troisiesme. ^^7 

que N O eflant I , &: N Qeftant :?^, QJl eft ^^, & R S eft 
:^': Et a caiife qu'il s'en faut feulement R S, ou ^'^ que la 
ligne N P, qui eft q, ne foit triple de N Q^ qui eft :^, ou 
à^30 3 ^"^^ oubieu, 

Puis la Parabole F A G eftant defcrite , & C A la moi- 
tie^defbncofte'droit principal eftant^, fîon prent CD 
a)|,&laperpendiculaireDEcso^^, & que du centre E, 
par AjOndefcriuelecercleFA^G, ilcouppe cete Pa- 
rabole aux trois poins F, ^, & G , fans conter le point A 
qui en eft le fomm et. Ce qui mouftre qu'il y a trois raci- 
nes en cete Equation, à fçauoir les deux G K , 5<:g ^, qui 
font vrayes; & la troifîefme qui eft fauffe , a fçauoir F L. 
Et de ces deux vrayes c*eft ^/^ laplus petite qu'il faut 
prendre pour la ligne N 9 qui eftoit cherchée. Car l'au- 
tre G K, eftefgaleàN V, lafubtendue de la troifîefme 
partie de l'arc N V P, qui auec l'autre arc N QJ? achcue 
le cercle. Et lafaufte F L eft efgaîe a ces deux enfemble 
QJ^ & N V, ainfi qu'il eft ayfé a voir par le calcul. 

Ilferoitfuperflusqueiem'areftaiïeadonner icy d'au- Quetouî 
très exemples- car tous les Problefmes qui ne font que biefmes. 
foHdes fe peuuent réduire a tel point.qu'on n'a aucun be- ^°'"!" ^^ 
foin de cete reigle pour les conftruire.fînon entant qu'el- rcduire a 
le fert a trouuer deux moyennes proportionelles,oubien J",^^J^^. 
îidiuifervn angle en trois partiesefgales. Ainfi que vous tions. 
connoiftres en confiderant, que leurs difficuke's peuuent 
toufiours eftre comprifes en des Equations , qui ne mon- 
tent que iufque au quarré de quarre', ou au cube : Et que 
toutes celles qui montent au quarré de quarrd , fe redui- 
fent au quarre', par le moyen de quelques autres , qui ne 

Ddd 3 montent 

209 



39B La Géométrie. 

montent que infques au cube: Et enfin qu'on peut ofter 
le fécond terme de celles cy. En forte qu'il n'y en a point 
qui ne fe puiffe réduire a quelq^ vne de ces trois formes. 

Or fi on a ;^ ' 30 ^** .-;? ;^-i- ^, la reigle dont Cardan at- 
tribue l'inuention a vn nommd'Scipio Ferreus , nous ap- 
prent que la racine eft. 



V C.-4- \q -\' V'^ qq -^kjp' -^V ^ C.\ \ q -h y\ qq-^trP' 

Comme auffy lorfqu'on a ^^^ oo ^*" -H/; :^4- ^, &. que le 

qparrd de la moitié du dernier terme eft plus grand que 

le cube du tiers de la quantité' connue du penultiefme, 

vne pareille reigle nous apprent que la -racine eft. 

D'où il paroift qu'on peut conftruire tous les Probief- 
mes, dont les difficulteisfereduifent alVne de ces deux 
formes, fans auoir befoin des fecStions coniques pour au- 
tre chofe, que pour tirer les racines cubiques de quel- 
ques quantité'» données, c*eft a dire, pour trouuer deux 
moyennes proportionelles entre ces quantite's & IVnite. 

Puisiîona^'30*-f-/?^H-^, & que le quatre de k 
moitié du dernier terme nefoit point plus grand que le 
cube du tiers delà quantite'connuë du penultiefme, en 
fuppofant le cercle N Q P V,dont le demidiametre NO 
foit Vjp, c'eftadirela moyenne proportionelle entre 
le tiers de la quantité donnée/^ & l'vnitéj & fuppofant 

auffy la ligne N P iufcrite dans ce cercle qui foit y 

c'eft 



210 



THIRD BOOK 

expressed by equations of the third or fourth degree ; that all equa- 
tions of the fourth degree can be reduced to quadratic equations by- 
means of other equations not exceeding the third degree ; and finally, 
that the second terms of these equations can be removed ; so that every 
such equation can be reduced to one of the following forms : 

^3 ^ —pz+q s' = +ps-\-q s' = -\-ps—q 

Now, if we have 2^ = — p2-\-q, the rule, attributed by Cardan^^'^ to one 
Scipio Ferreus, gives us the root 



Similarly, when we have s^ = -\-pz-\-q where the square of half the 
last term is greater than the cube of one-third the coefficient of the 
next to the last term, the corresponding rule gives us the root 



It is now clear that all problems of which the equations can be 
reduced to either of these two forms can be constructed without the 
use of the conic sections except to extract the cube roots of certain 
known quantities, which process is equivalent to finding two mean pro- 
portionals between such a quantity and unity. Again, if we have 
z^ = -\-pz-\-q, where the square of half the last term is not greater 
than the cube of one-third the coefficient of the next to the last term, 

describe the circle NQPV with radius NO equal to \hy-_fi, that is to 

the mean proportional between unity and one-third the known quantity 

p. Then take NP = ~ , that is, such that NP is to q, the other known 
P 

[216] (3ai-(jan ; Liber X, Cap. XI, fol. 29 : "Scipio Ferreus Bononiensis iam annis 
ab hinc triginta fermé capitulum hoc inuenit, tradidit uero Anthonio Marise Flor- 
ido Veneto, qui cû in certamen z\x Nicolao Tartalea Brixellense aliquando uenisset, 
occasionem dedit, ut Nocolaus inuenerit & ipse, qui cum nobis rogantibus tradidis- 
ser, suppressa demonstratione, freti hoc auxiho, demonstrationem quseliuimus, 
eamque in modos, quod dififciHimum fuit, redactam sic subjecimus." 

See also Cantor, Vol. II (1), p. 444; Smith, Vol. II, p. 462. 

'^'^ Descartes wrote this : 



Vc.+|^+Vl^^+2> + Vc-iWi^^+è^' 



211 



GEOMETRY 

quantity, as 1 is to —p, and inscribe NP in the circle. Divide each of 

the two arcs NQP and NVP into three equal parts, and the required 
root is the sum of NQ, the chord subtending- one-third the first arc, and 
NV, the chord subtending one-third of the second arc."'"' 

Firially, suppose that we have z^ = pz—q. Construct the circle NQPV 

whose radius NO is equal to^/-— ^, and let NP, equal to-^, be in- 

scribed in this circle ; then NO, the chord of one-third the arc NQP, 
will be the first of the required roots, and NV, the chord of one-third 
the other arc, will be the second. 

An exception must be made in the case in which the square of half 
the last term is greater than the cube of one-third the coefficient of the 
next to the last term ;'^''^ for then the line NP cannot be inscribed in 
the circle, since it is long-er than the diameter. In this case, the two 

'^'^^ It may be noted that the equation z^ ^Zz — q may be obtained from the 
equation ^r^ =: Ss + g by transforming the latter into an equation whose roots have 
the opposite signs. Then the true roots of .s^ = 3r — (7 are the false roots of 
2^ ^Zz-\-q and vice-versa. Therefore FL = NQ + NP is novir the true root. 

[238] 'pj^g so-called irreducible case. 



212 



Livre Troisiesme. 



W 




c'eftadirequifoit à l'autre quantité donnée q comme 
IVnite eft au tiers de/?; il ne faut que diuifer chafcun des 
deux arcs NQP&NVPen trois parties efgales , Se on 
auraNQ, la fubtendue du tiers de IVn , &N Vlafub- 
tenduedu tiersderautre,quîiointes enferable compo- 
fèront la racine cherchée. 

Enfin fî on a i^ ao*;? ^-q , en fuppofant derechef le 

cercle N QP V, dont le rayon N O foit ^^/?,& l'infcri- 
te NPfoit ^^, NQ^Ia fubtendue du tiers de Tare NQP fe- 
ralVne des racines cherchées, & NV la fubtendue du 
tiers de Tautre arc fera l'autre. Au moins fi le quarré de 
la raoitiédu dernier terme, n'eft point plus grand,que le 
cube du tiers de la quantité connue du penultiefme. car 
s'il eftoit plus grand,la ligne N P ne pourroit eftre infcri- 
te dans le cercle , a caufe quelle feroit plus longue que 
fon diamètre: Ce qui feroit caufe que les deux vray es ra- 
cines 

213 



4^ La GeometrI'E. 

cines de cete Equation ne feroient qu'imaginaires , & 
qu'il ny en auroit de réelles que la faufle , qui fuiuant la 
reigle de Cardan feroit, 

la façon ^^ ^. | ? -f- / i^^-^T -^ '^ C. J y - J^T^p^T. 

d'exprt" -^^ ^^^^ ^^ ^^ ^ remarquer que çete façon d'exprimer 
merhva- la valeur dcs racines par le rapport qu'elles ont aux co- 
toures les ft^s de Certains cubes dont il n'y a;qu e le contenu qu'on 
racines counoilTe, n'eft en rien plus intelligible , ny plus fîmple, 
quations que de Ics exprimer par le rapport qu'elles ont aux fub- 
^^^^jj;"^ tenduësdecertainsarcs, ouportionsde cercles , dont 
de coures [e triple eft donne. En forte que toutes celles des Equa- 
te mor* tions cubiques qui ne peuuent eftre exprimées par \ç,% 
tent que rgigles dc CardaH, le peuuent eftre autant ou plus claire- 

iufquesau ° , - . {-, 

quarré de tii€nt parla façoDicy propoiee. 

^"^^"^' Car fî par exemple , on penfe connoiftrc la racine de 
cete Equation, ^^30 * „ ^ ^ ■+-/'• ^ caufe qu'on fçait 
qu'elle eft corapofee de deux lignes. <}ont IVne eft le 
coftéd'vn cube, duquel le contenu eft | q, adiouftc^au 
cofte''d'v^a quarre" , duquel derechef le contcnn eft 
ï^^— _i^ 'j Et l'autre eft le cofte'd'vn aut^e cube, dont 
le contenu eftla difference, 'qui cft entre |^, &:Iecoftc 
de ce quarre dont le contenu eft \ qq - -^p \ qui eft tout 
ce qu'on enapprent par la reigle de Cardan. Il ny a point 
de doute qu'on ne connoiffe auçant ou plus diftiudte- 
mcntlaracine de celle cy, ^{.^^o'^-i-^-]?, enlaconfî- 
derant infcrite dans vn cercle, dont le dqinidiametre eft 
y f ^& fçachant qu'elle y eft la fubtenduë cj'vn arc 
dont le triple a pour fafubtendue y. Mefme ces ter» 

mes 

214 



THIRD BOOK 



roots that were true are merely imaginary, and the only real root is the 
one previously false, which according to Cardan's rule is 



s/4^+>/i.-.vWi-vi 



/-2V'- 



Furthermore it should be remarked that this method of expressing the 
roots by means of the relations- which they bear to the sides of certain 
cubes whose contents only are known'""' is in no respect clearer or 
simpler than the method of expressing them by means of the relations 
which they bear to the chords of certain arcs (or portions of circles), 
when arcs three times as long are known. And the roots of the cubic 
equations which cannot be solved by Cardan's method can be expressed 
as clearly as any others, or more clearly than the others, by the method 
given here. 

For example, grant that we may consider a root of the equation 
z^ == — çz-\-p known, because we know that it is the sum of two lines 

of which one is the side of a cube whose volume is ^j^ ^ increased by the 
side of a square whose area is — /— :^ p^, and the other is the side of 
another cube whose volume is the difference between -^^ q and the side 

of a square whose area is ^ ç'^— -^ p^. This is as much knowledge of 

the roots as is furnished by Cardan's method. There is no doubt that 
the value of the root of the equation z^ = -\-qz—p is quite as well 
known and as clearly conceived when it is considered as the length of a 

chord inscribed in a circle of radius ^^^p and subtending an arc that 

is one-third the arc subtended by a chord of length — . 

'*"' Descartes here makes use of the geometrical conception of finding the cube 
root of a given quantity. 



215 



GEOMETRY 

Indeed, these terms are much less compHcated than the others, and 
they might be made even more concise by the use of some particular 
symbol to express such chords,'^"' just as the symbol \^ '"'^' is used to 
represent the side of a cube. 

By methods similar to those already explained, we can express the 
roots of any biquadratic equation, and there seems to me nothing fur- 
ther to be desired in the matter : for by their very nature these roots 
cannot be expressed in simpler terms, nor can they be determined by 
any constuction that is at the same time easier and more general. 

It is true that I have not yet stated my grounds for daring to declare 
a thing possible or impossible, but if it is remembered that in the method 
I use all problems which present themselves to geometers reduce to a 
single type, namely, to the question of finding the values of the roots 
of an equation, it will be clear that a list can be made of all the ways of 
finding the roots, and that it will then be easy to prove our method the 
simplest and most general. Solid problems in particular cannot, as I 
have already said, be constructed without the use of a curve more com- 
plex than the circle. This follows at once from the fact that they all 
reduce to two constructions, namely, to one in which two mean pro- 

(2411 -phis is another indication of the tendency of Descartes's age toward sym- 
bolism. This suggestion was never adopted. 

'""' In Descartes's notation, | C. 



216 



Livre Troisiseme. 4oi 

mes font beaucoup moins embarafTés que les autres , & 
ils fetrouueront beaucoup plus cours fî on veut vfêr de 
quelque chiffre particulier pour exprimer ces fubten- 
dûés, ainii qu'on fait du chiffre T^C* pour exprimer le 
codé des cubes. 

Et on peut aufTy en fuite de cecy exprimer les racines 
de toutes les Equations qui montent iufques au quarre 
de quarre'', par les reigles cy deffus- expliquées. En forte 
queienefçacheriendeplus a defirer en cete matière. 
Car enfin la nature de ces racines ne permet pas qu'on 
les exprime en termes plus fîmples, ny qu'on les deter- 
mine par aucune conftrudtion qui foit enfemble plus gé- 
nérale & plus facile. 

Il eft vray que ie xi'ay pas encore dit fur quelles raifons Po^i'^- 
ie me fonde, pour ofer ainfi afîurer, fi vne chofe eft polîî- probiêr^ 
ble, ouneTeftpas. Mais fîonprent garde comment, par ™" ^°^'- 
la méthode dont ieraefers, tout ce qui tombe fous kpcuucnc 
confîderation des Géomètres , fe reduift a vn mefme f-"^^ ^°"' 
genre de Problefmes , qui eft de chercher la valeur des CinTksfc- 
racines de quelque Equation • on iu^era bien qu*il n eft ^^°°^ 

1 r/j r • j/ t coniques, 

pas malayie de taire vn dénombrement de toutes les vo- ny ceux 
yesparlefquelles on les peut trouuer, qui foit ^^ifîîfant JJ^J^^" m- 
pourdemonftrer qu'on a choifi la plus générale, & la plus pofcsfans 
firaple. Et particulièrement pour cequi eft des Probief- ^u«cT u- 
mes foHdes, que lay dit ne pouuoireftre conitruis , fans g"" P^^s 
qu'on y employe quelque hgne plus compofée que lafe°cT/°" 
circulaire , c'eft chofe qu'on peut affés trouuer, de ce 
qu'ils fereduifent tous a deux con ft rudions j en i'vne 
defquelles il faut auoir tout enfemble les deux poins,qui 
déterminent deux moyenes proportionelles entre deux 

Eee lignes 

217 



^oz La Géométrie. 

lignes données- & en l'autre les deux peins , qui diuifent 
en trois parties efgales vn arc donné: Car d'autant que la 
courbure du cercle ne depend , que d'vn iîmple rapport 
de toutes fes parties, au point qui en eft le centre • on ne 
peut aufly s'en feruir qu a determiner vn feul point entre 
deux extremes, comme a trouuer vne moyenne propor- 
tionelle entre deux lignes droites données, ou diuifer en 
deux vn arc donne : Au lieu que la courbure des fecStions 
coniques, dependant toufîoursde deux diuerfes chofes, 
peut aufly feruir a determiner deux poins difFerens. 

Mais pour cete mefme raifon il eft impoffible , qu'au- 
cun des Problefmes qui font dVn degré plus compofés 
que les folides, & qui prefuppofent l'inuention de quatre 
moyennes proportionelles,ou la diuifion d'vn angle en 
cinq parties efgales, puiffenteftrecoîiftruitsparaucune 
des fecStions coniques. Ceft pourquoy ie croyray faire en 
cecy tout le mieux qui fc pui{fe,lî ie donne vne reigle gé- 
nérale pour les conftruire, en y employant la ligne cour- 
be qui fe defcrit par l'interfedlriô dVne Parabole & d'vne 
ligne droite en lafaçoncydeflfus expliquée, car i ofe af- 
furerqu'ilnyenapointdeplusfimpleenla nature, qui 
puifle feruir a ce mefme eff'eétj & vous aués vu comme 
elle fuît immédiatement les fedtions coniques, en cete 
queftion tant cherchée par les anciens , dont la folutiou 
enfeigne par ordre toutes les ligues courbes, qui doiuenc 
lacoaL eftrereceuës en Géométrie. 

neraïc Vousfçaucs deflacommcnt , lorfqu'on cherche les 

5!î*uirc quantités qui font requifes pour la conftrudtion de ces 
tousles pfoblefmes, on les peut toufiours réduire a quelque E- 

problef . f ^ /J U 

mes rc- quation,qui ne monte que lulques au quatre de cube, ou 

duics a ^y 

218 



THIRD BOOK 

portionals are to be found between two given lines, and one in which 
two points are to be found which divide a given arc into three equal 
parts. Inasmuch as the curvature of a circle depends only upon a sim- 
ple relation between the center and all points on the circumference, the 
circle can only be used to determine a single point between two 
extremes, as, for example, to find one mean proportional between two 
given lines or to bisect a given arc ; while, on the other hand, since 
the curvature of the conic sections always depends upon two different 
things. '"*^^ it can be used to determine two different points. 

For a similar reason, it is impossible that any problem of degree more 
complex than the solid, involving the finding of four mean proportion- 
als or the division of an angle into five equal parts, can be constructed 
by the use of one of the conic sections. 

I therefore believe that I shall have accomplished all that is possible 
when I have given a general rule for constructing problems by means 
of the curve described by the intersection of a parabola and a straight 
line, as previously explained ;'"^''' for I am convinced that there is noth- 
ing of a simpler nature that will serve this purpose. You have seen, 
too, that this curve directly follows the conic sections in that question 
to which the ancients devoted so much attention, and whose solution 
presents in order all the curves that should be received into geometry. 

'^^' As, for example, the distance of any point from the two foci. Descartes 
does not say "all points on the circumference," but "toutes ses parties." 
'-"^ See page 84. 



219 



GEOMETRY 

When quantities required for the construction of these problems are 
to be found, you already know how an equation can always be formed 
that is of no higher degree than the fifth or sixth. You also know how 
by increasing the roots of this equation we can make them all true, and 
at the same time have the coefficient of the third term greater than the 
square of half that of the second term. Also, if it is not higher than 
the fifth degree it can always be changed into an equation of the sixth 
degree in which every term is present. 

Now to overcome all these difficulties by means of a single rule, I 
shall consider all these directions applied and the equation thereby 
reduced to the form : 

y'^_py5j^qy*—ry^j^sy-—ty-{-u = 

in which q is greater than the square of ^ p. 



220 



Livre Troisiesme. 4^5 

au fiirfblide. Puis vous fçau^aufTy comment, enaug- yneEqua. 
mentant k valeur desracines de cete Equation, on peut "°° S"^ 
toufiours faire qu'elles deuienent toutes vrayesj & auec plus de"*^ 
cela qu« la quâtitd connue du troifîefme terme foitplus ^**,i^'- 
graiîde que lequarré de la moitié de celle du fecond:Et 
enfin comment, fi elle ne monte que iufques au furfolî- 
de, on la peut hauffer iufques au quatre de cube j & fai- 
re que la place d'aucun de fes termes ne manque deftre 
remplie. Or aiîîn que toutes les difficultés , dont il eft 
icy queftion , pui/Tent eflre refoluè's par vne mefme rei- 
gle^ ie délire qu'on face toutes ces chofes, & par ce 
moyen qu'on les reduife toufiours a vne Equation de 
telle forme, 

& en laquelle la quantité nommée q foit plus grande 
qucJe quarré de la moitié de celle qui eft nommée /r. 



£ e e 2 Puis 



221 



404 



La Géométrie. 

Puis ayant faft a 
ligne B K indefî- 
niement longue 
des deux coftes; 
6c du point B 
ayant tiré la per- 
pendiculaire A B, 
dontia longueur 
foir^/^jil faut dans 
vn plan lepare de- 

fcrire vne Para- 
bole , comme C 
D F dont le cofté 
droit principalfoit 




ri. 



• 

que ie nommeray 
n pour abréger. 
Après cela il faut 
pofer le plan dans 
lequel eft cete Parabole fur celuy ou font les lignes AB & 
BK, en forte que fonaiffieuDEfe rencontre iuftement 
au deflus de la ligne droite BK: Et ayant pris la par- 
tie de cet aiffieu , qui eft entre les poins E & D , efgale à 

— ^, il faut appliquer fur ce point E vne longue reigle, 

en telle façon queftantaufTy appliquée fur le point A 
du plan de deffbus, elle demeure toufîours iointe a ces 
deux poins, pendant quonhaufleraoubaiflera la Para- 
bole 



222 



THIRD BOOK 



Produce BK indefinitely in both directions, and at B draw 
AB perpendicular to BK and equal to ^ p. In a separate plane^""' 
describe the parabola CDF whose principal parameter is 



Vw 



-\-ç- —P 



which we shall represent by n. 

Now place the plane containing the parabola on that containing the 
lines AB and BK, in such a way that the axis DE of the parabola falls 

along the line BK. Take a point E such that DE == and place a 

pn 

ruler so as to connect this point E and the point A of the lower plane. 

Hold the ruler so that it always connects these points, and slide the 

parabola up or down, keeping its axis always along BK. Then the 

[245] -pi-ijg (jQgg j^Q^ mean in a fixed plane intersecting the first, but, for exam- 
ple, on another piece of paper. 



223 



GEOMETRY 

point. C, the intersection of the parabola and the ruler, will describe 
the curve ACN, which is to be used in the construction of the proposed 
problem. 

Having thus described the curve, take a point L in the line BK on the 

2 4u 

concave side of the parabola, and such that BL = DE== ; then lay 

p)i 

t 
off on BK, toward B, LH equal to ^ ;— , and from H draw HI 

In \ u 

perpendicular to LH and on the same side as the curve ACN. Take 
HI equal to 

which we may, for the sake of brevity, set equal to ~. Join L and I, and 

71' 

describe the circle LPI on LI as diameter; then inscribe in this circle 



the line LP equal to J^±É^ijL. Finally, describe the circle PCN about 

I as center and passing through P. This circle will cut or touch the 
curve ACN in as many points as the equation has roots ; and hence the 
perpendiculars CO, NR, OO, and so on, dropped from these points 
upon BK, will be the required roots. This rule never fails nor does it 
admit of any exceptions. 

For if the quantity j were so large in proportion to the others, p, q, 
r, t, n, that the line LP was greater than the diameter of the circle 



224 



Livre Troisiesme. ^^^ 

bole tout le long de la ligne B K , fur laquelle Ton aifïîeii 
eft applique au moyen dequoy Tinterfedtion de cete Pa- 
rabole, & de cete reigle, qui fe fera au point C , defcrira 
la ligne courbe A C N, qui eft celle dont nous auons be- 
fbinde nous feruir pour la conftruétion du Problefme 
propofé. Car après qu'elle eft ainfîdefcrite, fi on prent 
le point L en la ligne B K, du coftc vers lequel eft tourné 
lefbmmet de la Parabole , Se qu'on face B L efgalc à D 

E, c'eft àdireà : Puis du point L , vers B , quon 
prcne en la mefme ligue BK , la ligne LH, efgale à 
^~y:;i & que du point H ainfi trouue, ou tire à angles 
droits, du cofte'qu'eft la courbe A CN, la ligne HT, 
dont la longcur foit £;4- -V 7^^, qui pour abréger 

fera nommée — : Et après, ayant ioint les poins L & I, 

qu'on defcriue le cercle L P I , dont I L foit le diamètre; 
& qu'on infcriueen ce cercle la ligne LP dont la lon- 

geur fbit ^ ~~;~ - Puis enfin du centre I, par le point P 

ainfi trouué, qu'on defcriue le cercle P C N. Ce cercle 
couppera ou touchera la ligne courbe A C N , en autant 
de ppins qu'il y aura de racines eu l'Equation ; En forte 
que les perpendiculaires tirées de ces poins fur la ligne 
B K, comme C G, N R, Q^O , & fembîablcs , feront les 
racines cherchées. Sans qu'il y ait aucune exception ny 
aucun defFàut en cete reigle. Car fi la quantité/ cftoic 
fi grande, à proportion des autres)^, q, r, /■, & Vy que la li- 
gne LP fetrouuaft plus grande que le diamètre ducer- 

Eee 3 cle 



225 



406 La Géométrie. 

ciel L, en forte qu'elle n y puft eftre iufcritejil ny auroit 
aucune racine en l'Equation propofee qui.ne fuft imagi- 
naire: Non pJus que û le cercle I P eftoit li petit, qu'il ne 
coupait la courbe A C N en aucun point. Et il la peut 
couper en fix diflPerens , ainfi qu'il peut y auoir fix 
diuerfes racines en l'Equation. Mais lorfqu'il la coupe 
en moins , cela tefmoigne qu'il y a queloues vnes de 
ces racines qui font efgales entre elles , oubienquine 
font qu'imaginaires. 




Que 



226 



THIRD BOOK 

LI/^*' so that LP could not be inscribed in it, every root of the pro- 
posed equation would be imaginary ; and the same would be true if the 
circle IP'-*'' were so small that it did not cut the curve ACN at any 
point. The circle IP will in general cut the curve ACN in six differ- 
ent points, so that the equation can have six distinct roots/"**' But if 
it cuts it in fewer points, this indicates that some of the roots are equal 
or else imaginary. 

'^'*'That is, the circle I PL, of which the diameter is t, page 222. 
^'"^ That is, the circle PCN. 

'"*"' The points determining these roots must be points of intersection of the 
circle with the main branch of the curve obtained, that is, of the branch ACN. 



227 



GEOMETRY 



If, however, this method of tracing the curve ACN by the transla- 
tion of a parabola seems to you awkward, there are many other ways 
of describing it. We might take AB and BL as before (page 226), and 
BK equal to the latus rectum of the parabola, and describe the semi- 
circle KST with its center in BK and cutting AB in some point S. 
Then from the point T where it ends, take TV toward K equal to BL 
and join S and V. Draw AC through A parallel to SV, and draw SC 
through S parallel to BK ; then C, the intersection of AC and SC will 
be one point of the required curve. In this way we can find as many 
points of the curve as may be desired. 



228 



Livre Troisiesme. 407 

Que fî la façon de tracer la ligne A C N par le mouue- 
inent dVne Parabole volts femble incommode , il eft ay- 
fe'de trouuer plufieurs autres moyens pour la defcrire. 
Comme fî ayant les mefmcsquantité's que deuant pour 
A B & B L; & la mefme pour B K,qu on auoit pofce pour 
le cofte droit principal de la Parabolcjondefcrit le demi- 
cercle K S T dont le centre foit pris a difcretion dans la 
ligne B K, en forte qu'il couppe quelq; part la ligne A B, 
comme au point S, & que du point T, du il fînift,on pre- 
ne vers K la ligne T V, efgale à B L- puis ayant tiré la li- 
gne S V, qu'on en tire vne autre , qui luy foit parallèle, 
par le point A, comme A C- & qu'on en tire aufly vne 
autre par S, qui foit parallèle a B K, comme S C; le point 
C,ou ces deux parallèles fè rencontrent,fera l'vn de ceux 
delaligne courbe cherchée. Et on en peut trouuer, en 
mefme forte,autant d'autres qu'on en délire. 



Or 



229 



4Q8 La GEOMETRIE. 

Or la demonftration de tout cecy eft affes facile, car 
appliquant lareigle A E auec la Parabole EJXfur le point 
Gj comme il eft certain quelles peuuent y eftre appli- 
quées enfemble , puifque ce point C eft en la courbe 
A C N,qui eft defcrite par leur interfedion ; lî C G fe 

yv 
nomme ^, G D fera ^ » à caufe que le cofte" droit , qui 

crt«,eftàCG,commeCGaGD.6coftanc DE,, quieft 

iVx' J'y 2 V'i' 

— ,de GD, onà^— -^,pourGE. Puis à caufe que. 

A B eft a B E, comme 
CGeftaGE ^ AB 
eftant ^p , B E eft 

zn ' ny* 

Et tout de mefme 
en fuppofant que le 
point C de la courbe à 
efte'trouuié par l'inter- 
feétiôdes lignes droi- 
tes , S C parallèle à B 
K, & AC parallèle a 
SV. SBquieftefgalc 
àCG, eft y : & BK 
eftant efgale au coftjé' 
droit de la Parabole, 
que iay nommé « , B 

T eft -. car comme 

n 

KBeftaBS, ainfiBS 

eft a B T. Et TV 

eftant 




230 



THIRD BOOK 

The demonstration of all this is very simple. Place the ruler AE 
and the parabola FD so that both pass through the point C. This can 
always be done, since C lies on the curve ACN which is described by 
the intersection of the parabola and the ruler. If we let CG=y, GD 
will equal —, since the latus rectum n is to CG as CG is to GD. Then 



n 



2^71 y"^ 2 Vz7 

DE= , and subtracting DE from GD we have GE== — ——, — . 

pn n pn 

Since AB is to BE as CG is to GE, and AB is equal to \ p, therefore 

BE =^^— — ~, Now let C be a point on the curve generated 
2« ?iy 

by the intersection of the line SC, which is parallel to BK, and 
AC, which is parallel to SV. Let SB = CG = y, and BK = n, the 

latus rectum of the parabola. Then BT = "*-, for KB is to BS as BS is 

n 



231 



GEOMETRY 

to BT, and since TV = BL = -^— ^ we have BV = ^ - ^^. Also SB 

p7i n pn 

is to BV as AB is to BE, whence BE =^^ — — ^ as before. It is evi- 

dent, therefore, that one and the same curve is described by these two 
methods. 

Furthermore, BL = DE, and therefore DL = BE ; also LH = — ^ 

2n'\u 

and DL=^-^ ^^^ 



2n ny 

therefore DH = LH + DL = f^ - — + ;; 1= 

In 7iy 2 7/ "V 71 

Also, since GD= — , 
n 

GH = DH-GD = ^^ -^^—~. 
c n ny in \ 2i 

which may be written 

GH= ^—^ 



-y'+ ii^^+^^^- V7 



ny 
and the square of GH is equal to 



y" 



-py'+{\p'' ±y+{^ ^+ 2^^y + it, -f ^y~'y+" 



n^'f 



Whatever point of the curve is taken as C, whether toward N or 
toward Q, it will always be possible to express the square of the seg- 
ment of BH between the point H and the foot of the perpendicular 
from C to BH in these same terms connected by these same signs. 



232 



Livre Troisiesme. "^^^ 

pn 



eftant la mefme que BL , c'eû a dire— -^ , B V eft 



-„"—"' &:comraeSBeftaBV, ainfiABeftàBEjqui 

p y Vf 

eft par confequent ^- — - comme deuant,d où on voit 

que c'eftvne mefme ligne courbe qui fe defcrit en ces 
deux façons. 

Après cela, pourceque B L & D E font efgales, D L & 
B E le font aufty: de façon qu'adiouftat L H, qui eft —^ 

p y Vf 

àDL, qui«ft- — .^, on à la toute DH , qui cft 
^- ;j -^ ^Tv 5 &: en oftant G D , qui eft f 
on à GH, qui eft{-{ - ^V "V; - 1^ C^^^^ i'^^^"^ 
par ordre en cete forte G H so — j? -H ^ /?yy -H ^^ — 1/^v* 



y 



ny 
Et le quàrre de G H eft, 

nn yy 
Et en quelque autre endroit de cete ligne courbe qu'on 
veuille imaginer le point G, comme vers N, ou vers Q, 
ontrouueratoufîours que le quarré de là ligne droite, 
qui eft entre le point H & celuy où tombe la perpendicu- 
laire du point C fur BH, peut eftreexprime^en ces mef- 
mcs termes, & auec les mefmes fignes H- & -- . 

pe plus I H cftant £ , & L H eftant ^-^> I L eft 

^ -f- i'^>à caufe de l'angle droit I H L^ &: LP eftât 

Fff V^ 

233 



^to 



LA GEOMETRIE, 




nn ' 



nn 
'i/'mm 



IPoulCefl, 



T^v --"£ "Tn ^ 3 caufeauûTy de 1 angle 
droit I P L. Pois ayant fait C M perpendiculaire fur I H, 
I Meft la difference qui eftentrel H, &HM011 CG, 

c*eft a dire entre ^-, &^ , en forte que Ton quarre* 

_ -, mm 1 »»y . 

eft toufiours — ^ -- -^ -^yy, qui citant ofte du quatre 

de 



234 



THIRD BOOK 



A • TTT ^^ i 

Again, IH = -2, LH = o„ / , whence 






IL = J^ + 



/2 



since the angle IHL is a right angle ; and since 



n n 

and the angle IPL is a right angle, 



Now draw CM perpendicular to IH, and 

IM = HI-HM = HI-CG="f,-j'; 



whence the square of IM is . — — ^ +i/^. 

?r n' 



235 



GEOMETRY 

Taking this from the square of IC there remains the square of CM, or 

• /^ ^ p\7i 2my 2 

4 2 i 2 ~r 2 — y 1 

n u 71 71 n 

and this is equal to the square of GH, previously found. This may be 
written 

— 71^ y'^ + 2 77iy>^ —p '\ 71 y'^ — sy>^ + , y'. 

4:7 ( 



Now, putting 
for n~y*, and 



n-y 



^y+9y-|/y 



2\7l 



for 2my^, and multiplying both members by n-y-, we have 

.«-//+ (i/- ^),<+ (2 VV+ :^y+ (£ -/ v7)/-/.+. 

equals 

or 

y — /)y^-[-çy*— rj/'+^j;-— fy+w ^ 0, 

whence it appears that the lines CG, NR, QO, etc., are the roots of this 
equation. 

If then it be desired to find four mean proportionals between the 
lines a and h, if we let x be the first, the equation is x^—a*h = or 
x^—a*bx = 0. Let y—a = x, and we get 

/-6ay^^+15a=y— 20aV+15ay— (6fl^+o^&)y+a«+a-'&=0. 
Therefore, we must take AB -= 3a, and BK, the latus rectum of the 



236 



Livre Tkoisiesme. 4*' 

delC, il refte — - ~ •.- h— --vy. 

pour le quarrede CM, qui cft efgal au quarre de G H dé- 
fia trouue'. Oubien en failànt que cete fomme foit diui- 

fee comme l'autre par nnyy^ on a 

tt 
— miy ^ -H 2 my ^ " pV v yy —syy -f- -j'y. Puis 

t 
remettant ~ y *" -^ ^y*" -- i ppy"" , pour nny* j & 

rj' ' -H 2 y i' ^ ' -4- ^^^ ', pour miy^ : & multipliant 
iVne & l'autre Ibmme par 7in vy, on a 

y'"py'' --v~Jy 




C'eftadirequ'ona, 
y^^-py^-^qy^^-^ry^-^^-syy-ty-^-vloo. 
D'où il paroilt'que les lignes C G, N R, QO, & fembla- 
bles font les racines de cete Equation, qui eft ce qu'il fal- 
loitdemonftrer. 

Ainfidonclîon veut trouuer quatre moyennes pro- 
portionelles entre les bgnes/2 &^, ayant pofe'-vpour la 
premiere , l'Equation eft a; '*'''*'*-- ^^^3oo oubien 
:v'^'**'^*-V«-^a;*30(?. Et taifant^-.i^ooA-ilvient 

y'-6af-V'ijaay^--2oa^y^-\-\^a^yy\:';^,}yllli^^o, 
C'eft pourquo y il fa ut prendre 5 a pour la ligne A Bj &: 

,, -r-i- 6 a a pour B K, ou le cofte'' droit de laPa- 

r f f 2 rabole 



237 



4ii La Géométrie. 



rabolequeiaynommé?;. 3cY^'^^^~^ ^^ ?^^^ D E ou 
B L. Et après auoir defcrit la ligne courbe A C N fur 
la meiure de ces trois , il raut taire L H , 33 — -" 

& HI 30 — -{--Vaa-h-ab-T- —====-& £ P 33 

s-—^^ — Car le cercle qui ayant Ion centre 

au point Ipaflera par le point Painfitrouue, couppera la 
courbe aux deux poins C&Nj defquels ayant tiré les 
perdeudiculaires N R &: C G, fi la moindre, N R, eft 
oftee delapIusgrande,CG,lerefte fera, .v, la premiere 
des quatre moyenne s proportionellescherché'es. 

Il eft ayfe en mefme façon de diuifer vn angle en cinq 
parties efgales, &d'infcrire vne figure d'vnze ou treze 
coftc'scfgauxdansvn cercle, &de trouucr vnc infinite' 
d'autres exemples de cete reigle. 

Toutefois il eft a remarquer, qu'en plufieurs de ces 
exemples, il peut arriuer que le cercle couppe fi obli- 
quement la parabole du fécond genre; que le point de 
leur interfed:ionfoit difficile a reconnoiftre: &ainfiquc 
cete conftrudtion ne foit pas commode pour la pratique. 
A quoy il feroit ayfcde remédier en compofant d'autres 
règles, à limitation de celle cy , comme on en peut 
compofer de mille fortes. 

Maismondcffeinn'eftpas défaire vn gros liure, & 
ie tafche plutoft de comprendre beaucoup en peu de 
mots: comme on iugera peuteftre que iay fait , fi on con- 
fidere, qu'ayant réduit à vne mefmc conftru(ition tous 

les 



238 



THIRD BOOK 

parabola must be 



^a^ + ad 
which I shall call n, and DE or BL will be 



— \a'-\-ab. 

671 

Then having described the curve ACN, we must have 

\^ri= J—, 

2?i ^a'-\-ab 
and 

and 



LP= ^" . /l5«'+6aV«'+«ô. 

71 \ 



For the circle about I as center will pass through the point P thus 
found, and cut the curve in the two points C and N. If we draw the 
perpendiculars NR and CG, and subtract NR, the smaller, from CG, 
the greater, the remainder will be x, the first of the four required mean 
proportionals.'"^"' 

This method applies as well to the division of an angle into five equal 
parts, the inscription of a regular polygon of eleven or thirteen sides 
in a circle, and an infinity of other problems. It should be remarked, 
however, that in many of these problems it may happen that the circle 
cuts the parabola of the second class so obliquely'"'"' that it is hard to 
determine the exact point of intersection. In such cases this construc- 
tion is not of practical value.""' The difficulty could easily be overcome 
by forming other rules analogous to these, which might be done in a 
thousand dift'erent ways. 

[2«] 'pj^g |-^Q roots of the above equation in y are NR and CG. But we know 
that a is one of the roots of this equation, and therefore NR, the shorter length, 
must be a, and CG must be 3'. Then x —. y ■ — a = CG — NR, the first of the 
required mean proportionals. Rabuel, p. 580. 

[250] 'ppj^^ jg^ makes so small an angle with it. 

[2oi] -pj^jg jg especially noticeable when there are six real positive roots. 

239 



GEOMETRY 

But it is not my purpose to write a large book. I am trying rather 
to include much in a few words, as will perhaps be inferred from what 
I have done, if it is considered that, while reducing to a single construc- 
tion all the problems of one class, I have at the same time given a 
method of transforming them into an infinity of others, and thus of 
solving each in an infinite number of ways ; that, furthermore, having 
constructed all plane problems by the cutting of a circle by a straight 
line, and all solid problems by the cutting of a circle by a parabola ; and, 
finally, all that are but one degree more complex by cutting a circle by 
a curve but one degree higher than the parabola, it is only necessary to 
follow the same general method to construct all problems, more and 
more complex, ad infinitum ; for in the case of a mathematical progres- 
sion, whenever the first two or three terms are given, it is easy to find 
the rest. 

I hope that posterity will judge me kindly, not only as to the things 
which I have explained, but also as to those which I have intentionally 
omitted so as to leave to others the pleasure of discovery. 



[the end] 



240 



Livre Trois 1 ES ME/ ^^^ 

les Problefmes dVn mefme genre , iay tout enfemble 
donne la façon de les réduire à vne infinité d'autres di- 
uerfesj & ainfi de refoudre chafcun deux en vne infinité 
de façons. Puis outre cela qu'ayant conftruit tous ceux 
qui font plans, en coupant d'vn cercle vne ligne droite- 
& tous ceux qui font folides , en coupant aufly d'vn cer- 
cle vne Parabole^ & enfin tous ceux qui font d'vn degré 
plus compofcs, en coupant tout de mefme d'vn cercle 
vne ligne qui n eft que d'vn degré" plus compofçe que la 
Parabole; il ne faut que fuiure la mefme voye pour con- 
ftruire tous ceux qui font plus compofcs a l'infini. Car en 
matière de progreiîîons Mathématiques ^lorfqu on a les 
deux ou trois premiers termes, il n'eft pas malayfe'de 
trouuer les autres. Eti*efpere que nos neueux me fçau- 
ront gré , non feulement des chofes que iay icy expli- 
quées; mais aufly de celles que iay omifes volontaire- 
rement^^affin de leurlaifler leplaifîrdelesinuenter.- 



F I N, 



241 



PAr graced priuilege du Roy très chre- 
ftien il eft permis a T Autheur du liure in* 
titule DîfcouYs delà Méthode Ç^c, plm la Dio^ 
ptriqueJesMet€ores^& la Geornetrie&c. de le 
faire imprimer en telle part que bonkiyfem. 
bl^ra dedans 6^ dehors le royaume de France, 
&: ce pendant le terme de dix années confe- 
quutiues, a conter du iour qu'il fera parache- 
ué d'imprimer, fans qu'aucun autre que le li- 
braire qu'il aura choifî le puifTe imprimer , ou 
faire imprimer^en tout ny en partie, fous quel- 
que prétexte ou deguifèment que ce puifle 
eftre^ ny en vendre ou débiter d'autre impref- 
fion que de celle qui aura efté faite par fa per- 
miiTion^a peine de mil liures d'amande, con- 
fifcation de tous les exemplaires &c. Ainfi 
qu il eft plus amplement déclaré dans les let- 
tres données a Paris le 4 iour de May 1637. fi- 
gnees par le Roy en (on confeil Ceheret &C 
feellees du grand fceau de cire iau ne fur fîmple 
queue. 

l'A utheur a permis a lan Maire marchand 
libraire a Leyde^ d'imprimer le dit liure S>C de 
iouir du dit priuilege pour le tenis 6c aux con- 
ditions entre eux accordées, 

Jlcheué d'imprimer le 8. icur de luin 1 657. 



242 



By the grace and privilege of the very Christian King, it is per- 
mitted to the author of the book entitled Discourse on Method, etc., 
together with Dioptrics, Meteorology, and Geometry, etc., to have 
printed wherever he wishes, within or without the Kingdom of France, 
and during the period of ten consecutive years, beginning on the day 
when the printing is completed, without any publisher (except the one 
whom he selects) printing it, or causing it to be printed, under any pre- 
text or disguise, or selling or delivering any other impression except 
that which has been allowed, under penalty of a fine of a thousand 
livres, the confiscation of all the copies, etc. This is more fully set forth 
in the letters given at Paris, on the fourth day of May, 1637, signed 
by the King and his counsel, Ceberet, and sealed with the great seal of 
yellow wax on a simple ribbon. 

The author has given permission to Jan Maire, bookseller at Leyden, 
to print the said book and enjoy the said privilege for the time and 
under the conditions agreed upon between them. 

The printing is completed the eighth day of June, 1637. 



243 



INDEX 



The numbers refer to the pages of the present edition, not to those at the top 
of the facsimiles. 



PAGE 

Abscissa 88 

Adam, C 10,17 

Agnesi, M. G 2 

Alembert, J. le R. d' 40 

Angle, division of 219,239 

Apollonius.... 17-22, 26, 68, 72, 75, 96 

Applicate 67 

Arithmetic and geometry 2 

Axes 95 

Ball, W. W. R 6 

Beaune, F. de 2 

Beman, W. W 13,26 

Biquadratic equation. 195 seq., 216 seq. 

Boncompagni, B 159 

Bouquet, J. C 55, 67, 71 

Boyd, J. H 55 

Briot, C 55, 67, 71 

Cantor, M.44, 91, 92, 160, 175, 179, 211 

Cardan, H. (G., or J.) 

159, 160, 211. 215 

Catoptrics 115 

Cavalieri, B 26 

Cissoid 44 

Clairaut, A. C 147 

Class of curves 48, 56 

Commandinus, F 6, 17, 19 

Complex curves 43, 48, 56 

Conchoid 44, 55, 113 

Conic sections 44 

Coordinates, transformation of.. 51 



PAGE 

Cousin, V 10, 19, 63, 72, 112, 135 

Cubic equation 195 seq., 208 seq. 

Curved lines 40 

D'Alembert, J. le R 40 

Diderot, D 40 

Dioptrics 115, 124, 135 

Division 2 

Enriques, F 13 

Equality, symbol of 6 

Equating to zero 9.6 

Equations.l3, 34, 37, 156, 159, 192, 195 

Equations, transformation of 

163, 164, 166 

Euclid 17, 19. 22 

False (negative) roots 159,200 

Fermât, P 25, 26, 112 

Fibonacci, L 159 

Finie, K 26 

Focus 128 

Fundamental theorem 160 

Geometric curves 40, 48 

Guisnée 156 

Harriot, T 160 

Heath, T. L 26, 44, 96, 155 

Heiberg, J. L 68 

Horner's Method 179 

Hultsch, F. 6, 19 



245 



INDEX 



PAGE 

Hutton, C 67 

Imaginary roots 175, 187 

Irreducible cubic 212 

Kepler, J 128 

Klein, F 13 

Leibniz, G. W 40 

Lenses 124-147 

Leonardo Pisano 159 

L'Hospital, G. F. A., de 156 

Loci, plane and solid 79 

Mascheroni, L 13 

Mechanical curves 40, 91 

Mean proportionals 47, 155,219 

Mersenne, Marin 10, 63 

Mikami, Y 179 

Mirrors 127-136 

Multiplication 2, 33 

Negative numbers 63, 111 

Normals 112 

Order of curves 48 

Ordinate 67, 88 

Oresme, N 26 

Ovals 116-131, 143 

Pappus 

6, 17, 19, 21, 26, 40, 59, 63, 156, 188 

Pappus, problem of 19, 21, 63 

Parent, A 147 

Plato 6 

Pliny 135 

Polygon, regular . 239 

Problem solving 6 

Ptolemy, C 135 

Quadratic equation 13, 34 

Quadratrix 44 



PAGE 

Rabuel, C 2, 6, 9, 17, 

33, 40, 47, 55, 56, 59, 68, 79, 88, 
107, 111, 112, 120, 135, 191, 208, 239 

Remainder Theorem 179 

Riccati, V 2 

Roberval, G. P., de 26 

Roots 5 

Roots increased or diminished... 163 

Roots multiplied or divided 172 

Rudolph, C 159 

Rule of Signs (equations) 160 

Russell, B 91 

Saladino, G 2 

Scipio Ferreus 211 

Signs, Rule of (equations) 160 

Smith, D. E.. . .13, 26, 44, 92, 179, 211 

Solid analytic geometry 147 

Spirals 44 

Steiner, J 13 

Stifel, M 159 

Supersolids (sursolids).. . .56, 80, 152 

Symbolism 5. 6, 175, 180 

Synthetic division 179 

Tangents 112 

Tannery, P 10, 17, 21 

Tartaglia, N 211 

Taylor, C 44 

Three-dimensional space 147 

Transcendental curves 91 

Transformation of roots. .. .164, 166 
True roots 159 

Van Schooten, F 2, 6, 9, 55, 147 

Vieta, F 10, 26, 43 

Weber, H 13 

Wellstein, J 13 

Zeuthen, H. G 17 



246 



Sf 



DUE DATE 


DEC a 


6bbZ 




















































































































































Printed 
in USA 



QA 33.D5 




3 9358 00024562 8 



QA33 

D5 



I 



Descartes, Rene, ISSé^léSO. 

The geometry of Rene Descartes, 
translated from the French and Latin by 
David Eugene Smith and Marcia L« 
Latham; with a facsimile of the first 
edition, 1637« Chicago, The Open Court 
Pub. Co. , 1 S25. 

xiii, 246 p. front, (port.) diaêrs. 
24 cm* 

24562 



3 



lidBNU 



24 FEB 78 



635157 NEDDbp 



25-17 2S2 



QA 33.D5 



3 9358 00024562 8 




•i^