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THE GEOMETRY
OF
RENÉ DESCARTES
TRANSLATED FROM THE FRENCH
AND LATIN
BY
DAVID EUGENE SMITH
AND
MARCIA L. LATHAM
WITH A FACSIMILE OF THE FIRST EDITION, 1637
CHICAGO • LONDON
THE OPEN COURT PUBLISHING COMPANY
1925
C
D6
Copyright by
THE OPEN COURT PUBLISHING COMPANY
1925
PRINTED IN THE UNITED STATES OF AMERICA
THE GEOMETRY OF RENE DESCARTES
ZA-5Q.Z
Preface
If a mathematician were asked to name the great epoch-making
works in his science, he might well hesitate in his decision concerning
the product of the nineteenth century ; he might even hesitate with
respect to the eighteenth century ; but as to the product of the sixteenth
and seventeenth centuries, and particularly as to the works of the
Greeks in classical times, he would probably have very definite views.
He would certainly include the works of Euclid, Archimedes, and
Apollonius among the products of the Greek civilization, while among
those which contributed to the great renaissance of mathematics in the
seventeenth century he would as certainly include La Gcomcfrie of
Descartes and the Principia of Newton.
But it is one of the curious facts in the study of historical material
that although we have long had the works of Euclid, Archimedes,
Apollonius, and Newton in English, the epoch-making treatise of Des-
cartes has never been printed in our language, or, if so, only in some
obscure and long-since-forgotten edition. Written originally in French,
it was soon after translated into Latin by Van Schooten, and this was
long held to be sufficient for any scholars who might care to follow
the work of Descartes in the first printed treatise that ever appeared
on analytic geometry. At present it is doubtful if many mathemati-
cians read the work in Latin ; indeed, it is doubtful if many except the
French scholars consult it very often in the original' language in which
it appeared. But certainly a work of this kind ought to be easily access-
ible to American and British students of the history of mathematics,
and in a language with which they are entirely familiar.
On this account, The Open Court Publishing Company has agreed
with the translators that the work should appear in English, and with
such notes as may add to the ease with which it will be read. To this
organization the translators are indebted for the publication of the
book, a labor of love on its part as well as on theirs.
As to the translation itself, an attempt has been made to give the
meaning of the original in simple English rather than to add to the dif-
ficulty of the reader by making it a verbatim reproduction. It is
believed that the student will welcome this policy, being content to go
to the original in case a stricter translation is needed. One of the
translators having used chiefly the Latin edition of Van Schooten, and
the other the original French edition, it is believed that the meaning
which Descartes had in mind has been adequately preserved.
Table of Contents
BOOK I
Problems the Construction of which Requires Only Straight
Lines and Circles
How the calculations of arithmetic are related to the operations of geometry. . 297
How multiplication, division, and the extraction of square root are performed
geometrically 293
How we use arithmetic symbols in geometry 299
How we use equations in solving problems 300
Plane problems and their solution 302
Example from Pappus 304
Solution of the problem of Pappus 307
How we should choose the terms in arriving at the equation in this case 310
How we find that this problem is plane when not more than five lines are given 313
1 It should be recalled that the first edition of this work appeared as a kind of
appendix to the Discours de la Méthode, and hence began on page 297. For con-
venience of reference, the original paging has been retained in the facsimile. A
new folio number, appropriate to the present edition, will also be found at the foot
of each page. For convenience of reference to the original, this table of contents
follows the paging of the 1637 edition.
VI
TABLE
'Des matières de U
GEOMETRIE.
L'mre Tremier,
DES PROBLESMES QJCJ'ON PEUT
conflruire fans y employer que des cercles &
des lignes droites.
O M M E N T /^ calcul d' Ay'ithmeticjtie p rapporte auxopC"
rations de (geometric. 2çj
Comment fê fint (jcometricjuement la Multiplication , U
_ Dmifion^Cr lextra^ion de laracine c^Harree, 29S
Comment on pent vfer de chiffres en Géométrie, j.çç
Comment il jkut venir aux Equations qui f entent a re foudre les pro-
blefmes^ ^00
^uels font les probief mes plans -^ Et comment tlsfe refoluent. ^02
Exemple tiré de Pappus. j 04
Tiejpon/ê a la cjueflion de Pappus. ^o/
Cornent on doitpofer les termes pottr venir a C Equation en cet exeple.^i 0
K k k Corn
Vlï
BOOK II
On the Nature of Curn'ed Lines
What curved lines are admitted in geometry 315
The method of distinguishing all curved lines of certain classes, and of know-
ing the ratios connecting their points on certain straight lines 319
There follows the explanation of the problem of Pappus mentioned in the pre-
ceding book 323
Solution of this problem for the case of only three or four lines 324
Demonstration of this solution : 332
Plane and solid loci and the method of finding them 334
The first and simplest of all the curves needed in solving the ancient problem
for the case of five lines 335
Geometric curves that can be described by finding a number of their points... 340
Those which can be described with a string 340
To find the properties of curves it is necessary to know the relation of their
points to points on certain straight lines, and the method of drawing
other lines which cut them in all these points at right angles 341
General method for finding straight lines which cut given curves and make
right angles with them 342
Example of this operation in the case of an ellipse and of a parabola of the
second class 343
Another example in the case of an oval of the second class 344
Example of the construction of this problem in the case of the conchoid 351
Explanation of four new classes of ovals which enter into optics 352
The properties of these ovals relating to reflection and refraction 357
Demonstration of these properties 360
Table.
Comment on trciéue cjue ceprohhfwe cflplan lorpja'tl n' eft point propofé
en plm de s lignes. ^ , ,
Di [cours Second.
DE LA NATURE DES LIGNES
COURBES.
Q V clic i font tes lignes conrbes <!jt4 on pent receuoiren Cjeometne. ^\ ;
La façon de dtjlwgPicY tentes ces lignes courbes en certains aenres:
ht de connoiflre le rapport qti ont toHS leurs poins a ceux des lianes
droites. j i ç
Suite de l' expliCAtion de la c^uejlion de Pappu4 wife au hure preeedenr..
3^3-
Sobttion de ceteqmjïion quand elle nejl proposée' cjh en j ou ^Ifrnes.
' 3^4.
Demonflration de ccte folution. ^^2
^els font les lieux plans & fohdes & la façon de les trouucr tous. ^^4.
^elle efi la premiere & la plu^ fimple de toutes les lignes courbes cjni
feruent a la cjuejîion des anciens cjuandelle ef propofé e en cinq lignes,
33S'
Celles font les lignes courbes qu'on defcnt en trouuant plufeurs de leurs
poins qui peuuent eflre receucs en Géométrie. ^4.0
Celles font au fjt celles qu on defcnt aueç vne chorde,qui peuuent y eflre
receues, 2 ^o
^epour troHuer toutes les proprietez^des lignes courber , il fufflt defca-
uoir le rapport quoht tous leurs poins a ceux des lignes droites ; cr U
façon de tirer a autres lignes qui les coupent en tous ces poms a angles
droits. j^;
Façon générale pour trouuer des lignes droites qui couppent les courbes
données yOU leurs contingentes a angles droits. Z4.z
Exemple de ce te operation en vne Ellipfe : Et en vne parabole du fecoiid
geure, ^^^
tAutre exemple en vne ouale du fc-condgeure. 3 44
Exemple de la conflruRion de ce probief me en la conchoide. 5 j r,
Explication de 4, nouueaux geures d*Ouales qm feruent a l'Optique, ^sï
Les propriete'^de ces Ouales touchant Icsreflextons cr les réfractons.
357
DemonjlrAtion de ces proprie tez., ^60
IX
TABLE OF CONTENTS
How it is possible to make a lens as convex or concave as we wish, in one of
its surfaces, which shall cause to converge in a given point all the rays
which proceed from another given point 363
How it is possible to make a lens which operates like the preceding and such
that the convexity of one of its surfaces shall have a given ratio to the
convexity or concavity of the other 366
How it is possible to apply what has been said here concerning curved lines
described on a plane surface to those which are described in a space of
three dimensions, or on a curved surface 368
BOOK III
On the Construction of Solid or Supersolid Problems
On those curves which can be used in the construction of every problem 369
Example relating to the finding of several mean proportionals 370
On the nature of equations 371
How many roots each equation can have 372
What are false roots Z12
How it is possible to lower the degree of an equation when one of the roots
is known 2>12
How to determine if any given quantity is a root ZTh
How many true roots an equation may have yii
How the false roots may become true, and the true roots false 373
How to increase or decrease the roots of an equation 374
That by increasing the true roots we decrease the false ones, and vice versa. . 375
How to remove the second term of an equation 376
How to make the false loots true without making the true ones false 2)11
How to fill all the places of an equation 378
How to multiply or divide the roots of an equation 379
How to eliminate the fractions in an equation 379
How to make the known quantity of any term of an equation equal to any
given quantity 380
De La Géométrie.
Comment 0» peut faire vn verre autant connexe ou concatig en l*vne de
fes fuperficieStCju on voudra, ^uirajfemble a vn point donné tout les
rayons cjiti vienent d*vn autre point donné. ^ 6^
(Comment on en peut fkire vn t^ut fhce le mefme , 6r cjue la conaexite de
i'vne de fs ftperfictes ait la proportion donnée ausc la conuexité ou
conçauité de II autre. ^.6 6,
Comment on peut rapporter tout ce quia ejlé dit des lignes courbes dé-
faites ^r vne fuperficte_plate,a celles <jui fe defcriuent dans vn ejpaee
(jui a s dimenJtonSf oubien fur vne ftperficie courbe» }6i
Liure Troijtefme
DE LA CONSTRUCTION DES
problefmes roIides,ou plufque folides.
DE cfuelles lignes courbes on peut fe jèruir en la conJlruBion de chaf-
cjue probUfme. 5 6ç
Exemple touchant l' muent ion deplufiems moyenes proportionelles, 57 e
De la nature des Ecjuations. ^71
Combien il peut y auoir de racines en chafque EcjHation, S7Z
.Celtes font les fnuffes racines. ^yZ
Comment on peut diminuer le nombre des dimenfions dtvne Equation,
lorfquon connoifhcjuelcju'vne defes racines, 37 z
Comment on peut examiner fi quelque quantité donnée efi la valeur
d' vne racine, ^7 s
Combien il peut y auoir de vrajes racines en chafque Equation. 57^
Comment omfkit que les fnujfes racines deuienent vrayes , & les vrajes
fautes, . ^'7^
-Comment on peut augmenter ou diminuer les racines d'vneSquation.^74.
j:£^V/2 augmentant aw files vrayes racines on diminue lesfhuffes , ou au
contraire, 375
Comment on peut ofler le pcond terme dvne Equation, 376
Comment on fan que les fauffes racines deuienent vrajes fins que les
vrayes deuienent faujfes, S77
(Comment on fait que toutes les places d'vneEquationfoient remplies ^78
(Comment on peut multiplier ou diuifer les racines êH vne Equation, 3 jç
(Comment on ofle les nombres rompus d'vne Equation, 379
(fomment on rend la quaiîtité connue de l'vn des t<rmes d'vne Equation
efgale a telle autre qu'on veut. J ^ "
Kkk z ^^c
TABLE OF CONTENTS
That both the true and the false roots may be real or imaginary 380
The reduction of cubic equations when the problem is plane 380
The method of dividing an equation by a binomial which contains a root 381
Problems which are solid when the equation is cubic 383
The reduction of equations of the fourth degree when the problem is plane.
Solid problems 383
Example showing the use of these reductions 387
General rule for reducing equations above the fourth degree 389
General method for constructing all solid problems which reduce to an equa-
tion of the third or the fourth degree 389
The finding of two mean proportionals 395
The trisection of an angle 396
That all solid problems can be reduced to these two constructions 397
The method of expressing all the roots of cubic equations and hence of all
equations extending to the fourth degree 400
Why solid problems cannot be constructed without conic sections, nor those
problems which are more complex without other lines that are also more
complex 401
General method for constructing all problems which require equations of de-
gree not higher than the sixth 402
The finding of four mean proportionals 411
Table. De LA Géométrie.
^^e les racines tant vrayes que fknjfes peunent eftre réelles ou imaginai-
res, ^so
La rediiEîion des Equations cubiques lorfque le problefme efl plan, ^So
La façon de diuifer vne Equation par ijn binôme qui contient [à racine.
^j4els problefmes font jéhdes lorfque l'Equation efl cubique. ^S^
La redutlion des Equations qui ont quatre di wen fions lorfqne le problef-
me efl plan. Et quels [ont ceux qui. font folides. 5 S^
exemple de L^vftge de ces reduBions. ^ s 7
'^gle gêner aie pour réduire toutes les Equations quipaffentle quarré de
quarré. ^ g ^
Façon générale pour confîruire tous les problefmes jôltdes réduits a vne
Equation de trois ou quatre dimenfions, 2Sç
Vinuenticn de deux moyenes proportionelles. 2Çf
La diuifion de l'angle en trois , ^o^
^e tous les problefmes folides fe peutient réduire a ces deux confiru-
61 ions. ^çy^
La façon d! exprimer la valeur de toutes les racines des Equations cubi-
ques: Et en fuite de toutes cell&s qui ne montent que lufques au quar-
ré de quarrè. ^00
T^ourquov les problefmes folides ne peuuent eflre conflruits (ans les fe-
rlions coniques y ny ceux qui font plus compofés [ans quelques autres
lignes plus co?npfeés. ^ot
Façon générale pour confirutre tous les problefmes réduits a vne Eq [■ra-
tion qui n'a point plus de fx dimenftons. ^02
L'inuention de quatre moyenes proportionelles.
411
F I N.
Les
BOOK FIRST
The Geometry of Rene Descartes
BOOK I
Problems the Construction of Which Requires Only Straight
Lines and Circles
ANY problem in geometry can easily be reduced to sucb terms that
a knowledge of the lengths of certain straight lines is sufficient
for its construction.''' Just as arithmetic consists of only four or five
operations, namely, addition, subtraction, multiplication, division and the
extraction of roots, which may be considered a kind of division, so in
geometry, to find required lines it is merely necessary to add or subtract
other lines ; or else, taking one line which I shall call unity in order to
relate it as closely as possible to numbers/"' and which can in general be
chosen arbitrarily, and having given two other lines, to find a fourth
line which shall be to one of the given lines as the other is to unity
(which is the same as multiplication) ; or, again, to find a fourth line
which is to one of the given lines as unity is to the other (which is
equivalent to division) ; or. finally, to find one, two, or several mean
proportionals between unity and some other line (which is the same
''' Large collections of problems of this nature are contained in the following
works: Vincenzo Riccati and Girolamo Saladino, Institutioncs AnaIyticae,'Bo\ogna,
1765; Maria Gaetana Agnesi, Istltusioni Analitkhc, Milan. 1748; Claude Rabuel,
Commentaires sur la Géométrie de M. Descartes, Lyons, 1730 (hereafter referred
to as Rabuel) ; and other books of the same period or earlier.
'"'Van Schooten, in his Latin edition of 1683, has this note: "Per unitatem
intellige lineam quandam determinatam, qua ad quamvis reliquarum linearum talem
relationem habeat, qualem unitas ad certum aliquem numerum." Geotnetria a
Renato Des Cartes, una cum notis Flori)nondi de Beanne, opera aiqne studio
Francisci à Schooten, Amsterdam, 1683, p. 165 (hereafter referred to as Van
Schooten).
In general, the translation runs page for page with the facing original. On
account of figures and footnotes, however, this plan is occasionally varied, but not
in such a way as to cause tlie reader any serious inconvenience.
{ pri'iUrrvf^^M-tJ^ayTl^
297
L A
GEOMETRIE.
LIVRE PREMIER.
^es problefmes qu'on peut conftruire [ans
y employer que des cercles 0^ des
lignes droites.
^<S^^^ O u s les Problefmes de Géométrie fè
peuucnt facilement réduire a tels termes,
% qu'il n'eft befoin par après que de connoi-
ftre la longeur de quelques lignes droites,
'pour les conftruire.
Et comme toute l'Arithmétique n'eft compofée, que Commcc
de quatre ou cinq operations, qui font l'Addition, la|p, "j*=^^
Souftradion, la Multiplication , la Diuifîon , & l'Extra- thJeti-
•<Stion des racines , qu'on peut prendre pour vne efpece '^^^ ^^
de Diuifion : Ainfî n'at'on autre chofe a faire en Geo- auxope-
metrie touchant les lignes qu'on cherche , pour les pre- ^"0 "' ^^
parer a eftre connues, que leur en adioufter d'autres , ou t"e.
en ofter, Oubicn en ayant vne, que le nommeray l'vnite'
pour la rapporter d'autant mieux aux nombres , & qui
peut ordinairement eftre pnfe a dircretion,puis en ayant
encore deux autres, en trouuer vne quatriefme , qui foit
à r vne de ces deux, comme l'autre eft a IVnitc, ce qui eft
le mefme que la Multiplication i oubien en trouuer vne
quatriefme, qui foit al' vne de ces deux, comme rvnite'
Pp eft
3
«eome-
LaMulti-
plicatioD.
29% La Géométrie.
eft a l'autre, ce qui eft le mefme que la Diuifiorij ou enfin
trouuer vne,ou deux ,ou plufieurs moyennes proportion-
nelles entre l'vnité, & quelque autre ligne j ce qui eft le
mefme que tirer la racine quarrée^ on cubiqu Cj&c. Et ie
ne craindray pas d'introduire ces termes d'Arithméti-
que en la Géométrie , afEn de me rendre plus intel-
ligibile.
Soit pai* exemple
ABlVnite', & qu'il fail-
le multiplier B D par
C B G, ie n ay qu'a ioindre
les poins A & C, puis ti-
rer D E parallèle a C A,
&, B E eft le produit de
cete Multiplication.
Oubiens'il faut diuifer BE par BD, ayant ioint les
poins E & D , ie tire A C parallèle a D E, & B G eft le
produit de cete diuifîon.
Ou s'il faut tirer la racine
quarree de G H , ie luy ad-
ioufte en ligne droite F G,
qui eft rvnite'^o.: diuifànt F H
H en deux parties efgales au
point K, du centre K ie tire
le cercle F I H, puis eiîeuant du point G vne ligne droite
iufquesà I,à angles droits fur FH, c'eft GI la racine
cherchée. le ne dis rien icy de la racine cubique, ny des
autres, à caufe que l'en parleray plus commodément cy
après.
^^^'peut^ Mais fouuent on n'a pas befoin de tracer ainfî ces li-
gne
La Divi-
flOQ.
TExtra-
éliondela
racine
quarrcc.
FIRST BOOK
as extracting the square root, cube root, etc., of the given hne.'" And
I shall not hesitate to introduce these arithmetical terms into geometry,
for the sake of greater clearness.
For example, let AB be taken as unity, and let it be required
to multiply BD by BC. I have only to join the points A and C, and
draw DE parallel to CA ; then BE is the product of BD and BC.
If it be required to divide BE by BD, I join E and D, and draw AC
parallel to DE ; then BC is the result of the division.
If the square root of GH is desired, I add, along the same
straight line, EG equal to unity ; then, bisecting EH at K, I describe
the circle EIH about K as a center, and draw from G a perpendicular
and extend it to I, and GI is the required root. I do not speak here of
cube root, or other roots, since I shall speak more conveniently of them
later.
Often it is not necessary thus to draw the lines on paper, but it is
sufficient to designate each by a single letter. Thus, to add the lines
BD and GH, I call one a and the other b, and write a + b. Then a — b
will indicate that b is subtracted from a; ab that a is multiplied by b;
a
^ that a is divided hy b ; aa or a- that a is multiplied by itself ; a^ that
this result is multiplied by a, and so on, indefinitely.''' Again, if I wish
to extract the square root of ar^b-, I write ^Ja--\-b"; if I wish to
extract the cube root of a^ — b^-\-ab~, I write ^a^ — b^-^ah'^, and sim-
ilarly for other roots. '^' Here it must be observed that by a", b^, and
similar expressions, I ordinarily mean only simple lines, which, how-
ever, I name squares, cubes, etc., so that I may make use of the terms
employed in algebra.'*'
''' While in arithmetic the only exact roots obtainable are those of perfect
powers, in geometry a length can be found which will represent exactly the square
root of a given line, even though this line be not commensurable with unity. Of
other roots, Descartes speaks later.
'■*' Descartes uses a", a*, œ', a'"', and so on. to represent the respective powers
of a, but he uses both aa and a- without distinction. For example, he often has
aabb, but he also uses -rr^.
4b-
'°^ Descartes writes : ^JC.à^' — d'^-j-abd. See original, page 299, line 9.
'*' At the time this was written, a- was commonly considered to mean the sur-
face of a square whose side is a, and b'^ to mean the volume of a cube whose side
is b; while b*, b'', . . . were unintelligible as geometric forms. Descartes here says
that a~ does not have this meaning, but means the line obtained by constructing a
third proportional to 1 and a, and so on.
GEOMETRY
It should also be noted that all parts of a single line should always
be expressed by the same number of dimensions, provided unity is not
determined by the conditions of the problem. Thus, a^ contains as
many dimensions as ab' or b^, these being the component parts of the
line which I have called ^a^ — b^-\-ab-. It is not, however, the same
thing when unity is determined, because unity can always be under-
stood, even where there are too many or too few dimensions ; thus, if
it be required to extract the cube root of a-b- — b. we must consider the
quantity a^b" divided once by unity, and the quantity b multiplied twice
by unity. ^''
Finally, so that we may be sure to remember the names of these lines,
a separate list should always be made as often as names are assigned
or changed. For example, we may write, AB=1, that is AB is equal
to 1 ;'" GH = a, BD = 6. and so on.
If, then, we wish to solve any problem, we first suppose the solution
already effected.'^' and give names to all the lines that seem needful for
its construction, — to those that are unknown as well as to those that
are known. ''"^ Then, making no distinction between known and unknown
lines, we must unravel the difficulty in any way that shows most natur-
'"' Descartes seems to say that each term must be of the third degree, and that
therefore we must conceive of both a-b- and b as reduced to the proper dimension.
'*' Van Schooten adds "seu unitati," p. 3. Descartes writes, AB 00 1. He
seems to have been the first to use this symbol. Among the few writers who fol-
lowed him, was Hudde (1633-1704). It is very commonly supposed that 00 is a
ligature representing the first two letters (or diphthong) of "aequare."' See. for
example, M. Aubry's note in W. W. R. Ball's Recreations Mathématiques et Prob-
lèmes des Temps Anciens et Modernes, French edition, Paris, 1909, Part III, p. 164.
'" This plan, as is well known, goes back to Plato. It appears in the work of
Pappus as follows: "In analysis we suppose that which is required to be already
obtained, and consider its connections and antecedents, going back until we reach
either something already known (given in the hypothesis), or else some fundamen-
tal principle (axiom or postulate) of mathematics." Pappi Ale.yandrini Collectiones
quae supcrsimt e Hbris manu scripfis edidit Latina interpcllatione ct commentariis
instni.vit Frcdericus Hulisch. Berlin, 1876-1878; vol. II, p. 635 (hereafter referred
to as Pappus). See also Commandinus, Pappi Alexandrini Mathcmaticae Collec-
tiones, Bologna, 1588, with later editions.
Pappus of Alexandria was a Greek mathematician who lived about 300 A.D.
His most important work is a mathematical treatise in eight books, of which the
first and part of the second are lost. This was made known to modern scholars
by Commandinus. The work exerted a happy influence on the revival of geometry
in the seventeenth century. Pappus was not himself a mathematician of the first
rank, but he preserved for the world many extracts or analyses of lost works, and
by his commentaries added to their interest.
''"' Rabuel calls attention to the use of a, b, c, ... for known, and x, y, z, . . .
for unknown quantities (p. 20).
Livre Premier. 299
gnes fur le papier, & il fuffift de les defigner par quelques ^ç^^ ^^
lettres, chafcune par vue feule. Comme pour adioufter clnfFresea
la ligne B D a G H, ie nomme Tvne a & l'autre b,&c Qfcris tde!"^^'
a~h b-^Eta— ^,pour fouftraire b d' a-^ Et a ^,pour les mul-
tiplier IVne par l'autre; Et ^,pourdiuifer^zpar^j-Ec a a,
1 5
ou a, pour multipliera par foymefmc; Et/^, pour le
multiplier encore vne fois par a , &:ainfl a rinfini ^ Et
'il z z
^ ^-j- b y pour tirer la racine quarrce d' a -h b -^Et
* Ca-'b-i^abbj pour tirer la racine cubique d' a—b
-h abb, & ainfi des autres.
Où il cil a remarquer que par a ou b ou femblables,
ie ne conçoy ordinairement que des lignes toutes fîm--
pies, encore que pour me feruir des noms vfités en l'Al-
gèbre, ie les nomme des quarre's ou des cubes, ôcc,
Ileltaufly a remarquer que toutes les parties dVne
mefmeligne,fedoiuent ordinairement exprimer par au*
tant de dimenfions l'vne que l'autre, lorfque IVnite'n'eil:
point déterminée en la queftion, comme icy a en con»-
tientautantqu'^^^ ou b dont fecompofe la ligne que
Tay nommée ^C. a- b -i- abb: mais que ce n'eft
pas de mefine lorfque Tynite eft déterminée, a caufo-
qu'elle peut eftre foulèntendue par tout ou il y a trop ou
trop peu de dimenfions : comme s'il faut tirer la racine
cubique de aabb — b j il faut penfer que la quantité
aabbcd diuifee vne fois par l'vnite', & que l'autre quan-
tité b eft multipliée deux fois par la mefme,
P p a Au
^^^ La Géométrie.
Au refte affin de ne pas manquer a fe fauuenir des
noms de ces lignes, il en faut toufîours faire vn regiftrc
fèpare'' , à mefure qu'on les pofe ou qu'on les change,
cfcriuant par exemple .
A B 30 I , c'eft a dire, A B efgal à t.
GH 30 ^
BD 00 b, ''zc,
Cemmct Ainfî voulautrefoudre quelque problefînc, on doit d'à-
nir^rux^^ bord le confiderer comme delîa fair, & donner des noms
Equatiôs a toutcs les lignes, qui femblent necefTaires pour le con-
uent are- ûruifc^ auffy bien a celles qui font inconnues , qu'aux
foudre les autres. Puis fans confiderer aucune difference entre ces
mes. lignes connu es, & mconnues , on doit par counr la diffi-
culté, felon l'ordre qui monftre le plus naturellement
de tous en qu'elle forte elles dependent mutuellement.
les vnes des autres, iufques a ce qu'on ait trouue moyen
'd'exprimer vne mefme quantite^'en deux façons : ce qui
le nomme vneEquationj car les terme s de l'vnc de ces
deux façons font efgaux a ceux de l'autre. Et on doit
trouuer autant de telles Equations,qu'ona fuppofc de li-
gnes, qui eftoient inconnuë:t. Oubien s'il ne s'en trouue
pas tant, & que nonobflant on n'omette rien de ce qui ell
defiré en la queftion,cela tefmoigne qu'elle n*eft pas en-
tièrement déterminée. Et lors on peut prendre a difcre-
tion des lignes connues, pour toutes les inconnues auf.
qu'elles ne correfpond aucune Equation. Après cela s'il
enrefte encore plufieurs , il fe faut feruir par ordre de
chafcune des Equations qui refteut aufly , foit en la con-
fiderant toute feul^,foit en la comparant auec lés autres,
pour expliquer chafcune de ces lignes inconnues; & faire
ainfî
FIRST BOOK
ally the relations between these lines, until we find it possible to express
a single quantity in two ways.'"^ This will constitute an equation, since
the terms of one of these two expressions aie together equal to the
terms of the other.
We must find as many such equations as there are supposed to be
unknown lines ;''"' but if, after considering everything involved, so many
cannot be found, it is evident that the question is not entirely deter-
mined. In such a case we may choose arbitrarily lines of known length
for each unknown line to which there corresponds no equation."''
If there are several equations, we must use each in order, either con-
sidering it alone or comparing it with the others, so as to obtain a value
for each of the unknown lines ; and so we must combine them until
there remains a single unknown line"*' which is equal to some known
line, or whose square, cube, fourth power, fifth power, sixth power,
etc., is equal to the sum or difference of two or more quantities, "°' one
of which is known, while the others consist of mean proportionals
between unity and this square, or cube, or fourth power, etc., multiplied
by other known lines. I may express this as follows :
or s-= — aa-\-b-,
or c^= a::.- -\-b-jj — c'^
or ::*=-ac^ — ■c^.c-\-d'^, etc.
That is, 2, which I take for the unknown quantity, is equal to b; or,
the square of ^ is equal to the square of b diminished by a multiplied
by 2; or, the cube of a is equal to a multiplied by the square of s, plus
the square of b multiplied by ^. diminished by the cube of c ; and sim-
ilarly for the others.
'"^ That is, we must solve the resulting simultaneous equations.
'^"' Van Schooten (p. 149) gives two problems to illustrate this statement. Of
these, the first is as follows : Given a line segment AB containing any point C,
required to produce AB to D so that the rectangle AD.DB shall be equal to the
square on CD. He lets AC = a, CB = b, and BD = x. Then AD = a + b+x,
and CD =zb 4- x, whence ax -\- bx + x- ^b~-'r 2b x + x- and x = 7- .
a — b
^"' Rabuel adds this note : "We may say that every indeterminate problem is an
infinity of determinate problems, or that every problem is determined either by
itself or by him who constructs it" (p. 21).
'"' That is, a line represented by x, x-, x^, x*, ....
'"^ In the older French, "le quarré. ou le cube, ou le quarré de quarré, ou le sur-
solide, ou le quarré de cube &c.," as seen on page 11 (original page 302).
GEOMETRY
Thus; all the unknown quantities can be expressed in terms of a sin-
gle quantity/"' whenever the problem can be constructed by means of
circles and straight lines, or by conic sections, or even by some other
curve of degree not greater than the third or fourth.'^''
But I shall not stop to explain this in more detail, because I should
deprive you of the pleasure of mastering it yourself, as well as of the
advantage of training your mind by working over it, which is in my
opinion the principal benefit to be derived from this science. Because,
I find nothing here so difficult that it cannot be worked out by any one
at all familiar with ordinary geometry and with algebra, who will con-
sider carefully all that is set forth in this treatise.''^'
'"' See line 20 on the opposite page.
^"' Literally, "Only one or two degrees greater."
'^^' In the Introduction to the 1637 edition of La Geometric, Descartes made
the following remark : "In my previous writings I have tried to make my mean-
ing clear to everybody; but I doubt if this treatise will be read by anyone not
familiar with the books on geometry, and so I have thought it superfluous to repeat
demonstrations contained in them." See Oeuvres de Descartes, edited by Charles
Adam and Paul Tannery, Paris, 1897-1910, vol. VI, p. 368. In a letter written
to Mersenne in 1637 Descartes says: "I do not enjoy speaking in praise of myself,
but since few people can understand my geometry, . and since you wish me to
give you my opinion of it, I think it well to sav that it is all I could hope for,
and that in La Dwptriquc and Les Météores, I have only tried to persuade people
that my method is better than the ordinary one. I have proved this in my geom-
etry, for in the beginning I have solved a question which, according to Pappus,
could not be solved by any of the ancient geometers.
"Moreover, what I have given in the second book on the nature and properties
of curved lines, and the method of examining them, is, it seems to me, as far
beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond
.the a, b, c of children. . . .
"As to the suggestion that what I have written could easily have been gotten
from Vieta, the very fact that my treatise is hard to understand is due to my
attempt to put nothing in it that I believed to be known either by him or by any
one else. ... I begin the rules of my algebra with what Vieta wrote at the
very end of his book. De eincndatioiic acquationutn. . . . Thus, I begin where
he left off." Oeuvres de Descartes, publiées par llctor Cousin, Paris, 1824, Vol.
VI, p. 294 (hereafter referred to as Cousin).
In another letter to Mersenne, written April 20, 1646, Descartes writes as
follows: "I have omitted a number of things that might have made it (the geom-
etry) clearer, but I did this intentionally, and would not have it otherwise. The
only suggestions that have been made concerning changes in it are in regard to
rendering it clearer to readers, but most of these are so malicious that I am com-
pletely disgusted with them." Cousin, Vol. IX, p. 553.
In a letter to the Princess Elizabeth, Descartes says : "In the solution of a
geometrical problem I take care, as far as possible, to use as lines of reference
parallel lines or lines at right angles ; and I use no theorems e.xcept those which
assert that the sides of similar triangles are i)roportional, and that m a right
triangle the square of the hypotenuse is equal to the sum of the squares of the
sides. I do not hesitate to introduce several unknown quantities, so as to reduce the
question to such terms that it shall depend only on these two theorems." Cousin,
Vol. IX, p. 143.
10
Livre Premier. 5oi
ain{îenlesdemefjant, qu'il n'en demeure quVne feule,
efgale a quelque autre, qui foit connue , oubiea dont le
quarré, oulecube,oulequarredequarré', ouïe furfbli-
de, ouïe quarre''de cube, &c. foit efgal a ce, qui fe pro-
duift par l'addition, ou fouflradtion de deux ou plufieurs
autres quantités ^dontlVne foit connue , & les autres
foient compofe'es de quelques moyennes proportion»
Belles entre rvnite', & ce quarré, ou cube , ou quarre de
quarre',&c. multipliées par d'autres connues. Ce que i'e-
fcris en cete forte.
;{_ 30 ^. ou
i.
^30 — a ^-^bb. ou
s^ 00 'i-a ^-^bb^s^-'C, ou
4 } î 4
^ 30 ^J5 î^ " c :^-H d. &c.
C'eftadire, ^ que ieprens pour la quantité* inconnue,
eftefgaléa^, ou le quarré de ^ eft efgâl au quarre de b
moins « multiplié par ^. ou le cube de ^ eft efggl à a
multipliépar le quarre de i^plus le quarre' de ^ multiplie
par ;^moins le cube de c, & ainfi des autres.
Et on peut toufîours réduire ainfi toutes les quantités
inconnues à vne feule, lorfque le Problefme fe peut con-
ftruire par des cercles & des ligues droites, ou aufîy par
des fedtions coniques,ou mefme par quelque autre ligne
qui ce foit que d'vn ou deux degrés plus compofce. Mais^
ie ne m'areft^e point a expliquer cecy plus en detail ,^a
caufe que ie vous ofterois le plaifir de l'apprendre de
vous mefme, & l'vtilité de cultiuer voftrc efpric en vous
y exerceant, qui eft a mon auis la principale, qu'on puifle
Pp 3 tirer
11
Quels
fondes
problef-
3°^ La Géométrie.
tirer de cetefcience. Aufîy que ien y remarque rien de
Il difficile, que ceux qui feront vn peu verfé's en la Géo-
métrie commune, & en l'Algèbre, & qui prendront gar-
de a tout ce qui eil en ce traite, ne puifTent trouuer.
C'eftpourquoyieme contenteray icy de vous auer-
tir, que pourvu qu'en demcflant ces Equations on ne
manque point a feferuir de toutes les diuifîons, qui fe-
ront poffibles , on aura infalliblemcnt les plus fimples
termes, aufquels la queftion puifTe eftre réduite.
Et que 11 elle peut eftre refolue par la Géométrie ordi-
naire, c eft a dire, en ne fe feruant que de lignes droites
mes plans ^ circulaires tracées furvnefuperficie plate , lorfque la
dernière Equation aura efté entièrement déo]eflee,iln y
reftera tout au plus qu'vn quatre inconnu, efgal a ce qui
fe produift de l'Addition , ou fouftradtion de fa racine
multipliée par quelque quantité connue , & de quelque
autre quantité' auiTy connue
Et lors cete racine, ou ligne inconnue fetrouue ayfe-
ment. Car (î i*ay par exemple
1.
.,. loo a :{-i'bb
iefais le triangle re(5tan-
gle N L M, dont le co-
fte'L M eft efgal à b ra-
cine quarrée de la quan-
tité connue bb, 8c l'au-
j^ trcLNeft ^ ^, la moi-
tié de l'autre quantité'
connue, qui eftoit multipliée par ^que ie fuppofe eftre la
ligne inconnue, puis prolongeant M N la baze de ce tri-
angle,
Com-
ment ils
fe refol-
uenc.
12
FIRST BOOK
I shall therefore content myself with the statement that if the stu-
dent, in solving these equations, does not fail to make use of division
wherever possible, he will surely reach the simplest terms to which
the problem can be reduced.
And if it can be solved by ordinary geometry, that is, by the use of
straight lines and circles traced on a plane surface,''"' when the last
equation shall have been entirely solved there will remain at most only
the square of an unknown quantity, equal to the product of its root
by some known quantity, increased or diminished by some other quan-
tity also known. '^°' Then this root or unknown line can easily be found.
For example, if I have 2- = a3 -{- &-/"' I construct a right triangle NLM
with one side LM, equal to b, the square root of the known quan-
tity b-, and the other side, LN, equal to ^ a, that is, to half the
other known quantity which was multiplied by a, which I supposed to
be the unknown line. Then prolonging MN, the hypotenuse'"' of this
triangle, to O, so that NO is equal to NL, the whole line OM is the
required line z. This is expressed in the following way:'^'
But if I have y' = — ay-\-b-, where y is the quantity whose value
is desired, I construct the same right triangle NLM, and on the hypote-
''"' For a discussion of the possibility of constructions by the compasses and
straight edge, see Jacob Steiner, Die gcometrischen Constructionen ausgefiihrt
fnittelst dcr gcradcn Linic und cincs fcstcn Krciscs, Berlin, 1833. For briefer
treatments, consult Enriques, Fragcn dcr Elemcntar-Gcomctric, Leipzig, 1907 ;
Klein, Problems in Elementary Geometry, trans, by Beman and Smith, Boston,
1897; Weber und Wellstein, Ëncyklopddie der Elementarcn Géométrie, Leipzig,
1907. The work by Mascheroni, La gcometria del compasso, Pavia, 1797, is inter-
esting and well known.
'^^ That is, an expression of the form z- ^= a::± b. "Esgal a ce qui se produit
de l'Addition, ou soustraction de sa racine multiplée par quelque quantité connue,
& de quelque autre quantité aussy connue," as it appears in line 14, opposite page.
'"^' Descartes proposes to show how a quadratic may be solved geometrically.
'^' Descartes says "prolongeant MN la baze de ce triangle," because the hypote-
nuse was commonly taken as the base in earlier times.
i^^'From the figure OM.PM = ^M^ If OM = .3, PM = s — a, and since
LM := t, we have .Î (.? — o) ^ fc- or r- 3= ar-f-b-. Again, MN = \/~o- + fc-j whence
OM = 3=ON-(-MN = -a-f\/ja=-|-6-. Descartes ignores the second root, which
is negative.
13
GEOMETRY
mise MN lay off NP equal to NL. and the remainder PM is y, the
desired root. Thus I have
■'= -9'' + \h''' + ^'-
In the same way, if I had
.v* = — ax"" -\- i>%
PM would he x- and I should have
and so for other cases.
Finally, if I have ;:- = as—h~, I make NL equal to ^ a and LM equal
to b as before ; then, instead of joining the points M and N, 1
draw MOR parallel to LN, and with N as a center describe a circle
through L cutting MQR in the points Q and R ; then .c, the line sought,
is either MQ or MR, for in this case it can be expressed in two ways,
namely :'^^'
^ = r + \/^^-'^^
and
' = i"-Vr'-*=-
^-" Since MR.MQ^zLM". then if R = -, we have \iQ = a — s, and so
s {a — a)=: b- or r- := «r — b-.
If, instead of this, MQ = .3, then MR = a — ^, and again, .s- = a^ — b'-. Further-
more, letting O be the mid-point of QR,
1
MQ = OM - OQ = - « - Jl a-.-_ ^2,
and
MR
= MO + OR= j'^+yjl a^--b^-
Descartes here gives both roots, since both are positive. If MR is tangent to the
circle, that is, if è = — a, the roots will be equal; while if t» > — a, the line MR
will not meet the circle and both roots will be imaginary. Also, since RM.OM:=LM',
c.^2., = b^,andRM + QU = s^ + 3^ = a.
14
Livre Premier. 3^3
angle , iufques a O , en forte qu'N O foit efgale a N L,
la toute O M eft :^ la ligne cherchée Et elle s'exprime
en cete forte
;^ x> ^ « -h- t^~ aa -{- bb.
Que fi i^jyy :xi — a y H- bbjSc qu'y foit la quantité
qu'il faut trouuer , ie fais le mefme triangle rectangle
NLM, &defabazeMNi'ofteNPefgalea NL, &Ie
refte P M eft ^ la racine cherchée. De façon que iay
b b. Et tout de mefme fî i'a-
30 - ^ ^
V'jaa
uois X :x> — a X H- b. P M feroit x . & i'aurois
X :ù ^ - ^
4
^^: &ainfî des autres.
Enfin il i'ay
2^ CO a^'- bb:
ie fais N L efgale à | ^, & L M
efgale à b corne deuât, pusis,au lieu
de ioindre les poins M N , ie tire
M QJl parallèle a L N. & du cen-
tre N par L ayant defcrit vn cer-
cle qui la couppe aux poins Q 8c
R, la ligne cherchée ;{ eft M Q?
oubië M R, car en ce cas elle s'ex-
prime en deux façons, a fçauoir \:x:)'^a»r-V ^aa-bb^
&c ^ 7G~a— x/'^aa-'bb.
Et fi le cercle, qui ayant fon centre au point N , pafîe
par le point L, ne couppe ny ne touche la Hgne droite
MQ^, il n'y a aucune racine en l'Equation, de fa^n
qu'on peut affurer que la conftru^tion du problefms
propofé eft impoffible .
Au
15
304 La GEOMETRIE.
Au refle ces mefmes racines fe peuuent trouuer par
vne infinité d'autres moyens , & i'ay feulement veulu
mettre ceux cy, comme fort fimples, aîHn défaire voir
qu'on peut conftruire tous les Problefmes de la Géomé-
trie ordinaire, fans faire autre chofe que le peu qui efl
compris dans les quatre figures que i'ay expliquées. Ce
queienecroy pas que les anciens ayent remarqué, car
autrement ils n'eufTent pas prisJa peine d'en efcrire tant
de gros liures, ou le fèul ordre de leurs propofîtions nous
fait connoiftre qu'ils n'ont point eu lavraye méthode
pourles trouuer toutes,mais qu'ils ont feulement ramaf^
fe celles qu'ils ont rencontrées.
exemple Et on le peut voir aufTy fort clairement de ce que Pap-
Pappus. pus amis au commencement de fonfeptiefme liure, ou
après s'eftre arefte'' quelque tems a dénombrer tout ce
qui auoit efté efcrit en Géométrie par ceux qui l'auoient
precede', il parle enfin d vne queftion , qu'il dit que ny
Euclide,ny Apollonius, ny aucun autre n'auoient fceu
entièrement refoudre. & voycy fes mots.
Je cite Quem autem àicit [Apollonius) in tertio lihro locum ad
Jcrfionh- i^^^i ^ quatuor Uneas ah Eucliâe perfeBum non ejje , ne que
tine que le 2pJ"e perficere poterat , neque aliqui; alius'-: fed neque fau-
affin que lulum quidaddere iî5 , quœ Euclides {cripfityper ea tantum
chafcun çQ^jQii ^ qj^^ ufquc ad Eudidù t empara prtvmonjirata
plu4 ayfe- Juntj^C.
ment. £j. ^^^ ^^ aprc^s il explique ainfi qu'elle eft cete que-
Hion.
At locus ad très ^ ^ quatuor linens , in quo (Apolloîiius)
magnifiée fe iaBat i & oftentat^nulla habita gratia ei , qui
prius fcripferat , cflbujufmodi. Sipofitione datùtnbus
reïlis
16
FIRST BOOK
And if tlie circle described about N and passing through L neither
cuts nor touches the Hne MOR, the equation has no root, so that we
may say that the construction of the problem is impossible.
These same roots can be found by many other methods ,'''^ I have
given these very simple ones to show that it is possible to construct
all the problems of ordinary geometry by doing no more than the little
covered in the four figures that I have explained.'""' This is one thing
which T believe the ancient mathematicians did not observe, for other-
wise they would not have put so much labor into writing so many books
in which the very sequence of the propositions shows that they did not
have a sure method of finding all,'""' but rather gathered together those
propositions on which they had happened by accident.
This is also evident from what Pappus has done in the beginning of
his seventh book,'"'' where, after devoting considerable space to an
enumeration of the books on geometry written by his predecessors,'""'
he finally refers to a question which he says that neither Euclid nor
Apollonius nor any one else had been able to solve completely ;''" and
these are his words :
"Quern autem dicit (Apollonius) in tertio libro locum ad très, &
quatuor tineas ah Euclide perfectum non esse, neque ipse perficere
poterat, neque aliquis alius; sed neque paululum quid addere its, quœ
'"^^ For interesting contraction, see Rabuel, p. 23, et seq.
'-"' It will be seen that Descartes considers only three types of the quadratic
equation in s, nan^ely, S' + as — b~ = 0, z- — as — b- =^ 0, and s- — o5 + &- = 0.
It thus appears that he has not been able to free himself from the old traditions
to the extent of generalizing the meaning of the coefficients, — as negative and
fractional as well as positive. He does not consider the type z- + as + b- = 0,
because it has no positive roots.
'^^ "Qu'ils n'ont point eu la vraye méthode pour les trouuer toutes."
'='1 See Note [9].
1='^ See Pappus, Vol. II, p. 637. Pappus here gives a list of books that treat
of analysis, in the following words : "Illorum librorum, quibus de loco, 'ava\v6^ei>os
sive resoluto agitur, ordo hie est. Euclidis datorum liber unus, Apollonii de pro-
portionis sectione libri duo, de spatii sectione duo, de sectione determinata duo, de
tactionibus duo, Euclidis porismatum libri très, Apollonii inclinationum libri duo,
eiusdem locorum planorum duo, conicorum octo, Aristaci locorum solidorum libri
duo." See also the Commandinus edition of Pappus, 1660 edition, pp. 240-252.
'^"^ For the history of this problem, see Zeuthen : Die Lchrc von den Kegel-
schnitten im AUerthum, Copenhagen, 1886. Also, Adam and Tannery, Oeuvres de
Descartes, vol. 6, p. 723.
17
GEOMETRY
Enclides scripsit, per ea tantum conica, qucc usque ad Eiiclidis tcmpora
prœmonstrata sunt, arc." '"'
A little farther on, he states the question as follows :
"At locus ad très, & quatuor lincas, in quo {Apollonius) niagnifice
se jactat, & ostentat, nulla habita gratia ei, qui prins scripserat, est
hujusmodi.^^"^ Si positione dotis tribus redis lineis ab tino & eodem
piincto, ad très lineas in datis angulis rectœ lincœ ducantur, & data sit
proportio rcctanguli contcnti duabiis dnctis ad quadratiim reliqiiœ:
piinctnm contingit positione datum solidum locum, hoc est unam ex
tribus conicis sectionibus. Et si ad quatuor rectas lincas positione datas
in datis angulis linecc ducantur; & rectanguli duabns ductis contenti ad
contcntum duabns reliqitis proportio data sit; similiter punctum datum
coni sectioncm positione continget. Si quidem igitur ad duas tantum
locus planus ostensus est. Quod si ad plures quam quatuor, punctum
continget locos non adhuc cognitos, sed lincas tantum dictas; quales
autem sint, vel quam habcant proprietatem, non constat; earum unam,
neque primam, & qucc manifestissima videtur, composucrant osten-
dentes utilem, esse. Propositiones autem ipsarum hce sunt.
"Si ab aliqiio puncto ad positione datas rectas lineas quinque ducantur
rectœ linecc in datis angulis, & data sit proportio solidi parallèle pip edi
rectanguli, quod tribus dnctis lineis continctur ad solidum parallelepipe-
dum rectangulum, quod continctur rcUquis duabus, & data quapiam
tinea, punctum positione datani lincaui continget. Si auteui ad sex, &
data sit proportio solidi tribus lineis contcnti ad solidum, quod tribus
reliquis continctur; rursus puncturn continget positione datant lineam.
Quod si ad plures quam sex, non adhuc habent diccre, an data sit pro-
portio cnjiispiam contenti quatuor lineis ad id quod reliquis continctur,
'^'' Pappus, Vol. II, pp. 677, et seq., Commandimis edition of 1660, p. 251.
Literally, "Moreover, he (Apollonius) says that the problem of the locus related
to three or four lines was not entirely solved hy Euclid, and that neither he him-
self, nor any one else has been able to solve it completely, nor were they able to
add anything at all to those things which Euclid had written, by means of the
conic sections only which had been demonstrated before Euclid." Descartes arrived
at the solution of this problem four years before the publication of his geometry,
after spending five or six weeks on it. See his letters, Cousin, Vol. VI, p. 294,
and Vol. VI, p. 224.
''^' Given as follows in the edition of Pappus by Hultsch, previously quoted:
"Sed hie ad très et quatuor lineas locus quo magnopere gloriatur simul addens ei
qui conscripserit gratiam habendam esse, sic se habct."
18
Livre Premiek. Boi*
reBis lineis ah uno & eodem'punBe, ad très lineas in àatis art"
gulis reU^ Uneœ ducantur , (3 data fit proportio reUanguli
contenti duahu^ duBis ad quadrutum reliquœ : punUum con-*
tingîtpofitione datum folidum locum , hçc efl unam ex tribus
conicisfeBionihus. Et fi ad quatuor reBas lineas pojîtione
datas in datis angulis linece ducantur i ^ reBanguli duabus
duBis contenti ad content um duabus reliquis proportio data
fit: fi militer punBum, datum coni feBionem pofitione cwitin-
get. Si quidem igituradduas tantum locus planus ojlenfus
cfl, ^luodfi adplures quam quatuor, punBum continget /«-
cos non adhuc cognitos^ fed lineas tantum diBas s quales au-*
temfntj velquam habeant proprietatem, non confiât: earum
unam, nequeprimam^ & quœmanifefiijfimavidetur, compO'
jueruntofi}endentes utilemefe. propoftiones autemipfarum
hce funt.
Si ab aliquo punBo adpoftione datas reBds lineas quin-
que ducantur reBce linete in datis angulis , ^ data fit propor^
tio falidiparallelepipêdi reBanguli-, quod tribus duBis lineis
continetur ad folidum par allelepipedum reBangulum , quod
continetur reliquis duabus j (3 dataquapiamlinea^ punBum
p opt ion e datam Une am continget . Si autem adfex , S? data
fit propo rtio folidi tribus lineis contenti ad folidum, quod
tribus reliquis continetur i rurfus punBum continget pofitione
datam lineam. ^hiodfiadplures quamfex, non adhuchabent
dicere^an data fit proportio cuiufpiâ contenti quatuor lineis
ad id quod reliquis continetur, quoniam non efi aliquid con*
tentum pluribus quam tribus dimenfionibus.
Ou ie vous prie de remarquer en paffant, que lefcru-
pulcj que faifoient les anciens dV fer. des termes del'A-
rithmetiqueen la Géométrie, qui ne pouuoit procéder,
O q que
19
306 La Géométrie.
que de ce qu'ils ne voyoient pas afTes clairement leur
rapport, caufoit beaucoup dobfcuritc, & d'embaras, es
la façon dont ils s'expliquoient. car Pappus pourfuit en
ce te forte..
jicquiefcuntaufem his , quipaulo ante talia interf retail
fimt. 7ieque unum ali quo pact 0 comprehenfibîlefigniJïca?itcs
quodhîs co7itinetur.Licehit aute per coniunïïas prop orti ones
hc£C, (3 âiceret ^ demonflrare univerfe m diPcis proport ion?-
biis, atque his in hune modum. Si ah aliquo pwiBo adpojl-
tione datas-reBas Iméas ducanturrecl(Ç lineœ in datis angu-
lis, ^ data fît proportio cotiiunUa ex e<i, quam habet una du'
Rarum adunam, (3 altera adalteram^^ alia adaliam^^ te*
liqua ad datant lineam, fifint feptemj fivero oFio, ^ r cliqua
a d reliquam: pun&um continget pofitione datas lineas. Et
fimiliter quotcumque fmt impares vel pares multitudine,
€um hœcy ut dixi, loco ad quatuor lineas refpondeant^ nullum
igiturpofuerwntita utlinea not a fit » ^c,
La queftion donc qui auoit elle commencée a rofou*
dreparEucIide, &:pourfuiuieparApolloDius, fans auoir
eftèacheuéeparperfonoe , eftoit telle. Ayant trois oa
quatre ou plus grand nombre de lignes droites données
par pofîtioHj premièrement on demande vn point, du-
quel on puifle tirerautant d'autres lignes droites, vne fur
chafcune des données, qui façent auec elles des angles
donnes, & que le redangle contenu en deux de celles,
qui feront ainfi tirées d'vn mefme point., ait la propor-
tion^ donnée auec le quarré de la troifiefme , s'il n'y en a
que trois; oubien auec le redangle des deux autres, s'il y
en a quatreiOubien,s'il y en a cinq, que le parallélépipède
compofede trois ait la proportion donnée auec le parais
lelepipede
20
FIRST BOOK
quoniam non est aliquid contcntnm plurihns quavi tribus dimensioni-
hus." '"'
Here I beg you to observe in passing that the considerations that
forced ancient writers to use arithmetical terms in geometry, thus mak-
ing it impossible for them to proceed beyond a point where they could
see clearly the relation between the two subjects, caused much obscur-
ity and embarrassment, in their attempts at explanation.
Pappus proceeds as follows :
"Acqiiiescimt aiitem his, qui paulo ante talia interpretati sunt ; neque
nnum aliquo pacto comprehensibile significantes quod his continetur.
Licebit autem per conjiinctas proportiones hœc, & dicere & demonstrare
universe in dictis proportionibus, atque his in hunc modum. Si ab
aliquo puncto ad positione datas rectas lineas ducantur rectœ lineœ in
datis angulis, & data sit proportio conjuncta ex ea, quam habet nna
ductarum ad iinam, & altera ad alteram, & alia ad aliani, & rcliqua ad
datam lineam, si sint septcm; si vcro octo, & reliqua ad reliquam:
punctum continget positione datas lineas. Et similiter quotcumque sint
'"' This may be somewhat freely translated as follows : "The problem of the
locus related to three or four lines, about which he (Apollonius) boasts so proudly,
giving no credit to the writer who has preceded him, is of this nature: If three
straight lines are given in position, and if straight lines be drawn from one and
the same point, making given angles with the three given lines; and if there be
given the ratio of the rectangle contained by two of the lines so drawn to the
square of the other, the point lies on a solid locus given in position, namely, one
of the three conic sections.
"Again, if lines be drawn making given angles with four straight lines given
in position, and if the rectangle of two of the lines so drawn bears a given ratio
to the rectangle of the other two; then, in like manner, the point lies on a conic
section given in position. It has been shown that to only two lines there corre-
sponds a plane locus. But if there be given more than four lines, the point gen-
erates loci not known up to the present time (that is, impossible to determine by
common methods), but merely called 'lines'. It is not clear what they are, or
what their properties. One of them, not the first but the most manifest, has been
examined, and this has proved to be helpful. (Paul Tannery, in the Oeuvres de
Descartes, differs with Descartes in his translation of Pappus. He translates as
follows : Et on n'a fait la synthèse d' aucune de ces lignes, ni montré qu'elle servit
pour ces lieux, pas même pour celle qui semblerait la première et la plus indiquée.)
These, however, are the propositions concerning them.
"If from any point straight lines be drawn making given angles with five
straight lines given in position, and if the solid rectangular parallelepiped contained
by three of the lines so drawn bears a given ratio to the solid rectangular paral-
lelepiped contained by the other two and any given line whatever, the point lies
on a 'line' given in position. Again, if there be six lines, and if the solid con-
tained by three of the lines bears a given ratio to the solid contained by the other
three lines, the point also lies on a 'line' given in position. But if there be more
than six lines, we cannot say whether a ratio of something contained by four
lines is given to that which is contained by the rest, since there is no figure of
more than three dimensions."
21
GEOMETRY
impares vel pares mult itn dine, cum hœc, ut dixi, loco ad quatuor lineas
respondeant, nullum igitur posuerunt ita ut linea nota sit, &c}^*^
The question, then, the solution of which was begun by Euclid and
carried farther by Apollonius, but was completed by no one, is this :
Having three, four or more lines given in position, it is first required
to find a point from which as many other lines may be drawn, each
making a given angle with one of the given lines, so that the rectangle
of two of the lines so drawn shall bear a given ratio to the square of
the third (if there be only three) ; or to the rectangle of the other two
(if there be four), or again, that the parallelepiped''"^ constructed upon
three shall bear a given ratio to that upon the other two and any given
line (if there be five), or to the parallelepiped upon the other three (if
there be six) ; or (if there be seven) that the product obtained by mul-
tiplying four of them together shall bear a given ratio to the product
of the other three, or (if there be eight) that the product of four of
them shall bear a given ratio to the product of the other four. Thus
the question admits of extension to any number of lines.
Then, since there is always an infinite number of different points
satisfying these requirements, it is also required to discover and trace
the curve containing all such points.'^"' Pappus says that when there
are only three or four lines given, this line is one of the three conic
sections, but he does not undertake to determine, describe, or explain
the nature of the line required'"' when the question involves a greater
number of lines. He only adds that the ancients recognized one of
them which they had shown to be useful, and which seemed the sim-
'^^' This rather obscure passage may be translated as follows : "For in this are
agreed those who formerly interpreted these things (that the dimensions of a
figure cannot exceed three) in that they maintain that a figure that is contained by
these lines is not comprehensible in any way. This is permissible, however, both
to say and to demonstrate generally by this kind of proportion, and in this man-
ner : If from any point straight lines be drawn making given angles with straight
lines given in position; and if there be given a ratio compounded of them, that
is the ratio that one of the lines drawn has to one, the second has to a second,
the third to a third, and so on to the given line if there be seven lines, or, if there
be eight lines, of the last to a last, the point lies on the lines that are given in
position. And similarly, whatever may be the odd or even number, since these,
as I have said, correspond in position to the four lines ; therefore they have not
set forth any method so that a line may be known." The meaning of the passage
appears from that which follows in the text.
'^^^ That is, continued product.
'^^^ It is here that the essential feature of the work of Descartes may be said
to begin.
'^'^ See lino 19 on the opposite page.
22
Livre Premier. 3^7
felepipedecoinpofedes deux qufreftcntj&dVne antre
ligncdonnée. Ou s'il y en a fîx, que le parallélépipède
côpofédetroisaitla proportion donnée auec le parafle-
lepipcde des trois autres. Ou s'il y en a fept^que ce qui fe
produid lorfqu'on en multiplie quatre Tvne par l'autre,
aitlaraifon donnée auec ce qui feproduift par [a multi-
plication des trois autres, & encore d'vne autre ligne
donnec; Ou s'il y en a huit, que le produit de la multi-
plication de quatre ait la proportion donne'e auec le-pro-
duit des quatre autres. Et ainfi cete queftiou fe peuc
eftendre a tout autre nombre de lignes. Puis a caufe qu'il
y Tw toufiours vneinfînite'dediuerspoins qui peuucnt fa-
tisfaireacequi eft icy demande, il eft aufly requis de
connoiftre, & de tracer la ligne,dans laquelle ils doiuent
tousfe trouuer. & Pappus dit que lorfqu'il n'y a que
trois ou quatre lignes droites données , c'eft en vne des
trois feétions coniques, mais il n'entreprend point de la
determiiier, nyde la defcrire. non plus que d'expli-
quer celles ou tous ces poins fe doiuent trouuer, lorfquc
laqueftioneftpropofeeenvnplus grand nombre de li-
gnes. Seulement ilaioufte que les anciens en auoient
imagine vne qu'ils monftroient y eftrevtile , mais qui
fembloit la plus manifefte, & qui n'eftoit pas toutefois la
premiere. Ce qui m'a donne' occafion d'effayer fî par la
méthode dont ie me ièrs on peut aller aulTy loin qu'ils
ont efte'.
Et premièrement i'ay connu que cete queftion n'eftant Rcfponfc
propofee qu'en trois, ou quatre,ou cinq lignes , on peutl'^^T
r I 11/ ^ '"on de
toufiours trouuer les poms cherches par la Géométrie Pappus.
fimplci c*eft a dire en ne fe feruant que de la reigle & du
Q,q 2 compas.
23
3©^ JLa Géométrie,
compas, uj ne fmfàm auti;echofe, t|ue ce qui a défia efte.
dit; excepteTeuIement lorfqu'il y a cinq lignes données,
fi elles font toutes parallèles. Auquel cas, comme aufly
lorfquela queftion eft propofee en fix, ou 7, ou 8, ou 9
lignes, on peuttoufiourstrouuer les poins cherchés par
la Géométrie des folides j c'eft a dire en y employar«t
quelqu^vne des trois fediions coniques. Excepte' feule-
ment lorfqu'il y a neuf lignes données, fi elles font toutes
parallèles. Auquel cas derechef, 8c encore en 10,11,12,
ou 13 hgnes on peut trouuer les poins cherchés par le
moyen d'vne hgne courbe qui foit d'vn degré plus cora-
pofée que les fed;ions coniques. Excepte'' en treize fi el-
les font toutes parallèles , auqueîcas , & en quatorze, i y,
16, & 17 il y faudra employer vne ligne courbe encore
d'va degré' plus compofca que la précédente. & ainfî
a l'infini.
Puisiay trouuc'auffy, que lorfqu'il ny a que trois ou
quatre hgnes données, les poins cherchés fe rencontrent
tous , non feulement en l'vnedes trois fedions coni-
ques , mais quelquefois aulTy en la circonférence d'vu
cercle , ou en vne Hgne droite. Et que lorfqu'il y en a
cinq, ou fix, ou fept, ou huit, tous ces poins fe rencon-
trent en quelque vne des lignes, qui font dVn degré plus
corapofées que les fecStions coniques , & il eft impofîîble
d'en imaginer aucune qui ne foit vtile a cete queftioU;
mais ils peuuent aufTy derecheffe rencontrer en vne fe-
(Stion conique, ou en vn cercle, ou en vne ligne droite.
Et s'il y en a neuf, ou i o, ou n , ou 1 2,, ces poins fe ren-
contrent en vne hgne, qui ne peut eftrc que d'vn degr^
plus compofée que les précédentes 3 mais toutes celles
qui
24
FIRST BOOK
plest, and yet was not the most important. '''' This led me to try to find
out whether, by my own method, I could go as far as they had gone.'''"'
First, I discovered that if the question be proposed for only three,
four, or five lines, the required points can be found by elementary
geometry, that is, by the use of the ruler and compasses only, and the
application of those principles that I have already explained, except
in the case of five parallel lines. In this case, and in the cases where
there are six, seven, eight, or nine given lines, the required points can
always be found by means of the geometry of solid loci,''"' that is, by
using some one of the three conic sections. Here, again, there is an
exception in the case of nine parallel lines. For this and the cases of
ten, eleven, twelve, or thirteen given lines, the required points may be
found by means of a curve of degree next higher than that of the conic
sections. Again, the case of thirteen parallel lines must be excluded,
for which, as well as for the cases of fourteen, fifteen, sixteen, and
seventeen lines, a curve of degree next higher than the preceding must
be used ; and so on indefinitely.
Next, I have found that when only three or four lines are given, the
required points lie not only all on one of the conic sections but some-
times on the circumference of a circle or even on a straight line.'"'
When there are five, six, seven, or eight lines, the required points
lie on a curve of degree next higher than the conic sections, and it is
impossible to imagine such a curve that may not satisfy the conditions
of the problem ; but the required points may possibly lie on a conic
section, a circle, or a straight line. If there are nine, ten, eleven, or
twelve lines, the required curve is only one degree higher than the pre-
ceding, but any such curve may meet the requirements, and so on to
infinity.
'"*' See lines 5-10 from the foot of page 23.
'^^' Descartes gives here a brief summary of his solution, which he amplifies
later.
[40] -pj^jg term was commonly applied by mathematicians of the seventeenth cen-
tury to the three conic sections, while the straight line and circle were called plane
loci, and other curves linear loci. See Fermât, Isagoge ad Locos Pianos et Solidos,
Toulouse, 1679.
'"' Degenerate or limiting forms of the conic sections.
25
GEOMETRY
Finally, the first and simplest curve after the conic sections is the
one generated by the intersection of a parabola with a straight line in
a way to be described presently.
I believe that I have in this way completely accomplished what
Pappus tells us the ancients sought to do, and I will try to give the
demonstration in a few words, for I am already wearied by so much
writing.
Let AB, AD, EF, GH, ... be any number of straight lines
given in position,''"' and let it be required to find a point C, from which
straight lines CB, CD, CF, CH, . . . can be drawn, making given angles
CBA, CDA, CFE, CHG, . . . respectively, with the given lines, and
'*'' It should be noted that these lines are given in position but not in length.
They thus become lines of reference or coordinate axes, and accordingly they
play a very important part in the development of analytic geometry. In this con-
nection we may quote as follows: "Among the predecessors of Descartes we
reckon, besides Apollonius, especially Vieta, Oresme, Cavalieri, Roberval, and
Fermât, the last the most distinguished in this field; but nowhere, even by Fermât,
had anv attempt been made to refer several curves of difïerent orders simultane-
ously to one system of coordinates, which at most possessed special significance
for one of the curves. It is exactly this thing which Descartes systematically
accomplished." Karl Fink, A Brief History of Mathematics, trans, by Beman and
Smith, Chicago, 1903, p. 229.
Heath calls attention to the fact that "the essential difference between the
Greek and the modern method is that the Greeks did not direct their efforts to
making the fixed lines of a figure as few as possible, but rather to expressing
their equations between areas in as short and simple a form as possible." For fur-
ther discussion see D. E. Smith, History of Mathematics, Boston, 1923-25, Vol. II,
pp. 316-331 (hereafter referred to as Smith).
26
Livre Premier. ^^^
qui font dVn degré plus compofees y peuuentferuir, &
ainfî a l'infini.
Au refle la premiere, & la plus fimple de toutes après
les fednons coniques , eft celle qu'on peut defcrirepar
i'interfeétiond'vne Parabole, &dVne ligne droite, en la
façon qui fera tantoft explique'e. En forte que ie penfè
auoir entièrement fatisfait a ceque Pappus nous dit auoir
efte'chetché'en cecy par les anciens. & ic tafcheray d en
mettre la demonftration en peu de mots.car il m'ennuie
défia d'en tant efcrire.
Soient A B, A D, E F, G H, &c. plufieurs lignes don-
nées par pofition, &: qu'il faille trouuer vn point, comme
C, duquel ayant tire'd'autres lignes droites fur les don-
nées, comme C B, C D, C F, & C H , en forte que les
anglesCBA,CDA,CFE,CHG,&c.foientdonnds,
Qq 3 &
27
3^^ La Géométrie.
&que ce qui eft produit par la multiplication d' vue par-
tic de ces lignes, foit efgal a ce qui eft produit par la mul-
tiplication des autres, oubien qu'ils ayent quelque autre
proportion donnée, car cela ne rend point la queftion
pius difficile.
Commet Premièrement ie fuppofe la chofe comme defîa faite^
^'ofedes ^-pour me demeller de la côfulion de toutes ces- lignes,
termes ie confidcre l'vne des donne'es, & IVne de celles qu'il
«n VE- fauttrouuer, parexemple A B, & C B , comme lesprin-
quation cipalcs, & aufquclles ie tafche de rapporter ainfi toutes
exemple. Ics autrcs. Quc le fegment de la ligne A B, qui eft entre
les poins A & B, foit nommé x. & que B C foit nomme'
y, & que toutes les autres lignes données foient prolon-
gées, iufques a ce qu'elles couppent ces deux, aufly pro-
longées s'il eft befoin, ôcCi elles ne leur font point paral-
lèles, comme vous voy es icy qu'elles couppent la ligne
A B aux poins A, E, G, & B C aux poins R,S,T. Puis a
caufequetousles angles du triangle A RB font donne''s3
la proportion, qui eft entre les coftés A B, & B R, eft auf-
fy donnée, & ie la pofe comme de ^ à ^, de façon qu' A B
eftant x, R B fer i *:' &: la toute C R fera y -+- ~ ' à caufe
que le poin t B tombe entre C & R^ car fi R tomboit en-
tre C & B,C R feroit ;/---{'& fi C tomboit entre B & R,
CR feroit — ^-i-"7* Tout de mefme les trois angles
du triangle D R C font donnel; , & par confequent aufiy
la proportion qui eft entre les cofte's C R, & C D , que ie
pofe comme de ;^à r; de façon que C R eftant y -^- -*
CD
28
FIRST BOOK
such that the product of certain of them is equal to the product of the
rest, or at least such that these two products shall have a çiven ratio,
for this condition does not make the problem any more difficult.
First, I suppose the thing done, and since so many lines are confus-
ing, I may simplify matters by considering one of "the given lines and
one of those to be drawn (as, for example, AB and BC) as the prin-
cipal lines, to which I shall try to refer all the others. Call the segment
of the line AB between A and B, x^ and call BC, y. Produce all the
other given lines to meet these two (also produced if necessary) pro-
vided none is parallel to either of the principal lines. Thus, in the
figure, the given lines cut AB in the points A, E, G, and cut BC in the
points R, S, T.
Now, since all the angles of the triangle ARB are known,'"' the ratio
between the sides AB and BR is known.'"' If we let AB :BR = r :b,
since AB = x, we have RB = — ; and since B lies between C and R '"',
z
/>x
we have CR^v + -— • (When R lies between C and B, CR is equal
to y — —, and when C lies between B and R, CR is equal to — y + — )
Again, the three angles of the triangle DRC are known,'*"' and there-
fore the ratio between the sides CR and CD is determined. Calling this
ratio z : c, smce CR = y -{--;:> we have CD = " -f- ^:;^- i hen, smce
'"' Since BC cuts AB and AD under given angles.
'^' Since the ratio of the sines of the opposite angles is known.
'"' In this particular figure, of course.
'*"' Since CB and CD cut AD under given angles.
29
GEOMETRY
the lines AB, AD, and EF are given in position, the distance from A
to E is known. If we call this distance k, then EB = A- -f- x ; although
EB = fe — X when B lies between E and A, and E=^- — k -{- x when E
lies between A and B. Now the angles of the triangle ESB being
given, the ratio of BE to BS is known. We may call this ratio a : d.
Then BS = '^^^ + ^^' and CS = ^-L+^''^i±_^^'.i-] ^j^^^ g y^^^ between B
G 2
and C we have CS = , and when C lies between B and S
z
we have CS = ~ — — — . The angles of the triangle ESC are
known, and hence, also the ratio of CS to CF, or s : e. Therefore,
ezy -^ de/; -\- i/fx t -i • \ r- i • • ^ T^r^ i
LP = -^ — . Likewise, AG or / is given, and B(j = / — x.
Also, in triangle BGT, the ratio of BG to BT, or ,z : f, is known. There-
fore, BT =-^^ ~-^'^" and CT = ^-'' "^-^'^~^\ In triangle TCH, the ratio
z z
of TC to CH, or z : g, is known,'''' whence CH ^ '^^^ ^ ^^- .
i^'i We have
, dk-\-dx
= y + ~
^y-\-dk^dx
and similarly for the other cases considered below.
The translation covers the first eight lines on the original page 312 (page 32
of this edition.
'"' It should be noted that each ratio assumed has ^ as antecedent.
30
5^x
CD fera t^
hex
-. Apres cela pourceque les lignes A F,
A D, &: E F font données par pofition, la diftance qui eft
entre les poins A & E eft au fTy donnée, & fi onlanom-
me K, on aura E B efgal a k^ -{- x-^ mais ce feroit /^— x , fi
le point B tomboit entre E & A;& -- >^-f- .r^fi E tomboit
entre A &B. Et pourceque les angles du triangle ESB
font tous donnés, la proportion de BE a BS eftaufly
donnée, & ie la pofe comme :^à^ , fibienque BS eft
dk>i< dx „ , ^ ^ f^ zy 'i* dk <i>d x
& la toute C S eft
mais ce feroic
\y •- dk -- dx
file point s tomboit entre B &C5& ce feroic
■ - z.y >i* d k 'i* dx
K.
, fi C tomboit entre B^ & S. De plus les
trois angles du triangle F S Cfont donne's, 6c en fuite îa
pro-
31
^^* La Géométrie.
proportion de C S à C F, qui foie comme de ^kc, 5c1â
toute C F fera ^^ . En meime taçon AG
que ie nomme /eft donnée, &B G eft /-- x\ & acaufe
dutriangleBGTlaproportion de BG la BTefraufîy
fl'-fx
donnée, quifoit comme de :^ à /! &B Tfera — ^ ,&
C T co ^•'^'^{"^ . Puis derechef la proportion de T C a
C H eft donnée , acaufe du triangle T C H , & lapofant
comme de^agy on aura C H 30 — .
EtainfivousvoyeX qu'en tel nombre de lignes don-
nées par pofition qu'on puifîeauoir, toutes les lignes ti-
rées defTus du point C a angles donne's fuiuant la teneur
delaqueftion ,fepeuuent toujours exprimer chafcune
par trois termes j dont l'vn eft compofe'de la quantité in-
connue j', multipliée , ou diuifce par quelque autre
connue^ & l'autre de la quantité' inconnue x, aufly mul-
tiplie'e ou diuifce par quelque autre connue , & le trolîel^
me d'vne quantité toute connue. Excepte feulement lî
elles fontparalleles joubien a la ligne AB, auquel cas le
terme compofe de la quantité AT fera nul ; oubien a la li-
gne C B, auquel cas celuy qui eft compofe'de la quantité"
y fera nulj ainfi qu'il eft trop manifeftc pour que ie m are-
fte a l'expliquer. Et pour les fignes 4-, &: -, qui fe ioi-
gnent à ces termes, ilspeuuent eftre changes en toutes
les façons imaginables.
Puis vous voyés aufly, que multipliant plufîeurs de
ces lignes l'vne par l'autre, les quantités x3cy, qui fe
trouuent dans le produit, n'y peuuentauoir que chafcu-
ne autant de dimenfions, qu'il y a eu deligues, al'expli-
cation
32
FIRST BOOK
And thus you see that, no matter how many Hues are given in posi-
tion, the length of any such hne through C making given angles with
these lines can always be expressed by three terms, one of which coh-
sists of the unknown quantity y multiplied or divided by some known
quantity ; another consisting of the unknown quantity .r multiplied or
divided by some other known quantity ; and the third consisting of a
known quantity.''"' An exception must be made in the case where the
given lines are parallel either to AB (when the term containing .r van-
ishes), or to CB (when the term containing 3' vanishes). This case is
too simple to require further explanation. '°"' The signs of the terms
may be either + or — in every conceivable combination.''''
You also see that in the product of any number of these lines the
degree of any term containing x or y will not be greater than the num-
ber of lines (expressed by means of .r and y) whose product is found.
Thus, no term will be of degree higher than the second if two lines
be multiplied together, nor of degree higher than the third, if there be
three lines, and so on to infinity.
'^"^ That is, an expression of the form ax + by + c, where a, b, c, are any real
positive or negative quantities, integral or fractional (not zero, since this exception
is considered later).
[50] Yj-jg following problem will serve as a very simple illustration : Given three
parallel lines AB, CD, EF, so placed that AB is distant 4 units from CD, and CD
is distant 3 units from EF ; required to find a point P such that if PL, PM, PN
be drawn through P, making angles of 90°, 45°, 30°, respectively, with the
parallels. Then PM-= PL.PN.
Let PR = y, then PN = 2y, PM = V2 ( v + 3) , PL = j + 7. If PM " = PN . PL,
we have V^^ i' + >^) | = 2v ( J + 7) , whence :v = 9. Therefore, the point P lies on
the line XY parallel to EF and at a distance of 9 units from it. Cf. Rabuel, p. 79.
'°'' Depending, of course, upon the relative positions of the given lines.
2>l
GEOMETRY
Furthermore, to determine the point C, but one condition is needed,
namely, that the product of a certain number of hues shall be equal to,
or (what is quite as simple), shall bear a given ratio to the product of
certain other lines. Since this condition can be expressed by a single
equation in two unknown quantities,'"'' we may give any value we please
to either .v or y and find the value of the other from this equation. It
is obvious that when not more than five lines are given, the quantity x,
which is not used to express the first of the lines can never be of degree
higher than the second.'"^'
Assigning a value to 3', we have x- =-^ ± a.v ±: h-, and therefore x
can be found with ruler and compasses, by a method already explained.'"'
If then we should take successively an infinite number of different
values for the line y, we should obtain an infinite number of values for
the line .r, and therefore an infinity of different points, such as C, by
means of which the required curve could be drawn.
This method can be used when the problem concerns six or more
lines, if some of them are parallel to either AB or BC, in which case
'""' That is, an indeterminate equation. "De plus, à cause que pour determiner
le point C, il n'y a qu'une seule condition qui soit requise, à sçavoir que ce qui est
produit par la multiplication d'un certain nombre de ces lignes soit égal, ou (ce qui
n'est de rien plus mal-aisé) ait la proportion donnée, à ce qui est produit par la
multiplication des autres ; on peut prendre à discretion l'une des deux quantitez
inconnues x ou y, & chercher l'autre par cette Equation." Such variations in the
texts of different editions are of no moment, but are occasionally introduced as
matters of interest.
''^^' Since the product of three lines bears a given ratio to the product of two
others and a given line, no term can be of higher degree than the third, and there-
fore, than the second in x.
'^^' See pages 13, et seq.
34
Livre Premier. 5^5 .
cation defquelles elles feruent , qui ont elle'' ainfî multi-
pliées: enforce qu'elles n'auront iaraais plus de deux dî-
menfious, en ce qui ne fera produit que par la multipli-
cation de deux lignes; ny plus de trois , en ce qui ne fera
produit que par la multiplication de trois , & ainfi a l'in-
fini .
De plus, a caufe que pour determiner le point C, il o^^^ou^c
n'ya qu'vne feule condition qui foitrequife , à fçauoir que ce
que ce qui eft produit par la multiplication d'vn certain ^^°^^ '
nombre de ces lignes foit efgal , ou Ccequi n eft de rien plan lorji
plus malayfe] ait la proportion donnée , à ce qui eft pro- "^l-'^^ ^
duit par la multiplication des autres; on peut prendre api^opofé
*"^ , 1 1 . , . - en plus de
difcretion T vne des deux quantités mconnues x ou y , & j lignes.
chercher l'autre par cete, Equation, en laquelle il eft eui-
dent que lorfque la queftion n eft point propofee en plus
decinqlignes, la quantité a: qui ne ferc point a Icxpref-
(îon de la premiere peut toufîours n'y auoir que deux di-
menfious. de façon que prenant vne quantité connue
pourjy, il ne reftera que xxyi-hou-- ax-{- ou — bb, &c
ainfî on pourra trouuer la quantité x auec la reigle &le
compas, en la façon tantoft explique'e. Mefme prenant
faccelîîuement infinies diuerfes grandeurs pour la ligne
y y onentrouneraauffyiniSnies pourlahgne Ar,&ain{ion
auravncinfiniteMediuerspoins , tels que celuy qui eft
marqué C , par le moyen defquels on defcrira la ligne
courbe demandée.
11 fe peut faire aufTy, la queftion eftant propofe^e en fîx,
ou plus grand nombre de lignes^ s'il y en a entre les don-
nées, qui foient parallèles a B A, ou B C , quel'vne des
deux quantités x ou y n'ait que deux dimenfîons en
Rr TEqua-
35
^14 i-A GEOMETRIE^
TEquation, Se ainfî qu'on puifTe trouuuer le point C aaec
lareigle &: le compas. Mais au contraire fi elles font tou-
tes parallèles , encore que la queftion ne foit propofee
qu'en cinq lignes, ce point C ne pourra ainfi eftre trou-
ue', a caufe que la quantité x ne fe trouuant point en tou-
te rEquation,il ne fera plus permis de prendre vne quan-
tité connue pour celle qui eft nommeej' , mais ce fera
elle qu'il faudra chercher. Et pource quelle aura trois di-
menfions,on nelapourra trouuer qu'en tirant la racine
dVn€ Equation cubique, cequi ne fe peut généralement
faire fans qu'on y employe pour le moins vne fedion co-
nique. Et encore qu'il y ait iufques a neuf lignes don-
nées,pourvûqu'elles ne foient point toutes parallèles, oiî
peut toufiours faire que l'Equation ne monte que iufques
auquarrédequarré. au moyen dequoy on lapeutauffy
toufiours refoudre par les fedtions coniques, en la façon
que i'expliqueraycy après. Et encore qu'il y en ait iuf^
ques a treize , on peut toufiours faire qu'elle ne nionte
que iufques au quarré de cube, en fuite de quoy on la
peut refoudre par le moyen d'vne ligne , qui n'eft que
d'vn degré' plus compofée que les feétions coniques, en
la façon que i'exphquerayauflycy après. Et cecy eft la
premiere partie de cequei'auoisicyademonftrer^ mais
auant que ie pafi^e a la féconde il eft befoin que ie- die
quelque chofe en general delà nature des lignes cour-
bes.
LA
36
FIRST BOOK
either x or y will be of only the second degree in the equation, so that
the point C can be found with ruler and compasses.
On the other hand, if the given lines are all parallel even though a
question should be proposed involving only five lines, the point C can-
not be found in this way. For, since the quantity x does not occur at
all in the equation, it is no longer allowable to give a knowni value to y.
It is then necessary to find the value of 3'.'^'^ And since the term in y
will now be of the third degree, its value can be found only by finding
the root of a cubic equation, which cannot in general be done without
the use of one of the conic sections.'^"'
And furthermore, if not more than nine lines are given, not all of
them being parallel, the equation can always be so expressed as to be
of degree not higher than the fourth. Such equations can always be
solved by means of the conic sections in a way that I shall presently
explain.'"'
Again, if there are not more than thirteen lines, an equation of degree
not higher than the sixth can be employed, which admits of solution by
means of a curve just one degree higher than the conic sections by a
method to be explained presently.'^*'
This completes the first part of what I have to demonstrate here, but
it is necessary, before passing to the second part, to make some general
statements concerning the nature of curved lines.
''"''' That is, to solve the equation for y.
''"' See page 84.
i="i See page 107.
^^^ This line of reasoning may be extended indefinitely. Briefly, it means that
for every two lines introduced the equation becomes one degree higher and the
curve becomes correspondingly more complex.
37
BOOK SECOND
Geometry
BOOK II
On the Nature of Curved Lines
THE ancients were familiar with the fact that the problems of geom-
etry may be divided into three classes, namely, plane, solid, and linear
problems.'^"' This is equivalent to saying that some problems require
only circles and straight lines for their construction, while others
require a conic section and still others require more complex curves.'*''
I am surprised, however, that they did not go further, and distinguish
between different degrees of these more complex curves, nor do I see
why they called the latter mechanical, rather than geometrical.'"'
If we say that they are called mechanical because some sort of instru-
ment'"*' has to be used to describe them, then we must, to be consistent,
[59] (-£ Pappus, Vol. I, p. 55, Proposition 5, Book Til : "The ancients consid-
ered three classes of geometric problems, which they called plane, solid, and linear.
Those which can be solved by means of straight lines and circumferences of circles
are called plane problems, since the lines or curves by which they are solved have
their origin in a plane. But problems whose solutions are obtained by the use of
one or more of the conic sections are called solid problems, for the surfaces of solid
figures (conical surfaces) have to be used. There remains a third class which is
called linear because other 'lines' than those I have just described, having diverse
and more involved origins, are required for their construction. Such lines are the
spirals, the quadratrix, the conchoid, and the cissoid, all of which have many impor-
tant properties." See also Pappus, Vol. I, p. 271.
'""^ Rabuel (p. 92) suggests dividing problems into classes, the first class to
include all problems that can be constructed by means of straight lines, that is,
curves whose equations are of the first degree ; the second, those that require curves
whose equations are of the second degree, namely, the circle and the conic sec-
tions, and so on.
'"^ Cf. Encyclopedic on Dictionnaire Raisonne des Sciences, des Arts et des
Metiers, par une Société de gens de lettres, mis en ordre et publiées par M . Diderot,
et quant à la Partie Mathématique par M. d'Alcmbert, Lausanne and Berne," 1780.
In substance as follows : "Mechanical is a mathematical term designating a con-
struction not geometric, that is, that cannot be accomplished by geometric curves.
Such are constructions depending upon the quadrature of the circle.
The term, mechanical curve, was used by Descartes to designate a curve that
cannot be expressed by an algebraic equation." Leibniz and others call them
transcendental.
1"'^ "Machine."
40
Geome-
tric.
Livre Secokd. Sif
GEOMETRIE.
LIVRE SECOND.
^e la nature des lignes courhes,
T E s anciens ont fore bien remarque , qu'entre les
-■— 'Problefmes de Géométrie, les vns font plans , les au- Quelles
tresfolidesj&lesautreslineaircs, c'eil adire^queles vns ["J^^^f
peuuenteftreconflruits, eu ne traçant que des lignes courbes
droites, &:descerclesjau lieu que les autres ne le peu- peuTV
uent eftre, qu'on n'y employe pour le moins quelque fe- ^^uoir en
d:ion conique, ni enfin les autres , qu on n'y employe '^""^"
quelque autre ligne plus compofee. Mais ie m'eftonne
de ce qu'ils n'ont point outre cela difliugué diuers de-
grees entre ces lignes plus compofées, & ie ne fçaurois
comprendre pourquoy ils les ont nommées mecl^ni-
ques, plutoft que Géométriques. Carde dire que c'ait
efte'', a caufe qu'il efV befoin de fe fèruir de quelque ma-
chine pour les defcrire, il faudroit reietter par melrne
raifon les cercles & les lignes droitesjvû qu'on ne les de-
fcrit fur le papier qu'auec vn compas, & vne reigle, qu'on
peut auffy nommer des machines. Ce n'eft pas non plus,
a caufe que les inftrumens, quiferuent a les tracer^eftanc
plus compofe's que la reigle & le compas , ne peuueut
eftre fî iuftes; car il Eiudroit pour cete raifon les reietter
des Mechaniques, où la iultelTe des ouurages qui fortent
delamaineftdefirec; plutoft que de la Géométrie , ou
c'cft feulement la iufteile du raifonnemct qu'on recher-
Rr 2 che,
41
3'<^ La Géométrie.
che, & qui peut fans doute eftre^ufly parfaite touchant
CCS lignes , que touchant les autres. le ne diray pas aufly,
que ce foit a caufe qu'ils n*ont pas voulu augmenter le
nombre de leurs demandes , & qu'ils fe fontcontentés
qu'on leur accordaft , qu*ils puflent ioindre deux poins
donnés par vne ligne droite , & defcrire vn cercle d'wn
centre donne, qui pafîaft par vn point donne.carils n'ont
point fait de fcrupule de fuppofer outjr e ceIa,pour traiter
des fedîions coniques , qu*on puft coupper tout cône
donnd'parvn plan donne. &iln*eft befoin de rien fup-
pofer pour tracer toutes les lignes courbes , que ie pre-
tens icy d'introduire; finon que deux ou plulîeurs lignes
pniflent eftre meues IVne par l'autre , & que leurs inter-
férions éo marquent d'autres ^; ce qui ne me paroift en
rien plus difficile. Il eft vray qu'ils n ont pas aufly entiè-
rement receu les fed:ions coniques en leur Géométrie,
& ie ne veux pas entreprendre de changer les noms qui
ont efte^approaue's par Ivfàge; mais il eft, ce me fèmble,
très clair, que prenant comme on fait pour Géométri-
que ce qui eft precis & exad: , & pour Mechanique
ce qui ne Teft pas ; & confiderant la Géométrie comme
vne fcience, qui enfeigne généralement a connoiftre les
mefures de tous les cors, on n'en doit pas plutoft exclure
les lignes les plus com pofees que les plus limples, pourvu
qu'on les puiflc imaginer eftre defcrites par vn mouue-
ment continu, ou par plufieurs qui s'entrcfuiuent & dont
les derniers foient entièrement règles par ceu:: qui les
precedent, car par ce moyen on peut toufîotirs auoir
vue connoiftance exaéte de leur mefure. Mais peuteftre
que ce qui a empefche' les anciens Géomètres de reçe-
uou:
42
SECOND BOOK
reject circles and straight lines, since these cannot be described on
paper without the use of compasses and a ruler, which may also be
termed instruments. It is not because the other instruments, being
more complicated than the ruler and compasses, are therefore less
accurate, for if this were so they would have to be excluded from
mechanics, in which accuracy of construction is even more important
than in geometry. In the latter, exactness of reasoning alone'""' is
sought, and this can surely be as thorough with reference to such lines
as to simpler ones.'"' I cannot believe, either, that it was because they
did not wish to make more than two postulates, namely, (1) a straight
line can be drawn between any two points, and (2) about a given center
a circle can be described passing through a given point. In their treat-
ment of the conic sections they did not hesitate to introduce the assump-
tion that any given cone can be cut by a given plane. Now to treat all
the curves which I mean to introduce here, only one additional assump-
tion is necessary, namely, two or more lines can be moved, one upon
the other, determining by their intersection other curves. This seems
to me in no way more difficult. '°^'
It is true that the conic sections were never freely received into
ancient geometry, '°°' and I do not care to undertake to change names
confirmed by usage ; nevertheless, it seems very clear to me that if we
make the usual assumption that geometry is precise and exact, while
mechanics is not f^ and if we think of geometry as the science which
furnishes a general knowledge of the measurement of all bodies, then
we have no more right to exclude the more complex curves than the
simpler ones, provided they can be conceived of as described by a con-
tinuous motion or by several successive motions, each motion being
completely determined by those which precede ; for in this way an exact
knowledge of the magnitude of each is always obtainable.
'"'^ An interesting question of modern education is here raised, namely, to what
extent we should insist upon accuracy of construction even in elementary geometry.
'"' Not only ancient writers but later ones, up to the time of Descartes, made
the same distinction ; for example, Vieta. Descartes's view has been universally
accepted since his time.
'"^' That is, in no way less obvious than the other postulates.
'*°' Because the ancients did not believe that the so-called constructions of the
conic sections on a plane surface could be exact.
'"' Since it is not possible to construct an ideal line, plane, and so on.
43
GEOMETRY
Probably the real explanation of the refusal of ancient geometers to
accept curves more complex than the conic sections lies in the fact that
the first curves to which their attention was attracted happened to be
the spiral, '""' the quadratrix,'"*' and similar curves, which really do
belong only to mechanics, and are not among those curves that I think
should be included here, since they must be conceived of as described
by two separate movements whose relation does not admit of exact
determination. Yet they afterwards examined the conchoid/"" the
cissoid/''' and a few others which should be accepted; but not knowing
much about their properties they took no more account of these than
of the others. Again, it may have been that, knowing as they did only
a little about the conic sections,'"' and being still ignorant of many of
the possibilities of the ruler and compasses, they dared not yet attack
a matter of still greater difficulty. I hope that hereafter those who are
clever enough to make use of the geometric methods herein suggested
will find no great difficulty in applying them to plane or solid problems.
I therefore think it proper to suggest to such a more extended line of
investigation which will furnish abundant opportunities for practice.
Consider the lines AB. AD, AF, and so forth (page 46), which we
may suppose to be described by means of the instrument YZ. This
instrument consists of several rulers hinged together in such a way that
YZ being placed along the line AN the angle XYZ can be increased or
decreased in size, and when its sides are together the points B, C, D,
E, F, G, H, all coincide with A ; but as the size of the angle is increased,
'"^' See Heath, History of Greek Mathematics (hereafter referred to as Heath),
Cambridge, 2 vols., 1921. Also Cantor, Vorlesungen ilber Geschichte der Mathe-
niatik, Leipzig-, Vol. I (2), o. 263, and Vol. H (1), pp. 765 and 781 (hereafter
referred to as Cantor).
'«°i See Heath, I, 225 ; Smith, Vol. H, pp. 300, 305.
'™i See Heath, I, 235, 238 ; Smith, Vol. H, p. 298.
"'' See Heath, I, 264; Smith, Vol. U, p. 314.
'"^ They really knew much more than would be inferred from this statement.
In this connection, see Taylor, Ancient and Modern Geometry of Conies, Cam-
bridge, 1881.
44
Li vre Second. ^^^
uoir celles qui eftoient plus compofees que lesfedions
coniques, c eft que les premieres qu'ils ont confiderees,
ayant par hafard efte la Spirale, la Quadratrice , & fein-
blables , qui n'appartienent véritablement qu'aux Me-
chaniquesj&nefont point du nombre de celles que ie
penfe deuoir icy eftre receues, a caufe qu'on les imagine
defcrites par deux mouuemens fepares, & qui n*ont en-
tre eux aucun raport qu'on puifTe raefurer exadtement,
bienqu'ils ayent après examiné la Conchoide , la Ciflbi-
de, & quelque peu d'autres qui en font, toutefois a cau-
fe qu'ils n'ont peuteftre pas afles remarqué leurs pro-
priete's , ils n'en ont pas fait plus d'eftat que des premie-
res. Oubien c'eft que voyant , qu'ils ne connoiffoient
encore , que peu de chofes touchant les ferions coni-
ques, &qu 'illeur enreftoitmefme beaucoup, touchant
ce qui fe peut faire auec la reigle & le compas , qu'ils
ignoroient, ils ont creu ne deuoir point entamer de ma-
tière plus difficile. Mais pourceque i'efpere que d'orena-
uant ceux qui auront Tadreffe de fe feruir du calculGeo-
metriqueicy propofe'', netrouueront pas aire's dequoy
s'arefter touchant les problefmes plans, ou folidesj ie
croy qu'il eft a propos que ie lesinuite a d'autres re-
cherches , où ils ne manqueront iaraais d'exercice.
Voyesleslignes AB,A D, A F, & ferablables queie
fuppofe anoir efté defcrites par l'ayde de l'inftrumenc
Y Z, qui eft compofé de plufîeurs reigles tellement ioin-
tes, que celle qui eft marquee YZ eftant areftée fur la
ligne A N,on peut ouurir & fermer l'angle X Y Z; & que
lorfqu'ileft tout fermé , les poins B, C, D, F, G, H font
tous aflemblés au point A ; mais qu'a mefure qu'on
Rr 5 l'oaure.
45
5IS
La Geometrte.
Tomire, la reigle B C, qui eft iointe a angles droits auec
XYau point B, poufTevers Z la reigle CD, qui coule
fiirY Zenfaifant toufiours des angles droits auec elle, 82
C D poufle D H, qui coule tout de mefme fur Y X en de-
meurant parallèle a B Q D E poufTe EF,E F poufTe F G,
cellecy poufTe G H. & on en peut conceuoir vne infinite
d'autres*, qui fe pouflent confequutiuement en mefme
façon, & dont les vnesfacent toufiours les mefmes an-
gles auec Y X, & les autres auec Y Z. Or pendant qu'on
ouureainfi l'angle XYZ,le point B dcfcritlaligne AB,
qui eft vn cercle, &les autres poins D^F, H, ou fe font
les interfe(5tions des autres reigles , defcriuent d'autres
lignes courbes AD, A F, A H, dont les dernières font
par ordre plus copofc'es que la premiere, & cellecy plus
que le cercle, mais ie ne voy pas ce qui peut empefcher,
qu'on ne concoiueauffy nettement j & auflTy diftindte-
ment la defcripcion de cete premiere^que du cercle , ou
du
46
SECOND BOOK
the ruler BC, fastened at right angles to XY at the point B, pushes
toward Z the ruler CD which slides along YZ always at right angles.
In like manner, CD pushes DE which slides along YX always parallel
to BC ; DE pushes EF ; EF pushes EG ; EG pushes GH, and so on.
Thus we may imagine an infinity of rulers, each pushing another, half
of them making equal angles with YX and the rest with YZ.
Now as the angle XYZ is increased the point B describes the curve
AB, which is a circle ; while the intersections of the other rulers,
namely, the points D, E, H describe other curves, AD, AE, AH, of
which the latter are more complex than the first and this more complex
than the circle. Nevertheless I see no reason why the description of
the first'"^ cannot be conceived as clearly and distinctly as that of the
circle, or at least as that of the conic sections ; or why that of the sec-
ond, third,'''' or any other that can be thus described, cannot be as
clearly conceived of as the first; and therefore I see no reason why
they should not be used in the same way in the solution of geometric
problems.™
'"' That is, AD.
"*i That is, AF and AH.
'^^' The equations of these curves may be obtained as follows: (1) Let
YA = YB = a, YC = .r, CD —y, YD = ^; then z : x = x : a, whence s = — •
Also s-==x- + y-; therefore the equation of AD is x* = a"(x- + y-). (2) Let
YA = YB = a, YE = x, EF = v, YF = :r. Then z : x = x : YD, whence
YD = ^. Also
.r : YD = YD : YC, whence YC == '— -^ x = — . •
z- z .
But YD : YC = YC : a, and therefore
4i
Also, z' = x- + y^. Thus we get, as the equation of AF,
'd! = X- + y-, or x^ = a- (x- + y- ) \
(3) In the same way, it can be shown that the equation of AH is
.r'- = a"(x- + y-)^.
See Rabuel, p. 107.
47
GEOMETRY
I could give here several other ways of tracing and conceiving a
series of curved lines, each curve more complex than any preceding
one,™ but I think the best way to group together all such curves and
then classify them in order, is by recognizing the fact that all points of
those curves which we may call "geometric." that is, those which admit
of precise and exact measurement, must bear a definite relation'"' to
all points of a straight line, and that this relation must be expressed by
means of a single equation.''"' If this equation contains no term of
higher degree than the rectangle of two unknown quantities, or the
square of one, the curve belongs to the first and simplest class,''"' which
contains only the circle, the parabola, the hyperbola, and the ellipse ;
but when the equation contains one or more terms of the third or fourth
degree'*"' in one or both of the two unknown quantities'"'' (for it
requires two unknown quantities to express the relation between two
points) the curve belongs to the second class ; and if the equation con-
tains a term of the fifth or sixth degree in either or both of the unknown
quantities the curve belongs to the third class, and so on indefinitely.
[78] "Qui seroient de plus en plus composées par degrez à l'infini." The French
quotations in the footnotes show a few variants in style in different editions.
'"' That is, a relation exactly known, as, for example, that between two straight
lines in distinction to that between a straight line and a curve, unless the length
of the curve is known.
'™' It will be recognized at once that this statement contains the fundamental
concept of analytic geometry.
''"' "Du premier & plus simple genre," an expression not now recognized. As
now understood, the order or degree of a plane curve is the greatest number of
points in which it can be cut by any arbitrary line, while the class is the greatest
number of tangents that can be drawn to it from any arbitrary point in the plane.
'*"' Grouped together because an equation of the fourth degree can always be
transformed into one of the third degree.
'"' Thus Descartes includes such terms as .r-^', .v-.v-, . . as well as x^^, y*
48
Livre Second. 519
du moms que des fedtions coniques- ny ce qui peut em-
pefcher, qu'on ne concoiue la féconde , & la troifiefme,
& toutes les autres, qu'on peut defcrire, aufTy bien que
lapremi&re; ny par consequent qu'on ne les recoiue
toutes en mefme façon, pour feruir aux fpeculations de
Géométrie.
le pourrois mettre icy plufieurs autres moyens pour La ùcoa
tracer &conçeuoir des liraes courbes, qui feroient <Je "^^ "^'^^"^"
plus en plus compolées par degrés a 1 infini, mais pour tes les li-
comprendreenfèmble toutes celles, qui font en la natu- ^""'^^^"'^'
re , & les diftiuguer par ordre en certains genres j ie ne certains
fçache rien de meilleur que de dire que tous les poins, de ^Tcol'. "^
celles qu'on peut nommer Géométriques, c'eft a dirc"°'^^^ ^^
qui tombent fous quelque meflire precife ôc exad:e, ont wont
necefTairement quelque rapport a tous les poins dVne-^°"^ '^""^
1- j- • n • / polos a
hgne droite, qui peut eirre exprime par quelque equa-^euxdes
tion, en tous par vnemefme. Et que lorfque ceteequa^ Jj^SJ'.^^g
tion ne monte que iufques au recftangle de deux quanti-
tés indéterminées, oubien au quarréd'vnemefine, la li-
gne courbe eft du premier & plus fîmpie genre, dans le-
quel il ny a que lé cercle, la parabole, l'hyperbole , &
TEllipfe qui foient comprifes. mais que lorfque l'équa-
tion monte^iufques a la trois ou quatriefme dimenfion
des deux, ou de Tvne des deux quantite^s indéterminées,
car il en faut deux pour expliquer icy le rapport d\n
point a vn autre, elle eft du fecondrSc que lorfque l'équa-
tion monte iufques a la y ou fixiefme dimenfion, elle-
eft du troifiefme; & ainli des autres a l'infini.
Comme fi ie veux fçauoir de quel genre eft la ligne
E C;, que l'imagine eftre defcrite par i'interfedion de la-
reigîe-
49
320
La GEOMETRIE.
reigle G L, & du plan rediligne G N K L, dont le cofté
K N eft indefiniement prolongé vers G , & qui eftant
meu fur le plan de deflbus en ligne droite , c'eft a dire en
telle forte que fon diamètre. KL fe trouue toufîours ap-
pliqueTur quelque endroit de la ligne B A prolongée; de
part & d'autre, fait mouuoir circulairement cete reigle
G L autour du point G, a caufe quelle luy eft tellement
iointe quelle pafle toufîours par le point L. le choiiîs
vne ligne droite, comme A B,pour rapporter a fes diuers
poinstousceuxdecetelignecourbeEG, &en cete li-
gne A B ie choifis vn point, comme A, pour commencer
par luy ce calcul. le dis que ie choifis &rvn& l'autre, a
caufe qu'il eft libre de les prendre tels qu'on veult. car
encore qu il y ait beaucoup de choix pour rendre l'équa-
tion plus courte, &: plus ayfécj toutefois en quelle façon
qu'ouïes prene, on peut toufîours faire que la ligne pa-
roiflè de meûne genre, ainfî qu'il eft ayfe^ a demonftrer.
Apres
50
SECOND BOOK
Suppose the curve EC to be described by the intersection of
the ruler GL and the rectihnear plane figure CNKL, whose side
KN is produced indefinitely in the direction of C, and which, being
moved in the same plane in such a w^ay that its side'^'' KL always coin-
cides with some part of the line BA (produced in both directions),
imparts to the ruler GL a rotary motion about G (the ruler being
hinged to the figure CNKL at L)."" If I wish to find out to what
class this curve belongs, I choose a straight line, as AB, to which to
refer all its points, and in AB I choose a point A at which to begin the
investigation.'"' I say "choose this and that," because we are free to
choose what we will, for, while it is necessary to use care in the choice
in order to make the equation as short and simple as possible, yet no
matter what line I should take instead of AB the curve would always
prove to be of the same class, a fact easily demonstrated.''"'
^^^ "Diamètre."
^*^^ The instrument thus consists of three parts, (1) a ruler AK of indefinite
length, fixed in a plane ; (2) a ruler GL, also of indefinite length, fastened to a
pivot, G, in the same plane, but not on AK; and (3) a rectilinear figure BKC, the
side KG being indefinitely long, to which the. ruler GL is hinged at L, and which
is made to slide along the ruler GL.
'*^^ That is, Descartes uses the point A as origin, and the line AB as axis of
abscissas. He uses parallel ordinates, but does not draw the axis of ordinates.
'*^' That is, the nature of a curve is not affected by a transformation of
coordinates.
51
GEOMETRY
Then I take on the curve an arbitrary point, as C, at which we will
suppose the instrument applied to describe the curve. Then I draw
through C the line CB parallel to GA. Since CB and BA are unknown
and indeterminate quantities, I shall call one of them y and the other x.
To the relation between these quantities I must consider also the known
quantities which determine the description of the curve, as GA, which
I shall call a ; KL, which I shall call h ; and NL parallel to GA, which
I shall call c. Then I say that as NL is to LK, or as c is to h, so CB, or
y, is to BK, which is therefore equal to - y. Then BL is equal to
- y — h, and AL is equal to x -\- -y — h. Moreover, as CB 13 to LB,
b . l> , .
that is, as -v is to - T — h, so AG or a is to LA or x -\- - y — h. Multi-
ah
plying the second by the third, we get — y — ah equal to
b , ,
xy^- y — by,
which is obtained by multiplying the first by the last. Therefore, the
required equation is
ex
y '= cy 7- 3' + ^v — <3^.
52
Livre Second.
321
A près cela prenant vn point a difcretion dans la courbe,
comme C, fur lequel ie fuppofe que l'inflrument qui ferc
a la defcrire eft applique', ie tire de ce point C- la ligne
C B parallèle a G A, &:pourceque C B & B A font deux
quantités indéterminées & inconnues , ie les nomme
Tvne^ & l'autre a;, maisaffin de trouucr le rapport de
IVneàrautrcjieconfidere aufTy les quantités connues
qui déterminent la defcription de ccte ligne courbe,
comme G A que ie nomme ^, K L que ie nomme b , &
N L parallele'a G A que ie nofnme c. puis ie dis^ comme
NLeftàLK,oucà/^,ainriCB,ou;^, eftàBK, qui eft
^ b b
parconfequent-;;': ôcBLeft— y-b, &c A Left a: -H
b h
~y — b, de plus comme C B eft à L B, ou j/ à -jy-b, ainfî
a^ ou G A, eft a L A, ou a: -^ -^y -b, de façon que mul-
S f tipliant
S3
J^ La Géométrie.
tip liant la féconde par la troifrefme on produit 77 - ai^
qui eft efgale à xy-h^^yy - by qui fe produit en multi-
pliant la premiere par la dernière. & âinfî Tequation qu'il
faUoittrouuereft .
y y 30 cy- ^y -h ay - ae.
de laquelle onconnoift que la ligne EC eft da premier
genre , comme en effedl elle n eft autre qu vne Hy-
perbole.
Que fî en Tinftrument qui fèrt a la defcrire on fait
qu'au lieu de la ligne droite C N K, ce fdit cete Hyper-
bole, ou quelque autre ligne courbe du premier genre,
qui termine le plan C NKL; Tinterfedtion de cetc ligne
& de la reigle G L defcrira, au lieu de l'Hyperbole E C,
vne. autre ligne courbe, qui fera du fécond genre. Com^
me fî C N K eft vu cercle, dont L fôit le centre , on de-
fcrira la premiere Conchoidedes anciens j &fî ceft vne
Parabole dont le diamètre foit K B , oii defcrira la ligne
courbe, que i'ay tantoft diteftre la premiere, & Ia*plus
fîmplè pourla^eftion dePappus,lorfqu'il n'y a que cinq
lignes droites données par pofîtion. Mais lî au lieu d vne
de ces lignes courbes du premier genre , c'en eft vue du
fécond, qui termine le plan C N K L, on en defcrira par
fon moyen vne du troifîefme, ou fi c'en eff vne du troifi-
cfme, onen defcrira vne du quatriefme, & ainfi a l'infini,
comme il eft fort ayfea connoiftr^ par le calcul. Et en
quelque autre façon, qu'on imagine la defcriptiou d'vne
ligne courbe , pourvûqu'elle foit du nombre de celles
qucictiomme Géométriques , on pourra toufiourstrou-
uer
54
SECOND BOOK
From this equation we see that the curve EC belongs to the first class,
it being, in fact, a hyperbola.'""
If in the instrument used to describe the curve we substitute for the
rectilinear figure CNK this hyperbola or some other curve of the first
class lying in the plane CNKL, the intersection of this curve with the
ruler GL will describe, instead of the hyperbola EC, another curve,
which will be of the second class.
Thus, if CNK be a circle having its center at L, we shall describe
the first conchoid of the ancients, '^^ while if we use a parabola having
KB as axis we shall describe the curve which, as I have already said,
is the first and simplest of the curves required in the problem of Pappus,
that is, the one which furnishes the solution when five lines are given
in position.'"*'
^^^ Ci. Briot and Bouquet, Elements of Analytical Geometry of Two Dimen-
sions, trans, by J. H. Boyd, New York, 1896, p. 143.
The two branches of the curve are determined by the position of the triangle
CNKL with respect to the directrix AB. See Rabuel, p. 119.
Van Schooten, p. 171, gives the following construction and proof: Produce
AG to D, making DG =: EA. Since E is a point of the curve obtained when
GL coincides with GA, L with A, and C with N. then EA = NL. Draw DP
parallel to KG. Now let GCE be a hyperbola through E whose asymptotes
are DP and PA. To prove that this hyperbola is the curve given by the instru-
ment described above, produce BC to cut DP in I, and draw DH parallel to AF
meeting BC in H. Then KL : LN = DH : HL But DH = AB = x, so we may
write b : c = x : HI, whence HI = ^, IB — a + c ^, IC = o + c — -7 y.
000
But in any hyperbola IC.BC = DE.EA, whence we have (a + c i- —y)y^ac,
cxy
or y^ ^ cy -r' + ay — ac. But this is the equation obtained above, which is
therefore the equation of a hyperbola whose asymptotes are AP and PD.
Van Schooten, p. 172, describes another similar instrument : Given a ruler
AB pivoted at A, and another BD hinged to AB at B. Let AB rotate about A
so that D moves along LK ; then the curve generated by any point E of BE will
be an ellipse whose semi-major axis is AB + BE and whose semi-mmor axis is
AB — BE.
'"^ See notes 59 and 70.
'**' Por a discussion of the elliptic, parabolic, and hyperbolic conchoids see
Rabuel, pp. 123, 124.
55
GEOMETRY
If, instead of one of these curves of the first class, there be used a
curve of the second class lying in the plane CNKL, a curve of the third
class will be described ; while if one of the third class be used, one of
the fourth class will be obtained, and so on to infinity.'""' These state-
ments are easily proved by actual calculation.
Thus, no matter how we conceive a curve to be described, provided
it be one of those which I have called geometric, it is always possible
to find in this manner an equation determining all its points. Now I
shall place curves whose equations are of the fourth degree in the same
class with those whose equations are of the third degree ; and those
whose equations are of the sixth degree'""' in the same class with those
whose equations are of the fifth degree""' and similarly for the rest.
This classification is based upon the fact that there is a general rule for
reducing to a cubic any equation of the fourth degree, and to an equa-
tion of the fifth degree'"'' any equation of the sixth degree, so that the
latter in each case need not be considered any more complex than the
former.
It should be observed, however, with regard to the curves of any
one class, that while many of them are equally complex so that they
may be employed to determine the same points and construct the same
problems, yet there are certain simpler ones whose usefulness is more
limited. Thus, among the curves of the first class, besides the ellipse,
the hyperbola, and the parabola, which are equally complex, there is
also found the circle, which is evidently a simpler curve ; while among
those of the second class We find the common conchoid, which is
described by means of the circle, and some others which, though less
'*°^ Rabuel (p. 125), illustrates this, substituting for the curve CNKL the semi-
cubical parabola, and showing that the resulting equation is of the fifth degree,
and therefore, according to Descartes, of the third class. Rabuel also gives (p. 119),
a general method for finding the curve, no matter what figure is used for CNKL.
Let GA = a, KL=b, AB = .r, CB = y and KB = r; then LB = s—b, and
AL = x + c—b. Now GA:AL = CB:BL, or a : x + s — b — y : :: — b,
, xy — by-^-ab
whence r = * .
a — y
This value of .:: is independent of the nature of the figure CNKL. But given
any figure CNKL it is possible to obtain a second value for :: from the nature of
the curve. Equating these values of z we get the equation of the curve.
[90] "ÇgUes dont l'équation monte au quarré de cube."
'"' "Celles dont elle ne monte qu'au sursolide."
""' "Au sursolide."
56
Livre Second. Î^J
uer vne equation pour déterminer tous Tes poins en cere
forte.
Au refteie mecs les lignes courbes qui font monter
cete equation iufques au quarre de quatre , au mefme
genre que celles qui ne la font monter que iufques au
cube. & celles dont Tequation monte au quarrédecu-
be,au mefme genre que celles dont elle ne monte qu'au
furfolide. &ain(î des autres. Dontlaraifoneft, qu'iîy a
reigle générale pour réduire au cube toutes lesdifScul-
te's qui vont au quarre'de quarre , &au furfolide toutes
celles qui vont au quarre de cube , de façon qu'on ne les
doit point eftiraer plus compofees.
Mais il eft a remarquer qu'entre les lignes de chafque
genre, encore que la plus part foient efgalement compo-
sées , en forte qu'elles peuuentferuir a déterminer les
mefmes poins. Su conftruire lesmefmes problefmes ,il y
eoa toutefois aufly quelques vues , qui font plus fimplcs,
&qui n'ont pas tant d'eftendue en leur puilfance. cora-
mcentre celles du premier genre outre l'Ellipfe l'Hyper-
bole & la Parabole qui font efgalement compofees ,Ic
cercle y eft aufiy compris , qui mauifeftement eft plus
fimplcr & entre celles du fécond genre il y a la Conchoi-
de vulgaire, qui afon origine du cercle^ &il y en a en-
core quelques autres, qui bien qu'elles n ayentpas tant
d'eftendue que la plus part de celles du mefme genre,
nepeuuenr toutefois eftre mifes dans le premier.
Or après auoirainfî réduit toutes les lignes courbes a J^,"!"], J^
certains genres , il m eft ayfe'de pourfuiure en la de- ^ion delà
ppus
monftrationdelarefponfe,qiiei'ay tanroftfaite alaque- Tzf,,^
ftion de Pappus. Car preaierement ayant fait voir cy ^'^^ 'ii
Ol Z dcliuS, ccJep-
57
3*4 La GEOMETRIE.
delTus , que lorfqu'il n'y a que trois ou 4 lignes droites
données, l'équation qui fert a determiner les poins cher-
chés, ne monte quciufqucs au qnarréj il efVeuidcntjque
la ligne courbe ou fetrouuent ces poins, eft neceflaire-
ment quelquVpe de ceîles du premier genre:a eaufe que
cete mefme equation explique le rapport , qu'ont tous
les poins deshgnes du premier genre a ceux d'vne ligne
droite^ Et que lorfqu'il n'y a point plus de 8 lignesdroi-
tes données , cete equation ne monte que iufques au
quarredequarré'tputauplus, 5c que par confequent la
hgne cherchée ne peut eftre que du fécond genre , ou au
deffous.Et que lorfqu'il n'y a point plus de 1 2 lignes don-
nées , l'équation ne monte que iufques au quarre'de cu-
be, & que par confequent la hgne cherchée n'cft que du
troifîefmegenre, ouaudeffous. &ainfi des autres. Et
mefme a caufe que la pofition deslignes droites données
peut varier en toutes fortes, & par confequent faire châ-
ger tant les quantités connues, que les fîgnes H- & -- de
l'équation, eu toutes les façons imaginables j il eft eui-
dentqn*iln'ya aucune ligne courbe du premier genre,
qui ne (bit vtilea cete queftion, quand elle eftpropofeh
en4hgnesdroitesjnyaucunedufecondqui nyfoit vti-
le, quand elle eft propofee en huit; ny du troifîefme,
quand elle eft propofee en douze: ôc ainfi des autres. En
forte qu'il n'y a pas vne Hgne courbe qui tombe fous le
Solution calcul&puifleeftre recede en Géométrie , quin'yfoit
^^ ^ftioti ^^^^ P°^^ quelque nombre de hgnes.
quandeiie Maisil faut icy plus particuHeremeut queiedetermi-
pofée^^° ne, & donne la façon de trouuer la ligne cherchée * qui
qu'en î fçf i; eu chafque cas, lorfqu'il ny a que 3 ou 4 lignes droi-
58
SECOND BOOK
complicated''"'' than many curves of the same class, cannot be placed
in the first class. '"^
Having now made a general classification of curves, it is easy for me
to demonstrate the solution which I have already given of the prob-
lem of Pappus. For, first, I have shown that when there are only three
or four lines the equation which serves to determine the required
points'*^' is of the second degree. It follows that the curve containing
these points must belong to the first class, since such an equation
expresses the relation between all points of curves of Class I and all
points of a fixed straight line. When there are not more than eight
given lines the equation is at most a biquadratic, and therefore the
resulting curve belongs to Class II or Class I. When there are not
more than twelve given lines, the equation is of the sixth degree or
lower, and therefore the required curve belongs to Class III or a lower
class, and so on for other cases.
Now, since each of the given lines may have any conceivable posi-
tion, and since any change in the position of a line produces a corre-
sponding change in the values of the known quantities as well as in
the signs + and — of the equation, it is clear that there is no curve
of Class I that may not furnish a solution of this problem when it
relates to four lines, and that there is no curve of Class II that may not
furnish a solution when the problem relates to eight lines, none of
Class III when it relates to twelve lines, etc. It follows that there is*
no geometric curve whose equation can be obtained that may not be
used for some number of lines."*'
It is now necessary to determine more particularly and to give the
method of finding the curve required in each case, for only three or
'^'' "Pas tant d'étendue." Cf. Rabuel, p. 113. "Pas tant d'étendue en leur
puissance."
^"^^ Various methods of tracing curves were used by writers of the seventeenth
century. Among these there were not only the usual method of plotting a curve
from its equation and that of using strings, pegs, etc., as in the popular construc-
tion of the elHpse, but also the method of using jointed rulers and that of using
one curve from which to derive another, as for example the usual method of
describing the cissoid. Cf. Rabuel, p. 138.
'*^^ That is, the equation of the required locus.
[96] «-gj^ sorte qu'il n'y a pas une ligne courbe qui tombe sous le calcul & puisse
être receuë en Géométrie, qui n'y soit utile pour quelque nombre de lignes."
59
GEOMETRY
four given lines. This investigation will show that Class I contains
only the circle and the three conic sections.
Consider again the four lines AB, AD, EF, and GH, given before,
and let it be required to find the locus generated by a point
C, such that, if four lines CB, CD, CF, and CH be drawn through it
making given angles with the given lines, the product of CB and CF
is equal to the product of CD and CH. This is equivalent to saying
that if
CB = y,
„„ ezy + dek -\- dex
z^ '
and ç^^^g"^y-\-f9\-fg--_
z^
then the equation is
, {cfglz — dcks^)y — (dez^ -\- cfgz — hcgz)xy -\- hcfglx — bcfgx^
ez^ — cgz'
60
Livre Second. ^*^
res données; & enverra par mefme moyen que le pre-
mier genre des lignes courbes n'en contient aucunes au-
tres, queles trois fecStions coniques, (Se le cercle.
Reprenons les 4 ligues AB, AD, EF,&GH don-
nées cy deflus, & qu'il failletrouuer vne autre ligne , ea
laquelleilfe rencontre vne infinite de poins tels que C,
duquel ayant tireles 4 lignes CB,CD,CF, & CH,a
«igles donnes, fur les données, CE multipliée parCF,
produift une fomme efgale a C D , multiplie'e par C H.
c z. y >i< b c X,
c'eft a dire ayant fait C B so j , C D oo —
GF^ ^- ,, &CH3Q ^ ,: lequatioeft
-dekzz, "^t "dezzx ^ >i>bifglx
^ i -- W t ^ <, A
i- ^fê^^ j ^ -cfgz^x U ..bcfgxx
>i' hcgzx J
}
Sf î
au
61
ja^ La GeometriEo
au moins en fuppofant e i^plus grand que f ^.car s'il eftoit
moindre, il faudroit changer tous les fîgnes H- & — . Et
il la quantité j' fe trouuoit nulle, ou moindre que rien en
ceteequationjlorrqa'onafupporé'Ie point C en l'angle
D AG, il faudroitle fuppofer au jGTy en l'angle D A E, on
E A R, ou R A G, en changeant les lignes 4- & — felon
qu'il feroit requis a cet effect. Et (i en toutes ces 4 po-
fitions la valeur d'j/ fe trouuoit nulle , la queftion feroit
impoffible au cas propofé. Mais fuppofons la icy eftrc
poffible, 5c pour en abréger les termes, au lieu des quan-
titcs ^ elcriuons ±m , ôc au heu de
ez,-- cgzz
dezz^i* cfgz--bc^7 . tn ^
efcnuons — ; & ainli nous au-
î z
ez-cgzl^
rons
^ ^" ... 'i^bcfgîx-.bcfgxx jont la raci-
yy^zmy- 7- xy -,
€ Z— CgZZ
ne. eft
nx •//" imnx nnxx*^ bcfglx -■ bcfgxx,
y^m- --t- mrïï ^ h-^~^ 7:~TZ^
abréger, au lieu de
efcriuonso,&:àu lieu de-
ez-- CgZZ
ô^ derechef pour abréger, au lieu de
tmn bcfgl - . , . I- , nn -bcfn
ez-cgzz e.-cgzz
efcriuons ^. car ces quantite's eftant toutes données,
nous les pouuons nommer comme il nous plaift, 6r
ainfi nous auons
y TOm-'-X'^-'^ mm-^- oa:-- -.vAr,quidoit cftrela
longeur delà ligne B C, en laiffaut A B, ou .v indeter-
raince.
62
SECOND BOOK
It is here assumed that cz is greater than eg ; otherwise the signs +
and — must all be changed.""' If y is zero or less than nothing in this
equation/"*' the point C being supposed to lie within the angle DAG,
then C must be supposed to lie within one of the angles DAE, EAR,
or RAG, and the signs must be changed to produce this result. If for
each of these four positions y is equal to zero, then the problem admits
of no solution in the case proposed.
Let us suppose the solution possible, and to shorten the work let us
write 2w instead of — ^- s—, and — mstead of ~ ^r^-
ez^ — cgz^ 2 ez^ — cgz^
Then we have
^ 2« hcfqlx — hcfqx^
^ ^ z ' ' ez^ — cgz-
of which the root"*' is
"•^ , / , 2mnx n-x- hcfqlx — hcfqx'
2 \ z z^ ez^ — cgz-
A • r , 1 r 1 • 2w« hcfql , , ,
Again, for the sake of brevity, put + -^ — ^ equal to o, and
«^ bcfg . p
-ly — — 1. -v equal to—; for these quantities being given, we can
z ez — cgz"- f'l
represent them in any way we please.''""' Then we have
y = m — - X + Lfi2 I o.r + - x^.
This must give the length of the line BC, leaving AB or x undeter-
[""J When cs is greater than eg, then ez^' — eg a- is positive and its square root
is therefore real.
'**' Descartes uses "moindre que rien" for "negative."
'*®^ Descartes mentions here only one root ; of course the other root would fur-
nish a second locus.
'""'In a letter to Mersenne (Cousin, Vol. VII, p. 157), Descartes says: "In
regard to the problem of Pappus, I have given only the construction and demon-
stration without putting in all the analysis ; ... in other words, I have given the
construction as architects build structures, giving the specifications and leaving
the actual manual labor to carpenters and masons."
63
GEOMETRY
mined: Since the problem relates to only three or four lines, it is obvi-
ous that we shall always have such terms, although some of them may
vanish and the signs may all vary.'""'
After this, I make KI equal and parallel to BA, and cutting off on
BC a segment BK ecjual to m (since the expression for BC contains
-|- m; if this were — m, I should have drawn IK on the other side of
AB,"°'' while if m were zero, I would not have drawn IK at all). Then
I draw IL so that IK : KL =- ^ : n; that is, so that if IK is equal to x,
KL is equal to ~x. In the same way I know the ratio of KL to IL,
which I may call n : a, so that if KL is equal to - x, IL is equal to
a
-X. I take the ponit K between L and C, since the equation contains
z
— -.V ; if this were -1 — .r, I should take L between K and C ;'""'' while if
z z
- X were equal to zero, I should not draw IL.
This being done, there remains the expression
LC= x/;n.- + o.r + -A-2,
from which to construct LC. It is clear that if this were zero the point
'^"'^ Having obtained the value of BC algebraically, Descartes now proceeds to
construct the length BC geometrically, term by term. He considers QC equal to
BK+KL + LC, which is equal to BK — LK + LC which in turn is equal to
~ -^' +\/ Mî2 + OX + —
\ m
1'"=' That is, take I on CB produced.
'"'^ That is, on KB produced. C is not yet determined.
64
Livre Second.
3*7
îBinée. Et il eft euident que la queftion n'eftantpro-
pofce qu'en trois ou quatre lignes , on peut toufîours
auoirdetels termes, excepfe que quelques vns d'eux
peuuenteftrenuls, & que les figues t1- Ôc -- peuuent di-
uerfement eftrechangés.
Après celaie fais Kl efgalc & parallèle aB A, en forte
qu'elle couppe de B C la partie B K efgale à /w , à caufe
qu'il y a icy -f- m; & ielauroisadioufteeentirantcete
ligne I K de l'autre code, s'il y ^uoit QU — m; & ie ne Tau-
rois point du tout tirée, fi la quantité" ttî euftefte'' nulle.
Puis ie tire aufiy I L , en forte que la ligne I K efi: à K L,
comme Z eft a «. ceft adiré que IK efiantA:, KL eft
-.V. Et par raefme moyen ieconnois au fly la proportion
qui
65
52$» I^A GEOMETRIE.
qui ell: entre KL, & I L, que ie pofe comme entre n Se a:
fibienque K L eftant -x, I L eft - x; Et ie fais que Ie
point K foit entre L &: C , a caufe qu'il y a icy — - x-,
au lieu que i'aurois mis LentrcK & Cjfi i'eulTe en ^- - .r,-
& ien'eufTe pointtiré'ceteIigneIL,fi^A;euft efte'nulle.
OrceIafait,iInemereftepluspourlaligne LC, que
ces termes, LCoo m'm'^r ox "-^^^. doùievoy
<5ue s'ils eftoient nuls, ce point C fe trouueroit en la li-
gne droite I L3& que s'ils eftoient tels que la racine s'en
ft
pufttirer,c'eftadirequew2/»&;^A; :v eftant marqués
dVn mefme figne 4- ou — , 00 fuftergalà4^;7?,oubien
queIestermes/ww&oA:,ouoA; &- xx fuflent nuls, ce
point C fe trouuerpit en vne autre ligne droite qui ne fe-
roit pas pins malayfee a trouuer qu' I L. Mais lorfque
cela n'eft pas, ce point C eft toufiours en l'une des trois
fedions coniques , ou en vn cercle , dont l'vn des dia-
mètres eft en la ligne I L,&: la ligne L C eft l'vne de cel-
les qui s'appliquent par ordre à ce diamètre j ou au con-
traire L C eft parallèle au diamètre , auquel celle qui efc
«n la ligne I L eft appliquée par ordre. A fçavoir fi le ter«
me ^xx, eft nul cete fe£tion.conique efi vne Parabole-
& s'il eft marqué du fîgne -f- , c'eft vne Hyperbole ; &
enfin's'il eft marque du fîgne — c'eft vne Ellipfe. Excepte"
feulement fi la quantité' aam eft efgale à pw & que l'an-
gle ILC foit droit ; auquel cas on à vn cercle au lieu
d'vne
66
SECOND BOOK
C would lie on the straight line IL ;'""' that if it were a perfect square,
P
that is if «r and — x- were both -1-'"^' and o- was equal to Apm, or if
m
m' and ox, or ox and -- x-, were zero, then the point C would lie on
another straight line, whose position could be determined as easily
as that of IL.'^"*'
If none of these exceptional cases occur,'""^ the point C always lies
on one of the three conic sections, or on a circle having its diameter
in the line IL and having LC a line applied in order to this diameter,^'"*'
or, on the other hand, having LC parallel to a diameter and ÎL applied
in order.
In particular, if the term — x- is zero, the conic section is a parabola ;
if it is preceded by a plus sign, it is a hyperbola; and, finally, if it is
preceded by a minus sign, it is an ellipse. '^°*^ An exception occurs when
[104] -pj^g equation of IL is j) := m — ~x.
tio6] -phere jg considerable diversity in the treatment of this sentence in differ-
ent editions. The Latin edition of 1683 has "Hoc est, ut, mm & — xx signo +
p
notalis." The French edition, Paris, 1705, has "C'est à dire que mm et —xx étant
-m
marquez d'un môme signe + ou ■ — ." Rabuel gives "C'est a dire que mm and
^ XX k.\.2,x\\ marquez d'un même signe +." He adds the follov^ring note: "Il y a
dans les Editions Francoises de Leyde, 1637, et de Paris, 1705, 'un même signe +
ou — ', ce qui est une faute d'impression." The French edition, Paris, 1886, has
"Etant marqués d'un même signe + ou — ."
[i°8] Note the difficulty in generalization experienced even by Descartes. Cf.
Briot and Bouquet, p. 72.
'""' "Mais lorsque cela n'est pas." In each case the equation giving the value
of ;y is linear in x and y, and therefore represents a straight line. If the quantity
under the radical sign and x are both zero, the line is parallel to AB. If the
quantity under the radical sign and m are both zero, C lies in AL.
[los] «^j^ ordinate." The equivalent of "ordinition application" was used in the
16th century translation of Apollonius. Hutton's Mathematical Dictionary, 1796,
gives "applicate." "Ordinate applicate," was also used.
''•^J Cf. Briot and Bouquet, p. 143.
Gl
GEOMETRY
a'm is equal to p2^ and the angle ILC is a right angle/""' in which case
we get a circle instead of an ellipse. ''"'
If the conic section is a parabola, its latus rectum is equal to — and
a
its axis always lies along the line IL.'"'' To find its vertex, N, make
IN equal to — ;^, so that the point I lies between L and N if m^ is posi-
tive and ox is positive; and L lies between I and N if wr is posi-
tive and o.v negative ; and N lies between I and L if in- is negative and
ox positive. It is impossible that nr should be negative when the terms
are arranged as above. Finally, if m- is equal to zero, the points N and
I must coincide. It is thus easy to determine this parabola, according
to the first problem of the first book of Apollonius'"".
If, however, the required locus is a circle, an ellipse, or a hyper-
bola,'"'' the point M, the center of the figure, must first be found. This
'""' Rabuel (p. 167) adds "If a-m^^pz- or if m=^p the hyperbola is equi-
lateral."
'"'' In this case the triangle ILK is a right triangle, whence IK^ = LK'^ -|- K?;
but by hypothesis IL : IK : KL = a : s : 7i; then a'-\-n'^ = s-. Now the equa-
tion of the curve is
:>' = '«-? + '^\m^ + 02-^ x\
^ \ nt
and therefore the term in x"^ is
and if a^m=^ pz-, then — = -r;, and this term in x- becomes
^'+"' .,2 2
Therefore, the coefficients of x- and ■v" are unity and the locus is a circle.
iu2] "pi^is ffjay ijg ggçj^ 2s follows : From the figure, and by the nature of the
parabola LC^= LN./) and LN = IL-(-IN. Let IN — 4>; then since IL = -x, we
Û 71 ft {I
have LN = - .r -|- 0 and LC = j' — in+—x; whence (;y — ni-\- — x)- ^ (-x-\-<P) p.
But {y — m-\- —x)~ =^ m- -\- ox from the equation of the parabola; therefore
- .r/i -|- 0/> =: m^ -(- o.r. Equating coefficients, we have -pr=o; p ^ ~^; <pp=:m^;
02 „ , a»r
a 02
'"'' ApoUonii Pcrgaeii Quae Graece exstant edidit I. L. Heiberg, Leipzig, 189L
Vol. I, p. 159. Liber I, Prop. LII. Hereafter referred to as Apollonius. This
may be freely translated as follows : To describe in a plane a parabola, having
given the parameter, the vertex, and the angle between an ordinate and the corre-
sponding abscissa.
'^"' Central conies are thus grouped together by Descartes, the circle being
treated as a special form of the ellipse, but being mentioned separately in all cases.
68
Livre Second.
329
d'y ne Ellipfe. Que fi cete fedion eft vne Parabole , fon
colle droit eft efgal à -^, & fon diamètre eft toujours en
la ligne IL. &: pour trouuer le point N, qui en eft le
fommet, il faut faire I N efgale a 7^,- & que le point I
fait entre!. & N,fî les termes font -j-mm-^ox; oubien
que le point L foit entre I & N, s'ils font -^ mm — ox;
oubien il faudroit qu' N fuft entré I & L , s'il y auoit
" m m -^ 0 X , Mais il ne peut iamais y auoir
— m m, en la façoaque les termes ont icy cfte' pofe^s. Et
enfin le point N feroit le mefme que le point I (î la quan-
tité w;7ze(xoit nulle. Au moyen dequoy il dt ayfé de
trouucrcereParaboleparlei^^^Problefrae du i^r. jiure
d'Apollonius.
Tt QLie
69
J5o La GEOMETRIE.
Que (î la ligne demâdee efc vn cercIe,ou vne eIlipfe,ou
vnc Hyperbole, il faut premièrement chercher le point
M, qui en eft le centre , & qui eft toufiours en la ligne
ao m
droite IL, ou on le trouue en prenant ~ pour IM. en
forte que fi la quantité o eft nulle, ce centre eft iuftement
au point I. Et fi la ligne cherchée eft vn cercle, ou vne
ElHpfej on doit prendre lé point Mdumefme*'cofté que
lepointL, aurefpedi du point I, lorfqu'on a -H oatj &
lorfqu'on à — o a; , on le doit prendre de l'autre. Mais
tout au contraire en l'Hyperbole, fi on a — ox, ce centre
MdoiteftreversLj&fîona-^-oAT, il doit eftrede l'au-
tre cofte. Après cela le cofte' droit de la figure doit eftre
—jj- H 7^ lorfqu'on a H- w wî , &: que la ligne
cherchée eft vn cercle, ou vne EUipfè ; oubien lorfqu'on
a— mm, & que c'eft vne Hyperbole. & il doit eftre
't/'ûozz, A^P^^'p^ I- T. 1 » n f
~~r. Jr~"la hgne cherchée eftant vn cercle,
ou vneElîipfe,ortà->7;2 77?;DTibien fi eftant tne Hyper-
bole & la quantité'o o eftant plus grande que 4 mp, on à
-f- m m. Qiie fi la quantitcTW m eft nulle, ce cofte droit
eft"^, & fi (? :c eft nulle ,il eft: f^^.^^^. Puis pour le cofté
a, a a, ^
travcrfant, il fauttrouuer vne ligne, qui foita ce cofte'
droit, corne «<îw2 eft à^ :^:^,àfçauoir fi ce cofte droit t^t
%
'U' 0 0 zz 4 w P^^> t r f^ 'i/ a a.0 omm ^ aam
""7^"'^" — — — letrauerianteit -— — — ■ -r-— ■
Et en tous ces cas le diamètre de la fedion ek en la ligne
I M, & L Ceft l'vnede celles qui luy cft appliquée par
ordre; Sibienque £iifant M N efgale a la moitié du cofte
trauer*
70
SECOND BOOK
will always lie on the line IL and may be found by taking I M equal to
-^ — .'"^^ If 0 is equal to zero M coincides with I. If the required locus
is a circle or an ellipse, M and L must lie on the same side of I when
the term ox is positive and on opposite sides when ox is negative. On
the other hand, in the case of the hyperbola, M and L lie on the same
side of I when ox is negative and on opposite sides when ox is positive.
The latus rectum of the figure must be
4
if m^ is positive and the locus is a circle or an ellipse, or if m^ is nega-
tive and the locus is a hyperbola. It must be
if the required locus is a circle or an ellipse and m^ is negative, or if it
is an hyperbola and o^ is greater than 4mp, mr being positive.
oz
But if m' is equal to zero, the latus rectum is — ; and if o^ is equal to
;ro'"''^ it is
4
lAmpz^
For the corresponding diameter a line must be found which bears
the ratio —-5- to the latus rectum: that is, if the latus rectum is
4
o'^s- Anips-
the diameter is
4
a^o^m^ 4a^m^
+
p-z"" ^ pz^
In every case, the diameter of the section lies along IM, and LC is one
of its lines applied in order. '"'^ It is thus evident that, by making MN
equal to half the diameter and taking N and L on the same side of M,
'"'^ Cf. Briot and Bouquet, p. 156.
'"*' Some editions give, incorrectly, ox for oc.
["'J See note 108.
71
GEOMETRY
the point N will be the vertex of this diameter.''"' It is then a simple
matter to determine the curve, according to the second and third prob-
lems of the first book of Apollonius.'"*'
When the locus is a hyperbola'^^' and in- is positive, if o- is equal to
zero or less than 4pm we must draw the line MOP from the center M
parallel to LC, and draw CP parallel to LM, and take MO equal to
4/ '
while if o.v is equal to zero, MO must be taken equal to m. Then con-
sidering O as the vertex of this hyperbola, the diameter being OP and
the line applied in order being CP, its latus rectum is
and its diameter'"'' is
4w2
''^*'If the equation contains — m" and +nx, then n^ must be ;?reater than
4mp, otherwise the problem is impossible.
'""' Cf. Apollonius, Vol. I, p. 173, Lib. I, Prop. LV : To describe a hyperbola,
given the axis, the vertex, the parameter, and the angle between the axes. Also
see Prop. LVI : To describe an ellipse, etc.
'"*' Cf. Letters of Descartes, Cousin, Vol. VIH, p. 142.
[ini "Qf^^Q traversant."
72
Livre Second. 35i
traucrfant 6c le prenant du piefme coCté du point M,
qu efc le point L, on a le point N pour le fommet de ce
diamètre .en fuite dequoy il eCt ayfeMe trouuer la fedtion
par le fécond ôc 3 prob. du i", liu. d'Apollonius-
Mais quand cote fedion eftant vne Hyperbole , on à
•4- m W5 & que la quantité 0 0 eft nulle ou plus petite que
4;? m, on doit tirer du centre M la ligne MOP parallèle a
L C ,' & C P parallèle à L M; & faire M O efgale a
^ ww--^^.oubien la faire efgale à m fila quantite'orc
eft nulle. Puis confiderer le point O, corne le fommet
de cete HyperbolCi dont le diamètre eft O P , & C P la
Tt 2 lign^^
73
332- La- Géométrie.
ligne qui Iqy eft appliquée par ordres fori coftedroireft
-— ; — 77^:;^ & Ion coite trauersat elc *^ ^mjn-
Excepte'quand o x eft nulle.car, alors le cofte droit db
— ^77~. ^letrauerfanteft iw. &ainfî il çft ay/c de la
trouuer par le 3 prob.du i^^, ijy^ d'Apollonius.
UraTimi Et Ics demonftrations de tout cecy font euidentes.car
detoutcccompofant vn efpace des quantités que iay afîign ces
^^cft^'e^^'^pourlecoftedroit, & Je trauerfant, ôcpourlefegment
cipiiquc. dudiametreNL,ouOP,fuiuâtlateneurderii,du ii,&
d:u 13 theorefraes du i", liure d'Apollonius, on trouuera
tous les mefmes termes dont eft compofé lé quarrè de
îaligne C P,ou C L,qui eit appliquée par ordre a ce dia-
mètre. Gomme en cet ex'emple oftantlM , qui eft
TTT, de N M, qui eft -— 0 0 -I- 4 mpy iay I N, a laquel-
le aiouftant IL, qui eft ~^, lay N L ,^qui eft - X' — -^ — -
•H JT^"^ 0 0 -h 4 ;» /> , ôd cecy eftant multiplie^ par
;^<^ 0-1- 4 »2/?, qui eft le cofte droit de la figure, il vient
rvy 0 0-^4 j?z^ "' ,"; ^ oo-j- ^mp -h ~7 -h z m ?n.
pour le rectangle, duquel il Faut oftet vn efpace qui foi t
au quatre de N L comme le cofté'droit eft au trauerfant.
& ce quarré de N L eft ^^f^:- -— -.^-
<l « o * 7» rt nam.
_ ^ L
74
SECOND BOOK
An exception must be made when ox is equal to zero, in which case the
latus rectum is , ^ and the diameter is 2;;;. From these data the
p2-
curve can be determined in accordance with the third problem of the
first book of Apollonius/'^^
The demonstrations of the above statements are all very simple, for,
forming the product'^^^ of the quantities given above as latus rectum,
diameter, and segment of the diameter NL or OP, by the methods of
Theorems 11, 12, and 13 of the first book of Apollonius, the result will
contain exactly the terms which express the square of the line CP or
CL, which is an ordinate of this diameter.
In this case take IM or -^^—- from NM or from its equal
am
Yfz
9.„ Vo'H-4w/).
To the remainder IN add IL or— jt, and we have
2
a aom am
z Ipz Ipz ' '^
Multiplying this by
the latus rectum of the curve, we get
for the rectangle, from which is to be subtracted a rectangle which is
to the square of NL as the latus rectum is to the diameter. The square
of NL is
^'-"-1 See note 113.
1123] "Composant un espace."
GEOMETRY
Divide this by a-m and multiply the quotient by pc-, since these terms
express the ratio between the diameter and the latus rectum. The result is
P 1 1-^1 i 1 ^^"^ ^^ l-n 1 9
— x'^ — ÛX -I- X -yJo^ -I- 4Mp 4- -— — — — — V^ + -if/ip -I- Tfr.
m ^ ^ ^ ^ 2/ 2/ ^ ^ ^
This quantity being subtracted from the rectangle previously obtained,
we get
CL, =tn^ Jr-ox — — x'^.
m
It follows that CL is an ordinate of an ellipse or circle applied to NL,
the segment of the axis.
Suppose all the given quantities expressed numerically, as EA=3,
AG = 5, AB = BR, BS= |- BE, GB = BT, CD= |cR, CF-2CS, CH =
— CT, the angle ABR=60° ; and let CB . CF=CD . CH. All these quan-
ties must be known if the problem is to be entirely determined. Now
let AB^,r, and €6=3». By the method given above we shall obtain
3;^==2y — xy-\-^x — ,r^ ;
whence BK must be equal to 1. and KL must be equal to one-half KI ;
and since the angle IKL = angle ABR ^ 60° and angle KIL (which is
one-half angle KIB or one-half angle IKL) is 30°, the angle ILK is a
right angle. Since IK = AB = ;»:, KL = -.v-, IL = ;f a/-, and the quantity
/3 3
represented by z above is 1 , we have a = \\-, ?fi = l, c? = 4, / = -, whence
\ 4 4
IM = a/ ~, NM = a/ — -; and since a^w (which is .) is equal to ps^ , and
76
Livre Secon^d. ?33
i/^M~ ^ (?o H- 4;»/? qu'il fautdiuiferpar^tf^ôc
multiplier par;j^^,acaufe que ces termes expliquent la
proportion qui eft entre le cofté trauerfant & le droit, &
0 0 in
il s\^\A-xx--oX'\'xV 00 -^ ± mp -.
tn ■' i />
«-^-^ "/ oo-^-A-mp -fr m;«.cequ'il faut ofler du red:anele
precedent, ôcontrouue ?w;»-Hoa; — - ATArpourlequar-
redeCL, qui par confequent eft vne ligne appliquée
p^r ordre dans vne Ellipfe,oudans vn cercle,au lègment
du diamètre NL.
Et Convent expliquer toutes les quantite's données
par nombres, en faifant par exemple EAa)^, A God y,
AB:»BR,BSfX)iBE,GB30 BT, CDco ^CR,CF
002CS, CHx>f CT, & quel'angle ABR foit de 60
degrésj & enfin que le redtangle des deux C B , & C F,
foit efgai au re&ngle des deux autres C D ôrC Hj car il
faut auoir toutes ces chofesaffin que la queftion foit en-
tièrement déterminée. & auec cela fappofànt A B do .v,
& G B 30^, on trouue par la façon cy deflus expliquée
y y 30 2 j " X y -^ ^ X " X X Sc y CO j .. L.x -h"
/'i-f-4A;'-|^':v: fi bienqueB Kdoit eftre i,& KL
doit eftre la moitié de Kl, & pourceqae Tangle I Kli
ou A BR eft de ($0 degrés, &îKILquieftla moitic'de
K I B ou I K L, de 30, 1 L K eft droit. Et pourceque I K
ou ABeftuomme:c,KLeft^A;, Ôc IL- eft a:^|, &lâ
quantité qui eftoit tantoft nomm^ ^ eft i , celle qui
eftoit a cft î^^ |, celle qui eftoit m eft r, celle qui eftoit 0
eft 4, & celle qui eftoit p eft |,de façon qu'on à / '|
Tt i powr.
77
3M
La GEOMETRIE.
Quels
font les
lieux
plans, &
lblides:&
la façon
de les
Uouuer.
pour I M, Se V ^^ pour N M, & pourceque aam qui
eft I eft icy efgâl à ps^::^ & que Tangle I L C eft droit , oa
trouue que la ligne courbe N C eft vn cercle. Et on
peut facilement examiner tousles autres cas en mcfme
forte .
Aurefte acaufe que les equations, qui ne montent
que iufques au quarre^, font toutes comprifes en ce que ie
viens d*expliquer ; non feulement le problefine des an-
ciens en 5 & 4 lignes eft icy entièrement acheue'j mais
aufly tout ce qui appartient à ce qu'ils nommoient la
compolîtion des lieux folides- Ôcparconfèquent auffya
celle des lieux plans» a caufe qu'ils font compris dans les
folides. Car ces lieux ne font autre chofe, fînon que lors
qu'il eft queftion de trouuer quelque point auquel il
manque
78
SECOND BOOK
the ang-Ie ILC is a right angle, it follows that the curve NC is a circle.
A similar treatment of any of the other cases ofïers no difficulty.
Since all equations of degree not higher than the second are included
in the discussion just given, not only is the problem of the ancients
relating to three or four lines completely solved, but also the whole
problem of what they called the composition of solid loci, and conse-
quently that of plane loci, since they are included under solid loci.'^"'
For the solution of any one of these problems of loci is nothing more
than the finding of a point for whose complete determination one con-
'^' Since plane loci are degenerate cases of solid loci. The case in which
neither x^ nor y- but only xy occurs, and the case in which a constant term occurs,
are omitted by Descartes. The various kinds of solid loci represented by the equa-
tion y=i±ni±—x±: — ± \ ± m- ± ox ± —x may be summarized as follows :
(1) If all the terms of the right member are zero except -7, the equation repre-
sents an hyperbola referred to its asymptotes. (2) If — is not present, there are
several cases, as follows: (a) If the quantity under the radical sign is zero or a
perfect square, the equation represents a straight line; (b) If this quantity is not
a perfect square and if — .r- = 0, the equation represents a parabola; (c) If it is
not a perfect square and if — x^ is negative, the equation represents a drcle or an
ellipse; (d) If — x~ is positive, the equation represents a hyperbola. Rabuel, p. 248.
79
GEOMETRY
ditioii is wanting, the other conditions being such that (as in this exam-
ple) all the points of a single line will satisfy them. If the line is
straight or circular, it is said to be a plane locus ; but if it is a parabola,
a hyperbola, or an ellipse, it is called a solid locus. In every such case
an equation can be obtained containing two unknown quantities and
entirely analogous to those found above. If the curve upon which the
required point lies is of higher degree than the conic sections, it may
be called in the same way a supersolid locus, ''"^' and so on for other
cases. If two conditions for the determination of the point are lacking,
the locus of the point is a surface, which may be plane, spherical, or
more complex. The ancients attempted nothing beyond the composition
of solid loci, and it would appear that the sole aim of Apollonius in his
treatise on the conic sections was the solution of problems of solid loci.
I have shown, further, that what I have termed the first class of
curves contains no others besides the circle, the parabola, the hyperbola,
and the ellipse. This is what I undertook to prove.
I12EJ u^j^ jjgy sursolide."
80
Livre Second. 33/
manquevne condition poureflre entieretncnt determi-
ne, ainfî qu'il arritie en cete exemple, tous les poins d'Vne
mefme ligne peuuent eftre pris pour celuy qui efl de-
mande'. Et fî cete ligne eft droite, ou circulaire , on la
nomm^vn lieu plan. Mais fi c'eftvne parabole, ouvne
hyperbole, ou vne cUipfè, on la nomme vn lieu folide. Et
toutefois & quantes que cela eft, on peut venir a vne E-
quationqui contient deux quantite's inconnues, & eft
pareille a quelqu'vne de celles que ie viens de refoudre.
Que fi la ligne qui determine ainfi lè point cherché , eft
d'vndegre'pluscompofeequeles fciflions coniques, on
la peut nommer, en mefme façon , vn heu furfohde , &
ainfi des autres. Et s'il manque deux conditions a la de-
termination de ce point, le heu ou il fè trouue eft vne fu-
perficie, laquelle peut eftre tout de mefme ou plate, ou
fpherique, ou plus compofee. Mais le plus haut but
qu'ayent eu les anciens en cete matière a efte deparue-
niralacompofîtiondes lieux folides: Et il femble que
tout ce qu'Apollonius a efcrit des fedlions coniques n'a
efte'qu'àdefleinde la chercher. ^ u n.
^ Quellcclt
De plus on voit icy que ceque iay pris pour le premier '^ prcmie-
genredeshgnes courbes,n en peut comprendre aucunes pîu? fim-
autres que le cercle, la parabole, l'hyperbole, &rellipfe.P^'''^*= ,
. /• -, . . , ^ toutes les
qui eit tout ce quel auois entrepris de prouuer. lignes
Que fi la queftion des anciens eftpropofee en cinq li- '°"^^^"
gnes, qui foîent toutes parallèles ; ilefteuidentque le uent^en la
point chercheTeratoufîours en vne ligne droite. Maisfi ]lf^Z"
elle eftpropofee en cinq lignes, dont ilyenait quatre ciens
qui foient parallèles, Sequela cinquiefme les couppe a S pro-
angles droits, & mefme que toutes les limes tirées duP°f^^"*
. cinqli-
pOintgncs.
81
53<^ La Géométrie.
point cherche les rencontrent aufîy a angles droits, &
enftn que le parallélépipède compofè de trois, des lignes
ainfî tirées fur trois de celles qui font paralleles/oit efgal
au parallélépipède compofé des deux hgnes tirées Tvne
fur Ja quatriefme de celles qui font parallèles & l'autre
fur celle qui les couppe a angles droits, & dVne troifîcf.
me ligne donnée, ce qui eft ce femble le plus ûm-
pic cas qu'on puiflb imaginer après le precedent j le
point cherche fera en ja ligne courbe , qui eft defcnte
parle raouuementd'vne parabole en la façon cy deffus
expliquée.
Soient
82
SECOND BOOK
If the problem of the ancients be proposed concerning five hnes, all
parallel, the required point will evidently always lie on a straight line.
Suppose it be proposed concerning five lines with the following condi-
tions :
(1) Four of these lines parallel and the fifth perpendicular to each
of the others ,
(2) The lines drawn from the required point to meet the given lines
at right angles ;
(3) The parallelepiped"""' composed of the three lines drawn to meet
three of the parallel lines must be equal to that composed of three lines,
namely, the one drawn to meet the fourth parallel, the one drawn to
meet the perpendicular, and a certain given line.
This is, with the exception of the preceding one, the simplest pos-
sible case. The point required will lie on a curve generated by the
motion of a parabola in the following way:
[120] Yhat is, the product of the numerical measures of these lines.
83
GEOMETRY
Let the required lines be AB, IH, ED, GF, and GA. and
let it be required to find the point C, such that if CB, CF, CD, CH, and
CM be drawn perpendicular respectively to the given lines, the paral-
lelepiped of the three lines CF, CD, and CH shall be equal to that of
the other two, CB and CM, and a third line AI. Let CB=3;, CM=jr.
AI or AE or GE=a; whence if C lies between AB and DE, we have
CF=2a— V, CD==a— 3;, and CH=v-fa. Multiplying these three to-
gether we get y^~2ay-—a-y^2a'' equal to the product of the other
three, namely to axy.
I shall consider next the curve CEG, which I imagine to be described
by the intersection of the parabola CKN (which is made to move so
that its axis KL always lies along the straight line AB) with the ruler
GL (which rotates about the point G in such a way that it constantly
lies in the plane of the parabola and passes through the point L). I
take KL equal to a and let the principal parameter, that is, the par-
ameter corresponding to the axis of the given parabola, be also equal to
a, and let GA=2a, CB or MA=y, CM or AB=.r. Since the triangles
GMC and CBL are similar, GM (or 2a— y) is to MC (or x) as CB
(.ovy) is to BL, which is therefore equal to ^ - - . Since KL is a, BK
2a— y
^y 2a — ay — xy
IS a — - or . Finally, since this same BK is a segment
2a— y 2a— y
of the axis of the parabola, BK is to BC (its ordinate) as BC is to a
(the latus rectum), whence we get y^—2ay-—a-y-^2a"^=axy, and there-
fore C is the required point.
84
Livre Sicokb. 337
Soient par exemple les lignes cherchées A B,I H,E D,
G F, & G A. & qu'on demande le point C, en forte que
tirant C B, C F, C D, C H, & C M a angles droits fur les
données, le parallélépipède des trois CF, CD, & CH
foit efgal a celuy des 2 autres C B, & C M, & d'vne troi-
fiefme qui foit A I. le pofè C B y3y. C M O) x\ A I, ou
A E, ou G E 00 ^,de façon que le point C eflant entre les
lignes A B, &DE, iayCFooa^ —y, C D :» ^ — ^. &
C H 30^ H- ^. & multipliant ces trois l'y ne par l'autre,
lay y —layy-- a ay -^ ia efgal au produit des trois
autres quieft^ATj/. Après cela icconfidere ta ligne cour-
be C E G, que i'imaginc eftre defcrite par l'interfedion,
de la Parabole C K N, qu'on fait mouuoir en telle forte
que fon diamètre KL eft toufiours fur la ligne droite
A B, & de la reigle G L qui tourne cependant autour du
point G en telle forte quelle pafle toufiours dans le plan
de cete Parabole par le point L. EticfaisKLoo «, &le
coftd'droit principal, c'eft adiré celuy qui fè rapporte a
l'aiflieudeceteparabole^auflyefgalà^, &GA30 2^7, &
CB ou M A 30 j^, & C M ou A B 30 AT. Puis a cau/è des
triangles femblables GM C & C B L,G M qui eft 2 ^ -y,
eft à M C qui eft ^, ,comme C B qui efty, eft à B L qui eft
X y
par confequent -^^. ^ pourceque L K eft ^, B K eft ^
- xy laa -• ay - xy
— -,oubien — ^^^ — . Et enfin pourceque ce mef-
mcB Keftant vn fegment du diamètre de la Parabole
eft à B C quiluy eft appliquée par ordre , comme cel-
iecyeft au cofté droit qui eft a, le calcul monilre que
y "Zayy —aay -h z-a, eft efgal à a xy. &par confè-»
V V quenc
85
33»
La Géométrie.
quent que le point C eft celuy qui eftoit demande. Et il
peut eftre pris en tel endroir de la ligne C E G qu'on ve-
uille choifîr, ou aufTy en Ton adiointe ^ E G ^ qui fe de-
fcri t en mefme façon, excepté que le fommet de laPara-
bol e eft tourne vers l'autre cofté , ou enfin en leurs con-
trepofe'es Nlo,nl 0,qui font defcrites par l'interfeétion
que fait la ligne G L en l'autre cofté de la Parabole
KN.
Or encore que les parallèles donné'cs A B , 1 H, E D,
& G F ne fuficnt point efgalement distantes, & que G A
ne les couppaft point a angles droits, ny aafly les lignes
tirées
86
SECOND BOOK
The point C can be taken on any part of the curve CEG or of its
adjunct cEGc, which is described in the same way as the former, except
that the vertex of the parabola is turned in the opposite direction ; or
it may He on their counterparts""'' NIo and «lO, which are generated
by the intersection of the hue GL with the other branch of the para-
bola KN.
Again, suppose that the given parallel lines AB, III, ED, and GF are
not equally distant from one another and are not perpendicular to GA,
and that the lines through C are oblique to the given lines. Tn this case
the point C will not always lie on a curve of just the same nature. This
may even occur when no two of the given lines are parallel.
[i2'] "£j^ leurs contreposées."
87'
GEOMETRY
Next, suppose that we have four parallel lines, and a fifth line cutting
them, such that the parallelepiped of three lines drawn through the
point C (one to the cutting line and two to two of the parallel lines)
is equal to the parallelepiped of two lines drawn through C to meet the
other two parallels respectively and another given line. In this case
the required point lies on a curve of different nature/^^*^ namely, a
curve such that, all the ordinates to its axis being equal to the ordinates
of a conic section, the segments of the axis between the vertex and
the ordinates'^^"' bear the same ratio to a certain given line that this
line bears to the segments of the axis of the conic section having equal
ordinates. ''^°'
I cannot say that this curve is less simple than the preceding ; indeed,
I have always thought the former should be considered first, since its
description and the determination of its equation are somewhat easier.
I shall not stop to consider in detail the curves corresponding to the
other cases, for I have not undertaken to give a complete discussion of
the subject ; and having explained the method of determining an infinite
number of points lying on any curve, I think I have furnished a way
to describe them.
It is worthy of note that there is a great difference between this
method'"^^ in which the curve is traced by finding several points upon
I12S] Yi^e general equation of this curve is axy — xy~ -\-2a-x ^ a-y — ay-.
Rabuel, p. 270.
112»] That is, the abscissas of points on the curve.
[ISO] -pi^g thought, expressed in modern phraseology, is as follows : The curve is
of such nature that the abscissa of any point on it is a third proportional to the
abscissa of a point on a conic section whose ordinate is the same as that of the
given point, and a given line. Cf. Rabuel, pp. 270, et seq.
'"'' That is, the method of analytic geometry.
88
Livre Second, 33?
tirées du point C vers elles, ce point (j ne IaiÏÏ*eroit pas
de fe trouuer toufiours en vne ligne courbe, qui feroit
de cete mefme nature. Et il s'y peut aufly trouuer quel-
quefois, encore qu'aucune des lignes données uefoienc
parallèles. Maisfî lorfqu'ilyena 4 ainfî parallèle s, & vne
ciuquiefme qui les trauerlê: 6c que le parallélépipède de
trois des lignes tire'cs du point cherche, l'vne fur cete
cinquiefme, &: lès 1 autres fiir 2 de celles qui font paral-
lèles; foitefgal a celuy, des deux tirées fur les deux au-
tres parallèles , Ôcd'vne autre Hgne donnée. Ce point
cherchcf'eften vne ligne courbe d'vue autre nature, â
fçauoir en vne qui eft telle, que toutes les lignes droites
appliquas parordre a fon diamètre eftant efgales a cel-
les dVne fe<SÎ:ion conique, les fegmens de ce diamètre,
quifoDteptrelefommet&ces lignes , ont mefme pro-
portion a vne certaine ligne donnée, que cete ligne don-
née a aux fegmens du diamètre de la fêd:ion conique,
aufquels les pareilles lignes font appliquas par ordre. Et
ie ne fçaurois véritablement dire que cete ligne foit
moins fîmple que la précédente, laquelle iay creu toute-
fois deuoir prendre pour la premiere, acaufêquela de-
fcription , & le calcul en font en quelque façon plus
faciles.
Pour les lignes qui feruent aux autres cas, ienc mare-
fteray point aies diftinguer par efpeces. car ie n'aypas
entrepris de dire tout ; &: ayant explique la faconde
trouuer vne infinite de poins par ou elles paffectjie pçnfç
âuoir aflcs donné le moyen de les defcrire.
Mefm€ ileft a propos de remarquer, qu'il y a grande
diflference entre cete façon de trouuer plufieurs poins
Vv 2 pour
89
340 La Géométrie.
font les pour tracer vue ligne courbe, & celle dont on le lert pour
l^g"es j.^ fpirale, & fes femblablés. car par cete dernière on ne
courbes ^ t
qu'on de- trouue pas indiffère ment tous les poins dé la ligne qu'on
trouu" cherche, maisfèulernent ceux qui peuuent eftre dcter-
piuficurs mines par quelque mefurephisfimple, que celle qui eft
poin7,qyirequifepourlacomporer, & ainfî a proprement parler
peuucnc on ne trouue pasjvude {ç,% poins. c'eft a dire pas vn de
ceuL^eû ceux qui luy font tellement propres, qu'ils ne puifîcnt
Gcoine- eftre trouuc's que par elle: Au lieu qu'il ny a aucun point
dans.les lignes qurferuent a la queftion propofé'e , qui ne
fe puifTe rencontrer entre ceux qui fe déterminent par la
façon tahtoft expliquée. Et pourceque cete façon de
tracer une Hgne courbe, en trouuant indifferêment plu-
iîeurs de fês poins , ne s'eftend qu'a celles qui peuuent
aufly eftre defcrites par vnmouuement régulier & con-
tinu, on ne la doit pas entièrement reietter de la Géo-
métrie.
Sft^ufly Et on n'en doit pas reietter non plus, celle ou on fe
celles fert d'vn fil, ou d'vne chorde repliée, pour determiner
?crit auec ^^g^^^î^ OU là difference de deux ou plufieurs lignes
vnechor- droitcs quipeuugnt eftre tirées de chafque point de la
pc'ui?e"nc courbe qu'on cherche, a certains autres poins ^ ou fur
y eftre Certaines autrcs lignes a certains aneles. ainfî que nous
auons fait en la Dioptrique pour expliquer rEllipie &:
THyperbole. car encore <]u'on n'y puiiTe reçeuoir au-
cunes lignes qui femblent a dès chordes , c'eft a dire qu]
deuienent tantoft droites &: tantoft courbes, a cauie que
la proportion, qui eft entre les droites &■ les courbes,
n'eftant pas connue, & mefme ie croy ne le pouuant eftre
par les hommes, on ne pourroit rien conclure de là qui-
fuft
90
.Tcceucs.
SECOND BOOK
it, and that nsed for the spiral and similar curves.'"'' In the latter not
any point of the required curve may be found at pleasure, but only such
points as can be determined by a process simpler than that required for
the composition of the curve. Therefore, strictly speaking, we do not
find any one of its points, that is, not any one of those which are so
peculiarly points of this curve that they cannot be found except by
means of it. On the other hand, there is no point on these curves which
supplies a solution for the proposed problem that cannot be determined
by the method I have given.
But the fact that this method of tracing a curve by determining a
number of its points taken at random applies only to curves that can
be generated by a regular and continuous motion does not justify its
exclusion from geometry. Nor should we reject the method"^" in which
a string or loop of thread is used to determine the equality or difference
of two or more straight lines drawn from each point of the required
curve to certain other points.''"' or making fixed angles with certain
other lines. We have used this method in "La Dioptrique" '"'' in the
discussion of the ellipse and the hyperbola.
On the other hand, geometry should not include lines that are like
strings, in that they are sometimes straight and sometimes curved, since
the ratios between straight and curved lines are not known, and I
believe cannot be discovered by human minds,'""' and therefore no con-
clusion based upon such ratios can be accepted as rigorous and exact.
'^"' That is, transcendental curves, called by Descartes "mechanical" curves.
I133J ç-£ j.j^g familiar "mechanical descriptions" of the conic sections.
'"^' As for example, the foci, in the description of the ellipse.
'"'' This work was published at Leyden in 1637, together with Descartcs's
Discours de la Méthode.
1136] Yhis is of course concerned with the problem of the rectification of
curves. See Cantor, Vol. II (1), pp. 794 and 807, and especially p. 778. This
statement, "ne pouvant être par les hommes" is a very noteworthy one, coming as
it does from a philosopher like Descartes. On the philosophical question involved,
consult such writers as Bertrand Russell.
91
GEOMETRY
Nevertheless, since strings can be used in these constructions only to
determine lines whose lengths arc known, they need not be wholly
excluded.
When the relation between all points of a curve and all points of a
straight line is known. '"'^ in the way I have already explained, it is easy
to find the relation between the points of the curve and all other given
points and lines ; and from these relations to find its diameters, axes,
center and other lines'"**^ or points which have especial significance for
this curve, and thence to conceive various ways of describing the curve,
and to choose the easiest.
By this method alone it is then possible to find out all that can be
determined about the magnitude of their areas,"'""' and there is no need
for further explanation from me.
''^'^ Expressed by means of the equation of the curve.
[138] Pqj. example, the equations of tangents, normals, etc.
I"»] Por the history of the quadrature of curves, consult Cantor, Vol. II (1),
pp. 758, et seq.. Smith, History, Vol. II, p. 302.
92
Livre Se CONI5. 3fi
fufirexad&afTuré. Toutefois a caufe qu'orrnefe ferr
de chordcs en ces conftrud:ions , que pour détermine^
des lignes droites, dont on connoift parfaitement la lon^
geur, cela ne doit point faire qu'on les reîette.
Orde cela feul qu'on fçait le rapport, qu'ont tousles Q^e pont,
poins d'vne ligne courbe a tous ceux d'vne ligne droite, J^'^'J^j^'iç
en la façon queiay expliqueej il eft ayfé de trouuer auffy proprié-
té rapport qu'ils ont a tous les autres poins, & lignes don- ^^^^
nées: & en fuite de connoiftreles diamètres , les aiffieux, couibcs,
le^ centres, &: autres lignes , ou poins ^ a qui cliaique ii- ddcaudr
gne courbe aura quelque rapport plus particulier , ou^erapporc
plus fimple, qu'aux autres: & ainfî d'imaginer diuers toutîeuis
moyens pour les defcnre,& d'en choilîr les plus faciles. P°''^^
Et mefme on peut aufTy par cela feul trouuer quafï tout lignes
cequipeut^ftre déterminé' touchant la grandeur de Te- «^'^J''^"»
fpace quelles comprenent, fans qu'ilfoit befbin- que i-en de cirer
donne plus d'ouuerture. Et enfin pour cequi eH detou-j!^"^""
tes les autres propriete's qu'on peut attribuer aux lignes qui les
courbes, elles ne dependent que de la grand,eur des an- ^^"JJj"^
gles qu'elles font auec quelques autres figues. Mais lorA "s poins
qu on peut tirer des lignes droites qui les couppent a an- droifs.
gles droits, aux poins ou elles fpnt rencontrées par cel-
lésauec qui elles font les angles qu'on veut mefurer, oiî,
cequeie prensicy pour le mefme, qui couppent leurs
contingentes- la grandeur de ces- angles ireftpas plus
malayfée a trouuer, que s'ils eftoient compris entre deux
lignes droites. C'eftpourquoy ie croyray auoir miS' iey
tout ce qui ell requis pour les elemens des lignes cour-
bes, lorfque i*auray généralement donne' la façon de ti-
rer des lignes droites, qui tombent a angles droits fur
Vr 5 tels
9>i
Façon
générale
pour
trouuer
des lignes
droites»
qui coup-
pent les
courbes
données,
ou leurs
coBtia-
ger^tcs>a
angles
droits.
^^^ La Géométrie.
tels déleurs poins qu'on voudra choifîr. Et i'ofe dire
que c'eft cccy le problefme le plus vtilc , & le plus gene-
ral non feulement que iefçache, mais rnefme que l'aye
iamais defîré de fçauoir en Géométrie.
Soit G E
la ligne courbe,
& qu'il faille ti-
rer vne ligne
droite par le
point C, qui fa-
ce auec elle des angles droits. le fùppofc la chofe defîa
faite, & que la ligne cherchée eft C P , laquelle ie pro-
longe iufques au point P, ou elle rencontre la ligne droi-
te G A, que ie fuppoiè eftre celle aux poins de laquelle
on rapporte tous ceux de la hgne C E : en forte que fai-
fant M A ou C B 30^^, & G M, ou B A X) at, iay quelque
equation, qui explique le rapport, qui eft entre x ôç^y*
PuisiefaisPCoo/, &PA»r^ouP M y> v -y, &c a
caufe du triangle redtangle P M C iay//, qui eft h quar-
re de la baze efgal à xx'hvv-'ivy-hyy , qui font
les quarrés des deux coftes . c'eft a dire iay x oa
f^sx'-vv-h ivy-^yy^ oubien ^ ao t/ -H V ss — xx,8c
parie moyen de cete equation, i'ofte de l'autre equa-
tion qui m'explique le rapport qu'ont tous les poins de la
courbe C E a ceux de la droite G A,rvue des deux quan-
tités indéterminés X ou y. ce qui eft ayfé a faire en
mettant partout V ss — vv-i^ ivy-- yy au lieu d'.r , Se
le quatre de cete fomme au lieu d^xx^ &fon cube au heu
d'x, &ainudesautres,ficeft;cqueie veuille oûerj ou-
bien
94
SECOND BOOK
Finally, all other properties of curves depend only on the angles
which these curves make with other lines. r>ut the angle formed by
two intersecting curves can be as easily measured as the angle between
two straight lines, provided that a straight line can be drawn making
right angles with one of these curves at its point of intersection with
the other. '"°^ This is my reason for believing that I shall have given
here a sufficient introduction to the study of curves when I have given
a general method of drawing a straight line making right angles with
a curve at an arbitrarily chosen point upon it. And I dare say that
this is not only the most useful and most general problem in geometry
that I know, but even that I have ever desired to know.
Let CE be the given curve, and let it be required to draw
through C a straight line making right angles with CE. Suppose the
problem solved, and let the required Hne be CP. Produce CP to meet
the straight line GA, to whose points the points of CE are to be
related.'"'' Then, let MA=CB=y ; and CM=BA=.r. An equation
must be found expressing the relation between .r and y.''''' I let PC=i',
PA=7', whence FM^v—y. Since PMC is a right triangle, we see that
s", the square of the hypotenuse, is equal to s--\-v-—2vy-\-y-, the sum
of the squares of the two sides. That is to say, x= ^s-—v'^-{-2z>y—y-
or y= V + '^s^ —X' . By means of these last two equations, I can elimi-
nate one of the two quantities x and 3' from the equation expressing
the relation between the points of the curve CE and those of the straight
line G A. If .r is to be eliminated, this may easily be done by replacing
.r wherever it occurs by ^s' — v^ -\-2vy — yr , x' by the square of this ex-
pression, x^ by its cube, etc., while if y is to be eliminated, y must be
replaced by v -\- V/— .^-'^ and y',y^, ... by the square of this expres-
'^*"' That is, the angle between two curves is defined as the angle between the
normals to the curve at the point of intersection.
'""' That is, the line GA is taken as one of the coordinate axes.
''^-' This will be the equation of the curve. See also the figure on page 97.
95
SECOND BOOK
sion, its cube, and so on. The result will be an equation in only one
unknown quantity, .i' or 3'.
For example, if CE is an ellipse, MA the segment of its
axis of which CM is an ordinate, r its latus rectum, and q its trans-
verse axis,'""' then by Theorem 13, Book I, of Apollonius,'"'' we have
x^ = ry — -y' . Eliminating x' the resulting equation is
2
Ç" ' "" - ■ g-r
Î 2 , o 2 ^ 2 ^^ 2 I qyy-2qvy + gv'-gs'
s —V +ivy—y =ry — - y , or y -\ = 0.
In this case it is better to consider the whole as constituting a single
expression than as consisting of two equal parts.'""'
If CE be the curve generated by the motion of a parabola (see pages
47, et seq.) already discussed, and if we represent GA by b, KL by c,
and the parameter of the axis KL of the parabola by d, the equation
u"] "Le traversant."
'"'^Apollonius, p. 49: "Si conus per axem piano secatur autem alio quoque
piano, quod cum utroque latere trianguli per axem posita concurrit, sed neque basi
coni parallelum ducitur neque e contrario et si planum, in quo est basis coni,
planumque secans concurrunt in recta perpendicular! aut ad basim trianguli per
axem positi aut ad earn productam quselibet recta, quae a sectione coni communi
sectioni planorum parallela ducitur ad diametrum sectiones sumpta quadrata aequalis
erit spatio adplicato rectje cuidam, ad quam diametrus sectionis rationem habet,
quam habet quadratum rectse a vertice coni diametro sectionis parallels ducts usque
ad basim trianguli ad rectangulum comprehensum rectis ab ea ad latera trianguli
abscissis, latitudinem rectam ab ea e diametro ad verticem sectionis abscissam et
figura deficiens simili similiterque posita rectangulo a diametro parametroque com-
prehenso; vocetur autem talis sectio ellipsis." Cf. Apollonius of Perga, edited by
Sir T. L. Heath, Cambridge, 1896, p. 11.
■'"' That is, to transpose all the terms to the left member.
96
Livre Second.
?<f3
bien fîc'cft^, en mettant en fon lieu j/^- i^ss-xx , 6c
le quarré, ou le cube,&c. de cete (bmme, au lieu dyy,o\x
y &c. De façon qu'il rcfte toufîours après cela vne equa-
tion, en laquelle il ny a plus quVne feule quantité" indé-
terminée, a;, ou^.
Comme fi C E eft vne Ellipfe , 6c que M A foit le
fegment de fon diamètre, auquel G M foit appliquée par
ordre, & qui ait r pour fon cofté droit , & ^ pour le tra-
uerfantjonàparle 15 th.
du I liu. d'Apollonius.
S6XX>ry"^y y , d*oa
oftant XX, il refte fS"-
.r
- vv-b-zvy-yy X) ry--yy,
oubien,
y y ^ ^^"^V^,^ ^^"^efgala rien, car il cft mieux eu
cet endroit de confîderer ainfî enfemble toute la fbm-
me y que d'en faire vne partie efgale a l'autre.
Tout de mcûne fî C
E eft la ligne courbe
defcrite par le mou-
uement d'vne Parabole
en la façon cy deiTuj
expliquc'e, ôc qu'on ait
pofë^pourGA, c^oax
KL, & ^ pour le cofte
droit du diamètre KL
e n JUparabole : l'equatio
qui explique le rApport
qui
97
344-
La Géométrie.
r-^
- "1 C.d-K "Y ■-• i,h h c d-\
i?yi>i*hb\^^Ahcd K y^ ccdd(
^tidJ ^ - ^ddv-y - ddssC
gui éft entr-e oc Uy, c^y — hyy — c dy H- b c d ^ d xy x> o*
d'où oltant x , on a j — byy — ^ây-hbcd-^Ay
V ss—vp-^z.vy—yy, & remetrant en ordre ces
termes parle moyen de la multiplication, il vient
- i^b b c d-\
yy -- zb c cddy >ii bb ccddxio
<i(d.d Tj V
Et ainfi des autres.
Mefme encore que les poins de la ligne courbe ne fê
rapportafTentpasenlafaçonqueiay ditte a ceux d'vne
ligne droite, mais en tCKite autre qu'on fçauroit imagi-
j]er, on ne laifle pas de pouuoir toufîour s auoir vne telle
equation- ^ Comme fi Ç E eft vne ligne , qui ait tel rap-
port aux trois poins F, G, &: A, que les lignes droites ti-
rées de chafcun de fes poins comme C^iufques au point
F, furpafTent la ligne F A d'vne quantité, qui ait certaine
proportiôdon-
Ql^^^-s?^^ nce a vne autre
^ quantité' dont
GA furpafleles
lignes tire'es
des mcfmes
poins iufques à G. Faifons GAoo^, AFoor, & prenant
àdifcretionlepoint C dans la courbe, que la quantité
dont CF furpaflfe FA. foit à celle dont G A furpaffe
GC, commè^à^, en ibrteque fi cete quantité qui eft
indéterminée fe nomme .^iFC eftcH-:{,&GCeft^ — ^:{.
PuispofantMAcoy, G -Aedb-y, ScFM eft^^-;', &
iicaufe du triangle rWlmgle CM G, oftant le quarré
de
98
SECOND BOOK
expressing the relation between x and v is y^ — by^ — cdy-^bcd-\-d.Y\=0.
Eliminating x, we have
y^—l7y-—cdy + [h'd+ dy \s-—v'^-\-2vy—y-=0.
Arranging the terms according to the powers of y by squaring/'"' this
becomes
y<^-2hy''-^{h--2cd-\-d-)y*^{Ahcd—2d-v)y'^
-{-(c"d-—d~s--ird-v-—2b-cd)y-—2bc-d-y+b-c-d-=0,
and so for the other cases. If the points of the curve are not related
to those of a straight line in the way explained, but are related in some
other way,''^'' such an equation can always be found.
Let CE be a curve which is so related to the points F, G, and A,
that a straight line drawn from any point on it, as C. to F exceeds
the line FA by a quantity which bears a given ratio to the excess of GA
over the line drawn from the point C to G.''**' Let GA=&, AF=c, and
taking an arbitrary point C on the curve let the quantity by which CF
exceeds FA be to the quantity by which GA exceeds GC as d is to e.
Then if we let c represent the undetermined quantity, FC=c+:: and
GC = l>--,z. Let MA=;', GM = ô-y, and FM = r+j'. Since CMG is a
d
right triangle, taking the square of GM from the square of GC we have
i"«i "j7n remettant en ordre ces termes par moyen de la multiplication."
'"'' "Mais en toute autre qu'on saurait imaginer."
^''" That is the ratio of CF — FA to GA — CG is a constant.
99
GEOMETRY
r' 2be
left the square of CM, or --^z^ —j- z-\-2by—y'^. Again, taking the
square of FM from the square of FC we have the square of CM
expressed in another way, namely : z--\-2cz — 2cy — y-. These two expres-
sions being equal they will yield the value of y or MA, which is
2bd^-\-2cd'
Substituting this value for y in the expression for the square of CM,
we have
——2 bd^z--\-ce^z--\-2bcd-z—2bcdez
^^ = b¥+7d-' y-
If now we suppose the line PC to meet the curve at right angles at C,
and let PC=j and FA^î' as before, PM is equal to v—y\ and since
PCM is a right triangle, we have s^-—z>--\-2vy—y- for the square of
CM. Substituting for y its value, and equating the values of the square
of CM, we have
2 2bcd'z-2bcdez-2c(Pvz-2bdevz-bd'^s'' + bd''i?-cd''s'^cd'^v^
^ ^ bd'^-^ce'+e'v-d'v
for the required equation.
Such an equation having been found'""' it is to be used, not to deter-
mine X, y, or z, which are known, since the point C is given, but to
find V or s, which determine the required point P. With this in view,
observe that if the point P fulfills the required conditions, the circle
about P as center and passing through the point C will touch but not
cut the curve CE ; but if this point P be ever so little nearer to or far-
ther from A than it should be, this circle must cut the curve not only
[119] 'pj^ree such equations have been found by Descartes, namely those for the
ellipse, the parabolic conchoid, and the curve just described.
100
Livre Second. 345"
de G M du quarre de G C, on a le quarre de C M, qui eft
'' ^..L!o^_l-2 3y--j/j. puis oftant le quarre' de F M
du quarre'de F C, on a encore le .quarre de C M en d'au-
tres termes, a fçauoir:^:^ 4-2 <: :^— 2 fj'— y j', & ces ter-
mes eftantefgaux auxprecedens, ils font connoiftrej,
ouMA,quicfl;— --TT^j^rr^ -&fubftituantce-
te forame au lieu d)' dans le quarfede C M , ontrouue
qu'il s'exprime en ces termes.
bddz.z. »^ ceez.z <^ i bcddz.-- i bcdcz.
bdd ^ cdd ^ " "jy*
Puis fuppofant que la ligne droite PC rencontre la
courbe à angles droits au point C, Scfaifant PC 30x, &
V k-Xiv comme deuant, PMeftr-y j & a caufe du
trîangle redangle P C M,on à //- vv -I- 2 vy-yy pour
le quarre de C M, ou derechef ayant au lieu d)' fubftitue
la fomme qui luy eft efgale, il vient
►f 1 bcddz. -- 1 bcdez.— i cdd-vz. -- i bdevz. — bddss ►{« bddw-
x{, ' bdd >¥ cee ee v --^df
-- cddss^cddvv. 00 opourTequation que nous cherchions.
Orapre's qu'on à trouuevne telle equation , auliea
des'enferuirpourconnoiftrelcsquantite's .v,ou7, ou ^,
qui font défia donne'es, puifque le point C eft donne, on
la doit employer a trouuert;, ou / , qui déterminent le
point P, qui eft demande'. Et a cet effed il faut confide-
rer,que fi ce point P eft telqu'on le defire, le cercle dont
il fera le centre, &: qui paflera par le point C, y touchera
la ligne courbe C E, fans la coupper: mais que fi ce point
P, eft tant foit peu plus proche, ou plus efloigné du point
Xx A, qu'il
101 ■
^^^ La Géométrie.
A, qu'il ne doit, ce cercle couppera la courbe , non feu-
lement au point C, mais aufîy neeefTairement en quel-
que autre. Puis il faut aufïyconfîderer, que lorfque ce
cercle couppe la ligne courbe C E, l'équation par laquel-
le on cherche la quantité' :v, ou 7, ou quelque autre fem-
blable, en fuppofant P A & P C eftre connues, contient
neceffairement deux racines, qui font inefgales. Car par
exemple fi ce cercle couppe la courbe aux poins C & H,
ayant tire E Qjparallele a CM, les noms des quantités
indéterminées x 5f^, conuiendront aufly bieii aux lignes
EQ^&:QA,quaCM, &MAj puis PEeft efgale a
PC,.acaufe du cercle, fi bien que cherchant les hgnes
EQ & QA, parPE &
P A qu'on fuppofe com-
me données , on aura la
mefme equation , que fi
on cherchoic C M &
M A par PC,PA. d'où
il fuit euidcmment,que la
valeur d'AT, ou d'/, ou de
telle autre quantité qu'on aura fuppofee , fera double en
cete equation, cell a dire qu'il y aura deux racines ineL
gales entre elles; ocdontl'vue feraCM, l'autre EQ, fi
c'eft X qu'on cherche- oubien l'vne fera M A , & l'autre
Q Ajfic'efty. &ainfi des autres. Il eft vray que fi le
point Ene fe trouue pas du mefinecofte de la courbe
que le point Cj il ny aura que l'vne de ces deux racines
qui fait vraye, & l'autre fera renuerfec, ou moindre que
rien: mais plus ces deux poins, C, & E, font proches l'vn
de l'autre, moins il y a de difference entre ces deux raci-
nes;
p M
QjS
102
SECOND BOOK
at C but also in another point. Now if this circle cuts CE, the equation
involving x and y as unknown quantities (supposing PA and PC
known) must have two unequal roots. Suppose, for example, that
the circle cuts the curve in the points C and E. Draw EQ paral-
lel to CM. Then x and 3' may be used to represent EQ and QA respec-
tively in just the same way as they were used to represent CM
and MA; since PE is equal to PC (being radii of the same circle),
if we seek EQ and QA (supposing PE and PA given) we shall get the
same equation that we should obtain by seeking CM and MA (suppos-
ing PC and PA given). It follows that the value of x, or y, or any
other such quantity, will be two-fold in this equation, that is, the equa-
tion will have two unequal roots. If the value of x be required, one of
these roots will be CM and the other EQ ; while if y be required, one
root will be MA and the other QA. It is true that if E is not on the
same side of the curve as C, only one of these will be a true root, the
other being drawn in the opposite direction, or less than nothing.''""^ The
nearer together the points C and E are taken however, the less differ-
ii^o] "j7^ l'autre sera renversée ou moindre que rien."
103
GEOMETRY
ence there is between the roots ; and when the points coincide, the roots
are exactly equal, that is to say, the circle through C will touch the
curve CE at the point C without cutting it.
Furthermore, it is to be observed that when an equation has two
equal roots, its left-hand member must be similar in form to the expres-
sion obtained by multiplying by itself the difiference between the
unknown quantity and a known quantity equal to it ;^'"^ and then, if the
resulting expression is not of as high a degree as the original equation,
multiplying it by another expression which will make it of the same
degree. This last step makes the two expressions correspond term by
term.
For example, I say that the first equation found in the present dis-
cussion,'"^' namely
a , çn' — "^çvy + q'v^ — qs^
y + ,
q-r
must be of the same form as the expression obtained by making ^=y
and multiplying y — e by itself, that is, as 'f- — 2ey-\-e'. We may then
compare the two expressions term by term, thus : Since the first term,
nyv '2,p'vv
•f , is the same in each, the second term,'"^' ^-^ ^-^, of the first is
q—r
equal to —2ey, the second term of the second ; whence, solving for v,
r 1
or PA, we have v = e—~e-\-~r, or, since we have assumed e equal to;',
q 2
r 1
v=y — -y-\-~ r. In the same way, we can find ^ from the third term,
q I
"^^'^ That is, the left-hand member will be the square of the binomial x — a
when ;ir = a.
'^'^'■'^ See page 96. The original has "first equation," not "first member of the
equation."
[163] That is, the second term in ;y.
104
Livre Secokd. 347
nesj &: enfin elles font entièrement efgales, s'ils font tous
denxioins en vn^ c*eft adiré fi le cercle, qui palTe par C,
y touche la courbe CE fans la coupper.
De plus il faut confiderer, que lorfqu'ily a deux raci-
nes efgales en vue equation, elle a neceflairement la
mefme forme,que fi on multiplie par foy mcfme la quan-
tité" qu'on y fuppofe eftre inconnue moins la quantité
connue qui luy^ft cfgale, & qu'après cela fi cetc dernière
fommen'apas tant de dimenfions que la précédente,
on la multiplie par vne autre fomme qui en ait autant
qu'il luy en manque^ afiîn qu'il puiffe y auoir feparement
equation entre chafcun des termes de l'vne , & chafcun
des termes de l'autre.
Comme par exemple ic dis que la premiere equation
trouuee cy deflus, afçauoir
y y — ; — aoitauoirlamefine forme que
celle qui feproduift en faifànt^ efgala/, & multipliant
ye par (by mefiiie,d'où il vient ^y — zey-^-e e, en forte
qu'on peut comparer fèparement chafcun de leurs ter-
mes, & dire que puifque le premier qui eft; ; eft tout le
mefme en Tvne qu'en l'autre, le fécond qui eftenlVnc
qr y - -z (i v y,
—TTr — ^ft €%^^ ^" fecôd de l'autre qui eft - 2 ey ,d'où
cherchant la quantité' v qui eft la ligne P A , on à
v'Xie — ~^-H ï?*, oubie
a caule que nous auons
fuppofe' e efgal a; , oti a
Xx a ainfi
105
^4& l'A GEOMETRIE.
ainlî on pourroit trouuer s par le troifîefine reime
ee co^^^^^^^^^T^'maispourceque la quantité t/ determine
affés le point P,qiiî eft le feul que nous cherchions,on n'a
pas befoin de pafTer outre.
Tout de mefme la féconde equation trouuée cy dç(-
fus, a fçauoif,
i^i dd-^ - idd-uJ '- d d ssC
>itd d V -v^
doit auoir mefme forme , que la fomme qui fe produifir
lorfqu'on multiplie ^^ '-^ei -A- ee par
4 î 5 4
y -^fj '-^ggn^^^y-^ -i, qui eft
- "^^^ >hee,-' ^eef Ç ^eeggJ ^ e e t?ij
de façon que de ces deux equations i'en tire fix autres,
qui feruent a connoiftre les fix quantite^s /^ g, h, \, v, & j :
D'où il eft fort ayfe' a entendre, que de quelque genre,
qucpuiffe eftrela ligne courbe propofee, il vient tou-
fiours par cete façon de procéder autant d'équations,
qu'on cft obligé de fuppofer de quantités , qui font in-
connues. Mais pour demeller par ordre ces equations,
& trouuer enfin la quantité z^, qui eft la feule dont on a
befoin, & à l'occafion de laquelle on cherche les autres:
Il faut premièrement par le fécond terme chercher/, la
premiere des. quantités inconnues de la dernière fom-
me, & on trouue/:» ze— ib.
Vu\s par le dernier il faut chercher /^1a dernière des
quantite's inconnues de la mefme fomme, ôc on trouuc
bbccdd.
/•^30—
^ ee
Puis
106
SECOND BOOK
2 Of' — qs'
e — ; but since v completely determines P, which is all that is
q—r
required, it is not necessary to go further.''"''
In the same "way, the second equation found above, '''^' namely,
4- (rV^ - 2/)-r./+ d'-i- - d's' )/ - 2âr'dy + /; W' ,
must have the same form as the expression obtained by multiplying
_v-— 2^3'+^- by y^-\-fy'''+g-y--\-lry-\-k*,
that is, as
y'-^(f-2e)y'-\-(cf--2ef^c~)y*Jr(Ji'-^eg"-+e-f)y'
-\-(k'—2eJr-\-e-g-)y--{-(e-h"-2ek')y^e'kK
From these two equations, six others may be obtained, which serve to
determine the six quantities /, g, h, k, v, and s. It is easily seen that
to whatever class the given curve may belong, this method will always
furnish just as many equations as we necessarily have unknown quan-
tities. In order to solve these equations, and ultimately to find v, which
is the only value really wanted (the others being used only as means
of finding îO. we first determine /. the first unknown in the above
expression, from the second term. Thus, f=2e — 2b. Then in the last
terms we can find k, the last unknown in the same expression, from
'"'' That is, to construct PC we may lay off AP = 7' and join P and C. If
instead we use the value of e, taking C as center and a radius CP = r, we con-
struct an arc cutting AG in P, and join P and C. Rabuel, p. 309. To apply
Descartes's method to the circle, for example, it is only necessary to observe that
all parameters and diameters are equal, that is, q^r; and therefore the equation
7' = y v-|- — ;- becomes z'= _, ^ = — diameter. That is, the normal passes
through the center and is a radius of the circle. Rabuel, p. 313.
''■''^' See page 99. As before, Descartes uses "second equation" for "first mem-
ber of the second equation."
107
GEOMETRY
which fe*^ — ^ — . From the third term we get the second quantity
g--=Ze-—Ahe—2cd^h-^d-.
From the next to the last term we get h, the next to the last quantity,
which is'"°'
2^VV2 2^rV2
h' =
ê '
In the same way we should proceed in this order, until the last quantity
is found.
Then from the corresponding term (here the fourth) we may find
V, and we have
le" T^be^ b'^e 2ce 2bc b^ l^V\
a add dee
or putting y for its equal <f, we get
2y^ ^by"" b'^y 2cy 2bc b^ bh^
for the length of AP.
""1 Found from.
108
L I V R E s E C O N D. 34P
Puis par le troifiefme rerme il faut chercher a la féconde
quantité, &ona^^30 ^ ee — ^^be — z cd'r' bb-i-dd.
Puis par le pcnukiefnie il faut chercher /j la penultiefîne
quantité, qui eft Z» ' oo
ib b c cdd 1 bccdd .
ei
Etaiiiiî il fau-
droit continuer fuiuant ce mefme ordre iufques a la der-
nière, s'il y en auoit d'auantage en cete fomme • car c'eft
chofe qu'on peut toufîours faire en mefme façon.
Puis par le terme qui fuit en ce mefme ordre, qui eft
icy le quatriefrae, il faut chercher la quantité' v, & On a
vX>-
h b e 1 ce i bc
bec hh c c^
? bee
dd ~'~dd" '' dd~~ d ' " ' d -■ ee
©u mettant/ au lieu d'^ qui luy cft efgal on a
-y t ^^yy ^^y -^y* ^^^ bec bbcc.
.... ~~^ -
f/30
d
7
dd d4i ' dd
pour la ligne A P,
Etainfila troifiefme equation; qui eft
Xx 3
yy
y'
K.^'
109
iSO La GEOMETRIE.
tft zbcddz'- xbcdex.--z cddvz, — ibdevK •• bddss ifi b ddvv-
K\-
bdd i^t6t^ eev'
■ ' cdds s >î< c ddvv ,
a la mefme forme que
^^'-if^-^ff, en fuppofant/efgal a ;^, fi bienque il
y a derechef equation entre— 2/, ou — 2 :{, &
>i* 1 b c dd -' 1 hc d e — î. cddv --1 hdcTJ .
' Tdd>i<cee>i.eev..ddv d OÙ OU COmioift qUC
« . / /1 bcdd-bcde>i* bddz. ^ ceez
ia quantité v eft -7di:^JJ7..ee^^dd^
C'eftpourquoy
composant la
ligne A P , de
cete fbmme ef^
gale à V dont
toutes les quan-
tite's font connues, ôc tirant du point Painfî trouue", vne
ligne droite vers C, elle y couppe la courbe CE a an-
gles droits, qui eft ce qu'il falloit faire. Et ie ne voy rien
qui empefche, qu'on n'eftende ce problefme en mefme
façon a toutes les lignes courbes, qui tombentfous quel-
que calcul Géométrique.
Mefme il eft a remarquer touchant la dernière fom-
me, qu'on prent a difcretion , pour remplir le nombre
des dimenlîons de l'autre fomme , lorfqu 'il y en man-
que , comme nous auons pris tantoft
y ''^ fy ' "^Zg, y y -h /^ '^ -+- >^^ 5 que les lignes -^ & —
ypeuuenteftrefuppofestels, qu'on veut, fans que la \U
gne Vf ou A P, fe trouue diuerfè pour cela , comme vous
pourresayfement voir par experience, car s'il falloit que
icm'areftalTeademonftrertous les theorefmes dont ie
fais
110
SECOND BOOK
Again, the third''"' equation, namely,
Ibcd^'z - 2bcdez - 2cdh'2 - 2bdevz - bd^-s' + bd^-v'—cd's^ + cd'h^
2' + -
bd^'+ce^+e'v-d'^v
is of the same form as zr—2fc-\-f- where /=r, so that —2/ or —2z
must be equal to
2bcd'^ - 2bcdc - 2cd'^v - Zbdev
bd^+ce'' + é\'-d\'
whence
bcd"^ - bcde -\-bd'^z+ ce^z
^'~ cd''-^bde-€''z\d''z '
Therefore, if we take AP equal to the above value of v, all the
terms of which are known, and join the point 1' thus determined
to C, this line will cut the curve CE at right angles, which was required.
I see no reason why this solution should not apply to every curve to
which the methods of geometry are applicable.''"'
It should be observed regarding the expression taken arbitrarily to
raise the original product to the required degree, as we just now took
that the signs + and — may be chosen at will without producing dif-
ferent values of V or AP.'''°' This is easily found to be the case, but if
I should stop to demonstrate every theorem I use, it would require a
'"'' First member of the tliird equation.
'"*' Let us apply this method to the problem of constructing a normal to a para-
bola at a given point. As before, s^ — x- -^ v- — 2vy ^ y- . If we take as the
eciuation of the parabola .r- = ry, and suljstitute, we have
j= =: rv 4- e'= — 2tt + J- or v^ + (r — 2zO.V + ^'- — ^" = 0-
Comparing this with y- — 2cy^ c- — '^, we have r — 2v = — 2c\ v~ — s- = e- ;
t;=J + f. Since e = y, v^^- + y. Let AM = r. and 7' = AP ; then
AM — AP = MP = one-half the parameter. Rabuel, p. 314.
['"^ It will be observed that Descartes did not consider a coefficient, as a, in the
general sense of a positive or a negative quantity, but that he alwavs wrote the
sign intended. In this sentence, however, he suggests some generalization.
Ill
GEOMETRY
much larger volume than I wish to write. I desire rather to tell you
in passing that this method, of which you have here an example, of sup-
posing two equations to be of the same form in order to compare them
term by term and so to obtain several equations from one, will apply
to an infinity of other problems and is not the least important feature
of my general method.'""^
I shall not give the constructions for the required tangents and nor-
mals in connection with the method just explained, since it is always
easy to find them, although it often requires some ingenuity to get short
and simple methods of construction.
[160] Yhe method may be used to draw a normal to a curve from a given point,
to draw a tangent to a curve from a point without, and to discover points of
inflexion, maxima, and minima. Compare Descartes's Letters, Cousin, Vol. VI,
p. 421. As an illustration, let it be required to find a point of inflexion on the
first cubical parabola. Its equation is y" = a-x. Assume that D is a point of
inflexion, and let CD = y, AC = x, PA ^ s, and AE =: r. Since triangle PAE is
similar to triangle PCD we have -^. — =-, whence .v = " . Substituting in
A' + j 5 r
the equation of the curve, we have \'^ — — ^+a-j^O. But if D is a point of
r
inflexion this equation must have three equal roots, since at a point of inflexion
there are three coincident ixjints of section. Compare the equation with
y^ — Zey- + Zc-y — e^ = 0.
Then Ze"^ = 0 and e ^0. But c ^ y, and therefore y ^^ 0. Therefore the point of
inflexion is (0, 0). Rabuel, p. 321.
It will be of interest to compare the method of drawing tangents given by
Fermât in Methodus ad disquircndam maxiniam et minimam, Toulouse, 1679,
which is as follows : It is required to draw a tangent to the parabola BD from a
point O without. From the nature of the parabola > -, since O is without the
DI tj i^
curve. But by similar triangles 5£. = ^l^. Therefore —>£^. Let CE = a,
CI = e, and CD = ^; then DI = d — e, and -; — — >7 ^- : whence
a — c (a — e)^
de- — 2ade > — a-e.
Dividing by e, we have dc — 2ad > — a-. Now if the line BO becomes tangent to
the curve, the point B and O coincide, de — 2ad = — a-, and e vanishes ; then
2ad — a- and a — 2d in length. That is CE = 2CD.
112
Livre Secokd. $fx
fais quelque mention, ie ferois contraint d'efcrire vn vo-
lume beaucoup plus gros que ie ne defîre. Mais 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. ^ cddz,. ^ ^ 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
bbcde 4« bccde — bceex, - - cceez "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 0 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 = 0 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 := 0 and multiply this by
X- — Sx-\-6 = 0, we have x^'—9x--\-26.v — 24 = 0 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 = 0 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 = 0
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 0
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 = 0
we vv^rite
_|_,t-4^4^-n_i9_^.2_io6.r-120 = 0
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 = 0
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 = 0 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 0 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 = 0 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* = 0 ; 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 0 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 = 0
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/
= 0
0 _ 16/ - 128y2
- 16 - 16
+ /+ 8/+ 4 =
= 0
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- = 0
we find that the division can be performed as follows :
+ /+ aM 4 -«4 » 2- «' 1
0 - 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 0
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 0
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 = 0
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 = 0
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
= 0
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 = 0 instead of .r* — I7x- — 20a' — 6 = 0 we
find that y- = 16; then, instead of the original equation
-f x' — 17 X- — 20.r — 6 = 0
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 = 0 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 = 0
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 = 0 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 pourquoy il faut 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
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