910 



SCIENCE. 



[N. S. Vol. VIII. No. 208. 



that of the side of a square to its diagoual, as 1 : 

 V'2. Theodorus of Gyrene added to this the 

 fact that the sides of squares represented in 

 length by -v/S, -v/S, etc., up to \/17, and 

 Thesetetus, that the sides of any square, repre- 

 sented by a surd, are incommensurable with the 

 linear unit." Now in fact Theodorus and 

 Thesetetus made no representation whatever of 

 the length of these sides, simply saying, e. g., 

 that the side of the square whose area is 3 is in- 

 commensurable with the side of the square 

 whose area is one. For Euclid there was no 

 such ratio as 1 : \/2 ; for 1 is a number and so 

 if it could have had a ratio to \/2 this would 

 have been a number. But Euclid, Boole X., 

 Proposition 7 is: "Incommensurable magni- 

 tudes are not to one another in the ratio of one 

 number to another number." 



The Hindus were the first to recognize the 

 existence of irrational numbers. Even through 

 the Middle Ages and the Renaissance they were 

 absurd fictions, ' numeri surdi,' a designation 

 attributed to Leonardo of Pisa. The first writer 

 to treat them genuinely was Stifel (1544), and 

 even he had not completely freed himself from 

 the older terminology, since he says : "sic irra- 

 tionalis uumerus non est verus numerus atque 

 lateat sub quadam iufinitatis nebula." 



In reference to the next step, the conceiving 

 of ratio as number, Pringsheim says, page 51: 

 ' ' Hatte schon Descartes beliebige Streckenver- 

 hdltnisse mit einfachen Buchstaben bezeichnetj 

 und damit wie mit Zahlen gerechnet," etc. But 

 here I think the careful German has slipped. 



In regard to just this point a common error 

 is still widespread, which we see in the follow- 

 ing, read before Sections A and B of the Ameri- 

 can Association for the Advancement of Science, 

 1891 : 



"The doctrine of Descartes was that the algebraic 

 symbol did not represent a concrete magnitude, but 

 a mere niimber or ratio, expressing the relation of the 

 magnitude to some unit. Hence that the product of 

 two quantities is the product of ratios, * * * ; that 

 the powers of a quantity are ratios like the quantity 

 itself, ' ' etc. 



That every statement here quoted is a mis- 

 take will be instantly seen from the following, 

 taken from pages numbered 297-9 of the original 

 edition cf Descartes' Geometric, 1637, a copy of 



which (perhaps unique on this continent) I have 

 had the good fortune to possess since my stu- 

 dent days (1876). 



"Etoomme toute I'Arithmetique n'est composee 

 que de quatre ou cinq operations, que sont I'Addi- 

 tion, la .Soustraotion, la Multiplication, la Diuision, 

 & I'Extraetiou des raoines, qu'on peut prendre pour 

 vne espece de IMuision : Ainsi n' at' on autre chose 

 a faire en Geometrie touchant les lignes qu'on oherche, 

 pour les preparer a estre connues, que leur en adi- 

 ouster d'autres, ou en oster; Oubien en ayant vne, 

 que ie nommeray I'vnite pour la rapporter d'autant 

 mieux aux norahres, & qui peut ordinairement estre 

 prise a discretion, puis en ayant encore deux autres, 

 en trouuer vne quatriesme, qui soit a I'vne de ces 

 deux, comme I'autre est a I'vnite, ce qui est le 

 niesme que la Multiplication; oubien en trouuer vne 

 quatriesme, qui soit a I'vne de ces deux, comme 

 l'vnit(5 est a I'autre, ce qui est le mesme que la Diui- 

 sion ; ou enfin trouuer vne, ou deux, ou plusieurs 

 raoyennes proportiounelles entre I'vnite, & quelque 

 autre ligue ; ce qui est le mesme que tirer la raciue 

 quarr^'e, ou cabiqne, &c. Et ie ne craindray pas 

 d'introduire ces termes d'Arithmetique en la Geo- 

 metrie, afSn de me reudre plus intelligible. * * * 



" Mais souuent on n'a pas lesioin de tracer ainsi 

 ces lignes sur Ie papier, & il sufiSst de les designer par 

 quelques lettres, chascune par vne seule. Comme 

 pour adiouster la ligne BD a GH, ie nomrae I'vne a 

 & 1' autre b, & esoris a -\- b ; Et a — b, pour soustraire 

 b d' a; Et ai, pour les multiplier I'vne par 1' autre ; 

 EtJ, pour diuiser a par b ; Et aa, ou a'^, pour multi- 

 plier a par soy mesme ; Et a", pour le multiplier en- 

 core vne fois par a, & ainsi a 1' infini ; Et v'a^ + l>^, 

 pour tirer la raciue quarr^e d' a' + 6- ; Et 

 l/C. tt' — b^-\-abb, pour tirer la racine cuhique d' 

 a' — 6'' + abb, & ainsi des autres. 



" Ou il est a reraarquer que par a^ ou b^ ou sem- 

 hlables, ie ne couooy ordinairement que des lignes 

 toutes simples, encore que pour me seruir des noms 

 vsites en 1' Algebre, ie les nommedesquarres, ou des 

 cubes, &c. " 



Thus what Descartes really did was to make 

 a geometric algebra, in which, however, the 

 product of two sects (Strecken) was not a rect- 

 angle but a sect ; the product of three sects not 

 a cuboid but a sect. Here Descartes represents 

 by the single letters a, b, sects, Strecken, not 

 Streckenverhdltnisse. Descartes does not here 

 pass beyond Euclid's representation of the ratio 

 of two magnitudes by two other magnitudes, 

 does not reach the conception of the systematic 

 representation of the ratio of two magnitudes 



