A L G 



trafk on angular fertions ; and concUuies the work -\viih the 

 uiimei^l rdolutloii uf affccttJ equations, in thf manner of 

 Viita, but more t\iil;i.it. 



1» 1637 Dos Caitcs lirtt piibliftieil liis geometry, wliicli 

 may be conliJtrtd as an ap^jlicatioo of algi bra to geometry, 

 a;id not as a ftpanite trcittift oii cither of liicfe fcieiices. As 

 Dr. Wallis has manifellcd too great a degri-e of partiality to 

 our touiitryman Han lot, and afe-ribed to him difcovtrits which 

 had been made by Victa and others ; and as liombelli and 

 M. dt Gua, in the Memoirs of the Academy of Sciences lor 

 1741, cited i'l ihe lall edition of the Encycloptdie, have de- 

 viated far into the other extreme, in unduly extolling the dif- 

 rcoveries of Victa, and thofe of Des Cartes, to the prejudice 

 of Harriot, we ihall avail ourfelvcs of the analyfis of Dr. 

 Hutton in giving a particular account of the improvements 

 and inventions of Des Cartes, that our readers may be able 

 to form their own judgment in this controverfy. Mon- 

 tucia indeed feems to liave given an im])artial account 

 of tile elifcoveries both of Harriot and Des Cartes, in- 

 termixed with reflections, which fome may think lefs 

 candid than they ought to have been, on our illuftrious coun- 

 tr)-man Dr. Wallis. Hid. des Mathem, torn. ii. p. 106 — 

 186. This excellent hidorian of the mathematical fcicuces 

 acknowledges, that Des Cartes might poffibly have been in- 

 di.btcdto Harriot, though lie thinksil very probable that the 

 principal difcoveries of his geometiy were anterior to the 

 date of the work of the Englilh analyft. It ouglit however 

 to be recoUeited, that the work of Harriot was pofthumous, 

 that he lived to the age of 60, and that his difcoveries, at a 

 period when the fpirit of enquiry was excited, might have 

 been communicated to men of Icience, between wliom an in- 

 tercourfe fubfilled, long before he died. Moiitiicla, by way 

 of balancing the account between Des Carles and Harriot, 

 or rather between Wallis and the partial advocates of Des 

 Cartes, intimates, that if Des Cartes was indebted to Harriot, 

 the latter was under no lefs important obligations to Vieta, 

 vhofe works were publilhed before the year 1 600. To 

 ftrengthen the prefumption that this might have been the 

 cafe, he alledges, on the authority of Sherburii, the tranfla- 

 tor of Manilius, that Vieta had for fome time employed an 

 Engli(h fecretary, or amaiiuenlis, whofe name was Nathaniel 

 Torporlev : and as this Torporley was frequently in familiar 

 uitercourfc with Harriot at the table of the Duke of Nor- 

 thumberland, he fuggells the probability of his having com- 

 municated the ideas and manufcripls of Vieta, of which he 

 was the depofitar)' to Harriot. 



The geometry of Des Cartes (Apud Opera, torn. iii. 

 Francof. ad Moeiuim, 1695, 410.) confifts of three books. 

 The firfl is entitled, " De Problematibus, qux conftrui pof- 

 funt, adhibendo tantum rectas lineas et circulos." In this 

 book the author flievvs how to accommodate arithmetical 

 computation to geometrical operations. For this purpofe 

 he affumes a line to reprefent unity, and then, by means of 

 proportionals, teaches the method of multiplying, dividing, 

 and extracting of roots by lines. He proceeds to explain his 

 mode of notation, which is not diflerent from that of other 

 authors. AITuming a and h for two quantities, their fum is 

 (xpreffed by a -f- />, their difference by a — b, their produft 



by ab, their quotient by -j- , the fquare of a by aa or a', its 

 b 



cube by a', &c. the fquare root of a' -\- b' by ^/^i 1 p 



and the cube root by ^yc — a^ — b^-j-ab/), &c. He then 

 fhews, as Stifclius had done, that there muft be as many equa- 

 tions as there are unknown lines or quantities, and that all 

 of them mull be reduced to one final equation, by exteiini- 



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nating all the unknown letters except ojie ; fo that the final 

 equation u ill appear in the following forms, the character 50 

 being fubilituted for =: or equality, and the highell term or 

 power being on one iide of the equation, and the other term* 

 with theii proper figns on the other lidc ; 



« 30 b, or, 



z' x> — /72 -}- b', or, 



K^ X -f- fls'-f- A z — c', or, 



%*x -\- az.'-\-b'z' -cz -J- fl\ &c. 

 Having defined plane problems, or fuch as can be re- 

 folved by right lines and circles, defcribed on a plane fuper- 

 ficies, and having in the final equation only the 2d power of 

 the unknown quantity, he conllructs fuch equations or qua- 

 dratics by means of the circle, and tluis geometrically invcf- 

 tigates the pofitive root or roots. But if the lines, by which 

 the roots arc determined, neitlier cut nortoucli, he obferves 

 that the equation in this cafe has no pofflble root, or that the 

 problem is impoffible. This book clofes with the algcbrai- 

 cal folution of the celebrated problem, confidered bv the an- 

 cients, which is that of finding a point, or the lociis of all 

 the points, from which if a line be drawn to meet any num- 

 ber of given lines in given angles, the product of tlie fes:- 

 ments of fome of them ihall have a given ratio to that of the 

 reft. 



The fecond book is entitled, " De Natura Linearum Ciir- 

 varum." This is the firft treatife of tlie kind on curve lines 

 produceil by the moderns. The nature of the curve is here 

 expreffed by an equation, containing two unknown or 

 variable lines, and others that are known or conltant, as 



ex V 



j' >3 fy f 1- ay — ac. See Curve. We have in this 



book a difcovery of importance, as it is the firft ftep towards 

 the arithmetic of infinites; and that is the metliod of tan- 

 gents, or of drawing a line perpendicular to a curve at any 

 point, which is an ingenious application of the general form 

 of an equation, generated in the method of Harriot, that has 

 two equal roots, to the equation of the curve. See Tan- 

 gent. 



Thethird book, entitled, "DeConftruftione Problematum 

 Solidorum, et Solida excedentium," commences with remarks 

 on the nature and roots of ecjuations ; and the author ob- 

 ferves, that they liave as many roots as dimenfions ; and he 

 ftiews, after Harriot, that they may be obtained by multi- 

 plying a certain number of fimple binomial equations toge- 

 ther, as .\- — 2 » o, .V — 3 » o, and x — 4 x> o, which pro- 

 duce x' — C)xx -j- 26.V — 24 » o, in which equation x has 

 three dimenfions, andalfo three values, ij;e. 2, 3, and 4. He 

 here remarks, that fome equations have their roots fo/ff, or, 

 as he expreffes it, lefs than nothing, called by us negative, 

 and thefe he contradiftinguifties to thofe that are true or po- 

 fitive, which Cardan had before done. E. G. L.et .x-f-jjoo 

 be multiplied by x' — gxx -f- 26.V — 24 » o, and we Ihall 

 have .V — 4.v^ — 19.V.V -f- io6.v — 120 30 o, in which equa- 

 tion three roots, -viz. 2, 3, and 4 are true, and one, -viz. 5, 

 ftilfe. From the generation or compofition of equations by 

 multiplication Des Cartes naturally deduces their rcfolu- 

 tion, depreffion, or decompofition, by dividing them by the 

 binomial factors which compofed them ; and hence he ob- 

 ferves, that this divifor is one of the binomial roots, and that 

 there can be no more roots than dimenfions, or than fuch as 

 form with the unknown quantity .v binomials that will ex- 

 aftly divide the equation, as Harriot had before {hewn. Our 

 author adverts to other properties, moft of which had been 

 noticed before; e.^. that equations may have as many true roots 

 as the terms have changes of the figns -l-and — , and as many 

 I falie 



