A L G 



A L G 



falfe onfs ns faccefTions of the fame figiis ; which hail before 

 been paitly flicwii by Cardan and V'icta from tlie relation of 

 the co-efficients niid their fitjns, and more fully by Harriot. 

 Hence Ues Cartes was led to adopt Cardan's method of 

 changing the true roots to falfe, and the falfc to true, by 

 merely changing the figns of the even terms. He then di- 

 redls his attention to other rcdudlions or tranfnuitations 

 taught by Cardan, Vieta, and Harriot ; fuch as increafmg 

 ordiminilhing the roots by any quantity, taking away thefe- 

 cond term, and altering the roots in any pioportion, and thus 

 extricating the equation from fraftions and radical;;. Having 

 obfervtd (p. 76.) that the roots of equations, both true and 

 falfe, may be cither real or imaginary, which imaginary roots 

 were firll noticed by Albert Girard, as in the equation 

 «' — 6xx 4" I J*' — 10 » o, that has only one real root, 

 Wz. 2 ; he proceeds to the depreffion of a cubic equation to 

 a quadratic or plane problem, &c. that it maybe conftrucled 

 by the circle, by dividing it by one of the binomial faftors, 

 which, in Harriot's method, compofe the equation. As Pe- 

 letarius had fliewn that the fimple root is one of the divifors 

 of the known term of the equation, and Harriot had ob- 

 ferved that this term is the continual produft of all the 

 roots ; Des Cartes tries all the fimple divifors of that term, 

 till he finds one of them, which coiincfted with the unknown 

 quantity .x by + or — , will exaftly divide the equation : and 

 the fame procefs ferves for higher powers than the cube. But 

 when a divifor cannot be found in this way, in order to de- 

 prefs a biquadratic equation into a cubic one, he gives a new 

 rule for diffolving it into two quadratics, by means of a cu- 

 bic equation, in the following manner (p. 79, &c.) : Let 

 the given biquadratic equation be -\- x^ * .J>xx . gx . r x> o ; 



ip. -i-X) 



And fuppofe it com- 1 + •■^■•■*-' ~ J'*' 4" iyj 



pofed of thefe two \ 



others, uiz. I -j- xx -{-yx •}- iyy . \p . -!— » o : 



L 2y 



in which two quadratic equations the fign of Ip mud be the 



fame with the fign of^ in the given equation ; and in the 



firil of them, having— j-.y, the fign of — ^ muft be the fame 



with that of ^or-j-; and in the fecond quadratic, having -|-v.y, 

 its fign mull be — ; and -vice verf,i. Then find the root 



-^-:^ — of the following cubic equation, viz. y'' . 2py^ -j- 



_ HV ~ ?? X' Oj in which the fign of 2/1 is the fame with that 



4'". 

 of ^ in the given biquadratic, but the fign of ^r contrary' to 



that of r in the fame equation ; and the value ofj', deduced 

 hence, and fubllituted for it in the two quadratic equations, 

 and their two pairs of roots being taken, thefe will be the 

 four roots of the propofed biquadratic. E. G. Let the bi- 

 quadratic be .V' * — 4V.V — 8.V + 35 30 o, for which mull be 

 fubllituted y'' — 8)'' — 124v_v — 64 30 o ; becaufe the quan- 

 tity called p being in this cafe — 4, — Sj' mull be fublli- 



+ 16 

 tuted for .2/v'> and r being -|" 35' .vv> <"■ ~ '^4.)'J'> 



muft be fubllituted for i^ 17 



-4'' 



be — 64. And fo of others. In the fame manner, fays 

 Des Cartes, may equations of the 6th power be reduced to 

 thofe of the 5th, and thofe of the Sith power to thole of the 

 7th, &c. The invclligation of this rule is not given by Des 

 Cartes ; but it has been evidently done by affuming indcter- 

 mlnate quantities after the manner of Ferrari and Cardan, as 

 co-effieients of the terms of the two quadratie equations, and 

 after multiplying the two together, determining tlicir values 

 by comparing the rcfulting terms with thofe of the propofed 



140 

 and q being 8, — qq will 



biquadratic equation. Des Cartes, after thefe rcduolions, 

 in order to limijlily and deprels the equations as much ab pof. 

 fible, proceeds to give tliVconllrnclion of folid and other 

 higher pniblems, or of cubre; and iiigher equations by means 

 of parabolas and circles ; obferving, that tlie falfe roots are 

 denoted by the ordinates to the paiabola lying on the con- 

 trary fide of the axis to the true roots: and he cUfes the book 

 witli illulli-ating thefe conllruftions by various problems 

 concerning the tiifeftion of an angle, and the invclligation 

 of two or four mean proportionals. 



Of the improvements contained in this work, it is ob- 

 ferved by Dr. Hutton, that Des Cartes, with a view to the 

 more eafy application of equations to the conllrudtion of 

 problems, mentions many pai ticulars concerning the nature 

 and redtiftion of equations, and ftates them in his own lan- 

 guage and manner, which is ufually more clear and explicit 

 than that of others, and frequently accompanied with his 

 own improvements. Here he chiefly followed Cardan, 

 Vieta, and Harriot, and efpecially the lall ; explaining fome 

 of their rules and difcoveries more dillinftly, and with fome 

 little variation in the notation, in which he puts the firlt 

 letters of the alphabet for known, and the latter letters 

 for unknown quantities, a ' for a a a, &c. and 30 for =. 

 But Herigone had two years before ufed the fame numeral 

 exponents. Des Cartes explained or improved moil parts of 

 the reduftion of equations, in their various tranfmutations, 

 the number and nature of their roots, true and falfe, real 

 and imaginary, as he calls them, or as they are denominated by 

 Girard, involved ; and alfo the depreffion of equations to lower 

 degrees. His inventions and dilcoveries comprehend the appli- 

 cation of algebra to the geometry of curve lines, theconlliuc- 

 tion of equations of the higher orders, and a rule forrefolving 

 biquadratic equations by means of a cubic and two qua- 

 dratics. 



Fermat, who publirtud Diophantus's arithmetic with 

 valuable notes, was a contemporaiy of Des Cartes, and 

 alio a competitor for fome of his moll valuable dif- 

 coveries. This ingenious mathematician, before the pub- 

 lication ol Des Cartes's geometry, had applied al- 

 gebra to curve lines, exprefled them by an algebraic 

 equation, and by them conftrucled equations of the 3d and 

 4tli orders ; and he had alfo difcovered a method of tan- 

 gents, and a method de maximis et minimis, approaching 

 very nearly to the inethod of fluxions or increments, in the 

 manner of treating the problems as well as in the algebraic 

 notation and procefs. Fermat was alio diflinguilhtd by his 

 knowledge of the Diophantine problems. 



At the period to which we have now rcfen-ed, algebra 

 had acquired a regular and permanent form ; and from this 

 time the wiiters on the whole, or detached parts of this 

 fcience, became (o numerous, that the limits of this article 

 will fcareely admit our reoiting their names and publica- 

 tions, and much lets doing jullice to the improvements 

 which this branch of mathematical fcience derived from 

 their performances. In the courfe of our biographical ar- 

 ticles, and on other occafions, we lliall endeavour to fupply 

 the defefts of the prefent curfory notice. 



The geometry of Des Cartes engaged the attention of feveral 

 mathematicians in Holland, when it waspubhihed ; and alfo 

 in France :md England. Francis Sehootrn, profeflor of ma- 

 tliematics at Lcyden, was one of the firll cultivators of the 

 new geometiy ; and in 1649 he publillied a tranflation of 

 Des Cartes's geonntry, from tlie French into Latin, with his 

 own commentary and notes by M. de Beaune. In 1659, 

 appeared an eiilargeii edition in two volumes, with fevend 

 additional pieces by De Beaune, Hudde, Van Hcuract, De 

 Witt, with fome tracts by Schooten the editor. Rabuel, 

 a jefuit, publiflRd an elaborate commcntai'v on the fame 

 4 1' 2 work i 



