A L G 



^ccogn't!one, et Emen;latione ;" and were not' publidied 

 till the ye.ir 1615, hy vVlexander AndeiTon, an inreniotis 

 Scotfnian, with Tarioiis corrcftions and additions. Vieta^s 

 improvements compvcheiid the following particulars. He firll 

 introduci-d the general ufe of the letters of liie al])hiibet to de- 

 note indefinite given qiiantities. Accordingly he expicfl'es un- 

 k.;o\vn qnantities by the vowels A, E, I, O, U, Y, and the 

 known ones by the confonants B, C, D, &c. Tie alio invented 

 many terms and forms of expredion which are in prefent 

 ufe ; as co-efileient, afRrniative and negative, pure and nd- 

 feC^ed or aflefted, unciie, homogcneum adleelionis, homo- 

 geneum comparationis, and the line or vinculum over com- 

 pound quaritilies, thus A -}- B : And his method of arrange- 

 inent is to pl.Ace the homogcneum comparationis, or ahfolute 

 known term on the right-hand fide alone, and all the terms 

 that contain the unknown quantity, with their proper figns, 

 on the other fide. He iomewhat improved the rules and 

 modes of reduction for cubic and other equations ; he (hewed 

 how to change the root of an equation in a given propor- 

 tion ; he deduced the cubic and bicpiadratic, &c. equations 

 from quadratics, not in Harriot's way by compoiUiou, but 

 by fquaring and othei-wife nuiltiplying certain parts of the 

 quadratic ; and as fome quadratic equations have two roots, 

 the cubic and other equations railed from tliem will alio have 

 two roots and no more. In this way Vieta perceived the relation 

 which the two roots bear to the co-efficients of the two lowell 

 terms of cubic and other equations, when they have only 

 tliree terms, namely, by comparing them with ijmilar equa- 

 tions thus raifed from quadratics ; and, -vice verj'ii, what the 

 roots are in tenns of fucli co-efficients. He alfo made fome 

 obfervations on the limits of the two roots of certain equa- 

 tions ; he ftatcd the general relation between the roots of 

 certain equations and the co-efficients of the terms, when the 

 terms are alternately plus and minus, and none of them are 

 wanting, or the roots all pofuive. He extra<5led the rootg of 

 affefted equations by a method of approximation fimilar to 

 that for pure powers ; and moreover, he gave the conilruc- 

 tion of certain equations, and exhibited their roots by 

 means of angular feftions. 



In the Hiilory of Algebra, Albert Girard, an ingenious 

 Dutch or Flemilh mathematician, already mentioned, as the 

 editor of Stevinus's arithmetic, who died about the year 

 1633, deferves particular notice, on account of his work, en- 

 titled, " Invention Nouvelle en I'Algcbre, tant pour la So- 

 lution des Equations, que pour recoignolire le 1, ombre des 

 Solutions qu'elles regoivent, avec plulieures cl.:)fcs qui font 

 necelTiiires al a perfection de cefte divine Science ;" printed at 

 Amfterdam in 1629, 4to. From an analyfis of this work,»it 

 appears that Girard was the firit peifon who undcrftood the 

 general doftrine of the formation t)f the co-efficients of the 

 powers from the funis of the roots, and their produfts, &c. 

 He was alfo the firll: who underflood the »iie of negative 

 roots in the folation of geometrical problems ; who fpoke of 

 the imaginary roots, and underftood that every equation 

 might have as many roots real and imaginary, and no more, 

 as there are units in the index of the higheil power, and 

 who applied the denomination of quaiililia lefs than nothing 

 to the negative : and he was the firll; perfon who difeovered 

 the rules for fumming the powers of the roots of any equation. 



The next perfon who claims particular notice in the hif- 

 toiy of this fcience is Thomas Harriot, wdio died at the age 

 of 60 years in 1 62 1 , and vvhofe Algebra was publilhed by his 

 friend Walter Warner, in 163 1. The book is a folio vo- 

 lume, and entitled, " Artis Analytics Praxis, ad ^qua- 

 tiones Algcbraicas nova, exjiedita, et gene;-ali methodo, re- 

 folveridas ;" a work, fays Dr. Hutton, in all parts of it. 



Vol. I. 



A L G 



fliewing mai'ks of gi-eat genius and originality, and the firfl 

 inllance of the niodcrji form of algebra in which it has ever 

 fince appeared. On the lotmdation laid by Harriot, lays 

 Dr. Wailis (Aljj(br;i, p. 126.) Des Cartes, without naminij 

 him, hath briilt the j-rreateft part, if not the whole, of Lu 

 algebra or geometiy ; witliout which, as he adds, " that 

 whole luperilniitureof DeaCaites ( [ doubt) had never been." 

 A fummai-y of Harriot's improvements is as follou-s : He 

 introduced tiie uniform ufe of the fmall letters <7, /;, r, ci, &c. 

 expreffing the unknown quaritities by the vowels a, e, &c. 

 and the known ones by the confonants b. c, (i,f, &c. join- 

 ing them together in the form of a word to reprcfen* ihc 

 prodnOvt of any number of tliefe literal quantities ; and i)rc- 

 fixingthe numeral co-efficient, feparated from the quantity con- 

 nctted with it bv a point, thus 5. bbc. For a root, he placed 

 the index of the root after the radical mark ^, as \''3), for 

 the cube root. He alfo introduced the charafters y and Z. 

 for greater and lefs ; and in the reduClion of equations, lie 

 ananged the operations in feparate fteps or lines, felting the 

 explanations in the margin on the left hand, for each line. 

 In thefe rcfpefts he introduced and ellabliflied the form of 

 algebra as it now exills. He alfo fliewed the univerfal ge- 

 neration of all the conipouird or afTedled equations, by the 

 continual multiplication of fo many limple ones, or binomial 

 roots ; tlius pkiinly exhibiting to the eye all the circumllances 

 of tlie nature, niyileiy and number of tlic routs of equations, 

 with the compofition and relations of tlie co-efficients of the 

 terms ; from whicii many of the moll important properties 

 have been fuice deduced. He alfo improved the numeral exc- 

 gelis, or extrafiijnof the roots of all equations, by clear and 

 explicit rules and methods, drawn from the foregoing gene- 

 ration orcompofition of atiecled cq.iations of all degrees. 



Oughtred, contemporary with Harriot, was born about 

 the ye.ar 1573, and died in 1660. His " Clavis" 

 was publilbed in 1631. In this work he chiefly follow* 

 Victa, in the notation by the capitals A, B, C, D, &c. and in 

 the defignation ofprojutls, powers and roots, with lomcfew 

 variations. To him we owe the feparation of decimals front 

 the integers after this manner, 21I — 56, and having the de- 

 cimals annexed without a denominator. In algebraical mul- 

 tiplitatlon Oughtred either joins the letters in a word, or con- 

 nects them by the fign X , introducing for the firll time thi» 

 character of multiplication ; thus, A X A, or A A, or A y; 

 but he omits the vinculum of Vieta. He alfo introduces 

 many ufeful contractions in the multiplication and divifion of 

 decimals ; fuch as that of inverting the multiplier for reduc- 

 ing the number of decimals, and abridging the work, that of 

 omitting one figure at a time, of the divifor, and that of di- 

 viding by the factors of a number inftcad of the number 

 itfelf, and many others^ He Rates proprirlion thus, 

 7.9 :: 28.36; and denotes continued proportion by -jt-. 

 With rcfpeA to the genefis and analyfis of powers he fol- 

 lows Vieta ; and he furnlfhes a table of the powers of the bi- 

 nomial A -f-E as far as the loth power, with all their teitiiS 

 and co-efficients, or ur.cicc, an cxpreffion which he adopts 

 from Vieta. He gives particular directions for the reduction 

 of equations, correfponding to their vai-ious forms: he ufcs 

 the letter » after ^/, for univerfal, inftcad of the vinculum 

 of Vieta : and he obfcrvej, that the figns of all the terms of 

 the powers of A-|-E are pofitive, a.idthofc of A — E alter- 

 nately pofitive and negative. He fubjoins many propcrliea 

 of triangles and other geometrical figures, and the firll in- 

 llance of applying algebra to geometry, fo as to inveftigate 

 new geometrical properties ; and after the algebraical refo- 

 lution of each problem, he commonly deduces and gives a 

 geometrical conltrutlion adapted to it. He gives alfo- a goixi 

 3, P trasi 



