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3, 4, 5, J:c. He treated all the higher orders of quadratics by 

 the 1.11111: ijeni-i-al rule. He introduced the iiuincial expo- 

 nents of the powers, - 3, - *#- I, O, i, 2, 3, &c. both 

 polUivc and iug:itive, as far a|ffntegial numbers, but not 

 fiMt^ional ones; calLd them by tlut nanie exponens,, expo- 

 nent ; and taught the ul'c uf cxpo.ieiits in the opera- 

 tions of powers : and he uf'ed the hieral notation A, 13, 

 C, D, &c. for fo many diifcrciit unknown and general 



quantities. 



|ohn Scheubclius, profcfTor of mathematics at Tubingen, 

 in Germany, publiihcd feveral tieatilcs on arithmttic and al- 

 gebra. Trom one of them, entitled, " Algebroe compcn- 

 diofa facihfqne Dekiiptio, qua dcpromuntiir magna Aiith- 

 metices miracul.i," pniiled at Paris in 1552, which Dr. 

 Hutton has analyfed, it appears, that he was the firll mo- 

 dern algebraiftwhomcntioiicd Diophantus, to whom writers, 

 as he fays, afcri!)e this art ; that his chaiaCkrs and opera- 

 tions are much the fame with thofe of Stifclius, but that he 

 iifcd 6^ for I or the o power ; and prctixts the numeral co- 

 cfikieiits. He treats merely of two orders ot equations, i<n. 

 iimple and quadratic equations, though, he fays, they may 

 be of infinite degrees ; and he ufes for the fquare root ^ :, for 

 the cube root w v'' :, and v^/ :, for the 4th root. He gives the 

 four fundamental rules in the arithmetic of finds ; in fquar- 

 ing the fum or difl^rcnce of the fuids he fcts the root to the 

 whole compound ; and this root, called by Cardan, "radix uiii- 

 vcrfalis," he denominates "radix coUccti:" but when they may 

 be reduced to acommon fuid, he unites them into one number. 

 He proceeds in a fimilar manner with cubic finds and 4th 

 roots. He remarks the different kinds of binomial and refi- 

 diial furds, correfponding to the feveral inational lines in 

 the loth book of Euclid's Elements ; and gives the follow- 

 ing general rule for extracting the root of any binomial or 

 rcfidual a + b, in which one or both parts are luids, and a 

 the greater quantity, -ciz. that the fquare root of it is 



^l a + -ya--b-- _^ J a-^a- 



which he ilUiftrates 

 2 — 2 • 



by examples. As he takes no notice of cubic equations, it 

 is probable that though they were known in Italy he had 

 not heard of them in Germany. 



Robert Recorde, in England, publiflied the firft part of 

 liis aritlimetic in 1552, and the fecond part in 1557, under 

 the title of " The Whetllone of Witte, which is the fe- 

 conde parte of Arithmetike ; containing the Extraction of 

 Rootes, the Coffike Pradife, with the Rule of Equation ; 

 and the Woikes of Surde Numbers." What is princi- 

 pally new in this work comprehends the extraftion of the 

 roots of compound algebraic quantities, the uie of the 

 terms binomial and refidual, and the ule of =:, as the fign 

 of equality. 



The Algebra of Pelctarius was printed at Paris in 410. in 

 155S, under this title, " Jacobi Pcletarii Cenoniani, de Oc- 

 culta Parte Numerornm, quam Algebram vocant. Lib. duo." 

 This work, containing an account of rational and irrational 

 or furd quantities, is an ingenious treatiie on thofe parts of 

 the fcience that were then known, cubic equations excepted ; 

 and the difcoveries or improvements of the author are the 

 following, vix. that the root of an equation is one of the di- 

 vifors of the abfolute term ; that trinomials may be reduced 

 to fimple tenns by multiplying them by compound faftors ; 

 and that a feries both of fquare and cube numbers may be 

 cjnllrui^ed by addition only, that is, by adding fucceflively 

 their feveral orders of differences. 



Peter Ramus wrote his Arithmetic and Algebra about 

 the year 1560. He exprefles the powers by /, q, c, b 

 J, Uie initials of latus, quadratus, cubus, and biquadra- 



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tus ; and he treats only of fimplc and quadratfc equations. 

 In 1567, PetcT Nonius, or Nunez, a Portuguefe, puh- 

 lifhed his Algebra in Spaniih, though he informs us in 

 an cpiftle, dated 1564, that it had then been written 

 30 yiars before in Poituguefe. He proceeds no fur- 

 ther than quadratic equations. The Algebra of Raphael 

 Bombelli wris publiflied ;'t Bologna, in 1579, in the Itahan 

 language, but was written fome time before, as the dedica- 

 tion bears the date of 1572. Among other writers on this 

 fcience he particularly mentions Diophantus, whofe Greek 

 work had been found in the V.itit ^n hl;iary ; and he adds, 

 that he and Antc^iio Maria Pa/.7,i Ivegjnarro, profellbr of ma- 

 thematics at Rome, lu'.d tranflated five out of the fix books 

 which were then extant ; and that tiiey had found in the 

 faid work frequent citations of the Indian authors. Hence, 

 thty inferred, that this fcience was known among the In- 

 dians before the Arabians became acquainted with it. Such 

 references, if they aftually exiiled, would ferve to detennine 

 the controverfy relating to the origin of tfiis fcience ; but 

 they do not now remain in the work, nor are they mentioned 

 by any other writer. In his work pombelli has very well 

 explained the rules and methods of former writers ; but, ck- 

 cept the trifeftion of angles by means of a cubic equation, 

 and his mode of notation, he has not introduced any new in- 

 vention or improvement. In this notation he ufes the initial 

 R for root, with q or c after it for quadratic or cubic, &c. 

 root ; p for plus, and m for minus. He calls the unknown 

 quantity tanto, and marks it thus I ; the 2d power 2, its 



cube 3, and the higheft powers 4, 5, &c. denoting all the 



powers, which he denominates dignita, or dignity, by their 

 exponents fet over the common eharafter ■^. Chrillopher 

 Clavius, who follows Stifehus and Schubelius in his notation 

 and method, without fcarcely any variation, wrote his Alge- 

 bra about the year 1580, which was publiCied at Orleans in 

 1608. iSimon Stevinus, of Bruges, publilhed his Algebra 

 foon after his Arithmetic, which appeared in 1585 ; and both 

 were printed in an edition of his works in 1634, with notes 

 and additions by Albert Girard. The pecuhar inventions 

 contained in this ingenious and original work are as foUow. 

 The author invented a new charafter,'Diz, a fmall circle O for 

 the unknown quantity ; and he alio improved the notation of 

 powers by numeral indices, firft applied to integral expo- 

 nents by Stifelius, which Stevinus inclofed within a circle, thus, 



©' (i)» ©' CD' ^"' ""^ ^^^ °' '^'' ^^' 3"^' ^'^' P°^^''s of 

 the quantity O; and he further extended them to fradlional 



and all other forts of exponents ; fo that (\), (!}, QV &c. 

 are the fquare, cube, 4th roots, &c. and (f) is the cube root 

 of the fquare, and (7) is the fquare root of the cube. Sec. 

 Stevinus alfo extended the ufe and notation of co-efficients, 

 making them to comprehend fraftions, radicals, and all forts 

 of numbers. He diftinguiilied a quantity of feveral terms 

 by the general appellation of a multinomial ; and denoted all 

 iiomials whatever by particular names, expreffing the number 

 of their terms as binomial, trinomial, quadrinomial, S:c. He 

 alfo propofed one general method for a numeral reiolution 

 of all equations whatever. 



About the fame time with Stevinus appeared Francis 

 Vieta, who contributed more to the improvement of alge- 

 braic equations than any former author. His algebraical 

 works were written about the year 1600; fome of them 

 were not publiflied till after his death in 1603 ; and all his 

 mathematical works were colleftedby Francis Schooten, and 

 printed in 1646 in folio. The two books, which contain his 

 thief improvements in algebra, are iutitled " De .£quationum 

 I Recogiiitione, 



