A L G 



work ; which vns enriched with notes by James Bemouilli, 

 anJ printed ;it Bnlil. Huygciis alfi> Jirn^li.d his attention 

 to the algebraic analylis, and his inventions arc cited by 

 Schootcn, who was his pupil.< Shilius, canon of Liege, 

 publiihcd in 1659, " Mefolahum, feu dua; medix propor- 

 tionales per CircuUira et EUiplin, vel Hypcrbolain, infinitis 

 B.jJii eihibiue," a acv.' edition of whicli appeared in 1668, 

 containing much vjJuablc matter relating both to algebra 

 and geomitr)-. 



Bnt brfrtie the time of Dcs Cartes, as well as after the 

 p\iblic.itinii of his gcoinetry, algcbia engaged the atten- 

 tion of ;r..itheinatieians. In 1619 feveral pieces of Van 

 Collen, or Cen'en, were tranllated from Dnlch into Latin, 

 and publiihed at Leyden, by W. Snell ; one of which is a 

 p::it:vnl:irtrcatirc on Surds. In 1621, Bachet publilhed an 

 edition i f Dioj li mtns with notes, and Fcrmat's edition, 

 with additions, appeared in 1670. The fame author pub- 

 lilhed, in 1624, a trealife of mathematical recreations under 

 the title of " ProWemea plaifans et deleftables." Heri- 

 gone, in 1634, ptibliihed at Paris tlie tirll courfe of mathe- 

 matics, in 5 vols. Svo ; containing a treatlfe on algebra, and 

 bearing, fays Hctton, evident marks of originality and in- 

 genuity, in which he ufes the notation by fmall letters, in- 

 troduced three years before by Harriot ; he alfo exprefies 

 flushy +, minui by :/) , and | for equality, with other abbre- 

 viations. In his notation of powers and roots, he annexes 

 to the letter the numeral exponents. Cavalerius, in 1635, 

 publiflied his " Indivifibles," and introduced a new xra in 

 analytical feience and new modes of c<imputation. He 

 was followed in 1 640 by Roberval, whofe improvements in 

 analytics were publilhed in the early volumes of the me- 

 moirs of the Academy of Sciences, by De Billy, who 

 publifhedin 1643, " Nova Geometric clavis Algebra," and 

 in 1670, " Dioj)hantus redivivus ;" and by Renaldine, 

 ■who, in 1665, publiflied in 410, " Opus Mathematicum," 

 both ancient and modern, with mathematical refulution and 

 coinpofitipn, enlarged and republiflied in folio, in 1662, 

 1667, and 1682, under the title of " Ars analytica Mathe- 

 matum, in trcs partes dillributa, &c." This author ufes 

 the parenthcfes (a' + ^V ^^ a vinciJum. In 1655, Dr. 

 Wallis publilhed his " Arithmetica Infinitorum," which 

 greatly improved the Indivifibles of Cavalerius, and led the 

 way to infinite feries, the binomial theorem, and the me- 

 thod of fluxions. -The " Algebra Rhonii, (or Rahnii) 

 Gcrmanicc," was publiflied in 1659, and tranflated into 

 Engliflt in 1668, by Mr. Thomas Brancker, with altera- 

 tions and additions, by Dr. John Pell, v.-ho ufed a pecuhar 

 method of regiftenng the ftcps of an algebraic procefs by 

 means of marks and abbreviations in the margin, explain- 

 ing each line or Hep, as Harriot had before done in words 

 at length. HemeUng was alfo the author of a German 

 work, refolving 600 queftions, pubUfhed in 1684. Mr. 

 Kinckhuyfen, in 1661, pubhflied a treatife of algebra in 

 Dutch, which Sir Ifaac Newton, when profcflbr of mathe- 

 matics at Cambiidge, ufed and improved, and which he de- 

 ligned to republidi, with his method of fluxions and infinite 

 feries, but was prevented by the accidental burning of fome 

 of his papers. In 1667 Jacob Fergufon publilhed his 

 " Labyrinthus Algebra"," in 410, Dutch; and in 1679, 

 De Graaf gave a courfe of mathematics, in the fame lan- 

 guage and fize. In 1665 or 1666, Sir Ifaac Newton made 

 fcveral of his moft valuable difcoveries, though they were 

 not publilhed till a later period ; fuch as the binomial the- 

 orem, the method of fluxions and infinite feries, the quadra- 

 ture, reftification, &c. of curves, the invcftigation of the 

 roots of all forts of equations, buth numeral and literal, 

 in infinite converging feries, the_ reverCoa of fejics, &c. 



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M. Freniclc, in r666, communicated feveral trafls concern- 

 ing combinations, magic fquares, trixmgular number;, &c. 

 which were printed in the early volumes of the Memoirs of 

 the Academy of Sciences. In 166S, Mereator publiflitd 

 his " Logarithmoteehnia," in which he gives the quadra- 

 ture of the hyperbola, by means of an infinite feries of al- 

 gebraic terms, found by dividing a fimple algebraic quan- 

 tity by a compound one ; which operation was now firll 

 made public, though Newton had before expanded all forts 

 of compound algebraic quantities into infinite feries. The 

 demonftration ot Mcrcator's quadrature of the hyperbola 

 by ihe fame feries, was publilhed in this year, by James 

 Gregory, in his " Exercitationes Geometries ;" and in 

 the fame year Lord Brouncker publifiied in the Philofophi- 

 cal Tranfaftions, his quadrature of the hyperbola by an- 

 other infinite feries of fimple rational terms, of which he 

 had been in poffelHon fince the year 1657, when Dr. Wallis 

 announced it to the public. His feries for the quadrature 

 of the circle had been puliliflied by Wallis in his " Arith- 

 metica Infinitorum." In 1669, Dr. Barrow publiflied his 

 " Optical and Geometrical Leiitures," aboundir.g with 

 profound refearches on the dimenlions and properties of 

 curve lines, and containing his method of tangents, by a 

 mode of calculation fimilar to that of fluxions or incre- 

 ments, and little difl'ering from it, except in the notation. 

 In the J3lh ledlure, (p. 277. Stow's Edit.) the fubjedt of 

 which is equations, he adopts a new method of explaining 

 tlieir nature, diff^erent from that of Vieta, who illuftrates 

 it by the analogy of the terms, or that of Hairiot and Des 

 Cartes, by multiplying them into one another. His m.e- 

 thod of explaining them depends upon the defeription of 

 hnes adapted to each ; and thus he inveftigates the nature 

 and number of their roots, and the hmits of their magni- 

 tudes, coulidering the fubjeft as a branch of the maxima 

 and minima. 



The " Elements of Algebra" were pubhihed by John 

 Kerfey in 1675, in 2 vol. folio, containing the illullration 

 of the feience and of the nature of equations, the explicn- 

 tion of Diophantus's problems, and many additions con- 

 cerning mathematical compofition and refolution, from 

 Ghctaldus. This work, fays Hutton, is very ample and 

 complete. The firft part appeared in 1673, ^'"^ ''^^ f<^' 

 cond in 1674. In 1675 Prellet publilhed his " Nouveaux 

 Elemens des Mathematiques," to which the author, with a 

 prefumption hardly excufable, has prefixed a dedication of 

 the work to God Almighty. In 1677 Leibnitz difcovered 

 his " Methodus Differentialis," or made a variation in 

 Newton's fluxions or extended Barrow's method, of which 

 he gave the firft inftance in the Leipfic acts for 1684. See 

 Fluxions. In the fame afts for 1682, he communicated 

 an improvement of infinite feries, and a fimple feries for 

 the quadrature of the circle. An amplification of Wallis's 

 arithmetic of infinites was publiflied in folio, in 1682, by 

 Ifmael Bulliald, entitled, " Opus novum ad Aiithnieticani 

 Infinitorum." Tchirnhaufen, in 1683, communicated a 

 memoir in the Leipfic afts, propofing the extraction of th*; 

 roots of all equations in a general way ; but his method 

 did not fucceed. Baker's " Clavis Geometrica Cathohc.', 

 Geometrical Key, or Gate of Equations unlocked," wis 

 publifiied in Latin and Englifli in 1684. This was an im- 

 provement of Des Cartes's conftruftion of all equations under 

 the 5th degree, by m.eans of a circle and paiabola for all 

 equations, any diameter being ufed inftead of the axis of 

 the parabola. Dr. Wallis's " Treatife of Algebra, bot'i 

 Hifl:(»rical and Pradlical, fliewing the original, progrei-', 

 and advancement of it from time to time," Avas publiflv.d 

 in 1685, in folio. In 1687, Dr Halley communicated iii 



6 il..' 



