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eitlier quantities or operations, excepting fomc ftw abbre- 

 viations of the words or names ; and that ti)c art was merely 

 cni]jloyed in rcfolving ccrta:n numeral problcnis. If the 

 fcicace had b'cn carried further in Africa than qujidratic 

 equations, which was probably the cafe, as we may inter 

 from an Arabic manufcrlpt, faid to be on cubic equations, 

 dcpofited in the librai-)' of the univerfity of I .eyden, by War- 

 ner, the Europeans had at this period obtaii^.d only an im- 

 erfcfl knowh.dge of it. The publication of the works of 

 Aicas dc Burgo promoted the Ihidy, and extended the 

 knowledge of algebra; fo tlrat about the y:ar 1505, Scipio 

 I'cnxus, profellor of mathematics at Bonoiiia, in Italy, dif- 

 covcred the firft rule for rcfolving one cafe of a compound 

 cubic equation. The next Italian, who dillinguifhed him- 

 fclf by the cultivation and improvement of algebra, was ilie- 

 ronymns Cardanus, of Bononia, who publiilied nine books 

 of his arithmetical writings, in 1539, in Latin, at Milan, 

 wliere he praetifed phyllc, and read leClmes on mathematics; 

 a'ld in 1545, a tenth book, containing the whole doctrine of 

 cubic equations. Cardan dcnomuiates algebra, after I^ncas 

 <]e Burgo and others, " Ars Magna quam vulgo Cofliim 

 vocant," or " Regulx Algebraicae," and afcribes the in- 

 vention of it, on the authority of Leonard of Pifa, to Ma- 

 homet, the fon of Mofe», an Arabian. He adds, that this 

 fnppofrd inventor left four rules or cafes, which perhaps 

 onl) included quadratic equations ; that afterwards three de- 

 rivatives were added by an unknown author, fuppoftd by 

 fome to have been Lucas Paeiolus, and afterwa'ds three 

 other derivatives for tlie cube and lixth power, by another 

 unknown author ; all which were refolved like quadratics; 

 that then Scipio Ferreus, about 1505, found out tlit rule 

 for the cafe " cubum et rerum nuraero a:qualium," or, as it 

 is now written, x' -\- l/x = c, which he reprefents as a thing 

 admirable; that the fame difcovery was made in 1535, by 

 Tartalea, who, aft^ earned intreaties, difclofed it to him 

 (Cardan); and that he aud his former pupil, Lewis Fer- 

 rari, much augmented and extended the cafes ; and tl(at 

 all the demonltrations of the rules arc his own, except 

 three of Mahomet for quadratics, and two of Ferrari for 

 cubics. 



To Cardan's treatife on cubic equations is annexed, " Li- 

 bellus de Aliza Regula," or the Algebraic Logiftics, in 

 which he treats of fome of the more abflrufe parts of arith- 

 metic and algebra, efpecially cubic equations, with many 

 additional attempts for the folution of the irreducible cafe, 

 x' = 3x -f- c. 



From a minute and accurate detail of the contents of Car- 

 dan's treatife, given by Dr. Hutton, it appears, that the im- 

 provements in algebra, communicated by this author, are as 

 follow. To the rules furniihed-by Tartalea for rcfolving thefe 

 three cafes of cubic equations, vi-z, x' -f- /'.v = r, x^ = i.\'-^c, 

 and .v' + t = ix ; he has added rules for all forms and varie- 

 ties of cubic equations, dcmonftratiug thefe rules geo- 

 metrically, and fully difculfrng almoil all forts of trans- 

 formations of eq\iations in a manner before unknown. Car- 

 dan alfo appears to have been well acquainted with all the 

 If al roots of equations, both pofitive and negative, or, as he 

 calls them, true and tiftitious, both of which he occafionally 

 ul'cd. He has alfo fhewn that the even roots of pofitive 

 quanlities are either politive or negative ; that the odd roots 

 cf negative quantities are real and negative ; but that the 

 even roots of them are impoffible, or nothing as to common 

 life. He alfo well knew the number and nature of the roots 

 of an equation, partly from the figns of the terms, and partly 

 from the magnitude and relation of the co-efitcicnts. He 

 aifg knew that the number of pofitive roots is equal to the 



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number of chang'es of the figns of the terms ; that the co- 

 elTicient of the 2d term of the equal on is the difference be- 

 tween the pofitive and negative rods ; that when tlie fecoiid 

 term is wanting, the fum of tile negative roots is equal to tiie 

 fum of the pofitive roots ; how to compofc equations that 

 fliall have given roots ; that changing the figns of the even 

 terms changes the figns of all the roots ; that the number of 

 roots failed in pairsj or that the impolfible roots, as they are 

 now called, were always in pairs ; to cliange the equation 

 friiTii one form to another, by taking away any term from it ; 

 and to increafe or dimiiiilh the loots by a given quantity. It 

 appears alio, thatCardanhad arulefor extrafting thecnbe root 

 of fuch binomials as admit of extraction ; that he often ufed 

 the literal notation a, It, c, il, &c. ; that he gave a rule for 

 biquadratic equations, luiting all their cafes; and that, in the 

 invelligation of that rule, he made ufe of an affumed indeter- 

 minate quantity, and aitcrwards found its value by the ar- 

 bitrary alTumption of a relation between the terms ; that he 

 applied algebra to the rcfolutlon of geometrical problems ; 

 and that he was well acquainted with the difficulty of the ir- 

 reducible cafe, and that he devoted much time and attention 

 to the folution of it ; and that, though he did not completely 

 fucceed, he laid down rules for many patticular forms of it, 

 aud ihewed how to approximate very nearly to the root in 

 all cafes whatever, 



Tartalea, or Tartaglia, of Brefcia, was a contemporaiy of 

 Cardan, and publilhed his book of algebra, entitled, " Que-. 

 fiti e Invenzioni diverfe," in 1546, at Venice, whtTe he re- 

 fided as public lecturer in mathematics. This work was de- 

 dicated to Henry VIII. of England, and confifts of nine 

 books, the lalt of which contains all tliofe queftions that re- 

 late to arithmetic and algebra. Thefe queftions comprehend 

 exercifes of fimple and quadratic equations, with eompleit 

 calculations of radical quantities, evincing the ikill of the au- 

 thor in the fcience of algebra. He retained the notation or 

 forms of expredion ufed by Luca> de Burgn, calhng the ift- 

 power of the unknown quantity " cofa," the 2d power 

 " cenfa," the third " cubo," &c. ; and he cxpreflcd ths 

 names of all the operations in words, without any contraction?, 

 except the initial \\. for root, or radicalitv. What is moil 

 remarkable in this collection of queftions is the difcovery of 

 the rules for cubic equations, together with the various cir- 

 cumllances that attended it. 1'he firit two of thefe were 

 difcovered by Tartalea in 1530, -viz. thofe for .v'-|-a.\' = ..-, 

 and x^ = ax' -\- c : and the rules for the other two eafe^, 

 1']%. x^-\-bx := c, and .v^ = bx -j- c, were difcovered, in 1535, 

 at Venice. Under queftion 31, we have an account of the 

 correfpondence between Tartalea and Cardan on the fubjeft 

 of cubic equations, and on the manner in which Cardan drew 

 from him his difcoverics relating to them ; for a more parti- 

 cular account of wliich, fee the biographical article Car- 

 dan. Tartalea publifhed at Venice, in 1 556, &c. a very- 

 large work, in folio, on aritiimetic, geometry, and algebra ; 

 the latter of which is impcrfeft, and extended no farther 

 than quadratic equations, his death having prevented his 

 completion of it. 



The contemporaries of Tartalea and Cardan were Michad 

 Stifelius and Scheubelius. The"Aritl!metiea Integra" of Sti- 

 felius was printed atNorimbergin 15^4, and is, favs Dr. Hut- 

 ton, an excellent treatife on arithmetic and aigibra. The 

 invention of the fcience is afcribed by this author to Geber, 

 an Arabian aflronomer. The improvements of Stifelius and 

 other Germans beyond thofe of tlie Italians, recited in C-ar. 

 dan's book of 1539, were" as follow. He introduced the 

 charafters ■\-, — , y/, for plus, minus, and root, or radix ; 

 and the initials ?-, Zi rf j 3 3>/i» &e-. for the powers i, i. 



