A L G 



tninfpofition, fo ss to bring the equation to fimple tci-ms, 

 and then dcprcfling it to ;> lower degree by equal divifion, 

 wlien the powers of tlic unknown quantity are in cvciytcmi: 

 Vkliich preparation or reduCtlcn of the complex equation, b<|- 

 iwf; made, or reduced to what we call a final equation, this 

 author proceeds no further, but merely fays, \vl\at the root, 

 or rf. ignola, is, without giving any rules for finding it, or for 

 the refolution of equations ; thus intimating, that rules for 

 this purpofe were to be found in fome other work, either of 

 his own or of fome other pcrfon. The chief excellence of 

 Diophantus's coUeclion of quciliona, which feems to be a 

 feries of exercifes for rules wliich had been elfewhere given, 

 is the neat mode of fubllitution or notation, which being 

 once made, the reduction to the final equation is ealy and ob- 

 vious. This work indicates much accurate knowledge of 

 the fcience of algebra, in fome of its mod abftrufe parts. 

 But as the author reduces all his notations to a fimple equa- 

 tion, or a fimple quadratic, it docs not appear how far his 

 knowledge extended to the refolution of compound or aiFccl- 

 cd equations. 



Algebra, however, feems not to have been wholly unknown 

 to the ancient mathematicians, long before the age of Dio- 

 phantus. We obferve the traces and efFedls of it in many 

 places, though it feems as if they had intentionally concealed 

 It. Something of it appears in Euclid, or at Icall in Theon 

 upon Euclid, who obferves that Plato had begun to teach 

 it. And there are other inftances of it in Pappus, and more 

 in Archimedes and ApoUonius. But it fhould be obferved, 

 that the analyfis ufcd by thefe authors is rather geometrical 

 than algebraical ; this appears from the examples of it that 

 occur in their works ; and therefore, Diop'iantus is the hrtt 

 and only author among the Greeks, who has treated profeff- 

 «dly of algebra. Our knowledge of the fcience w as derived, 

 not from DiopliantUJ, but from the Moors or Arabians ; but 

 whether the Greek's or Arabians v.eif the inventors of it has 

 bten a fubjedt of diipute. It is probable^ however, that it 

 »as much more ancient than Diuphantus, hecauft his trea- 

 tife feems to refer to works fimilar and prior to his own. 

 Abulfaiagius, an Arabic hiltorian, in one phue afeiibes the 

 ir.vention, or rather the arrangement of the principles and 

 rules of the fcience, to Diophaiitus ; and from him we learn 

 that the Arithmetic of Diophantus was tranflated into Arabic 

 by Mahomet Ben-Yahya Baziani : but in another place he 

 feems to alcribe it to Mahomet Ben Mufa, who is faid to 

 have lived about the year 850 or 900, and who was the iirft 

 of the Arabs by whom this Icience was cultivated. Cardan 

 attributes the invention of it to this Arabian, and apprehends 

 that he obtained the appellation of Geber from this art. Sec 

 Bib. .'\rab. ct Hifp.tom. i. p. 370. cited by Rufrelinhis Hift. 

 of Aleppo, vol. ii. p. 409. Stcvinus is of opinion that this 

 fcience, and otlier parts of mathematics, were much more 

 ancient among the Orientals, than any learning they derived 

 from the Greeks. Dr. Walhs adopts the femiments of thofe 

 wlio think that the Arabs derived this fcience, as well as the 

 knowledge of numeral figures, from the Perfians, and origi- 

 nally by their means from the Indians ; and he alledges, as a 

 prefumptive evidence of their not having derived it from the 

 Greeks, that the name they give it, viz. al-guibr iv'al-moio- 

 laln, feems to liave no affinity with any Greek name. We 

 may here add, that fome veftiges of algebraical calcula- 

 tion have been dilcovered among the Brahmins ; particularly 

 rules for the folution of certain arithmetical queftions, with 

 which it would fcem that nothing but algebra could have 

 furniihed them. Afiatic Refearchcs, voL ii. p. 468. note. 

 4.S7, 495. But wherever algebra was invented or firll cul- 

 tivated, the fcience, and alfo the name of it, wiere tranfmittcd 



A L G 



to Europe, and particularly to Spain, by the Arabians or S.v 

 racens, about the year 1100, or fomewhat fooner. Italy 

 feems to have taken the lead in the cultivation of this fcience, 

 after its introdudion into Europe : and I.ucas Paciolus, or 

 Lucas de I'uigo, a minorite I'rancifcan friar, was the firft 

 author on the fubjeft, who wrote feveral treatifes in the 

 years 1476, 1481, 1470, 1487, and 1509; but his principal 

 work, entitled, " Summa Arithmcticx et Geometnx, Pro- 

 portionumque et Proportionalitatum," was publilhed in Ita- 

 lian at Venice, in 1494, and again in 1523. In tins work he 

 mentions fevci-al writers, and particularly i^eonardus Pifanus, 

 placed bv VoiTius about the year 1400, or a little fooner, and 

 faid to be thefirft of the moderns who wrote of algebra, from 

 whom he derived his knowledge of thofe fciences ; and from 

 the treatife of Leonard, not now extant, the contents of that 

 «f Lucas were chiefly collefted. The age of Leonard of 

 Pifa has been ufually fixed to the end of the 14th century. 

 But it now appears by a manufcript of this algebraill, difco- 

 vered in a library of Italy, by M. Targioni Tozzeti, and 

 communicated to M. Coffali, a canon regular of Parma, that 

 he lived two centuries before this period, or at the com- 

 mencement of the 13th ccntuiy : and of courfe that Italy 

 is indebted to him for its firll knowledge of algebra. His 

 pro.per name was Bonacci, and he was a merchant, who 

 traded in the fea ports of Africa, and the Levant. Being 

 ambitious of obtaining an acquaintance with the fciences that 

 flouriflicd araongll the Arabs, and particularly that of al- 

 gebra, he travelled into their country. Accordingly his 

 arithmetic was publirtiyd in i2C2,and a new enlarged edition 

 of it appeared in 1228. At this time, however, algebra was 

 not a part of arithmetic, but was dillinguilhed from it by the 

 title of " Ars Magna," or " Arte Maggiore." From the 

 manufcript above-mentioned it appears, according to Col-, 

 fall's account of it, that Leonard had penetrated deeply into 

 the fecrets of the algebraic analyfis ; that he was particu- 

 larly acquainted with the analyfis of problems finiilar in 

 kind to thofe of Diophantus, and with the refolution of 

 equations of the fecond degree ; and that he had written a 

 treatife, entitled " De' Numeri Qjradrati," which is not ex- 

 tant, but which ColFali has rellored from fome fragments of 

 Lucas del Burgo. This I.eonard, therefore, mull not be 

 confounded with another called Camillus Leonardus of Pcfa- 

 ro, author, as it is faid, of a book entitled, " Liber defideratun 

 canonum aequatorii motuum ccelellium fine calculo, &c." 

 Pifaur. 1496, 4to. Montucla HilL Math. torn. ii. p. 716. 

 This Leonard of Pifa made long voyages into Arabia and 

 other eaftern countries, in order to gain the knowledge of the 

 mathematics. Montucla (torn. i. p. 536.) mentions two 

 other perfons who previouily to this diicovery were thought 

 to have preceded I^eonard in this department of fcience, viz. 

 Paul dell'Abaco, who lived towards the end of the 14th cen- 

 tury, and who is fuppofed by Ximenes, to have been the firll 

 perlon in Italy who ufed algebiaic equations ; and alfo Prof- 

 docimo Belmando, or Beldomando, of Padua, who was fup- 

 pofed to have fhared with Leonard the honour of introducing 

 into Italy the knowledge of algebra. His book, entitled, "Dell 

 Algorithmo," was printed in 1483, but dated at the begin- 

 ning of the 15th century. Lucas informs us,, that algebra 

 came originally from the Arabs, and never mentions Dio- 

 phantus ; from which circumihmce it has been infeired that 

 this Greek author was not then known in Europe. From 

 the book of Lucas de Burgo, we learn, that the knowledge 

 of the Europeans in his time, or about the year 1500, ex- 

 tended no further than to quadratic equations, of which they 

 ufed only the pofitive roots ; that they admitted only one 

 unknow 11 quantity ; that they had no marks er figns for 

 3 either 



I 



