ALGEBRA. 



invented for expressing the quantities 

 that are the objects of this science, and 

 also their mutual relation and depen- 

 dence. These quantities might proba- 

 bly, in the infancy of the science, be de- 

 noted by their names at full length; 

 these, being found inconvenient, were 

 succeeded by abbreviations, or by their 

 mere initials; and, at length, certain let- 

 ters of the alphabet were adopted as ge- 

 neral representations of all quantities; 

 other symbols or signs were introduced, 

 to prevent circumlocution, and to facili- 

 tate the comparison of various Quantities 

 with one another ; and, in consequence of 

 the use of letters or species, and other 

 general symbols, or indeterminate quan- 

 tities, algebra obtained the appellation of 

 specious, literal, and universal arithmetic. 

 The origin of Algebra, like that of other 

 sciences of ancient date and gradual pro- 

 gress, is not easily ascertained. The 

 most ancient treatise on that part of ana- 

 lytics, which is properly called algebra, 

 now extant, is that of Diophantus, a 

 Greek author of Alexandria, who flou- 

 rished about the year of our Lord 350, 

 and who wrote 13 books, though only 

 six of them are preserved, which were 

 printed, together with a single imperfect 

 book on multangular numbers, in a Latin 

 translation by Xylander, in 1575, and 

 afterwards in Greek and Latin, with a 

 comment, in 1621 and 1670, by Gaspar 

 Bachet, and M. Fermat, Tolosje, fol. 

 These books do not contain a treatise on 

 the elementary parts of algebra, but 

 .merely collections of some difficult ques- 

 tions relating to square and cube num- 

 bers, and other curious properties of 

 numbers, with their solutions. Algebra, 

 however, seems not to have been wholly 

 unknown to the ancient mathematicians, 

 long before the age of Diophantus. We 

 observe the traces and effects of it in 

 many places, though it seems as if they 

 had intentionally concealed it. Something 

 of it appears in Euclid, or at least in 

 Theon upon Euclid, who observes that 

 Plato had begun to teach it. And there 

 are other instances of it in Pappus, and 

 more in Archimedes and Appollonius. 

 But. it should be observed, that the ana- 

 lysis used by these authors is rather ge- 

 ometrical than algebraical ; this appears 

 from the examples that occur in their 

 works ; and, therefore, Diophantus is the 

 first and only author among the Greeks 

 who has treated professedly of algebra. 

 Our knowledge of the science was deri- 

 ved, not from Diophantus, but from the 

 rs or Arabians; but whether the 



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Greeks or Arabians were the inventors of 

 it has been a subject of dispute. It is 

 probable, however, that it is much more 

 ancient than Diophantus, because his 

 treatise seems to refer to works similar 

 and prior to his own. 



Algebra is a peculiar kind of arithme- 

 tic, which takes the quantity sought, 

 whether it be a number, or a line, or any 

 other quantity, as if it were granted ; 

 and by means of one or more quantities 

 given, proceeds by a train of deductions, 

 till the quantity at first only supposed to 

 be known, or at least some power of it, is 

 found to be equal to some quantity or 

 quantities which are known, and conse- 

 quently itself is known. 



Algebra is of two kinds, numeral and 

 literal. 



ALGEBRA, numeral or vulgar, is that 

 which is chiefly concerned in the resolu- 

 tion of arithmetical questions. In this, the 

 quantity sought is represented by some 

 letter or character; but all the given 

 quantities are expressed by numbers. 

 Such is the algebra of the more ancient 

 authors, as Diophantus, Paciolus, Stifeli- 

 us, &c. This is thought by some to have 

 been an introduction to the art of keep- 

 ing merchants' accounts by double en- 

 try. 



ALGEBRA, specious or literal, or the new 

 algebra, is that in which all the quanti- 

 ties, known and unknown, are express- 

 ed or represented by their species, or 

 letters of the alphabet. There are in- 

 stances of this method from Cardan, and 

 others about his time ; but it was more 

 generally introduced and used by Vieta. 

 Dr. Wallis apprehends that the name of 

 specious arithmetic, applied to algebra, is 

 given to it with a reference to the sense in 

 which the Civilians use the w r ord species. 

 Thus, they use the names Titus, Sempro- 

 nius, Cair.s, and the like, to represent in- 

 definitely any person in such circumstan- 

 ces ; and cases so propounded, they call 

 species. Vieta, accustomed to the lan- 

 guage of the civil law, gave, as Wallis 

 supposes, the name of species to the let- 

 ters, A, B, C, &c. which he used to re- 

 present indefinitely any number or quan- 

 tity so circumstanced, as the occasion 

 required. This mode of expression frees 

 the memory and imagination from that 

 stress or effort, which is required to keep 

 several matters, necessary for the disco- 

 very of the truth investigated, present to 

 the mind; for which reason this art may 

 be properly denominated metaphysical 

 geometry. Specious algebra is not, like 

 the numeral, confined to certain kinds of 



