ALGEBRA. 



el for expressing- the quantities 



iliut are the objects of this science, and 

 :ilso their mutual relation and depni- 

 Thcsc quantities might, proba- 

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

 notrd hy their names at full length; 

 :id inconvenient, were 

 succeeded b\ abbreviations, or by their 

 Jin-re initials; and, at length, certain Ict- 



i'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- 

 is not easily ascertained. The 

 most ancient treatise on that part of ana- 

 Is tics, which is properly called algebra, 

 now extant, is that of Diophantus, a 

 author of Alexandria, who flou- 

 rished about the \car of our Lord o50, 

 and \\lio \\n;te ].! books, though only- 

 six of them are preserved, which \\ en- 

 printed, together with a single imperfect 

 book on multangular numbers, in a Latin 

 translation by \\lander, in 1575, and 



;rds in divek and Latin, with a 

 comment, in Kw?l and lf>7<), b\ < 

 Hachct, and .M. Fennat, Tolosac, fol. 

 I books do not contain a treatise on 



llu- elementary parts of algebra, but 

 merely collections of some difficult ques- 

 tions relating to square and cube num- 



.iiul other curious properties of 

 numbers, with their solutions. Algebra, 

 houever, seems not to have been wholly 

 unknown to the ancient mathematicians, 

 long before the age of Diophantus. \\ e 

 observe the traces and effects of it in 

 many places, though it seems as if they 

 hadintcntionalh concealed it. Something 

 of it appears in Kuclid, or at least in 

 Theon upon Kuclid, who obsi rves that 

 I'lato had begun to teach it. And there 

 are oilu-r instances of it in Pappus, and 

 more in Archimedes and Appollonius. 

 Hut it should be observed, that the ana- 

 Is sis 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 Creeks 

 v, hohas treated professcdh of algebra. 

 Our knowledge of the science was deri- 

 ved, not from Diophantus, but from the 

 Moor* or Arabians* but whether the 



re the in 1 



of it has been a suhj--,-t of dispute. It is 

 probable, howe\er, t'i I > more 



ancient than DiophauUisJu < ;m>< iii^trea- 

 mi to iv fv. r 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 an\ 

 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 (> 

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

 found to be equal to some quantity o: 

 quantities which are known, and 

 queutly itself is known. 



Algebra is of two kinds, numeral and 

 literal. 



A'.nf:nn.4, innnniil or viilyar, is (ha* 

 which is chiefly concerned in t'.ie res-ih;- 

 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, Pacioltis, Stifeli- 

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

 been an introduction to the art of keep- 

 ing merchants' accounts by double 

 entry. 



ALKKHIIA, sficciwiit or literal, or the new 

 algebra, is that in which nil the quanti- 

 ties, known and unknown, are express- 

 ed or represented by their species, orlet- 

 u r^ i.i'the alphab. . are instan- 

 ces 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 .- 

 which the Civilians use the word species. 

 Thus, they use the name* Titus, Sempro- 

 nius, Caius, and the lik- . ent in- 



definitely any pel-son in such 



propounded, t!i 



speci' s. Yieta, ac -iisiomed to the lan- 

 guage of the civil la\v , . \Vallis 

 supposes, the name <>i 

 ters, A, 15, (', 8tc. v. b to re- 

 present indefinitely any number or quan- 

 tit\ so < iivumstauccd, as i : 

 required. This mode of expression fre.-s 

 the memory and imagination from that 



OP cflbrt, w hieh is required 

 severul mat'. 



i i he truth ii ', present to 



the mind; for which ivi-.>u 0. s 

 be properly (h-noin'm 

 geometry. Specious algebra is not, like 

 the numeral, c-iutinvd U> <-. -tain kimls of 



