igS The fame Methods of Reafoning common t'i 



demonftrate, that the three angles of any triangle are equal 

 to two right angles; what elfe is this, but to fliow that the 

 three angles ot a triangle arc one fpecies of the fubdivifions 

 into which the magnitude of two right angles maybe marked 

 out? Can a finale cafe of mathematical inquiry be pointed 

 out, in which the reafoning (hall not appear to have tlie moft 

 perfect analoin' with that which is ufcd in every other branch 

 of fcience, and, indeed, in the whole bufiuefs ot common 

 life? Impoflible. 



But the mathematician reprcfents all inagnitudes by com- 

 binations of lines incapable of exhibiting folid contents, or 

 even, with accuracy, the divcrfilics of furface ; and in his de- 

 TOonftrations employs the letters of the alphabet to denomi- 

 nate thofe lines. He does. But the abftractions of the fancy 

 of the poet are thus reprefentcd, and but in part, by the 

 imagery of the fculptor and the pointer. The naluralift thus 

 employs delineations of thofe objects which he cannot keep 

 conltantly under his aftual infpeclion. The letters of the 

 alphabet, and other figns not in any other way fignilicant, 

 are emploved in our reafonings in every other branch of 

 knowledge, juft as well as in mathematics. If a few tech- 

 nical terms be peculiar to this branch of the fcience of the 

 magnitudes of figure; arc not certain fets of technical terms, 

 in almoft every other inftance. much more numerous than 

 thofe of mathematics, peculiar refpc6fivelv to every different 

 fcience, and to every dift'erenf branch ot art ? 



It may iVill, however, be alleged, that there is a peculiarity 

 in alocbra, that branch of mathematical reafoning in which 

 the moderns chiefly delight. But, no : the reafoning is, even 

 in alijcbra, fiill the fame. The figns only, and the abbre- 

 viations, are ibmcv.hat different. Algebra refines even upon 

 the abCtraftions of arithmetic, and enables the mind to pro- 

 ceed through a long train of reafoning refi)e6ting numerical 

 quantities, magnitudes, or dirtin(!^l values of any I'ort, without 

 the intervention of connnon language, or even of the com- 

 mon figns of number. But the objecl is ftill to arrange in- 

 dividuals under their fpecies, fpecies under their genera, ge- 

 nera imdcr abftracStions Itill more comprehcnfive, &c. The 

 reafoning is (lili bv an arialvtical collection ^f particular fiicts, 

 and a conclufion to a general truth from their agreement. 

 The fcries of abbreviated fvllogifms may be juft as clearly 

 traced in demonftrations by algebraic fynibols, as in thofe in 

 which more tedious forms of words are emploved. Algebra 

 is but the brachvgraphy of mathematics ; and fliort-hand 

 writing implies no peculiaritv of reafoning, but merely a vo- 

 luntary and conventional connexion of certain figns with 



certaiu 



