Mathematics and to Natural H'lf.ory. 297 



full flower in natural hlftory, that 2 and 2 are, as well as 4, 

 but combinations of the fame fpecles, genus, order, or claf^ 

 of unity. It is alfo an example of analyfis and indudlion. 

 You analyfe when you confider, as one unavoidably does, 

 the conftitucnt units leparately; you draw your general in- 

 ference when vou perctive the equalities. Any ail in mul- 

 tiplication might, as to the reafoning of it, be in the lame 

 manner illuftrated. Subtra6lion realons exadlly in the fame 

 manner as addition; but, inftead of equalities, difcovers dif- 

 iimilitude, and of that diihmilitude gives a numerical defi- 

 nition. Divifion combines the reafonings of addition or mul- 

 tiplication with thofc of fubtraftion. The rule of three is, 

 moft obviouily, a chain of abridged fyllogifms; a procefs of 

 regular analylis and induftion ; an afcertaining of the proper 

 fpecie;;, £cc. to which certain individuals belong. Purfue 

 the reafonings which refpcil numbers through every other 

 variety of the accuflomed operations; and you ihalt find, ftill, 

 nothing but the fame a6ts of ratiocination conftantly repeated. 

 A lyllogiltic comparifon of ideas; to difcover equality or dif- 

 ference ; a procefs of analyfis and induction ; a reference of 

 individuals to their rcfpective fpecics, of fpecies to their ge- 

 nera, Sec. &c. ; are ftill the only atSls of intelkft which are 

 pertbrmed : and ftill the mind is confcious of no ideas but 

 what have their origin in the obfervation of material exift.- 

 enceSj and in abftraiStion from thefe. 



Thus it is in that part of mathematics which abftrafls 

 quantity from extenfion, and concerns itfelf onlv about the 

 relations and properties of number confidercd exclufively of 

 figure. — Examine, on the other hand, any of the propofitions 

 and dcmonftralions of Euclid, or any other mathematician who 

 inveftigates the qualities of figures in the antient form of ma- 

 thematical reafoning. Does he demonltrate a theorem ? This 

 is only to eftablilli, bv analyfis and induction, a general faft 

 before unknown. Is he to folve a problem ? This is to explain 

 the fecret caufes of a general truth before known, but unac- 

 counted for ; to invent a rule, and prove it to be founded on 

 a due knowledge of the relations of things ; or to adopt a 

 rule from others, and to fhow by what means it is that com- 

 pliance with this rule perfectly aceomp-liflies the end propofed. 

 A theorem and a problem differ from one another merely as 

 two diflerent modes for fuggefling the fame queliion or quef- 

 tions of the fame fpecies. In both, the mind analyfes and 

 abflracts juft as in the profcculion of any difcovery in natural 

 hiftory. The train of demnnlfration in the works of all the 

 antient mathematicians who treat of figure, is a fcries of cn- 

 thymcmes or abbreviated fyllogifms. When I so about to 



ucmonflrate. 



