296 The fame Methods of Reafoiiing connnon to 



diftinguifh lines into the fpecies of curved and ftraight, iinlefs 

 •we had fcen imperfe£l examples of both in the extremities 

 or outlines of material bodies. Thus, not only our fmiplcll 

 mathematical ideas of unity and of extcnfion are abftrac- 

 tions from material exiftence, juft as exaftly as are the ge- 

 nera, fpecies, &c. in natural hiftory ; but our notions of the 

 combining of unities and cxtenfions, by multiplication and 

 addition, fo as to produce new varieties of them, are ccrtainlv 

 in their origin produced by a farther obfervation of material 

 nature, and without this, moft probably, could not arife. 

 The mathematician who purfues, in arithmetic, unity 

 through all its combinations, or who combines, in extcn- 

 fion, lines into all the imaginable diverfities of curved and 

 reftilinear figures, employs, in thefe afts, a mental procefs 

 perfeftly analogous to that of the poet who feigns, ad liiiiinn, 

 charafters of deities, men, and dasmons, by new combina- 

 tions of thofc moral, intelleftual, and bodily qualities which 

 he has had occafion to behold in real life. This mathema- 

 tician does, in faft, a thing precifely fimilar to that which is 

 done by the common maibn or carpenter, who frames, of 

 Hone or wood, many new figures of building, furniture, and 

 ulenfils, not exaftly fimilar perhaps to any that he may have 

 fcen in fimple unviolated nature. The chcmill a6ls in the 

 fame manner, in every trial of the unknown combinations 

 of any chemical fubftance of which he defires to invcftigate 

 the charafter. And it is exadly thus, too, that the zoolo- 

 gift, to elucidate the natural hiftory of any animal, places it 

 in every diverfity of circumftances in which he can imagine 

 that a new quality may be difplayed by it. 



Compare the fimpler operations in arithmetic with thofc 

 of logic. Multiplication is only an abbreviation of addition : 

 addition, in every acl of it, fimply compares feveral different 

 individuals, and thus afcertains that they belong to the fame 

 fpecies : every a6l, whether of addition or of multiplication, 

 is precifely that elliptical fyllogifm which is named an enthv- 

 nieme. In every a6f, whether of the one or of the other, 

 there is, in truth, an analyfis of particulars, and a general 

 induftion from them. I fay, for inftance, in addition, 3 

 and 2 are 4. This, as a fyllogifm, affirms, that all numbers 

 containing equal units are equal : 4 cxpreftc-3 a certain com- 

 bination of unities : 2 and 2 make together exaftly the fame 

 combination of unities; 3 and 2 are therefore equal to 4. 

 Let any one attend vigilantly to what palles in his mind iu 

 this aft of addition ; and he will find it to be clearly what is 

 here ftatcd. Again, this aft of addition fliows, juft like the 

 obfervation of a rofe-tree only in leaf, and of a rofe-trce in 



full 



