TRAINS OF REASONING. 287 



pair of angles out of another, while each pair shall be 

 corresponding angles of triangles which have two 

 sides and the intervening angle equal. It is by this 

 happy contrivance that so many different inductions 

 are brought to bear upon the same particular case. 

 And this not being at all an obvious idea, it may be 

 seen from an example so near the threshold of mathe- 

 matics, how much scope there may well be for scien- 

 tific dexterity in the higher branches of that and other 

 sciences, in order so to combine a few simple induc- 

 tions, as to bring within each of them innumerable 

 cases which are not obviously included in it; and how 

 long, and numerous, and complicated may be the 

 processes necessary for bringing the inductions toge- 

 ther, even when each induction may itself be very 

 easy and simple. All the inductions involved in all 

 geometry are comprised in those simple ones, the 

 formulae of which are the Axioms, and a few of the 

 so-called Definitions. The remainder of the science 

 is made up of the processes employed for bringing 

 unforeseen cases within these inductions ; or (in syllo- 

 gistic language) for proving the minors necessary to 

 complete the syllogisms ; the majors being the defini- 

 tions and axioms. In those definitions and axioms are 

 laid down the whole of the marks, by an artful combi- 

 nation of which men have been able to discover and 

 prove all that is proved in geometry. The marks 

 being so few, and the inductions which furnish them 

 being so obvious and familiar; the connecting of several 

 of them together, which constitutes Deductions, or 

 Trains of Reasoning, forms the whole difficulty of the 

 science, and, with a trifling exception, its whole bulk ; 

 and hence Geometry is a Deductive Science. 



5. It will be seen hereafter that there are weighty 



