TRAINS OF REASONING. 



143 



by the equality of the angles, AE will 

 Ue on AD. But AB and AC, AE 

 and AD are equals ; therefore they 

 will coincide altogether, and of course 

 at their extremities, D, E, and B, C. 



Third Fobmvla.— Straight lines, hav- 

 ing their extremities coincident, coin- 

 cide. 



B E and CD have been brought 

 within this formula by the preceding 

 induction ; they will, therefore, coin- 

 cide. 



Fourth Formula. — Angles, having 

 their sides coincident, coincide. 

 The third induction having shown 

 that BE and CD coincide, and the 

 second that AB, AC, coincide, the 

 angles ABE and ACD are thereby 

 brought within the fourth formula, 

 and accordingly coincide. 



Fifth Formula. — Things which coin- 

 cide are equal. 

 The angles ABE and ACD are 

 brought within this formxila by the 

 induction immediately preceding. 

 This train of reasoning being also 

 applicable, mutatis mutandis, to the 

 angles EBC, DCB, these also are 

 brought within the fifth formula. 

 And, finally. 



Sixth Formula. — The differences of 

 equals are equaL 



The angle ABC being the differ- 

 ence of ABE, CBE, and the angle 

 ACB being the difference of ACD, 

 DCB ; which have been proved to be 

 equals ; ABC and ACB are brought 

 within the last formula by the whole 

 of the previous process. 



The difficulty here encountered is 

 chiefly that of figuring to ourselves 

 the two angles at the base of the 

 triangle ABC as remainders made by 

 cutting one pair of angles out of an- 

 other, while each pair shall be corre- 

 sponding angles of triangles which 

 have two sides and the intervening 

 angle equal It is by this happy con- 

 trivance that so many different iqduc- 



tions are brought to bear upon the 

 same particular case. And this not 

 being at all an obvious thought, it 

 may be seen from an example so near 

 the threshold of mathematics how 

 much scope there may well be for 

 scientific dexterity in the higher 

 branches of that and other sciences, 

 in order so to combine a few simple 

 inductions 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 for bringing the 

 inductions together, 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 fornmlse 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 forbringingunforeseen cases 

 within these inductions ; or (in syllogis- 

 tic language) for proving the minors 

 necessary to complete the syllogisms ; 

 the majors being the definitions and 

 axioms. In those definitions and 

 axioms are laid down the whole of 

 the marks, by an artful combination 

 of which it has been found possible 

 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 De- 

 ductions or Trains of Reasoning, forms 

 the whole difficulty of the science, and, 

 with a trifling exception, its whole 

 bulk ; and hence Geometry is a De- 

 ductive Science. 



§ 5. It will be seen hereafter * that 

 there are weighty scientific reasons 

 for giving to every science as much of 

 the character of a Deductive Science 

 as possible ; for endeavouring to con- 

 struct the science from the fewest and 

 the simplest possible inductions, and 

 to make these, by any combinations 

 however complicated, suffice for prov- 



* Infra, book lii. ch. iv. $ 3, and else- 

 where. 



