FOURTH FORMULA. _Angles, having their sides coincident, coincide_. The third induction having shown that B E and C D coincide, and the second that A B, A C, coincide, the angles A B E and A C D are thereby brought within the fourth formula, and accordingly coincide. FIFTH FORMULA. _Things which coincide are equal_. The angles A B E and A C D are brought within this formula by the induction immediately preceding. This train of reasoning being also applicable, _mutatis mutandis_, to the angles E B C, D C B, these also are brought within the fifth formula. And, finally, SIXTH FORMULA. _The differences of equals are equal_. The angle A B C being the difference of A B E, C B E, and the angle A C B being the difference of A C D, D C B; which have been proved to be equals; A B C and A C B 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 A B C, as remainders made by cutting one 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 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 necessary 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 formulæ 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 syllogistic 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 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.