Second Formula. Equal straight lines being applied to one another coincide.
AC, AB, are within this formula by supposition; AD, AE, have been brought within it by the preceding step. Both these pairs of straight lines have the property of equality; which, according to the second formula, is a mark that, if applied to each other, they will coincide. Coinciding altogether means coinciding in every part, and of course at their extremities, D, E, and B, C.
Third Formula. Straight lines, having their extremities coincident, coincide.
BE and CD have been brought within this formula by the preceding induction; they will, therefore, coincide.
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 coincide are equal.
The angles ABE and ACD are brought within this formula 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.