remember, also, that we have to prove that AE is equal to E B. Now, we cannot fail to see that the data involve the equality of the triangles ACE, CE B, by Euc. I., 4, and therefore that A E is equal to E B. This solution is synthetical, notwithstanding the prior reference to the relation which has to be established. For we proceed from the data-A C, CE, equal to BE, CE, and the included angles equal -to the equality of the triangles ACE, BCE in all respects, and thence to the equality of A E, EB. In the analytical solution we should argue thus:We have to show that A E is equal to E B. Now, if AE is equal to EB, then since AC, CE are respectively equal to BC, CE, the angles ACE and BCE will be equal (Euc. I., 8); but these angles are equal, being right angles; hence we are led to reverse the steps as a probable method of solving our problem; and, on trial, we find that the proof of the equality of AE, EB, is complete by this method. We shall presently see that the mere fact of obtaining by the analytical method a result corresponding to certain data of a proposition is no certain test that the problem is correct; and I will at once show that it is no certain proof that the reversal of the process will give at once a satisfactory solution of a problem.

Suppose that we have given to us AC equal to C B, and the angle CA E equal to the angle CBE, and that we have to show from these data that CE is at right angles to A B. We proceed analytically thus: If CE is at right angles to A B, then AC,

CE, being equal, respectively, to B C, C E, the triangles AC E, B C E are equal in all respects ; therefore the angle CA E will be equal to the angle CBE. Now, these angles are equal; therefore we might expect the reversal of the process to lead at once to the solution of our problem. This, however, is not the case we have AC, CE equal to BC, CE, and the angles CAE, CBE, opposite to the common side, C E, equal to each other; but there is no proposition in Euclid which enables us to assert from these data that the triangles CAE and CBE are equal in all respects.

Of course, there is no difficulty in the above problem. The equality of the angles CAE and CBE give us immediately A E equal to E B (Euc. I., 6), and thence the equality of the triangles, A CE, BCE, follows at once. But it is well to notice that analysis may lead to a result involved in our data which yet does not involve the immediate solution of our problem.

Let us take next a less obvious proposition :

Ex. 2.—In the figure to Euc. I., 5 (Fig. 7), if B G, CF intersect in H, show that AH bisects the angle BAC.

Let us go over our data ::- -We have A B equal to A C, the angle ABC equal to the angle B C A, and also (see the proof of Euc. I., 5), the angle ABG equal to the angle A C F, and the angle G B C equal to the angle BCF. There are other relations which seem unlikely to aid us, so we content ourselves with

these. Remembering that we have to prove the equality of the angles BAH and C A H, we are at once led to notice that our data point to the equality of the triangles HBA and HCA. For we have the angle AB H equal to the angle A CH, and also AB, AH equal to CA, AH, respectively. But these relations are not sufficient. Seeing, however, the probability that the solution of our problem lies in this particular direction, we search for some new equality in the elements of the triangles ABH, CAH.

8

FIG. 7.

Can we, for instance, show that the angle AHB is equal to the angle AHC? This seems no easier than to establish the equality of the angles HA B, HAC. Can we, then, prove the equality of the sides HB, HC? This would involve the equality of the angles HBC, HCB (Euc. I., 6); and this is one of our data. Hence we see our way at once to the solution of the problem, which runs thus :—

Since the angle HBC is equal to the angle HCB, H B is equal to HC. Hence in the triangles

BAH, CAH, we have BA, AH equal to CA, A H, each to each, and the base BH equal to the base C H. Therefore the angle BAH is equal to the angle CA H. (Euc. I., 8.)

It will be noticed that the reasoning from which this solution is obtained is partly synthetical and partly analytical. We apply our data to obtain a result which very nearly gives us what we want; then we inquire analytically how the missing link is to be supplied; and finally, having seen our way to the solution, we run over such portions of our reasoning as are required for the complete proof of the proposition. The mental process is, of course, considerably longer than the solution which results from it-the mind runs rapidly over the elements given and required, selecting and rejecting this or that relation until the path to the complete solution has been traced out. I have only followed one such process of reasoning-that which seems to me most natural. Others might readily be conceived. Thus the equality of the lines A F, A G, and the angles AFC, AGB (see the proof of Euc. I., 5) might occur as the most obvious data for selection. would, then, be seen that before we can establish the equality of the triangles, FAH, GAH, we must prove that FH is equal to HG; but we know that FC is equal to BG; therefore, we must prove that the remainder, H C, is equal to the remainder, H B. This requires the equality of the angles, HBC, HCB. We know these angles to be equal; there

It

fore, H C is equal to H B, and thence FH to HG; and since AF, AH are equal to G A, AH, the angle FAH is equal to the angle G AH. It is probable, however, that the geometrician, being led in this way to the equality of HB and HC, would not retrace the steps he had followed, but would immediately notice the shorter proof depending on the equality of B A, A H to C A, A H, respectively. Thus we gather an important rule. Having tracked out, analytically or synthetically, a complete proof of a proposition, it is well before writing down the solution to notice whether the relations which have presented themselves in the process of reasoning suggest a shorter proof, or whether any of the steps of the reasoning may be omitted, or so varied as to be reduced in number. The value of a proof is, of course, much enhanced by brevity and conciseness.

IV. THEOREMS.

We have hitherto taken theorems involving exact results as our illustrative examples, and we have seen that to such theorems, analytical or synthetical methods, or combinations of both, are applicable with equal advantage. We shall presently discuss other propositions of this sort, and of greater complexity. But we must now notice the fact that in certain propositions we have no choice as to the method of solution. This is almost always the case with theorems involving general results, and with problems properly so called—that is, with propositions in which some