364 



♦ KNOWLEDGE ♦ 



[June 15, 1883. 



Tlio pntellitcs of Jiipitcr ore alwavs spoken of by nstroQomcrs by 

 their mimbors, in tho order of their distance from Jupiter. It 

 wunld l)0 well if tlie same were done with Saturn's. I cannot give 

 yon all the names, being away from my books, and never using the 

 names myself, except iu the case of Japetus, Hyperion, and Titan, 

 the Sth, 7tU, and Cth of Saturn's satellites. — J. Buown. The heat 

 would bo greater, for that side of any object exposed to tho sun, 

 much loss for the other, and tlicreforo the extremes greater — as in 

 cnso of the moon, tho temperature of whose surface varies some 

 600 degrees (Fahrenheit). 



©ur illatftrmatiral Column, 



GEOMETEICAL PROBLEMS. 

 By Bicn.\RD A. Peoctor. 

 Analysis and Synthesis. 



THERE are two general modes of treatment applicable to pro- 

 blems, termed, conventionally, the sytithetical and the analytical, 

 or synthesis and analysis. In the former, we study what is given 

 and work up to what is sought ; in the latter, we examine what is 

 sought and work back to what is given. I am not concerned here 

 with tho correct applicability of the names synthesis and analysis 

 to these processes, and shall therefore content myself with dis- 

 cussing the processes themselves under the names usually given to 

 them. 



It is a mistake to suppose that, as some have asserted, analysis is 

 the method always employed — consciously or unconsciously — in the 

 solution of problems. Of course, we are compelled to consider 

 what it is we liave to do or prove, and thus far the analytical method 

 cannot but enter into otir processes. But in the solution of a 

 problem, we may proceed, as may be most convenient, by either the 

 synthetical or the analytical process, or — which in complex problems 

 is far more commonly the case — by an alternation of both methods. 

 As an illustration of my meaning, I may compare geometrical 

 problems to those examples in algebra, trigonometry, &c., in which 

 wo have to establish the identity of two expressions. In such 

 cases we may either take one expression, and try to work it into 

 the same form as the other, or vice versCi, we may select the latter 

 to work upon, or — which is the surer process — we may work both 

 down to a common form. 



However, it will be better to select a few examples of geometrical 

 problems, and to exhibit the application of different processes to 

 them, than to discuss general rules. I begin with very simple 

 examples. 



Suppose we have given to us the following deduction : — 



Ex. 1. — The line A B (Fig. 6), is hisected in C, and C D is drann 

 at right angles to AB. From any point E in CD lines are drau-n 

 to A and B. Shoxc that E A is equal to E B. 



Having constructed a figure in accordance with these data, we 

 go over the data thus : we have A C equal to C B, and C E at right 

 angles to A B. We remember, also, that we have to prove that 

 A E is equal to E B. Xow, we cannot fail to see that the data in- 

 volve the fiqnahty of the triangles, A C E, C E B, by Euc. I., 4, and 

 therefore that A E is oi|Ual 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, C E, equal 

 to B E, C E, and the included angles equal — to the equality 

 of the triangles, ACE, B C E, in all respects, and thence 

 to the equality of A E, E B. In the analytical solution we 

 should argue thus :— Wo have to show that A E is equal to E B. 

 Xow, 1/ A E is ctinal to E B, then since A C, C E are respectively 

 equal to B C, C E, the angles ACE and B C E will be equal 

 (Euc. I., 8) ; but these angles are equal, being rfght 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 tho equality of 

 A E, E B, is complete by this method. We shall pressntly see that 

 the mere fact of obtaining by the analytical method a result corre- 

 sponding 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 

 liroof that the reversal of the process will give at once a satisfactory 

 solution of a problem. 



Suppose that we have given to us A C equal to C B, and tho 

 angle A E equal to the angle C B E, and that we have to shoir 

 from these data that C E is at right angles to A B. We proceed 

 analytically thus : // C E is at right angles to A B, then A C, C E 

 being equal, respectively, to B C, C E, the triangles A C E, B C E 

 are equal in all respects ; therefore the angle C A E will be eqtial 

 to tho angle C B E. Xow, these angles are equal ; therefore we 

 might exiicct the reversal of the process to lead at once to the solu- 

 tion of our problem. This, however, is not the case — we have A C, 

 C E equal to B C, C E, and the angles C A E, C B E 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 C A E and B A E are equal in all respects. 



Of course, there is no difficulty in the above problem. The equality 

 of the angles, C A E and QBE, give us immediately A E equal to 

 E B (Euc. I. G), and thence the equality of the triangles, ACE, 

 B C E, follows at once. But it is well to notice that analysis may 

 lead to a result included 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 Fig. to Euc. I., 5 (Fig. 7), if B G, CF intersect in 

 H, show that AR bisects the angle BAG. 



Let us go over our data : — We have A B equal to A C, the angle 

 ABC equal to the angle BCA, and also (see the proof of Euc. I., 

 5), the angle A B G 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. Remember- 

 ing 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, H B A and H C A. For we have the angle 

 A B H equal to the angle A C H, and also A B, A H, equal to CA, AH, 

 respectively. But these relations are not suificient. Seeing, how- 

 ever, the probability that the solution of our problem lies in this 

 particular direction, we search for some new equality in the 

 elements of the triangle, A B H, C A H. Can we, for instance, 

 show that the angle A H B is equal to the angle A H C ? This 

 poems no easier than to establish the equality of the angles, 

 II A B, HAC. Can we, then, prove the equality of the sides, 

 H B, H C .^ This would involve the equality of the angles, H B C, 

 HOB (Euc. I., 6) ; and this is one of our data. Hence we see 

 our way at once to the solution of the problem, which rtms 

 thus : — 



Since the angle H B C is equal to the angle H C B, H B is equal 

 to H C. Hence in the triangles B A H, C A H, we have B A, A H 

 equal to C A, A H, each to each, and the base B H equal to the 

 base C H. Therefore the angle B A H is equal to the angle C A 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 wo want ; 

 then we inquire analj-tically how the missing link is to be supplied ; 

 and finally, having seen oar 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 thati tho solution which results from it — the mind runs 

 rapidly over the elements given and required, selecting and reject- 

 ing this or that relation until the path to the complete solution 

 has been traced out. 1 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, AG, and the angles AFC, A G B (see the 

 proof of Euc. I., 5) might occur as the most obvious data 

 for selection. It would, then, be seen that before we can establish 

 the equality of the triangles, F A H, G A H, we must prove that 

 F H is equal to H G ; but we know that F C is equal to B G ; there- 



