REMARKS OS BOOK I.] 



MATHEMATICS. PLANE GEOMETRY. 



657 



useless tautology ; but precision requires that they should 

 always be retailed. If you were to say that two sides 

 of one triangle are equal to two sides of another, your 

 meaning might be taken to be, that the aggregate or sum 

 of the two sides of one triangle is equal to the aggregate 

 or sum of the two sides of the other ; but the addition of 

 the words "each to each" would preclude the possibility 

 of such a mistake, and would show that the sides, taken 

 separately and individually in the one triangle, were 

 affirmed to be equal to corresponding sides in the other. 



The four propositions next following are problems. 

 You may be pretty sure that Euclid has postponed them 

 till they became indispensable. We do not think Euclid 

 liked problems ; at all events, there is less careful finish 

 about them than in his theorems. Proposition IX., for 

 instance, needs mending a little ; it professes to teach 

 how to bisect an angle, of whatever magnitude it may 

 be. Now, suppose that the triangle A D E, in the book, 

 is an equilateral triangle, and that we want to bisect the 

 angle DAE. Euclid tells us to construct an equilateral 

 triangle on 1) E ; and without the diagram before our 

 eyes, where the construction is exhibited in its completed 

 state, we should naturally describe the equilateral triangle 

 he directs, abort D K, and not below ; in which case we 

 should get nothing ; for our new equilateral triangle 

 would simply cover the one already there, and the point, 

 F, falling on A, would have no separate existence ; so 

 that there would be no guide to the drawing of A F, the 

 bisecting line ; it should have been distinctly stated, 

 fore, that the equilateral triangle, to be described 

 on 1 ) E, should have its vertex, F, on tho opposite of 

 IN to the point A. This restriction is introduced in 

 the present edition. 



In going over Euclid's proposition without the book, 

 as we have recommended above, always refrain from co/nj- 

 inyt/ii' diai/rn i/i.. We know that such is the usual practice ; 

 but it should be condemned. The progress of the diagram 

 should just keep pace with that of the text, and no line 

 should be introduced till it is actually demanded by the 

 text. It would be nearly as faulty as to write out the 

 whole text, and then to supply the diagram (as the boy 

 did who said he would tell the story first, and draw the 

 picture afterwards), aa to commence with the completed 

 diagram, and then supply the text. In a printed book, 

 the diagram must, of course, be presented cmnpleted ; but 

 in your own private practice you should make it grow to 

 maturity along with the text. In the whole course of 

 your geometrical studies, let us urge upon you never to 

 allow your judgment or conviction to be in the slightest 

 degree biassed by your visual impressions from the dia- 

 gram. Let two lines look ever so like two equal lines, 

 do not forestal the reasoning, and conclude them equal 

 from their appearance ; remember always that you are 

 engaged in a purely intellectual process, and that you are 

 not to be allured by the matter from the minil. Graphical 

 accuracy, in the figured form, is of no moment ; logical 

 accuracy; in the abstract reasoning, is all that you have 

 to attend to ; and therefore we think it worse than waste 

 of time to be over-scrupulous with scale and compasses, 

 in reference to the lines introduced into Euclid's demon- 

 strations ; we have already given you some hints on this 

 matter. 



There is nothing that calls for special notice till we 

 reach Proposition XVI. This is easy enough, aa far as 

 Euclid carries the demonstration ; but when, at the close, 

 he says, as in other editions of the Elements he is made 

 to do, " in the same manner it may be demonstrated," a 

 beginner is likely to feel a difficulty. There is really a 

 good deal to do before the proof can be completed ; and, 

 when completed, " in the same manner," there is a need- 

 less amount of complication. We would advise you to 

 finish the reasoning rather differently. By carefully 

 looking at the argument, you will see that this truth is 

 established, and nothing more ; namely, that if one side 

 of a triangle (any side, of course) be produced, the ex- 

 terior angle is greater thnn that interim- nmjle which is 

 o/iponite In the side thus mat/need ; the angle A C D is thus 

 greater than A. Let now A C be produced to G ; then, 

 nice the exterior angle is greater than that interior one 



which is opposite to the side produced, the angle B C G 

 is greater than ABC; but B C G is equal to A C D (by 

 the fifteenth), therefore A C D is greater than ABC; 

 but A C D was shown to be greater also than BAG; 

 therefore A C D is greater than either of the interior and 

 opposite angles, B A C, A B C. It is this form of com- 

 pleting the demonstration that has been adopted in the 

 present work. 



Proposition XVII. would seem, at first sight, to have 

 been introduced without any object. The truth of it is 

 clearly implied in Proposition XXXII., and it is not 

 required in any of the intervening propositions. But 

 that Euclid had an object is not to be questioned ; and 

 it seems to have been this : It was desirable that, at 

 some convenient place, before the introduction of the 

 theorems respecting parallel lines, something should be 

 established by demonstration that would diminish the 

 repugnance, very properly felt at the outset, to the 

 twelfth axiom. We have recommended you (p. 540) to 

 keep this axiom in the background till you arrive at 

 Proposition XXIX. , where a reference to it becomes in- 

 dispensably necessary. The axiom is no other than the 

 converse of this seventeenth proposition ; this shows that 

 if two meeting or non-parallel lines, B A, C A, be cut by 

 a third line, 15 1), the two interior angles, C B A, BC A, 

 on the same side of it, are together less than two right 

 angles ; and the twelfth axiom asserts, conversely, that 

 if a straight line cutting two others make the two interior 

 angles on the same side of it less than two right anyles, 

 those must be non-parallel or meeting lines. 



The seventeenth proposition, therefore, enables us to 

 see more clearly the exact amount of assent demanded 

 of us by the twelfth axiom, and prevents our overrating 

 that amount. If two lines out by a third meet, the two 

 interior angles are less than two right angles : this is 

 proved. If two lines cut by a third make the two interior 

 angles less than two right angles, tltey meet: this is 

 assumed. 



Passing over, for the present, the intermediate pro- 

 positions, let us suppose Proposition XXIX. to be 

 reached. The two propositions immediately preceding 

 sufficiently show that the lines called parallel lines exist ; 

 the twenty-ninth demonstrates a property of them, 

 admitting the truth of the axiom just mentioned. Geo- 

 meters without number have tried, some to evade this 

 axiom altogether, and others to prove it by establishing 

 the converse of Proposition XV11. ; but all have failed. 

 What can be the cause of this failure ? Is it not in the 

 imperfect definition of a straight line ? Our conception 

 of a straight line, independently of all formal definition, 

 necessarily involves two ideas ; namely, that of lem/th, 

 and that of uniformity of direction. Lenyth is implied 

 in the word line ; and invariability of direction in the 

 term straight. A line which changes its direction is a 

 crooked line or a curved line ; a line that never changes 

 its direction is a straight line. Now it necessarily 

 follows, from this uniformity of direction, that if two 

 straight lines, however far prolonged, can never meet, 

 then at no part of their course can either make any 

 approach towards the other ; for if two lines, proceeding 

 in any two directions, approach and continue undeviat- 

 ingly to pursue those directions, they cannot fail even- 

 tually to meet. It follows, therefore, that parallels must 

 thrmiyKout be equidistant ; but two distinct straight 

 lines, through the same point, cannot throughout bo 

 equidistant from a third ; so that two straight lines 

 through a point cannot both be parallel to the same 

 straight lins. Proposition XXVIII. shows that one 

 (C D) will be parallel to another (A B), provided a line, 

 cutting both, make the interior angles together equal 

 to two right angles ; a second line through H, which 

 causes the interior angles to be lens than two right 

 angles, being distinct from CD, must therefore meet 

 A B, if prolonged ; and this is the assertion of the twelfth 

 axiom. 



What is here said, remember, is not a demonstration 

 of this axiom. An axiom, you know, is an indemon- 

 strable truth. All we wish to show is, that it is an 

 axiom ; that is, a truth necessarily implied intlio correct 



