S38 



♦ KNOWLEDGE ♦ 



[Oct. 24, 1884. 



might be put tlius : If two straight lines falling upon a 

 third straight lino meet, the two inferior angles made by 

 tlietu with tliat third line, on the side towards which they 

 oieet, are together less than two right angles. 



A. That is obviously the converse of the twelfth so- 

 called axiom. And the 32nd proposition may obviously 

 be regarded as proving hy hov) much the two inferior angles 

 fall short of two right angles — viz., by the angle at which 

 the two straight lines are inclined to each other. But how 

 could all this be corrected % 



M. It is manifestly an incongruity to have definitions 

 which imply propositions, and propo.sitions which are the 

 conTerse of axioms ; and it is always undesirable to increase 

 the number of axioms. It appears to me that the number 

 of simple propositions should rather have been increased. 

 After all, the ideas underlying the definition of parallel 

 lines and axiom twelfth are simple enough. By accepting as 

 admisaible the definition of a right angle (for clearly there 

 mast be some position in which the angles on either side of 

 the finite line are equal), and the axiom that two straight 

 liaes cannot enclose a sjiace as really axiomatic, we can 

 arrive, I think, at both the definition of parallel straight 

 lines and at the proposition involved in the twelfth axiom, 

 without any difficulty — without even assuming that new 

 axiom which Simsou proposed. Through a given point only 

 one parallel can be drawn to a given straight line. 



A. How would you do this % 



M. I would deline parallel straight lines as straight lines 

 at right angles to the same straight line ; and I would 

 ^rove their fundamental property thus : — Let AB (Fig. 1) 



Fig. 1. 



(je a straight line, D C E a line at right angles to A B ; 

 and suppose that on A B, D E lines have been superposed 

 in the same way that a triangle is supposed to be super- 

 posed on another in the fourth proposition of the first book. 

 Now suppose the superposed lines shifted, so that while 

 the line superposed on D C E still coincides with this line, 

 the line superposed in A C B is shifted to some other 

 |X5sition as F H G. Then, when we note the equality of 

 the angles F H D, D H G, A D, D C B, we see that any 

 line of reasoning by which it could be proved that H G, 

 O B meet if produced towards G, B, would equally prove 

 that these same lines meet if produced towards F, A. 

 (We see this at once if we suppose the whole figure 

 turned round H E as an axis till H F comes to the 

 position H G, and A to the position C B.) Hence if 

 F G and A B, produced both w ays, meet at all, they 

 meet both towards F A, and towards G B, and so enclose 

 a space But two straight lines cannot enclose a space. 

 Hence F G and A B do not meet, however far they may 

 be produced both ways. 



A. Does not this demonstration indicate also some pro- 

 perties of parallels % 



M. It indicates several, among others the equidistance of 

 parallels. Thus suppose the points K and ]M on A B 

 carried to L and N by the shifting motion described, it is 

 evident that K L=C H = M N. 



A. And how would you deal with the present twelfth 

 axiom % 



M. Somewhat in this way. (I do not enter fuUy into 

 these matters because you really want information on a 

 different subject.) The lines D H E, A C B, F H G, Fig. 2, 

 being the same as in Fig. 1 , let K H L be any straight line 

 falling within the angle C H G, (so that the angles HOB 

 and C H L are together less than two right angles). Then 

 if we take any point L on K H L, and make L M = H L= 

 M N = N 0, &c., and draw perps. L G, M P, N Q, E, 

 &c., to HR, it can be readily shown that MP = 2 LG, 

 NQ=3LG, 0R=4LG and so on. Hence, however 



small L G may be, we must at last arrive in this way at a 

 distance — as S T, greater than H C, or such that S lies on 

 the side of A C B remote from F H G. Hence at some 

 point — as V — between H and S the line H S must inter- 

 sect the line A B produced. This proves what Simson 

 wanted as an axiom, — that through the same point H there 

 cannot be two straight lines H T and H M, both parallel 

 to another straight line A B. Then, as Simson showed, we 

 can deduce the 12th axiom as the general proposition of 

 which the above is a special case. But you will find all 

 that you need on that point in the Notes to Todhunter's 

 School Euclid. 



A. Let me see ; how did we get upon this subject of the 

 12th axiom 1 



M. It is a case of a limiting value. The axiom asserts, 

 and as we have seen, it may be jtroveH, that, no matter how 

 small the angle by which the " two angles taken together " 

 fall short of two riglit angles, the lines will meet if pro- 

 duced far enough. Now in all the problems we shall have 

 to consider, if we are to discuss the subject you have svg 

 gested, we have in fact to infer the exact value of som 

 curve, surface, volume, or the like by showing that such 

 value does not differ from that definite value by any 

 quantity that can be assigned, — however small. 



A. Before we leave this question of Euclid's manner, 

 may I ask whether he ever directly uses the method of 

 limits. I see that he indirectly does so in the twelfth 

 axiom. 



J/. He uses it in showing that a straight line at right 

 angles to a diameter of a circle, through its extremity, is a 

 tangent to the circle. For there he shows in effect that no 

 matter how near a point be taken on the circumference, to 

 the point through which the perpendicular is drawn, the 

 line joining the two points is not at right angles to the 

 diameter : we only get the right angle when the second 

 point actually merges into the first. 



A. Still, that is not the way Euclid puts it. 



J/. True. But in the twelfth book (which I should like 

 to see much more read than it is) Euclid definitely adopts 

 the method of limits, both in proving that circles are to 

 each other as the squares, and that spheres are to each other 

 as the cubes, of their diameters. His method is as follows : 

 If a sphere (for example) does not bear to another sphere 

 the ratio of the cube of the diameter of the first to the cube 

 of the diameter of the second, then must the first sphere 

 bear this ratio to another sphere differing from the second. 



