Dec. 26, 1884.] 



♦ KNOWLEDGE ♦ 



531 



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EUCLID'S THEORY OF PARALLELS. 



[1542] — In letter 1,495, Mr. C. L. Dodgson seems to imply that 

 a proposition is not truly the converse of another unless it follows 

 immediately from that other. This is by no means the case. The 

 proposition that " all animals are men '' is the converse of the pro- 

 position that " all men are animals ; " yet one is true the other false. 

 With regard to my comparison, at p. 337, between Euclid I., 17, and 

 the 12th axiom, the point to be noticed is that the two propositions 

 are equally far from being axiomatic ; or if there is any difference 

 Prop. 17 is the more nearly axiomatic; for it is only a general pro- 

 position, whereas Axiom 12 is exact, implying as it does that no 

 matter how minute the deficiency of the two augles from two right 

 angles, the two lines will meet. The best proof that Prop. 17 is 

 the simpler of the two is found by comparing the results when the 

 method of superposition is applied. The truth of Axiom 12 is only 

 proved — even in mere sketch I presented — by a rather long pro- 

 cess ; but here is the proof for Prop. 17 : — 



Fig. 1. 



Prodnce the side B C of the triangle ABC (Pig. 1) to c, so that 

 C c is equal to BC. Apply the triangle ABC to Cc so that BC 

 coincides with C c. TVe may suppose this done by sliding the whole 

 triangle ABC along B C c. Then obviously the line B A will fall 

 into such a position as C a, wholly on the right of A B. Hence the 

 two angles ABC, A C B together are equal to the angles a C c, 

 A C B together, — or fall short of two right angles by the angle 

 AC a. 



In like manner Prop. IG may be proved, — or made obvious. 



As to Mr. Dodgson's second objection, he does not show that KL 

 is not obviously equal to C H. This is obvious from a consideration 

 of the method of superposition employed. He goes on to a proof 

 depending on the proposition that if two triangles have the three 

 sides of the one equal respectively to the three sides of the other, 

 they are equal in all respects. But I wished to use only consi- 

 derations based on the effect of superposition. Of course Prop. 32 

 can easily be proved by a simple method of superposition. Indeed 

 Prop. 32 has almost as good a right as Prop. 16 or Prop. 17 to be 

 regarded as axiomatic. If we adopt the idea of an angle as pro- 

 duced by the turning df a straight line round a point, the line 

 moving in one plane, Prop. 32 comes out frora very simple consi- 

 derations, thus : — 



1. Suppose A B (Fig. 2) to represent two coincident straight lines, 

 and one of these to turn round the point C, moving in one plane, 

 to the position D C E : then B C D is the angle thus swept out. 



2. It matters nothing so far as the angle between the two lines 

 is concerned whether the turning be round the point C or another 

 point as F (to position G P H), so that the amount of turning is the 



Fig. 2. 



same, — that is, the angle G F B (tested by superposition) equal to 

 the angle D C B. 



Fig. 3. 



3. Xow let ABC be any triangle, and let its sides be all pro- 

 duced both ways, as in Fig. 3, the line DACE so formed being 

 supposed double. Let one of these lines turn around the point A 

 to the position F A B G, sweeping ont the angle B A C. F A B G is 

 now double. Let our component of this double line turn around 

 the point B to the position H C B K, sweeping ont the angle A B C. 

 H C B K is now double. Let one of its components be turned 

 around the point C to the position E C A D, sweeping out the angle 

 B C A. Then a straight line originally coinciding with DACE 

 has been so turned as to measure in succession the three angles of 

 the triangle ABC, and is found at the end of the process to coin- 

 cide again with the line DACE, but so that the ends which had 

 been towards D and E are now towards E and D respectively. 

 In other words, it has made half a rotation, or has swept ont two 

 right angles. Hence the angles of the triangle A B C are together 

 equal to three right angles. 



The third objection in Mr. Dodgson's letter is apparently based 

 on the idea that Simson's axiom, " There cannot be two parallels, 

 through a given point, to the same straight line," refers to 

 parallels as negatively defined by Euclid. It this had been the 

 case, Simson's axiom would be less axiomatic even than Axiom 12. 

 It would, in fact, be simply another form of Axiom 12, only with- 

 out the definite statement which at least enables the student to 

 grasp the meaning of the statement. My definition of parallels in- 

 cludes, as a consequence, the property which Euclid assigns to 

 parallels, and since I prove that not more than one line fulfilling the 

 positive property I assign to parallels can be drawn through a point, 

 so as to be parallel to a given line, I in effect prove that not more 

 than one parallel as defined by Euclid (i.e., negatively) can be 

 so drawn. Richabd A. Pkoctob. 



New York, Xov. 27, 1884. 



MATTER. 



[15431— In Letter 1,534 your correspondent F. W. H. gives an 

 extract "from Haeckel's " Pedigree of Man," in which it is said 

 that " we must hold that atoms are the smallest particles of 

 masses, having an iinalterahle nature, separated from one another 

 by hvpothetical ether." On reading this passage the question wiU 

 naturally occur to every thoughtful person, What basis can there 

 be for that part of the statement which I have underlined ? What 

 claim has it, even if uttered by the highest authority, upon our 

 acceptance? The answer seems to me to be contained in some 

 sentences written by the late Professor Clifford in his " Ethics of 

 Belief." There he says : — " He [referring to a person making a 

 statement similar to" the one just quoted] may quite honestly 

 believe that this statement is a fair inference from his experiments 

 but in that case his judgment is at fault." '' I have no right to 



