382 



♦ KNOVS/'LEDGE ♦ 



[Nov. 7, 1884. 



are less than the curved lines A C, C D, D E, ic, together, 

 — that is, less than the arc A E B — they are greater than 

 A B, and so come nearer to the arc in length. So, if we 

 divide the arc A C into a number of small ones, the chords 

 of these together will be nearer in value than the chord 

 A C to the arc A C. And so of the other arc D, D E, 

 E F, &c. Thus we get nearer and nearer to the length of 

 A B, by taking a greater and greater number of chords 

 along its length. Still this does not prove that we can 

 approach the length of the arc within any quantity, how- 

 ever small, that may be indicated. For you may draw 

 nearer and nearer to a quantity yet always remain separated 

 from in by a finite quantity. 



A. That sounds paradoxical. 



M. WpII, take the quantity -3. Suppcse you take 1, add 

 i to that, I to the sum, J to that sum, and so on con- 

 tinually. Are you not continually approaching the value 

 3 (or indeed 4, or 5, or any number over 2) 1 Yet as your 

 sum never exceeds 2, you are always separated from 3 by 

 the finite quantity 1. 



A. Still in the case illustrated by Fig. 3 it really looks 

 as though you could get as near as you please to the length 

 of the arc A E B, by increasing sufficiently the number of 

 chords. 



J/. It looks so, and it is so ; ^but you cannot easily 

 prove it. 



A. I observe in your figure a number of straight lines 

 AH, H K, K L, L M, M N, NO, OB outside the arc 

 A B. What do they mean 1 



M. They are tangents at A, C, D, E, F, G and B. It 

 is tolerably obvious not only that A H, H K, K L, L M, 

 M N, N O, and O B together are greater than the arc A B, 

 but that by increasing the number of such tangent lines, 

 we can approach the length of the arc A B as nearly as we 

 plpase. 



^1. Yes ; only you approach it from the other side, from 

 outside, as it were. The real length of the arc lies between 

 tlie sum of the chords and the sum of the tangents. 



M. Precisely ; and if we can show that these sums draw 

 nearer and nearer to each other, and are finally separated 

 by a length which may be made less than any that can be 

 assigned we shall have obtained a satisfactory way of 

 measuring arc lengths. 



A. And I notice that so far as the set of outside lines 

 are concerned we may be said to be following the natural 

 way of measuring. For if we wished to measure the 

 length of such a curved arc as A E B we should apply to it 

 a fine string or a delicate chain, extended from A to B ; 

 and diminishing the number of our tangents AH, H B, 

 &c., corresponds exactly to making the links of our 

 measuring-chain shorter and shorter, which would naturally 

 make the chain a more and more delicate measurer. But 

 now can you prove that the outer and inner sets of short 

 straight lines approach each other indefinitely X 



Fig. 4. 



M. It will go near to be thought so presently. Let 

 A C, Fig. 4, represent the A of Fig. 3, A H the tangents 

 at A and C (not necessarily equal). Draw H P square to 

 A C, and with centres, A C, describe circular arcs, P k^ P /, 

 to the lines AH, H C. Then obviously the two lines H A, 

 H C together exceed A C, by the sum of the two short 

 lines H h, and H I. But since the arc A C, by the very 

 nature of a curved arc, changes continuously through alj 



the directions included within its length, the angles H A 0, 

 II C A, may be made as email as we please by bringing C 

 near enough to A in Fig. 3. We can always magnify it to 

 any size we please in Fig. 4. But P H is a tangent to the 

 circles of which Vk and P^ are arcs ; and it is obvious that 

 the nearer we bring A H and C H to A C, the smaller h H 

 and I H must become, and they can be made as small as 

 we please, while A H and H C never become less than 

 A P and P C. Therefore the excess of A H, H C together 

 over A C may be made indefinitely small compared with 

 A C. So that the sum of the excesses of AH, H K, K L, 

 L M, ic, in Fig. 3, over A C, C D, D E, E F, &c., may be 

 neglected compared with the sum either of A B, C D, D E, 

 E F, ic, or of A H, H K, K L, L M, i-c. 



A. But may not these little " excesses," as you call them, 

 though each is very small, made up an appreciable quantity 

 through their number. As you increase the number of 

 divisions you get more of these little bits. 



M. That makes no difference. We make the ratio of 

 each such pair as H ^, HZ in Fig. 4, to each part as A 0, 

 exceedingly small, — as small in fact as we please. The 

 ratio of all of them together to all such lines as A C, C D, 

 ic, together, cannot be greater than the ratio of the 

 largest pair, like H A, H Z to the chord corresponding to 

 AC. 



A. Illustrate this numerically. • 



M. Suppose you break up the number 1,000,000 into 

 a million small parts pretty nearly equal, each therefore 

 about equal in value to unity. Now, if from each of 

 these quantities you take off a very minute fraction, say, 

 one-millionth, you have a very large number — a million — 

 of these millionths of unity (or thereabouts) ; but all 

 together they cannot amount to more than one, a very 

 small proportion of your original number. If you increase 

 the number of parts of your original number to a billion, 

 and take oflT a trillionth of each part, the sum of these 

 billion trillionths will be just a trillionth of the original 

 number. It matters not how great the number of these 

 parts you take ; if the ratio of each to that from which 

 it is taken be very small, the sum of all of them will bear 

 just that small ratio to the original total. 



A. Is there any assumption at all, then, in your reason- 

 ing about arc lengths ? 



M. There is ; though no one can really doubt its fair- 

 ness. I have assumed that A H and H C together, are 

 greater than the arc A Q G within these lines and touching 

 them at A and 0. 



Fig. 5. 



A. But obviously they are greater. Why, you can draw 

 a tangent T Q T' to some point Q in the arc A Q C (Fig. .5). 

 Then the broken line A T T' C is nearer to the arc A Q 

 than the broken line A H C, and is manifestlv less than 

 A H C (since T Q T' is less than T H, H T' together). Then 

 we can draw the shorter tangents < R <', T S T' ; and so on, 

 getting continually nearer and nearer to the arc A R Q S C, 

 and at each step diminishing the total length of the tangent 

 chords. 



J/. Your reasoning is accurate. But still you require 

 finally an assumption, when your drawing of tangents 

 comes to an end. However I think we may fairly regard 

 the assumption that any tangents as A H, H C are together 

 less than the arc A t,) C as to all intents and purposes 

 axiomatic. 



