Nov. 21, 1884.] 



KNOWLEDGE ♦ 



417 



They may not do so at the beginning of our approach, but 

 they must do so to the end. For by similar figures these 

 ratios are the same respectively as A B to A K, A r to A K, 

 A rf to A K ; and eventually we get a line as A d but in- 

 clined to A T at an indefinitely small angle. Such a line 

 would be actually equal to A K, so long as the angle B K T 

 is finite. Hence such a ratio A B to A K becomes even- 

 tually one of equality when the point B is about to merge 

 into A. Nor does it matter if the lines drawn from C and 

 D, or have such positions as Gl, D in, — that is make 

 varying angles with A T, — so long as the angles B K T, 

 C ^ L, D 9»i M remain finite, no matter how small they may 

 become, the ratio of the chord like A D to the part like 

 A m cut off the tangent must eventually be one of equality. 

 For a finite angle, however small, is infinitely greater than 

 an evanescent angle. Thus from B, Fig. 7 on the line A B, 

 let a line B K L be drawn making a very small but still a 

 finite angle L B A with A B ; then it is manifest that if an 



Fig. 7. 



indefinite line A T be turned round the point A B (in the 

 plane of the figure) till it coincides with A B, the point of 

 intersection K will move up to and eventually coincide 

 with B, no matter how small the angle L K B : that is 

 A K, and A B will eventually be equal, — ^just as A T, in 

 its motion around A, is merging into A B. 



A. Now tell me how this helps us. 



M. Why, instead of the chord A B in Fig. G, we may 

 take A K, a part of the tangent cut ofi" by any line such as 

 B K making a finite angle {let it he what it may) with the 

 tangent A T. 



A. Can you illustrate the advantage of this? 



M. In other words, can I show how the length of a 

 curved arc may be obtained, in some given instance, by this 

 method ? I can ; and much more readily than by taking 

 chords or tangents fitting round the curve as in Fig. 3. 



A. Will you begin with the arc of an ellipse ? 



M. Not quite. 



A. And why not? 



M. Simply because the case is too difficult. I will begin 

 with the arc of a cycloid. 



A . I thought the cycloid was a curve of higher order 

 than the ellipse. 



M. So it is. In fact, it is of infinitely higher order, 

 since, if expressed in the form of an equation between 

 X and y, the powera of these variables would be infinite. 

 But that need not concern us. It happens that the arc can 

 be easily obtained by a geometrical method. 



Pig. 8. 



A. To the charge, then ! 



M. Please observe that I am not going to do more, in 

 this or any other case, than give such suificient outline of 

 the proof as may enable you to satisfy yourself that the 



relation dealt with is really demonstrable on the lines fol- 

 lowed. You must fill in details for yourself. 



A. I will endeavour so to do. 



M. Let A P D be a half cycloid, A B its axis, Aqp'R 

 half the generating circle. Through P, Q, two neighbour- 

 ing ]ioints on the cycloid, draw Pyj L, T Q (7 i- M. Join A q; 

 draw A k n p ; the tangent P T ; and the arc q n round A 

 as centre. Then, by a known property of the cycloid, T P 

 is parallel and equal to kp : q n is eventually perp. to A p, 

 when Q comes close np to P; and triangle (7/7 A' is even- 

 tually isosceles { /. q k p ■= L k p\i-=z A q p k, standing even- 

 tually on equal arc); hence A?i = 71/) ; or kp, i.e. F'T:='2np. 

 But P T is eventually equal to chord or to arc P Q since 

 Q T is inclined at a finite angle to the tangent P T ; and 

 np is the excess of chord A^) over chord A q. Thus if we 

 suppose the arc of the cycloid measured from A, while as 

 we take successively small increments of the arc (as we 

 have just taken Q P) we keep on taking the chord of Aqp B 

 farther and farther towards B, (as we have just passed 

 from the chord A 7 to the chord A})) the growth of the arc 

 of the cycloid will always be double the] gro\\ th of the 

 chord from A, to advancing points along the semicircle 

 A (? /) B. Since they start together from naught, then, the 

 cycloidal arc must always be just double the corresponding 

 chord. Thus, 



Arc A Q P=2 chord A /) ; 

 Arc A P D = 2 diameter A B ; 

 And any arc QPE = 2 K r, obtained by describing the 

 circular arc </ ?i K round A, to meet chord A r. 



A. That is strange, — a curved arc like Q R equal to 

 twice a straight line like K r, and that, too, in the case of 

 such a curve as the cycloid ! 



^f. Do you see any flaw in the proof 1 



A. No; except that when P is very near A it can hardly 

 be said that the angle between P T and Q T is finite. 



J/. You are right. But consider how much or how little 

 this affects our result. It amounts merely to this, that for 

 an indefinitely small part of the arc A P D, near A, our 

 proof fails ; that is, for a portion of this arc near A, which 

 may be made less than any distance which can be assigned, 

 the proof fails, — which is as much as to say that it does 

 not fail at all. 



{To be continited.) 



THE CHEMISTRY OF COOKERY. 



By W. Mattieo Williams. 

 XLVII.— THE COLOURING OF WINE. 



SO^IE years ago, while resident in Birmingham, an 

 _ enterprising manufacturing druggist consulted me on 

 a practical difficulty which he was unable to solve. He 

 had succeeded in producing a very fine claret (Chateau 

 Digbeth, let us call it) by duly fortifying with silent spirit 

 a solution of cream of tartar, and flavouring this with a 

 small quantity of orris root. Tasted in the dark it was all 

 that could be desired for introducing a new industry to 

 Birmingham ; but the wine was white, and every colouring 

 material that he had tried producing the required tint 

 marred the flavour and bouquet of the pure Chateau Dig- 

 beth. He might have used one of the magenta dyes, but 

 as these were prepared by boiling aniline over dry arsenic 

 acid, and my Birmingham friend was burdened with a 

 conscience, he refrained from thus applying one of the 

 recent triumphs of chemical science. 



This was previous to the invasion of France by the 

 phylloxera. During the early period of that visitation, 

 French enterprise being more powerfully stimulated and 



