44 



♦ KNOWLEDGE ♦ 



[Jax. 16, 1885. 



outwards into the bedroom. A little examination of the 

 way in which the flat plates of the butt are countersunk 

 into the frame and the door&tile respectively, will teach 

 him more than a page of description. A couple of cup- 

 board handles with a small flat bolt may be easily fixed by 

 the aid of the centre-bit and a firmer chisel. Finally, a 

 piece of planking 4 ft. 6 in. long, 9 in. wide, and .V in. thick 

 may be nailed on to cover the gap between the top of the 

 door-frame and the top of the cupboard {tt. Fig. 34), this 

 may be moulded as recommended for the upper part of the 

 bookshelves, whose construction I described in Vol. VI., 

 p. 479, and finally the whole structure should be painted 

 to match the rest of the rcom. 



CHATS ON 

 GEOMETEICAL MEASUREMENT. 



{Continued from p. 11.) 



By PacHARD A. Pkoctor. 

 THE HYPERBOLA. 



A. I do not feel quite satisfied yet, with regard to the 

 conic sections. We have done very little with the hyjier- 

 bola. Can you not show how it is to be dealt with 

 geometrically ? 



J/. I would not trouble about it if I were you. You 

 will not regard the result, or much of the work as 

 geometrical. 



A. Still, T should like to have an idea of its nature. The 

 area of the parabola came out so neatly, that I want to see 

 why the hyperbola cannot be conveuiently dealt with. 



M. I have no objection. Let K A K' (Fig. 1) be a 

 rectangular hyperbola ; C its centre ; A L its axis ; C A, 

 C B its semidiametera ; and ON, C N' the asymptotes. I 

 suppose what j-ou want is to determine the area of such a 

 space as A P K L, where K L is drawn perpendicular to 

 the axis 1 



Fig. 1. 



A. That was what we did with the parabola. 



M. Note first, then, that drawing X K K' N' as in the 

 figure, C A, A L being given lengths, K L and K K are 

 known. For K K . K N'=sq. on A h' or B C. Thus, if, on 

 N N', a semicircle N k N' is described, and L li being taken 

 equal to A V, k h k' is drawn parallel to N X', then k K, 

 /.■ K' perp. to N N', give points K and K' on the hyperbola; 

 and K L is thus seen to be equal to a side of a right-angled 

 triangle L K k, whose hypothenuse is equal to L N (or C L) 



its other .side to B C or C A. Thus, in what follows we 

 may deal with X K, K L as assigned lengths. (We note 

 also in passing a way of obtaining any number of points on 

 the hyperbola by the simple geometrical construction just 

 indicated.) 



A. And now, can we determine the area A P K L ^ 



J/. Xot directly. For the lengths corresponding to 

 K L are not so simply related to A C as to make a series 

 of thin rectangles, perp. to C L, conveniently summable (if 

 I may invent such a word). 



A. How then are we to proceed? 



J/. The simplest rectangular relations of the hyperbola, 

 — those, therefore, which seem most likely to suit our 

 purpose, — belong to the space between the hyperbola and 

 its asymptotes. As you know, if from P, P ^ and P V 

 are drawn perp. to the asymptotes, the rectanoje P / C ^' is 

 constant, and equal to the square ADC D'. (This, by the 

 way, gives another and a still more convenient way of 

 obtaining points on the hyperbola : thus draw any straight 

 line CoO cutting D A in o, and D'A, produced, in O; 

 then 0^ parallel to C N', meets I' o parallel to C N, pro- 

 duced, in P, a point on the hyperbola. For the parallelo- 

 grams U' and D D' ai'e equal, since l.o and oD' are 

 complementary.) 



A. Do you think it is well to go off on these side 

 tracks 1 



J/. For me the chief charm of mathematics resides in 

 side views ; and wo are not in such a hurry, here, that 

 we need always " 'ammer, 'ammer, 'ammer along the 'ard 

 'igh road." To return to the main track, however, — 

 suppose we deal with such an area as P Z E K, obtained 

 by drawing perps. P I and K E to C N. Now, if we try 

 drawing Q m, E n, and so on, perp. to C N, at equal dis- 

 tances from each other, we shall find we make no progress ; 

 for the rectangles Q I, Pi in, io., are not conveniently pro- 

 portioned to each other, or to any other set of rectangles, 

 as in the case of the ellipse and parabola. Suppose, then, 

 we try, next, such an arrangement that the rectangles Q /, 

 R m, ifcc, shall all be equal. In this case we have 

 I in :mn:: Hn : Q M 



: : C m : C n (since C m . m Q = C n . n R) 

 Cl:Cm:: Cm : Cn 

 so that C/:C« ::(CO-:(Cm)■- 

 A similar relation holding for all the points of division 

 along I E, it follows that if there are n spaces like I m, m n, 

 between I acd E, — 



C/:CE::(CO":(Cto)» 

 CE /Cots". 



wherefore, taking logarithms 



A. Oh, come ! I say, you know ! These are geometrical 

 chats, are they n it ] 



M. And taking logarithms is not a geometrical process, 

 you would imply 1 



A. Well, I think Euclid would have been rather sur- 

 prised if you had as-ked him to take a logarithm. 



J/. We have arrived at a stage where we cannot help 

 ourselves. Just as the area of the circle (and therefore 

 that of the ellipse) can only be determined by arithmetical 

 operations, so it is with the area of hyperbolic sections. 

 Arithmetic is, however, very closely related to geometry. 

 Observe, the relation we have just obtained, means that 

 the length of C E bears to the length of C ? (these lengths 

 being expressed as numbers) is the same ratio which the nth 

 jiower of the length of C m bears to the nth. power of the 

 length of C m (these lengths being also represented as 

 numbers). Now what we want to determine, in reality, is 

 the length C m, such that n spaces measured off from I on 



