10 



* KNO^A^LEDGE ♦ 



[November 1, 1886. 



MATHEMATICAL RECREATIONS. 



[fE interest taken iu the quasi-matbemafcical 

 jnizzles which have recently appeared in 

 Knowledge has suggested the idea that 

 pleasant recreation might be combined with 

 useful instruction in the properties of 

 mathematical curves, figures, and so forth, 

 in a series of papers (not necessarily con- 

 tinuous), to be called Mathematical Eecreations. 



As examples, 1 ttike the three puzzles given in the October 

 number of Knowledge. 



I. The Parabola.— Tho properties of this curve must 

 of course be studied in treatises on conic sections to be 

 thoroughly understood ; but many who do not ctxre to enter 

 into such complete study of the curve as this may be inter- 

 ested to know how they may pleasantly and prettily con- 

 struct true parabolas, while others who really are students 

 of mathematics will gain clearness of insight into the 

 parabola's properties by constructing the curve in various 

 attractive yet exact ways. 



Of all recreative ways of presenting geometrical curves 

 none are so pleasing, either in actual construction or in 

 their effect, as those which show a series of enveloping 

 tangents, as is required in the puzzle on the parabola. I 

 give for the present only methods of this kind. (The curves 



Fig. 1 



may be dealt with in other ways, some of which are perhaps 

 in a geometrical sense more effective.) All the conic sec- 

 tions may be represented by enveloping tangents, but the 

 construction for the parabola is the simplest of all. 



Draw lightly in pencil on a card, preferably an enamelled 

 white or tinted card — only in this case draw the lines on the 



unenamelled sides — two straight lines, ab and Ac, fig. 1, at 

 some such angle as is shown iu the figure. Opening a pair 

 of dividere to any convenient small distiince, ad or ae, 

 measure off equal distances along ab and ca as shown. 

 There is no occasion to divide a given line, ab, into a 

 number of equal small parts ; simply the measurements go 

 on from a till a point is reached conveniently near b, and 

 then similar measurements are made along AC. Tbere 

 should be the same number of divisions along ab and AC, 

 and the number should be even, for a reason presently 

 recognised. Now pierce fine holes through the card at the 

 division marks, and taking a needleful of tine silk (of any 

 dark colour if the card is white, but if the card has tint 

 then it is best to have the silk of the com[ilenientary 

 colour) carry it from b (where it may be knotted on the 

 under side) to A, thence under the card to come out at e 

 and from e to f, thence under the card to c;, and from g to 

 H, and so on. If the number of divisions was originally 

 even, yon arrive in this way at a line km, and going on, 

 you finish with the lines dn and CA, completing the series 

 of thwart lines. Each of these lines is a tangent to a 

 parabola having its axis on the bisector of the angle bac, 

 and the lines drawn as described show the true shape of a 

 parabola very effectively by giving a .series of enveloping 

 tangents. You may now draw bc and bisect in n, con- 

 necting Are with a line of silk. This line bisects km as 

 at a, which is the vertex of your silken parabola. If 

 your work has been con-ectly done, all the correspondiug 

 tangent lines from au and Ac cro.-<s on Are. 



To mark in the focus of your jiarnbola, the construction 

 indicated in the figure is all that is necessary. Di-aw any 

 straight line, as Ikl', i)arallel to km or bc, and take kl and 

 kl', each equal to twice uk. Then drawing hi, W , you have 

 lines intersecting the parabola at the extremities of the lahis 

 rectum, lsl', cutting an in the focus s. These lines of con- 

 struction should be drawn in pencil, lightly, before the 

 parabola is obtained from its silken envelopes. Afterwards 

 ])ierce the card at h and l', and take a silken line lsl'. 

 Around the focus s a little star may be made (with yellow 

 rays) as shown in the figure. 



The following features may be noticed : — 



(1) Such intersection points as h, h, h', (fee, p;)//, &c., lie on 

 lines parallel to \n. By an optical illusion, it will be found 

 that if we run the eye from any division point (as ii) on ab 

 AC to the series (as h,h', itc.) really lying on a line parallel 

 to AW, the idea is conveyed that the line so followed curves 

 inwards towards Are. 



(2) If we take points P, /;, c, kc, or m,;j, //, </, itc, running 

 from any division point along AC or ab athwart the other 

 diagonals of the quadrangular divisions, we get a series 

 of points lying on a parabola having its axis in the 

 line Are. 



(3) The alternate parallels thus obtained are diameters 

 through o,o', L, l', etc., the points in which the enveloping 

 tangents of our figure touch the parabola. 



(4) If we take any point, as o, in which one of the 

 tangents touches the parabola, then a series of lines, nL', 

 o'm, Lq, rr , d-c, obtained by joining the successive points of 

 tangential contact on either side of o, will be parallel to 

 each other and to the tangent at o ; so that they will be 

 ordinates to the diameter through o. 



II. The Ellipse. — This, of all the conic sections, is the 

 least convenient to deal with in the manner we are con- 

 sidering. Here, however, is a construction, the reason and 

 demonstration of which I leave the student to deduce for 

 himself : — 



Let CA, CB be the half axes of the ellipse we want to 

 draw. Complete the rectangle cbda, and the quadrant fde. 

 Divide bf and ae at G, h, k, l, etc., and p, Q, R, kc, so that 



