May 2, 1887.] 



♦ KNOWLEDGE ♦ 



153 



points B, E, F, meet at a point in h vertically above the centre 

 of the sphere, and have the apparent forms oaKh, Kh'hb, 

 hcob, while those on the farther side, if seen through the 

 sphere, would have the forms avJ)h, bb'hc, and abco. Now 

 obviously all these are equal rhombuses ; so that as two of 

 tliose on the upper side are rhombuses of a size and shape 

 already determined, so also is the third k6l6'. But this can 

 easily be independently demonstrated. For the diameter 

 KL is obviously equal to Ac or 2r, while bb' , which appears 



to be equal to -^^, is really seen at an angle equal to that 



between the perpendicular from an angle of a regular tetra- 

 hedron on an opposite face and an edge through that angle 

 (consider fig. 3 to see that this must be so). Therefore bb' 



is really equal to "—, increased in the proportion which an 



edge of a regular tetrahedron bears to such a perpendicular, 



or as 2 to * / ^. Hence the shorter diagonal of the 



rhombus bKb'h is really equal to 



V3 



Wi = 



s/2, 



>; 



as before. 



In fact, the semiregular solid formed by the expansion of 

 the elastic surfaces of the equal globes is no other than the 

 solid considered in the solution of Problem XXIT., in our 

 last number. 



The rhombuses shown in perspective in fig. ^ are identical 

 in shape with those seen at the ends of the cells of the 

 honeycomb, when these cells are regular. 



III. THE ALPENSTOCK PUZZLE. 



Puzzle XXVII. I must deal briefly with this, having 

 already given more space than I intended to the solutions of 

 these problems. (I would remark, however, that the study 

 of all these problems will be found to afford excellent exer- 

 cise in tridimensional geometry.) 



The "ingenious beast " having explained his ideas en 7-oute 

 to his two masculine fellow-travellers, they proceeded at once 

 on their arrival at their resting-place to the following con- 

 struction : — 



Laying down the six sticks ab, de, A'c, in the position 

 shown in fig. 6, Ljr, no, and tq lying upon the others, and 



^W 



Fig. C. 



all crossing at their middle points, they tied them with stout 

 cord at db and OE, and at c, f, ii. Then two of them brought 

 K over to A, tying the sticks ab and gk together by the ends 

 AK, while the third brought the ends L, x, m together and 

 stoutly tied them. Thus they had the sticks in the form 

 shown in fig. 7. Then they used the ropes which as careful 

 mountaineers they used in travelling to " seat " the tri- 

 angular spaces ACH, bcf, efh. As an end of one of these 

 ropes had already been used in fastening the ends N, p, l 

 together, it was easy to carry the rest of this rope down 

 under c, f, and h outside the slant alpenstocks, giving the 

 lines CH, he, and CF, which admit of being tautened to any 

 necessary extent, by carrying the rope in the directions 

 shown by the • dotted lines within- the •triangle Chf. This 



makes the ropes within the seat-triangles taut and strong, 

 while shawls or wraps thrown over these seats make them 

 sufliciently comfortable for an rd fresco meal. Anyone 

 sitting alone at ak, db, or eg, can tip the whole concern 



Fifi, 7. 



over ; but tlie seats are not meant to be used in that way. 

 But two people sitting even so absurdly as this— that is to 

 say, one at ak, the other at eg — would not disturb the 

 equilibrium of the seat. When two sit on each side of c, 

 two on each side of F, and two on each side of h, all six are 

 safe, assuming the alpenstocks sufficiently (and the six 

 people not too) stout. 



OUR PUZZLES 



OR 



^L\THEJ^ATIC.A.L RECREATIONS FOR MAT. 



ROBLEM XXVIII. A nialhematicalhj disposed 

 person is troubled by the circumstance, that he 

 sees himself in an ordinary mirror, not as his 

 friemls see him, but with his ri{/ht side where 

 they see his left, ami vice-versa. Abo, it seems 

 to him a nuisance that he has always to go in 

 front of his mirror to see himself in it at all. 

 He therefore divides his mirror into tivo parts, and so fits 

 these that icherever he may be in his room (his eyes not being, 

 however, above or below certain convenient limits of level), he 

 can see himself in the mirror as newly adjusted, and always 

 as his friends see him, not as one usually sees oneself in a 

 mirror. Hoxo did he manage this ? 



Problem XXIX. Supposing the eight corners of a room 

 lined with mirror glass, agreeing exactly ivith the plane sur- 

 faces of the walls, ceiling, and floor {the room having all its 

 faces perfectly rectangular), what would one see on looking 

 into any corner ? 



Problem XXX. Supposing the tirelve edges of the sam^ 

 room lined with mirror glass, how many images of himself 

 could any one loithin the room see, and in what positions ami 

 aspects 1 



Curious Observatioxs on the Growth of the' Heart. — 

 Dr. Benecke, of Marburg, has made known Iiis curious observations 

 on the growth of the human heart, the fact appearing that the 

 increase is greatest and most rapid during the first and second 

 years of life, its bulk at the end of the second year being exactly 

 double what it originally was ; between the second and seventh 

 years it is again almost doubled. A slower rate of growth now sets 

 in, until about the fifteenth year, the augmentation of volume 

 during the intervening seven or eight years being only about two- 

 thirds. In the period of maturity which now approaches the 

 growth of the heart again makes progress, the increase keeping 

 pace with the advance toward maturity of the other portions of the 

 system. After the fifteenth year, up to the fiftieth, the annual 

 growth is about 061 of a cubic inch, the increase ceasing with the 

 fiftieth year, a slight diminution then ensuing. Again, in child- 

 hood the male and female heart are alike ; after maturity, the male 

 heart develops more than the female, and the difference between 

 the two that is thus established— one and a half to two cubic 

 inches— is Faid to be maintained throughout the remainder of life. 



