Oct. 24, 1884.] 



♦ KNOWLEDGE ♦ 



339 



— say, SQialler than it. Now let this other or third sphere, 

 and the second, be supposed concentric, the second being the 

 outermost. Euclid shows then (in his spider-web problem, 

 which no one now ever seems to read) that there may be 

 •enclosed within the outer sphere a polyhedron (call it P^) 

 not touohiog or cutting the inner one. Let a similar poly- 

 hedron be enclosed within the first sphere, and call this 

 polyhedron P,. Then if S„ S., S^ be the volumes, and Dj, 

 D.,, and D^ be the diameters of the three spheres, we have, 

 from an already established property of polyhedra, 



But 



P, : p., :: D/' 



S, : S3 :: B,^ 



Si : S3 :: P^ 



S^ : P, :: S3 



D/, by our hypothesis : 



P2 



P2 



But Si is greater than the enclosed polyhedron P, ; there- 

 fore S3 is greater than P^, or a sphere is greater than a 

 polyhedron within which it wholly lies : which is absurd. 

 Hence the sphere Sj bears to the sphere So the same ratio 

 which the cube of the diameter D| bears to the cube of the 

 diameter Do. 



J. Ah! Thanks. I think I follow you. But I hope 

 the demonstrations of arc-lengths, surfaces, volumes, &c., 

 will be simpler. 



M. Nay, if you have followed me at all attentively, the 

 proof is simplicity itself. However, the demonstrations 

 we have now to consider will be direct instead of indirect, 

 and simplified by the introJuction of certain considerations 

 not noted by Euclid. They will also, of course, be properly 

 illustrated. Next time we meet we will enter upon them. 



A. Willingly. 



(To be continued.) 



SEAT 



AND FOOT-BOARD 

 ROWING-BOATS. 



FOR 



THE slidiug-seat of the usual construction slides between 

 two tracks held on a suitable frame. From the back 

 •of the seat projects a rod whose rear end is pivoted to the 

 •apper end of an upright lever pivoted to a bar projecting 



irom the rear of the frame. A spiral spring, surrounding 

 the bar, is held between the rear of the seat and a cross- 

 piece. The foot-board is secured to a cross-piece sliding in 

 longitudinal grooves formed in plates in the boat. The 

 lower end of the kver is connected by rods with the foot- 



board. The pressure of the spring can be varied by a 

 collar on the rod back of the seat. 



When the oarsman makes a stroke, the seat is moved 

 back and the spring is compressed, and the rod is moved in 

 the same direction, when, by means of the lever, the foot- 

 board is moved in the opposite direction. As the oarsman 

 recovers, the spring expands, and pushes the seat back, 

 while the footboard is drawn forward, thereby relieving 

 the oarsman of the necessity of pulling back the seat, and 

 enabling him to expend all his force and power on the 

 stroke. The recovery being very rapid, fast rowing is 

 admissible. 



This invention has, says the Scientific American, been 

 patented by Mr. James J. Turpel, of Halifax, Novo Scotia. 



HOW TO RIDE A TEICYCLE. 



By John Browning, 



Chairman of the London Tricycle Club. 



HOW to ride a tricycle easily, that is what I mean. In 

 the early spring of this glorious summer, I was 

 walking with my wife on the esplanade at Brighton, and 

 looking towards Hove I saw a gentleman riding towards 

 us on a rear-steering tricycle. His legs were going up and 

 down like the beams of an engine, and his shoulders were 

 working as hard as his legs, fie was an immensely power- 

 ful man, standing little, if anything, short of 6 ft. in 

 height, and weighing about 16 st. I said to my wife "He 

 will tear that machine to pieces before long," but I had 

 little idea how soon my prediction would he fulfilled, for 

 within a minute of the words being spoken, he fell oS the 

 machine, and, on going to his assistance, I found that he 

 had torn the iron seat ofi' the spring. Part of the metal- 

 plate had given way, the bolts were wrenched ofi", and one 

 of the strong metal straps had broken. Yet, with all the 

 strength he was exerting, he was not travelling at more 

 than six miles an hour. Had the same amount of strength 

 been skilfully applied, it would have urged him along at 

 the rate of twelve miles an hour, at least I have never 

 before or since seen so bad a rider ; but there are very 

 few riders who apply their strength to the best advantage, 

 and obtain the speed they should get in proportion to the 

 work they are doing. 



Many riders have complained to me that, while they can 

 ride eight or nine miles in one hour, they cannot, without 

 great difiiculty, ride more than about forty miles in a day. 

 Should they be possessed only of the most moderate staying 

 power, if they will act on the following hints they will 

 soon find the distance they can travel greatly increased. 

 They will then know that they have tired so soon because 

 they have not studied how to ride easily : — 



First, I would recommend all those who wish to 

 thoroughly enjoy riding, and to obtain the utmost benefit 

 to their health from the exercise, to restrict their pace to 

 about five or six miles an hour on a Sociable, or six to 

 seven miles an hour on a single machine, and the distance 

 they ride to from thirty to forty miles in a day on a 

 Sociable, and from forty miles to fifty miles in a day on a 

 single machine. 



I have known of two or three cases of people giving up 

 riding, after a very short experience, because they found 

 that they could not ride at the rate of nine or ten miles an 

 hour " for even half a day." They had been misled into 

 believing that this would be easy work for them. Appa- 

 rently it had never occurred to them that, as they would 

 have to climb up many bills, and travel over rough roads, 

 and, at times, drive against the wind, their pace would 



