432 



♦ KNOWLEDGE ♦ 



[May 22, 1885. 



cylinder, while the adjacent side is equal to the cylinder's 

 height. 



A. How do you prove this 1 





Fig. 1. 



JI. Let A D F (Fig. 1) be a right cylinder on the circular 

 base, F B D, A B, C D, E F being generating lines. Then 

 we might divide the whole curved surface of the cylinder 

 into small areas like ABb a, and show that each of these 

 approaches in area to a rectangle having B b for base and 

 A B for height. Adding all such rectangles we should 

 obtain a rectangle having the circumference of the circle 

 F B D for base and A B for height. But it is simpler to 

 suppose the cylinder rolled on a plane, till its whole sur- 

 face has touched the plane. If the generating line A B 

 (Fig, 1), touches the plane at the beginning of the rolling 

 as in A B (Fig. 2), it is obvious that the surface rolled 

 over will be such a rectangle as A B' where B B' is the 

 straight line traced by the circle F B D, A A' the parallel 

 traced by the circle E A C, and A B, a 6, D, E F in Fig. 2 

 are the lines touched by the correspondingly lettered lines 

 of the cylindrical surface A F D in Fig. 1 ; A' B' being 

 where A B of Fig. 1 again reaches the plane. 



A. It seems tolerably obvious. But how do you }n-ove 

 that the circular edges of the top and bottom of the 

 cylinder will trace parallel lines t 



M. As you begin to roll the cylinder from contact with 

 A B, it is evident that as A T, B t, the tangents at A, B, to 

 the circles E A 0, K B D, are square to A B, the beginning 

 of the rolling starts the traces An, Bb, Fig. 2, square to AB, 

 the next trace a b being parallel to AB. Thence to the next 

 trace and the next ; so that startinr/ along the perpeu- 



B i 



diculars A a, B 6, to A B Fig. 2, and all the time movintr 

 squarely to A B, the successive points of the circles E A C, 

 F B D in Fig. 1 cannot move otherwise than along the 

 parallels A A', B B', Fig. 2. 



A. And now for the volume of a cylinder. 



M. In this case we are not limited to right circular 

 cylinders, but can deal with oblique cylinders, and cylinders 

 of any section, so only that we know the height and can 

 determine the area of the base. 



A. How do we proceed 1 



M. Let A D G X-, Fig. 3, be a cylinder, on the base 

 A C F H, A ffl, B i, C c, ifec, being generating lines ; and 

 therefore, by the general definition of a cylinder, all these 

 lilies equal and parallel to each other. In the curvilinear 

 area A B C D E F . . . , describe a polygon A B C D E F . . . 

 having a great number of sides, abcdef... being the 

 corresponding polygon in the curvilinear area abc d ef. . . 

 Then we have a pi ism ABC... abc. . . , enclosed within 



the cylinder A D G ^^ And manifestly we can make the 

 volume of the enclosed prism as nearly equal as we please 

 to the volume of the enclosing cylinder, by taking the 

 points A B .... L, sufficiently near to each other. For, 



the thin slices A B 6 a, B C c 6, &c., left outside the prism, 

 will together be less than a shell enclosed between the 

 cylindrical surface A D G A:, and another liaving the planes 

 aj\ A F, for its flat surfaces, and touching the planes 

 A B i a, B C c A, itc, and this shell may be made as thin as 

 we please, by making the sides A B, B C, C D, &c., of our 

 polygon as short as we please. 



A . Hence the cylinder A D G A; is ultimately equal to 

 the prism ABC . . . . ab cA 



M. Yes ; and this prism, since the rectilinear area 

 A B C . . . . L is ultimately equal to the curvilinear area 

 A D F H is equal, by what was shown in dealing with 

 prisms, to the volume of a rectangular parallelepiped having 

 its base equal in area to A D F H, and its height equal to 

 y'M, the height of the cylinder. 



A. Then in the case of a right circular cylinder, having 

 // for height, and r as the radius of its circular base, the 

 volume of the cylinder is represented by Trr~h 1 



M. Precisely ; while the volume of a right elliptical 

 cylinder, whose height is h, and the semi-axes of its ellip- 

 tical base a and i, is represented by Trabh. 



The curved surface in the former case is represented 

 by '2irrlt, and the tntal surface of the cylinder by 

 2;rr/(-|-2Tr-, or by l-r^r (r-\-/i) ; so that the total surface of 

 a right circular cylinder is equal to the rectangle under 

 the circumference of the base and the radius 2}lus the 

 height. 



(To be continued.) 



PiioTOGEAPHY. — The article in this month's Cornhill, on "The 

 lliee and Progress of Photography," is by our contributor, Mr. W. 

 Jerome Harrison, F.G.S. 



At a meeting of the Anthropological Institute on May 12 Mr. 

 J. H. Kerry-Nicholla read a paper on "The Origin, Physical Cha- 

 racteristics, and Manners and Customs of the Maori Race." What- 

 ever may have been their original course of migration, there can 

 be no donlit that the Maoris owe their origin to the Malay stock. 

 They are tall, well-built, and erect, with broad chests and massive 

 limbs, which usually display great muscular development. The 

 Maoris have longer bodies and arms, with shorter legs, than 

 Europeans of similar stature, the feet are short and broad, and the 

 hands small and tapering; the hair is coarse, black, and 

 straight, and the skin of a brown coffee-colour. Half-castes are not 

 uncommon, and are remarkable not only for their fine, well-formed 

 persons, but also for their intellectual jjowers. The race is rapidly 

 dying out, owing chiefly to diseases contracted by contact with 

 civilisation, and not a little to the immoderate use of tobacco by 

 young and old of both sexes. The native religion is a kind of 

 polytheism — a worship of elementary spirits and deified ancestors. 

 The priests held an exalted tribal rank, and were believed to jjossess 

 miraculous powers ; the Maoris acknowledge the existence of the 

 soul after death, Imt do not believe in corporeal resurrection, nor 

 in the transmigration of souls, and they seem to have some rather 

 indefinite ideas of a heaven and a hell. 



