JnLT 24, 1885.] 



♦ KNOWLEDGE ♦ 



CHATS ON GEOMETRICAL MEASUREMENT. 



Bt Richard A. Peoctok. 



THE SPHERE. 



(Continued from page 25.) 



II. Volume. 



A. And now we are to attack great Archimedean problem, — to 

 compare the volnme of the sphere and cylinder ! 



il. Wo have to all intents and purposes solved it already, in 

 determining the area of the sphere. 



A. That is strange ! 



M. It is as certain as that the quadrature of the circle is 

 accomplished so soon as we have determined the circle's circum- 

 ference. 



A. I think I see your drift. "We divide the circle into an in- 

 definitely largo number of sectors which in the limit may be re- 

 garded as triangles ; and I suppose you divide the sphere into an 

 indefinitely large number of spherical sectors (I know not what 

 wonld be the name for them) which in the limit may be regarded 

 as pyramids. 



Jlf. That is the l^ 

 way. Suppose 



small spherical 

 triangle, on the 

 flurfarie of the 

 sphere ABED, 

 Fig. 3, whose 

 centre is at C, 

 andioinCa,Cb, 

 C c. Then it is 

 evident that i 

 the sides of the 

 triangle ahc are 

 Bmall enough, 

 difference 



betwf 



lolid 



the 



gular base i 

 compared i 



ith 



Fig. 3. 

 either the solid 



sector or of the pyramid, and may therefore be neglected. By taking 

 other points d,e,f,g,h, &c. and making the triangles hdc, d g c, and 

 BO on, we can cover the whole surface of the sphere with small 

 spherical triangles, the solid sectors corresponding to which make 

 up the total volume of the pyramid. Now manifestly the sum 

 of the surfaces of the plane triangles ahc, hdc, &c., is equal 

 ultimately to the area of the sphere. Hence, Volnme of sphere 



=• sum of volumes of the pyramids C ah c, G hd c, &c. 



— pyramid having sum of areas ah c,h dc, &c., for base, and 

 radius of sphere for height. 



= pyramid haying surface of sphere for base, and radius of 

 sphere for height. 



= J rectangular parallelepiped having a base equal to the 

 area of the sphere, and a height equal to the sphere's radius. 



= * rectangular parallelopiped having a base equal to a great 

 circle of the sphere, and a height equal to radius of that circle. 



= 5 cylinder as ( T Y ;/ having base equal to great circle and 

 height equal to the diameter of sphere. 



= 3 of a cylinder enclosing the sphere. 

 A. That is certainly not a very difficult solution. It is singular, 

 by the bye, that the volume of the sphere bears the same relation 

 to the volume of the enclosing cylinder, that the surface of the 

 sphere bears to the complete surface of that cylinder. 



M. Yes, putting the radius of the sphere equal to r we have 



(1) Surface of sphere = 4)rr' 



(2) Surface of cylinder = CnT' 



!3) Volume of sphere = !»)■' 

 4) Volume of cylinder=2)rr' 

 Note also that if we have a cone with y Y one of the circular 

 ends of the enclosing cylinder t T Y y as base, and A as vertex, 



vol. of enclosed cono : vol. of enclosed sphere : vol. of 

 enclosing cylinder 



:: 1 : 2 : 3 



(ro he o 



Hnucd.) 



(Bwv JnbentorS' Column. 



Ve give here, ti'cel ly veek, a terse description o/ suc^i of the many 

 i7iventions as u-e thinh may he of use to our readers. Where it is 

 possihle, the numher of the patent is quoted, to enable those who 

 desire fuller information to procure the specification from the 

 Patent Oflce in Cursitor-street, Chancery-lane. We shall, gene- 

 rally speaking, confine ourselves to the more recent inventimt; but 

 it often happens that an article comes under our notice which, 

 although not quite novel, is worthy of mention for its utility an^ 

 ingenuity. In such a case we should not hesitate to refer our 

 readers to it. And while we thus increase the interest of our paget, 

 we at the same time assist the inventors by giving greater puhlicity 

 to their inventions (Knowledge heing a popular magazine) than 

 it accorded 6y the most excellent trade journals.. 



LAWN-TENNIS POLES. 



[Patent No. 10,776. 1884.]— This patent, taken out by Mr. B. 

 Pearce Brown, Thorganby Hall, Y'ork, claims simplicity, neatness, 

 and facility for fixing. Briefly described, it is a combination of 

 lawn-tennis poles jointed to plugs driven into the ground, and 

 capable of motion in the plane of the net, of toothed racks lying 

 on the ground and stays jointed to the poles, the lower chisel- 



pointed ends of which slide over the inclined teeth of the racks, 

 and abut against the vertical sides of the said teeth of the racks, 

 thus preventing the return motion of the poles. Thus, when it is 

 wished to tighten the net, it is only necessary to separate the poles 

 to the distance giving the required tightness to the net, when the 

 poles are automatically fixed in the positions to which they have 

 been brought, and the' net is maintained at the required degree of 

 tension. The makers are Messrs. Chas. JIalings & Co., 18, 

 Cockspur-street, W.C. 



STKREOSCOPIC PICTURES. 

 Me. W. E. Crowtuer, late of the Manchester Technical School, 

 has devised a scientific .arrangement for the production of stereo- 

 scopic effects in pictures thrown upon the screen by the lantern. 

 The method is as follows : two lanterns are used, each of which 

 projects one of the two corresponding stereoscopic transparencies 

 upon the screen, so that both fall together upon the same field. 

 The light from the lanterns, however, is not white, but of any two 

 complementary colours, as red and bluish green. The spectators 

 wear non-magnifying spectacles fitted with glasses of the same 

 tints as those used in the lantern, and the result is that each indi- 

 vidual sees one picture only with each eye, viz., the picture thrown 

 from the green lantern is alone seen by the eye wearins the green 

 glass, whilst the eye shielded by red glass sees ilie picture thrown 

 from the red lantern. In tliis way each eye is affected only by its 

 appropriate sutims, .;>:,■ .;,n. niilst the mind receiving simnltane- 

 ously tlie tw.) \ ; nry colours combines them into 



a light and sIklu. i :: which possesses some peculiar 



Th<' o.iiMl.ipc.] i . '1 l>y the raked eye, is for tho 



most |iir: ^^ i :■ ■ ' . f il.i' n .1 .-ind trrecu lights upon 



the srn^' ;: '. . ■ ■ !h,.' pictures or over- 



Iap,ii„.. ,.. : .^ i; slowly turnmg the 



lantern,' so ;v to ^::fi i ■ ■■ >.r-.:: r: i:;r.', sHl-IuIv to the wrong 

 side of each other, so tliat tho oi>tic axes of the spectator may 

 intersect in front of the screen, the picture appears to advance tmd 

 become suspended in the air before the screen. 



