♦ KNOWLEDGE ♦ 



[Aug. 7, 1885. 



CHATS ON GEOMETRICAL MEASUREMENT. 

 By Richard A. Proctor. 



THE SPHERE. 



(ConHnuedfrompagc75.) 



A. Can wo determine the rolume of a portion of the sphere cut 

 off by a plane as L M N, Fig. 3 ? 



M. Quite easily. Take first the volume, L A N, and let a plane 

 LON cat the enclosing cylinder in In. Then the volume LAN 

 is equal to the solid sector CLAN, diminished by the cone C L N ; 

 and we know by the method of our proof for the whole sphere, that 

 the volume of the solid sector C L A N is e<inal to ird the volume 

 of a rectangular parallelepiped having a base equal to the curved 

 surface of cylinder tlnT and height equal to the radius of the 

 sphere; while the cone C L N is also of known volume, when the 

 position of the point is known. But to get suitable e.xpressions 

 for the volumes of spherical segments we may conveniently proceed 



Jence vol. of sector CLAN = 3 ■ 27rr- vers 



vol. of cone C L X^^^^^" • ,rr- 



7. vol. of seguient L A N = -3- (2 vers 



= '^(2-3c. 



Similarly, vol. of segment LEN=-s- (2 + 3 e( 



Also, vol. of section B L N D = '^ (^ cos c 



i^— 



■ Y 





Fig. 3. 



This last result is all that need be remembered, and owinj 

 prevalence of 3's in it, it is very easily held in the memory. 



A. What is the volume of a slice of the sphere betwe 

 parallel planes, neither of which passes through the centre r 



M. We get the same volume whether the planes are parallel or 

 not so that they do not intersect within the sphere. Thus let one 

 be such that the angle corresponding to LCO =o, while the other 

 is farther from the centre, and has the corresponding angle =^-i. 

 Then, the volume of the space between these two planes, it they 

 do not meet within the sphere, is obviously 



,08a)-(cos=/3-cos'..) "!■ 



TTHf . 



0S-/3- 



(co 



>s«) 



iut we are getting a little outside of geometrical methods. 

 A. What proportion does a sphere bear to the enclosing cube ? 

 M. The volume of the enclosing cube is (2r)» or 8r'. Therefore 

 vol. of sphere : vol. of enclosing cube::-p- : 8::jr : 6 



or roughly, ::^ : 6::il : ?.i. 



A. I think the usual idea is that a sphere occupies a larger propor- 

 tion than this of the space within an enclosing cube. It is little more 

 than half ! So that if you have a box of spherical bodies, say 

 cannon balls, set so (that each row falls exactly alongside or 

 above the neighbouring rows, the lines joining the centres forming 

 right angles with each other, the spaces between the spheres 

 amount in all to only 10-2lst parts of the whole space ! 



M. That is so. But the balls can be more closely packed, as 

 when set in triangular or quadrangular pyramids. 



A. Which 



arrangement 



s the b, 



?r of those t 



I n 



nfor 



closeness of fitting ? 



M. They are in that respect precisely the same. I hope, here- 

 after, to extend our inquiries in that direction. 



A. Can we do anything with spheroids and ellipsoids ? 



M. Our results are very easily extended to them. Thus,— 



An oblate spheroid may be regarded as produced by shortening 

 every perpendicular to a certain great circular section of a sphere 

 in a certain proportion, while in the prolate sphere every perpen- 

 dicular is lengthened in a certain proportion. This circular section 

 is the equatorial section of the resulting spheroid. By taking any 

 plain perpendicular to this section and lengthening or shortening 

 in a given proportion all the ordinates perpendicular to it, our 

 spheroid becomes an ellipsoid. Ilence, manifestly, by regarding 

 these perpendicular ordinates as elements of the volume, we get the 

 following results : — 



Volume of an oblate spheroid having a for the radius of its prin- 

 cipal circular section, and b for its shortest semidiameter 



4--^b. (1) 



= -^nal\ (2) 



of an ellipsoid having semiaxes a, I, and c 

 = 'ial>c. (3) 



(To be continued.) 



The Great Glacier of Alaska.— According to the Son Francisco 

 Courier, the great glacier of Alaska is moving at the rate of a 

 quarter of a mile per annum. The front presents a wall of ice 

 500 ft. in thickness ; its breadth varies from three to ten miles, 

 and its length is about 150 miles. Almost every quarter of an hour 

 hundreds of tons of ice in large blocks fall into the sea, which they 

 agitate in the most violent manner. The waves are said to be such 

 that they toss about the largest vessels which approach the glacier as 

 if they were small boats, the ice is extremely pure and dazzling to 

 the eye ; it has tints of the lightest blue as well as of the deepest 

 ■ " ".. . • ery rough and broken, forming small hills, a ' 



n chai 



sof m 



iS of ic 



1 the sf 



This 

 rage of a thousand feet thick, adva 



MP.— The electric lamp used for ex- 

 amining General Grant's throat, manufactured by agents of the 

 Edison Light Company, is mounted on a hard rubber holder, about 

 7 in. long, having a reflector at the lamp end, by which the light 

 can be thrown to any desired angle. The holder is connected by 

 two silk-covered wires to a small storage battery carried in the 

 pocket of the physician. The light is turned on by simply pressing 

 a small button mi tht- rubber holder, and the quantity is governed 

 by an. .111. , l,,i;. -,, , ,.in, ni.-iit to the operator. The lamp in inserted 

 in tho n. . ■ ill., palate, with the reflector above the 



lamp, u' i_lit down the throat. The lamp has no 



unpKa^ai.; . r, r . Liv.salight equal to half a sperm candle. 

 The exucuic o,i.,|.lieiU uf the whole appliance makes it very 

 valuable to the physician and dentist. 



Colbubn's wood and paper brake shoe, a shoe consisting of alter- 

 nate layers of compressed paper and wood, of about J in. each, has, 

 the Railroad Gazette says, been recently tested on the New York 

 Elevated Railroad, three cars on the Third Avenue line having been 

 equipped. These cars are stated to have beeu in daily service for 

 thirteen weeks, making a run of 9,271 miles, against eight weeks 

 and 6,000 miles of the standard metal shoes of the road. This, it 

 is claimed, would equal a run of 200,000 miles on an ordinary road, 

 since the number of stops is about twenty times as many. Quicker 

 stops can be made than with metal shoes, it is said, and naturally 

 with much less wear to the wheel tread. The patentee is L. S. 

 Colburn, of Oberlin, 0. There is especial necessity for some other 

 than a metallic shoe on the elevated roads, if it can be had, to 

 avoid the annoyance and danger to eyesight of flying particles of 

 metal. 



