MENSURATION. 



Mensura- 

 lion of 

 Solids. 



Fig. 83. 



*52 



Ex. Suppose the diameters of the ends to be 8 and 

 C, and the length 10 ; required the content. 

 First 8x8'+3x6'+4x8x6=812, 



Then 8l2x 10 X -05236=425. 1632 the content. 

 PROBLEM XIV. 



To find the solid content of a spheroid, or solid ge- 

 nerated by the rotation of an ellipse about either axis. 



RULE. Multiply continually together the fixed axis, 

 the square of the revolving axis, and the number .5236 

 (or of 3.1416,) and the last product will be the solid 

 content. 



This rule has been investigated in FLUXIONS, Art. 

 161. 



Ex. 1. The greater axis AB of an oblong spheroid is 

 50, and the lesser axis CD 30, what is the solid con- 

 tent ? 



Here the greater is the fixed, and the lesser the re- 

 volving axis. 



Therefore the solid content is 



50 x 30 s x- 5236=23562 the content. 



Ex. 2. What is the content of an oblate spheroid, the 

 axes being as in last example ? 



30 X 50* X .5236=39270 the content. 

 PROBLEM XV. 



To find the solid content of the frustum of a sphe- 

 roid, its ends being perpendicular to the fixed axis, and 

 one of them passing through the centre. 



Fig. J4. ROLE. To the area of the less end, add twice that of 



the greater ; multiply the sum by the altitude of the 

 frustum, and ^ of the product will be the content. 



NOTE. This rule applies also to the frustum of a 

 sphere. 



Investigation. Let ABE be a quadrant of an ellipse, 

 C its centre, CAFE a rectangle circumscribed about it, 

 and CF the diagonal. Draw ^ny straight line DG pa- 

 rallel to CE, meeting AC, CF, ABE, and EFin D, H, 

 B, G. By CONIC SECTIONS, Part. I. Sect. 2. Prop. 16. 



DB : CE' : : CA 2 CD' : CA', 

 and by sim. trian. DH 2 : CE' : : DC 2 : CA 2 . 



Therefore, (GEOM. Sect. 3. Prop. 10.) 



DB*+ DH 1 : CE' : : CA 2 : CA*. 



Hence DB'+DH'=CE 2 =DG 2 . 



Suppose now the figure to revolve about AC as an 

 axis, so that the elliptic quadrant may generate the half 

 of a spheroid, the rectangle AE a cylinder, and the 

 triangle ACF a cone; it is evident, as in the case of 

 the sphere, ( Prob. 9.) that every section of the first of 

 these solids is equal to the difference of the sections of 

 the other two ; and consequently, that the frustum of 

 the spheroid between CE and DG is equal to the dif- 

 ference between the cylinder having DG or CE for the 

 radius of its base, and the cone having DH for the ra- 

 diiis of its base, and CE for its altitude. 



Put n=3.14l6, then (Prob. 4.) the content of the 

 cylinder is 47JxDG 2 xCD, and (Prob. 5.) the con- 

 tent of the cone is f n x DH 2 x DC : Therefore, the 

 difference, or the content of the spheroid, is 

 4 n x CD (DG' | DH 1 ). 



But it was shewn, that DH*=DG 2 DB', therefore 

 the content of the solid is equal to 



nxCD(2DG*+DB*), 

 and hence is derived the rule. 



When the altitude becomes the scmiaxis, the frustum 

 becomes half the spheroid, which is f of the circum- 

 scribing cylinder, agreeing with the rule of Prob. 12. 



Ex. Suppose the greater end of the frustum to be 

 15, the less 9, and the length 10 inches, required the 

 content ? 



The area of the gr. end= 1 5* x .7854. 

 The area of the less=9 8 x .7854. 

 The content = .7854(9'+ 2 x 15) X 1 J S=13 9- 158 

 cubic inches. 



GAUGING. 



The geometrical rules by which the content of any 

 cask may be computed, form a particular branch of 

 Mensuration called GAUGING. 



Casks are usually considered as having one or other 

 of these four forms : 



1. The middle frustum of a spheroid, (computed by 

 Prob. 15.) 



2. The middle frustum of a parabolic spindle, (Prob. 

 13.) 



3. The two equal frustums of a paraboloid, (Prob. 11.) 



4. The two equal frustums of a cone, (Prob. 6.) 



The content of any cask whatever may also be near- 

 ly found, in wine or ale gallons, by the following ge- 

 neral rule: 



RULE. Add into one sum 39 times the square of the 

 bung diameter, 25 times the square of the head diame- 

 ter, and 26 times the product of the diameters. Mul- 

 tiply the sum by the length, and the product by .00034 ; 

 then the last product, divided by Q, will give the wine 

 gallons, and divided by 11, will give the ale gallons. 



In investigating this rule, it is assumed as a hypo- pig. 

 thesis, that one-third of a cask at each end is nearly a 

 frustum of a cone, and that the middle part may be 

 taken as the middle frustum of a parabolic spindle. 

 This being supposed, let AB and CD be the two right 

 lined parts, and BC the parabolic part. Produce AB 

 and DC to meet in E, and draw lines as in the figure. 

 Let L denote the length of the cask, B the bung diame- 

 ter, and H the head diameter: Then, since AB and DC 

 have the same directions as the parabolic curve BFC ; 

 at B and C, they will be tangents to the curve : There- 

 fore FI =1 El. But BI = | AK, and hence by similar 

 triangles EI = fEK; consequently FI = El = 

 EK = f FK = -^ (B H) ; so that the common dia- 

 meter BL=FG 2 FI=B f (B H )=> (4B + H) 

 which call C. Now, by the rules for parabolic spindles 

 and conic frustums, we obtain (putting n for -7854:.) 

 * L_ 328 B'+ 44 BH + 3H 8 

 < ~~ ~~~ 





Mensura- 

 tion of 

 Solids. 



for the parabolic or middle part ; and 



_ 



'' : 



for the two ends, and the sum of these two, after pro- 

 per reduction, is 



(39 B 2 + 26 BH + 25 H*) x ^ nearly, 

 y(J 



for the length in inches ; and the factor or - 



being divided by 231 (the inches in a wine gallon) gives 



00034 



the multiplier for wine-gallons ; and since 231 

 y 



is to 252 as 9 to 1 1 nearly, - will be the multi- 



plier for ale-gallons, as in the rule. 



Ex. Suppose the bung and head diameters of a cask 

 to be 32 and 24 inches, and the length 40 inches, re- 

 quired the content in ale, also in wine gallons. 



Here (39- X 32' + 26 X 32 x 24 + 25 x 24') x 

 40 X '00034 = 1010-5, which being divided by 9 and 

 by 11, we obtain 112-3 wine gallons, or 91'9 ale gal- 

 lons for the content. (|.) 



