MENSURATION OF SOLIDS.] 



APPLIED MECHANICS. 



911 



hemisphere, it may be taken, as a paraboloid, and its 

 capacity may be found thus 



Ride. Square the diameter of the base, multiply by 

 the height and by -3927 (half of -7854). Or, multiply 

 the area of the base by half the height. 



Example 8. Required the solid content of a parabo- 

 loid, base 2 feet 6 inches diameter, height 1 foot 8 

 inches. 

 Area of a circle 2 feet 6 inches, or 



30 inches diameter 707 sq. ins., nearly. 



Half of 1 foot 8 inches, or 20 inches 10 



Solid content 7070 cubic inches. 



In general, the capacity of any solid of revolution may 

 be found by multiplying the area of its generating figure 

 by the circumference described by the centre of gravity 

 of that figure. In a ring, for instance, generated by the 

 revolution of a circle round some centre without it, if 

 we know the diameter of its generating circle or section, 

 and the distance of its centre from.that round which it 

 revolves, we can compute its solid contents, as we may 

 show by an example. 



Example 9. A ring of circular section measures 10 

 ins. external diameter, and 6 ins. internal : required 

 its solid contents. Subtracting 6 from 10, there 

 are left 4 ins., which must bo double the diameter 

 of the ring's section, giving 2 ins. as the actual 

 diameter of the section, which has therefore an area 

 2 X 2 X -7854 = 3-1416sq. ins. Again, adding 6 to 10, and 

 taking half, we get 8 ins. as the distance between A and B 

 (Fig. 272), the centres of the circular sections, and the cir- 



Fig. 27:. 



10 





cnmfercnceof acircleS ins. diam. is 8 X 3-1416=251328 

 ins. The solid content of the ring is therefore 

 3 -1416 sq. ins. X 25 -1328 ins. = 78 9572 cub. ins. In this 

 particular case, it happens that the centre of gravity of 

 the section coincides with the centre of figure. When this 

 is not the case, the position of the centre of gravity may 

 readily be found by the following mechanical process : 

 Draw upon a card or piece of thin wood, or plate of 

 metal such as zinc, the figure, either of its full size or to 

 any convenient scale ; and having cut it out to the shape, 

 make two small holes in it, such as A and C (Fig. 273). 

 Suspend it by a thread from one of the holes, and along 

 the face of it let a plumb-line hang and mark B, the 

 point where the plumb-line crosses its edge, tracing a 

 line A B coinciding with the plumb-line. Suspend it 

 now by the other hole C, with the plumb-line crossing 

 the line formerly traced at D. The point where the lines 

 cross is the centre of gravity, or that round which all 

 parts are balanced. The position of the centre of gravity 

 thus ascertained may be laid down upon the drawing of 



the work to be estimated, and the solid content may 

 be found by the rule we have given. 



For the measurement of a solid of irregular form, the 

 most convenient method is to suppose it divided by 



Fig. 273; 



numerous parallel planes into sections of equal thick- 

 ness to measure the area of each of those sections, and 

 find the mean or average of them by summing them all 

 and dividing by their number. This average area mul- 

 tiplied by the total thickness, measured perpendicularly 

 across the planes of section, will give the solid content. 

 In many cases of the measurement of cylindrical bodies, 

 such as timber or round bars of metal, it is more con- 

 venient to use the girth or circumference as a known 

 dimension for estimating the cubic contents. A near 

 approximation to the capacity may be made thus : 



Rule. Multiply the square of the girth by tho length, 

 and by 8, and point off two decimal places. 



Example 10. Required the cubic contents of a round 

 log, having an average girth of 6 ft. 3 ins. , and 12 ft. long. 



ft. 3 ins., decimally expressed, is 6 '25, and 

 C- 25 X 6 -25 X 12 X 8 = 3750. Pointing off two decimal 

 places, the solid content is 37 '5 cub. ft. The ordinary 

 rule for timber gives a considerably smaller result, and 

 is intended, not to furnish the accurate contents, but the 

 contents estimated according to trade custom. It is, 



Jln/e. Multiply the square of Jth tho girth by the 

 length. 



According to this rule, the contents of the log. in 

 Example 10 would be (Jth of 6 -25 being 1-5023) 

 1-6025 X 1-5625 x 12=29-3 cubic feet. 



DUODECIMALS. 



In the mensuration of surfaces and solids, a system of 

 computation is frequently adopted, called the duodecimal 

 system, because the notation is reckoned by twelves 

 (Latin, duodecim), instead of the ordinary decimal scale 

 of tens (Latin, decem). 



According to the duodecimal system, a square foot is 

 supposed to be divided into 12 equal parts, each con- 

 taining 12 square inches : and each of those parts again 

 into twelfths, each 1 square inch ; and these again into 

 twelfths. So, also, a cubic foot is divided into 12 equal 

 parts, each 144 cubic inches ; each of those into 12 parts, 

 each 12 cubic inches ; and each of those into 12 parts, 

 each 1 cubic inch. To multiply a certain number of 

 feet aud inches by some other number of feet and inches, 

 the one quantity is written below the other, as in 

 ordinary multiplication. 



1. The inches are multiplied by inches, the product 

 divided by twelve, the remainder written one place 

 back from the inches, and the quotient carried to the 

 next operation. 



2. The feet are multiplied by inches, and the number 

 carried from the former operation added, the result 

 divided by twelve, the remainder written in the place of 

 inches, and the quotient written in the place of feet. 



3. The inches are multiplied by feet, the result divided 

 by twelve, tho remainder written in the place of inches, 

 and the quotient carried. 



4. The feet are multiplied by feet and the carried 

 number added, and the whole written in the place of 

 feet. 



