lon answering to it should be 268 cubical inches; 2 gallons PART III. make a peck, and 32 pecks, or 8 bushels, a quarter.

The Scots wheat-firlot contains 21 pints, or 2199 cubical inches, and the barley-firlot 31 pints, or 3208 cubical inches; 4 firlots make a boll, and 16 bolls a chalder; a peck is the fourth part of a firlot, and a lippie the fourth part of a peck. A Paris pint is nearly equal to 2 English pints, and contains 48 cubical Paris inches.

PROB. I.

To find the content of a prism.

If it be a rectangular parallelopiped, of which the altitude is one inch, it is evident, that there will be as many cubical inches in its content, as there are fquare inches in its base; and if its altitude be any number of inches, the content will be as many times the number of cubical inches; therefore find the area of the bafe, and multiply it by the height, and the product is the folid content: and every prifm is equal to a rectangular paral- a 2. Cor. lelopiped, of the fame base and altitude with it. Therefore, if 32.11.E. the base found by the 4th or 5th Problem of the 2d Part be 96 inches, and the height 11 inches, the content is 1056 folid inches.

a

To find the folid content of a pyramid.

Find the area of the base by the 4th or 5th Problem of the 2d Part, and multiply it by of the height, the product is the content, by 4th Prop. 12th Book E.

COR. If the folid content of a fruftum of a pyramid be required, find the content of the entire pyramid, and of the part that is wanting, and their difference is the content of the fruftum or find the areas of the bafes, and multiply them by correfponding fides of these bases; then, as the difference of these fides to the height of the fruftum, so is the difference a 4. 6. and of the products to the content.

:

PROB. III.

To find the folid content of a cylinder.

Uu2

Find

19.5. E

PART III.

Find the area of the bafe, by Prob. 6. or 7. of the 2d Part, and multiply it by the height, the product is the folid content a: a 6. 12. E. Thus, if the diameter of the bafe be 14, and the height 12 inches, the area of the bafe is 153-9384, which, multiplied by 12, gives 1844.2603 folid inches for the content.

C

and 1. of

tais.

Pl. III.

COR. If the bung-diameter EF of a cak ABCD be not much F. 13. greater than the head-diameter AB; find the areas of the circles at the head and bung, and take half the fum for a mean bafe, which, multiplied by the length, gives the content; only the dimenfions are to be taken within the flaves.

11. E.

To find the folid content of a cone.

Find the area of the bafe, by Prop, 6. 2d Part, and multiply 1.Cor.7. it by of the height, to get the folid content 2. For example, if the diameter of the base be 8 inches, and the height 9 inches, the area of the base is 50.2656, which, multiplied by 9, gives 452.39 folid inches for the content.

Pl. III. COR. 1. If the content of the fruftum of a cone be required, Fig. 14. let it be ABCD; draw AG, parallel to DE; then GC: CD:: b31. 1. E. AC: CE :: AH: EF, the altitude of the complete cone; and 4. 6. E. if the content of the complete cone ECD, and of the part EAB, be found, their difference is ABCD. Which may be done, by fubtracting the product of the diameters CD and AB from the fquare of their fum, the remainder, multiplied by.7854, and then by of the height AH, gives the content of the fruftum.

PI. III. Fig. 15.

COR. 2. Cafks, of which the ftaves are very much bended towards the middle, and ftraight towards the ends, as ABEF, may be taken for two fruftums of a cone.

PROB. V.

To find the folid content of a sphere.

Because the sphere is of its circumfcribed cylinder ", find the area of a great circle of the fphere, and multiply it by of the axis, the product is the content: or, which is the fame thing, multiply the cube of the axis by of .7854, that is, .5236. For example, let the axis be 9 inches, the cube of it is 729, which, multiplied by .5236, gives 381.7044 for the folid

content.

COR. I.

I.

COR. 1. It may be proved, as was done in the 9th Propofition PART III. of the 12th Book, that the folid described by the revolution of the part BGMC of the circle, together with the cone defcribed by the triangle BGP, is equal to the cylinder defcribed by the rectangle BH; and therefore the I content of the zone defcribed by BGMC is got by fubtracting of the fquare of GP or GB the height, from the fquare E of BC, and multiplying the remainder by 3.1416, and then by the height BG. COR. 2. Hence the content of a feg

G

B

K

P

F

I

b3. 12. E.

ment of the sphere may be found, by subtracting the remaining zone from the hemifphere.

To find the folid content of a spheroid.

A fpheroid is a folid described by the revolution of a semiellipfe about one of its axis.

G

if ACR be a femiellipfe, and the preparation be made as in the 9th Propofition A of the 12th Book, it may be proved, that the fpheroid defcribed by the femiellipfe I ACR, about the axis AR, is of the circumfcribing cylinder: For, by the nature of the ellipfe, the square of EL is to the E rectangle AE, ER, as the fquare of DA is to that of AB, that is, as the fquare of EO to that of EB; therefore the fquares of EL, EO are to the rectangle AE, ER, and the square of EB, as the square of DA to that of AB'b; but the rectangle AE, ER, with the fquare of EB, is equal to the fquare of AB; therefore the squares of LE, EO are equal to that of AD or EF ;

a

c

B

R

6. 11. E.

and the cylinders defcribed by the revolution of the rectangles BL, BO, are therefore equal to the cylinder defcribed by EF. e . Cor. From which it may be proved, that the cone described by BAD, together with the hemifpheroid described by AMC, is equal to the cylinder defcribed by AC, as was done in the 9th Prop. of the 12th Book. And the cone is a third part of the cylinder f; f 1. Cor. 7. therefore the hemifpheroid is 3 of it. Therefore, find the area of the circle defcribed by the revolving axis, and multiply it by of the fixed axis.

11. E.

COR. I.

PART III.

COR. I. In the fame manner, it may be proved, that the part of the fpheroid defcribed by the revolution of BGMC, together with the cone defcribed by BGP, is equal to the cylinder defcribed by BGHC. Now, the circle defcribed by GP is the excefs of the circle defcribed by GH or BC, above that defcribed by GM; therefore, from three times the area of the greater base, fubtract the excess of this area above the area of the lefs bafe; or, which is the fame thing, to twice the area of the greater base add the area of the lefs base, and the fum is the f1. Cor.7. base of a cone, equal f in height and content to the zone of the fpheroid.

12. E.

Pl. III.

COR. 2. When the ftaves of a cafk are much bended, it is Fig. 16. fuppofed to be the middle zone of a fpheroid: Let the length of fuch a cafk be 40 inches, and its bung-diameter 32 inches, and its head diameter 26 inches; to 2048, twice the fquare of 32, add 676, the fquare of 26, and multiply the fum 2724 by 40, the length, and then by .2618, of .7854, and the product 28525.728 is the folid content in inches, which, divided by 282, gives 101 gallons I pint of ale; or if it be divided by 231, the quotient is 123 gallons I pint of wine.

To find the folid content of any regular body.

The folidity of the tetrahedron, which is a pyramid, is found by the 2d Problem of this Part. The hexahedron, or cube, is found, Prob. 1. of this. The octahedron, being two pyramids of equal heights, upon the fame square for a bafe, is measured by Prob. 2. The dodecahedron confifts of 12 equal pyramids, upon regular pentagons for their bafes, and may be measured by Prob. 2. of this. The icofahedron confifts of twenty equal pyramids, upon triangular bafes, and may be measured by the fame problem. The bafes and heights of these pyramids may be either measured, or else found by Trigonometry.

To find the folidity of any body, however irregular.

Let the body be immerfed in water, in a veffel of the figure of a prifm, and take notice how much the water is raised by the immerfion of the body: for it is plain, that the space which

the

the water fills, after immerfing the body, exceeds the space it PART III, occupied before, by a space equal to the folid content of the body; and this excefs may be found, by Prob. 1. by multiplying the area of the mouth of the veffel, by the difference between the elevations of the water before and after immerfion. In the fame manner, the folidity of a part of a body may be found, by immerfing that part only.

Tir

PROB. IX.

find how much is contained in a cask, that is Pl. III. in part empty, of which the axis is parallel to Fig. 17. &

the horizon.

Let AGBH be the circle made by cutting the cask at the bung, by a plane perpendicular to the axis, AB the bung-diameter, and GBH the fegment of it, filled with liquor, of which the depth EB may be found by the guaging rod; and the diameter AB, being alfo known, the area of the fegment GBH may be found, by the 2d Cor. to Prob. 6. Part 2. or it may be found, from a table of fegments, fuch as that at the end of guaging.

Find the mean base of the cafk, which, if the cafk be not of the kind mentioned in Prob. 3. of this Part, may always be found, by dividing the content of the cafk by its length; let this be the circle CKDL, and let the fegment of it, fimilar to GBH, be KDL; this fegment may be found, for the circle AGBH is to CKDL, as the fegment GBH to the fegment KDL; and this fegment, multiplied by the length of the cafk, gives the quantity of liquor in it.

OF GUAGING.

To guage a veffel, is to find its content in the measures of any country; and this may be done, by means of the preceding problems, by firft finding the folid content in inches, and then dividing it by the number of inches in the measure required but the work may be shortened, by finding multipliers, which will give the content in that measure, instead of inches. For example, if the veffel be contained by planes, inftead of dividing by 231, multiply by the decimal .004329, and the product will be the content in wine-gallons; and instead of dividing by 282, multiply by .003546, to get the content expreffed in gallons of ale: But thofe multipliers are feldom ufed, becaufe they do not abridge the operation; they are found by annexing

cyphers

18.