EXAMPLE I.

Required the folidity of a triangular pyramid whofe height is 30, and each fide of the bafe 3.

First, 3×433013 = 3·897117 = area of base. Then 3.897117X10 = 38.97117 is the folidity.

Corol. 1. Since A: Jab: bx, it will be √

bb

fruftum, that content will become } x × (A+a+√/Aa).

Corol. 2. Or, if s be a fide, or any other line, in the greater end, and s a fimilar fide or line in the lefs, fince s-s: s::x : h,

hence, which being fubftituted instead of it, gives

AXX

S

SS

for the value of the faid fruftum.

Corol. 3. If x= If xb, then a o, and the general expreffion becomes Ab the whole pyramid; which is our rule.

Corol. 4. If the base be a regular figure; let s be one of its fides, sa fide of the lefs end of the fruftum, and n the area of a fimilar figure whofe fide is 1, to be found in the table in page 114: then A = nss, and a = nss; hence the folidity of the fruftum will be

83—53
S-S

xnx, and that of the pyramid

Corol. 5. Iffs+Is the half fum of the fides, and d=s-d the half dif. then the fruftum will be (3+ d) xnx..

=

×

Corol

EXAMPLE II.

Required the folidity of the fquare pyramid, each fide of whofe bafe is 30, and the flant height 25.

the base.

First, 30 × 30 = 900
But BCVAC2 — A B2 = √252-152

=√ 5a × 52 — 32 × 5a = √ 4a × 5a

=

4× 520, the perpendicular height. Then 300 X 20 6000 is the fo

lidity.

Corol. 6. If the bafe be a fquare; then # 1, and the last expreffions become ssh for the whole pyramid, and

123

Corol. 7. If the bafe be a circle; then #7854, 2618ssh =

the cone, and (ss+ss + ss) × •2618x, or

93-53

S-5

X2618.x,

or (32+d2) x 2618x the fruftum; where s is the diameter of the bafe, and s that of the top.

Corol. 8. If a = A; then (A + a + √^ a) × 3x becomes 3A X = Ax= the prifm.

Corol. 9. Hence a prifm is to a pyramid of the fame base and height, as 3 to 1, and to the fruftum of the fame bafe and height, as

Corol. 10. Similar pyramids are as the cubes of their like fides. For the pyramid is as Ab, or as h3, or as VA3 because A is as bb by fimilar triangles.

Corol. II. All fimilar folids are as the cubes of their like fides. This follows from their being compofed of fimilar pyramids.

EXAMPLE III.

Wanting to measure an oblique cone; and having taken the circumference of the bafe equal to 40 feet, and the outward angle which the fide of the cone, where it was shortest, made with the horizon, 85°; going thence in a direct line towards the part to which the vertex inclined, to the diftance of 100 feet, I found there the angle of elevation of the vertex of the cone to be 50°: required the folidity.

The LBLE = 85° +50° = 135°
which taken from 180

But

40

10

F

H

== =12732396 is the diameter. 3°1416 •7854

And 12.732396 X 10 = 127.32396 is the bafe. Theref. DC X 127.3239642°44132 × 107.9228 4580 386 is the content of the cone.

EXAMPLE IV.

There is a cone whofe perpendicular altitude is 40 feet, and the circumference of its bafe 30 feet, which is cut by a plane drawn from the vertex, and paffing through the bafe at the distance of 2 feet from the

center;

center; what is the folidity of the whole cone, and of the two pyramids, GHBC, GHAC, into which it is cut?

Therefore 4774648-2 = A

2 774648 is the versed fine of the lefs fegment of the circle.

F BD E

H

Then, to find its area by the table of circular fegments, 27746489549299

2905611 is the tabular verfed fine, to which anfwers the tabular fegment 18955709;

and this taken from 78539816

leaves the other tab. feg. 59584107.

Then each of thefe fegments multiplied by the fq. of the diam. give 17.2856 for the lefs bafe GBHG, and 543341 for the greater bafe GAHG. 17.2856 230°4743 the lefs pyramid, X5433417244543 the greater pyr.

Then and

30

4

their fum is 954929} the whole cone.

PROBLEM VI.

To find the Solidity of the Fruftum of a Pyramid.

Add into one fum the areas of the two ends and the mean proportional between them, multiply the fun by the perpendicular height, and of the pro13 duct will be the folidity.

That is, If A be the area of the greater end,

a that of the lefs, and b the height, Then (A+a+Aa) × will be the folidity.

Note

Note 1. If the ends be regular polygons, the particular rule for them will be eafier, thus: Add together the fquare of a fide of each end of the fruftum, and the product of thofe fides, multiply the fum by the height, and the product by the tabular area anfwering to the particular figure of the ends, and 183 of the laft product will be the content.

Or, Divide the difference of the cubes of the faid fides by their difference, and multiply the quotient by the height, and the tabular area, and take of the product.

Note 2. If the ends be circles, the fruftum will be that of a cone, and then multiply 2618, namely of 7854, by the height, and the product either by the quotient arifing from the divifion of the difference of the cubes of the diameters by the difference of the diameters, or by the fum arifing from the addition of the fquare of each diameter and the product of the diameters, or by the fum arifing from the fquare of the half difference of the diameters added to triple the fquare of the half fum.*

EXAMPLE I.

How many folid feet are there in a tree whofe bafes are squares, each fide of the one being 15 inches, and each fide of the other 6, and the length along the fide meafures 24 feet?

Here 15×15225, the greater bafe,

CA

*All these rules come from the demonftration of prob. 5, and

its corollaries.