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Theorem XYII.—To find the area of a parabola; the base,b, and the altitude, a, being given :

2 a b .

A ~~3

Theorem XYIII.—To find the area of the sinical curve;the radius, r, and the versed sine, v, of the segment, beinggiven :

A — r v.

Areas of Curved Surfaces.

Theorem I.—To find the area of a cylindric surface; theaxis, a, and the diameter, d, being given :

A = 3.1416do in a right cylinder.

Or A = p a in an oblique cylinder, where p is the peri-meter.

Theorem II.—To find the curved surface of a cylindric

(j[ § f*

ungula, A =--- h ; where h is the length of the part

that is cut, and v the versed sine of the segment which formsthe base, and d the diameter.

Theorem III.—To find the area of the curved surface ofa right cone ; the side, s, and the circumference, c, of the cir-cumference, being given :

sc

A ~ ~2'

Theorem IV.—To find the area of the segment of asquare dome.

A = 4(s“ + c 2 ), or 4 dv ; where s is the sine, and v theversed sine of the circular segment which forms the verticalsection, or d the diameter of the circle which forms thevertical section.

Theorem V.—To find the area of a segment of a hemi-sphere ; given the radius, s, of the base, the versed sine, v,and the diameter, d, of the axal section :

a = 3.1416 (s 2 + or 3.1416 dv.

Theorem YI.—To find the area of the frustum of thesegment of a hemisphere ; the circumference, c, of the greatcircle, and the distance, d, of the parallel planes, beinggiven :

A — c d.

Theorem VII. Rule 1.—To find the surface of a spheroid;given the axis, a, and the diameter, d, of the great circle :

A = .5236 X (4 ad + 2 d 2 ).

Or, A = 1.0472 (2 a d + d 2 ).

This is a near approximation.

Theorem VIII. Rule 2.a ! + 2 d 1

a = .8805 (2 a d + d 2 ) --- ; still nearer to the truth

than Rules 1 and 2.

Theorem IX. Rule 1.—To find the surface of anellipsoid ; the length, a, the breadth, b, and the thickness, c,being given:

2 X .5236 (a b + a c b c).

This rule is a near approximation.

Theorem X. Rule 2.

a Q -f- b* 4- c 2

a — 8805 (a J + « « + i c) + --- 5 itisanearer

o

approximation than Rule 1.

Theorem XI.—To find the surface of a semi-circulargroin; given the side b.

a = 1.1416 b\

Solidities of Bodies.

Theorem I.—To find the solidity of a cube.

s = s’, where s is the linear dimension.

Theorem II.—To find the solidity of a prism.s = a h ; where a is the area of the base, and h t‘ ie r

pendicular height. This also includes cylinders.

Theorem III.— To find the solidity of a pyramid.

a Jh b- a ,

s = —, or v X a,orv X J; where a is the a

of i:

the base, and h the perpendicular height. Thisincludes cones.

Theorem IY.—To find the solidity of a wedge, or P' t |, emoid ; the two adjoining edges, a and b, of theedge, c, in the same plane with a, and the heightgiven

2 a b + c b 7

s =

-h.

Theorem V.—To find the solidity of the frustumpyramoid.

of'

,it*

S = +_cd+ 4 ef h ; where a and c are the opp* 51

6 . d d t¥

terminations of a plane of one of the sides ; b anopposite terminations of the plane of one of tlieadjsides; a and b being adjoining sides of the base, c ,those of the top, e the half sum of a and c, and f the haof b and d.

Theorem VI.—To find the solidity of a cuneoid.

2 a b + c b

X .7854 X h. This formula is

the &

s = _ _ _

. r 78^'

as that of Theorem V. except the additional multiph er ii>this solid only differing in construction from the py rahaving elliptic sections instead of rectangular ones. 0 f *

Theorem VII.—To find the solidity of the frustu ,cuneoid.

ab+cd-\-4ef n 3 c ^

s =---- x .7854 x h ; where a. a n i

6 ^ t>* ; :

the two axes of the elliptic base, and c and rfthe axeselliptic top; c being opposite to a, and d opposite to ■solid only differs from the frustum of a pyramoiu 1 i

circular. e0 t ’

Theorem VIII.—To find the solidity of the sega cylinder. ^

*9*“

Rule 1.— s = a l -f- JL j ; where a is tlie ’ ^ r , 1sine, b the chord of the base, and l the length of the c 7

= l X ( !

a b + a*

Rule 2.—s'

Rule 1. when the segment is nearly a semi-cyl> n ^ er ' ^ of *

+ r6 ) ‘ ,,m ne,rer

Theorem IX. —To find the solidity of the seg c0square dome.

Rule 1.—s r= 4ri a -— ; where x is the heigb .

o . e ve>

radius of the circle, of which the segment formingsection is a part. See vol. i. p. 284 of this Work

ah 1

d

gflO**

Rule 2.—s = ; where s is the side of the * 0 {

Z * 1 i. P* ;

base, and a the altitude of the dome. See vol. • i

this Work. /