Examples.

1. What is the solid content of a dome, in the form of a hemisphere, the diameter of the base being 21 feet?

Here 212.7854-346.3614 the area of the base.

And 346.3614x 14-4849.0596 feet, the solidity.

2. Required the solidity of an octagonal dome, each side of the base being 12 feet, and the height 24 feet.

PROBLEM IV.

Answer, 7634.688 feet.

TO FIND THE SUPERFICIAL CONTENT OF A SPHERICAL DOME.

Rule. Multiply the area of the base by 2, and the product will be the superficial content.

Note. If the dome be elliptical, multiply the two diameters of the base together, and that product by 1.5708, and the last product will be the area, sufficiently near for practical purposes.

Examples.

1. Required the superficial content of an octagonal spherical dome, each side of the base being 10 feet.

4.828427

100 square of 10

482.842700

2

965.685400 Content.

2. The two diameters of an elliptical dome are 20 and 15 feet; required the superficial content.

PROBLEM V.

Answer, 471.240 feet.

TO FIND THE SOLID CONTENT OR VACUITY OF A SALOON.

Rule.-Multiply the area of a transverse section by the circumference of the solid part of the saloon taken in the

middle; subtract this product from the whole vacuity of the room, supposing the walls to go upright to the flat ceiling, and the difference will be the answer.

Note. In a rectangular, circular, or polygonal room, the base of the dome will be a square, a circle, or a polygon, consequently the vacuity on the whole upright space of the room will be a parallelopipedon, a cylinder, or a prism.

Examples.

1. The height A-B of a saloon is 3 feet, the chord ADC 5 feet, B-C 4 feet, D-E 1 foot, and the mean compass 60 feet; required the solidity of the saloon.

Now the area of the triangle ABC = 6.000
And the area of the segment=

3.343

Area of the transverse section ABCEA=2.657

And 2.657x 60=159.420 the content of the solid part, which deducted from the whole upright space will give the content of the vacuity within the room.

2. A circular building of 40 feet in diameter, and 25 feet high to the ceiling, is covered with a saloon, the circular arch of which is 5 feet radius; what is the solidity of the room in cubic yards?

Answer, 1139.499 yards.

PROBLEM VI.

TO FIND THE SUPERFICIAL CONTENT OF A SALOON.

Rule-Find the breadth of the curved part of the saloon by applying a tape close to it across the surface; and its length

by measuring along the middle of it, quite round the room; and the product of these two dimensions will be the surface as required.

Examples.

1. Required the superficial content of a saloon, the girt across the face of the saloon is 12 feet, and its mean compass 83 feet.

Here 83 x 12=996 feet, the content.

2. The mean compass of a saloon is 104 feet, and the girt across its face 9.5 feet, what is the area of its surface?

Answer, 988 feet.

PROBLEM VII.

ΤΟ FIND THE SOLIDITY OF THE VACUITY FORMED BY A GROIN ARCH, EITHER CIRCULAR OR ELLIPTICAL.

Rule-Multiply the area of the base by the height, and the product by .904, and it will give the solidity required.

Examples.

1. What is the solidity of the vacuity formed by a circular groin, springing from the sides of a square base, each side of which is 12 feet?

Here 12× 12× 6x .904=781.056 feet, the solidity.

2. Required the solid content of the vacuity formed by an elliptical groin, the side of its square base being 20, and its height 7 feet.

Answer, 2531.2 feet.

PROBLEM VIII.

TO FIND THE CONCAVE SURFACE OF A CIRCULAR OR AN

ELLIPTICAL GROIN.

Rule.-Multiply the area of the base by 1.1416, and the

product will be the superficies required.

Examples.

1. Required the concave surface of a circular groin arch, the side of its square base being 14 feet.

Here 14x 14=196, area of the base;

And 196x 1.1416=223.7536 feet, the surface

2. The base of a groin arch is a rectangle whose sides are 30 by 22 feet; required the concave surface of the arch?

Answer, 753.456 feet.

Note. The general rule adopted in measuring the brick or stone work of any arch, is, to deduct the vacuity contained between the springing of the arch and the under side of it, from the content of the whole, and the difference will be that of the solid part.

BOOK XII.

CONIC SECTIONS

AND THEIR SOLIDS.

DEFINITIONS.

1. Conic Sections are such surfaces as are formed by cutting a cone. They are five in number, viz., the triangle, circle, ellipse, parabola, and hyperbola.

2. If the cutting plane pass through the vertex A of the cone, to any part of the base, the section will be a triangle, as ABC.

3. If a cone be cut by a plane parallel to the base, the section will be a circle, as D—E.

4. If a cone be cut by a plane which passes through its two slant sides in an oblique direction, the section will be an ellipse, as F-G.

5. If a cone be cut by a plane parallel to one of the slant sides, the section will be a parabola, as H-K.