EXAMP. 1, To find the area of the segment ABC, whose chord AC is 172, the chord of half the arch ABC, viz. BC= 104, and the versed sine BD = 58.48.

RULE.-By Art. 23, find the length of the arch line ABC, and by Art. 24, the diameter FB; then multiply half the chord of the arch ABC by half the diameter, and the product will be the area of the sector ABCE: then find the area of the triangle AEC, whose base AC is 172, and perpendicular height 34, found by subtracting the versed sine BD from half the diameter; and the area of the triangle AEC, being subtracted from the area of

F

the sector ABCE, will leave the area of the segment ABC.

EXAMP. 2. In the segment ABCD greater than a semicircle, given the chord of the whole segment AĎ 136, the chord AC of half

=

the

ART. 27. To find the Area of an Ellipsis.

Definition. An ellipsis, or oval, is a curve which returns into itself like a circle, but has two diameters, one longer than the other, the longest of which is called the transverse, and the shortest the conjugate diameter.

RULE. Multiply the two diameters of the ellipsis together; then multipling the product by 7854, this last product will be the area of the ellipsis.

the con

Mensuration of Superficies is easily applied to Surveying thus, take the angles of the plot with a good compass, then measure the sides with Gunter's chain, which note down in links (or chains and links, which is done by separating the two right hand figures of your links by a comma, your chain being 100 links) then cast up tents, according to the rule of the figure, cutting off the five right hand figures of the product, and those at the left hand, if any, are acres; then multiply the five figures cut off, by 4, by 40, and by 2724, cutting off as before, and those at the left hand, will be roods, poles and feet, respectively.

SECTION II. OF SOLIDS.

Solids are measured by the solid inch, foot or yard, &c. 1728 of these inches, that is 12x12x12, make one cubick or solid foot. The solid content of every body is found by rules adapted to their particular figures,

ART. 28. To measure a Cube.*

Definition. A cube is a solid of six equal sides, each of which is an exact square,

The

* Here follows a Table of the Proportions, which the following Solids have to the Cube and Cylinder, having the fame Bafe and Altitude.

1. A Cube whofe fide is 12 inches, contains 2. A Prifm, having an equilateral triangle, whofe fide is 12 inch. es from its Bafe, and its Altitude 12 inches, contains

3 A Square Pyramid, whofe height and the fide of its bafe, are each 12 inches, is of the above cube, and therefore contains 4. A Triangular Pyramid, whofe height and fide of its triangular base are each 12 inches, is near of the cube and contains

1

5. A Cylinder, whofe diameter and height are each 12 inches, is of the above cube, and contains

Solid Inches.

1728

784.24

576

249'413

1357'17

6. A Sphere or Globe, whofe axis or diameter is 12 inches, equal

qual}

904.78

to the fide of the cube, is of it, and contains

7. A Cone, whose base and altitude are each 12 inches, equal to 452-38829

the fide of the cube, is of it, and contains

The solid foot is composed of 1728 inches; for a solid, that is { foot, or 12 inches every way, that is 12x12x12, contains 1728 inch

es.

8. A Parabolick Conoid, whofe diameter at the base and height,

RULE. Solid Inches.

each 12 inches, being its circumfcribing cylinder, contains are? 678-583

ΤΟ

9. A Hyperbolick Conoid, whofe height, and diameter at the base, are each 12 inches, 1s of its circumfcribing cylinder, and contains 10. A Parabolick Spindle, whofe height and middle diameter are each 12 inches, is of its circumfcribing cylinder, and contains Hence arifes a different method of finding their contents.

:

565-49

arc} 723·824

General Rule. If the bafe of the folid, whofe contents you would find, be reetilinear, confider it as Parallelopipedon; if curved. as a Cylinder, and find the content accordingly then take fuch a part of the content, thus, found, as is fpecified in the preceding Table, which if the parts be taken in inches, will be the folid content of the given figure, in inches, which, divided by 1728, will give the cubick feet.

EXAMP I. There is a triangular prifm, the fide of whose base is 48 inches, and whofe perpendicular height is 108 inches: what is its folid content?

The bafc being tight lined, I confider it as a parallelopipedon, the fide of whose bafe is 48 inches, and whofe length is 108 inches. and as 784.24 is contained 2-203407 12 times in a cubick foot; 2.20340712 is a divifor, to divide the content of the parallelopipedon by; therefore 48X48X108÷÷220340712=11293056 folid inches 65 353 folid feet.

Had the dimenfions been given in feet, it would have been 4X4 X9÷2-20340712 =65.353 feet.

EXAMP. 2. There is a square pyramid, whose height is 12 feet, and the fide of whose base is 3-5 feet; what is its content?

35X35X12-3-49 feet, Ans. EXAMT. 3. There is a triangular pyramid, whole height is 15 feet, and the fide of whofe bafe is 5 feet: what is its content?

5X5X15 7 53'57 feet, Ans. EXAMP. 4. There is a cylinder whofe diameter is 2.5 feet, and whofe length is 24 feet; what is its content?

Here, the diameter is to be confidered as the side of the base of a parallelo. pipedon. Therefore, 2.5X25X24X11÷14117857 feet, Ans. EXAMP. 5. There is a spherical balloon, whofe diameter is 50 feet; how many cubick feet of air does it contain?

Here, the diameter is to be confidered as the fide of a cube. Therefore,

50X50X50X11÷÷21=65476·19 feet, Ans. EXAMP. 6. There is a cone, whofe height is 15 feet, and the diameter of whose bafe is 5 feet; what is its content?

Here, the diameter of the bafe is to be confidered as the side of the base of a parallelopipedon, and its height, as the length. Therefore,

5X5×15×5÷19=98-684 feet, Ans.

EXAMP. 7. There is a parabolick conoid, whofe diameter at the base is 29 feet, and whofe height is 6 feet; what is the content?

This folid being of a cylinder; we must first find the content as of that of a cylinder, and then halve it. Therefore,

29×29×6×11÷÷14=39'647, and 39'647÷2=19·823, Ans. EXAMP. 8. There is a hyperbolick conoid, whofe diameter at the base is 2'9 feet, and whofe height is 6 feet; what is the content?

First, find the content of a cylinder,

2·9X2·9×6×11÷14=39'647, and 39'647X16-519 feet, Aos.

EXAMP. 9. There is a parabolick spindle, whose middle diameter is 2.9 feet, and whofe length is 6 feet; required the content?

Firft, find the content of a cylinder.

2·9X2′9X6X11÷÷14=59'647, and 39 647X8=21·145 feet, Ans,

RULE.-Multiply the side by itself and that product by the same side, and this last product will be the solid content of the cube.

EXAMP. The side of a cube AB, being 18 inches, or 1 foot and 6 inches, to find the con

A

I have done this two different ways, that the learner may see they come out the same. The content in inches is 5832, which being divided by 1728, the inches in a solid foot, and the division continued by annexing cyphers, it comes out the same as the decimal operation.

Note. The area of the surface, or superficial content of the cube and parallelopipedon is found by adding the areas of the several quadrilateral figures which compose them.

ART. 29. To measure a Parallelopipedon.

Definition. A parallelopipedon is a solid of three dimensions, length, breadth and thickness; as a piece of timber exactly squared, whose length is more than the breadth and thickness. The ends are called bases, which are equal.

RULE. Find the area of the base, then multiply that by the length, and it will give the solid content.

EXAMP. 1. The side AB is 1.75 foot, and the length AD 9.5 feet, to find the solid content?