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 the contents, 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 12×12×12, 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,

*Here follows a Table of the Proportions, which the following Solids have to the Cube and Cylinder, having the same Base and Altitude. Solid Inches. 1. A Cube whose side is 12 inches, contains

2. A Prism, having an equilateral triangle, whose side is 12 12}

inches from its Base, and its Altitude 12 inches, contains

3. A Square Pyramid, whose height and the side of its base, are each 12 inches, is of the above cube, and therefore contains

}

1728

784-24

576

249-413

4. A Triangula, Pyramid, whose height and side of its triangu lar base are each 12 inches, is near of the cube, and contains 5. A Cylinder, whose diameter and height are each 12 inches, 1357-17 is of the above cube, and contains

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

to the side of the cube, is 11 of it, and contains

7. A Cone, whose base and altitude are each 12 inches, equal }

to the side of the cube, is 5 of it, and contains

904.78

452-38829

3. A Parabolick Conoid, whose diameter at the base and height, 678-583 are each 12 inches, being its circumscribing cylinder, contains

9. A Hyperbolick Conoid, whose height, and diameter at the base, are each 12 inches, is of its circumscribing cylinder, and contains

12

10. A Parabolick Spindle, whose height and middle diameter are each 12 inches, is 8 of its circumscribing cylinder, and contains Hence arises a different method of finding their contents.

15

565.49

723-824

General Rule. If the base of the solid, whose contents you would find, be rectilinear, consider it as Parallelopipedon; if curved, as a Cylinder, and find the content accordingly: then take such a part of the content, thus found, as is specified in the preceding Table, which if the parts be taken in inches, will be the solid content of the given figure, in inches, which, divided by 1728, will give the cubick feet.

EKAMP. 1. There is a triangular prism, the side of whose base is 48 inches, and whose perpendicular height is 108 inches: what is its solid content?

The base being right lined, I consider it as a parallelopipedon, the side of whose base is 48 inches, and whose length is 108 inches, and as 704-21 is con

The solid foot is composed of 1728 inches; for a solid, that is 1 foot, or 12 inches every way, that is 12x12x12, contains 1728 inches.

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

Carried over.

tained 220340712 times in a cubick foot; 2-20340712 is a divisor, to divide the content of tire parallelopipedon by; therefore 48×48×108÷2-20340712= 112030-56 solid inches-65.353 solid feet.

Had the dimensions been given in feet, it would have been 4×4X92-20340712=65.353 feet.

EXAMP. 2. There is a square pyramid, whose height is 12 feet, and the side of whose base is 3.5 feet; what is its content?

3.5X35X12÷3-49 feet, Ans. EXAMP. 3. There is a triangular pyramid, whose height is 15 feet, and the side of whose base is 5 feet: what is its content?

5X5X157-53.57 feet, Ans. EXAMP. 4. There is a cylinder whose diameter is 2-5 feet, and whose length is 24 feet; what is its content?

Here, the diameter is to be considered as the side of the base of a parallelopipedon. Therefore, 2.5X25X24x11÷14=117-857 feet, Ans. EXAMP. 5. There is a spherical balloon, whose diameter is 50 feet; how many cubick feet of air does it contain?

Here, the diameter is to be considered as the side of a cube. Therefore,

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

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

5X5X15X519-98-684 feet, Ans. EXAMP. 7. There is a parabolick conoid, whose diameter at the base is 2.9 feet, and whose height is 6 feet; what is the content?

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

2.9x2.9x6x11÷÷14=-39-647, and 39-647÷÷÷2=19·823, Ans. EXAMP. 3. There is a hyperbolick conoid, whose diameter at the base is 2.9 feet, and whose height is 6 feet; what is the content?

First, find the content of a cylinder.

29×29×6×11-14—39-6-17, and 39.647 × 516.519 feet, Ans. EXAMP. 9. There is a parabolick spindle, whose middle diameter is 2.9 feet, and whose length is 6 feet; required the content?

First, find the content of a cylinder.

2·9×2·9×6×11÷14—39-647, and 39-647-21·145 feet, Ans.

15

+ Multiplying a side by itself, or squaring a side, gives the area of the base, or the number of square inches, feet, &c. in the base; whence one inch, foct, &c. in height would give as many solid inches, feet, &c. as there are squares in the base; two inches, &c. in height, twice as many, and so on, and is the rule, when the sides are equal to each other. In the same way, the rule for the content of the Parallelopipedon is proved.

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?

If a piece of timber, or any other thing, be of an equal bigness through its whole length, though there be a difference between the breadth and thickness, if the breadth and thickness are multiplied together, and that product multiplied by the length, this last product will be the solid content.

EXAMP. 3. A piece of timber being 1 foot and 6 inches, or 18 inches broad, 9 inches thick, and 9 feet 6 inches, or 114 inches long, to find the content?

Breadth 18 inches.

=

1 foot 6 inches

1.5 foot

9 inches.

Note. When the end is given in inches and the length in feet, find the area at the end in inches, multiply that by the length in feet, and divide this product by 144 (the square inches in a foot) and the quotient will be the feet.. Take the last example.

162 area in inches.
9.5 feet length.

810 1458

By the sliding Rule.

Set 12 inches on the girt line D to the side of the square end on C, then, against the length on D, you will have the answer on C.

By Gunter.

Extend the compasses from 12 inches to the length of the side of the square end; that distance, twice turned over from the length, 144)1539(10-6875=content, will reach to the content.

When the side of a square solid is given, in inches, to find how much in length will make a foot solid.

RULE. As the given side is to 12, so is 12 to a fourth number, and so is that fourth number to its required length. Or divide 1728 by the area at the end, and the quotient will be the length making a solid foot.

If the given side is in foot measure, then,

RULE. As the given side is to 1; so is 1 to a fourth number, and so is that fourth number to the required length.

When two sides of an equal square solid (that is, of unequal breadth) are given, to find what length will make any number of solid feet.

RULE. Multiply the proposed number of feet by 144: divide that product by the product of the breadth and depth, and the quotient will be the length required.

ART. 30. To measure a Cylinder.

Definition. A cylinder is a round body, whose bases are circles, like a round column, or a rolling stone of a garden.

RULE. The diameter of the base being given, find the area of the end by Art. 15, then, multiplying the area of the base by the length, that product will be the content of the cylinder.

If the square of the diameter of a cylinder be multiplied by 7854, and the solidity divided by that product, the quotient will be the length, and if the content be divided by the length, the quotient will be the area of the end, from which the diameter is found by Art. 13.