The learner may, for his practice, reduce all the dimensions to inches, and find the solid content in inches, which being divided by 1728, the quotient will be the solid content in feet: or, if he finds the area at the end in inches, and multiplies that by the length in feet, and divides by 144; the quotient will be feet.

This is a general rule for finding the content of any straight solid body, of equal bigness from end to end, of whatever form the bases are for, if the area of the base be multiplied by the length, the product will be the solid content.

By the Sliding Rule.

Set 135, the square root of 183·34 (which is a guage point arising from the division of 144 by 7854) found on D, to the diameter found on C, and opposite to the length, on D, you will find the content on C.

Or, as 42 54 is to the circumference; so is the length in feet to a fourth number, and so is that fourth number to the answer.

Note. The superficial content of a cylinder is found by multiplying the circumference of one of the bases into the length, and to the product adding the areas of the two bases, or ends.

When the diameter is given in inches, to find what length will make a solid foot.

RULE. As the given diameter is to 13 531: so is 12 to a fourth number, and so is that fourth number to the required length. If the diameter be given in foot measure: Rule, as the given diameter is to 1.128: so is 1 to a fourth number, and so is that fourth number to the required length. Or, divide 1728 by the area at the end in inches, and the quotient will be the required length.

To find how much a Cylindrick or round Tree, that is equally thick from end to end, will hew to, when made square.

RULE. Multiply twice the square of its semidiameter by the length, then divide the product by 144, and the quotient will be the answer.

If the diameter of a round stick of timber be 24 inches from end to end, and its length 20 feet: how many solid feet will it contain, when hewn square; and what will be the content of the slabs which reduce it to a square?

12×12×2×20

144 24×24X 7854×20

144

=40 feet, the solidity when hewn square.

=628 feet, or 2×2×·7854×20=62·8 the total solidity, whence 62.8--40-228 feet, the solidity of the slabs. Note. The rule of workmen for measuring round timber is to multiply the square of the quarter girt or one fourth of the cir cumference, by the length. This rule allows about one fifth, for the bark, waste in hewing, &c. The example above, in which the diameter of the cylinder is 1 foot 9 inches, and the length 12 feet

6 inches, will give the quarter girt 1-3744 feet, and the solid content is 1-3744"x12.5=23.61 feet, which is nearly four fifths of 30 G625, the content by the accurate rule.

A rule, nearly correct, is to multiply twice the square of one fifth of the circumference by the length. Thus, in the example, of the circumference is 1.0995, and 2x1.0995'x12.5= 30.22 feet.

ART. 31. To measure a Prism.

Definition. A prism is a body with two equal or parallel ends, either square, triangular, or polygonal, and three or more sides, which meet in parallel lines, running from the several angles at one end, to those of the other.

RULE. Prisms of all kinds, whether square, triangular or polygonal, are measured by one general rule, viz. Find the superficial content, or area at the base (or end) by the proper rule of Sect. 1. and this multiplied by the length, or height of the prism, will give the solid content.

EXAMP. The side of a stick of timber, AB, hewn three square, is 10 inches, and the length, AC, is 12 feet, to find the content?

Side 10 inches.

Perpendicular 4.33 inches.

43.3 area at the end.

12 feet length.

144)519-6(3-6 feet, content.

432

876

864

12

Note. The superficial content is found by adding the areas of the several quadrilateral and triangular figures which compose it. ART. 32. To measure a Pyramid.

Definition. Solids, which decrease gradually from the base till they come to a point, are generally called pyramids, and are of different kinds, according to the figure of their bases; thus, if it has a square base, it is called a square pyramid: if a triangular base, a triangular pyramid: If the base be a circle, a circular pyramid, or simply a cone. The point, in which the top of a pyramid ends, is called a Vertex, and a line drawn from the vertex, perpendicular to the base, is called the height of the pyramid.

RULE. Find the area of the base, whether triangular, square, polygonal or circular, by the rules in superficial measure: then, multiply this area by one third of the height, and the product will be the solid content of the pyramid.

EXAMP. 1. In a triangular pyramid, the height BE, being 48, and each side of the base 13: the base being a triangle, let the perpendicular height DE be 11; to find the content.

5.5

half ED.

EXAMP. 2. In a quadrangular pyramid, the height BE being 48, and each side of the base 13, to find the content.

E

B

EXAMP. 3. To measure a Cone.--The diameter

AC being 13, and the height BD 48, to find the

Note. The superficial content of all pyramids is found by taking the sum of the several areas, which compose them. That of a cone, by multiplying the circumference of the base into half the line joining the vertex and any point in that circumference, and adding the area of the base to the product.

ART. 33. To measure the Frustum of a Pyramid.

Definition. The frustum of a pyramid is what remains after the top is cut off by a plane parallel to the base, and is in the form of a log greater at one end than the other, whether round, or hewn three or four square, &c.

KULE. If it be the frustum of a square pyramid, multiply the side of the greater base by the side of the less; to this product add one third of the square of the difference of the sides, and the sum will be the mean area between the bases; but if the base be any other regular figure, multiply this sum by the proper multiplier of its figure in the Table, Art. 11. and the product will be the mean area between the bases: lastly, multiply this by the height, and it will give the height of the frustum.

EXAMP. 1. In the frustum of a square pyramid the

side of the greater base AD=15, the side of the less, B BC=6, and the height EF=40, to find the content.

F

4680 content.

3)81=square of the difference. A

27 of the square.

Or, if it be a tapering square stick of timber, take the girth of it in the middle; square of the girth (or multiply it by itself in inches) then say, as 144 (inches) to that product; so is the length, taken in feet, to the content in feet.

EXAMP. 2. What is the content of a tapering square stick of timber, whose side of the largest end is 12 inches, of the least end, 8, and whose length is thirty feet.

One fourth of the girth in the middle=10, and 10×10=100, the area in the middle; then, as 144 : 100 :: 30 feet: 20 83 feet the content.

By the Sliding Rule.

of the circumference on C, and against the

Set 12 on D to length on D is the answer on C.

By Gunter.

The extent from 12 to 4 of the circumference doubled, or twice turned over, will reach from the length to the content. EXAMP. 3. In the frustum of a triangular pyramid, the side of the greater base AC=15, as before, the B side of the less BD6, and the height EF-40, to find the content.

F

Or, if it be a tapering three square stick of timber, you may find the area midway from end to end, then, as 144 is to that area, so is the length, taken in feet, to the content in feet.

EXAMP. 4. To measure the Frustum of a Cone.

RULE. Multiply the diameters of the two bases together, and to the product add one third of the square of the difference of the diameters: then multiplying this sum by 7854, it will be the mean area between the two bases, which being multiplied by the length of the frustum, will give the solid content.

Or, to the areas of the top and bottom add the square root of the product of those areas, and the sum, multiplied by one third of the height of the frustum, will give the solidity.

When figures run uniformly taper; but not to a point (they being considered as portions of the cone or pyramid) we may find the solidity by supplying what is wanting to complete the figure, and then deducting the part cut off.

A general rule for completing every straight sided solid, whose ends are parallel and similar.

As the difference of the top and bottom diameters is to the perpendicular height, (or depth which is the same :) so is the longest diameter to the altitude of the whole cone or pyramid.