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THEORY AND PRACTICE OF ENGINEERING.

Book 11.

Fig.1209.

consequently a sixth of the heptagon, ABCDEF.G: there will remain the trapezoidBO FZ for the last sixth. Thus the irregular heptagon will be divided into six equalparts by the lines B, Z, O, 3, 4, 5.

To divide Figures of unequal Dimensions by similar Lines , as to divide the figureA B C D into three parts similar to three divisions on the plan EFGH; measure thelength E I of the plan, EFGH, with the scale K, as45 : measure likewise on the figure A B C D 45, fromA to L : then take with a protractor the angle E I M,

105°, and from the angle ALN also 105°, on the sideA B, at the point L. Measure the length F O, 40°,with the scale K, and set off 40° from B to P; at thepoint P draw the angle B P Q equal to F O M : then,having found that the distance G R is 70 parts, set offthe same number from C to S. At the point S drawthe angle C S T equal to GRM: the lines L, N, P, Qand S, T, intersecting at V, will divide the figureA B C D similarly to E F H G.

Mensuration of Solids. — Determine the value of thespaces included by contiguous surfaces, and the sum of themeasure of those including surfaces is the whole surface orsuperficies of the body; the rules for performing suchoperations we have already described. The measure of a solidis called its solidity, capacity, or content, and is usually measuredby cubes, whose sides are inches, feet, or yards: hence thesolidity of a body is said to be so many cubic inches, feet,yards, &c., as will fill its capacity or space, or another of anequal magnitude.

The cube A may be supposed to contain a solid foot, con-sequently 1728 cubic inches; for if the area of its base weredivided into square inches, it would contain 144, and as thecube is 12 inches in height, it would permit of 12 layers ofcubical inches being piled up to complete the figure, or 144cubes x 12 = 1728 : consequently, to find the solid content of acu!>e, we only have to multiply the area of its base by itsheight.

Of a ParaUelopipedon. — Multiply the area of one end by thelength, and the product is the solid content.

Inclined Parallelopipedons are cubed in a similar manner.

The solid content of a rectangular parallelopipcdon is said tobe equal to the product of its three dimensions, that is, asAB x AC x AD, when A B,

AC, AD, are the three edges:this expression being inter-preted in the same sense withthe product of the two dimen-sions or sides, which is said toconstitute the area of a rect-angle, viz. that the number of

Fig. )210.

rig. 1211 .

Fig. 1*212

Fig. 1213.

cubical units in the parallelopiped is equal to the product ofthe numbers which denote how often the corresponding linearunit is contained in the three edges: it is on this accountsaid that the solid content of a rectangular parallelopiped isequal to the product of its base and altitude. The cube isconsidered a unit in the mensuration of all other solids, theircontent being the same with the content of rectangularparallelopipcds equal to them.

Prisms in general have their solid content found by mul-tiplying the area of their base by their height.

To find the solid content of an irregular solid whose sidesare parallel, we must first measure it in detail, find the con-tent of the several parts, and add them together: the presentexample might be taken as three parallelopipedons; or the solid content of the partscut out might be ascertained, which, deducted from the solid content obtained by mul-tiplying the area of its base by its height, would give the content of the irregular solid.As all prisms and cylinders are equal to parallelopipedons of equal bases and altitudes,it is only necessary to multiply the base or end by the height to obtain the solidcontent.

Fig. 1214. Fig. 1215.