Problem 18.

To find the Solidity of a Wedge.

Rule.

To twice the Length of the Back add once the Length of the Edge; multiply this Sum by the Product of the Height multiplied into the Breadth of the Wedge, and (one fixth) of this laft Product will be the Solidity.

Example.

Suppose a Wedge whose Height AD is 8 Inches; its Edge A B, 3 Inches; the Length of the Back E F, 4 Inches, and the Breadth of the Back CE, 2 Inches; what is its Solid Content?

Twice the Length of the Back EF = 8+ once the Length of the Edge AB = 3 is = 11; this x by 16 (the Height AD = 8 multiplied by the Breadth CE = 2) gives 176; which by 6 gives 29.333, &c. Inches, the Solid

Content.

Note. When the Back and Edge are of the fame Length, the Wedge is equal to half a Prism of the fame Base and Altitude; and its Content may be found by multiplying the Area of the Back by half the Height of the Wedge.

Problem 19.

To cut a Tree through in such Manner, that the two Parts measured separately, shall produce a greater Solidity, than that of the whole Tree.

Rule.

Cut it through exactly in the Middle, or at half the Length, and the two Parts will measure (confiderably) more than the Whole did before.

Example.

Suppose a Tree to girth 14 Feet at the greater End, Feet at the less, and 8 Feet in the Middle, and that the Length is 32 Feet.

By the common Method, the whole Tree measures only 128 Feet. When cut through the Middle, the greater Part measures 121 Feet, and the less Part 25 Feet, which together make 146 Feet, and exceeds the whole by 18 Feet.

Problem 20.

To cut a Tree so that the Part next the greater End may measure the most poffible, and sometimes confiderably more than the whole Tree.

Rule.

Cut it through where the Girth is equal to (one third) of the greatest Girth, and the greater End will then measure the greatest possible.

To find where the Tree must be cut through from the less End, to make the Girth there equal to of the Girth of the greater End.

Rule.

From the Girth of the greater End, fubtract 3 Times the Girth of the less End, and divide the Remainder by the Difference between the greater and less Girths; this Quotient multiply by of the Length of the Tree, and the Product will give the Length of the Piece to be cut off from the less End, to leave a Remainder greater than the whole Tree.

Operation.

Taking here the fame Example as in the last Problem, we shall have 7.1 Feet the Length to be cut off; 24.9 Feet, the Length of the remaining Part; and 4.666 Feet, the Girt at the Place cut off. Now the Content of the. whole Tree as before is only 128 Feet; but the Content of the greater Part here is 135.5 Feet, which exceeds 128 by 7.5 Feet, and is the greatest poffible.

If the greatest Girth doth not exceed the less 3 Times, the Tree cannot be cut as required by the Problem; for when the leaft Girth is equal exactly to of the greater, the Tree then measures to the most possible.

Problem 21.

A Person, for a confiderable Wager, is to travel in a certain Time from the Town at A, to another at B, but he is obliged to call at a Place which is somewhere on the (Road or) Line CD: Now his Time being limited, he is defirous to know (by Geometry) the Situation of the Place upon the Line CD, from whence the Distance to A and B shall be the least possible ?

First, produce the Line A C downwards towards G, and make CG = CA. Next, draw the Line GB through the Line CD at E; draw also the Line from A to E; then will. GE = AE; and the Angle CEA CEG. But the Sum of GE and E B is the least possible, when they are in the same Direction, or make one right Line.. Consequently the Point E is the Place on the Line CD, at which the Traveller must must call to make the Journey the leaft poffible.

Problem 22.

A Farmer borrowed a Stack of Hay of his Neighbour, which measured 6 Feet every Way, * and paid him back again by 2 equal Cubical Pieces, each of whose Sides were 3 Feet. Query, whether the Lender was fully paid?

The Content of the larger Cube is 216 Feet. The Content of the smaller only 27 Feet. The 2 Payments, therefore, amounted to no more than 54 Feet, which is just (or fourth Part) of what was borrowed; consequently the Farmer is still indebted more to his Neighbour.

* That is, 6 Feet in Length, 6 in Breadth, and 6 in Depth.