line drawn through k, perpendicular to EQ, will evidently be the direction of the shadow which the rod AB projects on the vertical plane, at the same instant of time that it projects on the equinoctial dial the shadow CK; and as the hours are indicated on the equinoctial dial by the position of the revolving shadow CK, they will also be shown on the vertical plane EQNS, by the successive positions of the rectilineal shadow FkF, which will always be parallel to NS.

Now, as the plane aABb is perpendicular to the plane of the meridian, and passes through the poles, it must be the plane of the six o'clock hour circle, or that circle in the heavens passing through the poles of the world, in which the sun is always seen at six in the morning and six in the evening. Therefore the arc cK of the equinoctial dial, intercepted between the perpendicular Cc and Ck, the position of the shadow at any time will be the measure of the horary angle described by the sun in the heavens, between six o'clock and that time; and the straight line ck, the distance of the shadow of the rod AB from the line ab immediately under it, will be the tangent of that arc to the radius Cc.

40. Let the horary angle from six o'clock be denoted by E', and let ck, the distance of the hour-line from ab, be x; also let Cc, the height of the rod above the plane of the dial, be denoted by d, then because rad. tan. E' :: dx, the general formula expressing the position of the hour-Anes on an cast or west dial, in respect of the line ab, will be (supposing radius=1)

:

x=d tan. E.

(3)

from which it appears that, in these dials, the position of the hourlines in respect of each other is altogether independent of the latitude of the place. Indeed, the same thing might have been inferred from what has been said in art. 33 and 34, for a vertical east or west dial, for any place whatever, would manifestly be an horizontal dial at the equator.

MENSURATION.

MENSURATION is the art of computing the extension, superficies, a solidity of lines, surfaces, or solids, from given data.

B

D

Def. Every quantity is measured by some other quantity of the same kind; as a line by a line, a surface by a surface, and a solid by a solid; and the number which shows how often the lesser, called the measuring unit, is contained in the greater, or quantity measured, is called the content of the quantity so measured. Thus, if the quantity to be measured be the rectangle ABCD, and the little square E, whose side is one inch, be the measuring unit, then, as often as the said little square is contained in the rectangle, so many square inches the rectangle is said to contain: so that if the length DC be supposed 5 inches, and the breadth AD 3 inches, the content of the rectangle will be 3 times 5, or 15 square inches: because, if lines be drawn parallel to the sides, at an inch distance one from another, they will divide the whole rectangle ABCD into 3 times 5, or 15 equal parts, of one inch each. And, generally, whatever the measures of the two sides may be, it is evident that the rectangle will contain the square E, as many times as the base AB contains the base of the square, repeated as often as the altitude AD contains the altititude of the square.

Hence we have the following rule for any parallelogram whatever. Prob. 1. To find the area of a parallelogram, whether it be a square, a rectangle, a rhombus, or a rhomboides.

Multiply the length by the perpendicular height, and the product will be the area.

Ex. 1. What is the area of a square ABCD, whose side AB or BC is 2f. 3 in. ?

Ans.

By duodecimals.

2 3i

2 31

By decimals.

MENSURATION.

Ex. 2. What is the area of a rectangle ABCD, whose length AB seing 12f. 6i., and the breadth BC 2f. 9i.

By duodecimals.

12 6i

2 gi

946

25 0

An. 34 4 6

f. iii

By decimals.

f. 34 375

B

Ex. 3. What is the area of a rhombus ABCD, the length AB being

12f. 61. and the height BE 6f. 3i.?

By duodecimals.

12 6

By decimals.

f. iii

Ex. 4. What is the area of a rhomboides ABCD, whose length AB s 16f. 3i. and the height DE 5f. 6i.?

Multiply the base by the perpendicular height, and half the product

will be the area.

Ex. What is the area of a triangle ABC, the base AB being 12f. 31. and the height CD 8f. 6i. ?

Prob. 3. To find the area of a triangle, whose three sides only are given.

From the half sum of the three sides, subtract each side severally; multiply the half sum and the three remainders together, and the square root of the product will be the area required.

Ex. Requireth the area of a triangle ABC, whose three sides AB BC, and CA, are respectively 13, 14, and 15 feet.

Prob. 4. Any two sides of a right-angled triangle being given, to find a third side.

Case 1.-When the two sides are given, to find the hypothenuse.

Add the squares of the two legs together, and the square root of the sum will be the hypothenuse.

Ex. 1. Requireth the hypothenuse AC of the right angle ABC, the above AB being 4, and the perpendicular BC 3

Ex. 2. There is a roof of which the span AB is 30 feet, and the height CD 12 feet; required the length of a rafter AC or BC? This is only two right-angled triangles of one-continued base, joined together at their perpendicular.