REMARK.

In the application of this theorem, it must be carefully remembered that the cosine of an angle is positive when that angle is in either the first or fourth quadrants, and negative when it is in the second or third quadrants. For a demonstration of this beautiful theorem, see also, Gibson, by Trotter.

N. B. When the sum of the opposite angles is 180°, that is, when the trapezium can be inscribed in a circle, the above rule is simply: from half the sum of the given sides, subtract each side severally; multiply the four remainders continually together, and extract the square root, gives the

་

area.

EXAMPLE.

"One morning in May I went to survey,
As soon as bright Sol I espied;

I measured round a four cornered ground,
In the margin see the length of each side;
The angle at B, together with D,
An hundred and fifty degrees;

The meadow's content is all that I want,
Assist me kind youths, if you please."

Whence (s-A B)X(s-B C)X(s-C D)X(8–D A)= 32.4X16.8×19.2X22.4X6.4-46242.2016=-46242.

2016

And A B. B C. C D. D A.X(1+cos. 150°)

2

That is 15.60X13.20×10.00×26.00

2

-X0.1339746=3586.4464

Difference 42655.7552

The square root of 42655.7552 is 206, 5327=

A. R. P.

area in square four pole chains, or 20. 2. 24,55232. N. B. This problem is taken from Deighan's Arithmetic, vol. second, page 148, and the answer

A. R. P.

there given is 21. 2. 00,64, which is obtained by taking the trapezium to be inscribed in a circle, which is not the case.

When the opposite angles of a quadralateral are equal to two right angles, a circle can be described about it. The rule to find the area, then, is: multiply the half sum, and four remainders continually together, and extract the square root, for in that case 1+cos.(A+B)=0.

21st. To find the area of a circle having the diameter given. Rule-Square the diameter, and multiply by .7854, and you have the area.

22d. To find the area of an ellipsis. RuleMultiply the transverse and conjugate diameters together, and that product by .7854, and you have the area.

23d. To find the area of a parabola. RuleMultiply the height by the breadth, and take twothirds of the product; you have the area.

24th. To find the area of a segment of a parabola. Rule-Multiply the base of the segment by the altitude thereof, and two-thirds of the product gives the area.

25th. To find the area of a field or lot, which is found to be the frustum or zone of a parabola, included by two parallel right lines, and the intercepted curves of the parabola. Rule-Add the two

parallel ends, divide the square of either of these ends by this sum, add the quotient to the other end, multiply this sum by the altitude of the frustum or distance of the ends, take two-thirds of the product, and it gives the area.

TRIGONOMETRICAL SURVEYING.

26th. It was not my intention to say any thing concerning this branch of surveying, as it is too extensive a subject for this small work; but as some young readers may not have met with any thing on that subject, I will present them with an outline of how that grand operation is conducted.

When an entire country, or part of a country, containing one or more counties is to be surveyed, it is done by triangulation, and the application of the rule given in the 12th section of this work. A line of some miles in length is measured and remeasured in order to prove its accuracy, on some plane or heath which is nearly level, first having been traced by a transit instrument, and poles placed in an exact straight line, to guide the measurers, as A B in the annexed figure, which is assumed as the base of the operations. A number of hills and elevated spots are selected, on which signals can be placed, suitably distant and visible

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