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rectilineal boundaries: and this method, which is purely geometrical, does not require the use of any instrument constructed for the purpose of estimating the magnitudes of angles; its operations being performed, all of them, by means of a rule containing a scale of equal parts, a compass, and some standard measure of linear magnitude.

It is manifest, that any rectilineal plot of ground may be divided into triangles; and that the lengths of the sides of those several triangles may be found by the application of the standard measure, whether it be a foot, a yard, or any other standard measure of length, which has been chosen. Then, by the help of E. 22. 1, and of the scale of equal parts, an exact plan of the ground may be laid down on paper: and, lastly, a rectangle may (E. 45.1. Cor.) be described which shall be equal to the figure representing the plot of ground, and which shall have one of its sides equal to one, or to any given number, of the equal parts of the scale. If, therefore, one of its sides be made equal to one of the equal parts of the scale, it is plain that the number of such parts in the adjacent side, will shew the dimension of the plot of ground in square measure. For, it will indicate how many squares, each having one of the equal parts for its side, are contained in the rectangle that is equal to the plan of the ground; and so many squares, it is evident, each having the standard measure, that was used, for its side, will there be in the plot of ground itself.

If the side of the rectangle, constructed by

means of E. 45. 1, be taken any given multiple of one of the equal parts of the scale, then, if that multiple constitute any other standard measure of length, the dimension of the ground will still be found, as before, by finding how many of those multiples there are in the adjacent side; but it will be of a different denomination.

But if the multiple, assumed for one side of the equal rectangle, be not any standard measure of length, the number of equal parts contained in the adjacent side must be counted; and the product of this latter number, multiplied by the number of equal parts in the assumed side, will shew the dimension of the constructed rectangle, and of the plot of ground, also, which is required to be measured.

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For, although a rectangular surface can only be measured, in a direct manner, by the application of some lesser standard square, to its several parts in succession, yet, since it is evident, even from inspection, that any rectangle may be divided into a number of lesser squares, equal to the product of the numbers which shew how often a side of one of those lesser squares is contained in each of the two adjacent sides of the rectangle, that direct method of measurement is never employed. Hence, no such instrument as a standard square is wanted for the measurement of surfaces, and no such instrument is in use. By means of E. 35. 1, E. 41. 1, and E. 45. 1. the mensuration of parallelograms, triangles, and other rectilineal figures, is

reduced to the mensuration of a rectangle, which is effected, in the manner already described.

Thus a parallelogram is denoted, in square measure, by the product of the number of standard equal parts in its base multiplied by the number of such parts in its altitude: and a triangle is also denoted, by the half of the product of the number of standard equal parts in its base multiplied by the number of such parts in its altitude,

The plan of the piece of ground, required to be measured, having been previously drawn, if the deduction from E. 37. 1, set down in this book, be had recourse to, a triangle will be obtained, that is equal to the rectilineal figure so drawn: and the operations by which this result is arrived at, will be very easily and expeditiously performed, by the help of a parallel ruler, It will then only remain, to let fall, from the vertex of the triangle, a per pendicular on its base; to measure both the perpendicular and the base; and, lastly, to take the half of the product of the resulting numbers.

The mode of planning and measuring which has here been described, is not, it is true, sufficient for all practical purposes. It contains, however, the first principles of the mensuration of plane surfaces. It has the advantage of being very simple and very easy to be understood: and it may, perhaps, afford some degree of satisfaction to the mathematical student, to consider with how small a stock of Geometry he may be enabled to solve a problem of no small utility and importance.

PROP. XLVII.

(LXXI.)

If two triangles have two sides of the one equal to two sides of the other, each to each, and if the angles opposite to either pair of equal sides be each a right angle, the triangles shall be equal, and similar to each other.

(LXXII.)

To find a square which shall be equal to any number of given squares.

(LXXIII.)

Two unequal squares being given, to find a third square, which shall be equal to the excess of the greater of them above the less.

(LXXIV.)

If the side of a square be equal to the diameter of another square, the former square shall be the double of the latter.

(LXXV.)

In any right-angled triangle, the square which is described on the side subtending the right angle, as a diameter, is equal to the squares described upon the other two sides, as diameters.

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If two given straight lines be divided, each into any number of parts, the rectangle contained by the two straight lines, is equal to the rectangles contained by the several parts of the one and the several parts of the other.

PROP. V.

(11.)

If a straight line be divided into two unequal parts, in two different points, the rectangle con

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