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PROBLEM XLIX.

To make a square equal to the sum of any number of squares taken together.

Draw two indefinite lines A m, A n, perpendicular to each other' at the point A. On one of these set off AB the side of one of the given squares, and on the other AC the side of another of them. Join BC, and it will be the side of a square equal to the two together. Then take AD equal to BC, and AE equal to the side of the third given square. So shall DE be the

m

E

side of a square equal to the sum of the three given squares. And so on continually, always setting more sides of the given squares on the line An, and the sides of the successive sums on the other line Am.

NOTE.

And thus any number of any kind of figures may be added together.

PROBEM L.

To construct the lines of the plane scale.

The divisions on the plane scale are of two kinds; one kind having relation merely to right lines, and the other to the circle and its properties. The former are called lines, or scales, of equal parts, and are either simple or diagonal.

By the lines of the plane scale, we here mean the following lines, most of which commonly, and all of them sometimes, are drawn on a Plane Scale.

1. A LINE or SCALE of EQUAL PARTS, marked E. P.

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1. To construct plane diagonal scales.

Draw any line, as A B, of any convenient length. Dir vide it into 11 equal parts.* Complete these into rectangles of a convenient height, by drawing parallel and perpendicular lines. Divide the altitude into 10 equal parts, if it be for a decimal scale for common numbers, or into 12 equal parts, if it be for feet and inches; and through these points of division draw as many parrallel lines, the whole length of the scale. Then divide the length of the first division A C into 10 equal parts, both above and below; and connect these points of division by diagonal lines, and the scale is finished, after being numbered as you please.

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Only 4 parts are here drawn for want of room.

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Of the preceding three forms of scales for two figures the first is a decimal scale, for taking off common numbers consisting of two figures. The other two are duodecimal scales, and serve for feet and inches.

In order to construct the other lines, describe a circumference with any convenient radius, and draw the diameters A B, DE, at right angles to each other; continue BA at pleasure toward F; through D draw D G parrallel to BF; and draw the chords B D, B E, A D, A E. Circumscribe the circle with the square HMN, whose sides HM, MN, shall be parallel to AB, ED.

2. To construct the line of chords.

Divide the arc A D into 90 equal parts; mark the 10th divisions with the figures 10, 20, 30, 40, 50, 60, 70, 80, 90; on D, as a centre, with the compasses, transfer the several divisions of the quadrantal arc to the chord A D, which, marked with the figures corresponding, will be a line of chords.

NOTE. In the construction of this and the following scales, only the primary divisions are drawn; the intermediate ones are omitted, that the figure may not appear too much crowded.

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3. To construct the line of rhumbs.*

Divide the arc B E into 8 equal parts, which mark with the figures 1, 2, 3, 4, 5, 6, 7, 8; and divide each of those parts into quarters; on B, as a centre, transfer the divisions of the arc to the chord BE, which, marked with the corresponding figures, will be a line of rhumbs.

4. To construct the line of sines.t

Through each of the divisions of the arc AD draw right lines parallel to the radius AC; and CD will be divided into a line of sines, which are to be numbered from C to D for the right sines; and from D to C for the versed sines. The versed sines may be continued to 180 degrees, by laying the divisions of the radius CD from C to E.

5. To construct the line of tangents.

A rule on C, and the several divisions of the arc AD, will intersect the line DG, which will become a line of tangents, and is to be figured from D to G with 10, 20, 30, 40, &c.

6. To construct the line of secants.§

The distances from the centre C to the divisions on the line of tangents, being transferred to the line CF from the

* Rhumbs here are chords, answering to the points of the Mariners' Compass, which are 32 in the circle.

+ The sine of an arc is a right line, drawn from one end of an arc perpendicular to the radius, drawn to the other end. The versed sine is the part of the radius, included between the arc and its sine.

The tangent of an arc is a right line, touching that arc at one end, and terminated by a secant, drawn through the other end.

§ The secant of an arc is a right line drawn from the centre through one end of the arc, and terminated by the tangent, drawn from the other end.

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