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PROBLEM XLVI

To make a Square equal to a given Rectangle ABCD.

Produce one side A B, till BE be equal to the other side

B C. Bisect AE in o; on which as a centre, with radius Ao, describe a semicircle, and produce BC to meet it at F.

E

F

B

R

On B F make the square BFGH, and it will be equal to the rectangle ABCD, as required.

PROBLEM XLVII.

To make a Square equal to two given Squares P and Q

Set two sides AB, BC, of the given squares perpendicular to each other. Join their extremities AC; so shall the square R, constructed on A C. be equal to the two P and Q taken together.

P

B

R

NOTE. Circles, or any other similar figures, are added in the same manner. For if A B and B C be the diameters of two circles, AC equal to the other two. sides of any two similar figures, then A C will be the like side of another similar figure equal to the two former, and upon which the third figure may be constructed, by Problem xlii.

will be the diameter of a circle And if AB and BC be the like

PROBLEM

PROBLEM XLVIII

To make a Square equal to the Difference of two given

Squares P, R.

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To make a Square equal to the sum of any number of Squares taken together.

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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 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 A m.

NOTE.

NOTE. And thus any number of any kind of figure may be added together.

PROBLEM 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 follow ing 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. Di vide it into 11 equal parts. Complete these into rectan gles 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 parallel lines, the whole length of the scale. Then divide the length of the first division AC 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.

Of

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 DG parallel to BF; and draw the chords BD, BE, AD, AE. 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 AD 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 AD, 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.

3. To construct the Line of Rhumbs..

Divide the arc BE into 8 equal parts, which mark with the figures 1, 2, 3, 4, 5, 6, 7, 8; and divide each of

those

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