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(3.) Construct the points of the mariner's compass by drawing two lines at right angles to one another, and dividing the four right angles thus formed, each into eight equal parts.

(4.) A ladder being 20 feet long, construct a figure, and find by measurement how near a wall the foot of the ladder must be placed, that its top may just reach 17 feet up the wall. Ans. 10 feet 6 inches.

74. PROBLEM 5.-To let fall a perpendicular on a straight line from a given point without it.

METHOD I.-Open the points of the compasses to a distance somewhat greater than that of the point from the line, and from the given point A, Fig. 38, as centre, strike arcs cutting the line in the points B and C; again from B and C, as centres, with radius equal to about three-fourths of the distance between B and C, strike arcs intersecting in D; the straight line through A and D shall be the perpendicular required.

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Reason.-Identical with that given in Art. 73, Method I. METHOD II.-When the paint is so situated that the perpendicular would fall near the end of the line, and the line cannot conveniently be produced.

From the given point A, Fig. 39, draw the straight line A B, oblique to the given line BD; bisect A B in E; from E, as centre, with radius E A, or E B, strike an arc, cutting the given line in D; then the straight line passing through A and D shall be the perpendicular required.

Reason.-Identical with that in Art. 73, Method III., since

D is a point in the circumference of the semicircle of which A B is the diameter.

EXERCISES.

(1.) Construct any triangle, and let fall perpendiculars on the sides from the opposite angles. If accurately constructed, the three perpendiculars will intersect in one point.

(2.) A person walks 50 yards up a slope inclined at 15° to the horizon; construct a figure, and find by measurement, through what vertical distance he has ascended. Ans. 12.94 yds.

The problem of constructing right angles is of such constant occurrence in all Arts and Manufactures, that it becomes of great importance to possess the means of performing it with the greatest facility and precision. In the preparation of plans and designs the desired object is accomplished by the drawing-board and T-square, or by the straight-edge and setsquare.

*

75. PROBLEM 6.-To draw lines at right angles, by the straightedge and set-square.

The set-square is a thin piece of wood or other material cut into the form of a right-angled triangle.

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Fig. 40.

Let A B, Fig. 40, be a given line to which it is desired to draw a perpendicular through a given point, either in the line or without it.

METHOD I.-Place one of the sides, C D, of the set-square against the given line; place the straight-edge against the

* See T-square and Set-square in "Mathematical Instruments," by J. F. Heather, M.A., Weale's Series.

hypothenuse, CE; holding the straight-edge firm, slide the setsquare C D E along it into a new position C' D' E', until the side D' E' passes through the given point; then a line drawn along D' E' will be the perpendicular required.

METHOD II.-Place one of the sides CD, Fig. 41, of the

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set-square against the given line, and the straight-edge against the hypothenuse as before; holding the straight-edge firm, turn the triangle over, and place it in the new position C' D'E', with the other side D'E' against the straight-edge, and the hypothenuse C E', passing through the given point; then a line drawn along C'E' shall be the perpendicular required.

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76. PROBLEM 7.-To test the accuracy of a supposed right angle B A C.

From centre A, Fig. 42, strike a small arc; from a point in this arc, and with the same radius, strike arcs intersecting the sides in B and C; then if the points BDC are in one straight line, B A C is a correct right angle; but if not, draw the straight line B D E,

Fig. 42.

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through the points B and D, and the straight line C D F, through the points C and D; then the angle B D F, or C DE," is twice the error, that is twice the angle by which B A C either exceeds or falls short of a right angle.

Suppose B C joined, then if D falls within the triangle C B A, the error is one of defect, and if it falls without the triangle the error is one of excess.

77. PROBLEM 8.-To test the accuracy of a set-square. Place one side DE, Fig. 43, of the set-square in contact with a straight line A B, and draw a line along the other side C D; turn the triangle over into the new position C'D E', place the same side D E' in contact with A B, and draw a line along C'D; then if the lines C D and CD coincide, the setsquare is correct; but if not, the angle C D C measures twice the error of the supposed right angle C D E.

78. PROBLEM 9.-At a given point in a given straight line,

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to construct an angle equal to a given rectilineal angle.

METHOD I.-Let A, Fig. 44, be the given angle, BC the given line, and B the given point; from the angular point A, as centre, with any convenient radius, strike arcs cutting the sides

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of the angle in D, and E; from the centre B, with the same radius, strike an arc CF; take the distance from D to E, as radius, * The angles B D F and C D E are equal (Theor. 6).

and from C, as centre, strike an arc, cutting CF in F; through B and F draw the straight line B F; then shall the angle FBC be equal to the angle at A, as required.

Reason. Suppose straight lines to be drawn from D to E, and from C to F, then the triangles thus formed will have their sides equal each to each, and will, therefore (Theor. 2), be equal in all respects.

METHOD II.-Take a piece of paper, and cut one edge, PQ, Fig. 45, straight; place the edge P Q in contact with the side AD of the given angle, and mark upon the paper the points, A and G, where the other side cuts its edges; again, place

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the piece of paper with its straight edge P Q, in contact with the given line B C, and the point marked in this edge in coincidence with the given point, B; mark at H a point in coincidence with the mark on the other edge of the piece of paper; remove the piece of paper, and draw a straight line through the points B and H; then shall the angle F B C be equal to the given angle at A, as required.

Reason.-Identical with that given for the preceding method, the triangle H B Q being an exact copy of the triangle G A Q. This method indicates the means of constructing protractors, to set off angles, given by their measurements, or to measure angles already laid down.

79. PROBLEM 10.-To construct a plan of a protractor of the ordinary form, and graduate it at every 15°.

Construct a rectangle A B C D, Fig. 46, six inches long, and 1 inches broad; bisect A B at right angles by the straight line PQ, and mark the point where this line cuts the side CD of the rectangle 90°; on P B and PQ describe equilateral triangles PSB and PR Q, and mark the points where PS and PR cut

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