GEOMETRY.

773

Method of drawing Polygons on Right Lines by means

of Degrees- For the Pentagon divide 360 degrees by 5, and

the quotient gives 72 degrees for the angle in the centreof the pentagon ; then subtract these 72 degrees from 180,the value of the three angles of a triangle, there remains 108degrees for the angle of the figure, the half of which, 54,will be the angle of the demi-polygon. Then make at thepoint A on the given line A B the angle B A C of 54 degreesby placing the centre of a protractor on the point A, andmaking its diameter coincide with A B : count 54 degrees onthe circumference, which will terminate at D, and having re-moved the protractor, draw from the point A through thepoint D a line ADC, which will form an angle BAC of54 degrees. The same process must be repeated at the pointB, the other extremity of the given line A B, in order todraw the line B E, which will cut A C in F and form a centre,from which, with the radius FA, a circumference may be de-scribed, on which, if we apply the line A B five times, weshall form the pentagon A G H I B, and by a similar methodall the other polygons.

Method of describing Circles, Ovals, Parabolas, SpiralLines, etc. — The following table of factors is sometimes madeuse of to construct a circle geometrically: suppose c to be thecircumference of a circle, the diameter of which is unity, andx the side of a square equal in area to it, and let a be the area of

the circle : then c = 3*14159 ; 2c = 6-28319 ; -|=sl*57080;

___ =0’2618012360

— - = 0*00873 ;360

i =0-31831 j ? = 0*63662 ;

-1S~ = 114-59156 ; x = 0-88623; and a = 0-78540.

c

To describe a Circle _Place one foot of the compasses in the

place where the centre is to be, and with the radius requireddescribe the circle: to set out a circle on the ground, fix apiquet or staff* in the place where the centre is to be, as at A,and tie the end of a cord to the piquet by a slip knot, so as toturn round easily. Then to the other end of this cord attacha staff, B, at the distance required for the radius, and thendescribe the circle by stretching the cord A B equally, andbeing careful to keep the staff B perfectly perpendicular, andnot to allow the cord to touch the ground.

To divide a Circle. — Divide the diameter A B into as manyparts as it is required ; then from the point A, with the lengthof the diameter, describe B C, and from the point B, with thesame radius, describe AC . From their point of intersection,through the division marked 2, draw the right line C D untilit touches the circle: the line AD will be one-fifth of thecircumference, the diameter having been previously dividedinto five equal parts. By this means any circumference maybe divided, so as to inscribe any regular polygon, remem-bering that we must always draw the line C D through thesecond point of the division of the diameter A B, whateverthe diameter of the circle may be divided into: to form anequilateral triangle it must be drawn through both points ofdivision.

To draw a Circle through three given Points -It is necessary

that the three given points should not be in the same line.Let it be required to draw a circle through A.B, and C. Ifon the ground, take the distance between A B with a cord,or if on paper with the compasses, and describe above andbelow, from the points A C , the arcs D E and II I, andwith the same radius from the point B intersect them: thendraw lines through their points of intersection, and where theycross will be the centre, from which a circle may be described,that will pass through the three given points.

This method of describing circumferences of circles is of

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A

Fig. 837.

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Fig 838.

Fig. 839.

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Fig. 840.

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