Chap. VIII.

GEOMETRY.

767

to the fourth, as the difference of the first and second, is to the difference of the third andfourth; 9, 12, 16, 24, are such ; for 9 : 24 : : 3 : 8.

To trace on the ground a straight line equal in length to a circle, it is only necessaryto divide the diameter A C of the circle into eight equal parts, and prolonging the lineto F, upon which six of the divisions are to be set out.

Through the point C, at right angles, draw G H, and from F as a centre, with the radiusFA, describe I A K, when A K will be the length sought.

When it is required to draw a straight line equal to a portion of the curve, as that of L,it may be done by dividing it into small portions as shown, set out from M, and transferringthem to a straight line, as OP: by this means, it may be performed with sufficient accu-racy for ordinary purposes.

To draw either on the ground or on paper angles of any kind, a protractor is some-times made use of: this is a semicircular limb of metal, divided into 180°, and subtendedby a diameter, in the middle of which is a line and dot to mark the centre of the circle, towhich all the divisions of the degrees radiate.

By this simple instrument an angle of any number of degrees may be set out by layingits straight edge on a line previously ruled upon a sheet of paper, and then marking off theangle required; lines then drawn from the dot or centre will fully express it.

When this instrument is made use of for surveying on a large scale, it is formed into anentire circle, with four arms radiating from the centre. A circular disk of glass is placedover a hole in the centre, on which two lines are drawn,crossing each other at right angles : round this is a smallcircular ring of brass, which carries two arms, one ofwhich has attached to it a vernier, which moves over theouter circle, divided into degrees, and the other a head,which can be moved round, and made to turn a smallpinion that works in a toothed rack round the outer edgeof the protractor. The arms are moved by this rackand pinion entirely round the whole circumference, andthe vernier can be set to any particular angle. Thearms are made to extend over the outer rim, and eachcarries a fine point, which is pressed down when the in-strument is to be used, and these make a small dot or holein the paper.

To draw an angle of 30° on the line B C, for instance atthe point B, the demicircle has its dot or centre laid at B;the number of degrees are counted off from D to E, andthen, moving the protractor, a line is drawn from Bthrough E, and A B C is the angle required.

Or from the point G, on the right line I H, it is re-quired to set out an angle of 90°. Place the centre of theprotractor at the point G, and its diameter I K along theline IH; then counting off the number of degrees, andmark the point, as at L, remove the instrument, and drawthe line G F through L, and H G F will be the anglerequired.

The same may be done by placing the protractor at N,and its diameter along the line OP, and counting fromthis line O P on the circumference of the protractor 144°,as from R to S, draw then the right line NM through S,and the obtuse angle of 144° is obtained.

As this is a right angle it is only necessary to raise aline perpendicular to another to obtain it: the side ofa square makes an angle of 90°, and its diagonal oneof 45°, or the half.

The division of an angle into any number of equalparts is termed its angular section; and its trisectionrequires the aid of solid geometry, being equivalent tothe solution of a cubic equation; the general division ofan angle into any proposed number of equal parts, is aproblem which mathematicians have not yet solved.

To draw on a right line an angle equal to any givenangle, as on the line A B, from the point A, to draw anangle equal to the given angle CDE. Place one footof the compasses on the point D, and strike the arcF G : with the same radius, from the point A, strike a similar arc from II to F ; thentake the height from F to G, and transfer it from H to K: draw the line A L throughK, and you have what is required.

Fifi. 809.

1 O

K

Fig. 810.

M

X’a

tt N

Fig. 811.

c x

/

E K

.>

Fig. 812.

X

n ;

Fig. 813.