764

THEORY AND PRACTICE OF ENGINEERING.

Book II

To druw a perpendicular at the end of a line : place onefoot of the compasses in the point A, and the other in thepoint C; then from C as a centre, and with the radius C A,describe the arc D AE ; from the point F, where the circlecuts the right line A B, draw a line through the point Ctill it cuts DAE in G : a right line drawn from the pointA to the point G will be perpendicular to the line A B.When the perpendicular is to be raised in the middle of aline, or let fall from I and K, equally distant from H,strike curves intersecting at N and O, and a line drawnthrough these points of intersection will be the right lineor perpendicular required

Should the point not be in the middle, as is the case atX, on the line YZ, the points 1 and 2 must be set out atequal distances from it, and then arcs of intersection struck.

Or if it should be required to drop a perpendicular tothe line Q R , from a point situated at P, then from P as acentre strike the curve ST, and where it crosses the lineQ R, with the same radius form the intersecting curves atV, and draw the line P V, which will be the perpendicularrequired.

Scale is a term applied to a mathematical instrument,containing an assemblage of lines and figures, by means ofwhich certain proportional quantities can be taken ; inmensuration it signifies a line or rule of a definite length,divided into a given number of equal parts, and is used formeasuring other linear magnitudes. An ordinary scale isusually set out by stepping the compasses along a givenline, very lightly from one end to the other; the distancebetween the points being previously determined on. Some-times it is required to find out the scale by dividing thegiven line into a certain number of parts, and considerablenicety is necessary accurately to perform this :

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Fig. 798.

practice alone can effect it.

To construct a scale, considerable care is requisite that all the portions or divisions maybe set out equally : to form one of 80 feet forinstance, it is only required to step the com-passes so many times along the line; or after 10feet have been set out at one end, to take itsextent, and then set out the remaining 80 feet.

But when it is required to make a scale of 140feet upon a given line, as G H, then it is betterfrom G as a centre, first to strike the arc IIS,and from H as a centre the arc G O, and thendrawing the lines G B and H E, at the sameangle from the line G II; these may be sub-divided into the same number of equal parts ;then, by uniting the points K Y, L X, M V,

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Fig. 799.

NT, OS, PR, they will cross the line G II, and leave all the divisions equal.

When it is required to divide a long line into a consi-derable number of equal parts, it is best, if the numberwill admit of it, to resolve it into two factors, and first todivide the line into the number of equal parts indicated bythe small factor, then subdivide each of these parts into thenumber expressed by the larger ; thus, in dividing a line into150 equal parts, it is better to divide it first into fifties, andthen into tens; it is advisable always, w'hen the number ofdivisions is considerable, to adopt a small number at thecommencement, and continue to subdivide them into therequired portions and when all is performed, to verify themover the whole length of line.

To construct two scales on two lines of unequal length,as suppose it is required to divide each of the lines 1VI Nand Q R into ten equal parts; take, or step along the lineA K, ten parts all equal, but of any length ; then from thepoints A and K at the extremity of the line, strike arcswhich intersect at L ; from this point draw lines through C,

D, E, F, G, II, and S; then from the point L, with Mand N as radii, strike arcs which will cut at O and P, and with O and R as radii, other

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Fig. 800.