Chap. VIII.

GEOMETRY.

737

Plans always have a scale attached to them; in that ofthe fort F its length and divisions are made with refer-ence to one of its sides, as H or I, which may besupposed either 50 or 500 feet in extent. In the figuresIW, N, the scales K and L may be made to agree withthe length of either of its sides, and by reducing orenlarging the scale, the figures to be traced may be madelarger or smaller, care being taken to set out the anglescorrectly.

Working drawings are usually upon a large scale, asthey are intended for the artizans to set out their respectivelabours; they should be of a size to express accurately theparts of the machine or other object to be executed : an out-line is generally sufficient for the purpose; sometimes aquarter scale is required, where the parts are intricate orsmall, at other times a 12th or 24th part of the real size:for fortifications or earth works the plans are made upona scale of so many chains or feet to an inch, and whenlaid down accurately, they may be diminished or in-creased by adopting different scales. A very easy andsimple method of performing this is by covering thedesigns with squares or lines parallel and at right angleswith each other; when the plan is irregular, as thatshown at M and N, it is necessary to bound them by asquare, or, taking the longest sides for a base line, toconstruct one upon it, and then set out accurately all theangles, or, according to the degrees they measure, a dia-gonal line may be drawn from the points of the twoextreme angles, and then parallelograms and squares maybe constructed on each side of it to embrace the otherportions of the plan, and afterwards the figure whichbounds the whole may be reduced or enlarged; the sub-ordinate parts will all have the same relative value toeach other, and one will be a fac-simile of the other,though on a different scale.

Wavy or curved Lines rise and fall, as indicated atA C E D F B, and when a circle is struck from the centreV, the line N is a curve.

We may suppose the surface of the earth to present awavy or curved form, which it is required to ascertain;this is performed, as we shall hereafter see, by means of aninstrument called the level; the wavy line is measuredfrom another, which is set out by means of upright stavesand rods with great accuracy, and where the inequalitiesare numerous, it is requisite to make the measurementsfrequently; when the eye of an observer is placedon a level with the plane which presents an undu-latory surface, it is difficult to measure the rise and fallwithout establishing a number of fixed points, or placingin each hollow a perpendicular staff that can be seenfrom the station point; if these rods were all so cutthat the eye could see their tops in one continued line, thelevel might be established, or the inequalities measuredsufficiently for ordinary purposes, but where great nicetyis required, instruments carefully adjusted are necessary,as the eye is easily deceived in long distances.

Spherical Lines are those which may be traced on theglobe L.

Spiral Lines proceed with a regular and gradual en-largement of distance round the volute P to the ter-mination at O. This name is given to all those curveswhich have the peculiar property of receding fromthe centre while they continue to revolve about it;there are many varieties, as the equable, the hyperbolic,the logarithmic, spiral, and others: the first was ablytreated upon by Archimedes , who also showed the rulesby which it could be generated.

3 B

Fig. 627.

Scale K

Fig. 628.

Fig. 629.

Fig. 630.

Fig. 631.

Fig. 632.