Chap. VIII.

GEOMETRY.

749

divided into 180 degrees, and subtended by a diameter,in the middle of which is a notch to mark the centre :when this notch is placed at the angle of any figure, andthe diameter laid along a given straight line, an angle ofany number of degrees may be marked off.

For complicated or large surveys the protractor is inthe form of the entire circle, having its rim connected withthe centre by four radial bars; over the centre is a disc ofglass, on which two lines are drawn, crossing each other atright angles, their centre of intersection denoting that ofthe instrument. Round the centre, and concentric withthe circle, is fitted a collar, which carries two arms; oneof these has a vernier at its extremity adapted to thedivided circle, and the other a milled head, which turns apinion, working in a toothed rack round the exterior edgeof the instrument. The rack and pinion give motion to thearms, each of which carries a fine steel pricker, which ispressed down when the protractor is placed in its requiredposition.

The Quadrant contains 90 degrees, and is divided asshown at ABC. This instrument, when attached to anartificial globe, is made of a thin pliable slip of brass, whichwhen applied to its surface serves as a scale for measuringdistances between points in degrees: it is graduated intominutes and seconds, and at one end is a nutt furnished witha screw, by which it can be attached to the brass meridianof the globe at any point. This point being placed in thezenith, and the quadrant applied to the globe, its zerocoincides with the horizon, and consequently the altitudeof any point along its graduated edge is indicated by thecorresponding division.

A Segment is a less portion, as D E F G. The segmentof a circle is a part of the area comprised between an arcand its chord, and segments of different circles are saidto be similar when their arcs have the same ratio to the cir-cumference of their respective circles, or when they con-tain the same number of degrees.

A Sector is that portion which by two right linespassing through the centre, as K I, LI in the figureV N. A small sector is indicated by the dotted por-tion M, and the greater sector by N.

Such a figure is a portion of the area of a circle, boundedby two radii and the intercepted arc: sectors of differentcircles are said to be similar when the sides or radii in-clude equal angles. The area of a sector is equal to thatof a triangle whose base is equal to the length of the con-tained arc, and altitude equal to the radius of thecircle.

Cylindrical Ring is bounded by two circles, and void inthe middle, as QO: its section, when solid, is that of acircle, the area of which multiplied by the length of itsdiameter gives its contents.

Ovals have both a long and short diameter, which dividethem into four equal parts: the line CE is the conjugateand B D the transverse diameters ; F, F, are the foci.

The Oval or Ellipse somewhat resembles the transversesection of an egg, and a variety of forms are given toit: it is produced by cutting the cone by a plane passingobliquely through its opposite sides. The name of ellipse isderived from one of its properties ascertained, viz. that thesquares of the ordinates are less than the rectangles underthe respective abscissa and the parameter, or differ fromthem in defect.

The ellipse is the curve in which the planets performtheir several revolutions about the sun, and its propertiesenter into every investigation where physical astronomy isconcerned. The curve it forms is defined by means of an

Fig. 711.

4C \ V

Fig.712.

Fig.713.

Fig.714.

Fig. 715.

Fig. 716.