Chai*. VIII.

GEOMETRY.

735

than those on the outside; hence some have supposedthat such an arrangement was intended to produce abetter effect.

'Die line O I\ in the triangle O Q R, is a perpen-dicular, because it falls at right angles with the base;OPQ and OPR being both right angles. In prac-tice, a perpendicular or right angle is set out by thenumbers 3, 4, and 5; for example, if Q P is made 3feet, P0 4 feet, and QO 5 feet, OP will be perpen-dicular ; or if O P is made 3000 feet, P II 4000, and R O5000, the result will be the same.

ST is perpendicular to the side T, because it is atright angles with it; hence the radius of a polygon,when it falls on the middle of one of its sides, is alsoits perpendicular.

An inclined line, as VX, is that which is neither pa-rallel nor perpendicular with another, as that of Y Z.

Parallel lines are those like A II, C D, and E E, whichmay be drawn to any length, and yet never approach.All lines which preserve an equal distance from eachother are called parallel lines.

The two curved lines ST, VX are parallel, althoughthey are not of an equal length.

This subject has ever beeu considered difficult of ex-planation, and much has been written upon it. Somewriters have exerted themselves to demonstrate thattwo parallel lines, when they meet a third, are equallyinclined to it, or make the alternate angles with it equal.Euclid has shown that if a straight line meet twostraight lines, so as to make the interior angles on thesame side of it less than two right angles, these straightlines, being continually produced, will at length meet onthe side on which the angles are which are less than tworight angles; but this is not so evident, and manycelebrated geometricians have attempted to make ourauthor more clear upon this point. Some have as-serted “ that straight lines are parallel which preservealways the same distance from each other; ” but the

Fig. 615.

Fig. 616.

Fig. 617-

Fig. 618.

correct definition would be, that “ two straight lines are parallel when there are two pointsin the one from which the perpendicular drawn to the other, and on the same side of it, areequal.” The difficulty in such a statement consists inshowing that all the perpendiculars drawn from the oneof these lines to the other are equal.

Parallel lines by some are said to be those whichmake equal angles with a third line towards the sameparts, or make the exterior angle equal to the interiorand opposite; this definition requires only that it shouldbe proved that all the straight lines which are equallyinclined to one given straight line are equally inclined toall the other straight lines which fall upon them.

Ordinates are lines in a parabola, RPQ, which aredrawn parallel w ith the base, as G H, I K, L M, N O,and are derived from the Latin ordine. A straight linedrawn from any point in a curve perpendicularly toanother straight line, which is called the absciss, is anordinate. The absciss and ordinate together are calledthe co-ordinates of the point. The situation of a pointin a plane is determined, wdien its distances from twostraight lines in the same plane are known; and when aseries of points are so situated in respect of each otherthat the co-ordinates of each have the same mathematicalrelation, they form a curve, the nature of w hich is ex-pressed by the relation of the co-ordinates.

Horizontal line , or apparent line of level, is that whichcuts or touches at right angles a line supposed topass through the centre of the earth; the line ab,resting on the perpendicular cd, is horizontal, and all Fig. 620.lines parallel with this are deemed horizontal.

Fig. 619.