Chav. VIII.
GEOMETRY.
Equal Figures contain equal quantities: the square G,for example, contains as much as the parallelogram D.
Equilateral figures are those which have their sidesequal to each other, and such, when inscribed in circles,are consequently equiangular, but the converse does notalways hold true.
In the square F, by drawing lines across from thedivisions made on the respective sides, it may be madeinto nine equal parts, and if the length of one was set outequal to the divisions in G or D this figure would be inproportion of nine to four when compared with them.
hoperimetrieal figures are such as have equal perimetersor circumferences. Problems which relate to them areextremely difficult of solution, and require a peculiaranalysis ; as, among curves having the same length, todetermine that of which some assigned property is amaximum or a minimum ; for example, among thosehaving the same perimeter, to find that which has thegreatest area ; this constitutes one of the simplest ques-tions of the kind, and the curve to which the property be-longs is proved by elementary geometry to be a circle.
In the figure G, the circumference equals the threesides of that shown at E, as well as the four of F in thepreceding diagram.
Figures are similar when their relative angles areequal : the side L K is to K M as O N is to N P, andthey are said to be similar when their angles and sidesexactly agree, as in the figure Q.
Centres are the points in the middle of a figure : A,for example, in the pentagon B C D E F.
Centre , in geometry and mechanics, has a variety ofsignifications, and is numerously applied. The centre ofa circle or an ellipse is the middle of any diameter ; centreof a curve is the point where two diameters intersect eachother ; and in mechanics we have to treat of the centresof attraction, equilibrium, gravity, oscillation, &c.
The centre of conversion is the point in a body aboutwhich it turns when a force is applied to any part of it,or unequal forces to its different parts. A rod, struck atone of its extremities in the direction perpendicular toits length, will turn it round, but there ■will be one pointin it which remains at rest, or about which the other pointsturn; this is the centre. The point or fulcrum uponwhich a lever turns is its centre of equilibrium.
The Centre of an irregular figure is that marked by astar in the middle of HIKLMNOPQRS, or theremay be found the centre of each moiety of the figure, asG T. and then the star taken as a mean.
The Point V is the centre of the circle XZ910, itbeing equidistant from the circumference in every part.
The Foci or centres of an ellipsis are the points by whichit is described, as 1 and 2.
The focus of the parabola is a point in the axis, havingthis property, that a radius drawn from it to any point inthe curve makes the same angle with the tangent at thatpoint that the tangent makes with the axis. Hence aray of light proceeding from the focus, and reflected bythe curve, proceeds in a direction parallel to the axis; orif parallel rays fall on the concave side of a parabola, theyare reflected into the focus.
In the ellipse the two foci are situated in the greateraxis, at equal distances from the centre, and if from bothfoci straight lines be drawn to the same point in the cir-cumference, the two lines make equal angles with thetangent at that point: a ray of light, therefore, issuingfrom the one focus is reflected by the curve into theother. There is a similar property in the hyperbola, but
G
..
3
rig. G'n.
Fig. 6*4.
Fig. G»7.
F
Fig. 688.
IT
Fig.esy.
o
rig. 690 .
Fig. 691-