Chap. VIII.
GEOMETRY.
The Circle X is inscribed within til epentagon 1 2 3 4 5, whosesides are each the base of an isosceles triangle, the property ofwhich is to have each of its angles at the base double that atthe vertex; and if any two adjoining angles of a regularpolygon be bisected, the intersection of the bisecting lines willbe the common centre of the two circles, the one within, andthe other circumscribing the polygon.
The regular polygons, which have the same number of sides,are similar figures; for their angles are equal, each to each,because they are contained the same number of times in thesame number of right angles, and their sides about the equalangles are to one another in the same ratio: it will also beevident that the area of a regular polygon is equal to halfthe rectangle under its perimeter and apothem, which is a per-pendicular let fall from the centre to the middle of one of thesides; therefore the sum of the sides multiplied by the lengthof the apothem will be the area.
"When the pentagon, hexagon, or heptagon, or either ofthem, are formed into triangles uniting in the centre, thosemust have equal bases and equal altitudes, and consequentlyare equal one to another; and whatever the shape of thepolygon, it must contain as many of these triangles as it hassides, therefore it must be equal to half the rectangle underthe perimeter and apothem.
The Circle Z is inscribed within the hexagon Y.
The Circle B contains inscribed within it a regular heptagon.
The Axis is the straight line, real or imaginary, about whicha body turns, in which sense it is sometimes called the axisof rotation or of oscillation, according to the motion of thebody: it is, however, a straight line about which the parts ofa figure are symmetrically disposed.
The axis in pcritrochio is one of the five mechanical powers,consisting of a peritrochium or wheel fixed immovably to anaxle, so that both turn together round the axis of motion.The power is applied to the circumference, and the weightraised by the rope is wound round the axle : the powergained is the same as that gained by the lever, the longer armof which is equal to the radius of the wheel, and the shorterequal to the radius of the axle.
The Axis of a Circle is a right line, A B, drawn throughthe centre C, so as to divide X into two equal parts: E F isthe chord, and C D the radius.
The Axis of a Globe is a line passing through its centreI, on which it can move as on two pivots.
Axis of a Cylinder is the line 4 7 passing through itsmiddle vertically, and round which the plane 4 56 7 maytraverse to generate the figure.
The axis of a column or frustum of a cone is a straightline drawn through its centre, and in the middle of its solidmass: all weight placed upon it should have regard to itstrue position.
The straight line which divides a conic section symmetricallyis called the axis : in the ellipse and hyperbola the axis cuts thecurve in two points, which are termed the principal vertices ofthe ellipse or hyperbola: and a straight line intercepted be-tween them is called the principal diameter or transverse axis.
The axis of any circle of the sphere is that diameter whichis perpendicular to the plane of the circle; its extremities arecalled the poles.
It is therefore evident that parallel circles have the same axisand poles, for a straight line which is perpendicular to one oftwo parallel planes is perpendicular to the other likewise: itmay also be observed that two parallel circles cannot both ofthem pass through the centre of the sphere, or they cannot bothbe great circles of the sphere.
Axis of revolution may' be considered that straight lineabout which the figure revolves.
Fig. 722.
Fig. 723.
I'ig. 724.
Fig. 725.
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Fig. 727.