[The drawings on page 91 are to half scale.]

(123) To find6.

Find the mean proportional between 3" and 2". Measure the length of this line we find it about 2·45′′.

Therefore we take √6=2'45.

(124) To find √5.

I,

We can take √5 as the mean proportional between 5 and or else as in the figure make a right-angled triangle of sides 2" and 1", then the area of the square on the hypotenuse = 22+ 12 in square inches=5 square inches. Therefore the length of the hypotenuse √5"=2′23′′.

(125) To find /8.

√8=the mean proportional between 4 and 2, otherwise 8 is the difference between 9 and 1. Make a right-angled triangle hypotenuse 3" and one side 1", then the remaining side measures √8"=2.83".

(126) Another method of finding square root of a number is to treat it as the sum of several squares, and repeatedly use Euclid I. 47. For example to find the square root 21

21=16+4+I.

Make a right-angled triangle of sides 4′′ and 2", at one end of the hypotenuse draw a perpendicular 1" long. Join the end of this perpendicular to the other end of the hypotenuse.

Then the square on this line=16+4+1=21 square inches, therefore the line itself√21" =4′58.

IF a right-angled triangle revolve about one of the sides containing the right angle, the solid thus formed is called a right cone. This is only half of a complete cone as treated of in the theory of conic sections, another equal cone is placed with its vertex opposite to that of the first, and both cones are supposed so large as to extend without limit. Another method of describing such a complete double cone is to take a straight line intersecting a given fixed straight line called the axis at a given point V, and to suppose it to revolve round the axis always making a fixed angle with it, and passing through the given point V on the axis.

This cone when cut by a plane gives what is known as a conic section.

If the plane passes through V the section is two straight lines. The section made by a plane cutting the axis at right angles is circular (fig. 1).

An oblique section, not parallel to the edge of the cone, but cutting one part of it only is called an ellipse (fig. 2).

An oblique section cutting both parts of the cone is a hyperbola (fig. 3).

A section parallel to an edge of the cone is a parabola (fig. 4).

It is found that all these sections can be described by a certain law.

Take a fixed point S called the focus, and a fixed straight line MX called the directrix, then if P be a point on the curve, and PM be drawn perpendicular to the directrix the ratio of SP to PM is the same for all points on the curve. If this ratio is less than unity the curve is an ellipse; if equal to unity, a parabola; and if greater than unity, a hyperbola.

The ellipse and hyperbola have each two foci and two directrices, and there is a point midway between the foci called the centre.

The straight line joining the foci is called the major axis, or sometimes more shortly the axis.

In an ellipse the straight line through the centre at right angles to the major axis is called the minor axis. The major

axis is the greatest chord through the centre, and the minor axis the least.

(127) To describe an ellipse when the major axis and minor axis are known.

Let ACA' be the major axis of the ellipse and CB half the minor axis, C being the centre.

In an ellipse it is found that if any point on the curve be joined to the two foci the sum of these distances is constant, and is equal to AA'.

Take centre B, and distance equal to CA the circle must cut the axis AA' at F1 and F, the two foci.

2

From centres F1 and F, and distances equal to AI, and A'ı, make arcs to intersect one another, the points thus found are on the curve for the sum of their distances from F1 and F, is equal to AA'. Similarly making arcs from F1 and F2 as centres at distances respectively equal to A2, A'2; and A3, A'3 we find more points on the curve. I, 2, 3 are any convenient points

on the axis.

(128) To describe an ellipse by means of a trammel, the lengths of the major and minor axis being known.

Let PNM be the straight edge of a piece of paper, or of a card, on which are measured PN=CB and PM CA. Move the trammel about in any way always keeping M and N upon the minor and major axis, then P traces out the ellipse.

From this construction another is evident. Make circles centre C, and distances respectively CA and CB. Draw any radius CRQ; from R draw a parallel to CA, and from Q a parallel to CB meeting it at P. The point P is on the curve.

By taking different positions of the radius CRQ we can find any number of points on the curve.