« PreviousContinue »
coincide in the projection of P, PQ then becomes the tangent to the curve at P, and its projection pq becomes the tangent to the projection of the curve at the projection of P.
9. The object of the method of projections is to extend our knowledge of curves from those which we already know, or can easily investigate, to others into which they project. Thus the properties of the ellipse can be deduced from those of the circle, and the rectangular hyperbola helps us to ascertain the properties of the hyperbola of unequal axes. The preceding propositions serve to connect the diameters and other lines, and the areas, of the known curve with those of its projection.
1. If the inclination of two planes be the third of two right angles, and a line of 4 yards in one of the planes be inclined to the line of intersection at half the above angle, find the area of the square on the projected line.
2. In Art. 6 when D falls without AB, find the inclination of the planes in order that, in any given case, the angle ACB may not be altered by projection.
3. A triangle is projected on to a plane whose line of intersection with its own plane coincides with the base ; it is projected back again on to its original plane, and the inclination of the plane is such that after these two projections its vertical angle is unaltered. Shew that this inclination gives the greatest possible increase to the vertical angle at the first projection.
4. Find the altitude of the sun when the greatest shadow of the side of a square on a horizontal plane equals its diagonal.
5. In what position must a cube be placed so that its shadow on a plane that receives the sun's rays directly may be the greatest possible?
On the Cone and Sphere.
1. DEFINITIONS. If through o the centre of a circle ABC we draw a straight line OV at right angles to its plane, then the surface traced out by a straight line which
passes through any point V of this line and the successive points of the circle ABC is called a Cone; or, more distinctively, a
We repeat here Euclid's definition of a sphere:
A sphere is a solid figure described by the revolution of a semicircle about its diameter which remains fixed.
2. If VP be any position of the generating line of a cone, the angle OVP is the vertical angle of a right-angled triangle of which the height O V and the base OP are constant; therefore OVP is constant, and the angle between the
; axis and the generating line in all its positions is invariable. Also if any plane at right angles to the axis cuts the cone in the curve abc, abc will be a circle. Let o and p be the points in which the plane cuts the axis and the generating line VP;
Then op is the base of a right-angled triangle of which the height o V and the vertical angle o Vp are invariable for all positions of VP; hence op is constant, and the section abc is a circle, with centre o.
Every plane containing the axis cuts the cone in two right lines inclined to each other at a constant angle.
3. From the definition of a sphere it is evident that every point in the sphere is equidistant from the centre of the generating circle.
Every section of a sphere by a plane is a circle.
Let P be any point of the section ABC of a sphere whose centre is O made by a plane. Draw ON at right angles to the cutting plane, and join OP, NP.
Then for every point in the section the height ON and the hypotenuse of the right-angled triangle ONP are constant, and therefore the base NP is constant, and ABC is a circle of which N is the centre.
4. DEFINITION. A straight line is said to touch a sphere when it touches the circle in which the plane through the line and the centre of the sphere cuts the sphere.
It is evident that if a plane be drawn at right angles to the diameter of a sphere through its extremity, every line drawn in this plane through the extremity of the diameter
will touch the sphere, and therefore the plane is said to touch the sphere.
5. All the lines drawn from one point to touch a sphere are equal.
Let TP (fig. $ 3) be any line drawn from the point T to touch the sphere ABCD whose centre is O in the point P. Join OP. Then TP touches a circle of which OP is a radius, therefore OPT is a right angle. Join OT. Then TP is the base of a right-angled triangle of which the height OP and the hypotenuse OT are constant.
Therefore all the lines drawn from T to touch the sphere are equal. Also the angle OTP is constant, and the equal tangents will therefore generate a cone of which T is the vertex and OT the axis. Also the points common to the sphere and cone lie on a circle whose plane is at right angles to OT.
On the. Ellipse.
1. DEFINITION. Let AVA' be a cone whose axis VO is in the plane of the paper, and VA, VA' its generating lines in that plane. And let the cone be cut by a plane APA at right angles to the plane of the paper and intersecting it in the line AA such that A, A' are points in the generating line on opposite sides of the axis VO; then the section APA’ is called an Ellipse.
It is manifest that the ellipse will be a closed curve, and divided into equal parts by the line AA': every line in the plane of the section drawn at right angles to AA will meet the curve in two points on opposite sides of AA', and at equal distances from it.
AA' is called the axis major of the ellipse, and a line through C its middle point in the plane of the curve at right angles to AA' limited by the surface of the cone is called the axis minor.