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A line drawn from the centre of a circle through one extremity of an arc until intercepted by a tangent drawn from the other extremity is called the secant of that arc.

423. Can you make the secant of the arc of 60°?

Like the word tangent, secant has two meanings, one as applied to a circle, and the other as applied to an arc.

Can you on one line, and beginning at one point in that line, place the secants of all the arcs from 10 to 80°? In other words

424. Can you make a line of secants?

425. Make and measure a few angles by the line of secants.

426. Say which you consider the most convenient for plotting and for measuring angles, the line of chords, of tangents, of sines, of versed sines, or of secants.

427. Calculate the length of a link of a landchain, and give on paper an exact drawing of such a link.

You have determined how far up the perpendicular of an equilateral triangle the centre is.

428. What ratio have the two parts of an equilateral triangle which are made by a line drawn through the centre of the triangle parallel to the base?

429. Suppose the side of a hexagon to be 1, it is required to determine the sides of a rectangle that shall exactly inclose it, and to find the area of the hexagon and the area of the rectangle, and the ratio between them.

430. Make a figure that shall be equal to one formed by three squares placed at an angle, thus ; and say whether it is possible to divide such figure into four equal and similar parts.

A right-angled triangle made to revolve about one of the sides containing the right angle forms a body called a cone, which may be very properly called a circular pyramid.

431. Make in paper a hollow cone, and give a plan of your method.

When a cone is cut by a plane at right angles to the axis, the section produced is a circle.

When a cone is cut by a plane that makes with the axis an angle that is less than a right angle, but not so small an angle as the angle which the side of the cone makes with it, such section is an ellipse.

A section of a cone making an angle with the axis equal to that which the side makes is a parabola.

A section of a cone which makes, with the axis, an angle that is less than that which the side makes is an hyperbola.

A section of a cone with which the axis coincides is an isosceles triangle.

432. Cut from an apple or a turnip as accurate a cone as you can, and give a specimen of each of the five conic sections.

433. Give a sketch of a builder's trammel, and make an ellipse with a trammel; and show that you can, on the principle on which it acts, make an ellipse without one.

The long diameter of an ellipse is called the axis major, and the short one the axis minor, and the distance of either of the foci of the

ellipse from its centre is called the eccentricity of the ellipse.

434. Give a figure to show what is meant.

Two lines drawn from the foci of an ellipse to a point in the circumference make equal angles with a tangent to the ellipse at that point.

435. Can you at a particular point in the circumference of an ellipse draw a tangent to that circumference?

With a pair of compasses, and using different-sized circles, how nearly can you imitate an ellipse? In other words

436. How would you make an oval?

437. Can you, out of a circular piece of mahogany, and without any loss, make the tops of two oval stools, with an opening to lift it by, in the middle of each?

The solid formed by revolving on its minor axis a semi-ellipse is called an oblate spheroid.

438. Show how an oblate spheroid is formed, and say what the oblate spheroid reminds you of.

The solid formed by revolving on its axis major one-half an ellipse is called a prolate spheroid.

439. Show how a prolate spheroid is formed, and say what it reminds you of.

440. Supposing a room to be built in the form of a prolate spheroid, and a person to speak from one focus, show where his voice would be reflected.

441. Would there be the same effect produced in a room built in the form of an oblate spheroid.

Provided no notice is taken of the resistance of the air, a stone thrown horizontally from the top of a tower, at a velocity of 48 ft. in a second, and subject to the incessant action of the earth, which from nothing induces it to fall by a uniformly-increasing velocity through about 16 ft. in the first second, 48 ft. in the second second, 80 ft. in the third second, 112 ft. in the fourth second, and so on, makes in its progress a kind of curve. Now, the terms of the series 16, 48, 80, 112, 144, etc., increase in

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