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7. In every triangle, the square on the side subtending either of the acute angles is less than the squares on the sides containing that angle, by twice the rectangle contained by either of those sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

In any acute-angled triangle, lines are drawn through the angles perpendicular to the sides respectively opposite to them; prove that the rectangles contained by the segments into which the sides are respectively divided, are together less than half the sum of the squares on the sides of the triangle.

8. If two circles cut one another, they shall not have the same centre. 9. If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.

10. In equal circles, equal angles stand upon equal circumferences, whether they be at the centres or circumferences.

In equal circles, straight lines equidistant from the centre subtend equal angles at the centre.

11. From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.

12. The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are homologous sides.

FIRST DIVISION.-(B.)

1. FROM the greater of two given straight lines to cut off a part equal to the less.

2. The angles which one straight line makes with another, on the same side of it, are either two right angles, or are together equal to two right angles.

3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. the sides opposite to equal angles in each, then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

The perpendiculars let fall on two sides of a triangle, from any point in the line bisecting the angle between them, are equal to each other.

4. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.

The straight lines which join the extremities of two equal and parallel straight lines, not towards the same parts, bisect each other.

5. To describe a square upon a given straight line.

6. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part.

7. In every triangle, the square on the side subtending either of the acute angles is less than the squares on the sides containing that angle, by twice the rectangle contained by either of those sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

In any acute-angled triangle ABC, AD, BE, CF are drawn respectively perpendicular to the opposite sides; prove that twice the rectangles contained by BD and CD, CE and AE, AF and BF, are together less than the sum of the squares on the sides of the triangle.

8. If two circles touch one another internally, they shall not have the

same centre.

9. If a straight line touch a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle.

10. In equal circles, equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less.

In equal circles, straight lines subtending equal angles at the centre are equidistant from the centre.

11. From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.

12. The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are homologous sides.

THURSDAY, January 10, 1856. 12 to 3.

SECOND DIVISION.-(A.)

1. If two angles of a triangle be equal to each other, the sides also which subtend the equal angles, shall be equal to each other.

If one angle of a triangle be equal to the sum of the other two, the triangle can always be divided into two isosceles triangles.

2. The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles.

3. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another.

4. Parallelograms upon the same base and between the same parallels are equal to one another.

Of all parallelograms, that can be described on the same base and between the same parallels, which has the least perimeter?

5. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a right angle.

6. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square of the whole line.

7. To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part.

8. If a straight line drawn through the centre of a circle bisect a straight line in it, which does not pass through the centre, it shall cut it at right angles; and conversely, if it cut it at right angles, it shall bisect it.

9. If two circles touch each other externally in any point, the straight line which joins their centres, shall pass through that point.

Describe a circle of given radius which shall touch externally a given circle in a given point.

10. Straight lines in a circle which are equally distant from the centre are equal to one another.

11. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

No parallelogram except a rectangular one can be inscribed in a circle.

12. If the angle of a triangle be divided into two equal angles by a straight line which also cuts the base; the segments of the base shall have the same ratio which the other sides of the triangle have to one another.

SECOND DIVISION.-(B.)

1. To draw a straight line at right angles to a given straight line, from a given point in the same.

If two straight lines bisect each other at right angles, every point in either of them is equidistant from the extremities of the other.

2. Any two sides of a triangle are together greater than the third side. In any quadrilateral figure, the sum of the sides is greater than the sum of the diagonals.

3. If a straight line falling on two other straight lines, make the alternate angles equal to each other, these two straight lines shall be parallel.

4. Parallelograms upon equal bases, and between the same parallels, are equal to one another.

5. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to half a right angle.

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6. If a straight line be divided into any two parts, the rectangle contained by the whole and one of its parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part. Shew that the rectangle contained by the two parts is greatest when the given line is bisected.

7. To describe a square that shall be equal to a given rectilineal figure. To find the centre of a given circle.

8.

9.

If one circle touch another internally in any point, the straight line which joins their centres being produced shall pass through that point. Describe a circle of given radius which shall touch internally a given circle in a given point: the given radius being less than that of the given circle.

10. Equal straight lines in a circle are equally distant from the centre. 11. The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference.

12. If the vertical angle of a triangle be divided by a straight line which also cuts the base, and the segments of the base have the same ratio which the other sides of the triangle have to one another, the vertical angle of the triangle shall be divided into two equal angles.

MECHANICS AND HYDROSTATICS.

FRIDAY, January 11, 1856. 9 to 12.

FIRST DIVISION.—(A.)

1. IF two forces acting perpendicularly on a straight lever in opposite directions and on the same side of the fulcrum balance each other, they are inversely as their distances from the fulcrum; and the pressure on the fulcrum is equal to the difference of the forces.

If the distances from the fulcrum of two forces acting in opposite directions and keeping it at rest be three and five feet respectively, and the pressure on the fulcrum be three pounds, what are the magnitudes of the forces?

2. If the adjacent sides of a parallelogram represent the component forces in direction and magnitude, the diagonal which passes through the intersection of these sides will represent the resultant force in direction and magnitude.

If the resultant force be represented in direction and magnitude by the diameter of a circle, and one of the component forces by a given chord passing through one extremity of that diameter, give a geometrical construction for representing the other component.

3. In a system of pullies in which each pully hangs by a separate string and the strings are parallel, there is equilibrium when P: W:: 1: that power of 2 whose index is the number of moveable pullies.

If P be equal to six pounds and W to forty-eight pounds, how many moveable pullies will there be?

4. The weight (W) being on an inclined plane, and the force (P) acting parallel to the plane, there is equilibrium when P: W:: the height of the plane: its length.

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2

If P be equal to what will be the inclination of the plane?

5. Define velocity, and shew that if P and W balance each other in the manner described in the preceding question, and the whole be put in motion, P: W:: W's velocity : P's velocity.

If the number of moveable pullies be four (the system being that in which each hangs by a separate string) and P's velocity be 32 feet in a second, what will be W's velocity?

6. When a body is suspended from a point, it will rest with its centre of gravity in the vertical line passing through the point of suspension.

A right-angled triangle is suspended by its right angle, and the inclination of the hypothenuse to the horizon is forty degrees, find the acute angles of the triangle.

7. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid.

Find the height of a column, standing in water 30 feet deep, when the pressure at the bottom is to the pressure at the top as 3 to 2.

8. Explain the hydrostatic paradox.

9. When a body of uniform density floats on a fluid, the part immersed : the whole body:: the specific gravity of the body: the specific gravity of the fluid.

Find the specific gravity of a material such that a cylinder formed of it four inches long floats in water with three inches immersed.

10. Describe the common hydrometer, and shew how to compare the specific gravities of two fluids by means of it.

11. Having given the number of degrees on Fahrenheit's thermometer, find the corresponding number on the Centigrade thermometer.

What is the temperature when the number of degrees on the Centigrade thermometer is as much below zero, as that on Fahrenheit's is above?

FIRST DIVISION.-(B.)

1. Ir two weights acting perpendicularly on a straight lever on opposite sides of the fulcrum balance each other, they are inversely as their distances from the fulcrum, and the pressure on the fulcrum is equal to their sum.

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