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5. In any right-angled triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle.

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

7. In obtuse-angled triangles, if a perpendicular be drawn from any of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle by twice the rectangle contained by the side upon which when produced the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle.

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 if it cuts it at right angles it shall bisect it.

9. One circle cannot touch another in more points than one whether it touches it on the inside or the outside.

10. In equal circles the angles which stand upon equal circumferences are equal to one another whether they be at the centres or the circumferences.

11. If from any point without a circle two straight lines be drawn one of which cuts the circle and the other touches it, the rectangle contained by the whole line which cuts the circle and the part of it without the circle shall be equal to the square of the line which touches it.

Through two given points draw a circle which shall touch a given

circle.

12. The sides about the equal angles of equiangular triangles are proportional, and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios.

THURSDAY, January 15, 1857. 12 to 31.

SECOND DIVISION.-(A.)

1. To bisect a given rectilineal angle, that is, to divide it into two equal angles.

2. Any two sides of a triangle are together greater than the third side. 3. To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

4. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are equal to two right angles.

If two sides of any triangle be produced, the two exterior angles are together greater than two right angles.

5. In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the 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. 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 two circles cut one another, they shall not have the same centre. 9. If two circles touch each other externally, the straight line which joins their centres shall pass through the point of contact.

10. If a straight line touches a circle, and from the point of contact a straight line be drawn cutting the circle; the angles which this line makes with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.

11. Upon a given straight line to describe a segment of a circle which shall contain an angle equal to a given rectilineal angle.

If any three points in the circumference of a circle be joined so as to form a triangle, the segments external to the triangle contain angles which are together equal to four right angles.

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 bisect a given finite straight line, that is, to divide it into two equal parts.

2. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

3. If from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

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.

If a straight line which joins the extremities of two equal straight lines, not parallel, make the angles on the same side of it equal to each other; the straight line which joins the other extremities shall be parallel to the first.

5. If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.

6. If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines, is equal to the rectangles contained by the undivided line, and the several parts of the divided line.

7. To describe a square that shall be equal to a given rectilineal figure.

8. If two circles cut one another, they shall not have the same centre. 9. If two circles touch each other internally, the straight line which joins their centres being produced shall pass through the point of contact.

10. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

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

To divide a given circle into segments, one of which shall contain an angle double of that in the other.

12. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or these produced, proportionally.

MECHANICS AND HYDROSTATICS.

FRIDAY, January 16, 1857. 9 to 12.

FIRST DIVISION.-(A.)

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.

Two weights, 6 and 9 lbs. respectively, balance at the extremities of a horizontal rod 10 feet long, find where the fulcrum must be placed.

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

Two forces whose magnitudes are as 3 to 4, acting on a point at right angles to each other produce a resultant of 15 lbs. Find the forces.

3. The weight W being on an inclined plane and the force P being parallel to the plane, there is an equilibrium when P W the height of the plane

its length.

A smooth inclined plane rises 3 ft. 6 inches for every 5 feet, what force must a man exert parallel to the plane in order to prevent a weight of 200 lbs. from slipping down?

4. Define Velocity, and assuming that the arcs which subtend equal angles at the centres of two circles are as the radii of the circles, shew that if P and W balance each other on the wheel and axle, and the whole be put in motion, P: W:: W's velocity: P's velocity.

5. Find the centre of gravity of any number of heavy points and shew that the pressure at the centre is equal to the sum of the weights in all positions.

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. Why is it more difficult to balance a body with its centre of gravity above than below the point of suspension?

7. The surface of every fluid at rest is horizontal.

8.

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

The specific gravity of mercury is 13.5 and that of aluminium is 2.6, how deep will a cubic inch of aluminium sink in a vessel of mercury?

9. The elastic force of the air at a given temperature varies as the density.

10. Describe the construction of the forcing-pump and its mode of operation.

Why must the piston-rod of the forcing-pump be made stronger than that of the common pump?

11. Shew how to graduate a common thermometer.

The freezing point of a thermometer is the zero of the scale and the distance between the freezing and the boiling points is divided into 80 equal parts. Compare the graduation of such a thermometer with that of the Centigrade.

FIRST DIVISION.-(B.)

1. Ir 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.

A lever 7 feet long is supported in a horizontal position by props placed at its extremities. Where must a weight of 28 lbs. be placed so that the pressure on one of the props may be 8 lbs.?

2. If two forces represented in magnitude and direction by the sides of a triangle act upon a point they will keep it at rest.

Can a point be kept at rest by forces whose magnitudes are proportional to the numbers 3, 4, and 7?

3. In a system in which each pulley 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 pulleys.

If there are three pulleys in such a system and the weight of each is equal to w, shew that P-w: W-w:: 1 : 8.

4. Define Velocity; shew that if P and W balance each other on the inclined plane and the whole be put in motion, P: W :: W's velocity : P's velocity.

5. Find the centre of gravity of two heavy points, and shew that the pressure at the centre of gravity is equal to the sum of the weights in all positions.

6. When a body is placed on a horizontal plane it will stand or fall according as the vertical line drawn from its centre of gravity falls within or without the base.

A quadrilateral lamina which has all its sides equal will be in equilibrium if its plane be vertical and any one of its sides in a horizontal plane.

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

8. When a body is immersed in a fluid the weight lost the whole weight of the body :: the specific gravity of the fluid: the specific gravity of the body.

A compound of silver (S.G. =10·4) and aluminium (S.G.=26) floats half immersed in a basin of mercury (S.G13.5). What weight of silver is there in 10 lbs. of the compound?

9. The elastic force of the air at a given temperature varies as the density.

10. Describe the construction of the common pump and its mode of operation.

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