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SUPPLEMENT FOR 1899.

SHEFFIELD SCIENTIFIC SCHOOL.

GEOMETRY

JUNE, 1899. PLANE.

1. (a) The sum of the angles of a triangle equals two right angles.

(b) State and prove the theorem on the sum of the angles of any polygon.

2. If two circumferences intersect, the straight line joining their centers bisects their common chord at right angles.

State the corresponding theorem when the circumferences are tangent to each other.

3. A straight line parallel to one side of a tria: gle divides the other two proportionally.

4 When is a straight line divided in extreme and mean ratio?

Give and prove the construction for dividing a given straight line in extreme and mean ratio.

To the construction of what regular polygon does this construction apply?

5. What is the meaning of π in geometry?

Find the length of the side of an equilateral triangle whose aera equals that of a circle of radius R.

JUNE 1899. SOLID AMD SPHERICAL.

I. When is a straight line perpendicular to a plane? (a) All the perpendiculars to a given straight line at the same point lie in the plane perpendicular to that line at that point.

(b) The perpendicular frem a point to a plane is the shortest line that can be drawn from that point to the plane. 2. The sum of the face angles of any convex polyhedral angle is less than four right angles.

3. What is a cone of revolution?

State and prove the theorem regarding the lateral area of a cone of revolution.

4. Through two points on the surface of a sphere not the extremities of a diameter one and only one great circle may be drawn.

5. At what distance from the centre of a given sphere of radius R will one-quarter of the surface of that sphere become visible

SEPTEMBER 1899. PLANE.

1. What is the distinction between equal figures and equivalent figures?

State the various cases of two equal triangles, and prove any one of them.

2 When are two magnitudes commensurable? they incommensurable?

When are

In equal circles or in the same circle two angles at the center have the same ratio as their intercepted arcs, whether these arcs are commensurable or incommensurable.

3. If three or more straight lines drawn through a common point intersect two parallels, the corresponding segments of the parallels are porportional.

4. Show how to construct a square equivalent to a given polygon.

5. If the circumference of a circle be subdivided into three or more equal arcs:

(a) Their chords form a regular polygon, whose center is the center of the circle;

(b) The tangents at the points of division form a regular circumscribed polygon, whose center is the center of the circle.

SEPTEMBER, 1899. SOLID AND SPHERICAL.

1. Between two straight lines not in the same plane a common perpendicular can be drawn, and only one.

2. What is a rectangular parallelopiped?

The volume of a rectangular parallelopiped equals the product of its three dimensions, if the unit of volume is a cube whose edge is the linear unit.

3. The bases of a cylinder are equal.

4. The area of a zone of a sphere equals the altitude of the zone times the circumference of a great circle.

5. What must be the radius of a sphere if its surface and volume are numerically equal?

TRIGONOMETRY

JUNE. 1899.

1. Explain the two methods of measuring angles in ordinary use in trigonometry.

2, (a) What angles have the same sine as ÷3?

(b) What ones have the same cosine?

(c) What ones have the same tangent?

3. (a) Derive a formula for expressing 2 cos a sin b as a difference of sines,

(b) Derive one for expressing cos p-cos q as a double product of sines.

(4) Derive a formula for expressing tan 1⁄2a in terms of

cos a.

5. Transform the first member of the following identity into the second:

[blocks in formation]

6. Solve the equation sec2 cosec2 + 2 cosec10 = 8. 7. (a) Write the formulae for solving a triangle when two sides, a and b, and the angle A,opposite a, are given, (b) How many solutions would there be in each of the following cases:

(1) A = 30°,
(2) A = 30°,
(3) A = 30°,

a = 300 ft.,
a = 200 ft..
a = 400 ft.,

ft.

b = 400
b = 400 ft.

b = 300 ft.

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