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1. Define a plane. When is a straight line said to be perpendicular to a plane? How is the diedral angle formed by two intersecting planes measured?

2. Prove that parallel planes intersect equal segments of parallel lines intersecting them.

3. Define a polyhedral angle. When are two polyhedral angles said to be equal? When symmetrical? Draw a figure representing two symmetrical polyhedral angles.

4. Determine (1) the locus of all points in space equally distant from two given points; (2) the locus of all points in space at a given distance from each of the two given points.

What condition must the given distance fulfill in order that the last problem may be possible?

5. When is one spherical triangle called the polar triangle of another? Prove that if A'B'C' is the polar triangle of ABC, then, reciprocally, ABC is the polar triangle of A'B'C'. State an important property of two reciprocally polar triangles.

6. Define the spherical excess of a spherical triangle. How can you find the area of a spherical triangle when you have its spherical excess given? Illustrate your answer by a simple example.


1. Bisect a given angle and prove your construction.

2. Of lines passing through the end of any radius of a circle the perpendicular is a tangent to the circle and every other line is a secant.

3. If from the right angle of a right-angled triangle a perpendicular be dropped upon the hyhothenuse, the square of the perpendicular will be equal to the rectangle of the two segments into which it divides the hypothenuse. 4. Explain what is meant by dividing a line harmonically and illustrate the definition.

Prove that the hypothenuse of a right-angled triangle is divided harmonically by any pair of lines through the vertex of the right angle, making equal angles with one of its sides.

5. (a) Of two equiangular triangles the side of one is twice a side of the other; compare the areas of the triangles. (b) Compute the length of the sides and the area of an equilateral triangle inscribed in a circle and whose radius is 1.

6. Define the angle which a straight line makes with a plane. Prove that a straight line makes equal angles with parallel planes.

7. Define a parallelopiped. Prove that the opposite faces of a parallelopiped are identically equal parallelograms.

8. Define the poles of a circle of a sphere; also a great circle of a sphere. How do any two great circles of the same sphere divide each other?

Prove that the poles of any two great circles AB, CD of a sphere, lie on a third great circle whose poles are the points of intersection of the circles AB, CD.

JUNE 1888.* PLANE.

1. If, when a straight line crosses two straight lines, the alternate interior angles are equal, the two straight lines are parallel.

2. The radius perpendicular to a chord of a circle bisects the chord and the arc subtended by it.

3. The sum of the squares of the two diagonals of a parallelogram is equal to the sum of the squares upon the four sides.

4. To divide a straight line internally into two segments which shall be to each other as two given straight lines.

5. State the methods for inscribing in a circle: (a) a regular decagon, (b) a regular pentagon, (c) a regular hexagon.

6. Of all plane figures having the same perimeter, what one has the greatest area?

* The 1887 papers cannot be obtained.


1. All lines perpendicular to another line at the same point lie in the same plane.

2. If each of two planes is perpendicular to a third plane, their line of intersection is also perpendicular to that third plane.

3. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles.

4. Calculate the surface and volume of a sphere whose radius is one foot, to four decimal places.

5. Calculate the area of a triangle on the preceding sphere whose angles are 90°, 75°, and 60°.


1. From a point outside a straight line only one perpendicular can be drawn to the line, and this perpendicular is the shortest distance from the point to the line.

2. If two secants be drawn from a point outside a circle, the angle between them is measured by half the difference of the intercepted arcs.

3. Given an equilateral triangle circumscribed about a circle whose radius is R, to express its altitude, and the length of either of its equal sides, in terms of R; also, to compare its area with the area of the circle.

4. To construct a parallelogram equivalent to a given square, and having the sum of its base and altitude equal to a given line.

5. If two planes be perpendicular to the same straight line they are parallel or coincident.

6. The sum of the face angles of any convex polyhedral angle is less than four right angles.

7. Compare the areas of two mutually equiangular spherical triangles, the one on a sphere of radius 1, the other on a sphere of radius 2.

8. Compare the volume of a sphere whose radius is R with that of a cone whose base is a great circle of the sphere and whose altitude is R.

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