Geometry.

Junior.

EUCLID I. and II.

1. What is meant by a definition, postulate, axiom. Give three examples of each. Does Euclid define a circle? Can a figure, having more than three sides, be properly called a parallelogram?

2. Distinguish between an oblong and a parallelogram, and between a rhombus and a square.

3. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angle contained by those sides also equal, the triangles are equal in every respect.

4. Draw a straight line at right angles to a given straight line of unlimited length, from a given point without it.

From two given points draw two equal straight lines which' shall meet in the same point of a given straight line.

5. Any two sides of a triangle are together greater than the third side.

Any side of a triangle is greater than the difference between the other two sides.

6. Construct a triangle whose sides shall be in the proportion of 3, 4, 6, and shew it to be obtuse-angled.

7. Find a point in a straight line such that its distance from two given points shall be equal.

8. Explain the terms proposition and demonstration. Distinguish between a direct and indirect demonstration, a direct and converse proposition; and give instances from Euclid I.

2. The angles at the base of an isosceles triangle are equal. Prove this by Euclid, and also practically.

3. Enunciate and prove Euclid I. 32, also I. 47.

Make a square equal to three given squares.

4. If the sides of a pentagon be produced to meet, the angles thus formed together equal two right angles.

5. If a straight line be divided into any two parts, the square of the whole line and of one of the parts are equal to twice the rectangle of the whole and that part together with the square of the other part.

Express in geometrical terms the formula a2 - b2 = (a + b) (a - b).

6. Describe a rectangular parallelogram which shall be equal to a given square, and have its adjacent sides together equal to a given line.

7. If a straight line be divided into any two parts, the square of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part together with the square of the other part.

Any rectangle is the half of the rectangle contained by the diameters of the squares on its two sides.

8. If ABCD be a square, and a circle be described with centre D, and radius DA, cutting the diagonal DB in E, and another circle be described with centre E, and radius EB, cutting AB in F; then FE is a tangent to the former circle at E.

Geometry.

Women.

1. Explain the terms proposition, and demonstration. Distinguish between direct and indirect demonstration, and between direct and converse propositions, with examples.

2. Comment upon Euclid's definition of an angle. What do you understand by a straight angle? Distinguish between the angle of and the angle in a segment of a circle.

3. Senior paper, No. 2.

4. Senior paper, No. 5.

5. Senior paper, No. 8.

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

Shew that if any side of the figure be produced, the exterior angle thus formed is equal to the opposite angle of the figure.

7. The two triangles formed by drawing lines from any point within a parallelogram to the extremities of the two opposite sides are together equal to half the parallelogram.

8. In equal circles angles, whether at the centres or circumferences, have the same ratios which the arcs on which they stand have to each other.

What proposition in Book III. is a particular case of the above? Apply it to shew that any two parallel straight lines in a circle intercept equal portions of the circumference.

Higher Mathematics.

Junior and Senior.

TRIGONOMETRY, MECHANICS, STATICS.

1. State the methods by which the magnitudes of angles are represented numerically in Trigonometry. Express in each method a right angle, the angle of an equilateral triangle, the angle of a regular pentagon.

2. Give the definition of the sine, cosine, tangent, and complement of an angle; and find an expression for the secant in terms of the cotangent. Give the proper sign for an angle between 180° and 225°.

3. From the top of a hill the angles of depression of two consecutive milestones were observed to be 30° and 15°. What was the height of the hill?

4. Name the simple machines, and give formulæ for the mechanical advantage of

(i) a lever of the third kind.

(ii) wheel and axle.

(iii) screw.

5. What is meant by the Centre of Gravity? How would you find the centre of gravity in

a block of wood of uniform thickness.

(ii) a tea-tray.

(iii) a pair of shears.

6. Enunciate the "parallelogram of forces," and prove its truth for the direction of the resultant of two forces.

7. Define acceleration. When is it said to be uniform, when variable? How is it measured in each case?

A train increases its speed in 10 min. from 25 to 35 miles an. hour. Compare its acceleration with that of gravity.

Higher Mathematics.

TRIGONOMETRY AND CONICS.

1. Define the sine, tangent, cotangent, and cosecant of an arc, and trace the changes in the magnitude and algebraical signs of the sine and cosine of an angle, as the angle increases from 0° to 360°.

3. Find the cosine and the sine of an angle of a plane triangle in terms of the sides.

4. The altitude of a cloud is 0°, and that of the sun in the same vertical plane as the cloud is 4°, and the distance of the shadow of the cloud from the observer was a yds.; find the height of the cloud.

5. Given the sides of a triangle are 12, 16, 20 feet, find the angles and the area.

6. Define a parabola, tangent to any curve, circle of curvature, and limit of a variable quantity.

7. Describe the different sections of a cone by a plane. Of which are the following particular cases: a point, circle, and straight line?

8. In the ellipse, the rectangle under the abscessæ of the axis major the square of the semi-ordinale :: the square of the axis major: the square of the axis minor.

9. Define the eccentric angle of any point of an ellipse.

10. Prove that the locus of the centre of a circle touching a given circle and a given straight line is a parabola.