12. Name the three kinds of angles; explain their names and accurately define them.

13. Name also the three kinds of triangles according to their angles. How many acute angles must every triangle have?

14. What is an axiom? and what a postulate?

15. Write down the axioms, or as many as possible.

16. Give familiar illustrations of the following axioms :—

A. Things which are equal to the same thing are equal to each other.

B. If equals be taken from equals the remainders are equal.

C. The whole is greater than its part.

D. Magnitudes which coincide, are equal.

17. Name the postulates.

18. Explain "data," "corollary," "enunciation," "hypothesis," "predicate," "analysis," and "synthesis."

19. Explain also,.., :, =, L, A, q. e. d., q. e. f.

2. EUCLID, BOOK I.

1. Distinguish accurately "problem" and "theorem," and give an instance of each.

2. Distinguish also "direct" from "indirect" proof; and give illustrations from common life showing that the latter, although indirect, is safe and accurate.

3. Distinguish accurately what is wrong from what is absurd, and give familiar instances.

4. How many parts or elements are there to every triangle? and what are they?

5. In what cases is the following proposition true, and in what case is it false? "If two plane triangles have three similar elements in the one respectively equal to three similar elements in the other, the triangles are equal in every respect.”

6. Show how an equilateral triangle may be constructed upon a given line, and analyse the various steps in the proof that it is equilateral.

7. Construct a figure, and prove the following:-"Two sides and the included angle in one triangle being equal to similar elements in another, the triangles are equal in every respect."

8. Illustrate by familiar examples the method of proof here adopted, howing its applicability and force.

9. Name other propositions in the first book of Euclid having reference to the equality of similar elements in two triangles, and compare them with the above.

10. Name propositions in Geometry, the proof of which depends upon this proposition of the equality of triangles when certain similar elements are respectively equal.

11. Show that, the base of a triangle being constant (i. e. given or measured in length), as the sides increase in length the vertical angle decreases, and vice versâ.

12. Name all the properties of parallel lines, and lines meeting them. How may it be known that lines are parallel?

13. "If a line meets two parallel lines the alternate angles will be equal.” Prove this, and the deductions from it.

14. Construct a square upon a given straight line.

15. Construct and prove the following proposition:- "In right-angled triangles, the square upon the side subtending the right angle is equal to the squares upon the other two sides."

16. Give some account of the discovery of this proposition and its discoverer, with some account also of the application of the principle in mathematical investigations.

17. Construct and write out the proof of the following propositions:

A. If two straight lines cut one another, the vertical angles are equal: and all the angles made by any number of lines meeting in a point are equal to four right angles.

B. The three angles of every triangle are equal to two right angles.

C. The angles at the base of an isosceles triangle are equal to one another.

D. Construct a parallelogram equal to a given rectilineal figure, with an angle equal to a given angle.

E. Two straight lines cannot have a common segment.

F. All the exterior angles of any rectilineal figure are equal to four right angles.

3. EUCLID, BOOK II.

1. With what kind of figures is the second book of Euclid mainly taken up? What is a rectangle, and what are said to contain it? Also what is a gnomon?

2. State all the features of "parallel lines" and "parallelograms" gained from the first book of Euclid.

3. "If a straight line be divided into any two parts, the square of the whole line is equal to the rectangles contained by the whole and each part." Prove this (1) by geometry; (2) by algebraic demonstration; and (3) by arithmetic, with a line 5 inches long divided into two parts, 3 inches and 2 inches in length.

4. Prove also in the same manner the following:

A. "If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts and twice their rectangle.

B. "If a straight line be divided into two equal and also into

two unequal parts, the squares of the unequal parts are together double of the square of half the line, and of the square of the line between the points of section.”

5. The value of the side opposite the right angle in a triangle being known (I. 47), state the value of the side opposite the obtuse angle in an obtuse-angled triangle; and also the value of the side opposite an acute angle.

6. Construct the figure and prove the property of an obtuse-angled triangle for the side opposite the obtuse angle, analogous to the property of a right-angled triangle for the side opposite the right angle.

7. In any triangle state the relation which exists between the side opposite an acute angle and the other two sides. Construct and prove.

8. Describe a square equal to a given rectilineal figure.

4. EUCLID, BOOKS III. & IV.

1. With what figures is the third book of Euclid mainly taken up? Define accurately "circle," "radius," "diameter; "chord," "arc."

also "segment,"

2. What is meant by cutting and touching a circle as applied to lines and curves?

3. Distinguish accurately the angle of a segment and the angle in a segment.

4. Define a sector. How is the distance of lines in a circle from the centre measured?

5. Show that the diameter is the longest straight line in a circle, and that other lines diminish as they recede from it.

6. How may the centre of a circle be exactly found? and if a straight line be drawn from the centre perpendicular to a chord, how does it cut it ?

7. State some properties of circles cutting each other, and touching each other, and the results.

8. State some conditions of lines cutting and touching circles.

9. Explain "the angle of a semicircle is greater than any acute rectilineal angle, and the remaining angle is less than any rectilineal angle."

10. Write out (symbolically if possible) the proof of the following:

A. The angle at the centre of a circle is double that of the angle at the circumference.

B. In any circle the angle in a semicircle is a right angle; the angle in a segment less than a semicircle is greater than a right angle; and the angle in a segment greater than a semicircle is less than a right angle.

C. For the proof of this proposition what ascertained facts of circles are needed? State also a corollary to this proposition, and prove it.

D. "If two straight lines cut each other in a circle, the rectangle of the segments is equal." Construct figures, and prove in every case, i. e. (1) when both lines pass through the centre; (2) when one passing through the centre cuts the other at right angles; (3) not at right angles; (4) when neither passes through the centre.

E. If from any point without a circle two straight lines be drawn, one cutting, the other touching the circle, the rectangle of the whole cutting line and the part without the circle is equal to the square of the touching line.

11. When is a figure said to be inscribed in another? and when described about another? Draw figures illustrating each case.

12. Perform the following:

A. Inscribe a square in a circle; and also inscribe a circle in a

square.

B. Describe a triangle about a circle; and also a circle about a triangle.

C. Inscribe a hexagon in a circle; and show the relation which exists between the radius of a circle, and the side of its inscribed hexagon.

D. Describe an isosceles triangle, having each of the angles at the base double of the third angle.

5. EUCLID, BOOKS V. & VI.

1. Define "ratio," and illustrate by arithmetic. Define "multiple," and state when one quantity may be called a multiple of another. 2. What does a compound ratio make, and what relation exists between the four terms composing it?

3. In the two proportions of three and four terms respectively

x:y:2
xy::a: b.

prove by Geometry (1) that x z = y2, and (2) that x b

4. In compound ratios (i.e. proportion with four terms) certain terms are made use of to express alteration of arrangement or combination of the ratios. Express the following ratio :

A. permutando.

C. componendo.

E. convertendo.

a: b::c: d.

B. invertendo.
D. dividendo.

5. Write out a geometrical proof of the last statement.

6. State some axioms for multiples, and give illustrations of them. 7. Define "similar rectilineal figures." Also, "altitude of a triangle." What are homologous sides? When is a line said to be cut in extreme and mean ratio?

8. Construct a figure, and cut the sides of a triangle proportionally. Prove and explain.

9. If the vertical angle of a triangle be bisected by a line cutting the