4. Enunciate and prove the corollaries of the last proposition. 5. PROP. XLIV.- To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

6. The quadrilateral figure whose diameters bisect each other is a parallelogram.

BOOK II.

7. PROP. VI.-If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced.

8. PROP. XII.-In obtuse-angled triangles, if a perpendicular be drawn from either 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.

9. In any isosceles triangle ABC, if AD be drawn from the vertex to any point in the base, show that the difference of the squares on AB and AD is equal to the rectangle of BD and CD.

BOOK III.

10. PROP. IX.-If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle.

11. PROP. XX.-The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference.

12. PROP. XXXI.—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.

13. ABC is a triangle of which the angle A is acute; show that the square of BC is less than the squares of AB, AC, by twice the square of the line drawn from A to touch the circle on BC as diameter.

14. If a quadrilateral is described about a circle, show that the angles subtended at the centre of the circle by two opposite sides of it are together equal to two right angles.

Used in l'oluntary Examinations.

Book IV.

1. PROP. IV. To inscribe a circle in a given triangle.

2. PROP. XII.. ·To describe an equilateral and equiangular pentagom about a given circle.

3. Inscribe (1) a square, (2) a circle in a given quadrant of a circle.

Воок ѴІ.

4. Give Euclid's definition of proportion.

5. PROP. I.-Triangles of the same altitude are one to the other as their bases.

6. PROP. VI.—If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.

7. PROP. XVIII.—Upon a given straight line to describe a rectilineal figure similar, and similarly situated, to a given rectilineal figure. 8. PROP. XXII. — If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals.

9. If two circles touch each other externally, the part of their common tangent between its points of contact is a mean proportional between the diameters.

BOOK XI.

10. PROP. IV. - If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.

11. PROP. VIII. — If two straight lines be parallel, and one of them is at right angles to a plane, the other also shall be at right angles to the same plane.

.

12. PROP. XX.—If a solid angle be contained by three plane angles, any two of them are greater than the third.

13. Two planes being perpendicular to each other, draw a third perpendicular to both.

(1) x3 + 2 x3y + 2 x y2 + y3 by s3 − 2 x2 y + 2 x y2 — y3

(2) a + a-4 - a2 - a-2 by a — -a-1

(3) xy + x3 y zł (x y z) — y z by x2 + (y z )ś

3. Show that the product of two quantities equals that of their greatest common measure and least common multiple.

Find the greatest common measure of

35 x2 + 47 x2 + 13 x + 1 and 42 xa + 41 æ3 —- 9 x2 – 9 x

2

-4

4. Extract the square root of ( x + 1 ) 2 — 4 ( x − 1 ),

Ꮖ

and explain how

the method of extracting the square root of numbers is deduced from the Algebraical rule.

5. Investigate a method for obtaining the cube root of an Algebraical expression, and find the cube root of

(1) a° — 6 a3 + 15 a1 — 20 a3 + 15 a2 — 6 a + 1

(2)

110592

2

=2ab

(5) (a + b)2 x + (~~) (+ - ) =

7. A watch gains as much as a clock loses; and 1798 hours by the clock are equivalent to 1802 hours by the watch: Find the rates of the clock and watch.

8. A certain number of sovereigns, shillings, and sixpences, together amount to 81. 6s. 6d., and the amount of the shillings is a guinea less than that of the sovereigns, and a guinea and a half more than that of the sixpences: Find the numbers of each coin.

9. Two minutes after a railway train has left a station A, where it had stopped 7 minutes, it meets an express train which set out from a station B when the former was 28 miles on the other side of A; the express travels at double the rate of the other, and performs the journey from B to A in 1 hour: Find the rate of each train,

and the distance from A to B.

10. Investigate an expression for the sum of n terms of an Arithmetic progression, and for the limit of the sum of an infinite Geometric progression.

12. When does one quantity vary as another?

If A & B, C, D, &c., when only one of the quantities is changed, show that ABCD. when all change,

Apply this principle to the following example:

If 10 men do a piece of work in 12 days of 12 hours each, in what time will 23 men do three times as much, each working 9 hours per day?

13. Find

(1) the number of permutations which can be formed from the letters of the word Sebastopol, taken all together.

(2) the number of combinations when three letters are taken together.

EUCLID, ALGEBRA, AND TRIGONOMETRY.

Set to Candidates for the Office of the Committee of Council on Education.

Note. In this Examination Mathematics are not prescribed, but may be selected by any candidate who has "made them his especial study," with the view of displaying his industry and intelligence.

1. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle shall be greater than the base of the other.

BOOK III.

2. PROP. 20.-The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is upon the same part of the circumference.

Book IV.

3. PROP. 11.-To inscribe an equilateral and equiangular pentagon in a given circle.

BOOK VI.

4. PROP. 18.-Upon a given straight line to describe a rectilineal figure similar, and similarly situated, to a given rectilineal figure.

5. A common tangent is drawn to two circles which touch externally; prove that if a circle be described on that part of it which lies between the points of contact, as diameter, it will pass through the point of contact of the two circles.

6. Inscribe a circle in a given quadrant of a circle.

7. Divide 4 ba3 + (4 c − a b) x2 — (4 d + a c) x + ad by 4x -a.

11. A and B have the same annual income, and occupy lodgings for 30 weeks in the year, the former at 14s., the latter at 21s. per week, all other expenses being exactly the same for both: B exceeds his income by as much as A comes short of his, and finds that he has spent one-tenth too much: Required the annual income and the whole expenditure of each.

12. Find the sum of the following series:

16. Having given the numerical value of sin A, find that of cos

; and

show that there ought to be four corresponding values. Determine which is the proper value when A lies between 180° and 270°.

17. In a plane triangle, having given two sides and the included angle, obtain the formulæ for solving the triangle.

Ex. Given a 205, b = 195, C = 4°, 102 = 30103, L cot 2°= 11.4569162, L cot 54° 20′ 9·8559376, L cot 54° 21′ 9·8556708; find the remaining angles.