3032
THE NEW ZEALAND GAZETTE.
|No. 73

(c.) An elementary knowledge of the geography of the lands bordering upon the Pacific and of the rest of the world (including great geographical discoveries and the chief trade routes of the world).

(15.) Mathematics.

(a.) Algebra.

Fundamental operations; easy fractions; factors of expressions that are the product of two binomial factors, and of such expressions as a³ ± b³ and a³ ± 3a²b + 3ab² ± b³, only numerical coefficients being used; common multiples and divisors to correspond; simple equations involving one or two unknown quantities; the solution by factors of easy quadratic equations involving one unknown quantity; easy problems; graphs of simple algebraical functions within the limits of the foregoing work, and graphical methods of solving simple equations involving two unknown quantities.

(b.) Geometry.

No formal proofs shall be required for examination purposes of the theorems in Part I. The truth of these should be established by intuition and experiment, and the teaching of formal geometry should be based upon the quasi-axiomatic acceptance of these results.

Every candidate shall be expected to answer questions in both theoretical and practical geometry. The questions in theoretical geometry shall consist of theorems contained in subsections (a), Part II, together with questions upon these theorems; easy deductions from them and arithmetical illustrations. The questions in practical geometry shall be set on the constructions, &c., contained in subsections (b) and (c), Part II, together with easy extensions of them. All figures should be drawn accurately. The constructions in subsections (b) are those of which formal proofs shall be expected.

Any proof of a proposition shall be accepted which appears to the examiners to form part of a systematic treatment of the subject.

PART I.

If a straight line stands on another straight line, the sum of the adjacent angles so formed is equal to two right angles; and the converse.

If two straight lines intersect, the vertically opposite angles are equal.*

When a straight line cuts two other straight lines, if a pair of corresponding angles are equal, the two straight lines are parallel; and the converse.

When a straight line cuts two other straight lines, if (i) a pair of alternate angles are equal, or (ii) a pair of interior angles on the same side of the cutting line are supplementary, then the two straight lines are parallel; and the converse.*

Straight lines which are parallel to the same straight line are parallel to one another.*

If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles included by those sides equal, the triangles are congruent.

If two triangles have two angles of the one equal to two angles of the other, each to each, and also a side of one equal to the corresponding side of the other, the triangles are congruent.

If two triangles have three sides of the one equal to three sides of the other, each to each, the triangles are congruent.

The area of a rectangle is measured by the product of the measures of its sides.

There is one circle and only one circle which passes through three given points not in the same straight line.

(NOTE.—The theorems marked * may serve in the class-room as a useful introduction to the process of logical deduction.)

PART II.

Section 1:—

(a.) The sum of the angles of a triangle is equal to two right angles.

If the sides of a convex polygon are produced in order, the sum of the exterior angles so formed is equal to four right angles.

If two sides of a triangle are equal, the angles opposite those sides are equal; and the converse.

If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to one pair of equal sides are right angles, the triangles are congruent.

(NOTE.—The ambiguous case should be discussed.)