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THE NEW ZEALAND GAZETTE.

No. 17

of a soil may be improved. The selection and testing of fertilisers).

The care of plants. The principles of pruning. The enemies of plants. The life-histories of the commoner animal pests. How to preserve specimens of plants.

Group III.

(14.) Elementary Mathematics.- (a.) Algebra :— Fundamental operations; easy fractions involving the knowledge of the factors of expressions that are the product of two binomial factors, and of such expressions as $a^{3} \pm b^{3}$ and $a^{3} \pm 3 a^{2} b+3 a b^{2} \pm b^{3}$, only numerical coefficients being used; common multiples and divisors to correspond; simple equations involving one or two unknown quantities, and 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 :—

The examination in geometry shall include questions on practical and on theoretical geometry. Every candidate shall be expected to answer questions in both branches of the subject.

The questions on practical geometry shall be set on the constructions contained in Section A, together with easy extensions of them. In cases where the validity of a construction is not obvious, the reasoning by which it is justified may be required. Every candidate shall provide himself with a ruler graduated in inches and tenths of an inch and in centimeters and millimeters, set-squares, a protractor, compasses, and a hard pencil. All figures should be drawn accurately. Questions may be set in which the use of the set-square or of the protractor is forbidden.

The questions in theoretical geometry shall consist of theorems contained in Section B, together with questions upon these theorems, easy deductions from them, and arithmetical illustrations. Any proof of a proposition shall be accepted which appears to the examiners to form part of a systematic treatment of the subject; the order in which the theorems are stated in Section B is not imposed as the sequence of their treatment.

In the proof of theorems and deductions from them, the use of hypothetical constructions shall be permitted. Proofs which are applicable only to commensurable magnitudes shall be accepted.

SECTION A.—PRACTICAL GEOMETRY.

Bisection of angles and of straight lines.

Construction of perpendiculars to straight lines.

Construction of an angle equal to a given angle.

Construction of parallels to a given straight line.

Simple cases of the construction from sufficient data of triangles and quadrilaterals.

Division of straight lines into a given number of equal parts or into parts in any given proportions.

Construction of a triangle equal in area to a given polygon.

Construction of tangents to a circle and of common tangents to two circles.

Simple cases of the construction of circles from sufficient data.

Construction of a fourth proportional to three given straight lines and a mean proportional to two given straight lines.

Construction of regular figures of 3, 4, 6, or 8 sides in or about a given circle.

Determination by measurement of the ratio of the circumference of a circle to its diameter.

Approximate determination of the area of a circle.

SECTION B.—THEORETICAL GEOMETRY.

Angles at a Point.

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

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

Parallel Straight Lines.

When a straight line cuts two other straight lines, if

(i.) A pair of alternate angles are equal ; or

(ii.) A pair of corresponding angles are equal; or

(iii.) A pair of interior angles on the same side of the cutting line are together equal to two right angles;

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.

Triangles and other Rectilinear Figures.

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 angles so formed is equal to four right angles.

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

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

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

If the two right-angled triangles have their hypotenuses equal, and one side of the one equal to one side of the other, the triangles are congruent.

If two sides of a triangle are unequal, the greater side has the greater angle opposite to it; and the converse.

Of all the straight lines that can be drawn to a given straight line from a given point outside it, the perpendicular is the shortest.

The opposite sides and angles of a parallelogram are equal, each diagonal bisects the parallelogram, and the diagonals bisect one another.

If there are three or more parallel straight lines, and the intercepts made by them on any straight line that cuts them are equal, then the corresponding intercepts on any other straight line that cuts them are also equal.

Areas.

Parallelograms on the same or equal bases and of the same altitude are equal in area.

Triangles on the same or equal bases and of the same altitude are equal in area.

Equal triangles on the same or equal bases are of the same altitude.

Illustrations and explanations of the geometrical theorems corresponding to the following algebraical identities :—

$k(a+b+c+\ldots)=k a+k b+k c+\ldots$

$(a+b)^{2}=a^{2}+2 a b+b^{2},$

$(a-b)^{2}=a^{2}-2 a b+b^{2},$

$a^{2}-b^{2}=(a+b)(a-b)$.

Loci.

The locus of a point which is equidistant from two fixed points is the perpendicular bisector of the straight line joining the two fixed points.

The locus of a point which is equidistant from two intersecting straight lines consists of the pair of straight lines which bisect the angles between the two given lines.

(15.) Greek.—Candidates will be expected to show such a knowledge of the language and of its vocabulary and grammar as may be gained by the study of Xenophon's Anabasis, Book II; but candidates will not be expected to have read that particular book, nor will the passages for translation necessarily be taken from it. Great importance will be attached to translation from Greek, and writing easy passages or sentences in Greek.

(16.) Latin.—Candidates will be expected to show such a knowledge of the language and its vocabulary and grammar as may be gained by the study of Caesar's Gallic War, Book II; but candidates will not be expected to have read that particular book, nor will the passages for translation necessarily be taken from it. Great importance will be attached to translation from Latin, and to writing easy passages or sentences in Latin.

(17.) French.—Candidates will be expected to show such a knowledge of the language and of its vocabulary and grammar as may be gained (1) by easy conversation in French about the facts of every-day life, (2) by the study of Jules Verne's Le Tour du Monde en quatrevingts jours (Siepmann's French Course); but candidates will not be expected to have read that particular book, nor will the passages for translation necessarily be taken from it. Great importance will be attached to translation from French, and to the writing of easy passages and sentences in French.

(18.) German.—Candidates will be expected to show such a knowledge of the vocabulary and grammar of the language as may be gained (1) by easy conversation in German about the facts of every-day life, (2) by the study of Von Wildenbruch's Das edle Blut (Siepmann's German Course); but candidates will not be expected to have read that particular book, nor will the passages for translation necessarily be taken from it. Great importance will be attached to translation from German, and to the writing of easy passages or sentences in German. German script will not be insisted upon.

(19), (20). Italian, Spanish.—The syllabus in any of these subjects will be supplied on application to the Inspector-General of Schools, Wellington.