Aug. 8.] THE NEW ZEALAND GAZETTE. 2449

A general knowledge of the external structure, of the bony skeleton, where present, and of the mode of life, of an earthworm, a crayfish, a spider, a beetle, a honey-bee, a butterfly, a garden snail, a frog, a fish, and a bird (comparisons should be made wherever possible). A knowledge of the chief characters of the classes to which the animals named in this syllabus belong. The reference to these classes of commonly occurring members thereof.

A knowledge of the life-history subsequent to hatching of a butterfly and a frog.

Candidates may be required to recognize or describe from actual specimens or photographs any of the above-mentioned animals or typical parts of them.

The candidate will be required to forward before the date of examination a certificate in the prescribed form that he has carried out satisfactorily a course of practical work based on the above syllabus.

GROUP III.

(12.) 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 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, 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: Every candidate shall be expected to answer questions in both practical and theoretical geometry. The questions on practical geometry shall be set on the constructions contained in Section A, together with easy extensions of them. All figures should be drawn accurately. 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.

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, and of the solution of triangles by drawing to scale.

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

Experimental determination of the areas of plane rectilineal figures; experimental proof of the proposition of Pythagoras.

Construction of tangents to a circle.

Simple cases of the construction of circles from sufficient data.

Construction of regular figures of 3, 4, 6, or 8 sides in 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 lines are parallel to one another.

D