Aug. 11.] THE NEW ZEALAND GAZETTE. 1855

Section 4:—

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

Determination (approximately) of the area of a circle.

Construction of tangents to a given circle.

Simple cases of the construction of circles from sufficient data.

The inscription in or circumscription about a given circle, by geometrical methods, of regular figures of three, four, six, and eight sides.

(15A) Alternative Mathematics.

(a) General idea of number as a quantity having unlimited positive and negative range; rational and irrational numbers.

Quantities that can be wholly represented by numbers—e.g., length, cost, area, angle, time. Relationships between such quantities treated graphically. Ideas of variation, variable, and function. Graphical work with two variables to give practice in curve tracing only. Ideas of continuity and limit. Linear relation between two variables—i.e., the function Y = aX + b, treated symbolically and graphically with oblique as well as rectangular axes.

Dimensions of length, area, and volume functions. Simple functions stated in algebraic symbols, and commonly known as formulae. Practice in evaluation, including cases where it is necessary to change the independent variable in formulae such as a = (v - u)/t, T = 2π√(l/g).

Simple manipulative exercises in the addition, subtraction, multiplication, division, and factorization of algebraical expressions, limited, as far as possible, to trinomials of no higher degree than the second. The methods used must invariably show work written in lines, with a full understanding of the meaning and use of the sign of equality. Simple equations involving one or two unknown quantities, and problems thereon, provided that the examples set must bear the stamp of reality, whether they be expressed in symbols or stated in words.

(b) Ideas of direction and of similarity of space. The straight line defined as one of uniform direction, and its minimum treated as a consequence of its “directness.” All straight lines similar to one another: application to making of straight edges. Plane surfaces: application to surface plates. The angle treated as a quantitative change in direction. Rotational proofs of the simple angular properties of figures, especially figures of three and four sides.

Parallels treated as straight lines in the same direction. Angular properties of figures with parallel sides.

(c) Ideas of symmetry and superposition. Uniqueness of figure formed by three given straight lines in the same plane. Congruence of triangles deduced from this result, and consequently such geometrical results in reference to triangles and parallelograms as are usually proved in an elementary course from congruent triangles.

Use of scales: scale ratio. Simple cases of proportionality, including similarity of triangles and other plane figures, treated with reference to scale ratio. The recognition and use of the sine, cosine, and tangent of an angle, regarded both as a function of the angle and as a ratio between the sides. Graphs and functional scales of sine, cosine, tangent.

Areas of rectangles, triangles, and parallelograms. Proofs by dissection, combined with symbolic expression, as in the geometrical truths illustrated by the expressions—

Area of triangle = ½ bh = ½ bc sin A = ½ ca sin B
(a + b + c) x = ax + bx + cx
(a + x) x = ax + x²
(a ± b)² = a² ± 2ab + b²
(a + b)(a - b) = a² - b²

Areas of similar figures proportional to the squares of the ratios of corresponding sides. Pythagoras’ theorem deduced from this result together with the fact that the perpendicular from the right angle to the hypotenuse divides the triangle into two parts both similar to it.

(d) The simplest treatment of directed quantities, such as displacements, to show that certain quantities cannot be fully represented by numbers alone. Ideas of addition and subtraction for such quantities applied in simple diagrams drawn to scale, such as are usually met with in elementary practical questions on heights and distances.