Oct. 28.] THE NEW ZEALAND GAZETTE. 3033

If two sides of a triangle are unequal, the greater side has the greater
angle opposite to it; and the converse.
Of all straight lines that can be drawn to a given straight line from a
given point outside it, the perpendicular is the shortest.
The opposite angles of a parallelogram are equal; and the converse.
The opposite sides of a parallelogram are equal and each diagonal bisects
the parallelogram; and the converse of the first part.
The diagonals of a parallelogram bisect one another; and the converse.
If a pair of opposite sides of a quadrilateral are equal and parallel, it
is a parallelogram.
The straight line drawn through the middle point of one side of a triangle
parallel to another side bisects the third side.
The straight line joining the middle points of two sides of a triangle is
parallel to the third side and equal to one-half of it.
If three or more parallel straight lines make equal intercepts on any
transversal, they make equal intercepts on any other transversal.
(b.) To bisect a given angle.
To bisect a given straight line.
To construct a perpendicular to a given straight line (i) from a given
point in the line, (ii) from a given point outside the line.
To construct an angle equal to a given angle.
To draw a straight line parallel to a given straight line.
(c.) To divide a straight line into any number of equal parts or in a given
ratio.
The construction of angles of 60°, 45°, 30°.
The construction of triangles and quadrilaterals from sufficient data
and the solution of triangles by drawing to scale.

Section 2:—

(a.) The area of a parallelogram is equal to the area of a rectangle on
the same base and between the same parallels; and is therefore measured
by the product of the measures of its base and its altitude.
Parallelograms on the same or equal bases and of the same altitude
are equal in area.
The area of a triangle is equal to one-half of the area of a rectangle on
the same base and between the same parallels; and is therefore measured
by one-half of the product of the measures of its base and its altitude..
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.
The square of the hypotenuse of a right-angled triangle is equal to the
sum of the squares on the other two sides; and the converse.
Geometrical proofs of the following algebraic identities:—

k (a + b + c) = ka + kb + kc
(a + b)² = a² + 2 ab + b²
(a — b)² = a² — 2ab + b²
a² — b² = (a + b) (a — b).

(b.) To construct a triangle equal in area to a given quadrilateral.
To construct a rectangle equal in area to a given triangle.
(c.) The determination by measurement of the areas of plane rectilineal
figures.
The experimental proof of the theorem of Pythagoras.

Section 3:—

(a.) The locus of points equidistant from two fixed points is the
perpendicular bisector of the line joining the two fixed points.
The locus of points equidistant from two intersecting straight lines
consists of the pair of straight lines which bisect the angles between the
two given lines.
(b.) The construction or plotting of the loci of points subject to simple
geometrical conditions.

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.

D