planed his board, and made it appear as smooth, and free from irregularities as possible, he applies the edge of his straight rule in many different positions on the surface, and sees that it touches the board throughout its length in each position. But, even if he could do this in each case with mathematical accuracy, he could never be sure that the surface of his board was plane, because he could never apply his rule between every two points, there being no limit to the number of points.

An angle is called a plane angle when each of the lines forming it is in the same plane superficies; and a figure is called a plane figure when it is entirely in the same plane superficies.

Whenever we talk of what can be more or less, we talk of a magnitude. A line is a magnitude and space; a superficies is a magnitude and space; but a point is neither a magnitude nor space. An angle is a magnitude but not space; it is the magnitude formed by the turning of one line about a point in another, the lengths of the lines having nothing to do with the size of the angle; we can compare one angle or opening with another, and determine which is more or less. Euclid defines an angle in such a manner that it is possible for an angle by becoming greater to become non-existent; for, if an angle increase until the lines forming it are in the same direction, the angle ceases to exist. Euclid does not consider the possibility of one angle being equal to or greater than two right angles. Of two coincident straight lines let one remain fixed whilst the other revolves about one of its extremities; in whatever position the second stops the inclination of the two lines is an angle (according to his definition) only until the two straight lines are in the same straight line. Euclid's angle may be the inclination of either straight or curved lines, but we have nothing to do with angles contained by curved lines. All the angles which are considered in Euclid's first book are rectilineal angles, and all rectilineal angles are plane angles, for any two straight lines which meet must be in the same plane superficies.

In the eighth axiom mention is made of magnitudes coinciding, or exactly filling the same space.

The meaning of the word filling, as used here, must be carefully considered. A geometrical figure occupies space, only in imagination, and an infinite number of figures may, in this sense of the word, fill at the same time, part or the whole of the same extent of space.

Points can coincide in position, but, having no magnitude, they can fill no space. Lines can coincide in position and length only. Plane figures are assumed to coincide when their boundaries coincide each

with each. Angles are said to coincide when the lines forming them are terminated at the same angular points, and coincide in direction.

Euclid never measures any of the magnitudes he treats of, by an unit, as we do in Arithmetic; he does not talk of a line being so many inches, or other units of length; he simply compares lines, angles, or figures, and then concludes their equality or inequality. The vast majority of such magnitudes are incommensurable with one another, that is, they cannot be measured by the same unit. It would be impossible to find an unit of length, however small, such that it should be contained an exact number of times both in the circumference and diameter of a circle. Euclid does not use any unit to measure angles with, but, having defined his right angle, and taken for granted that all right angles are equal, is able to talk of any other angle as being greater or less than a right angle.

The extremity of anything is not a part of that thing; the boundaries of a figure are not parts of the figure but merely enclose it. A triangle does not consist of the lines called the sides, but of the superficial area enclosed. When Euclid speaks of plane figures, such as circles, triangles, parallelograms etc., he means the surfaces contained within the boundaries of these plane figures. Definition 14 might be read: "A figure is a completely bounded space, the limits or bounds of which are either lines or superficies."

Euclid's definition of parallel straight lines is a negative definition. It would be impossible to determine from it whether lines were parallel; for, however far they had been produced and found not to meet, the test would not have been fully applied, because they could still be produced further, and then might meet. One point can prove the intersection of two lines, but millions of points cannot prove that they are parallel. The definition does not assert that there are straight lines which, however far produced, never meet; but says that if there be such straight lines, in the same plane, they are to be called parallel. Lines not in the same plane, although they do not meet when produced, are not parallel according to Euclid's definition.

The postulates may be said to allow Euclid the use of an imaginary ruler and compasses, but the ruler cannot be used for measuring, and the compasses close of themselves immediately they are taken from the superficies.

In assuming these postulates Euclid of course undertook not to make any other geometrical construction without showing how he could do so.

It is usual in works on Geometry to denote the relative positions of points by letters; and if A and B thus represent the positions of two points, the line joining these points is called AB; or, if necessary, more points in the same line may be mentioned. Angles are named with three letters, the first being a point in one of the lines containing the angle, the second the point of intersection of the two lines, and the third a point in the second line; the point of intersection of the two lines is called the vertex, or angular point. If AB and BC be two lines meeting at the point B, and not in the same direction, and if P be a point in the line AB, and be a point in the line BC, then ABC, ABQ, PBQ, or CBP each represents the same angle. In describing the position of a circle, three points in its circumference are named; thus we talk of the circle ABC, A, B and C representing the positions of three points in its circumference. Figures having more than one boundary are named by mentioning in order the letters at each of their angular points, as the triangle XYZ, or the square PRBV.

The following questions should be answered with the aid of diagrams to assist the imagination and memory.

(1) AB is a straight line, and C a point in it; name the parts into which C divides AB.

(2) AC is a straight line perpendicular to another straight line BD, and meeting it at C; name the right angles.

(3) Two straight lines, AB and CD, intersect at E; name the four angles thus formed.

(4) The diagonals of a rhombus ABCD intersect at E; name the eight triangles thus formed.

(5) PQR is an angle, and PQS is a part of it; name the other part.

(6) A straight line cuts the circumference of the circle ABC in the points B and C, ABC is a triangle of which the side AC passes through the centre of the circle; name the semicircle thus formed, and the segments of a circle greater than a semicircle.

(7) ABC and DBC are two triangles on the same base and on the same side of it, the vertex of each triangle being without the other; if the vertices be joined name the quadrilateral figure thus formed.

(8) A straight line AB is perpendicular to another straight line CD, meeting it at the point B; the angle ABD is divided into two parts by the straight line BE; name the right angles, the acute angles, and the obtuse angle thus formed.

(9) ABCD is an oblong; on the side AB, and remote from CD'

an equilateral triangle ABE is described; name the polygon thus formed.

(10) ABC is a triangle, D a point in AB, and E a point in BC; denote by eight different arrangements of the letters A, B, C, D, and E, taken three at a time, the angle opposite to the side AC.

(11) If the straight line AB fall upon the two other straight lines CD and EF, cutting CD at G and EF at H, which are the interior angles on the same side as D and F of the line AB?

(12) Define a plane figure, and the centre of a circle. State the meanings of the words postulate and axiom, explaining the difference between them.

(13) A straight line divides a circle into two parts; what is the general name of each of these two parts and what the particular name if the straight line pass through the centre?

(14) ABC is a triangle such that AB is greater than BC, and AC less than BC; what kind of triangle is ABC?

(15) Which of the following statements are true :—
(a) Every plane angle is a rectilineal angle.
(b) Every rectilineal angle is a plane angle.
(c) Every semicircle is a segment of a circle.
(d) Every segment of a circle is a semicircle.

(e) Every equilateral triangle is an isosceles triangle.
(f) Every isosceles triangle is an equilateral triangle.

(g) Every plane figure which has all its sides equal, but has not
all its angles right angles, is a rhombus.

(h) All straight lines which, if produced ever so far both ways, do not meet, are parallel.

(i) Every trapezium is a quadrilateral figure.

(5) Every quadrilateral figure is a trapezium.

(k) All magnitudes which coincide are equal to one another. (7) All magnitudes which are equal to one another, coincide. (16) Define a parallelogram and the radius of a circle.

(17) If a plane quadrilateral figure be rectilineal, equilateral, and rectangular, what is it called?

(18) A rectilineal figure enclosed by the least possible number of boundaries, has all its sides equal; what is it called?

(19) What plane figures contained by only two lines are defined in Euclid's first book?

(1) AC and CB.

ANSWERS.

(2) ACB and ACD.

(3) CEA, AED, DEB and CEB.

(4) AEB, BEC, DEC, AED, ABD, BCD, ABC, and ADC. (5) SQR.

(6) Semicircle ABC, segment greater than semicircle BCA.

(7) ABCD or ACBD.

(8) Right angles ABC and ABD, acute angles ABE and DBE,

obtuse CBE.

(9) ADCBE.

(10) ABC, ABE, DBC, DBE, CBA, EBA, CBD, EBD.

(11) DGH, GHF.

(13) Segment of a circle; semicircle.

(14) Scalene.

(15) b, c, e, i, k.

(17) Square.

(18) Equilateral triangle.

(19) Semicircle and segment of a circle.