The Surfaces, Lines, and Points of Geometry may be regarded as mental pictures of the surfaces, lines, and points which we know from experience.

It is, however, to be observed that Geometry requires us to conceive the possibility of the existence

VIII.

of a Surface apart from a Solid body,

of a Line apart from a Surface,

of a Point apart from a Line.

When two straight lines meet one another, but are not in the same straight line, the inclination of the lines to one another is called an ANGLE.

When two straight lines have one point common to both, they are said to form an angle (or angles) at that point. The point is called the vertex of the angle (or angles), and the lines are called the arms of the angle (or angles).

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Thus, if the lines OA, OB are terminated at the same point O, they form an angle, which is called the angle at O, or the angle AOB, or the angle BOA,-the letter which marks the vertex being put between those that mark the arms.

Again, if the line CO meets the line DE at a point in the line DE, so that O is a point common to both lines, CO is said to make with DE the angles COD, COE; and these (as having one arm, CO, common to both) are called adjacent angles.

Lastly, if the lines FG, HK cut each other in the point 0, the lines make with each other four angles FOH, HOG, GOK, KOF; and of these GOH, FOK are called vertically opposite angles, as also are FOH and GOK.

When three or more straight lines as OA, OB, OC, OD have a point common to all, the angle formed by one of

them, OD, with OA may be regarded as being made up of the angles AOB, BOC, COD; that is, we may speak of the angle AOD as a whole, of which the parts are the angles AOB, BOC and COD.

Hence we may regard an angle as a Magnitude, inasmuch as any angle may be regarded as being made up of parts which are themselves angles.

The size of an angle depends in no way on the length of the arms by which it is bounded.

We shall explain hereafter the restriction on the magnitude of angles enforced by Euclid's definition, and the important results that follow an extension of the definition.

IX. When a straight line (as AB) meeting another straight line (as CD) makes the adjacent angles equal to one another, each of the angles. is called a RIGHT ANGLE; and each line is said to be a PERPENDICULAR to the other.

X. An OBTUSE ANGLE is one which is greater than a right angle.

XI. An ACUTE ANGLE is one

which is less than a right angle.

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XII. A FIGURE is that which is enclosed by one or more boundaries.

XIII. A CIRCLE is a plane figure contained by one line, which is called the CIRCUMFERENCE, and is such, that all straight lines drawn to the circumference from a certain point (called the CENTRE) within the figure are equal to one another.

XIV. Any straight line drawn from the centre of a circle to the circumference is called a RADIUS.

XV. A DIAMETER of a circle is a straight line drawn through the centre and terminated both ways by the circumference.

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Thus, in the diagram, O is the centre of the circle ABCD, OA, OB, OC, OD are Radii of the circle, and the straight line AOD is a Diameter. Hence the radius of a circle is half the diameter.

XVI. A SEMICIRCLE is the figure contained by a diameter and the part of the circumference cut off by the diameter.

XVII. RECTILINEAR figures are those which are contained by straight lines.

The PERIMETER (or Periphery) of a rectilinear figure is the sum of its sides.

XVIII. A TRIANGLE is a plane figure contained by three straight lines.

XIX. A QUADRILATERAL is a plane figure contained by four straight lines.

XX. A POLYGON is a plane figure contained by more than four straight lines.

When a polygon has all its sides equal and all its angles equal it is called a regular polygon.

XXI. An EQUILATERAL Triangle is one which has all its sides equal.

XXII. An ISOSCELES Triangle is one

which has two sides equal.

The third side is often called the base of the triangle.

The term base is applied to any one of the sides of a triangle to distinguish it from the other two, especially when they have been previously mentioned.

XXIII. A RIGHT-ANGLED Triangle is

one in which one of the angles is a right angle.

The side subtending, that is, which is opposite the right angle is called the Hypotenuse.

XXIV. An OBTUSE-ANGLED Triangle is one in which one of the angles is obtuse.

It will be shewn hereafter that a triangle can have only one of its angles either equal to, or greater than, a right angle.

XXV. An ACUTE-ANGLED Triangle is

one in which ALL the angles are acute.

XXVI. PARALLEL STRAIGHT LINES are such as, being in the same plane, never meet when continually produced in both directions.

Euclid proceeds to put forward Six Postulates or Requests that he may be allowed to make certain assumptions on the construction of figures and the properties of geometrical magnitudes.

Let it be granted

POSTULATES.

I. That a straight line may be drawn from any one point to any other point.

II. That a terminated straight line may be produced to any length in a straight line.

III. That a circle may be described from any centre at any distance from that centre.

IV. That all right angles are equal to one another. V. That two straight lines cannot inclose a space. VI. That if a straight line meet two other straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced shall at length. meet upon that side, on which are the angles, which are together less than two right angles.

The word rendered "Postulates" is in the original airnμara, "requests."

In the first three Postulates Euclid states the use, under certain restrictions, which he desires to make of certain instruments for the construction of lines and circles.

In Post. I. and II. he asks for the use of the straight ruler, wherewith to draw straight lines. The restriction is, that the ruler is not supposed to be marked with divisions so as to measure lines.

In Post. III. he asks for the use of a pair of compasses, wherewith to describe a circle whose centre is at one extremity of a given line and whose circumference passes through the other extremity of that line. The restriction is, that the compasses are not supposed to be capable of conveying distances,

Post. IV. and v. refer to simple geometrical facts, which Euclid desires to take for granted.

Post. VI. may, as we shall shew hereafter, be deduced as a Theorem from a more simple Postulate. The student must defer the consideration of this Postulate, till he has reached the 17th Proposition of Book I.

Euclid next enumerates, as statements of fact, nine Axioms, or, as he calls them, Common Notions, applicable (with the