VIII. [A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.] This definition is put in brackets, as useless, and unnecessary to be remembered. IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. When two straight lines meet at a point, so that if produced they would intersect (cross) each other, the indefinite space between them is called an angle. D N.B. "When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other straight line: thus B the angle which is contained by the straight CE lines, AB, CB, is named the angle ABC, or CBA; that which is contained by AB, DB, is named the angle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E." This explanation is put in inverted commas, as being Dr. Simson's addition; it is very necessary to be remembered. X. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of these angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. By borrowing the terms vertical and horizontal from the language of Physics, we may define a right angle to be that which is formed by the meeting of a vertical and horizontal line. XI. An obtuse angle is that which is greater than a right angle. XII. An acute angle is that which is less than a right angle. The term oblique angles is applied both to obtuse angles and to acute angles. XIII. [A term or boundary is the extremity of any thing.] This definition is unnecessary, being merely verbal. XIV. A figure is that which is enclosed by one or more boundaries. XV. A circle is a plane figure contained [or, bounded] by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. The circumference of a circle is its boundary. The space con- XVI. And this point is called the centre of the circle. The straight line drawn from the centre to the circumference, is called the radius. XVII. A diameter of a circle is a straight line drawn through the centre, anc terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. [A segment of a circle is the figure contained by a straight line, anc the part of the circumference it cuts off.] This definition is repeated in Book III. Definition VI. XX. Rectilineal [or, rectilinear, that is, formed of straight lines,] figures are those which are contained [or, bounded] by straight lines. XXI. Trilateral [that is, three-sided] figures, or triangles, [that is, three angled, are those which are contained] by three straight lines. XXII. Quadrilateral, [that is, four-sided; or, quadrangles, that is, four angled, are those which are contained] by four straight lines. XXIII. Mutilateral [that is, many-sided] figures, or polygons, [that is, many. angled, are those which are contained] by more than four straight lines. XXIV. Of three-sided figures, an equilateral [that is, equal-sided] triangle is that which has three equal sides. XXV. An isosceles [that is, equal-legged] triangle is that which has only two sides equal. XXVI. A scalene [that is, unequal] triangle is that which has three unequal sides. XXVII. A right-angled triangle is that which as a right angle. XXVIII. An obtuse-angled triangle is that which has an obtuse angle. XXIX. An acute-angled triangle is that which has three acuteangles. The acute angled triangle `must have three acute angles, because XXX. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. This definition is redundant. If the general definition annexed to XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal. This definition is also redundant; for an oblong [or rectangle, that is, a right-angled parallelogram] is that which has one angle a right angle, and the sides which contain it equal. XXXII. A rhombus is that which has all its sides equal, but its angles are not right angles. This is also redundant; for a rhombus is that which has one XXXIII. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. This definition may be replaced by that of a parallelogram above XXXIV. All other four-sided figures besides these, are called Trapeziums. Quadrilateral figures whose opposite sides are not parallel, are called trapeziums; but if one opposite pair be parallel and the other pair not, the figure is called a trapezoid. XXXV. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet. The meaning of this definition is, that the space between the lines is always of the same breadth. POSTULATES. Let it be granted, that a straight line may be drawn from any one point to any other point. When a straight line is drawn from one point to another point, the points are said to be joined. The points are understood to be in the same plane. II. That a terminated straight line may be produced to any length in a straight line. By terminated here, is meant of a definite length; and by produced is meant lengthened or extended indefinitely, in the same plane. III. And that a circle may be described from any centre, at any distance from that centre. By describing a circle at any distance, is meant drawing a circle in a plane with any given radius. Various other postulates (that is, demands of common sense,) are tacitly assumed by Euclid; as, that one figure or angle may be applied to another, for the purpose of comparison. See Prop. IV. of this book. AXIOMS. Things which are equal to the same thing, are equal to one another. II. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal. VI. Things which are double of the same, are equal to one another. VII. 'Things which are halves of the same, are equal to one another. VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. IX. The whole is greater than its part. Dr. Thomson, in his edition of Euclid, has added to this axiom, another, viz., "the whole is equal to all its parts taken together." X. Two straight lines cannot enclose a space. XI. All right angles are equal to one another. XII. If a straight line meets two 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 less than two right angles. that It is admitted by Dr. Simson that this axiom is not self-evident, which all axioms ought to be. Accordingly he demonstrates the truth of it as a proposition in his notes, by help of five different propositions! Though not considered free from objection, the substitute for this axiom, given in Playfair's edition of Euclid, is to be preferred: viz., "If two straight lines intersect each other, they cannot be both parallel to the same straight line." The number of axioms is in this book limited to twelve; but Euclid has tacitly assumed the truth of various other axioms, which will be noticed in the sequel. |