Book I. VI.-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. 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 line. Thus, the angle which is contained by the straight 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 BD, 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. VII. When a straight line AB, standing on another straight line CD, makes the adjacent angles ABC and ABD equal to one another, each of the angles is called a right angle, and the straight line, which stands on the other, is called a perpendicular to it. Cor. The angles which one straight line AB makes with another CD, upon one side of it, are either two right Book I. angles, or are, together, equal to two right angles. For, if AB be perpendicular to CD, then each of the angles ABC, ABD, is a right angle; and, therefore, are, together, equal to two right angles. But, if one of them, ABC, exceed a right angle, the other, A'BD, is manifestly as much less than a right angle, and, consequently, the two are together equal to two right angles. VIII. An obtuse angle is that which is greater than a right angle. IX. An acute angle is that which is less than a right angle. X.-A figure is that which is inclosed by one or more boundaries. XI.-Figures are said to be equivalent, when, without coinciding, they contain the same space. XII.-A circle is a plane figure, described by the revolution of a straight line, about one of its extremities, which remains fixed. XIII. The fixed point is called the centre of the circle, the describing line its radius, and the boundary traced by the remote end of that line its circumference. Book I. XIV. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. Cor. All radii of the same circle are equal to each other, and to a semi-diameter. XV. A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter. XVI.-Rectilineal figures are those which are contained by straight lines. XVII.-Trilateral figures, or triangles, by three straight lines. XVIII. Quadrilateral, by four straight lines. XIX.-Multilateral figures, or polygons, by more than four straight lines. XX. Of three sided figures, an equilateral triangle is that which has three equal sides. XXI.-An isosceles triangle, is that which has two sides equal. XXII. A scalene triangle, is that which has three unequal sides. XXIII. A right angled triangle, is that which has a right angle. XXIV. An obtuse angled triangle, is that which has an obtuse angle. XXV. An acute angled triangle, is that which has three acute angles. Book I. XXVI. Of four sided figures, a square is that which has all its sides equal, and all its angles right angles. XXVII.-An oblong, is that which has all its angles right angles, but has not all its sides equal. XXVIII-A rhombus, is that which has all its sides equal, but its angles are not right angles. XXIX.-A rhomboid, is that which has its opposite sides equal to one another. XXX.-All other four sided figures besides these, are called Trapegiums. XXXI.-Straight lines, which are in the same plane, and being produced even so far both ways, do not meet, are called Parallel lines. A PROPOSITION, is the enunciation of something for construction, or demonstraction, and is either a Problem or Theorem. In a PROBLEM, Something is required to be performed. Book I. A Lemma is a subsidiary truth, employed in the demonstration of a theorem, or the solution of a Problem. A Corollary is an obvious consequence that arises from a proposition. A Scholium is a remark on the nature and application of a proposition. An Axiom is a self-evident truth. AXIOMS. I. 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 doubles of the same thing, are equal to one another. VII. Things which are halves of the same thing, 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, and equal to all its parts taken together. X.-All right angles are equal to one another. XI. Two straight lines, which intersect one another, cannot be both parallel to the same straight line. XII. The shortest distance between two points is a straight line. Cor. Any two sides of a triangle are together greater than the third side. |