XXVI. A scalene triangle is that which has three unequal sides. XXVII. A right-angled triangle is that which has a right angle; and the side opposite the right angle is called the hypothenuse. XXVIII. An obtuse-angled triangle is that which has an obtuse angle. XXIX. An acute-angled triangle is that which has three acute angles. XXX. Of four-sided figures, a square is that which has all its angles right angles, and all its sides equal. XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal. XXXII. A rhombus is that which has all its sides equal, but its angles are not right angles. 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. XXXIV. All other four-sided figures, besides these, are called trapeziums. XXXV. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways, do not meet. POSTULATES. I. Let it be granted, 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. AXIOMS. I. Things which are equal to the same or equal things, 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. ས. If equals be taken from unequals, the remainders are unequal. VI. Things which are double of the same, or equal things, are equal to one another. VII. Things which are halves of the same, or equal things, 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. X. Two straight lines cannot enclose a space. All right angles are equal to one another. XII. If a straight line meet 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. A proposition is something either proposed to be done, or to be demonstrated, and is either a problem or a theorem. A problem is something proposed to be done; as the construction of a figure, &c. A theorem is something proposed to be demonstrated. A corollary is a consequence easily deduced from a proposition. An axiom is an evident truth, but which admits of no demonstration. A postulate is a request to admit the possibility of performing an operation. To describe an equilateral triangle upon a given straight line. Let AB be a given str. line, it is required to descr. an equilat. on AB. From cr. A, at dist. A B, descr. from cr. DBC; Post. 3. B, at dist. BA, descr. O ACE; from point C, in which the Os cut each other, Post. 1. draw CA, CB. Then ACB is an equilat.. C B E A is cr. BCD, B is cr. ACE, .. ACAB = BC. Wherefore on AB has been descr. an equilat. A ACB. From a given point to draw a straight line equal to a given straight line. Let A be the given pt., and BC the given str. line; it is required to draw from the pt. A, a str. line BC. Def. 15. Def. 15. Ax. 1. |