XIII. An Acute angle is one which is less than a right angle. XIV. An Obtuse angle is one which is greater than a right angle. XV. When the sum of two angles is equal to a right angle, they are called Complementary angles; each being the complement of the other. XVI. When the sum of two angles is equal to two right angles, they are called Supplementary angles; each being the supplement of the other. XVII. A Plane is a surface which is straight in every direction, or one with which a straight line, joining any two of its points, will coincide. XVIII. When a surface is not a plane surface it is called a Curved surface. XIX. A plane figure is a limited portion of a plane. When it is limited by straight lines, the figure is called a rectilineal figure, or a polygon; and the limiting lines, taken together, form the contour or perimeter of the polygon. XX. The simplest kind of polygon is one having only three sides, and is called a triangle. A polygon of four sides is called a quadrilateral; that of five sides is called a pentagon; that of six sides is called a hexagon; one of seven sides is called a heptagon; one of eight sides an octagon; one of nine sides a nonagon, and so on for figures of a greater number of sides. XXI. A triangle having the three sides equal, is called an equilateral triangle; one having two sides equal, is called an isosceles triangle; and one having no two sides equal, is called a scalene triangle. XXII. A triangle having a right angle, is called a right-angled triangle. The side opposite the right angle is called the hypotenuse. XXIII. A triangle having its three angles acute, is called an acute-angled triangle. XXIV. A triangle having an obtuse angle, is called an ob tuse-angled triangle. XXV. When the opposite sides of a quadrilateral are parallel, the figure is called a parallelogram. XXVI. When the four angles of a parallelogram are right angles, the figure is called a rectangle. XXVII. When the four sides of a rectangle are equal, the figure is called a square. XXVIII. When the four sides of a parallelogram are equal, and the angles not right, the figure is called a rhombus. XXIX. When only two sides of a quadrilateral are parallel, the figure is called a trapezoid. XXX. A diagonal of a polygon is a line joining the vertices of two angles, not adjacent. DEFINITIONS OF TERMS. I. A demonstration is a logical train of reasoning employed to establish an asserted truth. II. A proposition is a statement of a truth to be demonstrated, or of an operation to be performed. III. An axiom is a self-evident proposition, of which the simple mention carries a conviction of its truth. IV. A postulate is a proposition, the truth of which is required to be admitted without demonstration, notwithstanding it does not present the same degree of evidence as in the case of the axiom. V. A theorem is a proposition which requires a demonstra tion. VI. A corollary is an immediate consequence of one or more propositions. If any new course of reasoning is required to establish it, this reasoning is so simple that it may be supplied without much inconvenience. VII. A problem is a question proposed, which requires a solution. VIII. A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem. IX. A scholium is a remark on one or several preceding propositions, which tends to point out their connection, their use, their restriction, or their extension. X. An hypothesis is a supposition, made in the statement of a proposition, or in the course of its demonstration. REMARKS IN REFERENCE TO SIGNS, SYMBOLS, ETC. The signs and symbols which we shall employ, have the same signification in Geometry as in Algebra, to which, for their full explanation, we shall refer the student. For the doctrine of ratios and proportions, we will also refer the student to the method explained in the Algebra. There is this difference between geometrical ratios of magnitudes, and ratios of numbers: All numbers are commensurable; that is, their ratio can be accurately expressed but many magnitudes are incommensurable; that is, their ratio can be expressed only by approximation; which approximation may, however, be carried to any extent we desire. Such is the ratio of the circumference of a circle to its diameter, the diagonal of a square to its sides, etc. Hence many have deemed the arithmetical method not sufficiently general to apply to geometry. This would be a safe inference, were it necessary in all cases to assign the specific ratio between the two terms compared. But this is not the case. Such ratios themselves may be unknown, indeterminate, or irrational, and still their equality or inequality may be as completely determined by the arithmetical method as by the more lengthy method of the Greek geometers. AXIOMS. I. Things which are equal to the same thing, are equal to each other. II. When equals are added to equals, the wholes are equal. III. When equals are taken from equals, the remainders are equal. IV. When equals are added to unequals, the wholes are unequal. V. When equals are taken from unequals, the remainders are unequal. VI. Things which are double of the same or equal things, are equal. VII. Things which are halves of the same thing, are equal. VIII. Every whole is equal to all its parts taken together, and greater than any of them. IX. Things which coincide, or fill the same space, are identical. X. All right angles are equal to one another. XI. A straight line is the shortest distance between two points. XII. Through the same point only one straight line can be drawn parallel to another. XIII. Only one straight line can be drawn joining two given points. XIV. Straight lines which are parallel to the same line are parallel to each other. POSTULATES. I. Let it be granted that a straight line may be drawn from any point to any other point. II. That a terminated straight line may be produced, in either direction, to any extent. III. That a straight line may be bisected or halved. IV. That from a point, either within or without a straight line, a perpendicular may be drawn to the line. V. That a line may be drawn, making any given angle with another line. VI. That a line may be drawn from the vertex of an angle, bisecting it. OF ANGLES. THEOREM I. When a straight line meets or crosses another, the adjacent angles are supplements; and the opposite angles are equal. A Ꮐ C B F For, drawing FG perpendicular to AB, we see that the angle AFC exceeds the right angle AFG, by the angle CFG; while the angle BFC is less than the right angle BFG by the same angle CFG. Hence, the sum of AFC, BFC is equal to two right angles, and they are therefore supplements of each other (D. XVI.).* D In the same way by drawing a line perpendicular to CD, it may be shown that the angles CFA and AFD are supplements; consequently, BFC + CFA = CFA+AFD (A. I.): from each taking CFA, we have BFC equal to its opposite angle AFD. In a similar manner we have CFA-DFB. Cor. I. If either of the four angles, formed by the intersection of two straight lines, is a right angle, the remaining three angles will each be right, and the two lines will be mutually perpendicular. When the angles are not right there will be two equal acute angles, and two equal obtuse angles. The acute angle and the obtuse angle will be supplementary. Cor. II. All the angles which can be made at any point D, by any number of lines on the same side of AB, are together equal to two right A angles. E F C B D * In the references we shall use the following abbreviations: A. for Axiom, B. for Book, C. for Corollary, D. for Definition, P. for Problem, S. for Scholium, T. for Theorem. |