tion, or opening, is called an angle. The point of meeting is called the vertex, and the lines are called the sides of the angle. If there is only one angle at a point, it may be denoted by a letter placed at the vertex, as the angle at A. A But if several angles are at one point, any one of them is expressed by three letters, of which the middle one is the letter at the vertex. Thus, the angle which is contained by the straight lines BC, CD, is called the angle BCD, or DCB. B E D Angles, like other quantities, may be added, subtracted, multiplied, or divided. Thus, the angle BCD is the sum of the two angles BCE, ECD; and the angle ECD is the difference between the two angles BCD, BCE. 10. When a straight line, meeting another straight line. makes the adjacent angles equal to one another, each of them is called a right angle, and the straight line which meets the other is called a perpendicular to it. 11. An acute angle is one which is less than a right angle. An obtuse angle is one which is greater than a right angle. 12. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. 13. A plane figure is a plane terminated on all sides by lines either straight or curved. If the lines are straight, the space they inclose is called a rectilineal figure, or polygon, and the lines themselves, taken together, form the perimeter of the polygon. 14. The polygon of three sides is the simplest of all, and is called a triangle; that of four sides is called a quadrilateral: that of five, a pentagon; that of six, a hexagon, &c. 15. An equilateral triangle is one which has its three sides equal. An isosceles triangle is that which has only two sides equal. A scalene triangle is one which has three unequal sides. 16. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse. An obtuse-angled triangle is one which has an obtuse angle. An acute-angled triangle is one which has three acute angles. 17. Of quadrilaterals, a square is that which has all its sides equal, and its angles right angles. A rectangle is that which has all its angles right angles, but all its sides are not necessarily equal. A rhombus is that which has all its sides equal, but its angles are not right angles. A parallelogram is that which has its opposite sides parallel. A trapezoid is that which has only two sides parallel. 18. The diagonal of a figure is a line B which joins the vertices of two angles not adjacent to each other. Thus, AC, AD, AE are diagonals. D A E F 19. An equilateral polygon is one which has all its sides equal. An equiangular polygon is one which has all its angles equal. 20. Two polygons are mutually equilateral when they have all the sides of the one equal to the corresponding sides of the other, each to each, and arranged in the same order. Two polygons are mutually equiangular when they have all the angles of the one equal to the corresponding angles of the other, each to each, and arranged in the same order. In both cases, the equal sides, or the equal angles, are called homologous sides or angles. 21. An axiom is a self-evident truth. 22. A theorem is a truth which becomes evident by a train of reasoning called a demonstration. A direct demonstration proceeds from the premises by a regular deduction. An indirect demonstration shows that any supposition contrary to the truth advanced, necessarily leads to an absurdity. 23. A problem is a question proposed which requires a solution. 24. A postulate requires us to admit the possibility of an operation. 25. A proposition is a general term for either a theorem, or a problem. One proposition is the converse of another, when the conclusion of the first is made the supposition in the second. 26. A corollary is an obvious consequence, resulting from one or more propositions. 27. A scholium is a remark appended to a proposition. 28. An hypothesis is a supposition made either in the enunciation of a proposition, or in the course of a demonstration. Axioms. 1. Things which are equal to the same thing are equal to each other. 2. If equals are added to equals, the wholes are equal. 3. If equals are taken from equals, the remainders are equal. 4. If equals are added to unequals, the wholes are unequal. 5. If equals are taken from unequals, the remainders are unequal. 6. Things which are doubles of the same thing are equal to each other. 7. Things which are halves of the same thing are equal to each other. 8. Magnitudes which coincide with each other, that is, which exactly fill the same space, are equal. 9. The whole is greater than any of its parts. 10. The whole is equal to the sum of all its parts. 11. From one point to another only one straight line can be drawn. 12. Two straight lines, which intersect one another, can not both be parallel to the same straight line. Explanation of Signs. For the sake of brevity, it is convenient to employ, to some extent, the signs of Algebra in Geometry. Those chiefly em ployed are the following: The sign denotes that the quantities between which it stands are equal; thus, the expression A=B signifies that A is equal to B. The sign is called plus, and indicates addition; thus A+B represents the sum of the quantities A and B. The sign-is called minus, and indicates subtraction; thus, A-B represents what remains after subtracting B from A. The sign indicates multiplication; thus, AXB denotes the product of A by B. Instead of the sign X, a point is sometimes employed; thus, A. B is the same as AXB. The same product is also sometimes represented without any intermediate sign, by AB; but this expression should not be employed when there is any danger of confounding it with the line AB. A parenthesis () indicates that several quantities are tʊ be subjected to the same operation; thus, the expression Ax (B+C-D) represents the product of A by the quantity B+C-D. The expression ding A by B. A B indicates the quotient arising from divi A number placed before a line or a quantity is to be re garded as a multiplier of that line or quantity; thus, 3AB de notes that the line AB is taken three times; A denotes the half of A. The square of the line AB is denoted by AB2; its cube by AB3. The sign indicates a root to be extracted; thus, 2 denotes the square root of 2; ✔AXB denotes the square root of the product of A and B. N.B.-The first six books treat only of plane figures, or fig ures drawn on a plane surface. PROPOSITION I. THEOREM. All right angles are equal to each other. A Di K BE H G Le: the straight line CD be perpendicular to AB, and GH to EF; then, by definition 10, each of the angles ACD, BCD, EGH, FGH, will be a right angle; and it is to be proved that the angle ACD is equal to the angle EGH. Take the four straight lines AC, CB, EG, GF, all equal to each other; then will the line AB be equal to the line EF (Axiom 2). Let the line EF be applied to the line AB, so that the point E may be on A, and the point F on B; then will the lines EF, AB coincide throughout; for otherwise two different straight lines might be drawn from one point to another, which is impossible (Axiom 11). Moreover, since the line EG is equal to the line AC, the point G will fall on the point C; and the line EG, coinciding with AC, the line GH will coincide with CD. For, if it could have any other position, as CK, then, because the angle EGH is equal to FGH (Def. 10), the angle ACK must be equal to BCK, and therefore the angle ACD is less than BCK. But BCK is less than BCD (Axiom 9); much more, then, is ACD less than BCD, which is impossible, because the angle ACD is equal to the angle BCD (Def. 10); therefore, GH can not but coincide with CD, and the angle EGH coincides with the angle ACD, and is equal to it (Axiom 8). Therefore, all right angles are equal to each other. PROPOSITION II. THEOREM. The angles which one straight line makes with another, upon one side of it, are either two right angles, cr are together equ to two right angles. Let the straight line AB make with CD, upon one side of it, the angles ABC, ABD; these are either two right angles, or are together equal to two right angles. For if the angle ABC is equal to ABD, each of them is a right angle (Def. 10); but C R D |