29. SPACE is indefinite extension. 30. A VOLUME is a limited portion of space. A Volume has three dimensions: length, breadth, and thickness. AXIOMS. 1. Things which are equal to the same thing, are equa to each other. 2. If equals be added to equals, the sums will be equal. 3 If equals be subtracted from equals, the remainders will be equal. 4. If equals be added to unequals, the sums will be unequal. 5. If equals be subtracted from unequals, the remainders will be unequal. 6. If equals be multiplied by equals, the products will be equal. 7. If equals be divided by equals, the quotients will be equal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. 10. All right angles are equal. 11. Only one straight line can be drawn joining two given points. 12. The shortest distance from one point to another is measured on the straight line which joins them. 13. Through the same point, only ene straight line can be drawn parallel to a given straight line. POSTULATES. 1. A straight line can be drawn joining any two points. 2. A straight line may be prolonged to any length. If two straight lines are unequal, the length of the less may be laid off on the greater. 4. A straight line may be bisected; that is, divided into two equal parts. 5. An angle may be bisected. 6. A perpendicular may be drawn to a given straight line, either from a point without, or from a point on the line. 7. A straight line may be drawn, making with a given straight line an angle equal to a given angle. 8. A straight line may be drawn through a given point, parallel to a given line. NOTE. In making references, the following abbreviations are employed, viz. A. for Axiom; B. for Book; C. for Corollary; D. for Definition; 1. for Introduction; P. for Proposition; Prob. for Problem; Post. for Postulate; and S. for Scholium. In referring to the same Book, the number of the Book is not given; in referring to any other Book, the number of the Book is given. PROPOSITION I. THEOREM. If a straight line meet another straight line, the sum of the adjacent angles will be equal to two right angles. Let DC meet AB at C: then will the sum of the angles DCA and DCB be equal to two right angles. A C, let CE be drawn perpendicular to AB (Post. 6); then, by definition (D. 12), the angles A E D -B ECA and ECB will both be right angles, and conse quently, their sum will be equal to two right angles. The angle DCA is equal to ECA and ECD (A. 9); hence, But, the sum of the angles DCA+ DCB ECA + ECD + DCB; ECD + DCB is equal to ECB (A. 9); hence, DCA + DCB = ECA + ECB. The sum of the angles ECA and ECB, is equal to two right angles; consequently, its equal, that is, the sum of the angles DCA and DCB, must also be equal to two right angles; which was to be proved. Cor. 1. If one of the angles DCA, DCB, is a right angle, the other must also be a right angle. Cor. 2. The sum of the angles BAC, CAD, DAE, EAF, formed about a given point on the same side of a straight line BF, is equal to two right angles. For, their sum is equal to D E B -F the sum of the angles EAB and EAF; which, from the proposition just demonstrated, is equal to two right angles. DEFINITIONS. If two straight lines intersect each other, they form four angles about the point of intersection, which have received different names, with respect to each other. 1o. ADJACENT ANGLES are those which lie on the same side of one line, and on opposite sides of the other; thus, ACE and ECB, or ACE and ACD, are adjacent angles. D B 20. OPPOSITE, or VERTICAL ANGLES, are those which lie on opposite sides of both lines; thus, ACE and DCB, or ACD and ECB, are opposite angles. From the pro position just demonstrated, the sum of any two adjacent angles is equal to two right angles. PROPOSITION II. THEOREM. If two straight lines intersect each other, the opposite or vertical angles will be equal. Let AB and DE intersect at C: then will the opposite or vertical angles be equal. The sum of the adjacent angles ACE and ACD, is equal to D E B two right angles (P. I.): the sum of the adjacent angles ACE and ECB, is also equal to two right angles. But things which are equal to the same thing, are equal to each other (A. 1); hence, In like manner, we find, ACD + ACE = ACD + DCB ; and, taking away the common angle ACD, we have, ACE DCB. Hence, the proposition is proved. Cor. 1. If one of the angles about all of the others will be right angles also. each of its adjacent angles will be a right angle; and from the proposition just demonstrated, its opposite angle will also be a right angle. A C is a right angle, For, (P. I., C. 1), Ꭰ -B E Cor. 2. If one line DE, is perpendicular to another AB, then will the second line AB For, the angles DCA definition (D. 12); and be perpendicular to the first DE and DCB are right angles, by from what has just been proved, the angles ACE and БСЕ are also right angles. Hence, the two lines mutually perpendicular to each other. Cor. 3. The sum of all the angles ACB, BCD, DCE, ECF, FCA, that can be formed about a point, is equal to four right angles. B * 租 aie |
