PROP. 17. THEOR. Any two angles of a triangle are together less than ACD + ≤ ACB :. B+ But B and = 2 L1 ́s. (I. 13); ACB are < 2 Ls. ACB are any two angles, therefore any two angles of a triangle are together < 2 Ls. Q.E.D. Note. The converse of this proposition, that "if a straight line meeting two other straight lines make the two interior angles on the same side of it taken together less than two right angles, the two lines will, if produced far enough, meet on that side on which are the angles less than two right angles," is Euclid's twelfth Axiom, which we have replaced by Axiom 11. (See I. 29, Cor.) PROP. 18. THEOR. The greater side of a triangle has the greater angle opposite to it. Given ▲ ABC, having AC>AB. To prove ABC > ▲ ACB. In AC cut off AD = AB. (Post. 3). Join BD. AB AD (cons.), = .. ABD = 2 ADB (I. 5). But B ADB > ▲ C (I. 16), :. ≤ ABD > ≤ C Much more, then, is ABC > ≤ C. Q.E.D. C PROP. 19. THEOR. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it. Given ▲ ABC, having <B> AC is not would be AB, for then B B A = ▲ C (I. 5), which it is not. And AC is not <AB, for then (I. 18), which it is not. Since AC is neither B would be < ≤ C AB nor <AB, .. AC >AB. But BA and AC are any two sides; therefore any two sides of a triangle are together greater than the third. Q.E.D. OXFORD PREFACE. WHILE engaged in teaching Geometry, I have often been asked by my pupils to write on the black-board, to be copied by them, the proposition that we were working out together for next day's lesson. This led me to think of printing (in the first instance for their use) the first two Books of Euclid in the style that they found, even in their imperfect copy, easier to learn from than the ordinary text. The symbols and contractions so largely used are readily understood and easily acquired; and, as they enable the steps of a proposition to be brought closer, the connexion of the related parts becomes the more obvious. The diagrams and the arrangement of the matter are also designed to render such help to the beginner as mechanical means can give. A section on Analysis and Loci has been added to the First Book. Numerous easy Exercises have been introduced under the Propositions, and some more difficult ones in the continuation at the end. Those numbered in bolder type are important propositions, which form together a fairly complete sequel to this part of Euclid. The numbering of the Propositions has been retained as in Euclid; and, for the sake of those preparing for examinations in which only Euclid is accepted, the Euclidean Propositions of Book I., omitted in the text, are supplied in an Appendix. I beg to tender my thanks for much valuable help to Mr A. Y. Fraser, M. A., George Watson's College, Edinburgh, and to Mr Wm. Raitt, M.A., B.Sc., College of Science and Arts, Glasgow. GEORGE WATSON'S COLLEGE, A. J. G. BARCLAY. Exercises on preceding Propositions. 39. Prove that any two angles of a triangle are less than two right angles by drawing a line from the third angle to the opposite side, without producing any of the sides. 40. Only one line perpendicular to a given line can be drawn from a point outside it (I. 17). 41. Every right-angled triangle has two acute angles; so also has every obtuse-angled triangle. 42. The greatest side of a triangle has the greatest angle opposite to it; or, the least side of a triangle has the least angle opposite to it. 43. As I. 6 is not required by Euclid in any proposition before II. 4, it might have been displaced from its present position and proved in other ways.~ Prove it as an exercise depending on I. 18. 44. In a right-angled triangle the side opposite the right angle is the greatest side of the triangle. (This side is called the hypotenuse.) 45. The distance of a point from a line is the length of the perpendicular from the point to the line. This is the shortest distance from the point to the line; and of all other lines from the point to the line (called obliques) the one nearer the perpendicular is less than one more remote; and one, and only one, oblique can be drawn equal to any given oblique. 46. In I. 20 it can be proved that AB+ AC is > BC in other ways, not by putting BA and AC together, but by dividing BC into two parts, and comparing BA and AC with the parts: (1.) By dropping a perpendicular from A on BC; (2.) By bisecting BAC; (3.) By cutting off a part from BC one of the sides, and producing that side. 47. Any side of a triangle is greater than the difference of the other two sides. 48. The sum of the distances of any point from the three vertices of a triangle is greater than half the sum of the sides of the triangle. 49. The four sides of any quadrilateral are together greater than the two diagonals together, and less than the double of the two diagonals. 50. Two sides of a triangle are together greater than twice the line joining the vertex to the middle point of the base. This line is called a median. |