THEOR. 4. Rectangles of equal altitude are to one another in the same ratio as their bases. [Let AC, BC be two rectangles having the common side OC and their bases OA, OB on the same side of OC. In the line OAB indefinitely produced, take OM=m.OA and ON=1.OB, and complete the rectangles MC and NC. Then MC =m. AC and NC=n.BC, and it is plain that as OM is greater than, equal to, or less than ON, so is MC greater than, equal to, or less than AC, whence the rectangle AC : the rectangle BC :: base OA : base OB.] COR. Parallelograms or triangles of the same altitude are to one another as their bases. THEOR. 5. In the same circle or in equal circles angles at the centre and sectors are to one another as the arcs on which they stand. [In Book III. it was only necessary to consider arcs less than the 4 BOOK V. PROPORTION. INTRODUCTION. [For the use of those for whom it may be thought well to defer the study of the complete, but more difficult, mode of treatment of Proportion in Book IV., the following Definitions and Propositions referred to in this Book are here collected, with an indication of the principles of an incomplete mode of treatment by which they may be established for commensurable magnitudes.) Der. 1. One magnitude is said to be a multiple of another magnitude when the former contains the latter an exact number of times. According as the number of times is 1, 2, 3 ...m, so is the multiple said to be the Ist, 2nd, 3rd...mth. DEF. 2. One magnitude is said to be a measure or part of another magnitude when the former is contained an exact number of times in the latter. DEF. 3. If a magnitude can be found which is a measure of two or more magnitudes, these magnitudes are said A SYLLABUS OF PLANE GEOMETRY. 51 Def. 4. The ratio of one magnitude to another of the same kind is the relation of the former to the latter in equal to one another.] DEF. 5. When the ratio A:B is equal to the ratio P:Q, i.e. either sequents. DEF. 6. If A, B, C are three magnitudes of the same kind 52 A SYLLABUS OF such that A:B::B:C, B is said to be the mean proportional between A and C, and C the third pro portional to A and B. DEF. 7. If there are two ratios A :B, P: Q, and C be taken such that B:C::P:Q, then A is said to have to C a ratio compounded of the ratios A:B, P: Q. Thus if there are three magnitudes A, B, C, then A has to C the ratio compounded of the ratios A : B, B : C. DEF. 8. A ratio compounded of two equal ratios is called the duplicate of either of these ratios. GENERAL PROPOSITIONS OF PROPORTION. (1.) Ratios that are equal to the same ratio are equal to one another. (2.) Equal magnitudes have the same ratio to the same or to equal magnitudes. (3.) Magnitudes that have the same ratio to the same or equal magnitudes are equal. (4.) The ratio of two magnitudes is equal to that of their halves or doubles. (5.) If A:B::P: Q, then B: A::Q:P. (invertendo) (6.) If A:B::C: D, all the four being of the same kind, then A:C::B: D. (alternando) (7.) If A:B::P:Q, then A+B:B:: P+Q: Q, (componendo) and A-B:B:: P-Q : Q. (dividendo) (8.) If A:B :: C:D :: E:F, then A+ C + E : B + D + F :: A:B. (addendo) (9.) If A:B::P:Q and B:C:: Q : R, then A:C::P:R. (ex æquali) THEOR. 1. If two straight lines are cut by three parallel straight lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other. COR. 1. If the sides of a triangle are cut by a straight line parallel to the base, the segments of one side are to one another in the same ratio as the segments of the other side. COR. 2. If two straight lines are cut by four parallel straight lines the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other. Theor. 2. A given finite straight line can be divided internally into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality : and in each case there is only one such point of division. THEOR. 3. A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle. THEOR. 4. Rectangles of equal altitude are to one another in the same ratio as their bases. Cor. Parallelograms or triangles of the same altitude are to one another as their bases. THEOR. 5. In the same circle or in equal circles angles at the centre and sectors are to one another as the arcs on which they stand. SECTION 1. DEF. I. SIMILAR FIGURES. Similar rectilineal figures are those which have their angles equal, and the sides about the equal angles proportional. |