there are a large number of important propositions which depend upon this relation. Some criticism may also be advanced of Euclid's order of proof. Thus the properties of parallel lines and planes in space do not depend upon the properties of lines perpendicular to planes, and are easily deduced without them. It is hoped that the fuller treatment and different order of proof given in this work will render the subject much clearer to the average student. The difficulty of representing solid figures in plano is also a serious hindrance to many students. For this reason a simple account of a few elementary rules in Perspective Drawing has been inserted. All the modern extensions of Euclid, Books VI and XI which are usually taught in schools have been included, such as the Nine-Point Circle, Radical Axes, Harmonic Ranges, Inversion, Poles and Polars, and the elementary properties of the Parallelopiped, Sphere, and Tetrahedron. In this, as in the previous sections of the work, the authors' aims have been to give an exhaustive and accurate account of the subjects under treatmentindicating by asterisks those portions which should be omitted in ordinary school use--and to satisfy the legitimate modern demand for a more practical and concrete treatment of Geometry than can be obtained by the use of Euclid's Elements. CONTENTS. PAGE CHAPTER XIX.–RATIO AND PROPORTION CHAPTER XXI.-PROPORTION THEOREMS ON AREAS 434 CHAPTER XXII.-RECTANGLE THEOREMS ON TRI- CHAPTER XXIV.-HARMONIC RANGES : CENTRES CHAPTER XXV.—MISCELLANEOUS THEOREMS IN CHAPTER XXVI.-IRRATIONAL NUMBERS CHAPTER XXVII.-STRAIGHT LINES AND PLANES 510 CHAPTER XXVIII.—PARALLELS AND PARALLELO- CHAPTER XXIX. LINES PERPENDICULAR TO CHAPTER XXXI.-TRIHEDRAL AND POLYHEDRAL CHAPTER XIX. RATIO AND PROPORTION. 281. We shall now proceed to those properties of geometrical figures which involve Ratio and Proportion. The theory of Ratio and Proportion is essentially numerical, and is fully treated in books on Arithmetic and Algebra ; accordingly we shall assume a knowledge of this subject, at least so far as commensurable magnitudes are concerned, and shall merely give a brief recapitulation of the necessary definitions and theorems, with a few illustrative exercises. The question of incommensurable magnitudes is too difficult for the student at this stage: we shall therefore assume for the present that all magnitudes are commensurable, and shall attempt to satisfy the requirements of strict logic by dealing with incommensurables in Chapter XXVI.7 We have already stated ($ 133) that two quantities are commensurable if it is possible to find a third quantity which is contained an exact number of times in each of them. Thus the lengths 3.27" and 2.8" are commensurable, since the length •01" is contained 327 times in the first and 280 times in the second. Though the assumption that all magnitudes are commensurable is unsatisfactory from the logical standpoint, it should be noted that it is quite satisfactory from the practical standpoint. For in practical life all geometrical calculations are based upon measurements, and the measurements (which are of course not perfectly accurate) are always expressed in rational numbers, i. e. in whole numbers and vulgar or decimal fractions. + Euclid in his Fifth Book-a masterpiece of mathematical ingenuity-gives a treatment of proportion which is logically complete and yet avoids the necessity of distinguishing between commensurable and incommensurable magnitudes. His method of treatment is, however, too difficult for ordinary school use. 282. Ratio.—If A and B are two quantities of the same kind, e.g. two areas, we know from Arithmetic that the one can be expressed as a fraction of the other. DEFINITION.—The ratio of two quantities of the same kind is the fraction which expresses their relative magnitude. The ratio of A to B is expressed by either of the A notations A:B or B In the ratio A:B, A and B are called the terms of the ratio; A is called the antecedent and B the consequent. Example.—Evaluate the ratio 9d. : 2s. We must find what fraction 9d. is of 2s. Since 2s. = 24d. the required fraction is t or š. Thus the ratio of 9d. to 2s. is g. Note the importance of the order. The ratio 2s. : 9d. would be ş. It is very important that the student should understand clearly the difference between a ratio and a fraction. The numerator and denominator of a fraction must be numbers; but the antecedent and consequent of a ratio may be either two numbers or two concrete quantities of the same kind. For example, 3 yds. is not a fraction but a ratio, and it would not be 10 ft. advisable to perform any arithmetical operation with this ratio until we had reduced it to the equivalent fraction 10 ; on the other hand , may be regarded either as a fraction or as a ratio, and has exactly the same meaning in either case. The ratio compounded of two or more given ratios is the fraction obtained by multiplying the corresponding fractions. Thus if a, b, c, d, e, f, are numbers, then the ratio compounded of the three ratios a:b, c:d, and e:f is the ratio ace : bdf; for the fraction obtained by multiplying the fractions, is ū fbdf If a and b are numbers, then the duplicate ratio of a :b is the ratio a?:12 the sub-triplicate ratio of a :b is the ratio Ja: 31. SIM. GEO. D D ace EXERCISES CVIII. Reduce the following ratios to their equivalent vulgar fractions :1. 5 ft. : 2 yds. 2. 50 ins. : 2.5 ft. 3. 600 sq. ins. : 5 sq. ft. 4. 300 sq. metres : 4 sq. decametres. 5. 5 sq. ft. : 2.5 ft. 6. 4 right angles : 100°. 7. 10 right angles : 1000°. 8. 20 ft. :10 secs. Reduce the following ratios to their equivalent decimal fractions, correct to 3 significant figures 9. 1 metre : 273 cms. 10. 9 kiloms, :13 kiloms. 11. 132 ft. : 10 sq. yds. 12. 21R:10,000°. Express as vulgar fractions :13. The duplicate ratio of 4 ft. : 2 yds. 14. The sub-duplicate ratio of 8 ft. : 6 yds. 15. The triplicate ratio of 10 yds. : 40 ft. 16. The sub-triplicate ratio of 1 cub. in. :1 cub. ft. Express in decimals correct to 3 significant figures the ratio compounded of the ratios :17. 2:5 and 25:8. 18. 3 yds. : 2 ft. and 5 ft. : 2 yds. 19. 10 sq. ft. : 2 sq. yds. and 10 ins. : 2 ft. 20. 1 in. : 1 ft., ld. :18., and 1 hour :1 min. 21. The ratio of a inches to 2 ft. is 8; find a. 22. The ratio of 3 sq. ft. to x sq. inches is 1.2; find X. 283. Proportion. If the ratio of A to B is equal to the ratio of C to D, the four quantities A, B, C, D are called proportionals or are said to be in proportion. The statement of a proportion is best written in the form A C B D' but may also be written- A:B::C:D. In the proportion A:B = C:D, the quantities A, D are called the extremes, and the quantities B, C are called the means. In any proportion the first two quantities must be of the same kind (for otherwise they do not constitute a ratio), and the last two quantities must be of the same kind; but the first two need not be of the same kind as the last two. |