PROPOSITION XXIX. (Eucl. v. 25.) If four magnitudes of the same kind be proportionals, the greatest and least of them together must be greater than the other two together. Let A be to B as C is to D, and let A be the greatest of the four magnitudes, and consequently D the least. V. 18, and V. 14. Then must A, D together be greater than B, C together. Hence P, B, D together are greater than Q, B, D together. I. Ax. 4. QE D. .. A, D together are greater than B, C together. PROPOSITION XXX. (Eucl. v. C.) If the first be the same multiple of the second, or the same submultiple of it, that the third is of the fourth, the first must be to the second as the third is to the fourth. First, let A be the same multiple of B, that C is of D. Take of A and C any equimultiples mA, mC, Then mA = mpB and mC = mpD. V. 3. Now if mpB be greater than nB, and if equal, equal; if less, less. That is, if m.A be greater than nB, mC is greater than nD ; and if equal, equal; and if less, less. A is to B as C is to D. V. Def. 5. Next, let A be the same submultiple of B, that C is of D. Then must A be to B as C is to D. For. A is the same submultiple of B, that C is of D, .. B is the same multiple of A, that D is of C, .. B is to A as D is to C, by the first case, and ... A is to B as C is to D. V. 12. Q. E. D. PROPOSITION XXXI. (Eucl. v. E.) If four magnitudes be proportionals, they must also be proportionals by conversion; that is, the first must be to its excess above the second as the third is to its excess above the fourth. Let A, B together be to B as C, D together is to D. For A, B together is to B as C, D together is to D, .. A is to B as C is to D, V. 25. V. 12. and .: B is to A as D is to C, and.. A, B together is to A as C, D together is to C. V. 16. QE. D. BOOK VI. INTRODUCTORY REMARKS. THE chief subject of this Book is the Similarity of Rectilinear Figures. DEF. I. Two rectilinear figures are called similar, when they satisfy two conditions : I. For every angle in one of the figures there must be a corresponding equal angle in the other. II. The sides containing any one of the angles in one of the figures must be in the same ratio as the sides containing the corresponding angle in the other figure: the antecedents of the ratios being sides which are adjacent to equal angles in each figure. Thus ABC and DEF are similar triangles, if the angles at A, B, C be equal to the angles at D, E, F, respectively, and if BA be to AC as ED is to DF, and AC be to CB as DF is to FE, The sides adjacent to equal angles in the triangles are thus homologous, that is, BA, AC, CB are respectively homologous to ED, DF, FE. It will be shown in Prop. IV. that in the case of triangles the second of the above_conditions follows from the first. In the case of quadrilaterals and polygons both conditions are necessary: thus any two rectangles have each angle of the one equal to each angle of the other, but they are not necessarily similar figures. N.B.-The very important Prop. xxv. (Eucl. vi. 33) is independent of all the other Propositions in this Book, and might be placed with advantage at the very commencement of the Book. PROPOSITION I. THEOREM. Triangles of the same altitude are to one another as their Let the As ABC, ADC have the same altitude, that is, the perpendicular drawn from A to BD. Then must A ABC be to ▲ ADC as base BC is to base DC. In DB produced take any number of straight lines BG, GH each-BC. I. 3. In BD produced take any number of straight lines DK, KL, LM each-DC. I. 3. Join AG, AH; AK, AL, AM. Then CB, BG, GH are all equal, .. As ABC, AGB, AHG are all equal. I. 38. ..A AHC is the same multiple of ▲ ABC that HC is of BC. So also, ▲ AMC is the same multiple of ▲ ADC that MC is of DC. And A AHC is equal to, greater than, or less than ▲ AMC, according as base HC is equal to, greater than, or less than base MC. Now A AHC and base HC are equimultiples of A ABC and base BC, and ▲ AMC and base MC are equimultiples of A ADC and base DC. I. 38. .. A ABC is to ▲ ADC as base BC is to base DC. V. Def. 5. Q. E. D. |