The proof of the proposition for triangles instead of parallelograms is effected in a similar manner, the equimultiples of the triangles and their bases being constructed as in Art. 20. Art. 72. EXAMPLES. 20. Given two rectilineal areas, and a straight line, find another straight line such that the scale of the areas is the same as that of the lines. 21. Given two straight lines, and a rectilineal area, find another rectilineal area such that the scale of the lines is the same as that of the areas. 22. Given three rectilineal areas, find a fourth such that the scale of the first and second area is the same as that of the third and fourth. 23. Prove that the scale of the areas of two triangles on equal bases is the same as the scale of their altitudes. 24. If ABC be a triangle, and O any point in its plane, and if AO cut BC at D, prove that BD : DC – ▲ AОВ: ▲ AОC. Art. 73. PROPOSITION XVIII. (Euc. VI. 33.) ENUNCIATION 1. In the same circle or in equal circles (i) angles at the centre are proportional to the arcs on which they stand. (ii) angles at the circumference are proportional to the arcs on which they stand. (iii) angles at the centre are proportional to the sectors bounded by the sides of the angles and the arcs on which they stand. ENUNCIATION 2. In the same circle or in equal circles (i) the scale of two angles at the centre is the same as that of the arcs on which they stand. (ii) the scale of two angles at the circumference is the same as that of the arcs on which they stand. (iii) the scale of two angles at the centre is the same as that of the sectors bounded by the sides of the angles and the arcs on which they stand. If the angles are in the same circle, the figure may be drawn twice over, so that it is sufficient to consider the case where there are two equal circles. (i) Let A, B, Figs. 67-72, be the centres of two equal circles. Let CAD, EBF be two angles at the centres standing on the arcs CD, EF. It is required to prove that [CÂD, EÊF]= [arc CD, arc EF]. Take any integer r in the first column and any integer s in the second column of the scale of CÂD, EBF. Then there are three alternatives shown by the figures :— Then in order that the scale of CÂD, EBF may be the same as the scale of arc CD, arc EF, it is necessary to show that in these several cases r (CÂD) < 8(EÊF). Now make an angle CAG equal to r (CÂD). Then it is known by Art. 21 that the arc CG is equal to r (arc CD). Then it is known by Art. 21 that the arc EH is equal to s (arc EF). Hence the scale of CÂD, EBF is the same as that of arc CD, arc EF. (ii) An angle at the centre of a circle is double the angle at the circumference standing on the same arc. Hence the scale of two angles at the centre of the same or of equal circles is the same as that of the angles at the circumference on the same arcs. [Art. 42. Hence, by case (i), the scale of two angles at the circumference of the same or of equal circles is the same as that of the arcs on which they stand. [Prop. 10. (iii) The proof of this is derivable from that of (i) by replacing therein each arc by the corresponding sector. * This follows immediately from the proposition that in equal circles equal angles at the centres stand on equal arcs. H. E. 6 Art. 74. Def. 15. RECIPROCAL SCALES. The scale of B, A is called the reciprocal of the scale of A, B. If two reciprocal scales are the same, prove that each must be the identical scale. The ratios A:B and B: A are called reciprocal ratios. Art. 77. EXAMPLE 26. If two reciprocal ratios are equal, prove that each of them is a ratio of equality. Art. 78. PROPOSITION XIX. (Corollary to Euc. V. 4.)* ENUNCIATION 1. If two ratios are equal their reciprocal ratios are equal, ENUNCIATION 2. If the scale of A, B is the same as that of C, D, then the scale of B, A is the same as that of D, C, Let the scale of A, B be formed. Let the 1st and 2nd columns be interchanged. Then the result will be the scale of B, A. For it is the scale which would have been formed, if after the multiples of A and those of B had been arranged in a vertical line in a single series in ascending order of magnitude, the multiples of A had been moved to the right instead of the left as in Art. 30. Since the original scale was also the scale of C, D, it follows that after the interchange of the 1st and 2nd columns the new scale is that of D, C. Hence the new scale is the scale of B, A and also that of D, C. .. [B, A]~ [D, C]. * Simson numbers this proposition Euc. V. B. Art. 79. PROPOSITION XX. (i). (Euc. V. 7, 1st Part.) ENUNCIATION 1. Equal magnitudes have the same ratio to the same magnitude, then i.e. if A, B, C be three magnitudes of the same kind, and if A be equal to B, A:CB: C. ENUNCIATION 2. If A, B, C be three magnitudes of the same kind, and if A be equal to B, then the scale of A, C is the same as that of B, C, Art. 80. PROPOSITION XX. (ii). (Euc. V. 7, 2nd Part.) ENUNCIATION 1. The same magnitude has the same ratio to equal mag nitudes, then i.e. if A, B, C be three magnitudes of the same kind, and if A be equal to B, C: A= C:B. ENUNCIATION 2. If A, B, C be three magnitudes of the same kind, and if A be equal to B, then the scale of C, A is the same as that of C, B, |