if, on the contrary, we would find one of the means, we must multiply the two extremes, and divide by the known mean. 180. This property of the equality between the product of the extremes and that of the means, cannot appertain to four quantities which are not in geometrical proportion. In effect, if we have four quantities which are not in geometrical proportion, in multiplying the consequents by the ratio of the two first, there will only be the first antecedent, which will become equal to its consequent. For example, if we have 3, 12, 5, 10, in multiplying the consequents 12 and 10 by the ratio of the two first, terms 3 and 12, we shall have 3, 3, 5, 1 in which it is evident that the product of the extremes is not equal to that of the means; therefore, these products could not any more be equal, even though we had not multiplied the consequents by the ratio, and it is plain that this reasoning is applicable to every case. Therefore, if four quantities are such that the product of the extremes is equal to the product of the means, these quantities are in proportion. Hence we shall deduce this second property of proportions. 181. If four quantities be in proportion, they will also be in proportion if we put the extremes in the place of the means, and the means in the place of the extremes. 182. The same will be the case, that is to say, the proportion will subsist if we change the places of the extremes or those of the means. In effect, in every case, it is easy to see that the product of the extremes will be equal to that of the means. Thus the proportion 3: 8:: 12: 32 furnishes all the following proportions by the simple permutation of its terms. 3: 8:12:32 3:12:: 8:32 32:12:: 8: 3 32: 8:12: 3 8: 3:: 32:12 8:32 : 3:12 12: 3::32: 8 12:32 : 3: 8 And it is the same in every other proportion. 183. Since we can put the third term in the place of the second, and reciprocally, we thence conclude that we can, without disturbing a proportion, multiply or divide the two antecedents by the same number, and that it is the same with regard to the consequents; for in making this permutation, the two antecedents of the given proportion will form the first ratio; and the two consequents the second. Therefore, to multiply the two antecedents of the first proportion becomes the same as to multiply the two terms of a ratio each by the same number, which (170) does not change this ratio. For example, if I have the proportion 3: 7 :: 12: 28, I can, in dividing the two antecedents by 3, say 1: 7:4: 28, because from the proportion 3: 7:: 12: 28 we conclude (182) that 3:12:: 7:28; and in dividing the two terms of the first ratio by 3, we have 1:4:: 7:28, which (182) may be changed to 17:: 4:28. 184. Every change made in a proportion, so that the sum of the antecedent and consequent, or their difference, may be compared to the antecedent or to the consequent, in the same manner in each ratio, will always form a proportion. For example, if we have the proportion 12:3:: 32: 8, we can thence deduce the following proportions: 12+3 3:32+8: 8 12-3: 3 : 32-8: 8 12+3 12:32+8:32 For if it be to the consequent that we compare, it is easy to see that the antecedent, augmented or diminished by the consequent, will contain this consequent a time more or a time less than before; and as this comparison is made in the same manner for the second ratio, which, by the nature of the proportion, is equal to the first, it follows that the two new ratios will also be equal to one another. If it be to the antecedent that we compare, the same reasoning will still apply, in conceiving that in the proportion in which we make this change, we have put the antecedent. of each ratio in the place of its consequent, and the consequen in the place of the antecedent, which may be done.(181) 185. Since, in putting the third term of a proportion in the place of the second, and reciprocally, there is still a propor tion,(182) we shall conclude that the two antecedents contai each other as many times as the consequents also contain one another. Therefore, the sum of the two antecedents of every proportion contains the sum of the two consequents, or is contained in it as many times as one of the antecedents contains its consequent, a is contained in it. For example, in the proportion 12:3:32 : & 12+32:38:: 32: 8, which is evident. But to satisfy ourselves generally, we have only to observe that if the first antecedent contain the second four times, for example, the sum of the two antecedents will contain the second five times, and, for the same reason, the sum of the consequents will contain the second consequent five times therefore, the sum of the antecedents will contain that of the consequents as the quintuple of one of the antecedents con tains the quintuple of its consequent, that is to say,(170) as one of the antecedents contains its consequent. We might, in the same manner, prove that the difference of the antecedents is to the difference of the consequents, as one antecedent is to its consequent. 186. It is evident that the proposition which we have demonstrated leads to this: that if we have two equal ratios, for example, We shall still have the same ratio, in adding antecedent to antecedent, and consequent to consequent. Therefore, if we have several equal ratios, the sum of all the antecedents is to the sum of all the consequents as any one of the antecedents is to its consequent. For example, if we have the equal ratios 4: 12:: 7:21:: 2:6, we can say, that 4+7+2: 12+21+6:4: 12 or 7: 21, etc. For, after having added the antecedents and the consequents of the two first ratios together, the new ratio which, as we have seen, will be the same as each of the two first, will also be the same as the third: consequently, we can add it to this, and we shall still have the same ratio, and so on for any number of ratios. 187. We call compound ratio that which results from two or more ratios, the antecedents of which we multiply together, and the consequents together. For example, if we have the two ratios 12: 4 and 25: 5, the product of the antecedents 12 and 25 will be 300; that of the consequents 4 and 5 will be 20; the ratio of 300 to 20 is what we call the ratio composed of the ratios of 12 to 4 and of 25 to 5. 188. This ratio is the same as if we had estimated separately each of these composing ratios, and multiplied together the numbers which express these ratios. In effect, the ratio of 12 to 4 is 3, that of 25 to 5 is 5; now 3 times 5 is 15, which is the ratio of 300 to 20; and we may perceive that this is general, in observing that the ratio is measured (168) by a fraction which has the antecedent for numerator, and the consequent for denominator: therefore, the compound ratio should be a fraction having the product of the two antecedents for its numerator, and for its denominator the product of the two consequents; it is then (106) the product of the two fractions which express the two composing ratios. 189. If the ratios which we multiply are equal, the compound ratio is called double ratio if we have multiplied only two ratios; triple ratio if we have multiplied three; quadru ple if we have multiplied four, and so on. For example, if we multiply the ratio of 2 to 3 by that of 4 to 6, which is equal to it, we shall have the compound ratio 8: 18, which is called double of the ratio of 2 to 3, or of 4 to 6. 190. If we have two proportions, and if we multiply them in order, that is to say, the first term of the one by the first term of the other, the second by the second, and so on, the four products thence resulting will be in proportion. For in thus multiplying two proportions, we multiply two equal ratios by two equal ratios; (172) therefore, the two compound ratios which result should be equal, and consequently the four products should be in proportion.(172) 191. Let us thence conclude that the squares, the cubes, and in general, similar powers of four quantities in proportion, are also in proportion, since to form these powers we have only to multiply the proportion by itself several times successively. 192. The square roots, cube roots, and in general, similar roots of four quantities in proportion, are also in proportion; for the ratio of the square roots of the two first terms is nothing else than the square root of the ratio of these two terms;(142 and 167) and it is the same with regard to the ratio of the square roots of the two last terms; therefore, since the two primitive ratios are supposed equal, their square roots are equal; therefore, the ratio of the square roots of the two first terms will be equal to the ratio of the square roots of the two last. The same may be proved of the cube roots, fourth roots, etc. USES OF THE PRECEDING PROPOSITIONS. 193. The propositions which we have demonstrated, and which we call the Rules of proportion, have continual application in all parts of the mathematics. We shall confine our attention here to those which belong to arithmetic, and we shall commence with those which we can deduce from what has been established,(179) which is also the basis of almost all the others. OF THE RULE OF THREE DIRECT AND SIMPLE. 194. We distinguish several kinds of Rule of Three; they all have for their object to make known a term of a propor tion of which three are given. That which we call Rule of Three Direct and Simple, is called simple, because the declaration of the questions to which we apply it never contains more than four quantities, three of which are given, and the fourth is to be found. We call it direct, because that of the four quantities which we there consider, there are always two which are not only relative to the two others, but which depend upon them, so |