part, of the second? The consequence would have been, that we should have had a term, or number, fifteen times larger than that required. But this would be a matter of no difficulty; for it would, as you will see, be set right at once, and our purpose be gained, by dividing the over-large product by 15. Let us write this process down: 405 x 4518225, and 18225 15 = 1215. Which 1215 bears the same proportion to 405 as does 45 to 15. AND THIS IS THE RULE; this, when the terms are properly placed, this MULTIPLYING THE SECOND AND THIRD TERMS TOGETHER, AND DIVIDING THE PRODUCT BY THE FIRST, avoids all the difficulties arising from the occurrence of fractions in the course of the process, and gives us, in all cases, any proportional terms we may require. This is the RULE of PROPORTION, commonly called the RULE of THREE; and, in their admiration of it, and in testimony of their sense of its great value, the learned of former times bestowed on it the name of GOLDEN RULE; a title which it richly merits, as you will see, when you become acquainted with its great and various uses. 272. It is almost superfluous to employ another word on this subject, and quite unnecessary to give in this lesson, any further examples, or to make any further experiments. I have stated that the Rule laid down is easily applicable to all cases, to every degree of proportion amongst numbers and quantities; and this will be seen in the ensuing lessons, which, under the names of Rule of Three, Single, Double, and Inverse; of the Chain Rule, or Compound Proportion, apply the principles of proportion to affairs of business. THE RULE OF THREE; DIRECT AND INVERSE. 273. This is the Rule towards which I have been conducting my pupil throughout the whole of the preceding dissertation. And, that this rule, to say nothing of the other rules which immediately follow, and which grow out of the same Principle; that this rule is worthy of such a preparation, worthy of so careful a development of its Principles, will soon become apparent. 274. Numbers, which are expressed by figures, are employed to describe quantities. As, for instance, if we would describe, in writing, a quantity of any thing that is measured by the foot of twelve inches, the party to whom we would describe it, being previously acquainted with the length of the foot, requires only to be informed of the number of feet, which number we state in figures. So that figures describe numbers, and numbers describe quantities; and quantities, too, of every description, whether of weight, of measure, of extent, or of time. 275. This being the office, that is to say, the use of numbers, and having fully learned, in the former rules, not only this use, but how to join together, and to separate these numbers, in every possible mode, almost all that it may be desirable for you to learn further, is the method of duly proportioning quantities towards each other; as, for instance, suppose you have purchased a lot of goods, a piece of land, or any other species of property, for a certain sum, and that you wish to know, at what rate you should retail your purchase, in order to gain a certain sum by the whole transaction; or, suppose you have to adjust the claims of a number of parties, whether legatees, partners, or creditors, to certain effects, or to their value in money; or, that, if, according to a certain scale, you have to lay in provisions, for a specified time, for a certain number of persons; or, that, having a certain piece of work to get done, requiring a number of men, and that it were your wish to know, according to a certain rate, or scale, with which you may be provided, what number of men ought to be employed, in order to have the work done within a given time; to solve problems of this description, that is, to adjust proportions, is almost all that you have now to learn; and these, such problems as these, and a countless number of others of a similar description, which are for ever occurring, you are enabled, by this rule, to treat with ease and success. Having stated this, let us proceed to examples; the several methods of treating which, I shall state as occasion and opportunity occur. 276. First example. A man dies, having, by will, ordered his effects to be sold, and legacies, amounting altogether to £755. to be paid: that is to say, to A he willed £200; to B £270 4s. 6d.; and to C £284 15s. 6d. But, after paying necessary expences, it is found, that the effects produce only £604; and the question arises. How much of this £604 should be paid to each of the legatees, in order that each may have his just proportion? And, first, what is A to receive, instead of £200 ? 277. The proportion in which the money is to be divided, is, of course, the same as that which the produce of the effects bears to the sum of the legacies named in the will; and the question, therefore, must be stated thus: As that sum which ought to be £755, proves to be only £604, what, according to the same proportion, ought £200 to be? But, stated with the signs, instead of words, as taught in paragraph 264, the question would stand and the rule, in order to bring out the answer, as shown in the last lesson, paragraph 271, is, "to multiply the second and third terms together, and to divide the product by the first ;" and this, in the case before us, gives 160, which is the number of pounds due to the first-named legatee, A. Instead, however, of shares in the effects of a deceased person, these several sums might be shares in a partnership concern, or shares of creditors in an insolvent estate. And this rule would, with perfect correctness, adjust the claims of the several parties. 278. To return to the example: In fixing the first legacy, I purposely avoided the introduction of small money, or fractions of a pound, in order to keep the question as simple as possible. But these simple numbers, as before stated, are things not to be looked for in matters of real business; and, therefore, for cases in which fractions do occur, we must be provided with a suitable method; which method forms a part of the rule of three; and which, therefore, I now proceed to explain, and to join to the great, or main rule already laid down. 279. The first legacy is £200. and the question arising thereon, as before stated, stands thus; 755 : 604 : : 200. Now, you know, by the Rule, that we are to multiply the second and the third terms together; but, observe, if we were to reduce the third term, which now represents pounds, into shillings; that is to say, if we were to multiply it by twenty, and then to multiply this number of shillings by the second term, we should have, as : : the product, 2416000, which, divided by the first term, would give 3200, a sum just twenty times as great as it ought to be; which excessive sum, as you will immediately perceive, arises from the circumstance of our having, in reducing it into shillings, made the third term twenty times larger than its just proportion; for the proportion was 755 : 604 4000. But, mark; had we, on reducing the third term to shillings, reduced the first term, likewise, to the same denomination; had we done this, we should have preserved the proportion, and the answer would have come out correct; and correct, too, would it have been, had these two terms been reduced into pence, or into farthings; that is to say, if both of them had been so reduced, and the proportion between them had thereby been preserved. Hence, then, arises the second branch of the rule of which we are treating; that is to say, that the first and third terms are to be of the same denomination. And you must bear in mind, that if they be not so before you begin to work your question, the first thing you have to do is, to reduce these two terms to the same denomination. 280. There remains one other branch of this rule to be observed on here: you will find, if you have not already noticed it, that you cannot multiply sums consisting of pounds, shillings, and pence; or hundreds weight, quarters, and pounds, or any such compound terms, by certain large numbers; neither can you multiply large numbers by compound sums of this description. The consequence is, that when any of your terms in this rule consist of such sums, you must almost always reduce them into one even and simple number, or denomination, as you will see done in the case of the third term in the following example, which term, being £270. 4s. 6d. is reduced into pence; and, in order to balance it, nd preserve the proportion, as I have just taught, 7 |