THREE, RULE OF, the technical name of the rule in arithmetic by which, three quantities being given, the first and second of one kind, a fourth is found such that the four are in proportion, or that the first is the same multiple part, or parts, of the second, which the third is of the fourth.
In the earliest modern treatises are found the explanatory headings of this process, from which the denomination rule of three has been formed by abbreviation. Almost all such abbreviations date from tho time when systems of commercial arithmetic began to be written—that is, about the beginning of the 16th century. Before that time, such books as were written always contained demonstrations from full definitions; and it was not judged necessary to provide the simple case of finding a fourth proportional to three given numbers with a separate name, or to divide the rule for doing it from others. This, however, was done by traders in their daily practice, who separated the rule of three from the other parts of arithmetic, and called it the golden rule, an older term, probably, than rule of three. Bishop Tonstal, (' Ars supputandi; 1522) begins his chapter on the " Regula do tribes notis quattum ignotum cehhmonstmutibus" in this manner : " Prw.cipna onmilim regal est mite de trihus notis quartum ignotuni in notictam educentibus ab Arithmeticis traditur. Vulgus regular' aureate vocal; quia ince ceteris Arithmeticar regulis velut ceteris metallis minim pneatct.^ Robert Recorde (1510) calls it tho " hate of the rule of proportions, which° for his excellencie is called the golden rule." Humphrey Baker (1562) uses the phrase "rule of three," and says that " the philosophers did name it the golden rule, . . . but nowe iu these latter dales, by us it is called the rule of three." Tho immense variety of questions which are to be solved by finding a fourth proportional defies classification ; but they may all be reduced to one form, though it may in particular eases not be easy to see the modo of reduction. That form is :—A produces a; what will c produce! It may be that it is money which produces goods, or goods which produce money, or money which produces interest, or money of one country which produces money of another, or time which produces distance travelled, &e., &c., &c. The difficulty to beginners is the reduction of the question given to the above simple form, which must be done before what ie (or used to be) called the statement of the question can be made—namely, the writing down tho numbers A, 9, c, in the proper order, with the marks of proportion between them : It is proper enough to say that this is a question of tirorxtrtiou when numbers only are considered, but absurd when the things represented by the numbers are used instead of the numbers. Thus, if 5 pence buy 10 apples, 7 pence will buy 14 apples, and tho number 5 is to 7 as 10 is to 14, or 5 is the same fraction of 7 as 10 is of 14. But it is absurd to say that 5 pence bear the name proportion to 10 apples that 7 pence bear to 14 apples : simply because 5 pence are not any assign able fraction of 10 apples. That there is a relation is true; but that relation is not proportion. Thus, it is not absurd to say, in the
common language of the rule, As 5 pence are to 10 apples, so are 7 pence to 14 apples; for the first does stand to the second in the same relation as the third to the fourth : 5 pence must, at all rates, do as much towards the purchase of 10 apples as 7 pence towards that of 14 apples. With this understanding, there is no objection to the common mode of statement, and the proof of the rule is as follows :—If A of the first produce B of the second, then, at the same rate of production, 1 of the first must of the second; whence c of the first must A produce c x '2, or of the second.
A A The importance of the rule of three induced arithmeticians to attach two other rules to it : the inverse rule of three (called by Recorde, Baker, &c., the backer rule) ; and the double rule of three. Some of the writers of Cocker's school, 'apparently by an abbreviation of his words, tells us that the rule of three inverse is used "when less requires more and more requires less ;" meaning that the greater the third of the given numbers, the less will be the answer, and rice verse/. Thus, suppose that 10/. has been lent me for 3 months, and I want to know how long I ought to lend a given sum (other than 101.) in return : evidently the more I lend, the less the time for which I ought to lend it. If the sum be I5/., then 3 months is to the time required, not as 10 to 15, but in its inverse ratio, as 15 to 10, or 15 : 10 : : 3 : 3 x 10+15, or 2; and 2 months is the answer required.
The double rule of three (at least in the class of questions which are usually considered as falling under it) is applied where time is an element in the production which the question supposes. For example: supposing it known that A men can pave n square feet in c days, it may be asked how many men can pave 6 square feet in c days, or bow many square feet can a men pave in c days, or how many days will it take a men to pave 6 square feet. If we write down the data and answer in two lines, and in the following order—force employed—effect produced—time of production—thus : the rule is—Take such an answer as will make the extremes of each line multiplied by the mean of the other, the same in both. That is, let Abo=ase, and according as a, 6, or c, is to be found, the mode of working is as follows : Abe anc Abo a = = — 'c se' Ao' an r The proof is as follows:—One man in o days could pave t. square feet, and in one day square feet. By similar reasoning one man in one day could pave square feet. B b AC ac' • or anc = The principal caution which a beginner requires is, not to suppose that the rule of three (or the rule of finding a fourth quantity which, which three others, shall constitute a proportion) is to be applied In all cases in which three quantities are given to find a fourth. That such a caution is necessary arises from the defect of works on arithmetic, which frequently exhibit this rule without any mention of proportion, and leave it to be inferred that there is but one way of obtaining a fourth quantity from three others.