PROPORTION, in arithmetic and geometry, is a particular species of relation subsisting between groups of numbers or quantities. Notwithstanding that the idea of proportion is found to exist in perfection in the mind of every one, yet a good definition of it is a matter of extreme difficulty. The two definitions which, on the whole, are found to be least objectionable are that of Euclid, and the ordinary arithmetical definition. The lat ter states proportion to be the "equality of ratios," and throws us back on the definition of time term ratio (q.v.) ; which may most be considered as the relation of two numbers to each other, shown by a division of the one by the other. Thus, the ratio of 12 12 to 3, expressed by or 4, denotes that 12 contains 3 four times; and the ratio of 8 to 2 being also 4, we have from our definition a statement that the four numbers, 12, 3, 8 and 2, are in proportion, or, as it is commonly expressed, 12 bears to 3 the same ratio that 8 does to 2, or 12:3::8:2. In the same way, it is shown that 3:8: :13i:36; for 3 13-i- 27 3 expresses the ratio of the first to the second, and — = _7 It will be gathered — from the two arithmetical proportions here given,"and from any others that can be formed, that "the product of the first and last terms (the extremes) is equal to the product of the second and third terms (the means);" and upon this property of proportional numbers directly depends the arithmetical rule called "proportion," etc. The object of this rule is to find a fourth proportional to three given numbers, i.e., a number to which the third bears the same ratio that the first does to the second, and the number is at once found by multiplying together the second and third terms, and dividing the product by the first. Proportion is illustrated arithmetically by such problems as, "If four yards cost six shillings, what will ten cost ?" Here, 15 being the fourth proportional to 4, 6, and 10, fifteen shillings is the answer. The distinction of proportion into direct and inverse is not only quite unnecessary, but highly mischievous, as it tends to create the idea, that it is possible for more than one kind of proportion to subsist. Continued proportion, indi cates a property of every three consecutive or equidistant terms in a " geometrical pro gression" (q.v.)—for instance, in the series, 2, 4, 8, 16, 32 .. , 2: 4::4: 8, 4: 8:: 8:16,
etc., or 2:8::S:32, etc. In the above remarks, all consideration of incommensurable quantities (q.v.) has been omitted.—The definition given by Euclid is as follows : Four magnitudes are proportional, when, any equi-multiples whatever being taken of the first and third, and any whatever of the second and fourth, according as the multiple of the first is greater, equal to, or less than that of the second, the multiple of the third is also greater, equal to, or less than that of the fourth—i.e.. A, B, C, D are pmpor tionals, when, if mA is greater than nB, inC is greater than nD; if mA is equal to nil, m0 is equal to irD; if mA is less thanda MC is less than nD; in and n being any multi ples whatsoever. The apparent cumbrousness and circumlocution in this definition arise from Euclid's endeavor to include incommensurable quantities; throwing them out of account, it is sufficient to say that four magnitudes are proportional, if, like multi ples being taken of the first and third, and like of the second and fourth, when the multiple of the first is equal to the multiple of the second, the multiple of the third is equal to the multiple of the fourth. Abundance of illustrations of the general definition will be found in the fifth book of Euclid, and of the particular one in the notes appended to some of the later editions of the same work; it will be sufficient here to give an arith metical instance of the working of the particular definition. Taking the four numbers of a previous example-12, 3, 8, 2i of 12 and 8 take multiples by 4, mid of and 2 by 16, then 12 X 4 (the multiple of the hrst)=.3 x 16 (time multiple of the second), and 8 X 4 (the multiple of the third)=2 X 16 (the multiple of the fourth). In this example, the two multiples were so taken that the multiple of the first would be equal to time multiple of the second, and when it was found that the multiple of the third was also equal to the multiple of the fourth, the proportionality of the four numbers was established.