Proportion,

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 latter states proportion to be the 'equality of ratios,' and throws us back on the definition of the term Ratio (q.v.), which may most simply 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 to 3, expressed by $1\backslash frac\{2\}\{3\}$, 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 $3:8::13\backslash frac\{1\}\{2\}:36$; for $\backslash frac\{3\}\{8\}$ expresses the ratio of the first to the second, and $\backslash frac\{13\backslash frac\{1\}\{2\}\}\{36\}\; =\; \backslash frac\{27\}\{72\}\; =\; \backslash frac\{3\}\{8\}$. 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,' &c. 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 indicates a property of every three consecutive or equidistant terms in a 'Geometrical Progression' (q.v.)—for instance, in the series 2, 4, 8, 16, 32, ..., $2:4::4:8$, $4:8::8:16$, &c., or $2:8::8:32$, &c. In the above remarks all consideration of incommensurable quantities has been omitted. The definition given by Euclid is as follows: Four magnitudes are proportional when, any equimultiples 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 proportionals when, if $mA$ is greater than $nB$, $mC$ is greater than $nD$; if $mA$ is equal to $nB$, $mC$ is equal to $nD$; if $mA$ is less than $nB$, $mC$ is less than $nD$; $m$ and $n$ being any multiples whatsoever. The apparent cumbrousness and circum- locution in this definition arise from Euclid's endeavour to include incommensurable quantities; throwing them out of account, it is sufficient to say that four magnitudes are proportional if, like multiples 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. For example, take the numbers 12, 3, 8, and 2; multiply 12 and 3 respectively by such numbers as will give equal products, say by 4 and 16, the product being then 48 in both cases; the products of the remaining numbers, 8 and 2, by these multipliers are equal to one another, being both 32; and therefore these four numbers are proportional.