PROPORTION

, in Arithmetic &c, the equality| or similitude of ratios. As the four numbers 4, 8, 15, 30 are proportionals, or in proportion, because the ratio of 4 to 8 is equal or similar to the ratio of 15 to 30, both of them being the same as the ratio of 1 to 2.

Euclid, in the 5th definition of the 5th book, gives a general definition of four proportionals, or when, of four terms, the first has the same ratio to the 2d, as the 3d has to the 4th, viz, when any equimultiples whatever of the first and third being taken, and any equimultiples whatever of the 2d and 4th; if the multiple of the sirst be less than that of the 2d, the multiple of the 3d is also less than that of the 4th; or if the multiple of the first be equal to that of the 2d, the multiple of the 3d is also equal to that of the 4th; or if the multiple of the first be greater than that of the 2d, the multiple of the 3d is also greater than that of the 4th. And this definition is general for all kinds of magnitudes or quantities whatever, though a very obscure one.

Also, in the 7th book, Euclid gives another definition of proportionals, viz, when the first is the same equimultiple of the 2d, as the 3d is of the 4th, or the same part or parts of it. But this definition appertains only to numbers and commensurable quantities.

Proportion is often confounded with ratio; but they are quite different things. For, ratio is properly the relation of two magnitudes or quantities of one and the same kind; as the ratio of 4 to 8, or of 15 to 30, or of 1 to 2; and so implies or respects only two terms or things. But Proportion respects four terms or things, or two ratios which have each two terms. Though the middle term may be common to both ratios, and then the Proportion is expressed by three terms only, as 4, 8, 64, where 4 is to 8 as 8 to 64.

Proportion is also sometimes confounded with progression. In fact, the two often coincide; the difference between them only consisting in this, that progression is a particular species of Proportion, being indeed a continued Proportion, or such as has all the terms in the same ratio, viz, the 1st to the 2d, the 2d to the 3d, the 3d to the 4th, &c; as the terms 2, 4, 8, 16, &c; so that progression is a series or continuation of Proportions.

Proportion is either continual, or discrete or interrupted.

The Proportion is continual when every two adjacent terms have the same ratio, or when the consequent of each ratio is the antecedent of the next following ratio, and so all the terms form a progression; as 2, 4, 8, 16, &c; where 2 is to 4 as 4 to 8, and as 8 to 16, &c.

Discrete or interrupted Proportion, is when the consequent of the first ratio is different from the antecedent of the 2d, &c; as 2, 4, and 3, 6.

Proportion is also either Direct or Inverse.

Direct Proportion is when more requires more, or less requires less. As it will require more men to perform more work, or fewer men for less work, in the same time.

Inversc or Reciprocal Proportion, is when more requires less, or less requires more. As it will require more men to perform the same work in less time, or fewer men in more time. Ex. If 6 men can perform a piece of work in 15 days, how many men can do the same in 10 days. Then, the answer. reciprocally - as 1/15 to 1/10 so is 6 : 9 or inversely - as 10 to 15 so is 6 : 9

Proportion, again, is distinguished into Arithmetical, Geometrical, and Harmonical.

Arithmetical Proportion is the equality of two arithmetical ratios, or differences. As in the numbers 12, 9, 6; where the difference between 12 and 9, is the same as the difference between 9 and 6, viz 3.

And here the sum of the extreme terms is equal to the sum of the means, or to double the single mean when there is but one. As .

Geometrical Proportion is the equality between two geometrical ratios, or between the quotients of the terms. As in the three 9, 6, 4, where 9 is to 6 as 6 is to 4, thus denoted ; for 9/6 = 6/4, being each equal 3/2 or 1 1/2.

And in this Proportion, the rectangle or product of the extreme terms, is equal to that of the two means, or the square of the single mean when there is but one. For .

Harmonical Proportion, is when the first term is to the third, as the difference between the 1st and 2d is to the difference between the 2d and 3d; or in four terms when the 1st is to the 4th, as the difference between the 1st and 2d is to the difference between the 3d and 4th; or the reciprocals of an arithmetical Proportion are in harmonical Proportion. As 6, 4, 3; because ; or because 1/5, 1/4, 1/3 are in arithmetical Proportion, making . Also the four 24, 16, 12, 9 are in harmonical Proportion, because .

See Proportionals.

Compass of Proportion, a name by which the French, and some English authors, call the Sector.

Rule of Proportion, in Arithmetic, a rule by which a 4th term is found in Proportion to three given terms. And is popularly called the Golden Rule, or Rule of Three.