RATIO

, according to Euclid, is the habitude or relation of two magnitudes of the same kind in respect of quantity. So the ratio of 2 to 1 is double, that of 3 to 1 triple, &c. Several mathematicians have found fault with Euclid's definition of a Ratio, and others have as much defended it, especially Dr. Barrow, in his Mathematical Lectures, with great skill and learning.

Ratio is sometimes confounded with proportion, but very improperly, as being quite different things; for proportion is the similitude or equality or identity of two Ratios. So the Ratio of 6 to 2 is the same as that of 3 to 1, and the Ratio of 15 to 5 is that of 3 to 1 also; and therefore the Ratio of 6 to 2 is similar or equal or the same with that of 15 to 5, which constitutes proportion, which is thus expressed, 6 is to 2 as 15 to 5, or thus 6 : 2 :: 15 : 5, which means the same thing. So that Ratio exists between two terms, but proportion between two Ratios or four terms.

The two quantities that are compared, are called the terms of the Ratio, as 6 and 2; the first of these 6 being called the antecedent, and the latter 2 the consequent. Also the index or exponent of the Ratio, is the quotient of the two terms: so the index of the Ratio of 6 to 2 is 6/2 or 3, and which is therefore called a triple Ratio.

Wolfius distinguishes Ratios into rational and irrational.

Rational Ratio is that which can be expressed between two rational numbers; as the Ratio of 6 to 2, or of 6√3 to 2√3, 3 to 1. And

Irrational Ratio is that which cannot be expressed by that of one rational number to another; as the Ratio of √6 to √2, or of √3 to root √1, that is √3 to 1, which cannot be expressed in rational numbers.

When the two terms of a Ratio are equal, the Ratio is said to be that of equality; as of 3 to 3, whose index is 1, denoting the single or equal Ratio. But when the terms are not equal, as of 6 to 2, it is a Ratio of inequality.

Farther, when the antecedent is the greater term, as in 6 to 2, it is said to be the Ratio of greater inequality: but when the antecedent is the less term, as in the Ratio of 2 to 6, it is said to be the Ratio of | less inequality. In the former case, if the less term be an aliquot part of the greater, the Ratio of greater inequality is said to be multiplex or multiple; and the Ratio of the less inequality, sub-multiple. Particularly, in the first case, if the exponent of the Ratio be 2, as in 6 to 3, the Ratio is called duple or double; if 3, as in 6 to 2, it is triple; and so on. In the second case, if the Ratio be 1/2, as in 3 to 6, the Ratio is called subduple; if 1/3, as in 2 to 6, it is subtriple; and so on.

If the greater term contain the less once, and one aliquot part of the same over; the Ratio of the greater inequality is called superparticular, and the Ratio of the less subsuperparticular. Particularly, in the first case, if the exponent be 3/2 or 1 1/2, it is called sesquialterate; if 4/3 or 1 1/3, sesquitertial; &c. In the other case, if the exponent be 7/3, the Ratio is called subsesquialterate; if 3/4, it is subsesquitertial.

When the greater term contains the less once and several aliquot parts over, the Ratio of the greater inequality is called superpartiens, and that of the less inequality is subsuperpartiens. Particularly, in the former case, if the exponent be 5/3 or 1 2/3, the Ratio is called superbipartiens tertias; if the exponent be 7/4 or 1 3/4, supertripartiens quartas; if 11/7 or 1 4/7, superquadripartiens septimas; &c. In the latter case, if the exponent be the reciprocals of the former, or 3/5, the Ratio is called subsuperbipartiens tertias; if 4/7, subsupertripartiens quartas; if 7/11, subsuperquadripartiens septimas; &c.

When the greater term contains the less several times, and some one part over; the ratio of the greater inequality is called multiplex superparticular; and the Ratio of the less inequality is called submultiplex subsuperparticular. Particularly, in the former case, if the exponent be 5/2 or 2 1/2, the ratio is called dupla sesquialtera; if 13/4 or 3 1/4, tripla sesquiquarta, &c. In the latter case, if the exponent be 2/5, the Ratio is called subdupla subsesquialtera; if 4/13, subtripla subsesquiquarta, &c. Lastly, when the greater term contains the less several times, and several aliquot parts over; the Ratio of the greater inequality is called multiplex superpartiens; that of the less inequality, submultiplex subsuperpartiens. Particularly, in the former case, if the exponent be 8/3 or 2 2/3, the Ratio is called dupla superbipartiens tertias; if 25/7 or 3 4/7, tripla superbiquadripartiens septimas, &c. In the latter case, if the exponent be 3/8, the Ratio is called subdupla subsuperbipartiens tertias; if 7/25, subtripla subsuperquadripartiens septimas; &c.

These are the various denominations of rational Ratios, names which are very necessary to the reading of the ancient authors; though they occur but rarely among the modern writers, who use instead of them the smallest numeral terms of the Ratios; such 2 to 1 for duple, and 3 to 2 for sesquialterate, &c.

Compound Ratio, is that which is made up of two or more other Ratios, viz, by multiplying the exponents together, and so producing the compound Ratio of the product of all the antecedents to the product of all the consequents.

Thus the compound Ratio of 5	to  3,
	and 7	to  4,
is the Ratio of	35	to 12.

Particularly, if a Ratio be compounded of two equal Ratios, it is called the duplicate Ratio; if of three equal Ratios, the triplicate Ratio; if of four equal Ratios, the quadruplicate Ratio; and so on, according to the powers of the exponents, for all multiplicate Ratios. So the several multiplicate Ratios of

the simple Ratio of	3 to 2, are thus, viz.
the duplicate Ratio	9 :  4,
the triplicate Ratio	27 :  8,
the quadruplicate Ratio	81 : 16, &c.

Properties of Ratios. Some of the more remarkable properties of Ratios are as follow:

1. The like multiples, or the like parts, of the terms of a Ratio, have the same Ratio as the terms themselves. So a : b, and na : nb, and a/n : b/n are all the same Ratio.

2. If to, or from, the terms of any Ratio, be added or subtracted either their like parts, or their like multiples, the sums or remainders will still have the same Ratio. So a : b, and a ± na : b ± nb, and a ± a/n : b ± b/n are all the same Ratio.

3. When there are several quantities in the same continued Ratio, a, b, c, d, e, &c. whatever Ratio the first has to the 2d, the 1st to the 3d has the duplicate of that Ratio, the 1st to the 4th has the triplicate of that Ratio, the 1st to the 5th has the quadruplicate of it, and so on. Thus, the terms of the continued Ratio being 1, r, r², r³, r⁴, r⁵, &c, where each term has to the following one the Ratio of 1 to r, the Ratio of the 1st to the 2d; then 1 : r² is the duplicate, 1 : r³ the triplicate, 1 : r⁴ the quadruplicate, and so on, according to the powers of 1 : r.

For other properties see Proportion.

To approximate to a Ratio in smaller Terms.—Dr. Wallis, in a small tract at the end of Horrox's works, treats of the nature and solution of this problem, but in a very tedious way; and he has prosecuted the same to a great length in his Algebra, chap. 10 and 11, where he particularly applies it to the Ratio of the diameter of a circle to its circumference. Mr. Huygens too has given a solution, with the reasons of it, in a much shorter and more natural way, in his Descrip. Autom. Planet. Opera Reliqua, vol. 1, pa. 174.

So also has Mr. Cotes, at the beginning of his Harmon. Mensurarum. And several other persons have done the same thing, by the same or similar methods. The problem is very useful, for expressing a Ratio in small numbers, that shall be near enough in practice, to any given Ratio in large numbers, such as that of the diameter of a circle to its circumference. The principle of all these methods, consists in reducing the terms of the proposed Ratio into a series of what are called continued fractions, by dividing the greater term by the less, and the less by the remainder, and so on, always the last divisor by the last remainder, after the manner of finding the greatest common measure of the two terms; then connecting all the quotients &c together in a series of continued fractions; and lastly collecting gradually these fractions together one after another. So if b/a be any fraction, or exponent of any Ratio; then dividing thus, | gives c, e, g, i, &c, for the several quotients, and these, formed in the usual way, give the approximate value of the given Ratio in a series of continued fractions; thus, . Then collecting the terms of this series, one after another, so many values of b/a are obtained, always nearer and nearer; the first value being c or c/1, the next , the 3d value ; in like manner, the 4th value is ; the 5th value is ; &c. From whence comes this general rule: Having found any two of these values, multiply the terms of the latter of them by the next quotient, and to the two products add the corresponding terms of the former value, and the sums will be the terms of the next value, &c.

For example, let it be required to find a series of Rations in lesser numbers, constantly approaching to the Ratio of 100000 to 314159, or nearly the Ratio of the diameter of a circle to its circumference. Here first dividing, thus, there are obtained the quotients 3, 7, 15, 1, 25, 1, 7, 4. Hence 3 or 3/1 = c, the 1st value; , the 2d value; , the 3d value; , the 4th value; and so on; where the successive continual approximating values of the proposed Ratio are 3/1, 22/7, 333/106, 355/113, &c; the 2d of these, viz. 22/7, being the approximation of Archimedes; and the 4th, viz 355/133, is that of Metius, which is very near the truth, being equal

	to 3.1415929,
the more accurate Ratio being	" 3.1415927.

The doctrine of Ratios and Proportions, as delivered by Euclid, in the fifth book of his Elements, is considered by most persons as very obscure and objectionable, particularly the definition of proportionality; and several ingenious gentlemen have endeavoured to elucidate that subject. Among these, the Rev. Mr. Abram Robertson, of Christ Church College, Oxford, lecturer in geometry in that university, printed a neat little paper there in 1789, for the use of his classes, being a demonstration of that definition, in 7 propositions, the substance of which is as follows. He first premises this advertisement:

“As demonstrations depending upon proportionality pervade every branch of mathematical science, it is a matter of the highest importance to establish it upon clear and indisputable principles. Most mathematicians, both ancient and modern, have been of opinion that Euclid has fallen short of his usual perspicuity in this particular. Some have questioned the truth of the definition upon which he has founded it, and, almost all who have admitted its truth and validity have objected to it as a definition. The author of the following propositions ranks himself amongst objectors of the last mentioned description. He thinks that Euclid must have founded the definition in question upon the reasoning contained in the first six demonstrations here given, or upon a similar train of thinking; and in his opinion a definition ought to be as simple, or as free from a multiplicity of conditions, as the subject will admit.”

He then lays down these four definitions:

“1. Ratio is the relation which one magnitude has to another, of the same kind, with respect to quantity.”

“2. If the first of four magnitudes be exactly as great when compared to the second, as the third is when compared to the fourth, the first is said to have to the second the same Ratio that the third has to the fourth.”

“3. If the first of four magnitudes be greater, when compared to the second, than the third is when compared to the fourth, the first is said to have to the second a greater Ratio than the third has to the fourth.”

“4. If the first of four magnitudes be less, when compared to the second, than the third is when compared to the fourth, the first is said to have to the second a less Ratio than the third has to the fourth.”

Mr. Robertson then delivers the propositions, which are the following:

“Prop. 1. If the first of four magnitudes have to the second, the same Ratio which the third has to the fourth; | then, if the first be equal to the second, the third is equal to the fourth; if greater, greater; if less, less.”

“Prop. 2. If the first of four magnitudes be to the second as the third to the fourth, and if any equimultiples whatever of the first and third be taken, and also any equimultiples of the second and fourth; the multiple of the first will be to the multiple of the second as the multiple of the third to the multiple of the fourth.”

“Prop. 3. If the first of four magnitudes be to the second as the third to the fourth, and if any like aliquot parts whatever be taken of the first and third, and any like aliquot parts whatever of the second and fourth, the part of the first will be to the part of the second as the part of the third to the part of the fourth.”

“Prop. 4. If the first of four magnitudes be to the second as the third to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and fourth; if the multiple of the first be equal to the multiple of the second, the multiple of the third will be equal to the multiple of the fourth; if greater, greater; if less, less.”

“Prop. 5. If the first of four magnitudes be to the second as the third is to a magnitude less than the fourth, then it is possible to take certain equimultiples of the first and third, and certain equimultiples of the second and fourth, such, that the multiple of the first shall be greater than the multiple of the second, but the multiple of the third not greater than the multiple of the fourth.”

“Prop. 6. If the first of four magnitudes be to the second as the third is to a magnitude greater than the fourth, then certain equimultiples can be taken of the first and third, and certain equimultiples of the second and fourth, such, that the multiple of the first shall be less than the multiple of the second, but the multiple of the third not less than the multiple of the fourth.”

“Prop. 7. If any equimultiples whatever be taken of the first and third of four magnitudes, and any equimultiples whatever of the second and fourth; and if when the multiple of the first is less than that of the second, the multiple of the third is also less than that of the fourth; or if when the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth; or if when the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth: then, the first of the four magnitudes shall be to the second as the third to the fourth.”

And all these propositions Mr. Robertson demonstrates by strict mathematical reasoning.