Part I. Section III. Chapter 6. “Of Geometrical Ratio.”

440 The Geometrical ratio of two numbers is found by resolving the question, How many times is one of those numbers greater than the other? This is done by dividing one by the other; and the quotient will express the ratio required.

441 We have here three things to consider;

  1. the first of the two given numbers, which is called the antecedent;
  2. the other number, which is called the consequent;
  3. the ratio of the two numbers, or the quotient arising from the division of the antecedent by the consequent.

For example, if the relation of the numbers 18 and 12 be required, 18 is the antecedent, 12 is the consequent, and the ratio will be ¹⁸⁄₁₂ = 1½; whence we see that the antecedent contains the consequent once and a half.

442 It is usual to represent geometrical relation by two points, placed one above the other, between the antecedent and the consequent. Thus, \(a:b\) means the geometrical relation of these two numbers, or the ratio of \(a\) to \(b\).

We have already remarked that this sign is employed to represent division, and for this reason we make use of it here; because, in order to know the ratio, we must divide \(a\) by \(b\); the relation expressed by this sign being read simply, \(a\) is to \(b\).

443 Relation therefore is expressed by a fraction, whose numerator is the antecedent, and whose denominator is the consequent; but perspicuity requires that this fraction should be always reduced to its lowest terms: which is done, as we have already shewn, by dividing both the numerator and denominator by their greatest common divisor. Thus, the fraction ¹⁸⁄₁₂ becomes ³⁄₂, by dividing both terms by 6.

444 So that relations only differ according as their ratios are different; and there are as many different kinds of geometrical relations as we can conceive different ratios.

The first kind is undoubtedly that in which the ratio becomes unity. This case happens wlien the two numbers are equal, as in 3:3∷4:3∷\(a:a\); the ratio is here 1, and for this reason we call it the relation of equality.

Next follow those relations in which the ratio is another whole number. Thus, 4:2 the ratio is 2, and is called double ratio; 12:4 the ratio is 3, and is called triple ratio: 24:6 the ratio is 4, and is called quadruple ratio, etc.

We may next consider those relations whose ratios are expressed by fractions; such as 12:9, where the ratio is ⁴⁄₃, or 1⅓; and 18:27, where the ratio is ⅔, etc. We may also distinguish those relations in which the consequent contains exactly twice, thrice, etc. the antecedent: such are the relations 6:12, 5:15, etc. the ratio of which some call subduple, subtriple, etc. ratios.

Farther, we call that ratio rational which is an expressible number; the antecedent and consequent being integers, such as 11:7, 8:15, etc. and we call that an irrational or surd ratio, which can neither be exactly expressed by integers, nor by fractions, such as √5:8, or 4:√3.

445 Let \(a\) be the antecedent, \(b\) the consequent, and \(d\) the ratio, we know already that \(a\) and \(b\) being given, we find \(d=\frac{a}{b}\); if the consequent \(b\) were given with the ratio, we should find the antecedent \(a = bd\), because \(bd\) divided by \(b\) gives \(d\): and lastly, when the antecedent \(a\) is given, and the ratio \(d\), we find the consequent \(b = \frac{b}{d}\); for, dividing the antecedent \(a\) by the consequent \(\frac{a}{d}\), we obtain the quotient \(d\), that is to say, the ratio.

446 Every relation \(a:b\) remains the same, if we multiply or divide the antecedent and consequent by the same number, because the ratio is the same: thus, for example, let \(d\) be the ratio of \(a:b\), we have \(d = \frac{a}{b}\); now the ratio of the relation \(na : nb\) is also \(\frac{na}{nb}=d\), and that of the relation \(\frac{a}{n}:\frac{b}{n}\) is \(\frac{na}{nb}=d\).

447 When a ratio has been reduced to its lowest terms, it is easy to perceive and enunciate the relation. For example, when the ratio \(\frac{a}{b}\) has been reduced to the fraction \(\frac{p}{q}\), we say

\[a : b = p : q,\]


\[a : b :: p : q,\]

which is read, \(a\) is to \(b\) as \(p\) is to \(q\). Thus, the ratio of 6:3 being ²⁄₁, or 2, we say 6:3∷2:1. We have likewise 18:12∷3:2, and 24:18∷4:3, and 30:45∷2:3, etc. But if the ratio cannot be abridged, the relation will not become more evident; for we do not simplify it by saying 9:7∷9:7.

448 On the other hand, we may sometimes change the relation of two very great numbers into one that shall be more simple and evident, by reducing both to their lowest terms. Thus, for example, we can say, 28844:14422∷2:1; or, 10566:7044∷3:2; or, 57600:25200∷16:7.

449 In order, therefore, to express any relation in the clearest manner, it is necessary to reduce it to the smallest possible numbers; which is easily done, by dividing the two terms of it by their greatest common divisor. Thus, to reduce the relation 57600:25200 to that of 16:7, we have only to perform the single operation of dividing the numbers 57600 and 25200 by 3600, which is their greatest common divisor.

450 It is important, therefore, to know how to find the greatest common divisor of two given numbers; but this requires a Rule, which we shall explain in the following chapter.


  1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
  2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.