In mathematics, especially geometry, a ratio expresses the magnitude of quantities relative to each other rather. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Mathematically, a proportion is defined as the equality of two ratios. However in common usage the word proportion is used to indicate a ratio, especially the ratio of a part to a whole.
Notation and terminology
The ratio of quantities A and B can be expressed as:
- the ratio of A to B
- as B is to A
The quantities A and B are sometimes called terms with A being the antecedent and B being the consequent.
The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D. Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion.
History and etymology
It would be impossible to trace the origin of the concept of ratio since the ideas from which it developed would have been familiar to preliterate cultures. For example the idea of one village being twice as large as another or a distance being half that of another are so basic that they would have been understood in prehistoric society. However, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος (logos) appearing in Book V of Euclid's Elements. Early translators rendered this into Latin as ratio, meaning "reason". However a more modern interpretation of Euclid's meaning is more akin to computation or reckoning. Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.
Euclid collected the results appearing the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers. The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.
The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seem from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold. First, there was the previously mentioned reluctance to accept irrational numbers as true numbers. Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.
Man power is indispensable to the spread of civilization. All things equal, a numerous people will dominate the civilization of a smaller race. Hence failure to increase in numbers up to a certain point prevents the full realization of national destiny, but there comes a point in population increase where further growth is suicidal. Multiplication of numbers beyond the optimum of the normal man-land ratio means either a lowering of the standards of living or an immediate expansion of territorial boundaries by peaceful penetration or by military conquest, forcible occupation.
- "Ratio" The Penny Cyclopædia vol. 19, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
- "Proportion" New International Encyclopedia, Vol. 19 2nd ed. (1916) Dodd Mead & Co. pp270-271
- "Ratio and Proportion" Fundamentals of practical mathematics, George Wentworth, David Eugene Smith, Herbert Druery Harper (1922) Ginn and Co. pp. 55ff
- The thirteen books of Euclid's Elements, vol 2. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Press. pp. 112ff. https://books.google.com/books?id=lxkPAAAAIAAJ&pg=RA2-PA112.
- D.E. Smith, History of Mathematics, vol 2 Dover (1958) pp. 477ff