The doctrine of ratios and proportion is introduced by Euclid as a part of his system of geometry ; and the student seldom fails to remark, that in the treatises on algebra, the same subject is presented under a considerably different form; though he is usually quite unable to determine wherein the essential difference consists; and would probably find but few teachers who could precisely point out the distinction to him.
In many different fields, empirical phenomena appear to obey a certain general law, which can be called the Law of Large Numbers. This law states that the ratios of numbers derived from the observation of a very large number of similar events remain practically constant, provided that these events are governed partly by constant factors and partly by variable factors whose variations are irregular and do not cause a systematic change in a definite direction.siméon denis poisson
On an attentive examination of the methods adopted by modern elementary writers, in laying down the first principles of ratios and proportion, and especially in commenting upon Euclid, I long since experienced a conviction of the extremely unsatisfactory nature of most of their views; and this chiefly, as appearing to me to involve inadequate ideas of Euclid's real principle in treating of proportionals in his 5th book, and of the nature of the quantities which form the subject of investigation.
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