Note 31 — BOOK II.

It is not true, that if we choose any quantity as a unit, any other quantity of the same kind can be exactly represented either by a certain number of units, or of parts of a unit. To understand how this is proved, the pupil would require more knowledge than he can be supposed to have; but we can shew him that, for any thing he knows to the contrary, there may be quantities which are neither units nor parts of the unit. Take a mathematical line of one foot in length, divide it into ten parts, each of those parts into ten parts, and so on continually. If a point A be taken at hazard in the line, it does not appear self-evident that if the decimal division be continued ever so far, one of the points of division must at last fall exactly on A: neither would the same appear necessarily true if the division were made into sevenths, or elevenths, or in any other way. There may then possibly be a part of a foot which is no exact numerical fraction whatever of the foot; and this, in a higher branch of mathematics, is found to be the case times without number. What is meant in the words on which this note is written, is, that any part of a foot can be represented as nearly as we please by a numerical fraction of it; and this is sufficient for practical purposes.