[26] This is a very incorrect name, since the term ‘arithmetical’ applies equally to every notion in this book. It is necessary, however, that the pupil should use words in the sense in which they will be used in his succeeding studies.
[27] The same remark may be made here as was made in the note on the term ‘arithmetical proportion,’ page 101. The word ‘geometrical’ is, generally speaking, dropped, except when we wish to distinguish between this kind of proportion and that which has been called arithmetical.
[28] A theorem is a general mathematical fact: thus, that every number is divisible by four when its last two figures are divisible by four, is a theorem; that in every proportion the product of the extremes is equal to the product of the means, is another.
[29] If bx be substituted for a in any expression which is homogeneous with respect to a and b, the pupil may easily see that b must occur in every term as often as there are units in the degree of the expression: thus, aa + ab becomes bxbx + bxb or bb(xx + x); aaa + bbb becomes bxbxbx + bbb or bbb(xxx + 1); and so on.
[30] The difference between this problem and the last is left to the ingenuity of the pupil.
[31] 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.
[32] Since this was first written, the accident has happened. The standard yard was so injured as to be rendered useless by the fire at the Houses of Parliament.
[33] The minute and second are often marked thus, 1′, 1″: but this notation is now almost entirely appropriated to the minute and second of angular measure.
[34] The measures in italics are those which it is most necessary that the student should learn by heart.
[35] The lengths of the pendulums which will vibrate in one second are slightly different in different latitudes. Greenwich is chosen as the station of the Royal Observatory. We may add, that much doubt is now entertained as to the system of standards derived from nature being capable of that extreme accuracy which was once attributed to it.