Four Meanings of Equality.
Although it might seem that there are few terms more free from ambiguity than the term equal, yet scientific men do employ it with at least four meanings, which it is desirable to distinguish. These meanings I may describe as
By absolute equality we signify that which is complete and perfect to the last degree; but it is obvious that we can only know such equality in a theoretical or hypothetical manner. The areas of two triangles standing upon the same base and between the same parallels are absolutely equal. Hippocrates beautifully proved that the area of a lunula or figure contained between two segments of circles was absolutely equal to that of a certain right-angled triangle. As a general rule all geometrical and other elementary mathematical theorems involve absolute equality.
De Morgan proposed to describe as sub-equal those quantities which are equal within an infinitely small quantity, so that x is sub-equal to x + dx. The differential calculus may be said to arise out of the neglect of infinitely small quantities, and in mathematical science other subtle distinctions may have to be drawn between kinds of equality, as De Morgan has shown in a remarkable memoir “On Infinity; and on the sign of Equality.”20
Apparent equality is that with which physical science deals. Those magnitudes are apparently equal which differ only by an imperceptible quantity. To the carpenter anything less than the hundredth part of an inch is non-existent; there are few arts or artists to which the hundred-thousandth of an inch is of any account. Since all coincidence between physical magnitudes is judged by one or other sense, we must be restricted to a knowledge of apparent equality.
In reality even apparent equality is rarely to be expected. More commonly experiments will give only probable equality, that is results