It might seem as if, in the consideration of the term zero, we had commenced an explanation of negative quantities, and had obtained a justification of the phrase less than zero, if not of less than noticing. This may be true to a certain extent, too limited however for the pur poses of algebra, and not sufficiently expressive of the actual meaning of the words. When the distinction of positive and negative quantities is explained and adopted, the terms greater and less are no longer used in their simple arithmetical meaning, but take a wider signification, such as will allow old theorems of arithmetic to remain true under the same phraseology as before. After an express extension of signification has been accorded to these terms, it is not wonderful that uses of them should be perfectly allowable which could not be made if we retained the old significations. Those who use the extended meanings, without fully understanding and admitting them, will make a mystery of algebra : those who refuse to make the extensions, and yet charge others who do not refuse with falling into all the absurdities which extended uses without extended meanings present to themselves, are precisely in the condition of the honest tar who asserted that the French were such fools as not to know the difference between a cabbage and a shoe (chow). But those again who, professing to use extended meanings, do not take care to make their logic conformable to them, but neglect to distinguish between premises which are true of one set of meanings and not of the other, will fall into such mistakes as would be made by him who should conclude that blood is salt water, because both circulate in arms (of the human body and of the sea).
Admitting the scale of positive and negative numbers, it is obvious that on the right of 0, on which we have quantities common to pure arithmetic and algebra, we pass from the greater to the less by moving our eyes from right to left ; while on the left we have no meaning at all of greater and less yet established. Let us agree then that we are to pass from what we will call the greater to what we will call the less by passing from right to left in all cases; and there is no mystery in our meaning when we say that all negative quantities are less than 0, and that —10 is less than —5.
But is this convention a purely arbitrary one ? We answer that it rather bears the character of interpretation [INTERPRETATION] than of convention. Having new modes of quantity, with corresponding extensions of addition and subtraction, we are rather to ask what greater and less ought to mean than what, with liberty of choice, we shall make them mean. The great characteristics of greater and less (or more and less) in arithmetic are, that the more you add the more you get, and that the more you take away the less you leave, and rice versa. The preceding extensions of greater and less are the only ones which will allow of these theorems remaining universally true. Thus and 8 is greater than 7.
It is however to be remembered, though no rule has been laid down upon the subject, that it may be gathered from the practice of writers that the terms smaller and diminution do not accompany less in its extended meaning. The former term is particularly used in the Diffe rential Calculus to denote an approach to 0, which in a negative quantity is algebraic increase, and in a positive quantity diminution. And many, perhaps all, writers on the Differential Calculus, are lax hi their use of all the comparative terms, sometimes employing them in the algebraical and sometimes only in the arithmetical sense. The inconvenience is not very great, as a student must have learned to contend with greater difficulties than those of an unexplained use of dubious terms, before he is able to make his way to the higher mathe matics. But it may be useful to give him a hint that, in reading works of analysis, he would do well at first always to stop for a moment when the word greater or less occurs, and ask himself whether the problem requires and allows the extended signification or not, and to make some mark of distinction in every place. This will at once ensure the sound ness of the first reading, and facilitate the second.