Elements of Algebra

22 ELEMENTS SECT. I. 77. From this manner of considering fractions, the expressions numerator and denominator are derived. For, as in the preceding fraction $7/12$, the number under the line shews that 12 is the number of parts into which unity is to be divided; and as it may be said to denote, or name, the parts, it has not improperly been called the denominator. Farther, as the upper number, viz. 7, shews that, in order to have the value of the fraction, we must take, or collect, 7 of those parts, and therefore may be said to reckon or number them, it has been thought proper to call the number above the line the numerator. 78. As it is easy to understand what $2/3$ is, when we know the signification of $1/3$, we may consider the fractions whose numerator is unity, as the foundation of all others. Such are the fractions, $1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12$, &c. and it is observable that these fractions go on continually diminishing: for the more you divide an integer, or the greater the number of parts into which you distribute it, the less does each of those parts become. Thus, $1/100$ is less than $1/10$; $1/1000$ is less than $1/100$; and $1/10000$ is less than $1/1000$; &c. 79. As we have seen that the more we increase the denominator of such fractions the less their values become, it may be asked, whether it is not possible to make the denominator so great that the fraction shall be reduced to nothing? I answer, no; for into whatever number of parts unity (the length of a foot, for instance) is divided; let those parts be ever so small, they will still preserve a certain magnitude, and therefore can never be absolutely reduced to nothing. 80. It is true, if we divide the length of a foot into 1000 parts, those parts will not easily fall under the cognisance of our senses; but view them through a good microscope, and each of them will appear large enough to be still subdivided into 100 parts, and more. At present, however, we have nothing to do with what depends on ourselves, or with what we are really capable of performing, and what our eyes can perceive; the question is rather what is possible in itself: and, in this sense, it is certain, that however great we suppose the denominator, the fraction will never entirely vanish, or become equal to $0$. 81. We can never therefore arrive completely at $0$, or nothing, however great the denominator may be; and, consequently, as those fractions must always preserve a certain quantity, we may continue the series of fractions in the