This definition equally applies whether A and n be commensurable or incommensurable, since no attempt is made to measure B by an aliquot part of A. The two questions which must be asked, and satis factorily answered, previously to its reception, are as follows : 1. What right had Euclid, or any one else, to expect that the preceding most prolix and unwieldy statement should be received by the beginner as the definition of a relation the perception of which is one of the most common acts of his mind, since it is performed on every occasion where similarity or dissimilarity of figure is looked for or presents itself I 2. If the preceding question should be clearly answered, how can the definition of proportion ever be used ; or how is it possible to com pare every one of the infinite number of multiples of e with every one of the multiples of B ? To the first question we reply, that not only is the test proposed by Euclid tolerably simple, when more closely examined, but that it is, or might be made to appear, an easy and natural consequence of those fundamental perceptions with which it may at first seem difficult to compare it. To elucidate this, suppose the following case : There N a straight colonnade composed of columns at equal distances from each other, the first being distant from a bounding wall by a length equal to the distance between any two successive columns. In front of the colonnade let there be a row of railings * equidistant from each other, the first being at the same distance from the wall at which the railings are from each other. Let the columns be numbered from the wall, and also the railings; remember also that it is not supposed that there goes any exact number of railings to the interval of two columns, but that the interval of the columns may be to the interval of the railings in any ratio, commensurable or incommensurable.
If we may suppose this construction carried on to any extent, it Is easily shown that a spectator, by mere inspection, without measure ment, may compare the column-distance (c) with the railing-distance (a), to any degree of accuracy. For example, since the tenth railing falls between the fourth and fifth coluinns, it follows that 10 R is greater than 4 c and less than 5 o, or that R Iles between of o and of C. To get a more accurate notion, he may examine the 10,000th railing : if it fall betwecu the 4674th and 4075th colunme, it follows that 10,000a lies between 4074a and 4075 0, or R lies between and of a. There is no limit to the degree of accuracy thus obtain able ; and it can also be shown that the ratio of 0 and a is determined when the order of distribution of the railings among the columns is assigned ad infinitum; or, which is the same thing, when the position of any given railing eon be found, as to the numbers of the columns between which it lies. Any alteration, however small, in the place of the first railing, must at last affect the order of distribution. Suppose, for instance, that the first railing is moved farther from the wall by one part in a thousand of the distance between the columns, the second railing must then be pushed forward twice as Much, the third three times as much, and so on : those after the thousandth are pushed forward more than a thousand times as much,—that is, by more than the interval between the columns,—or the order with respect to the columns Is disarranged.
Let it now be proposed to make a model of the preceding con struction, in which c shall be the distance between the columns, arid r that between the railings. It needs no definition of proportion, her any thing more than the conception which we have of that term prior to definition (and with which we must allow the agreement of any definition we may adopt), to assure us that 0 must be to is in the same proportion as e to r, if the model be truly formed. Nor is it drawing too largely on that conception of proportion if we assert that the distribution of the railings among the columns in the model must be everywhere the same as in the original : for example, that the model would be out of proportion if its 56th railing fell between the 18th and 19th columns, while the 56th railing of the original fell between the 17th and 18th columns. Here, then, the question as to the dependence of Euclid's definition upon common notions is settled ; for the obvious relation between the construction and its model which has just been described contains the collection of conditions, the fulfilment of which, according to Euclid, constitutes proportion. According to Euclid, whenever as a exceeds, equals, or falls short of ten, then in c must exceed, equal, or fall short of n r by the most obvious property of the preceding constructions, according as the nab column comes after, opposite to, or before the nth railing, in the original, th mth column must come after, opposite to, or before the nth railing, in the correct modeL That the test proposed by Euclid is necessary, appears from the i preceding; and also that it is sufficient. For, admitting that, to a given original, with a given column-distance in the model, there is one correct model railing-distance (which must therefore be the one which distributes the railings among the columns as in the original), we have seen that any other railing-distance, however slightly different, would . .
at last give a different distribution : that is, the correct distance, and the correct distance only, satisfies all the conditions required by 1:ueliirs definition.
Let us now, by the distribution of one set of magnitudes among those of another set, agree to mean the placing of tho first magnitudes among those of the second set, the latter having been previously arranged in ascending order of magnitude. Thus, in the following Instance, we distribute the multiples of 3 among those of 8, the latter being in Roman numerals:— point of view, to define disproportion, and to make proportion consist in the absence of all disproportion. Similarly, obvious as is the notion of parallelism, and the connection of the non-intersection of two straight lines with that of their always keeping the same distance with each other, it Is more may to define this relation by the absence of all inter section than by any of Its positive properties.