The mere existence of incommensurables, to any nothing of their frequent occurrence, and the impossibility of avoiding them, renders the arithmetical theory of proportion inexact in its very definition. If we would say, for instance, that the diagonal of a larger square is to its side as the diagonal of a smaller square is to its side, we enunciate a proposition the meaning of which is unsettled. For if we mean to assert that the larger diagonal contains the larger side as many times or fractions of times as the einaller diagonal contains the smaller, it is answered, by those who wish for precise notions, that neither diagonal contains its side any exact number of times or parts of times. If we should say that the larger diagonal lies between 1.414213 and P414214 times the larger side, and that the smaller diagonal also lies between I'414213 and 1.414214 times the smaller aide, and if we should show this to be true, we certainly show that we could produce lines very nearly equal to the diagonals, which are, under the arithmetical definition, proportional to the sides ; and that this might be done without altering either diagonal by so much as the millionth part of the side. And the ten-millionth, hundred millionth, or any aliquot part of the aide, however small, might be substituted for the millionth, without detriment to our power of showing the truth of the proposition. If we use the means by which this process may be carried on ad infinitunt, we may perhaps be said to have established the truth of the proposition that the diagonals of squares are as their sides. But if we in any manner stop short of this, we destroy the rigorous character of geometry, and produce a system of mathematics which, like a common table of logarithms, is true to a certain number of places of decimals, and not farther. It is obvious that such a system of mathematics, like the table with which we have compared it, is sufficient for the purposes of practical application; nor have we the least quarrel with those who, desiring an instrument only, are content with one which is sharp enough for their purpose. 11'e only complain of them when they assert and teach others that their tool has an edge keen enough to separate the minutest truth from the minutest falsehood ; whereas, on examining it with a powerful micro scope, we find the so-called sharp edge capable of being magnified into a plane of any dimensions, though it may appear a sharp edge to the unassisted senses.
The imperfection of the definite arithmetical definition of proportion is universally admitted, while the compleiity of the rigorous definition by which Euclid supplied its place is aimed as universally felt to be a grievance. Many attempts have been made to avoid the trouble without incurring the reproach of inaccuracy. Ono Or two of these we shall notice.
Legendre, in his otherwise excellent wdrk on geometry, refers the student to works on arithmetic fir the theory of proportion; and, having stated that when A is to B ado to n, it LS known that e x D=13 x c, adds (twelfth edition, page 61), " This is certainly true for numbers; it is true also for any magnitudes, provided they can be expressed, or that we imagine them expressed by numbers, and this we may always A eystem of geometry which tells the learner in so many words that he may always suppose that which is not true, needs no further comment, even though Legendre were its author. It is true that in subsequent parte of the work we find demonstrations adapted to the case of incommensurable quantities, but they want that most important element of a proposition involving proportions, namely, a definition of what the term means ; these demonstrations turn upon the theorem that when four quantities are proportional, the first is greater than, equal to, or less than the second, according as the third is greater than, equal to, or less than the fourth ; but it has not been previously stated what the author means by four quantities being proportional. In the English translation of the pre
ceding work, a preliminary chapter is added on proportion, in which the definition given as to incommensurable magnitudes amounts to the following :—when A and B are incommensurable, and also o and D, the four are said to be proportional when A' and c' can be found, as near as we please to A and c, and which, being commensurable with B and D, are proportional to them, in the arithmetical sense of the term. This is a sufficient definition; but it really amounts to that of Euclid (as do all sufficient definitions which we have seen), and is not so easily used.
M. Lacroix (' El6mens do 06omdtrie,' p. 5) makes the approximate finding of a common measure stand in place of an exact process, and, fairly stating that the error of the process may be made too small to be visible, rests the exactness of his geometry on its not being sensibly erroneous.
The author of the ' Elements of Geometry,' in the 'Library of Useful Knowledge,' states the proportion of incommensurable mag nitudes to consist in " their ratios admitting of being approximately represented by the same numbers, to how great an extent soarer the degree of approximation may be carried." In virtue of the words in italics, this definition may be considered as being, when properly used, capable of forming the basis of an exact theory ; and that it is properly used in the work cited we fully admit, since its first application is to the establishment of the definition of Euclid. The only objection we should make to the work in question is that its expressions (page 98) would lead the student to imply that commensurability is the general rule, and incommensurability the exception ; an extended theory is given, because magnitudes are not always commensurable. Now it is important the student should know, and shOuld always bear in mind, that of two magnitudes of the same kind taken at hazard, or one being given and the other deduced by a geometrical construction, it is very much more likely that the two should be inconunensurable than that they should he commensurable. So that the apparently cumbrous theory of proportion is not introduced to meet a few cases which sometimes occur, but to prevent the majority of instances from being treated incorrectly.
The definition • of proportional quantities given by Euclid is as follows :—" Magnitudes are said to have the same ratio to one another, the first to the second, and the third to the fourth, when equlmultiples of the first and third, and equimultiples of the second and fourth, whatever the multiplications may be, yield a multiple of the first, greater than, equal to, or less than, that of the second, according as the multiple of the third is greater than, equal to, or less than, that of the fourth." That is, if A, B, C, and D be the four magnitudes, and nt and is any two whole numbers whatsoever, vs. A must be greater than, equal to, or leas than n B, according as in C is greater than, equal to, or lees than n b. Otherwise thus : whatever whole numbers in and n may be, must exceed, equal, or fall short of n-inths of B, according as c exceeds, equals, or falls short of n-inths of D. A person practised in algebra would comprehend the definition most easily when stated thus: nt A —n B must have the same sign as m —n D, for all whole values of ra and n.