Let D and s be two iucommensurable magnitudes : how are we to describe their relative manitudes ? That they have a definite relation is certain ; suppose, for precision, that s is the side of a square, and D its diagonal; any alteration of D, or any error in D, s being given, would make the figure cease to be a square. There are many mathematical notions in which accuracy is not attainable in finite terms, but is the limit towards which we approach when number or magnitude, as the case may be, is increased or diminished without limit. In the present ease the expression of ratio or relative magnitude, which is not accu rately attainable by one or more relations, can be continually amended by adding one or more relations, until the inaccuracy of the mode of expression is rendered as small as we please; in such a case, accuracy must be imagined to reside in the supposition of an infinite number of given, or at least of attainable, relations.
To explain our meaning• suppose that the person whom we address is altogether ignorant of the relative magnitude of the diagonal n and the side s. He asks for a relation, and knowing the mode of dealing with the ratios of commensurables, naturally desires to know how many diagonals make an exact number of sides. If we could answer this question, if for instance we could say that 100 diagonals make 142 sides exactly, the question would be settled : for an arithmetical rule would always deduce the diagonal when the side is given. But we are obliged to reply, that no number of diagonals whatsoever will make an exact number of sides. He then asks how he is to form a perfect conception of the diagonal ; we answer by placing two equal sides at right angles to one another, and joining the extremities. This, he replies, and properly, is not a mode of finding the relative magnitude, which is something connected with magnitudes only, and that the permission given by Euclid to join two given or determined points is not any real determination of the included length. We then tell him that it is at his pleasure to name a fraction of the side, and wo can express the diagonal with an error not so great as that fraction ; he names, say one millionth of the side, and we give him the promised information in telling him that 1,000,000 of diagonals exceed 1,414,213 sides, but fall short of 1,414,214 sides. The consequence is, that the diagonal lies between 1'414213 and of the side : these differ from one another by one-millionth of the side, and the error of the diagonal is of course less. If ho should ask how he is to carry this process yet further for himself, we give him the arithmetical symbol .,/2, and instruct him how to perform the arithmetical operation of approxima ting to its value. In this we show him how to find between what number of sides any number of diagonals lies ; and in so doing we give the ratio of the diagonal to the side, so far as the nature of the case will admit of its expression.
The relative magnitude, then, of two magnitudes is given, when the place of any multiple whatsoever of the one among the multiples of the other can be found from the data. For example, we carry on the multiple scale of the side and diagonal of a square, in the power of extending which ad infinitum lien that of expressing the ratio, so far as expression is possible, and of absolutely comparing the ratio with others, in as accurate a manner as if expression had been perfect.
In this table we see, for instance, that 10 diagonals are more than 14 awl less than 15 sides, and so on. The only doubt that can possibly remain may be thus expressed the preceding scale a property of the diagonal and of nothing else ? May there not be a length so near to the diagonal that its multiples shall never fall out of the same intervals as those of the diagonal I Let lc be a given quantity, no matter how small ; we say that it is impossible that all the multiples of D + K can lie in the same places among the multiples of a as the multiples of D. Take its times both ; then m (D + g) and ma differ by mg. Now however small K may be, it is possible to take ta so great that mg shall exceed s, or any multiple of s previously named : whence the thing asserted is evident. The definition of the ratio of s to D lies, then, in this scale ; or rather, whatever the definition may be, the mode of finding all relations between 8 and n lies in the formation of this scale so far as may be necessary for the purpose in hand. The definition of proportion is then contained in sameness of multiple scales ; that is, v is to 8 as ? to B, when any multiple whatever of D is contained between the same two multiples of s, that the same multiple of A is coat ained between those of B. We here come to the subject of PROPORTION, which the reader should now consult as a continuation of the prey :nt one.
It is we:I known that the word Sosor came to mean fraction, the expression of commensurable ratio. Fractional arithmetic was called logistic arithmetic even down to the 17th century. It is not so well known th it this use of the word is as old as the time of Euclid. Aristotle •rives, as his instances of discontinuous quantity, tipt01.46s Airyor, nu: nber and fraction. The logicians have always taken this to mean nun iber and speech, some transcriber having interpolated a sub sequent passage to this effect in the text of Aristotle. Plato tells us that the Egyptian Thenth invented ltin0aosre sal Acryloads, meaning, no doubt, integer and fractional reckoning. The idea implied in com position of ratio is very imperfectly treated in Euclid : and yet upon the correct understanding of it depends whether the boasted victory over the difficulties of incommensurables which the fifth book gives is real or imaginary.