The negative character of the definition of parallels does not prevent it from being very cagily proved that such lines exist ; and an exami nation of the first or last propositions of the sixth book of Euclid will show that the exiatenco of proportional quantities is as easily esta blished on the definition given as on any other. To take an instance, however, In which nothing but linos shall be the objects of considera tion, we shall here prove, in a different manner, the second proposition of Enclid'e sixth book, or one to the same effects Let o • is be a triangle, to one side of which a b is drawn parallel, nud in o A produced set off A A, A,, Sm., equal to o e, and a a„ equal to 0 a. Through every one of the points so obtained draw parallels to meeting oB produced in b, b„ n„ Sm. Then it is easily proved that b 6,, S:c., are severally equal to 0 by and n D„ Sm., to 0 B. Consequently a distribution of the multiples of o A among those of o a is made on one line, and of o a among those of 0 b on the other. The examination of this distribution in all its extent (which is impossible, and hence the apparent difficulty of using the definition) is rendered unnecessary by the known property of parallel lines. For since A, lies between cs, and a„ n must lie between b, and b„; for if not, the line A, n, would cut either or a,b,. Hence, without inquiring where does fall, we know that if it fall between and as fall between b. and : or if rn x A fall in imagui tudobetween nx o a and (n +1) x o a, then at x o B must fall between n x ob and (n + 1) x o b. Thus it is established that o A is to o a as o to o b.
The propositions of the fifth book become very simple when the definition is fully elticidated,and symbolic expression is substituted for the words at length of Euclid. They will be found thus treated in I'layfair's or Lardner's editions of Euclid, and in the Connection of Number and Magnitude' (London, 1838).
When quantities aro commensurable, a multiple of one may be found which is exactly equal to a multiple of the other thus if A = n, 13 Ass43B. In this case the arithmetical definition of proportion is sufficient, and the other may be shown to follow from it. Let A = 3t,n, and c= so that, arithmetically speaking, A is to B as a to D. Let e lie between n D and (n+ 1) n or x B lies between no and (n + 1)n. Then the number or OA ns must lie between n and n + 1; whence lies between n D and (n + 1)D, or,m c lies between n D and (n + 1) D.
It is however perfectly allowable to leave out of sight the possible case in which a multiple of A is exactly equal to a multiple of n ; since if the test be true in all other cases, it is therefore true in this. For, if possible, let 4 A=7 B, and 4 be (say) greater than 7o. Then at (4 c) exceeds in (7D) by ta times this difference, which may be made as great as we please, or 4 m o, and multiples succeeding it, may be made to fall in an interval as many intervals removed from that of Imo and (7m +1)D as we please. But A is equal to 7 m whence (4m + 1) e, must fall among the multiples of B in intervals of given nearness • to the interval of 7 m n and (7 m+ 1)B. Consequently the
multiples of A following 4 m A cannot always fall among the multiples of B in the same intervals as the same multiples of C among those of n ; and the rest of the test cannot be true, unless 4c =7D; that is, if the rest of tho test be true, then 4 Is = 7 D.
The following question will enable the reader to see for himself how far he is able to apply the method of Euclid :—Returning to the illus tration, suppose that the columns, instead of being mathematical lines, are of a given thickness, and that the columns in the model are of a proportionate thickness; let it also be supposed that when a 'Idling is projected upon the column, there are no means of determining on which side of the axis of the column it falls. It is to be shown that if the distribution be according to the definition as to every railing which is not so projected, and about which there is therefore no doubt, It must also be true as to every case in which the doubt exists.
Tho advantages of the study of proportion in the manner laid down by Euclid, or some other equivalent in extent and strictness, are pre cisely the advantages of accuracy over inaccuracy, and of real demon• etration over a false and slovenly appearance of it, which, though a close resemblance, is therefore all the more dangerous. And it must be remembered what we mean by demonstration, namely, the process of obtaining a conclusion by sound logic: from premises known to be true. If the prolixity of Euclid's method could be avoided by an assumption, we should not object, provided the assumption were both true and cagily seen to be true. For instance, if the theory of PADA', LIMB could be established without its axiom by means of a hundred This use of the word distribution having been well learnt, the following way of stating the definition will be easier than that of Euclid :—" Four magnitudes—A and B of one kind, and c and D of the same or the mine other kind—are proportional when all the multiples of A are distributed among the multiples of n in the same intervals as the corresponding multiples of o among those of D." Or, whatever numbers st and n may be, if ma. lie between ft n and (n + 1) n, m C lies between Iv n and (n + 1)D.
If the preceding test be always satisfied from and after any given multiples of A and c, it must be true before those multiples. For instance, let the test be always satisfied from and after 100 e and ] 000; and let, 5 e and 5 o be instances for examination, falling before 100 e • and 100 c. Take some multiple of 5 which will exceed 100, say 50 times, and let it be found on examination that 250 A lies between 678 B and 679n; then 250o Hot between 678 n and 679 D. Divide these by 50, and it follows that 5 A lies between 13 En and 13 is n, and still more between 13 D and 14 B. Similarly, 5 c lies between 13r, and 13 Is n, and still more between and 14 D. Or 5 e lies in the Larne interval •among the multiples of n in which 5 c lies among the multiples of n; and the same demonstration applies to any other instance.