magnitudes A and B, providing they be of the same kind, or such that the one can be multiplied so as to exceed the other. For, suppose that B is the least of the two; take B out of A as oft as it can be found, and let the quotient be noted, and also the remainder, if there be any; multiply this remainder by 10, or 100, &c. so as to exceed B, and let B be taken out of the quantity produced by this multiplication as oft as it can be found; let the quotient be noted, and also the remainder, if there be any. Proceed with this remainder as before, and so on continually; and it is evident, that we have an operation that is applicable to all magnitudes whatsoever, and that may be performed with respect to any two lines, any two plane figures, or any two solids, &c.

Now, when we have two magnitudes and two others, and find that the first divided by the second, according to this method, gives the very same series of quotients that the third does when divided by the fourth, we say of these magnitudes, as we did of the numbers above described, that the first is to the second as the third to the fourth. There are only two more circumstances necessary to be considered, in order to bring us precisely to Euclid's definition.

First, It is known from arithmetic, that the multiplication of the successive remainders each of them by 10, is equivalent to multiplying the quantity to be divided by the product of all those tens; so that multiplying, for instance, the first remainder by 10, the second by 10, and the third by 10, is the same thing, with respect to the quotient, as if the quantity to be divided had been at first multiplied by 1000; and therefore, our standard of the proportionality of numbers may be expressed thus: If the first multiplied any number of times by 10, and then divided by the second, gives the same quotient as when the third is multiplied as often by 10, and then divided by the fourth, the four magnitudes are proportionals.

Again, it is evident, that there is no necessity in these multiplications for confining ourselves to 10, or the powers of 10, and that we do so, in arithmetic, only for the conveniency of the decimal notation; we may therefore use any multipliers whatsoever, providing we use the same in both cases. Hence, we have this definition of proportionals, When there are four magnitudes, and any multiple whatsoever of the first, when divided by the second, gives the same quotient with the like multiple of the third, when divided by the fourth, the four magnitudes are proportionals, or the first has the same ratio to the second that the third has to the fourth.

We are now arrived very nearly at Euclid's definition; for, let A, B, C, D be four proportionals, according to the definition just given, and m any number; and let the multiple of A by m, that is mA, be divided by B; and first, let the quotient be the number n exactly, then also, when mC is divided by D, the quotient will be n exactly. But, when mA divided by B gives n for the quotient, mA=nB by the nature of division, so that when mAnB, mC=nD, which is one of the conditions of Euclid's definition,

Again, when mA is divided by B, let the division not be exactly performed, but let n be a whole number less than the exact quotient, then nB mA, or mA7nB; and, for the same reason, mC7nD, which is another of the conditions of Euclid's definition.

Lastly, when mA is divided by B, let n be a whole number greater than the exact quotient, then mA ZnB, and because n is also greater than the quotient of mC divided by D, (which is the same with the other quotient), therefore mCnD.

Therefore, uniting all these three conditions, we call A, B, C, D, proportionals, when they are such, that if mA7nB, mC7nD; if mA =nB, mC=nD; and if mA ▲nB, mCnD, m and n being any numbers whatsoever. Now, this is exactly the criterion of proportionality established by Euclid in the 5th definition, and is derived here by generalising the common and most familiar idea of proportion.

It appears from this, that the condition of mA containing B, whether with or without a remainder, as often as mC contains D, with or without a remainder, and of this being the case whatever value be assigned to the number m, includes in it all the three conditions that are mentioned in Euclid's definition; and hence, that definition may be expressed a little more simply by saying, that four magnitudes are proportionals, when any multiple of the first contains the second, (with or without remainder,) as oft as the same multiple of the third contains the fourth. But, though this definition is certainly, in the expression, more simple than Euclid's, it is not, as will be found on trial, so easily applied to the purpose of demonstration. The three conditions which Euclid brings together in his definition, though they somewhat embarrass the expression of it, have the advantage of rendering the demonstrations more simple than they would otherwise be, by avoiding all discussion about the magnitude of the remainder left, after B is taken out of mA as oft as it can be found. All the attempts, indeed, that have been made to demonstrate the properties of proportionals rigorously, by means of other definitions than Euclid's, only serve to evince the excellence of the method followed by the Greek Geometer, and his singular address in the application of it.

The great objection to the other methods is, that if they are meant to be rigorous, they require two demonstrations to every proposition, one when the division of mA into parts equal to B can be exactly performed, the other when it cannot be exactly performed, whatever value be assigned to m, or when A and B are what is called incommensurable; and this last case will in general be found to require an indirect demonstration, or a reductio ad absurdum.

M. D'Alembert, speaking of the doctrine of proportion, in a discourse that contains many excellent observations, but in which he has overlooked Euclid's manner of treating this subject entirely, has the following remark : "On ne peut démontrer que de cette maniere, (la réduction à absurde,) la plupart des propositions qui regardent "les incommensurables. L'idée de l'infini entre au moins implicite"ment dans la notion de ces sortes de quantités; et comme nous n'a

"vons qu'une idée negative de l'infini, on ne peut démontrer directe"ment, et a priori, tout ce qui concerne l'infini mathématique.” (Encyclopédie, mot Géométrie.)

This remark sets in a strong and just light the difficulty of demonstrating the propositions that regard the proportion of incommensurable magnitudes, without having recourse to the reductio ad absurdum; but it is surprising, that M. D'Alembert, a geometer no less learned than profound, should have neglected to make mention of Euclid's method, the only one in which the difficulty he states is completely overcome. It is overcome by the introduction of the idea of indefinitude, (if I may be permitted to use the word,) instead of the idea of infinity; for m and n, the multipliers employed,are supposed to be indefinite,or to admit of all possible values, and it is by the skilful use of this condition that the necessity of indirect demonstrations is avoided. In the whole of geometry, I know not that any happier invention is to be found; and it is worth remarking, that Euclid appears in another of his works to have availed himself of the idea of indefinitude with the same success, viz. in his books of Porisms, which have been restored by Dr. Simson, and in which the whole analysis turned on that idea, as I have shewn at length, in the Third Volume of the Transactions of the Royal Society of Edinburgh. The investigations of those propositions were founded entirely on the principle of certain magnitudes admitting of innumerable values; and the methods of reasoning concerning them seem to have been extremely similar to those employed in the fifth of the Elements. It is curious to remark this analogy between the different works of the same author; and to consider, that the skill, in the conduct of this very refined and ingenious artifice, acquired in treating the properties of proportionals, may have enabled Euclid to succeed so well in treating the still more difficult subject of Porisms.

Viewing in this light Euclid's manner of treating proportion, I had no desire to change any thing in the principle of his demonstrations. I have only sought to improve the language of them, by introducing a concise mode of expression, of the same nature with that which we use in arithmetic, and in algebra. Ordinary language conveys the ideas of the different operations supposed to be performed in these demonstrations so slowly, and breaks them down into so many parts, that they make not a sufficient impression on the understanding. This indeed will generally happen when the things treated of are not represented to the senses by Diagrams, as they cannot be when we reason concerning magnitude in general, as in this part of the Elements. Here we ought certainly to adopt the language of arithmetic or algebra, which, by its shortness, and the rapidity with which it places objects before us, makes up in the best manner possible for being merely a conventional language, and using symbols that have no resemblance to the things expressed by them. Such a language, therefore, I have endeavoured to introduce here; and I am convinced, that if it shall be found an improvement, it is the only one of which the fifth of Euelid will admit. In other respects I have followed Dr. Simson's edi

tion, to the accuracy of which it would be difficult to make any addition.

In one thing I must observe, that the doctrine of proportion, as laid down here, is meant to be more general than in Euclid's Elements. It is intended to include the properties of proportional numbers as well as of all magnitudes. Euclid has not this design, for he has given a definition of proportional numbers in the seventh Book, very different from that of proportional magnitudes in the fifth; and it is not easy to justify the logic of this manner of proceeding; for we can never speak of two numbers and two magnitudes both having the same ratios, unless the word ratio have in both cases the same signification. All the propositions about proportionals here given are therefore understood to be applicable to numbers; and accordingly, in the eighth Book, the proposition that proves equiangular parallelograms to be in a ratio compounded of the ratios of the numbers proportional to their sides, is demonstrated by help of the propositions of the fifth Book.

On account of this, the word quantity, rather than magnitude, ought in strictness to have been used in the enunciation of these propositions, because we employ the word Quantity to denote not only things extended, to which alone we give the name of Magnitudes, but also numbers. It will be sufficient, however, to remark, that all the propositions respecting the ratios of magnitudes relate equally to all things of which multiples can be taken, that is, to all that is usually expressed by the word Quantity in its most extended signification, taking care always to observe, that ratio takes place only among like quantities. (See Def. 4.)

DEF. X.

The definition of compound ratio was first given accurately by Dr. Simson; for, though Euclid used the term, he did so without defining it. I have placed this definition before those of duplicate and triplicate ratio, as it is in fact more general, and as the relation of all the three definitions is best seen when they are ranged in this order. It is then plain, that two equal ratios compound a ratio duplicate of either of them; three equal ratios, a ratio triplicate of either of them, &c. It was justly observed by Dr. Simson, that the expression, compound ratio, is introduced merely to prevent circumlocution, and for the sake principally of enunciating those propositions with conciseness that are demonstrated by reasoning ex æquo, that is, by reasoning from the 22d or 23d of this Book. This will be evident to any one who considers carefully the Prop. F. of this, or the 23d of the 6th Book.

An objection which naturally occurs to the use of the term compound ratio, arises from its not being evident how the ratios described in the definition determine in any way the ratio which they are said to compound, since the magnitudes compounding them are assumed at pleasure. It may be of use for removing this difficulty, to state the matter as follows: if there be any number of ratios (among magnitudes of the same kind) such that the consequent of any of them is

the antecedent of that which immediately follows, the first of the antecedents has to the last of the consequents a ratio which evidently depends on the intermediate ratios, because if they are determined, it is determined also; and this dependence of one ratio on all the other ratios, is expressed by saying that it is compounded of them. Thus, A B C D if be " series of ratios, such as described above, the any BC'D'E

Α ratio or of A to E is said to be compounded of the ratios E'

The ratio is evidently determined by the ratios &c. because

A
E

А В
B'C'

if each of the latter is fixed and invariable, the former cannot change. The exact nature of this dependence, and how the one thing is determined by the other, it is not the business of the definition to explain, but merely to give a name to a relation which it may be of importance to consider more attentively.

BOOK VI.

DEFINITION II.

This definition is changed from that of reciprocal figures, which was of no use, to one that corresponds to the language used in the 14th and 15th propositions, and in other parts of geometry.

PROP. XXVII, XXVIII, XXIX.

As considerable liberty has been taken with these propositions, it is necessary that the reasons for doing so should be explained. In the first place, when the enunciations are translated literally from the Greek, they sound very harshly, and are, in fact, extremely obscure. The phrase of applying to a straight line, a parallelogram deficient, or exceeding by another parallelogram, is so eliptical, and so little analogous to ordinary language, that there could be no doubt of the propriety of at least changing the enunciations.

It next occurred, that the Problems themselves in the 28th and 29th propositions are proposed in a more general form than is necessary in an elementary work, and that, therefore, to take those cases of them that are the most useful, as they happen to be the most simple, must be the best way of accommodating them to the capacity of a learner. The problem which Euclid proposes in the 28th is, "To a given "straight line to apply a parallelogram equal to a given rectilineal fi"gure, and deficient by a parallelogram similar to a given parallelogram;" which may be more intelligibly enunciated thus : To cut "a given line, so that the parallelogram which has in it a given angle, "and is contained under one of the segments of the given line, and a