Again, it is evident, that there is no neceffity in thefe mul. Book V. tiplications for confining ourselves to 10, or the powers of 10, and that we do fo, in arithmetic, only for the conveniency of the decimal notation; we may therefore ufe any multipliers whatsoever, providing we use the fame in both cafes. 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 fame quotient with the like multiple of the third, when divided by the fourth, the four magnitudes are proportionals, or the firft has the fame ratio to the fecond 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 alfo, when mC is divided by D, the quotient will be n exactly. But, when mA divided by B gives n for the quotient, mAnB by the nature of divifion, fo that when mAnB, mCnD, which is one of the conditions of Euclid's definition,

Again, when mA is divided by B, let the divifion not be exactly performed, but let n be a whole number less than the exact quotient, then nB <mA, or mA> nB; and, for the fame reason, mC> nD, which is another of the conditions of Euclid's definition.

Laftly, when mA is divided by B, let n be a whole number greater than the exact quotient, then mA <nB, and be. caufe n is also greater than the quotient of mC divided by D, (which is the fame with the other quotient), therefore mC <nD.

Therefore, uniting all these three conditions, we call A, B, C, D, proportionals, when they are fuch, that if mA > nB, mC > nD; if mA nB, mCnD; and if mA> B, mC <nD, 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 generalifing 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 mČ contains D, with or without a remainder, and of this being the cafe whatever value be affigned to the number mi, includes in it all the three conditions that are mentioned in Euclid's definition;

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Book V. definition; and hence, that definition may be expreffed a little more fimply by faying, that four magnitudes are proportionals, when any multiple of the first contains the fecond, (with or without remainder), as oft as the fame multiple of the third contains the fourth. But, though this definition is certainly, in the expreffion, more fimple than Euclid's, it is not, as will be found on trial, fo eafily applied to the purpose of demonstration. The three conditions which Euclid brings together in his definition, though they fomewhat embarrass the expreffion of it, have the advantage of rendering the demonftrations more fimple than they would otherwise be, by avoiding all difcuffion 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 demonftrate the properties of proportionals rigorously, by means of other definitions than Euclid's, only ferve to evince the excellence of the method followed by the Greek Geometer, and his fingular 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 propofition, one when the divifion of mA into parts. equal to B can be exactly performed, the other when it cannot be exactly performed, whatever value be affigned to m, or when A and B are what is called incommensurable; and this laft cafe will in general be found to require an indirect demonftration, or a reductio ad abfurdum.

M. D'Alembert, fpeaking of the doctrine of proportion, in a discourse that contains many excellent obfervations, but in which he has overlooked Euclid's manner of treating this fubject entirely, has the following remark: "On ne peut démontrer que de cette maniere, (la réduction à l'absurde), la "plupart des propofitions qui regardent les incommenfura

bles. L'idée de l'infini entre au moins implicitement dans "la notion de ces fortes des quantités; et comme nous n'a"vons qu'une idée negative de l'infini, on ne peut démontrer "directement, et a priori, tout ce qui concerne l'infini mathé"matique." (Encyclopédie, mot, Géométrie.)

This remark fets in a strong and just light the difficulty of demonftrating the propofitions that regard the proportion of incommenfurable magnitudes, without having recourse to the reductio ad abfurdum; but it is furprising, that M. D'Alembert, a geometer no lefs learned than profound, should have neglected to make mention of Euclid's method, the only one

in

in which the difficulty he ftates is completely overcome. It is Book V. 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 poffible values, and it is by the skilful use of this condition that the neceffity of indirect demonftrations 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 fame fuccefs, viz. in his books of Porifms, which have been restored by Dr Simfon, and in which the whole analysis turned on that idea, as I have fhewn at length, in the Third Volume of the Tranfactions of the Royal Society of Edinburgh. The investigations of thofe propofitions were founded entirely on the principle of certain magnitudes admitting of innumerable values; and the methods of reafoning concerning them feem to have been extremely fimilar to those employed in the fifth of the Elements. It is curious to remark this analogy between the different works of the fame author; and to confider, that the fkill, in the conduct of this very refined and ingenious artifice, acquired in treating the properties of proportionals, may have enabled Euclid to fucceed fo well in treating the ftill more difficult fubject of Porifms.

Viewing in this light Euclid's manner of treating proportion, I had no defire to change any thing in the principle of his demonftrations. I have only fought to improve the language of them, by introducing a concife mode of expreffion, of the fame 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 thefe demonftrations fo flowly, and breaks them down into fo many parts, that they make not a fufficient impreffion on the understanding. This, indeed, will generally happen when the things treated of are not reprefented to the fenfes by Diagrams, as they cannot be when we reafon 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 hortnefs, and the rapidity with which it places objects before us, makes up in the beft manner poffible for being merely a conventional language, and ufing fymbols that have no refemblance to the things expreffed by

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them.

Book V. 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 Euclid will admit. In other refpects I have followed Dr Simfon's edition, 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 defign, for he has given a definition of proportional numbers in the feventh Book, very different from that of propor tional magnitudes in the fifth; and it is not eafy to justify the logic of this manner of proceeding; for we can never fpeak of two numbers and two magnitudes both having the fame ratios, unless the word ratio have in both cafes the fame fignification. All the propofitions about proportionals here given are therefore understood to be applicable to numbers; and accordingly, in the eighth Book, the propofition that proves equiangular parallelograms to be in a ratio compounded of the ratios of the numbers proportional to their fides, is demonftrated by help of the propofitions 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 propofitions, 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 fufficient, however, to remark, that all the propofitions refpecting the ratios of magnitudes relate equally to all things of which multiples can be taken, that is, to all that is ufually expreffed by the word Quantity in its most extended fignification, taking care always to obferve, that ratio takes place only among like quantities. (See Def. 4.)

DEF. X.

The definition of compound ratio was firft given accurately by Dr Simfon; for, though Euclid used the term, he did fo without defining it. I have placed this definition before thofe of duplicate and triplicate ratio, as it is in fact more general,

and

and as the relation of all the three definitions is beft feen Book V. 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 expreffion, compound ratio is introduced merely to prevent circumlocution, and for the fake principally of enunciating those propofitions with concifeness that are demonftrated by reafoning ex aquo, that is, by reafoning from the 22d or 23d of this Book. This will be evident to any one who confiders 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, arifes from its not being evident how the ratios defcribed in the definition determine in any way the ra tio which they are faid to compound, fince the magnitudes compounding them are affumed, at pleasure. It may be of ufe to remove this difficulty, to state the matter thus: If there be any number of ratios (among magnitudes of the fame kind) fuch that the confequent of any of them is the antecedent of that which immediately follows, the first of the antecedents has to the laft of the confequents a ratio which evidently depends on the intermediate ratios, because if they are determined, it is determined alfo; and this dependence of one ratio on all the other ratios, is expreffed by saying that it is C D

compounded of them. Thus, if AB, be any feries

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of ratios, such as described above, the ratio, or of A to E, is

E

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 confider more attentively.