1 tient as when the third is muliplied 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 mA=nB, 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 mA 7nB; 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 /nB, 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 mA 7nB, 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 generalizing 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 follow ed 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 wil generally 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 manière, (la réduction à absurde,) la plupart des propositions qui regardent les incommensurables. L'idée de l'infini entre au moins implicitemens dans la notion de ces sortes de quan"tités; et comme nous n'avons qu'une idée negative de l'infini, on ne peut "démontrer directement, et a priori, tout ce qui concerne l'infini mathéma"tique." (Encyclopédie, mot Geométrie.) 66 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 shown at length in the Third Volume of the Transactions of the Royal Society of Edinburgh. The investigations of these 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 Euclid will admit. In other respects I have followed Dr. Simson'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 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 Magnitude, 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 be any series of ratios, such as described above, the ratio B'C' D'E if A B A E , or of A to E, is said to be compounded of the ratios &c. The ratio A &c. because if each of the A B is evidently determined by the ratios E' 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. A, B, C, &c. Nine propositions are added to this Book on account of their utility and their connection with this part of the Elements. The first four of them are in Dr. Simson's edition, and among these Prop. A is given immediately after the third, being, in fact, a second case of that proposition, and capable of being included with it, in one enunciation. Prop. D is remarkable for being a theorem of Ptolemy the Astronomer, in his Meyaλn Zuvragis, and the foundation of the construction of his trigonometrical tables. Prop. E is the simplest case of the former; it is also useful in trigonometry, and, under another form, was the 97th, or, in some editions, the 94th of Euclid's Data. The propositions F and G are very useful properties of the circle, and are taken from the Loci Plani of Apollonius. Prop. H is a very remarkable property of the triangle; and K is a proposition which, though it has been hitherto considered as belonging particularly to trigonometry, is so often of use in other parts of the mathematics, that it may be properly ranked among elementary theorems of Geometry. SUPPLEMENT. BOOK I. PROP. V. and VI, &c. THE demonstrations of the 5th and 6th propositions require the method of exhaustions, that is to say, they prove a certain property to belong to the circle, because it belongs to the rectilineal figures inscribed in it, or described about it according to a certain law, in the case when those figures approach to the circles so nearly as not to fall short of it or to exeeed it, by any assignable difference. This principle is general, and is the only one by which we can possibly compare curvilineal with rectilineal spaces, or the length of curve lines with the length of straight lines, whether we follow the methods of the ancient or of the modern geometers. It is therefore a great injustice to the latter methods to represent them as standing on a foundation less secure than the former; they stand in reality on the same, and the only difference is, that the application of the principle, common to them both, is more general and expeditious in the one case than in the other. This identity of principle, and affinity of the methods used in the elementary and the higher mathematics, it seems the most necessary to observe, that some learned mathematicians have appeared not to be sufficiently aware of it, and have even endeavoured to demonstrate the contrary. An instance of this is to be met with in the preface of the valuable edition of the works of Archimedes, lately printed at Oxford. In that preface, Torelli, the learned commentator, whose labours have done so much to elucidate the writings of the Greek Geometer, but who is so unwilling to acknowledge the merit of the modern analysis, undertakes to prove, that it is impossible, from the relation which the rectilineal figures inscribed in, and circumscribed about, a given curve have to one another, to conclude any thing concerning the properties of the curvilineal space itself, except in certain circumstances which he has not precisely described. With this view he attempts to show, that if we are to reason from the relation which certain ectilineal figures belonging to the circle have to one another, notwithstanding that those figures may approach so near to the circular spaces within which they are inscribed, as not to differ from them by any assignable magnitude, we shall be led into error, and shall seem to prove, that the circle is to the square of its diameter exactly as 3 to 4. Now, as this is a conclusion which the discoveries of Archimedes himself prove so clearly to be false, Torelli argues, that the principle from which it is deduced must be false also; and in this he would no doubt be right, if his former conclusion had been fairly drawn. But the truth is, that a very gross paralogism is to be found in that part of |