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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.)


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'




or of A to E, is said to be compounded of the ratios


A B B'C' &c. because if each of the

&c. The ratio


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.



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 Zuvrağıs, 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.


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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 rectilineal 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

his reasoning, where he makes a transition from the ratios of the small rectangles, inscribed in the circular spaces, to the ratios of the sums of those rectangles, or of the whole rectilineal figures. In doing this, he takes for granted a proposition, which, it is wonderful, that one who had studied geometry in the school of Archimedes, should for a moment have supposThe proposition is this: If A, B, C, D, E, F, be any number of magnitudes, and a, b, c, d, e, f, as many others; and if

ed to be true.

A: Bab,

C: D::c:d,

E: Fef, then the sum of A, C and E will be to the sum of B, D and F, as the sum of a, c and e, to the sum of b, d and f, or A+C+E : B+D +F::a+c+e: b+d+f. Now, this proposition, which Torelli supposes to be perfectly general, is not true, except in two cases, viz. either first, when A Ca: c, and


A: Eae; and consequently,

B: D::b: d, and

B: Fb:f; or, secondly, when all the ratios of A to B, C to D, E to F, &c. are equal to one another. To demonstrate this, let us suppose that there are four magnitudes, and four others,

thus AB: a: b, and

CD: cd, then we cannot have A+C: B+D::a+c: b+d, unless either A: C::a: c, and B : D::b: d; or A: C::b: d, and consequently a : b :: c : d.

K, A, B, L,

Take a magnitude K, such that a:c:: A: K, and another L, such that b:d:: B: L; and suppose it true, that A+C: B+D:: a+c: b+d. Then, because by inversion; K: A::c: a, and, by hypothesis, A: B::a: b, and also B: L::b:d, ex æquo, K: L::c:d; and consequently, K: L:: C: D.

Again, because A: K:: a: c, by addition,

c, a, b, d.

A+K: K::a+c: c; and for the same reason,
B+L: L::b+d: d, or, by inversion,

L: B+Ld: b+d. And, since it has been shewn, that

K: L::c:d; therefore, ex æquo,

A+K, K, L, B+L,

a+c, c, d, b+d.

A+K: B+La+c: b+d; but by hypothesis,

A+C: B+D::a+c: b+d, therefore

A+K: A+C:: B+L: B+D.

Now, first, let K and C be supposed equal, then it is evident that L and D are also equal; and therefore, since by construction a: c:: A: K, we have also ac:: A: C; and, for the same reason, b: d:: B: D, and these analogies from the first of the two conditions, of which one is affirmed above to be always essential to the truth of Torelli's proposition. Next, if K be greater than C, then, since

A+K: A+C:: B+L: B+D, by division,

A+K: K-C:: B+L: L-D. But, as was shewn,

K: L:: C: D, by conversion and alternation,

K-C: K:: L-D: L, therefore, ex æquo,

A+K: K::B+L: L, and lastly, by division,

A: K:: B: L, or A: B:: K: L, that is,

A: BC: D.

Wherefore, in this case the ratio of A to B is equal to that of C to D, and consequently, the ratio of a to b equal to that of c to d. The same may be shewn, if K is less than C; therefore in every case there are conditions necessary to the truth of Torelli's proposition, which he does not take into account, and which, as is easily shewn, do not belong to the magnitudes to which he applies it.

In consequence of this, the conclusion which he meant to establish respecting the circle, falls entirely to the ground, and with it the general inference aimed against the modern analysis.

It will not, I hope, be imagined, that I have taken notice of these circumstances with any design to lessen the reputation of the learned Italian, who has in so many respects deserved well of the mathematical sciences, or to detract from the value of a posthumous work, which by its elegance and correctness, does so much honour to the English editors. But I would warn the student against that narrow spirit which seeks to insinuate itself even into the abstractions of geometry, and would persuade us, that elegance, nay, truth itself, are possessed exclusively by the ancient methods of demonstration. The high tone in which Torelli censures the modern mathematics is imposing, as it is assumed by one who had studied the writings of Archimedes with uncommon diligence. His errors are on that account the more dangerous, and require to be the more carefully pointed out.


This enunciation is the same with that of the third of the Dimensio Circuli of Archimedes; but the demonstration is different, though it proceeds like that of the Greek Geometer, by the continual bisection of the 6th part of the circumference.

The limits of the circumference are thus assigned; and the method of bringing it about, notwithstanding many quantities are neglected in the arithmetical operations, that the errors shall in one case be all on the side of defect, and in another all on the side of excess (in which I have followed Archimedes,) deserves particularly to be observed, as affording a good introduction to the general methods of approximation.



SOLID angles, which are defined here in the same manner as in Euclid, are magnitudes of a very peculiar kind, and are particularly to be remarked for not admitting of that accurate comparison, one with another, which is

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