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ratios

A B
B'C'

&c., because 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.

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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 enuneiations 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 elliptical, 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

Book V, 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 parallelo"gram equal to a given rectilineal figure, 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 straight line which "has a given ratio to the other segment, may be equal "to a given space;" instead of which problem I have substituted this other: "To divide a given straight "line so that the rectangle under its segments may "equal to a given space." In the actual solution of problems, the greater generality of the former proposition is an advantage more apparent than real, and is fully compensated by the simplicity of the latter, to which it is always easily reducible.

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The same may be said of the 29th, which Euclid enunciates thus: "To a given straight line to apply a pa"rallelogram equal to a given rectilineal figure, exceed"ing by a parallelogram similar to a given parallelo66 gram. This might be proposed otherwise: "To "produce a given line, so that the parallelogram having "in it a given angle, and contained by the whole line produced, and a straight line that has a given ratio to "the part produced, may be equal to a given rectilineal "figure." Instead of this, is given the following problem, more simple, and, as was observed in the former instance, very little less general: "To produce a given "straight line, so that the rectangle contained by the

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segments, between the extremities of the given line, "and the point to which it is produced, may be equal to ❝ a given space."

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 edi- Book IV. tion, 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 Meyan Eurai, 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 the elementary theorems of Geometry.

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Supplemeut

SUPPLEMENT.

T

BOOK I.

PROP. V. and VI. &c.

HE 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, even when those figures approach to the circle so nearly, as not to fall short of it, or to exceed 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 more necessary to observe, that some learned mathematicians have appeared not to

be sufficiently aware of it, and have even endeavoured to Supplement 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 defined. With this view he attempts to shew, 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 supposed to be true. 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 A: B::a: b,

C: D: c: d,

The

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

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