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between C and a multiple of D is equal to E, E must measure C (prop. i. cor.); it must therefore measure a multiple of C; but the difference between B and a multiple of C is D, which E has already been shown to measure; hence E must measure B. Therefore E must measure a multiple of B; and since the difference between A and a multiple of B is C, which E has been shown to measure, E must measure A. Consequently E measures both A

and B.

Now it has been shown above that every common measure of A and B measures the last remainder; and it has been just proved that the last remainder must measure A and B. Consequently the last remainder is the greatest common measure of A and B.

COR. Hence, if the last remainder can never be arrived at, that is, if the above process is interminable, the proposed magnitudes cannot have a common measure-in other words, they are incommensurable.


If one magnitude contain another, and leave a remainder such that the greater of the two magnitudes is to the smaller as the smaller to this remainder, then the two magnitudes will be incommensurable.

Let A, the greater of two magnitudes, contain the smaller, B, any number of times, leaving for a remainder a magnitude C, such that A: B: B: C; then A and B cannot have a common measure.

For let C, D, E, &c. be the successive remainders, in í the process of finding the common measure (last prop.) Then C cannot measure B, otherwise B would measure A (prop. xii., Book v.), and there could be no remainder; but C is contained as often in B as B is contained in A (def. 7, Book v.) Let then P be the greatest multiple of

B, which is contained in A; and let Q be an equimultiple of C, which must therefore be the greatest multiple of C contained in B. Then (prop. vi., Book v. cor.),

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Hence D cannot measure C, inasmuch as C cannot measure B.

Let now P' be the greatest multiple of C in B; and Q' the like multiple of D: then, pursuing the same course as before, there results the proportion

CD: D: E;

so that E cannot measure D; and so on, for each succeeding remainder. It appears, therefore, that it is impossible for any remainder ever to measure the preceding remainder. Consequently the process for finding the common measure of A and B will be interminable; that is, the two magnitudes are incommensurable.

COR. If a line be divided in extreme and mean proportion (prop. xi., Book 11., or prop. xxx., Book vi.), the two parts will be incommensurable.


The diagonal and side of a square are incommensurable. Let ABCD be a square; the diagonal AC is incommensurable with its side AB.

From the point C as a centre, with the radius CB, describe the semicircle FBE.



Then the angle at B being right, AB touches the circle; consequently (pr. xxxvi. B. III., and pr. xvii. B. vi.) AE: AB:: AB: AF; and therefore (prop. iii., Supp.), AE, AB are incommensurable. Hence AC, AB are also incommensurable; for if these had a common measure, the same would also measure their sum AE, which has been proved to be incommensurable with AB. Therefore the diagonal of a square is incommensurable with its side.


It appears, from the above demonstration, that it would be in vain to attempt to express accurately by numbers the side and diagonal of a square; and, from the demonstration of proposition iii., we infer that the attempt to give numerical values to the segments of a line divided in extreme and mean proportion would be equally fruitless. We are thus furnished with conclusive evidence of the insufficiency of numbers to answer rigorously all the demands of geometry. We cannot, for instance, take upon ourselves to say that any two lines that may be promiscuously chosen shall be susceptible of accurate numerical representation, without first enquiring whether these lines are commensurable or yot; since, for ought we know to the contrary, one of the

proposed lines may be equal to the side, and the other to the diagonal of the same square; or the two lines may be the segments of a third line divided in extreme and mean proportion, in either of which cases an accurate numerical representation of the two lines is, as we have just seen, demonstrably impossible. It follows, therefore, that all those treatises on proportion in which numbers are employed to represent lines are quite inadequate to answer rigorously all the purposes of geometry.





Explanation of Fundamental Principles.


(Article 1.) PLANE TRIGONOMETRY is one of the most interesting applications of the Elements of Geometry. It is also a subject of very considerable practical importance; as its object is to investigate theorems, and to furnish methods of computation, by the aid of which any side, or any angle of a plane triangle may be accurately determined, whenever the numerical values of certain other parts of the triangle are ascertained by actual observation and measurement.

Of these parts, as they are called, there are six-the three sides and the three angles. The surface inclosed by the three sides, though determinable by aid of trigonometry, is not included under this technical term.

As all plane rectilinear figures may, by means of

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