therefore E measures A, B, C ; I say next that it is also the greatest. For, if possible, let there be some magnitude F grea E, and let it measure A, B, C. Now, since F measures A, B, C, it will also measure A, B, and will measure the greatest common measure of A, But the greatest common measure of A, B is D ; therefore F measures D. But it measures C also; therefore F measures C, D; therefore F will also measure the greatest common of C, D. But that is E; therefore F will measure E, the greater the less: which is impossible. Therefore no magnitude greater than the magn will measure A, B, C; [ therefore E is the greatest common measure of A, B do not measure C, and, if it measure it, D is itself the greatest common Therefore the greatest common measure of the thr commensurable magnitudes has been found. PORISM. From this it is manifest that, if a ma measure three magnitudes, it will also measure their common measure. Similarly too, with more magnitudes, the greatest measure can be found, and the porism can be extende Q. E The argument in the proof that e, the greatest common measure the greatest common measure of a, b, c, is the same as that in vII. 3 The Porism contains the extension of the process to the cas or more magnitudes, corresponding to Heron's remark with regar similar extension of vII. 3 to the case of four or more numbers. PROPOSITION 5. Commensurable magnitudes have to one another th which a number has to a number. Let A, B be commensurable magnitudes; I say that A has to B the ratio which a number h number. For, since A, B are commensurable, some magnit measure them. Let it measure them, and let it be C. And, as many times as C measures A, so many there be in D; and, as many times as C measures B, so many units be in E. Since then C measures A according to the units while the unit also measures D according to the units therefore the unit measures the number D the same of times as the magnitude C measures A; [v therefore, as C is to A, so is the unit to D; therefore, inversely, as A is to C, so is D to the unit [cf. Again, since C measures B according to the unit while the unit also measures E according to the unit x. 5] PROPOSITIONS 4, 5 25 therefore the unit measures E the same number of times as C measures B; therefore, as C is to B, so is the unit to E. But it was also proved that, as A is to C, so is D to the unit; therefore, ex aequali, [v. 22] as A is to B, so is the number D to E. Therefore the commensurable magnitudes A, B have to one another the ratio which the number D has to the number E. Q. E. D. The argument is as follows. If a, b be commensurable magnitudes, they have some common measure <, and It will be observed that, in stating the proportion (1), Euclid is merely expressing the fact that a is the same multiple of c that m is of 1. In other words, he rests the statement on the definition of proportion in VII. Def. 20. This, however, is applicable only to four numbers, and c, a are not numbers but magnitudes. Hence the statement of the proportion is not legitimate unless it is proved that it is true in the sense of v. Def. 5 with regard to magnitudes in general, the numbers 1, m being magnitudes. Similarly with regard to the other proportions in the proposition. There is, therefore, a hiatus. Euclid ought to have proved that magnitudes which are proportional in the sense of vII. Def. 20 are also proportional in the sense of v. Def. 5, or that the proportion of numbers is included in the proportion of magnitudes as a particular case. Simson has proved this in his Proposition C inserted in Book v. (see Vol. II. pp. 126-8). The portion of that proposition which is required here is the proof that, Take any equimultiples pa, pc of a, c and any equimultiples qb, qd of b, d. Now pa = pmb | But, according as pmb >=< qb, pmd >=< qd. Therefore, according as pa >=< qb, pa >=< qd. And pa, pc are any equimultiples of a, c, and qb, qd any equimultiples of b, d. Therefore a: b = c: d. [v. Def. 5.] IO 15 20 ratio which the number D has to the number E; 5 I say that the magnitudes A, B are commensurable. For let A be divided into as many equal parts are units in D, and let C be equal to one of them; and let F be made up of as many magnitudes equal there are units in E. Since then there are in A as many magnitudes e as there are units in D, whatever part the unit is of D, the same part is C of therefore, as C is to A, so is the unit to D. But the unit measures the number D; therefore C also measures A. [v And since, as C is to A, so is the unit to D, therefore, inversely, as A is to C, so is the number unit. [ Again, since there are in F as many magnitu to C as there are units in E, therefore, as C is to F, so is the unit to E. But it was also proved that, as A is to C, so is D to the unit; 25 therefore, ex aequali, as A is to F, so is D to E. But, as D is to E, so is A to B ; therefore also, as A is to B, so is it to Falso. [ Therefore A has the same ratio to each of the m B, F; 30 therefore B is equal to F. But C measures F; therefore it measures B also. Further it measures A also ; therefore C measures A, B. And, if a mean proportional be also taken between as B, as A is to F, so will the square on A be to the square that is, as the first is to the third, so is the figure on th 45 to that which is similar and similarly described on the s [VI. But, as A is to F, so is the number D to the numb therefore it has been contrived that, as the number the number E, so also is the figure on the straight lin the figure on the straight line B. Q. E. 15. But the unit measures the number D; therefore C also mea These words are redundant, though they are apparently found in all the MSS. The same link to connect the proportion of numbers with the pr of magnitudes as was necessary in the last proposition is necessary he being premised, the argument is as follows. so that measures b n times, and a, b are commensurable. The Porism is often used in the later propositions. It follows (1 a be a given straight line, and m, n any numbers, a straight line x found such that |