The same thing as VIII. 26 with cubes. It is proved in the same way except that VIII. 19 is used instead of vIII. 18. The last note of an-Nairīzi in which the name of Heron is mentioned is on this proposition. Heron is there stated (p. 194-5, ed. Curtze) to have added the two propositions that, 1. If two numbers have to one another the ratio of a square to a square, the numbers are similar plane numbers ; 2. If two numbers have to one another the ratio of a cube to a cube, the numbers are similar solid numbers. The propositions are of course the converses of VIII. 26, 27 respectively. They are easily proved. then, since there is one mean proportional (cd) between c2, d2, there is also one mean proportional between a, b. Therefore a, b are similar plane numbers. [VIII. II or 18] (2) is similarly proved by the use of VIII. 12 or 19, vIII. 8, VIII. 21. [VIII. 8] [VIII. 20] The insertion by Heron of the first of the two propositions, the converse of vIII. 26, is perhaps an argument in favour of the correctness of the text of IX. 10, though (as remarked in the note on that proposition) it does not give the easiest proof. Cf. Heron's extension of vII. 3 tacitly assumed by Euclid in VII. 33. BOOK IX. PROPOSITION I. If two similar plane numbers by multiplying one another make some number, the product will be square. Let A, B be two similar plane numbers, and let A by multiplying B make C; I say that C is square. For let A by multiplying itself make D. Therefore D is square. A B C Since then A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. And, since A, B are similar plane numbers, [VII. 17] therefore one mean proportional number falls between A, B. [VIII. 18] But, if numbers fall between two numbers in continued proportion, as many as fall between them, so many also fall between those which have the same ratio; [VIII. 8] so that one mean proportional number falls between D, C also. And D is square; therefore C is also square. The product of two similar plane numbers is a square. Now And between a, b there is one mean proportional. [VIII. 22] Q. E. D. [VIII. 18] Therefore between a2: ab there is one mean proportional. [VIII. 81 And a2 is square; therefore ab is square. [VIII. 22] PROPOSITION 2. If two numbers by multiplying one another make a square number, they are similar plane numbers. Let A, B be two numbers, and let A by multiplying B make the square number C; I say that A, B are similar plane numbers. For let A by multiplying itself make D; therefore D is square. A B C Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. And, since D is square, and C is so also, therefore D, C are similar plane numbers. [VII. 17] Therefore one mean proportional number falls between D, C. [VIII. 18] And, as D is to C, so is A to B ; therefore one mean proportional number falls between A, B also. [VIII. 8] But, if one mean proportional number fall between two numbers, they are similar plane numbers; therefore A, B are similar plane numbers. [VIII. 20] Q. E. D. If ab is a square number, a, b are similar plane numbers. (The converse of IX. 1.) For a: b = a2: ab. [VII. 17] And a2, ab being square numbers, and therefore similar plane numbers, they have one mean proportional. [VIII. 18] Therefore a, b also have one mean proportional, [VIII. 8] [VIII. 20] whence a, b are similar plane numbers. PROPOSITION 3. If a cube number by multiplying itself make some number, the product will be cube. For let the cube number A by multiplying itself make B ; I say that B is cube. H. E. II. 25 For let C, the side of A, be taken, and let C by multiplying itself make D. It is then manifest that C by multiplying D has made A. Now, since C by multiplying itself has made D, A B D therefore C measures D according to the units in itself. But further the unit also measures C according to the units in it; therefore, as the unit is to C, so is C to D. [VII. Def. 20] Again, since C by multiplying D has made A, therefore D measures A according to the units in C. But the unit also measures C according to the units in it; therefore, as the unit is to C, so is D to A. But, as the unit is to C, so is C to D; therefore also, as the unit is to C, so is C to D, and D to A. Therefore between the unit and the number A two mean proportional numbers C, D have fallen in continued proportion. Again, since A by multiplying itself has made B, therefore A measures B according to the units in itself. But the unit also measures A according to the units in it; therefore, as the unit is to A, so is A to B. [VII. Def. 20] But between the unit and A two mean proportional numbers have fallen; therefore two mean proportional numbers will also fall between A, B. But, if two mean proportional numbers fall between two numbers, and the first be cube, the second will also be cube. And A is cube; therefore B is also cube. The product of a3 into itself, or a3 . a3, is a cube. 1: a = a: a2 = a2 ; a3. : [VIII. 23] Q. E. D. Therefore between 1 and a3 there are two mean proportionals. I a3 = a3: a3, a3. Therefore two mean proportionals fall between a3 and a3, a3. [VIII. 8] (It is true that VIII. 8 is only enunciated of two pairs of numbers, but the proof is equally valid if one number of one pair is unity.) And a is a cube number: therefore a3. a3 is also cube. [VIII. 23] PROPOSITION 4. If a cube number by multiplying a cube number make some number, the product will be cube. For let the cube number A by multiplying the cube number B make C; I say that C is cube. For let A by multiplying itself make D; therefore D is cube. [IX. 3] And, since A by multiply A B ing itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. And, since A, B are cube numbers, A, B are similar solid numbers. [VII. 17] Therefore two mean proportional numbers fall between A, B; [VIII. 19] so that two mean proportional numbers will fall between D, C also. [VIII. 8] And D is cube; therefore C is also cube. [VIII. 23] Q. E. D. a3: b3 = a3. a3 : a3. b3. [VII. 17] And two mean proportionals fall between a3, b3 which are similar solid numbers. [VIII. 19] Therefore two mean proportionals fall between a3 . a3, a3 . b3. [VIII. 8] But a3. a3 is a cube: [IX. 31 therefore a3. b3 is a cube. [VIII. 23] PROPOSITION 5. If a cube number by multiplying any number make a cube number, the multiplied number will also be cube. For let the cube number A by multiplying any number B make the cube number C; |