Squares, Cubes, and Similar Numbers

Theorem 95 (Euclid VIII.11). Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.

Proof: Let A, B be squares with sides C, D. Let C by multiplying D make E. C by multiplying C, D makes A, E, so C : D = A : E (Theorem 56). C, D by multiplying D make E, B, so C : D = E : B (Theorem 57). So A : E = E : B. And since A, E, B are three numbers in proportion, A has to B the ratio duplicate of A to E, that is, of C to D.

Theorem 96 (Euclid VIII.12). Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

Proof: Let A, B be cubes with sides C, D. Let C by multiplying itself make E and by multiplying D make F; let D by multiplying itself make G; and let C, D by multiplying F make H, K. Then C by multiplying E makes A, and D by multiplying G makes B. As in Theorem 95, C : D = E : F = F : G. C by multiplying E, F makes A, H, so E : F = A : H. C, D by multiplying F make H, K, so C : D = H : K. D by multiplying F, G makes K, B, so F : G = K : B. So A : H = H : K = K : B, and A has to B the ratio triplicate of C to D.

Theorem 97 (Euclid VIII.13). If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products will be proportional; and, if the original numbers by multiplying the products make certain numbers, the latter will also be proportional.

Proof: Let A, B, C be in continued proportion. Let them by multiplying themselves make D, E, F, and by multiplying D, E, F make G, H, K. Let A by multiplying B make L, and A, B by multiplying L make M, N; let B by multiplying C make O, and B, C by multiplying O make P, Q. As in Theorems 95 and 96, D, L, E and G, M, N, H are in the ratio of A to B, and E, O, F and H, P, Q, K in the ratio of B to C. And A : B = B : C. So, ex aequali, D : E = E : F and G : H = H : K (Theorem 54).

Theorem 98 (Euclid VIII.14). If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.

Proof: Let A, B be squares with sides C, D, and let C by multiplying D make E. A, E, B are in continued proportion in the ratio of C to D (Theorem 95). If A measures B, then A measures E (Theorem 86), and since A : E = C : D, C measures D (Definition 18). If C measures D, then A measures E, and E measures B, so A measures B.

Theorem 99 (Euclid VIII.15). If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.

Proof: As Theorem 98, by the two means of Theorem 96 and by Theorem 86.

Theorem 100 (Euclid VIII.16). If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.

Proof: If the side measured the side, the square would measure the square (Theorem 98); and conversely.

Theorem 101 (Euclid VIII.17). If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.

Proof: By Theorem 99, as Theorem 100.

Theorem 102 (Euclid VIII.18). Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

Proof: Let A be plane with sides C, D, and B with sides E, F, and C : D = E : F. Alternately, C : E = D : F (Theorem 53). Let D by multiplying E make G. D by multiplying C, E makes A, G, so C : E = A : G (Theorem 56). E by multiplying D, F makes G, B, so D : F = G : B. So A : G = G : B. And A has to B the ratio duplicate of A to G, that is, of C to E.

Theorem 103 (Euclid VIII.19). Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.

Proof: Let A be solid with sides C, D, E, and B with sides F, G, H, and C : D = F : G, D : E = G : H. Let C by multiplying D make K, and F by multiplying G make L. K, L are similar plane numbers, so they have one mean M, the product of D, F, and K : M = M : L = C : F (Theorem 102). Also C : F = D : G = E : H (Theorem 53, Theorem 54). E by multiplying K makes A, and H by multiplying L makes B. Let E, H by multiplying M make N, O. E by multiplying K, M makes A, N, so K : M = A : N. E, H by multiplying M make N, O, so E : H = N : O. H by multiplying M, L makes O, B, so M : L = O : B. So A : N = N : O = O : B, and A has to B the ratio triplicate of C to F.

Theorem 104 (Euclid VIII.20). If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

Proof: Let C fall between A, B. Take D, E the least in the ratio of A to C (Theorem 72). D measures A the same number of times that E measures C (Theorem 59); let there be so many units in F. Then F by multiplying D makes A, so A is plane with sides D, F. Again D measures C the same number of times that E measures B; let there be so many units in G. Then G by multiplying E makes B, so B is plane with sides E, G. And F, G by multiplying E make C, B, so F : G = C : B = D : E (Theorem 57). Alternately, D : F = E : G. So A, B are similar plane numbers.

Theorem 105 (Euclid VIII.21). If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

Proof: Let C, D fall between A, B. Take E, F, G the least in the ratio of A, C, D (Theorem 81). E, G are prime to one another (Theorem 82), and one mean falls between them, so they are similar plane numbers (Theorem 104): let H, K be the sides of E, and L, M of G, so that E, F, G are in the ratio of H to L and of K to M. Ex aequali, E : G = A : D, so E measures A the same number of times that G measures D (Theorem 59); let there be so many units in N. Then A is solid with sides H, K, N. Likewise E : G = C : B, so E measures C the same number of times that G measures B; let there be so many units in O. Then B is solid with sides L, M, O. And N, O by multiplying E make A, C, so N : O = A : C = E : F = H : L = K : M. So A, B are similar solid numbers.

Theorem 106 (Euclid VIII.22). If three numbers be in continued proportion, and the first be square, the third will also be square.

Proof: One mean falls between the first and third, so they are similar plane numbers (Theorem 104). The first is square, so the third is square.

Theorem 107 (Euclid VIII.23). If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.

Proof: Two means fall between the first and fourth, so they are similar solid numbers (Theorem 105). The first is cube, so the fourth is cube.

Theorem 108 (Euclid VIII.24). If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.

Proof: Let A : B = C : D, with C, D squares and A square. One mean falls between C, D (Theorem 95), so one mean falls between A, B (Theorem 87). A is square, so B is square (Theorem 106).

Theorem 109 (Euclid VIII.25). If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.

Proof: By Theorem 96, Theorem 87 and Theorem 107, as Theorem 108.

Theorem 110 (Euclid VIII.26). Similar plane numbers have to one another the ratio which a square number has to a square number.

Proof: Let A, B be similar plane numbers. One mean C falls between them (Theorem 102). Take D, E, F the least in the ratio of A, C, B (Theorem 81). The extremes D, F are squares (Theorem 81, Porism). Ex aequali, D : F = A : B.

Theorem 111 (Euclid VIII.27). Similar solid numbers have to one another the ratio which a cube number has to a cube number.

Proof: Two means fall between them (Theorem 103). Of the least four numbers in that ratio the extremes are cubes (Theorem 81, Porism), and ex aequali they have the ratio of the two solids.

Theorem 112 (Euclid IX.1). If two similar plane numbers by multiplying one another make some number, the product will be square.

Proof: Let A, B be similar plane numbers, and A by multiplying B make C. Let A by multiplying itself make D. A by multiplying A, B makes D, C, so A : B = D : C (Theorem 56). One mean falls between A, B (Theorem 102), so one falls between D, C (Theorem 87). D is square, so C is square (Theorem 106).

Theorem 113 (Euclid IX.2). If two numbers by multiplying one another make a square number, they are similar plane numbers.

Proof: Let A by multiplying B make the square C, and by multiplying itself make D. A : B = D : C (Theorem 56). D, C are squares, so one mean falls between them (Theorem 95), and so one between A, B (Theorem 87). So A, B are similar plane numbers (Theorem 104).

Theorem 114 (Euclid IX.3). If a cube number by multiplying itself make some number, the product will be cube.

Proof: Let the cube A by multiplying itself make B. Let C be the side of A, and C by multiplying itself make D; then C by multiplying D makes A. The unit is to C as C to D, and as D to A; so two means, C and D, fall between the unit and A. And since A by multiplying itself makes B, the unit is to A as A to B. So two means fall between A, B (Theorem 87). A is cube, so B is cube (Theorem 107).

Theorem 115 (Euclid IX.4). If a cube number by multiplying a cube number make some number, the product will be cube.

Proof: Let the cube A by multiplying the cube B make C, and by multiplying itself make D. D is cube (Theorem 114). A : B = D : C (Theorem 56). Two means fall between the cubes A, B (Theorem 96), so two fall between D, C (Theorem 87). D is cube, so C is cube (Theorem 107).

Theorem 116 (Euclid IX.5). If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.

Proof: Let the cube A by multiplying B make the cube C, and by multiplying itself make D. D is cube (Theorem 114). A : B = D : C. Two means fall between D, C (Theorem 96), so two between A, B (Theorem 87). A is cube, so B is cube (Theorem 107).

Theorem 117 (Euclid IX.6). If a number by multiplying itself make a cube number, it will itself also be cube.

Proof: Let A by multiplying itself make the cube B, and by multiplying B make C. C is cube. A by multiplying A, B makes B, C, so A : B = B : C (Theorem 56). Two means fall between the cubes B, C (Theorem 96), so two between A, B (Theorem 87). B is cube, so A is cube (Theorem 107).

Theorem 118 (Euclid IX.7). If a composite number by multiplying any number make some number, the product will be solid.

Proof: Let the composite A by multiplying B make C. Some number D measures A; let there be as many units in E. Then D by multiplying E makes A, and C is the product of D, E, B.