Elementa

Euclid

Elementa

Euclid · Elem 8 · perseus-eng2 · More by Euclid

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Original / primary: Perseus Eng2

8.prop.1.p2400

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

8.prop.1.p2401

Let there be as many numbers as we please, A, B, C, D, in continued proportion, and let the extremes of them A, D be prime to one another; I say that A, B, C, D are the least of those which have the same ratio with them.

8.prop.1.p2402

For, if not, let E, F, G, H be less than A, B, C, D, and in the same ratio with them.

8.prop.1.p2403

Now, since A, B, C, D are in the same ratio with E, F, G, H, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers E, F, G, H, therefore, ex aequali, as A is to D, so is E to H. [VII. 14]

8.prop.1.p2404

But A, D are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [VII. 20]

8.prop.1.p2405

Therefore A measures E, the greater the less: which is impossible.

8.prop.1.p2406

Therefore E, F, G, H which are less than A, B, C, D are not in the same ratio with them.

8.prop.1.p2407

Therefore A, B, C, D are the least of those which have the same ratio with them. Q. E. D.

8.prop.2.p2408

To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.

8.prop.2.p2409

Let the ratio of A to B be the given ratio in least numbers; thus it is required to find numbers in continued proportion, as many as may be prescribed, and the least that are in the ratio of A to B.

8.prop.2.p2410

Let four be prescribed; let A by multiplying itself make C, and by multiplying B let it make D; let B by multiplying itself make E; further, let A by multiplying C, D, E make F, G, H, and let B by multiplying E make K.

8.prop.2.p2411

Now, since A by multiplying itself has made C, and by multiplying B has made D, therefore, as A is to B, so is C to D. [VII. 17]

8.prop.2.p2412

Again, since A by multiplying B has made D, and B by multiplying itself has made E, therefore the numbers A, B by multiplying B have made the numbers D, E respectively.

8.prop.2.p2413

Therefore, as A is to B, so is D to E. [VII. 18]

8.prop.2.p2414

But, as A is to B, so is C to D; therefore also, as C is to D, so is D to E.

8.prop.2.p2415

And, since A by multiplying C, D has made F, G, therefore, as C is to D, so is F to G. [VII. 17]

8.prop.2.p2416

But, as C is to D, so was A to B; therefore also, as A is to B, so is F to G.

8.prop.2.p2417

Again, since A by multiplying D, E has made G, H, therefore, as D is to E, so is G to H. [VII. 17]

8.prop.2.p2418

But, as D is to E, so is A to B.

8.prop.2.p2419

Therefore also, as A is to B, so is G to H.

8.prop.2.p2420

And, since A, B by multiplying E have made H, K, therefore, as A is to B, so is H to K. [VII. 18]

8.prop.2.p2421

But, as A is to B, so is F to G, and G to H.

8.prop.2.p2422

Therefore also, as F is to G, so is G to H, and H to K; therefore C, D, E, and F, G, H, K are proportional in the ratio of A to B.

8.prop.2.p2423

I say next that they are the least numbers that are so.

8.prop.2.p2424

For, since A, B are the least of those which have the same ratio with them, and the least of those which have the same ratio are prime to one another, [VII. 22] therefore A, B are prime to one another.

8.prop.2.p2425

And the numbers A, B by multiplying themselves respectively have made the numbers C, E, and by multiplying the numbers C, E respectively have made the numbers F, K; therefore C, E and F, K are prime to one another respectively. [VII. 27]

8.prop.2.p2426

But, if there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, they are the least of those which have the same ratio with them. [VIII. 1]

8.prop.2.p2427

Therefore C, D, E and F, G, H, K are the least of those which have the same ratio with A, B. Q. E. D.

8.prop.2.p2428

Porism. From this it is manifest that, if three numbers in continued proportion be the least of those which have the same ratio with them, the extremes of them are squares, and, if four numbers, cubes.

8.prop.3.p2429

If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.

8.prop.3.p2430

Let as many numbers as we please, A, B, C, D, in continued proportion be the least of those which have the same ratio with them; I say that the extremes of them A, D are prime to one another.

8.prop.3.p2431

For let two numbers E, F, the least that are in the ratio of A, B, C, D, be taken, [VII. 33] then three others G, H, K with the same property; and others, more by one continually, [VIII. 2] until the multitude taken becomes equal to the multitude of the numbers A, B, C, D.

8.prop.3.p2432

Let them be taken, and let them be L, M, N, O.

8.prop.3.p2433

Now, since E, F are the least of those which have the same ratio with them, they are prime to one another. [VII. 22]

8.prop.3.p2434

And, since the numbers E, F by multiplying themselves respectively have made the numbers G, K, and by multiplying the numbers G, K respectively have made the numbers L, O, [VIII. 2, Por.] therefore both G, K and L, O are prime to one another. [VII. 27]

8.prop.3.p2435

And, since A, B, C, D are the least of those which have the same ratio with them, while L, M, N, O are the least that are in the same ratio with A, B, C, D, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers L, M, N, O, therefore the numbers A, B, C, D are equal to the numbers L, M, N, O respectively; therefore A is equal to L, and D to O.

8.prop.3.p2436

And L, O are prime to one another.

8.prop.3.p2437

Therefore A, D are also prime to one another. Q. E. D.

8.prop.4.p2438

Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.

8.prop.4.p2439

Let the given ratios in least numbers be that of A to B, that of C to D, and that of E to F; thus it is required to find numbers in continued proportion which are the least that are in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F.

8.prop.4.p2440

Let G, the least number measured by B, C, be taken. [VII. 34]

8.prop.4.p2441

And, as many times as B measures G, so many times also let A measure H, and, as many times as C measures G, so many times also let D measure K.

8.prop.4.p2442

Now E either measures or does not measure K.

8.prop.4.p2443

First, let it measure it.

8.prop.4.p2444

And, as many times as E measures K, so many times let F measure L also.

8.prop.4.p2445

Now, since A measures H the same number of times that B measures G, therefore, as A is to B, so is H to G. [VII. Def. 20, VII. 13]

8.prop.4.p2446

For the same reason also, as C is to D, so is G to K, and further, as E is to F, so is K to L; therefore H, G, K, L are continuously proportional in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F.

8.prop.4.p2447

I say next that they are also the least that have this property.

8.prop.4.p2448

For, if H, G, K, L are not the least numbers continuously proportional in the ratios of A to B, of C to D, and of E to F, let them be N, O, M, P.

8.prop.4.p2449

Then since, as A is to B, so is N to O, while A, B are least, and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; therefore B measures O. [VII. 20]

8.prop.4.p2450

For the same reason C also measures O; therefore B, C measure O; therefore the least number measured by B, C will also measure O. [VII. 35]

8.prop.4.p2451

But G is the least number measured by B, C; therefore G measures O, the greater the less: which is impossible.

8.prop.4.p2452

Therefore there will be no numbers less than H, G, K, L which are continuously in the ratio of A to B, of C to D, and of E to F.

8.prop.4.p2453

Next, let E not measure K.

8.prop.4.p2454

Let M, the least number measured by E, K, be taken.

8.prop.4.p2455

And, as many times as K measures M, so many times let H, G measure N, O respectively, and, as many times as E measures M, so many times let F measure P also.

8.prop.4.p2456

Since H measures N the same number of times that G measures O, therefore, as H is to G, so is N to O. [VII. 13 and Def. 20]

8.prop.4.p2457

But, as H is to G, so is A to B; therefore also, as A is to B, so is N to O.

8.prop.4.p2458

For the same reason also, as C is to D, so is O to M.

8.prop.4.p2459

Again, since E measures M the same number of times that F measures P, therefore, as E is to F, so is M to P; [VII. 13 and Def. 20] therefore N, O, M, P are continuously proportional in the ratios of A to B, of C to D, and of E to F.

8.prop.4.p2460

I say next that they are also the least that are in the ratios A : B, C : D, E : F.

8.prop.4.p2461

For, if not, there will be some numbers less than N, O, M, P continuously proportional in the ratios A : B, C : D, E : F.

8.prop.4.p2462

Let them be Q, R, S, T.

8.prop.4.p2463

Now since, as Q is to R, so is A to B, while A, B are least, and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent, [VII. 20] therefore B measures R.

8.prop.4.p2464

For the same reason C also measures R; therefore B, C measure R.

8.prop.4.p2465

Therefore the least number measured by B, C will also measure R. [VII. 35]

8.prop.4.p2466

But G is the least number measured by B, C; therefore G measures R.

8.prop.4.p2467

And, as G is to R, so is K to S: [VII. 13] therefore K also measures S.

8.prop.4.p2468

But E also measures S; therefore E, K measure S.

8.prop.4.p2469

Therefore the least number measured by E, K will also measure S. [VII. 35]

8.prop.4.p2470

But M is the least number measured by E, K; therefore M measures S, the greater the less: which is impossible.

8.prop.4.p2471

Therefore there will not be any numbers less than

8.prop.4.p2471

N

8.prop.4.p2471

,

8.prop.4.p2471

O

8.prop.4.p2471

,

8.prop.4.p2471

M

8.prop.4.p2471

,

8.prop.4.p2471

P

8.prop.4.p2471

continuously proportional in the ratios of

8.prop.4.p2471

A

8.prop.4.p2471

to

8.prop.4.p2471

B

8.prop.4.p2471

, of

8.prop.4.p2471

C

8.prop.4.p2471

to

8.prop.4.p2471

D

8.prop.4.p2471

, and of

8.prop.4.p2471

E

8.prop.4.p2471

to

8.prop.4.p2471

F

8.prop.4.p2471

;

8.prop.4.p2471

therefore

8.prop.4.p2471

N

8.prop.4.p2471

,

8.prop.4.p2471

O

8.prop.4.p2471

,

8.prop.4.p2471

M

8.prop.4.p2471

,

8.prop.4.p2471

P

8.prop.4.p2471

are the least numbers continuously proportional in the ratios

8.prop.4.p2471

A

8.prop.4.p2471

:

8.prop.4.p2471

B

8.prop.4.p2471

,

8.prop.4.p2471

C

8.prop.4.p2471

:

8.prop.4.p2471

D

8.prop.4.p2471

,

8.prop.4.p2471

E

8.prop.4.p2471

:

8.prop.4.p2471

F

8.prop.4.p2471

. Q. E. D.

8.prop.4.p2471

1

8.prop.5.p2472

Plane numbers have to one another the ratio compounded of the ratios of their sides.

8.prop.5.p2473

Let A, B be plane numbers, and let the numbers C, D be the sides of A, and E, F of B; I say that A has to B the ratio compounded of the ratios of the sides.

8.prop.5.p2474

For, the ratios being given which C has to E and D to F, let the least numbers G, H, K that are continuously in the ratios C : E, D : F be taken, so that, as C is to E, so is G to H, and, as D is to F, so is H to K. [VIII. 4]

8.prop.5.p2475

And let D by multiplying E make L.

8.prop.5.p2476

Now, since D by multiplying C has made A, and by multiplying E has made L, therefore, as C is to E, so is A to L. [VII. 17]

8.prop.5.p2477

But, as C is to E, so is G to H; therefore also, as G is to H, so is A to L.

8.prop.5.p2478

Again, since E by multiplying D has made L, and further by multiplying F has made B, therefore, as D is to F, so is L to B. [VII. 17]

8.prop.5.p2479

But, as D is to F, so is H to K; therefore also, as H is to K, so is L to B.

8.prop.5.p2480

But it was also proved that, as G is to H, so is A to L; therefore, ex aequali, as G is to K, so is A to B. [VII. 14]

8.prop.5.p2481

But

8.prop.5.p2481

G

8.prop.5.p2481

has to

8.prop.5.p2481

K

8.prop.5.p2481

the ratio compounded of the ratios of the

8.prop.5.p2481

sides; therefore

8.prop.5.p2481

A

8.prop.5.p2481

also has to

8.prop.5.p2481

B

8.prop.5.p2481

the ratio compounded of the ratios of the sides. Q. E. D.

8.prop.5.p2481

1

8.prop.6.p2482

If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.

8.prop.6.p2483

Let there be as many numbers as we please, A, B, C, D, E, in continued proportion, and let A not measure B; I say that neither will any other measure any other.

8.prop.6.p2484

Now it is manifest that A, B, C, D, E do not measure one another in order; for A does not even measure B.

8.prop.6.p2485

I say, then, that neither will any other measure any other.

8.prop.6.p2486

For, if possible, let A measure C.

8.prop.6.p2487

And, however many A, B, C are, let as many numbers F, G, H, the least of those which have the same ratio with A, B, C, be taken. [VII. 33]

8.prop.6.p2488

Now, since F, G, H are in the same ratio with A, B, C, and the multitude of the numbers A, B, C is equal to the multitude of the numbers F, G, H, therefore, ex aequali, as A is to C, so is F to H. [VII. 14]

8.prop.6.p2489

And since, as A is to B, so is F to G, while A does not measure B, therefore neither does F measure G; [VII. Def. 20] therefore F is not an unit, for the unit measures any number.

8.prop.6.p2490

Now F, H are prime to one another. [VIII. 3]

8.prop.6.p2491

And, as F is to H, so is A to C; therefore neither does A measure C.

8.prop.6.p2492

Similarly we can prove that neither will any other measure any other. Q. E. D.

8.prop.7.p2493

If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.

8.prop.7.p2494

Let there be as many numbers as we please, A, B, C, D, in continued proportion; and let A measure D; I say that A also measures B.

8.prop.7.p2495

For, if A does not measure B, neither will any other of the numbers measure any other. [VIII. 6]

8.prop.7.p2496

But A measures D.

8.prop.7.p2497

Therefore A also measures B. Q. E. D.

8.prop.8.p2498

If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.

8.prop.8.p2499

Let the numbers C, D fall between the two numbers A, B in continued proportion with them, and let E be made in the same ratio to F as A is to B; I say that, as many numbers as have fallen between A, B in continued proportion, so many will also fall between E, F in continued proportion.

8.prop.8.p2500

For, as many as A, B, C, D are in multitude, let so many numbers G, H, K, L, the least of those which have the same ratio with A, C, D, B, be taken; [VII. 33] therefore the extremes of them G, L are prime to one another. [VIII. 3]

8.prop.8.p2501

Now, since A, C, D, B are in the same ratio with G, H, K, L, and the multitude of the numbers A, C, D, B is equal to the multitude of the numbers G, H, K, L, therefore, ex aequali, as A is to B, so is G to L. [VII. 14]

8.prop.8.p2502

But, as A is to B, so is E to F; therefore also, as G is to L, so is E to F.

8.prop.8.p2503

But G, L are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [VII. 20]

8.prop.8.p2504

Therefore G measures E the same number of times as L measures F.

8.prop.8.p2505

Next, as many times as G measures E, so many times let H, K also measure M, N respectively; therefore G, H, K, L measure E, M, N, F the same number of times.

8.prop.8.p2506

Therefore G, H, K, L are in the same ratio with E, M, N, F. [VII. Def. 20]

8.prop.8.p2507

But G, H, K, L are in the same ratio with A, C, D, B; therefore A, C, D, B are also in the same ratio with E, M, N, F.

8.prop.8.p2508

But A, C, D, B are in continued proportion; therefore E, M, N, F are also in continued proportion.

8.prop.8.p2509

Therefore, as many numbers as have fallen between

8.prop.8.p2509

A

8.prop.8.p2509

,

8.prop.8.p2509

B

8.prop.8.p2509

in continued proportion with them, so many numbers have also fallen between

8.prop.8.p2509

E

8.prop.8.p2509

,

8.prop.8.p2509

F

8.prop.8.p2509

in continued proportion. Q. E. D.

8.prop.8.p2509

1

8.prop.9.p2510

If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.

8.prop.9.p2511

Let A, B be two numbers prime to one another, and let C, D fall between them in continued proportion, and let the unit E be set out; I say that, as many numbers as fall between A, B in continued proportion, so many will also fall between either of the numbers A, B and the unit in continued proportion.

8.prop.9.p2512

For let two numbers F, G, the least that are in the ratio of A, C, D, B, be taken, three numbers H, K, L with the same property, and others more by one continually, until their multitude is equal to the multitude of A, C, D, B. [VIII. 2]

8.prop.9.p2513

Let them be taken, and let them be M, N, O, P.

8.prop.9.p2514

It is now manifest that F by multiplying itself has made H and by multiplying H has made M, while G by multiplying itself has made L and by multiplying L has made P. [VIII. 2, Por.]

8.prop.9.p2515

And, since M, N, O, P are the least of those which have the same ratio with F, G, and A, C, D, B are also the least of those which have the same ratio with F, G, [VIII. 1] while the multitude of the numbers M, N, O, P is equal to the multitude of the numbers A, C, D, B, therefore M, N, O, P are equal to A, C, D, B respectively; therefore M is equal to A, and P to B.

8.prop.9.p2516

Now, since F by multiplying itself has made H, therefore F measures H according to the units in F.

8.prop.9.p2517

But the unit E also measures F according to the units in it; therefore the unit E measures the number F the same number of times as F measures H.

8.prop.9.p2518

Therefore, as the unit E is to the number F, so is F to H. [VII. Def. 20]

8.prop.9.p2519

Again, since F by multiplying H has made M, therefore H measures M according to the units in F.

8.prop.9.p2520

But the unit E also measures the number F according to the units in it; therefore the unit E measures the number F the same number of times as H measures M.

8.prop.9.p2521

Therefore, as the unit E is to the number F, so is H to M.

8.prop.9.p2522

But it was also proved that, as the unit E is to the number F, so is F to H; therefore also, as the unit E is to the number F, so is F to H, and H to M.

8.prop.9.p2523

But M is equal to A; therefore, as the unit E is to the number F, so is F to H, and H to A.

8.prop.9.p2524

For the same reason also, as the unit E is to the number G, so is G to L and L to B.

8.prop.9.p2525

Therefore, as many numbers as have fallen between A, B in continued proportion, so many numbers also have fallen between each of the numbers A, B and the unit E in continued proportion. Q. E. D.

8.prop.10.p2526

If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.

8.prop.10.p2527

For let the numbers D, E and F, G respectively fall between the two numbers A, B and the unit C in continued proportion; I say that, as many numbers as have fallen between each of the numbers A, B and the unit C in continued proportion, so many numbers will also fall between A, B in continued proportion.

8.prop.10.p2528

For let D by multiplying F make H, and let the numbers D, F by multiplying H make K, L respectively.

8.prop.10.p2529

Now, since, as the unit C is to the number D, so is D to E, therefore the unit C measures the number D the same number of times as D measures E. [VII. Def. 20]

8.prop.10.p2530

But the unit C measures the number D according to the units in D; therefore the number D also measures E according to the units in D; therefore D by multiplying itself has made E.

8.prop.10.p2531

Again, since, as C is to the number D, so is E to A, therefore the unit C measures the number D the same number of times as E measures A.

8.prop.10.p2532

But the unit C measures the number D according to the units in D; therefore E also measures A according to the units in D; therefore D by multiplying E has made A.

8.prop.10.p2533

For the same reason also F by multiplying itself has made G, and by multiplying G has made B.

8.prop.10.p2534

And, since D by multiplying itself has made E and by multiplying F has made H, therefore, as D is to F, so is E to H. [VII. 17]

8.prop.10.p2535

For the same reason also, as D is to F, so is H to G. [VII. 18]

8.prop.10.p2536

Therefore also, as E is to H, so is H to G.

8.prop.10.p2537

Again, since D by multiplying the numbers E, H has made A, K respectively, therefore, as E is to H, so is A to K. [VII. 17]

8.prop.10.p2538

But, as E is to H, so is D to F; therefore also, as D is to F, so is A to K.

8.prop.10.p2539

Again, since the numbers D, F by multiplying H have made K, L respectively, therefore, as D is to F, so is K to L. [VII. 18]

8.prop.10.p2540

But, as D is to F, so is A to K; therefore also, as A is to K, so is K to L.

8.prop.10.p2541

Further, since F by multiplying the numbers H, G has made L, B respectively, therefore, as H is to G, so is L to B. [VII. 17]

8.prop.10.p2542

But, as H is to G, so is D to F; therefore also, as D is to F, so is L to B.

8.prop.10.p2543

But it was also proved that, as D is to F, so is A to K and K to L; therefore also, as A is to K, so is K to L and L to B.

8.prop.10.p2544

Therefore A, K, L, B are in continued proportion.

8.prop.10.p2545

Therefore, as many numbers as fall between each of the numbers A, B and the unit C in continued proportion, so many also will fall between A, B in continued proportion. Q. E. D.

8.prop.11.p2546

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.

8.prop.11.p2547

Let A, B be square numbers, and let C be the side of A, and D of B; I say that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D.

8.prop.11.p2548

For let C by multiplying D make E.

8.prop.11.p2549

Now, since A is a square and C is its side, therefore C by multiplying itself has made A.

8.prop.11.p2550

For the same reason also D by multiplying itself has made B.

8.prop.11.p2551

Since then C by multiplying the numbers C, D has made A, E respectively, therefore, as C is to D, so is A to E. [VII. 17]

8.prop.11.p2552

For the same reason also, as C is to D, so is E to B. [VII. 18]

8.prop.11.p2553

Therefore also, as A is to E, so is E to B.

8.prop.11.p2554

Therefore between A, B there is one mean proportional number.

8.prop.11.p2555

I say next that A also has to B the ratio duplicate of that which C has to D.

8.prop.11.p2556

For, since A, E, B are three numbers in proportion, therefore A has to B the ratio duplicate of that which A has to E. [V. Def. 9]

8.prop.11.p2557

But, as A is to E, so is C to D.

8.prop.11.p2558

Therefore A has to B the ratio duplicate of that which the side C has to D. Q. E. D.

8.prop.12.p2559

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.

8.prop.12.p2560

Let A, B be cube numbers, and let C be the side of A, and D of B; I say that between A, B there are two mean proportional numbers, and A has to B the ratio triplicate of that which C has to D.

8.prop.12.p2561

For let C by multiplying itself make E, and by multiplying D let it make F; let D by multiplying itself make G, and let the numbers C, D by multiplying F make H, K respectively.

8.prop.12.p2562

Now, since A is a cube, and C its side, and C by multiplying itself has made E, therefore C by multiplying itself has made E and by multiplying E has made A.

8.prop.12.p2563

For the same reason also D by multiplying itself has made G and by multiplying G has made B.

8.prop.12.p2564

And, since C by multiplying the numbers C, D has made E, F respectively, therefore, as C is to D, so is E to F. [VII. 17]

8.prop.12.p2565

For the same reason also, as C is to D, so is F to G. [VII. 18]

8.prop.12.p2566

Again, since C by multiplying the numbers E, F has made A, H respectively, therefore, as E is to F, so is A to H. [VII. 17]

8.prop.12.p2567

But, as E is to F, so is C to D.

8.prop.12.p2568

Therefore also, as C is to D, so is A to H.

8.prop.12.p2569

Again, since the numbers C, D by multiplying F have made H, K respectively, therefore, as C is to D, so is H to K. [VII. 18]

8.prop.12.p2570

Again, since D by multiplying each of the numbers F, G has made K, B respectively, therefore, as F is to G, so is K to B. [VII. 17]

8.prop.12.p2571

But, as F is to G, so is C to D; therefore also, as C is to D, so is A to H, H to K, and K to B.

8.prop.12.p2572

Therefore H, K are two mean proportionals between A, B.

8.prop.12.p2573

I say next that A also has to B the ratio triplicate of that which C has to D.

8.prop.12.p2574

For, since A, H, K, B are four numbers in proportion, therefore A has to B the ratio triplicate of that which A has to H. [V. Def. 10]

8.prop.12.p2575

But, as A is to H, so is C to D; therefore A also has to B the ratio triplicate of that which C has to D. Q. E. D.

8.prop.13.p2576

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.

8.prop.13.p2577

Let there be as many numbers as we please, A, B, C, in continued proportion, so that, as A is to B, so is B to C; let A, B, C by multiplying themselves make D, E, F, and by multiplying D, E, F let them make G, H, K; I say that D, E, F and G, H, K are in continued proportion.

8.prop.13.p2578

For let A by multiplying B make L, and let the numbers A, B by multiplying L make M. N respectively.

8.prop.13.p2579

And again let B by multiplying C make O, and let the numbers B, C by multiplying O make P, Q respectively.

8.prop.13.p2580

Then, in manner similar to the foregoing, we can prove that D, L, E and G, M, N, H are continuously proportional in the ratio of A to B, and further E, O, F and H, P, Q, K are continuously proportional in the ratio of B to C.

8.prop.13.p2581

Now, as A is to B, so is B to C; therefore D, L, E are also in the same ratio with E, O, F, and further G, M, N, H in the same ratio with H, P, Q, K.

8.prop.13.p2582

And the multitude of D, L, E is equal to the multitude of E, O, F, and that of G, M, N, H to that of H, P, Q, K; therefore, ex acquali, as D is to E, so is E to F, and, as G is to H, so is H to K. [VII. 14] Q. E. D.

8.prop.14.p2583

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.

8.prop.14.p2584

Let A, B be square numbers, let C, D be their sides, and let A measure B; I say that C also measures D.

8.prop.14.p2585

For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D. [VIII. 11]

8.prop.14.p2586

And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [VIII. 7]

8.prop.14.p2587

And, as A is to E, so is C to D; therefore also C measures D. [VII. Def. 20]

8.prop.14.p2588

Again, let C measure D; I say that A also measures B.

8.prop.14.p2589

For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D.

8.prop.14.p2590

And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20]

8.prop.14.p2591

And A, E, B are continuously proportional; therefore A also measures B.

8.prop.14.p2592

Therefore etc. Q. E. D.

8.prop.15.p2593

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.

8.prop.15.p2594

For let the cube number A measure the cube B, and let C be the side of A and D of B; I say that C measures D.

8.prop.15.p2595

For let C by multiplying itself make E, and let D by multiplying itself make G; further, let C by multiplying D make F, and let C, D by multiplying F make H, K respectively.

8.prop.15.p2596

Now it is manifest that E, F, G and A, H, K, B are continuously proportional in the ratio of C to D. [VIII. 11, 12]

8.prop.15.p2597

And, since A, H, K, B are continuously proportional, and A measures B, therefore it also measures H. [VIII. 7]

8.prop.15.p2598

And, as A is to H, so is C to D; therefore C also measures D. [VII. Def. 20]

8.prop.15.p2599

Next, let C measure D; I say that A will also measure B.

8.prop.15.p2600

For, with the same construction, we can prove in a similar manner that A, H, K, B are continuously proportional in the ratio of C to D.

8.prop.15.p2601

And, since C measures D, and, as C is to D, so is A to H, therefore A also measures H, [VII. Def. 20] so that A measures B also. Q. E. D.

8.prop.16.p2602

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.

8.prop.16.p2603

Let A, B be square numbers, and let C, D be their sides; and let A not measure B; I say that neither does C measure D.

8.prop.16.p2604

For, if C measures D, A will also measure B. [VIII. 14]

8.prop.16.p2605

But A does not measure B; therefore neither will C measure D.

8.prop.16.p2606

Again, let C not measure D; I say that neither will A measure B.

8.prop.16.p2607

For, if A measures B, C will also measure D. [VIII. 14]

8.prop.16.p2608

But C does not measure D; therefore neither will A measure B. Q. E. D.

8.prop.17.p2609

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.

8.prop.17.p2610

For let the cube number A not measure the cube number B, and let C be the side of A, and D of B; I say that C will not measure D.

8.prop.17.p2611

For if C measures D, A will also measure B. [VIII. 15]

8.prop.17.p2612

But A does not measure B; therefore neither does C measure D.

8.prop.17.p2613

Again, let C not measure D; I say that neither will A measure B.

8.prop.17.p2614

For, if A measures B, C will also measure D. [VIII. 15]

8.prop.17.p2615

But C does not measure D; therefore neither will A measure B. Q. E. D.

8.prop.18.p2616

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.

8.prop.18.p2617

Let A, B be two similar plane numbers, and let the numbers C, D be the sides of A, and E, F of B.

8.prop.18.p2618

Now, since similar plane numbers are those which have their sides proportional, [VII. Def. 21] therefore, as C is to D, so is E to F.

8.prop.18.p2619

I say then that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E, or D to F, that is, of that which the corresponding side has to the corresponding side.

8.prop.18.p2620

Now since, as C is to D, so is E to F, therefore, alternately, as C is to E, so is D to F. [VII. 13]

8.prop.18.p2621

And, since A is plane, and C, D are its sides, therefore D by multiplying C has made A.

8.prop.18.p2622

For the same reason also E by multiplying F has made B.

8.prop.18.p2623

Now let D by multiplying E make G.

8.prop.18.p2624

Then, since D by multiplying C has made A, and by multiplying E has made G, therefore, as C is to E, so is A to G. [VII. 17]

8.prop.18.p2625

But, as C is to E, so is D to F; therefore also, as D is to F, so is A to G.

8.prop.18.p2626

Again, since E by multiplying D has made G, and by multiplying F has made B, therefore, as D is to F, so is G to B. [VII. 17]

8.prop.18.p2627

But it was also proved that, as D is to F, so is A to G; therefore also, as A is to G, so is G to B.

8.prop.18.p2628

Therefore A, G, B are in continued proportion.

8.prop.18.p2629

Therefore between A, B there is one mean proportional number.

8.prop.18.p2630

I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to E or D to F.

8.prop.18.p2631

For, since A, G, B are in continued proportion, A has to B the ratio duplicate of that which it has to G. [V. Def. 9]

8.prop.18.p2632

And, as A is to G, so is C to E, and so is D to F.

8.prop.18.p2633

Therefore A also has to B the ratio duplicate of that which C has to E or D to F. Q. E. D.

8.prop.19.p2634

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.

8.prop.19.p2635

Let A, B be two similar solid numbers, and let C, D, E be the sides of A, and F, G, H of B.

8.prop.19.p2636

Now, since similar solid numbers are those which have their sides proportional, [VII. Def. 21] therefore, as C is to D, so is F to G, and, as D is to E, so is G to H.

8.prop.19.p2637

I say that between A, B there fall two mean proportional numbers, and A has to B the ratio triplicate of that which C has to F, D to G, and also E to H.

8.prop.19.p2638

For let C by multiplying D make K, and let F by multiplying G make L.

8.prop.19.p2639

Now, since C, D are in the same ratio with F, G, and K is the product of C, D, and L the product of F, G, K, L are similar plane numbers; [VII. Def. 21] therefore between K, L there is one mean proportional number. [VIII. 18]

8.prop.19.p2640

Let it be M

8.prop.19.p2641

Therefore M is the product of D, F, as was proved in the theorem preceding this. [VIII. 18]

8.prop.19.p2642

Now, since D by multiplying C has made K, and by multiplying F has made M, therefore, as C is to F, so is K to M. [VII. 17]

8.prop.19.p2643

But, as K is to M, so is M to L.

8.prop.19.p2644

Therefore K, M, L are continuously proportional in the ratio of C to F.

8.prop.19.p2645

And since, as C is to D, so is F to G, alternately therefore, as C is to F, so is D to G. [VII. 13]

8.prop.19.p2646

For the same reason also, as D is to G, so is E to H.

8.prop.19.p2647

Therefore K, M, L are continuously proportional in the ratio of C to F, in the ratio of D to G, and also in the ratio of E to H.

8.prop.19.p2648

Next, let E, H by multiplying M make N, O respectively.

8.prop.19.p2649

Now, since A is a solid number, and C, D, E are its sides, therefore E by multiplying the product of C, D has made A.

8.prop.19.p2650

But the product of C, D is K; therefore E by multiplying K has made A.

8.prop.19.p2651

For the same reason also H by multiplying L has made B.

8.prop.19.p2652

Now, since E by multiplying K has made A, and further also by multiplying M has made N, therefore, as K is to M, so is A to N. [VII. 17]

8.prop.19.p2653

But, as K is to M, so is C to F, D to G, and also E to H; therefore also, as C is to F, D to G, and E to H, so is A to N.

8.prop.19.p2654

Again, since E, H by multiplying M have made N, O respectively, therefore, as E is to H, so is N to O. [VII. 18]

8.prop.19.p2655

But, as E is to H, so is C to F and D to G; therefore also, as C is to F, D to G, and E to H, so is A to N and N to O.

8.prop.19.p2656

Again, since H by multiplying M has made O, and further also by multiplying L has made B, therefore, as M is to L, so is O to B. [VII. 17]

8.prop.19.p2657

But, as M is to L, so is C to F, D to G, and E to H.

8.prop.19.p2658

Therefore also, as C is to F, D to G, and E to H, so not only is O to B, but also A to N and N to O.

8.prop.19.p2659

Therefore A, N, O, B are continuously proportional in the aforesaid ratios of the sides.

8.prop.19.p2660

I say that A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, or D to G, and also E to H.

8.prop.19.p2661

For, since A, N, O, B are four numbers in continued proportion, therefore A has to B the ratio triplicate of that which A has to N. [V. Def. 10]

8.prop.19.p2662

But, as A is to N, so it was proved that C is to F, D to G, and also E to H.

8.prop.19.p2663

Therefore A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, D to G, and also E to H. Q. E. D.

8.prop.20.p2664

If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

8.prop.20.p2665

For let one mean proportional number C fall between the two numbers A, B; I say that A, B are similar plane numbers.

8.prop.20.p2666

Let D, E, the least numbers of those which have the same ratio with A, C, be taken; [VII. 33] therefore D measures A the same number of times that E measures C. [VII. 20]

8.prop.20.p2667

Now, as many times as D measures A, so many units let there be in F; therefore F by multiplying D has made A, so that A is plane, and D, F are its sides.

8.prop.20.p2668

Again, since D, E are the least of the numbers which have the same ratio with C, B, therefore D measures C the same number of times that E measures B. [VII. 20]

8.prop.20.p2669

As many times, then, as E measures B, so many units let there be in G; therefore E measures B according to the units in G; therefore G by multiplying E has made B.

8.prop.20.p2670

Therefore B is plane, and E, G are its sides.

8.prop.20.p2671

Therefore A, B are plane numbers.

8.prop.20.p2672

I say next that they are also similar.

8.prop.20.p2673

For, † since F by multiplying D has made A, and by multiplying E has made C, therefore, as D is to E, so is A to C, that is, C to B. [VII. 17]

8.prop.20.p2674

Again, † since E by multiplying F, G has made C, B respectively, therefore, as F is to G, so is C to B. [VII. 17]

8.prop.20.p2675

But, as C is to B, so is D to E; therefore also, as D is to E, so is F to G.

8.prop.20.p2676

And alternately, as D is to F, so is E to G. [VII. 13]

8.prop.20.p2677

Therefore

8.prop.20.p2677

A

8.prop.20.p2677

,

8.prop.20.p2677

B

8.prop.20.p2677

are similar plane numbers; for their sides

8.prop.20.p2677

are proportional. Q. E. D.

8.prop.20.p2677

1

8.prop.21.p2678

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

8.prop.21.p2679

For let two mean proportional numbers C, D fall between the two numbers A, B; I say that A, B are similar solid numbers.

8.prop.21.p2680

For let three numbers E, F, G, the least of those which have the same ratio with A, C, D, be taken; [VII. 33 or VIII. 2] therefore the extremes of them E, G are prime to one another. [VIII. 3]

8.prop.21.p2681

Now, since one mean proportional number F has fallen between E, G, therefore E, G are similar plane numbers. [VIII. 20]

8.prop.21.p2682

Let, then, H, K be the sides of E, and L, M of G.

8.prop.21.p2683

Therefore it is manifest from the theorem before this that E, F, G are continuously proportional in the ratio of H to L and that of K to M.

8.prop.21.p2684

Now, since E, F, G are the least of the numbers which have the same ratio with A, C, D, and the multitude of the numbers E, F, G is equal to the multitude of the numbers A, C, D, therefore, ex aequali, as E is to G, so is A to D. [VII. 14]

8.prop.21.p2685

But E, G are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio with them the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A the same number of times that G measures D.

8.prop.21.p2686

Now, as many times as E measures A, so many units let there be in N.

8.prop.21.p2687

Therefore N by multiplying E has made A.

8.prop.21.p2688

But E is the product of H, K; therefore N by multiplying the product of H, K has made A.

8.prop.21.p2689

Therefore A is solid, and H, K, N are its sides.

8.prop.21.p2690

Again, since E, F, G are the least of the numbers which have the same ratio as C, D, B, therefore E measures C the same number of times that G measures B.

8.prop.21.p2691

Now, as many times as E measures C, so many units let there be in O.

8.prop.21.p2692

Therefore G measures B according to the units in O; therefore O by multiplying G has made B.

8.prop.21.p2693

But G is the product of L, M; therefore O by multiplying the product of L, M has made B.

8.prop.21.p2694

Therefore B is solid, and L, M, O are its sides; therefore A, B are solid.

8.prop.21.p2695

I say that they are also similar.

8.prop.21.p2696

For since N, O by multiplying E have made A, C, therefore, as N is to O, so is A to C, that is, E to F. [VII. 18]

8.prop.21.p2697

But, as E is to F, so is H to L and K to M; therefore also, as H is to L, so is K to M and N to O.

8.prop.21.p2698

And H, K, N are the sides of A, and O, L, M the sides of B.

8.prop.21.p2699

Therefore A, B are similar solid numbers. Q. E. D.

8.prop.22.p2700

If three numbers be in continued proportion, and the first be square, the third will also be square.

8.prop.22.p2701

Let A, B, C be three numbers in continued proportion, and let A the first be square; I say that C the third is also square.

8.prop.22.p2702

For, since between A, C there is one mean proportional number, B, therefore A, C are similar plane numbers. [VIII. 20]

8.prop.22.p2703

But A is square; therefore C is also square. Q. E. D.

8.prop.23.p2704

If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.

8.prop.23.p2705

Let A, B, C, D be four numbers in continued proportion, and let A be cube; I say that D is also cube.

8.prop.23.p2706

For, since between A, D there are two mean proportional numbers B, C, therefore A, D are similar solid numbers. [VIII. 21]

8.prop.23.p2707

But A is cube; therefore D is also cube. Q. E. D.

8.prop.24.p2708

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.

8.prop.24.p2709

For let the two numbers A, B have to one another the ratio which the square number C has to the square number D, and let A be square; I say that B is also square.

8.prop.24.p2710

For, since C, D are square, C, D are similar plane numbers.

8.prop.24.p2711

Therefore one mean proportional number falls between C, D. [VIII. 18]

8.prop.24.p2712

And, as C is to D, so is A to B; therefore one mean proportional number falls between A, B also. [VIII. 8]

8.prop.24.p2713

And A is square; therefore B is also square. [VIII. 22] Q. E. D.

8.prop.25.p2714

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.

8.prop.25.p2715

For let the two numbers A, B have to one another the ratio which the cube number C has to the cube number D, and let A be cube; I say that B is also cube.

8.prop.25.p2716

For, since C, D are cube, C, D are similar solid numbers.

8.prop.25.p2717

Therefore two mean proportional numbers fall between C, D. [VIII. 19]

8.prop.25.p2718

And, as many numbers as fall between C, D in continued proportion, so many will also fall between those which have the same ratio with them; [VIII. 8] so that two mean proportional numbers fall between A, B also.

8.prop.25.p2719

Let E, F so fall.

8.prop.25.p2720

Since, then, the four numbers A, E, F, B are in continued proportion, and A is cube, therefore B is also cube. [VIII. 23] Q. E. D.

8.prop.26.p2721

Similar plane numbers have to one another the ratio which a square number has to a square number.

8.prop.26.p2722

Let A, B be similar plane numbers; I say that A has to B the ratio which a square number has to a square number.

8.prop.26.p2723

For, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [VIII. 18]

8.prop.26.p2724

Let it so fall, and let it be C; and let D, E, F, the least numbers of those which have the same ratio with A, C, B, be taken; [VII. 33 or VIII. 2] therefore the extremes of them D, F are square. [VIII. 2, Por.]

8.prop.26.p2725

And since, as D is to F, so is A to B, and D, F are square, therefore A has to B the ratio which a square number has to a square number. Q. E. D.

8.prop.27.p2726

Similar solid numbers have to one another the ratio which a cube number has to a cube number.

8.prop.27.p2727

Let A, B be similar solid numbers; I say that A has to B the ratio which a cube number has to a cube number.

8.prop.27.p2728

For, since A, B are similar solid numbers, therefore two mean proportional numbers fall between A, B. [VIII. 19]

8.prop.27.p2729

Let C, D so fall, and let E, F, G, H, the least numbers of those which have the same ratio with A, C, D, B, and equal with them in multitude, be taken; [VII. 33 or VIII. 2] therefore the extremes of them E, H are cube. [VIII. 2, Por.]

8.prop.27.p2730

And, as E is to H, so is A to B; therefore A also has to B the ratio which a cube number has to a cube number. Q. E. D.

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