Elementa

Euclid

Elementa

Euclid · Elem 5 · perseus-eng2 · More by Euclid

Choose a text id="mobileReaderAuthor" class="mobile-reader-author" href="/authors/euclid" aria-label="More works by Euclid"> â€”

â‹¯

Primary version Secondary version

Original / primary: Perseus Eng2

5.def.1.p1277

A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

5.def.2.p1278

The greater is a multiple of the less when it is measured by the less.

5.def.3.p1279

A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

5.def.4.p1280

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

5.def.5.p1281

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

5.def.6.p1282

Let magnitudes which have the same ratio be called proportional.

5.def.7.p1283

When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

5.def.8.p1284

A proportion in three terms is the least possible.

5.def.9.p1285

When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

5.def.10.p1286

When four magnitudes are ltcontinuouslygt proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

5.def.11.p1287

The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

5.def.12.p1288

Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.

5.def.13.p1289

Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.

5.def.14.p1290

Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.

5.def.15.p1291

Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.

5.def.16.p1292

Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.

5.def.17.p1293

A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;

5.def.17.p1294

Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.

5.def.18.p1295

A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

5.prop.1.p1296

If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.

5.prop.1.p1297

Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude; I say that, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.

5.prop.1.p1298

For, since AB is the same multiple of E that CD is of F, as many magnitudes as there are in AB equal to E, so many also are there in CD equal to F.

5.prop.1.p1299

Let AB be divided into the magnitudes AG, GB equal to E, and CD into CH, HD equal to F; then the multitude of the magnitudes AG, GB will be equal to the multitude of the magnitudes CH, HD.

5.prop.1.p1300

Now, since AG is equal to E, and CH to F, therefore AG is equal to E, and AG, CH to E, F.

5.prop.1.p1301

For the same reason

5.prop.1.p1302

GB is equal to E, and GB, HD to E, F; therefore, as many magnitudes as there are in AB equal to E, so many also are there in AB, CD equal to E, F; therefore, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.

5.prop.1.p1303

Therefore etc. Q. E. D.

5.prop.2.p1304

If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.

5.prop.2.p1305

Let a first magnitude, AB, be the same multiple of a second, C, that a third, DE, is of a fourth, F, and let a fifth, BG, also be the same multiple of the second, C, that a sixth, EH, is of the fourth F; I say that the sum of the first and fifth, AG, will be the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.

5.prop.2.p1306

For, since AB is the same multiple of C that DE is of F, therefore, as many magnitudes as there are in AB equal to C, so many also are there in DE equal to F.

5.prop.2.p1307

For the same reason also, as many as there are in BG equal to C, so many are there also in EH equal to F; therefore, as many as there are in the whole AG equal to C, so many also are there in the whole DH equal to F.

5.prop.2.p1308

Therefore, whatever multiple AG is of C, that multiple also is DH of F.

5.prop.2.p1309

Therefore the sum of the first and fifth, AG, is the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.

5.prop.2.p1310

Therefore etc. Q.E.D.

5.prop.3.p1311

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.

5.prop.3.p1312

Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples EF, GH be taken of A, C; I say that EF is the same multiple of B that GH is of D.

5.prop.3.p1313

For, since EF is the same multiple of A that GH is of C, therefore, as many magnitudes as there are in EF equal to A, so many also are there in GH equal to C.

5.prop.3.p1314

Let EF be divided into the magnitudes EK, KF equal to A, and GH into the magnitudes GL, LH equal to C; then the multitude of the magnitudes EK, KF will be equal to the multitude of the magnitudes GL, LH.

5.prop.3.p1315

And, since A is the same multiple of B that C is of D, while EK is equal to A, and GL to C, therefore EK is the same multiple of B that GL is of D.

5.prop.3.p1316

For the same reason

5.prop.3.p1317

KF is the same multiple of B that LH is of D.

5.prop.3.p1318

Since, then, a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D, and a fifth KF is also the same multiple of the second B that a sixth LH is of the fourth D, therefore the sum of the first and fifth, EF, is also the same multiple of the second B that the sum of the third and sixth, GH, is of the fourth D. [V. 2]

5.prop.3.p1319

Therefore etc. Q. E. D.

5.prop.4.p1320

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.

5.prop.4.p1321

For let a first magnitude A have to a second B the same ratio as a third C to a fourth D; and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D; I say that, as E is to G, so is F to H.

5.prop.4.p1322

For let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.

5.prop.4.p1323

Since E is the same multiple of A that F is of C, and equimultiples K, L of E, F have been taken, therefore K is the same multiple of A that L is of C. [V. 3]

5.prop.4.p1324

For the same reason M is the same multiple of B that N is of D.

5.prop.4.p1325

And, since, as A is to B, so is C to D, and of A, C equimultiples K, L have been taken, and of B, D other, chance, equimultiples M, N, therefore, if K is in excess of M, L also is in excess of N, if it is equal, equal, and if less, less. [V. Def. 5]

5.prop.4.p1326

And K, L are equimultiples of E, F, and M, N other, chance, equimultiples of G, H; therefore, as E is to G, so is F to H. [V. Def. 5]

5.prop.4.p1327

Therefore etc. Q. E. D.

5.prop.5.p1328

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.

5.prop.5.p1329

For let the magnitude AB be the same multiple of the magnitude CD that the part AE subtracted is of the part CF subtracted; I say that the remainder EB is also the same multiple of the remainder FD that the whole AB is of the whole CD.

5.prop.5.p1330

For, whatever multiple AE is of CF, let EB be made that multiple of CG.

5.prop.5.p1331

Then, since AE is the same multiple of CF that EB is of GC, therefore AE is the same multiple of CF that AB is of GF. [V. 1]

5.prop.5.p1332

But, by the assumption, AE is the same multiple of CF that AB is of CD.

5.prop.5.p1333

Therefore AB is the same multiple of each of the magnitudes GF, CD; therefore GF is equal to CD.

5.prop.5.p1334

Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD.

5.prop.5.p1335

And, since AE is the same multiple of CF that EB is of GC, and GC is equal to DF, therefore AE is the same multiple of CF that EB is of FD.

5.prop.5.p1336

But, by hypothesis,

5.prop.5.p1337

AE is the same multiple of CF that AB is of CD; therefore EB is the same multiple of FD that AB is of CD.

5.prop.5.p1338

That is, the remainder EB will be the same multiple of the remainder FD that the whole AB is of the whole CD.

5.prop.5.p1339

Therefore etc. Q. E. D.

5.prop.5.p1339

1

5.prop.6.p1340

If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.

5.prop.6.p1341

For let two magnitudes AB, CD be equimultiples of two magnitudes E, F, and let AG, CH subtracted from them be equimultiples of the same two E, F; I say that the remainders also, GB, HD, are either equal to E, F or equimultiples of them.

5.prop.6.p1342

For, first, let GB be equal to E; I say that HD is also equal to F.

5.prop.6.p1343

For let CK be made equal to F.

5.prop.6.p1344

Since AG is the same multiple of E that CH is of F, while GB is equal to E and KC to F, therefore AB is the same multiple of E that KH is of F. [V. 2]

5.prop.6.p1345

But, by hypothesis, AB is the same multiple of E that CD is of F; therefore KH is the same multiple of F that CD is of F.

5.prop.6.p1346

Since then each of the magnitudes KH, CD is the same multiple of F, therefore KH is equal to CD.

5.prop.6.p1347

Let CH be subtracted from each; therefore the remainder KC is equal to the remainder HD.

5.prop.6.p1348

But F is equal to KC; therefore HD is also equal to F.

5.prop.6.p1349

Hence, if GB is equal to E, HD is also equal to F.

5.prop.6.p1350

Similarly we can prove that, even if GB be a multiple of E, HD is also the same multiple of F.

5.prop.6.p1351

Therefore etc. Q. E. D.

5.prop.7.p1352

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

5.prop.7.p1353

Let A, B be equal magnitudes and C any other, chance, magnitude; I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B.

5.prop.7.p1354

For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.

5.prop.7.p1355

Then, since D is the same multiple of A that E is of B, while A is equal to B, therefore D is equal to E.

5.prop.7.p1356

But F is another, chance, magnitude.

5.prop.7.p1357

If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less.

5.prop.7.p1358

And D, E are equimultiples of A, B, while F is another, chance, multiple of C; therefore, as A is to C, so is B to C. [V. Def. 5]

5.prop.7.p1359

I say next that C also has the same ratio to each of the magnitudes A, B.

5.prop.7.p1360

For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude.

5.prop.7.p1361

If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less.

5.prop.7.p1362

And F is a multiple of C, while D, E are other, chance, equimultiples of A, B; therefore, as C is to A, so is C to B. [V. Def. 5]

5.prop.7.p1363

Therefore etc.

5.prop.7.p1364

Porism. From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely. Q. E. D.

5.prop.8.p1365

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.

5.prop.8.p1366

Let AB, C be unequal magnitudes, and let AB be greater; let D be another, chance, magnitude; I say that AB has to D a greater ratio than C has to D, and D has to C a greater ratio than it has to AB.

5.prop.8.p1367

For, since AB is greater than C, let BE be made equal to C; then the less of the magnitudes AE, EB, if multiplied, will sometime be greater than D. [V. Def. 4]

5.prop.8.p1368

[Case I.]

5.prop.8.p1369

First, let AE be less than EB; let AE be multiplied, and let FG be a multiple of it which is greater than D; then, whatever multiple FG is of AE, let GH be made the same multiple of EB and K of C; and let L be taken double of D, M triple of it, and successive multiples increasing by one, until what is taken is a multiple of D and the first that is greater than K. Let it be taken, and let it be N which is quadruple of D and the first multiple of it that is greather than K.

5.prop.8.p1370

Then, since K is less than N first, therefore K is not less than M.

5.prop.8.p1371

And, since FG is the same multiple of AE that GH is of EB, therefore FG is the same multiple of AE that FH is of AB. [V. 1]

5.prop.8.p1372

But FG is the same multiple of AE that K is of C; therefore FH is the same multiple of AB that K is of C; therefore FH, K are equimultiples of AB, C.

5.prop.8.p1373

Again, since GH is the same multiple of EB that K is of C, and EB is equal to C, therefore GH is equal to K.

5.prop.8.p1374

But K is not less than M; therefore neither is GH less than M.

5.prop.8.p1375

And FG is greater than D; therefore the whole FH is greater than D, M together.

5.prop.8.p1376

But D, M together are equal to N, inasmuch as M is triple of D, and M, D together are quadruple of D, while N is also quadruple of D; whence M, D together are equal to N.

5.prop.8.p1377

But FH is greater than M, D; therefore FH is in excess of N, while K is not in excess of N.

5.prop.8.p1378

And FH, K are equimultiples of AB, C, while N is another, chance, multiple of D; therefore AB has to D a greater ratio than C has to D. [V. Def. 7]

5.prop.8.p1379

I say next, that D also has to C a greater ratio than D has to AB.

5.prop.8.p1380

For, with the same construction, we can prove similarly that N is in excess of K, while N is not in excess of FH.

5.prop.8.p1381

And N is a multiple of D, while FH, K are other, chance, equimultiples of AB, C; therefore D has to C a greater ratio than D has to AB. [V. Def. 7]

5.prop.8.p1382

[Case 2.]

5.prop.8.p1383

Again, let AE be greater than EB.

5.prop.8.p1384

Then the less, EB, if multiplied, will sometime be greater than D. [V. Def. 4]

5.prop.8.p1385

Let it be multiplied, and let GH be a multiple of EB and greater than D; and, whatever multiple GH is of EB, let FG be made the same multiple of AE, and K of C.

5.prop.8.p1386

Then we can prove similarly that FH, K are equimultiples of AB, C; and, similarly, let N be taken a multiple of D but the first that is greater than FG, so that FG is again not less than M.

5.prop.8.p1387

But GH is greater than D; therefore the whole FH is in excess of D, M, that is, of N.

5.prop.8.p1388

Now K is not in excess of N, inasmuch as FG also, which is greater than GH, that is, than K, is not in excess of N.

5.prop.8.p1389

And in the same manner, by following the above argument, we complete the demonstration.

5.prop.8.p1390

Therefore etc. Q. E. D.

5.prop.9.p1391

Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.

5.prop.9.p1392

For let each of the magnitudes A, B have the same ratio to C; I say that A is equal to B.

5.prop.9.p1393

For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; [V. 8] but it has; therefore A is equal to B.

5.prop.9.p1394

Again, let C have the same ratio to each of the magnitudes A, B; I say that A is equal to B.

5.prop.9.p1395

For, otherwise, C would not have had the same ratio to each of the magnitudes A, B; [V. 8] but it has; therefore A is equal to B.

5.prop.9.p1396

Therefore etc. Q. E. D.

5.prop.10.p1397

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

5.prop.10.p1398

For let A have to C a greater ratio than B has to C; I say that A is greater than B.

5.prop.10.p1399

For, if not, A is either equal to B or less.

5.prop.10.p1400

Now A is not equal to B; for in that case each of the magnitudes A, B would have had the same ratio to C; [V. 7] but they have not; therefore A is not equal to B.

5.prop.10.p1401

Nor again is A less than B; for in that case A would have had to C a less ratio than B has to C; [V. 8] but it has not; therefore A is not less than B.

5.prop.10.p1402

But it was proved not to be equal either; therefore A is greater than B.

5.prop.10.p1403

Again, let C have to B a greater ratio than C has to A; I say that B is less than A.

5.prop.10.p1404

For, if not, it is either equal or greater.

5.prop.10.p1405

Now B is not equal to A; for in that case C would have had the same ratio to each of the magnitudes A, B; [V. 7] but it has not; therefore A is not equal to B.

5.prop.10.p1406

Nor again is B greater than A; for in that case C would have had to B a less ratio than it has to A; [V. 8] but it has not; therefore B is not greater than A.

5.prop.10.p1407

But it was proved that it is not equal either; therefore B is less than A.

5.prop.10.p1408

Therefore etc. Q. E. D.

5.prop.11.p1409

Ratios which are the same with the same ratio are also the same with one another.

5.prop.11.p1410

For, as A is to B, so let C be to D, and, as C is to D, so let E be to F; I say that, as A is to B, so is E to F.

5.prop.11.p1411

For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.

5.prop.11.p1412

Then since, as A is to B, so is C to D, and of A, C equimultiples G, H have been taken, and of B, D other, chance, equimultiples L, M, therefore, if G is in excess of L, H is also in excess of M, if equal, equal, and if less, less.

5.prop.11.p1413

Again, since, as C is to D, so is E to F, and of C, E equimultiples H, K have been taken, and of D, F other, chance, equimultiples M, N, therefore, if H is in excess of M, K is also in excess of N, if equal, equal, and if less, less.

5.prop.11.p1414

But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less; so that, in addition, if G is in excess of L, K is also in excess of N, if equal, equal, and if less, less.

5.prop.11.p1415

And G, K are equimultiples of A, E, while L, N are other, chance, equimultiples of B, F; therefore, as A is to B, so is E to F.

5.prop.11.p1416

Therefore etc. Q. E. D.

5.prop.12.p1417

If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.

5.prop.12.p1418

Let any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C to D and E to F; I say that, as A is to B, so are A, C, E to B, D, F.

5.prop.12.p1419

For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.

5.prop.12.p1420

Then since, as A is to B, so is C to D, and E to F, and of A, C, E equimultiples G, H, K have been taken, and of B, D, F other, chance, equimultiples L, M, N, therefore, if G is in excess of L, H is also in excess of M, and K of N, if equal, equal, and if less, less; so that, in addition, if G is in excess of L, then G, H, K are in excess of L, M, N, if equal, equal, and if less, less.

5.prop.12.p1421

Now G and G, H, K are equimultiples of A and A, C, E, since, if any number of magnitudes whatever are respectively equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. [V. 1]

5.prop.12.p1422

For the same reason L and L, M, N are also equimultiples of B and B, D, F; therefore, as A is to B, so are A, C, E to B, D, F. [V. Def. 5]

5.prop.12.p1423

Therefore etc. Q. E. D.

5.prop.13.p1424

If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.

5.prop.13.p1425

For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F; I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth F.

5.prop.13.p1426

For, since there are some equimultiples of C, E, and of D, F other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D, while the multiple of E is not in excess of the multiple of F, [V. Def. 7] let them be taken, and let G, H be equimultiples of C, E, and K, L other, chance, equimultiples of D, F, so that G is in excess of K, but H is not in excess of L; and, whatever multiple G is of C, let M be also that multiple of A, and, whatever multiple K is of D, let N be also that multiple of B.

5.prop.13.p1427

Now, since, as A is to B, so is C to D, and of A, C equimultiples M, G have been taken, and of B, D other, chance, equimultiples N, K, therefore, if M is in excess of N, G is also in excess of K, if equal, equal, and if less, less. [V. Def. 5]

5.prop.13.p1428

But G is in excess of K; therefore M is also in excess of N.

5.prop.13.p1429

But H is not in excess of L; and M, H are equimultiples of A, E, and N, L other, chance, equimultiples of B, F; therefore A has to B a greater ratio than E has to F. [V. Def. 7]

5.prop.13.p1430

Therefore etc. Q. E. D.

5.prop.14.p1431

If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.

5.prop.14.p1432

For let a first magnitude A have the same ratio to a second B as a third C has to a fourth D; and let A be greater than C; I say that B is also greater than D.

5.prop.14.p1433

For, since A is greater than C, and B is another, chance, magnitude, therefore A has to B a greater ratio than C has to B. [V. 8]

5.prop.14.p1434

But, as A is to B, so is C to D; therefore C has also to D a greater ratio than C has to B. [V. 13]

5.prop.14.p1435

But that to which the same has a greater ratio is less; [V. 10] therefore D is less than B; so that B is greater than D.

5.prop.14.p1436

Similarly we can prove that, if A be equal to C, B will also be equal to D; and, if A be less than C, B will also be less than D.

5.prop.14.p1437

Therefore etc. Q. E. D.

5.prop.15.p1438

Parts have the same ratio as the same multiples of them taken in corresponding order.

5.prop.15.p1439

For let AB be the same multiple of C that DE is of F; I say that, as C is to F, so is AB to DE.

5.prop.15.p1440

For, since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to F.

5.prop.15.p1441

Let AB be divided into the magnitudes AG, GH, HB equal to C, and DE into the magnitudes DK, KL, LE equal to F; then the multitude of the magnitudes AG, GH, HB will be equal to the multitude of the magnitudes DK, KL, LE.

5.prop.15.p1442

And, since AG. GH, HB are equal to one another, and DK, KL, LE are also equal to one another, therefore, as AG is to DK, so is GH to KL, and HB to LE. [V. 7]

5.prop.15.p1443

Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [V. 12] therefore, as AG is to DK, so is AB to DE.

5.prop.15.p1444

But AG is equal to C and DK to F; therefore, as C is to F, so is AB to DE.

5.prop.15.p1445

Therefore etc. Q. E. D.

5.prop.16.p1446

If four magnitudes be proportional, they will also be proportional alternately.

5.prop.16.p1447

Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D; I say that they will also be so alternately, that is, as A is to C, so is B to D.

5.prop.16.p1448

For of A, B let equimultiples E, F be taken, and of C, D other, chance, equimultiples G, H.

5.prop.16.p1449

Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is E to F.

5.prop.16.p1450

But as A is to B, so is C to D; therefore also, as C is to D, so is E to F. [V. 11]

5.prop.16.p1451

Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H. [V. 15]

5.prop.16.p1452

But, as C is to D, so is E to F; therefore also, as E is to F, so is G to H. [V. 11]

5.prop.16.p1453

But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less. [V. 14]

5.prop.16.p1454

Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less.

5.prop.16.p1455

Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D; therefore, as A is to C, so is B to D. [V. Def. 5]

5.prop.16.p1456

Therefore etc. Q. E. D.

5.prop.16.p1456

1

5.prop.17.p1457

If magnitudes be proportional componendo, they will also be proportional separando.

5.prop.17.p1458

Let AB, BE, CD, DF be magnitudes proportional componendo, so that, as AB is to BE, so is CD to DF; I say that they will also be proportional separando, that is, as AE is to EB, so is CF to DF.

5.prop.17.p1459

For of AE, EB, CF, FD let equimultiples GH, HK, LM, MN be taken, and of EB, FD other, chance, equimultiples, KO, NP.

5.prop.17.p1460

Then, since GH is the same multiple of AE that HK is of EB, therefore GH is the same multiple of AE that GK is of AB. [V. 1]

5.prop.17.p1461

But GH is the same multiple of AE that LM is of CF; therefore GK is the same multiple of AB that LM is of CF.

5.prop.17.p1462

Again, since LM is the same multiple of CF that MN is of FD, therefore LM is the same multiple of CF that LN is of CD. [V. 1]

5.prop.17.p1463

But LM was the same multiple of CF that GK is of AB; therefore GK is the same multiple of AB that LN is of CD.

5.prop.17.p1464

Therefore GK, LN are equimultiples of AB, CD.

5.prop.17.p1465

Again, since HK is the same multiple of EB that MN is of FD, and KO is also the same multiple of EB that NP is of FD, therefore the sum HO is also the same multiple of EB that MP is of FD. [V. 2]

5.prop.17.p1466

And, since, as AB is to BE, so is CD to DF, and of AB, CD equimultiples GK, LN have been taken, and of EB, FD equimultiples HO, MP, therefore, if GK is in excess of HO, LN is also in excess of MP, if equal, equal, and if less, less.

5.prop.17.p1467

Let GK be in excess of HO; then, if HK be subtracted from each, GH is also in excess of KO.

5.prop.17.p1468

But we saw that, if GK was in excess of HO, LN was also in excess of MP; therefore LN is also in excess of MP, and, if MN be subtracted from each, LM is also in excess of NP; so that, if GH is in excess of KO, LM is also in excess of NP.

5.prop.17.p1469

Similarly we can prove that, if GH be equal to KO, LM will also be equal to NP, and if less, less.

5.prop.17.p1470

And GH, LM are equimultiples of AE, CF, while KO, NP are other, chance, equimultiples of EB, FD; therefore, as AE is to EB, so is CF to FD.

5.prop.17.p1471

Therefore etc. Q. E. D.

5.prop.18.p1472

If magnitudes be proportional separando, they will also be proportional componendo.

5.prop.18.p1473

Let AE, EB, CF, FD be magnitudes proportional separando, so that, as AE is to EB, so is CF to FD; I say that they will also be proportional componendo, that is, as AB is to BE, so is CD to FD.

5.prop.18.p1474

For, if CD be not to DF as AB to BE, then, as AB is to BE, so will CD be either to some magnitude less than DF or to a greater.

5.prop.18.p1475

First, let it be in that ratio to a less magnitude DG.

5.prop.18.p1476

Then, since, as AB is to BE, so is CD to DG, they are magnitudes proportional componendo; so that they will also be proportional separando. [V. 17]

5.prop.18.p1477

Therefore, as AE is to EB, so is CG to GD.

5.prop.18.p1478

But also, by hypothesis, as AE is to EB, so is CF to FD.

5.prop.18.p1479

Therefore also, as CG is to GD, so is CF to FD. [V. 11]

5.prop.18.p1480

But the first CG is greater than the third CF; therefore the second GD is also greater than the fourth FD. [V. 14]

5.prop.18.p1481

But it is also less: which is impossible.

5.prop.18.p1482

Therefore, as AB is to BE, so is not CD to a less magnitude than FD.

5.prop.18.p1483

Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to FD itself.

5.prop.18.p1484

Therefore etc. Q. E. D.

5.prop.19.p1485

If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.

5.prop.19.p1486

For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted; I say that the remainder EB will also be to the remainder FD as the whole AB to the whole CD.

5.prop.19.p1487

For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16]

5.prop.19.p1488

And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17] that is, as BE is to EA, so is DF to CF, and, alternately, as BE is to DF, so is EA to FC. [V. 16]

5.prop.19.p1489

But, as AE is to CF, so by hypothesis is the whole AB to the whole CD.

5.prop.19.p1490

Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [V. 11]

5.prop.19.p1491

Therefore etc. [

5.prop.19.p1492

Porism. From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.

5.prop.19.trailer

] Q. E. D.

5.prop.20.p1494

If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less.

5.prop.20.p1495

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, so that, as A is to B, so is D to E, and as B is to C, so is E to F; and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and, if less, less.

5.prop.20.p1496

For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [V. 8] therefore A has to B a greater ratio than C has to B.

5.prop.20.p1497

But, as A is to B, so is D to E, and, as C is to B, inversely, so is F to E; therefore D has also to E a greater ratio than F has to E. [V. 13]

5.prop.20.p1498

But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; [V. 10] therefore D is greater than F.

5.prop.20.p1499

Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

5.prop.20.p1500

Therefore etc. Q. E. D.

5.prop.21.p1501

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.

5.prop.21.p1502

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E, and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and if less, less.

5.prop.21.p1503

For, since A is greater than C, and B is some other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8]

5.prop.21.p1504

But, as A is to B, so is E to F, and, as C is to B, inversely, so is E to D. Therefore also E has to F a greater ratio than E has to D. [V. 13]

5.prop.21.p1505

But that to which the same has a greater ratio is less; [V. 10] therefore F is less than D; therefore D is greater than F.

5.prop.21.p1506

Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

5.prop.21.p1507

Therefore etc. Q. E. D.

5.prop.22.p1508

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.

5.prop.22.p1509

Let there be any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F; I say that they will also be in the same ratio ex aequali, ltthat is, as A is to C, so is D to Fgt.

5.prop.22.p1510

For of A, D let equimultiples G, H be taken, and of B, E other, chance, equimultiples K, L; and, further, of C, F other, chance, equimultiples M, N.

5.prop.22.p1511

Then, since, as A is to B, so is D to E, and of A, D equimultiples G, H have been taken, and of B, E other, chance, equimultiples K, L, therefore, as G is to K, so is H to L. [V. 4]

5.prop.22.p1512

For the same reason also, as K is to M, so is L to N.

5.prop.22.p1513

Since, then, there are three magnitudes G, K, M, and others H, L, N equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if G is in excess of M, H is also in excess of N; if equal, equal; and if less, less. [V. 20]

5.prop.22.p1514

And G, H are equimultiples of A, D, and M, N other, chance, equimultiples of C, F.

5.prop.22.p1515

Therefore, as A is to C, so is D to F. [V. Def. 5]

5.prop.22.p1516

Therefore etc. Q. E. D.

5.prop.23.p1517

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali.

5.prop.23.p1518

Let there be three magnitudes A, B, C, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely D, E, F; and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E; I say that, as A is to C, so is D to F.

5.prop.23.p1519

Of A, B, D let equimultiples G, H, K be taken, and of C, E, F other, chance, equimultiples L, M, N.

5.prop.23.p1520

Then, since G, H are equimultiples of A, B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is G to H.

5.prop.23.p1521

For the same reason also, as E is to F, so is M to N. And, as A is to B, so is E to F; therefore also, as G is to H, so is M to N. [V. 11]

5.prop.23.p1522

Next, since, as B is to C, so is D to E, alternately, also, as B is to D, so is C to E. [V. 16]

5.prop.23.p1523

And, since H, K are equimultiples of B, D, and parts have the same ratio as their equimultiples, therefore, as B is to D, so is H to K. [V. 15]

5.prop.23.p1524

But, as B is to D, so is C to E; therefore also, as H is to K, so is C to E. [V. 11]

5.prop.23.p1525

Again, since L, M are equimultiples of C, E, therefore, as C is to E, so is L to M. [V. 15]

5.prop.23.p1526

But, as C is to E, so is H to K; therefore also, as H is to K, so is L to M, [V. 11] and, alternately, as H is to L, so is K to M. [V. 16]

5.prop.23.p1527

But it was also proved that, as G is to H, so is M to N.

5.prop.23.p1528

Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. [V. 21]

5.prop.23.p1529

And G, K are equimultiples of A, D, and L, N of C, F.

5.prop.23.p1530

Therefore, as A is to C, so is D to F.

5.prop.23.p1531

Therefore etc. Q. E. D.

5.prop.24.p1532

If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.

5.prop.24.p1533

Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F; and let also a fifth BG have to the second C the same ratio as a sixth EH has to the fourth F; I say that the first and fifth added together, AG, will have to the second C the same ratio as the third and sixth, DH, has to the fourth F.

5.prop.24.p1534

For since, as BG is to C, so is EH to F, inversely, as C is to BG, so is F to EH.

5.prop.24.p1535

Since, then, as AB is to C, so is DE to F, and, as C is to BG, so is F to EH, therefore, ex aequali, as AB is to BG, so is DE to EH. [V. 22]

5.prop.24.p1536

And, since the magnitudes are proportional separando, they will also be proportional componendo; [V. 18] therefore, as AG is to GB, so is DH to HE.

5.prop.24.p1537

But also, as BG is to C, so is EH to F; therefore, ex aequali, as AG is to C, so is DH to F. [V. 22]

5.prop.24.p1538

Therefore etc. Q. E. D.

5.prop.25.p1539

If four magnitudes be proportional, the greatest and the least are greater than the remaining two.

5.prop.25.p1540

Let the four magnitudes AB, CD, E, F be proportional so that, as AB is to CD, so is E to F, and let AB be the greatest of them and F the least; I say that AB, F are greater than CD, E.

5.prop.25.p1541

For let AG be made equal to E, and CH equal to F.

5.prop.25.p1542

Since, as AB is to CD, so is E to F, and E is equal to AG, and F to CH, therefore, as AB is to CD, so is AG to CH.

5.prop.25.p1543

And since, as the whole AB is to the whole CD, so is the part AG subtracted to the part CH subtracted, the remainder GB will also be to the remainder HD as the whole AB is to the whole CD. [V. 19]

5.prop.25.p1544

But AB is greater than CD; therefore GB is also greater than HD.

5.prop.25.p1545

And, since AG is equal to E, and CH to F, therefore AG, F are equal to CH, E.

5.prop.25.p1546

And if, GB, HD being unequal, and GB greater, AG, F be added to GB and CH, E be added to HD, it follows that AB, F are greater than CD, E.

5.prop.25.p1547

Therefore etc. Q. E. D.

Reader

Word analysis

Intertext edge

Note