G.388 incommensurable magnitudes added
X.16
If two
incommensurable magnitudes be added together, the whole is incommensurable with each of them; and, if the whole be incommensurable with one of them, the original magnitudes is incommensurable.
incommensurable_magnitudes_added ()
prove / A C / is also incommensurable with each of the magnitudes / A B / , / B C / ; / A B / , / B C / are also incommensurable; < A C D E > is equal to the rectangle contained by / A C / , / C B /
A
B
C
D
A
D
C
B
construct Let the two incommensurable magnitudes / A B / , / B C / be added together
assert If
/ A C / ,
/ A B / are not incommensurable, some magnitude will
measure them
assert Let it measure them, if possible, and let it be D
assert then D measures / A C / , / A B / ,
assert it will also measure / B C /
assert It measures / A B / also
assert D measures / A B / , / B C /
assert / A B / , / B C / are commensurable
assert But they were also, by hypothesis, incommensurable:
assert ⇒⇐
assert Similarly we can prove that / A C / , / B C / are also incommensurable
assert / A C / is incommensurable with each of the magnitudes / A B / , / B C /
assert Next, let / A C / be incommensurable with one of the magnitudes / A B / , / B C /
assert First, let it be incommensurable with / A B /
assert If they are commensurable, some magnitude will measure them
assert Let it measure them, and let it be D
assert then D measures / A B / , / B C / ,
assert it will also measure / A C /
assert It measures / A B / also
assert D measures / A C / , / A B /
assert / A C / , / A B / are commensurable
assert But they were also, by hypothesis, incommensurable:
assert ⇒⇐
assert LEMMA. ~~~~~~
construct Let there be applied to
/ A B / the parallelogram
< A C D E > deficient by the
square figure < C B F D > assert This is indeed at once manifest
assert For, since < C B F D > is a square,
assert / C D / = / C B /
∴ < A C D E > is the rectangle / A C / , / C D / , that is, the rectangle / A C / , / C B /
If to any
straight line there be applied a parallelogram deficient by a square figure, the applied parallelogram is equal to the
rectangle contained by the segments of the straight line resulting from the application.
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Required for
G.390 G.400 G.407 G.408 G.409 G.410 G.411 G.412 G.413 G.414 G.415 G.416 G.417 G.418 G.419 G.420 G.421 G.423 G.425 G.427 G.431 G.433 G.434 G.435 G.436 G.437 G.438 G.439 G.440 G.441 G.442 G.443 G.444 G.445 G.446 G.447 G.448 G.449 G.450 G.451 G.452 G.453 G.455 G.457 G.458 G.459 G.460 G.461 G.462 G.463 G.464 G.465 G.466 G.467 G.468 G.470 G.472 G.474 G.476 G.477 G.478 G.479 G.480 G.481 G.482 G.483 G.484 G.485 G.486 G.487 G.488 G.489 G.490 G.491 G.492 G.493 G.494 G.495 G.496 G.497 G.498 G.499 G.500 G.584 G.589 G.597 G.600 G.601
