G.178 magnitude multiple magnitude¶
If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder is the same multiple of the remainder that the whole is of the whole.
magnitude_multiple_magnitude()
prove / E B / is also the same multiple of / F D / that / A B / is of / C D /
assert Let the magnitude / A B / be the same multiple of the magnitude / C D / that the part / A E / subtracted is of the part / C F / subtracted
construct Whatever multiple / A E / is of / C F /, let / E B / be made that multiple of / C G /.
∵ there_number_magnitudesassert / A E / is the same multiple of / C F / that / E B / is of / C G / → therefore / A E / is the same multiple of / C F / that / A B / is of / G F /
assert But, by the assumption, / A E / is the same multiple of / C F / that / A B / is of / C D /
assert / A B / is the same multiple of each of the magnitudes / G F /, / C D /
assert / G F / = / C D /
construct Let / C F / be subtracted from each;
assert / C G / = / F D /
assert since / A E / is the same multiple of / C F / that / E B / is of / C G /,
assert / C G / = / F D /,
assert / A E / is the same multiple of / C F / that / E B / is of / F D /
assert But, by hypothesis,
assert / A E / is the same multiple of / C F / that / A B / is of / C D /
assert / E B / is the same multiple of / F D / that / A B / is of / C D /
∴ That is, / E B / will be the same multiple of / F D / that / A B / is of / C D /