Proof
Discrete analogue of VI.16: cross-multiplication preserves and
detects proportionality. Use VII.17 in one direction and the
converse argument (proportion from equal products via VII.14) in the
other.
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Depends on (3)
- VII.14Proposition VII.14If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the…
- VII.17Proposition VII.17If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the…
- VII.18Proposition VII.18If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the…
Required by (dependents) (8)
- VII.20Proposition VII.20The least numbers of those which have the same ratio with them measure those which have the same ratio the same number…
- VIII.18Proposition VIII.18Between two similar plane numbers there is one mean proportional number, and the plane number has to the plane number…
- VIII.20Proposition VIII.20If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.
- VIII.22Proposition VIII.22If three numbers be in continued proportion, and the first be square, the third will also be square.
- IX.16Proposition IX.16If two numbers be prime to one another, the second will not be to any other number as the first is to the second.
- IX.18Proposition IX.18Given two numbers, to investigate whether it is possible to find a third proportional to them.
- IX.19Proposition IX.19Given three numbers, to investigate when it is possible to find a fourth proportional to them.
- IX.35Proposition IX.35If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last…
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