Euclid's Elements, Book IX, Proposition 19

Proposition 19

[The Greek text of this Proposition is corrupt. However, analogously to Proposition 18 the condition that a fourth proportional to A, B, and C exists is that A measure the product of B and C. ]

Given three numbers, to investigate when it is possible to find a fourth proportional to them.
	Let A, B, and C be the given three numbers. It is required to investigate when it is possible to find a fourth proportional to them.
Q.E.D.

Note that a fourth proportional d to a, b and c has to satisfy a:b = c:d, so d would have to be bc/a. So the third proportional exists when a divides bc. No doubt that is what Euclid concludes in the missing part of this proposition.

Next proposition: IX.20

Previous: IX.18

Book IX introduction