Proposition 31. If there be these three quantities, and those three quantities, such as the ratio of the first and the second of these is greater than that between the first and the second of those, while the ratio of the second and the third of these is greater than that of the second and third of those, if arranged in order. then ex equali the ratio between the first and the third of these will also be greater than the ratio between the first and the third of those. If a:b>d:e and b:c>e:f then a:c>d:f.¹