proportionals

Eratosthenes had himself invented an instrument to determine mean proportionals, the mesolab (“mean-getter”), and he tells the story of the Delian problem precisely to emphasize that earlier solutions, including that of Archytas, were in the form of geometrical demonstrations, which could not be employed for practical purposes.

Archytas Huffman, Carl 2007

After Hippocrates the problem of doubling the cube was always seen as the problem of finding two lines such that they were mean proportionals between G, the length of the side of the original cube, and D, a length which is double G.

Archytas Huffman, Carl 2007

Tradesmen without hesitation multiply the second by the third, and divide the product by the first; either because they have not forgotten the rule which they received from a master without any proof, or because they have often made trial of it with simple numbers, or by virtue of the proof of the nineteenth proposition of the seventh book of Euclid, namely, in virtue of the general property of proportionals.

The Ethics 2007

By employing physical instruments, which “required much common handicraft,” and in effect constructing machines to determine the two mean proportionals, Archytas was focusing not on the intelligible world but on the physical world and hence destroying the value of geometry.

Archytas Huffman, Carl 2007

Hippocrates recognized that if we could find two mean proportionals between the length of the side of the original cube G, and length D, where D = 2G, so that G: x:: x: y::

Archytas Huffman, Carl 2007

The meaning of the words that ‘solid bodies are always connected by two middle terms’ or mean proportionals has been much disputed.

Timaeus 2006

Between fire and earth, the two extremes, he remarks that there are introduced, not one, but two elements, air and water, which are compared to the two mean proportionals between two cube numbers.

Timaeus 2006

Having reflected on the singular numerical phenomena of the existence of one mean proportional between two square numbers are rather perhaps only between the two lowest squares; and of two mean proportionals between two cubes, perhaps again confining his attention to the two lowest cubes, he finds in the latter symbol an expression of the relation of the elements, as in the former an image of the combination of two surfaces.

Timaeus 2006

An instance of (2) would be the law that proportionals alternate.

Posterior Analytics Aristotle 2002

But the mean distance of the moon moving around a movable centre, is to the same mean distance when the centre of attraction is fixed, as the sum of the masses of the two bodies, to the first of two mean proportionals between this sum and the largest of the two bodies inversely.

Outlines of a Mechanical Theory of Storms Containing the True Law of Lunar Influence T. Bassnett