The body of the Kosmos was required to be both visible and tangible: it could not be visible without fire: it could not be tangible without something solid, nor solid without earth. But two things cannot be well put together by themselves, without a third to serve as a bond of connection: and that is the best bond which makes them One as much as possible. Geometrical proportion best accomplishes this object. But as both Fire and Earth were solids and not planes, no one mean proportional could be found between them. Two mean proportionals were necessary. Hence the Demiurgus interposed air and water, in such manner, that as fire is to air, so is air to water: and as air is to water, so is water to earth.[25] Thus the four elements, composing the body of the Kosmos, were bound together in unity and friendship. Of each of the four, the entire total was used up in the construction: so that there remained nothing of them apart, to hurt the Kosmos from without, nor anything as raw material for a second Kosmos.[26] [Footnote 25: Plato, Tim. pp. 31-32. The comment of Macrobius on this passage (Somn. Scip. i. 6, p. 30) is interesting, if not conclusive. But the language in which Plato lays down this doctrine about mean proportionals is not precise, and has occasioned much difference of opinion among commentators. Between two solids (he says), that is, solid numbers, or numbers generated out of the product of three factors, no one mean proportional can be found. This is not universally true. The different suggestions of critics to clear up this difficulty will be found set forth in the elaborate note of M. Martin (Études sur le Timée, vol. 1, note xx. pp. 337-345), who has given what seems a probable explanation. Plato (he supposes) is speaking only of prime numbers and their products. In the language of ancient arithmeticians _linear numbers_, _par excellence_ or properly so-called, were the prime numbers, measurable by unity only; _plane numbers_ were the products of two such linear numbers or prime numbers; _solid numbers_ were the products of three such. Understanding solid numbers in this restricted sense, it will be perfectly true that between any two of them you can never find _any one_ solid number or any whole number which shall be a mean proportional, but you can always find _two_ solid numbers which shall be mean proportionals. One mean proportional will never be sufficient. On the contrary, one mean proportional will be sufficient between two plane numbers (in the restricted sense) when these numbers are squares, though not if they are not squares. It is therefore true, that in the case of two _solid_ numbers** (so understood) one such mean proportional will never be sufficient, while two can always be found; and that between two _plane_ numbers** (so understood) one such mean proportional will in certain cases be sufficient and may be found. This is what is present to Plato's