Why is definition 5 of Book V:

*“Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of the latter equimultiples respectively taken in corresponding order.”*

different from definition 20 of Book VII?

*“Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.”*

VII.20 deals in number, and V.5 does not. Number creates (or perhaps is) a concise language with which to describe the basic concept of proportionality. As with any translation from one language to another, the new statement will imply more or less than the old one. To take these two definitions as equivalent implies that all magnitudes which can be in the same ratio can be expressed as numbers—or that magnitudes and numbers are equivalent in some way.

This makes some sense. Any magnitude which we can bisect can be expressed as a multitude of units—as a number—so long as we are free to define our base unit as we choose. However, despite this, numbers, magnitudes, and the mathematical entities that have magnitudes are all different. Numbers, while less complicated than magnitudes to talk about, introduce the problem of incommensurability; it is possible for there to be two magnitudes which are not possible to measure in number using the same unit, no matter how small we make that unit. This problem can't arise if we aren't talking about how to describe the magnitudes in terms of units. More specifically, it can’t arise if we aren’t insisting that more than one magnitude ought to be measured using the same unit. The presence of incommensurability as a problem may be the most important distinction that this “translation” out of the language of magnitudes (the language of all ratios?) and into the language of number has brought.

Some of the apparent simplicity of definition VII.20 comes from its use of the terms “part” and “parts.” A “part” is basically the same whether we’re dealing with magnitudes or numbers. A magnitude or number is “part” of another magnitude or number, the less of the greater, when the smaller measures the greater; this is from V.1, V.2 (which clarifies measure), and VII.3. For one number to be “parts” of another, however, means that the lesser number does not measure the greater number(VII.4). Why aren’t these terms used in V.5? We don’t have a definition of “parts” that applies to magnitude. If we extended the definition of “parts” from numbers onto magnitudes, we could say that any magnitude that was shorter than a second magnitude but did not measure it was “parts” of the second magnitude.

However, I think this equivalent definition is absent from Book V for a reason. Any group which *only* includes all numbers which are “the same multiple, or the same part, or the same parts” of some other number* can’t *include all possible ratios of magnitudes. This is, again, because some magnitudes which are in the same ratio are not commensurable with one another, and thus can’t be expressed at all within the same system (using the same sized base units) of number. Having no definition of “parts” which applies to magnitude highlights this difference, however subtlety.

NO! NON-EUCLIDEAN GEOMETRIES!!

ReplyDeleteThose are for senior year. :)

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