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My understanding is that magnitudes to ancient Greeks meant the actual line segments and plane regions (not the size of the line segment or the area of the plane region), the concept of ratio was then used to talk about the 'size' of different magnitudes. But Euclid defines ratio as 'sort of a relation in respect of size between two magnitudes of the same kind'. It seems this definition attaches no numerical meaning to the concept of ratio.

My questions are:

  1. Was there a particular reason that the magnitudes were defined this way?
  2. With no numerical meaning, how was this concept of ratio used to talk about size?
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  • $\begingroup$ I think the short answer is that you're imposing modern concepts on ancient systems of thought. In most modern school textbooks, the measure of an object (line segment, angle, area) is defined as a real number, which is separate from the object itself. The ancient Greeks did not have a separate concept of a real number system. They would say that two angles were "isos" (the same), whereas today we would say that their measures were the same. $\endgroup$ – Ben Crowell May 4 '17 at 2:26
  • $\begingroup$ See Def.V.1: "A magnitude is a part of a magnitude, the less of the greater, when it measures the greater." We have to try to abstract (in principle) from numbers. Consider a segment $AB$ and a less one $AC$: put $AC$ on $AB$ and tip over $AC$ around $C$ and repeat until you reach the point $B$. In this way, you have "measured" $AB$ with $AC$ and $AC$ is part of $AB$. If insted, with the last move, $AC$ "fall outside" $AB$, this means that $AC$ is not part of $AB$. $\endgroup$ – Mauro ALLEGRANZA May 4 '17 at 9:57
  • $\begingroup$ Now, by Def.V.2: "The greater is a multiple of the less when it is measured by the less", we have that $AB$ is a multiple of $AC$. Of course, we can count the numbers of times we have "tipped over" $AC$, and this is the "usual" measure of $AB$ in term of $AC$ assumed as unit of measure. $\endgroup$ – Mauro ALLEGRANZA May 4 '17 at 9:59
  • $\begingroup$ In the same way, you can imagine this procedure applying to rectangles, and so on. Thus, by Def.V.3: "A ratio is a sort of relation in respect of size between two magnitudes of the same kind", we may express the result of the above "measuring process" (between magnitudes of the same kind: line with line, planar shapes with planar shapes) as a relation called ratio. If the size of $AB$ is $b$ and the size of $AC$ is $c$, their ratio will be $\dfrac b c$, taht obviously corresponds to the number $n$ resulting from the above "measuring process". $\endgroup$ – Mauro ALLEGRANZA May 4 '17 at 10:03
  • $\begingroup$ @MauroALLEGRANZA Thanks it makes sense but why were they trying to abstract from numbers? I understand that they did not accept real numbers because the lack of logic but I do not understand what made the concepts of measurement and ratio as presented in Def V.2 and V.3 acceptable (logical) to them. $\endgroup$ – abk May 4 '17 at 23:09
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Fowler's Ratio in Early Greek Mathematics is a standard reference on the subject, see also his book Mathematics Of Plato's Academy (both are freely available).

Book V, Definition 3 of Euclid's Elements reads: "A ratio (logos) is a sort of relation in respect of size between two magnitudes of the same kind". Thus, ratios of magnitudes were indeed non-numbers, but they could be related to measurements. In the simplest form, originally used by Pythagoreans, one could talk about a ratio of magnitudes $a:b$ being equal to a ratio of numbers $m:n$ if there was a common measure $h$, which reproduced $m$ times equals $a$, and reproduced $n$ times equals $b$ (this makes particularly simple sense for line segments). The common measure could be found through the anthyphairesis, the Euclidean algorithm in geometric form, which produces the sequence of partial quotients of the continuous fraction for $m/n$. Of course, Greeks did not talk of fractions or anything like "rational numbers". Here is Fowler:

"Operations corresponding to the multiplication of ratios do occur within the Elements, with treatments that are worthy of note. First observe that although the idea of taking the product of two ratios is never defined — nor could it be since ratio itself is not defined, but only the equality of two ratios — the operation is used in VI, 23... There could not be a general proposition in Book V on the compound of two proportions between magnitudes, since the result might seem to involve the product of the magnitudes, which would not in general be defined. Particular cases do however occur... At no point in the Elements, or in the surviving corpus of classical Greek mathematics does anything remotely related to a formal treatment of addition of ratios of numbers occur."

The situation was different in practical use, and later Hellenistic astronomy, where fractions and even sexagesimals, were arithmetically manipulated. Fowler believes that experimentation with the anthyphairesis might have led to the discovery that it does not terminate for some pairs of segments (like the sides and diagonals in squares and pentagons), and hence to the discovery that some magnitudes are incommensurable. This led to the first, "constructive", theory of incommensurable ratios developed by Theaetetus based on non-terminating anthyphairesis sequences. It was later replaced by the technically more convenient device of Eudoxus, presented in Book V of Elements, which defines when two potentially incommensurable ratios are equal by cleverly abstracting and modifying the original Pythagorean definition. Some results of Theaetetus's theory, although established by different methods, are presented in Book X of Elements, where Euclid surprisingly does not use the general Eudoxian theory of Book V.

The Eudoxian definition was the basis for the double reductio arguments later termed the method of exhaustion, which go back to Eudoxus himself (volume of the pyramid), and were extensively used by Archimedes. In modern translation, it declares two ratios $a:b$ and $c:d$ to be equal if for any pair of numbers $m,n$ (meaning positive integers, of course) the equality/inequality between $na$ and $mb$ always goes in the same direction as the one between $nc$ and $md$. Archimedes's results also provide nice examples of how the theory of ratios was used to talk about sizes. Instead of expressing areas and volumes as numbers, the way we do today, he states in On Sphere and Cylinder that the volume of a sphere is to the volume of the circumscribed cylinder as $2:3$, see details in Who calculated for the first time the volume (and surface area) of the sphere exactly?

How much the attitudes have changed by the end of 17th century is attested to by Newton's Arithmetica Universalis (1707), where he writes:

"We understand a number not as a set of units, but as the abstract ratio of one magnitude to another magnitude of the same kind taken for that unit."

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  • $\begingroup$ you mean '...that it does not terminate for some...' $\endgroup$ – Marius Kempe May 3 '17 at 23:41
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For the Greeks, the number (or ratio) was what we call rational number nowadays. (They did not have the concept of real number which was rigorously defined only in the end of 19th century).

It was a great discovery that some segments (constructed by ruler and compass) do not have a ratio. To deal with this the Greeks invented proportions. Proportion says that two ratios (in the modern sense of the word) are equal. The Greeks discovered that one can develop the theory of proportions (without defining the ratios themselves!) which works even in the case when the ratios are not rational numbers. This theory is essentially equivalent to our modern theory of real numbers. It was influential, and until the end of 18th century physicists were talking about proportions instead of real numbers.

Even when I was at school (in 1960th) all physical laws (like Hooke's law or gravitation law or Coulomb law) were stated in the form of proportions rather then equalities between real numbers.

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