Introduction
By tradition, the Pythagoreans had a theory of proportion which they used strongly. For example, VI.31 is claimed to be a Pythagorean proof of the Pythagorean theorem that is much simpler than I.47, but it uses triangle similarity which is based on proportion. The Pythagorean theory of proportion was probably based on the "axiom" that all lengths are commensurable (which reduces the theory of length proportion to the theory of numerical proportion), which was proven to be false. Only a century or so after this discovery did Eudoxus provide a new definition of proportion of magnitudes (which is equivalent to Dedekind's definition of real numbers). This definition is great, and Mathematicians went back and re-proved the basic properties of proportion and similarity based on this definition, which are collected in book V and book VI of Euclid (V - proportion, VI - similarity).
BUT!
Eudoxus' definition assumes a different axiom, that although was not (read: can not) be proven to be false, I see no reason to assume it to be true: Archimedes' Axiom - in the sense that for any two lengths, if one is multiplied by a large enough natural number it will exceed the other. This axiom is used implicitly in some of Euclid's proofs on proportion (see V.Def 4), for example V.8 which is used to prove the very important theorem: V.9, which must be true if we want any true theory of proportion.
Paul Ellis and Humberto Montalvan suggested a different definition of proportion: A:B = C:D iff AD = BC, where AD is the area of a rectangle with sides A and D. This can even be constructively (with straight-edge and compass) verified using II.14 (which can convert a rectangle into a square with equal area, which allows us to compare areas of rectangles -- without assuming Archimedes' Axiom).
So now we need to go through all of books V and VI to reprove the theorems. Let's try. Whichever we can't do might not be exactly true, but only true modulo infinitesimals. But we will probably succeed in most (we won't succeed in XII.2 - the proof of the existence of
). Book V contains results similar to proving the basic results of the ordered field of real numbers (since basically that's what it is -- for example V.23 is cryptically equivalent to commutativity of multiplication of real numbers). Book VI is geometrical. So let's start with book V. Here are all the propositions of book V. Let's go through them and start re-proving them with our new theory: (the relevant ones -- for example Proposition 1 is just a technical lemma used for Eudoxus' defintion of proportion that we don't need anymore). Take a look at the logical structure of book V to see the dependencies.
Proposition 1 [Avital: Irrelevant, technical lemma for Eudoxus' Definition]
If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.
Proposition 2 [Avital: Irrelevant, technical lemma for Eudoxus' Definition]
If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth.
Proposition 3 [Avital: Irrelevant, technical lemma for Eudoxus' Definition]
If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth.
Proposition 4 [Avital: We need to prove this. Proof of V.4]
If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
Proposition 5 [Avital: Irrelevant, technical lemma for Eudoxus' Definition]
If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole.
Proposition 6 [Avital: Irrelevant, technical lemma for Eudoxus' Definition]
If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them.
Proposition 7 [Avital: We need to prove this. Proof of V.7]
Equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio.
Corollary If any magnitudes are proportional, then they are also proportional inversely.
Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.
Magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal.
Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.
Ratios which are the same with the same ratio are also the same with one another.
If any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.
If a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth.
If a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less.
Parts have the same ratio as their equimultiples.
If four magnitudes are proportional, then they are also proportional alternately.
If magnitudes are proportional taken jointly, then they are also proportional taken separately.
If magnitudes are proportional taken separately, then they are also proportional taken jointly.
If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole.
Corollary. If magnitudes are proportional taken jointly, then they are also proportional in conversion.
If there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less.
If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less.
If there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali.
If a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth.
If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two.