Modern vs Euclid's definition of equivalent ratios

[logicgate]

TL;DR Summary

I'm trying to understand the relationship between Euclid's definition of equivalent ratios and our modern understanding of it.

Euclid defines equivalent ratios as the following : "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."
I can understand Euclid's definition of equivalent ratios but what I don't understand is how it relates to our current definition of equivalent ratios which states that two ratios a/b and c/d are equivalent if and only if their cross products are equal.

 

[FactChecker]

Two consecutive "equimultiplies", first by $b\ne 0$, then by $d\ne 0$.
$a/b=c/d \iff a=bc/d \iff da=bc$

 

[mathwonk]

@logicgate: The problem Euclid is addressing is to compare the ratios of lengths of line segments. These lengths are not represented in Euclid by numbers, so there is no way for him to compare the quotients of numbers that represent these lengths, by way of cross multiplication. I.e. the only numbers available to Euclid are integers and their ratios, rational numbers, and in general the length of a line segment is a non rational real number, and those had not been invented yet. So in Euclid, two line segments X,Y have a relative size, since they can be compared as to which is longer, but he wants to measure more precisely how much longer one is than the other, or rather what is the ratio of their lengths, without assigning that ratio a number.

What he does is compare that ratio to every possible rational number. I.e. he can take multiples of his line segments and compare those, so not only can he say whether X is less than Y, but for every pair of positive integers n,m, he can say whether mX is less than nY. Thus the ratio X/Y is less than the rational number n/m if and only if the ratio mX/nY is less than 1, if and only if the multiplied segment mX is shorter than nY. Thus given two pairs of lengths X,Y and Z,W, he says they have the same ratio if and only if, for every pair of positive integers n,m, we have mX < nY whenever mZ < nW, and mX > nY whenever mZ > nW, and also mX = nY whenever mZ = nW. I.e. the ratios X/Y and Z/W are the same if the family of rational numbers n/m that are less than X/Y is the same family that are less than Z/W. This is the precursor of Dedekind’s definition of a real number as a “cut” in the family of rational numbers.

When I taught this course to children in a 2 week summer session, I did not have time to treat Euclid's theory of ratios, and so I made up a definition of equal ratios that does use “cross multiplication”. I.e. one has to make up a definition of the statement that XW = YZ, where X,Y and Z,W are pairs of line segments. Since Euclid had treated (equality of) areas of plane polygons, just define the product XxW to be the rectangle with sides X,W. Thus two ratios X/Y and Z/W are equal if and only if the rectangles XxW and YxZ are equal in Euclid’s sense of equidecomposability. Note there is no number assigned to their area; i.e. there is a precise notion that the rectangles have equal size, but that size is not assigned a number. In this sense then, Euclid’s equality of ratios is given a familiar looking form as equality of cross multiplication. Again there is a precise notion of when two pairs of segments have the same ratio, but that ratio is not assigned a number. (Today we might say the ratio is being assigned an infinite collection of approximating rational numbers, i.e. a single real number.)

Then one can deduce the important Prop. 4, Book 6, that similar (equiangular) triangles have proportional sides, as a corollary of Prop.35, Book 3, that line segments formed by intersecting secants in a circle (hence forming equiangular triangles) define rectangles of equal size. This allows one to treat similar triangles from Book 6, without covering Book 5 on ratios.

 

[mathwonk]

To put it another way, Euclid is in fact using cross multiplication, except he is using it in the broader sense that measures inequality as well as equality. I.e. fractions A/B and C/D are equal if and only if AD=BC, but if we put them over a common denominator, A/B = AD/BD, and C/D = BC/BD, then they are related exactly as their numerators are related. I.e. AD/BD < BC/BD if and only if AD<BC. Thus A/B < C/D if and only if, after cross multiplying, AD<BC.

Next, Euclid can only use cross multiplication on quantities that he knows how to multiply. He can only multiply line segments by positive integers, so he can only compare a ratio of line segments X/Y, to a ratio of integers n/m, and then he can say that X/Y < n/m if and only if mX < nY. Thus Euclid can identify all rational numbers that are greater than the ratio X/Y, and can say that two ratios of segments X/Y and Z/W, are equal if and only if they are less than exactly the same family of rational numbers, and this is in fact what he says in post #1 above.

One further remark: this method of Euclid only works if every ratio of segments does separate the rational numbers into two families, i.e. if and only if for every pair of lengths X,Y, there are some integers m,n such that 1/m < X/Y < n, i.e. X < nY and Y < mX. (This is called the Archimedean axiom since Archimedes made it precise, but Euclid already makes clear at the beginning of Book 5 that he only defines segments to "have a ratio" when they satisfy this property.)

Hence Euclid's theory of similar triangles works only with this extra hypothesis, i.e. in a so called Archimedean geometry. But there are also non -Archimedean geometries, in which there are "infinitely short" as well as "infinitely long" line segments, or rather pairs of line segments whose ratio is infinitely large or small. (This phenomenon comes up when studying "hyperreal" numbers, as is done elsewhere on this forum in the setting of analysis.)

Finally observe that the Archimedean axiom is not needed for the geometric definition of cross multiplication given above, since Euclid's theory of comparing sizes of rectangles does not use it. Thus our alternate treatment of similar triangles works also in the non - Archimedean setting.

 

[mathwonk]

@logicgate: I would appreciate any feedback on whether these answers were useful, and clear.