Quandaries and Queries


Name: Kenneth

Who is asking: Other
Level: All


Here is my question:

The terms of a ratio in a proportion are often expressed as a is to b as c is to d.

Example: 2/4 = 6/12 this proportion represents that 2 is to 4 as 6 is to 12.

What does the "a is to b as c is to d" really represent or indicate in ratios?

I thank you for your reply.



Hi Kenneth,

Euclid gives the precise meaning in definition 5 of book 5 of the elements

"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 are 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

Thus, to apply it to your example "2 is to 4 as 6 is to 12", let's take "equimultiples" of 2 and 6, say 112 = 22 and 116 =66, and equimultiples of 4 and 12, say 5 4 = 20 and 512 = 60.

If we compare the multiple of 2 with the multiple of 4 we get 22 > 20, so we will get the same inequality when we compare the multiple of 6 with the multiple of 12: 66 > 60. This will work no matter what "equimultiple" we take of 2 and 6, and no matter what equimultiple we take of 4 and 12.

Nowadays, we would simply say that 2 is to 4 as 6 is to 12 because 212 = 46; why did Euclid choose such a complicated way of putting it? To understand this, we need to go back 2300 years in the time of Euclid. In those days 4 types of "magnitudes" were known: number, length, area, volume. People knew that it made sense to say things like "The area of this triangle is to the area of that circle as the volume of that cube is to the volume of that sphere", but they needed a definition to make the meaning

The "numbers" were just natural numbers, with no negatives and no fractions. They could multiply numbers by lengths (to make "multiples" of the length), by area or by volume, multiply lengths by lengths or by areas, but they could not multiply areas by areas or volumes, or volumes by volumes. Furthermore, they knew that some ratios were not expressible by ratios of numbers. For instance, if we try to come up with numbers such that "The length of the diagonal of this square is to the length of its side as this number is to that number", we will not find any that fit the bill.

So, with all these difficulties, they had to come up with the above definition for "magnitudes that are in the same ratio".



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