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Definitions 1 and 2

Def. 1. A unit is that by virtue of which each of the things that exist is called one.

Def. 2. A number is a multitude composed of units.

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These 23 definitions at the beginning of Book VII are the definitions for all three books VII through IX on number theory. Some won't be used until Books VIII or IX.

These first two definitions are not very constructive towards a theory of numbers. The numbers in definition 2 are meant to be whole positive numbers greater than 1, and definition 1 is meant to define the unit as 1. The word "monad," derived directly from the Greek, is sometimes used instead of "unit."

Throughout these three books on number theory Euclid exhibits numbers as lines. In the diagram above, if A is the unit, then BE is the number 3. But, just because he draws them as lines does not mean they are lines, and he never calls them lines.

It is not clear what the nature of these numbers is supposed to be. But their nature is irrelevant. Euclid could illustrate the unit as a line or as any other magnitude, and numbers would then be illustrated as multiples of that unit.

There is a major distinction between lines and numbers. Lines are infinitely divisible, but numbers are not, in particular, the unit is not divisible into smaller numbers.

Euclid has no postulates to elaborate the concept of number (other than the Common Notions which are meant to apply to numbers as well as magnitudes of various kinds). Indeed, mathematicians did not develop foundations for number theory until the late nineteenth century. Peano's axioms for numbers are the best known. The most important of Peano's axioms is the principle of mathematical induction which states that if (1) a property of numbers holds for 1, and (2) for any number n, if the property holds for n then it holds for n + 1, then (3) the property holds for all numbers. Euclid does not use the principle of mathematical induction, but he does implicitly use a similar property of numbers, namely, that any decreasing sequence of numbers is finite. That property is known variously as the "well-ordering principle" for numbers and the "descending chain condition." We will discuss it later in more detail.

Book VII Introduction - Definitions 3 through 5.

© 1997
D.E.Joyce
Clark University