---
book: 7
part: "definitions"
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7"
license: "CC-BY-SA-4.0"
---

# Book VII Definitions

## Definition 1

An *unit* is that by virtue of which each of the things that exist is called one.

## Definition 2

A *number* is a multitude composed of units.

## Definition 3

A number is *a part* of a number, the less of the greater, when it measures the greater;

## Definition 4

but *parts* when it does not measure it.

## Definition 5

The greater number is a *multiple* of the less when it is measured by the less.

## Definition 6

An *even number* is that which is divisible into two equal parts.

## Definition 7

An *odd number* is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

## Definition 8

An *even-times even number* is that which is measured by an even number according to an even number.

## Definition 9

An *even-times odd number* is that which is measured by an even number according to an odd number.

## Definition 10

An *odd-times odd number* is that which is measured by an odd number according to an odd number.

## Definition 11

A *prime number* is that which is measured by an unit alone.

## Definition 12

Numbers *prime to one another* are those which are measured by an unit alone as a common measure.

## Definition 13

A *composite number* is that which is measured by some number.

## Definition 14

Numbers *composite to one another* are those which are measured by some number as a common measure.

## Definition 15

A number is said to *multiply* a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

## Definition 16

And, when two numbers having multiplied one another make some number, the number so produced is called *plane,* and its *sides* are the numbers which have multiplied one another.

## Definition 17

And, when three numbers having multiplied one another make some number, the number so produced is *solid*, and its *sides* are the numbers which have multiplied one another.

## Definition 18

A *square number* is equal multiplied by equal, or a number which is contained by two equal numbers.

## Definition 19

And a *cube* is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

## Definition 20

Numbers are *proportional* when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.

## Definition 21

*Similar plane* and *solid* numbers are those which have their sides proportional.

## Definition 22

A *perfect number* is that which is equal to its own parts.