# Definition:Euclid's Definitions - Book VII

## Euclid's Definitions: Book $\text{VII}$

These definitions appear at the start of Book $\text{VII}$ of Euclid's *The Elements*.

- An
**unit**is that of which each of the things that exist is called one. - A
**number**is a multitude composed of units. - A number is
**a part**of a number, the less of the greater, when it measures the greater; - but
**parts**when it does not measure it. - The greater number is a
**multiple**of the less when it is measured by the less. - An
**even number**is that which is divisible into two equal parts. - An
**odd number**is that which is not divisible into two equal parts, or that which differs by an unit from an even number. - An
**even-times even number**is that which is measured by an even number according to an even number. - An
**even-times odd number**is that which is measured by an even number according to an odd number. - An
**odd-times odd number**is that which is measured by an odd number according to an odd number. - A
**prime number**is that which is measured by an unit alone. - Numbers
**prime to one another**are those which are measured by an unit alone as a common measure. - A
**composite number**is that which is measured by some number. - Numbers
**composite to one another**are those which are measured by some number as a common measure. - 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. - 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. - 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. - A
**square number**is equal multiplied by equal, or a number which is contained by two equal numbers. - And a
**cube**is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers. - 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. **Similar plane**and**solid**numbers are those which have their sides proportional.- A
**perfect number**is that which is equal to its own parts.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Definitions