Definition:Euclid's Definitions - Book VII

These definitions appear at the start of Book VII of by Euclid.


 * 1) A unit is that of which each of the things that exist is called one.
 * 2) A number is a multitude composed of units.
 * 3) A number is a part of a number, the less or the greater, when it measures the greater;
 * 4) but parts when it does not measure it.
 * 5) The greater number is a multiple of the less when it is measures by the less.
 * 6) An even number is that which is divisible into two equal parts.
 * 7) An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.
 * 8) An 'even-times even number is that which is measured by an even number according to an even number.
 * 9) An 'even-times odd number is that which is measured by an even number according to an odd number.
 * 10) An 'odd-times odd number is that which is measured by an odd number according to an odd number.
 * 11) A prime number is that which is measured by a unit alone.
 * 12) Numbers prime to one another are those which are measured by a unit alone as a common measure.
 * 13) A composite number is that which is measured by some number.
 * 14) Numbers composite to one another are those which are measured by some number as a common measure
 * 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.
 * 16) And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another.
 * 17) And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another.
 * 18) A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
 * 19) And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
 * 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.
 * 21) Similar plane and solid numbers are those which have their sides proportional.
 * 22) A perfect number is that which is equal to the sum its own parts.