14 proportion, the third number from unity is a square number and so are all forward, leaving one between, the fourth number from unity is a cube number and so are all forward, leaving bwo between, and the seventh number from unity is both a square and a cube and so are all forward, leaving five between. If the given prime numbers measure a certain least number, no other prime number shall measure that least number. If bhree least numbers be in the same ratio, then the sum of any two of them shall be incommensurable with the third.' + If there be two incommensurable numbers other than unity, there shall be no bhird number in the ratio of these two.' To find a bhird number in bhe ratio of two number, if possible.' * To find a fourth number in the ratio of three numbers, if possible
- The sum of any number of even numbers shall be even.
• The sum of an even number of odd numbers shall be even. 'The sum of an odd number of odd numbers shall be odd.
- If an even number be taken from an even number. the re
mainder shall be an even number .' + If an odd number be taken from an even number, the remainder shall be an odd number. If an even number be taken from an odd number the remainder shall be an odd number .' * If an odd number be taken from an odd number, the remainder shall be an even number.' & 'The product of an odd number and an even number shall be an even number. ' The product of two odd numbers is odd.' * An odd number measures an odd number with an odd quotient. • Numbers beginning with two in which each succeeding number is double of the preceding number shall be evenly even’ A number whose half is an odd number is an evenly odd number. The number, which is not in the series beginning with two in which each succeed- ing number is double bhe preceding one and of which the half is not an odd number, is evenly-even and also evonly-odd. f In a series of numbers beginning with unity, in which each suc ceeding number is double of the prece one, if the sum of Bhe terms be a prime number, then the product of this sum and the last number shall be a perfect number.* The book
- tProps, 1, 3, 4, 7, 8, 15, 16, 17, 18, 2021, 2, 23, 4, 25, 28, 2, 28, 29, 31,
84, 35, 86 and 88, Book x.
