18 ightbh Book To elucidate this a few propositions may be en nciated. If in a certain series of numbers in a certain ratio, the first and the last are incommensurable, then these are the lowest numbers in the series in the same ratio.’ ‘To find the lowest numbers in a certain ratio. The ratio of a plane or superficial number with another plane number shall be the product of the radios of the sides of those plane numbers' + If in a certain series in a certain ratio, the first number measures the last number, then the first number shall also measure she second number .' * If there are two square numbers and if there is a mean proportional number between thembhe ratio of the square numbers to one another shall be equal to the square of the ratio of the sides of the square numbers.' + The squares and cubes of those numbers which are in a certain radio shall also be in the same ratio.
- If a number falls between two numberE, and if bhe
three numbers are in the same ratio, bhen the two num bers shall be like plane numbers' If between two numbers there fal WO other numbers so that the four numbers are in the same ratio, then the bwo numberg (between which two obher numbers fall) shall be like solid numbers. १If three numbers be in one ratio, and if the first be a square number, the third shall also be a square number'. If four numbers are in one ratio and if the first be a cube number, the fourth shall also be a cube number. ' + Two like plane numbers are in bhe ratio of their squares.' + Two like solid numbers are in the ratio of their cubes'* There are in all 27 propositions in this book. The Ninth Bcok continues the treatment of square and cube numbers, takes up odd and even numberE, not hitherto dealt with, and treats of their properties, as the follow: ing enunciations of some of the propositions will show. The product of two like plane number is a square number ' / 'The square of a cube number is a cube number. The product of bwo cubes shall be a cube. A composite number, multiplied by a certain number, becomes a solid number . If in a series beginning with unity, there be numbers in the same continual
- Props. 1, 2, 5, 7, 11, 18, 18, 18, 20, 21, 26, and I respectively, Book III
