Posts tagged 'book-viii'

G.296 there many numbers

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

G.297 find numbers in continued proportion

Find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.

G.298 many numbers please

If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.

G.299 many ratios please

Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.

G.301 plane numbers ratio

Plane numbers have to one another the ratio compounded of the ratios of their sides.

G.303 there many numbers

If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.

G.304 there many numbers

If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.

G.305 numbers there fall

If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.

G.306 numbers prime numbers

If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.

G.307 numbers fall numbers

If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.

G.309 square numbers there

Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.

G.312 cube numbers there

Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

G.313 there many numbers

If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products is proportional; and, if the original numbers by multiplying the products make certain numbers, the latter is proportional.

G.314 square measure square

If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.

G.315 cube number measure

If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.

G.316 square number measure

If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.

G.317 cube number measure

If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.

G.319 similar plane numbers

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

G.321 similar solid numbers

Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.

G.322 mean proportional number

If one mean proportional number fall between two numbers, the numbers is similar plane numbers.

G.323 mean proportional numbers

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

G.324 numbers continued proportion

If three numbers be in continued proportion, and the first be square, the third is square.

G.325 numbers continued proportion

If four numbers be in continued proportion, and the first be cube, the fourth is cube.

G.326 numbers ratio square

If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second is square.

G.327 numbers ratio cube

If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second is cube.

G.328 similar plane numbers

Similar plane numbers have to one another the ratio which a square number has to a square number.

G.329 similar solid numbers

Similar solid numbers have to one another the ratio which a cube number has to a cube number.