Posts tagged 'book-ix'

G.330 similar plane numbers

If two similar plane numbers by multiplying one another make some number, the product is square.

G.331 numbers multiplying make

If two numbers by multiplying one another make a square number, they are similar plane numbers.

G.332 cube number multiplying

If a cube number by multiplying itself make some number, the product is cube.

G.333 cube number multiplying

If a cube number by multiplying a cube number make some number, the product is cube.

G.334 cube number multiplying

If a cube number by multiplying any number make a cube number, the multiplied number is cube.

G.335 number multiplying itself

If a number by multiplying itself make a cube number, it will itself also be cube.

G.336 composite number multiplying

If a composite number by multiplying any number make some number, the product is solid.

G.337 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit is square, as will also those which successively leave out one; the fourth is cube, as will also all those which leave out two; and the seventh is at once cube and square, as will also those which leave out five.

G.338 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be square, all the rest is square. And, if the number after the unit be cube, all the rest is cube.

G.339 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two.

G.340 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the proportional numbers.

G.342 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit is measured by the same.

G.343 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be prime, the greatest will not be measured by any except those which have a place among the proportional numbers.

G.344 number least measured

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

G.345 numbers continued proportion

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together is prime to the remaining number.

G.346 numbers prime number

If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

G.347 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 last will not be to any other number as the first to the second.

G.348 numbers investigate whether

Given two numbers, to investigate whether it is possible to find a third proportional to them.

G.349 numbers investigate possible

Given three numbers, to investigate when it is possible to find a fourth proportional to them.

G.350 prime numbers more

Prime numbers are more than any assigned multitude of prime numbers.

G.351 many even numbers

If as many even numbers as we please be added together, the whole is even.

G.353 many odd numbers

If as many odd numbers as we please be added together, and their multitude be even, the whole is even.

G.354 many odd numbers

If as many odd numbers as we please be added together, and their multitude be odd, the whole is odd.

G.355 even number even

If from an even number an even number be subtracted, the remainder is even.

G.356 even number odd

If from an even number an odd number be subtracted, the remainder is odd.

G.357 odd number odd

If from an odd number an odd number be subtracted, the remainder is even.

G.358 odd number even

If from an odd number an even number be subtracted, the remainder is odd.

G.359 odd number multiplying

If an odd number by multiplying an even number make some number, the product is even.

G.360 odd number multiplying

If an odd number by multiplying an odd number make some number, the product is odd.

G.361 odd number measure

If an odd number measure an even number, it will also measure the half of it.

G.362 odd number prime

If an odd number be prime to any number, it is prime to the double of it.

G.364 numbers continually doubled

Each of the numbers which are continually doubled beginning from a dyad is even-times even only.

G.366 number half odd

If a number have its half odd, it is even-times odd only.

G.367 number continually doubled

If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both even-times even and even-times odd.

G.368 many numbers please

If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.

G.370 many numbers please

If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product is perfect.