Posts tagged 'multiply'

G.269 multiplication of numbers

A:term:number <G.248> is said to:term:**multiply** <multiply> a number when that which is multiplied is added to itself as many times as there are:term:units <G.247> in the other, and thus some number is produced.

G.270 numbers multiplying make

If two numbers by multiplying one another make certain numbers, the numbers so produced are equal.

G.271 number multiplying numbers

If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

G.272 numbers multiplying number

If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

G.273 numbers proportional number

If four numbers be proportional, the number produced from the first and fourth will equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, the four numbers is proportional.

G.276 least numbers ratio

The least numbers of those which have the same ratio with them are prime to one another.

G.278 numbers prime number

If two numbers be prime to any number, their product also is prime to the same.

G.279 numbers prime product

If two numbers be prime to one another, the product of one of them into itself is prime to the remaining one.

G.280 numbers prime numbers

If two numbers be prime to two numbers, both to each, their products also is prime to one another.

G.281 numbers prime multiplying

If two numbers be prime to one another, and each by multiplying itself make a certain number, the products is prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter is prime to one another [and this is always the case with the extremes].

G.285 numbers multiplying make

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

G.289 many numbers please

Given as many numbers as we please, to find the least of those which have the same ratio with them.

G.290 numbers find least

Given two numbers, to find the least number which they measure.

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.300 plane and solid numbers

And, when two:term:numbers <G.248> having:term:multiplied <G.269> one another make some number, the number so produced is called:term:**plane** <plane-number>, and its:term:**sides** <side-of-number> are the numbers which have multiplied one another.

G.301 plane numbers ratio

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

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.308 cube number

A:term:**square number** <square-number> is equal:term:multiplied <G.269> by equal, or a:term:number <G.248> which is contained by two equal numbers.

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.311 square number

And a:term:**cube** <cube-number> is equal:term:multiplied <G.269> by equal and again by equal, or a:term:number <G.248> which is contained by three equal numbers.

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.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.320 square and cube numbers

And, when three:term:numbers <G.248> having:term:multiplied <G.269> one another make some number, the number so produced is:term:**solid** <solid-number>, and its:term:**sides** <side-of-number> are the numbers which have multiplied one another.

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.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.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.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.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.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.

G.371 unequal magnitudes set

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there is left some magnitude which is less than the lesser magnitude set out.