Posts tagged 'book-vii'

G.247 unit

An:term:**unit** <unit> is that by virtue of which each of the things that exist is called one.

G.248 number

A:term:**number** <number> is a multitude composed of:term:units <G.247>.

G.249 numbers prime to one another

Numbers prime to one another are those which are:term:measured <G.253> by an:term:unit <G.247> alone as a common measure.

G.250 unequal numbers set

Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers is prime to one another.

G.251 numbers prime find

Given two numbers not prime to one another, to find their greatest common measure.

G.252 numbers prime find

Given three numbers not prime to one another, to find their greatest common measure.

G.253 part of a number

A:term:number <G.248> is a:term:**part** <part-aliquot> of a number, the less of the greater, when it:term:measures <part-aliquot> the greater;

G.254 parts of a number

But:term:**parts** <parts-non-aliquot> when it does not:term:measure <G.253> it.

G.255 number part parts

Any number is either a part or parts of any number, the less of the greater.

G.256 multiple of a number

The greater:term:number <G.248> is a:term:**multiple** <multiple> of the less when it is:term:measured <G.253> by the less.

G.257 number part number

If a number be a part of a number, and another be the same part of another, the sum is the same part of the sum that the one is of the one.

G.258 number parts number

If a number be parts of a number, and another be the same parts of another, the sum is the same parts of the sum that the one is of the one.

G.259 number part number

If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder is the same part of the remainder that the whole is of the whole.

G.260 number parts number

If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder is the same parts of the remainder that the whole is of the whole.

G.261 number part number

If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

G.262 number parts number

If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.

G.263 proportional numbers

Numbers are:term:**proportional** <G.180> when the first is the:term:same multiple <G.256>, or the:term:same part <G.253>, or the:term:same parts <G.254>, of the second that the third is of the fourth.

G.264 whole whole number

If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder is to the remainder as whole to whole.

G.265 there many numbers

If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.

G.266 numbers proportional proportional

If four numbers be proportional, they is proportional alternately.

G.267 there many numbers

If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they is in the same ratio ex aequali.

G.268 unit measure number

If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.

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.274 least numbers ratio

The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.

G.275 numbers prime least

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

G.276 least numbers ratio

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

G.277 numbers prime number

If two numbers be prime to one another, the number which measures the one of them is prime to the remaining number.

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.282 numbers prime sum

If two numbers be prime to one another, the sum is prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers is prime to one another.

G.283 prime number

A:term:**prime number** <prime-number> is that which is:term:measured <G.253> by an:term:unit <G.247> alone.

G.284 prime number prime

Any prime number is prime to any number which it does not measure.

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.286 composite number

A:term:**composite number** <composite-number> is that which is:term:measured <G.253> by some:term:number <G.248>.

G.287 composite number measured

Any composite number is measured by some prime number.

G.288 number prime measured

Any number either is prime or is measured by some prime number.

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.291 numbers measure number

If two numbers measure any number, the least number measured by them will also measure the same.

G.292 numbers find least

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

G.293 number measured number

If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.

G.294 number part whatever

If a number have any part whatever, it is measured by a number called by the same name as the part.

G.295 find the number which is the least that

Find the number which is the least that will have given parts.

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.302 even number

An:term:**even number** <even-number> is that which is divisible into two equal parts.

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.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.318 similar plane and solid numbers

Similar plane and solid numbers are those which have their sides:term:proportional <G.180>.

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.341 numbers composite to one another

Numbers composite to one another are those which are:term:measured <G.253> by some:term:number <G.248> as a common measure.

G.352 odd number

An:term:**odd number** <odd-number> is that which is not divisible into two equal parts, or that which differs by an:term:unit <G.247> from an:term:even number <G.302>.

G.363 even-times even number

An:term:**even-times even number** <even-times-even-number> is that which is:term:measured <G.253> by an:term:even number <G.302> according to an even number.

G.365 even-times odd number

An:term:**even-times odd number** <even-times-odd-number> is that which is:term:measured <G.253> by an:term:even number <G.302> according to an:term:odd number <G.352>.

G.369 perfect number

A:term:**perfect number** <perfect-number> is that which is equal to its own parts.