Euclid's Elements, Book VII

Clay Mathematics Institute Historical Archive

Euclid's Elements, Book VII

elem.7.1	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 will be prime to one another.	image
elem.7.2	Given two numbers not prime to one another, to find their greatest common measure.	image
elem.7.3	Given three numbers not prime to one another, to find their greatest common measure.	image
elem.7.4	Any number is either a part or parts of any number, the less of the greater.	image
elem.7.5	If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.	image
elem.7.6	If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.	image
elem.7.7	If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder that the whole is of the whole.	image
elem.7.8	If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.	image
elem.7.9	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.	image
elem.7.10	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.	image
elem.7.11	If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.	image
elem.7.12	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.	image
elem.7.13	If four numbers be proportional, they will also be proportional alternately.	image
elem.7.14	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 will also be in the same ratio ex aequali.	image
elem.7.15	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.	image
elem.7.16	If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.	image
elem.7.17	If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.	image
elem.7.18	If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.	image
elem.7.19	If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.	image
elem.7.20	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.	image
elem.7.21	Numbers prime to one another are the least of those which have the same ratio with them.	image
elem.7.22	The least numbers of those which have the same ratio with them are prime to one another.	image
elem.7.23	If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.	image
elem.7.24	If two numbers be prime to any number, their product also will be prime to the same.	image
elem.7.25	If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.	image
elem.7.26	If two numbers be prime to two numbers, both to each, their products also will be prime to one another.	image
elem.7.27	If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes].	image
elem.7.28	If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.	image
elem.7.29	Any prime number is prime to any number which it does not measure.	image
elem.7.30	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.	image
elem.7.31	Any composite number is measured by some prime number.	image
elem.7.32	Any number either is prime or is measured by some prime number.	image
elem.7.33	Given as many numbers as we please, to find the least of those which have the same ratio with them.	image
elem.7.34	Given two numbers, to find the least number which they measure.	image
elem.7.35	If two numbers measure any number, the least number measured by them will also measure the same.	image
elem.7.36	Given three numbers, to find the least number which they measure.	image
elem.7.37	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.	image
elem.7.38	If a number have any part whatever, it will be measured by a number called by the same name as the part.	image
elem.7.39	To find the number which is the least that will have given parts.	image

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