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.
number_part_number_vii_9()
prove whatever part or parts A is of D, the same part or the same parts is / B C / of / E F / also
assert Let the number A be a part of the number / B C /, and another, D, the same part of another, / E F /, that A is of / B C /
assert I say that, alternately also, whatever part or parts A is of D, the same part or parts is / B C / of / E F / also
assert For since, whatever part A is of / B C /, the same part also is D of / E F /,
assert as many numbers as there are in / B C / equal to A, so many also are there in / E F / equal to D
assert Let / B C / be divided into the numbers = to A, namely / B G /, / G C /, and / E F / into those = to D, namely / E H /, / H F /
assert the multitude of / B G /, / G C / will be equal to the multitude of / E H /, / H F /
assert the numbers / B G /, / G C /=, and the numbers / E H /, / H F /=, while the multitude of / B G /, / G C / is equal to the multitude of / E H /, / H F /,
assert whatever part or parts / B G / is of / E H /, the same part or the same parts is / G C / of / H F / also
∵number_part_number, number_parts_numberassert So that, in addition, whatever part or parts / B G / is of / E H /, the same part also, or the same parts, is the sum / B C / of the sum / E F /
assert/ B G /=A, and / E H / to D
∴ whatever part or parts A is of D, the same part or the same parts is / B C / of / E F / also
