The History of Mathematical Proof in Ancient Traditions

342 reviel netz

To two given ςς: to add to the smaller of them, and to take away from the greater,

and to make the resulting <number> have a given ratio to the remainder.

Let it be set forth to add to 20, and to take away from 100 the same ς, and to

make the greater 4-times the smaller.

(a) Let the <number> which is added and taken away from each ς <sc. of

the two given numbers> be set down, <namely> number, one. (1) And if it is

added to twenty, results: ς 1 Μ ο 20. (2) And if it is taken away from 100, results:

Μ ο 100 lacking number 1. (3) And it shall be required that the greater be 4- tms

the smaller. (4) Th erefore four- tms the smaller is equal to the greater; (5) but

four- tms the smaller results: Μ ο 400 Ψ ς4; (6) these equal ς 1 Μ ο 20

(7) Let the subtraction be added <as> common, (8) and let similar

<terms> be taken away from similar <terms>. (9) Remaining: numbers, 5,

equal Μ ο 380. (10) And the ς results: monads, 76.

To the positions. I put the added and the taken away on each ς, ς 1; it shall

be Μ ο 76. And if Μ ο 76 is added to 20, result: monads, 96; and if it is taken

away from 100, remaining: monads, 24. And the greater shall stand being

4- tms the smaller.

Diophantus the deuteronomist: systematization and the general

To understand the function of the text above, I move on to compare it

with  three other, hypothetical texts. I argue that all were possible in the

late  ancient Mediterranean. However, only the fi rst two had existed,

while  the third remained as a mere logical possibility, never actualized in

writing.

Text 1:

A: I have a hundred and a twenty. I take away a number from the greater and add it

to the smaller. Now the smaller has become four times that which was greater. How

much did I take away and add?

B:?

A: Seventy six! Check for yourself.

Text 2:

Hundred and twenty. I took away from the greater and added the same to the

smaller, and the smaller became four times that which had been greater.

Take the greater, a hundred. Its four times is four hundred. Take away the

smaller,  twenty. Left is three hundred eighty. Four plus one is fi ve. Divide three

hundred eighty by fi ve: seventy six. Seventy six is the number taken away and

added.