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We know that when a number is added and subtracted, the effect is null (same number of course).

I want to know the first occurrence of this documented method. Is it from Euclid's elements in Book $5$?

the relationships of equivalence and of excess are compatible with addition and subtraction in the sense that if equivalent magnitudes are added to (or taken from) each of two others, the resulting magnitudes will be in the same relation as the originals.


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up vote 2 down vote accepted

Book V of Euclid's Elements deals not with number, but with ratio between magnitudes.

The interpretative issues are thorny; see :

Jacob Klein, Greek Mathematical Thought and the Origins of Algebra, German original ed 1936 (Dover reprint)

Wilbur R. Knorr, The Evolution of the Euclidean Elements, 1975

Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements, 1981 (Dover reprint)

Ivor Grattan-Guinness, Numbers, Magnitudes, Ratios, and Proportions in Euclid’s Elements, 1996 in Historia Mathematica.

Magnitudes for Euclid are not numbers; for ancient Greeks, numbers are (our) natural numbers.

Lenght and area are magnitudes. The discovery of the irrationality of $\sqrt 2$ forced ancient Greek mathematics to leave the Pythagorean "axiom" that every natural phenomenon can be described with (natural) numbers : the diagonal of the unit square cannot.

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