Occasionally I see the claim, that mathematics was constructive before the rise of formal logic and set theory. I'd like to understand the history better.

1. When did these methods of proof become accepted/standard? Can one find them for instance in classical works (Archimedes, Euclid, Euler, Gauss, etc.)?
2. Was there ever a debate about their validity before Brouwer?

My interest was sparked after reading the following "proof" from Newton's Principia that seems to use contradiction:

LEMMA I.

Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.

If you deny it, suppose them to be ultimately unequal, and let $D$ be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference $D$; which is against the supposition.

Occasionally I see the claim, that mathematics was constructive before the rise of formal logic and set theory. I'd like to understand the history better.

1. When did proofs by contradiction or by excluded middle become accepted/standard? Can one find them for instance in classical works (Archimedes, Euclid, Euler, Gauss, etc.)?
2. Was there ever a debate about their validity before Brouwer?

My interest was sparked after reading the following "proof" from Newton's Principia that seems to use contradiction:

LEMMA I.

Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.

If you deny it, suppose them to be ultimately unequal, and let $D$ be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference $D$; which is against the supposition.