Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
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"Magnitudes which can be made to coincide are equal."
"And the whole is greater than the part."
"If a straight line be cut in extreme and mean ratio, the greater segment is also cut in extreme and mean ratio by the lesser segment."
"A plane surface is a surface which lies evenly with the straight lines on itself."
"Rectilineal figures are those which are contained by straight lines..."
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A triangle is classified by its angles: if one angle equals exactly 90 degrees, it is right-angled; if one angle exceeds 90 degrees, it is obtuse-angled; if all three angles fall below 90 degrees, it is acute-angled. This is a precise definitional statement establishing mutually exclusive categories for every possible triangle, giving geometry a shared vocabulary for classification.
Euclid built mathematics on rigorous definitions before proofs, and this quote comes directly from his Elements, the foundational geometry text he authored around 300 BC. As the Father of Geometry, Euclid believed clarity of definition was the bedrock of logical reasoning. He systematically defined every term before using it, reflecting his disciplined, axiomatic approach to mathematical truth.
Around 300 BC in Alexandria, Egypt, Greek intellectual culture prized logical systematization over intuition. Euclid worked under Ptolemy I, whose patronage at the Library of Alexandria encouraged rigorous scholarship. Prior geometric knowledge from Babylon and Egypt existed but lacked formal structure. Euclid's definitional precision transformed scattered practical measurements into a unified logical science that would dominate mathematics for over two thousand years.
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