The purpose of this note is to remind readers of information at times well-known and at times almost forgotten, namely that for several centuries in the modern West Euclid’s Elements was simultaneously regarded as the epitome of knowledge and as flawed and confused. It is well known that many mathematicians brought up on Euclid and other Greek geometers complained that they found themselves compelled to accept the conclusions but not instructed in how to do geometry, and the long struggle with the parallel postulate has also been frequently discussed. The confusion discussed here is different, and relates to the concepts of straightness and shortest distance. It will also be suggested that the growing awareness of the defects in Euclid’s presentation by the end of the 18th century contributed to the creation of the new geometries of the 19th century: projective geometry and non-Euclidean geometry.

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Book De Risi, Vincenzo (2015) Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age.

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Article
Krol, Z.;
(2006)

Ancient Geometry and Plato's Philosophy on the Base of Pappus' “Comment on the Xth Book of Elements of Euclid”
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Ambrosi, Gerhard Michael;
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Pre-Euclidean Geometry and Aeginetan Coin Design: Some Further Remarks
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Book
De Risi, Vincenzo;
(2015)

Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age
(/isis/citation/CBB843280666/)

Chapter
Vitrac, Bernard;
(2012)

The Euclidean Ideal of Proof in *The Elements* and Philological Uncertainties of Heiberg's Edition of the Text
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Conventionalism in Reid's “Geometry of Visibles”
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Hjelmslev's geometry of reality
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Mateusz Hohol;
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Cognitive Artifacts for Geometric Reasoning
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Article
Carlos Alvarez;
(2023)

Some remarks concerning the trisection of an angle. Euclid between Pappus and Viète
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David S. Richeson;
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Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity
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Vincenzo De Risi;
(2022)

Euclid’s Fourth Postulate: Its authenticity and significance for the foundations of Greek mathematics
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Marcella Giulia Lorenzi;
Mauro Francaviglia;
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Dal cosmo al numero e alla geometria euclidea
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Book
Berlinski, David;
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The King of Infinite Space: Euclid and His Elements
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Article
Daniele Pasquazi;
Benedetto Scoppola;
(2021)

Aritmetica euclidea e filosofia stoica
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Article
Roy Wagner;
Reviel Netz;
(2023)

Between music and geometry: a proposal for the early intended application of Euclid’s Elements Book X
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Article
Webster, Colin;
(2014)

Euclid's Optics and Geometrical Astronomy
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Article
Mueller, Ian;
(2003)

Remarks on Euclid's *Elements* I, 32 and the Parallel Postulate
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Article
Mode, ;
(2007)

A Study of the Editions of Euclid's Elements (II)
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Article
Aboav, David;
(2008)

Euclid's Book on Divisions of Figures: A Conjecture as to Its Origin
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Chapter
Renato Migliorato;
(2016)

Il paradigma euclideo e la sua eclissi
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Article
Saito, Ken;
(2012)

Traditions of the Diagram, Tradition of the Text: A Case Study
(/isis/citation/CBB001211476/)

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