Abstract
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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Notes
- 1.
See Heath ’s commentary at this point (1956, vol. 1, pp. 249–250).
- 2.
See De Risi ’s analysis for what this means, but roughly speaking points not on the line cannot be unique in situation with respect to A and B because they cannot be distinguished from their mirror images in the line.
- 3.
See Bonola (1906, 54) who cites Seance de l’Ecole Normale, 1, pp. 28–33, reprinted in Mathesis 9, pp. 139–141 (1883).
- 4.
Translation from Ewald (1996 vol. 1, 159).
- 5.
In Gauss Werke, X.1, 483–574. There is an English translation in Dunnington (2004, 469–496).
- 6.
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Gray, J. (2015). A Note on Lines and Planes in Euclid’s Geometry. In: De Risi, V. (eds) Mathematizing Space. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12102-4_3
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