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Trinity College Dublin

MA232A - Euclidean and non-Euclidean Geometry
Michaelmas Term 2015
Dr. David R. Wilkins
The Theory of the Circle in Book III of Euclid's Elements of Geometry

Work in Progress: Transcription of Statements and Proofs of Propositions in Heath's Edition of Euclid

The material currently available is here

of Book III of Euclid in the Editions of Heath and Casey

Comparison of the presentations of Book III of Euclid in the Editions of Heath and Casey

T.L. Heath's edition of Euclid is a translation of the definitive text of Heiberg. John Casey's edition is based on the tradition of English-language textbooks based on Robert Simson's edition, ultimately deriving from the Latin translation of Commandinus. The statements and proofs differ on occasion between the two editions.

Euclid, Book III, Proposition 1
Proposition 1 of Book III of Euclid's Elements provides a construction for finding the centre of a circle. The statements and proofs of this proposition in Heath's Edition and Casey's Edition correspond except that the labels C and D have been interchanged.
Euclid, Book III, Proposition 2
Proposition 2 of Book III of Euclid's Elements shows that any straight line joining two points on the circumference of a circle falls within the circle. The statements and proofs of this proposition in Heath's Edition and Casey's Edition differ, though the proofs are related. The original proof of Euclid, presented by Heath, showed that a contradiction would arise were some point of the line segment outside the circle; and claimed that a contradiction would also arise if there were some point between the endpoints of the line segment lying on the circle. Casey gives a related direct proof.
Euclid, Book III, Proposition 3
Proposition 3 of Book III of Euclid's Elements shows that a straight line passing though the centre of a circle cuts a chord not through the centre at right angles if and only if it bisects the chord. The statements and proofs of this proposition in Heath's Edition and Casey's Edition correspond except for the labelling of the construction points.
Euclid, Book III, Proposition 4
Proposition 4 of Book III of Euclid's Elements shows that two chords of a circle, not passing through the centre, cannot bisect one another. The statements and proofs of this proposition in Heath's Edition and Casey's Edition correspond except for the labelling of the construction points.
Euclid, Book III, Proposition 5
Proposition 5 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 6
Proposition 6 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 7
Proposition 7 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 8
Proposition 8 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 9
Proposition 9 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 10
Proposition 10 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 11
Proposition 11 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 12
Proposition 12 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 13
Proposition 13 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 14
Proposition 14 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 15
Proposition 15 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 16
Proposition 16 of Book III of Euclid's Elements, as formulated by Euclid, introduces horn angles that are less than any rectilineal angle. The horn angle in question is that between the circumference of a circle and a line that passes through a point on a circle perpendicular to the radius at that point. Euclid proves that this line at right angles to the radius touches the circle, and also that no line can be interposed between this line and the circumference of the circle. Euclid also states that the angle of the semicircle (i.e., that between the arc of the circle and the diameter that divides the circle is greater than any acute rectilineal angle. Euclid's proofs employ reductio ad absurdum, together with the Pons Asinorum and various consequences of the basic result of Proposition 16 of Book I, which asserts that the exterior angle of a triangle is greater than either of the opposite internal angle. In contrast, Casey's edition merely states that the perpendicular to the radius at a point of the circle touches the circle, and that any other line through that point cuts the circle. Moreover the proof found in Casey's edition, makes use of Pythogoras's Theorem, and is thus differs substantially from Euclid's original proof. The statements and proofs of this proposition in Heath's Edition and Casey's Edition thus differ substantially.
Euclid, Book III, Proposition 17
Proposition 17 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 18
Proposition 18 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 19
Proposition 19 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 20
Proposition 20 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 21
Proposition 21 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 22
Proposition 22 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 23
Proposition 23 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 24
Proposition 24 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 25
Proposition 25 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 26
Proposition 26 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 27
Proposition 27 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 28
Proposition 28 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 29
Proposition 29 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 30
Proposition 30 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 31
Proposition 31 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 32
Proposition 32 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 33
Proposition 33 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 34
Proposition 34 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 35
Proposition 35 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 36
Proposition 36 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.
Euclid, Book III, Proposition 37
Proposition 37 of Book III of Euclid's Elements is to be considered. The statements and proofs of this proposition in Heath's Edition and Casey's Edition are to be compared.

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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.