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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
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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.
Back to Resources related to Euclid's Elements of Geometry
Back to Module MA232A
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.