# The diagrams in Codex P of Euclid s Elements

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```					       The diagrams in Codex P of Euclid’s
Elements
Ken Saito

General Principles for the Diagrams
The diagrams in manuscripts, which seem to be drawn in an awfully incorrect
manner, follow some speciﬁc principles, though these principles are not so
rigid. This is pointed out ﬁrst by Netz, in his book of 1999, The Shaping
of Deduction in Greek Mathematics. In my communication, I present some
peculiar feature of the diagrams, and try to ﬁnd underlying principles.
The following result is based on my preliminary research in the ﬁrst four
books of the Elements, based almost only one manuscript, that is, codex P
(Vat. Gr. 190). However, there is no sign that we should assume that other
medieval codices have very diﬀerent principles about the diagrams.
The most conspicuous principles are the following:
1. The diagrams are ”standardized”:
(a) an angle becomes a right angle if it can be:
—though it can be otherwise – e.g., line ∆Γ in III-16;
(b) two lines are drawn equal if they can be equal:
e.g., in I-47, the rectangular triangle becomes rectangular and
isosceles; in II-1, BD=DE=EΓ in the diagram;
(c) a segment of a circle becomes a semicircle if it can be:
e.g., III-24.
Thus, there appear more squares, rectangles, isosceles and equilateral
triangles in the diagrams of manuscripts than in their texts. Heiberg
“generalized” these diagrams, for example, making two lines diﬀerent
if they are not always equal, followed by all the current translations.
However, the “generalization” seems to be a recent tendency, for Bar-
row’s edition preserves the standardized diagrams fairly well.
2. Metrical correctness is not so important.
(a) Less important part of the diagram is disregarded:
e.g., in II-5, the line AΓ, the left-half of the bisected line AB, is
drawn much shorter than BΓ (right-half), because (I believe) the
argument is developped mainly in the square constructed on BΓ.

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(b) Even the distinction between straight lines and curved lines is not
of absolute importance:
In IV-16, in a diagram in the margin, the sides of ﬁfteen-angled
ﬁgure are represented by concave arcs, so that they do not coincide
with the circumference of the circle in which the ﬁgure is inscribed.
This is also the case in some Archimedean codices, as reproduced
in Netz’ translation of the Sphere and Cylinder.
There are also some cases which seem to contradict the principle of “stan-
dardization”. The square in II-4 is oblong. A possible explanation is that
the scribe did not want to leave too much space around the diagram—a sort
of horror vacui.

An Intriguing Case: Elements III-25
The diagrams for the proposition III-25 (to complete a given segment of
a circle) present an interesting problem. We are accustomed to the three
diagrams in Heiberg’s edition (followed by all the translations, of course),
which correspond to the three cases in the proof: the given segment being
less, equal, greater than a semicirle. However, the manuscripts presents only
one diagram in the column where the text and the diagram appear, and the
three diagrams according to three cases appear in margin.
What is intriguing in the diagram in the column (not in margin) is that
though it describes the given segment of circle as a semicircle, it marks two
other points E and Θ, one in the semicircle and the other out of it, so that
the diagram seems to serve for all the three cases. Other diagrams in the
margin are likely to have benn added afterwards, according to the three cases
in the text.
We cannot exclude the possibility that the division of the proof into three
cases is a result of later intervention, and the original proposition presented
the proof only once in the text, accompanied by one ﬁgure applicable to three
cases (another manuscript, codex B, presents one in the column, and two in
the margin. In fact, the diﬀerence of the cases, which is apparent in the
diagram, is not important in the text. The same procedure is repeated three
times, and if one is indiﬀerent to the place where the center of the circle falls
(whether in the segment ABΓ, or on the chord AΓ, or out of the segment),
one has no need of distinguishing the three cases.
It would be too hasty to try to give some conclusion to this problem, for
the tradition of Book III is very complicated, accompanied by alternative
proofs of several propositions.

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