# [FOM] Understanding Euclid

Vaughan Pratt pratt at cs.stanford.edu
Fri Dec 5 04:13:25 EST 2008

```Bill,

Derivable from what?  Proposition 32?  In my message I fingered
Proposition 32 as a counterexample to spherical geometry along with
Proposition 47.  We're in agreement that Proposition 32 does not hold in
spherical geometry.

The difficulty I'm having is that I don't see how any of Euclid's five
postulates could fail in spherical geometry.  My point was that
propositions such as 32 and 47, which clearly don't hold in spherical
geometry, can't follow logically from postulates all of which hold in
spherical geometry.

Which one of Euclid's five postulates fails in spherical geometry?

Vaughan

William Tait wrote:
> Vaughan,
>
> The converse of Postulate 5 is derivable in Euclid. He does define two
> lines to be parallel when they do not intersect. The converse of
> Postulate 5 then is: If two lines are perpendicular to a third, then
> they are parallel. If they intersected, the three line segments would
> form a triangle. Two of its interior angles are, by assumption, equal to
> two right angles and so the three interior angles are more than two
> right angles,  contradicting Proposition 32 of Book 1.
>
> Proposition 32 does not hold in spherical geometry.
>
> Bill
> On Dec 3, 2008, at 5:15 PM, Vaughan Pratt wrote:
>
>> Euclidean geometry is standardly understood as the geometry of the
>> plane, more generally of flat or uncurved space.  To make this stick
[...]
>>
>> 2.  Proposition 47 of Book 1 of the Elements states and proves
>> Pythagoras's theorem.  This is clearly false on the sphere, which
>> contains an equilateral triangle all three of whose angles are right.
>> This is not the earliest counterexample: Proposition 32 shows that the
>> angles of a triangle sum to two right angles.  Euclid is evidently not
>> playing by his own rules, and is appealing to unstated postulates.  Does
>> he appeal somewhere to the converse of his fifth postulate?  If so how
>> did he fail to notice he was doing so?  [...]
```