A history of Greek mathematics - Wilbourhall.org

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THE COLLECTION. BOOKS V, VI 397
The Sphaerica of Theodosius is dealt with at some length
(chaps. 1-26, Props. 1-27), and so are the theorems of
Autolycus On the moving Sphere (chaps. 27-9), Theodosius
On Days and Nights (chaps. 30-6, Props. 29-38), Aristarchus
On the sizes and distances of the Sun and Moon (chaps. 37-40,
including a proposition, Prop. 39 with two lemmas, which is
corrupt at the end and is not really proved), Euclid's Optics
(chaps. 41-52, Props. 4 2-54), and Euclid's Phaenomena (chaps.
53-60, Props. 55-61).
Problem arising out of Euclid's
'Optics'.
There is little in the Book of general mathematical interest
except the following propositions which occur in the section on
Euclid's Optics.
Two propositions are fundamental in solid geometry,
namely
(a) If from a point A above a plane AB be drawn perpendicular
to the plane, and if from B a straight line BD be
drawn perpendicular to any straight line EF in the plane,
then will AD also be perpendicular to EF (Prop. 43).
(b)
If from a point A above a plane AB be drawn to the plane
but not at right angles to it,
to the plane (i.e.
and AM be drawn perpendicular
if BM be the orthogonal projection of BA on
the plane), the angle ABM is the least of all the angles which
AB makes with any straight lines through B, as BP, in the
plane ; the angle ABP increases as BP moves away from BM
on either side ; and, given any straight line BP making
a certain angle with BA, only one other straight line
plane will make the
line BP /
in the
same angle with BA, namely a straight
on the other side of BM making the same angle with
it that BP does (Prop. 44).
These are the first of a series of lemmas leading up to the
main problem, the investigation of the apparent form of
a circle as seen from a point outside its plane. In Prop. 50
(= Euclid, Optics, 34) Pappus proves the fact that all the
diameters of the circle will appear equal if the straight line
drawn from the point representing the eye to the centre of
the circle is either (a) at right angles to the plane- of the circle
or (b), if not at right angles to the plane of the circle, is equal