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. 42-54), 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 A B be drawn perpen 
dicular 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 A B be drawn to the plane 
but not at right angles to it, and AM be drawn perpendicular 
to the plane (i.e. 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 A BP increases as BP moves away from BM 
on either side; and, given any straight line BP making 
a certain angle witli BA, only one other straight line in the 
plane will make the same angle with BA, namely a straight 
line BP' 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