A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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, 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 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
in 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
398 PAPPUS OF ALEXANDRIA
in length to the radius of the circle. In all other cases
(Prop. 51 = Eucl. Optics, 35) the diameters will appear unequal.
Pappus's other propositions carry farther Euclid's remark
that the circle seen under these conditions will appear
deformed or distorted (Trapecnracrfiii'os), proving (Prop. 53,
pp. 588-92) that the apparent form will be an ellipse with its
centre not, ' as some think ', at the centre of the circle but
at another point in it, determined in this way. Given a circle
ABDE with centre 0, let the eye be at a point F above the
plane of the circle such that FO is neither perpendicular
to that plane nor equal to the radius of the circle. Draw FG
perpendicular to the plane of the circle and let ADG be the
diameter through G. Join AF, DF, and bisect the angle AFD
by the straight line FG meeting AD in C. Through G draw
BE perpendicular to AD, and let the tangents at B, E meet
AG produced in K. Then Pappus proves that G (not 0) is the
centre of the apparent ellipse, that AD, BE are its major and
minor axes respectively, that the ordinates to AD are parallel
to
BE both really and apparently, and that the ordinates to
BE will pass through K but will appear to be parallel to AD.
Thus in the figure, G being the centre of the apparent ellipse,
it is proved that, if LGM is any straight line through G, LG is
apparently equal to CM (it is practically assumed— a proposition
proved later in Book VII, Prop. 156—that, if LK meet
the circle again in P, and if PM be drawn perpendicular to
AD to meet the circle again in M, LM passes through G).