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

THE ANALEMMA OF PTOLEMY 287
288 TRIGONOMETRY
and arcs of the heavenly sphere by means of orthogonal
the diagram is one position of it, coinciding with the equator),
projection upon three planes mutually at right angles, the
and it was called eKrrjfjiopos kvkXo? (' the circle in six parts ')
meridian, the horizon, and the prime vertical ' '. The definite
because the highest point of it above the horizon corresponds
problem attacked is that of showing the position of the sun at
to the lapse of six hours ; the second, called the hour-circle, is
any given time of the day, and the use of the method and
the circle represented by any position, as BSQA, of the circle
of the instruments described in the book by Ptolemy was
of the horizon as it revolves round BA as axis.
connected with the construction of sundials, as we learn from
The problem is, as above stated, to find the position of the
Vitruvius. 1 There was 'another dvdXrj/i/ia besides that of
sun at a given hour of the day. In order to illustrate
Ptolemy ; the author of it was Diodorus of Alexandria, a contemporary
of Caesar and Cicero (' Diodorus, famed among the
the method, it is sufficient, with A. v. Braunmuhl, 1 to take the
simplest case where the sun is on the equator, i.e. at one of
makers of gnomons, tell me the time ! ' says the Anthology 2 ),
the equinoctial points, so that the hectemoron circle coincides
and Pappus wrote a commentary upon it in which, as he tells
with the equator.
us, 3 he used the conchoid in order to trisect an angle, a problem
Let S be the position of the sun, lying on the equator MSC,
evidently required in the Analemma in order to divide any
P the pole, MZA the meridian, BOA the horizon, BSQA the
arc of a circle into six equal parts (hours). The word
hour-circle, and let the vertical great circle ZSV be drawn
'
dpdXrjizfxa evidently means taking up ' (* Aufnahme ') in the
through S, and the vertical great circle ZQC through Z the
sense of making a graphic representation ' ' of something, in
zenith and G the east-point.
this case the representation on a plane of parts of the heavenly
We are given the arc SO = 90° — t, where t is the hourangle,
and the arc MB = 90° — (p, where f
/S>)\
now available in William of Moerbeke's own words, Heiberg
E / \
having edited it from Cod. Vaticanus Ottobon. lat. 1850 of the
thirteenth century (written in William's own hand), and included
it with the Greek fragments (so far as they
G
exist) in
/V: \
parallel columns in vol. ii of Ptolemy's works (Teubner,