Vitruvius  The Ten Books of Architecture
Extract from Book V, Chapter VI  Plan of the Theatre
1.
The plan of the theatre is to be constructed as follows. Having fixed
upon the principal centre, draw a line of circumference equivalent to
what is to be the perimeter at the bottom, and in it inscribe four
equilateral triangles, at equal distances apart and touching the
boundary line of the circle, as the astrologers do in a figure of the
twelve signs of the zodiac, when they are making computations from the
musical harmony of the stars. Taking that one of these triangles whose
side is nearest to the scaena, let the front of the scaena be determined by the line where that side cuts off a segment of the circle (A  B), and draw, throught the centre, a parallel line (C  D) set off from that position, to separate the platform of the stage from the space of the orchestra.
Plan of a Roman Theatre according to Vitruvius
2.
The platform has to be made deeper than that of the Greeks, because all
our artists perform on the stage, while the orchestra contains the
places reserved for the seats of senators. The height of this platform
must not be more than five feet, in order that those that sit in the
orchestra may be able to see the performances of all the actors. The
sections (cunei) for spectators in the theatre should be so divided,
that the angles of the triangles which run about the circumference of
the circle may give the direction for the flights of steps between the
sections, as far as up to the first curved crossaisle. Above this, the
upper sections are to be laid out, midway between (the lower sections),
with alternating passageways.
3. The angles at the bottom, which give the directions for the flights of steps, will be seven in number (C,E,F,G,H,I,D); the other five angles will determine the arrangement of the scaena: thus, the angle in the middle ought to have the 'royal door' (K) opposite to it; the angles to the right and left (L,M) will designate the position of the doors for guest chambers; and the two outermost angles (A,B)
will point to the passages in the wings. The steps for the spectators'
places, where the seats are arranged, should not be less than a foot
and a palm in height, nor more than a foot and six fingers; their depth
should be fixed at not more than two and a half feet, nor less than two
feet.
4. The roof of the colonnade to be built at the top of the rows of seats, should be level with the top of the scaena,
for the reason that the voice will then rise with equal power until it
reaches the highest rows of seats and the roof. If the roof is not so
high, in proportion as it is lower, it will check the voice at the
point which the sound first reaches.
5.
Take one sixth of the diameter of the orchestra between the lowest
steps, and let the lower seats at the ends on both sides be cut away to
a height of that dimension so as to leave entrances (O,P). At the point where this cutting away occurs, fix the soffits of the pasages. Thus their vaulting will be sufficiently high.
6. The length of the scaena ought to be double the diameter of the orchestra. The height of the podium....
