The puzzling circle of constellations
The constellations marked on the innermost rings of Oughtred's instrument are called the 'inner nocturnal'. I found them hard to understand. The radial lines associated with these constellations were aligned to the clock scale and apparently also to the calendar scale.
When performing the time calculation, you first find a constellation that is closest to the meridian and find the time for that constellation on the hour circle. If the meridian is midway between two constellations, you find the time midway between them on the hour scale.
Next, you find the time on the clock scale that is adjacent to the day of the month for the reading. That is added to the previous value to give the actual time.
Oughted gave a list of the constellations he used, which I have attempted to identify in modern terms.
List of Constellations from Oughtreds "The Circles of Proportion..."
"The bright star in the head of ♈". This refers to the star Hamal in the constellation Ares. The inscription on the scale is Luc and the zodiac symbol for Aries is beneath it.
"The Bull's eye". This should be the star Aldebaran in the constellation Taurus. The inscription on the scale is Ocu (from Latin: Oculus) and the zodiac sign for the bull is displayed below it.
"The latter shoulder of Orion". I suppose this to be Betelgeuse. The inscription on the scale is Seg Or (OR-ion)
"the little dogge". The star is most likely Procyon, and the inscription for this constellation is Can mi, for Canis Minor.
"the heart of the Lyon". The inscription is Cor - the Latin word for heart. Underneath, there is a squiggly line that was used as the zodiac symbol for Leo.
"The tayle of the Lyon". The inscription is Cau (Latin: Caudis) and this also has the zodiac symbol for Leo.
"Spica Virginis". This is the star Spica in the Virgo constellation, and the inscription is Spi. The 17th-century zodiac symbol for Virgo is under it.
"The North ballance". This term refers to the constellation Libra. The inscription is Lanx, literally 'Pan', which, I suppose, refers to the pans of the scales. The star would therefore either be Gamma Librae or Sigma Librae
"The head of Ophiuchus". I think this is alpha Ophiuchi (the star RasalHague), and the inscription for this is Cap (Latin: Caput) Ophi
"The heart of the Vultur". The inscription for this is Vultur, which I gather is an old name for Lyra, whence the star would surely be Vega
"The mouth of Pegasus". The inscription is OS Peg (Latin Oris = mouth). I think the star is Enif (epsilon Pegasi)
"The tip of the wing of Pegasus". The inscription is ext ala, possibly from the Latin for the 'extremity of the winged (Pegasus)', I guess. I think the star is Algenib (gamma Pegasi).
How does this actually work
My current information comes from the excellent animated pages on this instrument by the University of Lisbon, text for which was developed by Simon Gessler According to the University of Lisbon site (text by Simon Gessler), these stars are arranged along the clock in terms of Right Ascension). The operation of the calculation involved the Right Ascension of the stars closest to the meridian and the Right Ascension of the sun
However, I could see that the locations appear at different locations than the reported RA for the stars, seemingly by a constant offset of about one hour. Eventually, with the help from Simon and information on the University of Lisbon website, I think I've got it figured out, see here.
As for the constellations on the scale: Oughtred seemingly intended you to use the brightest stars in the constellations that he chose for this instrument, and the constellations were selected so that the sky was more or less evenly divided into hours. The list above is substantially the same as proposed on the University of Lisbon site.
The image to the left is the test build for the Circles of Proportion. The printer needed a new toner cartridge (hence the worn-looking scales).
My testing of the calculating scales was minimal (3 x 2 = 6, etc.), possibly because I've grown over-confident. Anyway, the logarithmic scales have easy-to-find markers that I was able to compare to photos of the originals.
I spent most of the time trying to understand and validate the nocturnals.
Because I live in an area swamped with light pollution, it made no sense to try to test this thing outside. For that reason, I didn't bother to use a hollow rivet (or grommet) in the centre of the disc in my test piece.
To evaluate the nocturnals, I used the sky and telescope interactive sky chart to provide a view of the night sky. The website allows you to print the sky, and when I needed to measure angles I would take a hardcopy and do that.
For my tests, I used my own longitude and latitude coordinates; I needed to use an online Local to Solar time calculator. I found that the computations worked for the examples given by Oughtred, for current dates as well as 17th century.