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Tycho made two kinds of angular measurements: altitudes and separations. Altitudes are relatively straightforward; one needs to know where the horizon is and where the star is, then the angle between them is the altitude. Horizons are actually sort of difficult, unless you are observing from a large, flat, plane such as the Midwest US or the quiet ocean. In my area, Eastern US, the horizon is difficult to find because of all the trees. Tycho had decent horizons but chose a more elegant method, the plumb bob. This is simply a hanging weight, and it points straight down, directly opposite to the zenith. The horizon is then that direction perpendicular to the string that is supporting the weight.
We now know that that method has some weaknesses, but it was completely adequate for Tycho's time. I couldn't see an easy way to build a plumb line (the string) into a telescope, so I started thinking about using the surface of a liquid to define the local horizontal. Modern sextants, used in navigation, use bubble levels if no true horizon is available, and they thus do measure the altitude above the horizon. I settled on making a mirror with the liquid metal mercury to let me locate the zenith.
Separation angles are a different beast altogether. Modern telescopes cannot measure the angle, simultaneously, between two stars that might be tens of degrees apart. They tend to look in one direction only. Cameras are different, though, they can be designed to take in a field of view of tens of degrees in one image. One can then measure the separation between the two stars in pixels, but what is needed is the actual angle.
I spent many years observing asteroids and measuring the angles between the asteroid and nearby stars, but this relied on first calibrating the camera to find what is called the 'plate scale' (degrees per pixel). This is actually not a single number but a 2-D function that varies across the image. The calibration method is to upload a list of star positions from a precision star catalog and then use these precise positions to compare to where the detected stars are actually found. I used the popular program 'Astrometrica' for that task, and it provided excellent plate scale functions for asteroid position determinations.
But this technique uses existing star catalogs as a basis for calibration, and Tycho had no accurate catalogs to fall back on, much less imaging detectors. So I did not pursue an imaging approach, instead I followed Tycho's lead by examining his 'trigonal sextant'.
His trigonal sextant needed two observers, one for each star. Each observer looked through a set of slits, near his face, toward the center post on the far end of the instrument. As each observer fixed on his star, a ruled arc was used to measure the angle between the two lines of sight.
The most straightforward way to do this today would be to have two telescopes, each looking at a separate star, with some device to measure the angle between the two telescope tubes.
I did not have any convenient pair of telescopes, so I thought of the idea of using a single telescope that could look in two directions at the same time. For that purpose, I did have a candidate telescope in my collection of astronomy equipment, an old 5" NexStar telescope. The idea was to place a cover, over the telescope aperture, that had two smaller openings. This created two optical paths to the focal plane, each of which could function as an independent telescope. By placing a rotatable flat mirror in front of one of the small apertures, I could change the view direction of that path. Each optical path would put its own image on the focal plane. These images would overlap. If the two optical paths were looking at different stars, then I could bring the stellar images to fall at the same place on the detector. By measuring the angle of the flat mirror, I could measure the separation of the two optical paths, and thus the separation of the two stars. A more complete description follows on the next page.
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