# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Bris Sextant Math.**

**From:**Alexandre Eremenko

**Date:**2005 Oct 31, 20:18 -0500

Dear John, My math estimate from the observationbs is that the two angles between the adjacent glass panes are about 5.2d and 6.5d. (The rest of this message is pure math. I apologise to the list members who don't like it. Just skip it then). Here is the theory of the Bris sextant. The ray from the Sun to your eye experiences on its way an even number of reflections. Two reflections are equivalent to a rotation by twice the angle between the two reflection planes. (This is a theorem we teach in our linear algebra courses; the ordinary sextant is based exactly on this theorem: the ray from the Sun experiences exactly two reflections, one in the Index mirror and another in the Horison mirror). Now we have 3 mirrors. Say 1, 2, 3. Let the angle between 1 and 2 be A/2, and the angle between 2 and 3 be B/2. (The mirrors are enumerated in the natural order, from the Sun to your eye, for example. Consider first the three bright Suns. They correspond to the rays which experience 2 reflections. The possible combinations of 2 reflections are (21), (32) and (31). The angles of deflection are A, B and A+B. Now consider the dim Suns. They correspond to the rays experiencing 4 reflections. There are 8 combinations of 4 reflections possible, (2121), (2132), (2131), (3232), (3231), (3121), (3132), (3131). Some of them will produce the same angles. (These Suns will look slightly brighter). The possible angles of deflectio of the ray are only 5: 2A, 2B, 2A+B, 2B+A and 2A+2B. Now, assuming without loss of generality, that A