NavList:

Brian V., you wrote:
"My question relates to 'noon' sights, and making sights before/after. When observing the sun in its daily path (over just one day), it is making a near circle in the celestial sphere (from the perspective of Earth),  but for an observer on the surface, with a horizon, etc., would it not appear to be an ellipse (or semi-ellipse)?"

Hmmm... I'm not sure what you're trying to picture, but I will take a stab at it. First you refer to the "celestial sphere" as being "from the perspective of Earth" instead of for an observer on the surface. That isn't so. While physical celestial sphere models are often built centered on a small earth globe, that construction isn't relevant. Wherever you are, the celestial sphere is centered on you, the observer. Sometimes we like to consider a geocentric observer, but that only has a significant impact for the closest astronomical objects, especially the Moon. When we describe the apparent motions of astronomical objects "on the celestial sphere", this is a description of what an observer actually sees which is then imagined (for the sake of convenience, if any!) to be painted on the inside of a very large sphere centered on the observer.

The path of the Sun across the sky is certainly not an ellipse. And ignoring those small daily changes that I mentioned in my previous email, it closely corresponds to a circle in the sky. Except on the equinoxes, this is a small circle. Another way to imagine the Sun's motion is to imagine pointing at it all day long with a stick that has its base at the bridge of your nose (as close as possible to your eyes. On any day, during the course of the day, the tip of that stick will trace out a circle in front of you. That circle will be tilted depending on your latitude and the latitude of the Sun on that date (the Sun's "declination"). Now picture the 3-dimensional surface swept out by that stick during the day. The base of the stick is stationary on the bridge of your nose. The end of the stick tracks the Sun. On the equinoxes the stick sweeps out a flat, plane surface in front of you during the course of the year. On any other day the stick sweeps out a rather flat cone. The point of the cone is the base of the stick, and the surface of the cone tracks the Sun. Again, on any day, the tip of the stick traces out a circle in the observer's aky --equivalently, "on the celestial sphere".

I'm guessing that perhaps you're trying to picture a sort of graph or diagram of the altitude of the Sun as a function of time (or maybe as function of the Sun's compass bearing during the day. On some days, in some latitudes, that graph may look somewhat like a portion of an ellipse. But in general, that's not the case at all. That graph of altitude versus time is no simple shape, and it depends a great deal on the observer's latitude and the Sun's latitude. Picture the graph of the Sun's altitude at the equator on one of the equinoxes: it's a straight-up steady climb for six hours, followed immediately by a straight-down steady fall. The graph of that versus time is just two straight lines joined by a corner at the top. Or for another case, picture the graph of the Sun's altitude at the summer solstice for an observer just south of the Arctic circle. It looks vaguely like a bell curve (it's not, but there'e a superficial resemblance).

One thing you can count on: in most latitudes on most dates, the graph of the altitudes of the Sun near noon for some short enough period near local noon will look like an "inverted parabola". This is true of anything that we can measure, with a sextant or not, as the measured quantity approaches a maximum and then falls away again. With few exceptions it will closely resemble a parabola. Maybe that's all you're looking for? If you want a name for the curve of altitudes of the Sun as plotted against time before/after noon, the name for that curve is a "parabola" (and it may be useful to refer to it as an "inverted parabola" as I did here since parabolas are usually drawn with the "open" side up --concave upward).

Frank Reed