NavList:

Brian Walton, you wrote:
"Fact: the rise and set time of the Sun at a place 90° different in latitude to your latitude, is also the time the Sun is on an azimuth of 090 or 270 at your position, that is, on the Prime Vertical."

Yes, it is a fact. And it's a fun fact! And a fact with good reason behind it. But it needs to be interpreted carefully. First, it's not exactly the tabulated time. Instead the time of sunrise that matches perfectly is the time when the Sun's true altitude (altitude of center, no refraction) is exactly 0°. The tabulated sunrise times in the Nautical Almanac (and in most other resources) are calculated under the assumption of 34' of refraction and 16' of Sun semi-diameter. This places the center of the Sun 50' below the true horizon. It takes the Sun in mid-latitudes about five minutes to climb 50' in altitude, so if you use the tabulated time of sunrise for a latitude 90° away to compute the time when the Sun is on the prime vertical (e.g. use sunrise time at 50°S for prime vertical time at 40°N), you'll be wrong by five minutes. So you can use the sunrise time as an estimate. Or you can adjust it, correct that estimate, using a little correction table. Or you can ignore all this and just calculate the time when the Sun is on the prime vertical directly, which is certainly not much work. 

You wrote:
"Look up your correction table, and find the correction to be made to your sunrise, to get the sunrise time at a place 90° different in latitude. Check out the azimuth of the Sun at your position, at that time, using the NavList Data app. If it is not 090 or 270, it is because your correction table is rather crude.  It doesn’t matter much, but it tells you when to get your Sungun out."°

I'm not entirely sure what you mean by the NavList Data app. Do you mean the USNO data page (linked on the main NavList webpage)? Regardless, any tool for calculating sun azimuth will work, so yes, we can do this. The problem with the time derived from the table is not that the correction is "rather crude" but, as I described above, it's computed for the standard civil definition of sunrise which will be wrong by about five minutes for "true" horizon sunrise in mid-latitudes. As you note, this estimate can be used as a planning tool to decide when the Sun is on the prime vertical if this matters to you.

Next, you wrote:
"What use is a shot on the PV? The trigonometry now becomes sin Alt=sin Dec/sin Lat. Set your Sungun up to that figure, with normal corrections reversed, and go outside just before the time of PV." 

All this just to preset the sextant? I use an index card, and that's what I teach, too. Hold a common index card at arm's length (3x5 inches at a distance of about 25 inches), and you can easily count off in multiples of 11.5° for the long side of the card or 7° for the short side of the card. Reversing the corrections is pointless since the field of view of your sextant is much larger than that, and also because it's a bad habit that does not in general work correctly (when you're actually trying to do things accurately).

And you continued:
"If you were doing a time sight 150 years ago, you could enter your epitome and take logsinlat away from logsindec, and look up that figure in the same logsin table and take out an angle(Hc) and time(LHA). You have just precalculated your shot just as well as you did with sandwich fixes."

You could do that, yes, but are you projecting? Did they do this historically? Some few examples perhaps? Did any significant number of navigators work this way, or are you just projecting back on them a technique that might have been clever if they had done it?

You concluded:
"Those old guys were smart."

But were they? Most of them were just trained in standard methodologies. How did they proceed? We can only know by consulting their actual work and sometimes their memoirs. The idea that navigators went to great lengths to shoot time sights (sights for local time, converting their sextants into sundials...) when the Sun was right on the prime vertical is a late spin on things, a very popular description of the methodology after it had become obsolescent, but in the period the advantages were minor --except in those rare cases where there was unusually large uncertainty in latitude.

The best "timing" for a time sight corresponds quite closely to the rules for a running fix in modern celestial navigation and for exactly the same reasons. Suppose I'm sailing from New England to Bermuda in mid-June, and I shoot the Sun at local noon. I get, let's say 39°05', and my "southing" is 5 knots (so my estimated latitude is dropping 5' per hour). Should I wait until the prime vertical to get a sight for longitude? No. All you need to do is wait until there has been some significant change in the Sun's azimuth, which could happen as little as an hour after noon. If the Sun has shifted from azimuth 180° to azimuth 225°, then we have enough change to cross two lines of position with reasonable confidence. Or, equivalently, we can have reasonable confidence in a calculated longitude from a time sight. It doesn't matter that the Sun has not reached the prime vertical. 

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA