NavList:

Well, more in the tradition of George, but let's not quibble about this ..

For all practical purposes the sun 'hangs in the sky' at LAN.  We all agree about this.  While it means that there is no practical difference in altitude over several minutes around LAN, and therefore the quibble is truly pedantic if not pointless in terms of this practical exercise, it also results in a useful effect; the potential to improve the accuracy of the noon sight quite simply.

Rather than just take one random sight and accept its unknown errors (in this case, apparently amounting to 1.2' ), we can take as many sights as possible over a few minutes on either side of LAN. Then graph them; altitude on the vertical axis, time on the horizontal.  Then compare the pattern of those sights with a horizontal line, disregarding any that clearly don't match the others (outlier = gross error), and thinking hard about the others, while bearing in mind those closest to the instant of LAN will least reflect any change in altitude.  An intuitive process.

Of course, this technique will only be useful in reducing the effects of random error, it won't do anything about systemic error.  That is another topic.