NavList:

Back in May I wrote,
"cos(HA) = sin(Alt)/cos(Dec)/cos(Lat) - tan(Dec)/tan(Lat)"

And Tony, you suggested that there were several typos and wrote:
"cos(HA) = sin(Alt)/[cos(Dec)·cos(Lat)] - tan(Dec)·tan(Lat)"

You're absolutely correct that there was one typo: the divide between the two tangents at the end should have been a multiply. I've written this out so many hundreds of times for my classes in the past ten years I was bound to make that typo eventually. Quite correct! However the other changes are un-necessary. Here's the proper version:
   cos(HA) = sin(Alt)/cos(Dec)/cos(Lat) - tan(Dec)·tan(Lat).

Divisions are done left to right, in order. So 100/4/5 is equal to 5. Note that you can add parentheses and modify this expression as (100/4)/5 or (100/5)/4 or 100/(4·5). Only the last requires parentheses. In the other cases, the parentheses are optional and you're back to the original form: 100/4/5. You don't need the extra parentheses. This isn't just my idiosyncratic understanding of order of operations and math rules, by the way. Try it any spreadsheet or any programming language. 

Note that historically there was a tendency to prefer writing this relationship as
   cos(HA) = [sin(Alt) - sin(Dec)·cos(Lat)] / [cos(Dec)·cos(Lat)].
There were historical reasons for this preference which are now gone.  For example, it used to be considered "frugal" to limit an equation to sines and cosines only. But if you're tapping on a calculator, sin and cos and tan are all first-class citizens, any of the three is as easy to tap as the others. On the other hand, using parentheses on a calculator can be quite a nuisance and, in fact, scary and confusing for beginning students. 

The result of this calculation, the Hour Angle, HA, gets added or subtracted to the GHA to produce the longitude. How do you decide to add or subtract? If the celestial body is in your western sky, then subtract, else add. But if you can't remember that rule, no matter --just try them both and pick the one closer to your DR. Historically, in some periods in the 20th century, this was called "working a time sight" but it's perhaps a misleading name for a modern navigator. We're just calculating a longitude directly from the altitude at some specified latitude. We're plucking a single point off the circle of position for a given latitude.

Once you have a single point of the circle of position with a latitue and longitude, what do you do with it? One option is to repeat the calculation for a slightly different latitude. You get a different longitude. You plot those two points on any graph paper, draw a line segment through them, and you're all set. You have a "two point" line of position. A historical name for a "two point" line of position is a "Sumner line", and Greg Rudzinski's been posting some examples recently. I usually avoid the Sumner name because, like the expression "time sight", it's associated in the minds of many navigators with antiquated practice. It shouldn't be -- it just is. It all depends on your audience.

Frank Reed