John, you wrote:
"Old navigation was Lat. by noon shot and Long. by time sight. New navigation was Line of Position."
There are many other factors which distinguished the Old Navigation from the New Navigation. It's worth considering the history of these expressions, too. They appear in the 1930s at least, maybe somewhat earlier. The purpose seems to have been a sort of pedagogic "marketing". Which would you learn: the Olde Stuff or the sparkling New Science!
"It did not matter if you had time on a watch ( cronometer ) or time by a luner - did same thing. You assumed a Lat. and from your shot figured a local apparent time then compared the LAT to GMT and the difference was longitude. The tables in Bowditch were in time, not degrees, minutes, and seconds. Longitude was in time."
Sometimes. Most navigators in the 19th century were well aware that the end result would be a longitude in degrees. The conversion was simple. But it's quite true that longitudes, even on maps and in atlases, were sometimes listed in time instead of degrees. If you look through logbooks and other primary source navigation documents, however, this was rare at sea. They liked their longitudes in degrees as much as we do.
Then you wrote:
"Now came Sumner. Do the time sight twice, assuming two different lat. to get two different points. Draw a line connecting the two points and you get a Line of position - New Navagation."
This is where we run into "cartoon history". The introductory pages in navigation manuals are awash in these sorts of unhistorical histories. And indeed, they often use phrases like "now came Sumner" as if he revolutionized everything. But this didn't happen. Retro-actively, authors of navigation histories, especially American authors, have taken some of the marketing materials (taken right from the opening pages of the second edition of Sumner's book) and imagined Sumner as a turning point. But it's not so. Sumner's work largely flopped. Lines of position were almost unheard of until the 1880s in British shipping and the early 20th century in American shipping. This colossal myth imagining that the world changed after Sumner has led to great confusion when armchair historians have tried to understand late 19th and early 20th century navigation practice. They have been taught that it all changed with Sumner starting in the 1840s, but where are the lines of position?? They are, in fact, almost non-existent until those much later dates. Sumner's method was considered an exotic experiment, not a standard technique.
"It was the LOP that was new, not the way you got the point. Rember, you could have done a lunar distance to get GMT and worked up the Long. or used a watch to get GMT - same thing. Long. by Lunar or Long by cronometer give you the same thing, only how you get GMT is different."
That's right. And in the early part, through the middle of the 19th century, you see "lon by lunar" and "lon by chro" side-by-side. The distinction made here was between the two diffferent tools and methodologies for acquiring Greenwich Time. In both cases, you need a time sight for local time. Hence the attachment of the time sight in later decades in British "slang" to the expression "lon by chro" was a historical accident, and it didn't make any logical sense.
"Along comes St. Hilarie. He figured out a different way to get a LOP (new nav) He assumed a Lat. and a Long. (the AP) did different math to figure how far away you were (intercept), figured an azimuth and then drew a line of position. The difference between that and Sumner was how you got the LOP. Both were new navigation."
Except for the un-historical emphasis on Sumner (as I described above), your comparison here is spot on. The "Sumner" algorithm and the "Intercept" algorithm both produce a celestial line of position from any sight. So what gives? Why do we need both? In fact, you can and should generate a line of position from two points along it (usually considered a "Sumner" approach) just to see that they are the same thing. We don't need both! They both work, and you can use either one. Historically the computation work was slightly longer by the two-point approach than by the intercept method. Otherwise they produce the same line of position. And that's the key thing to remember: every celestial altitude yields a nearly linear segment on a huge celestial circle of position with some error band around it. This is the underlying "geometric reality" of any sight no matter what mathematical procedure we use to find its extent. The intercept method did not "invent" lines of position, nor did Sumner. They were always there. If you worked a single time sight in the early 19th century from some DR latitude, you were simply "plucking off" a single point from the line of position. And indeed, unlike the modern philosophy, that single point may well be completely adequate for your navigational needs.
You also wrote:
"Long. by cronometer ( or long. by Lunar ) gave you a point at the assumend lat. A Sumner gave you a Line of Position from two points. An intercept gives you a LOP from an assumed point and an azimuth."
And there are plenty of other options that bridge across these possibilities. For example, you can calculate a single time sight to get one point on a line of position, and then you can use azimuth tables (or graphs). You then draw the line of position through the single position perpendicular to the azimuth. Note that there is no "intercept method" in this anywhere, yet it makes use of some of the more modern tricks. This technique unfortunately throws out one of the great advantages of the two-point (Sumner) calculation of the line of position, which is that you can plot it on common graph paper or equivalent, no special plotting sheets required.
"The new formulas did replace the old tables and formulas but more because we started using angles and not time."
No, that's not why. It's a good starting point, but it's not why.