NavList:

Just my two cents.

Old navigation was Lat. by noon shot and Long. by time sight.  New navagation was Line of Position.

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.

Now came Summer.  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.  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.

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.

Chris, you said we do not figure local apparent time anymore.  Yes, we do - we call it Local Hour Angle.  In the 20th C. and 21st C, we work in degrees, minutes, and seconds.  Local Hour Angle is the same thing as Local Apparent Time.  One is in degrees ( angle ) and the other is in hours, minutes, and seconds (  time ).  Both give the difference between you - your meridian, your longitude, and the ground point of the sun - the sun's meridian, the sun's GHA.  When you figure the sun,s LHA is 45 degrees west that is the same as saying the local apparent time is 3:00 PM

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.  The new formulas did replace the old tables and formulas but more because we started using angles and not time.

IMHO    -    YMMV      -     :-)

John H.   41N  100 W      or 41 N,  6 hours 40 minutes west of Greenwitch