A lunar observation from March of 1825

The observation consists of the angular distance between the Sun and the Moon and the altitudes of both objects. The dead reckoning position of the vessel at the time of the observation was about 440 miles north of Puerto Rico (approximately in the latitude of Miami, Florida). The Moon was nearly due east 59 degrees high. The Sun was in the west-southwest 39 degrees high. The arc of the lunar was nearly straight across the zenith so the net distance correction is very close to the sum of standard altitude corrections. The time of day is a little after three in the afternoon.

The original scan of the page from the logbook on the Seaport Library's web site is difficult to read. Here's a copy of the original image:

(click image to enlarge)

For easier reading, I've re-written the calculations long-hand using exactly the same layout, copying out everything I could read and keeping as much as possible the style of writing out angles and logarithms from the original logbook entry. I've also marked off each section of the calculation and labeled them A through L (there is no section I). Here's the result:

(click image to enlarge)

Analysis

This logbook page is a complete lunar observation and the corresponding time sight worked by Bowditch's method. This was the "First Method" in Bowditch for most of the 19th century. The entire calculation fits on a single page with room to spare. Notice that if the navigator had had a chronometer, only the time sight portion of the calculation would be required (sections J-L). So the math work for a lunar using this method was approximately 3.5 times longer than would be the case using a chronometer. Figuring five minutes of work for a time sight, a lunar could be worked in less than 20 minutes (which matches my experience).

Section A:

This is a calculation of reduced time, marked "RT". Reduced time was the name Bowditch used for the best estimate of Greenwich Time at the time of the observation. The reduced time was used to enter the Nautical Almanac. The time from the watch (3:06:43) is added to the DR longitude converted to hours and minutes (4:28:00 = 67° 00') .

Section B:

The Moon's measured altitude was apparently 59° 47'. Following the recommended practice in Bowditch, 20 minutes is subtracted from the altitude. That's an average correction for dip + semi-diameter for an Upper Limb sight. The Moon's semi-diameter (SD) and horizontal parallax (here labeled P) are listed at 12 hour intervals in the old almanacs, so there is a process of interpolation to bring them up to the "reduced time". Finally, 4 seconds are added to the Moon's semi-diameter to account for augmentation.

Section C:

In old logbooks, the Sun is almost always refered to by its old astronomical symbol: a circle with a dot in the center. The Sun's measured altitude was 38° 45' 30". As with the Moon, this altitude is corrected using the simple 12/20 rule (whether it's Sun or Moon, add 12 minutes for a Lower Limb sight, subtract 20 minutes for an Upper Limb sight).

Section D:

The measured lunar distance was 80° 02' 45". This is limb-to-limb (always near limbs when using the Sun) so the semi-diameters are added to the distance to yield the center-to-center distance of 80° 34' 21". This distance and the altitudes of the Sun and Moon are added up and then divided by two. This yields the "half sum" of Bowditch's Principal Method  [(d+h1+h2)/2 = 89° 29' 30"]. Then the distance is subtracted from the half sum to yield the "first remainder" and the Sun's altitude is subtracted from the half sum to yield the "second remainder".

Section E:

This is the logarithmic and table lookup section of the calculation. The first line in each column is logsin(d) where d is the center-to-center lunar distance. The second line in each column is logcsc(R2) where R2 is the second remainder (calculated in section D). The third line in the left-hand column is logsec(R1) where R1 is the first remainder (section D). The third line in the right-hand column is logsec(H.S.) where H.S. is the half sum (section D). The fourth line in each column is found in Bowditch's Tables XVIII and XIX. In both cases, the entries in these tables are calculated from log(1/2)+log(h)+P.L.(dh) where h is the object's altitude, dh is the altitude correction for that altitude, and "P.L.(x)" means proportional log of x [defined by log(3°/x)]. The sum of each column yields the proportional logs of the first and second corrections. Looking up these in the table of proportional logs yields the corrections 2' 06" and 47".

Section F:

The corrections are added to the distance. Notice that 2° is subtracted from the altitude right at the top. The next two lines are the altitude corrections for the Sun and Moon, but in the case of Bowditch's method, they are subtracted from constant values of 60' and 59' 42" respectively. That is, the Sun's actual altitude correction in this case is 1' 03" but in Bowditch's table XVIII, the value listed is 60' - 1' 03" or 58' 57". This trick helps to keep the corrections additive. The Moon's correction is subtracted from the odd value 59' 42" in order to accomodate a similar change in the quadratic correction. After the actual altitude corrections we find the corrections calculated in step E and finally the last line before the sum is the quadratic correction, 18", from Bowditch's Table XX. This 18" value actually represents a "zero" correction in Bowditch's method. It's paired with the 59' 42" base for the Moon's correction to make one whole degree.

Section G:

These are the predicted geocentric lunar distances from the Nautical Almanac. Without them there would be no way to guess the date of this lunar observation except that it is marked March 27 (by sea account) and probably in the 1820s. By using the online lunar distance almanac calculator on this web site, it was easy to find lunar distances at 6 hours and 9 hours Greenwich Time that closely matched the values in the logbook. Note that it is necessary to select Greenwich Apparent Time since the almanacs in this era listed lunar distances as a function of GAT. This fixed the date of this observation as March 26, 1825. Here's a link to the lunars almanac preset for the right date and DR position. Note that the distances in the almanac as recorded in 1825 match the modern values with a discrepancy of only 6 seconds of arc. This would lead to an error of about 3 minutes of longitude.

Section H:

The interpolation to find the difference in Greenwich Time. The values on the right are the differences in the distances. The four digit numbers on the left are the proportional logs of those differences. The time difference 1:36:14 is added onto the "6", the time of the lower lunar distance from the almanac in section G.

Section J:

This is the setup for the time sight. In sections A through H, the Greenwich Apparent Time was found. In this and the following sections, the Local Apparent Time is found. The Sun's altitude is cleared of semi-diameter (16' 10") and then dip and refraction combined (4' 43"). Notice that this is the second time the Sun's altitude has been corrected on this page of calculation. First the altitude was cleared using the "rough" 12/20 rule which is adequate for altitudes in a lunar. Here it is cleared again, accurate to the nearest second of arc for the purpose of the time sight. In the next lines, the DR latitude and the polar distance (90°-declination) are added to the Sun's altitude. Then the "half sum" is calculated: 76° 12' 11". Finally a "remainder" is calculated. These numbers are used in the logarithmic calculation in the next section.

Section K:

The first line in the column is logsec(L) where L is the latitude. The second line is logcsc(p) where p is the polar distance. The third line is logcos(H.S.) where H.S. is the half sum from section J. The fourth line is logsin(R) where R is the remainder from section J. These four logarithms are added up and then divided by two finally yielding 9.60274. This last value is the logsin(LHA/2) where LHA is the Sun's Local Hour Angle. This is tabulated in Bowditch in the same table as the logsin, directly in hours and minutes of time.

Section L:

The Local Hour Angle of the Sun derived from the time sight in the previous section is 3:08:56. Expressed in time units, this is the Local Apparent Time exactly. The Sun's Local Hour Angle is the Local Apparent Time. Note that the "time on the watch" at the very top of the page (in the calculation of "reduced time" in section A) differed from this calculated Local Apparent Time by 2 minutes and 16 seconds. If the time sight had been ignored and the time from the watch used instead, the difference in Local Time would have led to an error of 34' in the longitude, independently of any error in the lunar itself. Getting the longitude from the time sight's Local Apparent Time and the lunar's Greenwich Apparent Time is trivial: subtract the times and convert to degrees by multiplying by 15. So the final longitude is 10' 30" east of the DR longitude that was noted in section A.