Nevertheless, after I had studied Bruce's Tables and his explanatory texts, I was astonished by their ingenuity. They don't repeat old solutions mechanically, but are significantly better than renowned works of the past, although they don't misuse the modern technical possibilities and go the fully traditional way of tabular and paper solution. It had to be an intellectual adventure to compose them and it is a delight to study them.

Let me consider them in the historical perspective. I won't repeat information already published elsewhere (you can now read it at http://web.dkm.cz/kalivoda/LunDistClass.htm). Here would I only remind that two classes of methods for clearing Lunar Distances (LD's) existed:

The "approximate" methods grew ripe relatively quickly and 50 years after the first volume of the Nautical Almanac had been published, they had reached the state of perfection with David Thomson in 1824. After that date no significant development in this field took place. These methods were very popular at sea during the whole 19th century for their speed, simplicity and for the important fact that they required the use of 4-digit logs only. Moreover, in spite of it, they permitted the (nearly) same accuracy as their counterparts - see an exception immediately below.

(At http://web.dkm.cz/kalivoda/Thomson.pdf you can read the detailed description and commentary on Thomson's "Lunar and Horary Tables for new and concise Methods of performing the Calculations necessary for ascertaining the Longitude by Lunar Observations or Chronometers...", London 1824 and subsequent sixty seven editions up to 1880).

Approximate methods had two great drawbacks. Firstly, the most popular and most widely used ones didn't allow the user to take the effect of non-standard refraction upon the measured distance into account, or they allowed it only by very bothersome procedures that would have deprived them of all their advantages, if used. This gap could only exceptionally create an error greater than 30" in the cleared distance, which was not a tragedy. Nevertheless, with these methods and in tropical (or Arctic) latitudes, the navigator had always to doubt of the reliability of his LD a bit, if he used the Moon or the other distance body in a lower altitude than some 20 degrees.

Secondly, the auxiliary tables necessary for use of these methods were very scarce in giving details of their structure and genesis. The sailor had to use them or reject them, but he could not make his own opinion about them. Some of these tables were checked by mathematicians, but only many years after their publication. Some were found very accurate (Thomson), some rather inaccurate (Elford), but without any impact on the sea practice.

It is no wonder that teachers of navigation hid the most popular "approximate" methods from their learners and that sailors with less fatalistic point of view sought another solutions. Such solutions were offered by the second class of methods for clearing LD's, by the "rigorous" methods. These methods were absolutely lucid for men that wanted to understand them. They gave the full control of the calculation, allowed every sort of corrections, the correction of refraction necessary for real atmospheric conditions, needless to say, included. But their drawback was their relative complexity and above all the necessity to use the 6-digit logs in computing and to switch from log values to natural values of trig functions alternately while solving them.

Old astronomers and arithmeticians used to say that each further digit of logs used in calculation increased its length and tediousness by a half at least. If so, the difference between the work with 4-digit or 6-digit log tables was palpable. In our days, when we have the accuracy of a calculation up to 10 digits and more at our disposal within the reach of one button of a hand calculator, we cannot imagine what a burden everyday logarithmic calculations created for ordinary navigators of 19th century.

Therefore, new rigorous methods for clearing LD's arose repeatedly during the 19th century and none of them was fully successful. They were pressed upon students of navigational courses, but in the sea practice probably only few fans and some snooty navy officers used them. Their main drawbacks mentioned above remained.

Up to Bruce Stark in 1995/1997.

Above all, Bruce derived and uses the very apt formula for reducing LD. Here it goes:

hav D = hav (M~S) + (cos M cos S sec m sec s) × SQRT{hav [d-(m~s) hav [d+(m~s)]}

M,S,D - true geocentric altitudes of the Moon and the Sun/star and distance of them

m,s,d - apparent, i.e. observed values

(Maybe it would be useful to consult the excellent article of George Huxtable on logarithmic computations published at http://fer3.com/arc/m2.aspx/Using-logs-nautical-tables-Huxtable-jun-2003-w10330 while reading the following text.)

The formula seems horrible, as all "rigorous" formulae do, but with Bruce’s comfortable tables and work sheets, only a sharp pencil is needed for quickly resolving it. Its extraordinary advantage (never achieved before) is evident: the term (cos M cos S sec m sec s) excepted (which is taken from tables by inspection), only one trig function - haversine is needed for computing!

And more: the haversine is extraordinary suitable at this place, as 5-digit log tables of it suffice to obtain the accurate result within the range of some arc-seconds. As you know, the haversine of an angle is the squared sine of the half angle. The squaring beneficially enlarges the differences of log mantissas between subsequent function values and the halving moves the used angle arguments farther from the right angle, where the sine would be very unreliable. Thank to both these features, the use of only 5-digit log haversine tables can be accepted. It would be impossible with the sine or cosine, so frequently used in old rigorous formulae. See bellow the third reason permitting the use of only 5-digit values.

The second Bruce's deed is the manner how he had solved the problem with the addition in his formula. Such addition makes the straightforward logarithmic solution of the equation impossible (see George Huxtable's text mentioned above). Additions, mostly inevitable in rigorous formulae for clearing LD even after torturing them by the most sophisticated trigonometric transformations, used to be overcome by jumping between log and natural values of trig functions. Of course, each such jump enlarged the time and effort demanded by the method and increased the maximal possible error of the result.

Bruce Stark goes another way. He uses the Gaussian logarithms that make possible to remain in world of logarithms all the time of calculation and transform an addition of natural numbers to the addition and subtraction of their common and special logarithmic values by use of a special table. It is much easier than to convert logs to their natural values, to add them and again to convert them to logs. Moreover, Gaussian logs yield greater accuracy of result than the traditional computing method and help 5-digit log values to be sufficiently accurate for this method.

The use of "Gaussians" by Bruce is original in the field of navigation. I don't know another example of using them by seamen or aviators - with the exception of Soviet navigators, which had Gaussians in their standard table sets up to cca 1960. The Gaussians were probably regarded by Stalin's commissars as opponents of Anglo-Saxon cosmopolitan and aggressive haversine that was not allowed to the Soviet navigational practice. However, in Bruce's hands, Gaussians coact peacefully with haversines in rationalizing the LD procedure to the level unknown so far.

The third asset of Bruce is his method of obtaining reference lunar distances that are to be compared with the cleared distance for obtaining G.M.T. One would say that after these distances had disappeared from nautical almanacs in 1907-1924, the death of lunars was imminent. Who was bold enough to tell sailors to compute reference distances by hand?

However, Bruce Stark changed this handicap to the contrary. He proposed the formula for obtaining the reference distances to be compared that is conformal with the notoriously known haversine formula for finding the altitude in Marc St.Hilaire’s method. Therefore, with the prepared work sheet the time and effort for computing them is pressed to an absolute minimum possible. And because with modern almanacs at sailor's disposal one can compute such reference distances for each hour without any interpolation of GHA and declination, the interpolation of G.M.T. from them is much more accurate that in the times when 3-hours almanac intervals were common for tabulated distances. For an user of Bruce's Tables this makes possible to evaluate even very short distances that would have unusable second differences in three hours intervals. In addition, as Bruce Stark emphasizes, such short distances are the easiest ones to be observed from small sailing ships of archeonavigators riding their hobby of the celestial navigation.

Other advantages of Bruce Stark's tables I can mention only briefly, so that I could end this article soon enough. They are e.g.:

• Shifting from arc-seconds to hundredths of arc-minutes. This agrees with the custom of modern seamen
• Very handy "inside-out" tables reducing the demand for place
• Combining the corrections of altitudes for dip and semidiameters in one table
• If the user does not care about the principles, he need not even understand the idea of logarithm

After Bruce Stark had published his tables, every sailing navigator (fondling the GPS in his pocket) can revert to the sea history in his practice very easy, if he likes it. He can be sure that with these Tables, the history of Lunar Distances is consummated now and the long line of rigorous methods for clearing them ends successfully - only in our days.

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