A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 Aug 3, 16:49 -0700
Chauvenet fell into the same trap that caught a number of late nineteenth century mathematicians with navigational interests. He believed that a "rigorous" solution was required to fix the methods which he found insufficient. Bessel's "rigorous" method was un-necessary. There was no call for an exact solution to the lunars problem incorporating the oblateness of the Earth. That could be managed with a small correction at the end of the work using any of various efficient methods to clear lunars. Chauvenet's theory that Bessel's method did not gain traction among sailors because there were too many cases is a mathematician's conceit. It's much more likely that it was simply considered pointless. There were beautiful, slick, accurate, efficient methods for clearing lunars already universally available, and the tiny improvement (around a tenth of a minute of arc difference, on average) that could be gained from including the Earth's oblateness in the problem just wasn't sufficiently interesting.
The distinction between "rigorous" solutions to the lunars clearing problem and "approximative" solutions is misleading today. These names reflect a shift in meaning of words and/or growing maturity in mathematical concepts. The methods which were formerly called "approximative" were no less rigorous in principle than the methods that were tagged as "rigorous". A better name for those so-called rigorous solutions today might be "closed form". There are closed form solutions, which give a single fixed set of equations (possibly rather lengthy) which solve a given model problem in all cases, and there are series solutions which provide a potentially infinite (but practically short) set of terms which are derived by standard calculus from any closed form solution.
My favorite example of an exact closed form solution compared to a series solution is simple ballistic particle motion, also known as "throwing a ball". Suppose we go to the Moon and toss a ball back and forth (no air up there so that's a major factor out of the way). The equations for the motion of a tossed ball that any high school student can write down would be:
x = v0x·t + x0,
y = -(1/2)g·t2 + v0y·t + y0.
Supplied with the right input values, these equations will provide highly accurate representations of a projectile tossed back and forth on the lunar surface. The resulting shape is an inverted parabola.
These equations, yielding a parabolic arc, are a series solution, an approximation, to a problem in orbital motion. When we throw that "ball" on the lunar surface, we should "properly" (in what sense?) treat it as an object in a Keplerian elliptical orbit with one focus of the ellipse at the center of the Moon. The arc above the lunar surface is the very top end of a long elliptical orbit. To derive the arc, we should calculate the mean anomaly, M, then solve Kepler's equation:
M = E - e·sin(E)
for the eccentric anomaly, E (which used to quite difficult in practical cases since E is inside the sine function also outside standing by itself, and still has computational interest in special cases!). Then convert that to the true anomaly (no need to list more equations here, I think), which is a polar angle. Next convert polar coordinates to rectangular coordinates, and finally apply a little calculus to get a series solution for the case where distances and heights are not large compared to the radius of the Moon. And whaddaya know, for small distances and heights, you find that the elliptical orbit is almost perfectly represented by the x and y equations that we wrote down at the beginning. Or in geometric terms, the tiny bit at the top end of an elliptical orbit is identical to an inverted parabola. Whenever we solve the problem of throwing a ball back and forth, we are solving an orbit problem, but we know in advance that the full baggage of that solution is almost always irrelevant. Besides, other modifications to the problem have to be applied before those differences matter, no matter how we look at it.
This complicated "orbital ellipse" versus simple "parabolic arc" in the analysis of ballistic projectile motion is analogous to the "closed form" (formerly known as "rigorous") solutions to the problem of working lunars, popular with the mathematically-inclined, versus the simple "series" solutions which were widely practical and provided exactly the same practical results except in rare, special cases. A series solution is every bit as rigorous as a closed form solution.
It's either in this article or one the follow-ups that Chauvenet shows his hand and demonstrates his own ignorance of lunars math, or at least its recent history. He complains rather bitterly about a method of "one Mendoza Rios", recently published, or so he believes. In fact, Jose de Mendoza y Rios (also known as Joseph Mendoza Rios) had been dead for decades when Chauvenet wrote his dismissive comments, and the method he was attacking was, in fact, Nathaniel Bowditch's original method, the very one praised so effusively in modern editions of Bowditch, which was known for various reasons, including some slightly obscure prior publication and also a little anti-American sentiment, as the method of Mendoza Rios in British resources.
Chauvenet should have, and could have, produced some modest improvements in one of the various methods already available, but instead he travelled down the road that so many mathematical enthusiasts have followed and developed his own highly idiosyncratic method for lunars which he, like so many others, bragged was better than all of the other methods combined... and then some!
Lunars were already dead in maritime practice, and the inclusion of Chauvenet's method in the greatly revised Logan-Bowditch in 1881 (the greatly improved government-issue version that deleted almost everything in the original Bowditch) was an anachronism, possibly in deference to Chauvenet's standing as one of the original teachers at the US Naval Academy decades earlier. I doubt that more than a handful of lunars were ever cleared for maritime use outside the classroom using Chauvenet's method. I have found no evidence in primary source materials of anyone clearing lunars using Chauvenet's method.