It's an impossible challenge, as those formulae are not meant to be used to calculate when there's an eclipse. The title is TO FIND THE TIME, MAGNITUDE, AND DURATION OF A LUNAR ECLIPSE. Rowbotham instead tells us to find patterns in eclipse tables like the ancients to predict when there's an eclipse.
The simplest method of ascertaining any future eclipse is to take the tables which have been formed during hundreds of years of careful observation; or each observer may form his own tables by collecting a number of old almanacks one for each of the last forty years: separate the times of the eclipses in each year, and arrange them in a tabular form. On looking over the various items he will soon discover parallel cases, or "cycles" of eclipses; that is, taking the eclipses in the first year of his table, and examining those of each succeeding year, he will notice peculiarities in each year's phenomena; but on arriving to the items of the nineteenth and twentieth years, he will perceive that some of the eclipses in the earlier part of the table will have been now repeated--that is to say, the times and characters will be alike. If the time which has elapsed between these two parallel or similar eclipses be carefully noted, and called a "cycle," it will then be a very simple and easy matter to predict any future similar eclipse, because, at the end of the "cycle," such similar eclipse will be certain to occur; or, at least, because such repetitions of similar phenomena have occurred in every cycle of between eighteen and nineteen years during the last several thousand years, it may be reasonably expected that if the natural world continues to have the same general structure and character, such repetitions may be predicted for all future time. The whole process is neither more nor less--except a little more complicated--than that because an express train had been observed for many years to pass a given point at a given second--say of every eighteenth day, so at a similar moment of every cycle or eighteenth day, for a hundred or more years to. come, the same might be predicted and expected. To tell the actual day and second, it is only necessary to ascertain on what day of the week the eighteenth or "cycle day" falls.
Tables of the places of the sun and moon, of eclipses, and of kindred phenomena, have existed for thousands of years, and w ere formed independently of each other, by the Chaldean, Babylonian, Egyptian, Hindoo, Chinese, and other ancient astronomers. Modern science has had nothing to do with these; farther than rendering them a little more exact, by averaging and reducing the fractional errors which a longer period of observation has detected.
I wrote a mathematically rigorous walkthrough for the duration formula, with modern notations, to prove that Rowbotham was not just being cryptical.
The small balls represent the moon, the big ball is the
eclipse shadow.
α := |SB| [euclidian distance from S to B]
μ := the moon's semi-diameter = |BM|
λ := the moon's latitude in opposition = |So|
With a basic trigonometry formula, we can calculate cos n:
cos n = adjacent / hypotenuse = |Sm| / λ
=> |Sm| = λ.cos n
The Pythagoras' theorem yields |SM|² = |Mm|² + |Sm|²
=> |Mm|² = |SM|² - |Sm|²
= (|SB| + |BM|)² - |Sm|²
= (α + μ)² - (λ cos n)²
= (α + μ)² - λ² cos² n
=> |Mm| =
h := speed of the moon (in longitude)
duration of the eclipse = (|Mm| + |Nm|) / h
= 2 |Mm| / h (symetry, |Nm| = |Mm|)
And Rowbotham wrote:
Note that an α was lost, but it's an obvious typo.
The formulæ above quoted are entirely superfluous, because they add nothing to our knowledge of the causes of eclipses, and would not enable us to predict anything which has not hundreds of times already occurred. Hence all the labour of calculation is truly effort thrown away, and may be altogether dispensed with by adopting the simple process referred in the previous quote.