# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Errors in Cotter's "History of Nautical Astronomy, once more**

**From:**Jan Kalivoda

**Date:**2003 Mar 26, 13:03 +0100

Hello, all, some months ago several members of the group were prepared to search for the old, but very valuable book "A History of Nautical Astronomy" from Charles H. Cotter (London 1968). George Huxtable has published the list of its errors and misprints in this group. In the last days I have ascertained that the interest in this book awoke again in the group. So I send to you the list of other errors I have found or I suppose to have found in this title. Perhaps it can be useful for some of you. If not, neglect the rest. Maybe in some cases I am wrong, not Cotter. If so, you will make me wiser. Both lists, the mine and George's (I will insert this last to the end of this message for your convenience), seem to be very long together. I don't want to detract from Cotter's merits. His book was a goldmine of invaluable information about the history of navigation for me. The author really had read the old handbooks from 18th - 19th centuries (not older, I guess) and compiled very precious text from them. But he made it a bit hastily and the junctures are to be seen somewhere. So let's go. The second number marks the line of the page - excuse me, if I had missed the numbering a bit. -------------------------- 83,10 from above Alae sui -> Alae sive -------------------------- 84,15 from below the longest side -> the lower shorter side ??? --------------------------- 101,1 from below the true zenith distance theta_1 -> the apparent zenith distance theta_1 at the point O_1 (i.e. "O with the index 1") ---------------------------- 107,5 from below To the formula at this line a remark would be useful: "(mi - 1) = U". Otherwise the deduction from the formulas at the page 105 is not clear. ---------------------------- 108,3 from above Here I am not certain. But the deduction of this formula from the formulas III and IV on the page 105 seems to be wrong. Cotter proceeds, as "2 PZ" on page 105 would equal "lambda" (geographical latitude !) in the picture 2, page 104. But this not the case! Can anybody help with this derivation? Maybe it's my fault. ----------------------------- 110,11 from above the first edition (of Maskelyne's "Requisite Tables") -> the second edition ----------------------------- 113,6 from above One would add to this line: "And h much less than R" (otherwise R/(R+H) would not equal to (R-h)/R) ----------------------------- 121,11 from below 1/4 (approx.) -> 4/1 (approx.) ----------------------------- 135,9 15th century -> 16th century ----------------------------- 151, 5-7 from above The words from "Moreover, ..." to the end of paragraph seem to be wrong to me. When reducing the measured altitude of the Sun to the time of the first observation by the run and azimuth of the Sun, the observer should not make any other reductions acording to the run between observations? ------------------------------ 158,16 from above ZY is the great circle -> XY is the great circle ------------------------------ 163,12 from below after the meridian altitude -> before the meridian altitude ??? ------------------------------- 217,16-18 from below This was true only if the navigator used the table of "logarithmic differences", giving the value of the equation X from the page 216 by inspection. Such tables were in use, but introduced an error of 3-6 arc-seconds into result. This was tolerable in the times when lunar positions itself (hence the true lunar distances, too) were tabulated with the error of 15-30 arc-seconds in alamanacs, owing to deficiences of the used theories of lunar motion. ------------------------------- 225 below, 231 passim Cotter explains the fundamentals of the lunar distances clearly and sufficiently. But his historical sketch of their evolution is very unsatisfactory. With exception of Borda's method and his direct successors, he describes only the methods of ending 19th century and neglects the most esteemed methods from the first half of 19th century when the importance of lunars was the greatest. Elford, Bowditch, Turner, Thomson, Cambridge Tables - all are missing. Maybe, I will write a modest supplement to this subject. In particular, the Dunthorne's method mentioned on these pages was not a indirect method, but the first and the most succesful direct method, preceding the Borda's in time and a simpler one. It was the most common method of lunars in the continental Europe, apart from France. -------------------------------- 239,6 from below prop. log SMALL delta being given in the almanac -> prop.log. GREAT delta being given in the almanac !!! -------------------------------- 241,8 from above the value 50' of error when neglecting the second differences is grossly exaggerated - George Huxtable emphasized it rightly some time ago. -------------------------------- 248,5 from above vers (PX � PX) -> vers (PZ � PX) (in the numerator on the right side of the equation) -------------------------------- 265,7 from below delta d -> delta h -------------------------------- 265,3 from below tan l cos h -> tan l csc h -------------------------------- 266,2 from above cos h -> csc h -------------------------------- 266,9-13 from below This text seems completely confused - either by Cotter or by the author of original. One could try to correct it, but it would better to verify it in the Nautical Magazine for 1848, which I haven't at my disposal. -------------------------------- 272,19 from below if the two hour angles -> if the difference of two hour angles -------------------------------- 308,3 from below Now the least important -> Not the least important -------------------------------- 310 The manual of Pedro Nunez (mentioned elsewhere in the book) "De arte et ratione navigandi" from 1522 is missing - it was very important and very early publication, the example for many other authors -------------------------------- 311 The manual of Martin Cortes appeared in 1551 -------------------------------- 366 Nos 238 and 240 of bibliography seem uncomplete to me. The first (1767?) and fourth (1811?) editions of Maskelyne's "Requisite Tables" are not mentioned. -------------------------------- Jan Kalivoda ======================================= EARLIER CORRECTIONS FROM GEORGE HUXTABLE ======================================= Here's an updated list of some things I suspect are wrong in Charles H Cotter's otherwise-excellent book "A History of Nautical Astronomy". Jan Kalivoda has kindly helped by confirming some earlier diagnoses and adding others of his own. If any listmember has others to add, I would be pleased to receive them. I doubt if my list, below, is at all exhaustive: I have not deliberately searched out errors; these are just the ones I have stumbled across in my own reading. The paging corresponds to my Hollis & Carter 1968 edition, the last page of its index being 387.. I think there was also a US edition: the paging was probably identical. I doubt if there were any later editions. Here goes- =========================== page 49. The third paragraph starts- "The civil day at sea commenced at midnight", which is correct. In the next paragraph Cotter states "The civil day commenced when the Mean Sun culminated at noon." which is contradictory, and wrong ========================== page 118, foot of. Cotter states "Augmentation = Moon's semidiameter x sine apparent altitude". This is wrong. It would be roughly true to state instead- Augmentation (in minutes) = Moon's semidiam. (in degrees) x sine apparent altitude but more accurate to say- Augmentation (in minutes) = Moon''s semidiam. (in minutes) x sine apparent altitude / 55. ========================== page 120, 2nd line, Cotter says- "... body Y, which has the same APPARENT place as body X". but fig 5 shows body Y at the same TRUE place as body X. ========================== page 210-212, Borda's method. Here I think Cotter has got into a real mess with his trig. The equation that precedes equation (Y) is given as - (sin D/2)^2 = sin{(M+S)/2 + theta} sin {(M+S)/2 - theta} Here, he has got the last term the wrong way round and it should be- (sin D/2)^2 = sin{(M+S)/2 + theta} sin {theta- (M+S)/2} so in consequence, in equation (Y), the second sine term in the product of two sines is also reversed. Similarly in the last equation on page 210, for log sin D/2, the last term in the sum should end up as log sin (theta- (M+S)/2), not log sin ((M+S)/2 - theta), as Cotter gives it. If you slavishly follow Cotter's steps, you will end up taking the log of a negative quantity, which is an impossibility. I think Cotter has realised there's something wrong, without being sure what it is, because on page 211 he states the rules in words for clearing the distance, and in rule 5 he says- "Find the sum of and difference between theta and phi". Because he hasn't defined here which way round to take that difference, the navigator will presume that he should subtract in such a direction as to give a positive answer. So that bit of "fudging" has got Cotter out of his problem. In fact the subtraction should ALWAYS be theta - phi, and NEVER as stated at the foot of 210, phi - theta. On line 3 of page 212, that's what he has written down in the calculation, theta - phi, just as it should be. ========================= There's an additional error in Rule 5, page 211, in that the last sentence should not read- "The result is the sine of half the true lunar distance, that is D/2.", but instead- "The result is the LOG sine of half the true lunar distance, that is D/2." ======================== page 226, Dunthorne's method. Dunthorne's is in fact a rigorous method of clearing the lunar distance, but Cotter has included it in his list of approximate methods ======================== page 237. For a navigator, it may be useful to know that alpha Aquilae is more familiar as Altair, alpha Arietis as Hamal, and alpha Pegasi as Markab. ========================= page 250. The four equations shown on this page all use the quantity s, but I cannot find any definition of s. I presume that s is half the perimeter of the PZX triangle, so- s = 1/2 (ZX + PZ + PX) In the third expression, for cos P/2, a quantity s with a subscript 2 appears. That little 2 appears to be a misprint should be erased. ======================== page 264. Cotter says, about finding the moment of noon by equal Sun altitudes- "By taking the equal-altitude sights shortly before and after noon the necessity for applying a correction for the change in the Sun's declination in the interval is obviated, since any such change will be trifling." I disagree with Cotter's analysis here. It seems to me that the correction necessary for a changing declination does not reduce as the interval chosen gets closer to noon. =========================== page 265. Re Hall's rule. Cotter introduces delta-d as the correction in seconds of time.. This ought to be delta-h and in the equation near the foot of the page, cos h should be cosec h. page 266. Similarly, in the equation at the top of this page cos h should be cosec h. ========================== page 354. Napier's rule. I suspect that the second expression, shown as- cos x = cos y cos z is wrong, and should be- cos y = cos x cos z ========================== It's rather disappointing that so many errors can be found in Cotter, and indicates some degree of carelessness in the checking and proofreading. But we can all get things wrong, as I have good reason to know... The prevalence of these detected errors leaves me suspecting that there may be many more, lurking as-yet unseen. They detract somewhat, but not a lot, from the value of Cotter's book, which remains by far the best source I know to treat the development of astronavigation, and is a wonderful goldmine of references. George Huxtable.