224 



NATURE 



[January 4, 1894 



navigation, and in advocating the merits of his own. For 

 the precise object of the new method let our author speak 

 for himself In page iv. of the preface we read : " It is 

 here claimed for the semi-azimuth method that it renders 

 double altitudes unnecessary ; that a better result than 

 they have been supposed to yield can be obtained from 

 either observation singly, and this also with a great 

 saving of time and trouble." 



Again on page 27 : " The range of the General Method 

 exlepds, either from the meridian, or from the limit at 

 which the direct method becomes uncertain, viz., three 

 points from the meridian, up to an azimuth of seven points, 

 or more, that is to say to within less than one point from 

 the prime vertical, which may be considered the practical 

 limit of the semi-azimuth method. ... It may safely be 

 affirmed that nothing like this extent of range in the 

 computation of latitude has ever been attained by the 

 current systems of navigation." 



The feats which our new teacher claims to have ac- 

 complished are two in number. 



(i) The discovery of anew formula for reduction to 

 the meridian. 



(2) By a somewhat tedious system of approximations to 

 obtain by the semi-azimuth method from either of two 

 observations for double altitude the same or a better re- 

 sult than would under the usual double altitude method 

 be derived from the two combined, subject only to the 

 limitation that the body must not be within a point of 

 azimuth from the prime vertical. 



Let us consider (i) and (2) in detail. 



First with regard to (i), the formula obtained is this : 



Reduction = h cos / x arc i azimuth. 



The formula is a simple one, and now that hour angle 

 is so easily converted into azimuth by the Burdwood and 

 Davis' tables, there seems no objection to its adoption 

 by those who prefer to work out their correction rather 

 than to take it direct from the tables. 



Mr. BuUer deduces the formula from a somewhat ela- 

 borate construction on the Mercator's chart. It is, how- 

 ever, merely an old friend masquerading in a new dress, 

 and he might, if he pleased, have easily obtained it from 

 the expression in ordinary use. 



Thus in the formula 



sin - = sm / sin 

 2 



cosec z sin 



-h 



where is the correction,/ the polar distance, c the co- 

 latitude, r the zenith distance, and h the hour angle, if 

 we assume that 



= i sin c . A . A . sin i', 



the BuUer formula in a logarithmic shape. 



The use of this formula is first exemplified by its 

 application to examples from Jeans' " Navigation," and 

 so long as the azimuth is small it answers well enough. 

 But our author is tempted further afield, and so soon as 

 he gets well away from the meridian his formula begins 

 to give trouble, and he is only able to obtain an accurate 

 result by a process of approximation so cumbrous as to 

 be quite useless for practical purposes. As an instance 

 of this, witness the example of which the results, but not 

 the full work, are shown on p. 14. Were the work to be 

 shown in full, it would occupy more nearly two pages 

 than one page of the text. 



Now let us suppose it had not been given to Mr. Duller 



NO. 1262, VOL. 49J 



to discover the methods lately presented to the nautical 

 world, and that he had been content to follow mere 

 ordinary processes. 



Imagine, for instance, that he had selected the well- 

 known furmula 



vers Mer Zen Dist = vers z - sin / sin c vers k 



in order to work out example § 16, p. 12. 



Then, upon his own assumption that the latitude was 

 51°, or 12' in error, he would obtain as a first result 

 lat. 50 49' 42". Repeating the process with this new 

 latitude, a second approximation would be 50' 48' 16", 

 while a third repetition would result in 50" 48' 3", the 

 true value and this with less than one-third of the 

 trouble. 



The semi-azimuth had better, therefore, be confined to 

 observations within a point or so of the meridian. 



We pass on to the second and more important part of 

 the task which Mr. Buller has set himself, namely, to 

 show that the latitude within certain wide limits of 

 azimuth may be obtained with accuracy from a single 

 altitude without waiting for a second. 



And this may be at once conceded, that by making 

 the necessary adjustments for change of azimuth, and by 

 successive approximations, an altitude may be reduced to 

 the meridian, even when the azimuth is considerable. 



But the same result may be obtained from the versine 

 formula given above with far less trouble but greater 

 accuracy. 



The question, then, narrows itself to this : Why should 

 we not in all cases of observation within seven points of 

 the meridian, reduce the altitude to the meridian at once, by 

 one method or another, and so obtain the latitude without 

 waiting for hours to take a second altitude, and then 

 making lengthy calculations? 



Why have Robertson, and Raper, and Inman,and the 

 other giants failed to hit the bull's-eye, while it is left to 

 Mr. Buller to put his finger on the "blind spot".? 



The answer is easily stated. 



The " blind spot " is to be found not in the accepted 

 custom of mariners, but in the author himself. Mr. Buller 

 in arriving at his conclusions leaves out of consideration 

 the real vital point which attaches to every observa- 

 tion taken ashore or at sea, but especially to the latter 

 class, namely, what is well defined by Raper as the 

 " Degree of dependence" to be placed on the observation. 



In every observation off the meridian we have to deal 

 in one form or another with a spherical triangle, in which 

 three elements being given we have to find a fourth. 



Now the three given elements are in general known 

 only approximately, and it behoves us to find under what 

 conditions an observation should be taken that the smallest 

 possible error maybe produced in the final result. In the 

 problem under consideration, treated by the Buller process, 

 the data are the polar distance, the approximate 

 colatitude, and the zenith distance. 



Of these the polar distance may fairly be regarded as 

 accurately known, since the difference can only be at most 

 but a few seconds. 



The latitude is required to be known only approxi- 

 mately, the object of the observation being to find the 

 amount it is in error, and it has already been admitted 

 that the new process will suffice to obtain the correct 

 latitude, always supposing that the observed altitude is 

 correct. 



What reason is there then to consider that the altitude 

 is correct, and what will be the effect if it is incorrect ? 



When we take into account the haziness and uncer- 

 tainty of the horizon, the difficulty of accurate observation 

 on board a rolling ship, the varying effects of refraction, 

 the imperfections of the sextant, and the personal error 

 of the observer, it is probable that an average error of 

 2' is a very moderate estimate. 



Taking 2', therefore, as an average value, let us see 



