May 5, 1898] 



NATURE 



II 



angle, but is a table which is made use of daily in the 

 calculations which belong to Mercator sailing, and which 

 is consequently to be found in every collection of nautical 

 tables. It is known as the table of " meridional parts," 

 or, as the French call it, " latitudes croissantes." The 

 meridional parts for a given latitude are defined by some 

 writers as " the value in minutes of a great circle of the 

 line on the Mercator's chart, into which the true difference 

 of latitude is expanded." 



For a given latitude / the meridional parts represent 

 the sum of the series 



seco' + seci' + sec 2' + sec 3' + . . . + sec (/° - i') 



which is found by the integral calculus to be 



log, tan (45° + -^ 

 «°°log.tan(4S° + 



10800 



when r is expressed in minutes. 



In the table of meridional parts we have then a series 



of logarithms to the base^'°^°°, which has been found to 

 lend itself in a remarkable manner to the purpose which 

 we have in view. 



It should be mentioned here that M. Guyou's general 

 method is to deduce his formuhe from a study of the 

 properties of the curves of equal altitude on a Mercator's 

 chart. To other writers, especially in Italy, where con- 

 siderable attention has been bestowed upon the new 

 formula?, it has appeared more satisfactory, while ac- 

 cepting the expressions, to deduce them directly from 

 fundamental trigonometrical formulic. 



Shortly before the issue of M. Guyou's second work 

 there was published, in the numbers of the Nautical 

 Magazine for November and December 1895, » system 

 of formul;e, for the solution of all the ordinary problems of 

 nautical astronomy, by the aid of this table of meridional 

 parts alone, the general principle adopted being to break 

 up the spherical triangle, or "triangle of position," as it 

 is generally called in nautical astronomy, into two right- 

 angled triangles, and thus obtain expressions which, 

 containing three terms only, would be more manageable 

 than the general formuhc involving four terms. 



This treatment of the subject was based upon certain 

 easily established lemmas, the most important of which 

 may be thus stated. (The abbreviation MP will be 

 adopted for meridional parts throughout.) 



MP(i8o' - fl) = MP(e) 



MP( - e) = - Mr(e) . 



tan X = sin d, 



MP(2.v) ■■= 2MP(fl) . . 



tan a = tan b tan c . . . 



(3) 



It 

 then will 



If 



then will 



MP(2a - 90°) = MP(2/' - 90) + MP(2f - 90°) . (4) 



With regard to (i) it may be stated that from the form 

 of the expression 



MP for lat /" = rlog ^ tan (^45° + —V 



the meridional parts in the first instance have reference 

 to angles in the first quadrant only. The lemma 

 enables us to pass to angles in the second quadrant. 



Similarly by lemma (2) we can introduce negative 

 angles also. 



The result involved in (3) is exceedingly important, 



NO. 1488, VOL. 58] 



for it follows from this that if we have a logarithmic 

 formula connecting the sines and cosines of parts of a 

 spherical triangle, we may pass by means of auxiliary^ 

 angles to other logarithmic formuhe, involving only the 

 meridional parts of the angles employed, and that not 

 only for right-angled and quadrantal triangles, as in the 

 Nautical Magazine^ but for any spherical triangle 

 whatever. 



As an example we may take one of the family of 

 formuhe which express a function of an angle of a 

 spherical triangle in terms of functions of the sides, 

 supposed known. These expressions are perhaps, from 

 a navigator's point of view, the most important which 

 spherical trigonometry presents ; for in the problem of 

 finding the hour angle of a body, and thence the longi- 

 tude of the place, such a formula may have to be brought 

 into requisition on board a fast steam-ship as many as 

 four or five times in the course of twenty-four hours. 

 And while many of the problems of navigation may be, 

 to some extent, " dodged " or evaded by the use of some 

 of the many tables which ingenious persons have devised, 

 there is no getting away from the hour-angle problem, 

 because in that case the necessary degree of accuracy is 

 more minute than any table of reasonable size could be 

 expected to afford, unless we are content to spend more 

 time and trouble in interpolating for variations in the 

 values of the elements from the arguments given in the 

 tables, than would suffice for thie actual calculation by 

 logarithms. 



Let us assume that in the spherical triangle ABC we 

 have to deal with the expression 



Assume that 



/sin (j - b) sin (j - 

 \' sin s sin (y - a) 



So that 



sin {s - b) = tan x 

 sin (j - (■) — tan y 



sin s = tan w, 



sin (j - a) = tan z. 



2 V 1 



tan X tan J/ 

 tan IV tan z 



By lemma (3) we have 



MP(2-«:) = 2MP(i-(J), 



and so on for j/, w^ z \ a system of equations which wilt 

 determine 2^, 2j, 2w^ 2s. 

 Then by lemma (4) 



MP(A - 90°) = i!MP(2x - 90") -I- MP(2j - 90') 



- MP{2w - 90°) - MP(23 - 90°);, 



whence A is readily determined. 



The formula here established is only given as an 

 illustration of the ease with which by the aid of lemma 

 (3) we may pass from a sine or cos ne formula to one 

 involving meridional parts only by the simplest possible 

 transformations. 



The processes deduced by M. Guyou from the curves 

 of altitude upon the Mercator's chart are probably some- 

 what shorter, and more likely, therefore, to be adopted 

 for general use. His methods of procedure however^ 

 although, as has been well said of them by an Italian 

 critic, "of high scientific interest for their originality and 

 rigorous analysis," may be found somewhat subtle and 

 difficult to follow by any but expert mathematicians. At 

 all events, although, as has been said, the Ciuyou 

 formulae were received in Italy with much favour, 

 mathematicians in that country lost no time in setting to 

 work to establish them upon a basis purely trigono- 

 metrical. 



An interesting article in the Rir'ista A/aritti//ia (Rome) 

 for January 1897, by Signor P. L. Cattolica, "Capitano 

 di corvetta," gives a summary of the work done in 1896 

 by Signor Molfino and other writers, whence it appears 



