SECTION XL 

 NAVIGATION AND NAUTICAL ASTRONOMY, 



CHAPTER I. 

 INTRODUCTION". 



THE principles of Navigation rest almost entirely upon 

 that part of Plane Trigonometry which is limited to the 

 doctrine of plane triangles. A person unacquainted 

 with the mathematical theory upon which the practical 

 rules followed by the navigator are based, would natu- 

 rally imagine, as the track of a ship is a path marked out 

 on the surface of a sphere, that to calculate, from the 

 necessary data, the length of this track, the aid of 

 Spherical Trigonometry would be required. But, in 

 general, such is not the case ; and for this reason : 

 spherical trigonometry is wholly concerned with the arcs 

 of great circles of the sphere, and with the angles formed 

 by such arcs ; whereas, the course of a ship at sea, unless 

 it sail on the equator, or on a meridian, is never a great 

 circle ; it is either a small circle, a parallel to the equa- 

 else a line cutting the successive meridians over 

 it sails, obliquely, and at the same invariable angle, 

 so long as its course remains unchanged. A ship, con- 

 tinuing on this unchanged course, would trace out, on 

 the surface of the sphere, a winding or spiral curve, 

 called in navigation a rhumb line, and which is widely 

 different in figure from a circle. If a vessel were to start 

 from any point between the equator and either pole, and 

 on a course inclined ever so little towards that pole, it 

 would wind round the globe in this spiral path, approach- 

 ing nearer and nearer to the pole, but actually arriving 

 at it only after it had circulated round it an infinite 

 number of times. 



It is the length of a portion of such a spiral line that 

 is one of the objects of navigation to calculate ; and it is 

 pretty obvious that rules and formulae, supplied by 

 iphericul trigonometry, can give no aid in such a calcula- 

 tion ; since the latter science has no.thing to do with the 

 spiral curves, which vessels at sea trace out. 



It would seem, however, that these spirals on a sphere, 

 are equally external to the proper objects of plane trigo- 

 nometry, which recognises only straight lines, drawn on 

 a flat surface. But it must be observed that navigators 

 are not interested in any investigations respecting the 

 form or shape of the spiral path of a ship ; but only in 

 its length, and in the angle it makes with the meridians 

 crossed by it. Lengths, and the angles formed by them, 

 are, of course, the proper objects of consideration in plane 

 trigonometry ; and some notion may, therefore, be formed 

 as to how it happens that a ship's course and distance 

 sailed, are matters for investigation by plane, and not by 

 spherical trigonometry. In the latter subject, the form 

 iji the lines concerned cannot possibly be disregarded, 

 any more than their lengths ; it would not be spherical 

 trigonometry unless the lines of which it treats were all 

 of them portions of great circles of the same sphere. 

 Plane trigonometry regards lengths and angles only ; 

 and these alone are all that navigators require to be cal- 

 culated. 



It is obvious, therefore, that, in order to understand 

 the science of navigation, the theory and practice of plane 

 trigonometry as far, at least, as the triangle is con- 

 cerned must be previously acquired ; but modern books 

 on Trigonometry, with but few exceptions, are very defi- 

 cient in this needful preparatory instruction ; so that it 

 becomes almost imperative upon a writer on navigation, 

 who wishes to be clear and intelligible, to conduct his 

 reader through some amount of preliminary matter be- 



fore formally entering upon the special object of his 

 work. We shall, therefore, introduce the subject by an 

 article on the calculation of plane triangles ; referring to 

 the chapters on Trigonometry, in the Section of MATHE- 

 MATICS, for the necessary theoretical principles ; except 

 in a few cases where it may be thought expedient to use 

 rules of operation not expressly provided for in that part 

 of this work : in such cases the theoretical investigation 

 will precede the practical rule. 



Most persons who read this Section, will very likely do 

 so for the sole purpose of becoming acquainted with the 

 general principles of a most valuable part of practical 

 mathematics ; and more especially for learning in what 

 way a few elementary theories, in pure mathematics, are 

 made available for objects of such commercial and na- 

 tional importance, as the conducting a ship across the 

 ocean from one port to another ; the daily registry of its 

 position on the globe ; the determination of its direct 

 distance from any preceding position, however irregular 

 it* actual course between the two positions may have 

 been ; of its distance from the equator at any time, and 

 from the meridian of Greenwich, <tc., <tc. These things 

 are well worth inquiring into even by the non-profes- 

 sional student of science ; as they give a practical value, 

 of the highest kind, to what might otherwise seem to be 

 but barren speculations. 



Nothing can more forcibly illustrate the importance of 

 the theoretical principles developed in the Mathematical 

 Section above referred to, than the application of those 

 principles to Navigation and Nautical Astronomy ; for, 

 aided by the Compass and the Log, and by a simple 

 optical instrument the construction of which those 

 principles have suggested they instruct the mariner to 

 read aright the intelligence of the stars those faithful 

 finger-posts of heaven, which direct him unerringly on 

 his course across a vast and dreary expanse, where no 

 human skill can erect a signal to guide, or a beacon to 

 warn ; and where even the busy traffic of ages has left no 

 beaten pathway behind it. 



Our object is to describe, in simple language, but in a 

 strictly scientific manner, how all this is brought about ; 

 and while we shall explain everything that can be reason- 

 ably desired by the mathematical student, we shall also 

 furnish, to the practical seaman, the reasons on which 

 his rules of calculation are founded, and the principles 

 on which his tables are constructed ; and will thus, we 

 hope, afford a useful accompaniment to whatever book 

 of rules and tables he may choose to consult in the or- 

 dinary exercise of his profession. It is in the highest 

 dt-gree desirable that persons engaged in a calling so re- 

 sponsible, should be enabled to judge for themselves of 

 the soundness and accuracy of the directions which guide 

 them, and not deliver themselves blindly up to authori- 

 tative rules. The amount of mathematical knowledge 

 necessary to convert their art into a science is really very 

 small ; and the satisfaction arising from knowing a thing 

 as an ascertained truth, instead of receiving it upon 

 trust, is alone sufficient to compensate for the time ex- 

 pended on the study of it. Besides, mere printed di- 

 rections can never suffice for unforeseen emergencies ; and 

 in no occupation are these more likely to arise than 

 amidst the hazardous duties of the sailor's life. In such 

 circumstances a little science may often be of more avail 



