I Louis Wedle Go. MAUTICAL msTRHMBSTO, CHARTS & B(H)K& 6 CaUfornla St sail FaiNcisco, calipopjiia UNITED STATES IIYDKOGKAPHIC OFFICE. No. yo: THE DEVELOPMENT OF GREAT CIRCLE SAILING-, BY Gr. ^V. X.-ITTLEHALES, U. S. HYDROGRAPHIC OFFICE. SECONJ> KDITION. WASHINGTON: GOVERNMENT PRINTING OFFICE. 1899. c () isr T 1^] N T s . Page. Prekack to the Fikst P^dition 5 Prei'ack To tiik Skcond Edition 7 Introduction v 9 Section I. — Methods l!eqnirinutation of the Great Circle Course and Distance, and the Latitudes and Longitudes of Points on Great C'ircular Arcs 53 Asnius's Method for the Construction of a Great Circle on the ^lercator Pro- jection 55 Zescevich's Method for Finding the Positions of the Points of the Arc of a Great Circle 59 The Computation of the Latitude at the Middle Longitude 62 List of Literature upon the Subject of Great Circle Sailing 63 3 PREFACE TO THE FIRST EDITION. Tliis i)ablicatiou lias for its object tlie furtlierauce of tlie effort of tlie Bureau of Xavigatiou of the Xavy IJepartment to keep i)ace with the progress of the nautical sciences. It consists of an exposition of giaphical and analytical methods embodying cardinal priuciples relat- ing to the great circle, as applied to navigation, and gives publicity for the first time to several of the most convenient and useful methods yet devised. The actual state of the science of great circle sailing is here pre- sented, so as to give a clear conception of each method, and to furnish references where more extended information can be found. George L. Dyer, Hydrographer. PREFACE TO THE SECOND EDITION. During tha decade that has elapsed since the first edition of tins publication was printed, a fuller recognition of the place of the great circle route in the problem of accelerating ocean transit has stimulated an advance to methods by which great circle courses cau be taken from the Solar Azimuth Tables or measured from the chart compass with very great facility. All the parts of the original work have been retained in this edition, and explanations of the new developments have been incorporated into those sections of the book in which they are appropriate. J. E. Cratg, Captain, U. S. N., Hydrographer. IlYimOGKAlMlIC OKriCK, Bureau of Equii'ment, May 1, 1899. TIIK DEVKLOPMEFr OF (IREAT CIRCLK SAILING. INTRODUCTION. It is not seiicrally recognized that science, employing the mathe- nuitician and the engineer ali lie in the problem of shortening the dura- tion of ocean transit, has acconiplislied as much by causing ships to travel fewer miles as by causing them to travel faster. In the age of steam propulsion the route of minimum distance in ocean trausit is coming- to take the place which was held by the route through the regions of favorable winds in the age of sail i)ropalsion. A knowledge of the i)riuciples of tlie great circle must have been coeval with the knowledge of the spherical form of the earth, but in the early days of ocean navigation great circle sailing was doubly impracticable, for seamen were without the means of tinding longitude and-, moreover, the course of ships was controlled by the wind. Inasmuch as great circle courses alter continuously in proceeding along tlie track, it becomes necessary to know the latitude and longitude of tlie ship in order to determine the course to be followed. At the present day there are convenient means for determining at sea the longitude as well as the latitude with considerable i)recisiou, but in the early ])art of the present century these means did not exist and the principles of great circle sailing could not be applied. Besides increasing the rate of travel, modern motive power, by making possible a departure from the old meteorological routes, has had another and a greater eftect in the progress of the universal policy of civilized nations to accelerate transit from place to place to the utmost possible extent, because, under steam, even if they go no taster, ships may j^et get I'arther toward the poit of destination in a giv^en time since they may be navigated along arcs of great circles of the earth. The increasing rec'ognition among mariners of the sound prin- ciple of conducting a ship along the arc of the great circle Joining the points of departure and destination, and the expanding sense of the advantages to be gained by a knowledge of this branch of nautical science, have greatly heightened the value of methods which place the benefits of the knowledge and use of the great circle trade at the serv- ice of the mariner without the labor of the cahuilations which are nec- essary to find the series of courses to be steered. The general lack of 9 10 the application of tlie principles of the great circle in later times, and even in the jireseut generation, seems to have resulted not from the want of recognizing that the shortest distance between any two places on the earth's surface is the distance along the arc of the great circle passing through them, nor that the great circle course is the only true course and that the courses in Mercator and parallel sailing are cir- cuitous, but to the tedious operations which have been necessary and to the want of concise methods for rendering these benefits readily available. A knowledge of the great circle route is important in working to windward with sailing ships, especially when far from the port of des- tination. A seaman not bearing in mind his great circle course, which is the only one that heads the vessel for her port, may unwittingly sail away from that port by taking the wrong tack. The great circle course to a far distant port may vary three or four points from the rhumb course. For example, on the route from Yokohama (Cape King) to Cape Flattery the great circle course at Cape King is NF., while the rhumb course is E. by N., a diiierence of three points. In this case for a wind directly ahead on the rhumb route an uninformed mariner would lay his vessel on either tack indifferently. If on the port tack, his vessel would head SE. by S., nine points away from the bearing of Cape Flattery. On the starboard tack she would head IST. by E., only three points away. This is perhaps an extreme case, but it may serve to show how important it is that the master of a vessel should know his great circle course even when not distinctly i^ursuing a continuous great circle route. It has been i^ointed out that the rhumb line, although appearing as a straight line on the Mercator chart, and thus giving a false idea that it represents the shortest route, is in reality a roundabout track, and that it is only when a vessel's course is shaped by the great circle pass- ing through the places of departure and destination that she has the shortest possible distance to make and heads for her port as if it were in sight 'throughout the voyage. In the case of the great circle track between Yokohama and Cape Flattery, the vertex, or point of highest latitude reached, is shown to lie within Bering Sea, and the track is consequently obstructed by the Aleutian Islands. It frequently occurs, when laying out an extended great circle route, that the lay of the land, or the extreme of climate, or dangers to navigation, when weighed in connection with the saving of distance that is made good on the great circle, leads the mariner to limit his track by a given parallel of latitude higher than which he decides not to go. Under these conditions the shortest route to follow is made up of an arc of the limiting parallel of latitude and two arcs of great circles which pass, respectively, through the points of depar- ture and destination and whose vertices, or points of highest latitude reached, lie on the limiting parallel. In such a case the vessel's course is laid from the i)oint of departure along the first great circle arc until its highest latitude, which is the latitude of the limiting parallel, is 11 reached; thence along the limiting parallel to the point of the highest latitude of the second great circle; and finally along the second great circle to the point of destination. tSuch a route is called composite. With the exception of the miscellaneous methods of Airy, Chau- venet, Harris, Fisher, Sigsbee, and Proctor, the development of grai)liical methods in this branch of nautical science has proceeded in two distinct lines-, the history of one is the history of the development of the principles of the gnomonic projection, and that of the other is an account of the various devices wliich have been contrived to find the vertex of a particular great circle. SECTION I METHODS EEQUIRIFG A KNOWLEDGE OF THE VERTEX. By a system of great circles is meant all great circles whose common diameter is a diameter of the equator. If we consider such a system of circles to be drawn upon the surface of a sphere, and imagine tbe com- mon diameter to revolve in the plane of the equator, all possible great circles Avill be described, for we have at first drawn all the great circles of one system and then revolved this system into all possible positions. From these considerations it is obvious that a distinguishing feature of every great circle is its inclination to the plane of the equator, or the latitude of its vertex. All properties of spherical courses, latitudes, and longitudes and distances from the vertex of a certain great circle of one system are identical in the corresponding great circle of every other system. Therefore, by tabulating these properties for each great circle of one system, tables of the properties of all great circles are formed. towson's method. In 1850 Mr. John T. Towson, examiner to the Mercantile Marine Board at Liverpool, proposed a method for facilitating great circle sailing, con- sisting in the use of tables, such as described above, and a diagram, by means of which the whole operation of finding the successive spherical courses and distances on any great circle is reduced almost to an affair of inspection. These tables, as published by the British Admiralty, give the latitudes, spherical courses, and distances on great circles of the earth corresponding to each degree of longitude reckoned from the meridian of the great circle's vertex. Mr. Towson's diagram, the whole object of which is to find the vertex of the particular great circle which passes through any two places, is constructed as follows: -p -P a I, Let A and B represent two points on the surface of the globe, one being the place of deiiarture and the other that of destination. Let P represent the pole, PA and PB the meridians passing through A and B. Let a great circle pass through A and B. The points a, v, h are on the equator. From P draw PV perpendicular to the great circle, then Y will be the vertex. In the spherical triangle APV, tan(90-Lv) cot L^ ,,> cos APv =. ,n,\ T N or ^ ■ T- (1) tan (00 — L|) cot ui ^ ' 12 13 111 the spl)eri»-al triiiiijulo I>PV, cos BPV = . j" (2) cot L ^ ' cot Lv U These two eciuations remain the same, whetlier the perpeiulicular falls within or withont the triangle, for all possible values of the parts named. In either eiiuation, any two of the terms being given, the third becomes known; and from the similarity of (1) and (2) it is obvious that each of these equations comi)uted for all possible values of Lv and Li or L^ will give the same series of results for the arc APV or BP\', and embraces all the values that the arc can have. The successive values of this arc were computed for all the integral values of Lv and Li from 1° to 89°, inclusive. The results form the dis- tance column of Towson's tables. They were also projected as ordiiiates to the axes named Meridians of Vertex in the linear index, which is appended, and the curves were drawn through the extremities of these ordiiiates. Supjiose the vertex of the great circle passing through two places, A and ii, is to be found; with a pair of dividers take out their differ- ence of longitude from the scale of differences of longitude, and, if the latitudes of A and B have the same name, set one point of the dividers in the division of the diagram nearest to the right hand on that curve whose number corresponds to the latitude of either A or B, whichever is the nearer to theecjuator, and keeping the line joining the points of the dividers horizontal, follow the curve up or down till the other point meets the curve corresponding to the latitude of the other place. The index linepassing through the points of the dividers in this position will indicate the latitude of the vertex. If that point of the dividers which stands on the curve corresponding to the latitude of departure be kept fixed, and the other be moved in a horizontal direction until the nearest meridian of vertex is reached, the dividers, being applied to the scale, will indicate the longitude of the point of departure from the vertex. With the latitude of the vertex and longitude from the vertex, thus found, the tables are entered and the distance from the vertex and the course are picked out. If the latitudes of A and B are of different names, the vertical line named Equator in the linear index must be kept between the points of the dividers in finding the latitude of the vertex, but the rest of the operation remains the same as before described. DEICHMAN'S METHOD. In 1857 Mr. A. H. Deicuman, in full knowledge of and with a view to improving what had already been done by Towson, devised a dia- gram called a Scale of Great Circles to be used in connection with a table for finding great circle courses and distances. The table is the same as Towson's, except that the longitudes and distances are reckoned from the intersection of the great circle with the equator, instead of the vertex. The Scale of Great Circles is a device for finding the vertex of any great circle, which is all that is necessary, for when the vertex of a great circle is known all courses and distances on that great circle are virtually known. Deichman's diagram or Scale of Great Circles con- 14 sists of a Mercator projection, of such scale as to make tbe representa- tion of oiie-fourtli of the earth's surface of convenient size, with a series of great circles projected upon it. His method of finding the vertex consists in cutting out of paper or card-board, on the same scale as that of the Scale of Great Circles, an " Index Model," whose two upper edges shall represent the positionsof the two places through which the great circle jiasses in their approxi- mate latitudes and longitudes, and sliding this (see figure) over the Scale of Great Circles until both the upper edges shall lie on the same great circle. The latitude of the vertex is then read off from the Scale of Great Circles. 70° 6(f Kit 70° y ^ ^ \ X 60° 50° 40° 30° 20° 10° / ^ ^ b < \ a/t /. y- ^ ^ ^-" ^ ^ \ \ V 3(f 20" 10° A / X ^ -^ ■^ ^^ N^ x\ // ;;> ^ ^ ^ ~ ^^ H '^ \ \\ b ^ ^ -^ ■ — ' ^ :s y^l,^ ■— — — 180° 90' 180° brevoort's method. Mr. J. Carson Brevoort submitted to the Hydrographic OfiBce, for an opinion, in July, 1887, a method for facilitating great circle sailing which closely resembles the methods of Towson and Deichman. His tables are the same as Towson's, except that the column of distances is omitted and a column of compass courses, reading to one-eighth of a 90° 80' 70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° SO' 60° 70° S0° 90_ point, is added. The index diagram is a Mercator projection, laid down ou transparent material, with a system of great circles projected 15 upon it, like Beiclinian's Scale of Great Circles. This is designed to slide over a Mercutor chart on the same scale as that of the index diagram, with the equators of the diagram and the chart in coincidence, until the two places between wliicli tlie great circle track passes lie on the same great circle of the diagram. Jn the figure the lines rei)resented by dashes are those which occur on the transparent diagram. It will thus be seen that, while Deichman's method causes any part of the earth to revolve over one system of great circles, Brevoort's causes one system of great circles to revolve over any part of the earth. BERGEN'S METHOD. In 1857 W. C. Bergen, master in the British mercantile marine, pub- lished a set of spherical tables of general adaptability to the i)roblems of nauti(!al astronomy, and a diagram, by the aid of which the results ill the tables are rendered ap]>licable to the problem of great circle sailing. Although this method can not be used with the same degree of facil- ity as some of the others of the same character, it is well worth atten- tion on account of the radical principles involved. The diagram, which is appended, represents a quadrant of the earth's surface with the parallels of latitude and the meridians of longitude from degree to degree. Each fifth parallel and fifth meridian is num- bered and drawn stronger than the others, and in order further to distin- guish thein the alternate parallels between them are dotted. Over this net- work of parallels and meridians a system of great circles is drawn. Under the equator the degrees from the meridian of the vertex are numbered, thus showing the longitude from the vertex, and the half differ- ence of longitude; and under this again these numbers are doubled, indicating the difference of longitude; and, to obviate the necessity of taking the supplement of the difference of longitude, in the case of the places being on opposite sides of the equator, the third line of numbers is given, containing the difference of longitude reckoned from the point where the great circle crosses the eqnator. The object of the diagram is to find the latitude of the vertex of the particular great circle which passes through the points of departure and destination, and the longitude of the point of departure from the meridian passing through the vertex. In the figure, let C, the center of the primitive PELR, be the projec- tion of a point on the earth's equator, PL that of a meridian i^assing through the point 0, and ER that of the equator. Let P represent the nearer pole, and Vmh and Pnh meridian circles equally inclined to the primitive. Then, it is evident that PL bisects the angle mPn, and therefore viFn is double the angle CPw. Again, let a great circle EAR be drawn through EU, cutting Pwj, PL, and Vn in points y, A, z. And through yz let the parallel of latitude ^y^i'^o be drawn. 16 Now, if the quadraut PCE be couceived to turn around the radius PC, wliicb remains fixed, and to be placed on the quadrant PCK, the projections of the one will fall upon the corresponding projections of the other, and will coincide with them; Pm will therefore coincide with Pn, AE with AE, p.v with ox, and the point ij with the point s. A is the vertex of the great circle EAR, PL is the meridian of the vertex, and CP» is the longitude of the point z from the vertex. Hence, when two places are given on the same parallel of latitude, to find the latitude of the vertex and the longitude from the vertex, let Cw be equal to half the difference of longitude, and ox represent part of the parallel of latitude passing through the places ; with a pointed in- strument trace up the meridian P??, and with another trace along the parallel ox until the curves meet in the point z, then trace the great circle K;jAE, passing through z, until it meets the meridian of vertex in the point A. Then CA is the latitude of the vertex, and Cn is the lon- gitude from the vertex. Again, if the places are on the same side of the equator and ou differ- ent parallels of latitude, ox and rq, let Pn, Ps, and P^ represent mericV- ians equidistant from one another, and let Cs be equal to half the dif- ference of longitude. With the right hand trace up the meridian I's until it cuts the lower parallel of latitude, rq, then, with the left hand, trace it up again until it meets the higher parallel of latitude ox, then trace the lower latitude to the right hand and the higher to the left, being careful to move an equal number of degrees on each side of the meridian P.s, until the curve of the great circle passes through both points. Let these points be represented by z and u, and let livzA rep- resent the great circle, then AC will represent the latitude of the ver- tex, and C?i and (M the longitudes of the points z aud u from the ver- tex. It is evident that \( y be the place of the ship and u that of her des- tination, or conversely, the meridian of vertex falls between them, ana' 17 'POLE t) c 17 that win added to nt is the diflEereuce of longitude; but C?t is equal to hiilt' o( inn, and therefore twice Cn added to nt is equal to the difference of longitude. IS^ow, ns is half of nt, and when twice C.s is considered as the difference of longitude, when the hand is moved to the left from s to n, twice ns is taken from the difference of longitude, but the other hand being moved from s to f, they are distant from each other nt, which is equal to twice ns, and 7it being added to twice Cn, the difference of longitude is the same as at first. If the meridian of the vertex fall outside of the portion of the great circle track which lies between the points of departure and destination, as in the case in which z and u are taken to represent those points, let CH be equal to ns, which is equal to the half difference of longitude between z and n, and let Pfl be traced uj) as before. In this case the left hand will meet the meridian of vertex without meeting the great circle, and the right hand must be moved until it is at a distance from the meridian of the vertex equal to the difference of longitude. Both hands must then be moved equally until they arrive at the points z and u where the great circle passes through them. For the explanation of the case in which the points of departure and des- tination are on opposite sides of the equator, imagine the opposite pole of the primitive to be taken as the projecting i)oiut, and equal circles to be projected upon the lower right hand quadrant LCR, and con- ceive it to be turned upon the radius CR as an axis until it shall coin- cide with the quadrant PCR. The projections of one will then coincide with the corresponding projections of the other ; and if the places are in equal latitudes, as z and ^', then hR will be half the difference of long- itude, and'C« will be half its supplement. Therefore by taking the sup- plement of the difference of longitude and proceeding exactly as in the other cases the latitude of the vertex AC or A'C and the longitude of the vertex Cn or C^ will be found. The latitude of the vertex and the longitude from the vertex having been found from the diagram by inspection, a table, of which the fol- lowing is a specimen i)age heading, is entered, and the course and the distance from the vertex are picked out. Latitude of vertex, 19°. Longittido from vertex. Distance from vertex. Course. 1° 0° ",' 0O19' In the actual construction of the tables the author greatly abridged his work by observing the trigonometric principle that, if the comple- ments of the hypotenuse and base of a right-angled spherical tri- 19862 2 18 angle be taken as the base and hypotenuse of another to the same angle, the perpendicular of each triangle is equal to the complement of the arc which measures the remaining angle of the other triangle. Let the figure represent a projection on the plane of the primitive. Let P and P' be the poles, EQ the equator, EVQV any great circle, whose northern and sonthern vertices are at V and V, respectively, and PLMP' the meridian passing through the i»oint L on the great circle EVQV, at which the course and distance are required. The angle LQM, between the plane of tlje equator and the plane of the great circle, is numerically equal to the arc VO, or the latitude of the vertex V. In the triangle LMQ, the side LQ is the complement of the distance of the point L from the vertex, the side MQ is the com- plement of the longitude of L from the vertex, the sideLM is a measure of the latitude of L, the angle MLQ represents the course, and the angle LQM the latitude of the vertex. In the construction of the tables, the triangle LMQ was computed to every degree of the angle Q, and also of the arc LQ, until LV was equal to or next less than MQ. Then the com- plements of LQ and MQ were taken, and the trigonometric principle already mentioned was applied. With regard to the degree of dependence to be placed upon this method, Mr. Bergen remarks, in his writings on the subject, that, by means of an extensive induction, he arrived at the conclusion that for a diflerence of longitude equal to about 15^, the course may be de^ pended upon generally to less than one-eighth i)oint; in some instances, ■when the diflerence of longitude is small and the latitude high, the course may be erroneous nearly one-quarter of a point ; but this will not happen with a difference of longitude more than 15° and a latitude less than 600. 19 THE TERRESTRIAL GLOBE. The Strict practical solution of the problem of great circle sailing, which consists in using a terrestrial globe, is of ecpuil convenience and of at least equal accuracy with the foregoing niethotl. A great circle passing through any two places upon the earth's surface may be traced upon a terrestrial globe by elevating or depressing the i)olar axis and, at the same time, turning the globe until the places coincide with the upper edge of the wooden horizon. In this position of the globe the upper edge of the wooden hori/on represents the great circle, and can be used as a ruler for tracing it upon the surface of the globe. Having thus traced upon the globe the great circle passing through the two given jtlaces, their distance apart is shown by tlio number of degrees intercepted between them on the Avooden horizon, the latitude and longitude of the vertex can be read oft', and the various places tlirough which the great circle passes are shown. It will thus be seen that, instead of tinding the latitude of the vertex and longitude from the vertex by the diagrams which have been de- vised by Towson, Deichman, Bergen, and Brevoort, a terrestrial globe can be used for finding these elements with quite as nnich efficiency. After which the spherical tables, which have been arranged by any of these authors can be used for finding the courses and distances. Or the latitudes and longitudes of a sufiicient number of points or places on the track of a great circle, thus traced upon a terrestrial globe, n)ay be read oif and the corresponding positions plotted on a Mercator sailing chart. Then, connecting the points by drawing a curved line through them, a graphical representation of a great circle upon the chart is obtained without any computation. THE DIRECT TRACK SCALE. This was recently designed by Mr. Gustave Herrle, of the U. S. Hy- drogTai)hic OfiBce, as an appendage to the Mercator chart, with a view of making the chart more directly available in pursuing a great-circle track. It consists of a gnomonic projection, with the point of contact on the equator, contracted in longitude. From this the position of any point of a great-circle track can be read oft" and transferred to the Mercator chart. There are also logarithmic scales of sines and cosines for the measurement of courses and distances. The gnomonic projection, with the point of contact on the equator, is that which is used by Lieutenant Ililleret, French navy, whose great circle sailing charts will be referred to. Let r represent the radius of the sphere; f^ represent the longitude of any meridian reckoned from the meridian passing through the point of contact, or, as it is gen- erally called, the middle meridian; cp represent the latitude of any parallel. Then the successive distances of the parallel straight lines of the pro- 20 jection, which represent the meridians, from the middle meridian, will be r . tan 6; and the points at which the successive parallels of latitude cut the middle meridian will be r . tan (p, measured from the equator. The distance above the equator at which any i^arallel whose latitude is (p cuts a meridian whose longitude from the middle meridian is 6 will be r . tan cp . sec 6. By the application of the preceding expressions the gnomonic equa- torial projection of the Northern Hemisphere up to latitude G0°, shown in the above figure, has been constructed. In order to economize area and yet preserve all the properties of the projection in relation to direct tracks the first projection is projected orthogra|)hicany upon a second plane, which j)ayses through a me- ridiiin of the first plane of projection aud makes with it any angle, a. The degree of contraction will then be directly proportional to the cosine of the angle a, so that the contracted projection could be con- structed at once by inserting in the expression for distances of the suc- cessive meridians from the middle meridian the cosine of the angle of inclination of the two i)rojecting planes. This expression would there- fore be r . tan 6 . cos a. The expression for constructing the parallels of latitude will remain the same, i. e., r . tan ^ sec 0. THE MEASUREMENT OF COURSES AND DISTANCES ON THE DIRECT TRACK SCALE. In the figure, let AB represent any great circle of the earth, A the place of departure or the ship's position at any time, and B the place of destination. Let C represent the course at A, Y represent the vertex of the great circle AVB, d represent the distance between A and B, Lg represent the latitude of A, Lv represent the latitude of V, Lp represent the latitude of B, A, represent the difference of longitude between A and B. COURSE a DISTANCE SCALES. dJnnjJTTbiffliiirWi.iffliJtm^y^ji'liffl^la atsq ^ , Distance Scale; r 21 From the right-angled spherical triangle APV, siu C : sin 90° :: cos Lv : cos Lg sin C=«£iilr cos Lg log sin C=log cos L,.— log cos Lg (1) From the spherical triangle APB, sin C : sin A :: cos L,, : sin d log sin (?=log sin A+log cos L^— log sin C. (2) In order to solve the logarithmic equations (1) and (li) graphically, and thns find the coarse at any point and the distance from that point to the place of destination without computation, a length is adopted corresponding to a certain number of units of the mantissa of a log sin or log cos, and a linear scale is thus constructed giving the necessary angles. The Direct Track Scale and its accompanying logarithmic scales are here shown reduced to one-half the size which it is proposed to use in practice. The following rules for using this method are given by Mr. Herrle: On the meridian of the diagram marked 0", plot the latitude of the place of departure, and on the meridian whose longitude from the me- ridian marked 0°, is equal to the difference of longitude between the places of departure and destination, plot the latitude of the place of destination, connect the two points thus plotted by a straight line and find by inspection on the diagram the highest latitude which it reaches. This latitude will be L,., the latitude of the vertex of the great circle which passes through the places of departure and destination. Find Lg and L^ by the lower readings on the logarithmic scale ; take their difference and lay it off irom the right end of the scale toward the left. The upper reading of the point laid off corresponds to C. Courses greater than 28° may be measured on either scale, courses less than 28° on the lower scale only. For the measuremeut of distances, find C at L^ and A on the upper readings and L,, on the lower readings ; take the difference between C and A and lay it off from Lp toward the right or left, according as the place of destination is to the right or left of the ship's position. The upper reading of tlie point laid oft' corresponds to d, the distance. Distances less than 28° or 1,G80 nautical miles can be measured on the lower scale only. SECTION II. METHODS J)EPENDmG UPON THE (INOMONIC CHART. The object of cLiirts is to exhibit, by suitable representation, on a reduced scale and on a plane surface, the relative positions of points, lines, or objects on the earth's surface; and since such positions are usually defined by spherical coordinates, the primary object of the pro- iection upon which the chart is based is the delineation of these circles of reference according to certain assumed or fixed geometrical laws. Any point, line, or object intended for representation may then be laid down by means of its known coordinates, and, conversely, the coordi- nates of any plotted point may be ascertained. The gnomonic chart is based upon a system of projection in which the jilaue of projection is tangent to tlie surface of the sphere, and the eye of the spectator is supposed to be situated at the center of the sphere, where, being at once in the plane of every great circle, it will see tliese circles i^rojected as straight lines where the visual rays, passing through them, intersect the plane of projection. godfray's cinomonic chart. In 1858 Hugh Godfray, M. A., Fellow of the Cambridge Thil. Soc, prepared, for the jjurpose of great circle sailing, two gnomonic charts covering the greater part of the world — one for the Northern and one for the Southern Hemisphere. He used as points of contact the geo- graphical poles of the earth. The i)arallels of latitude are tlierefore represented by a series of con- centric circles wiiose radii are equal to r tan (90° — hit.) or r cot lat., where r is any convenient linear magnitude, and represents the radius of the sphere. The meridians are straight lines drawn from the com- mon center or pole, dividing each circumference into three hundred and sixty (Hjual parts. Any one of these being selected for the prime meridian, the coast lines of different countries may then be traced in the usual manner by means of the longitudes from the i^rime meridian 22 GREAT CIHCLE SAIUNd lOITnSKSSDISTAMES 23 and the latitudes of the diiferent i)oints. Mr. Godfray also devised means tending toward convenience for the measurement of courses and distances on the grc^at circle track. His diagram fur tliis jmrimse, con- structed upon a separate slieet, measuring 1") by 19 inches, consists of a series of concentric curves corresponding to the ])ara]lels of latitude, bounded by a vertical line, AC, and a horizontal line, AB. The hori- zontal line is divided into seventy equal parts, representing the degrees of latitude from 20° to 90^. The vertical line AC is a scale of distances from the highest lati- tude and is divided into twenty-one equal parts, each representing 200 nautical miles. Through the various points of division of these scales are drawn horizontal and vertical straight lines over tlie whole diagram. The concentric curves corresponding to the parallels of latitude are traced from the following considerations: Let ABC represent any great circle, B represent its vertex, P represent the pole, Lv represent the latitude of the vertex, Li represent any lower latitude on the great circle, <1 represent the distance from the vertex, we have from the right angled spherical triangle L'BC, by Napier's analogies : cos (90^^— L,)=cos d cos (90°— L,.) cos ole is least a(la})ted. No complete gnomonic chart of the Indian Ocean could be made on this i)lan, no 27 means could be devised by which the advantajies of the great circle route could be apijliel in passing from one i)olar hemisphere to the otlier. liiasmiK-li as the radius with whicli each parallel is desiMibed iu the gnomonic projection with the point of contact at the pole is e([ual to the tangent of the co-latitude of that parallel, and as the co latitude of the e(|uator is 90°, the radius with whi(;h the e<{uator would be described is tan O0°=cc . Tlie delineation of tlie ecpiator and of regions near the equator is therefore impossible when the point of contact is the pole. hekrle's method. Within the present century many attemi)ts have been made so to devise tiie gnomonic i)rojection as to treat conveniently the problem of great circle sailing. V^arious successful applications were made from time to time, as seen in the charts of Mr. (xodfray and Lieutenant llil- leret. No one suc('eeded, however, in making the projection itself an embodiment of the means for measuring both courses and distances until Mr. (Justave Herrle designed his great circle compass and sug- gested the application of the perpendi(;ular distance from the point of contact to the great circle track as an argument in measuring distances on the chart itself. The princiides of the Mercator projection are such that great circles on the spliere will not generally api)ear as straiglit lines on the chart, but any straight line on the chart, excepting a meridian of longitude or a parallel of latitude, represents a rhumb-liue and indicates a partic- ular mutual bearing of two places so connected. On the sphere such a line is known as a loxodromic curve, and it jiossesses the property of cutting the meridians at equal angles, so that, iu pursuing a straight line on the Mercator chart, the course is constant, though the route is really circuitous. The simplicity of the methods necessary for navigat- ing this circuitous track and the long duration of its usage have so intrenched it in tlie estimation of mariners that no method of handling charts not analogous to these has found favor with them. Another essential consideration in the construction of charts for great circle sailing is a method that afllbrds facilities for measuring the course and distance from the actual i)Osition of the vessel indei)endent]y of any great circle track that may have been previously laid down. Just as the rhumb course and distance are measured on the Mercator chart from the actual ]>ositioii in which the vessel is found to be. Both of these principles are recognized by Mr. Herrle in the construction of his great circle sailing charts. With a notable degree of i)ublic spirit he gave to the Hydrographic Ottice his plans, as they existed in 1881, and he has contributed largely to the subseic nuiy vary from 0" to 90", a great number of such nets would be lecpiired to meet every case. A method of combining them was therefore devised, which has resulted in the production of the great circle (compasses or course indicators which appear on the great circle sailing charts of the Ilydro- grai)hic Office. This consists of a series of eijuidistant concentric cir- cumferences, which, in the gnomonic chart of the Indian Ocean, has for its center the i)oint of contact. I'^ach circumference is divided into 3(>() degrees, in such a manner that if each division of any circumfer- ence, as for example the one marked latitude 36°, were connected by radii with the common center, the comidete radiation of the meridians from degree to degree would be shown for a gnomonic projection having its i)oint of contact in latitude (90'^— 30°) or 54°. The circumferences are marked in the direction of tho radius from 3 to 3 degrees for values of i/', and by reading from the chart the latitude of a the circumference which is to be used in the measurement of the angle )=cot C sin A + cos (90O — 9^,) cos A (1) or cos qjx tan ^2= cot C sin A + sin q)\ cos A (2) whence cot q^c os 9^ tan ^2-sin y, cos A sin A Dividing both terms of the fraction of the second member by cos <^,, cot C— ^^° (^2— t an ^1 cos A ^. sec (p I sin A , t^ tan (B2 — tancoiCosA ,r\ or cot 0= /Tf . — T~-^ — (^) sec q)i sm A — Putting 2/= tan cpx cos A ^ y'=Uucp2 I ,Q. ir=sec<^i sin A j ^ '' x'=o J equation (5) becomes cotO=-^"~^' (7) x — o Therefore G is the angle made with the axis of Y by a straight line joining the points {,v, y) and (o, y'). From equation (G) . - = sec r« 1 and - ,= tan 9 25 64 19 50 34 Opposite .-'5 05 66 25 09 36 38 name. 50 36 46 00 68 70 ' Same iiaiue. > 30 30 35 52 40 41 18 72 41 14 42 36 31 74 46 37 44 31 38 76 52 01 46 26 42 78 57 25 48 21 42 80 62 51 50 16 39 1 34 35 The above table is taken from '' Notes on Navigation aud the Deter- mination of Meridian Distances," by Commander P. F. Ilarrington, U. S. Navy. The center of the arc to be described frequently falls beyond the limits of the chart which may be employed, and the track may occupy more than one chart. In these cases diflficnlties arise in drawing the re- quired arc. It is of value to project a great circle track to which it is intended to adhere throughout a passage, but if a vessel be diverted from the pro- jected great circle a new track must be laid down with the attendant inconveniences. Great circle distances can not be measured closely on the approximate arc. fisher's method for circular arc sailing. Besides the foregoing there is another method for describing circular arcs as substitutes for actual great circle tracks on sailing charts. This method was proposed by the Rev. George Fisher, M. A., F. R. S., and described in the article on circular arc sailing in Riddle's Naviga- tion, published in 18G4. Unfortunately it fiuds its most extensive and easy application in those cases in which the vertex of the gr«at circle is in so high a latitude as to make the navigation of it dangerous. In the tropics the method is impracticable, because great circular arcs are there represented on sailing charts nearly as straight lines ; and in the region between 40° and 00° of latitude the curvature of the great circular arcs on sailing charts is so small that it is ditficult to describe substitutes for them on account of the length of the radius required. The curve PRQAVB shows the direction of the great circle which passes through the points A and B, which represent the Capo of Good Hope and the south part of Van Diemen's Land. Although in sailing 36 from one place to another the navigator is only coucernetl with that portion of the great circle upon which he means to travel, the curve is nevertheless continued through both hemispheres in the figure, in order that its general form may be the better comprehended. The northern and southern portions of this curve are equal and similar to each other; and the curve cuts the equator at two points, P and Q, at a distance of 180° of longitude from each other, an ; the bisecting perpendicular to Aa passes through C; then the bisecting perpendiculars to A;;, pB give us, by theii- intersections with those to Aa and B6, D and F at once.) What we want is to determine the angle Aj>B, between the arcs A/>, /^B ; and it is obvious that this is the supplement of the angle V>p^, which is easily 19862 4 50 measured with a protractor. The number of degrees, multiplied hy 60, gives the number of geographical miles or hnots in the distance AVB. An example of these methods is given in the stereographic chart, which has been reduced from Proctor's great circle sailing chart for the Soutlieru Hemisphere. In this chart the same letters are used as in Figs. 2 and 3. * SPHERICAL TRAVERSE TABLES. Under the methods of Bergen and Towson the general computation of the parts of spherical triangles and the arrangement of the results in the form of spherical tables were treated of. In liaper's excellent work on navigation spherical traverse tables are given. Tliey are adapted to great circle sailing in the same manner as the ordinary traverse tables are adapted to sailing on a rhumb line. Practical rules covering all cases are there given for the guidance of navigators. GREAT CIRCLE COURSES FROM THE SOLAR AZIMUTH TABLES. At page 8 of a treatise on Azimuth ^ by Lieut, Commander Joseph Edgar Craig, U. S. Kavy, the following equation is stated for the solu- tion of the time-azimuth problem : , rr COS Li tan d— sin Li cos t ,-. . cot n = -. — ; (1) sin < from which are derived tan ^ = co8 /. cot d (2) and cot Z = cot t. cos (^-f Li) . cosec (/> (3) in which /, d, and Z represent respectively the hour-angle, declination, and azimuth of the observed celestial body, and Li the geographical latitude of the observer. At page 54 of the present work on the Development of Great Circle Sailing, the equations stated for finding the great circle course are: tan (p = cos A cot Lj (4) and cot C = cot A cos (L, + ^) cosec (/> (5) in which Li represents the latitude of the ship, L^ the latitude of the l)lace of destination, A the difference of longitude between the meridian of the ship and the meridian of the place of destination, and C the course. If, in equations (2) and (3), A be substituted for t and L^ for d, their second members will be identical with those of ecpiations (4) and (5), and therefore, when the difference of longitude and the latitude of the place of destination in the great-circle problem are equal respec- tively to the hour-angle and declination in the time-azimuth ])roblem; the course C resulting from equation (5) will be identical with the azimuth Z resulting from equation (3), Now the values of Z resulting ' Azimuth. A Treatise on this Subject, with .i study of the Astronomical Triangle, and of the Effect of Errors in the Dat.a. Ilhistrated by Loci of Maxininra and Mini- mum Errors. By .Joseph Edgar Craig, Lieutenant-{;oinniandcr, U. S. Navy. New York : John Wiley & Sous, 1887. Chart for great circle sailing, sho the grea Face page 50. 51 from equation (3) have been computed for declinations between 23° N. and 23° S., which represent the range of declinations of the sun, and arranged, for stated values of the hour-angle, declination, and latitude, in the Solar Azimuth Tables, whicli have been well known among marijiers for many years. It is therefore evident that, when the i)ort of destination is situated within the Tropics, the Solar Azimuth Tables may be used for ascertaining great circle courses by simply regarding the latitude of the port hound to as declination, and the difierence of longitude, converted into time, as the hour-angle. The latitude of the shi}) remains the latitude of the observer as in taking out values of the azimuth from the tables. The identity of the time-azimuth problem and the great-circle course problem may also be graphically illustrated. j-i^j. Fiff.2. Let Fig. 1 represent the astronomical triangle, I* ~ X, projected on the plane of the celestial meridian of the observer, P^ P' z' ^ and let P represent the elevated pole, ~ the zenith of the observer, whose latitude is Li, and X the position of the observed celestial body, whose declina- tion is fZ, hour-angle #, and azimuth Z ; and let Fig. 2 represent a pro jection, on the plane of the terrestrial meridian, P^ P' ~', passing through the point of departure, of the sidierical triangle P ^ X, whose ver- tices are the place of departure ~, the place of destination X, and the elevated i)ole P, and whose sides are arcs of the meridian of the place of departure, of the meridian of the place of destination, and of the great circle i^assing through the places of departure and destina- tion. Then it will be apparent that if the values of L, are identical in the two figures, and if A, the difference of longitude is equal to #, the hour angle, and L.,, the latitude of the place of destination, is equal to d^ the declination, the great-circle course C must be identical with the azimuth, Z. Captain Craig, the author above mentioned, and others of the fore- most navigators of the Tnited States Navy, had noticed the ready adaptability of the Solar Azimuth Tables to the finding of great-circle courses, and had for years made use of the knowledge in practical 52 navigatiou ; but no formal disclosure of the method appears to have been made uutil refereuce was made to it in the ninth edition of Lecky's work on navigation, entitled Wrinkles in Practical Naviga- tion. It was the impression of this author, as the Azimuth Tables extend only to 23 degrees of declination, that this method would only be aiiplicable where the latitude of the place of destination is within 23 degrees of the equator, or within the Tropics. It will be valuable, therefore, to point out to navigators that the Solar Azimuth Tables are universally applicable for finding great-circle courses with very great facility, because all great circles pass into the Tropics; and, if the problem of finding the courses is with reference to a great-circle track between a point of departure and a point of destination, both lying outside of the Tropics, it is only necessary to find a iioint lying on the prolongation of the great-circle arc beyond the point of actual destination and within the Tropics, and treat this point as the place of destination in finding the courses from the Azimuth Tables.* To illustrate, take the problem of finding the initial course on a voy- age from Bergen, in latitude 60"^ N. and longitude 5° E., and the Strait of Belle Isle, in latitude 52° 12' N. and longitude 55° W. On a copy of a gnomonic chart, such as Godfray's, which accompanies this work, draw a straight line between the geographical positions above stated and extend it beyond the latter into the Tropics. It will be found to intersect the twentieth degree parallel of latitude in longitude 90° W., or 9.5° from the meridian of the point of departure. Entering the Azi- muth Tables at latitude 60°, under declination 20° and opposite hour- angle 95° or 6'^ 20'", we find the required course to be N. 75° 31' W. *See "The ' Ex-Meridian ' treated as a problem in Dynamics," by H. B. Goodwin, M. A., formerly examiner in nautical astronomy at the Royal Naval College, Green- wich, England. London : George Philip &. Won, 32 Fleet street, E.G., 1894. i SECTION IV. METHODS REQUIRING COMPUTATIJN. This section is devoted to the exposition of those methods for find- ing great circle courses and distances, and the latitudes and longitudes of points on great circular arcs which require computation. THE COMPUTATION OF GREAT CIRCLE DISTANCES. A'^ ^ r^ Let AB represent the arc of a great circle passing through the two points A and B, whose difference of longitude is A and whose latitudes are (p\ and tp-i , respectively. And let d be the distance between A and B, measured along the arc AB. From the fundamental equations ot spherical trigonometry we have, in the triangle ABP, cos d=cos (90O— i) sin (90O— ^2) cos A cos (Z=sin q)x sin (7>2+cos q)\ cos (pi cos A =sin cpi sin cpi (1+cot cp^ cot cp-i cos A) THE COMPUTATION OF GREAT CIRCLE COURSE AND DISTANCE, AND THE LATITUDES AND LONGITUDES OF POINTS ON GREAT CIRCU- LAR ARCS. Fig. 1. Let WE represent the equator, A and B the gix^en points, PA and PB the meridians passing through A and B, respectively, and 53 54 AB the required arc. Let BF represent the great circle drawn from one of the given points perpendicular to the meridian passing through the other. Let A =A.PB denote the difference of longitude between A and B. Let Li represent the latitude of A, L2 the latitude of B, qj the arc PP. Then AF will equal 90° — (L, + 97), and we have — tan cp = cos A cot L2 ' (1) cot C = cot A cos (Li4-^) cosec qt (2) cot f? = cos O tan (Li + <^) (3) in which C represents the course from A; but, should the course from B, be desired iutercbange Li and L^ in each of the above formultB. In drawing the diagram, let fall the perpendicular upon the meridian passing through that point from which the course is desired. The signs of the functions must be carefully noted. That branch of the great circle which corresponds to A < 180° is sought. cp may be taken in the first or second quadrant, according to the sign of its tangent; but it will be found convenient to restrict it to a value less than 90°, marking it positive or negative, according to the sign of its tangent. The latitude of one of the points being regarded as positive that of the other point, when of the opposite name, must be marked negative. Thus, in Pig. 2, PB is numerically 90°+ L2. To find the position of the vertex, iu the triangle PAC*> (Fig. 1), C" being the vertex, cos Lv=siu C cos L,, (4) cot Ay =tan C sin Li, (5) If the angles PAB and PBA are both less than 90°, the vertex will be between the points A and B ; but if the course from one given point 55 is less than QO'^ and that from the other greater than 90°, the vertex will be upon the arc AB extended. The latitude of the vertex sought will be of the same name as that of the given point, which is nearer to a Xiole than the other point. To find other points of the curve, as C and 0", assume differences of longitude from ths vertex, usually at intervals of 5*^ or 10°; Ihen tan L'=tan L^ cos A/ tan L" = tan L„cos >s A/ ) s X" \ (G) Each of these forrauhe will determine two points of the curve, symmet- rically situated on opposite sides of the vertex; bat only one of these will be Uiied when the vertex falls outside of the required arc. Likewise successive values of the latitudes may be assumed, and the corresponding differences of longitude found by the formuhe : cos A'= cot L, cos A"= cot L, tan L' tan L ;.! {-: ASMUS'S METHOD FOR THE CONSTRUCTION OF A GREAT CIRCLE ON THE MERCATOR PROJECTION. Fig. 1. Let ABMN (Fig. 1) represent the equator, C the center, k.C=(i the radius, and D the pole of the earth. Let AD represent the prime meridian, as for instance the meridian of Greenwicli ; MI), any other meridian; the angle ACM=A, the longitude; the angle MCP=^, the latitude of the point P. Let BP represent a great circle cutting the equator at an angle MBP = ;)/, and let the point of intersection B be determined by its longi- tude, /^, the angle ACB. Suppose, now, that BP takes the infinitesimal increment PQ, then A will be increased by the angle MCN = fZA, and q) by the angle \lCQ,—d(f). PK denotes a part of the parallel passing through P. The radius of the parallel PR is a . cos cp. 56 Since PR is described with the radius a . cos 9>, and belongs to the central angle ^A, the arc PR=a . cos q) . dX, and since QK is described with the radius a, and corresponds to the central angle dcp, QR=a . d(p. Since infinitesimal arcs may be considered straight lines it follows that the triangle PQR may be considered a plane one. In the Mercator projection the lines PR and MN (Fig. 1) are equal to each other. Since this is really not the case, and since it is desirable that the spherical figure and its representation be similar, if not as a whole yet in their smallest parts, and since it is necessary that the angle QPR be the same on the sphere and the chart, w^e imagine, instead of the triangle PQR a similar one, P'Q'R' (Fig. 2), whose sides are sec q) times as large as those of PQR. Therefore, in the triangle P'Q'R', Fig. 2. P'R'=PR sec q)=a . cos

Whence tan T=tan y cos (A— /i) cos a^, or since by (3) tan y= . ' '1 ^ — ^ ^ •' V / -^ sm (A— ytf) tan r = cot (A—/:/) sin ^ (13) which does not contain y. The positions of the tangents at Pi and P2 are now determined by the formulae tan Ti = cot (Aj— /?) sin (pi and tan T-z = cot {A-,—/3) sin cpz With these two tangents two circular arcs may be constructed passing through the points Pi and P2. The centeus Oi and O3 of these circles- are the intersections of the normals at Pi and P2 with the perpendicular to the middle point of the chord joining Pi and P2. The real curve is contained in the meniscus between the two arcs, and may be drawn suf- ficiently close with an irregular curve. This method is too tedious for purposes of navigation, and if used at all by the navigator it will be sufficient to determine the successive posi- tions through which the great circle passes, to plot these positions upon the chart, and connect them by circular arcs. The courses and dis- tances may then be taken directly from the chart. As far as the deter- mination of the positions of the successive points on a great circle is con- cerned this method is shorter than those ordinarily emploj'ed. It originated with J. Asmus, of the Hydrographic office of the German Empire, and was published in the Annalen der Hydrographie und ma- ritimen Meteorologie, 1879, Vol. lY, p. 151 et seq. ZESCEVICH'S. METHOD FOR FINDING THE POSITIONS OF THE POINTS OF THE ARC OF A GREAT CIRCLE. This method, which was published by Piofessor Gelcich in the " Mit- theilungen aus dem Gebiete des Seewesens," Nos. IX and X, 188G, is due to the late German hydrographer Zescevich. It consists in com- puting the latitude at the middle longitude of a great circular arc, and 60 has the advantages of simplicity, saving of time in calculation, and ex eluding the i)0ssibility of error in plotting. Let A be the point of departure and B that of destination. Let ACB be the arc of a great circle passing through A and B. Let P be the pole, ^0 the latitude of A, and cp^ the latitude of B. AP is therefore equal to 90° — ^o, and BP equal to 90° — ^i. Bisect the difference of longitude APB=JA, by the meridian PC and let the angle PCB be represented by x. Let PC=2/, AC=&, and BG=a, then, from the triangle PBC, sin (p\ = cos y cos a + sin y sin a cos x (1) cos a — cos y sin cpi + sin y cos cpi cos ^z/A (2) Substituting the value of f.os a from equation (2) in equation (1). we obtain : sin (pi sin y = cos y cos q)i cos ^z/A-|- sin a cos x and dividing by cos cpi . . 1 ^1 , sin a . cos x tan £0i sin y = cos y cos A^a -\ — ^ -^ -^ - cos q)i Since sin a sin ^z/A cos cpi ~ sin X tan q)i sin y = cos y cos ^/JX 4- sin ^ JA cot x (3) Similarly we obtain from the triangle APC tan q)^| sin y = cos y cos J JA — sin ^JX cot a? (4) Adding (3) and (4), sin y (tan c/j,) + tan ^,) = 2 cos y cos Jz^A or sinysin(yo+y >i)^2 cos y cos .i JA cos q)Q cos ^i 61 or tan V — 2co8^^ j;i cos (pa cos ^ ,gv "~ sin (y^-I- '•A'l) 2/ is the conii)lenieiit of the hititiule of tlie i)oiiit C, which lies on the great circle e(iui(listant in longitude l)etw<;en the two i)oints marking the extremities of the arc in question. Jf the latitude of this i)oiut be denoted by (/; , then (00°— = , sin(c^o+y..) ^6^ i L' cos ^/jA cos qjo cos (pi In this equation the sign of the cp^ will depend upon the sign of the numerator. If north latitude be denoted by + and south by — the resulting latitude cp will be north when the algebraic sum of cp^ and (pi is positiv e and south when it is negative. The latitude of that point on the great circle whose longitude is equi- distant from A and C is now found by the same formula, aud similarly the latitude of the point midway in longitude between C and B. According to the adopted notation these formuhe will be and s 3 cos ^ZJA cos cpo COS cp^ t^n cp, = - f'in^^^l 2 cos ^zJA cos q)i cos (^j The latitudes of these points of the great circle, which are midway in longitude between the points now known, can be computed in like man- ner until the positions of the requisite number of points are established. In computing these sets of formulic many of the logarithms are re- peated, aud the work is thus materially abridged, as may be seen from the following set: -Proceeding with the computation, it would next be necessary to de- termine ^j, (pg, (pi, and q)i^ for which the equations are , sin ((po-\- (Pi) tan fflj = —- vy»-rv-4/ 2 cos ^z/A cos ^0 cos (p^ t^u(p,=- — ^i^in^^^) — ^ 2 cos ^JA cos (pi cos (p^ tan cp,= smi(pi, + (p i) ^^ 2 cos iJA cos (ph cos (pi . sin ((pi+ (pi) tan (pi = ri r-A-i -^ ' 2 cos J JA cos q>i cos (pi 62 THE COMPUTATION OF THE LATITUDE AT THE MIDDLE LONGITUDE. A shorter and more direct method of arriving at Zescevich's results Is as follows: Let the great circle ab be gnomonically projected upon a plane tangent to the sphere at a point O on the equator, as indicated in the accompanying figure, in which the plane of projection is repre- sented as having been rebatted into the plane of the equator. ^qtudor Let OP — r — 1 represent the radius of the sphere, q) the latitude of the point «, projected at rtj, cp' the latitude of the point &, projected at i»2, q)^ the latitude of the point w, projected at m-i, A the difference of longitude between a and h. Then of = og = tan i A ftti = sec ^ A . tan cp (jhi — sec h A. tan ^^ om-i — tan f/>i =' ' '^ — i sec .] A (tan (/> + tan cp^). If the extremities of the great circular arc in question have latitudes of different name, the formula for the latitude at the middle longitude, (p being the greater latitude, is tan (p i = ^ sec ^ A (tan q^ — tan ,). 63 LIST OF LITEKATJRF. (POX THE SFIUKCT OF (IRE.IT CIRCLE SAJLLVO. Aiinalen der llydrograpbie uiid iiiaiitimen Meteorologie, Vol. XII, 18S(), p. r>;5G; also Vol. lY, 1871), \). 151 et seq. Azimuth Tables for parallels of latitude between (M'^ N. and Gl^ S., and Ibr declinations between 23'^ N. and 23^' S., by Lieuts. Seaton Schroeder and W. H. H. Southerland, U. S. N., 4to., 1897. Hydro- graphic Office publication No. 71. Einfoche Losung- der Probleme der Schitlahrt ini grossteu Kreise, Dr. F. Paugger, Triest, 1805. Freucli Hydrographic Chart, No. 3680, Lieutenant Hilleret, French Navy. Great Circle Sailing Chart on the Gnoraonic Polar Projection, Hugh Godfray, M. A., London: J. D. Potter, 1858. Great Circle Sailing IMade Easy, John Seaton, London, 1850. Great Circle Sailing: indicating the shortest sea routes and describ- ing maps for finding them in a few seconds, by Richard A. Proctor, London : Longmans, Green »S: Co., 1888. Hydrographic Office Charts, Nos. 904, 995, 1127, 1128, 1129, 1280, 1281, 1282, 1283, and 1284. Mittheiluugen aus dem Gebiete des Seewesens, Nos. IX and X, 1886. Navigation and Nautical Astronomy, John Kiddle, F. R. A. S,, Lon- don : Sinipkin and Marshall, 1857. Navigation in Theory and Practice, Henry Evers, London and Glas- gow, 1875. Navigation and Nautical Astronomy, Merrifield and PWers, London, 1868. Notes on Navigation and the Determination of Meridian Distances, Commander P. F. Harrington, U. S. Navy, Washington, 1882. Nouveau calculateur nautique pour efiectuer rapidemeut tons les problemes de la navigation, E. Lartigue, Paris, 1884. Nouvelles tables destinees a abreger les calculs nautiques, Perrin, Enseigne de vaisseau, Paris. Practice of Navigation and Nautical Astronomy, Lieut. Henry Kaper, E. N., London: J. D. Potter, 1866. Proceedings of the United States Naval Institute, Vol. XI, No. 2, 1885; also Vol. XXIII, No. 1, 1807. Spherical Tables and Diagram, with their application to Great Circle Sailing, W. C. Bergen, London: Simpkin and Marshall, 1857. Spherical Tables and Diagram, A. H. Deichnum, London: 1857. Tables for Facilitating Great Circle Sailing, John T. Towson, London: 1850. Tables pratiques pour la navigation, Bretel, lieutenant de vaisseau, Paris, 1880. Tableau des distances de port a port. Published by the French Gov- ernment, 1882. The Theory and Practice of Great Circle Sailing under one general rule, with examples, liev. P. Robertson, London: Bell and Dalby, 1855. C % (y 1^070 DATE DUE ^2070 ,.-, ., 1 w 19*T3 Rsnj HCi V 29 1973 nECl2i )74 OCT of ^' ' 6 1974 ^no*? -« A 100*^ hkct '■':'■ 1 'V OCT 1 6 198b -I ■. ,. . GAYLORD PRINTED IN U.S.A. II mil III III I I 3 1970 00487 5891 yQ cfjiiTHfH''; Rt':"'l'';i''L [iBRAR/ ^i.rlLITY AA 000 596 299 8