NAVIGATION is the art of conducting a ship across the ocean. It is here treated to the exclusion of seamanship, which forms a distinct subject. The present article will give, first, a view of the history of the art from the time of the epoch-making voyages of Columbus and the Portuguese, with special reference to advances made in England, and then a sketch of practical navigation as the art now stands.
Up to the time of the Portuguese exploring expeditions, sent out by Prince Henry, which led to the discovery of the Azores in 1419, and the rediscovery of the Cape Verd Islands in 1447 and of Sierra Leone in 1460, navigation had been conducted in the most rude, uncertain, and dangerous manner it is possible to conceive. Thousands of years had passed without the least improvement being introduced, except the magnetic needle about the beginning of the 14th century (see Compass and Magnetism). Prince Henry did all in his power to bring together and systematize the knowledge then obtainable upon nautical affairs; he also established an observatory at Sagres (near Cape St Vincent) in order to obtain more accurate tables of the declination of the sun. John II., who ascended the throne in 1481, followed up the good work of his grand-uncle. He employed Roderick and Joseph, his physicians, with Martin de Bohemia, from Fayal, to act as a committee on navigation. They calculated tables of the sun’s declination, and invented the astrolabe, or at least recommended it as more convenient than the cross-staff. The king established forts and settlements on the coast of Africa; that at St George de la Mina was on the Gold Coast, showing by the position a great geographical advance.
The backward state of navigation at this time is best understood from a sketch of the few rude appliances which the mariner had. He had a compass, a cross-staff or astrolabe, a moderately good table of the sun’s declination, a correction for the altitude of the pole star, and occasionally a very incorrect chart. The first map or sea chart seen in England was brought by Bartholomew Columbus in 1489, and the first map of England was made in 1520. Decimal arithmetic was invented by Simon Stevin about the end of the 16th century. Watches were unknown till 1530, and immediately Gemma Frizon or Frisius seized the idea for the purpose of ascertaining the difference of longitude between two places. They were too rough to be of use, and their advocate proposed to correct them by water-clocks or sand-clocks. Almanacs were first published in Poland in 1470, and in London three years later. These contained tables of the sun’s declination and that of many of the stars, and tables for finding the latitude by the pole star and the “pointers.” There was not till 1607 any means whatever of measuring a ship’s progress through the water, and none in general use till twenty or thirty years later (see Log).
The “cross-staff” appears to have been used by astronomers at a very early period for measuring heights and distances, more recently by seamen for measuring altitudes. It was one of the few instruments possessed by Columbus and Vasco da Gama. The old cross-staff, called by the Spaniards “ballestilla,” consisted of two light battens. The part we may call the staff was about an inch and a half square and 36 inches long. The cross was made to fit closely and to slide upon the staff at right angles; its length was a little over 26 inches, so as to allow the “pinnies” or sights to be placed exactly 26 inches apart. A sight was also fixed on the end of the staff for the eye to peep through at the other two sights and objects to be measured. It was made by describing the angles on a table, and laying the staff upon it (fig. 1). The scale of degrees was marked on. the upper face. Afterwards shorter crosses were introduced, so that smaller angles could be taken by the same instrument. These angles were marked on the sides of the staff.
Another primitive instrument in common use at the beginning of the 16th century was the astrolabe, which was more convenient than the cross-staff for taking altitudes, but was incapable of measuring distances. Fig. 2 represents an astrolabe as described by Martin Cortes. It was made of copper or tin, about one-fourth of an inch in thickness and 6 or 7 inches in diameter, and was perfectly circular except at one place, where a projection was provided for a hole by which it was suspended. Weight was considered desirable in order to keep it steady when in use. The face of the metal having been well polished, a plumb line from the point of suspension marked the vertical line, which when carefully subdivided gave the horizontal line and centre. The upper left quadrant was divided into degrees. The second part was a pointer pt of the same metal and same thickness as the circular plate, about 1∣ inches wide, and in length equal to the diameter of the circle. The centre was bored, and a line was drawn across it the full length, which was called the line of confidence. On the ends of that line were fixed plates s, s, having each a larger and a smaller hole, both exactly over the line of confidence, as sights for the sun or stars. The pointer moved upon a centre the size of a goose quill. When the instrument was suspended the pointer was directed by hand to the object, and the angle read on the one quadrant only. Some years later the other quadrant was also graduated, to give the benefit of a second reading.
Among the earliest writers who touched upon navigation was John Werner of Nuremberg, who in 1514, in his notes upon Ptolemy’s geography, describes the cross-staff as a very ancient instrument, but says that it was only then beginning to be introduced among seamen. He recommends measuring the distance between the moon and a star as a means of ascertaining the longitude.
Thirty-eight years after the discovery of America, when long voyages had become comparatively common, Gemma Frisius wrote upon astronomy and cosmogony, with the use of the globes. His book comprehended much valuable information to mariners of that day, and was translated into French fifty years later (1582) by Claude de Bossière. The system adopted is that of Ptolemy. The following are some of the points of interest for the subject before us. There is a good description of the sphere and its circles; the obliquity of the ecliptic is given as 23° 30'. The distance between the meridians is to be measured on the equator, allowing 15° to an hour of time; longitude is to be found by eclipses of the moon and conjunctions, and reckoned from the Fortunate Islands (Azores). Latitude should be measured from the equator, not from the ecliptic, “as Clarean says.” The use of globes is very thoroughly and correctly explained. The scale for measuring distances was placed on the equator, and 15 German leagues, or 60 Italian leagues, were to be considered equal to one degree. The Italian league was 8 stadia, or 1000 paces, therefore the degree is taken much too small. We are told that, on plane charts, mariners drew lines from various centres (i.e., compass courses), which were very useful since the virtue of the loadstone had become known; it must be remembered that parallel rulers were unknown. Such a confusion of lines has been continued upon sea charts till very recently. Frisius gives rules for finding the course and distance correctly, except that he treats difference of longitude as departure. For instance, if the difference of latitude and difference of longitude are equal, the course prescribed is between the two principal winds,—that is, 45°. He points out that the courses thus followed are not straight lines, but curved, because they do not follow the great circle, and that distances could be more correctly measured on the globe. The tide is said to rise with the moon, high water being when it is on the meridian and nadir. From a table of latitudes and longitudes a few examples are here selected, by which it appears that even the latitude was much in error. The figures in brackets represent the positions according to modern tables, counting the longitude from the western extremity of St Michael. Flores is 5° 8' farther west. [table]
In 1534 Gemma produced an “astronomical ring, which he dedicated to the secretary of the king of Hungary. He admitted that it was not entirely his own invention, but asserted that it could accomplish all that has been said of quadrants, cylinders, and astrolabes,—also that it was a pretty ornament, worthy of a prince. As it displayed great ingenuity, and was followed by many similar contrivances during two centuries, a sketch is here given (fig. 3). The description must necessarily be brief.
The outer and principal sustaining circle EPQπ represents the meridian, and is about 6 inches in diameter; P, π are the poles. The upper quadrant is divided into degrees. It is suspended by a fine cord or wire placed at the supposed latitude. The second circle EQ is fixed at right angles to the first, and represents the equinoctial line. The upper side is divided into twenty-four parts, representing the hours from noon or midnight. On the inner side of that circle are marked the months and weeks. The third ring CO is attached to the first at the poles, and revolves freely within it. On the interior are marked the months, and on another side the corresponding signs of the zodiac; another is graduated in degrees. It is fitted with a groove which carries two movable sights. On the fourth side are twenty-four unequal divisions (tangents) for measuring heights and distances. Its use is illustrated by twenty problems; it can do roughly all that any instrument for taking angles can. Thus, to find the latitude, set the sights C, C to the place of the sun in the zodiac, and shut the circle till it corresponds with 12 o’clock. Peep through the sights and alter the point of suspension till the greatest elevation is attained; that time will be noon, and the point of suspension will be the latitude. The figure is slung at lat. 40°, either north or south. To find the hour of the day, the latitude and declination being known: the sights C, C being set to the declination as before, and the suspension on the latitude, turn the ring CO freely till it points to the sun, when the index opposite the equinoctial circle will indicate the time, while the meridional circle will coincide with the meridian of the place. In this we may see the germ of the “equatorial” now used in the principal observatories.
There is at present in the museum attached to the Royal Naval College at Greenwich an instrument, ticketed as Sir Francis Drake’s astrolabe, prior to 1570. It is not an astrolabe, but may be a combination of astronomical rings as invented by Gemma with other things, probably of a later date. It has the appearance of a large gold watch, about 21 inches in diameter, and contains several parts which fall back on hinges. One part is a sun-dial, the gnomon being in connexion with a graduated quadrant, by which it could be set to the latitude of the place. There are a small compass and an hour circle. It is very neat, but too small for actual use; it may be simply an ornament representing a larger instrument. There is a table of latitudes engraved inside one lid; that given for London is 51° 34', about three miles too much.
In 1537 Pedro Nunez (Nonius), cosmographer to the king of Portugal, published a work on astronomy, charts, and some points of navigation. He recognized the errors in plane charts, and tried to rectify them. Among many astronomical problems is one for finding the latitude of a place by knowing the sun’s declination and the altitude when on two bearings, not less than 40° asunder. Gemma did a similar thing with two stars; therefore the problem now known as a “double altitude” is a very old one. They could do it by the globe within a degree. To Nuiïez has been erroneously attributed the present mode of reading the exact angle on a sextant, the scale of a barometer, and various other things, the credit of which is due to Vernier nearly a hundred years later. The mode of dividing the scale which Nunez published in 1542 was the following. The arc of a large quadrant was furnished with forty-five concentric segments, or scales, the outer one graduated to 90°, the others to 89, 88, 87, <fcc., divisions. As the fine edge of the pointer attached to the sights passed among those numerous divisions it touched one of them, suppose the fifteenth division on the sixth scale, then the angle was ⅜i of 90° = 15° 52' 56". This was a laborious method; Tycho Brahe tried it, but abandoned it in favour of the diagonal lines then in common use, and still found on all scales of equal parts.
In 1545 Dr Pedro de Medina published The Art of Navigation at Valladolid, dedicated to Don Philippo, prince of Spain. This appears to be the first book ever published professedly on navigation. It was soon translated into French and Italian, and many years after into English by John Frampton. Though this pretentious work came out two years after the death of Copernicus, the astronomy is still that of Ptolemy. The general appearance of the chart of the Mediterranean, Atlantic, and part of the Pacific is in its favour, but examination shows it to be very incorrect. A scale of equal parts, near the centre of the chart, extends from the equator to what is intended to represent 75° of latitude; by this scale London would be in 55° instead of 51∣°, Lisbon in 37∣ o instead of 38° 42'. The equator is made to pass along the coast of Guinea, instead of being over four degrees farther south. The Gulf of Guinea extends 14 degrees too far east, and Mexico is much too far west. Though there are many vertical lines on the chart at unequal distances they do not represent meridians; there is no indication of longitude. A scale of 600 leagues is given (German leagues, fifteen to a degree). By this scale the distance between Lisbon and the city of Mexico is 1740 leagues, or 6960 miles; by the vertical scale of degrees it would be about the same; whereas the actual distance is 4820 miles. Here two great wants become apparent,—a knowledge of the actual length of any arc, and the means of representing the surface of the globe on flat paper. There is a table of the sun’s declination to minutes; on June 12th and December 11th (o.s.) it was 23° 33'. The directions for finding the latitude by the pole star and pointers appear good. For general astronomical information the book is inferior to that of Gemma.
In 1556 Martin Cortes published at Seville The Art of Navigation. He gives a good drawing of the cross-staff and astrolabe, also a table of the sun’s declination for four years (the greatest being 23° 33'), and a calendar of saints’ days. The motions of the heavens are described according to the notions then prevalent, the earth being viewed as fixed. He recommends the height of the pole being found frequently, as the estimated distance run was imperfect. He devised an instrument whereby to tell the hour, the direction of the ship’s head, and where the sun would set. A very correct table is given of the distances between the meridians at every degree of latitude, whereby a seaman could easily reduce the difference of longitude to departure. In the rules for finding the latitude by the pole star, the star is supposed to be then 3° from the pole; it is now (1883) 1° 18' 54". Martin Cortes attributes the tides entirely to the influence of the moon, and gives instructions for finding the time of high water at Cadiz, when by means of a card with the moon’s age on it, revolving within a circle showing the hours and minutes, the time of high water at the place for which it was set would be indicated. He deplores the loss of the earl of Niebla and other valiant captains of Spain, before Gibraltar, in 1436, because the mariners kept no account of the tides. In this instance it was more probably the effect of current. There is a chapter upon the signs which prognosticate fair or foul weather, from Pliny and Aristotle; another upon “shining exhalations,” the “fire of St Elmo,” and other old superstitions. Directions are given for making a compass similar to those then in common use, also for ascertaining and allowing for the variation. The east is here spoken of as the principal point, and marked by a cross, after that the true north.
There is a table of difference of latitude and departure in proportion to the tangent of the course.
The third part of Martin Cortes’s work is upon charts; he laments that wise men do not produce some that are correct, and that pilots and mariners will use plane charts which are not true. In the Mediterranean and Channel of Flanders the want of good charts is (he says) less inconvenient, as there they do not navigate by the altitude of the pole.
As some subsequent writers have attributed to Cortes the credit of first thinking of the enlargement of the degrees of latitude on Mercator’s principle, his precise words may be cited. In making a chart, it is recommended to choose a well-known place near the centre of the intended chart, such as Cape St Vincent, which call 37°, “and from thence towards the Arctic pole the degrees increase; and from thence to the equinoctial line they go on decreasing, and from the line to the Antarctic pole increasing.” It would appear at first sight that the degrees increased in size as well as being called by a higher number, but a specimen chart in the book does not justify that conclusion. It is from 34° to 40°, and the divisions are unequal, but evidently by accident, as the highest and lowest are the largest. He states that the Spanish scale was formed by counting the Great Berling as 3° from Cape St Vincent (it is under 2∣ o ). Twenty English leagues are equal to 17⅜ Spanish or 25 French, and to 1° of latitude. Cortes was evidently at a loss to know the size of a degree, and consequently the circumference of the globe. The degrees of longitude are not laid down, but for a first meridian we are told to draw a vertical line “through the Azores, or nearer Spain, where the chart is less occupied.” It is impossible under such circumstances to understand or check the longitudes assigned to places at that period. Martin Cortes’s work was held in high estimation in England for many years, and appeared in several translations. One by Richard Eden in 1609 gives an improved table of the sun’s declination from 1609 to 1625—the greatest declination being 23° 30'. The declinations of the principal stars and the times of their passing the meridian, and other improved tables, are given, with a very poor traverse table for eight points. The cross-staff, he said, was in most common use; but he recommends Wright’s sea quadrant.
William Cuningham published in 1559 a book called his Astronomical Glass, in which he teaches the making of charts by a central meridional line of latitude in equal parts, with other meridians on each side, distant at top and bottom in proportion to the departure at the highest and lowest latitude, for which purpose a table of departures is given very correctly to the third place of sexagesimals. The chart would be excellent were it not that the parallels are drawn straight instead of being curved. In another example, which is one-fourth of the sphere, the meridians and parallels are all curved; it would be good were it not that the former are too long. The hemisphere is also shown upon a projection approaching the stereographic; but the eighteen meridians cut the equator at equal distances, instead of being smaller towards the primitive. He gives the drawing of an instrument like an astrolabe placed horizontally, divided into 32 points and 360 degrees, and carrying a small magnetic needle to be used as a prismatic compass, or even as a theodolite (fig. 4). A sketch is given of Ptolemy observing the sun with a primitive instrument, likely from its great size to give good results after being correctly fixed, except for the amount of error caused by the shrinkage or expansion of the parts.
Gerhard Mercator’s great improvements in charts have been noticed in the article Map, where a sketch is given of his map of the world, of 1569 (vol. xv. p. 521). From facsimiles of his early charts in Jomard, Les Monuments de la Géographie, the following measurements have been made. A general chart of 1569, of North America, 25° to 79°, is 2 feet long north and south, and 20 inches wide. Another of the same date, from the equator to 60° south is 15 ’ 8 inches. The charts agree with each other, a slight allowance being made for remeasuring. As compared with Dr Inman’s table of meridional parts, the spaces between the parallels are all too small. Between 0° and 10° the error is 8'; at 20° it is 5'; at 30°, 16'; at 40°, 39'; at 50°, 61'; at 60°, 104'; at 70°, 158'; and at 79°, 182',—that is, over three degrees upon the whole chart. As the measures are always less than the truth it is possible that Mercator was afraid to give the whole. In a chart of Sicily by Romoldus Mercator in 1589, on which two equal degrees of latitude, 36° to 38°, subtend 9∣ inches, the degree of longitude is quite correct at one-fourth from the top; the lower part is a mile too large. One of the north of Scotland, published in 1595, by Romoldus, measures 10∣ inches from 58° 20' to 61°; the divisions are quite equal and the lines parallel; it is correct at the centre only. A map of Norway, 1595, lat. 60° to 70° = 9∣ inches, has the parallels curved and equidistant, the meridians straight converging lines; the spaces between the meridians at 60° and 70° are quite correct.
Norman’s discovery of the dip (1576) has been spoken of at vol. XV. p. 221. He mentions and condemns the practice of each country having compass cards set to their variation, and sailors using them indiscriminately in any part of the world.
In 1581 Michael Coignet of Antwerp published sea charts, and also a small treatise in French, wherein he exposes the errors of Medina. He was probably the first who said that rhumb lines form spirals round the pole. He published also tables of declination, and observed the gradual decrease in the obliquity of the ecliptic. He described a cross-staff with three transverse pieces, which was then in common use at sea. Coignet died in 1623.
The Dutch published charts made up as atlases as early as 1584, with a treatise on navigation as an introduction.
In 1585 Roderico Zamorano, who was then the lecturer at the naval college at Seville, published a concise and clearly-written compendium of navigation; he follows Cortes in the desire to obtain better charts. Andres Garcia de Cespedes, the successor of Zamorano at Seville, published a treatise on navigation at Madrid in 1606. In 1592 Petrus Plancius published his universal map, containing the discoveries in the East and West Indies and towards the north pole. It possessed no particular merit; the degrees of latitude are equal, but the distances between the meridians are noted. He made London appear in 51° 32' N. (which is near enough) and long. 22°, by which his first meridian should have been more than 3° east of St Michael.
In 1594 Blundeville published a description of Mercator’s charts and globes; he confesses to not having known upon what rule the meridians were enlarged by Mercator, unless upon such a table as Wright had sent him (see below). Wright’s table of meridional parts is here published, also an excellent table of sines, tangents, and secants,—the former to seven places of figures, the latter to eight. These are the tables made originally by Regiomontanus, and improved by the Jesuit Clavius.
In 1594 the celebrated navigator John Davis published a pamphlet of eighty pages, in black letter, entitled The Seaman's Secrets, in which he proposes to give all that is necessary for seamen—not for scholars on shore. He defines three kinds of sailing :—horizontal, paradoxical, and great circle. His horizontal sailing consists of short voyages which may be delineated upon a plain sheet of paper. The paradoxical or cosmographical embraces longitude, latitude, and distance,—the getting together many horizontal courses into one “infallible and true,” i.e., what is now called traverse and Mercator’s sailings. His “paradoxical course” he describes correctly as a rhumb line which is straight on the chart and a curve on the globe. He points out the errors of the common or plane chart, and promises if spared to publish a “paradoxall chart.” It is not known whether such a chart appeared or not, but he assisted Wright in producing his chart a few years later. Great circle sailing is clearly described by Davis on a globe; and to render it more practicable he divides a long distance into several short rhumb lines quite correctly. His list of instruments necessary to a skilful seaman comprises the sea compass, cross-staff, chart, quadrant, astrolabe, an “instrument magnetical” for finding the variation of the compass, a horizontal plane sphere, a globe, and a paradoxical compass. The first three are sufficient for use at sea, the astrolabe and quadrant being uncertain for sea observations. The importance of knowing the time of the tides when approaching tidal or barred harbours is clearly pointed out, also the mode of ascertaining it by the moon’s age. A table of the sun’s declination is given for noon each day during four years, 1593-97, from the ephemerides of Stadius. The greatest is 23° 28'. Several courses and distances, with the resulting difference of latitude and departure, are correctly worked out. A specimen log-book provides one line only for each day, but the columns are arranged similarly to those of a modern log. Under the head of remarks after leaving Brazil, we read, “the compass varied 9°, the south point westward.” He states that the first meridian passed through St Michael, because there was no variation at that place; the meridian passed through the magnetic pole as well as the pole of the earth. He makes no mention of Mercator’s chart, nor of Cortes or other writers on navigation. Rules are given for finding the latitude by two altitudes of the sun and intermediate azimuth, also by two fixed stars, by the globe. There is a drawing of a quadrant, with a plumb line, for measuring the zenith distance, and one of a curious modification of a cross-staff with which the observer stands with his back to the sun, looking at the horizon through a sight on the end of the staff, while the shadow of the sun, from the top of a movable projection, falls on the sight box. This remained in common use till superseded by Hadley’s quadrant. The eighth edition of Davis’s work was printed in 1657.
Edward Wright, of Caius College, Cambridge, published in 1599 a valuable work entitled Certain Errors in Navigation Detected and Corrected. One part is a translation from Roderico Zamorano; there is a chapter from Cortes, and one from Nunez. A year after appeared his chart of the world, upon which both capes and the recent discoveries in the East Indies and America are laid down truthfully and scientifically, as well as his knowledge of their latitude and longitude would admit. Just the northern extremity of Australia is shown. Wright said of himself that he had striven beyond his ability to mend the errors in chart, compass, cross-staff, and declination of sun and stars. He considered that the instruments which had then recently come in use “could hardly be amended,” as they were growing to “perfection,”—especially the sea chart and the compass, though he expresses a hope that the latter may be “freed from that rude and gross manner of handling in the making.” He gives a table of magnetic declinations, and explains its geometrical construction. He states that Medina utterly denied the existence of variation, and attributed it to bad making and bad observations. Wright expresses a hope that a right understanding of the dip of the needle would lead to a knowledge of the latitude, “as the variation did of the longitude.” He gives a table of declinations for every minute of the ecliptic, and another for the use of English mariners during four years— the greatest being 23° 3Γ 30". The latitude of London he made 51° 32'. For these determinations a quadrant over 6 feet in radius was used. He also treats of the “dip” of the sea horizon, refraction, parallax, and the sun’s motions. With all this knowledge the earth is still considered as stationary,—although Wright alludes to Copernicus, and says that he omitted to allow for parallax. Wright ascertained the declination of thirty-two stars, and made many improvements or additions to the art of navigation, considering that all the problems could be performed arithmetically by the doctrine of triangles, without globe or chart. He devised sea rings for taking observations, and a sea quadrant to be used by two persons, which is in some respects similar to that by Davis. While deploring the neglected state which navigation had been in, he rejoices that the worshipful society at the Trinity House, under the favour of the king (James I.), had removed “many gross and dangerous enormities.” He joins the brethren of the Trinity House in the desire that a lectureship should be established on navigation, as at Seville and Cadiz; also that a grand pilot should be appointed, as Sebastian Cabot had been in Spain, who examined pilots (i.e., mates) and navigators. Wright’s desire was partially fulfilled in 1845, when an Act of Parliament paved the way for the compulsory qualification of masters and mates of merchant ships; but such was the opposition by shipowners that it was left voluntary for a few years. England was in this respect more than a century behind Holland. It has been said that Wright accompanied the earl of Cumberland to the Azores in 1589, and that he was allowed £50 a year by the East India Company as lecturer on navigation at Gresham College, Tower Street.
The great mark which Wright made in the world was the discovery of a correct and uniform method for dividing the meridional line and making charts which are still called after the name of Mercator. He considered his charts as true as the globe itself; and so they were for all practical purposes. He commenced by constructing a meridional line, upon the proportion of the secants of the latitude, for every ten minutes of the arc, and in the edition of his work published in 1610 his calculations are for every minute. His calculations were based upon the discovery that the radius bears the same proportion to the secant of the latitude as the difference of longitude does to the meridional difference of latitude—a rule strictly correct for small arcs only. One minute is taken as the unit upon the arc and 10,000 as the corresponding secant, 2' becomes 20,000, 3'= 30,000, &c., increasing uniformly till 49', which is equal to 490,001; 1° is 600,012. The secant of 20° is 12,251,192, and for 20° Γ it will be 12,251,192 + 10,642,—practically the same as that used in modern tables. The principle is simply explained by fig. 5, where b is the pole and bf the meridian. At any point a a minute of longitude: a min. of lat.:: ea (the semi-diameter of the parallel): kf (the radius). Again ea: kf:: kf: ki :·: radius: sec. α∕∙∕'(sec. of lat.). To keep this proportion on the chart, the points of latitude must increase in the same proportion as the secants of the arc contained between those points and the equinoctial, which was then to be done by the “canon of triangles.”
Subsequent writers, including Gunter, Norwood, and Bond, give Wright the credit of having been the first to establish a correct proportion between the meridians and parallels for every part of the chart. This great improvement in the principle of constructing charts was adopted slowly by seamen, who, putting it as they supposed to a practical test, found good reason to be disappointed. The positions of most places had been laid down erroneously, by very rough courses and estimated distances upon an entirely false scale, viz., the plane chart; from this they were transferred to the new projection.
When Napier’s Canon Mirificus appeared in 1614, Wright at once recognized the value of logarithms as an aid to navigation, and undertook a translation of the book, which he did not live to publish (see Napier). E. Gunter’s tables (1620) made the application of the new discovery to navigation possible, and this was done by T. Addison in his Arithmetical Navigation (1625), as well as by Gunter in his tables of 1624 and 1636, which gave artificial sines and tangents, to a radius of 1,000,000, with directions for their use and application to astronomy and navigation, and also logarithms of numbers from 1 to 10,000. Several editions followed, and the work retained its reputation over a century. Gunter invented the sector, and introduced the meridional line upon it, in the just proportion of Mercator’s projection.
The third edition of Gunter’s work was published in 1653, and the fifth edition in 1673, amended by Henry Bond, a practitioner in the mathematics, in the Bulwark near the Tower—a thick octavo. A table of meridional parts is given, with instructions to construct it by the addition of secants as Wright did. The table has been found upon examination to be very correct. The degree is divided into 1000 parts.
With the latitude left, course steered, and difference of longitude made good, Bond found the latitude of ship, by projection on the chart, by the sector, or by the following rule :—
tan of course × proportional diff. of lat. 1 . „ 1 i —⅛- — = dιff. long, radius °
And conversely, suppose*latitude left 50°, course 33° 45', difference of longitude 5⅛°=330'; then cot 30° 45'×330' . no , 1 ,
-p = 493 - 5, prop, dιff. lat., radius r r which, added to the meridional parts corresponding to 50°, will give the number opposite 55°, and 55° is the latitude. Various problems in sailing according to Mercator are solved arithmetically upon the tangents, without the table of meridional parts, which may also be done geometrically upon the tangent line of the cross-staff. The following important proposition is in Bond’s own words:—“First we must know that the logarithmic tangents from 45° upwards do increase in the same manner as the secants added together do, if we account every half degree above 45° to be a whole degree of Mercator’s meridional line; and so the table of logarithmic tangents is a table of meridional parts to every two minutes of the meridian line, leaving out the radius.”
The way of using this proposition is as follows. The table begins at 45°, and every 30 minutes is reckoned a whole degree; therefore, when both latitudes are given, take the half of each increased by 45°, subtract the tangent of the lesser sum from that of the greater, and divide the remainder by the tangent of 45° 30' (radius omitted); the quotient will be the equal, or equinoctial, degrees contained between the two latitudes. Or multiply the aforesaid remainder by ten and divide by half the tangent of 45° 30', and the quotient will be equal to the equinoctial leagues contained between the two latitudes. The logarithmic tangents are here treated as natural numbers, and the division done by logarithms. Bond lays no stress on the above solution as being new; it is merely used in lieu of a table of meridional parts.
The subsequent history of the problem of meridional parts may most conveniently be added here rather than in its chronological place. An important letter from Dr Wallis, professor of geometry at Oxford, is given with the Phil. Trans, for 1685, No. 176. The writer says that, the old inquiry about the sum or aggregate of secants having been of late renewed, he thought fit to trace it from its original, with such solutions as seemed proper to it. Archimedes and the ancients divided the curvilinear spaces as figs. 6 and 7. If they reckoned the first four it was too large; if the last four, too small. As the segments increased in number the error diminished. The degrees of longitude decrease as the cosine of the latitude (which is the semidiameter of such parallel) to the radius of the globe or equator. By the straight lines “each degree of longitude is increased above its due proportion, at such rate as the equator (or its radius) is greater than such parallel (or the radius thereof).” The old sea charts represented the degrees of latitude and longitude all equal. “Hereby, among other inconveniences (as Mr Edward Wright observed in 1599), the representation of places remote from the equator were distorted.” Wright advised that the degrees of latitude should be protracted in like proportion with those of longitude, “that is, everywhere in such proportion as is the respective seçant of such latitude to the radius” (see Wright’s explanation of this part, and fig. 5). Fig. 8 represents one quarter of the globe, the surface of which is opened out till the parallel LA becomes a straight line as la, and each of the four meridians reaches P, P, P, P. The equator is represented by EE; so that the position of each parallel on the chart should b at such distance from the equator “as are all the secants (taken at equal distances in the arc) to so many times the radius, . . . which is equivalent to a projection of the spherical surface on the concave surface of a cylinder, erected at right angles to the plane of the equator,” while each division of the meridian is equal to the secant of the latitude answering to such part, as fig. 9. This projection, if expanded into a plane, will be the same as a plane figure whose base is equal to a quadrantal arc extended (or a portion thereof), on which (as ordinates) are erected perpendiculars equal to the secants, answering to the respective points of the arc extended, as fig. 10. The first answers to the equator, the last to the pole infinite.
“For finding this distance answering to each degree and minute of latitude, Mr Wright added all the secants from the beginning to the position required. The sum of all except the greatest (answering to the figure inscribed) is too little. The sum of all except the least (answering to the circumscribed) is too great—which latter Mr Wright followed. It will be nearer the truth than either if we take the intermediate spaces; instead of minutes, take ⅛, 1⅜, 2⅜, &c., or the double of these, 1, 3, 5, 7, &c., which yet, because on the convex side of the curve, would be rather too little. Either of these ways will be exact enough for a chart. If we would be more exact, Mr Oughtred directs, as did Mr Wright before him, to divide the arc into parts yet smaller than minutes, and calculate secants thereto.” Wallis continued the subject and the doctrine of infinite series; but more than sufficient has been quoted for the purposes of navigation. At the end he adds that the same may be obtained in like manner by taking the versed sines in arithmetical proportion.
The next writer who made his mark upon this problem was Dr E. Halley {Phil. Trans., No. 219, 1695). He states that the tangential proportion between the latitude and the divisions of the meridional line was discovered by chance, and first published by H. Bond, in Norwood’s Epitome of Navigation. James Gregory furnished a demonstration in 1668; but it was long and tedious. Halley claimed for himself the honour of being the first to give a rule whereby the meridional parts between any two latitudes may be calculated at once by the relation of the logarithmic tangents; but it is practically the same as that published by Bond. Halley said that Wright’s table nowhere exceeded the truth by half a mile. Sir Jonas Moore’s system, he said, was nearer the truth, but the difference is not appreciable till beyond navigable waters.
A rather curious paper was read before the Royal Society, June 4, 1666, by Nicholas Mercator upon the meridional line; he proposes to divide it into the hundred-thousandth part of a minute. Roger Cotes wrote upon the same subject an exhaustive paper in Latin, called “Logometria,” Phil. Trans., No. 338, 1714. He gives an illustrative figure in which the rhumb line crosses the meridians at an angle of 45°. His demonstrations by the ratios arrive at similar conclusions to those clearly expressed by Halley.
All these rules assume the earth to be truly spherical, instead of spheroidal. For the history of inquiry into the exact figure of the earth, see Earth. It may be mentioned that a pamphlet on this subject by Murdoch, published in 1741, in which meridional parts are adapted to a (very exaggerated) spheroid, shows that plane charts and the roughly-divided Mercator’s charts were still used at that date. Plane charts, indeed are explained even later, as in Robertson’s Navigation, 1755.
The power of taking observations correctly, either at sea or on shore, was greatly assisted by the invention which bears the name of Pierre Vernier, which was published in Brussels in 1631 (see Vernier). As Vernier’s quadrant was divided into half degrees only, the sector, as he called it, spread over 14∣ degrees, and that space carried thirty equal divisions, numbered from 0 to 30. As each division of the sector contained 29 minutes of the arc, the vernier could be read to minutes. The verniers now commonly adapted to sextants can be read to 10 seconds. Shortly after the invention it was recommended by Bouguer and Jorge Juan, who describe it in a treatise entitled La Construction, &c., du quadrant nouveau. About this period Gascoigne applied the telescope to the quadrant (see Micrometer); and Hevelius invented the tangent screw, to give slow and steady motion when near the desired position. In 1635 Henry Gellibrand published his discovery of the change in variation of the needle, which was effected by his comparing the results of his own observations with those of Burrough and Gunter. The latter was his predecessor at Gresham College.
In 1637 Richard Norwood, a sailor, and reader in mathematics, published an account of his most laudable exertions to remove one of the greatest stumbling-blocks in the way of correct navigation, that of not knowing the actual size of a degree or nautical mile, in a pamphlet styled The Seaman's Practices. Norwood ascertained the latitude of a position near the Tower of London in June 1633, and of a place in the centre of York in June 1635, with a sextant of more than 5 feet radius, and, having carefully corrected the declination, refraction, and parallax, made the difference 2° 28'. He then measured the distance with a chain, taking horizontal angles of all windings, and he made a special table for correcting elevations and depressions. A few places which he was unable to measure he paced. His conclusion was that a degree contained 367,176 English feet; this gives 2040 yards to a nautical mile,—only about 12 yards too much. Norwood’s works went through numerous editions, and retained their popularity over a hundred years; the last which the writer has seen—a good book for the time, free from nonsense—is dated 1732. In it he says that, as there is no means of discovering the longitude, a seaman must trust to his reckoning. He recommends the knots on the log-line to be placed 51 feet apart, as the just proportion to a mile when used with the halfminute glass.
Dr Hooke read a paper at the Royal Society, in 1666, upon deep-sea sounding by means of a weight which became detached on striking the bottom, and allowed a float to ascend to the surface, while the time was carefully noted —basing his calculations upon performances in known depths. He was on the verge of a great success; he required Sir W. Thomson’s piano-wire instead of the float.
In the same year a paper was read by Dr Wallis (who had previously published a discourse on tides) showing that the modern theory was not then generally accepted. This was followed by a paper by Sir Robert Moray, who recommended frequent and extended observations, and proposed to form a table which embraced every circumstance that would appear to be desirable even at the present day. Sir Robert also spoke of the irregularities in the tides past the western islands of Scotland. In Phil. Trans., 1683, vol. xiii. No. 143, there is an account of Flamsteed’s tide table for London Bridge, which gave each high tide every day in the year. He justly condemns the old almanacs for deriving the moon’s age from the epact, and then allowing forty-eight minutes for every day. Brooker was the first to amend this reckoning, but in a rough manner. Henry Philips, well known by his works on navigation, was the first to bring the inequality to a rule, which was found more conformable to experience than was expected; but Flamsteed made corrections on his rule.
The necessity for having correct charts was equalled by the pressing need of obtaining the longitude by some simple and correct means available to seamen; and we have seen how many plans had already been thought of for this purpose. At one time it was hoped that the longitude might be directly discovered by the variation of the compass; in 1674 Charles II. actually appointed a commission to investigate the pretensions of a scheme of this sort devised by Bond, and the same idea appears as late as 1777 in S. Dunn’s Epitome. But the only real way of ascertaining the longitude is by knowing the difference of time at two meridians; and till the invention and perfecting of chronometers this could only be done by finding at two places the apparent time of the same celestial phenomenon. The most obvious phenomena to select were the motions of the moon among the sun and stars, which as we have seen were suggested as a means of finding the longitude by Werner in 1514, and continued to receive attention from later writers. But to make this idea practical it was necessary on the one hand to have better instruments for observation, and on the other to have such a theory of the moon’s motions as should enable its place to be predicted with accuracy, and recorded beforehand in an almanac. The very principles of such a theory were unknown before Newton’s great discovery, -when the lunar problem begins to have a chief place in the history of navigation; the places of stars were derived from various and widely discrepant sources; and almanacs gave little useful information beyond the declination of the sun, the age of the moon, and the time of high water. Another class of phenomena whose comparative frequency recommended them for longitude observations, viz., the occultations of Jupiter’s satellites, became known through Galileo’s discovery of these bodies (1610). Tables for these were published by Dominic Cassini at Bologna in 1688, and were repeated in a more correct form at Paris in 1693 by his son, who was followed by Pound, Bradley, Wargentin, and many other astronomers. But this method, though useful on land, is not suited to mariners; when Whiston, for example, in 1737 recommended that the satellites should be observed by a reflecting telescope, he did not sufficiently consider the difficulty of using a telescope at sea, or the infrequency of the occultations, and it is the lunar problem which will chiefly concern us in what follows.
The study of this problem was stimulated by the reward of 1000 crowns offered by Philip III. of Spain in 1598; the states-general followed with an offer of 10,000 florins. But for a long time nothing practical came of this; a proposal by J. Morin, submitted to Richelieu in 1633, was pronounced by commissioners appointed to judge of it to be incomplete through the imperfection of the lunar tables, and in like manner when the question was raised in England in 1674 by a proposal of St Piere to find the longitude by using the altitudes of the moon and two stars to find the time each was from the meridian, and when the king was pressed by St Piere, Sir J. Moore, and Sir C. Wren to establish an observatory for the benefit of navigation, and especially that the moon’s exact position might be calculated a year in advance, Flamsteed gave his judgment that the lunar tables then in use were quite useless, and the positions of the stars erroneous. The result was that the king decided upon establishing an observatory in Greenwich Park, and Flamsteed was appointed astronomical observer on March 4, 1675, upon a salary of £100 a year, for which also he was to instruct two boys from Christ’s Hospital. While the small building in the Park was in course of erection he resided in the Queen’s House (now the central part of the Greenwich Hospital school), and removed to the house on the hill, July 10, 1676, which came to be known as “Flamsteed House.” The institution was placed under the surveyor-general of ordnance,—perhaps because that office was then held by Sir Jonas Moore, himself an eminent mathematician. Though this was not the first observatory in Europe, it was destined to become the most useful, and has fulfilled the important duties for which it was established. It was established to meet the exigencies of navigation, as was clearly stated on the appointment of Flamsteed, and on several subsequent occasions; and we see now what an excellent foster-mother it has been to the higher branches of that science. This has been accomplished by much labour and patience; for, though the most suitable man in the kingdom was placed in charge, it was so starved and neglected that it was almost useless during many years. The Government did not provide a. single instrument. Flamsteed entered upon his important duties with an iron sextant of 7 feet radius, a quadrant of 3 feet radius, two telescopes, and two clocks, the last given by Sir Jonas Moore. Tycho Brahe’s catalogue of about a thousand stars was his only guide. In 1681 he fitted a mural arc which proved a failure. Seven years after another mural arc was erected at a cost of £120, with which he set to work in earnest to verify the latitude, and to determine the equinoctial point, the obliquity of the ecliptic, the right ascension and declination of the stars, till he numbered two or three thousand which appeared in the “British catalogue.” See Flamsteed and Astronomy.
Flamsteed died in 1720, and was succeeded by Halley, who paid particular attention to the motions of the moon with a view to the longitude problem. A paper which he published in the Phil. Trans., 1731, No. 421, shows what had been accomplished up to that date, and proves that it was still impossible to find the longitude correctly by the moon. He repeats what he had published twenty years before in an appendix to Street’s Caroline tables, which contained observations made by him (Halley) in 1683-84 for ascertaining the moon’s motion, which he thought to be the only practical method of “attaining” the longitude at sea. He found that it only needed a little patience to be able to manage a 5 or 6 foot telescope, capable of showing the occultations of the fixed stars by the moon, on shipboard in moderately fine weather (ships must have been very steady in his days). The Caroline tables of Street, though better than those before him, as well as those of Tycho, Kepler, Bullialdus, and Horrox, were uncertain; sometimes the equations would compensate one another; at other times when they fell the same way the result might be 100 leagues in error. He hopes that the tables will be so amended that an error may scarce ever exceed 3 minutes of motion (equal to l∣ o of longitude). Sir Isaac Newton’s tables, corrected by himself (Halley) and others up to 1713, would admit of errors of 5 minutes, when the moon was in the third and fourth quarters. He blames Flamsteed for neglecting that portion of the work, as he was at the observatory more than two periods of eighteen years.. He himself had at this time seen the whole period of the moon’s apogee—less than nine years—during which period he observed the right ascension of the moon at her transit, with great exactness, almost fifteen hundred times, or as often as Tycho Brahe, Hevelius, and Flamsteed together. He hoped during 1731 and a few future years to compute the moon’s position within 2 minutes of motion with certainty, which will be equal to 20 leagues at the equator and 15 in the Channel; and Hadley’s quadrant might be applied to taking distances at sea with the desired accuracy.
The last remark calls us to consider this great improvement in the instrumental means for taking observations at sea. Up to this time all instruments either depended on a plumb line or required the observer to look in two directions at once.
Their imperfections are clearly pointed out in a paper by Pierre Bouguer (1729) which received the prize of the Paris Academy of Sciences for the best method of taking the altitude of stars at sea.
Bouguer himself proposes a modification of what he calls the English quadrant, probably the one proposed by Wright and improved by Davis. Fig. 11 represents the instrument as proposed, capable of measuring fully 90° from E to N. A fixed pinule was recommended to be placed at E, through which a ray from the sun would pass to the sight C. The sight F was movable. The observer, standing with his back to the sun, would look through F and C at the horizon, shifting the former up or down till the ray from the sun coincided with the horizon. The space from E to F would represent the altitude, and the remaining part F to N the zenith distance. The English quadrant which this was to supersede differed in having about half the arc from E towards N, and, instead of the pinule being fixed at E, it was on a smaller arc represented by the dotted line eB, and movable. It was placed on an even number of degrees, considerably less than the altitude; the remainder was measured on the larger arc, as described.
Hadley’s instrument, on the other hand, described to the Royal Society in May 1731 (Phil. Trans., Nos. 420 and 421), embodies Newton’s idea of bringing the reflexion of one object to coincide with the other. He calls it an octant, as the arc is actually 45°, or the eighth part of a circle; but, in consequence of the angles of incidence and reflexion both being changed by a movement of the index, it measures an angle of 90°, and is graduated accordingly; the same instrument has therefore been called a quadrant. It was very slowly adopted, and no doubt there were numerous mechanical difficulties of centring, graduating, <fcc., to be overcome before it reached perfection. In August 1732, in pursuance of an order from the Admiralty, observations were made with Hadley’s quadrant on board the “Chatham” yacht of 60 tons, below Sheerness, in rough weather, by persons—except the master attendant—unaccustomed to the motion; still the results were very satisfactory. Two years later Hadley published (Phil. Trans., 1733) the description of an instrument for taking altitudes when the horizon is not visible. The sketch represents a curved tube or spirit-level, attached to the radius of the quadrant.
From the year 1714 the history of navigation in England is closely associated with that of the “commissioners for the discovery of longitude at sea,” a body constituted by Act 13 Anne c. 14 (commonly called 12 Anne c. 15), with power to grant sums not exceeding £2000 to assist experiments and reward minor discoveries, and also to judge on applications for much greater rewards which were offered to open competition. For a method of determining the longitude within 60 geographical miles, to be tested by a voyage to the West Indie's and back, the sum of £10,000 was offered; within 40 miles, £15,000; within 30, £20,000. £10,000 was also to be given for a method that came within 80 miles near the shores of greatest danger. No action seems to have been taken before 1737; the first grant made was in that year, and the last in 1815, but the board continued to exist till 1828, having disbursed in the course of its existence £101,000 in all. In the interval a number of other Acts had "been passed either dealing with the powers, constitution, and funds of the commissioners or encouraging nautical discovery; thus the Act 18 George II. (1745) offered £20,000 for the discovery by a British ship of the North-West Passage, and the Act 16 George III. (1776) offered the same reward for a passage to the Pacific either north-west or north-east, and £5000 to any one who should approach by sea within one degree of the North Pole. All these Acts were swept away in 1828, when the longitude problem had ceased to attract competitors, and voyages of discovery were nearly over. The suggestions and applications sent in to the commissioners were naturally very numerous and often very trifling; but they sometimes furnish useful illustrations of the state of navigation. Thus, in a memorial by Captain H. Lanoue (1736), which seems to be designed to commend a substitute for the log (a box with something, not fully explained, let into the sea), he records a number of recent casualties, which shows how carelessly the largest ships were then navigated. Several men-of-war off Plymouth in 1691 were wrecked through mistaking the Deadman for Berry Head. Admiral Wheeler’s squadron in 1694, leaving the Mediterranean, ran on Gibraltar when they thought they had passed the Strait. Sir Cloudesley Shovel’s squadron, in 1707, was lost on the rocks off Scilly, by erring in their latitude. Several transports, in 1711, were lost near the river St Lawrence, having erred 15 leagues in the reckoning during twenty-four hours. Lord Belhaven was lost on the Lizard in 1722, the same day on which he sailed from Plymouth.
One of the first points to which the attention of the commissioners was directed was the survey of the coasts of Great Britain, which was pressed on them by Whiston in 1737. He was appointed surveyor of coasts and headlands, and in 1741 received a grant for instruments. An Act passed in 1740 enabled the commissioners to spend money on the survey of the coasts of Great Britain and the “plantations.” At a later date they bore part of the expenses of Cook’s scientific voyages, and of the publication of their results. Indeed it is to them that we owe all that was done by England for surveys of coasts, both at home and abroad, prior to the establishment of the hydrographic department of the Admiralty in 1795. But their chief work lay in the encouragement they gave on the one hand to the improvement of timepieces, and on the other to the perfecting of astronomical tables and methods, the latter issuing in the publication of the Nautical Almanac. Before we pass on to these two important topics we may with advantage take a view of the state of practical navigation in the middle of last century as shown in two of the principal treatises then current.
Robertson’s Elements of Navigation passed through six editions between 1755 and 1796. It contains good teaching on arithmetic, geometry, spherics, astronomy, geography, winds and tides, also a small useful table for correcting the middle time between the equal altitudes of the sun,—all good, as is^ also the remark that “the greater the moon’s meridian altitude the greater the tides will be.” He states that Lacaille recommends equal altitudes being observed and worked separately, in order to find the time from noon, and the mean of the results taken as the truth. There is a sound article on chronology, the ancient and modern modes of reckoning time. A long list of latitudes, longitudes, and times of high water finishes vol. i. The second volume is said by the author to treat of navigation mechanical and theoretical; by the former he means seamanship. He gives instructions for all imaginary kinds of sailings, for marine surveying and making Mercator’s chart. There are two good traverse tables, one to quarter points, the other to every 15 minutes of the arc; the distance to each is 120 miles. There is a table of meridional parts to minutes, which is more minute than customary. Book ix., upon what is now called “the day’s work,” or deadreckoning, appears to embrace all that is necessary. A great many methods, we are told, were then used for measuring a ship’s rate of sailing, but among the English the log and line with a half-minute glass were generally used. Bouguer and Lacaille proposed a log with a diver to avoid the drift motion (1753 and 1760). Robertson’s rule of computing the equation of equal altitudes is as good as any used at the present day. He gives also a description of an equalaltitude instrument, having three horizontal wires, probably such a'one as was used at Portsmouth for testing Harrison’s timekeeper. The mechanical difficulties must have been great in preserving a perpendicular stem and a truly horizontal sweep for the telescope. It gave place to the improved sextant and artificial horizon. The second edition of Robertson’s work in 1764 contains an excellent dissertation on the rise and progress of modern navigation by Dr James Wilson, which has been greatly used by all subsequent writers.
Don Jorge Juan’s Compendio de Navegacion, for the use of mid·
shipmen, was published at Cadiz in 1757. Chapter i. explains what pilotage is, practical and theoretical. He speaks of the change of variation, “which sailors have not believed and do not believe now.” He described the lead, log, and sand-glass, the latter corrected by a pendulum, charts plane and spherical. Supposing his readers to be versed in trigonometry, he will explain what latitude and longitude are, and show a method for finding the latter different from what has been taught. He will show the error of middle latitude sailing, and show that the longitude found by it is always less than the truth. (It is strange that while reckoning was so rough and imperfect in many respects they should strain at such a trifle as that is in low latitudes. ) He promised to find the difference of longitude without a departure (a similar rule to that of Bond). After speaking of meridional parts, he offered to explain the English method, which was discovered by Edmund Halley, but omits the principles upon which Halley founded his theory, as it was too embarrassing. (He was not the first.) He gives instructions for currents and leeway, tables of declination, a few stars, meridional parts, &c. It is worthy of remark that, after giving a form for a log-book, he added that this had not been previously kept by any one, but he thought it should not be trusted to memory. He only required the knots, fathoms, course, wind, and leeway to be marked every two hours. Every hour is quite long enough, and that is often divided now. He gave a sketch of Hadley’s quadrant, in shape like those in use fifty years back, but without a clamping screw or tangent screw. Back glasses were much valued in those days,—the force of habit, no doubt. The book is quite free from all extraneous rubbish.
The introduction of timekeepers by which Greenwich time can be carried to any part of the world, and the longitude found with ease, simplicity, and certainty, is due to the invention of John Harrison. The idea of keeping time at sea was no novelty. Huygens (^.v.) made pendulum -watches for the purpose prior to 1665, at which date Major Holmes communicated to the Royal Society {Phil. Trans., i. 13) the fact of his having tried two of them on the coast of Guinea. He sailed from St Thomas, set his watches, sailed west 700 or 800 leagues, without changing course, then steered towards the coast of Africa N.N.E. 200 or 300 leagues. The masters of the other ships under his charge, fearing the want of water, wished to steer for Barbados. Holmes, on comparing the calculations, found them to differ from him from 80 to over 100 leagues. He considered that they were only 30 leagues from the Cape Verd Islands, where they arrived next afternoon. The vague manner of estimating distance is worth notice. William Derham published a scientific description of various kinds of timekeepers in The Artificial Clock-Maker, in 1700, with a table of equations from Flamsteed to facilitate comparison with the sun-dial. In 1714 Henry Sully, an Englishman, published a treatise at Vienna, on finding time artificially. He went to France, and spent the rest of his life in trying to make a timekeeper for the discovery of the longitude at sea. In 1716 he presented a watch of his own make to the Academy of Sciences, which was approved; and ten years later he went to Bordeaux to try his marine watches, and died before embarking. Julien le Roy was his scholar, and perfected many of his inventions in watchmaking.
Harrison’s great invention was the principle of compensation through the unequal contraction of two metals, which he first applied in the invention of the compensation pendulum, still in use, and then modified so as to fit it to a watch, devising at the same time a means by which the watch retains its motion while being wound up. To what has been said in the article Harrison on his successive attempts, and the success of the trial journey to Jamaica in 1761-62, it may be added that by the journal of the House of Commons we find that the error of the watch (as if there were only one) was ascertained by equal altitudes at Portsmouth and Barbados, the calculations being made by Short. The watch came greatly within the limits of the Act. At Jamaica it was only in error five seconds (assuming that the longitude previously found by the transit of Mercury could be so closely depended on, which as we now know, was not the case; the observations were too few in number, and taken with an untrustworthy instrument). Short found the whole error from November 6, 1761, till April 2, 1762, to be l m 54 a ∙5 = 18 geographical miles in the latitude of Portsmouth. He considered that a position determined by a transit of Mercury was liable to an error of 30 s only, and by Jupiter’s best satellite to 3 m 44 8 . During the passage home in the “Merlin” sloop-of-war the timekeeper was placed in the after part of the ship, because it was the dryest place, and there it received violent shocks which retarded its motion. It lost on the voyage home l m 49 s = 16 geographical miles
One might have supposed that Harrison had now secured the prize; but there were powerful competitors who hoped to gain it by lunars, and a bill was passed through the House in 1763 which left an open chance for a lunarian during four years. A second West Indies trial of the watch took place between November 1763 and March 1764, in a voyage to Barbados, which occupied four months; during which time it is said, in the preamble to Act 5 Geo. III. 1765, not to have erred 10 geographical miles in longitude. We only find in the public records the equal altitudes taken at Portsmouth and at Bridgetown, Barbados. William Harrison assumed an average rate of l s a-day gaining, as he anticipated that it would go slower by 1 3 for every 10° increase in temperature. The longitude of Bridgetown was determined by Maskelyne and Green by nine emersions of Jupiter’s first satellite, against five of Bradley’s and two at Greenwich Observatory, to be 3 h 54 m 20 8 west of Greenwich. In February 1765 the commissioners of longitude expressed an opinion that the trial was satisfactory, but required the principles to be disclosed and other watches made. Half the great reward was paid to Harrison under Act of Parliament in this year, and he and his son gave full descriptions and drawings, upon oath, to seven persons appointed by the commissioners of longitude. The other half of the great reward was promised to Harrison when he had made other timekeepers to the satisfaction of the commissioners, and provided he gave up everything to them within six months. The second half was not paid till 1773, after trials had been made with five watches. These trials were partly made at Greenwich by Maskelyne, who, as we shall see, was a great advocate of lunars, and was not ready to admit more than a subsidiary value to the watch. A bitter controversy arose, and Harrison in 1767 published a book in which he charges Maskelyne with exposing his watch to unfair treatment. The feud between the astronomer-royal and the watchmakers continued long after this date.
Even after Harrison had received his £20,000, doubts were felt as to the certainty of his achievement, and fresh rewards were offered in 1774 both for timekeepers and for improved lunar tables or other methods. But the tests proposed for timekeepers were very discouraging, and the watchmakers complained that this was due to Maskelyne. A fierce attack on the astronomer’s treatment of himself and other watchmakers was made by Mudge in 1792, in A Narrative of Facts, addressed to the first lord of the Admiralty, and Maskelyne’s reply does not convey the conviction that full justice was done to timekeepers. Maskelyne at this date still says that he would prefer an eclipse of a bright star by the moon and a number of correspondent observations by transits of the moon compared with those of fixed stars, made by two astronomers at remote places, to any timekeeper. The details of these controversies, and of subsequent improvements in timekeepers, need not detain us here. In England the names of Arnold and Earnshaw are prominent, each of whom received, up to 1805, .£3000 reward from the commissioners of longitude. It was Arnold who introduced the name chronometer. The French emulated the English efforts for the production of good timekeepers, and favourable trials were made between 1768 and 1772 with watches by Le Roy and Berthoud.
Meantime the steady progress of astronomy both by the multiplication and increased accuracy of observations, and by corresponding advances in the theory, had made it possible to construct greatly improved tables. In observations of the moon Greenwich still took the lead; and it was here that Halley’s successor Bradley made his two grand discoveries of aberration and nutation which have added so much to the precision of modern astronomy. Kepler’s Rudolphine tables of 1627 and Street’s tables of 1661, which had held their ground for almost a century, were rendered obsolete by the observations of Halley and his successor. At length, in 1753, in the second volume of the Commentarii of the Academy of Göttingen, Tobias Mayer printed his new solar and lunar tables, which were to have so great an influence on the history of navigation. Mayer afterwards constructed and submitted to the English Government in 1755 an improved body of MS. tables. Bradley found that the moon’s place by these tables was generally correct within 1 ', so that the error in a longitude found by them would not be much more than half a degree if the necessary observations could be taken accurately on shipboard. Thus the lunar problem seemed to have at length become a practical one for mariners, and in England it was taken up with great energy by N. Maskelyne— “the father,” as he has been called, “of lunar observations.”
In 1761 Maskelyne was sent to St Helena to observe the transit of Venus. On his voyage out and home he used Mayer’s printed tables for lunar determinations of the longitude, and from St Helena he wrote a letter to the Royal Society {Phil. Trans., vol. lii. p. 558, 1762), in which he described his observations made with Hadley’s quadrant of 20 inches radius, made by Bird, and the glasses ground by Dollond. He took the observations both ways to avoid the errors. The arc and index were of brass, the frame mahogany; the vernier was subdivided to minutes. The telescope was 6 inches long, magnified four times, and inverted. Very few seamen in that day possessed so good an instrument. He considered that ship’s time should be ascertained within twelve hours, as a good common watch will scarcely vary above a minute in that time. This shows that he must have intended the altitude of sun or star to be calculated—which would lead to new errors. He considered that his observations would each give the longitude within 1∣ degrees. On February 11th he took ten; the extremes were a little over one degree apart.
On his return to England Maskelyne prepared the British Mariner's Guide (1763), in which he undertakes to furnish complete and easy instructions for finding the longitude at sea or on shore, within a degree, by observing the distance between the moon and sun, or a star, by Hadley’s quadrant. How far that promise was fulfilled, and the practicability of the instructions, are points worth consideration, as the book took a prominent place for some years. The errors which he said were inseparable from the dead-reckoning “even in the hands of the ablest and most skilful navigators,” amounting at times to 15 degrees, appear to be overestimated.
On the other hand, the lunar equations, which were from Mayer’s tables, would, he believed, always determine the longitude within a degree, and generally to half a degree, if applied to careful observations. He recommends the two altitudes and distance being taken simultaneously when practicable. The probable error in a meridian altitude he estimated at one or two minutes, and in a lunar distance two minutes (equal to one degree of longitude). He then gave clear rules for finding the moon’s position and distance by ten equations, too laborious for seamen to undertake. Admitting the requisite calculations for finding the moon’s place to be difficult, he desired to see the moon’s longitude and latitude computed for every twelve hours, and hence her distance from the sun and from a proper star on each side of her carefully calculated for every six hours, and published beforehand.
In 1765 Maskelyne became astronomer-royal, and was able to give effect to his own suggestion by organizing the publication of the Nautical Almanac. The same Act of 1765 which gave Harrison his first £10,000 gave the commissioners authority and funds for this undertaking. Mayer’s tables, with his MS. improvements up to his death in 1762, were bought from his widow for <£3000; <£300 was granted to the famous mathematician Euler, on whose theory of the moon Mayer’s later tables were formed; and the first Almanac, that for 1767, was published in the previous year, at the cost and under the authority of the commissioners of longitude. This was not the first almanac in the country, perhaps by a hundred, as that name was applied to small periodical works, frequently of a frivolous character,—though the later and better description gave the sun’s declination and moon’s meridional passage approximately. In 1696 the French nautical almanac for the following year appeared, an improvement on what had been before issued by private persons, but it did not attempt to give lunar distances.. The time of twilight at Paris. The sun’s right ascension to hours and minutes. The sun’s declination at noon each day to seconds. The whole accompanied by necessary instructions.” ] In the English Nautical Almanac for 1767 we find everything necessary to render it worthy of confidence, and to satisfy every requirement at sea. The great achievement was that of giving the distance from the moon’s centre to the sun, when suitable, and to about seven fixed stars, every three hours. The mariner has only to find the apparent time at ship, and clear his own measured distance from the effects of parallax and refraction (for which at the end of the book are given Lyon’s and Dunthorn’s methods), and then by simple proportion, or proportional logarithms, find the time at Greenwich. The calculations respecting the sun and moon were made from Mayer’s last manuscript tables under the inspection of Maskelyne, and were so continued till 1804. The calculations respecting the planets are from Halley’s tables, and those of Jupiter’s satellites from tables made by Wargentin and published by Lalande in 1759, except the fourth satellite. It will be seen by the following outline of contents that the first Almanac contained all the principal points of information which the seaman required; and greater accuracy at that time was not desirable, or at least would not have been appreciated.
Page 1 of each month gave the Sundays and holidays, four phases of the moon, and positions of sun, moon, and planets in the signs of the zodiac; page 2, sun’s longitude, right ascension in time, declination, and equation for noon each day; page 3, sun’s semidiameter, time of passing the meridian, hourly motion of the sun, logarithm of sun’s distance, and place of the moon’s node, for every sixth day; also eclipses of Jupiter’s satellites, time of immersion; page 4, the positions of the four principal planets for every sixth day; page 5, the configuration of Jupiter’s satellites at 11 r.Μ. of every day; page 6, the moon’s longitude and latitude for noon and midnight of every day; page 7, the moon’s age, passage over the meridian, right ascension, and declination at noon and midnight; page 8, the moon’s semidiameter, horizontal parallax, and logistic logarithm—each at noon and midnight; pages 9 to 12, the moon’s centre from the sun and seven stars for every three hours, while within about 116 degrees. Then follow tables of refraction, moon’s parallax in altitude, a catalogue of stars, with their right ascension and declination, table for the “dip” of the sea horizon, and several other useful things, many of which are omitted in modern Nautical Almanacs, as they are included in and more properly belong to the permanent rules and requirements of navigation.
Various useful rules and tables were appended to early volumes of the Almanac. Thus the volume for 1771 contains a method and table for determining the latitude by two altitudes and the elapsed time first published by Cornelius Downes of Amsterdam in 174O. At the end of the Almanac for 1772 Maskelyne and Whichell gave three special tables for clearing the lunar distance; still their rule is neither short nor easily remembered. An improvement of Dunthorn’s solution is also given, and a problem in Mercator’s sailing by Halley solved by Israel Lyons, viz., the latitude of the point of departure given, distance sailed, and change of longitude, —required the course steered. In the edition for 1773 a new table for equations of equal altitudes is given by W. Whale. In those for 1797 and 1800 tables are added by John Brinkley for rendering the calculations for double altitudes easier.
From 1777 to 1788 inclusive, the moon’s place was calculated from improved tables by Charles Mason, founded on observations by Bradley, which were published in the Nautical Almanac for 1774. The difference then only amounted to 1" in longitude, the apogee 56", and the ascending node 45". From 1789 to 1804 the tables were further corrected by Mason, and calculated to tenths of a second. The distances between the moon and the stars were still further corrected by the use of Taylor’s logarithms to seconds, and their places by Bradley’s observations in 1756 and Maskelyne’s in 1809. The places of the planets at that time were from Lalande’s Astronomy (the 3d edition was published in 1792), more recently from vol. iii. of Professor Vince’s Astronomy. The places of the moon since the beginning of 1821 were calculated from Burckhardt’s tables. They are now taken from Hansen’s tables, completed with the aid of the English Government in 1857. The eclipses of Jupiter’s satellites for 1824 and following years were from De-
lambre’s new tables. In 1827 the positions of sixty of the principal stars were given for every tenth day, from the tables of Maskelyne and Dr Pearson. Since 1824 the work has been printed three and latterly four years in advance. The price was 5s. till 1855; but the Almanacs for that and subsequent years have been issued at 2s. 6d.
A book of Tables Requisite to be Used with the Nautical Ephemeris was published byMaskelyne at the same time as the first Almanac, and ten thousand copies were quickly sold. A second edition, prepared by W. Wales, appeared in 1781, an octavo of 237 pages, in the preface of which it is stated, with apparent truth, that it contains everything necessary for computing the latitude and longitude by observation. There are in all twenty-three tables, the traverse table and table of meridional parts alone being deficient as compared with modern works of the kind; deadreckoning Maskelyne did not touch. He gave practical methods for working several problems; the lunar especially is an improvement on those by Lyons and Dunthorn, though a rule there given for clearing the distance, called Dunthorn’s improved method, is remarkably short. The half sum of three logarithms gives an arc, and the half sum of other two gives half the true distance. The objection is the use of special logarithms. Maskelyne’s rule for finding the latitudes by two altitudes and the elapsed time is also good, but with the same objection. The third edition of the Tables was issued in 1802. It has been said that Maskelyne neglected the planets; be that as it may, he established the positions of sixty of the principal stars, and completed many other things. He had but one assistant, whereas there are now eight, and the Nautical Almanac is under another department.
As the necessary calculations for clearing the lunar distance from the effects of parallax and refraction were considered difficult to seamen, many efforts were made to shorten the process. Among others Whichell, master of the Royal Naval Academy, Portsmouth, conceived a plan whereby it could be taken from a table by inspection'. In October 1765 the commissioners of longitude awarded him £100 to enable him to complete and print 1000 copies of his table. On the following April they gave him £200 more. The work was continued on the same plan by Shepherd, the Plumian professor of astronomy, Cambridge, with some additions by the astronomer-royal. The total cost of the ponderous 4to volume up to the time of publication in June 1772 was £3100, after which £200 more was paid to the Rev. Thomas Parkinson and Israel Lyons for examining the errata. It is a very large and expensive volume,— very ill-adapted for ship’s use. Considerable sums were paid by the commissioners from time to time for other tables to facilitate navigation—not always very judiciously. It is sufficient to mention here the tables of Michael Taylor and the still esteemed tables of Mendoza, published in 1815. Here also may be mentioned a useful table by Stevens (1780) for finding the latitude by the altitude of the pole star, and Grossweil’s tables for facilitating the computation of lunars—partly new and partly after Maskelyne. These appear to be the first tables in which half the logarithmic sine, &c., is given, to save the trouble of halving a sum of four or more logarithms.
The plan of the Nautical Almanac was soon imitated by other nations. In France the Académie Royale de Marine had all the lunar distances translated from the British Nautical Almanac for 1773 and following years, retaining the Greenwich time for the three-hourly distances. The tables were considered excellent, and national pride was satisfied by their having been formed on the plan proposed by Lacaille. They did not imitate the mode given for clearing the distance, considering their own better.
Though the Spaniards were leaders in the art of navigation during the 16th and 17th centuries, it was not till November 4, 1791, that their first nautical almanac was printed at Madrid, having been previously calculated at Cadiz for the year 1792. They acknowledge borrowing from the English and French. The lunar distances were reduced from Greenwich meridian to that of Cadiz, by subtracting 25 m 9 8 . It is in larger and better print than the French almanac. In the book for 1803 the meridian of the royal observatory at Isla de Leon is placed 24 m 47 s- 5 west of Greenwich. In the English almanac for 1883 it is given as 24 m 49 s- 6; therefore they were very near the truth in 1791. The almanac for 1810, published at Madrid in 1807, was the first in which the lunar distances were reduced to the meridian of Isla de Leon—that is, giving the distance to the even hours 3, 6, 9, 12, &c. The Spanish almanacs for 1813 to 1816 were published in Fleet Street, the first-named only one year in advance. From 1820 to 1832 they were good octavos and clear type. Soon after that time they appeared in folio, giving nearly all the information found in the English almanac, but not neglecting the saints’ days and festivals. The excellent Berlin Astronomisches Jahrbuch began to appear in 1776, the American Ephemeris in 1849. These two ephemerides and the French Connaissance des Temps are independent and valuable works, and for astronomers at least in some respects superior to the English Almanac.
After Maskelyne’s death the correctness and reputation of the Nautical Almanac underwent a serious decline. The matter came before parliament in 1818, when the board of longitude was reconstructed, and the old Acts consolidated. Dr T. Young was appointed secretary to the commissioners, and superintendent of the Almanac. Ten years later, in 1828, the board was swept away, the Almanac was placed under the Admiralty, and Young, with Faraday as a chemist and Sabine as a practical observer, were appointed scientific advisers to the Admiralty, which ever since has spent a certain annual sum on scientific research. The Almanac still gave cause for dissatisfaction; a memorial to the House of Commons, dated January 28, 1829, states that the Nautical Almanac was for the good of astronomy as well as navigation, and that it is so declared in the first Almanac in 1767; that in 1818 fiftyeight errors were discovered, and a similar number in the Almanac for 1830, and that it had not kept pace with navigation or astronomy; that it did not give the moon’s distance from the four principal planets as the Portuguese and Danish ephemerides did, nor did it give the positions of those planets; that there was no list of the occultating stars which were ascertained to be visible in Halley’s time, but were neglected after the invention of Hadley’s sextant (they were in the Milan ephemerides); and that the tables of the sun were not correct. This was supported by a paper signed by J. F. W. Herschel, read at the board of longitude April 5, 1827, which stated that the moon’s meridian passage was not given at all, that of the sun roughly to the nearest minute. The right ascension and declination of the larger planets were not given with accuracy, as they should have been, as their theory was perfect. The moon’s right ascension in time and hourly motion should have been given, also the time of semidiameter passing the meridian, for use with moon-culminating stars. Young replied to this memorial and maintained that the fifty-eight errors were exaggerated; forty of them were in reality only one in the moon’s place, which would put a ship out 5 miles, and which was corrected in the next year’s book, “which every accurate navigator is bound to consult, to guard against possible minute accident.” The errors of 1830 were, he says, of less importance: the French Connaissance des Temps of 1821 was corrected by the English Almanac·, some errors were found in Taylor’s logarithms; the error in the solar tables, said to be 15 seconds, was really only one. The ultimate result of these controversies was the appearance of the new and reformed Nautical Almanac in 1834. It may be added that the last remnant of the old laws, the protection of the Almanac against competition by a penalty, was abolished by an Act passed August 6, 1861. The number of copies of the Nautical Almanac (for 1851) printed in 1847-48 was 10,000, and the number sold (of various years) 8638, at 5s. each; the gross produce of sale was <£2159, and the expense of preparation £3677. In 1882 15,071 copies were sold, of various years, extending to four years in advance; the cost was £3368, and the gross produce of sale at 2s. 6d. each £1900. The sale of the Nautical Almanac has lately decreased on account of the amount of information given in private publications.
Prior to 1795 the nautical portion of the British community, including the royal navy, were entirely dependent upon private industry and enterprise for charts and sailing directions. On August 12th of that year an order in council placed all such articles as were then in the possession of the Admiralty in charge of Mr Dalrymple, an eminent publisher of such things, who had long been employed by the East India Company, and whose catalogue in 1786 contained 347 charts between England, the Cape, India, and China; thus the germ of the hydrographic department was established. The expense was then limited to £650 a year, just one-tenth of what was allowed last year for drawing and engraving charts alone, besides £5500 for printing and mounting them. In 1881 there were 118,542 charts sold.
After the close of the long devastating war in 1815 both trade and science revived, and several Governments besides that of Great Britain saw the necessity of surveying the coasts in various parts of the globe; the greater portion of the work fell to the English hydrographical department, which took under its charge nearly every place where the inhabitants were not able to do it for themselves. The parts which received immediate attention were the shores of the United Kingdom, the lakes of Canada, Gulf of St Lawrence and Newfoundland, West India Islands, Sicily and the eastern part of the Mediterranean, west coast of Africa, Australia, Tierra del Fuego and Straits of Magellan, Arabia, Persia, Bay of Bengal from the Hooghly to Malacca Strait, and the Red Sea. Similar good work was carried on by the Governments of France, Norway, Denmark, Sweden, and Russia; Portugal was an exception. Spain had for her own shores the excellent work of Admiral Vincent Tofino, performed in the latter part of last century. The United States of America have surveyed both shores of that continent since 1832. Since 1837 the British surveyors have been continually employed in Australia and New Zealand, and since 1842 on China and Japan, and the whole west coast of America from Magellan to Behring’s Straits. British Columbia has been surveyed in detail, also the great barrier reef of Australia and Torres Strait, the coast of New Guinea and the Fiji group, coast of Arabia, and Persian Gulf. The Gulf of Suez and other parts of the Red Sea have been resurveyed. Fresh requirements are continually springing up, and greater precision is expected in the surveys, which are now continued, with the same zeal and unremitting attention, under the present hydrographer Captain Sir Frederick Evans, K.C.B., to whose paper for the British Association (1881) the reader is referred. The first official catalogue of charts was issued in 1830; the total number was then 962. In 1880 the number had increased to 2699. The greatest increase was on the British coasts, from 51 to 368, the Arctic seas and north-east coast of America, from 57 to 302, and the Pacific Ocean islands, from 11 to 100.
The question of the accuracy with which the sea charts now represent a portion of the globe is entirely set at rest by the possession of a correct knowledge of the figure and size of the earth, as well as the means of ascertaining the latitude and longitude of innumerable places (see Earth and Geography). They are now made as precisely as possible to represent the actual surface of the globe. The result of each survey is first plotted with the meridian lines inclined towards the pole, by which means all astronomical bearings coincide. It is afterwards opened proportionately to suit Mercator’s projection and the sailor’s use, as far as 70 or 80 degrees. For the polar regions a circular projection is used, which may comprise a radius of 30 or even 40 degrees. During 1882 sixty-one new plates of charts were engraved, and eighteen improved by the addition of new plans; 2700 plates received corrections by the engraver. The number of charts printed for the royal navy, Government departments, and to meet the requirements of the general public, during 1882 amounted to 229,700. The Admiralty catalogue of charts and sailing directions now contains 2680 of the former and 83 of the latter. The tide table for 1883 gives the time of high water at all the ports in Great Britain and Ireland, also for full and change days all over the globe,—upwards of 3200 in number. The cost of the hydrographic department was <£43,145 last year, of which £20,958 was for surveys in various parts of the world, hire of vessels, wages, Ac.
Many of the early writers and teachers of navigation regretted that no national care was bestowed upon the education of navigators; they might be rejoiced could they know that the Royal Naval College at Greenwich is maintained at a cost of £31,800; beside which there is the Naval College at Portsmouth, and the “Britannia” training ship for naval cadets, which latter costs over £17,000 a year, one half being paid by the parents. There is also a. school at Greenwich for a thousand boys, the sons of petty officers, seamen, and marines, all of whom (unless physically incapacitated) are destined for the sea. There are also several training vessels connected both with the royal navy and the mercantile marine. There can be no doubt that the educational qualification of seamen of all grades is well provided for, and as little doubt that the kindred qualification “seamanship” is neglected; but the examination for a lieutenant in the navy will in future include a knowledge of Channel pilotage.
A meteorological conference was held in Brussels in 1853 for devising a uniform system of observations at sea, upon winds and currents; and a meteorological department was established in England soon after, under the Board of Trade and Admiral FitzRoy. In 1866 it was placed under a committee selected by the Royal Society. It is hoped that before long much useful information may result from that source. Wind and current charts are now compiled by the Admiralty, which are very valuable aids in long voyages.
At the commencement of the present century all watches designed for the discovery of the longitude were called chronometers”; they were but slowly coming into general use; the Admiralty only supplied them to flagships, surveying ships, and exploring expeditions. Later every ship of war was allowed one, and an additional one if the captain possessed one of his own. The East India Company allowed their ships one each at an early date. Now flagships are allowed five, and ordinary ships of war three each. Chronometers were soon recognized as a certain and simple means of ascertaining the difference of longitude between two places; Captain Cook and others used them in the last century. Between 1817 and 1824 Captain Smyth used five chronometers in the survey of the Mediterranean. Dr Tiarks, 1822-23, used twenty-six to establish the longitude of Pendennis Castle, and for Funchal, Madeira, seventeen. Schubert in 1833 proceeded round the Baltic in a steam-vessel with fifty-six chronometers, reckoning his longitude from Altona. In 1826 King carried eleven in the “Adventure” and “Beagle ’; Foster in 1828, seventeen in the “Chanticleer”; FitzRoy, 1831, twenty-two in the “Beagle.” In 1839 Lieutenant Raper proposed that eighteen “secondary meridians” should be established, and, these being well spread over the globe, the minor places should be referred to them, and not to each other as formerly. There are now fifty such secondary meridians, and the electric wire has taken the place of the timekeeper, as the time-signal is the most perfect thing which human ingenuity can devise. The elaborate chronometric expeditions of English, Continental, and American astronomers belong rather to the history of astronomy. When the Atlantic cable was laid to Newfoundland in July 1866, and time-signals received direct from Greenwich Observatory to Heart’s Content, the Admiralty chart was found to be quite correct. In a similar manner all the principal places in North and South America and the West India Islands have been connected by time-signals with the United States observatories within the last seven years, and found to be fairly correct, the error seldom amounting to two seconds in time. The observatory at Lisbon was found to differ eight seconds of time. It may now be assumed that there is no place within the ordinary navigable parts of the ocean where an error in position exists of sufficient amount to affect a ship’s safety.
A reward was formerly given annually to the maker of the chronometer which stood the best test at the observatory; this was instituted in 1822, and discontinued after 1835; the recipients were Poole, Frodsham, Hutton, Hewett, Eiffe, and Dent. Since that time they are received at the observatory on trial during twenty-eight weeks, and those preserving the most equal rate are chosen for the public service at a price named by an official in charge of the department, and something above the ordinary price. Prior to 1849 as much as £62 was paid for one; in 1883 about six are recommended for purchase at £45 to £38 each. On Dent’s catalogue marine chronometers range from 35 to 45 guineas.
The following list of some writers of navigation whose works have not been already mentioned may be found useful :—Thomas Addison, Arithmetical Navigation, 1625 (he was the first to apply logarithms); Antonio de Najera (Lisbon, 1628) follows Nunez and Cespedes, but corrects the declination of sun and stars; Sir B. Dudley, L' arcano del mare, 1630-46, 2d ed., Florence, 1661, too ponderous for the use of seamen; Sir Jonas Moore (1681), one of the best books of the period; William Jones (1702), a useful compendium containing trigonometry applied to the various sailings, the use of the log, and tables of logarithms; Pierre Jean Bouguer, Traité Complet de la Navigation, folio, 1698, good but too large; Manuel Pimentai, L'arte de navegar, Lisbon, 1712; Pierre Bouguer, jun., Nouveau Traité de Navigation, 1753 (without tables), published at the request of the minister of marine, improved and shortened in 1769 under the superintendence of the astronomer Lacaille; Nathaniel Colson, The Mariner's Nero Calendar, 1735, a good book; Seller, Practical Navigation, a book very popular in its time (there was an edition as late as 1739); Samuel Dunn published good star charts and tables of latitude and longitude (1737), and framed concise rules for many problems on navigation (published by the board of longitude); John FL Moore, The Practical Navigator ami Seaman's New Daily Assistant, 1772, very popular, and generally used in the British navy,—the 18th and 19th editions (1810, 1814) were improved by J. Dessiou; W. Wilson (Edinburgh, 1773), a treatise of good repute at the time; Samuel Dunn, Nero Epitome of Practical Navigation, or Guide to the Indian Seas, 1777,—for the longitude he depends chiefly on a variation chart from observations on East Indiamen, and he still makes no mention of the Nautical Almanac or of parallel rulers; Samuel Dunn (probably a son of the last named), 1781, is the last writer who gives instructions for the use of the astrolabe; he also wrote on “lunars” (1783, 1793),. a word which was generally adopted about this time, and published an excellent traverse table (1785), and Daily Uses of the Nautical Sciences, 1790; Horsborough, Directory for East India Voyages, 1805; A. Mackay, The Complete Navigator, about 1791, 2d ed. 1810,—there is no instruction for finding longitude by the time.
keeper; Kelly, Spherical Trigonometry and Nautical Astronomy, 1796, 4th ed. 1813, clear and simple; N. Bowditch, Practical Navigator, 1800, passed through many editions, and is still much used in the United States navy; J. W. Norie, Epitome of Navigation, 1803, 21st ed. 1878, long a favourite in the mercantile marine from its simplicity, and because navigation can be learned from it without a teacher; T. Kerigan, The Young Navigator’s Guide to Nautical Astronomy, 1821; Inman, Epitome of Navigation, 1821,* with an excellent volume of tables—largely used in the British navy, 9th ed. 1854; E. Riddle, Navigation and Nautical Astronomy, 3d ed. 1824, 9th ed., by Escott, 1871, still worthy of its high reputation,— it does not include great circle sailing; J. T. Towson, Tables for Reduction of Ex-meridian Altitudes, 4th ed. 1854, very useful; H. Raper, Practice of Navigation, 1840, 10th ed. 1870, an excellent book, perhaps the best; H. Evers, Navigation and Great Circle Sailing, 1850, other works on the same subject by Merrifield and Evers (1868), and Evers (1875); R. Μ. Inskip, Navigation 'and Nautical Astronomy, 1865, a useful book, without tables; T. H. Sumner, A Method of finding a Ship's Position by two Observations and Greenwich Time by Chronometer,— this is set forth as a novelty, but was published by Captain R. Owen, R. N., early in the century, and practised by many officers; H. W. Jeans, Navigation and Nautical Astronomy, 1858, without tables, a good deal used in the British navy,—Jeans revised Inman’s tables in 1873; Harbord, Glossary of Navigation, 1863, enlarged ed. 1883, a very excellent book of reference; W. C. Bergen, Practice and Theory of Navigation, 1872; Sir W. Thomson, Navigation, a Lecture, 1876, well worth reading.
PRACTICAL OR MODERN NAVIGATION.
The following outline of navigation, as commonly practised in 1883, is sufficient to enable a person to conduct a ship from one port to another, or at least to show in plain simple terms the principle upon which it is done. There will be no tables or logarithms referred to which are not common to all works on navigation, and in all such books there are instructions for their use. Seamanship forms a separate subject, but there are a few points wherein it is inseparable from this; no man can be a good navigator unless he possesses sound judgment in seamanship.
It is necessary that the ship should be provided with good compasses, in suitable places, an ordinary log and line, as well as any patent log which may be desired, hand leads, deep-sea leads and lines, log-book, work-book, chronometer, tables of logarithms, &c., Nautical Almanac, sextant, artificial horizon, large parallel rulers, a box of mathematical instruments, charts of the ocean, and enlarged plans of the places the ship is likely to visit.
Before the ship leaves her anchorage or other security, the helm should be moved hard over each way, to prove that it is clear, and a leadsman stationed on each side of the ship. When clear of the harbour and adjacent head-, lands, the course is set towards her destination or to obtain an offing. At that time a bearing is taken of one known object and the distance estimated, or, two known objects being in a line, a bearing is taken of a third, or two cross bearings; the result in either case is entered in the logbook, with the exact time. This is called the departure (i.e., from the land), and should be secured before rain or fog intercept the view. Thereafter the rough or deck logbook should be marked every hour after the log has been hove (the hour being subdivided when necessary), and the estimated course steered, distance through the water, wind, and leeway (if there be any) carefully entered in their proper columns. The courses may be steered and expressed by degrees, if this is desired. The ship’s reckoning is invariably made up from noon to noon of each day; all the principal events are entered in the ship’s log-book, which is a very important document.
That part of a ship’s reckoning which is independent of observations of sun, moon, or stars is called dead-reckoning (D.R.), while the position by celestial objects is designated observed (Obs.).
The following illustration of a ship’s log and day’s reckoning will include all the “sailings” which are usually treated under separate heads. The force of the wind and state of weather are expressed according to a scale, from 1 to 12, and by symbols which have been in general use about fifty years. Variation and local deviation of the compass must be here attended to; the mode of ascertaining the amount will be explained further on. All problems in this part of navigation are solved by plane trigonometry, as if working on a plane, except in finding the proportion between difference of longitude and departure (the latter from the meridian in nautical miles).
The accompanying example of a ship’s log represents the greater portion of a page, which would contain the record of what had occurred during two civil days, and therefore embraces one complete astronomical day. While the ship remains on the same course the entry should not be repeated; the open space shows more clearly the amount of distance due to each. The “North Star” is here supposed to have been under way and clear of the harbour at noon, when the departure was taken from the Eddystone, the position of which is 50° 11' N. and 4° 15' W. The courses are intended to comprise many varieties of circumstances. If a ship be under steam or has a fair wind, these will be but few; but leeway will always exist according to the strength of the wind, if on the side, though scarcely perceptible by the wake if the ship is going fast. It is supposed that a ship either sailing or steaming even 15 miles an hour, with her broadside to the breeze, will be carried the same distance at a right angle to her course as if she had been stopped during the same period—surface of hull, sails, &c., exposed being similar in each case. It is difficult to estimate the drift of a ship during a gale, but it is very seldom overrated. Courses and distances within 50 miles are usually obtained by laying the edge of the parallel ruler on the two positions and running it to the compass, which is drawn on the chart; then take the space between the places in the dividers and apply it to the scale of latitude opposite them: the number of miles there indicated will be the distance. Distant places on Mercator’s) chart, when so treated, will give the course and distance approximately.
To Work the Reckoning Arithmetically from the Ship's Log.— The true bearing of the Eddystone is reversed and treated as a course. To correct the compass courses, westerly variation and deviation are both applied to the left of the course steered, that is, against the sun, the reverse when they are easterly. In the example the variation is taken as 21° westerly throughout the day’s run. The deviation is taken from a table of corrections for a compass to which a magnet had been applied (see Compass and Magnetism). The first course on the ship’s log is W. by S. ⅜ S. or S. 73° 8' W. Apply variation - 21°; deviation (5° 30') being easterly is +; leeway (5° 37') as the wind is south is also +: result S. 63° 15' W. In a similar manner deal with the other courses, marking well the change of sign east or west, and the direction of the wind with regard to the leeway. The direction of the current being true is entered as S. 79° W. After all the courses and distances for tho day have been correctly entered in a traverse table, the corresponding differences of latitude and departure are taken from the traverse table of the epitome, and placed in the proper columns. If the angle be less than 45°, read difference of latitude and departure at the top of the page; if greater than 45° it is reversed. As the table is arranged in decimals, 2 ‘5 can be treated as 25 or 250, or conversely, so that large distances may be embraced. This process is called working by inspection, and is sufficiently accurate for the purpose of finding the course and distance made good, or the course and distance, to places within 300 or 400 miles. [table]
As the difference of latitude is south, it must be subtracted from the latitude left.
Eddystone, 50° 1ΓΌ N. 4° 15' W.
-1 3∙1 . +2 27
49 7Ό —49° 7'54" 6 42
[table]
With middle latitude 49° 40' and departure 95'1, enter the traverse table.
Taking the complement of middle latitude as a course, and the departure in its column, the difference of longitude = 147 is found in the distance column; which being west is added to the Eddystone’s westerly longitude.
To find the Course and Distance Made Good by Inspection.— Difference of latitude 63 '1 and departure 95 ‘1 will be found opposite 56⅛ o and under 114; therefore it is S. 56⅜° W. 114 miles from the Eddystone.
To do the same more exactly by logarithms,—the multiplication and division of the quantities being performed by the addition and subtraction of their logarithms.
First, required the difference of longitude.
dep. 95 ’ 1 × rad. 1 ., τ , 1 ., i . . nθ . fv =<Iιff. long, cos mid. lat. 49 40 c,
95∙1 log +10 ... 11'978180
49° 40' log cos 9 - 811061
Diff. long. 146'9 log 2'167119
Or dep. 95 Ί × sec 49° 40' . .„.
— i -j = diff. long, rad. o
95'1 log 1'978180
49° 40' log sec-10 0Ί88939
Diff. long. 146'9 log 2Ί67119
The proportions which sines, tangents, and secants bear to the; course, distance, difference, latitude, and departure, and these to each other, form the principal points in plane trigonometry connected with navigation, which can be easily demonstrated by two figures. If the hypothenuse AB be made radius, as in fig. 12, the whole will be contained within the circle, and the sides are in proportion to the sines of their opposite angles; but if the base be made the radius, as in fig. 13, the other side BC is entirely without the circle, touching only at the point C; it is therefore a tangent subtending the angle A, and the hypothenuse AB becomes the secant of the angle A. In like manner BC may be made radius, then AC would become the tangent of angle B.
To find the course and distance made good by calculations. Making BC the radius, AC95∙1 log +10... 11’978180
rad. × dep. AC , BC 63Ί log 1’800029
-diff, lat BC ° tancourse ∙
B 56° 26' 7" log tan.... 10'178151
! BC63∙1 log 1’800029
BC X sec B 56° 26' 7" distancc B 56° 26' 7" log sec -10 0’257372 rad. ‘ I
AB dist. 114Ί log 2Ό57401
It is not necessary to work out a course to seconds; scarcely so to minutes. Distances are invariably expressed in miles and decimal parts. .
To find the Course and Distance to the Place of Destination.— Suppose it be the Strait of Belle Isle. Choose a point of land, or assume a position, near the entrance, say 51° 45' N. and 55° 10' W. A line drawn on a chart of the Atlantic from the ship’s position to the above will roughly indicate the course usually steered, which is a rhumb line cutting all the meridians at the same angle, and forming on the globe a spiral curve towards the pole. The exceptions are if it be east or west, north or south; in the former case it would be a small circle round the pole, and in the latter it would be on the meridian (which is a great circle) directly to one of the poles. The direction of the line on the chart is found by means of parallel rulers conveying it to the compass, or by a protractor placed on a meridional line; and the distance is measured on the scale of latitude at the side Between the two places. Long distances are calculated by “middle latitude,” or “Mercator’s sailing.” The former is most convenient in low latitudes and when the difference is small, but the latter is at all times more correct. Middle latitude may be made nearly as good a method by measuring from the chart half way between the two latitudes; or the same result may be obtained by taking the latitude corresponding to the mean of the meridian parts, as the table of these embodies the whole principle upon which Mercator’s charts are made.
Continuing the day’s work—
Lat. of ship 49° 8' Merid. pts. 3394’29
Near Belle Isle 51 45 3640’90
2 37 = 157' 2)7035’19
Mid. lat. 50° 27' 30". Mean 3517’59
which differs only one mile from the mean taken directly.
To find the departure corresponding to the difference of longitude {55° 10' - 6° 42'= 48° 28' = 2908') in mean latitude 50° 27' 30":—
diff. long. 2908 × cos 50° 27'30" 1 , or , o ., j =dep. 1851 ’3 miles.
rad. 1
With difference of latitude and departure to find the course :— τad, X dep. 1851’3 , rad. × diff. lat.
— i ! —=tan course or = = cot course,
αιπ. lat. 157 g5° g' 10"∙ dep.
The course is evidently 85 degrees to the west of north. To find the distance dep. 1851’3 X rad. , orn ..
-.— 'oκ° tv i a , -,=T858 miles, sin course 85 9 10
To find the course and distance by Mercator’s sailing. The difference between the meridional parts as above is 246 ’6, difference of longitude 2908 miles, and difference of latitude 157; whence rad. X diff. long. , diff. lat. × rad. ,. , , ..
., 1 . g , . f = tan course = dist. 1858 miles, mend. diff. lat gy> g, jθ,,. cos course
The result is the same by both methods when the difference of latitude is small. It is obvious that if we subtract the logarithm -corresponding to the meridional difference latitude from that of the difference longitude increased by ten in the index, the remainder will be the logarithmic tangent of the course. In like manner, if the logarithmic secant of the course be added to the logarithm of the true difference latitude, and ten in the index rejected, the sum will be the logarithm of the distance.
Under some circumstances, in high latitudes, the ordinary mode •of reducing the ship’s run during the twenty-four hours, by taking the difference between the northing and southing as the difference of latitude, and the excess of easting or westing as the departure, would not be correct. For example, suppose that from a position off the west coast of Ireland, in 52° N. and 11° W., a ship were to sail in such a manner as to make during the first eight hours 96 miles on a true N. W. course, during the next eight hours a similar distance due north, and during the last eight hours N.E. 96 miles. Here we should have the two oblique courses producing 67’9 miles each, both of difference of latitude and departure, the departures apparently cancelling each other and leaving the ship on the same meridian. But in reality the ship would not be on the same meridian, as the latter departure was performed in a higher latitude than the former, and they are of opposite names. Therefore this peculiar “day’s work” requires to be divided into two distinct parts; each may be taken by inspection from the traverse table. The first course N. 45° W. 96 miles will produce latitude 53° 7' 54", middle latitude 52° 34', which with departure 67’9 will give difference longitude 111° 6' W. The second course simply adds 96 miles to the latitude; the third, like the first, increases the latitude by 67’9 miles making it 55° 51' 48", and the middle latitude of the last section 55° 18', which with departure 67’9 will give difference longitude 119’2; this being east, and preponderating over the former by 7’6 miles, the ship will be 7’6 miles east of the meridian she first sailed from, or in 10° 52' 24" W.
Great Circle Sailing.— The course between the ship on May 10th and the Strait of Belle Isle, as ascertained by middle latitude and Mercator’s sailing, is a straight line on Mercator’s chart, but is in reality a curve upon the globe, and the nearer the positions are to either pole the greater the curvature and consequent increase of distance. The shortest distance between any two points on the globe is an arc of a great circle, the centre of which is also the centre of the sphere. The equator, ecliptic, and all meridians are great circles, between which an infinite number of others may be drawn in every conceivable direction, each dividing the surface of the globe into two equal parts. It is only of late years that this subject has been revived, since ships have made long voyages entirely under steam, thereby having the ability to steer any course desired. The practice of this mode of sailing requires but a small general knowledge of spherical trigonometry, and to that extent only will spherical problems be here introduced; though a clear understanding of the doctrine of the sphere is very desirable for all navigators. The formulae by which spherical problems are solved must here be postulated. They will be proved in Trigonometry. In all spherical triangles there are six parts, three sides and three angles, the sides as well as the angles being measured by degress; any three being given the remainder may be found. The angles measured between heavenly bodies, or their altitudes, are treated as spherical sides. When two places are on the same parallel of latitude, the shortest course is not east or west (except on the equator), as the parallels are small circles having th« pole as their common centre. In finding the true course and shortest distance, by calculating a segment of a great circle, the earth is treated as a sphere,— this being sufficiently accurate for the purpose.
The advantage of sailing on a great circle instead of a rhumb is best demonstrated by stretching a thread across a globe, cutting the two places under consideration; while by calculation it can be found numerically. Returning to the day’s work on May 10th, let us suppose it desired to find the great circle course and distance to Belle Isle. The co-latitude of each place gives the two sides, and the difference of longitude expresses the angle between them at the pole. Hence the two sides and included angle are given; required the third side and the angle at ship, which will be the distance and course (see fig. 14). P represents the pole, S the ship, and B Belle Isle. It is necessary first to find the angles Γ>SB and PBS. The formula is sin⅜diff sides X cot⅜ contained a⅛ tan , diff . ofother two angles - sin ⅜ sum of the two sides cos⅜diff. sides × - Cθt⅜containedangffi gtan ^ sumofother legι . cos ⅜ sum of sides
Instead of subtracting a sine or cosine, it is easier to add the cosecant or the secant and reject the index. The result is—
sin ⅜(PS - PB)[ = l o 18' 30"] × cot ⅜P[ = 24° 14']
sin ⅜(PB + BS)[ = 39° 33' 30"] ’
cos ⅜(PS - PB) X cot ⅜ P =tan 7 θ 0 , 3 θ,,.
cos⅜(PB + PS)
Hence 70° 51' 30" + 4° 33' 15"=75° 24' 45" is the greater angle B, opposite the greater side; and 70° 51' 30" — 4’ 33' 15" = 66° 18' 15" is angle S,—the ship’s course at-commencing,—say, N. 66° W. At the termination it will be S. 75° W. Having all the angles, we find the third side by the rule that the sines of the angles are proportional to the sines of the opposite sides.
sin PS 40° 52'X sin P 48° 28 ' = sin βg 3 θ 0 24 , 15 « = 1824 ∙ 2 distance; sin B 75 24 45
sin PB 38 Ί5 X sin P 48° 28' . pc , . o ,.
” ⅛8W∙18-iy s, ° bs30 24 15 ·
The distance by Mercator’s sailing was 1858, or 33’8 miles longer.
Should the sum of any two sides exceed 180°, the supplement of each must be used, and the supplement of the angles found. If a side be over 90° the logarithm found will be sine of its supplement or the cosine of its excess over 90°. It will be readily seen by the frequent alterations necessary in the course, to keep near the great circle, that the ship passes the pole, continually changing its bearing, as she would any other distant object on the globe. It is always desirable to ascertain the highest latitude a ship will pass when sailing on a great circle; this is done by imagining a perpendicular let fall upon the arc; the position is then found by right-angled spherical trigonometry, the rules for which are very brief. Excluding the right angle, the triangle consists of five parts, three sides and two angles. Two of those parts at least must be given to find the rest. If the two parts given and the one required are joined, the central one is called the “middle part.” If they are not joined (that is, an angle omitted between two sides, or a side between two angles, forming part of the equation) the part which is quite separate from the other two is called the ‘ ‘ middle part.” The solution is then by “Napier’s rules,” which have been embodied in the following rhymes :—
“The product of radius and middle part's sine Equals that of the tangents of parts that combine, And also the cosines of those that disjoin."
“Instead of the angles and hypothenuse
Their difference from ninety or complements use."
We know that the perpendicular will fall upon the arc between B and S, because both the angles already found are acute; thus the triangle is divided into two parts, each having one side and one angle known. Take the hypothenuse PS and angle S, to find the length of the perpendicular PR. Here are opposite sides and angles, therefore sin PS 40 o 52' × sin 8 66° 18'15 "^ sin pR 36° 48'30'', which is the rad complement of 53° 11'30", the highest latitude the ship will attain. To ascertain the longitude of that point it is necessary to find the angle RPS, which may be done according to the above rules in three ways, by varying the side or angle which is made the middle part. Thus, rad. + log cos PS *= log cot PSR + log cot SPR; rad. +log cos PSR = log cos PR +log sin SPR; rad. + log cos SPR = log tan PR + log cot PS .
As the part required is in each formula coupled with one which is known, the difference between that and the sum of the other two wi∏ give the angle SPR = 30° 7' 45"; which, added to the longitude at ship, will give 36° 49' 45" as the longitude of the highest point. As the original oblique triangle BPS has been divided into two distinct right-angled triangles, having the perpendicular PR common to both, the parts which form BPR may be treated in a similar manner for the purpose of finding the angle BPR and perpendicular PR, either in preference to the first, or as a proof. Other points on the arc may be found by taking a part of the angle at the pole, with the side and angle previously known, and thus forming another oblique spherical triangle, in which two angles and an included side are given to find the rest; or by assuming a distance on the arc (in degrees and minutes) with the angle and side before known. In that case two sides and an included angle will be given to find the other side (co-latitude), and the part of the angle P to indicate the longitude of the assumed point.
Given two angles and an included side, to find the other two sides :—
sin ⅜ diff. of angles × tan ⅜ side , 1 ,.~ « . 1
-—;—- ≡⅛—- ι — = tan ⅛ diff. of other sides; sin ⅜ sum of angles cos ⅜ diff. of angles × tan ⅜ side , . ~ ,, . 1
≤ ≡ =—- tan ⅜ sum of other sides.
cos ⅜ sum of angles
The second case where two sides and an included angle are given has been done. When the polar angle (difference of longitude) does not much exceed 50°, especially in loιv latitudes, an approximate curve may be drawn on Mercator’s chart by means of a pencil attached to a wire or batten making the curve to pass through three or four known points, after which it can be used from day to day as a guide to the ship’s course; she may not keep on it, but she can keep near it. This curve becomes very incorrect when the angle is large.
One other example of finding the course and distance by each method will be given. The position has been chosen to show the great distance an are of a great circle would carry a ship to the southward of her ordinary course, and also as touching upon the difficulty always experienced in finding a side or angle which is near 90°, in consequence of the very slight change in the value of large sines and small cosines.
A ship off the Cape of Good Hope in 34° 29' S. and 18° 24' E. is bound to Portland in Victoria. Assume a position outside the latter port to be in 38° 20' S. and 141° 40' E. The difference of latitude is 3° 51'=231 miles, the meridional difference of latitude 287Ί4, and difference of longitude 123° 16' = 7396.
rad. ×diff. long. 7396
merid. diff. lat. 287'14 = tan c0urse 87 46 37 > diff. lat. 231 ×rad. 1 . 1 cos course 87° 46' 37 " “dιstance 5955'1; diff. lat. 231 ×sec course 87° 46' 37" ,.
or , rad =distance 5955 Ί.
These two statements produce similar figures, because they are worked by the same course; an error of 10 seconds would produce oyer 7 miles in the distance, as the logarithm cosine and logarithm secant (arithmetical complements of each other) change very rapidly in large angles.
When the difference of latitude between two places is comparatively small, it is better to find the departure and use that in finding the distance. The table of meridional parts can be used for that purpose in two ways.
true diff. lat. 231 × diff. long. 7396 , „„„
merid. <iiif. lat. 287'14 = de P* 5950 ' 4 '
For this the logarithms of numbers alone are used. The middle latitude is best found by seeking the latitude corresponding to the mean of the meridional parts, which in this instance will be 2350 Ί3 and 36° 26'. The latter is the true central position on the chart, between the two parallels of latitude, the same as if measured on the side of the chart.
Radius bears the same proportion to the difference of longitude as the cosine of middle latitude does to the departure=5950 ’ 4. Then rad. ×dep. 5950 - 4 , „„ o
—,.. f 1 1 . oqi — =tan course 87 46 36; din. lat. 231
rad. X diff. lat. 231
or 5 rn - n . =cot course, dep. 5950 ’ 4
, dep. 5950’4 × rad. ... .... .
and -— i - o , ,>=distance 5954 ~9.
sin course 87 46 36
Having found by Mercator and middle latitude that a ship from the position off the Cape would, by steering S. 87° 46' E. for the whole distance, traverse 5954'9 miles before reaching the assumed point near Portland, it is now required to find an arc of a great circle passing through those points. The co-latitude of each place and the difference of longitude are given,—that is, two sides and an included angle; required the third side and angle C (fig. 15).
1st position 34° 29'; co-lat. = PC 55° 31' 18 24' E
2d „ 38° 20' „ =PV51°40' 141° 40 ' E
107° 11' 123° I6' = P
Half sum 53° 35' 30" 61° 38' = half P
3° 51'
Half difference 1° 55' 30"
8in⅜aiff. sides 1*55'30*x cotSP61·38' _ fan , /1iffAfjn . 1 . 1 y, a , sin ⅜ sum of sides 53 35 30
cos ⅜ diff. of sides ×cot⅞P^ tan sum of other tw θ angles 42 ° 16' 37 y cos ⅜ sum of sides
Consequently angle V is 43° 34' 5" and angle C is 40° 59 9', the larger angle being opposite the larger side. Having all the angles, the remaining side is found by the proportion which the sines of angles bear to their opposite sides; thus sin PC × βinP Γ sin VP × sin P Ί =sin yθ 90 ° 12 z .
sin V L ≡in C J
This example has been chosen because the side VC is so near to 90° that it can only be treated with certainty by letting fall a perpendicular from P to the arc, as represented by the. dotted line PH · which also gives the latitude of the highest point the snip would reach if she sailed on that arc. Thus the oblique-angled triangle, in which all the angles were known, is divided into two right-angled spherical triangles, each possessing one side and one angle previously known, and the side PH common to both is to be found. First, to find the side (or segment) HC. The angle at C is obviously the “middle part,” as it is between the known side and the one sought; they are joined or combined. Therefore rad. + log (co)sin C=log (co)tan PC + log tan HC; rad. ×cosC , ττr , ,_ o . n , “πStrc— tanHC4 ' 42 ∙
Similarly ⅛-⅜- ∣ m VH 42" 80 ’ . cotPv
The sum of the two segments will be 90° 12 ’, equal to the side VC, representing the distance of 5412 miles, which is 543 miles less than the rhumb line. Similar calculations must be made every day at sea to find the new course and remaining distance. It is necessary, before deciding upon a voyage following the arc of a great circle, to ascertain the shape of the curve and the highest latitude it will attain; for -which purpose the side PH must be found (fig. 15) by either of the segments as before explained; it will be 32° 43' 30", which is the complement of the latitude 57° 16' 30". To find the longitude of that point: by opposite sides and angles—
rad. ×sιn C H ^ sincpH 630 48 , 3 θ,, j sin PC
which added to the longitude of position off the Cape, 18° 24', gives 82° 12' 30" E. A glance at the chart will show that such a course would run the ship among the icebergs; therefore a position three or four hundred miles farther from the pole should be chosen, according to the time of year, and the course divided in two parts. It is easy to make a pencil curve on the chart in a lower latitude: the saving of distance would still be great.
The arc of a great circle from a position off Rio de Janeiro, 23° 5' S. and 43° 4' W., to the vicinity of Perth in Western Australia, 32° 2' S. and 115° 25' E., would measure 7270 miles. The rhumb line by Mercator (if it were possible) would be 8437 miles, or 1167 more. The highest latitude would be 70° 28', therefore impracticable. Such a curve as this could not be laid down on Mercator’s chart.
Soundings.— When approaching land in thick weather the precaution of taking frequent soundings should not be neglected, especially when the depth and nature of the bottom is clearly defined on the chart, as it is at the entrance of the English Channel. Loss of time should not be made an excuse for incurring a serious additional risk. Several ingenious devices have been provided by which the depth of 80 fathoms can be tested without stopping the ship, but the result is not so certain as the old plan of getting the line “up and down” and feeling the lead touch the bottom. If soundings are taken at ten or more miles apart, and do not coincide with those on the chart, it is a good plan to -write them up the edge of a strip of paper, preserving the distances the ship has run, according to the scale of the chart; when two or three are thus marked the slip should be moved about the chart, with the edge of the paper parallel to the course steered, till it coincides.
Observations. — The quadrant and sextant are practically the same instrument: the first is the eighth part of a circle, and by reflexion measures an angle of 90°; the second is the sixth part of a circle, and measures an angle of 120°. Both are fitted with verniers graduated in such a manner that the angle on the arc of the quadrant can be read to half a minute, and that on the sextant to ten or even six seconds. The handling of the instruments for five minutes will be better than a long description. The errors and adjustments are similar. The large movable mirror must stand perpendicular to the plane of the instrument. This is tested by placing the radius bar near the centre of the arc, and looking into the mirror at the reflexion of the uncovered part, which if all be right will continue in a straight line from the arc itself. If it appear broken, the screws must be moved. The error is difficult to remedy; therefore when practicable it is better to send the instrument to a maker. The fixed reflector, or horizon glass, has two adjustments. To make it perpendicular to the plane of the instrument, let the radius bar be placed near zero, and while the instrument is held quite vertical make the direct and reflected view of the horizon coincide, by moving the tangent screw. Then, if by sloping the instrument on one side or the other that line becomes broken, the glass is not vertical, and the screw for that purpose must be slightly moved till the lines coincide even when the instrument is held sixty degrees on one side or the other of the perpendicular. Also the reflected image of the sun or a star will cover the direct rays from them if the glass be vertical. If an horizon be not available, a distant hill or any sharp outline will do as well. The second adjustment is to make it parallel with the movable reflector. If at the end of the adjustment or test just described the radius stands at zero, or within two minutes, no adjustment is necessary. If it be much beyond that amount the radius should be set exactly at zero; and whil looking at the horizon, -which will appear broken, let the screw under the horizon glass be turned gently till the two parts coincide. To find the amount of index error accurately, the best method is that of measuring the diameter of the sun many times, which will give a number of small angles on and off the arc, as it is commonly called: that is, on the positive and negative side of zero, the mean of which will be the correction,—1- if it be off the arc, and - if on. Thus 32' 20" on and 30' 40" off would give a mean of - 50". The observer can at the same time test his own accuracy: thus, in the above example, the sun’s diameter appears to be 31' 30"; the Nautical Almanac will show if it were so on that day.
There is one error to which all sextants are liable that is seldom mentioned or attended to. It arises from the great difficulty of placing the centre of motion given to the radius bar and movable reflector exactly in the centre of the arc, or from the contraction or expansion of the metal. It has no connexion with the index error, and admits of no adjustment. Its existence and amount are not easily ascertained, but demand both time and patience. As it has no appreciable effect on small angles, it is advisable to use the artificial horizon, and take a set of altitudes, say ten, which will form a mean of about 100° on the arc, noting the time of each accurately by a trustworthy chronometer. Take similar altitudes in the afternoon, and work each set independently, as though to find the error of the chronometer (see below). Should the time so found coincide with the known rate of the chronometer, there is no error. Should the results differ several seconds of time, it may be assumed that the error of the instrument combined with personal error has caused it. By the rate at which the sun was rising or going down during the observations the amount of angle due to those seconds is-easily found. Half that amount will be the error of the sextant upon that angle. As an example, suppose the true reflected altitude to be 100° while the instrument made it 100° 1', the calculation would make it about three seconds later than the truth; in the afternoon a similar error would make it three seconds earlier. Thus a disagreement of six seconds arises for about one minute of altitude. By four or five such sets of altitudes at different parts of the arc sufficient data will be procured from which to form a table of corrections for all altitudes. This can be done by calculation, but a simple graphic method will be found sufficiently correct. Describe an arc of a circle with a radius of 6 or 8 inches. Let a line from the centre of projection indicate zero, from which lay off half the altitudes which have been observed, but give them the full numbers, c.g., mark 45° as 90°, to correspond with the motion of the radius bar and numbers on the sextant. From the points representing the altitudes lay off the errors towards the centre, from any convenient scale of equal parts, perhaps one hundred seconds to half an inch. Between zero and the centre of the arc will be found the centre of a second and smaller arc which will pass through all the points representing the corrections, or nearly so. The space between the two arcs may be remeasured at every 5° or 10° for the table of corrections (plus or minus, as the case may be).
Meridian Altitudes.— The first astronomical observation at sea. is usually a meridian altitude of the sun for the purpose of obtaining the latitude. The reflexion of the sun’s lower limb having, been brought by a sextant or quadrant to the edge of the visible horizontal on May 10, 1882, commencing a few minutes before noon, the greatest altitude that could be obtained was 58° 20' 15". The index correction was + 1' 30", and the eccentric error due to 58 degrees was-40". The Nautical Almanac shows the sun’s semidiameter on that day to be 15' 52"; as the lower limb was observed it is plus. The height of the observer’s eye above the sea was 20 feet, which on account of the spherical figure of the earth makes the altitude appear too great. The correction is taken from a small table for the “dip of the sea horizon,” which opposite 20 feet gives 4' 24"—always minus. The result will be the apparent altitude of the sun’s centre, which must still be corrected for refraction and parallax, the former on account of the rays of light from any object beyond the earth’s atmosphere being deflected upwards, and the latter because the sun does not appear to a person on the earth’s surface so high as it is in reference to the centre of the earth; the horizontal parallax is only nine seconds in the winter of the northern hemisphere, when the earth is nearest the sun. A table is usually arranged for refraction, minus the sun’s parallax; in one such table the altitude 58° 32' 33" requires a correction of 31 seconds—always minus. The result will be the sun’s true altitude, which taken from 90° gives the sun’s zenith distance 31° 27' 58", the sun being thus much south of the observer when on the meridian. By the Nautical Almanac it will be seen that the sun’s declination at apparent noon at Greenwich on May 10th was 17° 39'57"N., increasing 39" hourly. As the ship is in 7° W. long, nearly, the meridian passage was 28 m later, and about 18" must be added. This is of importance when the difference of longitude is great, as Greenwich time must always be found within a few minutes. The observer having found the distance he is north of the sun, and also that the sun is 17° 40' 15" north of the equator, the sum, or 49° 8' 13", -will obviously be his distance from the equator, which is the latitude.
Finding the latitude by a meridian altitude of a star differs from a similar operation for the sun only in so far as there is no semidiameter or parallax to be applied. Star observations are of great value after the sun has been obscured for one or more days, for which reason a navigator should know all the principal stars, so as to recognize them during a very partial clearance of the sky. The Naτdical Almanac gives the positions of all the principal fixed stars. The apparent right ascension of a heavenly body is its angular distance from the true equinox,—expressed by the abbreviation R.A. Sidereal time, given in page ii. of each month in the Nautical Almanac for mean noon each day, is the angular distance between the mean or imaginary position of the sun and the true equinox. Declination is the distance from the equinoctial or celestial equator. Mean time is that shown by a good clock, regulated upon the supposition that every day is of equal duration. Apparent time has reference to the time actually occupied by the sun between successive transits over any meridian. Equation of time is the difference between mean and apparent time. Sidereal time is measured by the transits of a star over any meridian. See Astronomy.
To find the time that any star will pass the meridian, subtract the sun’s R.A. from that of the star; or, in other words, find their distance apart as expressed in time by R.A. As an instance, on May 10, 1882, between 7 and 8 p.m. the sun’s R.A. was 3 h 10 lu and that of a Ursae Majoris 10 b 56 m ; consequently the star will pass the meridian of the ship in 7° W. at 7 h 46 m . The knowledge of the longitude is only nceessary for finding the Greenwich time (G.T.) and the sun’s R.A. The same night, the R.A. of Vega (α Lyrae) being 18 h 33 m and that of the sun 3 h ll m , the star was 15 h 22 nι after the sun, and would pass the meridian about 3 h 22 m a. μ. of next day. To know which suitable star will pass the meridian after a certain hour, add that hour to the sun’s R.A., the sum will be the R.A. of the meridian (decreased by twenty-four hours if necessary); the star table will then show the stars of that or greater R.A. Thus, on October 2, 1882, in 160° E. long., which bright star will pass the meridian after 10 p.m.? The difference of longitude in time being 10 h 40 m E., when the sun has passed the meridian ten hours it will evidently be forty minutes before noon at Greenwich, therefore the sun’s R. A. will be 12 h 33⅜ m at the time named. As it will be 10 p. Μ. at ship, the R.A. of any star then on the meridian must be (12 h 33⅜ πι + 10 h ) 22 h 33⅜ m . The first bright star found in the table having greater R.A. is Fomalhaut (α Pis. Aus.), the R.A. of which is 22 h 51 m , which is seventeen minutes more than the time sought; consequently that star will pass the meridian of the ship at 10.17 p.m., and Markab (α Pegasi) eight minutes later. Another ship on the same day being in 90° =6 hours west, the sun’s R.A. at her 10 p.m. w*ould be 12 h 36 m , and Fomalhaut would pass her meridian at 10.15 p.m., two minutes earlier by apparent time; the difference is due entirely to the change in the sun’s position. All doubt will be removed, when finding the approximate time of a star’s transit, if it be remembered that the sun (from which we reckon apparent time) indicates a position the R.A. of which is known; therefore by adding any time which it may have passed the meridian the R. A. of another meridian in the sky is obtained, called the R.A. of the meridian at that moment.
Having the true altitude of a star, and thus the Z. D. (zenith distance), we can determine the latitude without risk of mistake by remembering that the declination in the heavens corresponds with the latitude on the earth; therefore a star will pass the zenith of every place whose latitude corresponds with its declination. If a star having north declination passes south of the observer, the zenith distance must be added to the declination, as the latitude is then the greater. If such a star pass north of the observer, his north latitude will be less than the star’s declination, and the difference will be the latitude. The reverse holds good if the star has south declination and the observer is north of the equator. An old rule is also useful and simple :—If the zenith distances be invariably marked north or south according as the observer is north or south of the object observed, and the declination placed under it marked north or south, add like signs and take the difference of the unlike; the latitude will take the name of that which preponderates.
The oldest of all nautical observations is that of taking the height of the pole-star,—a star which everybody should know whether at sea or on shore. If it were exactly at the pole the corrected altitude would be the latitude, but it is not so. During 1882 its mean declination has been 88° 41', therefore it describes a diurnal circle round the pole with a radius of seventy-nine minutes. It was once far from the pole, and now advances nearly nineteen seconds annually. The mean R.A. has been during 1882 about l h 16 m , increasing about 28 s annually. The time it passes the meridian above and below the pole can be found as has been before described; and l o 19 ’ being subtracted from the upper transit or added to the lower one will give the latitude. The great consideration attached to the pole star is in consequence of the facility it affords, with very little calculation, for finding the latitude at all times when the star and horizon are visible. It is necessary to know the time at ship to the nearest minute. Apparent time is invariably kept on board ships at sea, corrected at noon each day. The change of longitude since noon must be roughly worked up in order to find the correct apparent time at ship, also the Greenwich time within half an hour, for the purpose of finding from the Nautical Almanac the sun s R.A. at that time, which added to the time at ship will give the R.A. of the meridian, rejecting twenty-four hours if necessary. Μ ith all nautical tables there is one for finding the latitude by the pole star, in which the correction is given for every ten minutes.
Enter the table with the year and R. A. of meridian. Opposite the latter, or by proportion, will be found a correction + or — to be applied to the true altitude; the result will be the latitude.
The principle of this table is that when the R.A. of the meridian coincides with that of the star the whole mean distance of the star from the pole for that year must be subtracted, while if the two R.A.’s differ by exactly twelve hours the same distance must be added, as the star is then on the meridian below the pole. If the star is six hours from the meridian on either side no correction is necessary. At other points the correction varies as the cosine of the time angle between the pole star and the meridian. Thus the R.A. of the pole star being l h 16 m for 1882, and distance 79 minutes, the correction corresponding to five hours is thus found:
5 h _ ih ι 6 m _ 3h 44m. 79×cos3 h 44 m = 44 , 1 θ w rad.
(Multiply the decimals taken from the logarithms of numbers by six, to bring them into seconds. ) As the star was less than six hours past the meridian the correction is -. A similar calculation having been made for every ten minutes of one quadrant, the result may be applied inversely to the one below it, and the remaining twelve hours written on the opposite side of the page with the sign + or - the reverse of the first column.
It is evident from the above that the table may be easily dispensed with; also, as annual tables must be founded on the mean position of the star during that period, greater accuracy would be obtained by taking the R. A. and declination for the day required from the Nautical Almanac, and finding the corrections as above.
A table for finding the latitude by the pole star when off the meridian is also given in the Nautical Almanac with instructions. It is used with sidereal and mean time; also an allowance is made for the spheroidal figure of the earth. When the latitude found by any star observation at sea is within one mile of the truth it may be considered satisfactory.
Latitude by the meridian altitude of the moon is obtained in a manner similar to that applied to the sun, but not so simple. First it is necessary to ascertain the approximate time of transit, by reference to p. iv. of each month in the Nautical Almanac. It is there given in mean time both for Greenwich and the antipodes; to one of those periods apply the proportion due to the ship’s longitude in time. It is necessary to guard against a false horizon often produced by clouds under the moon. Having obtained the meridian altitude, correct for errors of instrument and dip. From Nautical Almanac, p. iii., take the moon’s semidiameter (calculated from the centre of the earth) for the nearest noon or midnight, augmenting it by a table for the purpose, as it increases with the altitude (only a few seconds); add it if the lower limb were observed, and subtract the refraction due to that altitude. Take from the Almanac the horizontal parallax for the time of observation, correcting it for decrease in proportion to altitude by a table for the purpose, and add it to the apparent altitude. Tables generally give the parallax due to altitude, minus refraction, in one correction. Having the true altitude, take it from 90°, and mark the zenith distance N. or S. as it is north or south of the moon.
Take the declination for the nearest minute of Greenwich time. It is now given for every hour (mean time), and the change in ten minutes. Place the reduced declination marked N. or S. under the zenith distance; add if the signs are like, or take the difference if unlike, and the result will be the latitude.
Ex-mcridian Altitudes.— It is important to be able to get the latitude when the sun may have been obscured from a few minutes before noon till some minutes after. Such observations near the meridian are called ex-meridian altitudes. When the sky is cloudy the observer should keep the watch in hand in order to secure the apparent time with the altitude nearest the meridian. If the sun be rising fast or declining fast, it will be better to work the sight with two latitudes, and with the assistance of the chronometer, as described below. In a low latitude and less than half an hour from noon, the pamphlet by J. T. Towson, containing tables for the reduction of exmeridian altitudes, will be found very useful, as the correction or allowance for what the sun would further rise (or had risen) can be taken out entirely by inspection, which saves the labour of working four propositions. The principle is illustrated by fig. 16, which is projected with double the angle at P in order to make it more intelligible. PS represents the sun’s polar distance when observed, and Μ his position when he reaches the meridian. Consequently PS and PM are sides of an isosceles triangle, having the distance between the two positions as a base, and equal angles since the sides are equal. The thing sought is the length of ZM, or where PM is to be divided at noon; the lower part being the true zenith distance and the remainder co-latitude. The data are the same as those in Towson’s first example, which he there solves by his table only. Time from noon 28 m 14 s ; sun’s true altitude 24° 53' 53", and declination 18° 35' 16" S: to find the angles at the base we use the supplements of the legs, and get sec 71° 24' 44" (180° - 108° 35' 16") × cot ⅜P[ = 14 m 7 s ] = tan88° 52' 25", the supplement of each angle at the base; and sin ZS 65° 6' 7" × sin PS 108° 35' 16" . τ , r , c , ≡ι-⅛-u∙ smpzs ∙
As PZS is very obtuse, and the supplement is really required, it is taken out at once as the sine of 7° 22' 38", which is the angle MZS. The angle PSZ is next found; it is between two known sides, and the other angles are known, but not the opposite side :—
cos ⅜ sum of sides x tan ⅜ sum of angles , 1 tλγ , „ , j-. a ,—⅛ —=cot ⅜P⅛Z , cos ⅜ dιfi. sides which is 5° 22' 12"; taking this from PSM 91° 7' 35", we have the angle ZSM 85° 45' 23". Finally, sinZS×sinZSM . 0 „
— r > v ,-5 sin ZM 64 47 22 , sm ZMS
the true meridional zenith distance, which taken from the polar distance gives co-latitude 43° 47' 54", showing that the sun was 18' 45" higher when on the meridian. When the sun is on the equator, both the angles at the base are 90°, and the base MS will be the measure of the angle MPS. Norie’s method is short, but the latitude by dead-reckoning is used, which may cause the whole operation to be repeated. Also the use of a “log-rising” and natural versed-sines is objectionable. No one possessing a copy of Towson’s table need take the trouble to work the problem when in actual want of the latitude.
Double Altitudes.— Another mode of obtaining the latitude is by two altitudes of the sun or a star, with the interval between the observations carefully noted, or by simultaneous altitudes of two stars having a suitable difference in bearing. These are called ‘ ‘ double altitudes.” The principle is the same in each case. The elapsed time in the one and the difference between the right ascensions in the other indicate the angular distance at the pole. Fig. 17 represents two observations of the sun on the same side of the meridian,—the dark lines showing the parts which are given and the dotted lines those which are to be found.
On October 2, 1882, the estimated latitude was 45° 10' N. and longitude 35° 30' W. The first altitude was taken about 8.13 a.m. apparent time, and when corrected for errors of instrument, dip, refraction, and parallax was 20. The second altitude, corrected, was 39° 15', taken about 10.58 A.M. The elapsed time by a good watch was 2 h 44 m 33 s , which being mean time is reduced to apparent time by adding the proportionate increase in the equation taken from the Nautical Almanac, which for 2∣ hours will be two seconds additive. Therefore the change in apparent time was 2 h 44 m 35 s =SPβ.
Each altitude taken from 90° will give a zenith distance, 70° and 50° 45' respectively=ZS and Zs. Having the apparent time at ship and the longitude, which converted into time will be 2 h 22 m , we take the declination of the sun from the Nautical Almanac for apparent noon and correct it for the Greenwich time of each observation, obtaining 3° 36' 56" S. and 3° 39' 36" S. Those amounts added to 90° will give the sun’s polar distance at each period = PS and Ps. Thus four sides and one angle are known, which is all that is required to find the complement of the latitude PZ.
We shall work out this by the most simple method, in order to show the principle upon which it is done. First, in the triangle sPS two sides and the included angle are given, required the other two angles. Add together the sides Ps (93° 39' 36") and PS (93° 36' 56"), and find their half sum and half difference :—
sin ⅜ diff x cot ⅜P _ ^g∙ 0 ⅛er angles ≈ 0° 3' 36"; sin ⅜ sum i ° ’
cos⅜diff. xcot⅜P , , . x , 1 f , no ... n ,, i —=tan ⅛ sum of other angles = 88 38 9 .
cos ⅜ sum 2 5
The half difference taken from the half sum gives the smaller of the two angles = 88° 34' 33', and the two added give the greater. Had there been no change in declination, the two sides including the elapsed time would with the side Ss form an isosceles triangle, and the first equation would be unnecessary.
As the two sides were each more than 90° the angles found are the supplements of those required. One is sufficient for the present purpose; 180°-88° 34' 33" = 91° 25' 27", which is PSs opposite the larger side. That angle includes the two angles PSZ and ZSs, which must be separated in order to obtain PSZ. The side Ss, which is the spherical distance between the two positions of the sun, is given by the equation ri s P,x⅛,,P8
sin PSs
In the second part of the problem three sides are given to find an angle, namely ZSs. Write under each other the values of the two sides (SZ and Ss) enclosing the required angle and also the opposite side Zs. Find the half sum of the three sides and the difference between it and the opposite side (Zs). Add together the logarithm cosecant of the first two and logarithm sine of the last two; half the sum of those four logarithms (rejecting twenty from the index) will be the log. cosine of half the required angle. Thus ZSs =52° 36' 36", which taken from PSs will leave 38° 48'51" as the angle PSZ. The third part is similar to the first,—two sides and an included angle given, to find the rest, principally the side PZ
sin ⅜(PS^-.ZS) x c ot ⅜PSZ = tøu ^ gzp _ zpg) 3 θ o 24 , 21 „
A similar statement using two cosines instead of sines will give the half sum of the angles = 87° 3'51". By subtracting the half difference the smaller angle is obtained, ZPS = 56° 39' 30", which by the figure should evidently be placed opposite ZS. Finally, sin ZS× sin PSZ . π7 . . o co ,, -.—== is =sιn PZ 44 49 58 , sin ZPS
which is the complement of the latitude 45° 10' 2". If the first observation of the sun had been used to find the longitude by chronometer (the ship being stationary between the observations) the hour angle would coincide with the angle ZPS, and afford a test of its correctness. Simultaneous observations of two stars would be worked in a similar manner, with the advantage of having the difference of R.A., SPs, the spherical distance Ss, and the angles at S and s nearly permanent (requiring only a slight annual correction), consequently that part of the calculation might be used many times. Kerigan’s tables have been mentioned as giving the distances between the principal stars for this purpose. When two altitudes are taken of the same star, the interval by a watch keeping mean solar time must be reduced to sidereal time, as the stars pass the meridian quicker than the sun by 9≡ ’ 8 per hour.
The above illustration has been treated as if the ship were stationary, which at sea would seldom be the case. The bearing of the sun or star should be taken at the time of the first observation, and the run of the ship should be worked with reference to that bearing and also with regard to the change in longitude. Every mile which has been made good in a direct line towards the sun must be added to the first corrected altitude; if from the sun, it is subtracted. For every mile of longitude four seconds must be subtracted from the elapsed time if the ship ran west or added if she ran east. The result will then be the same as though the ship had been stationary at the place of the second observation.
Had the observations been taken on opposite sides of the meridian the calculations would be similar to the above, and the figure would differ only in having the triangle PsZ folded over, as it were, to the opposite side of the central line.
Other Double Observations.— The double altitude, though a very interesting problem in connexion with navigation, is at the present day of very little practical utility. When ships are provided with chronometers both latitude and longitude can be found more readily by two distinct observations of any of the heavenly bodies on different bearings. To find the longitude by means of a chronometer and also the variation of the compass should be one combined operation under ordinary circumstances, therefore they will be described together. It is desirable that every navigator should accustom himself to act independently of an assistant. A pocketwatch with a distinct second hand being taken to the chronometer, the hours and minutes about to be shown by each may be noted in the “deck-sight book,” and if the observer then counts the halfsecond beats of the chronometer for five seconds previous to the even minute, the tenth beat will answer the purpose of the word “’top” of the assistant, and more correctly. Let a similar comparison be made after the sights have been taken, and if there be any difference adopt the mean, or proportion. Take one or three bearings of the sun’s centre with the azimuth or standard compass in its usual place, noting the time, and then take three or five altitudes of the sun’s lower limb. The watch being held in tho palm of the left hand, the fingers are free to move the tangent screw till the moment the contact is formed between the sun’s reflected image and the edge of the horizon, when the eye is dropped from the telescope of the sextant to the face of the watch, and the second or even the half second noted; the minutes should be written before commencing, and seen to be correct on finishing the set. Take bearings again of the sun as before, noting the time. Make the mean of the sun’s bearings correspond with the mean of the time of sights (i.e., the sun’s altitudes).
On October 17, 1882, about 8.24 A.M. apparent time, in 38° 15' N. and 6° 50' E., observations were taken to ascertain the longitude and variation of the compass. The mean of the times by watch when the altitudes were taken was 8 h 25 m 35 s- 5. The result of comparison between the watch and chronometer was - 7 m 14 3 , the negative sign meaning that the watch was faster than the chronometer. The latter was known to have been fast on that day 36 ra 3 s . Therefore, both those amounts being subtracted, the mean time at Greenwich was 7 h 42 m 18 s , 5. The mean of five observations of the sun’s lower limb was 20° 42' 30". The corrections were index error + 2'; eccentric error (in proportion to altitude) - 15"; dip of the horizon corresponding to the height of eye, 24 feet, - 4' 50"; sun’s semidiameter + 16' 5"; refraction, minus parallax, - 2' 22"; hence the true altitude of sun’s centre was 20“ 53' 8" and zenith distance 69° 6' 52". The sun’s declination is more conveniently taken from the Nautical Almanac for mean noon, as the mean Greenwich time is always known by the chronometer, but the hourly variation is given on the opposite page. Thus, on the day in question, 9° 19' 29" S., having been corrected for 4 h 18 m before mean noon, gave 9° 15' 33' l as the sun’s declination at the time of observation. We have now to find the hour angle and azimuth from three sides :—viz., co-latitude 51° 45', co-declination or sun’s polar distance 99° 15' 33", and the zenith distance 69° 6' 52" (fig. 18). For convenience let these be placed under each other as mentioned, take their sum and half sum (110° 3' 42"); also the difference between the half sum and the side opposite the required angle, the zenith distance, = 40° 56' 50". Add together the logarithm cosecant of the first two and logarithm sine of the last two, remembering that when the degrees exceed ninety the supplement must be used, or the amount over 90° with reverse term. Half the sum of those four logarithms, rejecting twenty from the index, will be the cosine of half the angle required (3 h 35 m 48≡ ’ 4), the time from noon. This is the principle upon which the angle is found, but it is rather quicker to use the latitude and declination direct. Write the latitude, declination, and zenith distance under each other, marking the declination + if on the contrary side of the equator to the latitude, and-if they are on the same side. In the present instance the latitude being N. and declination S. the latter is additive. Take the difference between the half sum and the zenith distance, that still being the side opposite the required angle. Add together the secant of latitude, secant of declination, sine of half sum, and sine of difference; the sum of those four logarithms, rejecting twenty from the index, will be the sine of half the hour angle, l h 47 m 54 s, 2. That being doubled and taken from twelve hours shows that the apparent time at ship was 8 h 24 m 11≡ ’ 6. The equation at mean noon was 14 m 35 s- 1, the correction for four hours and eighteen minutes before noon was-2 s ; leaving 14 nι 33≡∙1 to be subtracted from apparent time (as the sun was then in advance of a mean clock). The result for ship mean time is 8 h 9∞ 38 s - 5, the difference between which and the Greenwich mean time as found by the chronometer, 27 m 20 s , is the longitude in time = 6° 50', and it is east, because ship time is the greater’ The azimuth would in practice be partly sought at the same time as the hour angle; as the sine of the polar distance or cosine of the declination can be taken at the same opening of the tables where the cosecant or secant was found. We have then sin P X sin PS . r , :—≈3 sin Z , sin ZS ’
the sun’s azimuth or angle from north, observing that when the angle is more than 9(Γ the supplement is found opposite the log. sine, and the amount over 90° opposite the cosine. Thus sinP3 h 35 m 48 s × sin 80° 45' (or cos 9° 15') . „„„ , λ , sinZS 69° 7' -sin 58 40, which is the supplement of PZS, and therefore reckoned from the south, which in this instance is most convenient, as the mean of bearings taken of the sun, by the standard compass, immediately before and after the altitudes for time, was S. 43° 30' E., which taken from the sun’s true bearing S. 58° 40' E. showed an error of 15° 10' westerly,—that is, that the north point of the compass was so much to the westward of the true north. It is unnecessary to reckon seconds.of angle in an azimuth. It is always necessary to note the.direction of the ship’s head at the time of observation, as there is in general an amount of local deviation caused by the iron in the ship. . In the instance given the ship’s head was W. N.W., on which point, as previously ascertained, the local deviation was 2 westerly; consequently the variation of the compass when freed from the ship’s influence was 13° 10' westerly.
When it is desirable to take an azimuth without finding the hour angle as above, there appears no clearer or better mode than that of treating it as a spherical triangle in which the three sides are given to find the angle opposite the sun’s polar distance.
From the above example take the zenith distance, co-latitude, and polar distance; place the degrees and minutes only under each other in the above order; take the difference between their half sum and the polar distance, which will be 10° 48'. Add together the logarithm cosecant of the first two terms and the logarithm sine of the last two; half the sum of those four logarithms will be the cosine of ⅜PZS. Thus PZS = 60° 40'× 2 = 121° 20'. This is evidently from north, consequently the supplement is S. 58° 40' E. as before. It is sometimes necessary to have recourse to this mode of finding the sun’s bearing even when the hour angle is known, as it may be difficult to decide which side of the east or west line the sun may be upon when the angle is near 90°. As the last method brings out the angle in two halves there can be no mistake.
Position by Cross-Bearings of tlιc Sun.— The most practical method of obtaining the greatest amount of information and immediate benefit from observations similar to those just described is by laying oft’ the position on the chart (especially if it be on a large scale), using the estimated latitude by which the sights were worked, and the longitude found by the chronometer, the accuracy of which depends on that of the latitude. Through that position draw a line at right angles with the sun’s bearing; in the above instance it would run N. 31° 20' E. and S. 31° 20' W. Though the latitude were wrong by many miles, the ship would, if all else were correct, be somewhere on that line. For the line thus drawn corresponds to a small arc tl in fig. 18, drawn at right angles to ZS, and all positions for which the sun’s zenith distance = ZS lie on that are.
As βn illustration, suppose that the above sights were worked by a dead-reckoning latitude, which the next observations about 10 h 14 nι will prove to be 17 miles too far south. The ship sailed during the interval W.N.W. 15 miles, -wind south, leeway half a point, variation + 13° W., and local deviation-2° W. Hence the true course was N. 77° W., which by inspection, under distance 15, gives diff. lat. 3'4 + , and dep. 14’6. The latter opposite the complement of the latitude (38°) gives 18' ’ 5 in the distance column, which is the difference of longitude, subtractive as it was westing and the longitude was east. The latitude was then assumed to be 38° 18' 24" and longitude 6° 31' 30". Draw the bearing taken at 8 h 24 m through the newly found point, which will represent the line the ship was on at 10 h 14 m . Five observations of the sun were then taken, and the mean of the times by watch was 10 h 17 m 39 s ; the comparison showed - 7 m 16 s , and the error of chronometer (fast) - 36 m 3 s . Greenwich mean time was 9 h 34 m 20 s . The equation was at that time 14 m 34 s + to mean time, and the corrected declination was 9° 17' 15" S. The mean of the altitudes when corrected gave a true zenith distance of the sun’s centre 53° 39' 52". All the data are thus given to find the hour angle as before described = l h 46 m 20 s , and apparent time at ship, 10 h 13 m 40 s ; subtract the apparent time at Greenwich, 9 h 48 m 54 s ; the difference, 24 nι 46 s , is the longitude in time, =6° 11' 30" E.
As the longitude carefully worked up from the former sights appeared to be 6° 31' 30", it is evident they have both been worked by an erroneous latitude, which is easily discovered.
The sun’s bearing was not taken at the time of the second set of observations, as it was too high for accuracy, but the true bearing was calculated as before described and found to be S. 33° 15' E., a right angle to which was E. 33° 15' N. If a chart on a sufficiently large scale be in use (or if one of similar latitude can be substituted) the most simple mode is to lay down the two positions (as A and B, fig. 19) on the same parallel and draw the two lines through them, as As: and Ba:. The point of intersection will be the ship’s true position, which ean be proved by working the second set of observations again with the corrected latitude. If the chart be too small, the diagram can be made on a piece of paper, the difference between the two longitudes being laid off from any scale of equal parts; e.g., BA is made to measure 20 and becomes the scale for the remainder of the figure. The lines at right angles from the sun’s bearing intersect at x, whence a perpendicular drawn with the protractor will give y on the base line. Then By is the total error in longitude to be applied to that found by the second observation, in this case 33'‘3, which added to 6° 11' 30" gives 6° 44' 48". The perpendicular xy represents the error in latitude, and would be correctly measured on the side of the chart as 17 - 2. But from the plain paper the numerical measurement would be too great in the ratio of the difference of longitude to the departure.
The measurement of xy, on the scale BA = 20, would be 21 - 8. By inspection (in traverse table) distance 21 - 8 opposite co-latitude 38° is found to give departure 17’2, which in this instance will represent the miles of latitude increased in size to suit the miles of longitude upon which the diagram is formed. That is the required correction to be added to the latitude assumed at the time of the second observation, and will give 38° 35' 36" as the true latitude at that time. The same corrections may be .found arithmetically by plane trigonometry, in which case it will be desirable to draw a rough figure showing how the lines cross. This mode of correcting the latitude admits of easy proof; by working the second set of sights with the latitude increased by 17' 12", the longitude will come out 6° 45'. Another good practical method is to work each set of sights by two latitudes, 10 or 20 miles apart. The logarithms can be taken out for the two with little more trouble than one, and they form a check on each other. When the two sets of sights are not worked by the same latitudes, but by latitudes in accord with each other by allowing for the run of the ship during the interval, and when the change of longitude due to 10 miles of latitude is known, the true position can be found by placing equivalent proportions under the first longitude brought forward and the second as found, till they agree.
By such double observations H.M.S. “Devastation,” in January 1847, was enabled, after having been set by current 28 miles past Goree, to which she was bound, to alter course at 11 a. μ. instead of increasing her distance by another hour’s steaming.
By this method the greatest accuracy of position may be maintained throughout the day and night, if the horizon be clear. The one line is very useful though the other may never be obtained, which is not so with the double altitude, where nothing is known till the whole calculation is completed.
In high latitudes during the winter the sun is of little use in finding the longitude; bright stars and planets must be used, taken near twilight when the horizon is clear, and also when they are near east or west if longitude only is required, and with sixty to ninety degrees difference in bearing at other times.
The only difference in the treatment of stars and planets from the treatment of the sun when finding the longitude consists in applying the difference of their right ascensions and that of the sun, reduced to the time of observation in order to obtain apparent time at ship, and from it mean time, to compare with that shown by the chronometer as Greenwich mean time. A mistake in the manner of applying the difference in right ascension to the hour angle found by a star is less likely to occur in actual practice than in a problem taken for exercise. The stars may be considered as marks in the heavens which pass the meridian so many hours and minutes after the sun, until it becomes more convenient to reckon how much they are before the sun,—remembering that, as the sun moves from west to east among the stars at a mean rate of nearly one degree or nearly four minutes of time daily, its right ascension must be carefully noted. It has been attempted to find the longitude by marking the time of sunset, without the aid of any instrument except the chronometer, but this process is very rough on account of the uncertain amount of refraction; and if the lower limb is allowed to touch the bright reflexion which appears to rise to meet it, the correct angle will be lost, as the centre of the sun will then be about 18 minutes below the horizon, in addition to the “dip of the horizon.”
Lunar Observation.— The moon is the least serviceable of all the heavenly bodies for the purpose of finding apparent time at ship and longitude by chronometer, in consequence of its rapid motion entailing more care in the corrections, though the Nautical Almanac gives its place with great accuracy for every hour, and the variation in ten minutes. That rapid and uniform motion rendered it the most valuable and at sea the only means of ascertaining the longitude after the construction of almanacs, and before the present perfection of chronometers. In presence of the latter lunar observations have fallen into the shade, and like double altitudes are found principally in examination papers. As the moon passes the stars at the mean rate of 33" of angle in one minute of time, it is obvious that an error to that amount in measuring the distance from a star would produce an error of 15 miles in longitude. As the moon’s motion with regard to the sun is nearly one degree a day less, a similar error in the distance would produce still more effect.
The Nautical Almanac gives the true distance between the centre of the moon and the sun while within range by the sextant, as it would be seen from the centre of the earth; and in like manner the distance from some of the principal fixed stars and planets, for every three hours of Greenwich mean time, with the proportional logarithm of the change in that time. It is essential that the star from which the lunar distance is measured should be near the ecliptic, in order to obtain the greatest amount of change. If the star is quite in the path of the moon the distance may be small, when the measurement will require less steadiness.
We take a star lunar as an example, which will introduce the problem of finding the time at ship by a star (fig. 20). On November 18, 1882, about 9 p.m. apparent time, the ship was in 38° 10' N. and 45° 15' W. by dead-reckoning; height of the eye 20 feet. There was not a chronometer on board. An assistant was employed to show a light on the arc of the sextant, to take time, and to write down; consequently the same sextant was used. An altitude of the moon’s lower limb was taken before and another after the other observations, and the mean was reduced by proportion to what it would have been if observed at the same instant as the mean of the distances, namely 35° 15'. Two or more altitudes of Aldebaran were taken before the distances and as many after, the mean of which when reduced to the same instant was 35° 6'. The mean of several measurements of distance from the moon’s farthest limb was 94° 40' 18". The index correction was +1' 20", eccentric error - 2' 9", semidiameter at midnight was 16' 9", which was augmented by 9" (taken from a table) in consequence of the moon being nearer the observer as it approaches the zenith. As the farthest side of the moon was used, the augmented semidiameter was subtracted. The corrected observed distance was then 94° 23' 11".
The observed altitude of Aldebaran was 35° 6', index correction + 1' 20", eccentric -25", dip of the horizon -4' 24"; giving apparent altitude 35° 2' 31" and Z.D. 54° 57' 29". Apply refraction - 1' 23"; then the true altitude is 35° 1' 8" and true Z.D. 54° 58' 52". Add the difference between latitude 38° 10' N. and declination (from Nautical Almanac, p. 327) 16° 16' 21" N. to true Z.D. 54° 58' 52", and note the difference betweeen the half sum and the true Z. D. Add together the logarithm secants of latitude and declination and logarithm sines of half sum and difference; half the sum of the four logarithms will be the logarithm sine of half angle SPZ = 3 h 51 m 43 s- 8, which is the time the star was east of the meridian. By adding the longitude in time to the time at ship it will appear that Greenwich apparent time was about midnight, when the sun’s right ascension was 15 h 37 m 6 s . That of the star being 4 h 29 m 14 s , it is evident that the star would pass the meridian l∙l h 7 m 52 s before the sun, and 12 h 52 m 8 s after it. The latter is more suitable at present. Taking the time the star is from the meridian from the time it is after the sun, it is seen that the sun had passed the meridian 9 h 0 m 24 s- 2, therefore that is the true apparent time at ship, provided the latitude was correct by which it was found. Hence the necessity of choosing a star when nearly east or west. The equation of time taken from the Almanac and reduced to apparent midnight was 14 m 32 s '5, and the result was ship mean time = 8 h 45 m 51 s *7.
The altitude of the moon’s lower limb was 35° 15', index correction +1' 20", eccentric error - 25", dip - 4' 24", semidiameter +16' 9", augmentation 9"; giving apparent altitude 35° 27' 49" and apparent Z.D. 54° 32' 11". To the latter add the refraction 1' 22", and subtract the parallax due to altitude, 48' 12". The result is true Z.D. =53° 45' 21". The horizontal parallax is the angle at the centre of the moon subtended by the semidiameter of the earth, and the parallax due to any altitude of the moon (that is, the amount by which the apparent altitude would increase if the observer were sunk to the centre of the earth) is given by the equation—
hor. parallax in seconds × cos of ap. alt. 11 . .,
£ t = parallax in alt.
It must always be added to the apparent altitude. With all navigation tables there is one for reducing the parallax, sometimes to every two minutes in altitude. The refraction is then included in the correction.
The most important part of this problem· is finding the true distance between the star and centre of the moon at the moment of observation, from the data already obtained; it is confined to the triangle ZSM (fig. 20). The two apparent zenith distances 54° 57' 29" and 54° 32' 11" with the apparent distance (that is, as measured) are drawn to scale, and from the three sides we find the angle MZS = 127° 53' 54".
As the correction applied to the altitudes of moon and star on account of parallax and refraction were in a direction to or from the zenith, the angle between the two zenith distances will be unaffected by any change in the length of those legs. We have therefore the true zenith distances Z?n 53° 45' 21" and Zs 54° 58' 52" with the included angle just found, to find the third side—a problem which has been already exemplified (see fig. 15 and the accompanying calculations). The true distance being over 90° will come out as a small cosine, the change in which (at that angle) is only one unit in the sixth place of figures for seven seconds. It is desirable to avoid angles of this description, which can be done by dropping a perpendicular from Z to the required side ms, which would then be found in segments, by two right-angled spherical triangles, as exemplified with fig. 15. The sum of the two segments, 93° 48' 36", is ten seconds more than the result obtained directly from the oblique-angled triangle, and is more likely to be correct.
Take the true distance as 93° 48' 30", and find in the Nautical Almanac an approximate distance, which was 93° 40' 45" at midnight of the 18th; the difference is 7' 45". As the moon was approaching the star the greater distance was before midnight. There are tables of proportional logarithms arranged for finding readily the proportion which three hours or three degrees bear to any part of either, or one part to another. The Nautical Almanac gives after each distauce the logarithm due to the amount of change during the next three hours, to four places of figures. As the distance in the present instance is so close to one of the periods, namely midnight, it is necessary to take the mean of the two logarithms before and after, which is - 2314. The difference between it and the logarithm for 7' 45" = l - 3659 is 1 - 1345, =13 m 12≡ , 5; or it may be found by simple proportion. Greenwich mean time was therefore ll h 46 m 47 s- 5, and ship mean time as found by the star 8 h 45 m 51 s ∙5; difference 3 h 0 m 56 s = 45° 14' W. long.
There are various graphic methods for finding the correction to the distance due to parallax and refraction, for which space cannot here be spared, as they are at best but approximations, and belong rather to the progress of the art. See Kelly’s Spherics.
The moon is occasionally used for finding the longitude with the aid of a chronometer, and the method is then similar to that with a planet or star, except that the parallax is so great and the change in right ascension and declination so rapid that the greatest care is necessary in taking out the corrections to the nearest minute of Greenwich time. A star appears to be far preferable for the purpose. The horizon under the moon sometimes assumes a bright sharp appearance which appears tempting; but caution is necessary, as a bank of cloud below the moon sometimes cuts off the more distant part of the water and presents a false horizon.
It should be mentioned that extraordinary refractions sometimes occur in the daytime tvhich make the horizon appear many minutes of angle higher than it really is. That has been observed in a very remarkable manner in the Baltic. The results of apparently good observations of the sun for the chronometer were found to be very erroneous, and, a short time after, the island Gottsko Sandö was seen presenting a second waterline above what had previously appeared to have been the horizon. On another occasion, April 21, 1854, the low islands and land east of Stockholm were seen from a distance of 12 miles as if inverted on the sky, presenting the appearance of level tableland, with dark cliffs, about half a degree above the horizon. On approaching, the upper line gradually disappeared, leaving the low islands covered with fir trees, in their natural state. It was calm at the time, and there was much ice in the Gulf of Bothnia. Fortunately such a state of the atmosphere is not frequent. In the Standard of August 12, 1882, mention is made of an extraordinary mirage at Alexandria on the previous evening, which caused the forts about Aboukir Bay, distant 15 miles, to be plainly visible.
Artificial Horizon.— The artificial horizon is a most useful contrivance whereby the effects of mirage or fog are avoided, but unfortunately it can only be used on solid ground, though many attempts have been made to overcome the effect of the ship’s motion. The artificial horizon in common use is composed of a wooden or copper trough 5 or 6 inches long by 3 to 3⅜ inches broad and ⅛ inch deep. This being placed on a solid stone or hard ground, 100^ yards from vehicular traffic, and at least 20 yards from foot passengers, mercury is poured into it from a small-necked bottle, till the bottom of the trough is covered. In pouring, the neck of the bottle should be kept well down in order to keep back the scum. A metal frame carrying two pieces of plate glass, which are fixed at 45° from the horizon, and at right angles to each other, is placed as a screen over the trough and mercury, to keep off the wind. By this arrangement of the glasses forming the screen, the refractions which a ray of light suffers when passing obliquely through the glasses destroy each other. The observer may be seated close to the horizon if the sun or star be high, and the whole disk of the sun will be seen; but with a low altitude the image passes into the box so obliquely that the observer must place himself at a distance and be content with seeing the upper half of the sun’s reflected image, to which is brought by the sextant the reflexion of the sun’s lower limb. An object with less altitude than 6° or 7° cannot be taken, nor a greater angle than 60°. By using suitable shades the two reflected disks will appear so sharp that a very accurate contact may be made, and repeated as desired.
Rating Chronometers.— Observations in port with the artificial horizon are used to rate chronometers. For this purpose a good watch should be compared with each before leaving the ship, and immediately on returning. The angle measured by the artificial horizon must be halved, after instrumental corrections have been made, and from the altitude thus obtained the mean time at the place can be found just as at sea. But in the present case the longitude is known, and Greenwich mean time will be found from the local mean time by adding or subtracting the longitude in time, according as the latter is W. or E. The difference between it and the time shown by each of the chronometers at the same instant will be their errors, which taken from the last error found and divided by the number of days and fractions of days will give the rate. The errors of chronometers should be ascertained at every port, and every ten days if a ship remains long at one port. The results being tabulated, the character of a watch during a year can be seen at a glance.
Chronometers should be kept in a dry place and equal temperature, free from vibration, not too softly padded (for fear that the motion of the balance-wheel should be imparted to the body of the watch); and they should not be carried about for any purpose.
“Equal altitudes” is a term which signifies a mode of obtaining the time by means of altitudes of the sun or a star taken at each side of the meridian; it is more accurate in result than any other method, as all errors, personal or instrumental, are neutralized by the repetition, as are also errors in latitude. If the object observed were perfectly stationary with regard to the celestial concave, as a star may be considered to be, one-half the elapsed time added to the time of the first observation would be the time of transit; and, the difference between the star and sun in right ascension being applied to this, the apparent time at place would be obtained. The motions of the planets vary in amount, and must be treated accordingly. Jupiter’s orbitual motion is so slow that the chauge during an interval of three or four hours may be disregarded, excepting where great accuracy is required.
When equal altitudes of the sun are observed, a considerable correction has usually to be made in consequence of the change of declination during the interval. Thus in the example illustrated by fig. 21, though the zenith distance was precisely the same before and after noon, the sun increased its distance from the pole, and instead of passing by the upper dotted line from S to s it is found to join the curve representing the zenith distance farther south, say at d. During three or four days at each solstice the change in declination is very slight, and may therefore be disregarded; also when the sun bears nearly east or west the change of a few minutes in declination will not affect the hour angle. If when on the meridian the sun be nearly vertical, and a sea horizon available, an altitude correctly timed ten miuutes before noon and ten after will give the time of transit without any calculation.
In order to secure a set of observations which shall have their mean exactly on either side the meridian, it is customary to fix the arc of the sextant in positions five or ten minutes apart (according to the motion of the sun, and whether the artificial horizon is used or not) in anticipation of the contact, calling “ ’top” when it takes place, and to repeat a similar operation in the afternoon with the same altitudes, but in the inverse order. It is more convenient to take the first set of sights, five or more, in the ordinary manner, that is, leaving the observer free to use the tangent screw and cry stop when he secures the contact. In the afternoon it will be desirable to begin a minute or two before the time for the greatest altitude, and to continue observations till past the lowest altitude of the forenoon. In other words, take one more at each end, the mean of which is likely to correspond nearly with the mean of the forenoon set. If desirable cut off one of the extremes and rectify any small disagreement by simple proportion till the exact altitude is obtained, with the equivalent time.
Example (fig. 21).—September 9, 1882. On the Mole of Palermo, 38° 8' 15" N., 13° 22' 50" E.( = 53 m 31 s, 3), at 10 a.μ. apparent time, the chronometer showed 9 h 43 m 15 s- 5 as a mean when the five altitudes of the sun were taken by the artificial horizon, which gave a mean (after all corrections) of 47° 20' 22". Precisely the same mean of altitudes was procured in the afternoon when the chronometer showed l h 42 m 51 3- 2. The chronometer was supposed to have been 39 m 30 s fast on Greenwich mean time, gaining l s- 8 daily.
This last information is only useful for reducing the equation of time and declination; it gives 9 h 4 m (nearest minute) Greenwich mean time for the first and l h 3 m p.m. for the second sight, or in Greenwich apparent time 10 h -long. 53 m = 9 h 7 m , and 2 h -53 m = l h 7 m . The declination at apparent noon was 5° 15' 52 ’ ' N., decreasing 56"'7 hourly, which for three hours less seven minutes before noon, gives a correction +2' 43"; for one hour and seven minutes after noon it is — 1' 3". The equation at apparent noon was 2 m 46 s, 4 (to be subtracted from apparent time), increasing 0 s- 85 hourly, which for the above periods gives - 2 s- 4 and +l s .
The latitude, the reduced declination (marked - because it is on the same side of the equator as the latitude), and the zenith distance can now be used as in the calculation with fig. 18 to find the hour angle, or time from apparent noon. For the forenoon it is exactly two hours (10 a.m.); for the afternoon, the declination being 3' 46" less, the latitude and zenith distance the same as before, the hour angle will be l h 59 m 38 s- 8.
In order to make the mode of arriving at the error of the chronometer perfectly clear, the figures will be given in full:— [table]
Here there is a perfect agreement in the result; as the chronometer was supposed to be gaining 1≡ ’ 8 daily, four hours would show an increase of three-tenths. As the altitude and latitude were assumed, they are necessarily faultless; but in actual practice such a result would rarely occur. The latitude of such a place as Palermo Mole, which was once used as a base from which to take chronometrical lines, is undoubted, but the error of one minute in the altitude would in this example produce a difference of fourteen seconds between the two results; the mean, however, would be quite correct.
There is no ambiguity in this mode of finding the errors of a dozen chronometers and of the sextant (if it has any) at the same time. The two separate sets of logarithms also afford a desirable check upon the work. The principal reason why ship’s chronometers are not always rated in this manner is that double the boat-work is required, perhaps from a distance. Various shorter methods have been devised: Norie’s is very short, but he uses a special table, with logarithms A and B, with rules respecting the correction being + or -; Inman’s rule is very confusing.
Chronometers can be rated with greater ease by means of a transit instrument, but that does not come within the reach of an ordinary navigator; nor can telescopes be used on board ship to find the longitude by Jupiter’s satellites, by occultations, or eclipses of sun or moon in a practical and satisfactory manner.
Variation.— The variation of the compass has been slightly touched upon when speaking of the most desirable mode of acquiring the longitude, altitude, and variation by the same daily observations. i In addition to azimuths taken casually with reference to the direction of the ship’s head, it is occasionally necessary to take a round of observations of some kind in order to test the change which altered circumstances have produced in the attraction of iron in the ship (compare Meteorology, section on Terrestrial Magnetism).
The standard compass by which the ship’s course is governed should always be placed in a position which is least affected by the ship’s iron, and whence azimuths and all other bearings can be conveniently taken. Whenever an azimuth or amplitude is taken the direction of the ship’s head should be noted and logged with the variation. The most simple mode of obtaining the variation is by amplitude—observing the bearing of the sun’s centre when it is more than the whole diameter above the horizon, as the refraction will be thirty-four minutes, and the dip four or five more.
To find the true bearing at sunset or sunrise, add together the log. secant of lat. and log. sine of declination; the sum will be the log. sine of the amplitude, that is, the number of degrees to the right or left of the east or west point according to the declination. To an observer on the equator the declination is also the bearing. Under the true bearing, however found, place the compass bearing; the difference will be the variation, which is called east or west according to the position of the north point, with regard to the meridian, or" true north. By keeping that point in view mistakes will be avoided; the same applies to local deviation. The true bearing may also be taken from the amplitude tables.
The best mode of ascertaining the amount of local attraction or deviation is by means of a second prismatic compass, placed on shore near the ship, free from guns or other metal. While the ship is slowly hove round by hawsers and stopped at each point, simultaneous bearings are taken, noted, and timed. The difference between the two compasses is attributed to error caused by the ship’s influence while her head was on each successive point. If landing cannot be effected, the thing may be done by observing a distant object, or by a succession of azimuths. If the latter case it may be found more convenient to use apparent time only for finding the true bearing about every twenty minutes, and the intermediate bearings by simple proportion.
In a time azimuth the apparent time from noon, the co-latitude and co-declination are given to find the angle opposite the latter. As the change in hour angle is the chief thing affecting the azimuth, two or three minutes in declination may be disregarded, and the declination may be treated as constant for two or three hours. Accurate bearings cannot be taken when the sun is more than 25° or 30° above the horizon. Burdwood’s azimuth tables are very useful; from them the true bearing can be taken out by inspection. When a round of azimuths have been taken for the purpose of ascertaining the deviation of the compass, the variation should also be obtained on shore, or taken from a recent authority, in order to separate the amount due to the ship. It has been found very convenient to gum a diamond of paper about ⅛ths of an inch broad on each point, bearing the number of degrees deviation to be allowed when the ship’s head is in that direction, westerly deviation being marked in blue or black and easterly in red. There is a very simple graphic method, when the deviation has been found upon six or eight scattered points only, of deducing therefrom the amount due to the intermediate points. Draw a central line through the length of a sheet of foolscap as a datum line, and about forty lines at right angles to it, three-tenths of an inch apart, to represent the thirty-two points of a compass, and a few repeated at each end. At the left side of the paper write on those lines the names of the points, commencing at north-west and so through north, east, south, and west to north and northeast. Lay off the amount of deviation found on any point upon the corresponding line, taking any convenient scale of equal parts, perhaps one or two tenths of an inch to a degree, according to the amount of deviation; there is no connexion between this scale and that by which the points are represented. Place the easterly deviation on the right of the datum line and the westerly on the left, till all the points secured have been marked. With a pencil form curves which will pass through those marks, or between them if they are close together and disagree, attributing the difference to errors in observation. Ultimately ink an approved curve and remeasure it, as the corrected scale of deviations. The result of this rough method will be found the same as that by a highly scientific calculation, which would be out of place here. See Admiralty Manual.
To insure the true bearing of a distant object it is necessary to take an angle from the sun (or a star) to the object, and find the bearing of the sun as above by amplitude or azimuth. If the former and the distant object be on the horizon, there is nothing to be done but apply the distance from the sun’s centre. Also if the sun has considerable altitude and the distance be measured by a theodolite or standard compass arc acting as such, or if the distance measured be 90° no correction is necessary. Otherwise the distance measured by the sextant is one side of a spherical triangle, the sun’s zenith distance another, and the distant object to the zenith the third; and from these the angle at the zenith must be found.
The position of a ship when in the vicinity of land is usually determined by cross-bearings; but if greater accuracy is required, and three objects are visible which are not in the circumference of a circle passing through the ship’s position, sextant angles should be taken between them, and laid off on a piece of tracing paper as a substitute for a station-pointer. When the three lines are brought to coincide with the three positions prick the point of junction, which will be the ship’s true position; or, without tracing paper or station-pointer, lay off the angle from 90° opposite each station towards the other which was observed with it; those lines will cross in the centre of the circle upon which the ship was situated. Repeat the operation from the centre to the third object, and where the two circles cross will be the ship’s position. The distance off shore can be ascertained by one such angle subtended by two known objects nearly abeam, which might be useful in keeping a ship outside or within a shoal. About 60° is a desirable spread. The distance from an island, the extremities of which do not spread sufficiently to give cross-bearings by the compass alone, or the height of a lighthouse, tower, or cliff, may be made useful in like manner.
Tides.— The ability to find the time of high water at any place during the interval between “full and change’’was once highly estimated, but in the present day, when almanacs are so plentiful, the information can be had without trouble. As a fact a modern navigator never does calculate the time of high water, because the rule only gives a uniform rate of change for all places on the globe, which is more or less incorrect; and, knowing the change which has taken place at London Bridge since the new moon by the Almanac, he has but to apply it to the standard of any other place.
The Nautical Almanac gives the mean time of two tides daily at London Bridge throughout the year, and the apparent time of high water on full and change days at about two hundred places in the United Kingdom and on neighbouring coasts. A more full account of the tides is published annually by the Admiralty. The times predicted in the Almanac are but approximations, and are frequently half an hour too soon or too late. A ship under sail only, working up or down the British Channel, will derive great assistance from the tides by making short tacks close to the headlands and a line joining them in the greatest strength of the tide, during the time it is favourable, and standing out towards the centre of the channel during the adverse tide.
Winds and Currents.— To make good passages a navigator should possess a general knowledge of the prevailing winds and principal currents which he is likely to encounter whether he be under sail or steam. When in the vicinity of land he should always be on his guard against unexpected effects from currents, for none are accurately known or free from change at certain or rather uncertain periods. The strength and direction of ocean currents can only be ascertained by comparing carefully kept dead-reckoning with the result of frequent observations, both of which are apt to be neglected when the navigator is free from land. A great impediment to a better knowledge of the currents is the assumption that they are well known and that the same water circulates across and round the seas, as it would in an enamelled basin. There is decidedly an indraught on the north-west coast of Africa, in the Bight of Biafra, and in many other places. A navigator should therefore be on the watch when steering parallel to a coast at night.
The advice here offered with regard to the choice of a track is intended for sailing ships only, steamships are more independent.
On leaving England for Gibraltar or Madeira the prevailing winds and currents will be found to set along the coast of Spain and Portugal; generally a passage to the south is easily effected. Westerly gales are frequently very strong; therefore ships should, while able, shape a course well outside of Cape Finisterre, towards which there is frequently an indraught. Should bad weather threaten from the south the ship (unless convenient to a port) should stand to the westward to gain an offing, before the force of the wind is felt or it turns to the westward. She should then wear and heave to on the starboard tack, which will enable her to bow the sea as the wind changes to west and north-west. This is an invariable rule when north of the equator; the reverse is the rule in the southern hemisphere.
Ordinary heavy gales frequently give warning of their approach by causing a set of the sea in the direction of their course, accompanied by a long swell, the day previous to the force of the wind being actually felt at the ship. The height of the waves from hollow to crest is sometimes over 40 feet.
Ships bound to the West Indies usually call at or pass near Madeira, thence continue south past the Canary Islands till they fairly catch the north-east trade wind, which may be expected in 30° N. lat., or sooner about the month of July. A fair wind and current will then be secured to the vicinity of the West India Islands, where it either falls calm, or a light wind from the shore is experienced every night. When bound to ports in the United States it is necessary to keep to the southward in a similar manner, skirting the northern edge of the trade wind. From either of those places or the Gulf of Mexico the homeward track is the same, that is, close to the banks of Newfoundland, in order to get the prevailing westerly winds.
Ships bound up the English Channel should enter near the centre, avoiding Ushant and Scιlly, especially in thick weather. Steamships are more under-command, and should be more accurately navigated, but the advantage thus gained in point of safety is more than counterbalanced by the unreasonable things required of them, such as communicating with the lighthouse on Bishop Rock, which caused the loss of the “Schiller” and many lives a few years back. The prevailing wind in the English Channel is south-west, and it will be found in the western half of the circle two days for every one in which it is in the eastern half.
Ships bound to Sierra Leone and the coast of Guinea should pass Madeira on either side, then outside or through the Canary Islands, and if it be about the month of March inside the Cape Verd Islands, where a strong north-east trade will be blowing. At other seasons pass outside these islands to avoid calms, thence close along shore with westerly and south-westerly winds. The current in this track will be found favourable from the Bay of Biscay to the Bight of Biafra, but not farther south. From Sierra Leone to Loando steer along the coast to Cape Palmas, then stretch across the south-east trade till able to return towards Africa on the opposite tack, which may be at 200 or 300 miles south of Ascension. It is especially necessary to keep well to the southward during July and August, when calms prevail near the shore. From Biafra to near the Cape of Good Hope the usual wind near the coast is from the south-west during the day, and either very light off shore or calm at night.
From England to South America the track will be similar to that to Sierra Leone as far as the Cape Verd Islands, when ships for the north-east coast may bear away to the westward, but always allowing for the strong current which may set them to leeward of their port during the calms to which they will be liable. Those bound to the southward of Cape St Roque must be careful not to make much westing till they have crossed the equator; the exact place most desirable for crossing has long been disputed, though the positions generally advocated lie between the meridians 18° and 26°. The argument in favour of the former is the certainty of weathering Cape St Roque, and for the latter less delay by calms between the two trade winds, i.e., in the space from about 10° north to the equator, or from 10° to 4° north while the sun is north of the equator; in this half of the year the S.E. trade comes farther north, and on the coast of Brazil it then blows from south-east and south, so that it is advisable to cross the equator at about 20° W. During December and January the wind frequently draws along the south-east coast of Brazil, past Bahia, from the E.N.E. At that season the line maybe crossed in 25° or 26°. All these rules have their exceptions: when the wind deviates from the usual direction, and the deviation is unfavourable, it is unquestionably better to put the ship on the opposite tack till it returns to the usual direction.
From the United States to the South Atlantic the course should be east, keeping to the north of Bermuda (32⅜ o ) till past 40° W., then south-easterly across the north-east trade, and joining the track of ships from Europe.
After passing the latitude of Cape St Roque, ships for Rio, Monte Video, or Cape Horn may steer direct courses, but for St Helena or Benguela they should keep to the port tack, standing to the southward till past the twentieth parallel, or a day longer if the wind should then be to the eastward of south-east. If bound to the Cape of Good Hope we must be content with making southing only till past the latitude of Rio; westerly winds may then be expected. The track indicated by an arc of a great circle from Rio to the Cape is not only the shortest, but about the best with regard to the wind.
When homeward bound from the Cape, ships should pass close to St Helena with a fair wind and current, and steer for about the western limit for crossing the line on the outward voyage, where they would meet the ships from the south-east coast of America, and running together across the centre of the Atlantic join the homeward bound West Indiamen between the Azores and Newfoundland, according to the winds experienced. From the Bight of Benin or any part of the Guinea coast it is necessary to stand south on the starboard tack to within a degree of the equator, then due west till past the meridian of Cape Palmas, and north-west into the middle of the Atlantic.
In the Mediterranean the prevailing wind is west, therefore there is no difficulty in running up that sea. A slight current sets from all directions towards Cephalonia. Off the coast of Algeria, during westerly gales, heavy hollow waves are experienced, running at the rate of 10 or 12 miles an hour. When making a passage out of the Mediterranean it is better to keep near the African coast while the wind is to the southward of west, to which it is sure to turn, and then stand over to the Spanish coast, where shelter can be found if required.
The destructive circular storms or cyclones in the West Indies called hurricanes are generally preceded by calm gloomy weather and a high barometer. It is' very desirable that the electric state of the atmosphere and the earth should be known, which is not the case at present. Their immediate approach is indicated by the barometer falling rapidly; which is the more remarkable, because the changes in the barometric readings are usually very slight within the tropics. If the ship be in a safe anchorage or harbour, the lower yards and topmasts should be struck and anchors laid out in the direction of the most probable danger. If the anchorage be not satisfactorily protected the sooner she is out of it the better. At sea it is necessary to get down the upper spars, and to run, if there be sea room, across the direction of the wind, from the centre of the storm,—that is, with the wind on the starboard side in north latitude, and on the port side in the southern hemisphere. If it be necessary to scud, it should be done with the wind as much on the starboard or port quarter as the ship will bear, in order to avoid running round or into the centre. The wind does not run in parallel circles round the centre, but rushes spirally towards it, especially when close (see fig. 22).
There is one dangerous position in which a ship may be placed with respect to an advancing cyclone for which no certain rules have been given, such as at A or D, or between those two positions, near the track upon which it is advancing. It is evident from the diagram that to run from A either before the wind, or with it on the starboard quarter, in the direction of the small arrow, would cause the ship to intercept the storm’s centre. It would be advisable to carry sail with the wind abeam as long as possible, and ultimately to heave to on the starboard tack (in north latitude) till the barometer begins to rise and the wind turns more to the south and west.
As the centre of the storm is supposed to move at a rate varying from 12 to 30 miles an hour, a ship sailing 10 or 12 miles an hour in a direct line from the centre, but on the track, would be overtaken by it. The great hurricane of 1830 extended from the West India Islands to Newfoundland at a mean rate of 18 knots an hour, and by the average time occupied in passing intermediate places (eighteen hours) it must have been about 330 nautical miles in diameter. The centre of the hurricane which devastated Barbados August 10, 1831, moved at a mean rate of about 14 knots an hour. It is evident from those quotations that a ship cannot depend upon running faster than the storm.
The navigator should be ready provided with a representation on tracing paper of a cyclone similar to fig. 22, of a radius equal to 150 miles on the scale of the chart upon which it is to be used. If made for one hemisphere it can be turned over to suit the other. By marking the ship’s position on the chart, and placing the tracing over it with one of the arrows correctly corresponding with the direction of the wind at the ship, the direction of the storm’s centre will be indicated with certainty, but the distance and track can only be conjectured,—the latter by assuming that it will be similar to those previously ascertained, which may be represented by a pencil line on the chart. If to the south of the 20° N. lat. it will be moving in a westerly direction, if about 30° to the northwest or north, and when about 35° and 40° to the north-east.
It is safer to assume the centre to be nearer than it really is than vice versa ; the pencil line will ran parallel to the storm’s course unless the ship be on the track, as at D. It is evidently her duty to run to the west if there be room till the wind veers to the northeast against the sun, when it will be certain that she is to the westward of the storm’s track. This is not likely to happen in the North Atlantic when as far north as 30°, as the centre usually passes along the coast or over the continent of America; but when farther south, and when the storm’s progress is to the westward, a ship may escape by running to the southward with the wind at north.
If the position with regard to the storm is supposed to be at B, she should run with the wind on the starboard quarter. If at C it will be optional either to run with the wind on starboard quarter or to heave to; for anything in that direction relative to the assumed track will have attained its nearest approach to the centre. When the direction of the wind remains unchanged, the force increasing and barometer falling, the centre must be advancing directly on the ship. When the centre begins to recede the wind will decrease in violence and change its direction, with the sun if the ship is east of the track and against the sun if the centre passed to the eastward of the ship; the barometer will also rise rapidly.
Fig. 22, and the argument upon it, are for the northern hemisphere; the typhoons and all circular storms south of the equator are similar, but the revolutions are reversed, and the track of centre curves off from west to south-west, south, and southeast.
In ordinary heavy gales in the North Atlantic a ship should heave to on the starboard tack, as the wind generally changes from south to west and north-west. On that tack she will be constantly “coming up” and bowing the sea as the wind veers, and will be less liable to be taken aback; and for similar reasons the third anchor for use should be carried on the starboard side. The reverse will hold good south of the equator.
The storm peculiar to the west coast of Africa is called a tornado. It is entirely different from a hurricane; it is not circular; it is very local and of short duration, seldom lasting more than ten or twenty minutes. These storms give good warning by a black cloud first appearing on the horizon, rising quickly. When the cloud is about twenty degrees high a bright arch will be seen under it. The faster it rises and the more distinct the arch the more severe will be the tornado, which bursts suddenly when at an angle of forty-five degrees. During its rise there will be ample time to shorten all sail and turn the ship’s stern towards it; she will be driven violently for a short time, when it will fall calm as suddenly as the wind came, and a deluge of rain will wash the decks.
Along the coast of California and the west coast of Mexico ami Central America voyages can be made to the south-east with westerly and south-westerly winds and favourable current, except during the latter part of May, June, and July, when the trade winds are strongest through the West India Islands; the wind is then off shore at the southern part of the track mentioned, and easterly, and passages along shore to the south-east are not easily accomplished. λVith the exception of the time last named, when ships are bound from Central America to California they must stand out into the trade wind, run down the meridian, and then make northing.
Along the west coast of South America light southerly winds and northerly current prevail, therefore ships bound south from Central America should stand well out from the land. On the southern part of Chili south-west winds prevail. On the west side of North America, beyond the tropic, westerly winds are most frequent, as in a similar belt across the Atlantic.
Ships bound to the East Indies will find, after rounding the Cape of Good Hope, the south-west monsoon from April till October giving them a fair wind and fine weather through the Mozambique to Bombay or Calcutta. The reverse is the case during the English winter months, that is, from October till April. In the Indian Ocean, as in the northern hemisphere, the most unsettled and stormy weather is about the time of each equinox, but more especially about the end of September, when circular storms of incredible violence called typhoons are sometimes experienced. When south of the equator, ships should always receive them on the port side, or run with the wind on the port quarter, if there be sea room; if north of the equator, keep the wind on the starboard side, as described above.
Bad weather is frequent on the coast of Malabar during May, and in the Bay of Bengal during October. When running up the Bay of Bengal in May or June before a south-west gale, with heavy rain and falling barometer, there will be reason to fear being overtaken by the centre of a typhoon; it will be advisable to heave to, or run to the eastward, till the barometer rises and weather improves.
In the Red Sea the wind is generally from the south, except during June, July, and August, when it is north, calm, or on shore by day, and off shore at night.
From the Cape of Good Hope to Australia and New Zealand ships have usually westerly winds and a favourable current, which is a good reason for returning to Europe by way of Cape Horn.
About 60 to 100 miles off the south-east extremity of Africa the Agulhas current sets to the westward, sometimes as much as 60 miles a day; its progress is not checked by a westerly gale, and consequently one of the most dangerous seas ever experienced is to be found there. About the year 1850, ships making long voyages were frequently navigated upon or approaching a great circle, which led the Australian ships into danger on account of the icebergs which were met in great numbers. In 1866 the Admiralty published ice-charts of the southern hemisphere for the guidance and warning of seamen. Every ship should be provided with wind and current charts for the sea to which she is going, and every officer should help to perfect these by means of careful observations.
λVhen a ship is to leeward of her port, or a chase, it is advisable to work up on a direct lino towards it, tacking as frequently as convenient, as then all flaws and changes in the wind will be in her favour. By the contrary mode of action ships have been known to sail round an island, e.g., the Bermudas.
A useful graphic method has been devised for shaping a course which will intercept another ship with the least loss of time, or for shaping a course across a current the direction and velocity of which can be estimated (see fig. 23, which for use should be more than twice as large, with a circle for every mile).
Ex. 1.—A ship bore N.E. 13 miles, steering S.S.E. 10 knots. Being nearly on a wind, which was E. byN., she could not alter course from her pursuer without loss of speed; the pursuer’ being a steamship only able to go 9⅛ knots, if he allows the chase to cross his bows she will escape.
Make a spot on the N.E. line, 13 miles from the centre, at A; place the parallel ruler to the course the chase is steering, S.S.E., and, with the distance representing her speed (10 knots) in the compasses, mark where it will reach on that bearing, from the N.E. line to a position between the circles representing 94 knots, which will be from a to δ. Then a line from the centre through ⅛ shows the course the pursuer must take, S. E. by E. ⅛ E., which continued will cross the S.S.E. line from A in E, where the vessels will meet. The distance and time of meeting may be measured on the diagram or calculated by trigonometry. The two ships will continue on the same relative bearings.
Ex. 2.—A point of destination bore VV.S.W. 10 miles; a current ran S.E. by S. 4 miles an hour; the boat can pull at the rate of 6 miles an hour. It is convenient to consider that the object is moving in the opposite direction to the current and at the same speed. As the diagram is so small the proportionate speeds are doubled. Eight miles in the compasses will extend on a line parallel to S. E. by S. from the 12-mile circle at d to the W.S.W. bearing at f. A line from the centre through d will indicate the course to be steered, W. by N. ∣N. From the point F on the 10 mile circle and S.S.W. bearing, a line drawn N.W. by N. will cut the former line in D, the place the boat would have reached had there been no current, 16 miles from the centre, occupying 2 h 40 m ; while DF represents the space through which the water has passed ∙=10∣ miles in the same time. When a distant object can be seen in a line with the point steered for, and if the strength of the current be equal all the distance, the same bearing should be preserved. As an example of crossing a river or channel where the tide will change at a known time, we may take that part of the Bristol Channel where it is 12 miles wide, say north and south, and the tide runs 5 knots. Suppose the boat can only pull 5 knots. By leaving one shore at an hour •and twelve minutes before high water, and steering directly north or south during 2 h 24 m , regardless of first being carried east and then west, the boat will reach its destination, having passed through 12 miles of water only.
Though so much has been done to increase the efficiency of ships, in point both of size and of equipment, a sea voyage is still attended with considerable peril, both by stranding and collision. In the year ending April 1882 there were 4367 vessels damaged, including 606 total wrecks, with the loss of 1097 lives, round the coast of Great Britain alone. The number entering or leaving British ports during the same period was 676,358, of 139,443,790 tons burthen. The number of vessels at the same time above 50 tons underthe English flag was over 22,000. Ships on long voyages were formerly content to make good one degree a day; now steamships occasionally run 400 miles in one day. In September 1883 the “Alaska” (Guion Line) made the voyage from Queenstown to New York in six days twenty-one and three-quarter hours, and returned in six days twenty-one hours. During the same month the “Oregon” made 20 knots an hour during her trial trip on the Clyde. The “Ionic” (White Star Line) arrived at Plymouth August 4, 1883, from New Zealand, through the Straits of Magellan, in forty-five days, including stoppages for coal. Her passage out, round the Cape of Good Hope, was performed in forty-three days. This voyage round the world is the quickest yet accomplished, having occupied less than eighty-eight days at sea. In May 1883 the “British King” returned from performing a similar voyage, and discharging or receiving 6000 tons of cargo, in less than four months.
Of late years, with the hope of diminishing the number of collisions, an international code of regulations has been drawn up by the Board of Trade and Admiralty, under the authority of the privy council, and sanctioned by nearly all maritime nations. The following is an abridgment of the law as now in force:—
Art. 1. Every ship under sail only is considered a sailing ship, but if any steam be used she is a steamship.
Art. 2. The lights hereafter mentioned, and none other, shall be carried from sunset to suni·ise.
Art. 3. A steamship under way shall carry a bright white light before the mast at least 20 feet above the deck, a green light on the starboard side, and a red light on the port side.
Art. 4. A steamship towing anything shall carry an extra bright light before the mast, i.e., two vertical lights.
Art. 5. A telegraph cable ship at work, or others not able to steer, to carry three red lights vertically, and by day three balls. Side lights with the above, only when making way through the water.
Art. 6. Sailing ship under way, or being towed, the green light on starboard side, and red on port side only.
Art. 7. Small vessels may carry portable lights similar to the above.
Art. 8. All ships at anchor shall carry a white light, not exceeding 20 feet above the deck.
Art. 9. Pilot vessels to carry a white light at the mast head only, and a flare-up every fifteen minutes.
In lieu of Art. 10. Open fishing boats or other open boats may carry one lantern, green one side and red the other. Stich boats when at anchor or fast to nets to exhibit a bright white light.
Art. 12. Steamships must be provided with an efficient steam whistle, foghorn, and bell; sailing ships with the last two. Turkish ships to carry a drum instead of a bell.
a. In fog, mist, or snow, by day or night, a steamship must make every two minutes or less a prolonged blast.
b. A sailing ship at similar intervals with fog-horn to make one blast on starboard tack, two on the port tack, and three if the wind be abaft the the beam.
c. All ships at anchor in a fog, <fcc., shall ring a bell every two minutes or less.
Art. 13. Every ship in fog, <fcc., to “go at a moderate speed.”
Art. 14. a. A ship running free shall make way for one which is close hauled. b. Ships on the port tack make way for those on the starboard tack. c. Similarly if running free.
d. When running free with the wind on the same side, the ship which is to windward shall keep out of the way.
e. A ship which has the wind aft shall keep out of the way of any other.
Art. 15. If two steamships are meeting nearly end on so as to involve risk, they shall both alter course to starboard.
Art. 16. When two steamships are crossing, the one which has the other on her port side is to keep out of the way.
Art. 17. A steamship shall keep out of the way of a sailing ship.
Art. 18. Steamships shall slacken speed, stop, or reverse if necessary.
Art. 19. Optional signals by the steam whistle:—one short blast means, “I am altering course to starboard”; two short blasts, “I am going to port’’; three short blasts, “I am going full speed astern."
Art. 20. When one ship overtakes another, the faster ship is to keep out of the way.
Art. 21. In narrow channels ships shall keep on the starboard side.
The books on navigation in general use are those of Norrie, Inman (especially his tables), Riddle, Raper, Jeans, and Evers, and. Harbord’s Glossary. (H. A. Μ.)
