See Brassica, Botany Index.

NAVIGATION

Is the art of conducting or carrying a ship from one port to another.

HISTORY.

The poets refer the invention of the art of navigation to Neptune, some to Bacchus, others to Hercules, others to Jason, and others to Janus, who is said to have made the first ship. Historians ascribe it to the Æginetans, the Phoenicians, Tyrians, and the ancient inhabitants of Britain. Some suppose, that the first hint was taken from the flight of the kite; others, as Oppian (De Piscibus, lib. i.), from the fish called nautilus; others ascribe it to accident.—Scripture refers the origin of so useful an invention to God himself, who gave the first specimen thereof in the ark built by Noah under his direction. For the rillery which the good man underwent on account of his enterprise shows evidently enough that the world was then ignorant of anything like navigation, and that they even thought it impossible.

However, profane history represents the Phoenicians, especially those of their capital Tyre, as the first navigators; being urged to seek a foreign commerce by the narrowness and poverty of a slip of ground they possessed along the coasts; as well as by the conveniency of two or three good ports, and by their natural genius for traffic. Accordingly, Lebanon, and the other neighbouring mountains, furnishing them with excellent wood for ship-building, in a short time they were masters of a numerous fleet; and constantly hazarding new navigations, and settling new trades, they soon arrived at an incredible pitch of opulence and populousness: insomuch as to be in a condition to send out colonies, the principal of which was that of Carthage; which, keeping up their Phoenician spirit of commerce, in time not only equalled Tyre itself, but vastly surpassed it; sending its merchant fleets through Hercules's Pillars, now the straits of Gibraltar, along the western coasts of Africa and Europe; and even, if we believe some authors, to America itself.

Tyre, whose immense riches and power are represented in such lofty terms both by sacred and profane authors, being destroyed by Alexander the Great, its navigation and commerce were transferred by the conqueror to Alexandria, a new city admirably situated for those purposes; proposed for the capital of the empire of Asia, which Alexander then meditated. And thus arose the navigation of the Egyptians; which was afterwards so much cultivated by the Ptolemies, that Tyre and Carthage were quite forgotten.

Egypt being reduced into a Roman province after the battle of Actium, its trade and navigation fell into the hands of Augustus; in whose time Alexandria was only inferior to Rome; and the magazines of the capital of the world were wholly supplied with merchandises from the capital of Egypt.

At length, Alexandria itself underwent the fate of Tyre and Carthage; being surprised by the Saracens, who in spite of the emperor Heraclius, overspread the northern coasts of Africa, &c. whence the merchants being driven, Alexandria has ever since been in a languishing state, though it still has a considerable part of the commerce of the Christian merchants trading to the Levant.

The fall of Rome and its empire drew along with it not only that of learning and the polite arts, but that of navigation; the barbarians into whose hands it fell, contenting themselves with the spoils of the industry of their predecessors.

But no sooner were the more brave among those nations well settled in their new provinces, some in Gaul, as the Franks; others in Spain, as the Goths; and others in Italy, as the Lombards; but they began to learn the advantages of navigation and commerce, and the methods of managing them, from the people they subdued; and this with so much success, that in a little time some of them became able to give new lessons, and set on foot new institutions for its advantage. Thus it is to the Lombards we usually ascribe the invention and use of banks, book-keeping, exchanges, rechanges, &c.

It does not appear which of the European people, after the settlement of their new masters, first betook themselves to navigation and commerce. Some think it began with the French; though the Italians seem to have the justest title to it; and are accordingly regarded as the restorers thereof, as well as of the polite arts, which had been banished together from the time the empire was torn asunder. It is the people of Italy then, and particularly those of Venice and Genoa, who have the glory of this restoration; and it is to their advantageous situation for navigation they in great measure owe their glory. In the bottom of the Adriatic were a great number of marshy islands, only separated by narrow channels, but those well screened, and almost inaccessible, the residence of some fishermen, who here supported themselves by a little trade in fish and salt, which they found in some of these islands. Thither the Veneti, a people inhabiting that part of Italy along the coasts of the gulf, retired, when Alaric king of the Goths, and afterwards Attila king of the Huns, ravaged Italy.

These new islanders, little imagining that this was to be their fixed residence, did not think of composing any body politic; but each of the 72 islands of this little archipelago continued a long time under its several masters, and each made a distinct commonwealth. When their commerce was become considerable enough to give jealousy to their neighbours, they began to think of uniting into a body. And it was this union, first begun in the sixth century, but not completed till the eighth, that laid the sure foundation of the future grandeur of the state of Venice. From the time of this union, their fleets of merchantmen were sent to all the ports of the Mediterranean; and at last to those of Egypt, particularly Cairo, a new city built by the Saracen princes on the eastern banks of the Nile, where they traded for the spices and other products of the Indies. Thus they flourished, increased their commerce, their navigation, and their conquests on the terra firma, till the league of Cambrai in 1508, when a number of jealous princes conspired to their ruin; which was the more easily effected by the diminution of their East India commerce, of which the Portuguese had got one part and the French another. Genoa, which had applied itself to navigation at the same time with Venice, and that with equal success, was a long time its dangerous rival, disputed with it the empire of the sea, and shared with it the trade of Egypt and other parts both of the east and west.

Jealousy soon began to break out; and the two republics coming to blows, there was almost continual war for three centuries before the superiority was ascertained; when, towards the end of the 14th century, the battle of Chioza ended the strife; the Genoese, who till then had usually the advantage, having now lost all; and the Venetians, almost become desperate, at one happy blow, beyond all expectation, secured to themselves the empire of the sea, and superiority in commerce.

About the same time that navigation was retrieved in the southern parts of Europe, a new society of merchants was formed in the north, which not only carried commerce to the greatest perfection it was capable of till the discovery of the East and West Indies, but also formed a new scheme of laws for the regulation therefore, which still obtain under the names of Usus and Customs of the Sea. This society is that famous league of the Hanse towns, commonly supposed to have begun about the year 1164. See Hanse Towns.

For the modern state of navigation in England, Holland, France, Spain, Portugal, &c. see Commerce, Company, &c.

We shall only add, that in examining the causes of commerce passing successively from the Venetians, Genoese, and Hanse towns, to the Portuguese and Spaniards, and from these again to the English and Dutch, it may be established as a maxim, that the relation between commerce and navigation, or if we may be allowed to say it, their union, is so intimate, that the fall of the one inevitably draws after it that of the other; and that they will always either flourish or dwindle together. Hence so many laws, ordinances, statutes, &c. for its regulation; and hence particularly that celebrated act of navigation, which an eminent foreign author calls the palladium or tutelar deity of the commerce of England; which is the standing rule, not only of the British among themselves, but also of other nations with whom they traffic.

The art of navigation has been exceedingly improved in modern times, both with regard to the form of the vessels themselves, and with regard to the methods of working them. The use of rowers is now entirely superseded by the improvements made in the formation of the sails, rigging, &c. by which means ships can not only sail much faster than formerly, but can tack in any direction with the greatest facility. It is also very probable that the ancients were neither so well skilled in finding the latitudes, nor in steering their vessels in places of difficult navigation, as the moderns. But the greatest advantage which the moderns have over the ancients is from the mariner's compass, by which they are enabled to find their way with as great facility in the midst of an immeasurable ocean, as the ancients could have done by creeping along the coast, and never going out of sight of land. Some people indeed contend, that this is no new invention, but that the ancients were acquainted with it. They say, that it was impossible for Solomon to have sent ships to Ophir, Tarshish, and Parvaim, which last they will have to be Peru, without this useful instrument. They insist, that it was impossible for the ancients to be acquainted with the attractive virtue of the magnet, and to be ignorant of its polarity. Nay, they affirm, that this property of the magnet is plainly mentioned in the book of Job, where the loadstone is mentioned by the name of topaz, or the stone that turns itself. But it is certain, that the Romans, who conquered Judea, were ignorant of this instrument; and it is very improbable, that such an useful invention, if once it had been commonly known to any nation, would have been forgot, or perfectly concealed from such a prudent people as the Romans, who were so much interested in the discovery of it.

Among those who admit that the mariner's compass is a modern invention, it has been much disputed who was the inventor. Some give the honour of it to Flavio Gioia of Amalfi in Campania*, who lived about the beginning of the 14th century; while others say that it came from the east, and was earlier known in Europe. But, at whatever time it was invented, it is certain, that the mariner's compass was not commonly used in navigation before the year 1420. In that year the science was considerably improved under the auspices of Henry duke of Viseo, brother to the king of Portugal. In the year 1485, Roderick and Joseph, physicians to John II. king of Portugal, together with one Martin de Bohemia, a Portuguese, native of the island of Fayal, and scholar to Regiomontanus, calculated tables of the sun's declination for the use of sailors, and recommended the astrolabe for taking observations at sea. Of the instructions of Martin, the celebrated Christopher Columbus is said to have availed himself, and to have improved the Spaniards in the knowledge of the art; for the farther progress of which a lecture was afterwards founded at Seville by the emperor Charles V.

The discovery of the variation is claimed by Columbus, and by Sebastian Cabot. The former certainly did observe this variation without having heard of it from any other person, on the 14th of September 1492, and it is very probable that Cabot might do the same. At that time it was found that there was no variation at the Azores, where some geographers have thought proper to place the first meridian; though it hath since been observed that the variation alters in time.—The use of the cross staff now began to be introduced among sailors. This ancient instrument is described by John Werner of Nuremberg, in his annotations on the first book of Ptolemy's Geography, printed in 1514. He recommends it for observing the distance between the moon and some star, in order thence to determine the longitude.

At this time the art of navigation was very imperfect, on account of the inaccuracies of the plane chart, which was the only one then known, and which, by its gross errors, must have greatly misled the mariner, especially in voyages far distant from the equator. Its precepts were probably at first only set down on the sea charts, as is the custom at this day; but at length there were two Spanish treatises published in 1545; one by Pedro de Medina; the other by Martin Cortes, which contained a complete system of the art, as far as it was then known. These seem to have been the oldest writers who fully handled the art; for Medina, in his dedication to Philip, prince of Spain, laments that multitudes of ships daily perished at sea, because there were neither teachers of the art, nor books by which it might be learned; and Cortes, in his dedication, boasts to the emperor that he was the first who had reduced navigation into a compendium, valuing himself much on what he had performed. Medina defended the plane chart; but he was opposed by Cortes, who showed its errors, and endeavoured to account for the variation of the compass, by supposing the needle to be influenced by a magnetic pole (which he called the point attractive), different from that of the world; which notion hath been farther prosecuted by others. Medina's book was soon translated into Italian, French, and Flemish, and served for a long time as a guide to foreign navigators. However, Cortes was the favourite author of the English nation, and was translated in 1561; while Medina's work was entirely neglected, though translated also within a short time of the other. At that time the system of navigation consisted of the following particulars, and others similar: An account of the Ptolemaic hypothesis, and the circles of the sphere; of the roundness of the earth; the longitudes, latitudes, climates, &c. and eclipses of the luminaries; a calendar; the method of finding the prime, epact, moon's age, and tides; a description of the compass, an account of its variation, for the discovering of which Cortes said an instrument might easily be contrived; tables of the sun's declination for four years, in order to find the latitude from his meridian altitude; directions to find the same by certain stars; of the course of the sun and moon; the length of the days; of time and its divisions; the method of finding the hour of the day and night; and lastly, a description of the sea chart, on which to discover where the ship is, they made use of a small table, that showed, upon an alteration of one degree of the latitude, how many leagues were run in each rhumb, together with the departure from the meridian. Besides, some instruments were described, especially by Cortes; such as one to find the place and declination of the sun, with the days, and place of the moon; certain dials, the astrolabe, and cross staff; with a complex machine to discover the hour and latitude at once.

About the same time were made proposals for finding the longitude by observations of the moon.—In 1539, Gemma Frisius advised the keeping of the time by means of small clocks or watches, then, as he says, newly invented. He also contrived a new sort of cross staff, and an instrument called the nautical quadrant; which last was much praised by William Cunningham, in his Astronomical Glass, printed in the year 1559.

In 1537 Pedro Nunez, or Nonius, published a book in the Portuguese language, to explain a difficulty in navigation proposed to him by the commander Don Martin Alphonso de Sousa. In this he exposes the errors of the plane chart, and likewise gives the solution of several curious astronomical problems; amongst which is that of determining the latitude from two observations of the sun's altitude and intermediate azimuth being given. He observed, that though the rhumbs are spiral lines, yet the direct course of a ship will always be in the arch of a great circle, whereby the angle with the meridians will continually change: all that the steersman can here do for the preserving of the original rhumb, is to correct these deviations as soon as they appear sensible. But thus the ship will in reality describe a course without the rhumb line intended; and therefore his calculations for assigning the latitude, where any rhumb line crosses the several meridians, will be in some measure erroneous. He invented a method of dividing a quadrant by means of concentric circles, which, after being much improved by Dr Halley, is used at present, and is called a nonius.

In 1577, Mr William Bourne published a treatise, in which, by considering the irregularities in the moon's motion, he shows the errors of the sailors in finding her age by the epact, and also in determining the hour from observing on what point of the compass the sun and moon appeared. He advises, in sailing towards the high latitudes, to keep the reckoning by the globe, as there the plane chart is most erroneous. He despairs of our ever being able to find the longitude, unless the variation of the compass should be occasioned by some such attractive point as Cortes had imagined; of which, however, he doubts: but as he had shown how to find the variation at all times, he advises to keep an account of the observations, as useful for finding the place of the ship; which advice was prosecuted at large by Simon Stevin, in a treatise published at Leyden in 1599; the substance of which was the same year printed at London in English by Mr Edward Wright, entitled the Haven-finding Art. In this ancient tract also is described the way by which our sailors estimate the rate of a ship in her course, by an instrument called the log. This was so named from the piece of wood or log that floats in the water while the time is reckoned during which the line that is fastened to it is veering out. The author of this contrivance is not known; neither was it taken notice of till 1607, in an East India voyage published by Purchas: but from this time it became famous, and was much taken notice of by almost all writers on navigation in every country; and it still continues to be used as at first, though many attempts have been made to improve it, and contrivances proposed to supply its place; many of which have succeeded in quiet water, but proved useless in a stormy sea.

In 1581 Michael Coignet, a native of Antwerp, published a treatise, in which he animadverted on Medina. In this he showed, that as the rhumbs are spi- rals, making endless revolutions about the poles, numerous errors must arise from their being represented by straight lines on the sea charts; but though he hoped to find a remedy for these errors, he was of opinion that the proposals of Nonius were scarcely practicable, and therefore in a great measure useless. In treating of the sun's declination, he took notice of the gradual decrease in the obliquity of the ecliptic; he also described the cross staff with three transverse pieces, as it is at present made, and which he owned to have been then in common use among the sailors. He likewise gave some instruments of his own invention; but all of them are now laid aside, excepting perhaps his nocturnal. He constructed a sea table to be used by such as sailed beyond the 60th degree of latitude; and at the end of the book is delivered a method of sailing on a parallel of latitude by means of a ring dial and a 24 hour glass. The same year the discovery of the dipping needle was made by Mr Robert Norman*. In his publication on that art he maintains, in opposition to Cortes, that the variation of the compass was caused by some point on the surface of the earth, and not in the heavens; he also made considerable improvements in the construction of compasses themselves; showing especially the danger of not fixing, on account of the variation, the wire directly under the fleur de luce; as compasses made in different countries have it placed differently. To this performance of Norman's is always prefixed a discourse on the variation of the magnetic needle, by Mr William Burrough, in which he shows how to determine the variation in many different ways. He also points out many errors in the practice of navigation at that time, and speaks in very severe terms concerning those who had published upon it.

All this time the Spaniards continued to publish treatises on the art. In 1585 an excellent compendium was published by Roderico Zamorano; which contributed greatly towards the improvement of the art, particularly in the sea charts. Globes of an improved kind, and of a much larger size than those formerly used, were now constructed, and many improvements were made in other instruments; however, the plane chart continued still to be followed, though its errors were frequently complained of. Methods of removing these errors had indeed been sought after; and Gerard Mercator seems to have been the first who found the true method of doing this, so as to answer the purposes of seamen. His method was to represent the parallels both of latitude and longitude by parallel straight lines, but gradually to augment the former as they approached the pole. Thus the rhumbs, which otherwise ought to have been curves, were now also extended into straight lines; and thus a straight line drawn between any two places marked upon the chart would make an angle with the meridians, expressing the rhumb leading from the one to the other. But though, in 1569, Mercator published an universal map constructed in this manner, it doth not appear that he was acquainted with the principles on which this proceeded; and it is now generally believed, that the true principles on which the construction of what is called Mercator's chart depends, were first discovered by an Englishman, Mr Edward Wright.

Mr Wright supposes, but, according to the general opinion, without sufficient grounds, that this enlargement of the degrees of latitude was known and mentioned by Ptolemy, and that the same thing had also been spoken of by Cortes. The expressions of Ptolemy alluded to, relate indeed to the proportion between the distances of the parallels and meridians; but instead of proposing any gradual enlargement of the parallels of latitude in a general chart, he speaks only of particular maps; and advises not to confine a system of such maps to one and the same scale, but to plan them out by a different measure, as occasion might require: only with this precaution, that the degrees of longitude in each should bear some proportion to those of latitude; and this proportion is to be deduced from that which the magnitude of the respective parallels bears to a great circle of the sphere. He adds, that in particular maps, if this proportion be observed with regard to the middle parallel, the inconvenience will not be great though the meridians should be straight lines parallel to each other. Here he is said only to mean, that the maps should in some measure represent the figures of the countries for which they are drawn. In this sense Mercator, who drew maps for Ptolemy's tables, understood him; thinking it, however, an improvement not to regulate the meridians by one parallel, but by two; one distant from the northern, the other from the southern extremity of the map by a fourth part of the whole depth; by which means, in his maps, though the meridians are straight lines, yet they are generally drawn inclining to each other towards the poles. With regard to Cortes, he speaks only of the number of degrees of latitude, and not of the extent of them; nay, he gives express directions that they should all be laid down by equal measurement on a scale of leagues adapted to the map.

For some time after the appearance of Mercator's map, it was not rightly understood, and it was even thought to be entirely useless, if not detrimental.—However, about the year 1592, its utility began to be perceived; and seven years after, Mr Wright printed his famous treatise entitled, The Correction of certain Errors in Navigation, where he fully explained the reason of extending the length of the parallels of latitude, and the uses of it to navigators. In 1610, a second edition of Mr Wright's book was published with improvements. An excellent method was proposed of determining the magnitude of the earth; at the same time it was judiciously proposed to make our common measures in some proportion to a degree on its surface, that they might not depend on the uncertain length of a barley corn. Some of his other improvements were, "The table of latitudes for dividing the meridian computed to minutes;" whereas it had been only divided to every tenth minute. He also published a description of an instrument which he calls the sea rings; and by which the variation of the compass, altitude of the sun, and time of the day, may be determined readily at once in any place, provided the latitude is known. He showed also how to correct the errors arising from the eccentricity of the eye in observing by the cross-staff. He made a total amendment in the tables of the declinations and places of the sun and stars from his own observations made with a six foot quadrant in the years 1594, 95, 96, and 97. A sea quadrant to take altitudes by a forward or backward observation; and likewise with a contrivance for the ready finding the latitude latitude by the height of the pole star, when not upon the meridian. To this edition was subjoined a translation of Zamorano's Compendium above mentioned, in which he corrected some mistakes in the original; adding a large table of the variation of the compass observed in very different parts of the world, to show that it was not occasioned by any magnetical pole.

These improvements soon became known abroad.—In 1628, a treatise entitled, Hypomnemota Mathematica, was published by Simon Stevin, for the use of Prince Maurice. In that part relating to navigation, the author having treated of sailing on a great circle, and shown how to draw the rhumbs on a globe mechanically, sets down Wright's two tables of latitudes and of rhumbs, in order to describe these lines more accurately, pretending even to have discovered an error in Wright's table. But all Stevin's objections were fully answered by the author himself, who showed that they arose from the gross way of calculating made use of by the former.

In 1624, the learned Wellebrodus Snellius, professor of mathematics at Leyden, published a treatise of navigation on Wright's plan, but somewhat obscurely: and as he did not particularly mention all the discoveries of Wright, the latter was thought by some to have taken the hint of all his discoveries from Snellius. But this supposition is long ago refuted: and Wright enjoys the honour of those discoveries which is justly his due.

Mr Wright having shown how to find the place of the ship on his chart, observed that the same might be performed more accurately by calculation: but considering, as he says, that the latitudes, and especially the courses at sea, could not be determined so precisely, he forbore setting down particular examples; as the mariner may be allowed to save himself this trouble, and only mark out upon his chart the ship's way, after the manner then usually practised. However, in 1614, Mr Raphe Handson, among his nautical questions subjoined to a translation of Pitiscus's trigonometry, solved very distinctly every case of navigation, by applying arithmetical calculations to Wright's table of latitudes, or of meridional parts, as it hath since been called. Though the method discovered by Wright for finding the change of longitude by a ship sailing on a rhumb is the proper way of performing it, Handson also proposes two ways of approximation to it without the assistance of Wright's division of the meridian line. The first was computed by the arithmetical mean between the cosines of both latitudes; the other by the same mean between the secants as an alternative, when Wright's book was not at hand; though this latter is wider from the truth than the first. By the same calculations also he showed how much each of these compendiums deviates from the truth, and also how widely the computations on the erroneous principles of the plane chart differ from them all. The method, however, commonly used by our sailors is commonly called the middle latitude; which, though it errs more than that by the arithmetical mean between the two cosines, is preferred on account of its being less operose: yet in high latitudes it is more eligible to use that of the arithmetical mean between the logarithmic cosines, equivalent to the geometrical mean between the cosines themselves; a method since proposed by Mr John Bassat. The computation by the middle latitude will always fall short of the true change of longitude; that by the geometrical mean will always exceed; but that by the arithmetical mean falls short in latitudes above 45 degrees, and exceeds in lesser latitudes. However, none of these methods will differ much from the truth when the change of latitude is sufficiently small.

About this time logarithms were invented by John Napier, baron of Merchiston in Scotland, and proved of the utmost service to the art of navigation. From which Mr Edmund Gunter constructed a table of logarithmic sines and tangents to every minute of the quadrant, which he published in 1620. In this work he applied to navigation, and other branches of mathematics, his admirable ruler known by the name of Gunter's scale *; on which are described lines of logarithms, of logarithmic sines and tangents, of meridional parts, &c. He greatly improved the sector for the same purposes. He showed also how to take a back observation by the cross staff, whereby the error arising from the eccentricity of the eye is avoided. He described likewise another instrument, of his own invention called the cross bone, for taking altitudes of the sun or stars, with some contrivances for the more ready collecting the latitude from the observation. The discoveries concerning logarithms were carried to France in 1624 by Mr Edmund Wingate, who published two small tracts in that year at Paris. In one of these he taught the use of Gunter's scale; and in the other, of the tables of artificial sines and tangents, as modelled according to Napier's last form, erroneously attributed by Wingate to Briggs.

Gunter's rule was projected into a circular arch by the Reverend Mr William Oughtred in 1633, and its uses fully shown in a pamphlet entitled, The Circles of Proportion, where, in an appendix, are well treated several important points in navigation. It has also been made in the form of a sliding ruler.

The logarithmic tables were first applied to the different cases of sailing by Mr Thomas Addison, in his treatise entitled, Arithmetical Navigation, printed in 1625. He also gives two traverse tables, with their uses; the one to quarter points of the compass, the other to degrees. Mr Henry Gellibrand published his discovery of the changes of the variation of the compass, in a small quarto pamphlet, entitled, A discourse mathematical on the variation of the magnetical needle, printed in 1635. This extraordinary phenomenon he found out by comparing the observations made at different times near the same place by Mr Burrough, Mr Gunter, and himself, all persons of great skill and experience in these matters. This discovery was likewise soon known abroad; for Father Athanasius Kircher, in his treatise entitled, Magnes, first printed at Rome in 1641, informs us, that he had been told it by Mr John Greaves; and then gives a letter of the famous Marinus Merseanus, containing a very distinct account of the same.

As altitudes of the sun are taken on shipboard by observing his elevation above the visible horizon; to obtain from thence the sun's true altitude with correctness, Wright observes it to be necessary that the dip of the visible horizon below the horizontal plane passing through the observer's eye should be brought into the account, account, which cannot be calculated without knowing the magnitude of the earth. Hence he was induced to propose different methods for finding this; but complains that the most effectual was out of his power to execute; and therefore contented himself with a rude attempt, in some measure sufficient for his purpose; and the dimensions of the earth deduced by him corresponded very well with the usual divisions of the log line; however, as he wrote not an express treatise on navigation, but only for the correcting such errors as prevailed in general practice, the log line did not fall under his notice. Mr Richard Norwood, however, put in execution the method recommended by Mr Wright as the most perfect for measuring the dimensions of the earth, with the true length of the degrees of a great circle upon it; and, in 1635, he actually measured the distance between London and York; from whence, and the summer solstitial altitudes of the sun observed on the meridian at both places, he found a degree on a great circle of the earth to contain 367,196 English feet, equal to 57,300 French fathoms or toises: which is very exact, as appears from many measures that have been made since that time. Of all this Mr Norwood gave a full account in his treatise called The Seaman's Practice, published in 1637. He there shows the reason why Snellius had failed in his attempt: he points out also various uses of his discovery, particularly for correcting the gross errors hitherto committed in the divisions of the log line. But necessary amendments have been little attended to by sailors, whose obstinacy in adhering to established errors has been complained of by the best writers on navigation. This improvement has at length, however, made its way into practice, and few navigators of reputation now make use of the old measure of 42 feet to a knot. In that treatise also Mr Norwood describes his own excellent method of setting down and perfecting a sea reckoning, by using a traverse table; which method he had followed and taught for many years. He shows also how to rectify the course by the variation of the compass being considered; as also how to discover currents, and to make proper allowance on their account. This treatise, and another on trigonometry, were continually reprinted, as the principal books for learning scientifically the art of navigation. What he had delivered, especially in the latter of them; concerning this subject, was contracted as a manual for sailors, in a very small piece called his Epitome; which useful performance has gone through a great number of editions. No alterations were ever made in the Seaman's Practice till the 12th edition in 1676, when the following paragraph was inserted in a smaller character: "About the year 1672, Monsieur Picart has published an account in French, concerning the measure of the earth, a breviate whereof may be seen in the Philosophical Transactions, No 112, wherein he concludes one degree to contain 365,184 English feet; nearly agreeing with Mr Norwood's experiment;" and this advertisement is continued through the subsequent editions as late as the year 1732.

About the year 1645, Mr Bond published in Norwood's Epitome a very great improvement in Wright's method, by a property in his meridian line, whereby its divisions are more scientifically assigned than the author himself was able to effect; which was from this theorem, that these divisions are analogous to the excesses of the logarithmic tangents of half the respective latitudes augmented by 45 degrees above the logarithm of the radius. This he afterwards explained more fully in the third edition of Gunter's works, printed in 1653; where, after observing that the logarithmic tangents from 45° upwards increase in the same manner that the secants added together do, if every half degree be accounted as a whole degree of Mercator's meridional line. His rule for computing the meridional parts belonging to any two latitudes, supposed on the same side of the equator, is to the following effect: "Take the logarithmic tangent, rejecting the radius, of half each latitude, augmented by 45 degrees; divide the difference of those numbers by the logarithmic tangent of 45° 30', the radius being likewise rejected; and the quotient will be the meridional parts required, expressed in degrees." This rule is the immediate consequence from the general theorem, That the degrees of latitude bear to one degree (or 60 minutes, which in Wright's table stands for the meridional parts of one degree), the same proportion as the logarithmic tangent of half any latitude augmented by 45 degrees, and the radius neglected, to the like tangent of half a degree augmented by 45 degrees, with the radius likewise rejected. But here was farther wanting the demonstration of this general theorem, which was at length supplied by Mr James Gregory of Aberdeen in his Exercitationes Geometricae, printed at London in 1668; and afterwards more concisely demonstrated, together with a scientific determination of the divisor, by Dr Halley in the Philosophical Transactions for 1695, No 219, from the consideration of the spirals into which the rhumbs are transformed in the stereographic projection of the sphere upon the plane of the equinoctial; and which is rendered still more simple by Mr Roger Cotes, in his Logometria, first published in the Philosophical Transactions for 1714, No 388. It is moreover added in Gunter's book, that if \( \frac{1}{\sqrt{2}} \) of this division, which does not sensibly differ from the logarithmic tangent of 45° 1' 30" (with the radius subtracted from it), be used, the quotient will exhibit the meridional parts expressed in leagues, and this is the divisor set down in Norwood's Epitome. After the same manner the meridional parts will be found in minutes, if the like logarithmic tangent of 45° 1' 30", diminished by the radius, be taken; that is, the number used by others being 12633, when the logarithmic tables consist of eight places of figures besides the index.

In an edition of the Seaman's Kalender, Mr Bond declared, that he had discovered the longitude by having found out the true theory of the magnetic variation; and to gain credit to his assertion, he foretold, that at London in 1657 there would be no variation of the compass, and from that time it would gradually increase the other way; which happened accordingly. Again, In the Philosophical Transactions for 1668, No 40, he published a table of the variation for 49 years to come. Thus he acquired such reputation, that his treatise, entitled, The Longitude Found, was in 1676 published by the special command of Charles II. and approved by many celebrated mathematicians. It was not long, however, before it met with opposition; and in 1678 another treatise, entitled, The Longitude not Found, made its appearance; and as Mr Bond's hy- pohesis did not in any manner answer its author's sanguine expectations, the affair was undertaken by Dr Halley. The result of his speculation was, that the magnetic needle is influenced by four poles; but this wonderful phenomenon seems hitherto to have eluded all our researches. In 1705, however, Dr Halley published a general map, with curve lines expressing the paths where the magnetic needle had the same variation; which was received with universal applause. But as the positions of these curves vary from time to time, they should frequently be corrected by skilful persons; as was done in 1644 and 1756, by Mr William Moun- taine, and Mr James Dodson, F. R. S. In the Philosophical Transactions for 1695, Dr Halley also gave a dissertation on the monsoons; containing many very useful observations for such as sail to places subject to these winds.

After the true principles of the art were settled by Wright, Bond, and Norwood, the authors on navigation became so numerous, that it would be impossible to enumerate them. New improvements were daily made, and every thing relative to it was settled with an accuracy not only unknown to former ages, but which would have been reckoned utterly impossible. The earth being found to be a spheroid, and not a perfect sphere, with the shortest diameter passing through the poles, a tract was published in 1741, by the Rev. Doctor Patrick Murdoch, wherein he accommodated Wright's sailing to such a figure; and Mr Colin Mac- laurin, the same year, in the Philosophical Transac- tions, No. 461, gave a rule for determining the meri- dional parts of a spheroid; which speculation is farther treated of in his book of Fluxions, printed at Edin- burgh in 1742.

Among the later discoveries in navigation, that of finding the longitude both by lunar observations and by time-keepers is the principal. It is owing chiefly to the rewards offered by the British parliament that this has attained the present degree of perfection. We are indebted to Dr Maskelyne for putting the first of these methods in practice, and for other important improve- ments in navigation. The time-keepers, constructed by Harrison for this express purpose, were found to answer so well, that he obtained the parliamentary re- ward.

The only works that have appeared of late in navi- gation are those on the longitude and navigation by Dr Mackay, of which the following account is transcribed from the Anti-Jacobin Review for September 1824.

"This publication, (Dr Mackay's Treatise on Na- vigation) and that on the longitude by the same author, form the most correct and practical system of naviga- tion and nautical science hitherto published in this country; they may be considered not only of individual utility, but of national importance."

THEORY OF NAVIGATION.

THE motion of a ship in the water is well known to depend on the action of the wind upon its sails, regula- ted by the direction of the helm. As the water is a resisting medium and the bulk of the ship very consider- able, it thence follows that there is always a great resistance on her fore-part; and when this resistance becomes sufficient to balance the moving force of the wind upon the sails, the ship attains her utmost degree of velocity, and her motion is no longer accelerated. This velocity is different according to the different strength of the wind; but the stronger the wind, the greater resistance is made to the ship's passage through the water; and hence, though the wind should blow ever so strong, there is also a limit to the velocity of the ship: for the sails and ropes can bear but a certain force of air; and when the resistance on the fore-part becomes more than equivalent to their strength, the velocity can be no longer increased, and the rigging gives way.

The direction of a ship's motion depends on the po- sition of her sails with regard to the wind, combined with the action of the rudder. The most natural di- rection of the ship is, when she runs directly before the wind, the sails are then disposed, so as to be at right angles thereto. But this is not always the case, both on account of the variable nature of the winds, and the situation of the intended port, or of intermediate head- lands or islands. When the wind therefore happens not to be favourable, the sails are placed so as to make an oblique angle both with the direction of the ship and with the wind; and the sails, together with the rudder, must be managed in such a manner, that the direction of the ship may make an acute angle with that of the wind; and the ship, making boards on different tacks, will by this means arrive at the intended port.

The reason of the ship's motion in this case is, that the water resists the side more than the fore-part, and that in the same proportion as her length exceeds her breadth. This proportion is so considerable, that the ship continually flies off where the resistance is least, and that sometimes with great swiftness. In this way of sailing, however, there is a great limitation: for if the angle made by the keel with the direction of the wind be too acute, the ship cannot be kept in that position; neither is it possible for a large ship to make a more acute angle with the wind than about 6 points; though small sloops, it is said, may make an angle of about 5 points with it. In all these cases, however, the velocity of the ship is greatly retarded; and that not only on account of the obliquity of her mo- tion, but by reason of what is called her lee-way. This is occasioned by the yielding of the water on the lee-side of the ship, by which means the vessel acquires a compound motion, partly in the direction of the wind, and partly in that which is necessary for attaining the desired port.

It is perhaps impossible to lay down any mathema- tical principles on which the lee-way of a ship could be properly calculated; only we may see in general that it depends on the strength of the wind, the roughness of the sea, and the velocity of the ship. When the wind is not very strong, the resistance of the water on the lee-side bears a very great proportion to that of the current of air; and therefore it will yield but very little; however, supposing the ship to remain remain in the same place, it is evident, that the water having once begun to yield, will continue to do so for some time, even though no additional force was applied to it; but as the wind continually applies the same force as at first, the lee-way of the ship must go on constantly increasing till the resistance of the water on the lee-side balances the force applied on the other, when it will become uniform, as both the motion of a ship sailing before the wind. If the ship changes her place with any degree of velocity, then every time she moves her own length, a new quantity of water is to be put in motion, which hath not yet received any momentum, and which of consequence will make a greater resistance than it can do when the ship remains in the same place. In proportion to the swiftness of the ship, then, the lee-way will be the less: but if the wind is very strong, the velocity of the ship bears but a small proportion to that of the current of air; and the same effects must follow as though the ship moved slowly, and the wind was gentle; that is, the ship must make a great deal of lee-way.—The same thing happens when the sea rises high, whether the wind is strong or not; for then the whole water of the ocean, as far as the swell reaches, has acquired a motion in a certain direction, and that to a very considerable depth. The mountainous waves will not fail to carry the ship very much out of her course; and this deviation will certainly be according to their velocity and magnitude. In all cases of a rough sea, therefore, a great deal of lee-way is made.—Another circumstance also makes a variation in the quantity of the lee-way; namely, the lightness or heaviness of the ship; it being evident, that when the ship sinks deep in the water, a much greater quantity of that element is to be put in motion before she can make any lee-way, than when she swims on the surface. As therefore it is impossible to calculate all these things with mathematical exactness, it is plain that the real course of a ship is exceedingly difficult to be found, and frequent errors must be made, which can only be corrected by celestial observations.

In many places of the ocean there are currents, or places where the water, instead of remaining at rest, runs with a very considerable velocity for a great way in some particular direction, and which will certainly carry the ship greatly out of her course. This occasions an error of the same nature with the lee-way: and therefore, whenever a current is perceived, its direction and velocity ought to be determined, and the proper allowances made.

Another source of error in reckoning the course of a ship proceeds from the variation of the compass.

There are few parts of the world where the needle points exactly north; and in those where the variation is known, it is subject to very considerable alterations. By these means the course of the ship is mistaken; for as the sailors have no other standard to direct them than the compass, if the needle, instead of pointing due north, should point north-east, a prodigious error would be occasioned during the course of the voyage, and the ship would not come near the port to which she was bound. To avoid errors of this kind, the only method is, to observe the sun's amplitude and azimuth as frequently as possible, by which the variation of the compass will be perceived, and the proper allowances can then be made for errors in the course which this may have occasioned.

Errors will arise in the reckoning of a ship, especially when she sails in high latitudes, from the spherical figure of the earth; for as the polar diameter of our globe is found to be considerably shorter than the equatorial one, it thence follows, that the farther we remove from the equator, the longer are the degrees of latitude. Of consequence, if a navigator assigns any certain number of miles for the length of a degree of latitude near the equator, he must vary that measure as he approaches towards the poles, otherwise he will imagine that he hath not sailed so far as he actually hath done. It would therefore be necessary to have a table containing the length of a degree of latitude in every different parallel from the equator to either pole; as without this a troublesome calculation must be made at every time the navigator makes a reckoning of his course. Such a table, however, hath not yet appeared; neither indeed does it seem to be an easy matter to make it, on account of the difficulty of measuring the length even of one or two degrees of latitude in different parts of the world. Sir Isaac Newton first discovered this spheroidal figure of the earth; and showed, from experiments on pendulums, that the polar diameter was to the equatorial one as 229 to 230. This proportion, however, hath not been admitted by succeeding calculators. The French mathematicians, who measured a degree on the meridian in Lapland, made the proportion between the equatorial and polar diameters to be as 1 to 0.891. Those who measured a degree at Quito in Peru, made the proportion 1 to 0.99624, or 266 to 265. M. Bouguer makes the proportion to be as 179 to 178; and M. Buffon, in one part of his theory of the earth, makes the equatorial diameter exceed the polar one by 1/25 of the whole. According to M. du Sejour, this proportion is as 321 to 320; and M. de la Place, in his Memoir upon the Figure of Spheroids, has deduced the same proportion. From these variations it appears that the point is not exactly determined, and consequently that any corrections which can be made with regard to the spheroidal figure of the earth must be very uncertain.

It is of consequence to navigators in a long voyage to take the nearest way to their port; but this is scarcely possible to be done. The shortest distance between any two points on the surface of a sphere is measured by an arch of a great circle intercepted between them; and therefore it is advisable to direct the ship along a great circle of the earth's surface. But this is a matter of considerable difficulty, because there are no fixed marks by which it can be readily known whether the ship sails in the direction of a great circle or not. For this reason the sailors commonly choose to direct their course by the rhumbs, or the bearing of the place by the compass. These bearings do not point out the shortest distance between places; because, on a globe, the rhumbs are spirals, and not arches of great circles. However, when the places lie directly under the equator, or exactly under the same meridian, the rhumb then coincides with the arch of a great circle, and of consequence shows the nearest way. The sailing on the arch of a great circle is called great circle sailing; and the cases of it depend all on the solution of problems in spherical trigonometry.

BOOK I. Containing the Various Methods of Sailing.

INTRODUCTION.

The art of navigation depends upon astronomical and mathematical principles. The places of the sun and fixed stars are deduced from observation and calculation, and arranged in tables, the use of which is absolutely necessary in reducing observations taken at sea, for the purpose of ascertaining the latitude and longitude of the ship, and the variation of the compass. The problems in the various sailings are resolved either by trigonometrical calculation, or by tables or rules formed by the assistance of trigonometry. By mathematics, the necessary tables are constructed, and rules investigated for performing the more difficult parts of navigation. For these several branches of science, and for logarithmic tables, the reader is referred to the respective articles in this work. A few tables are given at the end of this article; but as the other tables necessary for the practice of navigation are to be found in almost every treatise on that subject, it therefore seems unnecessary to insert them in this place.

CHAP. I. Preliminary Principles.

Sect. I. Of the Latitude and Longitude of a Place.

The situation of a place on the surface of the earth is estimated by its distance from two imaginary lines intersecting each other at right angles: The one of these is called the equator, and the other the first meridian. The situation of the equator is fixed, but that of the first meridian is arbitrary, and therefore different nations assume different first meridians. In Britain, we esteem that to be the first meridian which passes through the royal observatory at Greenwich.

The equator divides the earth into two equal parts, called the northern and southern hemispheres; and the latitude of a place is its distance from the equator, reckoned on a meridian in degrees and parts of a degree; and is either north or south, according as it is in the northern or southern hemisphere.

The first meridian being continued round the globe, divides it into two equal parts, called the eastern and western hemispheres; and the longitude of a place is that portion of the equator contained between the first meridian and the meridian of the given place, and is either east or west; according as it is in the eastern or western hemisphere, respectively to the first meridian.

Prob. I. The latitudes of two places being given, to find the difference of latitude.

Rule. Subtract the less latitude from the greater, if the latitudes be of the same name, but add them if of contrary; and the remainder or sum will be the difference of latitude.

Example 1. Required the difference of latitude between the Lizard, in latitude 49° 57' N. and Cape St Vincent, in latitude 37° 2' N?

| Latitude of the Lizard | 49° 57' N. | |-----------------------|------------| | Latitude of Cape St Vincent | 37° 2' N. |

Difference of latitude = 12° 55' = 775 miles.

Ex. 2. What is the difference of latitude between Funchal, in latitude 32° 38' N., and the Cape of Good Hope, in latitude 34° 29' S?

| Latitude of Funchal | 32° 38' N. | |---------------------|------------| | Lat. of Cape of Good Hope | 34° 29' S. |

Difference of latitude = 67° 7' = 4027 miles.

Prob. II. Given the latitude of one place, and the difference of latitude between it and another place, to find the latitude of that place.

Rule. If the given latitude and the difference of latitude be of the same name, add them; but if of different names, subtract them, and the sum or remainder will be the latitude required of the same name with the greater.

Ex. 1. A ship from latitude 39° 22' N. sailed due north 560 miles—Required the latitude come to?

| Latitude sailed from | 39° 22' N. | |----------------------|------------| | Difference of latitude 560' | = 9° 20' N. |

Latitude come to = 48° 42' N.

Ex. 2. A ship from latitude 7° 19' N. sailed 854 miles south—Required the latitude come to?

| Latitude sailed from | 7° 19' N. | |----------------------|------------| | Difference of latitude 854' | = 14° 14' S. |

Latitude come to = 6° 55' S.

Prob. III. The longitudes of two places being given, to find their difference of longitude.

Rule. If the longitudes of the given places are of the same name, subtract the less from the greater, and the remainder is the difference of longitude: but if the longitudes are of contrary names, their sum is the difference of longitude. If this exceeds 180°, subtract it from 360°, and the remainder is the difference of longitude.

Ex. 1. Required the difference of longitude between Edinburgh and New York, their longitudes being 3° 14' W. and 74° 10' W. respectively?

| Longitude of New York | 74° 10' W. | |-----------------------|------------| | Longitude of Edinburgh | 3° 14' W. |

Difference of longitude = 70° 56'

Ex. 2. What is the difference of longitude between Maskelyne's Isles in longitude 167° 59' E. and Olinde, in longitude 35° 5' W? PROB. IV. Given the longitude of a place, and the difference of longitude between it and another place, to find the longitude of that place.

RULE. If the given longitude and the difference of longitude be of a contrary name, subtract the less from the greater, and the remainder is the longitude required of the same name with the greater quantity; but if they are of the same name, add them, and the sum is the longitude sought, of the same name with that given. If this sum exceeds $180^\circ$, subtract it from $360^\circ$, the remainder is the required longitude of a contrary name to that given.

Ex. 1. A ship from longitude $9^\circ 54' E.$ sailed westerly till the difference of longitude was $23^\circ 18'$. Required the longitude come to?

Longitude sailed from $9^\circ 54' E.$

Difference of longitude $23^\circ 18' W.$

Longitude come to $13^\circ 24' W.$

Ex. 2. The longitude sailed from is $25^\circ 9' W.$ and difference of longitude $18^\circ 46' W.$ Required the longitude come to?

Longitude left $25^\circ 9' W.$

Difference of longitude $18^\circ 46' W.$

Longitude in $43^\circ 55' W.$

SECT. II. Of the Tides.

The theory of the tides has been explained under the article ASTRONOMY, and will again be farther illustrated under that of TIDES. In this place, therefore, it remains only to explain the method of calculating the time of high water at a given place.

As the tides depend upon the joint actions of the sun and moon, and therefore upon the distance of these objects from the earth and from each other; and as, in the method generally employed to find the time of high water, whether by the mean time of new moon, or by the epacts, or tables deduced therefrom, the moon is supposed to be the sole agent, and to have an uniform motion in the periphery of a circle, whose centre is that of the earth; it is hence obvious that method cannot be accurate, and by observation the error is sometimes found to exceed two hours. That method is therefore rejected, and another given, in which the error will seldom exceed a few minutes, unless the tides are greatly influenced by the winds.

PROB. I. To reduce the times of the moon's phases as given in the Nautical Almanac to the meridian of a known place.

RULE. To the time of the proposed phase, as given in the Nautical Almanac, apply the longitude of the place in time, by addition or subtraction, according as it is east or west, and it will give the time of the phase at the given place.

Ex. 1. Required the time of new moon at Salonique in May 1793? The nearest phase to 15th November is that of full moon.

Longitude of Funchal in time,

Time of full moon at Funchal,

Given day, November

Difference,

Time of high water at Funchal at full and change,

Equation from the Table to 2d 7h 38' before full moon,

Approx. time of high water, Nov.

Reduced time of full moon,

Interval,

Time of high water at full and change,

Equation to 1d 11h before full moon,

Time of high water,

Equation to 1d 11h + 12h = 1d 23h is 1h 15', and 12h 4' - 1h 15' = 10h 49' = time of high water in the forenoon.

Ex. 3. Required the time of high water at Duskey Bay, 24th October 1793?

The nearest phase to the 24th October is the last quarter.

Longitude of Duskey Bay in time,

Reduced time of first quarter of moon

Given day

Difference,

Time of high water at full and change,

Equation to 2d 16h 52' before last quarter,

Approximate time of high water,

Change of equation to app. time 1h 49'

Time of high water in the afternoon,

Change of equation to 12 hours,

Time of high water in the morning,

Sect. III. Of measuring a Ship's Run in a given Time.

The method commonly used at sea to find the distance sailed in a given time, is by means of a log-line and half-minute glass. A description of these is given under the articles Log and Log-line; which see.

It has been already observed, that the interval between each knot on the line ought to be 50 feet, in order to adapt it to a glass that runs 30 seconds. But although the line and glass be at any time perfectly adjusted to each other, yet as the line shrinks after being wet, and as the weather has a considerable effect upon the glass, it will therefore be necessary to examine them from time to time; and the distance given by them must be corrected accordingly. The distance sailed may, therefore, be affected by an error in the glass, or in the line, or in both. The true distance may, however, be found as follows.

Prob. I. The distance sailed by the log, and the seconds run by the glass, being given, to find the true distance, the line being supposed right.

Rule.—Multiply the distance given by the log by 30, and divide the product by the seconds run by the glass, the quotient will be the true distance.

Ex. 1. The hourly rate of sailing by the log is nine knots, and the glass is found to run out in 35 seconds. Required the true rate of sailing?

Ex. 2. The distance sailed by the log is 73 miles, and the glass runs out in 26 seconds. Sought the true distance?

Prob. II. Given the distance sailed by the log, and the measured interval between two adjacent knots on the line; to find the true distance, the glass running exactly 30 seconds.

Rule. Multiply twice the distance, sailed by the measured length of a knot, point off two figures to the right, and the remainder will be the true distance.

Ex. 1. The hourly rate of sailing by the log is five knots, and the interval between knot and knot measures 53 feet. Required the true rate of sailing?

Measured interval = 53

Twice hourly rate = 10

True rate of sailing = 5.30

Ex. 2. The distance sailed is 64 miles, by a log-line which measures 42 feet to a knot. Required the true distance?

Twice given distance = 120

Measured interval = 42

True distance = 53.76

Prob. III. Given the length of a knot, the number of seconds run by the glass in half a minute, and the distance sailed by the log; to find the true distance.

Rule. Multiply the distance sailed by the log by six times the measured length of a knot, and divide the product by the seconds run by the glass; the quotient, pointing off one figure to the right, will be the true distance.

Ex. The distance sailed by the log is 159 miles, the measured length of a knot is 42 feet, and the glass runs 33 seconds in half a minute. Required the true distance?

Distance by the log = 159

Six times length of a knot = 42 × 6 = 252

Seconds run by the glass = 33 × 4068 (121.4) = true distance. Plane sailing is the art of navigating a ship upon principles deduced from the notion of the earth's being an extended plane. On this supposition the meridians are esteemed as parallel right lines. The parallels of latitude are at right angles to the meridians; the lengths of the degrees on the meridians, equator, and parallels of latitude, are everywhere equal; and the degrees of longitude are reckoned on the parallels of latitude as well as on the equator.—In this sailing four things are principally concerned, namely, the course, distance, difference of latitude, and departure.

The course is the angle contained between the meridian and the line described by the ship, and is usually expressed in points of the compass.

The distance is the number of miles a ship has sailed on a direct course in a given time.

The difference of latitude is the portion of a meridian contained between the parallels of latitude sailed from and come to; and is reckoned either north or south, according as the course is in the northern or southern hemisphere.

The departure is the distance of the ship from the meridian of the place she left, reckoned on a parallel of latitude. In this sailing, the departure and difference of longitude are esteemed equal.

In order to illustrate the above, let \( \Delta \) (fig. 1.) represent the position of any given place, and AB the meridian passing through that place; also let AC represent the line described by a ship, and C the point arrived at. From C draw CB perpendicular to AB. Now in the triangle ABC, the angle BAC represents the course, the side AC the distance, AB the difference of latitude, and BC the departure.

In constructing a figure relating to a ship's course, let the upper part of what the figure is to be drawn on represent the north, then the lower part will be south, the right-hand side east, and the left-hand side west.

A north and south line is to be drawn to represent the meridian of the place from which the ship sailed; and the upper or lower part of this line, according as the course is southerly or northerly, is to be marked as the position of that place. From this point as a centre, with the chord of 60°, an arch is to be described from the meridian towards the right or left, according as the course is easterly or westerly; and the course, taken from the line of chords if given in degrees, but from the line of rhumbs if expressed in points of the compass, is to be laid upon this arch, beginning at the meridian. A line drawn through this point and that sailed from, will represent the distance, which if given must be laid thereon, beginning at the point sailed from. A line is to be drawn from the extremity of the distance perpendicular to the meridian; and hence the difference of latitude and departure will be obtained.

If the difference of latitude is given, it is to be laid upon the meridian, beginning at the point representing the place the ship left; and a line drawn from the extremity of the difference of latitude perpendicular to the meridian, till it meets the distance produced, will limit the figure.

If the departure is given, it is to be laid off on a parallel, and a line drawn through its extremity will limit the distance. When either the distance and difference of latitude, distance and departure, or difference of latitude and departure, are given, the measure of each is to be taken from a scale of equal parts, and laid off on its respective line, and the extremities connected. Hence the figure will be formed.

**Prob. I.** Given the course and distance, to find the difference of latitude and departure.

*Example.* A ship from St Helena, in latitude 15° 55' S., sailed S. W. by S. 158 miles. Required the latitude come to, and departure.

**By Construction.**

Draw the meridian AB (fig. 2.), and with the chord of 60° describe the arch m n, and make it equal to the rhumb of 3 points, and through n draw AC equal to 158 miles; from C, draw CB perpendicular to AB; then AB applied to the scale from which AC was taken, will be found to measure 131.4 and BC 87.8.

**By Calculation.**

To find the difference of latitude.

\[ \begin{align*} \text{As radius} & = 10,00000 \\ \text{is to the cosine of the course} & = 3 \text{ points} \\ \text{so is the distance} & = 158 \\ & = 2.19866 \\ \end{align*} \]

To the difference of latitude 131.4

\[ \begin{align*} \text{As radius} & = 10,00000 \\ \text{is to the sine of the course} & = 3 \text{ points} \\ \text{so is the distance} & = 158 \\ & = 2.19866 \\ \end{align*} \]

To the departure 87.8

\[ \begin{align*} \text{As radius} & = 10,00000 \\ \text{is to the sine of the course} & = 3 \text{ points} \\ \text{so is the distance} & = 158 \\ & = 2.19866 \\ \end{align*} \]

**By Inspection.**

In the traverse table, the difference of latitude answering to the course 3 points, and distance 158 miles, in a distance column is 131.4, and departure 87.8.

**By Gunter's Scale.**

The extent from 8 points to 5 points, the complement of the course on the line of sine rhumbs (marked SR.) will reach from the distance 158 to 131.4, the difference of latitude on the line of numbers; and the extent from 8 points to 3 points on sine rhumbs, will reach from 158 to 87.8, the departure on numbers (A).

Latitude St Helena = 15° 55' S.

Difference of latitude = 2 11 S.

Latitude come to = 18 6 S.

**Prob. II.** Given the course and difference of latitude, to find the distance and departure.

*Example.*

---

(a) For the method of resolving the various problems in navigation, by the sliding gunter, the reader is referred to Dr Mackay's Treatise on the Description and Use of that Instrument. Example. A ship from St George's, in latitude $38^\circ 45'$ north, sailed SE $\frac{1}{2}$ S: and the latitude by observation was $35^\circ 7'$ N. Required the distance run, and departure?

| Latitude St George's | $38^\circ 45'$ N | |----------------------|-----------------| | Latitude come to | $35^\circ 7'$ N |

Difference of latitude $3^\circ 38' = 218$ miles.

By Construction.

Draw the portion of the meridian AB (fig. 3.) equal to 218 m.: from the centre A with the chord of $60^\circ$ describe the arch mn, which make equal to the rhumb of $3\frac{1}{2}$ points: through A n draw the line AC, and from B draw BC perpendicular to AB, and let it be produced till it meets AC in C. Then the distance AC being applied to the scale, will measure 282 m. and the departure BC = 179 miles.

By Calculation.

To find the distance.

As radius $10.000000$ is to the secant of the course $3\frac{1}{2}$ points $10.111181$ so is the difference of latitude $218$ m. $2.33846$

to the distance $282$ $2.45027$

To find the departure.

As radius $10.000000$ is to the tangent of the course $3\frac{1}{2}$ points $9.91417$ so is the difference of latitude $218$ $2.33846$

to the departure $178.9$ $2.25253$

By Inspection.

Find the given difference of latitude 218 m. in a latitude column, under the course of $3\frac{1}{2}$ points; opposite to which, in a distance column, is 282 miles; a departure column is 178.9 m. the distance and departure required.

By Gunter's Scale.

Extend the compass from $4\frac{1}{2}$ points, the complement of the course, to 8 points on sine rhumbs; that extent will reach from the difference of latitude 218 miles, to the distance 282 miles in numbers; and the extent from 4 points to the course $3\frac{1}{2}$ points on the line of tangent rhumbs (marked T. R.) will reach from 218 miles to 178.9, the departure on numbers.

Prob. III. Given course and departure, to find the distance and difference of latitude?

Example. A ship from Palma, in latitude $28^\circ 37'$ N. sailed NW. by W. and made 192 miles of departure: Required the distance run, and latitude come to?

By Construction.

Make the departure BC (fig. 4.) equal to 192 miles, draw BA perpendicular to BC, and from the centre C, with the chord of $60^\circ$, describe the arch mn, which make equal to the rhumb of 3 points, the complement of the course; draw a line through C n, which produce till it meets BA in A: then the distance AC being measured, will be equal to 231 m. and the difference of latitude AB will be 128.3 miles.

By Calculation.

To find the course.

As the distance $285$ $2.45484$ is to the difference of latitude $198$ $2.29660$ so is the radius $10.000000$

to the cosine of the course $46^\circ 2'$ $9.84716$ Plane Sailing.

To find the departure.

As radius

is to the sine of the course

so is the distance

to the departure

By Inspection.

Find the given distance in the table in its proper column; and if the difference of latitude answering thereto is the same as that given, namely, 198, then the departure will be found in its proper column, and the course at the top or bottom of the page, according as the difference of latitude is found in a column marked lat. at top or bottom. If the difference of latitude thus found does not agree with that given, turn over till the nearest thereto is found to answer to the given distance. This is in the page marked 46 degrees at the bottom, which is the course, and the corresponding departure is 205 miles.

By Gunter's Scale.

The extent from the distance 285 to the difference of latitude 198 on numbers, will reach from 92° to 44°, the complement of the course on sines; and the extent from 92° to the course 46° on the line of sines being laid from the distance 285, will reach to the departure 205 on the line of numbers.

Prob. V. Given the distance and departure, to find the course and difference of latitude.

Example. A ship from Fort-Royal in the island of Grenada, in latitude 12° 0' N, sailed 260 miles between the south and west, and made 192 miles of departure: Required the course and latitude come to?

By Construction.

Draw BC (fig. 6.) perpendicular to AB, and equal to the given departure 192 miles; then from the centre C, with the distance 260 miles, sweep an arch intersecting AB in A, and join AC. Now describe an arch from the centre A with the chord of 60°, and the portion m n of this arch, contained between the distance and difference of latitude, measured on the line of chords, will be 47° the course; and the difference of latitude AB applied to the scale of equal parts, measures 177° miles.

By Calculation.

To find the course.

As the distance

is to the departure

so is radius

to the sine of the course

To find the difference of latitude.

As radius

is to the cosine of the course

so is the distance

to the difference of latitude

By Inspection.

Seek in the traverse table until the nearest to the given departure is found in the same line with the given distance 260. This is found to be in the page marked 47° at the bottom, which is the course; and the corresponding difference of latitude is 177°.

Vol. XIV. Part II.

By Gunter's Scale.

The extent of the compass, from the distance 260 to the departure 192 on the line of numbers, will reach from 92° to 47°, the course on the line of sines; and the extent from 92° to 43°, the complement of the course on sines, will reach from the distance 260 to the difference of latitude 177° on the line of numbers.

Latitude Fort Royal

Difference of latitude

Latitude in

Prob VI. Given difference of latitude and departure, sought course and distance.

Example. A ship from a port in latitude 7° 56' S., sailed between the south and east, till her departure is 132 miles; and is then by observation found to be in latitude 12° 3' S. Required the course and distance?

Latitude sailed from

Latitude in by observation

Difference of latitude

By Construction.

Draw the portion of the meridian AB (fig. 7.) equal fig. 7. to the difference of latitude 247 miles; from B draw BC perpendicular to AB, and equal to the given departure 132 miles, and join AC; then with the chord of 60° describe an arch from the centre A; and the portion m n of this arch being applied to the line of chords, will measure about 28°; and the distance AC, measured on the line of equal parts, will be 280 miles.

By Calculation.

To find the course.

As the difference of latitude

is to the departure

so is radius

to the tangent of the course

To find the distance.

As radius

is to the secant of the course

so is the difference of latitude

to the distance

By Inspection.

Seek in the table till the given difference of latitude and departure, or the nearest thereto, are found together in their respective columns, which will be under 28°, the required course; and the distance answering thereto is 280 miles.

By Gunter's Scale.

The extent from the given difference of latitude 247 to the departure 132 on the line of numbers, will reach from 45° to 28°, the course on the line of tangents; and the extent from 62°, the complement of the course, to 92° on sines, will reach from the difference of latitude 247, to the distance 280 on numbers.

Chap. III. Of Traverse Sailing.

If a ship sail upon two or more courses in a given time, the irregular track she describes is called a traverse; and to resolve a traverse, is the method of reducing these several courses, and the distances run, in... Traverse to a single course and distance. The method chiefly used for this purpose at sea is by inspection, which shall therefore be principally adhered to; and is as follows.

Make a table of a breadth and depth sufficient to contain the several courses, &c. This table is to be divided into six columns: the courses are to be put in the first, and the corresponding distances in the second column; the third and fourth columns are to contain the differences of latitude, and the two last the departures.

Now, the several courses and their corresponding distances being properly arranged in the table, find the difference of latitude and departure answering to each in the traverse table; remembering that the difference of latitude is to be put in a north or south column, according as the course is in the northern or southern hemisphere; and that the departure is to be put in an east column if the course is easterly, but in a west column if the course is westerly: Observing also, that the departure is less than the difference of latitude when the course is less than 4 points or $45^\circ$; otherwise greater.

Add up the columns of northing, southing, casting, and westing, and set down the sum of each at its bottom; then the difference between the sums of the north and south columns will be the difference of latitude made good, of the same name with the greater; and the difference between the sums of the east and west columns, is the departure made good, of the same name with the greater sum.

Now, seek in the traverse table, till a difference of latitude and departure are found to agree as nearly as possible with those above; then the distance will be found on the same line, and the course at the top or bottom of the page, according as the difference of latitude is greater or less than the departure.

In order to resolve a traverse by construction, describe a circle with the chord of $60^\circ$, in which draw two diameters at right angles to each other, at whose extremities are to be marked the initials of the cardinal points, north being uppermost.

Lay off each course on the circumference, reckoned from its proper meridian; and from the centre to each point draw lines, which are to be marked with the proper number of the course.

On the first radius lay off the first distance from the centre; and through its extremity, and parallel to the second radius, draw the second distance of its proper length; through the extremity of the second distance, and parallel to the third radius, draw the third distance of its proper length; and thus proceed until all the distances are drawn.

A line drawn from the extremity of the last distance to the centre of the circle will represent the distance made good; and a line drawn from the same point perpendicular to the meridian, produced, if necessary, will represent the departure; and the portion of the meridian intercepted between the centre and departure, will be the difference of latitude made good.

**Examples.**

I. A ship from Fyall, in lat. $38^\circ 32' N$, sailed as follows: ESE $163$ miles, SW $\frac{1}{2}$ W $110$ miles, SE $\frac{1}{2}$ S $180$ miles, and N by E $68$ miles. Required the latitude come to, the course, and distance made good.

| Course | Dist. | Diff. of Latitude | Departure | |--------|-------|------------------|-----------| | ESE | 163 | 62.4 | 150.6 | | SW $\frac{1}{2}$ W | 110 | 69.8 | 85.0 | | SE $\frac{1}{2}$ S | 180 | 144.5 | 107.2 | | N $\frac{1}{2}$ E | 68 | 66.7 | 13.3 | | | | 66.7 | 276.7 | | | | | 271.1 | | | | | 85.0 | | S $41\frac{1}{2}$ E | 281 | 210.0 | 186.1 | | Latitude left | - | 38° 32' N. | | Difference of latitude | - | 3 21 8 | | Latitude come to | - | 35 11 N. |

**By Construction.**

With the chord of $60^\circ$ describe the circle NE, SW (fig. 8.), the centre of which represents the place the Fig. 8. ship sailed from: draw two diameters NS, EW at right angles to each other; the one representing the meridian, and the other the parallel of latitude of the place sailed from. Take each course from the line of rhumbs, lay it off on the circumference from its proper meridian, and number it in order 1, 2, 3, 4. Upon the first rhumb C1, lay off the first distance $163$ miles from C to A; through it draw the second distance AB parallel to C2, and equal to $110$ miles; through B draw BD equal to $180$ miles, and parallel to C3; and draw DE parallel to C4, and equal to $68$ miles. Now CE being joined, will represent the distance made good; which applied to the scale will measure $281$ miles. The arch $S_n$, which represents the course, being measured on the line of chords, will be found equal to $41\frac{1}{2}$. From E draw EF perpendicular to CS produced; then CF will be the difference of latitude, and FE the departure made good; which applied to the scale will be found to measure $210$ and $186$ miles respectively.

As the method by construction is scarcely ever practised at sea, it, therefore, seems unnecessary to apply it to the solution of the following examples.

II. A ship from latitude $1^\circ 38' S.$ sailed as under. Required her present latitude, course, and distance made good?

| Course | Dist. | Diff. of Latitude | Departure | |--------|-------|------------------|-----------| | NW $\frac{1}{2}$ N | 43 | 35.8 | 23.9 | | WNW | 78 | 29.9 | 72.1 | | SE $\frac{1}{2}$ E | 36 | 31.1 | 46.6 | | WSW $\frac{1}{2}$ W | 62 | 18.0 | 59.3 | | N $\frac{1}{2}$ E | 85 | 84.1 | 12.5 | | | | 149.8 | 49.1 | | | | | 39.1 | | | | | 155.3 | | | | | 59.1 | | N $44^\circ W$ | 139 | 100.7 | 1° 41' | | Latitude left | - | 1 38 S. | | Latitude come to | - | 0 3 N. | III. Yesterday at noon we were in latitude $13^\circ 12'$ N, and since then have run as follows: SSE 36 miles, S 12 miles, NW $\frac{1}{2}$ W 28 miles, W 30 miles, SW 42 miles, WbN 39 miles, and N 20 miles. Required our present latitude, departure, and direct course and distance?

| Courses | Dist. | Diff. of Latitude | Departure | |---------|-------|------------------|-----------| | SSE | 36 | 33.3 | 3.8 | | S | 12 | 12.0 | | | NW $\frac{1}{2}$ W | 28 | 17.8 | 21.6 | | W | 30 | | 30.0 | | SW | 42 | 29.7 | 29.7 | | WbN | 39 | 7.6 | 38.2 | | N | 20 | | |

$S 74^\circ W$

Yesterday's latitude

Present latitude

IV. The course per compass from Greigness (B) to the May is SW $\frac{1}{2}$ S, distance 58 miles; from the May to the Staples S$\theta E \frac{1}{2} E$, 44 miles; and from the Staples to Flamborough Head S$\theta E$, 110 miles. Required the course per compass, and distance from Greigness to Flamborough Head?

| Courses | Dist. | Diff. of Latitude | Departure | |---------|-------|------------------|-----------| | SW $\frac{1}{2}$ S | 58 | 43.0 | 38.9 | | S$\theta E \frac{1}{2} E$ | 44 | 41.4 | 14.8 | | S$\theta E$ | 110 | 107.9 | 21.5 |

Hence the course per compass is nearly S $1^\circ W$, and distance 192$\frac{1}{2}$ miles.

CHAP. IV. Of Parallel Sailing.

The figure of the earth is spherical, and the meridians gradually approach each other, and meet at the poles. The difference of longitude between any two places is the angle at the pole contained between the meridians of those places; or it is the arch of the equator intercepted between the meridians of the given places; and the meridian distance between two places in the same parallel, is the arch thereof contained between their meridians. It hence follows, that the meridian distance, answering to the same difference of longitude, will be variable with the latitude of the parallel upon which it is reckoned; and the same difference of longitude will not answer to a given meridian distance when reckoned upon different parallels.

Parallel sailing is, therefore, the method of finding the distance between two places lying in the same parallel whose longitudes are known; or, to find the difference of longitude answering to a given distance, run in an east or west direction. This sailing is particularly useful in making low or small islands.

In order to illustrate the principles of parallel sailing, let CABP (fig. 9.) represent a section of one fourth part of the earth, the arch ABP being part of a meridian; CA the equatorial, and CP the polar semi-axis. Also let B be the situation of any given place on the earth; and join BC, which will be equal to CA or CP (c). The arch AB, or angle ACB, is the measure of the latitude of the place B; and the arch BP, or angle BCP, is that of its complements. If BD be drawn from B perpendicular to CP, it will represent the cosine of latitude to the radius BC or CA.

Now since circles and similar portions of circles are in the direct ratio of their radii; therefore,

As radius Is to the cosine of latitude; So is any given portion of the equator To a similar portion of the given parallel.

But the difference of longitude is an arch of the equator; and the distance between any two places under the same parallel, is a similar portion of that parallel.

Hence R : cosine latitude :: Diff. longitude : Distance.

And by inversion, Cosine latitude : R :: Distance : Diff. of longitude.

Also, Diff. of longitude : Distance :: R : cos. latitude.

PROB. I. Given the latitude of a parallel, and the number of miles contained in a portion of the equator, to find the miles contained in a similar portion of that parallel.

Ex. 1. Required the number of miles contained in a degree of longitude in latitude $55^\circ 58'$?

By Construction.

Draw the indefinite right line AB (fig. 10.); make the angle BAC equal to the given latitude $55^\circ 58'$, and AC equal to the number of miles contained in a degree of longitude at the equator, namely 60; from C draw CB perpendicular to AB; and AB being measured on the line of equal parts, will be found equal to 33.5; the miles required.

(b) Greigness is about 2$\frac{1}{2}$ miles distant from Aberdeen, in nearly a SEbE$\frac{1}{2}$E direction.

(c) This is not strictly true, as the figure of the earth is that of an oblate spheroid; and therefore the radius of curvature is variable with the latitude. The difference between CA and CP, according to Sir Isaac Newton's hypothesis, is about 17 miles. By Calculation.

As radius \( \frac{1}{\cos \text{latitude}} \) is to the cosine of latitude, so miles in a deg. of long. at eq. \( \frac{60}{\cos \text{latitude}} \).

To the miles in a deg. in the given par. \( \frac{33.58}{\cos \text{latitude}} = 1.52609 \).

By Inspection.

To \( 36^\circ \), the nearest degree to the given latitude, and distance 65 miles, the corresponding difference of latitude is 33.6, which is the miles required.

By Gunter's Scale.

The extent from \( 92^\circ \) to \( 34^\circ \), the complement of the given latitude on the line of sines, will reach from 60 to 33.6 on the line of numbers.

There are two lines on the other side of the scale, with respect to Gunter's line, adapted to this particular purpose; one of which is entitled chords, and contains the several degrees of latitude: The other, marked M. L. signifying miles of longitude, is the line of longitudes, and shows the number of miles in a degree of longitude in each parallel. The use of these lines is therefore obvious.

Ex. 2. Required the distance between Treguier in France, in longitude \( 3^\circ 14' \) W, and Gaspey Bay, in longitude \( 64^\circ 27' \) W, the common latitude being \( 48^\circ 47' \) N?

| Longitude Treguier | \( 3^\circ 14' \) W | |--------------------|-------------------| | Longitude Gaspey Bay | \( 64^\circ 27' \) W |

Difference of longitude \( 61^\circ 13' = 3673' \)

As radius \( \frac{1}{\cos \text{latitude}} \) is to the cosine of latitude, \( \frac{48^\circ 47'}{\cos \text{latitude}} = 9.81882 \)

So is the difference of longitude \( 3073' = 3.56502 \)

To the distance \( 2420' = 3.38384 \)

Prob. II. Given the number of miles contained in a portion of a known parallel, to find the number of miles in a similar portion of the equator.

Example. A ship from Cape Finisterre, in latitude \( 42^\circ 52' \) N, and longitude \( 9^\circ 17' \) W, sailed due west 342 miles. Required the longitude come to?

By Construction.

Draw the straight line AB (fig. 11.) equal to the given distance 342 miles, and make the angle BAC equal to \( 42^\circ 52' \), the given latitude: from B draw BC perpendicular to AB, meeting AC in C; then AC applied to the scale will measure 466', the difference of longitude required.

By Calculation.

As radius \( \frac{1}{\sec \text{latitude}} \) is to the secant of latitude, \( \frac{42^\circ 52'}{\sec \text{latitude}} = 10.13493 \)

So is the distance \( 342 = 2.53403 \)

To the difference of longitude \( 466' = 2.66896 \)

By Inspection.

The nearest degree to the given latitude is \( 43^\circ \); under which, and opposite to \( 171 \), half the given distance in a latitude column, is 234, in a distance column, which doubled gives 468, the difference of longitude.

If the proportional part answering to the difference between the given latitude and that used, be applied to the above, the same result with that found by calculation will be obtained.

By Gunter's Scale.

The extent from \( 47^\circ 8' \), the complement of latitude to \( 62^\circ \) on the line of sines, being laid the same way from the distance 342, will reach to the difference of longitude \( 466' \) on the line of numbers.

Longitude Cape Finisterre \( 9^\circ 17' \) W

Difference of longitude \( 7^\circ 47' \) W

Longitude come to \( 17^\circ 4' \) W

Prob. III. Given the number of miles contained in any portion of the equator, and the miles in a similar portion of a parallel; to find the latitude of that parallel.

Example. A ship sailed due east 358 miles, and was found by observation to have differed her longitude \( 8^\circ 42' \). Required the latitude of the parallel?

By Construction.

Make the line AB (fig. 12.) equal to the given distance; to which let BC be drawn perpendicular, with an extent equal to 522', the difference of longitude; describe an arch from the centre A, cutting BC in C; then the angle BAC being measured by means of the line of chords, will be found equal to \( 46^\circ 42' \), the required latitude.

By Calculation.

As the distance \( 358 = 2.53388 \)

Is to the difference of longitude \( 522 = 2.71767 \)

So is radius \( 10.00000 \)

To the secant of the latitude \( 46^\circ 42' = 10.16379 \)

By Inspection.

As the difference of longitude and distance exceed the limits of the table, let therefore the half of each be taken; these are 261 and 179 respectively. Now, by entering the table with these quantities, the latitude will be found to be between 46 and 47 degrees. Therefore, to latitude \( 46^\circ \), and distance 261 miles, the corresponding difference of latitude is \( 181' \), which exceeds the half of the given distance by \( 2' \). Again, to latitude \( 47^\circ \), and distance 261, the difference of latitude is \( 170' \), being \( 1' \) less than the half of that given: therefore the change of distance answering to a change of \( 1^\circ \) of latitude is \( 3' \).

Now, as \( 3':3 = 2':3 :: 1':42' \).

Hence the latitude required is \( 46^\circ 42' \).

By Gunter's Scale.

The extent from 522 to 358 on the line of numbers, will reach from \( 92^\circ \) to about \( 43^\circ \), the complement of which \( 46^\circ \) is the latitude required.

Prob. IV. Given the number of miles contained in the portion of a known parallel, to find the length of a similar portion of another known parallel.

Example. From two ports in latitude \( 33^\circ 58' \) N, distance 348 miles, two ships sail directly north till they are in latitude \( 48^\circ 23' \) N. Required their distance?

By Construction.

Draw the line CB, CE (fig. 13.), making angles Fig. with CP equal to the complements of the given latitudes, namely, $56^\circ 2'$ and $41^\circ 37'$ respectively; make BD equal to the given distance 348 miles, and perpendicular to CP; now from the centre C, with the radius CB, describe an arch intersecting CE in E; then EF drawn from the point E, perpendicular to CP, will represent the distance required; which being applied to the scale, will measure 278½ miles.

**By Calculation.**

As the cosine of the latitude left $33^\circ 58'$ 9.91874 is to the cosine of the lat. come to $48^\circ 23'$ 9.82226 so is the given distance $-348$ 2.54158

to the distance required $-278.6$ 2.44510

**By Inspection.**

Under 348, and opposite to 174, half the given distance in a latitude column is 210 in a distance column; being half the difference of longitude answering thereto. Now, find the difference of latitude to distance 210 miles over 48° of latitude, which is $145^\circ 5'$; from which $1^\circ 1'$ (the proportional part answering to 23 minutes of latitude) being subtracted, gives $139^\circ 4'$ which doubled is $278^\circ 8'$, the distance required.

**By Gunter's Scale.**

The extent from $36^\circ 2'$, the complement of the latitude left, to $41^\circ 37'$, the complement of that come to, on the line of sines, being laid the same way from 348, will reach to 278½, the distance sought on the line of numbers.

**Prob. V.** Given a certain portion of a known parallel, together with a similar portion of an unknown parallel; to find the latitude of that parallel.

**Example.** Two ships, in latitude $56^\circ 0'$ N., distant 180 miles, sail due south; and having come to the same parallel, are now 232 miles distant. The latitude of that parallel is required?

**By Construction.**

Make DB (fig. 14.) equal to the first distance 180 miles, DM equal to the second 232, and the angle DBC equal to the given latitude $56^\circ$; from the centre C, with the radius CB, describe the arch BE; and through M draw ME parallel to CD, intersecting the arch BE in E; join EC and draw EF perpendicular to CD; then the angle FEC will be the latitude required; which being measured, will be found equal to $43^\circ 53'$.

**By Calculation.**

As the distance on the known parallel 180 2.25527 is to the distance on that required 232 2.36549 so is the cosine of the latitude left $36^\circ 0'$ 9.74756

to the cosine of the latitude come to $43^\circ 53'$ 9.85778

**By Inspection.**

To latitude $56^\circ$, and half the first distance 90 in a latitude column, the corresponding distance is 161, which is half the difference of longitude. Now 161, and 116, half the second distance, are found to agree between 43 and 44 degrees; therefore, to latitude $43^\circ$ and distance 161, the corresponding difference of latitude is $117^\circ 7'$; the excess of which above $116'$ is $1^\circ 7'$; and to latitude $43^\circ$, and distance 161, the difference of latitude is $115^\circ 8'$; hence $117^\circ 7' - 115^\circ 8' = 1^\circ 9'$, Middle the change answering to a difference of $1^\circ$ of latitude.

Therefore, $1^\circ 9' : 1^\circ 7' :: 1^\circ : 53'$.

Hence, the latitude is $43^\circ 53'$.

**By Gunter's Scale.**

The extent from 180 to 232 on the line of numbers, being laid in the same direction on the line of sines, from 348, the complement of the latitude sailed from, will reach to $46^\circ 7'$, the complement of the latitude come to.

**Chap. V. Of Middle Latitude Sailing.**

The earth is a sphere, and the meridians meet at the poles; and since a rhumb-line makes equal angles with every meridian, the line a ship describes is, therefore, that kind of a curve called a spiral.

Let AB (fig. 15.) be any given distance sailed upon Fig. 15. an oblique rhumb, PBN, PAM the extreme meridians, MN a portion of the equator, and PCK, PEL two meridians intersecting the distance AB in the points CE infinitely near each other. If the arches BS, CD, and AR, be described parallel to the equator, it is hence evident, that AS is the difference of latitude, and the arch MN of the equator, the difference of longitude, answering to the given distance AB and course PAB.

Now, since CE represents a very small portion of the distance AB, DE will be the correspondent portion of a meridian: hence the triangle EDC may be considered as rectilineal. If the distance be supposed to be divided into an infinite number of parts, each equal to CE, and upon these, triangles be constructed whose sides are portions of a meridian and parallel, it is evident these triangles will be equal and similar; for, besides the right angle, and hypotenuse which is the same in each, the course or angle CED is also the same. Hence, by the 12th of V. Enc. the sum of all the hypothenuses CE, or the distance AB, is to the sum of all the sides DE, or the difference of latitude AS, as one of the hypothenuses CE is to the corresponding side DE. Now, let the triangle GHI (fig. Fig. 16.) be constructed similar to the triangle CDE, having the angle G equal to the course: then as GH : GI :: CE : DC :: AB : AS.

Hence, if GH be made equal to the given distance AB, then GI will be the corresponding difference of latitude.

In like manner, the sum of all the hypothenuses CE, or the distance AB, is to the sum of all the sides CD, as CE is to CD, or as GH to HI, because of the similar triangles.

The several parts of the same rectilineal triangle will, therefore, represent the course, distance, difference of latitude, and departure.

Although the parts HG, GI, and angle G of the rectilineal triangle GHI, are equal to the corresponding parts AB, AS, and angle A, of the triangle ASB upon the surface of the sphere; yet HI is not equal to BS, for HI is the sum of all the arcs CD; but CD is greater than OQ, and less than ZX: therefore HI is greater than BS, and less than AR. Hence the difference of longitude MN cannot be inferred from the departure reckoned either upon the parallel sailed from, or on that come to, but on some intermediate parallel TV. Middle TV; such that the arch TV is exactly equal to the departure; and in this case, the difference of longitude would be easily obtained. For TV is to MN as the sine PT to the sine PM; that is, as the cosine of latitude is to the radius.

The latitude of the parallel TV is not, however, easily determined with accuracy; various methods have, therefore, been taken in order to obtain it nearly, with as little trouble as possible: first, by taking the arithmetical mean of the two latitudes for that of the mean parallel; secondly, by using the arithmetical mean of the cosines of the latitudes; thirdly, by using the geometrical mean of the cosines of the latitudes; and lastly, by employing the parallel deduced from the mean of the meridional parts of the two latitudes. The first of these methods is that which is generally used.

In order to illustrate the computations in middle latitude sailing, let the triangle ABC (fig. 17.) represent a figure in plane sailing, wherein AB is the difference of latitude, AC the distance, BC the departure, and the angle BAC the course. Also, let the triangle DBC be a figure in parallel sailing, in which DC is the difference of longitude, BC the meridian distance, and the angle DCB the middle latitude. In these triangles there is, therefore, one side BC common to both; and that triangle is to be first resolved in which two parts are given, and then the unknown parts of the other triangle will be easily obtained.

**Prob. I.** Given the latitudes and longitudes of two places, to find the course and distance between them.

*Example.* Required the course and distance from the island of May, in latitude $56^\circ 12' N$, and longitude $2^\circ 37' W$, to the Naze of Norway, in latitude $57^\circ 50' N$, and longitude $7^\circ 27' E$?

| Latitude Isle of May | $56^\circ 12' N$ | |----------------------|----------------| | Latitude Naze of Norway | $57^\circ 50' N$ |

Difference of latitude $= 1^\circ 38' = 98'$

Middle latitude $= 57^\circ$

Longitude Isle of May $= 2^\circ 37' W$

Longitude Naze of Norway $= 7^\circ 27' E$

Difference of longitude $= 10^\circ 4' = 604'$

*By Construction.*

Draw the right line AD (fig. 18.) to represent the meridian of the May; with the chord of $62^\circ$ describe the arch mn, upon which lay off the chord of $32^\circ 59'$, the complement of the middle latitude from m to n; from D through n draw the line DC equal to $604'$, the difference of longitude, and from C draw CB perpendicular to AD; make BA equal to $98'$, the difference of latitude, and join AC; which applied to the scale will measure $343$ miles, the distance sought; and the angle A being measured by means of the line of chords, will be found equal to $73^\circ 24'$, the required course.

*By Calculation.*

To find the course (d).

As the difference of latitude $= 98'$

is to the difference of longitude $= 624'$

so is the cosine of middle latitude $= 57^\circ$

to the tangent of the cosine $= 73^\circ 24'$

To find the distance.

As radius $= 10,000,000$

is to the secant of the course $= 73^\circ 24'$

so is the difference of latitude $= 98'$

to the distance $= 343$

*By Inspection.*

To middle latitude $57^\circ$, and $\frac{1}{4}$ one-fourth of the difference of longitude in a distance column, the corresponding difference of latitude is $82.2$.

Now $24.5$, one fourth of the difference of latitude, and $82.2$, taken in a departure column, are found to agree nearest in table marked $6\frac{1}{2}$ points at the bottom, which is the course; and the corresponding distance $8\frac{1}{2}$ multiplied by 4 gives $343$ miles, the distance required.

*By Gunter's Scale.*

The extent from $98$ the difference of latitude, to $624$ the difference of longitude on numbers, being laid the same way from $33^\circ$, the complement of the middle latitude on sines, will reach to a certain point beyond the termination of the line on the scale. Now the extent between this point and $90^\circ$ on sines, will reach from $45^\circ$ to $73^\circ 24'$, the course on the line of tangents. And the extent from $73^\circ 24'$ the course, to $33^\circ$ the complement of the middle latitude on the line of sines, being laid the same way from $624$ the difference of longitude, will reach to $343$ the distance on the line of numbers.

The true course, therefore, from the island of May to the Naze of Norway is $N 73^\circ 24' E$, ENE$4\frac{1}{2}$ nearly; but as the variation at the May is $2\frac{1}{2}$ points west, therefore, the course per compass from the May is E$6S$.

**Prob. II.** Given one latitude, course, and distance sailed, to find the other latitude and difference of longitude.

*Example.* A ship from Brest, in latitude $48^\circ 23' N$, and longitude $4^\circ 30' W$, sailed SW$4\frac{1}{2}W$ $238$ miles. Required the latitude and longitude come to?

*By Construction.*

With the course and distance construct the triangle ABC (fig. 17.), and the difference of latitude $AB$ being measured, will be found equal to $142$ miles; hence the latitude come to is $46^\circ 1' N$, and the middle latitude $47^\circ 12'$. Now make the angle DCB equal to

(d) For R. : cosine mid. lat. :: Diff. of long. : Departure;

And diff. of lat. : Dep. :: R. : Tangent course.

Hence diff. of lat. : cosine mid. lat. :: diff. of long. : tang. course;

Or diff. of lat. : diff. of long. :: cosine mid. lat. : tang. course. By Calculation.

To find the difference of latitude.

As radius \( \frac{1}{4} \) points \( = 10,000,000 \) is to the cosine of the course \( = 977593 \) so is the distance \( = 238 \)

\( \text{Distance} = 2,37658 \)

To the difference of latitude \( = 141.8 \)

Latitude of Brest \( = 48^\circ 23' N \) Difference of lat. \( = 228 S \)

Lat. come to \( = 46^\circ 1' N \), Mid. lat. \( = 47^\circ 12' \)

To find the difference of longitude (E).

As the cosine of Mid. Lat. \( = 47^\circ 12' \) is to the sine of the course \( = 983215 \) so is the distance \( = 238 \)

\( \text{Distance} = 2,37658 \)

To the difference of longitude \( = 281.3 \)

Longitude of Brest \( = 4^\circ 30' W \) Difference of longitude \( = 4^\circ 41' W \)

Longitude come to \( = 9^\circ 11' W \)

By Inspection.

To the course \( \frac{1}{4} \) points, and distance \( 238 \) miles, the difference of latitude \( = 141.8 \), and the departure \( = 191.1 \). Hence the latitude come to is \( 46^\circ 1' N \), and middle latitude \( = 47^\circ 12' \). Then to middle latitude \( = 47^\circ 12' \), and departure \( = 191.1 \) in a latitude column, the corresponding distance is \( 281.3 \), which is the difference of longitude.

By Gunter's Scale.

The extent from \( 8 \) points to \( \frac{3}{4} \) points, the complement of the course on sine rhumbs, being laid the same way from the distance \( 238 \), will reach to the dif- ference of latitude \( = 142 \) on the line of numbers; and the extent from \( 42^\circ 48' \) the complement of the middle latitude, to \( 53^\circ 26' \), the course on the line of sines, will reach from the distance \( 238 \) to the difference of longi- tude \( = 281 \) on numbers.

Prob. III. Given both latitudes and course, requir- ed the distance and difference of longitude?

Example. A ship from St Antonio, in latitude \( 17^\circ 0' N \), and longitude \( 24^\circ 25' W \), sailed NW \( \frac{1}{4} N \), till by observation her latitude is found to be \( 28^\circ 34' N \). Required the distance sailed, and longitude come to?

Latitude St Antonio \( = 17^\circ 0' N \) Latitude by observation \( = 28^\circ 34' N \)

Difference of lat. \( = 11^\circ 34' = 69.4m. \)

Middle lat. \( = 22^\circ 47' \)

By Construction.

Construct the triangle \( ABC \) (fig. 19.), with the given course and difference of latitude, and make the angle \( BCD \) equal to the middle latitude. Now the distance \( AC \) and difference of longitude \( DC \) being

By Calculation.

To find the distance.

As the sine of the course \( = 3\frac{1}{4} \) points \( = 9,80236 \) is to radius \( = 10,000,000 \) so is the departure \( = 216 \)

\( \text{Distance} = 2,33445 \)

To the distance \( = 340.5 \)

This proportion is obvious, by considering the whole figure as an oblique-angled plane triangle. To find the difference of latitude.

As the tangent of the course \( \frac{3}{2} \) points \( 9.9147 \) is to radius \( 10.00000 \) so is the departure \( 216 \)

To find the difference of longitude.

As the cosine of middle latitude \( 42^\circ 6' \) is to the sine of the course \( 50^\circ 39' \) so is the distance \( 246 \)

to the difference of longitude \( 256.4 \) Longitude Cape Sable, \( 65^\circ 39'W \) Difference of longitude \( 4^\circ 16'E \)

Latitude come to \( 30^\circ 53'N \). Mid. lat. \( 28^\circ 42' \)

To find the difference of longitude.

As radius \( 10.00000 \) is to the secant of the mid. lat. \( 28^\circ 42' \) \( 10.09693 \) so is the departure \( 216 \)

to the difference of longitude \( 246.2 \) Longitude left \( 43^\circ 30'W \) Difference of longitude \( 4^\circ 6'E \)

Longitude come to \( 41^\circ 24'W \)

By Inspection.

The distance 246, and difference of latitude 156, are found to correspond above \( 4\frac{1}{2} \) points, and the departure is 190.1. Now, to the middle latitude \( 42^\circ \), and departure 192.1 in a latitude column, the corresponding distance is 256, which is the difference of longitude required.

By Gunter's Scale.

The extent from 246 miles, the distance, to 156, the difference of latitude on numbers, will reach from \( 90^\circ \) to about \( 39^\circ \), the complement of the course on the line of sines; and the extent from \( 48^\circ \), the complement of the middle latitude, to \( 50^\circ \), the course on sines, will reach from the distance 246m. to the difference of longitude 256m. on numbers.

Prob. VI. Given both latitudes and departure; sought the course, distance, and difference of longitude.

Example. A ship from Cape St Vincent, in latitude \( 37^\circ 2'N \), longitude \( 9^\circ 2'W \), sails between the south and west; the latitude come to is \( 18^\circ 16'N \), and departure 838 miles. Required the course and distance run, and longitude come to?

Latitude Cape St Vincent, \( 37^\circ 2'N \) Latitude come to \( 18^\circ 16'N \) Difference of latitude \( 18^\circ 46' = 2126 \text{ sum } 55^\circ 18' \) Middle latitude \( 17^\circ 39' \)

By Construction.

Make AB (fig. 21.) equal to 156 miles; draw BC perpendicular to AB, and make AC equal to 246 miles. Draw CD, making with CB an angle of \( 42^\circ 6' \) the middle latitude. Now DC will be found to measure 256, and the course or angle A will measure \( 50^\circ 39' \).

By Calculation.

To find the course.

As the distance \( 246 \) is to the difference of latitude \( 156 \) so is radius \( 10.00000 \)

to the cosine of the course, \( 50^\circ 39' \) \( 9.8019 \)

To find the distance.

As radius \( 10.00000 \) is to the secant of the course \( 35^\circ 39' \) \( 10.09566 \) so is the difference of latitude \( 1126 \)

to the distance \( 1403 \) To find the difference of longitude.

As radius is to the secant of mid. lat. \(27^\circ 39'\) so is the departure

\[ \begin{align*} \text{As radius} & = 10,000,000 \\ \text{is to the secant of mid. lat.} & = 47^\circ 25' \\ \text{so is the departure} & = 2,923,24 \end{align*} \]

to the difference of longitude \(946\) Longitude Cape St Vincent \(9^\circ 2'W\) Difference of longitude \(15^\circ 46W\)

Longitude come to \(24^\circ 48W\)

By Inspection.

One tenth of the difference of latitude \(112.6\) and of the departure \(83.8\), are found to agree under \(3\frac{1}{2}\) points, and the corresponding distance is \(140\), which multiplied by \(10\) gives \(1400\) miles. And to middle latitude \(27^\circ 37'\) and \(209.5\) one fourth of the departure in a latitude column, the distance is \(236.5\); which multiplied by \(4\) is \(946\), the difference of longitude.

By Gunter's Scale.

The extent from the difference of latitude \(112.6\) to the departure \(83.8\) on numbers, will reach from \(45^\circ\) to \(36^\circ\); the course on tangents; and the extent from \(53^\circ\); the complement of the course to \(90^\circ\) on sines, will reach from \(112.6\) to \(140.3\) the distance on numbers. Lastly, the extent from \(62^\circ\); the complement of the middle latitude, to \(90^\circ\) on sines, will reach from the departure \(83.8\) to the difference of longitude \(946\) on numbers.

Prob. VII. Given one latitude, distance, and departure, to find the other latitude, course, and difference of longitude.

Example. A ship from Bordeaux, in latitude \(44^\circ 50'N\), and longitude \(0^\circ 35'W\), sailed between the north and west \(374\) miles, and made \(210\) miles of westing. Required the course, and the latitude and longitude come to?

By Construction.

With the given distance and departure make the triangle \(ABC\) (fig. 23.). Now the course being measured on the line of chords is about \(34^\circ\); and the difference of latitude on the line of numbers is \(309\) miles; hence the latitude come to, is \(49^\circ 59'N\), and middle latitude \(47^\circ 25'\). Then make the angle \(BCD\) equal to \(47^\circ 25'\), and DC being measured will be \(310\) miles, the difference of longitude.

By Calculation.

To find the course.

\[ \begin{align*} \text{As the distance} & = 374 \\ \text{is to the departure} & = 210 \\ \text{so is radius} & = 10,000,000 \end{align*} \]

to the sine of the course \(34^\circ 10'\) \(9.74935\)

To find the difference of latitude.

\[ \begin{align*} \text{As radius} & = 10,000,000 \\ \text{is to the cosine of the course} & = 34^\circ 10' \\ \text{so is the distance} & = 374 \\ \text{to the difference of latitude} & = 309.4 \\ \text{Latitude of Bordeaux} & = 44^\circ 50'N \\ \text{Difference of latitude} & = 5^\circ 9N \text{ half } 2^\circ 33' \end{align*} \]

Latitude come to \(49^\circ 59'N\) Mid. lat. \(47^\circ 25'\)

Vol. XIV. Part II.

To find the difference of longitude.

As radius is to the secant of mid. lat. \(47^\circ 25'\) so is the departure

\[ \begin{align*} \text{As radius} & = 10,000,000 \\ \text{is to the secant of mid. lat.} & = 47^\circ 25' \\ \text{so is the departure} & = 2,923,24 \end{align*} \]

to the difference of longitude \(310.3\) Longitude of Bordeaux \(0^\circ 35'W\) Difference of longitude \(5^\circ 10W\)

Longitude come to \(5^\circ 45W\)

By Inspection.

The half of the distance \(187\), and of the departure \(105\), are found to agree nearest under \(34^\circ\), and the difference of latitude answering thereto is \(155\); which doubled is \(310\) miles.

Again, to middle latitude \(47^\circ 25'\), and departure \(105\) in the latitude column, the corresponding distance is \(155\) miles, which doubled is \(310\) miles, the difference of longitude.

By Gunter's Scale.

The extent from the distance \(374\) miles to the departure \(210\) miles on the line of numbers, will reach from \(90^\circ\) to \(34^\circ 10'\), the course on the line of sines; and the extent from \(90^\circ\) to \(55^\circ 50'\), the complement of the course on sines, will reach from the distance \(374\) to the difference of latitude \(309\) miles on numbers.

Again, the extent from \(42^\circ 35'\), the complement of the middle latitude, to \(90^\circ\) on sines, will reach from the departure \(210\) to the difference of longitude \(310\) on numbers.

Prob. VIII. Given one latitude, departure, and difference of longitude, to find the other latitude, course, and distance.

Example. A ship from latitude \(54^\circ 56'N\), longitude \(1^\circ 10'W\), sailed between the north and east, till by observation she is found to be in longitude \(5^\circ 26'E\), and has made \(220\) miles of easting. Required the latitude come to, course, and distance run?

Longitude left \(1^\circ 10'W\) Longitude come to \(5^\circ 26E\) Difference of longitude \(6^\circ 36'=396\)

By Construction.

Make BC (fig. 24.) equal to the departure \(220\), and Fig. 24. CD equal to the difference of longitude \(396\);—then the middle latitude BCD being measured, will be found equal to \(59^\circ 15'\); hence the latitude come to is \(57^\circ 34'\), and difference of latitude \(158^\circ\). Now make AB equal to \(158\), and join AC, which applied to the scale, will measure \(271\) miles. Also the course BAC being measured on chords will be found equal to \(54^\circ\).

By Calculation.

To find the middle latitude.

\[ \begin{align*} \text{As the departure} & = 220 \\ \text{is to the diff. of longitude} & = 396 \\ \text{so is radius} & = 10,000,000 \end{align*} \]

To the secant of mid. lat. \(56^\circ 15'\) Double, mid. lat. \(112^\circ 30'\) Latitude left \(54^\circ 56'\) Latitude come to \(57^\circ 34'\) Diff. of latitude \(2^\circ 38'=158\) miles To find the course.

As the difference of latitude \(158\) \(2.19866\) is to the departure \(220\) \(2.34242\) so is radius \(10.00000\)

to the tangent of the course \(54^\circ 19'\) \(10.14376\) To find the distance.

As radius \(10.00000\) is to the secant of the course \(54^\circ 19'\) \(10.23410\) so is the difference of latitude \(158\) \(2.19866\)

to the distance \(270.9\) \(2.43276\)

By Inspection.

As the differences of longitude and departure exceed the limits of the tables, let, therefore, their halves be taken; these are \(198\) and \(110\) respectively. Now these are found to agree exactly in the page marked \(5\) points at the bottom. Whence the middle latitude is \(56^\circ 15'\), and difference of latitude \(158\) miles.

Again, the difference of latitude \(158\) and departure \(220\) will be found to agree nearly above \(54^\circ\) the course, and the distance on the same line is \(271\) miles.

By Gunter's Scale.

The extent from the difference of longitude \(396\) to the departure \(220\) on numbers, will reach from \(90^\circ\) to \(33^\circ 45'\); the complement of the middle latitude on sines; and hence the difference of latitude is \(158\) miles. Now the extent from \(158\) to \(220\) on numbers, will reach from \(45^\circ\) to \(54^\circ 1'\) on tangents; and the extent from the complement of the course \(35^\circ 1'\) to \(90^\circ\) on sines, will reach from the difference of latitude \(158\) to the distance \(271\) on numbers.

Prob. IX. Given the course and distance sailed, and difference of longitude; to find both latitudes.

Example. A ship from a port in north latitude, sailed \(8^\circ E.\) \(438\) miles, and differed her longitude \(7^\circ 28'\). Required the latitude sailed from, and that come to?

By Construction.

With the course and distance construct the triangle \(ABC\) (fig. 25.), and make \(DC\) equal to \(448\) the given difference of longitude. Now the middle latitude \(BCD\) will measure \(48^\circ 58'\), and the difference of latitude \(AB\) \(324\) miles; hence the latitude left is \(51^\circ 40'\), and that come to \(46^\circ 16'\).

By Calculation.

To find the difference of latitude.

As radius \(10.00000\) is to the cosine of the course \(3\frac{1}{2}\) pts. \(9.86979\) so is the distance \(438\) \(2.64147\)

to the difference of latitude \(324.5\) \(2.51126\)

To find the middle latitude.

As the difference of longitude \(448\) \(2.65128\) is to the distance \(438\) \(2.64147\) so is the sine of the course \(3\frac{1}{2}\) pts. \(9.82758\)

to the cosine of mid. latitude \(48^\circ 58'\) \(9.81727\) half difference of latitude \(2^\circ 42'\)

Latitude sailed from \(51^\circ 40'\) Latitude come to \(46^\circ 16'\)

By Inspection.

To the course \(3\frac{1}{2}\) points, and half the distance \(219\) miles, the departure is \(147.9\), and difference of latitude \(162.2\); which doubled is \(324.4\). Again, to half the difference of longitude \(224\) in a distance column, the difference of latitude is \(149.9\) above \(48^\circ\), and \(146.9\) over \(49^\circ\).

Now, as \(30 : 29 :: 60' : 58'\).

Hence the middle latitude is \(48^\circ 58'\); the latitude sailed from is therefore \(51^\circ 40'\), and latitude come to \(46^\circ 16'\).

By Gunter's Scale.

The extent from \(8\) points to \(4\frac{1}{2}\) points, the complement of the course on sine rhumbs, will reach from the distance \(438\) miles to the difference of latitude \(324.5\) on numbers. And the extent from the difference of longitude \(448\), to the distance \(438\) on numbers, will reach from the course \(42^\circ 11'\) to the complement of the middle latitude \(41^\circ 2'\) on sines. Hence the latitude left is \(51^\circ 40'\), and that come to \(46^\circ 16'\).

Prob. X. To determine the difference of longitude made good upon compound courses, by middle latitude sailing.

Rule I. With the several courses and distances find the difference of latitude and departure made good, and the ship's present latitude, as in traverse sailing.

Now enter the traverse table with the given middle latitude, and the departure in a latitude column, the corresponding distance will be the difference of longitude, of the same name with the departure.

Example. A ship from Cape Clear, in latitude \(51^\circ 18'\) N., longitude \(9^\circ 46'\) W., sailed as follows:—SWbS \(34\) miles, WbN \(63\) miles, NNW \(48\) miles, and NEAE \(85\) miles. Required the latitude and longitude come to?

| Courses | Dist. | Diff. of Latitude | Departure | |---------|-------|------------------|-----------| | SWbS | 54 | — | 44.9 | | WbN | 63 | 12.3 | — | | NNW | 48 | 44.4 | — | | NEAE | 85 | 53.9 | 65.7 | | | | 110.6 | 65.7 | | | | 44.9 | — | | N \(34^\circ\) W | 79 | 65.7 | 6N | | Latitude of Cape Clear | 51 18N | | |

Latitude come to \(52^\circ 24'\) N. Sum \(103\) \(42\) Middle latitude \(51^\circ 51'\)

Now, to middle latitude \(51^\circ 51'\) or \(52^\circ\), and departure \(44.5\) in a latitude column, the difference of longitude is \(72\) in a distance column.

Longitude of Cape Clear \(9^\circ 46'\) W. Difference of longitude \(1^\circ 12'\) W.

Longitude come to \(10^\circ 58'\) W.

The above method is not always practised to find the difference of longitude made good in the course. Theorems might be investigated for computing the Mercator's errors to which the above method is liable. These corrections may, however, be avoided by using the following method:

**Rule II.** Complete the traverse table as before, to which annex five columns; the first column is to contain the several latitudes the ship is in at the end of each course and distance; the second, the sums of each following pair of latitude; the third, half the sums, or middle latitudes; and the fourth and fifth columns are to contain the differences of longitude.

Now find the difference of longitude answering to each middle latitude and its corresponding departure, and put them in the east or west difference of longitude columns, according to the name of the departure. Then the difference of the sums of the east and west columns will be the difference of longitude made good, of the same name with the greater.

**Example.** A ship from Halliford in Iceland, in lat. $64^\circ 30' N$, long. $27^\circ 15' W$, sailed as follows: SSW 46 miles, SW 61 miles, SSW 59 miles, SESE 86 miles, SSEE 79 miles. Required the lat. and long. come to?

| Traverse Table | Longitude Table | |----------------|-----------------| | Courses | Dist. | Diff. of Lat. | Departure | Successive Latitudes | Sums | Middle Latitudes | Diff. of Longitude | | | | N | S | E | W | | | E | W | | SSW | 46 | — | 42.5| — | 17.6| 64° 30'| — | — | — | | SW | 61 | — | 43.1| — | 43.1| 63° 48'| 128° 18'| 64° 9'| — | 40.4| | SSW | 59 | — | 57.9| — | 11.5| 63° 5'| 126° 53'| 62° 27'| — | 96.4| | SESE | 86 | — | 47.8| 71.5| — | 62° 7'| 125° 12'| 62° 36'| — | 25.0| | SSEE | 76 | — | 72.7| 22.0| — | 61° 19'| 123° 26'| 61° 43'| — | 150.9| | | | 264.0| 93.5| 72.2| | 60° 6'| 121° 25'| 60° 43'| — | 45.0|

By Rule I.

Latitude Halliford $64^\circ 30' N$ Difference of latitude $4^\circ 24' S$

Latitude in $60^\circ 6' N$ Sum $124° 36'$ Middle latitude $62° 18'$

Now, to middle lat. $62° 18'$, and departure $21.3$, the difference of long. is $46° E$.

Longitude Halliford $27° 15' W$

Longitude in $16° 29' W$

The error of comm. method, in this Ex. is $12'$.

**Chap. VI. Of Mercator's Sailing.**

It was observed in Middle Latitude Sailing, that the difference of longitude made upon an oblique rhumb could not be exactly determined by using the middle latitude. In Mercator's sailing, the difference of longitude is very easily found, and the several problems of sailing resolved with the utmost accuracy, by the assistance of Mercator's chart or equivalent tables.

In Mercator's chart, the meridians are straight lines parallel to each other; and the degrees of latitude, which at the equator are equal to those of longitude, increase with the distance of the parallel from the equator. The parts of the meridian thus increased are called meridional parts. A table of these parts was first constructed by Mr Edward Wright, by the continual addition of the secants of each minute of latitude.

For by parallel sailing,

\[ R : \cos \text{ of lat. } :: \text{ part of equat. } : \text{ similar part of parallel.} \] And because the equator and meridian on the globe are equal; therefore,

\[ R : \cos \text{lat.} :: \text{part of meridian} : \text{similar part of parallel}. \]

Or sec. lat. : R :: part of merid. : similar part of parallel.

Hence,

\[ \frac{\text{secant latitude}}{\text{part of meridian}} = \frac{\text{R}}{\text{part of parallel}}. \]

But in Mercator's chart the parallels of latitude are equal, and radius is a constant quality. If, therefore, the latitude be assumed successively equal to \(1^\circ, 2^\circ, 3^\circ\), &c., and the corresponding parts of the enlarged meridian be represented by \(a, b, c\), &c.; then,

\[ \frac{\text{secant } 1^\circ}{\text{part of mer. } a} = \frac{\text{secant } 2^\circ}{\text{part of mer. } b} = \frac{\text{secant } 3^\circ}{\text{part of mer. } c}, \] &c.

Hence secant \(1^\circ\) : part of mer. \(a\) :: secant \(2^\circ\) : part of mer. \(b\) :: secant \(3^\circ\) : part of mer. \(c\), &c.

Therefore by 12th V. Euclid.

Secant \(1^\circ\) : part of mer. \(a\) :: secant \(1^\circ\) + secant \(2^\circ\) + secant \(3^\circ\), &c.: parts of \(a + b + c\), &c.

That is, the meridional parts of any given latitude are equal to the sum of the secants of the minutes in that latitude (E).

Since CD : LK :: R : secant LD, fig. 15.

And in the triangle CED,

\[ ED : CD :: R : \text{tangent CED}; \]

Therefore, ED : LK :: R² : secant LD × tangent CED

Hence LK = \[ \frac{ED \times \text{sec. } \times LD \times \text{tang. CED}}{R^2} = \frac{ED \times \text{sec. } LD}{R} \times \frac{\text{tang. CED}}{R}. \]

But \[ \frac{ED \times \text{sec. } LD}{R} \] is the enlarged portion of the meridian answering to ED. Now the sum of all the quantities \[ \frac{ED \times \text{secant } LD}{R} \] corresponding to the sum of all the ED's contained in AS, will be the meridional parts answering to the difference of latitude AS; and MN is the sum of all the corresponding portions of the equator LK.

Whence MN = mer. diff. of lat. × tangent \[ \frac{CED}{R}. \]

That is, the difference of longitude is equal to the meridional difference of latitude multiplied by the tangent of the course, and divided by the radius.

This equation answers to a right-angled rectilineal triangle, having an angle equal to the course; the adjacent side equal to the meridional difference of latitude, and the opposite side the difference of longitude. This triangle is, therefore, similar to a triangle constructed, with the course and difference of latitude, according to the principles of plane sailing, and the homologous sides will be proportional. Hence, if, in fig. 26 the angle fig. 26. A represents the course, AB the difference of latitude, and if AD be made equal to the meridional difference of latitude; then DE, drawn perpendicular to AD, meeting the distance produced to E, will be the difference of longitude.

It is scarcely necessary to observe, that the meridional difference of latitude is found by the same rules as the proper difference of latitude; that is, if the given latitudes be of the same name, the difference of the corresponding meridional parts will be the meridional difference of latitude; but if the latitudes are of a contrary denomination, the sum of these parts will be the meridional difference of latitude.

**Prob. I.** Given the latitudes and longitudes of two places, to find the course and distance between them.

Ex. Required the course and distance between Cape Finisterre, in latitude \(42^\circ 52' N\), longitude \(9^\circ 17' W\), and Port Praya in the island of St Jago, in latitude \(14^\circ 54' N\), and longitude \(23^\circ 29' W\).

| Lat. Cape Finisterre | \(42^\circ 52'\) | Mer. parts | 2852 | |---------------------|---------------|------------|------| | Latitude Port Praya | \(14^\circ 54'\)| Mer. parts | 904 |

Difference of lat. \(= 27^\circ 58'\) Mer. diff. lat. \(1948\)

Longitude Cape Finisterre \(9^\circ 17' W\) Longitude Port Praya \(23^\circ 29' W\)

Diff. longitude \(= 14^\circ 12' = 852\).

**By Construction.**

Draw the straight line AD (fig. 26.) to represent the meridian of Cape Finisterre, upon which lay off AB, AD equal to 1678, and 1948, the proper and meridional differences of latitude; from D draw DE perpendicular to AD, and equal to the difference of longitude \(852\); join AE, and draw BC parallel to DE; then the difference AC will measure 1831 miles, and the course BAC \(23^\circ 37'\).

**By Calculation.**

To find the course.

As the meridian difference of lat. \(1948\) is to the difference of longitude \(852\) so is radius \(1000000\)

\[ \frac{1948}{852} = \frac{1000000}{x} \]

\[ x = 964085 \]

To find the distance.

As radius \(1000000\) is to the secant of the course, \(23^\circ 37'\) so is the difference of latitude \(1678\) to the distance \(1831\)

\[ \frac{1000000}{23^\circ 37'} = \frac{1678}{1831} \]

\[ x = 326277 \]

**By Inspection.**

As the meridian difference of latitude and difference of longitude are too large to be found in the tables, let the tenth of each be taken; these are 194.8 and 85.2 respectively. Now these are found to agree nearest under \(24^\circ\); and to 167.8, one-tenth of the proper difference of latitude, the distance is about 183 miles, which multiplied by 10 is 1830 miles.

**By Gunter's Scale.**

The extent 1948, the meridional difference of latitude, to 852, the difference of longitude on the line of numbers, will reach from \(45^\circ\) to \(23^\circ 37'\), the course on

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(E) This is not strictly true; for instead of taking the sum of the secants of every minute in the distance of the given parallel from the equator, the sum of the secants of every point of latitude should be taken. Prob. II. Given the course and distance, sailed from a place whose situation is known, to find the latitude and longitude of the place come to.

Example. A ship from Cape Hinlopen in Virginia, in latitude $38^\circ 47' N$, longitude $75^\circ 4' W$, sailed 267 miles NE$6N$. Required the ship's present place?

By Construction.

With the course and distance sailed, construct the triangle ABC (fig. 27); and the difference of latitude AB being measured, is 222 miles; hence the latitude come to is $42^\circ 29' N$, and the meridional difference of latitude 293. Make AD equal to 293; and draw DE perpendicular to AD, and meeting AC produced in E; then, the difference of longitude DE being applied to the scale of equal parts will measure 196; the longitude come to is therefore $71^\circ 48' W$.

By Calculation.

To find the difference of latitude.

As radius $10,000,000$ is to the cosine of the course, - 3 points $9,919,85$ so is the distance $267$ $2,420,51$

to the difference of latitude $222$ $2,346,36$

Lat. Cape Hinlopen $= 38^\circ 47' N$. Mer. parts $2528$

Difference of lat. $- 3 42 N.$

Latitude come to $- 42 29 N$. Mer. parts $2821$

Meridional difference of latitude $293$

To find the difference of longitude.

As radius $10,000,000$ is to tangent of the course, - 3 points $9,824,89$ so is the mer. diff. of latitude $293$ $2,466,87$

to the difference of longitude $195.8$ $2,291,76$

Longitude Cape Hinlopen $75^\circ 4' W$

Difference of longitude $- 3 16 E.$

Longitude come to $71 48 W.$

By Inspection.

To the course 3 points, and distance 267 miles, the difference of latitude is 222 miles; hence the latitude in is $42^\circ 29'$, and the meridional difference of latitude 293. Again, to course three points, and 146.5 half the mer. difference of latitude, the departure is 97.9, which doubled is 195.8, the difference of longitude.

By Gunter's Scale.

The extent from 8 points to the complement of the course 5 points, on sine rhumbs, will reach from the distance 267 to the difference of latitude 222 on numbers; and the extent from 4 points to 3 points on tangent rhumbs, will reach from the meridional difference of latitude 293 to the difference of longitude 196 on numbers.

Prob. III. Given the latitudes and bearing of two places; to find their distance and difference of longitude.

Prob. IV. Given the latitude and longitude of the place sailed from, the course and departure; to find the distance, and the latitude and longitude of the place come to.

Example. A ship sailed from Sallee in latitude $33^\circ 58' N$, longitude $6^\circ 20' W$, the corrected course was NW$6W4W$, and departure 420 miles. Required the distance run, and the latitude and longitude come to?

By Construction.

With the course and departure construct the triangle ABC (fig. 29); now AC and AB being measured, will be found to be equal to 476 and 224 respectively: hence By Calculation.

To find the distance.

As radius \( \frac{1}{\text{cosecant of the course}} = 5^{\circ} \) pts \( = 10.00000 \) so is the departure \( = 420 \) \( = 2.62325 \)

To the distance \( = 476.2 \) \( = 2.67782 \)

To find the difference of latitude.

As radius \( \frac{1}{\text{cotangent of the course}} = 5^{\circ} \) pts \( = 10.00000 \) so is the departure \( = 420 \) \( = 2.62325 \)

To the difference of latitude \( = 224.5 \) \( = 2.35121 \)

Lat. of Sallee \( = 33^{\circ} 58'N \) Mer. parts \( = 2169 \) Diff. of lat. \( = 3^{\circ} 44'N \)

Lat. in \( = 37^{\circ} 42'N \) Mer. parts \( = 2445 \)

Mer. difference of latitude \( = 276 \)

To find the difference of longitude.

As radius \( \frac{1}{\text{tangent of the course}} = 5^{\circ} \) pts \( = 10.00000 \) so is the mer. diff. of latitude \( = 276 \) \( = 2.44091 \)

To the difference of longitude \( = 516.3 \) \( = 2.71295 \)

Longitude of Sallee \( = 6^{\circ} 20'W \) Difference of longitude \( = 8^{\circ} 36'W \)

Longitude in \( = 14^{\circ} 56'W \)

By Inspection.

Above \( 5^{\circ} \) points the course, and opposite to \( 210 \) half the departure, are \( 238 \) and \( 112 \); which doubled, we have \( 476 \) and \( 224 \), the distance and difference of latitude respectively. And to the same course, and opposite to \( 138 \), half the meridional difference of latitude, in a latitude column, is \( 258 \) in a departure column; which being doubled is \( 516 \), the difference of longitude.

By Gunter's Scale.

The extent from \( 5^{\circ} \) points, the course on sine rhumbs, to the departure \( 420 \) on numbers, will reach from eight points on sine rhumbs to the distance \( 476 \) on numbers; and from the complement of the course \( 2^{\circ} \) points on sine rhumbs, to the difference of latitude \( 224 \) on numbers.

Again, the extent from difference of latitude \( 224 \) to the meridional difference of latitude \( 276 \) on numbers, will reach from the departure \( 420 \) to the difference of longitude \( 516 \) on the same line.

Prob. V. Given the latitudes of two places, and their distance, to find the course and difference of longitude.

Example. A ship from St Mary's, in latitude \( 36^{\circ} 57'N \), longitude \( 25^{\circ} 9'W \), sailed on a direct course between the north and east \( 1162 \) miles, and is then by observation in latitude \( 49^{\circ} 57'N \). Required the course steered, and longitude come to?

Latitude Aberdeen \( = 57^{\circ} 9'N \) Mer. parts \( = 4199 \) Latitude come to \( = 53^{\circ} 32'N \) Mer. parts \( = 3817 \)

Difference of latitude \( = 3^{\circ} 37' = 217' \) Mer. diff. lat. \( = 382 \)

By Construction.

With the difference of latitude \( 217' \) m. and departure Fig. 31. \( 146 \) m. construct the triangle ABC (fig. 31.), make AD By Calculation.

To find the course.

As the difference of latitude 217 is to the departure 146 so is radius

to the tangent of the course 33° 56' To find the distance.

As radius is to the secant of the course 33° 56' so is the difference of latitude

to the distance

To find the difference of longitude.

As the difference of latitude 217 is to the mer. diff. of latitude 382 so is the departure

to the difference of longitude

Longitude of Aberdeen

Difference of longitude

Longitude come to

By Inspection.

The difference of latitude 217, and departure 146, are found to agree nearest under 34°, and the corresponding distance is 262 miles. To the same course, and opposite to 190.7, the nearest to 101, half the meridional difference of latitude, is 128.6 in a departure column, which doubled is 257, the difference of longitude.

By Gunter's Scale.

The extent from the difference of latitude 217, to the departure 146 on numbers, will reach from 45° to about 34°; the course on the line of tangents; and the same extent will reach from the meridional difference of latitude 382 to 257, the difference of longitude on numbers.—Again, the extent from the course 34° to 90 on sines, will reach from the departure 146 to the distance 261 on numbers.

Prob. VII. Given one latitude, distance and departure; to find the other latitude, course, and difference of longitude.

Example. A ship from Naples, in latitude 40° 51' N, longitude 14° 14' E, sailed 252 miles on a direct course between the south and west, and made 173 miles of westing. Required the course made good, and the latitude and longitude come to?

By Construction.

With the distance and departure make the triangle ABC (fig. 32.) as formerly.—Now the course BAC being measured by means of a line of chords will be found equal to 43° 21', and the difference of latitude applied to the scale of equal parts will measure 183'; hence the latitude come to is 37° 48' N, and meridional difference of latitude 237.—Make AD equal to 237, and complete the figure, and the difference of longitude DE will measure 224': hence the longitude Meteor's Sailing.

Longitude come to

By Calculation:

To find the course.

As the distance 252 is to the departure 173 so is radius

to the sine of the course 43° 21'

To find the difference of latitude.

As radius is to the cosine of the course 43° 21' so is the distance

to the difference of latitude

Latitude of Naples 40° 51' N. Mer. parts 2690 Difference of latitude 3 3 8.

Latitude come to 37° 48' N. Mer. parts 2453

Meridional difference of latitude

To find the difference of longitude.

As radius is to the tangent of the course 43° 21' so is the mer. diff. of latitude

to the difference of longitude

Longitude of Naples

Difference of longitude

Longitude in

By Inspection.

Under 43° and opposite to the distance 252 m. the departure is 171.8, and under 44°, and opposite to the same distance, the departure is 175.0.

Then as 3.2 : 1.2 :: 60' : 22'.

Hence the course is 43° 22'.

Again, under 43° and opposite to 118.5, half the meridional difference of latitude in a latitude column, is 110.5 in a departure column; also under 44° and opposite to 118.5 is 114.4.

Then as 3.2 : 1.2 :: 3.9 : 1.5.

And 110.5 + 1.5 = 112, which doubled is 224, the difference of longitude.

By Gunter's Scale.

The extent from the distance 252 on numbers, to 90° on sines, will reach from the departure 173 on numbers, to the course 43° on sines; and the same extent will reach from the complement of the course 46° on sines will reach to the difference of latitude 183 on numbers.—Again, the extent from 45° to 43° on tangents will reach from the meridional difference of latitude 237, to the difference of longitude 224, on numbers.

Prob. VIII. Given one latitude, course, and difference of longitude: to find the other latitude and distance.

Example. A ship from Tercera, in latitude 38° 45' N, longitude 27° 6' W, sailed on a direct course, which, when corrected, was N 32° E, and is found by observation to be in longitude 18° 24' W. Required the latitude come to, and distance sailed? Make the right-angled triangle ADE (fig. 33.) having the angle A equal to the course 32°, and the side DE equal to the difference of longitude 522; then AD will measure 835, which added to the meridional parts of the latitude left, will give those of the latitude come to 48° 46′; hence, the difference of latitude is 601; make AB equal thereto, to which let BC be drawn perpendicular; then AC applied to the scale will measure 708 miles.

By Calculation.

To find the meridional difference of latitude.

As radius 10.00000 is to the co-tangent of the course 32° 0′ 10.20421 so is the difference of longitude 5 22 2.71767

to the mer. difference of latitude 8352 2.92188 Latitude of Tercera 30° 45′ N Mer. parts 2526 Mer. diff. of lat. 835

Latitude come to - 48 46N Mer. parts 3361

Difference of latitude 10 1=601 miles.

To find the distance.

As radius 10.00000 is to the secant of the course 32° 0′ 10.07118 so is the difference of latitude 601 2.77887

to the distance 707.7 2.85045

By Inspection.

To course 32°, and opposite to 130.5, one fourth of the given difference of longitude in a departure column, the difference of latitude is 208.8, which multiplied by 4 is 835, the meridional difference of latitude; hence the latitude in is 48° 46′ N, and difference of latitude 601.

Again, to the same course, and opposite to 200, one third of the difference of latitude, the distance is 236, which multiplied by 3 gives 708 miles.

By Gunter's Scale.

The extent from the course 32°, to 45° on tangents, will reach from the difference of longitude 522 to the meridional difference of latitude 835 on numbers.

And the extent from the complement of the course 58° to 90° on sines, will reach from the difference of latitude 601, to the distance 708 miles on numbers.

Prob. IX. To find the difference of longitude made good upon compound courses.

Rule. With the several courses and distances, complete the Traverse Table, and find the difference of latitude, departure, and course made good, and the latitude come to as in Traverse Sailing. Find also the meridional difference of latitude.

Now to the course and meridional difference of latitude, in a latitude column, the corresponding departure will be the difference of longitude, which applied to the longitude left will give the ship's present longitude.

Example. A ship from port St Julian, in latitude Mercator's 49° 10′ S, longitude 68° 44′ W, sailed as follows; Sailing ESE 53 miles, SEBS 74 miles, E by N 68 m. SESE 47 miles, and E 84 miles. Required the ship's present place?

| Course | Dist. | Diff. of Lat. | Departure | |--------|-------|--------------|-----------| | ESE | 53 | 20.3 | 49.0 | | SEbyS | 74 | 61.5 | 41.1 | | EbyN | 68 | 13.3 | 66.7 | | SEbyE½E | 47 | 22.1 | 41.5 | | E | 84 | 84.0 | |

13.3 103.9 282.3

S 72° E 197 90.6=1° 31′ Latitude left, 49 10 S m. pt. 3397 Latitude come to 50 41 S m. pt. 3539

Mer. difference of latitude 142 Now to course 72°, and opposite to 71, half the mer. difference of latitude in a latitude column, is 218.7 in a departure column, which doubled is 437, the difference of longitude.

Longitude of Port St Julian 68° 44′ W Difference of longitude 7 17 E Longitude come to 61 27 W

Although the above method is that usually employed at sea to find the difference of longitude, yet as it has been already observed, it is not to be depended on, especially in high latitudes, long distances, and a considerable variation in the courses, in which case the following method becomes necessary.

Rule II. Complete the Traverse Table as before, to which annex five columns. Now with the latitude left, and the several differences of latitude, find the successive latitudes, which are to be placed in the first of the annexed columns; in the second, the meridional parts corresponding to each latitude is to be put; and in the third, the meridional differences of latitude.

Then to each course, and corresponding meridional difference of latitude, find the difference of longitude, by Prob. VI. which place in the fourth or fifth columns, according as the coast is easterly or westerly; and the difference between the sums of these columns will be the difference of longitude made good upon the whole, of the same name with the greater.

Remarks.

1. When the course is north or south, there is no difference of longitude. 2. When the course is east or west, the difference of longitude cannot be found by Mercator's Sailing; in this case the following rule is to be used.

To the nearest degree to the given latitude taken as a course, find the distance answering to the departure in a latitude column; this distance will be the difference of longitude. ### NAVIGATION

**Ex. 1.** Four days ago we took our departure from Faro-head, in latitude $58^\circ 40'$ N, and longitude $4^\circ 50'$ W, Mercator's sailing, and since have sailed as follows: NW 32 miles, W 69 miles, WNW 93 miles, WSW 77 miles, SW 58 miles, and W$^{\frac{1}{2}}$S 49 miles.—Required our present latitude and longitude?

| Courses | Dist. | Diff. of Lat. | Departure | Successive Latitudes | Merid. Parts. | Merid. Diff. Lat. | Diff. of Longitude | |---------|-------|--------------|-----------|---------------------|---------------|------------------|-------------------| | NW | 32 | 22.6 | | | | | | | W | 69 | | | | | | | | WNW | 93 | 35.6 | | | | | | | WSW | 77 | 15.0 | | | | | | | SW | 58 | 41.0 | | | | | | | W$^{\frac{1}{2}}$S | 49 | 7.2 | | | | | | | W$^{\frac{1}{2}}$S | 58.2 | 63.2 | | | | | | | W$^{\frac{1}{2}}$S | 58.2 | | | | | | | | W$^{\frac{1}{2}}$S | 343 | 5.0 | | | | | |

Longitude of Faro head $4^\circ 50'$ W.

Difference of longitude $11^\circ 4$ W.

Longitude in $15^\circ 54'$ W.

**Ex. 2.** A ship from latitude $78^\circ 15'$ N, longitude $28^\circ 14'$ E, sailed the following courses and distances. The latitude come to is required, and the longitude, by both methods: the bearing and distance of Hacluit's head-land, in latitude $79^\circ 55'$ N, longitude $11^\circ 55'$ E, is also required?

| Courses | Dist. | Diff. of Lat. | Departure | Successive Latitudes | Merid. Parts. | Merid. Diff. Lat. | Diff. of Longitude | |---------|-------|--------------|-----------|---------------------|---------------|------------------|-------------------| | WNW | 154 | 58.9 | | | | | | | SW | 96 | | | | | | | | NW$^{\frac{1}{2}}$W | 89 | 56.4 | | | | | | | N$^{\frac{1}{2}}$E | 110 | 107.9 | | | | | | | NW$^{\frac{1}{2}}$N | 56 | 45.0 | | | | | | | S$^{\frac{1}{2}}$E$^{\frac{1}{2}}$E | 78 | 73.4 | | | | | | | | 268.2 | 141.3 | | | | | | | | 141.3 | | | | | | | | | 126.9 | | | | | | |

By Rule I.

Latitude left $78^\circ 15'$ N. Mer. pts. 7817

Diff. of latitude $2^\circ 7$ N.

Lat. come to $80^\circ 22$ N. Mer. pts. 8504

Meridional diff. of latitude $687$

As difference of lat. $126.9$

is to mer. diff. of lat. $687$

so is the departure $204.6$

to diff. of longit. $1432$

Longitude left $23^\circ 52'$ W.

Longitude in $28^\circ 14'$ E.

Longitude in $4^\circ 22'$ F.

The error of this method, in the present example, is therefore $1^\circ 23'$.

Now to $78.5$ half the meridional difference of latitude, and $185.0$ half the difference of longitude, the course $67^\circ$, and opposite to the difference of latitude $27$, the distance is $69$ miles.—Hence Hacluit's head-land bears $867^\circ$ E, distant $69$ miles.