NAVIGATION.

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 Jafon, and others to Janus, who is said to have made the first ship. Historians ascribe it to the Æginetæ, the Phœnicians, Tyrians, and the ancient inhabitants of Britain. Some will have it, the first hint was taken from the flight of the kite; others, as Oppian, (De píctibus, 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 raillery the good man underwent on account of his enterprise shews evidently enough the world was then ignorant of anything like navigation, and that they even thought it impossible.

However, history represents the Phœnicians, 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 to 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, which 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 Phœnician 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 in 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 cultivated by the Ptolemies, that Tyre and Carthage were quite forgot.

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 merchandizes from the capital of Egypt.

At length, Alexandria itself underwent the fate of Tyre and Carthage; being surprized 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 still it 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 ordinarily looked on as the restorers hereof, 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 of 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 gulph, 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 fifth 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 parts 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 their spices and other products of the Indies. Thus they flourished, increased their commerce. commerce, their navigation, and their conquests on the terra firma, till the league of Cambray 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 ere 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 thereof, which still obtain under the names of Uffs 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 reasons of commerce's 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 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 hath 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 the 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 do agree that the mariner's compass is a modern invention, it hath 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 Vifco, brother to the king of Portugal. In the year 1485, Roderic and Joseph, physicians to king John II. 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 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 shewed 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 shewed, 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 on the moon. In 1530, 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 Alfonso de Sufa. 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 steerman 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 shews 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 shewn 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 subject of which was the same year printed at London in English by Mr Edward Wright, intitled 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 shewed, that as the rhumbs are spirals, 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 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 owed 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 Forman*. 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; shewing especially the danger of not fixing, on account of the variation, the wire directly under the flower-de-luce; as compasses made in different countries have placed it differently. To this performance of Forman's is always prefixed a discourse on the variation of the magnetic needle, by Mr William Burrough, in which he shews 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 had 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 bear 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 tho' the meridians should be straight parallels 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 only been 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 shewed 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 by the height of the pole-star, when not upon the meridian. To this edition was subjoined a translation of Zenorano'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. parts of the world, to shew that it was not occasioned by any magnetical pole.

These improvements soon became known abroad. In 1608, a treatise intitled, Hypomnemata Mathematica, were 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 shewn how to draw the rhumbs on a globe mechanically, sets down Wright's two tables of latitude 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 Wellebrordus 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 co-sines 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 co-sines, 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 co-sines, equivalent to the geometrical mean between the co-sines themselves; a method since proposed by Mr John Balfat. 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 lower 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. They were first applied by Mr Edward Gunter in 1620. He constructed a table of artificial sines and tangents to every minute of the quadrant. These were applied according to Wright's table of meridional parts, and have been found extremely useful in other branches of the mathematics. He contrived also a most excellent ruler, commonly known by the name of Gunter's scale, on which were inscribed the logarithmic lines for numbers, and for sines and tangents of arches*. He also 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-bow, for taking altitudes of the sun or stars, with some contrivances for the more ready collecting the latitude from the observation. The discoveries concerning the 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 ruler was projected into a circular arch by the reverend Mr William Oughtred in 1633, and its uses fully shown in a pamphlet intitled, The circles of proportion; where, in an appendix, are well handled 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 Addisson, in his treatise intitled, 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, intitled, A discourse mathematical on the variation of the magnetic 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 intitled, 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 Merfennus, 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 collect from thence the sun's true altitude with correctness, Wright observes it to be necessary that the dip of the horizon below the observer's eye should be brought into the 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 so well with the usual divisions of the log-line, that 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 574,300 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. These necessary amendments, however, were little attended to by the sailors, whose obstinacy in adhering to established errors has been complained of by the best writers on navigation; but at length they found their 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 54 degrees; divide the difference of those numbers by the logarithmic tangent of 50° 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 or 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 \( \frac{1}{\pi} \) 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, betaken; 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 kalendar, 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, intitled, 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, intitled, The longitude not found, made its appearance; and as Mr Bond's hypothesis did not in any manner answer its author's sanguine expectations, the affair was undertaken by Dr Halley. The result of his speculations was, that the magnetic needle is influenced Part I.

Theory. influenced by four poles; but this wonderful phenomenon seems hitherto to have eluded all our researches. In 1700, 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 1744 and 1756, by Mr William Montague, and Mr James Dodson, F.R.S. In the Philosophical Transactions for 1690, 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; 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. Dr Patrick Murdoch, wherein he accommodated Wright's failing to such a figure; and Mr Colin MacLaurin, the same year, in the Philosophical Transactions, no. 461, gave a rule for determining the meridional parts of a spheroid; which speculation is farther treated of in his book of Fluxions, printed at Edinburgh in 1742.

Among foreign nations also many treatises were now published; but excepting the remarkable discovery of the longitude by Mr Harrison, no considerable improvement hath been made any-where. Indeed, the subject hath been so much canvassed and studied by men of learning and ingenuity in all nations, that there seems to be little room for farther improvements; and the art of navigation seems to be nearly brought to as much perfection as it is capable of.

PART I. 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, regulated by the direction of the helm. As the water is a resisting medium, and the bulk of the ship very considerable, 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 fiercely, 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 tackle gives way.

The direction of a ship's motion depends on the situation of her sails with regard to the wind. The most natural and easy position is, when she runs directly before it; but this is not often the case, on account of the variable nature of the winds, and the situations of the different ports to which the ship may be bound. When the wind therefore happens not to be favourable, the rudder and sails must be managed in such a manner that the ship may make an angle with the direction of the current of air, as represented Plate CCI, fig. 4. Thus, supposing a ship at D, bound for the port B. Supposing DG the length of the keel, it must be kept by the rudder in such a position as to make the acute angle EDB with the direction of the wind. If, when she arrives at B, it is required to sail to another port A, the keel must be kept in the position BF; and thus, by continually making the angle EBA with the direction of the wind, she will arrive at the desired port: and in this manner may a ship be steered to any other port, supposing to C or H.

The reason of the ship's motion in these cases is, on the lee side balances the force applied on the other, when it will become uniform, as doth 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, hath 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 only can 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 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 azimuths as frequently as possible, by which the difference of variation 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, especi- PART II. PRACTICE OF NAVIGATION.

The main end of all practical navigation is to conduct the ship in safety to her destined port; and for this purpose it is of the utmost consequence to know in what particular part of the surface of the globe she is at any particular time. This can only be done by having an accurate map of the sea-coasts of all the countries of the world, and, by tracing out the ship's progress along the map, to know at what time she approaches the desired haven, or how she is to direct her course in order to reach it. It is therefore a matter of great importance for navigators to be furnished with maps, or charts, as they are called, not only very accurate in themselves, but such as are capable of having the ship's course easily traced upon them, without the trouble of laborious calculations, which are ready to create mistakes.—The names of the two great divisions of navigation are taken merely from the kind of charts made use of. Plane sailing is that in which the plane chart is made use of; and Mercator's sailing, or globular sailing, is that in which Mercator's chart is used. In both these methods, it is easy to find the ship's place with as great exactness as the chart will allow, either by the solution of a case in plane trigonometry, or by geometrical construction.

§ 1. Of Plane Sailing.

As a necessary preliminary to our understanding this method of navigation, we shall here give the construction of the plane chart.

1. This chart supposes the earth to be a plane, and the meridians parallel to one another; and likewise the parallels of latitude at equal distance from one another, as they really are upon the globe. Tho' this method be in itself evidently false; yet, in a short run, and especially near the equator, an account of the ship's way may be kept by it tolerably well.

Having determined the limits of the chart, that is, how many degrees of latitude and longitude, or meridional distance (they being in this chart the same), it is to contain: suppose from the lat. of 20° N. to the lat. of 71° N.; and from the longitude of London in 0° deg. to the lon. of 50° W.; then choose a scale of equal parts, by which the chart may be contained within the size of the sheet of paper on which it is intended to be drawn. In the chart annexed, the scale is such, that each degree of latitude and longitude is \( \frac{1}{8} \) part of an inch.

Make a parallelogram ABCD, the length of which AB from north to south shall contain 51 degrees, the difference of latitude between the limits of 20° and 71°; and the breadth AD from east to west shall contain the proposed 50 degrees of longitude, the degrees being taken from the said scale of 8 degrees to an inch; and this parallelogram will be the boundaries of the chart.

About the boundaries of the chart make scales containing the degrees, halves and quarters of degrees (if the scale is large enough); drawing lines across the chart thro' every 5 or 10 degrees; let the degrees of latitude and longitude have their respective numbers annexed, and the sheet is then fitted to receive the places intended to be delineated thereon.

On a stiff slip of pasteboard, or stiff paper, let the scale of the degrees and parts of degrees of longitude, in the line AD, be laid close to the edge; and the divisions numbered from the right hand towards the left, being all west longitude.

Seek in a geographical table for the latitudes and longitudes of the places contained within the proposed limits; and let them be written out in the order in which they increase in latitude.

Then, to lay down any place, lay the edge of the pasteboard scale to the divisions on each side the chart, shewing the latitude of the place; so that the beginning of its divisions fall on the right-hand border AB; and against the division shewing the longitude of the given place make a point, and this gives the position of the place proposed; and in like manner are all the other places to be laid down.

Draw waving lines from one point to the other, where the coast is contiguous, and thus the representation of the lands within the proposed limits will be delineated.

Write the names to the respective parts, and in some convenient place insert a compass, and the chart will be completed.

2. The angle formed by the meridian and rhumb that a ship sails upon, is called the ship's course. Thus if a ship sails on the NNE rhumb, then her course will be 22°30'; and so of others.

3. The distance between two places lying on the same parallel counted in miles of the equator, or the distance of one place from the meridian of another counted as above on the parallel passing over that place, is called meridional distance; which, in plane sailing, goes under the name of departure.

4. Let A (no. 3.) denote a certain point on the Plate CCI. earth's surface, AC its meridian, and AD the parallel of latitude passing through it; and suppose a ship to sail from A on the NNE rhumb till she arrive at B; and through B draw the meridian BD, (which, according to the principles of plane sailing, must be parallel to CA,) and the parallel of latitude BC: then the length of AB, viz. how far the ship has sailed upon the NNE rhumb, is called her distance; AC or BD will be her difference of latitude, or nothing; CB will be her departure, or casting; and the angle CAB will be the course. Hence it is plain, that the distance sailed will always be greater than either the difference of latitude or departure; it being the hypotenuse of a right-angled triangle, whereof the other two are the legs; except the ship sails either on a meridian or a parallel of latitude: for if the ship sails on a meridian, then it is plain, that her distance will be just equal to her difference of latitude; and she will have no departure; but if she sail on a parallel, then her distance will be the same with her departure, and she will have no difference of latitude. It is evident also from the figure, that if the course be less than 4 points, or 45 degrees, its complement, viz. the other oblique angle, will be greater than 45 degrees. and so the difference of latitude will be greater than the departure; but if the course be greater than 4 points, then the difference of latitude will be less than the departure; and lastly, if the course be just 4 points, the difference of latitude will be equal to the departure.

5. Since the distance, difference of latitude, and departure, form a right-angled triangle, in which the oblique angle opposite to the departure is the course, and the other its complement; therefore, having any two of these given, we can (by plain trigonometry) find the rest; and hence arise the cases of plane-failing, which are as follow.

**Case I. Course and distance given, to find difference of latitude and departure.**

**Example.** Suppose a ship sails from the latitude of 30° 25' north, NNE, 32 miles, (No. 4.) Required the difference of latitude and departure, and the latitude come to. Then (by right-angle trigonometry,) we have the following analogy, for finding the departure, viz.

As radius - 10.00000 to the distance AC - 32 - 1.50515 so is the fine of the course A 22° 30' - 9.58284 to the departure BC - 12.25 - 1.08799

The ship has made 12.25 miles of departure easterly, or has got so far to the eastward of her meridian. Then for the difference of latitude or northing the ship has made, we have (by rectangular trigonometry) the following analogy, viz.

As radius - 10.00000 is to the distance AC - 32 - 1.50515 so is the co-fine of course A 22° 30' - 9.58284 to the difference of lat. AB - 29.57 - 1.47077

The ship has differed her latitude, or made of northing, 29.57 minutes.

And since her former latitude was north, and her difference of latitude also north; therefore,

To the latitude sailed from - 30° 25' N add the difference of latitude - 00° 29.57

and the sum is the latitude come to - 30° 54.57 N

By this case are calculated the tables of difference of latitude, and departure, to every degree, point, and quarter-point of the compass.

**Case II. Course and difference of latitude given, to find distance and departure.**

**Example.** Suppose a ship in the latitude of 45° 25' north, sails NE 5° 1/2 easterly (No. 5.) till she come to the latitude of 46° 55' north: Required the distance and departure made good upon that course.

Since both latitudes are northerly, and the course also northerly; therefore,

From the latitude come to - 46° 55' subtract the latitude sailed from - 45° 25' and there remains - 01° 30'

the difference of latitude, equal to - 90 miles.

And (by rectangular trigonometry) we have the following analogy, for finding the departure BD, viz.

As radius - 10.00000 is to the diff. of latitude A.B - 90 - 1.95424 so is the tangent of course A - 39° 22' - 9.91404 to the departure BD - 73 84' - 1.86828

So the ship has got 73.84 miles to the eastward of her former meridian.

Again, for the distance AD, we have (by rectangular trigonometry) the following proportion, viz.

As radius - 10.00000 is to the secant of the course - 39° 22' - 10.11176 so is the difference of latitude A.B 90 - 1.95424 to the distance AD - 116.4 - 2.06600

**Case III. Difference of latitude and distance given, to find course and departure.**

**Example.** Suppose a ship sails from the latitude of 56° 50' north, on a rhumb between south and west, 126 miles, and she is then found by observation to be in the latitude of 55° 40' north: Required the course she sailed on, and her departure from the meridian. No. 6.

Since the latitudes are both north, and the ship sailing towards the equator; therefore,

From the latitude sailed from - 56° 50' subtract the observed latitude - 55° 40' and the remainder - 01° 40' equal to 70 miles, is the difference of latitude.

By rectangular trigonometry we have the following proportion for finding the angle of the course F, viz.

As the distance sailed DF - 126 - 2.10037 is to radius - 10.00000 so is the diff. of latitude FD - 70 - 1.84510 to the co-fine of the course F - 56° 15' - 9.74473

Which, because she sails between south and west, will be south 56° 15' west, or SW 5° W. Then, for the departure, we have (by rectangular trigonometry) the following proportion, viz.

As radius - 10.00000 is to the distance sailed DF - 126 - 2.10037 so is the fine of the course F - 56° 15' - 9.91985 to the departure DE - 104.8 - 2.02022

Consequently she has made 104.8 miles of departure westerly.

**Case IV. Difference of latitude and departure given, to find course and distance.**

**Example.** Suppose a ship sails from the latitude of 44° 50' north, between south and east, till she has made 64 miles of easterly, and is then found by observation to be in the latitude of 42° 56' north: Required the course and distance made good.

Since the latitudes are both north, and the ship sailing towards the equator; therefore,

From the latitude sailed from - 44° 50' N take the latitude come to - 42° 56' and there remains - 01° 54' equal to 114 miles, the difference of latitude or southing.

In this case (by rectangular trigonometry) we have the following proportion to find the course KGL (No. 7.) viz.

As the diff. of latitude GK 114 - 2.05690 is to radius - 10.00000 so is the departure KL - 64 - 1.80618 to the tangent of course G - 29° 19' - 9.79428

Which, because the ship is sailing between south and east, will be south 29° 19' east, or SSE 1/2 east nearly.

Then for the distance, we shall have (by rectangular trigonometry) the following analogy, viz. Part II.

**NAVIGATION**

**Practice As radius**

is to the difference of latitude \( \text{GK} = 114 - 2.05690 \)

so is the secant of the course \( 29^\circ, 19' \) \( 10.05932 \)

to the distance \( GL = 130.8 - 2.11642 \)

consequently the ship has sailed on a SSE \( \frac{1}{4} \) east course 130.8 miles.

**Case V. Distance and departure given, to find course and difference of latitude.**

**Example.** Suppose a ship at sea sails from the latitude of \( 34^\circ 24' \) north, between north and west 124 miles, and is found to have made westing 86 miles: Required the course steered, and the difference of latitude or northing made good.

In this case (by rectangular trigonometry) we have the following proportion for finding the course \( ADB \), (\( N^\circ 8 \)) viz.

As the distance \( AD = 124 - 2.09342 \)

is to radius \( 10.00000 \)

so is the departure \( AB = 86 - 1.93450 \)

to the fine of the course \( D = 43^\circ 54' \) \( 9.84108 \)

so the ship's course is north \( 33^\circ 45' \) west, or NW\( \frac{1}{4} \) west nearly.

Then for the difference of latitude, we have (by rectangular trigonometry) the following analogy, viz.

As radius \( 10.00000 \)

is to the distance \( AD = 124 - 2.09342 \)

so is the co-fine of the course \( 43^\circ, 54' \) \( 9.85766 \)

to the diff. of latitude \( BD = 89.35 - 1.95108 \)

which is equal to 1 degree and 29 minutes nearly.

Hence, to find the latitude the ship is in, since both latitudes are north, and the ship sailing from the equator; therefore,

To the latitude sailed from \( 34^\circ, 24' \)

add the difference of latitude \( 1^\circ, 29' \)

the sum is \( 35^\circ, 53' \)

the latitude the ship is in north.

**Case VI. Course and departure given, to find distance and difference of latitude.**

**Example.** Suppose a ship at sea, in the latitude of \( 24^\circ 30' \) south, sails SE\( \frac{1}{8} \)S, till she has made of easting 96 miles: Required the distance and difference of latitude made good on that course.

In this case (by rectangular trigonometry and by case 2.) we have the following proportion for finding the distance, (\( N^\circ 9 \)) viz.

As the fine of the course \( G = 33^\circ, 45' \) \( 9.74474 \)

is to the departure \( HM = 96 - 1.98227 \)

so is radius \( 10.00000 \)

to the distance \( GM = 172.8 - 2.23753 \)

Then, for the difference of latitude, we have (by rectangular trigonometry) the following analogy, viz.

As the tangent of course \( 33^\circ, 45' \) \( 9.82489 \)

is to the departure \( HM = 96 - 1.98227 \)

so is radius \( 10.00000 \)

to the difference of latitude \( GH = 143.7 - 2.15738 \)

equal to \( 2^\circ, 24' \) nearly. Consequently, since the latitude the ship failed from was south, and she sailing still towards the south,

To the latitude sailed from \( 24^\circ, 30' \)

add the difference of latitude \( 2^\circ, 24' \)

and the sum \( 26^\circ, 54' \)

is the latitude she is come to south.

6. When a ship sails on several courses in 24 hours, the reducing all these into one, and thereby finding the course and distance made good upon the whole, is commonly called the resolving of a traverse.

7. At sea they commonly begin each day's reckoning from the noon of that day, and from that time they set down all the different courses and distances sailed by the ship till noon next day upon the log-board; then from these several courses and distances, they compute the difference of latitude and departure for each course (by Case 1. of Plane Sailing); and these, together with the courses and distances, are set down in a table, called the traverse table, which consists of five columns: in the first of which are placed the courses and distances; in the two next, the differences of latitude belonging to these courses, according as they are north or south; and in the two last are placed the departures belonging to these courses, according as they are east or west. Then they sum up all the northings and all the southings; and taking the difference of these, they know the difference of latitude made good by the ship in the last 24 hours, which will be north or south, according as the sum of the northings or southings is greatest: the same way, by taking the sum of all the eastings, and likewise of all the westings, and subtracting the lesser of these from the greater, the difference will be the departure made good by the ship last 24 hours, which will be east or west according as the sum of the eastings is greater or less than the sum of the westings; then from the difference of latitude and departure made good by the ship last 24 hours, found as above, they find the true course and distance made good upon the whole (by Case 4. of Plane Sailing), as also the course and distance to the intended port.

**Example.** Suppose a ship at sea, in the latitude of \( 48^\circ 24' \) north at noon any day, is bound to a port in the latitude of \( 43^\circ 40' \) north, whose departure from the ship is 144 miles east; consequently the direct course and distance of the ship is SSE \( \frac{1}{4} \) east 315 miles; but by reason of the shifting of the winds she is obliged to steer the following courses till noon next day, viz. SE\( \frac{1}{8} \)S 56 miles, SSE 64 miles, NW\( \frac{1}{4} \)W 48 miles, SW \( \frac{1}{4} \) west 54 miles, and SE\( \frac{1}{8} \)S \( \frac{1}{4} \) east 74 miles: Required the course and distance made good the last 24 hours, and the bearing and distance of the ship from the intended port.

The solution of this traverse depends entirely on the 1st and 4th Cases of Plane Sailing; and first we must (by Case 1.) find the difference of latitude and departure for each course. Thus,

1. Course SE\( \frac{1}{8} \)S distance 56 miles.

For departure.

As radius \( 10.00000 \)

is to the distance \( 56 - 1.74819 \)

so is the fine of the course \( 33^\circ, 45' \) \( 9.74474 \)

to the departure \( 31.11 - 1.49293 \)

For difference of latitude.

As radius \( 10.00000 \)

is to the distance \( 56 - 1.74819 \)

so is the co-fine of the course \( 33^\circ, 45' \) \( 9.91985 \)

to the diff. of latitude \( 46.57 - 1.66804 \)

2. Course SSE and distance 64 miles.

For departure.

As radius \( 10.00000 \)

is to the distance \( 64 - 1.80618 \)

so is the fine of the course \( 22^\circ, 30' \) \( 9.58284 \) From the above table it is plain, since the sum of the northings is 26.67, and of the southings 214.58, the difference between these, viz. 187.91, will be the southing made good by the ship the last 24 hours; also the sum of the eastings being 102.55, and of the westings 55.58, the difference 46.97 will be the easting or departure made good by the ship's last 24 hours; consequently, to find the true course and distance made good by the ship in that time, it will be (by Cafe 4. of Plane Sailing.)

As the difference of latitude - 187.91 is to the radius - 10.00000 so is the departure - 46.97 to the tangent of the course 14°, 03' - 9.39789 which is SbE ¼ east nearly. Then for the distance, it will be,

As radius - 10.00000 is to the difference of latitude - 187.91 so is the secant of the course - 14°, 03' - 10.01319 to the distance - 193.7 - 2.28712 consequently the ship has made good the last 24 hours, on a SbE ¼ east course, 193.7 miles; and since the ship is sailing towards the equator; therefore,

From the latitude sailed from - 48°, 24' N take the diff. of latitude made good - 3°, 08' S

there remains - 45°, 16' N the latitude the ship is in north. And because the port the ship is bound for lies in the latitude of 43° 40' N, and consequently south of the ship; therefore,

From the latitude the ship is in - 45°, 16' N take the latitude she is bound for - 43°, 40' N

and there remains - 1°, 36' or 96 miles, the difference of latitude or southing the ship has to make. Again, the whole easting the ship had to make being 144 miles, and she having already made 46.97 or 47 miles of easting; therefore the departure or easting she still has to make will be 97 miles; consequently, to find the direct course and distance between the ship and the intended port, it will be (by Cafe 4. of Plane Sailing.)

As the difference of latitude - 96 is to radius - 10.00000 so is the departure - 97 to the tangent of the course 45°, 19' - 10.00450

And

As radius - 10.00000 is to the difference of latitude - 96 so is the secant of the course - 45°, 19' - 10.15293 to the distance - 136.5 - 2.13020 whence the true bearing and distance of the intended port is SE, 136.5 miles.

§ 2. Of Parallel Sailing.

1. Since the parallels of latitude do always decrease Plate CCII., the nearer they approach the pole, it is plain a degree on any of them must be less than a degree upon the equator. Now in order to know the length of a degree on any of them, let PB (n° 10.) represent half the earth's axis, PA a quadrant of a meridian, and consequently A a point on the equator, C a point on the meridian, and CD a perpendicular from that point upon the axis, which plainly will be the fine of CP the distance of that point from the pole, or the co-fine of CA its distance from the equator; and CD will be to AB, as the fine of CP, or co-fine of CA, is to the radius. Again, if the quadrant PAB is turned round upon the axis PB, it is plain the point A will describe the circumference of the equator whose radius is AB, Part II.

Cor. I. Hence (because the circumference of circles are as their radii) it follows, that the circumference of any parallel is to the circumference of the equator, as the co-fine of its latitude is to radius.

Cor. II. And since the wholes are as their similar parts, it will be, As the length of a degree on any parallel is to the length of a degree upon the equator, so is the co-fine of the latitude of that parallel to radius.

Cor. III. Hence, as radius is to the co-fine of any latitude, so are the minutes of difference of longitude between two meridians, or their distance in miles upon the equator, to the distance of these two meridians on the parallel in miles.

Cor. IV. And as the co-fine of any parallel is to radius, so is the length of any arch on that parallel (intercepted between two meridians) in miles, to the length of a similar arch on the equator, or minutes of difference of longitude.

Cor. V. Also, as the co-fine of any one parallel is to the co-fine of any other parallel, so is the length of any arch on the first, in miles, to the length of the same arch on the other in miles.

2. From what has been said, arises the solution of the several cases of parallel sailing, which are as follow:

Case I. Given the difference of longitude between two places, both lying on the same parallel; to find the distance between those places.

Example I. Suppose a ship in the latitude of $54^\circ 20'$ north, sails directly west on that parallel till she has differed her longitude $12^\circ 45'$; required the distance sailed on that parallel.

First, The difference of longitude reduced into minutes, or nautical miles, is $765$, which is the distance between the meridian sailed from, and the meridian come to, upon the equator; then to find the distance between these meridians on the parallel of $54^\circ 20'$, or the distance sailed, it will be, by Cor. 3. of the last article,

$$\text{As radius } \quad \frac{10,0000}{\text{is to the co-fine of the lat.}} = \frac{54^\circ 20'}{\text{9.76572}}$$

so are the minutes of diff. long. $765$ $\text{2.88636}$ to the distance on the parallel $446.1$ $\text{2.64938}$

Example II. A degree on the equator being 60 minutes or nautical miles; required the length of a degree on the parallel of $51^\circ 32'$.

By Cor. 3. of the last article, it will be

$$\text{As radius } \quad \frac{10,0000}{\text{is to the co-fine of the latitude}} = \frac{51^\circ 32'}{\text{9.79383}}$$

so are the minutes in 1 degree on the equator $60$ $\text{1.77815}$ to $\text{37.32} \quad \text{1.57198}$ the miles answering to a degree on the parallel of $51^\circ 32'$

By this problem the following table is constructed, shewing the geographic miles answering to a degree on any parallel of latitude; in which you may observe, that the columns marked at the top with D. L. contain the degrees of latitude belonging to each parallel; and the adjacent columns marked at the top, Miles, contain the geographic miles answering to a degree upon these parallels.

A Table shewing how many Miles answer to a Degree of Longitude, at every Degree of Latitude.

| D. L. | Miles | D. L. | Miles | D. L. | Miles | D. L. | Miles | |-------|-------|-------|-------|-------|-------|-------|-------| | 1 | 59.99 | 19 | 56.73 | 37 | 47.92 | 55 | 34.41 | | 2 | 59.97 | 20 | 56.38 | 38 | 47.28 | 56 | 33.55 | | 3 | 59.92 | 21 | 56.01 | 39 | 46.62 | 57 | 32.68 | | 4 | 59.86 | 22 | 55.63 | 40 | 45.95 | 58 | 31.79 | | 5 | 59.79 | 23 | 55.23 | 41 | 45.28 | 59 | 30.90 | | 6 | 59.67 | 24 | 54.81 | 42 | 44.95 | 60 | 30.00 | | 7 | 59.56 | 25 | 54.38 | 43 | 43.88 | 61 | 29.09 | | 8 | 59.44 | 26 | 53.93 | 44 | 43.16 | 62 | 28.17 | | 9 | 59.26 | 27 | 53.46 | 45 | 42.43 | 63 | 27.24 | | 10 | 59.08 | 28 | 52.97 | 46 | 41.68 | 64 | 26.30 | | 11 | 59.09 | 29 | 52.47 | 47 | 40.92 | 65 | 25.36 | | 12 | 58.68 | 30 | 51.96 | 48 | 40.15 | 66 | 24.41 | | 13 | 58.46 | 31 | 51.43 | 49 | 39.36 | 67 | 23.45 | | 14 | 58.22 | 32 | 50.88 | 50 | 38.57 | 68 | 22.48 | | 15 | 57.95 | 33 | 50.32 | 51 | 37.76 | 69 | 21.50 | | 16 | 57.67 | 34 | 49.74 | 52 | 36.94 | 70 | 20.52 | | 17 | 57.36 | 35 | 49.15 | 53 | 36.11 | 71 | 19.54 | | 18 | 57.06 | 36 | 48.54 | 54 | 35.26 | 72 | 18.54 |

Though this table does only shew the miles answering to a degree of any parallel, whose latitude consists of a whole number of degrees; yet it may be made to serve for any parallel whose latitude is some number of degrees and minutes, by making the following proportion, viz.

As 1 degree, or 60 minutes, is to the difference between the miles answering to a degree in the next greater and next less tabular latitude than that proposed; so is the excess of the proposed latitude above the next tabular latitude, to a proportional part; which, subtracted from the miles answering to a degree of longitude in the next less tabular latitude, will give the miles answering to a degree in the proposed latitude.

Example. Required to find the miles answering to a degree on the parallel of $56^\circ 44'$.

First, The next less parallel of latitude in the table than that proposed, is that of $56^\circ$; a degree of which (by the table) is equal to $33.55$ miles; and the next greater parallel of latitude in the table, than that proposed, is that of $57^\circ$, a degree of which is (by the table) equal to $32.68$ miles; the difference of these is $87$, and the distance between these parallels is 1 degree, or 60 minutes; also the distance between the parallel of $56^\circ$, and the proposed parallel of $56^\circ 44'$, is $44$ minutes: then by the preceding proportion it will be, as $60$ is to $87$, so is $44$ to $638$, the difference between a degree on the parallel of $56^\circ$ and a degree on the parallel of $56^\circ 44'$; which therefore, taken from $33.55$, the miles answering to a degree on the parallel of $56^\circ$, leaves $32.912$, the miles answering to a degree on the parallel of $56^\circ 44'$, as was required.

Case II. The distance sailed in any parallel of latitude, or the distance between any two places on that parallel, being given; to find the difference of longitude.

Example. Suppose a ship in the latitude of $55^\circ 36'$ north. north sails directly east 685.6 miles: Required how much she has differed her longitude.

By Cor. 4. Art. 1. of this section, it will be

As the co-sine of the lat. - 55° 36' - 9.75202 is to radius - 10.00000 so is the distance sailed - 685.6 - 2.83607 to min. of diff. of long. - 1213 - 3.08405 which reduced into degrees, by dividing by 60, makes 20° 13', the difference of longitude the ship has made.

This also may be solved by help of the foregoing table, viz. by finding from it the miles answering to a degree on the proposed parallel, and dividing with this the given number of miles, the quotient will be the degrees and minutes of difference of longitude required.

Thus in the last example, we find, from the foregoing table, that a degree on the parallel of 55° 36' is equal to 33.89 miles; by this we divide the proposed number of miles 685.6, and the quotient is 20.13 degrees, i.e. 20° 13', the difference of longitude required.

CASE III. The difference of longitude between two places on the same parallel, and the distance between them, being given; to find the latitude of that parallel.

Example. Suppose a ship sails on a certain parallel directly west 624 miles, and then has differed her longitude 18° 46', or 1126 miles: Required the latitude of the parallel she sailed upon.

By Cor. 3. Art. 1. of this section, it will be,

As the min. of diff. long. - 126 - 3.05154 is to the distance sailed - 624 - 2.79518 so is radius - 10.00000 to the co-sine of the lat. - 56° 21' - 9.74364 consequently the latitude of the ship or parallel she sailed upon was 56° 21'.

From what has been said, may be solved the following problems.

PROB. I. Suppose two ships in the latitude of 46° 30' north, distant asunder 654 miles, sail both directly north 256 miles, and consequently are come to the latitude of 50° 46' north: Required their distance on that parallel.

By Cor. 6. Art. 1. of this section, it will be,

As the co-sine of - 46° 30' - 9.83781 is to the co-sine of - 50° 46' - 9.80105 so is - 654 - 2.81558 to - 601 - 2.77882 the distance between the ships when on the parallel of 50° 46'.

PROB. II. Suppose two ships in the latitude of 45° 48' north, distant 846 miles, sail directly north till the distance between them is 624 miles: Required the latitude come to, and the distance sailed.

By Cor. 5. Art. 1. of this section, it will be,

As their first distance - 846 - 2.92737 is to their second distance - 624 - 2.79518 so is the co-sine of - 45° 48' - 9.84334 to the cosine - 59° 04' - 9.71115 the latitude of the parallel the ships are come to.

Consequently to find their distance sailed,

From the latitude come to - 59° 04' subtract the latitude sailed from - 45° 48' and there remains - 13° 16'

equal to 796 miles, the difference of latitude or distance failed.

§ 3. Of Middle latitude Sailing.

I. When two places lie both on the same parallel, we showed in the last section, how, from the difference of longitude given, to find the miles of easting or westing between them, et cetera. But when two places lie not on the same parallel, then their difference of longitude cannot be reduced to miles of easting or westing on the parallel of either place: for if counted on the parallel of that place that has the greatest latitude, it would be too small; and if on the parallel of that place having the least latitude, it would be too great. Hence the common way of reducing the difference of longitude between two places, lying on different parallels, to miles of easting or westing, et cetera, is by counting it on the middle parallel between the places, which is found by adding the latitudes of the two places together, and taking half the sum, which will be the latitude of the middle parallel required. And hence arises the solution of the following cases.

CASE I. The latitudes of two places, and their difference of longitude, given; to find the direct course and distance.

Example. Required the direct course and distance between the Lizard in the latitude of 50° 00' north, and longitude of 5° 14' west, and St Vincent in the latitude of 17° 10' N. and longitude of 24° 20' W.

First, To the latitude of the Lizard 50° 00' N. add the latitude of St Vincent 17° 10'

The sum is 67° 10'

Half the sum or latitude of the middle parallel is 33° 35' N.

Also the difference of latitude is 33° 50'

Equal to 1970 miles of southing.

Again, From the longitude of St Vincent 24° 20' W. take the longitude of the Lizard 05° 14'

there remains 16° 06'

equal to 1146 min. of diff. of long. west.

Then for the miles of westing, or departure, it will be, (by Case 1. of Parallel Sailing)

As radius - 10.00000 is to the co-sine of the middle parallel - 33° 35' - 9.92069 so is min. diff. of long. - 1146 - 3.05918 to the miles of westing - 954.7 - 2.97987

And for the course it will be, (by Case 4. of Plane Sailing)

As the diff. of lat. - 1970 - 3.29447 is to radius - 10.00000 so is the departure - 954.7 - 2.97987 to the tang. of the course 25° 51' - 9.68540 which, because it is between south and west, it will be SSW ¼ west nearly.

For the distance, it will be, by the same case,

As radius - 10.00000 is to the diff. of lat. - 1970 - 3.29447 so is the secant of the course 25° 51' - 10.04579 to the distance - 2189 - 3.34026 whence the direct course and distance from the Lizard to St Vincent is SSW ¼ 2189 W miles.

CASE II. One latitude, course, and distance being Part II.

Practice being given; to find the other latitude and difference of longitude.

Example. Suppose a ship in the latitude of 50° north, sails south 50° west, 150 miles: Required the latitude she has come to, and how much she has differed her longitude.

First, For the difference of latitude, it will be, (by Case 1. of Plane Sailing.)

As radius 10.00000 is to the distance 150 2.17609 so is the co-fine of the course 50°, 06' 9.80716 to the diff. of latitude 96.22 1.98325 equal to 1°, 36'. And since the ship is sailing towards the equator; therefore,

From the latitude she was in 50°, 00' take the diff. of latitude 1°, 36'

and there remains 48°, 24' the latitude she has come to north. Consequently the latitude of the middle parallel will be 49° 12'.

Then for departure or westing it will be, by the same Case,

As radius 10.00000 is to the distance 150 2.17609 so is the fine of the course 50°, 06' 9.88489 to the departure 115.1 2.06098

As for the difference of longitude, it will be, (by Case 2. of Plane Sailing.)

As the co-fine of the middle parallel 49° 12' 9.81519 is to radius 10.00000 so is the departure 115.1 2.06098 to the min. diff. of longitude 176.1 2.24579 equal to 2° 56', which is the difference of longitude the ship has made westerly.

Case III. Course and difference of latitude given; to find the distance sailed, and difference of longitude.

Example. Suppose a ship in the latitude of 53° 34' north, sails SEbS, till by observation she is found to be in the latitude of 51° 12', and consequently has differed her latitude 2° 22', or 142 miles: Required the distance sailed, and the difference of longitude.

First, for the departure, it will be, (by Case 2. of Plane Sailing.)

As radius 10.00000 is to the diff. of latitude 142 2.15229 so is the tang. of course 33°, 45' 9.82489 to the departure 94.88 1.97718

And for the distance it will be, (by the same Case)

As radius 10.00000 is to the diff. of latitude 142 2.15229 so is the secant of the course 33°, 45' 10.08015 to the distance 170.8 2.23244

Then, since the latitude sailed from was 53° 34' north, and the latitude come to 51° 12' north; therefore the middle parallel will be 52° 23'; and consequently, for the difference of longitude, it will be (by Case 2. of Parallel Sailing.)

As the co-fine of the mid. parallel 52°, 23' 9.78560 is to the departure 94.88 1.97718 so is radius 10.00000 to min. of diff. of longitude 155.5 2.19158 equal to 2° 35' the difference of longitude easterly.

Case IV. Difference of latitude and distance fail-

Vol. VII. Required the course, distance, and difference of longitude.

First, By Case 4. of Plane Sailing, it will be for the course,

As the diff. of latitude - 193 - 2.28556 is to departure - 146 - 2.16137 so is radius - 10.00000 to the tang. of the course 36°, 55' - 9.87581 which, because the ship is sailing between south and east, will be south 36° 55' east, or SESE ¼ east nearly.

For the distance, it will be, by the same Case,

As radius - 10.00000 is to the diff. of latitude - 193 - 2.28556 so is the secant of the course 36°, 55' - 10.09718 to the distance - 241.4 - 2.38274

Then for the difference of longitude, it will be, by Case 2. of Parallel Sailing,

As the co-line of the mid. par. 45°, 00' - 9.84949 is to the departure - 146 - 2.16137 so is radius - 10.00000 to min. of diff. of longitude - 205 - 2.31188 equal to 3° 25', the difference of longitude easterly.

Case VII. Distance and departure given, to find difference of latitude, course, and difference of longitude.

Example. Suppose a ship in the latitude of 33° 40' north, sails between south and east 165 miles, and has then made of easting 112.5 miles: Required the difference of latitude, course, and difference of longitude.

First, for the course, it will be, by Case 5. of Plane sailing,

As the distance - 165 - 2.21748 is to radius - 10.00000 so is the departure - 102.5 - 2.05115 to the fine of the course 42°, 59' - 9.83367 which, because the ship sails between south and east, will be south 42° 59' east, or SESE ½ east nearly.

And for the difference of latitude, it will be, by the same Case,

As radius - 10.00000 is to the distance - 165 - 2.21748 so is the co-line of the course 42°, 59' - 9.86436 to the difference of lat. - 120.7 - 2.08184 equal to 2° 00'; consequently the latitude come to will be 31° 40' north, and the latitude of the middle parallel will be 32° 40'. Hence, to find the difference of longitude, it will be, by Case 2. of Parallel Sailing,

As the co-line of the mid. par. 32°, 40' - 9.92522 is to the departure - 112.5 - 2.05115 so is radius - 10.00000 to min. of diff. of long. - 133.6 - 2.12593 equal to 2° 13' nearly, the difference of longitude easterly.

Case VIII. Difference of longitude and departure given; to find difference of latitude, course, and distance failed.

Example. Suppose a ship in the latitude of 50° 46' north, sails between south and west, till her difference of longitude is 3° 12', and is then found to have departed from her former meridian 126 miles; required the difference of latitude, course, and distance failed.

First, For the latitude she has come to, it will be, by Case 3. of Parallel Sailing,

As min. of diff. of long. - 192 - 2.28330 is to departure - 126 - 2.10037 so is radius - 10.00000 to the co-line of the mid. par. 48°, 59' - 9.81707

Now since the middle latitude is equal to half the sum of the two latitudes (by art. 1. of this Sect.) and so the sum of the two latitudes equal to double the middle latitude; it follows, that if from double the middle latitude we subtract any one of the latitudes, the remainder will be the other. Hence from twice 48° 59', viz. 97° 58', taking 50° 46' the latitude failed from, there remains 47° 12' the latitude come to; consequently the difference of latitude is 3° 34', or 214 minutes.

Then for the course, it will be, by Case 4. of Plane Sailing,

As diff. of lat. - 214 - 2.33041 is to radius - 10.00000 so is the departure - 126 - 2.10037 to the tang. of the course 30°, 29' - 9.76996 which, because it is between south and west, will be south 30° 29' west, or SSW ¼ west nearly.

And for the distance, it will be, by the same Case,

As radius - 10.00000 is to the diff. of lat. - 214 - 2.33041 so is the secant of the course 30°, 29' - 10.06461 to the distance - 248.4 - 2.39502

2. From what has been said, it will be easy to solve a traverse, by the rules of Middle-latitude Sailing.

Example. Suppose a ship in the latitude of 43° 25' north, sails upon the following courses, viz. SWbS 63 miles, SSW ¼ west 45 miles, SbE 54 miles, and SWbW 74 miles: Required the latitude the ship has come to, and how far she has differed her longitude.

First, By Case 2. of this Sect. find the difference of latitude and difference of longitude belonging to each course and distance, and they will stand as in the following table.

| Courses | Distances | Diff. of Lat. | Diff. of Longit. | |---------|-----------|--------------|-----------------| | SWbS | 63 | 52.4 | 47.85 | | SSW ¼ W | 45 | 39.7 | 28.62 | | SbE | 54 | 53.0 | 14.75 | | SWbW | 74 | 41.1 | 81.08 |

Diff. of Lat. 186.2

Diff. of Long. 143.80

Hence it is plain the ship has differed her latitude 186.2 minutes, or 3° 6', and so has come to the latitude of 40° 19' north, and has made of difference of longitude 143.8 minutes, or 2° 23' 48" westerly.

3. This method of sailing, though it be not strictly true, yet it comes very near the truth, as will be evident, by comparing an example wrought by this method with the same wrought by the method delivered § 4. Of Mercator's sailing.

1. Though the meridians do all meet at the pole, and the parallels to the equator do continually decrease, and that in proportion to the co-fines of their latitudes; yet in old sea-charts the meridians were drawn parallel to one another, and consequently the parallels of latitude made equal to the equator, and so a degree of longitude on any parallel as large as a degree on the equator: also in these charts the degrees of latitude were still represented (as they are in themselves) equal to each other, and to those of the equator. By these means the degrees of longitude being increased beyond their just proportion, and the more so the nearer they approach the pole, the degrees of latitude at the same time remaining the same, it is evident places must be very erroneously marked down upon these charts with respect to their latitude and longitude, and consequently their bearing from one another very false.

2. To remedy this inconvenience, so as still to keep the meridians parallel, is plain we must protract, or lengthen, the degrees of latitude in the same proportion as those of longitude are, that so the proportion in easting and westing may be the same with that of southing and northing, and consequently the bearings of places from one another be the same upon the chart as upon the globe itself.

Plate CCH. Let ABD (No 11.) be a quadrant of a meridian, A the pole, D a point on the equator, AC half the axis, B any point upon the meridian, from which draw BF perpendicular to AC, and BG perpendicular to CD; then BG will be the fine, and BF or CG the co-fine of BD the latitude of the point B; draw D the tangent and CE the secant of the arch CD. It has been demonstrated in Sect. 2, that any arch of a parallel is to the like arch of the equator as the co-fine of the latitude of that parallel is to radius. Thus any arch as a minute on the parallel described by the point B, will be to a minute on the equator as BF or CG is to CD; but since the triangles CGB CDE are similar, therefore CG will be to CD as CB is to CE, i.e. the co-fine of any parallel is to radius as radius is to the secant of the latitude of that parallel. But it has been just now shown, that the co-fine of any parallel is to radius, as the length of any arch as a minute on that parallel is to the length of the like arch on the equator: therefore the length of any arch as a minute on any parallel, is to the length of the like arch on the equator, as radius is to the secant of the latitude of that parallel; and so the length of any arch, as a minute on the equator, is longer than the like arch of any parallel in the same proportion as the secant of the latitude of that parallel is to radius. But since in this projection the meridians are parallel, and consequently each parallel of latitude equal to the equator, it is plain the length of any arch as a minute on any parallel, is increased beyond its just proportion, at such rate as the secant of the latitude of that parallel is greater than radius; and therefore, to keep up the proportion of northing and southing to that of easting and westing, upon this chart, as it is upon the globe itself, the length of a minute upon the meridian at any parallel must also be increased beyond its just proportion at the same rate, i.e. as the secant of the latitude of that parallel is greater than radius. Thus to find the length of a minute upon the meridian at the latitude of 75° degrees, since a minute of a meridian is everywhere equal on the globe, and also equal to a minute upon the equator, let it be represented by unity; then making it as radius is to the secant of 75° degrees, so is unity to a fourth number, which is 3.864 nearly; and consequently, by whatever line you represent one minute on the equator of this chart, the length of one minute on the enlarged meridian at the latitude of 75° degrees, or the distance between the parallel of 75° 00' and the parallel of 75° 01', will be equal to 3 of these lines, and \(\frac{3}{100}\) of one of them. By making the same proportion, it will be found, that the length of a minute on the meridian of this chart at the parallel of 60°, or the distance between the parallel of 60° 00' and that of 60° 01', is equal to two of these lines. After the same manner, the length of a minute on the enlarged meridian may be found at any latitude; and consequently, beginning at the equator, and computing the length of every intermediate minute between that and any parallel, the sum of all these shall be the length of a meridian intercepted between the equator and that parallel; and the distance of each degree and minute of latitude from the equator upon the meridian of this chart, computed in minutes of the equator, forms what is commonly called a table of meridional parts.

If the arch BD (No 11.) represent the latitude of any point B, then (CD being radius) CE will be the secant of that latitude: but it has been shown above, that radius is to secant of any latitude, as the length of a minute upon the equator is to the length of a minute on the meridian of this chart at that latitude; therefore CD is to CE, as the length of a minute on the equator is to the length of a minute upon the meridian, at the latitude of the point B. Consequently, if the radius CD be taken equal to the length of a minute upon the equator, CE, or the secant of the latitude, will be equal to the length of a minute upon the meridian at that latitude. Therefore, in general, if the length of a minute upon the equator be made radius, the length of a minute upon the enlarged meridian will be everywhere equal to the secant of the arch contained between it and the equator.

Cor. 1. Hence it follows, since the length of every intermediate minute between the equator and any parallel, is equal to the secant of the latitude, (the radius being equal to a minute upon the equator), the sum of all these lengths, or the distance of that parallel on the enlarged meridian from the equator, will be equal to the sum of all the secants, to every minute contained between it and the equator.

Cor. 2. Consequently the distance between any two parallels on the same side of the equator is equal to the difference of the sums of all the secants contained between between the equator and each parallel, and the distance between any two parallels on contrary sides of the equator is equal to the sum of the sums of all the secants contained between the equator and each parallel.

3. By the tables of meridional parts given by all the writers on this subject, may be constructed the nautical chart, commonly called Mercator's chart. Thus, for example, let it be required to make a chart that shall commence at the equator, and reach to the parallel of 60 degrees, and shall contain 80 degrees of longitude.

Draw the line EQ representing the equator, (see No 12.) then take, from any convenient line of equal parts, 4800, (the number of minutes containing in 80 degrees), which set off from E to Q, and this will determine the breadth of the chart.

Divide the line EQ into eight equal parts, in the points 10, 20, 30, &c., each containing 10 degrees; and each of these divided into 10 equal parts, will give the single degrees upon the equator; then through the points E, 10, 20, &c., drawing lines perpendicular to EQ, these shall be meridians.

From the scale of equal parts take 4527.4, (the meridional parts answering to 60 degrees), and set that off from E to A and from Q to B, and join AB; then this line will represent the parallel of 60, and will determine the length of the chart.

Again, from the scale of equal parts take 603.1, (the meridional parts answering to 10 degrees), and set that off from E to 10 on the line EA; and through the point 10 draw 10, 10, parallel to EQ; and this will be the parallel of 10 degrees. The same way, setting off from E on the line EA, the meridional parts answering to each degree, &c., of latitude, and through the several points drawing lines parallel to EQ, we shall have the several parallels of latitude.

If the chart does not commence from the equator, but is only to serve for a certain distance on the meridian between two given parallels on the same side of the equator; then the meridians are to be drawn as in the last example: and for the parallels of latitude you are to proceed thus, viz. From the meridional parts answering to each point of latitude in your chart subtract the meridional parts answering to the least latitude, and set off the differences severally, from the parallel of the least latitude, upon the two extreme meridians; and the lines joining these points of the meridians shall represent the several parallels upon your chart.

Thus let it be required to draw a chart that shall serve from the latitude of 20 degrees north to 60 degrees north, and that shall contain 80 degrees of longitude.

Having drawn the line DC to represent the parallel of 20 degrees (see No 12.) and the meridians to it, as in the foregoing example; set off 663.3 (the difference between the meridional parts answering to 30 degrees, and those of 20 degrees) from D to 30, and from C to 30; then join the points 30 and 30 with a right line, and that shall be the parallel of 30. Also set off 1397.6 (the difference between the meridional parts answering to 40 degrees, and those of 20 degrees) from D to 40, and from C to 40; and joining the points 40 and 40 with a right line, that shall be the parallel of 40. And proceeding after the same way, we may draw as many of the intermediate parallels as we have occasion for.

But if the two parallels of latitude that bound the chart, are on the contrary sides of the equator; then draw a line representing the equator and meridians to it, as in the first example; and from the equator set off on each side of it the several parallels contained between it and the given parallels as above, and your chart is finished.

If Mercator's chart, constructed as above, hath its equator extended on each side of the point E 180 degrees, and if the several places on the surface of the earth be there laid down according to their latitudes and longitudes, we shall have what is commonly called Mercator's map of the earth. This map is not to be considered as a similar and just representation of the earth's surface; for in it the figures of countries are distorted, especially near the poles: but since the degrees of latitude are everywhere increased in the same proportion as those of longitude are, the bearings between the places will be the same in this chart as on the globe; and the proportions between the latitudes, longitudes, and nautical distances, will also be the same on this chart, as on the globe itself; by which means the several cases of navigation are solved after a most easy manner, and adapted to the meanest capacities.

N. B. Here you must take notice, that in all charts the upper part is the north side, and the lower part or bottom is the south side; also that part of it towards the right-hand is the east, and that towards the left-hand the west side of the chart.

4. Since, according to this projection, the meridians are parallel right lines; it is plain, that the rhumbs which form always equal angles with the meridians, will be straight lines; which property renders this projection of the earth's surface much more easy and proper for the use than any other.

5. This method of projecting the earth's surface upon a plane, was first invented by Mr Edward Wright, but first published by Mercator; and hence the failing by the chart was called Mercator's failing.

6. In No 13. let A and E represent two places upon Mercator's chart, AC the meridian of A, and CE the parallel of latitude passing through E; draw AE, and set off upon AC the length AB equal to the number of minutes contained in the difference of latitude between the two places, and taken from the same scale of equal parts the chart was made by, or from the equator, or any graduated parallel of the chart, and through B draw BD parallel to CE meeting AE in D. Then AC will be the enlarged difference of latitude, AB the proper difference of latitude, CE the difference of longitude, BD the departure, AE the enlarged distance, and AD the proper distance, between the two places A and E; also the angle BAD will be the course, and AE the rhumb-line between them.

7. Now since in the triangle ACE, BD is parallel to one of its sides CE; it is plain the triangles ACE, ABD, will be similar, and consequently the sides proportional. Hence arise the solutions of the several cases in this failing, which are as follow.

Case I. The latitudes of two places given, to find the meridional or enlarged difference of latitude between Of this case there are three varieties, viz. either one of the places lies on the equator, or both on the same side of it; or lastly, on different sides.

1. If one of the proposed places lies on the equator, then the meridional difference of latitude is the same with the latitude of the other place, taken from the table of meridional parts.

**Example.** Required the meridional difference of latitude between St Thomas, lying on the equator, and St Antonio in the latitude of $17^\circ 20'$ north. We look in the tables for the meridional parts answering to $17^\circ 20'$, and find it to be $1056.2$, the enlarged difference of latitude required.

2. If the two proposed places be on the same side of the equator, then the meridional difference of latitude is found by subtracting the meridional parts answering to the least latitude from those answering to the greatest, and the difference is that required.

**Example.** Required the meridional difference of latitude between the Lizard in the latitude of $50^\circ 00'$ north, and Antigua in the latitude of $17^\circ 30'$ north. From the meridional parts of $50^\circ, 00' = 3474.5$ subtract the meridional parts of $17^\circ, 30' = 1066.7$

there remains $2407.8$

the meridional difference of latitude required.

3. If the places lie on different sides of the equator, then the meridional difference of latitude is found by adding together the meridional parts answering to each latitude, and the sum is that required.

**Example.** Required the meridional difference of latitude between Antigua in the latitude of $17^\circ 30'$ north, and Lima in Peru in the latitude of $12^\circ 30'$ south.

To the merid. parts answering to $17^\circ, 30' = 1066.7$ add these answering to $12^\circ, 30' = 756.1$

the sum is $1822.8$

the meridional difference of latitude required.

**Case II.** The latitudes and longitudes of two places given, to find the direct course and distance between them.

**Example.** Required to find the direct course and distance between the Lizard in the latitude of $50^\circ 00'$ north, and Port-Royal in Jamaica in the latitude of $17^\circ 40'$; differing in longitude $70^\circ 46'$, Port-Royal lying so far to the westward of the Lizard.

**Preparation.**

From the latitude of the Lizard $50^\circ, 00'$ subtract the latitude of Port-Royal $17^\circ, 40'$

and there remains $32^\circ, 20'$ equal to $1940$ minutes, the proper difference of latitude.

Then from the meridional parts of $50^\circ, 00' = 3474.5$ subtract those of $17^\circ, 40' = 1077.2$

and there remains $2397.3$

the meridional or enlarged difference of longitude.

**Geometrically.** Draw the line AC (No. 14.) representing the meridian of the Lizard at A, and set off from A, upon that line, AE equal to $1940$ (from any scale of equal parts) the proper difference of latitude, also AC equal to $2397.3$ (from the same scale) the meridional or enlarged difference of latitude. Upon the point C raise CB perpendicular to AC, and make CB equal to $2426$, the minutes of difference of longitude.

Join AB, and through E draw ED parallel to BC: to the scale of equal parts the other legs were taken from, will give the direct distance, and the angle DAE measured by the line of chords will give the course.

**By Calculation.**

For the angle of the course EAD, it will be, (by rectangular trigonometry)

$$AC : CB :: R : T, BAC, i.e.$$ As the meridional diff. of lat. $2397.3 - 3.37970$ is to the difference of long. $4246.0 - 3.62798$ so is radius $10.00000$ to the tang. of the direct course $60^\circ 33'$ $10.34828$ which, because Port-Royal is southward of the Lizard, and the difference of longitude westerly, will be south $60^\circ 33'$ west, or SW $W \frac{1}{4}$ west nearly.

Then for the distance AD, it will be, (by rectangular trigonometry)

$$R : AE :: Sec. A : AD, i.e.$$ As the radius $10.00000$ is the proper diff. of lat. $1940 - 3.28780$ so is the secant of the course $60^\circ 33'$ $10.30833$ to the distance $3945.6 - 3.59613$ consequently the direct course and distance between the Lizard and Port-Royal in Jamaica, is south $60^\circ 33'$ $3945.6$ miles.

**Case III.** Course and distance given, to find difference of latitude and difference of longitude.

**Example.** Suppose a ship from the Lizard in the latitude of $50^\circ 00'$ north, sails south $35^\circ 40'$ west $156$ miles: Required the latitude come to, and how much she has altered her longitude.

**Geometrically.** 1. Draw the line BK (No. 15.) representing the meridian of the Lizard at B; from B draw the line BM, making with BK an angle equal to $35^\circ 40'$, and upon this line set off BM equal to $156$ the given distance, and from M let fall the perpendicular MK upon BK.

Then for BK the proper difference of latitude, it will be, (by rectangular trigonometry)

$$R : MB :: S, BMK : BK,$$ i.e. As radius $10.00000$ is to the distance $156 - 2.19312$ so is the co-tan. of the course $35^\circ, 40'$ $9.90978$ to the proper difference of lat. $127 - 2.10290$ equal to $2^\circ 07'$; and since the ship is sailing from a north latitude towards the south, therefore the latitude come to will be $47^\circ 53'$ north. Hence the meridional difference of latitude will be $193.4$.

2. Produce BK to D, till BD be equal to $193.4$; through D draw DL parallel to MK, meeting DM produced in L; then DL will be the difference of longitude: to find which by calculation, it will be, (by rectangular trigonometry)

$$R : BD :: T, LBD : DL,$$ i.e. As radius $10.00000$ is to the meridional diff. of lat. $193.4 - 2.28646$ so is the tangent of the course $35^\circ, 40'$ $9.85594$ to minutes of diff. of long. $138.8 - 2.14240$ equal to $2^\circ 18' 48''$, the difference of longitude the ship has made westerly.

**Case IV.** Given course and both latitudes, viz. the la- latitude sailed from, and the latitude come to; to find the distance sailed, and the difference of longitude.

**Example.** Suppose a ship in the latitude of 50° 20' north, sails south 33° 45' east, until by observa- tion she is found to be in the latitude of 51° 45' north: Required the distance sailed, and the difference of lon- gitude.

**Geometrically.** Draw AB (No 16.) to represent the meridian of the ship in the first latitude, and set off from A to B 155 the minutes of the proper differ- ence of latitude, also AG equal to 257.9 the minutes of the enlarged difference of latitude. Through B and G, draw the lines BC and GK perpendicular to AG; also draw AR, making with AG an angle of 33° 45', which will meet the two former lines in the points C and K; so the case is constructed; and AC and GK may be found from the line of equal parts: To find which,

**By Calculation;**

First, For the difference of longitude, it will be, (by rectangular trigonometry,) \[ R : AG :: T, A : CK, \] i.e. As radius \[ 10.00000 \] is to the enlarged diff. of lat. \[ 257.9 \] so is the tang. of the course \[ 33°, 45' \] to min. of diff. of longitude \[ 162.8 \] equal to \( 2° 42' 48'' \), the difference of longitude easterly.

This might also have been found, by first finding the departure BC, (by Case 2. of Plane Sailing), and then it would be \[ AB : BC :: AG : GK, \] the difference of longitude required.

Then for the direct distance AC, it will be, (by rec- tangular trigonometry,) \[ R : AB :: Sec. A : AC, \] i.e. As radius \[ 10.00000 \] is to the proper diff. of lat. \[ 155 \] so is the secant of the course \[ 33°, 45' \] to the direct distance \[ 186.4 \] consequently the ship has sailed south 33° 45' east 186.4 miles, and has differed her longitude \( 2° 52' 18'' \) easterly.

**Case V.** both latitudes, and distance sailed, given; to find the direct course, and difference of longitude.

**Example.** Suppose a ship from the latitude of 45° 26' north, sails between north and east 195 miles, and then by observation she is found to be in the latitude of 48° 6' north: Required the direct course and differ- ence of longitude.

**Geometrically.** Draw AB (no 17.) equal to 160 the proper difference of latitude, and from the point B raise the perpendicular BD; then take 195 in your compasses, and setting one foot of them in A, with the other cross the line BD in D. Produce AB, till AC be equal to 233.6 the enlarged difference of latitude. Through C draw CK parallel to BD, meeting AD produced in K: so the case is constructed; and the angle A may be measured by the line of chords, and CK by the line of equal parts: To find which,

**By Calculation:**

First, For the angle of the course BAD, it will be (by rectangular trigonometry,) \[ AB : R :: AD : Sec. A. i.e. \] As the proper diff. of lat. \[ 160 \] \[ 2.20412 \]

is to radius \[ 10.00000 \] so is the distance \[ 195 \] \[ 2.29003 \] to the secant of the course \[ 34°, 52' \] \[ 10.08591 \] which, because the ship is sailing between north and east, will be north 34° 52' east, or NE by N 1° 7' east- erly.

Then for the difference of longitude, it will be, (by rectangular trigonometry,) \[ R : AC :: T, A : CK, \] i.e. As radius \[ 10.00000 \] is to the merid.diff. of lat. \[ 233.6 \] so is the tang. of the course \[ 34°, 52' \] to min. of diff. of longitude \[ 162.8 \] equal to \( 2° 42' 48'' \), the difference of longitude easterly.

**Case VI.** One latitude, course, and difference of longitude, given; to find the other latitude, and dis- tance sailed.

**Example.** Suppose a ship from the latitude of 48° 50' north, sails south 34° 40' west, till her difference of longitude is 2° 42': Required the latitude come to, and the distance sailed.

**Geometrically.** 1. Draw AE (no 18.) to repre- sent the meridian of the ship in the first latitude, and make the angle EAC equal to 34° 40', the angle of the course; then draw FC parallel to AE, at the distance of 164 the minutes of difference of longitude, which will meet AC in the point C. From C let fall upon AE the perpendicular CE; then AE will be the enlarged difference of latitude. To find which by cal- culation, it will be, (by rectangular trigonometry,) \[ T, A : R :: CE : AE, \] i.e. As the tang. of the course \[ 34°, 40' \] is to the radius \[ 10.00000 \] so is min. of diff. longitude \[ 164 \] to the enlarged diff. of latitude \[ 237.2 \] and because the ship is sailing from a north latitude southerly, therefore

From the merid. parts of \[ 48°, 50' \] the latitude sailed from \[ 3366.9 \] take the merid. difference of latitude \[ 237.2 \] and there remains \[ 3129.7 \] the meridional parts of the latitude come to, viz. 46° 09'.

Hence for the proper difference of latitude, From the latitude sailed from \[ 48°, 50' N \] take the latitude come to \[ 46°, 09' N \] and there remains \[ 2°, 41' \] equal to 161, the minutes of difference of latitude.

2. Set off upon AE the length AD equal to 161 the proper difference of latitude, and thro' D draw DB parallel to CE: then AB will be the direct di- stance. To find which by calculation, it will be, (by rectangular trigonometry,) \[ R : AD :: Sec. A : AB, \] i.e. As radius \[ 10.00000 \] is to the proper diff of latitude \[ 161 \] so is the secant of the course \[ 34°, 40' \] to the direct distance \[ 195.8 \] \[ 2.29171 \]

**Case VII.** One latitude, course, and departure gi- ven; to find the other latitude, distance sailed, and dif- ference of longitude.

**Example.** Suppose a ship sails from the latitude of \[ 54° \] Part II.

GEOMETRICALLY. 1. Having drawn the meridian AB, (no. 19.) make the angle BAD equal to 42° 33'. Draw FD parallel to AB at the distance of 116, which will meet AD in D. Let fall upon AB the perpendicular DB. Then AB will be the proper difference of latitude, and AD the direct distance: to find which by calculation, first, for the distance AD it will be (by rectangular trigonometry.)

\[ S, A : BD :: R : AD. \]

i.e. As the sine of the course \( 42^\circ 33' \) - 9.83010 is to the departure - 116 - 2.06446 so is radius - 10.00000 to the direct distance - 171.5 - 2.23436

Then for the proper difference of latitude, it will be,

(by rectangular trigonometry)

\[ T, A : BD :: R : AB. \]

i.e. as the tang. of the course \( 42^\circ 33' \) - 9.96281 is to the departure - 116 - 2.06446 so is radius - 10.00000 to the proper difference of latitude - 126.4 - 2.10165 equal to \( 2^\circ 6' \): consequently the ship has come to the latitude of \( 52^\circ 30' \) north; and so the meridional difference of latitude will be 212.2.

2. Produce AB to E, till AE be equal to 212.2; and through E draw EC parallel to BD, meeting AD produced in C; then EC will be the difference of longitude; to find which by calculation, it will be, (by rectangular trigonometry)

\[ R : AE :: T, A : EC. \]

i.e. As radius - 10.00000 is to the merid diff. of latitude - 212.2 - 2.32675 so is the tang. of the course \( 42^\circ 33' \) - 9.96281 to the min. of diff. of longitude - 194.8 - 2.28956 equal to \( 3^\circ 14' 48'' \), the difference of longitude easterly.

This might have been found otherwise, thus: because the triangles ACE, ADB, are similar; therefore it will be,

\[ AB : BD :: AE : EC. \]

i.e. As the proper diff. of latitude - 126.4 - 2.10165 is to the departure - 116 - 2.06446 so is the enlarged diff of lat. - 212.2 - 2.32675 to min. diff. of longitude - 194.8 - 2.28956

CASE VIII. Both latitudes and departure given, to find course, distance, and difference of longitude.

EXAMPLE. Suppose a ship from the latitude of \( 46^\circ 20' \) north, sails between south and east 138 miles, and is then found by observation to be in the latitude of \( 43^\circ 35' \) north: Required the course and distance failed, and difference of longitude.

GEOMETRICALLY. Draw AK (no. 20.) to represent the meridian of the ship in her first latitude; set off upon it AC, equal to 165, the proper difference of latitude. Draw BC perpendicular to AC, equal to 126.4 the departure, and join AB. Set off from A, AK equal to 233.3, the enlarged difference of latitude; and through K draw KD parallel to BC, meeting AB produced in D; so the case is constructed, and DK will be the difference of longitude, AB the distance, and the angle A the course; to find

By Calculation:

First, For DC the difference of longitude, it will be,

\[ AC : CB :: AK : KD. \]

i.e. As the proper diff. of latitude - 165 - 2.21748 is to the departure - 126.4 - 2.10175 so is the enlarged diff. of latitude - 233.3 - 2.36791 to min. of diff. longitude - 178.7 - 2.25218 equal to \( 2^\circ 58' 42'' \), the difference of longitude westerly.

Then for the course it will be, (by rectangular trigonometry)

\[ AC : BC :: R : T, A. \]

i.e. As the proper diff. of latitude - 165 - 2.21748 is to departure - 126.4 - 2.10175 so is radius - 10.00000 to the tangent of the course \( 37^\circ 27' \) - 9.88427 which, because the ship sails between south and west, will be south \( 37^\circ 27' \) west, or SW/S \( 6^\circ 30' \) westerly.

Lastly, For the distance AB, it will be, (by rectangular trigonometry)

\[ S, A : BC :: R : AB. \]

i.e. As the sine of the course \( 37^\circ 27' \) - 9.78395 is to the departure - 126.4 - 2.10175 so is radius - 10.00000 to the direct distance - 207.9 - 2.31780

CASE IX. One latitude, distance failed, and departure given; to find the other latitude, difference of longitude, and course.

EXAMPLE. Suppose a ship in the latitude of \( 48^\circ 33' \) north, sails between south and east 138 miles, and has then made of departure 112.6: Required the latitude come to, the direct course, and difference of longitude.

GEOMETRICALLY. 1. Draw BD (no. 21.) for the meridian of the ship at B; and parallel to it draw FE, at the distance of 112.6, the departure. Take 138, the distance, in your compasses, and fixing one point of them in B, with the other cross the line FE in the point E; then join B and E, and from E let fall upon BD the perpendicular ED; so BD will be the proper difference of latitude, and the angle B will be the course; to find which, by calculation,

First, For the course it will be, (by rectangular trigonometry)

\[ BE : R :: DE : S, B. \]

i.e. As the distance - 138 - 2.13988 is to radius - 10.00000 so is the departure - 112.6 - 2.05154 to the fine of the course \( 54^\circ 41' \) - 9.91166 which, because the ship sails between south and east, will be south \( 54^\circ 41' \) east, or SE \( 0^\circ 41' \) easterly.

Then for the difference of latitude, it will be, (by rectangular trigonometry)

\[ R : BE :: Co S, B : BD. \]

i.e. As radius - 10.00000 is to the distance - 138 - 2.13988 so is the co-sine of the course \( 54^\circ 41' \) - 9.76200 to the difference of latitude - 79.8 - 1.90188 equal to \( 1^\circ 10' \). Consequently the ship has come to the latitude of \( 47^\circ 13' \). Hence the meridional difference of latitude will be 117.7.

2dly, Produce B to A, till BA be equal to 117.7; and and through A draw AC parallel to DE, meeting BE produced in C; then AC will be the difference of longitude; to find which by calculation, it will be,

\[ BD : DE :: BA : AC. \]

i.e. As the proper diff. of latitude \(79.8\) \(1.90180\) is to the departure \(112.6\) \(2.05154\)

so is the enlarged diff. of latitude \(117.7\) \(2.07078\) to the diff. of longitude \(166.1\) \(2.22044\)

equal to \(2^\circ 46' 06''\), the difference of longitude exactly.

10. From what has been said, it will be easy to solve a traverse according to the rules of Mercator's sailing.

**Example.** Suppose a ship at the Lizard in the latitude \(50^\circ 00'\) north, is bound to the Madeira in the latitude of \(32^\circ 20'\) north, the difference of longitude between them being \(11^\circ 40'\), the west end of the Madeira lying so much to the westward of the Lizard, and consequently the direct course and distance (by Case 2. of this Sect.), is south \(26^\circ 15'\) west \(1181.9\) miles; but by reason of the winds she is forced to fail on the following courses, (allowance being made for lee-way and variation, &c.), viz. SSW \(44\) miles, SWW \(1/2\) west \(36\) miles, SWbS \(56\) miles, and SbE \(28\) miles: Required the latitude the ship is in, her bearing and distance from the Lizard, and her direct course and distance from the Madeira, at the end of these courses.

The geometrical construction of this traverse is performed by laying down the two ports according to construction of Case 2. of this Section, and the several courses and distances according to Case 3. by which we have the following solution by calculation.

1. Course SSW, distance \(44\) miles.

For difference of latitude.

As radius \(10.00000\) is to the distance \(44\) \(1.64345\)

so is the co-sine of the course \(22^\circ 30'\) \(9.96562\) to the difference of latitude \(40.65\) \(1.60907\)

and since the course is southerly, therefore the latitude come to will be \(49^\circ 20'\) north, and consequently the meridional difference of latitude will be \(61.8\).

Then,

For difference of longitude.

As radius \(10.00000\) is to the enlarged diff. of lat. \(61.8\) \(1.79099\)

so is the tang. of the course \(22^\circ 30'\) \(9.61722\) to min. of diff. of longitude \(25.6\) \(1.40821\)

2. Course SWW \(1/2\) west, distance \(36\) miles.

For difference of latitude.

As radius \(10.00000\) is to the distance \(36\) \(1.55630\)

so is the co-sine of the course \(16^\circ 52'\) \(9.98090\) to the difference of latitude \(34.46\) \(1.53720\)

and since the course is southerly, therefore the latitude come to will be \(48^\circ 45'\). Hence the meridional difference of latitude will be \(53.4\). Then,

For difference of longitude.

As radius \(10.00000\) is to the enlarged diff. of lat. \(53.4\) \(1.72754\)

so is the tang. of the course \(16^\circ 52'\) \(9.48171\) to the difference of longitude \(16.19\) \(1.20925\)

3. Course SWbS, distance \(56\) miles.

For difference of latitude.

As radius \(10.00000\) is to the distance \(56\) \(1.74819\)

so is the co-sine of the course \(33^\circ 45'\) \(9.91985\) to the difference of latitude \(46.56\) \(1.66804\)

consequently the latitude come to is \(47^\circ 59'\); and therefore the enlarged difference of latitude will be \(69.2\).

Then,

For difference of longitude.

As radius \(10.00000\) is to the enlarged diff. of lat. \(69.2\) \(1.84011\)

so is the tang. of the course \(33^\circ 45'\) \(9.82489\) to the difference of longitude \(46.24\) \(1.66500\)

4. Course SbE, distance \(28\) miles.

For difference of latitude.

As radius \(10.00000\) is to the distance \(28\) \(1.44716\)

so is the co-sine of the course \(11^\circ 15'\) \(9.99157\) to the difference of latitude \(27.46\) \(1.43873\)

consequently the latitude come to will be \(47^\circ 31'\); and hence the meridional difference of latitude will be \(43.2\).

Then,

For difference of longitude.

As radius \(10.00000\) is to the enlarged diff. of lat. \(43.2\) \(1.63548\)

so is the tang. of the course \(11^\circ 15'\) \(9.26866\) to the diff. of longitude \(8.59\) \(0.93414\)

Now these several courses and distances, together with the difference of latitude and longitude belonging to each of them, being set down in their proper columns in the Traverse Table, will stand as follow.

| Courses | Distances | Diff. of Lat. | Diff. of Longit. | |---------|-----------|--------------|-----------------| | N. | S. | E. | W. | | SSW | 44 | 40.65 | 25.6 | | SWW \(1/2\) W | 36 | 34.46 | 16.16 | | SWbS | 56 | 40.56 | 49.24 | | SbE | 28 | 27.46 | 8.56 |

Diff. of Lat. \(149.13\) \(8.59\) \(88.03\) \(0.59\)

Diff. of Long. \(79.44\)

Hence it is plain that the ship has made of southing \(149.13\) minutes, and consequently has come to the latitude of \(47^\circ 31'\) north, and so the meridional difference of latitude between that and her first latitude will be \(226.1\): And since she has made of difference of longitude \(79.44\) minutes westerly; therefore, for the direct course and distance between the Lizard and the ship, it will be, (by Case 2. of this Section.

For the direct course.

As the merid. diff. of latitude \(226.1\) \(2.35430\) is to radius \(10.00000\)

so is the difference of longitude \(79.44\) \(1.90004\) to the tang. of the course \(19^\circ 22'\) \(9.54593\)

which, because the difference of latitude is southerly, and the difference of longitude westerly, will be south \(19^\circ 22'\) west, or SWW \(8^\circ 7'\) westerly. Then,

For the direct distance.

As radius \(10.00000\) is to the proper diff. of lat. \(149.13\) \(2.17349\)

so is the secant of the course \(19^\circ 22'\) \(10.02530\) Part II.

Practice to the direct distance - 158 - 2.19879

From the latitude the ship is in - 47° 31' N subtract the lat. of the Madera - 32° 20' N

and there remains - 15° 11'

equal to 911 minutes, the proper difference of latitude between the ship and the Madera.

Again, from the merid. parts answering to the latitude the ship is in - 3248.4

Take the meridional parts answering to the latitude of the Madera - 2052.0

and there remains - 1196.4

the enlarged difference of latitude between the ship and the Madera.

Also, from the diff. of long. between the Lizard and the Madera - 11° 40' W

Take the difference of lon. between the Lizard and the ship - 1° 19' 44" W

and there remains - 10° 20' 56" W

equal to 620.56 min. of difference of longitude between the ship and the Madera westerly.

Then for the direct course and distance between the ship and the Madera, it will be,

For the direct course: As the merid. diff. of latitude - 1196.4 - 3.07788 is to radius - 10.00000 so is the difference of longitude - 620.56 - 2.79278 to the tang. of the course - 27° 25' - 9.71493

For the direct distance: As radius - 10.00000 is to the proper diff. of latitude - 911 - 2.95952 so is the secant of the course - 27° 25' - 10.05174 to the direct distance - 1027 - 3.01126

11. It is very common in working a day's reckoning at sea, to find the difference of latitude and departure to each course and distance; and adding all the departures together, and all the differences of latitudes for the whole departure, and difference of latitude made good that day, from thence (by Case 8. of this Section) to find the difference of longitude, &c. made good that day. Now that this method is false, will evidently appear, if we consider that the same departure, reckoned on two different parallels, will give unequal differences of longitude; and consequently, when several departures are compounded together and reckoned on the same parallel, the difference of longitude resulting from that cannot be the same with the sum of the differences of longitude resulting from the several departures on different parallels; and therefore we have chosen, in the last example of a traverse, to find the difference of longitude answering to each particular course and distance, the sum of which must be the true difference of longitude made good by the ship on these several courses and distances.

12. We shewed, at Art. 3. of this Section, how to construct a Mercator's chart; and now we shall proceed to its several uses, contained in the following problems.

Prob. I. Let it be required to lay down a place upon the chart, its latitude, and the difference of longitude between it and some known place upon the chart being given.

Example. Let the known place be the Lizard lying on the parallel of 50° 00' north, and the place to be laid down St Katharine's on the east coast of America, differing in longitude from the Lizard 42° 36', lying so much to the westward of it.

Let L represent the Lizard on the chart, (see No. 12.) lying on the parallel of 50° 00' north, its meridian. Set off AE from E upon the equator EQ, 42° 36', towards Q, which will reach from E to F. Plate CCI. Through F draw the meridian FG, and this will be the meridian of St Katharine's; then set off from Q to H upon the graduated meridian QB, 28 degrees; and thro' H draw the parallel of latitude HM, which will meet the former meridian in K, the place upon the chart required.

Prob. II. Given two places upon the chart, to find their difference of latitude and difference of longitude.

Through the two places draw parallels of latitude; then the distance between these parallels, numbered in degrees and minutes upon the graduated meridian, will be the difference of latitude required; and thro' the two places drawing meridians, the distance between these, counted in degrees and minutes on the equator or any graduated parallel, will be the difference of longitude required.

Prob. III. To find the bearing of one place from another upon the chart.

Example. Required the bearing of St Katharine's at K (see No. 12.) from the Lizard at L.

Draw the meridian of the Lizard AE, and join K and L with the right line KL; then by the line of chords measuring the angle KLE, and with that entering the tables, we shall have the thing required.

This may also be done, by having compasses drawn on the chart, (suppose at two of its corners); then lay the edge of a ruler over the two places, and let fall a perpendicular, or take the nearest distance from the centre of the compass next the first place, to the ruler's edge; then with this distance in your compasses, slide them along by the ruler's edge, keeping one foot of them close to the ruler, and the other as near as you can judge perpendicular to it, which will describe the rhumb required.

Prob. IV. To find the distance between two given places upon the chart.

This problem admits of four cases, according to the situation of the two places with respect to one another.

Case I. When the given places lie both upon the equator.

In this case their distance is found by converting the degrees of difference of longitude intercepted between them into minutes.

Case II.- When the two places lie both on the same meridian.

Draw the parallels of those places; and the degrees upon the graduated meridian, intercepted between those parallels, reduced to minutes, give the distance required.

Case III. When the two places lie on the same parallel.

Example. Required to find the distance between the points K and N (see No. 12.) both lying on the parallel of 28° 00' north. Take from your scale the chord Practice chord of 60° or radius in your compasses, and with that extent on KN as a base make the isosceles triangle KPN; then take from the line of fines the co-fine of the latitude, or fine of 72°, and set that off from P to S and T. Join S and T with the right line ST, and that applied to the graduated equator will give the degrees and minutes upon it equal to the distance; which, converted into minutes, will be the distance required.

The reason of this is evident from the section of Parallel Sailing: for it has been there demonstrated, that radius is to the co-fine of any parallel, as the length of any arch on the equator, to the length of the same arch on that parallel. Now in this chart KN is the distance of the meridians of the two places K and N upon the equator; and since, in the triangle PNK, ST is the parallel to KN, therefore PN : PT :: NK : TS. Consequently TS will be the distance of the two places K and N upon the parallel of 28°.

If the parallel the two places lie on be not far from the equator, and they not far asunder; then their distance may be found thus: Take the distance between them in your compasses, and apply that to the graduated meridian, so as the one foot may be as many minutes above as the other is below the given parallel; and the degrees and minutes intercepted, reduced to minutes, will give the distance.

Or it may also be found thus: Take the length of a degree on the meridian at the given parallel, and turn that over on the parallel from the one place to the other, as oft as you can; then as oft as that extent is contained between the places, so many times 60 miles will be contained in the distance between them.

Case IV. When the places differ both in longitude and latitude.

Example. Suppose it were required to find the distance between the two places a and e upon the chart.

By

Prob. II. Find the difference of latitude between them; and take that in your compasses from the graduated equator, which set off on the meridian of a, from a to b; then thro' b draw bc parallel to de; and taking ac in your compasses, apply it to the graduated equator, and it will show the degrees and minutes contained in the distance required, which multiplied by 60 will give the miles of distance.

The reason of this is evident from Art. 6. of this Section: for it is plain ad is the enlarged difference of latitude, and ab the proper; consequently ae the enlarged distance, and ac the proper.

Prob. V. To lay down a place upon the chart, its latitude and bearing from some known place upon the chart being known, or (which is the same) having the course and difference of latitude that a ship has made, to lay down the running of the ship, and find her place upon the chart.

Example. A ship from the Lizard in the latitude of 50° 00' north, fails SSW till she has differed her latitude 36° 40'. Required her place upon the chart.

Count from the Lizard at L, on the graduated meridian downwards (because the course is southerly) 36° 40' to f; through which draw a parallel of latitude, which will be the parallel the ship is in; then from L draw a SSW line Lf, cutting the former parallel in f; and this will be the ship's place upon the chart.

Prob. VI. One latitude, course, and distance, sailed, given; to lay down the running of the ship, and find her place upon the chart.

Example. Suppose a ship at a in the latitude of 20° 00' north, sails north 37° 20', east 191 miles: Required the ship's place upon the chart.

Having drawn the meridian and parallel of the place a, set off the rhumb-line ae, making with ab an angle of 37° 20'; and upon it set off 191 from a to c; thro' c draw the parallel cb; and taking ab in your compasses, apply it to the graduated equator, and observe the number of degrees it contains; then count the same number of degrees on the graduated meridian from C to b, and through b draw the parallel be, which will cut ac produced in the point e, the ship's place required.

Prob. VII. Both latitudes and distance sailed, given; to find the ship's place upon the chart.

Example. Suppose a ship sails from a, in the latitude of 20° 00' north, between north and east 191 miles, and is then in the latitude of 45° 00' north: Required the ship's place upon the chart.

Draw de the parallel of 45°, and set off upon the meridian of a upwards, ab equal to the proper difference of latitude taken from the equator or graduated parallel. Through b draw bc parallel to de; then with 191 in your compasses, fixing one foot of them in a, with the other cross be in c. Join a in c with the right line ac; which produced will meet de in e, the ship's place required.

Prob. VIII. One latitude, course, and difference of longitude, given; to find the ship's place upon the chart.

Example. Suppose a ship from the Lizard in the latitude of 50° 00' north, sails SW by W, till her difference of longitude is 42° 36': Required the ship's place upon the chart.

Having drawn AE the meridian of the Lizard at L, count from E to F upon the equator 42° 36'; and through F draw the meridian EG; then from L draw the SW by W line LK, and where this meets FG, as at K, will be the ship's place required.

Prob. IX. One latitude, course, and departure, given; to find the ship's place upon the chart.

Example. Suppose a ship at a in the latitude of 20° 00' north, sails north 37° 20' east, till she has made of departure 116 miles: Required the ship's place upon the chart.

Having drawn the meridian of a, at the distance of 116, draw parallel to it the meridian kl. Draw the rhumb-line ac, which will meet kl in some point c; then through c draw the parallel cb, and ab will be the proper difference of latitude, and bc the departure. Take ab in your compasses, and apply it to the equator or graduated parallel; then observe the number of degrees it contains, and count so many on the graduated meridian from C upwards to b. Through b draw the parallel he, which will meet ac produced in some point as e, which is the ship's place upon the chart.

Prob. X. One latitude, distance, and departure, given; to find the ship's place upon the chart.

Example. Suppose a ship at a in the latitude of 20° 00' north, sails 191 miles between north and east, and then Part II.

Practice then is found to have made of departure 116 miles; Required the ship's place upon the chart.

Having drawn the meridian and parallel of the place \(a\), set off upon the parallel \(am\) equal to 116, and thro' \(m\) draw the meridian \(kl\). Take the given distance 191 in your compasses; setting one foot of them in \(a\), with the other crofs \(kl\) in \(c\). Join \(ac\), and through \(c\) draw the parallel \(cb\); so \(cb\) will be the departure, and \(ab\) the proper difference of latitude; then proceeding with this, as in the foregoing problem, you will find the ship's place to be \(e\).

Prob. XI. The latitude sailed from, difference of latitude, and departure, given; to find the ship's place upon the chart.

Example. Suppose a ship from \(a\) in the latitude of 20° 00' north, sails between north and east, till she be in the latitude of 45° 00' north, and is then found to have made of departure 116 miles; Required the ship's place upon the chart.

Having drawn the meridian of \(a\), set off upon it, from \(a\) to \(b\), 25 degrees, (taken from the equator or graduated parallel), the proper difference of latitude; then through \(b\) draw the parallel \(be\), and make \(bc\) equal to 116 the departure, and join \(ac\). Count from the parallel of \(a\) on the graduated meridian upwards to \(b\) 25 degrees, and through \(b\) draw the parallel \(be\), which will meet \(ac\) produced in some point \(e\), and this will be the place of the ship required.

13. In the section of Plane Sailing, it is plain, that the terms meridional distance, departure, and difference of longitude, were synonymous, constantly signifying the same thing; which evidently followed from the supposition of the earth's surface being projected on a plane in which the meridians were made parallel, and the degrees of latitude equal to one another and to those of the equator. But since it has been demonstrated (in this section) that if, in the projection of the earth's surface upon a plane, the meridians be made parallel, the degrees of latitude must be unequal, still increasing the nearer they come to the pole; it follows, that these terms must denote lines really different from one another.

§ 6. Of Oblique Sailing.

The questions that may be proposed on this head being innumerable, we shall only give a few of the most useful.

Prob. I. Coasting along the shore, I saw a cape bear from me NNE; then I stood away NWbW 20 miles, and I observed the same cape to bear from me NEbE: Required the distance of the ship from the cape at each station.

Plate CIII. Geometrically. Draw the circle NWSE (No 22.) to represent the compass, NS the meridian, and WE the east and west line, and let \(C\) be the place of the ship in her first station; then from \(C\) set off upon the NWbW line, CA 20 miles, and \(A\) will be the place of the ship in her second station.

From \(C\) draw the NNE line \(CB\), and from \(A\) draw \(AB\) parallel to the NEbE line \(CD\), which will meet \(CB\) in \(B\) the place of the cape, and \(CB\) will be the distance of it from the ship in its first station, and \(AB\) the distance in the second: to find which,

By Calculation;

In the triangle \(ABC\) are given \(AC\), equal to 20 miles; the angle \(ACB\), equal to 78° 45', the distance between the NNE and NWbW lines; also the angle \(ABC\), equal to BCD, equal to 33° 45', the distance between the NNE and NEbE lines; and consequently the angle \(A\), equal to 67° 30'.

Hence for \(CB\), the distance of the cape from the ship in her first station, it will be (by oblique trigonometry)

\[ S. ABC : AC :: S. BAC : CB, \]

i.e. As the fine of the angle \(B\) 33° 45' - 9.74473 is to the distance run \(AC\) 20 - 1.30103 so is the fine of \(BAC\) 67, 30 - 9.96562 to \(CB\) 33.26 - 1.52191

the distance of the cape from the ship at the first station. Then for \(AB\), it will be, by oblique trigonometry,

\[ S. ABC : AC :: S. ACB : AB, \]

i.e. As the fine of \(B\) 33° 45' - 9.74474 is to \(AC\) 20 - 1.30103 so is the fine of \(C\) 78° 45' - 9.99157 to \(AB\) 35.31 - 1.54786

the distance of the ship from the cape at her second station.

Prob. II. Coasting along the shore, I saw two headlands; the first bore from me NEbE 17 miles, the other SSW miles: Required the bearing and distance of these headlands from one another.

Geometrically. Having drawn the compass NWSE (No 23.) let \(C\) represent the place of the ship; set off upon the NEbE line \(CA\) 17 miles from \(C\) to \(A\), and upon the SSW line \(CB\) 20 miles from \(C\) to \(B\), and join \(AB\); then \(A\) will be the first headland, and \(B\) the second; also \(AB\) will be their distance, and the angle \(A\) will be the bearing from the NEbN line; to find which,

By Calculation;

In the triangle \(ACB\) are given, \(AC\) 17, \(CB\) 20, and the angle \(ACB\) equal to 101° 15', the distance between the NEbE and SSW lines. Hence (by oblique angular trigonometry) it will be

As the sum of the sides \(AC\) and \(CB\) 37 1.56820 is to their difference 3 0.47712 so is the tang. of \(A\) the sum of the angles \(A\) and \(B\) 39° 22' 9.91417 to the tang. of half their diff. 3° 49' 8.82309 consequently the angle \(A\) will be 43° 11', and the angle \(B\) 35° 34'; also the bearing of \(B\) from \(A\) will be SWbW 15° 49' westerly, and the bearing of \(A\) from \(B\) will be NEbE 1° 49' easterly.

Then for the distance \(AB\), it will be, (by oblique angular trigonometry),

\[ S. A : CB :: S. C : AB, \]

i.e. As the fine of \(A\) 43° 11' - 9.83527 is to \(CB\) 20 - 1.30103 so is the fine of \(C\) 101, 15 - 9.99157 to \(AB\) 28.67 - 1.45733

the distance between the two headlands.

Prob. III. Coasting along the shore, I saw two headlands; the first bore from me NWbN, and the second NNE; then standing away Ebn ¼ northerly 20 miles, I found the first bore from me WNW ¼ westerly, and the second NWbW ¼ westerly: Required the bearing and distance of these two headlands.

Geometrically. Having drawn the compass NWSE (No 24.) let \(C\) represent the first place of the ship;

Practice ship; from which draw the NWbN line CB, and the NNE line CD, also the EbN ¼ N line CA, which make equal to 20. From A draw AB parallel to the WNW ¼ W line, and AD parallel to the NsW ¼ W meeting the two first lines in the points B and D; then B will be the first and D the second headlands. Join the points B and D, and BD will be the distance between them, and the angle CDB the bearing from the NNE line: to find which,

By Calculation;

1. In the triangle ABC are given the angle BCA, equal to 104° 04', the distance between the NWbN line, and the ENE ¼ E line; the angle BAC, equal to 36° 34', the distance between the WSW ¼ W line and the WNW ¼ W line; the angle ABC equal to 39° 22', the distance between the ESE ¼ E line; and the SWbS line, also the side CA equal to 20 miles: whence for CB, it will be (by oblique trigonometry)

As the fine of CBA - 39°, 22' - 9.80228 is to AC - 20 - 1.30103 so is the fine of CAB - 36°, 34' - 9.77507 to CB - 18.79 - 1.27382 the distance between the first headland and the ship in her first station.

2. In the triangle ACD, are given the angle ACD, equal to 47° 49', the distance between the ENE ¼ E line, and the NNE line; the angle CAD, equal to 92° 49', the distance between the WSW ¼ W line; and the NsW ¼ W line, the angle CDA equal to 39° 22', the distance between the SSW line and the S¼E¼E line; also the leg CA equal to 20.

Hence for CD, it will be (by oblique trigonometry)

As the fine CAD - 39°, 22' - 9.80228 is to AC - 20 - 1.30103 so is the fine of CAD - 92°, 34' - 9.99960 to CD - 31.5 - 1.49835 the distance between the second headland and the ship in her first station.

3. In the triangle BCD are given BC 18.79, CD 31.5, and the angle BCD equal to 56° 15'; the distance between the NWbN line and the NNE line.

Hence for the angle CDB, it will be (by oblique trigonometry)

As the sum of the sides - 50.29 - 1.70148 is to the difference of sides 12.71 - 1.10415 so is tangent of ½ sum of \{61°, 51' - 10.27189 the unknown angles to tang. of half their diff. - 25°, 18' - 9.67458 consequently the angle CBD is 87° 10', and the angle CDB 36° 35'. Hence the bearing of the first headland from the second will be S 59° 8', W or SWsW ¼ W nearly; and for the distance between them, it will be,

As the fine of BDC - 36°, 35' - 9.77524 is to BC - 18.79 - 1.27382 so is the fine of BCD - 56°, 15' - 9.91985 to BD - 26.21 - 1.41843 the distance between the two headlands.

This, and the first problem, are of great use in drawing the plot of any harbour, or laying down any sea-coast.

Suppose a ship that makes her way good within 6½ points of the wind, at north, is bound to a port bearing east 86 miles distance from her: Required the course and distance upon each tack, to gain the intended port.

Geometrically. Having drawn the compass NE SW, (N° 25.) let C represent the ship's place, and set off upon the east line CA 86 miles, so A will be the intended port. Draw CD and CB on each side of the north line at 6½ points distance from it, and through A draw AB parallel to CD meeting CB in B; then the ENE ¼ E line CB, will be the course of the ship upon the starboard tack, and CB its distance on that tack; also the ESE ¼ E line Ab, will be the course on the larboard tack, and BA the distance on that tack: to find which,

By Calculation;

In the triangle ABC are given the angle ACB, equal to 16° 53', the distance between the east and ENE ¼ E line; the angle CBA, equal to 146° 14', the distance between the ENE ¼ E and the WNW ¼ W lines; the angle BAC equal to 16° 53', the distance between the east and ESE ¼ E lines; also AC 86 miles.

Hence, since the angle at A and C are equal, the legs CB and BA will likewise be equal; to find either of which (suppose CB) it will be (by oblique angled trigonometry)

As the fine of B - 146°, 14' - 9.74493 is to AC - 86 - 1.93450 so is the fine of A - 16.53 - 9.46303 to CB - 44.94 - 1.65260 the distance the ship must sail on each tack.

There is a great variety of useful questions of this nature that may be proposed; but the nature of them being better understood by practice at sea, we shall leave them, and go on to Great Circle Sailing.

§ 6. Great Circle Sailing.

A great many cases might be proposed in this kind of sailing; but as they serve rather for exercises in the solution of spheric triangles than for any real use towards the navigating of a ship, we shall only give the solution of one problem, as being the most generally useful.

Prob. Given the latitudes and longitudes of two places on the earth: Required the nearest distance on the surface, together with the angles of position (or that which a great circle, passing over both places, makes with the meridian of one of them) from either place to the other.

Case I. When both places lie under the same meridian, their difference of latitude shews their nearest distance.

Case II. When the two places lie under the equator, their distance is equal to the difference of longitude between them.

Case III. When the places lie under the same parallel of latitude.

Example. What is the least distance between St Mary's in Lat. 37° 00' N. Long. 25° 0' W. and Cape Henry, in Lat. 37° 00' Long. 76° 23' west?

Describe a circle PESQ representing the meridian of one of the places; suppose of the eastern one, as fig. 6. St Mary's; draw the line EQ representing the equator, and at right angles to it draw the line PS, for the axis of the earth, the extremity of which, P, is the north pole, and S the south pole; and on this circle lay off from P to A the complement of the lati- Part II.

The latitude of St Mary's, the eastern place. On the equator, from Q to C, lay off the difference of longitude between the two places; and through the points P, C, S, describe a circle, which will be the meridian of the other place Cape Henry; on which lay from P to B the co-latitude of this place, which is done by describing the arc Aa about the pole P according to the rules of projection, at the distance of the co-latitude. Through the points ABD describe a great circle; then will A represent St Mary's, and B Cape Henry; PA and PB are their co-latitudes; the angle APB, which is measured by the arc QC, is the difference of longitude; the arc AB is the nearest distance of these places; the angle PAB is the angle of position from A to B; and the angle PBA is the angle of position from B to A. The arc AB, and the angle PAB or PBA, may be measured according to the rules laid down under the article Projection. Now, the places having the same latitude, PA is equal to PB, and the angles PAB and PBA are likewise equal. Therefore if the arc PI be described, making the angle API = 25° 41½', the half of the difference of longitude; PI will be perpendicular to AB, and bisect it. And in the triangle API, right-angled at I, there will be given the hypotenuse AP = 53° 00' the angle API = 25° 41½'; to find the leg AI = half the distance sought, and the angle PAI = the angle of position. Then, for the distance: As radius is to the sine of the hypotenuse PA, so is the sine of the given angle API to the sine of the leg AI. Or,

As radius = 90° 00' = 10.00000 To col. lat. = 37° 00' = 9.90235 So fine ½ diff. long. = 25° 41½' = 9.63702 To fine ½ diff. = 20° 75½' = 9.53937

which doubled, gives 40° 31' for the distance; and this distance, reduced to nautical miles, is 2431; less by 31 than that given by parallel sailing.—For the angle of position, As radius is to the co-sine of the hypotenuse PA, so is the tangent of the given angle API to the co-tangent of the angle A. Or,

As radius = 90° 00' = 10.00000 To fine lat. = 37° 00' = 9.77946 So tang. ½ dif. long. = 25° 41½' = 9.68222 To co-tang. ang. posit. = 73° 51' = 9.46168

Hence it appears, that to sail from A to B, or from B to A, the ship must first steer N. 73° 51' W. or E. and then gradually increase her course till the comes to I, where it will be due west or east; and from thence the course is to be gradually diminished again till she comes to the other port, where it will be 73° 51', the same as she set out with.

CASE IV. When one place has latitude, and the other has none.

EXAMPLE. What is the nearest distance between the island of St Thomas, in lat. 0° 00', and long. 1°? The co-latitude of St Julian is 41° 09'; and the difference of longitude between the two places is 66° 10'.—Let the point A (plate CCI, fig. 7.) be St Thomas, and P and S the north and south poles. Make AC, the measure of the angle ASC, equal to 66° 10' the difference of longitude. Then, as Port St Julian is in south latitude, about S the south pole at the distance of Julian's co-latitude, describe the arc aa; cutting SCP, the meridian of Julian in B, through the points A, B, E, a great circle being described, the arc AB is the distance sought. The distances and angles may now be measured according to the rules of projection, or it falls under a case in spheric trigonometry: for, in the quadrant triangle ASB, there are given the co-latitude of St Thomas or AS = 90° 00'; the co-latitude of St Julian, or SB = 41° 09'; the difference of longitude, or the angle ASB = 66° 10', from whence all the rest may be found. Or, in the supplemental triangle, ACB, right-angled at C, there is given the latitude of St Julian's, or the leg CB = 48° 51'; the difference of longitude, or the leg CA = 66° 10', whence the rest may easily be found; and hence it will appear, that a ship sailing from the island of St Thomas must first shape her course south 51° 22' W.; and then, by constantly altering her course towards the west, so as to arrive at Port St Julian on a course S. 71° 36' W., she will have failed the shortest distance between these places.

CASE V. When the latitudes of the given places are both north, or both south.

EXAMPLE. What is the nearest distance between the Lizard and the island of Bermudas, and also the angles of position?—The difference of longitude of the two places is 58° 11'.

Make PA (Plate CCI. fig. 8.) = 57° 25', the co-latitude of Bermudas; PA = 40° 03', the co-latitude of the Lizard; and with the tangent of PA describe the arc aa. With the secant of 58° 11', the difference of longitude, describe arcs from P and S, which gives the centre of the circle PCS the meridian of the Lizard; its intersection with aa gives B, the place of the Lizard. The arcs of the circle and angles may be measured by spheric trigonometry as before. Had the eastern place, the Lizard, been put upon the primitive circle, the great circle AB would have been difficult to describe; and therefore the western place was put upon it, it being a matter of indifference which of the places are so taken.

CASE VI. When one of the given places has north latitude, and the other has south latitude.

EXAMPLE. What is the nearest distance from the island of St Helena to the island of Bermudas, and also the angles of position at each place; the difference of longitude between the two being 57° 43'?

Make QA (fig. 9.) = 15° 55', the lat. of St Helena; describe the arc aa about P, with the tangent of P = 57° 25', the co-latitude of Bermudas. Arcs described from P, S, with the secant of 57° 43', the difference of longitude, will give the centre of the circle PCS, the meridian of Bermudas; and its intersection B with aa, is the place of Bermudas. Describe a great circle through A, B, D; the intercepted arc AB is the distance sought; and the angles PAB, ABS, are the positions required, which must be measured according to the rules of spheric trigonometry. From the solutions of these triangles it will appear, that when a ship sails from St Helena to Bermudas on the arc of a great circle, she must first shape her course N. 48° 00' W., and gradually alter it from the north towards the west, so as to arrive at Bermudas on a course N 50° 01' west, after having run 73° 26', or 4406 miles. The course found by Mercator's sailing is N. 48° 45' W., and the distance is 4414 miles.—By this it appears

PRACTICE

pears, that when the places are one in N. latitude, and the other in S. latitude; neither of them being very far from the equator, there is but a small difference between the results found by Mercator's and great circle sailing: for, near the equator, the rhumb-lines do not differ much from great circles.

From the solution of the foregoing cases, it is plain, that to sail on the arc of a great circle, the ship must continually alter her course. But as this is a difficulty too great to be admitted into the practice of navigation, it has been thought sufficiently exact to effect this business by a kind of approximation, founded upon this principle, that, in small arcs, the difference between the arc and its tangent is so little, that they may be taken one for the other in any nautical operations. Upon this principle, the great circles of the earth are supposed to be made up of short right lines, each of which is a segment of a rhumb-line. And on this supposition the solution of the following problem is founded.

Having given the latitudes and longitudes of the places sailed from, and bound to; to find the successive latitudes on the arc of a great circle in those places where the alteration in longitude shall be a given quantity; together with the courses and distances between those places.

1. Find the angle of position at each place, and their distance by one of the preceding six cases.

2. Find the greatest latitude the great circle runs through; that is, find the perpendicular from the pole to that circle; and also find the several angles at the pole, made by the given alterations of longitude between this perpendicular and the successive meridians come to.

3. With this perpendicular, and the polar angles severally find as many corresponding latitudes, by saying:

As rad.: tan. greatest lat. :: cos. 1 polar ang.: tan. 1 lat. :: cos. 2 polarang.: tan. 2 lat. &c.

In the triangle PIB.

Given PB = 53° 00' the angle PBI = 73° 09'

To find PI.

Now the angle IPB = \(\frac{53° 27'}{2} = 26° 43\frac{1}{2}'\); the angle IPA = 21° 43\(\frac{1}{2}'\); the angle IPb = 16° 43\(\frac{1}{2}'\);

the angle IPC = 11° 43\(\frac{1}{2}'\); the angle IPd = 6° 43\(\frac{1}{2}'\), are the several polar angles.

Then rad. = 90° 00' 10.00000 10.00000 10.00000 10.00000 To co-tang. PI = 49° 51' 9.92612 9.92612 9.92612 9.92612 So co-sine polar angle = 9.96800 9.98123 9.99084 9.99700 To tang. lat. = 9.89412 9.90735 9.91696 9.92312

Which are 38° 05' 38° 56' 39° 33' 39° 57'

The degrees and min. set over each column, are the polar angles used in that proportion, and the corresponding latitudes stand at bottom.

The first term of these proportions being radius, and the second term constant, the operations may be very expeditiously performed thus.

On a slip of paper let the log. of the second or constant term be written of the same size with the printed figures; apply this log. co-tang. successively to the log. co-sines of the polar angles: Then the sum of the two logs being written down each time, will give the log tangents of the several latitudes arrived at.

By this method, each proportion will be worked by writing down only one line.

Hence it appears, the ship must first sail from the lat. $39^\circ 00'$ N. to lat. $38^\circ 05'$ N.; thence to lat. $38^\circ 56'$ N.; thence to lat. $39^\circ 33'$; thence to lat. $39^\circ 57'$ N.; thence to lat. $40^\circ 09'$ N., which is the greatest latitude she must go to; and from thence she must proceed through the latitudes $39^\circ 57'$, $39^\circ 33'$, $38^\circ$.

| Polar angles | Success. longs. | Success. lats. | Diff. long. | Diff. lat. | Merid. parts | Merid. diff. lat. | Cour. | Dist. | |--------------|----------------|----------------|------------|-----------|-------------|-----------------|-------|-------| | The angle IPB $26^\circ 43\frac{1}{2}'$ | $22^\circ 56'$ | $37^\circ 00'$ | $300$ | $65$ | $2392.6$ | $82.0$ | $74.43$ | $246.6$ | | IPa $21^\circ 43\frac{1}{2}'$ | $27^\circ 56'$ | $38^\circ 05'$ | $300$ | $51$ | $2474.6$ | $65.2$ | $77.44$ | $240.0$ | | IPb $16^\circ 43\frac{1}{2}'$ | $32^\circ 56'$ | $38^\circ 56'$ | $300$ | $37$ | $2539.8$ | $47.8$ | $80.57$ | $235.2$ | | IPc $11^\circ 43\frac{1}{2}'$ | $37^\circ 56'$ | $39^\circ 33'$ | $300$ | $24$ | $2587.6$ | $31.2$ | $84.04$ | $232.2$ | | IPd $6^\circ 43\frac{1}{2}'$ | $42^\circ 56'$ | $39^\circ 57'$ | $300$ | $12$ | $2618.8$ | $15.7$ | $87.46$ | $307.9$ |

In the first column are the angles at the pole contained between the perpendicular and the several meridians differing by $5^\circ$ of longitude.

In the second column, the departed longitude $22^\circ 56'$ being increased by the differences of longitude, make the successive longitudes come to.

In the third column are the successive latitudes passed thro' in sailing from the place set out from to the greatest latitude.

In the fourth and fifth columns are the differences between the longitudes and latitudes in the second and third columns.

In the sixth column are the meridional parts to the successive latitudes; and in the seventh column are the meridional diff. of latitudes.

The eight and ninth columns contain the courses and distances between the places answering to the second and third columns.

The numbers in the third, eighth, and ninth columns, are found by working the logarithmic proportions on a waste paper; but the work is here omitted, as it is so easily supplied.

Now the column of distances being summed up amounts to $1261.9$; which being doubled, gives $2523.8$ miles for the distance between the two places.

And the courses the ship must steer are, 1st, N. $74^\circ 43'W.$; 2nd, N. $77^\circ 44'W.$; 3rd, N. $80^\circ 57'W.$; 4th, N. $82^\circ 04'W.$; 5th, N. $87^\circ 46'W.$; 6th, S. $87^\circ 46'W.$; 7th, S. $84^\circ 04'W.$; 8th, S. $80^\circ 57'W.$; 9th, S. $77^\circ 44'W.$; 10th, S. $74^\circ 43'W.$; and on these courses she must run the respective distances standing against them.

Having now shown the method of solving the different cases of navigation mathematically, and supposing the course of the ship and distance run to be always exactly known, we shall now proceed to give an account of those mechanical methods by which the ship's course is observed, and the frequent variations and errors in it corrected at convenient times.

§ 7. Of the Log-line and Compass.

1. The method commonly made use of for measuring a ship's way at sea, or how far she runs in a given space of time, is by the Log-line, and Half-minute Glass.

2. The log, fig. 3, is generally about a quarter of Plate an inch thick, and five or six inches from the angular CLXTI. point a to the circumference b. It is balanced by a thin plate of lead, nailed upon the arch, so as to swim perpendicularly in the water, with about $\frac{1}{4}$ impressed under the surface. The line is fastened to the log by means of two legs a and b, fig. 2, one of which passes thro' a hole a at the corner, and is knotted on the opposite side; whilst the other leg is attached to the arch by a pin b, fixed in another hole, so as to draw out occasionally. By these legs the log is hung in equilibrium, and the line, which is united to it, is divided into certain spaces, which are in proportion to an equal number of geographical miles, as a half minute or quarter minute is to an hour of time.

3. These spaces are called knots, because at the end of each of them there is a piece of twine with knots in it, interwoven between the strands of the line, which shows how many of these spaces or knots are run out during the half minute. They commonly begin to be counted at the distance of about 10 fathom or 60 feet from the log; that so the log, when it is hove overboard, may be out of the eddy of the ship's wake before they begin to count; and for the more ready discovery of this point of commencement, there is commonly fastened at it a piece of red rag.

4. The log being thus prepared, and hove overboard from the poop, and the line veered out (by the help of a reel (fig. 4.) that turns easily, and about which it is wound) as fast as the log will carry it away, or rather as the ship sails from it, will show, according to the time of veering, how far the ship has run in a given time, and consequently her rate of sailing.

5. A degree of a meridian, according to the exactest measures, contains about 69,545 English miles; and each mile by the statute being 5280 feet, therefore a degree of a meridian will be about 367,200 feet; whence the $\frac{1}{60}$ of that, viz. a minute, or nautical mile, must contain 6120 standard feet; consequently, since $\frac{1}{60}$ minute... minute is the part of an hour, and each knot being the same part of a nautical mile, it follows, that each knot will contain the part of 6120 feet, viz. 51 feet.

6. Hence it is evident, that whatever number of knots the ship runs in half a minute, the same number of miles she will run in one hour, supposing her to run with the same degree of velocity during that time; and therefore it is the general way to heave the log every hour to know her rate of failing: but if the force or direction of the wind vary, and not continue the same during the whole hour; or if there has been more sail set, or any sail handed, that so the ship has run swifter or slower in any part of the hour than she did at the time of heaving the log; then there must be an allowance made accordingly for it, and this must be according to the discretion of the artist.

7. Sometimes, when the ship is before the wind, and there is a great sea setting after her, it will bring home the log, and consequently the ship will fail faster than is given by the log. In this case it is usual, if there be a very great sea, to allow one mile in ten, and less in proportion, if the sea be not so great. But for the generality, the ship's way is really greater than that given by the log; and therefore, in order to have the reckoning rather before than behind the ship, (which is the safest way), it will be proper to make the space on the log-line between knot and knot to consist of 50 feet instead of 51.

8. If the space between knot and knot on the log-line should happen to be too great in proportion to the half-minute glass, viz. greater than 50 feet, then the distance given by the log will be too short; and if that space be too small, then the distance run (given by the log) will be too great: therefore, to find the true distance run in either case, having measured the distance between knot and knot, we have the following proportion, viz.

As the true distance, 50 feet, is to the measured distance; so are the miles of distance given by the log, to the true distance in miles that the ship has run.

Example I. Suppose a ship runs at the rate of 6½ knots in half a minute; but measuring the space between knot and knot, I find it to be 56 feet: Required the true distance in miles.

Making it, As 50 feet is to 56 feet, so is 6.25 knots to 7 knots; I find that the true rate of failing is 7 miles in the hour.

Example II. Suppose a ship runs at the rate of 6½ knots in half a minute; but measuring the space between knot and knot, I find it to be only 44 feet: Required the true rate of failing.

Making it, As 50 feet is to 44 feet, so is 6.5 knots to 5.72 knots; I find that the true rate of failing is 5.72 miles in the hour.

9. Again, supposing the distance between knot and knot on the log-line to be exactly 50 feet, but that the glass is not 30 seconds; then, if the glass require longer time to run than 30 seconds, the distance given will be too great, if estimated by allowing one mile for every knot run in the time the glass runs; and, on the contrary, if the glass requires less time to run than 30 seconds, it will give the distance failed too small. Consequently, to find the true distance in either case, we must measure the time the glass requires to run out (by the method in the following article); then we have the following proportion, viz.

As the number of seconds the glass runs, is to half a minute, or 30 seconds; so is the distance given by the log, to the true distance.

Example I. Suppose a ship runs at the rate of 7½ knots in the time the glass runs; but measuring the glass, I find it runs 34 seconds: Required the true distance failed.

Making it, As 34 seconds is to 30 seconds, so is 7.5 to 6.6; I find that the ship fails at the rate of 6.6 miles an hour.

Example II. Suppose a ship runs at the rate of 6½ knots; but measuring the glass, I find it runs only 25 seconds: Required the true rate of failing.

Making it, As 25 seconds is to 30 seconds, so is 6.5 knots to 7.8 knots; I find that the true rate of failing is 7.8 miles an hour.

10. In order to know how many seconds the glass runs, you may try it by a watch or clock that vibrates seconds; but if neither of these be at hand, then take a line, and to the one end fastening a plummet, hang the other upon a nail or peg, so as the distance from the peg to the centre of the plummet be 39½ inches: then this put into motion will vibrate seconds; i.e. every time it passes the perpendicular, you are to count one second; consequently, by observing the number of vibrations that it makes during the time the glass is running, we know how many seconds the glass runs.

11. If there be an error both in the log-line and half-minute glass, viz. if the distance between knot and knot and the log-line be either greater or less than 50 feet, and the glass runs either more or less than 30 seconds; then the finding out the ship's true distance will be somewhat more complicate, and admit of three cases, viz.

Case I. If the glass runs more than 30 seconds, and the distance between knot and knot be less than 50 feet, then the distance given by the log-line, viz. by allowing 1 mile for each knot the ship fails while the glass is running, will always be greater than the true distance, since either of these errors gives the distance too great. Consequently, to find the true rate of failing in this case, we must first find (by Art. 8.) the distance, on the supposition that the log-line is only wrong, and then with this (by Art. 9.) we shall find the true distance.

Example. Suppose a ship is found to run at the rate of 6 knots; but examining the glass, I find it runs 35 seconds; and measuring the log-line, I find the distance between knot and knot to be but 46 feet: Required the true distance run.

First, (by Art. 8.) We have the following proportion, viz. As 50 feet : 46 feet :: 6 knots : 5.52 knots. Then (by Art. 9.) As 35 seconds : 30 seconds :: 5.52 knots : 4.73 knots. Consequently the true rate of failing is 4.73 miles an hour.

Case II. If the glass be less than 30 seconds, and the place between knot and knot be more than 50 feet; then the distance given by the log will always be less than the true distance, since either of these errors lessens the true distance.

Example. Suppose a ship is found to run at the rate of 7 knots; but examining the glass, I find it runs only 25 seconds; and measuring the space between Part II.

Between knot and knot on the log-line, I find it is 54 feet: Required the true rate of sailing.

Firstly, (by Art. 9.) As 25 seconds : 30 seconds :: 7 knots : 8.4 knots. Then (by Art. 8.) As 50 feet : 54 feet :: 8.4 knots : 9.072 knots. Consequently the true rate of sailing is 9.072 miles an hour.

Case III. If the glass runs more than 30 seconds, and the space between knot and knot be greater than 50 feet; or if the glass runs less than 30 seconds, and the space between knot and knot be less than 50 feet: then, since in either of these two cases the effects of the errors are contrary, it is plain the distance will sometimes be too great, and sometimes too little, according as the greater quantity of the error lies; as will be evident from the following examples.

Example I. Suppose a ship is found to run at the rate of 6½ knots per glass; but examining the glass, it is found to run 36 seconds; and by measuring the space between knot and knot, it is found to be 58 feet: Required the true rate of sailing.

Firstly, (by Art. 8.) As 50 feet : 58 feet :: 9.5 knots : 11.02 knots. Then (by Art. 9.) As 38 seconds : 30 seconds :: 11.02 knots : 8.7 knots. Consequently the ship's true rate of sailing is 8.7 miles an hour.

Example II. Suppose a ship runs at the rate of 6 knots per glass; but examining the glass, it is found to run only 20 seconds; and by measuring the log-line, the distance between knot and knot is found to be but 38 feet: Required the true rate of sailing.

Firstly, (by Art. 8.) As 50 feet : 38 feet :: 6 knots : 4.56 knots. Then (by Art. 9.) As 20 seconds : 30 seconds :: 4.56 knots : 6.84 knots. Consequently the true rate of sailing is 6.83 miles an hour.

But if in this case it happen, that the time the glass takes to run be to the distance between knot and knot, as 30, the seconds in half a minute, is to 50, the true distance between knot and knot; then it is plain, that whatever number of seconds the glass consists of, and whatever number of feet is contained between knot and knot, yet the distance given by the log-line will be the true distance in miles.

12. Though the method of measuring the ship's way by the log-line, described in the foregoing articles, be that which is now commonly made use of; yet it is subject to several errors, and these very considerable. For, first, the half-minute or quarter-minute glasses (by which and the log the ship's way is determined) are seldom or never true, because dry and wet weather have a great influence on them; so that at one time they may run more, and at another time fewer, than 30 seconds; and it is evident that a small error in the glass will cause a sensible one in the ship's way. Again, the chief property of the log is to have it swim upright, or perpendicular to the horizon: but this is too often wanting in logs, because few seamen examine whether it is so or not, and generally take it upon trust, being satisfied if it weigh a little more at the stern than the head. And from this there flows an error in the reckoning; for if the log does not swim upright, it will not hold water, nor remain steady in the place where it is heaved, since the least check in the hand in veering the line will make it come up several feet: this repeated will make the errors become fathoms, and perhaps knots, which, how insignificant forever they appear, are miles and parts of miles, and amount to a good deal in a long voyage.

Another inconvenience attending the log-line is its stretching and shrinking; for when a new line is first used, let it be ever so well stretched upon the deck, and measured as true as possible, yet after wetting it shrinks considerably; and consequently to be the better assured of the ship's way by the log-line, we ought to measure and alter the knots on it every time before we use it; but this is seldom done oftener than once a week, and sometimes not above once or twice in a whole voyage: also when the line is measured to its greatest degree of shrinking, it is generally left there; and when, by much use, it comes to stretch again, it is seldom or never mended, though it will stretch beyond what it first shrunk. These and many other errors, too well known, attending that method of measuring the ship's way by the log-line, plainly accounts for a great many errors committed in reckonings. So it is to be wished, that either this method were improved or amended, or that some other method less subject to error were found out.

13. The meridian and prime vertical of any place cuts the horizon in 4 points, at 90 degrees distance from one another, viz. North, South, East and West; that part of the meridian which extends itself from the place to the north point of the horizon is called the north line; that which tends to the south point of the horizon is called the south line; and that part of the prime vertical which extends towards the right hand of the observer, when his face is turned to the north, is called the east line; and lastly, that part of the prime vertical which tends towards the left hand is called the west line; the four points in which these lines meet the horizon are called the cardinal points.

14. In order to determine the course of the winds, and to discover their various alterations or shifting, each quadrant of the horizon, intercepted between the meridian and prime vertical, is usually divided into eight equal parts, and consequently the whole horizon into thirty-two; and the lines drawn from the place on which the observer standeth, to the points of division in his horizon, are called rhumb-lines; the four principal of which are those described in the preceding article, each of them having its name from the cardinal point in the horizon towards which it tends: the rest of the rhumb-lines have their names compounded of the principal lines on each side of them, as in the figure (Plate CII. No. 1.); and over whichever of these lines the course of the wind is directed, that wind takes its name accordingly.

15. The instrument commonly used at sea for directing the ship's way is called the Mariner's Compass; which consists of a card and two boxes. The card is a circle made to represent the horizon, whose circumference is quartered and divided into degrees, and also into thirty-two equal parts, by lines drawn from the centre to the several points of division, called points of the compass. On the back-side of the card, and just below the south and north line, is fixed a steel needle with a brass cupola, or hollow centre in the middle, which is placed upon the end of a fine pin, upon which the card may easily turn about; the needle is touched with a loadstone, by which a certain virtue is infused into it, that makes it (and consequently the south and north line on the card above it) hang nearly in the plane of the meridian; by which means the south and north The card is represented in No. 1., in which you may observe, that the capital letters N, S, E, W, denote the four cardinal points, viz. N the North, S the South, &c., and the small letter b signifies the word by. The rhumbs in the middle between any two of the cardinals are expressed by the letters denoting these cardinals, that which denotes the point lying in the meridian having the precedence; thus the rhumb in the middle between the north and east is expressed N. E. which is to be read North-east; also S. W. denotes the South-west rhumb, &c.: the other rhumbs are expressed according to their situation with respect to these middle rhumbs and the nearest cardinals, as is plain from the foreaid figure.

17. The card is put into a round box, made for it, having a pin erected in the middle, upon which the hollow centre of the needle is fixed, so as the card may lie horizontal, and easily vibrate according to the motion of the needle: the box is covered over with a smooth glass, and is hung in a brass hoop upon two cylindrical pins, diametrically opposite to one another; and this hoop is hung within another brass circle, upon two pins at right angles with the former. These two circles, and the box, are placed in another square wooden box, so that the innermost box, and consequently the card, may keep horizontal which way forever the ship heels.

18. Since the meridians do all meet at the poles, and there form certain angles with one another; and since, if we move ever so little towards the east or west, from one place to another, we thereby change our meridian, and in every place the east and west line being perpendicular to the meridian; it follows, that the east and west line in the first place will not coincide with the east and west line in the second, but be inclined to it at a certain angle; and consequently all the other rhumb-lines at each place will be inclined to each other, they always forming the same angles with the meridian. Hence it follows, that all rhumbs, except the four cardinals, must be curves or helihpherical lines, always tending towards the pole, and approaching it by infinite gyrations or turnings, but never falling into it. Thus let P (No. 2.) be the pole, EQ an arch of the equator, PE, PA, &c. meridians, and EFGHKL any rhumb: then because the angles PEF, PFG, &c. are by the nature of the rhumb-line equal, it is evident that it will form a curve-line on the surface of the globe, always approaching the pole P, but never falling into it; for if it were possible for it to fall into the pole, then it would follow, that the same line could cut an infinite number of other lines at equal angles, in the same point; which is absurd.

19. Because there are 32 rhumbs (or points in the compass) equally distant from one another, therefore the angle contained between any two of them adjacent will be $11^\circ 15'$, viz. $\frac{1}{32}$ part of $360^\circ$; and so the angle contained between the meridian and the NbE, will be $11^\circ 15'$, and between the meridian and the NNE will be $22^\circ 30'$; and so of the rest, as in the following table.

| North | South | Points | D. M. | North | South | |-------|-------|--------|-------|-------|-------| | N | S | | | | | | NE | SE | | | | | | NNE | SSE | | | | | | NEbN | SEbS | | | | | | NEbE | SEbE | | | | | | E | S | | | | | | EbN | EbS | | | | |

§ 8. Concerning Currents, and how to make proper allowances.

1. Currents are certain settings of the stream, by which all bodies (as ships, &c.) moving therein, are compelled to alter their course or velocity, or both; and submit to the motion impressed upon them by the current.

Case I. If the current sets just with the course of the ship, i.e. moves on the same rhumb with it; then the motion of the ship is increased, by as much as is the drift or velocity of the current.

Example. Suppose a ship sails SEbS at the rate of 6 miles an hour, in a current that sets SEbS 2 miles an hour: Required her true rate of sailing.

Here it is evident that the ship's true rate of sailing will be 8 miles an hour.

Case II. If the current sets directly against the ship's course, then the motion of the ship is lessened by... Example. Suppose a ship sails SSW at the rate of 10 miles an hour, in a current that sets NNE 6 miles an hour; Required the ship's true rate of sailing.

Here it is evident that the ship's true rate of sailing will be 4 miles an hour. Hence it is plain,

Cor. I. If the velocity of the current be less than the velocity of the ship, then the ship will get so much a-head as is the difference of these velocities.

Cor. II. If the velocity of the current be greater than that of the ship, then the ship will fall so much a-stern as is the difference of these velocities.

Cor. III. Lastly, If the velocity of the current be equal to that of the ship, then the ship will stand still; the one velocity destroying the other.

Case III. If the current thwarts the course of the ship, then it not only lessens or augments her velocity, but gives her a new direction compounded of the course she steers, and the setting of the current, as is manifest from the following

Lemma. If a body at A (No 26.) be impelled by two forces at the same time, the one in the direction AB capable to carry that body from A to B in a certain space of time, and the other in the direction AD capable to carry it from A to D in the same time; complete the parallelogram ABCD, and draw the diagonal AC; then the body at A, agitated by these two forces together, will move along the line BC, and will be in the point C at the end of the time in which it would have moved along AD or AB with the forces separately applied.

Hence the solution of the following examples will be evident.

Example I. Suppose a ship sails (by the compass) directly south 96 miles in 24 hours, in a current that sets east 45 miles in the same time: Required the ship's true course and distance.

Geometrically. Draw AD (see No 26.) to represent the south and north line of the ship at A, which make equal to 96; from D draw DC perpendicular to AD, equal to 45; and join AC. Then C will be the ship's true place, AC her true distance, and the angle CAD the true course. To find which,

By Calculation;

First, For the true course DAC, it will be, (by rectangular trigonometry),

As the apparent distance AD = 96 = 1.98227 is to the current's motion DC = 45 = 1.65321 so is radius = 10.00000 to the tangent of the true course DAC = 25° 07' 9.67094 consequently the ship's true course is S 25° 07' E, or SSE 2° 37' easterly.

Then for the true distance AC, it will be, (by rectangular trigonometry),

As the fine of the course A = 25° 07' = 9.62784 is to the departure DC = 45 = 1.65321 so is radius = 10.00000 to the true distance AC = 106 = 2.02537

Example. Suppose a ship sails SE 120 miles in 20 hours, in a current that sets WNW at the rate of 2 miles an hour: Required the ship's true course and distance sailed in that time.

Geometrically. Having drawn the compasses NESW (No 27.) let C represent the place the ship sails from; draw the SE line CA, which make equal to 120; then will A be the place the ship caped at.

From A draw AB parallel to the WNW line CD, equal to 40, the motion of the current in 20 hours, and join CB; then B will be the ship's true place at the end of 20 hours, CB her true distance, and the angle SCB her true course. To find which,

By Calculation;

In the triangle ABC, are given CA 120, AB 40, and the angle CAB equal to 34° 45', the distance between the EB and SE lines, to find the angles B and C, and the side CB.

First, For the angles C and B, it will be, (by oblique trigonometry),

As the sum of the sides CA and AB 160 = 2.04112 is to their difference = 80 = 1.90309 so is the tang. of half the sum of the angles B and C = 73° 07' 10.51783 to the tang. of half their diff. = 59° 45' 10.21680 consequently the angle B will be 131° 52', and the angle ACB 14° 23'. Hence the true course is S 30° 37' E, or SSE 2° 07' easterly.

Then for the true distance CB, it will be, (by oblique trigonometry),

As the fine of B = 131° 52' = 9.87193 is to AC = 120 = 2.07918 so is the fine of A = 33° 45' = 9.74474 to the true distance CB = 89.53 = 1.95194

Example III. Suppose a ship coming out from sea in the night, has sight of Scilly light, bearing NE by N distance 4 leagues, it being then flood tide setting ENE 2 miles an hour, and the ship running after the rate of 5 miles an hour: Required upon what course and how far she must sail to hit the Lizard, which bears from Scilly E by S distance 17 leagues.

Geometrically. Having drawn the compasses NESW (No 28.) let A represent the ship's place at sea, and draw the NE by N line AS, which make equal to 12 miles; so S will represent Scilly.

From S draw SL equal to 51 miles, and parallel to the E by S line; then L will represent the Lizard.

From L draw LC parallel to the ENE line, equal to 2 miles, and from C draw CD equal to 5 miles meeting AL in D; then from A draw AB parallel to CD meeting LC produced in B; and AB will be the required distance, and SAB the true course. To find which,

By Calculation;

In the triangle ASL are given the side AS equal to 12 miles, the side SL equal to 51, and the angle ASL equal to 118° 07', the distance between the NE by N and W by N lines; to find the angles SAL and SLA. Consequently, (by oblique trigonometry), it will be,

As the sum of the sides AS and SL = 63 = 1.79934 is to their difference = 39 = 1.59106 so is the tang. of half the sum of the angles SAL and SLA = 30° 56' 9.77763 to the tang. of half their diff. = 20° 21' 9.56935 consequently the angle SAL, will be 51° 17'; and to the direct bearing of the Lizard from the ship will be N 85° 02' E, or E by N 6° 17' E; and for the distance AL, it will be, (by oblique trigonometry),

30 H 2 As As the sine of SAL - 51° 17' - 9.8223 is to SL - 51° - 1.70757 so is the sine of ASL - 118° 07' - 9.94546 to AL - 57° 05' - 1.76080 the distance between the ship and the Lizard.

Again, in the triangle DLC, are given the angle L equal to 17° 32', the distance between the ENE and N 85° 02' E lines; the side LC, equal to 12 miles, the current's drift in an hour; and the side CD, equal to 5 miles, the ship's run in the same time. Hence for the angle D, it will be, (by oblique trigonometry), As the ship's run in 1 hour DC - 5° - 0.6987 is to the sine of L - 17° 32' - 9.47894 so is the current's drift LC - 2° - 0.30103 to the sine of D - 6° 55' - 9.08100 consequently, since by construction the angle LAB is equal to the angle LDC, the course the ship must steer is S 88°, 03' E.

Then for the distance AB, it will be, (by oblique trigonometry), As the sine of B - 155° 33' - 9.61689 is to AL - 57° 05' - 1.76080 so is the sine of L - 17° 32' - 9.47894 to AB - 41° 96' - 1.62285 consequently, since the ship is sailing at the rate of 5 miles an hour, it follows, that in sailing 8h 24m S 88° 03' E, she will arrive at the Lizard.

Example IV. A ship from a certain headland in the latitude of 34° 00' north, sails SE/S 12 miles in 3 hours, in a current that sets between north and east; and then the same headland is found to bear WNW, and the ship to be in the latitude of 33° 52' north: Required the setting and drift of the current.

Geometrically. Having drawn the compass NESW (No. 29.) let A represent the place of the ship, and draw the SE/S line AB, equal to 12 miles, also the ESE line AC.

Set off from A upon the meridian AD, equal to 8 miles, the difference of latitude, and through D draw DC parallel to the east and west line WE, meeting AC in C. Join C and B with the right line BC; then C will be the ship's place, the angle ABC the setting of the current from the SE/S line, and the line BC will be the drift of the current in 3 hours.

To find which,

By Calculation;

In the triangle ABC, right-angled at D, are given the difference of latitude AD equal to 8 miles, the angle DAC equal to 67° 30'. Whence for AC, the distance the ship has sailed, it will be

As radius - 10.00000 is to the diff. of latitude AD - 8° - 0.90309 so is the secant of the course DAC - 67° 30' - 10.41716 to the distance run AC - 20° 9' - 1.32025

Again, in the triangle ABC, are given AB equal to 12 miles, AC equal to 20° 9', and the angle BAC equal to 30° 45', the distance between the SE/S and ESE lines. Whence, for the angle at B, it will be,

As the sum of the sides AC and AB - 32° 9' - 1.51720 is to their difference - 8° 9' - 0.94930 so is the tang. of half the sum of the angles B and C - 73° 07' - 10.51806 to tang. of ½ their diff. - 41° 43' - 9.95025 consequently the angle B is 114° 51', and so the setting of the current will be N 81° 06' E, or E/E 2° 21' E. Then for BC, the current's drift in 3 hours, it will be,

As the sine of B - 114° 51' - 9.92700 is to the distance run AC 20° 9' - 1.32025 so is the sine of A - 33° 45' - 9.74474 to BC - 12° 8' - 1.10719 the current's drift in 3 hours; and consequently the current sets E/N 2° 21' E 4.266 miles an hour.

§. 9 Concerning the Variation of the Compass, and how to find it from the true and observed Amplitudes or Azimuths of the sun.

1. The variation of the compass is how far the north or south point of the needle stands from the true north or south point of the horizon towards the east or west; or it is an arch of the horizon intercepted between the meridian of the place of observation and the magnetic meridian.

2. It is absolutely necessary to know the variation of the compass at sea, in order to correct the ship's course; for since the ship's course is directed by the compass, it is evident that if the compass be wrong, the true course will differ from the observed, and consequently the whole reckoning differ from the truth.

3. The sun's true amplitude is an arch of the horizon comprehended between the true east or west point thereof, and the centre of the sun at rising or setting; or it is the number of degrees, &c. that the centre of the sun is distant from the true east or west point of the horizon, towards the south or north.

4. The sun's magnetic amplitude is the number of degrees that the centre of the sun is from the east or west point of the compass, towards the south or north point of the same at rising or setting.

5. Having the declination of the sun, together with the latitude of the place of observation, we may from thence find the sun's true amplitude, by the following astronomic proposition, viz.

As the co-sine of the latitude is to the radius So is the sine of the sun's declination to the sine of the sun's true amplitude which will be north or south according as the sun's declination is north or south.

Example. Required the sun's true amplitude in the latitude of 41° 50' north, on the 23d day of April 1731.

First, I find (from the tables of the sun's declination) that the sun's declination the 23d of April is 15° 54' north; then for the true amplitude, it will be, by the former analogy,

As the co-sine of the lat. 41° 50' - 9.87221 is to radius - 10.00000 so is the sine of the decl. 15° 54' - 9.43769 to the sine of the ampl. 21° 35' - 9.50548 which is north, because the declination is north at that time; and consequently, in the latitude of 40° 50' north, the sun rises on the 23d of April 21° 35' from the east part of the horizon towards the north, and sets so much from the west the same way.

6. The sun's true azimuth is the arch of the horizon intercepted between the meridian and the vertical circle passing through the centre of the sun at the time of observation. The sun's magnetic azimuth is the arch of the horizon, intercepted between the magnetic meridian and the vertical, passing through the sun.

8. Having the latitude of the place of observation, together with the sun's declination and altitude at the time of observation, we may find his true azimuth after the following method, viz.

Make it,

As the tangent of half the complement of the latitude is to the tangent of half the sum of the distance of the sun from the pole and complement of the altitude

So is the tangent of half the difference between the distance of the sun from the pole and complement of the altitude

To the tangent of a fourth arch which fourth arch added to half the complement of the latitude will give a fifth arch, and this fifth arch lessened by the complement of the latitude will give a sixth arch.

Then make it,

As the radius is to the tangent of the altitude so is the tangent of the fifth arch to the co-fine of the sun's azimuth which is to be counted from the south or north, to the east or west, according as the sun is situated with respect to the place of observation.

If the latitude of the place and declination of the sun be both north or both south, then the declination taken from 90° will give the sun's distance from the pole; but if the latitude and declination be on contrary sides of the equator, then the declination added to 90° will give the sun's distance from the nearest pole to the place of observation.

Example. In the latitude of 51° 32' north, the sun having 10° 39' north declination, its altitude was found by observation to be 38° 18': Required the azimuth.

By the first of the foregoing analogies, it will be

As the tangent of \(\frac{1}{2}\) the complement of the latitude \(19^\circ\), \(14'\) \(9.54269\)

is to the tangent of \(\frac{1}{2}\) the sum of the distance of the sun from the pole and complement of the altitude \(61^\circ\), \(01'\) \(10.25655\)

so is the tangent of half their difference \(9^\circ\), \(19'\) \(7.21499\)

to the tangent of a 4th arch \(40^\circ\), \(20'\) \(9.92885\)

which fourth arch \(40^\circ\), \(20'\), added to \(19^\circ\), \(14'\), half the complement of the latitude, gives a fifth arch \(59^\circ\), \(34'\); and this fifth arch lessened by \(38^\circ\), \(28'\), the complement of the latitude, gives the sixth arch \(21^\circ\), \(06'\); then for the azimuth, it will be, by the second of the preceding analogies,

As radius \(10.00000\)

is to the tangent of the altitude \(38^\circ\), \(18'\) \(9.89749\)

so is the tangent of the sixth arch \(21^\circ\), \(06'\) \(9.58644\)

to the co-fine of the azimuth \(72^\circ\), \(15'\) \(9.48393\)

which, because the latitude is north and the sun south of the place of observation, must be counted from the south towards the east or west; and consequently, if the altitude of the sun was taken in the morning, the azimuth will be \(S\) \(72^\circ\), \(15'\) \(E\), or ESE \(4^\circ\), \(45'\) \(E\); but if the altitude was taken in the afternoon, the azimuth will be \(S\) \(72^\circ\), \(15'\) \(W\), or WSW \(4^\circ\), \(45'\) \(W\).

9. Having found the sun's true amplitude or azimuth by the preceding analogies, and his magnetic amplitude or azimuth by observation, it is evident, if they agree, there is no variation; but if they disagree, then if the true and observed amplitudes at the rising or setting of the sun be both of the same name, i.e., either both north, or both south, their difference is the variation; but if they be of different names, i.e., one north and the other south, their sum is the variation. Again, if the true and observed azimuth be both of the same name, i.e., either both east or both west, their difference is the variation; but if they be of different names, their sum is the variation: And to know whether the variation is easterly, observe this general rule, viz.

Let the observer's face be turned to the sun: then if the true amplitude or azimuth be to the right-hand of the observed, the variation is easterly; but if be to the left, westerly.

To explain which, Let NESW (No 3c.) represent Plate CCIII a compass, and suppose the sun is really E&N at the time of observation, but the observer sees him off the east point of the compass, and so the true amplitude or azimuth of the sun is to the right of the magnetic or observed; here it is evident that the E&N point of the compass ought to lie where the east point is, and so the north where the N&W is; consequently the north point of the compass is a point too far east, i.e., the variation in this case is easterly. The same will hold when the amplitude or azimuth is taken on the west side of the meridian.

Again, let the true amplitude or azimuth be to the left-hand of the observed. Thus, suppose the sun is really E&N at the time of observation, but the observer sees him off the east point of the compass, and so the true amplitude or azimuth to the left of the observed: Here it is evident that the E&N point of the compass ought to stand where the east point is, and so the north where the N&E point is; consequently the north point of the compass lies a point too far westerly; so in this case the variation is west. The same will hold when the sun is observed on the west side of the meridian.

Example I. Suppose the sun's true amplitude at rising is found to be \(E\) \(14^\circ\), \(20'\) \(N\), but by the compass it is found to be \(E\) \(26^\circ\), \(12'\): Required the variation, and which way it is.

Since they are both the same way, therefore

From the magnetic amplitude \(E\) \(26^\circ\), \(12'\) \(N\),

take the true amplitude \(E\) \(14^\circ\), \(20'\) \(N\),

and there remains the variation \(11^\circ\), \(52'\) \(E\),

which is easterly, because in this case the true amplitude is the right of the observed.

Example II. Suppose the sun's true amplitude at setting is \(W\) \(34^\circ\), \(26'\) \(S\), and his magnetic amplitude \(W\) \(23^\circ\), \(13'\) \(S\): Required the variation, and which way it is.

Since they lie both the same way, therefore

From the sun's true amplitude \(W\) \(43^\circ\), \(26'\) \(S\),

take his magnetic amplitude \(W\) \(23^\circ\), \(13'\) \(S\),

there remains the variation \(11^\circ\), \(13'\) \(W\),

which

PRACTICE which is westerly, because the true amplitude, in this case, is to the left-hand of the observed.

EXAMPLE III. Suppose the sun's true altitude at rising is found to be $13^\circ 24'$ N., and his magnetic E $12^\circ 32'$ S.: Required the variation, and which way it lies.

Since the true and observed amplitudes lie different ways, therefore

To the true amplitude - $E 13^\circ 24'$ N. add the magnetic amplitude - $E 12^\circ 32'$ S.

the sum is the variation - $25^\circ 56'$ W. which is westerly, because the true amplitude is, in this case, to the left of the observed.

EXAMPLE IV. Suppose the sun's true altitude at setting is found to be W $8^\circ 24'$ N., but his magnetic amplitude is W $10^\circ 13'$ S.: Required the variation.

To the true amplitude - $W 8^\circ 24'$ N. add the magnetic - $W 10^\circ 13'$ S.

the sum is the variation - $18^\circ 37'$ E. which is easterly, because the true amplitude is to the right of the observed.

EXAMPLE V. Suppose the sun's true azimuth at the time of observation is found to be N $86^\circ 40'$ E., but by the compass it is N $73^\circ 24'$ E.: Required the variation, and which way it lies.

From the sun's true azimuth - N $86^\circ 40'$ E. take the magnetical - N $73^\circ 24'$ E.

there remains the variation - $13^\circ 16'$ E. which is easterly, because the true azimuth is to the right of the observed.

EXAMPLE VI. Suppose the sun's true azimuth is S $3^\circ 24'$ E., and the magnetical S $4^\circ 36'$ W.: Required the variation, and which way it lies.

To the true azimuth - S $3^\circ 24'$ E. add the magnetical azimuth - S $4^\circ 36'$ W.

the sum is the variation - $8^\circ 00'$ W. which is westerly, because the true azimuth is, in this case, to the left of the observed.

The variation of the compass was first observed at London, in the year 1580, to be $11^\circ 15'$ easterly, and in the year 1622 it was $6^\circ 05'$ E.; also in the year 1634, it was $4^\circ 05'$ E. still decreasing, and the needle approaching the true meridian, till it coincided with it; and then there was no variation; after which, the variation began to be westerly; and in the year 1672, it was observed to be $2^\circ 30'$ W.; also in the year 1683, it was $4^\circ 30'$ W.; and since that time the variation still continues at London to increase westerly; but how far it will go that way, time and observations will probably be the only means to discover.

Again, at Paris, in the year 1640, the variation was $3^\circ 00'$ E.; and in the year 1666, there was no variation; but in the year 1681, it was $2^\circ 30'$ W. and still continues to go westerly.

In short, from observations made in different parts of the world, it appears, that in different places the variation differs both as to its quantity and denomination, it being east in one place, and west in another; the true cause and theory of which, for want of a sufficient number of observations, has not as yet been fully explained.

§ 10. The Method of keeping a Journal at sea; and how to correct it, by making proper allowance for the Lee-way, Variation, &c.

1. Lee-way is the angle that the rhumb-line, upon which the ship endeavours to fall, makes with the rhumb she really falls upon. This is occasioned by the force of the wind or surge of the sea, when she lies to the windward, or is clove-hauled, which causes her to fall off and glide sideways from the point of the compass she capes at. Thus let NESW (N° 31.) represent the compass; and suppose a ship at C capes at, or endeavours to fall upon, the rhumb Ca; but by the force of the wind, and surge of the sea, she is obliged to fall off, and make her way good upon the rhumb Ch; then the angle aCb is the lee-way; and if that angle be equal to one point, the ship is said to make one point lee-way; and if equal to two points, the ship is said to make two points lee-way, &c.

The quantity of this angle is very uncertain, because some ships, with the same quantity of sail, and with the same gale, will make more lee-way than others; it depending much upon the mould and trim of the ship, and the quantity of water that she draws. The common allowances that are generally made for the lee-way, are as follow.

1. If a ship be clove hauled, has all her sails set, the water smooth, and a moderate gale of wind, she is then supposed to make little or no lee-way.

2. If it blow so fresh as to cause the small sails be handed, it is usual to allow one point.

3. If it blow so hard that the top-falls must be clove reefed, then the common allowance is two points for lee-way.

4. If one top-fall must be handed, then the ship is supposed to make between two and three points lee-way.

5. When both top-sails must be handed, then the allowance is about four points for lee-way.

6. If it blows so hard as to occasion the fore-course to be handed, the allowance is between 5½ and 6 points.

7. When both main and fore-courses must be handed, then 6 or 6½ points are commonly allowed for lee-way.

8. When the mizen is handed, and the ship is trying a-hull, she is then commonly allowed about 7 points for lee-way.

3. Though these rules are such as are generally made use of, yet since the lee-way depends much upon the mould and trim of the ship, it is evident that they cannot exactly serve to every ship; and therefore the best way is to find it by observation. Thus, let the ship's wake be set by a compass in the poop, and the opposite rhumb is the true course made good by the ship; then the difference between this and the course given by the compass in the binnacle, is the lee-way required. If the ship be within sight of land; then the lee-way may be exactly found by observing a point on the land which continues to bear the same way, and the distance between the point of the compass Part II.

Practice pafs it lies upon, and the point the ship capes at, will be the lee-way. Thus, suppose a ship at C, is lying up NWb, towards A; but instead of keeping that course, she is carried on the NNE line CB, and consequently the point B continues to bear the same way from the ship: Here it is evident, that the angle ACB, or the distance between the NAW line that the ship capes at, and the NNE line that the ship really sails upon, will be the lee-way.

4. Having the course steered, and the lee-way given; we may from thence find the true course by the following method, viz. Let your face be turned directly to the windward; and if the ship have her larboard tacks on board, count the lee-way from the course steered towards the right hand; but if the starboard tacks be on board, then count it from the course steered towards the left hand. Thus, suppose the wind at north, and the ship lies up within 6 points of the wind, with her larboard tacks on board, making one point lee-way; here it is plain, that the course steered is ENE, and the true course ESN; also suppose the wind is at NNW, and the ship lies up within 6½ points of the wind, with her starboard tack on board, making 1½ point lee-way; it is evident that the true course, in this case, is WSW.

5. We have shewed, in the last section, how to find the variation of the compass; and from what has been said there, we have this general rule for finding the ship's true course, having the course steered and the variation given, viz. Let your face be turned towards the point of the compass upon which the ship is steered; and if the variation be easterly, count the quantity of it from the course steered towards the right hand; but if westerly, towards the left hand; and the course thus found is the true course steered. Thus, suppose the course steered is NNE, and the variation one point easterly; then the true course steered will be NNE: Also suppose, the course steered is NESE, and the variation one point westerly; then, in this case, the true course will be NE; and so of others.

Hence, by knowing the lee-way variation, and course steered, we may from thence find the ship's true course; but if there be a current under foot, then that must be tried, and proper allowances made for it, as has been shown in the section concerning Currents, from thence to find the true course.

6. After making all the proper allowances for finding the ship's true course, and making as just an estimate of the distance as we can; yet by reason of the many accidents that attend a ship in a day's running, such as different rates of sailing between the times of heaving the log; the want of due care at the helm by not keeping her steady, but suffering her to yaw and fall off; sudden storms, when no account can be kept, &c.; the latitude by account frequently differs from the latitude by observation; and when that happens, it is evident there must be some error in the reckoning; to discover which, and where it lies, and also how to correct the reckoning, you may observe the following rules.

1st, If the ship fail near the meridian, or within 2 or 2½ points thereof; then if the latitude by account disagrees with the latitude by observation, it is most likely that the error lies in the distance run; for it is plain, that in this case it will require a very sensible error in the course to make any considerable error in the difference of latitude, which cannot well happen if due care be taken at the helm, and proper allowances be made for the lee-way, variation, and currents. Consequently, if the course be pretty near the truth, and the error in the distance run regularly through the whole, we may, from the latitude obtained by observation, correct the distance and departure by account, by the following analogies, viz.

As the difference of latitude by account is to the true difference of latitude, So is the departure by account to the true departure, And so is the direct distance by account to the true direct distance.

The reason of this is plain: for let AB (No. 33.) denote the meridian of the ship at A, and suppose the ship sails upon the rhumb AE near the meridian, till by account she is found in C, and consequently her difference of latitude by account is AB; but by observation she is found in the parallel ED, and so her true difference of latitude is AD, her true distance AE, and her true departure DE; then, since the triangles ABC ADE are similar, it will be AB : AD :: BC : DE, and AB : AD :: AC : AE.

Example. Suppose a ship from the latitude of 45° 20' north, after having sailed upon several courses near the meridian for 24 hours, her difference of latitude is computed to be upon the whole 95 miles southerly, and her departure 34 miles easterly; but by observation she is found to be in the latitude of 43° 10' north, and consequently her true difference of latitude is 130 miles southerly: then for the true departure, it will be, As the difference of latitude by account 95 is to the true difference of latitude 130; so is the departure by account 34 to the true departure 46.52, and so is the distance by account 100.9 to the true distance 138.

2dly, If the courses are for the most part near the parallel of east and west, and the direct course be within 5½ or 6 points of the meridian; then if the latitude by account differs from the observed latitude, it is most probable that the error lies in the course or distance, or perhaps both; for in this case it is evident, the departure by account will be very nearly true; and thence by the help of this, and the true difference of latitude, may the true course and direct distance be readily found by Cafe 4. of Plane Sailing.

Example. Suppose a ship from the latitude of 45° 50' north, after having sailed upon several courses near the parallel of east and west, for the space of 24 hours, is found by dead reckoning to be in the latitude of 42° 45' north, and to have made 160 miles of wetting; but by a good observation the ship is found to be in the latitude of 45° 35' north: Required the true course, and direct distance sailed.

With the true difference of latitude 75 miles, and departure 160 miles, we shall find (by Cafe 4. of Plane Sailing) the true course to be S. 64° 53' W. and the direct distance 176.7 miles.

3dly, If the courses are for the most part near the middle of the quadrant, and the direct course within 2 and 6 points of the meridian; then the error may be either in the course or in the distance, or in both, which. which will cause an error both in the difference of latitude and departure; to correct which, having found the true difference of latitude by observation, with this, and the direct distance by dead reckoning, find a new departure (by Case 3. of Plane Sailing); then half the sum of this departure, and that by dead reckoning, will be nearly equal to the true departure; and consequently with this, and the true difference of latitude, we may (by Case 4. of Plane Sailing) find the true course and distance.

Example. Suppose a ship from the latitude of 44° 38' north, sails between south and east upon several courses, near the middle of the quadrant, for the space of 24 hours, and is then found by dead reckonings to be in the latitude of 42° 15' north, and to have made of easting 136 miles; but by observation she is found to be in the latitude of 42° 04' north: Required her true course and distance.

With the true distance of latitude 154 miles, and the direct distance by dead reckoning 197.4, you will find (by Case 3. of Plane Sailing) the new departure to be 123.4, and half the sum of this and the departure by dead reckoning will be 123.7 the true departure; then with this, and the true difference of latitude, you will find (by Case 4. of Plane Sailing), the true course to be S. 39° 00' E. and the direct distance 198.2 miles.

7. In keeping a ship's reckoning at sea, the common method is to take from the log-book the several courses and distances sailed by the ship last 24 hours, and to transfer these together with the most remarkable occurrences into the log-book, into which also are inserted the courses corrected, and the difference of latitude and difference of longitude made good upon each; then the whole day's work being finished in the log-book, if the latitude by account agree with the latitude by observation, the ship's place will be truly determined; if not, then the reckoning must be corrected according to the preceding rules, and placed in the journal.

The form of the Log-book and Journal, together with an example of 2 days work, you have here subjoined.

Note, to express the days of the week, they commonly use the characters by which the sun and planets are expressed, viz. ☀ denotes Sunday, ⚛ Monday, ⚜ Tuesday, ⚝ Wednesday, ⚞ Thursday, ⚟ Friday, and ⚠ denotes Saturday.

§ II. The Form of the Log-Book, with the Manner of working Days Works at Sea.

| The Log-Book | |--------------| | H. K. ½ K. Courses. Winds. Observations and Accidents. Day of | | 1 North Fair weather, at four this afternoon I took my departure from the Lizard, in the latitude of 5° 00' north, it bearing NNE, distance five leagues. | | 2 SWbS NbE | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | | 10 SSW Ebs The gale increasing and being under all our sails. After three this morning, frequent showers with thick weather till near noon. | | 11 | | 12 | | 13 | | 14 | | 15 | | 16 | | 17 | | 18 | | 19 | | 20 | | 21 | | 22 | | 23 | | 24 | | 25 | | 26 | | 27 | | 28 | | 29 | | 30 | | 31 |

Hence the ship, by account, has come to the latitude of 47° 46' north, and has differed her longitude 2° 5' westerly; so this day I have made my way good S. 31° 31' W. distance 157.4 miles.

At noon the Lizard bore from me N. 31° 31' E. distance 157.4 miles; and having observed the latitude, I found it agreed with the latitude by account. ### The Log-Book

| H.K. | K. | Courses | Winds | Observations and Accidents | Day of | |------|----|---------|-------|---------------------------|--------| | 1 | 2 | SSW | W | This 24 hours, strong gale of wind and fore courses, lee-way 6 points. | | | 2 | 1 | | | | | | 3 | 1 | | | | | | 4 | 1 | | | | |

| SEbE | 32°5 | 17°8 | 37°7 | | ESE | 6 | 2°3 | 10°6 | | SbE | 9 | 8°9 | 1°3 |

| Diff. Lat. | Diff. Long. | |------------|-------------| | N | S | | 29°0 | 49°6 |

Hence the ship, by account, has come to the latitude of 47° 17' north, and has differed her longitude 49° easterly; consequently she has got 1° 16' to the westward of the Lizard, and has made her way good the last 24 hours 8.49° 08' E, distance 44.3 miles.

At noon the Lizard bore from me north 17° 7' east, distance 170.6 miles.

This day I had an observation, and found the latitude by account to disagree with the latitude by observation by 11 minutes, I being so much further to the southward than by dead reckoning, which by the third of the preceding rules I correct as in the Journal.

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**A Journal from the Lizard towards Jamaica in the ship Neptune, J. M. commander.**

| Week | Months | Month Days | Years | Days | Winds | Direct Course | Diff. Miles | Latitude Corrected | Whole Diff. Long. made | Bearing and Diff. from the Lizard | Remarkable Observations and Accidents | |------|--------|------------|------|------|-------|--------------|-------------|--------------------|--------------------------|----------------------------------|-------------------------------------| | D | | | | | NbE | S 31, 31 W | 157.4 | 47°, 46' | 2°, 5' W | At noon the Lizard bore N. P.M. I took my departure from the Lizard, it bearing NNE distance 5 leagues. | Fair weather at four | | | | | | | EbS | | | | | | | | | | | | | NNE | | | | | | | | | | | | | ENE | | | | | | | | | | | | | NEbE | | | | | | | | | | | | | Weft | S 34, 01 E | 8.2 | 47°, 06' | 1°, 55' W | At noon the Lizard bore S. 17° 55' W. Diff. 183 miles. | Strong gales of wind and variable. |

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Vol. VII. NAU

Navigation

Inland Navigation. See Canal, and Trade.