Navigation is the art of conducting a ship from one port or place to another.

HISTORY.

The profane poets refer the invention of the art of navigation to their heathen deities, though historians ascribe it to the Eginetans, the Phoenicians, the Tyrians, and the ancient inhabitants of Britain. Scripture refers the origin of so useful an invention to God himself, who gave the first specimen of navigation in the ark built by Noah under his direction.

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

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

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

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

The fall of Rome and its empire drew along with it not only the overthrow 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 braver amongst 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, than they began to learn the advantages of navigation and commerce, and the methods of managing them, from the people they had subdued; and this with so much success, that in a little time some of them became able to give new lessons, and set History on foot new institutions, for its advantage. Thus it is to the Lombards that 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, although the Italians seem to have the justest title to this distinction, and are accordingly regarded as the restorers of navigation, 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 that 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 these well screened, and almost inaccessible, the residence of some fishermen, who here supported themselves by a little trade in fish and salt, which they found in some of these islands. Thither the Veneti, a people inhabiting that part of Italy which stretches along the coasts of the gulf, retired, when Alaric king of the Goths, and afterwards Attila king of the Huns, ravaged Italy.

These new islanders, little imagining that this was to be their fixed residence, did not think of composing any body politic; but each of the seventy-two islands of this little archipelago continued a long time under its separate master, and each formed a distinct commonwealth. When their commerce had become considerable enough to occasion jealousy to their neighbours, they began to think of uniting into a body; and it was this union, first begun in the sixth century, but not completed till the eighth, that laid the sure foundation of the future grandeur of the state of Venice. From the time of this union, their fleets of merchantmen were sent to all the ports of the Mediterranean; and at last to those of Egypt, particularly Cairo, a new city, built by the Saracen princes, on the eastern bank of the Nile, where they traded for the spices and other products of the Indies. Thus they flourished, and increased their commerce, their navigation, and their conquests, till the league of Cambrai in 1508, when a number of jealous princes conspired to bring about 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. But jealousy soon began to break out; and the two republics coming to an open rupture, there was almost continual war for three centuries before the superiority was ascertained, when, towards the end of the fourteenth century, the battle of Chioga ended the strife; the Genoese, who till then had usually the advantage, having now lost all, and the Venetians, almost become desperate, having, by 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 of which it was capable till the discovery of the East and West Indies, but also formed a new scheme of laws for its regulation, which still obtain under the name of Uses and Customs of the Sea.

This society is that famous league of the Hanse Towns, commonly supposed to have been instituted about the year 1164. See Hanse Towns. For the modern state of navigation in England, Holland, France, Spain, Portugal, &c., see the articles Commerce, Company, &c.

We shall only add, that in examining the causes of commerce passing successively from the Venetians, Genoese, and Hanse Towns, to the Portuguese and Spaniards, and from these again to the English and Dutch, it may be established as a maxim, that the relation between commerce and navigation, or their union, is so intimate, that the fall of the one inevitably draws after it that of the other; and that they will always either flourish or decline together. Hence so many laws, ordinances, statutes, and edicts 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 was long considered as the standing rule, not only of the British amongst themselves, but also as that of other nations with whom they trafficked.

The art of navigation has been exceedingly improved in modern times, both with regard to the form of the vessels themselves, and also with respect to the methods of working them. The use of rowers is now entirely superseded by the improvements made in the formation of the sails, rigging, &c., by which means ships can not only sail much faster than formerly, but can tack in any direction with the greatest facility. It is also very probable that the ancients were neither so well skilled in finding the latitudes, nor in steering their vessels in places of difficult navigation, as the moderns. But the greatest advantage which the moderns possess over the ancients consists in the mariner's compass, by which they are enabled to find their way with more facility in the midst of an immeasurable ocean, than 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 imagine to have been 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 it had once been commonly known to any nation, would have been forgotten, or perfectly concealed from such a prudent people as the Romans, who were so deeply interested in the discovery of it.

Amongst those who admit that the mariner's compass is a modern invention, it has been much disputed who was the inventor. Some attribute the honour of the discovery to Flavio Gioia of Amalfi in Campania, who lived about the beginning of the fourteenth century; whilst others contend 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 Visco, brother to the king of Portugal. In the year 1485, Roderick and Joseph, physicians to John II. king of Portugal, together with one Martin de Bohemia, a Portuguese native of the island of Fayal, and scholar of 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 both by Columbus and by Sebastian Cabot. The former certainly did observe the variation without having heard of it from any other person, on the 14th of September 1492, and it is very probable that Cabot might have done 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 has since been observed that the variation alters in time. The use of the cross staff now began to be introduced amongst sailors. This ancient instrument is described by John Werner of Nuremberg, in his annotations on the first book of Ptolemy's Geography, printed in the year 1514. He recommends it for observing the distance between the moon and some star, in order thence to determine the longitude.

At this time the art of navigation was very imperfect, on account of the inaccuracies of the plane chart, which was the only one then known, and which, by its gross errors, must have greatly misled the mariner, especially in voyages far distant from the equator. Its precepts were probably at first only set down on the sea charts, as is the custom at this day; but at length two Spanish treatises were published in the year 1543, one by Pedro de Medina, and the other by Martin Cortes, which contained a complete system of the art, as far as it was then known. These seem to have been the oldest writers who fully handled the art; for Medina, in his dedication to Philip prince of Spain, laments that multitudes of ships daily perished at sea, because there were neither teachers of the art, nor books by which it might be learned; and Cortes, in his dedication, boasts to the emperor, that he was the first who had reduced navigation into a compendium, valuing himself much on what he had performed. Medina defended the plane chart; but he was opposed by Cortes, who showed its errors, and endeavoured to account for the variation of the compass, by supposing the needle to be influenced by a magnetic pole (which he called the point attractive), different from that of the world, which notion has been further prosecuted by others. Medina's book was soon translated into Italian, French, and Flemish, and for a long time served as a guide to foreign navigators. However, Cortes was the favourite author of the English nation, and was translated in the year 1561; whilst 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 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 that 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, in order to discover where the ship was, they made use of a small table, which showed, upon an alteration of one degree of the latitude, how many

---

1 See the articles Gioia and Magnetism. Navigation.

Leagues were run in each rhumb, together with the departure from the meridian. Besides, some other 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; together with a complex machine to discover the hour and latitude at once.

About the same time proposals were made for finding the longitude by observations of the moon. In 1530 Gemma Frisius advised the keeping of time by means of small clocks or watches, which were 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 the year 1537, Pedro Nunez, or Norius, published a book in the Portuguese language, to explain a difficulty in navigation proposed to him by the commander Don Martin Alphonso de Sousa. In this he exposed the errors of the plane chart, and likewise gave the solution of several curious astronomical problems, amongst which was that of determining the latitude from two observations of the sun's altitude and the intermediate azimuth. He observed, that although the rhumbs are spiral lines, yet the direct course of a ship will always be in the arc of a great circle, whereby the angle with the meridians will continually change; and hence all that the steersman can here do for the preserving of the original rhumb, is to correct these deviations as soon as they appear sensible. But in reality the ship will thus 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 having been much improved by Dr Halley, is used at present, and is called a nonius.

In the year 1577, Mr William Bourne published a treatise, in which, by considering the irregularities in the moon's motion, he showed the error 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 advised, in sailing towards the high latitudes, to keep the reckoning by the globe, as there the plane chart was most erroneous. He despaired 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 doubted; but as he had shown how to find the variation at all times, he recommended to keep an account of the observations, as useful for finding the place of the ship; and this advice was prosecuted at large by Simon Stevin, in a treatise published at Leyden in 1599, the substance of which was the same year printed at London in English by Mr Edward Wright, entitled the Haven-finding Art. In this ancient tract is also described the method 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 which floats in the water, whilst 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 that 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, although 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 the year 1581 Michael Coignet, a native of Antwerp, published a treatise, in which he animadverted on Medina. In this he showed, that as the rhumbs are spirals, making endless revolutions about the poles, numerous errors must arise from their being represented by straight lines on the sea charts; but although he hoped to find a remedy for these errors, he was of opinion that the proposals of Norius were scarcely practicable, and therefore in a great measure useless. In treating of the sun's declination, he took notice of the gradual decrease in the obliquity of the ecliptic; he also described the cross staff with three transverse pieces, which he admitted were then in common use amongst the sailors. He likewise described 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 sixtieth degree of latitude; and at the end of the hook is delivered a method of sailing upon a parallel of latitude by means of a ring dial and a twenty-four hour glass. The same year the discovery of the dipping needle was made by Mr Robert Norman. In his publication on that subject 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; and he also made considerable improvements on the construction of compasses themselves, showing especially the danger of not fixing, on account of the variation, the wire directly under the fleur de lis, as compasses made in different countries have it placed differently. To this performance of Norman's is prefixed a discourse on the variation of the magnetical needle, by Mr William Burrough, in which he shows how to determine the variation in many different ways, and also points out many errors in the practice of navigation at that time, speaking in very severe terms concerning those who had published upon it.

During this time the Spaniards continued to publish treatises on the art. In 1585 an excellent compendium was published by Roderico Zamorano, and 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; nevertheless, 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 effecting 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 formed an angle with the meridians, expressing the rhumb leading from the one to the other. But although in 1569 Mercator published an universal map constructed in this manner, it does not appear that he was acquainted with the principles upon 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 supposed, 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; with this precaution, however, that the degrees of longitude in each should bear some proportion to those of latitude, and this proportion was to be deduced from that which the magnitude of the respective parallels bore to a great circle of the sphere. He added, that, in particular maps, if this proportion be observed with regard to the middle parallel, the inconvenience will not be great, although the meridians should be straight lines parallel to each other. But here he is understood 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, although 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 in 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 afterwards, 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 thereof 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; and 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 barleycorn. Amongst his other improvements may be mentioned the Table of Latitudes for Dividing the Meridian computed to Minutes, whereas it had been only divided to every tenth minute. He also published a description of an instrument which he calls the sea rings, by which the variation of the compass, the altitude of the sun, and the time of the day, may at once readily be determined in any place, provided the latitude is known. He also showed how to correct the errors arising from the eccentricity of the eye in observing by the cross staff. In the years 1594, 1595, 1596, and 1597, he amended the tables of the declinations and places of the sun and stars from his own observations made with a six-feet quadrant, a sea quadrant to take altitudes by a forward or backward observation, and likewise with a contrivance for the ready finding of the latitude by the height of the pole star, when not upon the meridian. To this edition was subjoined a translation of Zamorano's Compendium, above mentioned, in which he corrected some mistakes in the original; adding a large table of the variation of the compass observed in different parts of the world, in order to show that it was not occasioned by any magnetical pole.

These improvements soon became known abroad. In 1608 a treatise entitled Hypomnemata Mathematica was published by Simon Stevin, for the use of Prince Maurice. In the portion of the work relating to navigation, the author having treated of sailing on a great circle, and shown how to draw the rhumbs on a globe mechanically, set down Wright's two tables of latitudes and of rhumbs, in order to describe these lines more accurately, pretending even to have discovered an error in Wright's table. But all Stevin's objections were fully answered by the author himself, who showed that they arose from the rude method of calculating made use of by the former.

In 1624 the learned Wellebrordus Snellius, professor of the 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 has been long ago refuted; and Wright now enjoys the honour of those discoveries, which is justly his due.

Mr Wright having shown how to find the place of the ship upon 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 to mark out upon his chart the ship's way, after the manner then usually practised. However, in 1614, Mr Raphe Hansdon, amongst the nautical questions he 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 has since been called. Although 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, Hansdon also proposes two methods of approximation without the assistance of Wright's division of the meridian line. The first was computed by the arithmetical mean between the cosines of both latitudes; and the other by the same mean between the secants as an alternative, when Wright's book was not at hand; although this latter is wider of the truth than the former. 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 generally used by our sailors, however, is commonly called the middle latitude, which, although it errs more than that by the arithmetical mean between the two cosines, is preferred on account of its being less oposse; yet in high latitudes it is more eligible to use that of the arithmetical mean between the logarithmic cosines, equivalent to the geometrical mean between the cosines themselves; a method since proposed by Mr John Bassat. The computation by the middle latitude will always fall short of the true change of longitude, that by the geometrical mean will always exceed; but that by the arithmetical mean falls short in latitudes of about 45 degrees, and exceeds in lesser latitudes. However, none of these methods will differ much from the truth when the change of latitude is sufficiently small.

About this period logarithms were invented by John Napier, baron of Merchiston in Scotland, and proved of the utmost service to the art of navigation. From these Mr Edmund Gunter constructed a table of logarithmic sines and tangents to every minute of the quadrant, which he published in 1620. In this work he applied to navigation, and other branches of mathematics, his admirable ruler known by the name of Gunter's scale, on which are described lines of logarithms, of logarithmic sines and tangents, of meridional parts, &c.; and he greatly improved the sector for the same purposes. He also showed how to take a back observation by the cross staff, by which the error arising from the eccentricity of the eye is avoided. He likewise described another instrument, of his own invention, called the cross bow, for taking altitudes of the sun or stars, with some contrivances for the more readily collecting the latitude from the observation. The discoveries concerning logarithms were carried into France in

---

1 See Gunter's Scale. 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, that of the tables of artificial sines and tangents, as modelled according to Napier's last form, erroneously attributed by Wingate to Briggs.

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

The logarithmic tables were first applied to the different cases of sailing by Mr Thomas Addison, in his treatise entitled Arithmetical Navigation, printed in the year 1625. He also gave two traverse tables, with their uses; the one to quarter-points of the compass, and the other to degrees. Mr Henry Gellibrand published his discovery of the changes of the variation of the compass, in a small quarto pamphlet, entitled A Discourse Mathematical on the Variation of the Magnetical Needle, printed in 1635. This extraordinary phenomenon he found out by comparing the observations which had been 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 Athanasius Kircher, in his treatise entitled Mognes, first printed at Rome in the year 1641, informs us, that he had been told of it by Mr John Greaves, and then gives a letter of the famous Marinus Mercennus, containing a very distinct account of the same.

As altitudes of the sun are taken on shipboard by observing his elevation above the visible horizon, to obtain from these the sun's true altitude with correctness, Wright observed it to be necessary that the dip of the visible horizon below the horizontal plane passing through the observer's eye should be brought into the account, which cannot be calculated without knowing the magnitude of the earth. Hence he was induced to propose different methods for finding this; but he complains that the most effectual was out of his power to execute, and therefore he contented himself with a rude attempt, in some measure sufficient for his purpose. The dimensions of the earth deduced by him corresponded very well with the usual divisions of the log line; nevertheless, 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 which measurement, and the summer solstitial altitudes of the sun observed on the meridian at both places, he found a degree on a great circle of the earth to contain 367,196 English feet, equal to 57,800 French fathoms or toises, which is very exact, as appears from many measurements 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 showed the reason why Snellius had failed in his attempt; and he also pointed out various uses of his discovery, particularly for correcting the gross errors hitherto committed in the divisions of the log line. But necessary amendments have been little attended to by sailors, whose obstinacy in adhering to established errors has been complained of by the best writers on navigation. This improvement, however, has at length made its way into practice, and few navigators of reputation now make use of the old measure of forty-two feet to a knot. In this treatise Mr Norwood also 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 likewise shows 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 abridged as a manual for sailors, in a very small piece called an Epitome, which useful performance has gone through a great number of editions. No alterations were ever made in the Seaman's Practice till the twelfth 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 of Wright's method, from a property in his meridian line, whereby the divisions are more scientifically assigned than the author himself was able to effect; it resulted from this theorem, that these divisions are analogous to the excesses of the logarithmic tangents of half the respective latitudes augmented by forty-five 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 he observed that the logarithmic tangents from $45^\circ$ upwards increase in the same manner as the secants do added together, 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 to be 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 forty-five degrees; divide the difference of those numbers by the logarithmic tangent of $45^\circ 30'$, the radius being likewise rejected, and the quotient will be the meridional parts required, expressed in degrees." This rule is the immediate consequence of the general theorem, that the degrees of latitude bear to one degree (or sixty 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 forty-five degrees, and the radius neglected, to the like tangent of half a degree augmented by forty-five degrees, with the radius likewise rejected. But here there was still wanting the demonstration of this general theorem, which was at length supplied by Mr James Gregory of Aberdeen, in his Exercitationes Geometricae, printed at London in 1668; and afterwards more concisely demonstrated, together with a scientific determination of the divisor, by Dr Halley, in the Philosophical Transactions for 1695 (No. 219), from the consideration of the spirals into which the rhumbs are transformed in the stereographic projection of the sphere upon the plane of the equinoctial, and which is rendered still more simple by Mr Roger Cotes, in his Logometria, first published in the Philosophical Transactions for 1714 (No. 388). It is moreover added in Gunter's book, that if $\frac{1}{\sqrt{2}}$th of this division, which does not sensibly differ from the logarithmic tangent of $45^\circ 1' 30''$, 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 $45^\circ 1' 30''$, diminished by the radius, be 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, although the wind should blow ever so strongly, 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 no longer be increased, and the rigging gives way.

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

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

It is perhaps impossible to lay down any mathematical principles on which the lee-way of a ship could be properly calculated; only we may observe in general that it depends on the strength of the wind, the roughness of the sea, and the velocity of the ship. When the wind is not very strong, the resistance of the water on the lee-side bears a very great proportion to that of the current of air, and therefore it will yield but very little; however, supposing the ship to remain in the same place, it is evident, that the water having once begun to yield, will continue to do so for some time, even though no additional force were applied to it; but as the wind continually applies the same force as at first, the lee-way of the ship must go on constantly increasing till the resistance of the water upon the lee-side balances the force applied on the other, when it will become uniform, as does the motion of a ship sailing... Mariner's before the wind. If the ship change 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 has 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 be very strong, the velocity of the ship will bear 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 be strong or not; for then the whole water of the ocean, as far as the swell reaches, has acquired a motion in a certain direction, and that to a very considerable depth. The mountainous waves will not fail to carry the ship very much out of her course; and this deviation will certainly be according to their velocity and their magnitude. In all cases of a rough sea, therefore, a great deal of lee-way is made. Another circumstance also occasions 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 occur, which can only be corrected by means of 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 considerable way in some particular direction, and which will certainly carry the ship greatly out of her course. This occasions an error of the same nature with the lee-way; and therefore, whenever a current is perceived, its direction and velocity ought to be determined, and the proper allowances made.

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

Errors will arise in the reckoning of a ship, especially when she sails in high latitudes, from the spheroidal figure of the earth; for as the polar diameter of our globe is found to be considerably shorter than the equatorial one, it thence follows, that the farther we remove from the Mariner's equator, the longer are the degrees of latitude. Of consequence, if a navigator assigns any certain number of miles for the length of a degree of latitude near the equator, he must vary that measure as he approaches towards the poles, otherwise he will imagine that he has not sailed so far as he has actually done. It would therefore be necessary to have a table containing the length of a degree of latitude in every different parallel from the equator to either pole; as, without this, a troublesome calculation must be made at every time the navigator makes a reckoning of his course. Such a table, however, has not yet appeared; neither indeed does it seem to be an easy matter to make it, on account of the difficulty of measuring the length even of one or two degrees of latitude in different parts of the world. Sir Isaac Newton first discovered this spheroidal figure of the earth; and showed, from theory originally suggested by experiments on pendulums, that the polar diameter was to the equatorial one as 229 to 230. This proportion, however, has not been admitted by succeeding calculators. The French mathematicians, who measured a degree of the meridian in Lapland, made the proportion between the equatorial and polar diameters to be as 1 to 0.9891; those who measured a degree at Quito in Peru made the proportion as 1 to 0.99624, or 265 to 265; M. Bouguer makes the proportion to be as 179 to 178; and M. Buffon, in one part of his theory of the earth, makes the equatorial diameter exceed the polar one by $\frac{1}{12}$th of the whole. According to M. du Sejour, this proportion is as 321 to 320; and M. de Laplace, in his Memoir upon the Figure of Spheroids, has deduced the same proportion. Later investigations, however, show that the polar axis is to the equatorial diameter in the ratio of 300 to 301 nearly. From these variations, it appears that the point is not exactly determined, and consequently, that any corrections which can be made with regard to the spheroidal figure of the earth must be very uncertain.

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

MARINER'S COMPASS.

A ship is enabled to keep her course at sea by means of an instrument called the mariner's compass. It consists of a magnetic steel bar attached to the under side of a card divided into points and quarter points, and supported by a fine pin, on which it turns freely within a box covered with glass. By reason of the directive property of the magnet, the north point, which is commonly denoted by a fleur de lis, is readily known. The circumference of the card is generally divided into thirty-two points, which in the best compasses are again subdivided into half points and quarters. These are reckoned sufficient for nautical purposes. On the inside of the box is drawn a dark vertical line called lubber's point. This point, or rather line, and the pin on which the card turns, are in the same line or plane with the keel of the ship; and hence the point on the circumference of the card opposite to lubber's point shows the angle which the ship's course makes with the magnetic meridian, called the course of the ship. The annexed diagram gives a general view of the compass. The names of the points, and the angles which they pass. The azimuth compass is the same instrument more nicely made. The circumference of the card is divided into degrees and parts by a vernier, and is fitted up with sight-vanes to take amplitudes and azimuths, for the purpose of determining the variation of the compass by observation. The variation is then applied to the magnetic course shown by the steering compass, whence the true course with respect to the meridian becomes known. The necessary rules for this purpose will be given in a succeeding part of this article.

Besides the variation, the needle is also affected by the dip, which is likewise fully explained in the article Magnetism, as well as Mr Barlow's method of correcting the effects of local attraction, arising from the effects of the iron, guns, &c., in the vessel itself. Having made these preliminary observations, we shall now proceed to the

BOOK I.

CONTAINING THE VARIOUS METHODS OF SAILING.

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

CHAP. I.—PRELIMINARY PRINCIPLES.

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

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

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

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

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

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

Example 1. Required the difference of latitude between the Lizard, in latitude 49° 58' N., and Cape St Vincent, in latitude 37° 3' N.

| Latitude of the Lizard | 49° 58' N. | |-----------------------|------------| | Latitude of Cape St Vincent | 37° 3' N. |

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

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

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

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

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

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

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

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

Latitude come to, 48° 42' N.

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

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

Latitude come to, 6° 55' S.

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

Rule. If the longitudes of the given places are of the same name, subtract the less from the greater, and the remainder is the difference of longitude; but if the longitudes are of contrary names, their sum is the difference of longitude. If this exceeds 180°, subtract it from 360°, and the remainder is the difference of longitude. Ex. 1. Required the difference of longitude between Edinburgh and New York, their longitudes being 3° 12' W. and 74° 2' W. respectively.

Longitude of New York, 74° 2' W. Longitude of Edinburgh, 3° 12' W.

Difference of longitude, 70° 50'

Ex. 2. What is the difference of longitude between Maskelyne's Isles, in longitude 167° 59' E., and Olinda, in longitude 34° 54' W.

Longitude of Maskelyne's Isles, 167° 59' E. Longitude of Olinda, 34° 54' W.

Sum, 203° 53' Subtract from, 360° 0'

Difference of longitude, 156° 2'

Prob. IV. Given the longitude of a place, and the difference of longitude between it and another place, to find the longitude of the latter place.

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

Ex. 1. A ship from longitude 9° 54' E. sailed westerly till the difference of longitude was 23° 18'. Required the longitude come to.

Longitude sailed from, 9° 54' E. Longitude come to, 13° 24' W.

Ex. 2. The longitude sailed from is 25° 9' W. and difference of longitude 18° 46' W. Required the longitude come to.

Longitude left, 25° 9' W. Difference of longitude, 18° 46' W.

Longitude in, 43° 55' W.

Sect. II.—Of the Tides.

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

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

Table I.—For determining the Time of High Water.

| Moon's Transit | Moon's Horizontal Parallax | |---------------|---------------------------| | | 60° | 59° | 58° | 57° | 56° | 55° | 54° | | h. m. | | | | | | | | | 0 0 | -4 | -3 | -2 | -1 | +2 | +4 | +6 | | 10 | 6 | 5 | 4 | 3 | 1 | 2 | 10 | | 20 | 8 | 7 | 6 | 5 | 4 | 3 | 20 | | 30 | 10 | 10 | 9 | 8 | 7 | 6 | 30 | | 40 | 12 | 12 | 11 | 10 | 9 | 8 | 40 | | 50 | 15 | 14 | 14 | 13 | 12 | 11 | 50 | | 1 0 | 17 | 17 | 16 | 16 | 15 | 15 | 13 | | 10 | 20 | 20 | 19 | 19 | 19 | 18 | 10 | | 20 | 22 | 22 | 22 | 22 | 22 | 22 | 20 | | 30 | 24 | 24 | 25 | 25 | 25 | 25 | 30 | | 40 | 27 | 27 | 28 | 28 | 28 | 28 | 40 | | 50 | 29 | 30 | 31 | 31 | 31 | 31 | 50 | | 2 0 | 31 | 32 | 33 | 33 | 34 | 35 | 36 | | 10 | 34 | 35 | 36 | 36 | 37 | 38 | 39 | | 20 | 36 | 37 | 38 | 39 | 40 | 42 | 43 | | 30 | 38 | 39 | 40 | 41 | 42 | 44 | 46 | | 40 | 40 | 41 | 43 | 44 | 46 | 48 | 50 | | 50 | 42 | 43 | 45 | 46 | 48 | 50 | 52 | | 3 0 | 44 | 45 | 47 | 49 | 51 | 53 | 55 | | 10 | 46 | 47 | 49 | 51 | 54 | 56 | 58 | | 20 | 48 | 49 | 51 | 53 | 56 | 58 | 61 | | 30 | 50 | 52 | 54 | 56 | 58 | 61 | 64 | | 40 | 52 | 54 | 56 | 58 | 61 | 64 | 67 | | 50 | 53 | 55 | 57 | 60 | 63 | 66 | 69 | | 4 0 | 55 | 57 | 59 | 62 | 65 | 69 | 72 |

Vol. XV. ### Table I.—(continued).

| Moon's Transit | 60° | 59° | 58° | 57° | 56° | 55° | 54° | |----------------|-----|-----|-----|-----|-----|-----|-----| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

### Table II.—For finding the Height of the Tide.

#### (PART I.)

| Time of Transit | Moon's Hor. Par. 60° | Moon's Hor. Par. 57° | Moon's Hor. Par. 54° | |-----------------|---------------------|---------------------|---------------------| | | Multipliers | Multipliers | Multipliers | | 0 | 12 0 | 0·995a + 0·1495 | 0·883a + 0·117b | | 0·40 | 12 40 | 1·014a + 0·0355 | 0·970a + 0·0305 | | 1·20 | 13 20 | 1·138a + 0·0005 | 1·000a + 0·0005 | | 2·0 | 14 0 | 1·104a + 0·038b | 0·970a + 0·0306 | | 2·40 | 14 40 | 0·995a + 0·1495 | 0·883a + 0·117b | | 3·20 | 15 20 | 0·853a + 0·3195 | 0·750a + 0·2506 | | 4·0 | 16 0 | 0·668a + 0·5276 | 0·587a + 0·4135 | | 4·40 | 16 40 | 0·440a + 0·7495 | 0·413a + 0·5876 | | 5·20 | 17 20 | 0·284a + 0·9586 | 0·250a + 0·7506 | | 6·0 | 18 0 | 0·133a + 1·277b | 0·117a + 0·883b |

#### (PART II.)

| Time from H.W. | Mult. | Time from H.W. | Mult. | |----------------|-------|----------------|-------| | 0 | 1·000 | 3 10 | 0·510 | | 0·10 | 0·993 | 3 20 | 0·460 | | 0·20 | 0·993 | 3 30 | 0·429 | | 0·30 | 0·985 | 3 40 | 0·389 | | 0·40 | 0·974 | 3 50 | 0·349 | | 0·50 | 0·959 | 4 0 | 0·311 | | 1·00 | 0·941 | 4 10 | 0·274 | | 1·10 | 0·921 | 4 20 | 0·238 | | 1·20 | 0·897 | 4 30 | 0·204 | | 1·30 | 0·871 | 4 40 | 0·173 |

### To find the time of High Water.

**Rule.** Let the approximate time of high water be found, by taking the corrections for the moon's horizontal parallax for the nearest noon or midnight from Table I. Again, to this time and the given longitude take from the Nautical Almanack the moon's horizontal parallax. Also to the time of the moon's transit over the meridian of Greenwich apply the variation answering to the longitude and daily variation between the given and preceding day if the longitude is east. Subtract it from the transit over the meridian of Greenwich, and the remainder will be the time of transit over the meridian of the given place. But if the longitude be west, the correction answering to the longitude and daily variation of transit between the given and following day must be added to the time of transit over the meridian of Greenwich to obtain the time of transit over the meridian of the given place. To the time of high water, if new and full moon at the given place, add the reduced time of transit over the meridian of the same place, and to the sum apply the equation from the table answering to the time of transit and horizontal parallax formerly found; the result will be the true mean time of high water required. The apparent time may be found by applying the equation of time, with its proper sign.

**Ex. I.** Required the time of high water at Leith on Wednesday the 10th of May 1837, in longitude 3° 11' W.

By the rule, the time of high water will be about six o'clock in the evening. In this case, the moon's horizontal parallax will be 54° 16', and the time of transit 4h 54m mean time, or 4h 54m apparent time by applying the equation of time 3m 50s by addition.

Apparent time of transit of upper meridian, 4h 54m

Equation from the table to horizontal parallax 54° 16', and transit 4h 54m, subtract

Remainder,

Time of high water at new and full moon,

Apparent time of high water,

Equation of time, subtract

Mean time of high water, If the sum exceed $12^h\ 25^m$, subtract this number from it; if it exceed $24^h\ 50^m$, subtract as before, and the remainder will be the time of high water in the afternoon of the given day nearly. The time of high water of the tide preceding may be found nearly by subtracting $25^m$ from it, and the succeeding tide by adding $25^m$ to it. In cases of great accuracy, however, a computation should be made for each tide in a manner similar to that above.

Ex. 2. Required the time of high water at Aberdeen on the 21st of June 1837, in longitude $2^\circ\ 6'W$. As before, the time of high water will readily be found to be about three o'clock.

Here the horizontal parallax of the moon will be $60^\circ\ 30'$, and the mean time of transit on the given day $15^h\ 32^m$. But as this transit exceeds $12^h$, it will be necessary to take the time of transit over the under meridian, or, what comes to the same thing, half the sum of the transits on the given and preceding days, or $\frac{1}{2}(14^h\ 32^m + 15^h\ 32^m) = 15^h\ 2^m$. Correction from the table, $-0\ 44$

Remainder, $14\ 18$ High water at new and full moon, $+1\ 10$ Sum exceeding $12^h$, $15\ 28$ By rule, subtract $12\ 25$ Apparent time of high water, $3\ 3$ Equation of time, $+0\ 1$ Mean time, $3\ 4$

Ex. 3. Required the depth at Aberdeen at the same time, the rise of spring tides being 19 feet, denoted by $a$ in Table II. part I, and that of the neap 15 feet, by $b$.

Now, by Table II. part I, to transit $15^h\ 2^m$, and horizontal parallax $60^\circ\ 30'$, will be obtained $0.917 \times 19 + 0.242 \times 14 = 20.8$ feet.

Ex. 4. Required the height of the tide at $3^h\ 15^m$ after high water.

By part 2, $20.8 \times 0.5 = 10.4$ feet.

In this manner, the time and rise of the tide may be readily obtained nearly, unless both are much influenced by the strength and direction of the wind.

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

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

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

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

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

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

$$\frac{9}{30} = \text{true rate of sailing.}$$

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

$$\frac{73}{30} = \text{true distance.}$$

$$26 \div 2190(84.2 \text{ the true distance.})$$

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

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

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

Measured interval, $= 53$ Twice hourly rate, $= 10$ True rate of sailing, $= 5.30$

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

Twice given distance, $= 128$ Measured interval, $= 42$ True distance, $= 5376$

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

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

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

Distance by the log, $= 159$ Six times length of a knot $= 42 \times 6 = 252$ Seconds run by the glass, $= 3340068(121.4 = \text{true distance.})$

CHAP. II.—OF PLANE SAILING.

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

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

The distance is the number of miles a ship has sailed on a direct course in a given time. The difference of latitude is the portion of a meridian contained between the parallels of latitude sailed from and come to; and is reckoned either north or south, according as the course is in the northern or southern hemisphere.

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

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

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

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

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

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

### Table III.—To reduce Points of the Compass to Degrees, and conversely.

| North-east Quadrant | South-east Quadrant | Points | D. M. S. | South-west Quadrant | North-west Quadrant | |---------------------|---------------------|--------|----------|---------------------|---------------------| | North | South | | | South | North | | N. E. | S. E. | 0 | 248 | 45 | S. W. | N. W. | | N. E. | S. E. | 0 | 37 | 30 | S. W. | N. W. | | N. E. | S. E. | 0 | 26 | 15 | S. W. | N. W. | | N. by E. | S. by E. | 1 | 11 | 15 | S. by W. | N. by W. | | N. by E. | S. by E. | 1 | 14 | 35 | S. by W. | N. by W. | | N. by E. | S. by E. | 1 | 16 | 52 | S. by W. | N. by W. | | N. by E. | S. by E. | 1 | 19 | 15 | S. by W. | N. by W. | | N. N. E. | S. S. E. | 2 | 22 | 30 | S. S. W. | N. N. W. | | N. N. E. | S. S. E. | 2 | 25 | 45 | S. S. W. | N. N. W. | | N. N. E. | S. S. E. | 2 | 28 | 30 | S. S. W. | N. N. W. | | N. N. E. | S. S. E. | 2 | 30 | 15 | S. S. W. | N. N. W. | | N. E. by N. | S. E. by S. | 3 | 33 | 45 | S. W. by S. | N. W. by N. | | N. E. | S. E. | 3 | 36 | 45 | S. W. | N. W. | | N. E. | S. E. | 3 | 39 | 30 | S. W. | N. W. | | N. E. | S. E. | 3 | 42 | 15 | S. W. | N. W. | | N. E. | S. E. | 4 | 45 | 0 | S. W. | N. W. | | N. E. | S. E. | 4 | 47 | 45 | S. W. | N. W. | | N. E. | S. E. | 4 | 50 | 30 | S. W. | N. W. | | N. E. | S. E. | 4 | 53 | 15 | S. W. | N. W. | | N. E. by E. | S. E. by E. | 5 | 56 | 15 | S. W. by W. | N. W. by W. | | N. E. by E. | S. E. by E. | 5 | 59 | 45 | S. W. by W. | N. W. by W. | | N. E. by E. | S. E. by E. | 5 | 61 | 30 | S. W. by W. | N. W. by W. | | N. E. by E. | S. E. by E. | 5 | 64 | 15 | S. W. by W. | N. W. by W. | | E. N. E. | E. S. E. | 6 | 67 | 30 | W. S. W. | W. N. W. | | E. by N. | E. by S. | 6 | 70 | 45 | W. by S. | W. by N. | | E. by N. | E. by S. | 6 | 73 | 30 | W. by S. | W. by N. | | E. by N. | E. by S. | 6 | 75 | 15 | W. by S. | W. by N. | | E. by N. | E. by S. | 7 | 78 | 45 | W. by S. | W. by N. | | E. by N. | E. by S. | 7 | 81 | 45 | W. by S. | W. by N. | | E. by N. | E. by S. | 7 | 84 | 30 | W. by S. | W. by N. | | E. by N. | E. by S. | 7 | 87 | 15 | W. by S. | W. by N. | | East | East | 8 | 90 | 0 | West | West | **NAVIGATION**

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

*Example.* A ship from St Helena, in latitude 15° 53' S., sailed S.W. by S. 158 miles. Required the latitude come to, and departure.

**By Construction.**

Draw the meridian AB, and with the chord of 60° describe the arch mn, and make it equal to the rhumb of three points, and through n draw AC equal to 158 miles; from C, draw CB perpendicular to AB; then AB applied to the scale from which AC was taken will be found to measure 131°4, and BC 87°8.

**By Calculation.**

To find the difference of latitude.

| As radius | 10-00000 | |-----------|----------| | is to the cosine of the course | 3 points | | so is the distance | 158 | | 9-01985 | 2-19866 |

To find the departure.

| As radius | 10-00000 | |-----------|----------| | is to the sine of the course | 3 points | | so is the distance | 158 | | 9-74474 | 2-19866 |

**By Inspection.**

In the traverse table, the difference of latitude answering to the course 3 points, and distance 158 miles, in a distance column, is 131°4, and departure 87°8.

**By Gunter's Scale.**

The extent from 8 points to 5 points, the complement of the course on the line of sine rhumbs (marked S.R.) will reach from the distance 158 to 131°4, the difference of latitude on the line of numbers; and the extent from 8 points to 3 points on sine rhumbs will reach from 158 to 87°8, the departure on numbers.

| Latitude St Helena, | 15° 53' S. | |---------------------|------------| | Difference of latitude, | 2 11 S. | | Latitude come to, | 18 6 S. |

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

*Example.* A ship from St George's, in latitude 38° 45' north, sailed S.E. by S.; and the latitude by observation was 35° 7' N. Required the distance run, and departure.

| Latitude St George's, | 38° 45' N. | |----------------------|------------| | Latitude come to, | 35° 7' N. | | Difference of latitude, | 3 38 = 218 miles |

**By Construction.**

Draw the portion of the meridian AB equal to 218 m.: from the centre A with the chord of 60° describe the arch mn, which make equal to the rhumb of 3½ points: through n draw the line AC, and from B draw BC perpendicular to AB, and let it be produced till it meets AC in C. Then the distance AC, being applied to the scale, will measure 282 m., and the departure BC 179 miles.

**By Calculation.**

To find the distance.

| As radius | 10-00000 | |-----------|----------| | is to the secant of the course | 3½ points | | so is the difference of latitude | 218 m. | | 10-11181 | 2-33846 |

To find the departure.

| As radius | 10-00000 | |-----------|----------| | is to the tangent of the course | 3½ points | | so is the difference of latitude | 218 | | 9-91417 | 2-33846 |

**By Inspection.**

Find the given difference of latitude 218 m. in a latitude column, under the course of 3½ points; opposite to which, in a distance column, is 282 miles; a departure column is 178°9 m. the distance and departure required.

**By Gunter's Scale.**

Extend the compass from 4½ points, the complement of the course, to 8 points on sine rhumbs; that extent will reach from the difference of latitude 218 miles to the distance 282 miles on numbers; and the extent from 4 points to the course 3½ points on the line of tangent rhumbs (marked T.R.) will reach from 218 miles to 178°9; the departure on numbers.

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

*Example.* A ship from Palma, in latitude 28° 37' N., sailed N.W. by W. and made 192 miles of departure. Required the distance run, and latitude come to.

**By Construction.**

Make the departure BC equal to 192 miles, draw BA perpendicular to BC, and from the centre C, with the chord of 60°, describe the arch mn, which make equal to the rhumb of 3 points, the complement of the course; draw a line through Cn, which produce till it meet BA in A: then the distance AC being measured, will be equal to 231 m., and the difference of latitude AB will be 128°3 miles.

**By Calculation.**

To find the distance.

| As the sine of the course | 5 points | |--------------------------|---------| | is to radius | 192 | | 9-91985 | 2-28330 |

To find the difference of latitude.

| As the tangent of the course | 5 points | |-----------------------------|---------| | is to radius | 192 | | 10-17511 | 2-28330 |

**By Inspection.**

Find the departure 192 m. in its proper column above the given course 5 points; and opposite thereto is the distance 231 miles, and difference of latitude 128°3, in their respective columns.

---

1 For the method of resolving the various problems in navigation by the sliding gunter, the reader is referred to Dr Mackay's Treatise on the Description and Use of that Instrument. By Gunter's Scale.

The extent from 5 points to 8 points on the line of sine rhumbs, being laid from the departure 192 on numbers, will reach to the distance 231 on the same line; and the extent from 5 points to 4 points on the line of tangent rhumbs will reach from the departure 192, to the difference of latitude 128° 3 on numbers.

Latitude of Palma, 28° 37' N. Difference of latitude, 2 8 N. Latitude come to, 30 45 N.

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

Example. A ship from a place in latitude 43° 13' N., sails between the north and east 285 miles; and is then by observation found to be in latitude 46° 31' N. Required the course and departure.

Latitude sailed from, 43° 13' N. Latitude by observation, 46° 31' N. Difference of latitude, 3 18 = 198 miles.

By Construction.

Draw BC perpendicular to AB, and equal to the given departure 190 miles; then from the centre C, with the distance 260 miles, sweep an arch intersecting AB in A, and join AC. Now describe an arch from the centre A with the chord of 60°, and the portion mn of this arch, contained between the distance and difference of latitude, measured on the line of chords, will be 47°, the course; and the difference of latitude AB, applied to the scale of equal parts, measures 177½ miles.

By Calculation.

To find the course.

As the distance 260 2-41497 is to the departure 190 2-27875 so is radius 10-00000

to the sine of the course 46° 57' 9-86378

To find the difference of latitude.

As radius 10-00000 is to the cosine of the course 46° 57' 9-83419 so is the distance 260 2-41497

to the difference of latitude 177-5 2-24916

By Inspection.

Seek in the traverse table until the nearest to the given departure is found in the same line with the given distance 260. This is found to be in the page marked 47° at the bottom, which is the course; and the corresponding difference of latitude is 177-5.

By Gunter's Scale.

The extent of the compass, from the distance 260 to the departure 190 on the line of numbers, will reach from 90° to 47°, the course on the line of sines; and the extent from 90° to 43°, the complement of the course on sines, will reach from the distance 260 to the difference of latitude 177½ on the line of numbers.

Latitude Fort-Royal, 12° 9' N. Difference of latitude, 177 = 2 57 S. Latitude in, 9 12 N.

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

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

Latitude sailed from, 7° 56' S. Latitude in by observation, 12° 3' S. Difference of latitude, 4 7 = 247. Draw the portion of the meridian AB equal to the difference of latitude 247 miles; from B draw BC perpendicular to AB, and equal to the given departure 132 miles, and join AC; then with the chord of 60° describe an arch from the centre A; and the portion mn of this arch, being applied to the line of chords, will measure about 28°; and the distance AC, measured on the line of equal parts, will be 280 miles.

By Calculation.

To find the course.

As the difference of latitude 247 is to the departure 132 so is radius 2:39270

To the tangent of the course 28° 7' so is radius 10:00000

To find the distance.

As radius 10:00000 is to the secant of the course 28° 7' so is the difference of latitude 247 to the distance 280

By Inspection.

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

By Gunter's Scale.

The extent from the given difference of latitude 247 to the departure 132 on the line of numbers, will reach from 45° to 28°, the course on the line of tangents; and the extent from 62°, the complement of the course, to 90° on sines, will reach from the difference of latitude 247 to the distance 280 on numbers.

CHAP. III.—OF TRAVERSE SAILING.

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

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

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

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

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

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

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

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

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

Ex. I. A ship from Fyall, in lat. 38° 32' N., sailed as follows: E. S. E. 163 miles, S. W. ½ W. 110 miles, S. E. ¾ S. 180 miles, and N. by E. 68 miles. Required the latitude come to, the course, and distance made good.

By Inspection.

| Course | Dist. | Diff.of Latitude | Departure | |--------|-------|-----------------|-----------| | E. S. E. | 163 | ... | 62° 4' | 150° 6' | | S. W. ½ W. | 110 | ... | 69° 8' | 85° 0' | | S. E. ¾ S. | 180 | ... | 144° 5' | 107° 2' | | N. by E. | 68 | 66° 7' | ... | 13° 3' |

| S. 41 ½ E. | 281 | 210° 0' | 186° 1' |

Latitude left ........................................... 38° 32' N. Difference of latitude .................................. 3° 21' S. Latitude come to ....................................... 35° 11' N.

By Construction.

With the chord of 60° describe the circle NE, SW (fig. 8), the centre of which represents the place the ship sailed from; draw two diameters NS, EW at right angles to each other, the one representing the meridian, and the other the parallel of latitude of the place sailed from. Take each course from the line of rhumbs, lay it off on the circumference from its proper meridian, and number it in order 1, 2, 3, 4. Upon the first rhumb C1, lay off the first distance 163 miles from C to A; through it draw the second distance AB parallel to C2, and equal to 110 miles; Traverse through B draw BD equal to 180 miles, and parallel to Sailing, C3; and draw DE parallel to C4, and equal to 68 miles. Now CE being joined, will represent the distance made good; which, applied to the scale, will measure 281 miles.

The arch Sa, which represents the course, being measured on the line of chords, will be found equal to 41°.

From E draw EF perpendicular to CS produced; then CF will be the difference of latitude, and FE the departure made good; which, applied to the scale, will be found to measure 210 and 186 miles respectively.

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

Ex. 2. A ship from latitude 1° 58' S. sailed as under. Required her present latitude, course, and distance made good.

| Course | Dist. | Diff.of Latitude | Departure | |--------|-------|-----------------|-----------| | N.W.by E... | 43 | 35°8 | ... | 23°9 | | W.N.W...... | 78 | 29°9 | ... | 72°1 | | S.E.by E... | 56 | ... | 31°1 | 46°6 | | W.S.W.by W.| 62 | ... | 18°0 | 59°3 | | N.4°E...... | 85 | 84°1 | ... | 12°5 | | N.44°W.... | 139 | 100°7=1°41' | ... | 96°2 |

Latitude left.............1° 58' S. Latitude come to.........0° 3° N.

Ex. 3. Yesterday at noon we were in latitude 13° 12' N., and since then have run as follows: S.S.E. 36 miles, S.12 miles, N.W. 1° W. 28 miles, W. 30 miles, S.W. 42 miles, W.by N. 39 miles, and N. 20 miles. Required our present latitude, departure, and direct course and distance.

| Course | Dist. | Diff.of Latitude | Departure | |--------|-------|-----------------|-----------| | S.S.E... | 36 | ... | 33°3 | 13°8 | | S...... | 12 | ... | 12°0 | ... | | N.W.1°W.| 28 | 17°8 | ... | 21°6 | | W...... | 30 | ... | ... | 30°0 | | S.W... | 42 | ... | 29°7 | 29°7 | | W.by N.| 39 | 7°6 | ... | 38°2 | | N...... | 20 | 20°0 | ... | ... | | S.74°W.. | 110 | ... | 29°6=0°30' | 105°7 |

Yesterday's latitude...........13° 12' N. Present latitude................12° 42' N.

Ex. 4. The course per compass from Greigness to the May is S.W. by S., distance 58 miles; from the May to the Staples, S.by E. 4° E., 44 miles; and from the Staples to Flamborough Head, S.by E., 110 miles. Required the course per compass, and distance from Greigness to Flamborough Head.

| Courses | Dist. | Diff.of Latitude | Departure | |--------|-------|-----------------|-----------| | S.W.by S... | 58 | ... | 43°0 | 38°9 | | S.by E.4°E.| 44 | ... | 41°4 | 14°8 | | S.by E.... | 11 | ... | 107°9 | 21°5 | | ... | ... | 192°3 | 36°3 | 38°9 | | ... | ... | 36°3 | ... | ... |

Hence the course per compass is nearly S. 1° W., and distance 192°3 miles.

CHAP. IV.—OF PARALLEL SAILING.

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

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

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

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

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

But the difference of longitude is an arch of the equa-

---

1 Greigness is about 2½ miles distant from Aberdeen, in nearly a S.E. by E. ¼ E. direction. 2 This is not strictly true, as the figure of the earth is that of an oblate spheroid; and therefore the radius of curvature varies with the latitude. The difference between CA and CP, according to Sir Isaac Newton's hypothesis, is about 17 miles. tor, and the distance between any two places under the same parallel is a similar portion of that parallel.

Hence \( R : \cosine \text{ latitude} :: \text{diff. longitude} : \text{distance} \).

And, by inversion,

\[ \cosine \text{ latitude} : R :: \text{distance} : \text{diff. of longitude}. \]

Also,

\[ \text{diff. of longitude} : \text{distance} :: R : \cosine \text{ latitude}. \]

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

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

**By Construction.**

Draw the indefinite right line \(AB\) (fig. 10); make the angle \(BAC\) equal to the given latitude \(55^\circ 58'\), and \(AC\) equal to the number of miles contained in a degree of longitude at the equator, namely, 60; from \(C\) draw \(CB\) perpendicular to \(AB\); and \(AB\), being measured on the line of equal parts, will be found equal to 33-5, the miles required.

**By Calculation.**

As radius

\[ \begin{array}{c} \text{is to the cosine of latitude} \\ \text{so is the distance} \end{array} \]

\[ \begin{array}{c} 55^\circ 58' \\ 974794 \end{array} \]

\[ \begin{array}{c} \text{to the difference of longitude} \\ \text{Longitude Cape Finisterre,} \\ \text{Difference of longitude,} \\ \text{Longitude come to,} \end{array} \]

\[ \begin{array}{c} 466-6 \\ 9^\circ 17' W. \\ 7^\circ 47' W. \\ 17^\circ 4 W. \end{array} \]

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

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

**By Inspection.**

To \(56^\circ\), the nearest degree to the given latitude, and distance 60 miles, the corresponding difference of latitude is 33-6, which is the miles required.

**By Gunter's Scale.**

The extent from \(90^\circ\) to \(34^\circ\), the complement of the given latitude on the line of sines, will reach from 60 to 33-6 on the line of numbers.

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

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

Longitude Treguier, \(3^\circ 14' W.\)

Longitude Gaspey Bay, \(64^\circ 27' W.\)

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

As radius

\[ \begin{array}{c} \text{is to the cosine of latitude} \\ \text{so is the difference of longitude} \end{array} \]

\[ \begin{array}{c} 48^\circ 47' \\ 981882 \end{array} \]

\[ \begin{array}{c} \text{to the distance} \\ \text{Probl. IV.} \text{ Given the number of miles contained in the portion of a known parallel, to find the length of a similar portion of another known parallel.} \end{array} \]

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

**By Construction.**

Draw the line \(CB\), \(CE\) (fig. 13), making angles with \(CP\) equal to the complements of the given latitudes, namely, \(56^\circ 2'\) and \(41^\circ 37'\) respectively; make \(BD\) equal to the given distance 348 miles, and perpendicular to \(CP\); now from the centre \(C\), with the radius \(CB\), describe an arch intersecting \(CE\) in \(E\); then \(EF\) drawn from the point \(E\), perpendicular to \(CP\), will represent the distance required; which being applied to the scale, will measure \(278\frac{1}{2}\) miles. By Calculation.

As the cosine of the latitude left is to the cosine of the lat. come to so is the given distance

\[ \frac{38^\circ 59'}{48^\circ 23'} = \frac{9\cdot 91874}{9\cdot 892926} \]

to the distance required

\[ 348 \cdot 2\cdot 54158 \]

\[ 278\cdot 6 \cdot 2\cdot 44510 \]

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

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

By Construction.

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

By Calculation.

As the distance on the known parallel 180 is to the distance on that required 232 so is the cosine of the latitude left 56° 0'

\[ \frac{2\cdot 25527}{2\cdot 36549} = \frac{9\cdot 74756}{9\cdot 85778} \]

CHAP. V.—OF MIDDLE LATITUDE SAILING.

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

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

Now, since CE represents a very small portion of the distance AB, DE will be the correspondent portion of a meridian; hence the triangle EDC may be considered as rectilineal. If the distance be supposed to be divided into an infinite number of parts, each equal to CE, and upon these, triangles be constructed whose sides are portions of a meridian and parallel, it is evident these triangles will be equal and similar; for, besides the right angle, and hypotenuse which is the same in each, the course or angle CED is also the same. Hence, by the 12th of V. Eucl. the sum of all the hypotenuses CE, or the distance AB, is to the sum of all the sides DE, or the difference of latitude AS, as one of the hypotenuses CE is to the correspond-

ing side DE. Now, let the triangle GIH (fig. 16) be constructed similar to the triangle CDE, having the angle G equal to the course; then, as GH : GI :: CE : DC :: AB : AS.

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

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

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

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

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

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

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

Example. Required the course and distance from the Island of May, in latitude 56° 12' N. and longitude 2° 37' W., to the Naze of Norway, in latitude 57° 50' N. and longitude 7° 27' E.

| Latitude Isle of May | 56° 12' N. | |----------------------|------------| | Latitude Naze of Norway | 57° 50' N. |

Difference of latitude, \( 1° 38' = 98'' \)

Middle latitude, \( 57° 1'' \)

Longitude Isle of May, \( 2° 37' W. \)

Longitude Naze of Norway, \( 7° 27' E. \)

Difference of longitude, \( 10° 4' = 60'' \) By Construction.

Draw the right line AD (fig. 18) to represent the meridian of the May; with the chord of 60° describe the arch mn, upon which lay off the chord of 32° 59', the complement of the middle latitude from m to n: from D through n draw the line DC equal to 60° the difference of longitude, and from C draw CB perpendicular to AD: make BA equal to 98' the difference of latitude, and join AC; which applied to the scale will measure 343 miles, the distance sought: and the angle A being measured by means of the line of chords, will be found equal to 73° 24', the required course.

By Calculation.

To find the course:

As the difference of latitude 98° 1-99123 is to the difference of longitude 60° 2-78104 so is the cosine of middle latitude 57° 1' 9-73591 to the tangent of the cosine 73° 24' 10-52572

To find the distance:

As radius 10-00000 is to the secant of the course 73° 24' 10-54411 so is the difference of latitude 98° 1-99123 to the distance 343 2-53584

The true course, therefore, from the Island of May to the Naze of Norway is N. 73° 24' E., E. N.E. ¾ E. nearly; but as the variation at the May is ¼ points west, therefore the course per compass from the May is E. by S.

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

Example. A ship from Brest, in latitude 48° 23' N. and longitude 4° 30' W., sailed S.W. ¾ W. 238 miles. Required the latitude and longitude come to.

By Construction.

With the course and distance construct the triangle ABC (fig. 17), and the difference of latitude AB, being measured, will be found equal to 142 miles: hence the latitude come to is 46° 1' N., and the middle latitude 47° 12'. Now make the angle DCB equal to 47° 12'; and DC, being measured, will be 281, the difference of longitude: hence the longitude come to is 9° 11' W.

By Calculation.

To find the difference of latitude:

As radius 10-00000 is to the cosine of the course 4½ 9-77503 so is the distance 238 2-37658 to the difference of latitude 141° 8' 2-15161

Latitude of Brest, 48° 23' N. 48° 23' N. Difference of lat. 2 22 S. half 1 11 S. Lat. come to, 46 1 N. Mid. lat. 47 12 N.

To find the difference of longitude:

As the cosine of mid. lat. 47° 12' 9-63215 is to the sine of the course 4½ points 9-90483 so is the distance 238 2-37658 to the difference of longitude 281° 3' 2-44926

Longitude of Brest, Difference of longitude, 4° 30' W. Middle Latitude Sailing: Longitude come to, 4 41 W. Latitude Sailing: 9 11 W.

Prob. III. Given both latitudes and course, required the distance and difference of longitude.

Example. A ship from St Antonio, in latitude 17° 0' N. and longitude 24° 25' W., sailed N.W. ¾ N., till by observation her latitude was found to be 28° 34' N. Required the distance sailed, and longitude come to.

Latitude St Antonio, 17° 0' N. 17° 0' N. Latitude by observation, 28 34 N. 28 34 N. Difference of lat. 11 34 = 69° 45 34 Middle Int. 22 47

By Construction.

Construct the triangle ABC (fig. 19), with the given course and difference of latitude, and make the angle BCD equal to the middle latitude. Now the distance AC and difference of longitude DC being measured, will be found equal to 864 and 558 respectively.

By Calculation.

To find the distance:

As radius 10-00000 is to the secant of the course 3½ points 10-09517 so is the difference of lat. 694 2-84136 to the distance 864 2-93653

To find the difference of longitude:

As the cosine of middle latitude 22° 47' 9-96472 is to the tangent of the course 3½ points 9-87020 so is the difference of latitude 694 2-84136 to the difference of longitude 558° 3' 2-74684

Longitude of St Antonio, 24° 25' W. Difference of longitude, 9 18 W. Longitude come to, 33 43 W.

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

Example. A ship from latitude 26° 30' N., and longitude 45° 30' W., sailed N.E. ½ N. till her departure is 216 miles. Required the distance run, and latitude and longitude come to.

By Construction.

With the course and departure construct the triangle ABC (fig. 20), and the distance and difference of latitude, being measured, will be found equal to 340 and 263 respectively. Hence the latitude come to is 30° 53', and middle latitude 28° 42'. Now make the angle BCD equal to the middle latitude, and the difference of longitude DC applied to the scale will measure 246'.

---

1 For R : cosine mid. lat. ii diff. of long. i departure; And diff. of lat. i dep. ii R : tangent course; Hence diff. of lat. i cosine mid. lat. ii diff. of long. i tang. course, Or diff. of lat. i diff. of long. ii cosine mid. lat. i tang. course.

* This proportion is obvious, by considering the whole figure as an oblique-angled plane triangle. By Calculation.

To find the distance.

As the sine of the course \( \frac{3}{4} \) points \( = 980236 \) is to radius \( = 1000000 \) so is the departure \( = 216 \) \( = 233445 \)

To the distance \( = 340.5 \) \( = 253209 \)

To find the difference of latitude.

As the tangent of the course \( \frac{3}{4} \) points \( = 991417 \) is to radius \( = 1000000 \) so is the departure \( = 216 \) \( = 233445 \)

To the difference of lat. \( = 263.2 \) \( = 242028 \)

Latitude sailed from, \( 26^\circ 30' N. \) Difference of latitude, \( 4^\circ 23' N. \) Mid. lat. \( = 28^\circ 42' N. \)

Latitude come to, \( 30^\circ 53' N. \) \( = 28^\circ 42' N. \)

To find the difference of longitude.

As radius \( = 1000000 \) is to the secant of the mid. lat. \( = 28^\circ 42' \) so is the departure \( = 216 \) \( = 233445 \)

To the difference of longitude \( = 246.2 \) \( = 239138 \)

Longitude left, \( 45^\circ 30' W. \) Difference of longitude, \( 4^\circ 6' E. \)

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

Prob. V. Given both latitudes and distance to find the course and difference of longitude.

Example. From Cape Sable, in latitude \( 43^\circ 24' N. \) and longitude \( 65^\circ 39' W. \), a ship sailed 246 miles on a direct course between the south and east, and was then by observation in latitude \( 40^\circ 48' N. \). Required the course and longitude in.

Latitude Cape Sable, \( 43^\circ 24' N. \) Latitude by observation, \( 40^\circ 48' N. \)

Difference of latitude, \( 2^\circ 36' = 156', \) sum \( 84^\circ 12' \) Middle latitude, \( 42^\circ 6' \)

By Construction.

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

By Calculation.

To find the course.

As the distance \( = 246 \) is to the difference of latitude \( = 156 \) so is radius \( = 1000000 \)

To the cosine of the course \( = 50^\circ 39' \) \( = 980219 \)

To find the difference of longitude.

As the cosine of middle latitude \( = 42^\circ 6' \) is to the sine of the course \( = 50^\circ 39' \) so is the distance \( = 246 \) \( = 239093 \)

To the difference of longitude \( = 256.4 \) \( = 240888 \)

Longitude Cape Sable, \( 65^\circ 39' W. \) Difference of longitude, \( 4^\circ 16' E. \)

Longitude come to, \( 61^\circ 23' W. \)

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

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

Latitude Cape St Vincent, \( 37^\circ 2' N. \) Latitude come to, \( 18^\circ 16' N. \)

Difference of latitude, \( 18^\circ 46' = 2126, \) sum \( 55^\circ 18' \) Middle latitude, \( 27^\circ 39' \)

By Construction.

Make AB (fig. 22) equal to the difference of latitude 1126 miles, and BC equal to the departure 838, and join AC; draw CD so as to make an angle with CB equal to the middle latitude \( 27^\circ 39' \). Then the course being measured on chords is about \( 36^\circ 5' \), and the distance and difference of longitude, measured on the line of equal parts, will be found to be 1403 and 946 respectively.

By Calculation.

To find the course.

As the difference of latitude \( = 1126 \) is to the departure \( = 838 \) so is radius \( = 1000000 \)

To the tangent of the course \( = 36^\circ 39' \) \( = 987170 \)

To find the distance.

As radius \( = 1000000 \) is to the secant of the course \( = 36^\circ 39' \) so is the difference of latitude \( = 1126 \) \( = 305154 \)

To the distance \( = 1403 \) \( = 314720 \)

To find the difference of longitude.

As radius \( = 1000000 \) is to the secant of mid. lat. \( = 27^\circ 39' \) so is the departure \( = 838 \) \( = 297590 \)

Longitude Cape St Vincent, \( 9^\circ 2' W. \) Difference of longitude, \( 15^\circ 46' W. \)

Longitude come to, \( 24^\circ 48' W. \)

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

Example. A ship from Bordeaux, in latitude \( 44^\circ 50' N. \), and longitude \( 0^\circ 35' W. \), sailed between the north and west \( 37\frac{1}{2} \) miles, and made 210 miles of westing. Required the course, and the latitude and longitude come to.

By Construction.

With the given distance and departure make the triangle ABC (fig. 23). Now the course being measured on the line of chords is about \( 341^\circ \), and the difference of latitude on the line of numbers is 309 miles; hence the latitude come to is \( 49^\circ 59' N. \), and middle \( 47^\circ 25' \). Then make the angle BCD equal to \( 47^\circ 25' \), and DC being measured will be 310 miles, the difference of longitude. NAVIGATION.

Middle Latitude Sailing:

By Calculation.

To find the course.

As the distance is to the departure so is radius

\[ \text{Distance} : \text{Departure} = \text{Radius} : \text{Distance} \]

\[ \frac{\text{Distance}}{\text{Departure}} = \frac{\text{Radius}}{\text{Distance}} \]

\[ \text{Distance}^2 = \text{Departure} \times \text{Radius} \]

\[ \text{Distance} = \sqrt{\text{Departure} \times \text{Radius}} \]

\[ \text{Distance} = \sqrt{374 \times 210} = 518 \text{ miles} \]

To find the difference of latitude.

As radius is to the cosine of the course so is the distance

\[ \text{Cosine} = \frac{\text{Distance}}{\text{Radius}} \]

\[ \text{Distance} = \text{Cosine} \times \text{Radius} \]

\[ \text{Distance} = 9.91772 \times 10^{-6} \times 374 = 3.69 \text{ miles} \]

Latitude of Bordeaux, \( 44^\circ 50' \text{ N.} \)

Difference of latitude, \( 5^\circ 9 \text{ N. half} \)

Latitude come to, \( 49^\circ 59' \text{ N. mid. lat.} \)

To find the difference of longitude.

As radius is to the secant of mid. lat. so is the departure

\[ \text{Secant} = \frac{\text{Departure}}{\text{Radius}} \]

\[ \text{Departure} = \text{Secant} \times \text{Radius} \]

\[ \text{Departure} = \frac{1}{\cos(47^\circ 25')} \times 210 = 232.22 \text{ miles} \]

Longitude of Bordeaux, \( 0^\circ 35' \text{ W.} \)

Difference of longitude, \( 5^\circ 10' \text{ W.} \)

Longitude in, \( 5^\circ 45' \text{ W.} \)

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

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

Longitude left, \( 1^\circ 10' \text{ W.} \)

Longitude come to, \( 5^\circ 26' \text{ E.} \)

Difference of longitude, \( 6^\circ 36' = 396 \text{ miles} \)

By Construction.

Make BC (fig. 24) equal to the departure 220, and CD equal to the difference of longitude 396; then the middle latitude BCD being measured, will be found equal to \( 56^\circ 15' \); hence the latitude come to is \( 57^\circ 34' \), and difference of latitude \( 158' \).

Now make AB equal to 158, and join AC, which, applied to the scale, will measure 271 miles. Also the course BAC, being measured on chords, will be found equal to \( 54^\circ 3' \).

By Calculation.

To find the middle latitude.

As the departure is to the difference of longitude so is radius

\[ \text{Departure} : \text{Difference of Longitude} = \text{Radius} : \text{Distance} \]

\[ \text{Distance} = \frac{\text{Departure} \times \text{Difference of Longitude}}{\text{Radius}} \]

\[ \text{Distance} = \frac{220 \times 396}{10^{-6}} = 8.7 \text{ miles} \]

To find the middle latitude.

Double middle latitude, \( 112^\circ 30' \)

Latitude left, \( 54^\circ 56' \)

Latitude come to, \( 57^\circ 34' \)

Difference of latitude, \( 2^\circ 38' = 158 \text{ miles} \)

To find the course.

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

\[ \text{Difference of Latitude} : \text{Departure} = \text{Radius} : \text{Distance} \]

\[ \text{Distance} = \frac{\text{Departure} \times \text{Difference of Latitude}}{\text{Radius}} \]

\[ \text{Distance} = \frac{220 \times 158}{10^{-6}} = 3.5 \text{ miles} \]

To find the distance.

As radius is to the secant of the course so is the difference of latitude

\[ \text{Secant} = \frac{\text{Distance}}{\text{Radius}} \]

\[ \text{Distance} = \text{Secant} \times \text{Radius} \]

\[ \text{Distance} = \frac{1}{\cos(54^\circ 19')} \times 10^{-6} = 2.43 \text{ miles} \]

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

Example. A ship from a port in north latitude, sailed S.E. \( \frac{1}{2} \text{ S.} 438 \text{ miles, and differed her longitude } 7^\circ 28'. \text{ Required the latitude sailed from, and that come to.} \)

By Construction.

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

By Calculation.

To find the difference of latitude.

As radius is to the cosine of the course so is the distance

\[ \text{Cosine} = \frac{\text{Distance}}{\text{Radius}} \]

\[ \text{Distance} = \text{Cosine} \times \text{Radius} \]

\[ \text{Distance} = 9.86979 \times 10^{-6} \times 438 = 4.3 \text{ miles} \]

To find the middle latitude.

As the difference of longitude is to the distance so is the sine of the course

\[ \text{Sine} = \frac{\text{Distance}}{\text{Radius}} \]

\[ \text{Distance} = \text{Sine} \times \text{Radius} \]

\[ \text{Distance} = \frac{1}{\sin(48^\circ 58')} \times 9.82708 = 2.6 \text{ miles} \]

Half difference of latitude, \( 2^\circ 42' \)

Latitude sailed from, \( 51^\circ 40' \)

Latitude come to, \( 46^\circ 16' \)

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

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

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

Example. A ship from Cape Clear, in latitude \( 51^\circ 15' \text{ N.}, \) longitude \( 9^\circ 46' \text{ W.}, \) sailed as follows:—S. W. by S. \( 54^\circ \text{ miles, W. by N. } 63 \text{ miles, N. N. W. } 48 \text{ miles, and N. E. } \frac{1}{2} \text{ E. } 85 \text{ miles. Required the latitude and longitude come to.} \) The above method is that always practised to find the difference of longitude made good in the course of a day's run, and will, no doubt, give the difference of longitude tolerably exact in any probable run a ship may make in that time, especially near the equator. But in a high latitude, when the distances are considerable, this method is not to be depended on. To illustrate this, let a ship be supposed to sail from latitude $57^\circ N.$, as follows: E. 240 miles, N. 240 miles, W. 240 miles, and S. 240 miles; then, by the above method, the ship will be come to the same place she left. It will, however, appear evident from the following consideration, that this is by no means the case; for let two ships, from latitude $61^\circ N.$, and distant 240 miles, sail directly south till they are in latitude $57^\circ N.$; now, their distance, being computed by Problem IV. of Parallel Sailing, will be 269.6 miles; and therefore, if the ship sailed as above, she will be 29.6 miles west of the place sailed from, and the error in longitude will be equal to $240 \times \secant 61^\circ - \secant 57^\circ = 29.6 \times \secant 57^\circ = 54.4$.

Theorems might be investigated for computing the errors to which the above method is liable. These corrections may, however, be avoided, by using the following method.

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

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

**Example.** A ship from Halliford in Iceland, in lat. $64^\circ 30' N.$, long. $27^\circ 15' W.$, sailed as follows: S. S. W. 46 miles, S. W. 61 miles, S. by W. 59 miles, S. E. by E. 86 miles, S. by E. $\frac{1}{2}$ E. 76 miles. Required the lat. and long. come to.

| Courses | Dist. | Diff. of Lat. | Departure | |---------|-------|--------------|-----------| | S. S. W.| 46 | 42-5 | 17-6 | | S. W. | 61 | 43-1 | 43-1 | | S. by W.| 59 | 57-9 | 11-5 | | S. E. by E.| 86 | 47-8 | 71-5 | | S. by E. $\frac{1}{2}$ E.| 76 | 72-7 | 22-0 |

| Courses | Dist. | Diff. of Lat. | Departure | |---------|-------|--------------|-----------| | Latitude Halliford | 64° 30' N. | 4 24 S. | | Latitude in | 60 6 N. | | Sum | 124 36 | | Middle latitude | 62 18 | | Now, to middle latitude 62° 18', and departure 21-3, the difference of longitude is | 46 E. | | Longitude Halliford | 27 15 W. | | Longitude in | 26 29 |

The error of common method in this example is 12'. It was observed, in Middle Latitude Sailing, that the difference of longitude made upon an oblique rhumb could not be exactly determined by using the middle latitude. In Mercator's Sailing the difference of longitude is very easily found, and the several problems of sailing are resolved with the utmost accuracy, by the assistance of Mercator's chart or equivalent tables.

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

For by parallel sailing,

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

And because the equator and meridian on the globe are equal; therefore,

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

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

Hence

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

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

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

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

Therefore, by 12th V. Euclid,

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

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

Since \(CD : LK :: R : \text{secant } LD\), fig. 15.

And in the triangle CED,

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

Therefore \(ED : LK :: R^2 : \text{secant } LD \times \text{tangent } CED.\)

Hence \(LK = \frac{ED \times \text{secant } LD \times \text{tangent } CED}{R^2} = \frac{ED \times \text{secant } LD}{R} \times \text{tangent } CED.\)

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

Whence \(MN = \text{mer. diff. of lat.} \times \text{tangent } CED.\)

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

This equation answers to a right-angled rectilineal triangle, having an angle equal to the course; the adjacent side equal to the meridional difference of latitude, and the Mercator's opposite side the difference of longitude. This triangle is, therefore, similar to a triangle constructed, with the course and difference of latitude, according to the principles of plane sailing, and the homologous sides will be proportional.

Hence, if, in fig. 26, the angle \(A\) represents the course, \(AB\) the difference of latitude, and if \(AD\) be made equal to the meridional difference of latitude, then \(DE\), drawn perpendicular to \(AD\), meeting the distance produced to \(E\), will be the difference of longitude.

The meridional parts on the terrestrial spheroid of \(\frac{1}{300}\) th of compression to any latitude \(l\), may be found from the following formula:

\[ P = 7915 \times 705 \log \tan (45^\circ + \frac{l}{2}) - 22^\circ 88 \sin l - 0^\circ 0508 \sin 3l, \ldots \]

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

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

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

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

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

1678 Longitude Cape Finisterre, \(9^\circ 17'\) W.

Longitude Port Praya, \(23^\circ 29'\) W.

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

**By Construction.**

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

**By Calculation.**

To find the course.

As the meridional difference of lat. \(1948\) is to the difference of longitude \(852\) so is radius \(10^\circ 00000\)

to the tangent of the course \(23^\circ 37'\) \(= 9^\circ 64085\)

To find the distance.

As radius \(10^\circ 00000\) is to the secant of the course \(23^\circ 37'\) \(= 10^\circ 03798\) so is the difference of latitude \(1678\) \(= 3^\circ 22479\)

to the distance \(1831\) \(= 3^\circ 26277\)

---

1 This is not strictly true; for instead of taking the sum of the secants of every minute in the distance of the given parallel from the equator, the sum of the secants of every point of latitude should be taken. **NAVIGATION**

**Prob. II.** Given the course and distance sailed from Sailing, a place whose situation is known, to find the latitude and longitude of the place come to.

*Example.* A ship from Cape Hinlopen in Virginia, in latitude 38° 47' N., longitude 75° 4' W., sailed 267 miles N. E. by N. Required the ship's present place.

**By Construction.**

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

**By Calculation.**

To find the difference of latitude.

| As radius | 10-00000 | |-----------|----------| | is to the cosine of the course | 3 points | | so is the distance | 267 | | to the difference of latitude | 222 |

Lat. Cape Hinlopen, = 38° 47' N. Mer. parts, 2528

Difference of lat. = 3 42 N.

Latitude come to, = 42° 29' N. Mer. parts, 2821

Meridional difference of latitude, = 293

To find the difference of longitude.

| As radius | 10-00000 | |-----------|----------| | is to tangent of the course | 3 points | | so is the mer. diff. of latitude | 293 | | to the difference of longitude | 195-8 |

Longitude Cape Hinlopen, = 75° 4' W.

Difference of longitude, = 3 16 E.

Longitude come to, = 71° 48' W.

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

*Example.* A ship from Port Canso in Nova Scotia, in latitude 45° 20' N., longitude 60° 55' W., sailed S. E. ¼ S., and, by observation, was found to be in latitude 41° 14' N. Required the distance sailed and longitude come to.

Lat. Port Canso, = 45° 20' N. Mer. parts, 3058

Lat. in, by observation, = 41° 14' N. Mer. parts, 2720

Difference of lat. = 4° 6' = 246 Mer. diff. lat. 338

**By Construction.**

Make AB (fig. 28) equal to 246, and AD equal to 338; draw AE, making an angle with AD equal to 3½ points, and draw BC, DE perpendicular to AD. Now AC being applied to the scale, will measure 332, and DE 306.

**By Calculation.**

To find the distance.

| As radius | 10-00000 | |-----------|----------| | is to the secant of the course | 3½ points | | so is the difference of latitude | 246 |

To the distance = 332

To find the difference of longitude.

| As radius | 10-00000 | |-----------|----------| | is to the tangent of the course | 3½ points | | so is the mer. diff. of latitude | 338 |

Longitude Port Canso, = 60° 53' W.

Difference of longitude, = 5° 6 E.

Longitude in, = 55° 49' W.

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

*Example.* A ship sailed from Saltee, in latitude 33° 58' N., longitude 6° 20' W., the corrected course was N.W. by W. ½ W., and departure 420 miles. Required the distance run, and the latitude and longitude come to.

**By Construction.**

With the course and departure construct the triangle ABC (fig. 29); now AC and AB being measured, will be found to be equal to 476 and 224 respectively; hence the latitude come to is 37° 42' N., and meridional difference of latitude 276. Make AD equal to 276; and draw DE perpendicular thereto, meeting the distance produced in E; then DE applied to the scale will be found to measure 516'. The longitude in is, therefore, 14° 56' W.

**By Calculation.**

To find the distance.

| As radius | 10-00000 | |-----------|----------| | is to the cosecant of the course | 5½ points | | so is the departure | 420 |

To the distance = 476-2

To find the difference of latitude.

| As radius | 10-00000 | |-----------|----------| | is to the cotangent of the course | 5½ points | | so is the departure | 420 |

to the difference of latitude = 224-5

Lat. of Saltee, = 33° 58' N. Mer. parts, 2169

Diff. of lat. = 3° 44' N.

Latitude in, = 37° 42' N. Mer. parts, 2445

Mer. difference of latitude, = 276

To find the difference of longitude.

| As radius | 10-00000 | |-----------|----------| | is to the tangent of the course | 5½ points | | so is the mer. diff. of latitude | 276 |

to the difference of longitude = 516-3 **NAVIGATION**

**Longitude of Sallee**, 6° 20' W. **Difference of longitude**, 8° 36' W.

**Longitude in**, 14° 56' W.

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

*Example.* A ship from St Mary's, in latitude 36° 57' N., longitude 25° 9' W., sailed on a direct course between the north and east 1162 miles, and was then by observation in latitude 49° 57' N. Required the course steered, and longitude come to.

Lat. of St Mary's, 36° 57' N. Mer. parts, 3470 Lat. come to, 49° 57' N. Mer. parts, 2389

**Difference of lat.** 13° 0' Mer. diff. lat. 1081

---

**By Calculation.**

To find the course.

As the distance 1162 3-06521 is to the difference of latitude 780 2-89209 so is radius 10-00000

to the cosine of the course 47° 50' 9-82688

To find the difference of longitude.

As radius 10-00000 is to the tangent of the course 47° 50' 10-04302 so is the mer. diff. of latitude 1081 3-03383

to the difference of longitude 1194 3-07685

Longitude of St Mary's, 25° 9' W. Difference of longitude, 19° 54' E.

Longitude in, 5° 15' W.

**Prob. VI.** Given the latitudes of two places, and the departure; to find the course, distance, and difference of longitude.

*Example.* From Aberdeen, in latitude 57° 9' N., longitude 2° 8' W., a ship sailed between the south and east till her departure was 146 miles, and latitude came to 53° 32' N. Required the course and distance run, and longitude come to.

Latitude Aberdeen, 57° 9' N. Mer. parts, 4199 Latitude come to, 53° 32' N. Mer. parts, 3817

**Difference of latitude,** 3° 37' = 217' Mer. diff. lat. 382

---

**By Construction.**

With the difference of latitude 217 m. and departure 146 m. construct the triangle ABC; make AD equal to 382, draw DE parallel to BC, and produce AC to E; then the course BAC will measure 33° 56', the distance AC 261, and the difference of longitude DE 257.

---

**By Calculation.**

To find the course.

As the difference of latitude 217 2-33646 is to the departure 146 2-16435 so is radius 10-00000

to the tangent of the course 33° 56' 9-82789

To find the distance.

As radius 10-00000 is to the secant of the course 33° 56' 10-08109 so is the difference of latitude 217 2-33646

to the distance 261-5 2-41755

To find the difference of longitude.

As the difference of latitude 217 2-33646 is to the mer. diff. of latitude 382 2-33206 so is the departure 146 2-16435

to the difference of longitude 257 2-40995

Longitude of Aberdeen, 2° 8' W. Difference of longitude, 4° 17' E.

Longitude come to, 2° 9' E.

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

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

**By Construction.**

With the distance and departure make the triangle ABC as formerly. Now the course BAC being measured by means of a line of cords, will be found equal to 43° 21'; and the difference of latitude applied to the scale of equal parts will measure 183'; hence the latitude come to is 37° 48' N., and meridional difference of latitude 237'. Make AD equal to 237, and complete the figure, and the difference of longitude DE will measure 224'; hence the longitude in is 10° 30' E.

**By Calculation.**

To find the course.

As the distance 252 2-40140 is to the departure 173 2-23805 so is radius 10-00000

to the sine of the course 43° 21' 9-86665

To find the difference of latitude.

As radius 10-00000 is to the cosine of the course 43° 21' 9-86164 so is the distance 252 2-40140

to the difference of latitude 183-2 2-26304

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

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

Meridional difference of latitude, 237 To find the difference of longitude.

As radius is to the tangent of the course so is the mer. diff. of latitude

10-00000 43° 21' 237 9-97497 2-37475

to the difference of longitude 223° 7' 2-34972

Longitude of Naples, 14° 14'E.

Difference of longitude, 3° 44 W.

Longitude in, 10° 30 E.

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

Example. A ship from Tercera, in latitude 38° 45' N., longitude 27° 6' W., sailed on a direct course, which, when corrected, was N. 32° E., and is found by observation to be in longitude 18° 24' W. Required the latitude come to, and distance sailed.

Longitude of Tercera, 27° 6' W.

Longitude in, 18° 24' W.

Difference of longitude, 8° 42' = 522

By Construction.

Make the right-angled triangle ADE, having the angle A equal to the course 32°, and the side DE equal to the difference of longitude 522; then AD will measure 835, which, added to the meridional parts of the latitude left, will give those of the latitude come to 48° 46'; hence the difference of latitude is 601; make AB equal thereto, to which let BC be drawn perpendicular; then AC applied to the scale will measure 708 miles.

By Calculation.

To find the meridional difference of latitude.

As radius is to the co-tangent of the course so is the difference of longitude

10-00000 32° 0' 5 22 10-20421 2-71767

to the mer. difference of latitude 8352 2-92188

Latitude of Tercera, 38° 45' N. Mer. parts, 2526 Mer. diff. of lat. 835

Latitude come to, 48° 46' N. Mer. parts, 3361

Difference of latitude, 10° 1' = 601 miles.

To find the distance.

As radius is to the secant of the course so is the difference of latitude

10-00000 32° 0' 601 10-07158 2-77887

to the distance 707° 7' 2-85045

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

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

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

Example. A ship from Port St Julian, in latitude 49° 10' S. longitude 68° 44' W., sailed as follows: E. S. E. 53 miles, S. E. by S. 74 miles, E. by N. 68 miles, S. E. by E. 1/2 E. 47 miles, and E. 84 miles. Required the ship's present place.

| Courses | Dist. | Diff. of Lat. | Departure | |---------|-------|--------------|-----------| | E. S. E. | 53 | 20° 3' | 49° 0' | | S. E. by S. | 74 | 61° 5' | 41° 1' | | E. by N. | 68 | 13° 3' | 66° 7' | | S. E. by E. 1/2 E. | 47 | 22° 1' | 41° 5' | | E. | 84 | | 84° 0' |

13° 3' 103° 9' 282° 3'

S. 72° E. 197° 90° 6' = 1° 31' 49° 10' S.m.pt. 3397

Latitude left................. 50° 41 S.m.pt. 3539

Latitude come to.............. 61° 27 W.

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

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

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

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

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

Ex. 1. Four days ago we took our departure from Faro Head, in latitude 58° 40' N. and longitude 4° 50' W. and since have sailed as follows: N. W. 32 miles, W. 69 miles, W. N. W. 93 miles, W. by S. 77 miles, S. W. 58 miles, and W. 3° S. 49 miles. Required our present latitude and longitude. ### NAVIGATION

#### Traverse Table

| Courses | Dist. | Diff. of Lat. | Departure | |---------|-------|---------------|-----------| | | | N. | S. | E. | W. | | N.W. | 32 | 22-6 | 22-6 | 58° 40' | 4370 | | W. | 69 | | 69-0 | 59 | 4415 | | W.N.W. | 93 | 35-6 | | 59 | 4415 | | W.by S. | 77 | 15-0 | | 59 | 4484 | | S.W. | 58 | 41-0 | | 59 | 4454 | | W. 4S | 49 | 7-2 | | 59 | 4374 |

#### Longitude Table

| Successive Latitudes | Merid. Parts | Merid. Diff. Lat. | Diff. of Longitude | |----------------------|--------------|-------------------|--------------------| | | | | E. | W. | | 58° 40' | 4370 | | | | 59 | 4415 | | | | 59 | 4415 | | | | 59 | 4484 | | | | 59 | 4454 | | | | 59 | 4374 | | |

Longitude of Faro Head: 4° 50' W.

Difference of longitude: 11° 4' W.

Longitude in: 15° 54' W.

---

**Ex. 2.** A ship from latitude 78° 15' N. longitude 28° 14' E. sailed the following courses and distances. The latitude come to is required, and the longitude, by both methods; the bearing and distance of Hacluit's headland, in latitude 79° 55' N. longitude 11° 55' E. is also required.

#### Traverse Table

| Courses | Dist. | Diff. of Latitude | Departure | |---------|-------|------------------|-----------| | | | N. | S. | E. | W. | | W.N.W. | 154 | 58-9 | 142-3 | | | S.W. | 96 | 67-9 | 67-9 | | | N.W. 1/2 W. | 89 | 56-4 | 68-8 | | | N.by E. | 110 | 107-9 | 21-5 | | | N.W. 3/4 N. | 56 | 45-0 | 33-4 | | | S.by E. 3/4 E. | 78 | 73-4 | 26-3 | |

By Rule I.

Latitude left: 78° 15' N. Mer. parts: 7817

Diff. of latitude: 2° 7 N.

Lat. come to: 80° 22 N. Mer. parts: 8504

Meridional diff. of latitude: 687

As difference of lat.: 126-2° 2-10346

is to mer. diff. of lat.: 687° 2-83696

so is the departure: 264-6° 2-42256

to diff. of longitude: 1432° 3-15606

23° 52' W.

Longitude left: 28° 14' E.

Longitude in: 4° 22' E.

The error of this method, in the present example, is therefore 1° 29'.

#### Longitude Table

| Successive Latitudes | Merid. Parts | Merid. Diff. of Lat. | Diff. of Longitude | |----------------------|--------------|---------------------|--------------------| | | | | E. | W. | | 78° 15' | 7817 | | | | 79 | 8120 | | | | 79 | 7774 | | | | 79 | 8056 | | | | 80 | 8676 | | | | 81 | 8970 | | | | 80 | 8504 | | |

Longitude left: 28° 14' E.

Difference of longitude: 22° 29' W.

Longitude in: 5° 45' E.

To find the bearing and distance of Hacluit's headland.

Lat H. H. = 79° 55' N. M.P. 8347 Lon. 11° 55' E.

Lat. ship = 80° 22 N. M.P. 8504 Lon. 5° 45' E.

Diff. lat. = 0° 27 M.D.L. 157 D.L. 6° 10'

Now to 78° 5 half the meridional difference of latitude, and 185° 0 half the difference of longitude, the course 67°, and opposite to the difference of latitude 27', the distance is 69 miles. Hence Hacluit's headland bears S. 67° E. distant 69 miles.

---

**CHAP. VII.—CONTAINING THE METHOD OF RESOLVING THE SEVERAL PROBLEMS OF MERCATOR'S SAILING, BY THE ASSISTANCE OF A TABLE OF LOGARITHMIC TANGENTS.**

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

**Rule.** To the arithmetical complement of the logarithm of the distance, add the logarithm of the difference of longitude in minutes, and the log. cosine of the given latitude; the sum, rejecting radius, will be the log. sine of the approximate course.

To the given latitude taken as a course in the traverse table, and half the difference of longitude in a distance column, the corresponding departure will be the first correction of the course, which is subtractive if the given latitude is the least of the two; otherwise additive.

In Table A, under the complement of the course, and... Method of opposite to the first correction in the side column, is the resolving second correction. In the same table find the number answering to the course at the top, and difference of longitude in the side column; and such part of this number being taken as is found in Table B opposite to the given latitude, will be the third correction. Now these two corrections subtracted from the course corrected by the first correction will give the true course.

Now, the course and distance being known, the difference of latitude is found as formerly.

| TABLE A. | TABLE B. | |----------|----------| | Arc. | Lat. | | 1° | | | 2° | | | 3° | | | 4° | | | 5° | | | 6° | | | 7° | | | 8° | |

Example. From latitude 50° N. a ship sailed 290 miles between the south and west, and differed her longitude 5°. Required the course, and latitude come to.

Distance, 290 nautical miles log. 7-53760 Diff. of longitude, 300 log. 2-47712 Latitude, 50° 0' co. 9-80807

Approximate course, 41 41 sine 9-82279 To lat. 50°, and half diff. long. 150 in a dist. col. the first correction in a dep. col. is 115, + 1 55 Approximate course, 41 41 Cor. 1 55 In Table A to co. course 48° and first corr. 1° 55', the second direction is To course 41° and diff. long. 5°, the number is 15, of which ½ (Table B) being taken, gives True course, S. 43 31 W.

To find the difference of latitude.

As radius 10-00000 is to the cosine of the course 43° 33' 9-86020 so is the distance 290 2-46240 to the difference of latitude 210-2 2-32260 Latitude left, 50° 0' N. Difference of latitude, 3° 30 S. Latitude come to, 46° 30 N.

This problem was proposed and resolved by Mr Robert Hues, in his Treatise on the Globes, printed at London in the year 1639, p. 181.

It was afterwards proposed by Dr Halley, in the second volume of the Miscellanea Curiosa, p. 35, in the following words:

A ship sails from a given latitude, and, having run a certain number of leagues, has altered her longitude by a given angle; it is required to find the course steered. And he then adds: The solution hereof would be very acceptable, if not to the public, at least to the author of this tract, being likely to open some further light into the mysteries of geometry.

Since that time, this problem has been solved in an indirect manner, by several writers on navigation, and others; as Monsieur Bouguer, in his Nouveau Traité de Navigation; Mr Robertson, in the second volume of his Elements of Navigation; Mr Emerson, in his Theory of Navigation, which accompanies his Mathematical Principles of Geography; Mr Israel Lyons, in the Nautical Almanac for 1772; and Monsieur Bezout, with the assistance of Mr Attwood, has given the first direct solution of this problem.

For a comparison of the various solutions which have hitherto been made of this problem, the reader is referred to that by Dr Mackay, in the fourth and sixth volumes of Baron Maseres' Scriptores Logarithmici.

CHAP. VIII.—OF OBLIQUE SAILING.

Oblique sailing is the application of oblique-angled plane triangles to the solution of problems at sea. This sailing will be found particularly useful in going along shore, and in surveying coasts and harbours, &c.

Ex. 1. At 11th A.M. the Giraffe Ness bore W. N. W., and at 2nd P.M. it bore N. W. by N.; the course during the interval S. by W. five knots an hour. Required the distance of the ship from the Ness at each station.

By Construction.

Describe the circle N, E, S, W, Fig. 34. and draw the diameters NS, EW at right angles to each other: from the centre C, which represents the first station, draw the W. N. W. line CF; and from the same point draw CH, S. by W., and equal to 15 miles, the distance sailed. From H draw HF in a N. W. by N. direction, and the point F will represent the Giraffe Ness. Now the distances CF, HF will measure 19-1 and 26-5 miles respectively.

By Calculation.

In the triangle FCH are given the distance CH 15 miles, the angle FCH equal to 9 points, the interval between the S. by W. and W. N. W. points, and the angle CHF equal to 4 points, being the supplement of the angle contained between the S. by W. and N. W. by N. points; hence CFH is 3 points; to find the distances CF, HF.

To find the distance CF.

As the sine of CFH 3 points 9-74474 is to the sine of CHF 4 points 9-84948 so is the distance CH 15 miles 1-17609 to the distance CF 19-07, 1-28083

To find the distance FH.

As the sine of CFH 3 points 9-74474 is to the sine of FCH points 9-99157 so is the distance CH 15 miles 1-17609 to the distance FH 26-48, 1-42292

Ex. 2. Running up Channel E. by S. per compass at the rate of 5 knots an hour. At 11th A.M. the Eddystone NAVIGATION.

By Construction.

Let the point C represent the first station, from which draw the N. by E. line CA, the N. E. by E. line CB, and the E. by S. line CD, which make equal to 25 miles, the distance run in the elapsed time; then from D draw the N. E. by N. line DA intersecting CA in A, which represents the Eddystone; and from the same point draw the N. E. line DB cutting CB in B, which therefore represents the Start. Now the distance AB applied to the scale will measure 22½, and the bearing per compass BAF will measure 73½°.

Many other examples might be given. These and all other cases which can occur in practice are to be resolved by plane trigonometry, from calculating the triangles which the data of the given case afford.

CHAP. IX.—OF WINDWARD SAILING.

Windward sailing is, when a ship by reason of a contrary wind is obliged to sail on different tacks in order to gain her intended port; and the object of this sailing is to find the proper course and distance to be run on each tack.

Ex. The wind at N. W., a ship bound to a port 64 miles to the windward proposes to reach it on three boards; two on the starboard and one on the larboard tack, and each within 5 points of the wind. Required the course and distance on each tack.

By Construction.

Draw the N. W. line CA (fig. 36) equal to 64 miles; from C draw CB W. by S., and from A draw AD parallel thereto, and in an opposite direction; bisect AC in E; and draw BED parallel to the N. by E. rhumb, meeting CB, AD in the points B and D; then CB = AD applied to the scale will measure 36½ miles, and BD = 2 CB = 72½ miles.

CHAP. X.—OF CURRENT SAILING.

The computations in the preceding chapters have been performed upon the assumption that the water has no motion. This may no doubt answer tolerably well in those places where the ebbsings and flowings are regular, as then the effect of the tide will be nearly counterbalanced. But in places where there is a constant current or setting of the sea towards the same point, an allowance for the change of the ship's place arising therefrom must be made. And the method of resolving these problems, in which the effect of a current or heave of the sea is taken into consideration, is called current sailing.

In a calm, it is evident a ship will be carried in the direction and with the velocity of the current. Hence, if a ship sails in the direction of the current, her rate will be augmented by the rate of the current; but if sailing directly against it, the distance made good will be equal to the difference between the ship's rate as given by the log and that of the current. And the absolute motion of the ship will be a-head if her rate exceeds that of the current; but if less, the ship will make sternway. If the ship's course be oblique to the current, the distance made good in a given time will be represented by the third side of a triangle, whereof the distance given by the log, and the drift of the current in the same time, are the other sides; and the true course will be the angle contained between the meridian and the line actually described by the ship.

Ex. 1. A ship sailed N. N. E. at the rate of 8 knots an hour during 18 hours, in a current setting N. W. by W. 2½ miles an hour. Required the course and distance made good.

By Construction.

Draw the N. N. E. line CA (fig. 37) equal to 18 × 8 = 144 miles; and from A draw AB parallel to the N. W. by W. rhumb, and equal to 18 × 2½ = 45 miles; now BC being joined will be the distance, and NCB the course. The first of these will measure 159 miles and the second 6° 23'.

Ex. 2. A ship from latitude 38° 20' N. sailed 24 hours in a current setting N. W. by N., and by account is in latitude 38° 42' N., having made 44 miles of easting; but the latitude by observation is 38° 58' N. Required the course and distance made good, and the drift of the current.

By Construction.

Make CE (fig. 38) equal to 22 miles, the difference of latitude by dead reckoning, and EA = 44 miles the departure, and join CA; make CD = 38 miles, the difference of latitude by observation; draw the parallel of latitude DB, and from A draw the N. W. by N. line AB, intersecting DB in B, and AB will be the drift of the current in 24 hours: CB being joined, will be the distance made good, and the angle DCB the true course. Now, AB and CB applied to the scale, will measure 19-2 and 50-5 respectively; and the angle DCB will be 41°.

CHAP. XI.—INSTRUMENTS PROPOSED TO SOLVE THE VARIOUS PROBLEMS IN SAILING, INDEPENDENT OF CALCULATION.

Various methods besides those already given have been proposed, to save the trouble of calculation. One of these methods is by means of an instrument composed of rulers, so disposed as to form a right-angled triangle, having numbers in a regular progression marked on their sides. These instruments are made of different materials, such as paper, wood, brass, &c. and are differently constructed, according to the fancy of the inventor. A number of other instruments, very differently constructed, have been proposed for the same purpose; of these, the rectangular instrument by the late A. Mackay, LL.D. F.R.S.E. &c. is one of the best. It is seldom, however, that any of them are used. The charts usually employed in the practice of navigation are of two kinds, namely, Plane and Mercator's Charts. The first of these is adapted to represent a portion of the earth's surface near the equator, and the last for all portions of the earth's surface. For a particular description of these, see the article Chart; and as these are particularly described under the above article, it is therefore sufficient in this place to describe their use.

Use of the Plane Chart.

Prob. I. To find the latitude of a place on the chart.

Rule. Take the least distance between the given place and the nearest parallel of latitude; now this distance applied the same way on the graduated meridian, from the extremity of the parallel, will give the latitude of the proposed place.

Thus the distance between Bonavista and the parallel of 15°, being laid from that parallel upon the graduated meridian, will reach to 16° 5', the latitude required.

Prob. II. To find the course and distance between two given places on the chart.

Rule. Lay a ruler over the given places, and take the nearest distance between the centre of any of the compasses on the chart and the edge of the ruler; move this extent along so as one point of the compass may touch the edge of the ruler, and the straight line joining their points may be perpendicular thereto; then will the other point show the course. The interval between the places, being applied to the scale, will give the required distance.

Thus the course from Palma to St Vincent will be found to be about S. S. W. 3 W. and the distance 134°, or 795 miles.

Prob. III. The course and distance sailed from a known place being given, to find the ship's place on the chart.

Rule. Lay a ruler over the place sailed from parallel to the rhumb, expressing the given course; take the distance from the scale, and lay it off from the given place by the edge of the ruler; and it will give the point representing the ship's present place.

Thus, suppose a ship had sailed S. W. by W. 160 miles from Cape Palmas; then, by proceeding as above, it will be found that she is in latitude 2° 57' N.

The various other problems that may be resolved by means of this chart require no further explanation, being only the construction of the remaining problems in Plane Sailing on the chart.

Use of Mercator's Chart.

The method of finding the latitude and longitude of a place, and the course or bearing between two given places by this chart, is performed exactly in the same manner as in the plane chart, which see.

Prob. I. To find the distance between two given places on the chart.

Case I. When the given places are under the same meridian.

Rule. The difference or sum of their latitudes, according as they are on the same or on opposite sides of the equator, will be the distance required.

Case II. When the given places are under the same parallel.

Rule. If that parallel be the equator, the difference or sum of their longitudes is the distance; otherwise, take half the interval between the places, lay it off upwards and downwards on the meridian from the given parallel, and Of finding the intercepted degrees will be the distance between the places.

Or, take an equal extent of a few degrees from the meridian on each side of the parallel, and the number of extents, and parts of an extent, contained between the places, being multiplied by the length of an extent, will give the required distance.

Case III. When the given places differ both in latitude and longitude.

Rule. Find the difference of latitude between the given places, and take it from the equator or graduated parallel; then lay a ruler over the two places, and move one point of the compass along the edge of the ruler until the other point just touches a parallel; then the distance between the place where the point of the compass rested by the edge of the ruler, and the point of intersection of the ruler and parallel, being applied to the equator, will give the distance between the places in degrees and parts of a degree, which multiplied by 60 will reduce it to miles.

Prob. II. Given the latitude and longitude in, to find the ship's place on the chart.

Rule. Lay a ruler over the given latitude, and lay off the given longitude from the first meridian by the edge of the ruler, and the ship's present place will be obtained.

Prob. III. Given the course sailed from a known place, and the latitude in, to find the ship's present place on the chart.

Rule. Lay a ruler over the place sailed from, in the direction of the given course, and its intersection with the parallel of latitude arrived at will be the ship's present place.

Prob. IV. Given the latitude of the place left, and the course and distance sailed, to find the ship's present place on the chart.

Rule. The ruler being laid over the place sailed from, and in the direction of the given course, take the distance sailed from the equator, put one point of the compass at the intersection of any parallel with the ruler, and the other point of the compass will reach to a certain place by the edge of the ruler. Now this point remaining in the same position, draw in the other point of the compass until it just touch the above parallel when swept round; apply this extent to the equator, and it will give the difference of latitude. Hence the latitude in will be known, and the intersection of the corresponding parallel with the edge of the ruler will be the ship's present place.

The other problems of Mercator's Sailing may be very easily resolved by this chart; but as they are of less use than those given, they are therefore omitted, and may serve as an exercise to the student.

BOOK II.

CONTAINING THE METHOD OF FINDING THE LATITUDE AND LONGITUDE OF A SHIP AT SEA, AND THE VARIATION OF THE COMPASS.

CHAP. I.—METHOD OF FINDING THE LATITUDE AT SEA.

Sect. I.—Of Hadley's Quadrant.

Hadley's quadrant is the chief instrument in use at present for observing altitudes at sea. The form of this instrument, according to the present mode of construction, is an octagonal sector of a circle, and therefore contains 45 degrees; but because of the double reflection, the limb... finding is divided into 90 degrees. See Astronomy and Quadrant. Fig. 39 represents a quadrant of the common construction, of which the following are the principal parts.

1. ABC, the frame of the quadrant. 2. BC, the arch or limb. 3. D, the index; a b, the subdividing scale. 4. E, the index-glass. 5. F, the fore horizon-glass. 6. G, the back horizon-glass. 7. K, the coloured or dark glasses. 8. HI, the vanes or sights.

Fig. 39.

Of the Frame of the Quadrant.

The frame of the quadrant consists of an arch BC, firmly attached to the two radii AB, AC, which are bound together by the braces LM, in order to strengthen it, and prevent it from warping.

Of the Index D.

The index is a flat bar of brass, and turns on the centre of the octant: at the lower end of the index there is an oblong opening; to one side of this opening the vernier scale is fixed, to subdivide the divisions of the arch; at the end of the index there is a piece of brass, which bends under the arch, carrying a spring to make the subdividing scale lie close to the divisions. It is also furnished with a screw to fix the index in any desired position. The best instruments have an adjusting screw fitted to the index, that it may be moved more slowly, and with greater regularity and accuracy, than by the hand. It is proper, however, to observe, that the index must be previously fixed near its right position by the above-mentioned screw.

Of the Index-Glass E.

Upon the index, and near its axis of motion, is fixed a plane speculum, or mirror of glass quicksilvered. It is set in a brass frame, and is placed so that its face is perpendicular to the plane of the instrument. This mirror being fixed to the index, moves along with it, and has its direction changed by the motion thereof; and the intention of this glass is to receive the image of the sun, or any other object, and reflect it upon either of the two horizon-glasses, according to the nature of the observation.

The brass frame with the glass is fixed to the index by the screw e; the other screw serves to replace it in a perpendicular position, if by any accident it has been deranged.

Of the Horizon-Glasses F, G.

On the radius AB of the octant are two small speculums: the surface of the upper one is parallel to the index-glass, and that of the lower one perpendicular thereto, when O on the index coincides with O on the limb. These mirrors receive the reflected rays, and transmit them to the observer.

The horizon-glasses are not entirely quicksilvered; the upper one F is only silvered on its lower half, or that next the plane of the quadrant, the other half being left transparent, and the back part of the frame cut away, that nothing may impede the sight through the unslivered part of the glass. The edge of the foil of this glass is nearly parallel to the plane of the instrument, and ought to be very sharp, and without a flaw. The other horizon-glass is silvered at both ends. In the middle there is a transparent slit, through which the horizon may be seen.

Each of these glasses is set in a brass frame, to which there is an axis passing through the wood-work, and is fitted to a lever on the under side of the quadrant, by which the glass may be turned a few degrees on its axis, in order to set it parallel to the index-glass. The lever has a contrivance to turn it slowly, and a button to fix it. To set the glasses perpendicular to the plane of the instrument, there are two sunk screws, one before and one behind each glass; these screws pass through the plate on which the frame is fixed, into another plate; so that by loosening one and tightening the other of these screws, the direction of the frame with its mirror may be altered, and set perpendicular to the plane of the instrument.

Of the Coloured Glasses K.

There are usually three coloured glasses, two of which are tinged red and the other green. They are used to prevent the solar rays from hurting the eye at the time of observation. These glasses are set in a frame, which turns on a centre, so that they may be used separately or together as the brightness of the sun may require. The green glass is particularly useful in observations of the moon; it may be also used in observations of the sun, if that object be very faint. In the fore observation, these glasses are fixed as in fig. 39; but when the back observation is used, they are removed to N.

Of the two Sight Vanes, H, I.

Each of these vanes is a perforated piece of brass, designed to direct the sight parallel to the plane of the quadrant. That which is fixed at I is used for the fore, and the other for the back observation. The vane I has two holes, one exactly at the height of the silvered part of the horizon-glass, the other a little higher, to direct the sight to the middle of the transparent part of the mirror.

Of the Divisions on the Limb of the Quadrant.

The limb of the quadrant is divided from right to left into 90 primary divisions, which are to be considered as degrees, and each degree is subdivided into three equal parts, which are therefore of 20 minutes each: the intermediate minutes are obtained by means of the scale of divisions at the end of the index. Of the Vernier, or Subdividing Scale.

The dividing scale contains a space equal to 21 divisions of the limb, and is divided into 20 equal parts. Hence the difference between a division on the dividing scale and a division on the limb is one twentieth of a division on the limb, or one minute. The degree and minute pointed out by the dividing scale may be easily found thus.

Observe what minute on the dividing scale coincides with a division on the limb; this division being added to the degree and part of a degree on the limb, immediately preceding the first division on the dividing scale, will be the degree and minute required.

Thus, suppose the fourteenth minute on the dividing scale coincided with a division on the limb, and that the preceding division on the limb to 0 on the vernier was $56^\circ$ $49'$; hence the division shown by the vernier is $56^\circ$ $54'$. A magnifying glass will assist the observer to read off the coinciding divisions with more accuracy.

Adjustments of Hadley's Quadrant.

The adjustments of the quadrant consist in placing the mirrors perpendicular to the plane of the instrument. The fore horizon-glass must be set parallel to the speculum, and the planes of the speculum and back horizon-glass produced must be perpendicular to each other when the index is at 0.

Adjustment I. To set the index-glass perpendicular to the plane of the quadrant.

Set the index towards the middle of the limb, and hold the quadrant so that its plane may be nearly parallel to the horizon; then look into the index-glass, and if the portion of the limb seen by reflection appears in the same plane with that seen directly, the speculum is perpendicular to the plane of the instrument. If they do not appear in the same plane, the error is to be rectified by altering the position of the screws behind the frame of the glass.

Adjustment II. To set the fore horizon-glass perpendicular to the plane of the instrument.

Set the index to 0; hold the plane of the quadrant parallel to the horizon; direct the sight to the horizon, and if the horizons seen directly and by reflection are apparently in the same straight line, the fore horizon-glass is perpendicular to the plane of the instrument; if not, one of the horizons will appear higher than the other. Now if the horizon seen by reflection is higher than that seen directly, release the nearest screw in the pedestal of the glass, and screw up that on the farther side, till the direct and reflected horizons appear to make one continued straight line. But if the reflected horizon is lower than that seen directly, unscrew the farthest, and screw up the nearest screw till the coincidence of the horizons is perfect, observing to leave both screws equally tight, and the fore horizon-glass will be perpendicular to the plane of the quadrant.

Adjustment III. To set the fore horizon-glass parallel to the index-glass, the index being at 0.

Set 0 on the index exactly to 0 on the limb, and fix it in that position by the screw at the under side; hold the plane of the quadrant in a vertical position, and direct the sight to a well-defined part of the horizon; then if the horizon seen in the silvered part coincides with that seen through the transparent part, the horizon-glass is adjusted; but if the horizons do not coincide, unscrew the milled screw in the middle of the lever on the other side of the quadrant, and turn the nut at the end of the lever until both horizons coincide, and fix the lever in this position by tightening the milled screw.

As the position of the glass is liable to be altered by fixing the lever, it will therefore be necessary to re-examine it; and if the horizons do not coincide, it will be necessary either to repeat the adjustment, or rather to find the error of adjustment, or, as it is usually called, the index error; which may be done thus:

Direct the sight to the horizon, and move the index until the reflected horizon coincides with that seen directly; then the difference between 0 on the limb and 0 on the vernier is the index error; which is additive when the beginning of the vernier is to the right of 0 on the limb, otherwise subtractive.

Adjustment IV. To set the back horizon-glass perpendicular to the plane of the instrument.

Put the index to 0; hold the plane of the quadrant parallel to the horizon, and direct the sight to the horizon through the back-sight vane. Now if the reflected horizon is in the same straight line with that seen through the transparent part, the glass is perpendicular to the plane of the instrument. If the horizons do not unite, turn the sunk screws in the pedestal of the glass until they are apparently in the same straight line.

Adjustment V. To set the back horizon-glass perpendicular to the plane of the index-glass produced, the index being at 0.

Let the index be put as much to the right of 0 as twice the dip of the horizon amounts to; hold the quadrant in a vertical position, and apply the eye to the back vane; then if the reflected horizon coincides with that seen directly, the glass is adjusted; if they do not coincide, the screw in the middle of the lever on the other side of the quadrant must be released, and the nut at its extremity turned till both horizons coincide. It may be observed, that the reflected horizon will be inverted; that is, the sea will be apparently uppermost and the sky lowermost.

This method of adjustment is esteemed troublesome, and is often found to be very difficult to perform at sea, on which account the method of observation by the back horizon-glass is seldom or never used.

Use of Hadley's Quadrant.

The altitude of any object is determined by the position of the index on the limb, when by reflection that object appears to be in contact with the horizon.

If the object whose altitude is to be observed be the sun, and if so bright that its image may be seen in the transparent part of the fore horizon-glass, the eye is to be applied to the upper hole in the sight-vane; otherwise, to the lower hole; and in this case the quadrant is to be held so that the sun may be bisected by the line of separation of the silvered and transparent parts of the glass. The moon is to be kept as nearly as possible in the same position; and the image of the star is to be observed in the silvered part of the glass adjacent to the line of separation of the two parts.

There are two different methods of taking observations with the quadrant. In the first of these the face of the observer is directed towards that part of the horizon immediately under the sun, and is therefore called the fore observation. In the other method, the observer's back is to the sun, and it is hence called the back observation. This last method of observation is to be used only when the horizon under the sun is obscured, or rendered indistinct by fog or any other impediment.

In taking the sun's altitude, whether by the fore or back observation, the observer must turn the quadrant about upon the axis of vision, and at the same time turn himself about upon his heel, so as to keep the sun always in that part of the horizon-glass which is at the same distance as the eye from the plane of the quadrant. In this way the reflected sun will describe an arch of a parallel circle round the true sun, the convex side of which will be downwards in the fore observation and upwards in the back; and consequently, when, by moving the index, the lowest point of Of finding the arch in the fore observation, or highest in the back, is made to touch the horizon, the quadrant will stand in a vertical plane, and the altitude above the visible horizon will be properly observed. The reason of these operations may be thus explained: The image of the sun being always kept in the axis of vision, the index will always show on the quadrant the distance between the sun and any object seen directly which its image appears to touch; therefore, as long as the index remains unmoved, the image of the sun will describe an arch everywhere equidistant from the sun in the heavens, and consequently a parallel circle about the sun as a pole. Such a translation of the sun's image can only be produced by the quadrant's being turned about upon a line drawn from the eye to the sun as an axis. A motion of rotation upon this line may be resolved into two, one upon the axis of vision, and the other upon a line on the quadrant perpendicular to the axis of vision; and consequently a proper combination of these two motions will keep the image of the sun constantly in the axis of vision, and cause both jointly to run over a parallel circle about the sun in the heavens; but when the quadrant is vertical, a line thereon perpendicular to the axis of vision becomes a vertical axis; and as a small motion of the quadrant is all that is wanted, it will never differ much in practice from a vertical axis. The observer is directed to perform two motions rather than the single one equivalent to them on a line drawn from the eye to the sun; because we are not capable, while looking towards the horizon, of judging how to turn the quadrant about upon the elevated line going to the sun as an axis, by any other means than by combining the two motions above mentioned, so as to keep the sun's image always in the proper part of the horizon-glass. When the sun is near the horizon, the line going from the eye to the sun will not be far removed from the axis of vision; and consequently the principal motion of the quadrant will be performed on the axis of vision, and the part of motion made on the vertical axis will be but small. Or the contrary, when the sun is near the zenith, the line going to the sun is not far removed from a vertical line, and consequently the principal motion of the quadrant will be performed on a vertical axis, by the observer's turning himself about, and the part of the motion made on the axis of vision will be but small. In intermediate altitudes of the sun, the motions of the quadrant on the axis of vision, and on the vertical axis, will be more equally divided.

Observations taken with the quadrant are liable to errors, arising from the bending and elasticity of the index, and the resistance it meets with in turning round its centre; whence the extremity of the index, on being pushed along the arch, will sensibly advance before the index-glass begins to move, and may be seen to recoil when the force acting on it is removed. Mr Hadley seems to have been apprehensive that his instrument would be liable to errors from this cause; and, in order to avoid them, gives particular directions that the index be made broad at the end next the centre, and that the centre, or axis itself, have as easy a motion as is consistent with steadiness; that is, an entire freedom from looseness, or shake as the workmen term it. By strictly complying with these directions the error in question may indeed be greatly diminished; so far, perhaps, as to render it nearly insensible, where the index is made strong, and the proper medium between the two extremes of a shake at the centre on one hand, and too much stiffness there on the other, is nicely hit: but it cannot be entirely corrected; for to more or less of bending the index will always be subject, and some degree of resistance will remain at the centre, unless the friction there could be totally removed, which is impossible.

To take Altitudes by the Fore Observation.

1. Of the Sun.

Turn down either of the coloured glasses before the horizon-glass, according to the brightness of the sun; direct the sight to that part of the horizon which is under the sun, and move the index until the coloured image of the sun appear in the horizon-glass; then give the quadrant a slow vibratory motion about the axis of vision; move the index until the lower or upper limb of the sun is in contact with the horizon at the lowest part of the arch described by this motion, and the degrees and minutes shown by the index on the limb will be the altitude of the sun.

2. Of the Moon.

Put the index to 0, turn down the green glass, place the eye at the lower hole in the sight-vane, and observe the moon in the silvered part of the horizon-glass; move the index gradually, and follow the moon's reflected image until the enlightened limb is in contact with the horizon at the lower part of the arch described by the vibratory motion as before, and the index will show the altitude of the observed limb of the moon. If the observation is made in the day-time, the coloured glass is unnecessary.

3. Of a Star or Planet.

The index being put to 0, direct the sight to the star through the lower hole in the sight-vane and transparent part of the horizon-glass; move the plane of the quadrant a very little to the left, and the image of the star will be seen in the silvered part of the glass. Now move the index, and the image of the star will appear to descend; continue moving the index gradually until the star is in contact with the horizon at the lowest part of the arch described, and the degrees and minutes shown by the index on the limb will be the altitude of the star. To take Altitudes by the Back Observation.

1. Of the Sun.

Put the stem of the coloured glasses into the perforation between the horizon-glasses; turn down either according to the brightness of the sun, and hold the quadrant vertically; then direct the sight through the hole in the back sight-vane, and the transparent slit in the horizon-glass to that part of the horizon which is opposite to the sun; now move the index till the sun is in the silvered part of the glass, and by giving the quadrant a vibratory motion, the axis of which is that of vision, the image of the sun will describe an arch the convex side of which is upwards; bring the limb of the sun, when in the upper part of this arch, in contact with the horizon, and the index will show the altitude of the other limb of the sun.

2. Of the Moon.

The altitude of the moon is observed in the same manner as that of the sun, with this difference only, that the use of the coloured glass is unnecessary unless the moon is very bright; and that the enlightened limb, whether it be the upper or lower, is to be brought in contact with the horizon.

3. Of a Star or Planet.

Look directly to the star through the vane and transparent slit in the horizon-glass; move the index until the opposite horizon, with respect to the star, is seen in the silvered part of the glass, and make the contact perfect, as formerly. If the altitude of the star is known nearly, the index may be set to that altitude, the sight directed to the opposite horizon, and the observation made as before.

Sect. II.—Of finding the Latitude of a Place.

The observation necessary for ascertaining the latitude of a place, is that of the meridional altitude of a known celestial object; or two altitudes when the object is out of the meridian. The latitude is deduced with more certainty and with less trouble from the first of these methods than from the second; and the sun, for various reasons, is the object most proper for this purpose at sea. It, however, frequently happens that, by the interposition of clouds, the sun is obscured at noon, and by this means the meridian altitude is lost. In this case, therefore, the method by double altitudes becomes necessary. The latitude may be deduced from three altitudes of an unknown object, or from double altitudes, the apparent times of observation being given.

The altitude of the limb of an object observed at sea requires four separate corrections in order to obtain the true altitude of its centre; these are for semidiameter, dip, refraction, and parallax. (See Astronomy, and the respective articles.) The first and last of these corrections vanish when the observed object is a fixed star.

When the altitude of the lower limb of any object is observed, its semidiameter is to be added thereto in order to obtain the central altitude; but if the upper limb be observed, the semidiameter is to be subtracted. The dip is to be subtracted from, or added to, the observed altitude, according as the fore or back observation is used. The refraction is always to be subtracted from, and the parallax added to, the observed altitude.

The dip in the preceding table answers to an entirely open and unobstructed horizon. It, however, frequently happens that the sun is over the land at the time of observation, and the ship nearer to the land than the visible horizon would be if unconfined. In this case, the dip will be different from what it would otherwise have been, and is to be taken from the subjoined table, in which the height is expressed at the top, and the distance from the land in the side columns in nautical miles. Seamen, in general, can estimate the distance of any object from the ship with sufficient exactness for this purpose, especially when that distance is not greater than six miles, which is the greatest distance of the visible horizon from an observer on the deck of any ship.

Table IV.—Dip of the Horizon.

| Height of Eye | Dip of Horizon | Height of Eye | Dip of Horizon | |--------------|---------------|--------------|---------------| | Feet | M.S. | Feet | M.S. | | 1 | 0.57 | 21 | 4.22 | | 2 | 1.21 | 22 | 4.28 | | 3 | 1.39 | 23 | 4.34 | | 4 | 1.55 | 24 | 4.40 | | 5 | 2.8 | 25 | 4.46 | | 6 | 2.20 | 26 | 4.52 | | 7 | 2.31 | 27 | 4.58 | | 8 | 2.42 | 28 | 5.3 | | 9 | 2.52 | 29 | 5.9 | | 10 | 3.1 | 30 | 5.14 | | 11 | 3.10 | 35 | 5.39 | | 12 | 3.18 | 40 | 6.2 | | 13 | 3.26 | 45 | 6.24 | | 14 | 3.34 | 50 | 6.44 | | 15 | 3.42 | 55 | 7.4 | | 16 | 3.49 | 60 | 7.23 | | 17 | 3.56 | 70 | 7.59 | | 18 | 4.3 | 80 | 8.32 | | 19 | 4.10 | 90 | 9.3 | | 20 | 4.16 | 100 | 9.33 |

Table V.—Dip of the Sea at different Distances from the Observer.

| Distance of Land in Sea Miles | Height of the Eye above the Sea, in Feet | |------------------------------|------------------------------------------| | | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | | Dip | M.M.| M.M.| M.M.| M.M.| M.M.| M.M.| M.M.| M.M.| | 0.01 | 11 | 22 | 34 | 45 | 56 | 68 | 79 | 90 | | 0.02 | 6 | 11 | 17 | 22 | 28 | 34 | 39 | 45 | | 0.04 | 4 | 8 | 12 | 15 | 19 | 23 | 27 | 30 | | 1.0 | 4 | 6 | 9 | 12 | 15 | 17 | 20 | 23 | | 1.04 | 3 | 5 | 7 | 9 | 12 | 14 | 16 | 19 | | 1.08 | 3 | 4 | 6 | 8 | 10 | 11 | 14 | 15 | | 2.0 | 2 | 3 | 5 | 6 | 8 | 10 | 11 | 12 | | 2.02 | 2 | 3 | 5 | 6 | 7 | 8 | 9 | 10 | | 3.0 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | | 3.02 | 2 | 3 | 4 | 5 | 6 | 6 | 7 | 7 | | 4.0 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 7 | | 5.0 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | | 6.0 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | ### Table VI

To reduce the Sun's Declination to any other Meridian, and to any given Time under that Meridian.

| Longitude | 10° | 20° | 30° | 40° | 50° | 60° | 70° | 80° | 90° | 100° | 110° | 120° | 130° | 140° | 150° | 160° | 170° | 180° | |-----------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|------|------|------|------|------|------|------|------| | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | Add. in W. | Sub. in E. | | 21 | 21 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 20 | 20 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 19 | 19 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 18 | 18 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 17 | 17 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 16 | 16 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 15 | 15 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 14 | 14 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 13 | 13 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | | 12 | 12 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 |

Time from Noon. Of finding the Latitude and Longitude at Sea.

**Prob. I.** To reduce the sun's declination to any given meridian.

**Rule.** Find the number in the table answering to the longitude in the table nearest to that given, and to the nearest day of the month. Now, if the longitude is west, and the declination increasing; that is, from the 20th of March to the 22d of June, and from the 22d of September to the 22d of December, the above number is to be added to the declination; during the other part of the year, or while the declination is decreasing, this number is to be subtracted. In east longitude the contrary rule is to be applied.

**Ex. 1.** Required the sun's declination at noon 16th April 1836, in longitude 81° W.

Sun's declination at noon at Greenwich, 10° 14' N. Number from table, + 5° Reduced declination, 10° 19'

**Ex. 2.** Required the sun's declination at noon 22d March 1836, in longitude 151° E.

Sun's declination at noon at Greenwich, 0° 45-9' N. Equation from table, - 9° Reduced declination, 0° 36-0' N.

**Prob. II.** Given the sun's meridian altitude, to find the latitude of the place of observation.

**Rule.** The sun's semidiameter is to be added to or subtracted from the observed altitude, according as the lower or upper limb is observed; the dip answering to the height from Table IV. or V. is to be subtracted if the fore observation is used; otherwise, it is to be added; and the refraction answering to the altitude from Table vol. IV. p. 100, art. Astronomy, is to be subtracted; hence the true altitude of the sun's centre will be obtained. Call the altitude south or north, according as the sun is south or north at the time of observation, which subtracted from 90°, will give the zenith distance of a contrary denomination.

Reduce the sun's declination to the meridian of the place of observation, by Prob. I.; then the sum or difference of the zenith distance and declination, according as they are of the same or of a contrary denomination, will be the latitude of the place of observation, of the same name with the greater quantity.

**Ex. 1.** October 19, 1836, in longitude 32° E., the meridian altitude of the sun's lower limb was 48° 53' S.; height of the eye 18 feet. Required the latitude.

Obs. alt. sun's low. limb, 48° 53' S. Sun's dec. noon, 9° 55' S. Semidiameter, +0° 16' Equation tab. = 2 Dip and refraction, -0° 5' Reduced dec. 9° 53' S. True alt. sun's centre, 49° 4' S. Zenith dist. 40° 56' N. Latitude, 31° 3' N.

**Prob. III.** Given the meridian altitude of a fixed star, to find the latitude of the place of observation.

**Rule.** Correct the altitude of the star by dip and refraction, and find the zenith distance of the star as formerly; take the declination of the star from the Nautical Almanac, and reduce it to the time of observation. Now, the sum or difference of the zenith distance and declination of the star, according as they are of the same or of a contrary name, will be the latitude of the place of observation.

**Ex. 1.** December 1, 1836, the meridian altitude of Sirius was 59° 50' S., height of the eye 14 feet. Required the latitude.

Observed altitude of Sirius, 59° 50' S. Dip and refraction, -0° 4' True altitude, 59° 46' S.

**Prob. IV.** Given the meridian altitude of a planet, to find the latitude of the place of observation.

**Rule.** Compute the true altitude of the planet as directed in last problem (which is sufficiently accurate for altitudes taken at sea); take its declination from the Nautical Almanac, and reduce it to the time and meridian of the place of observation; then the sum or difference of the zenith distance and declination of the planet will be the latitude, as before.

**Ex. 1.** August 7, 1836, the meridian altitude of Saturn was 68° 42' N., and height of the eye 15 feet. Required the latitude.

Observed altitude of Saturn, 68° 42' N. Dip and refraction, -0° 4' True altitude, 68° 38' N. Zenith distance, 21° 22' S. Declination, 9° 8' S. Latitude, 30° 30' S.

**Prob. V.** Given the meridian altitude of the moon, to find the latitude of the place of observation.

**Rule.** Take the proportional part of the daily variation of the moon's passing the meridian at Greenwich, answering to the ship's longitude; which being applied to the time of passage given in the Nautical Almanac, will give the time of the moon's passage over the meridian of the ship.

Reduce this time to the meridian of Greenwich; and by means of the Nautical Almanac find the moon's declination, horizontal parallax, and semidiameter at the reduced time.

Apply the semidiameter, dip, and refraction to the observed altitude of the limb, and the apparent altitude of the moon's centre will be obtained; to which add the moon's parallax in altitude (found from Problem VII. of next chapter), and the sum will be the true altitude of the moon's centre; which subtracted from 90°, the remainder is the zenith distance, and the sum or difference of the zenith distance and declination, according as they are of the same or of a contrary name, will be the latitude of the place of observation.

**Ex. 1.** December 19, 1836, in longitude 30° W., the meridian altitude of the moon's lower limb was 81° 15' N., height of the eye 12 feet. Required the latitude.

Time of pass. over mer. of Greenwich, 9° 30' Equation to this and long., +0° 4' Time of pass. over mer. ship, 9° 34' Longitude in time, 2° 0' Reduced time, 11° 34' Moon's dec. at 11h, 20° 9' N. Eq. to 34°, 0° 6' Reduced declination, 20° 15' N. Moon's hor. par., 54° 56' Moon's semidiameter, 15° 0' Observed altitude of the moon's lower limb, 81° 15' N. Semidiameter, +0° 15' Dip, -0° 3' Apparent altitude of the moon's centre, 81° 27' N. Refraction, 0° 0' Parallax in altitude, +0° 8' True altitude of the moon's centre, 81° 35' N. Zenith distance, 8° 25' S. Declination, 20° 15' N. Latitude, 11° 50' N. Remark. If the object be on the meridian below the pole at the time of observation, then the sum of the true altitude and the complement of the declination is the latitude of the same name as the declination or altitude.

Ex. 1. July 2, 1836, in longitude 15° W., the altitude of the sun's lower limb at midnight was 8° 58', height of the eye 18 feet. Required the latitude.

Observed altitude sun's lower limb, 8° 58' Semidiameter, +0 16 Dip and refraction, -0 10 True altitude of sun's centre, 9 4 N. Compl. declin. reduced to time and place, 66 57 N. Latitude, 76 1 N.

Prob. VI. Given the latitude by account, the declination and two observed altitudes of the sun, and the interval of time between them, to find the true latitude.

Rule. To the log. secant of the latitude by account, add the log. secant of the sun's declination; the sum, rejecting 20 from the index, is the logarithm ratio. To this add the log. of the difference of the natural sines of the two altitudes, and the log. of the half elapsed time from its proper column.

Find this sum in column of middle time, and take out the time answering thereto; the difference between which and the half elapsed time will be the time from noon when the greater altitude was observed.

Take the log. answering to this time from column of rising, from which subtract the log. ratio, the remainder is the logarithm of a natural number; which being added to the natural sine of the greater altitude, the sum is the natural cosine of the meridian zenith distance; from which and the sun's declination the latitude is obtained as formerly.

If the latitude thus found differs considerably from that by account, the operation is to be repeated, using the computed latitude in place of that by account.

Ex. In latitude 49° 48' N. by account, the sun's declination being 9° 37' S. at 6° 32' P.M. per watch, the altitude of the sun's lower limb was 28° 32'; and at 2h 41m it was 19° 25'; the height of the eye 12 feet. Required the true latitude.

First observed altit. 28° 32' Second altitude, 19° 25' Semidiameter, +0 16 Semidiameter, +0 16 Dip and refraction, -0 5 Dip and refraction, -0 6 True altitude, 28 43 True altitude, 19 35 Time per wat. Alt. N. Sines. Lat by acc. 49° 48' Secant, 0-19013 6h 32m 28° 43' 49° 48' Declination, 9° 37 Secant, 0-00615

| 2 41 | 19 35 | 33518 | Log. ratio, | 0-19628 | |------|-------|-------|------------|---------| | 2 9 | Diff. | 14530 | Log. | 4-16227 | | 1 4 | 30° | Half elapsed time, | 0-55637 | | 37 0 | Middle time, | 4-91492 | | 0 32 | Rising, | 3-00164 | | Natural number, | 639 | 2-80536 |

Mer. zenith dist. 60° 52' N. Cosine, 4-9667 Declination, 9° 37 S. Latitude, 51° 15 N.

As the latitude by computation differs 1° 27' from that by account, the operation must be repeated.

Computed latitude, 51° 15' Secant, 0-20348 Longitude at Sea. Declination, 9° 37 Secant, 0-00615

Logarithm ratio, Difference of nat. sines, 14530 Log. 4-16227 Half elapsed time, 1h 4m 30s Log. 0-55637

Middle time, 1 40 20 Log. 4-92827 Rising, 0 35 50 Log. 3-08630

Natural number, 753 2-87667 Gr. altitude, 28° 43' N. Sine, 48048

Mer. zen. dist. 60° 47 N. Cosine, 48801 Declination, 9° 37

Latitude, 51° 10 N.

As this latitude differs only 5' from that used in the computation, it may therefore be depended on as the true latitude.

Prob. VII. Given the latitude by account, the sun's declination, two observed altitudes, the elapsed time, and the course and distance run between the observations; to find the ship's latitude at the time of observation of the greater altitude.

Rule. Find the angle contained between the ship's course and the sun's bearing at the time of observation of the least altitude, with which enter the Traverse Table as a course, and the difference of latitude answering to the distance made good will be the reduction of altitude.

Now, if the least altitude be observed in the forenoon, the reduction of altitude is to be applied thereto by addition or subtraction, according as the angle between the ship's course and the sun's bearing is less or more than eight points. If the least altitude be observed in the afternoon, the contrary rule is to be used.

The difference of longitude in time between the observations is to be applied to the elapsed time by addition or subtraction, according as it is east or west. This is, however, in many cases so inconsiderable as to be neglected.

With the corrected altitudes and interval, the latitude by account and sun's declination at the time of observation of the greatest altitude, the computation is to be performed by the last problem.

Remark. If the sun come very near the zenith, the sines of the altitude will vary so little as to make it uncertain which ought to be taken as that belonging to the natural sine of the meridian altitude. In this case the following method will be found preferable.

To the log. rising of the time from noon found as before, add the log. secant of half the sum of the estimated meridian altitude, and greatest observed altitude; from which subtract the log. ratio, its index being increased by 10, and the remainder will be the log. sine of an arch; which added to the greatest altitude, will give the sun's meridian altitude.

This method is only an approximation, and ought to be used under certain restrictions; namely,

The observations must be taken between nine o'clock in the forenoon and three in the afternoon. If both observations be in the forenoon, or both in the afternoon, the interval must not be less than the distance of the time of observation of the greatest altitude from noon. If one observation be in the forenoon and the other in the afternoon, the interval must not exceed four hours and a half; and, in all cases, the nearest the greater altitude is to noon the better.

If the sun's meridian zenith distance be less than the latitude, the limitations are still more contracted. If the latitude be double the meridian zenith distance, the observations must be taken between half past nine in the morning and half past two in the afternoon, and the interval must not exceed three hours and a half. The observations must be taken still nearer to noon if the latitude exceed the zenith distance in a greater proportion. See Mackay's Treatises on the Longitude and Navigation, &c.; Requisite Tables, 3d edit.; Mendoza Rios's Tables; Norie's and Riddle's Treatises on Navigation; &c. Of finding CHAP. II.—METHOD OF FINDING THE LONGITUDE AT SEA

the Longitude at Sea by Lunar Observations.

Sect. I.—Introduction.

The observations necessary to determine the longitude by this method are, the distance between the sun and moon, the moon and a planet, or the moon and a fixed star near the ecliptic, together with the altitude of each. The planets used in the Nautical Almanac for this purpose are the following:—Venus, Mars, Jupiter, and Saturn. The stars are, α Arietis, Aldebaran, Pollux, Regulus, Spica Virginis, Antares, α Aquila, Fomalhaut, and α Pegasi; and the distances of the moon's centre from the sun, and from one or more of these planets and stars, are contained in the xiiiith—xvith pages of the month, at the beginning of every third hour mean time by the meridian of Greenwich. The distance between the moon and one of these objects is observed with a sextant; and the altitudes of the objects are taken as usual with a Hadley's quadrant.

In the practice of this method, it will be found convenient to be provided with three assistants. Two of these are to take the altitudes of the sun and moon, or moon and star, at the same time that the principal observer is taking the distance between the objects; and the third assistant is to observe the time, and write down the observations. In order to obtain accuracy, it will be necessary to observe several distances, and the corresponding altitudes, the intervals of time between them being as short as possible; and the sum of each divided by the number will give the mean distance and mean altitudes; from which the time of observation at Greenwich is to be computed by the rules to be explained.

If the sun or star from which the moon's distance is observed be at a proper distance from the meridian, the time at the ship may be inferred from the altitude observed at the same time with the distance. In this case the watch is not necessary; but if that object be near the meridian, the watch is absolutely necessary, in order to connect the observations for ascertaining the mean time at the ship and at Greenwich with each other.

An observer without any assistants may very easily take all the observations, by first taking the altitudes of the objects, then the distance, and again their altitudes, and reduce the altitudes to the time of observation of the distance; or, by a single observation of the distance, the time being known from which the altitudes of the bodies may be computed, the longitude may be determined.

A set of observations of the distance between the moon and a star or planet, and their altitudes, may be taken with accuracy during the time of the evening or morning twilight; and the observer, though not much acquainted with the stars, will not find it difficult to distinguish the star from which the moon's distance is to be observed. For the time of observation nearly, and the ship's longitude by account being known, the estimate time at Greenwich may be found; and by entering the Nautical Almanac with the reduced time, the distance between the moon and given star will be found nearly. Now set the index of the sextant to this distance, and hold the plane of the instrument so as to be nearly at right angles to the line joining the moon's cusps; direct the sight to the moon, and, by giving the sextant a slow vibratory motion, the axis of which being that of vision, the star, which is usually one of the brightest in that part of the heavens, will be seen in the transparent part of the horizon-glass.

Sect. II.—Of the Sextant.

This instrument is constructed for the express purpose of measuring with accuracy the angular distance between the sun and moon, or between the moon and a planet or fixed star, in order to ascertain the longitude of a place by lunar observations. It is, therefore, made with more care than the quadrant, and has some additional appendages that are wanting in that instrument.

Fig. 40 represents the sextant, so framed as not to be liable to bend. The arch AA is divided into 120 degrees; each degree is divided into three parts; each of these parts, therefore, contains twenty minutes, which are again subdivided by the vernier into every half minute or thirty seconds. The vernier is numbered at every fifth of the longer divisions, from the right towards the left, with 5, 10, 15, and 20; the first division to the right being the beginning of the scale.

In order to observe with accuracy, and make the images come precisely in contact, an adjusting screw B is added to the index, which may thereby be moved with greater accuracy than it can be by the hand; but this screw does not act until the index is fixed by the finger-screw C. Care should be taken not to force the adjusting screw when it arrives at either extremity of its adjustment. When the index is to be moved any considerable quantity, the screw C at the back of the sextant must be loosened; but when the index is brought nearly to the division required, this back screw should be tightened, and then the index may be moved gradually by the adjusting screw.

There are four tinged glasses D, each of which is set in a separate frame that turns on a centre. They are used to defend the eye from the brightness of the solar image and the glare of the moon, and may be used separately or together as occasion requires.

There are three more such glasses placed behind the horizon-glass at E, to weaken the rays of the sun or moon when they are viewed directly through the horizon-glass. The paler glass is sometimes used in observing altitudes at sea, to take off the strong glare of the horizon.

The frame of the index-glass is firmly fixed by a strong cock to the centre plate of the index. The horizon-glass F is fixed in a frame that turns on the axes or pivots, which move in an exterior frame; the holes in which the pivots move may be tightened by four screws in the exterior frame. G is a screw by which the horizon-glass may be set perpendicular to the plane of the instrument; should this screw become loose, or move too easy, it may be easily tightened by turning the capstan headed screw H, which The sextant is furnished with a plain tube without any glasses; and to render the objects still more distinct, it has two telescopes, one representing the objects erect, or in their natural position; the longer one shows them inverted; it has a large field of view, and other advantages, and a little use will soon accustom the observer to the inverted position, and the instrument will be as readily managed by it as by the plain tube alone. By a telescope the contact of the images is more perfectly distinguished; and by the place of the images in the field of the telescope, it is easy to perceive whether the sextant is held in the proper place for observation. By sliding the tube that contains the eye-glasses in the inside of the other tube, the object is suited to different eyes, and made to appear perfectly distinct and well defined.

The telescopes are to be screwed into a circular ring at K; this ring rests on two points against an exterior ring, and is held thereto by two screws: by turning one or other of these screws, and tightening the other, the axis of the telescope may be set parallel to the plane of the sextant. The exterior ring is fixed on a triangular brass stem that slides in a socket, and, by means of a screw at the back of the quadrant, may be raised or lowered so as to move the centre of the telescope to point to that part of the horizon-glass which shall be judged the most fit for observation. Tinged glasses are provided to screw on the eye end of either of the telescopes or the plain tube.

Adjustments of the Sextant.

The adjustments of a sextant are, to set the mirrors perpendicular to its plane and parallel to each other when the index is at zero, and to set the axis of the telescope parallel to the plane of the instrument. The three first of these adjustments are performed nearly in the same manner as directed in the section on the quadrant; as, however, the sextant is provided with a set of coloured glasses placed behind the horizon-glass, the index error may be more accurately determined by measuring the sun's diameter twice, with the index placed alternately before and behind the beginning of the divisions; half the difference of these two measures will be the index error, which must be added to or subtracted from all observations, according as the diameter measured with the index to the left of 0 is less or greater than the diameter measured with the index to the right of the beginning of the divisions. It will be more accurate to measure the sun's horizontal diameter, as the vertical diameter is often affected with refraction.

Adjustment IV.—To set the Axis of the Telescope parallel to the Plane of the Instrument.

Turn the eye end of the telescope until the two wires are parallel to the plane of the instrument; and let two distant objects be selected, as two stars of the first magnitude, whose distance is not less than 90° or 100°; make the contact of these objects as perfect as possible at the wire nearest the plane of the instrument; fix the index in this position; move the sextant till the objects are seen at the other wire, and if the same points are in contact, the axis of the telescope is adjusted; if not, Of finding proceed as at the other wire, and continue till no error remains.

It is sometimes necessary to know the angular distance between the wires of the telescope; to find which, place the wires perpendicular to the plane of the sextant, hold the instrument vertical, direct the sight to the horizon, and move the sextant in its own plane till the horizon and upper wire coincide; keep the sextant in this position, and move the index till the reflected horizon is covered by the lower wire, and the division shown by the index of the limb, corrected by the index error, will be the angular distance between the wires. Other and better methods will readily occur to the observer at land.

Use of the Sextant.

When the distance between the moon and the sun, a planet or a star, is to be observed, the sextant must be held so that its plane may pass through the eye of the observer and both objects; and the reflected image of the most luminous of the two is to be brought in contact with the other seen directly. To effect this, therefore, it is evident, that when the brightest object is to the right of the other, the face of the sextant must be held upwards; but if to the left, downwards. When the face of the sextant is held upwards, the instrument should be supported with the right hand, and the index moved with the left hand. But when the face of the sextant is from the observer, it should be held with the left hand, and the motion of the index regulated by the right hand.

Sometimes a sitting posture will be found very convenient for the observer, particularly when the reflected object is to the right of the direct one; in this case the instrument is supported by the right hand, the elbow may rest on the right knee, the right leg at the same time resting on the left knee.

If the sextant is provided with a ball and socket, and a staff, one of whose ends is attached thereto, and the other rests in a belt fastened round the body of the observer, the greater part of the weight of the instrument will by this means be supported by his body.

To observe the Distance between the Moon and any Celestial Object.

I. Between the Sun and Moon.

Put the telescope in its place, and the wires parallel to the plane of the instrument; and if the sun is very bright, raise the plate before the silvered part of the speculum; direct the telescope to the transparent part of the horizon-glass, or to the line of separation of the silvered and transparent parts, according to the brightness of the sun, and turn down one of the coloured glasses; then hold the sextant so that its plane produced may pass through the sun and moon, having its face either upwards or downwards, according as the sun is to the right or left of the moon; direct the sight through the telescope to the moon, and move the index till the limb of the sun is nearly in contact with the enlightened limb of the moon; now fasten the index, and by a gentle motion of the instrument make the image of the sun move alternately past the moon; and, when in that position where the limbs are nearest each other, make the coincidence of the limbs perfect by means of the adjusting screw; this being effected, read off the degrees and parts of a degree shown by the index on the limb, using the magnifying glass; and thus the angular distance between the nearest limbs of the sun and moon is obtained. 2. Between the Moon and a Planet or Star.

Direct the middle of the field of the telescope to the line of separation of the silvered and transparent parts of the horizon-glass; if the moon is very bright, turn down the lightest coloured glass, and hold the sextant so that its plane may be parallel to that passing through the eye of the observer and both objects; its face being upwards if the moon is to the right of the star, but if to the left the face is to be held from the observer; now direct the sight through the telescope to the star, and move the index till the moon appears by the reflection to be nearly in contact with the star; fasten the index, and turn the adjusting screw till the coincidence of the star and enlightened limb of the moon is perfect; and the degrees and parts of a degree shown by the index will be the observed distance between the moon's enlightened limb and the star.

The contact of the limbs must always be observed in the middle between the parallel wires.

It is sometimes difficult for those not much accustomed to observations of this kind, to find the reflected image in the horizon-glass; it will perhaps in this case be found more convenient to look directly to the object, and, by moving the index, to make its image coincide with that seen directly.

Sect. III.—Of the Circular Instrument of Reflection.

This instrument was proposed with a view to correct the errors to which the sextant is liable, particularly the error arising from the inaccuracy of the divisions on the limb. It consists of the following parts, a circular ring or limb, two moveable indices, two mirrors, a telescope, coloured glasses, &c.

The limb of this instrument is a complete circle of metal, and is connected with a perforated central plate by six radii; it is divided into 720 degrees, each degree being divided into three equal parts, and the division carried to minutes by means of the index scale as usual.

The two indices are moveable about the same axis, which passes exactly through the centre of the instrument—the first index carries the central mirror, and the other the telescope and horizon-glass, each index being provided with an adjusting screw for regulating its motion, and a scale for showing the divisions on the limb.

The central mirror is placed on the first index, immediately above the centre of the instrument, and its plane makes an angle of about 30° with the middle line of the index. The four screws in its pedestal for making its plane perpendicular to that of the instrument have square heads, and are therefore easily turned either way by a key for that purpose.

The horizon-glass is placed on the second index near the limb, so that as few as possible may be intercepted of the rays proceeding from the reflected object when to the left. The perpendicular position of this glass is rectified in the same manner as that of the horizon-glass of a sextant, to which it is similar. It has another motion, whereby its plane may be disposed so as to make a proper angle with the axis of the telescope, and a line joining its centre and that of the central mirror.

The telescope is attached to the other end of the index. It is an achromatic astronomical one, and therefore inverts objects; it has two parallel wires in the common focus of the glasses, whose angular distance is between two and three degrees, and which, at the time of observation, must be placed parallel to the plane of the instrument. This is easily done, by making the mark on the eye-piece coincide with that on the tube. The telescope is moveable by two screws in a vertical direction with regard to the plane of the instrument, but is not capable of receiving a lateral motion.

There are two sets of coloured glasses, each set containing four, and differing in shade from each other. The glasses of the larger set, which belongs to the central mirror, should have each about half the degree of shade with which the correspondent glass of the set belonging to the horizon-mirror is tinged. These glasses are kept tight in their places by small pressing screws, and make an angle of about 85° with the plane of the instrument, by which means the image from the coloured glass is not reflected to the telescope. When the angle to be measured is between 5° and 34°, one of the glasses of the largest set is to be placed before the horizon-glass.

The handle is of wood, and is screwed to the back of the instrument, immediately under the centre, with which it is to be held at the time of observation.

Fig. 41 is a plan of the instrument, wherein the limb is represented by the divided circular plate; A is the central mirror; aa, the places which receive the stems aa of the glass; EF the first or central index, with its scale and adjusting screw; MN the second or horizon-index; GH the telescope; IK the screws for moving it towards or from the plane of the instrument; C the plane of the coloured glass; and D its place in certain observations.

Fig. 42 is a section of the instrument, wherein several parts are referred to by the same letters as in fig. 41. The circular reflecting instrument was greatly improved by Messrs Mendoza Rios, Troughton, Dollond, &c.

Adjustments of the Circular Instrument.

I. To set the horizon-glass so that none of the rays from the central mirror shall be reflected to the telescope from the horizon-mirror, without passing through the coloured glass belonging to this last mirror.—Place the coloured glass before the horizon-mirror; direct the telescope to the silvered part of that mirror, and make it nearly parallel to the plane of the instrument; move the first index; and if the rays from the central mirror to the horizon-glass, and thence to the telescope, have all the same degree of shade with that of the coloured glass used, the horizon-glass is in its proper position; otherwise, the pedestal of the glass must be turned until the uncoloured images disappear. II. Place the two adjusting tools on the limb, about 35° of the instrument distant, one on each side of the division on the left, answering to the plane of the central mirror produced; then, the eye being placed at the upper edge of the nearest tool, move the central index till one half only of the reflected image of this tool is seen in the central mirror towards the left, and move the other tool till its half to the right is hid by the same edge of the mirror; then, if the upper edges of both tools are apparently in the same straight line, the central mirror is perpendicular to the plane of the instrument; if not, bring them into this position by the screws in the pedestal of the mirror.

III. To set the horizon mirror perpendicular to the plane of the instrument.—The central mirror being previously adjusted, direct the sight through the telescope to any well-defined distant object; then, if, by moving the central index, the reflected image passes exactly over the direct object, the mirror is perpendicular; if not, its position must be rectified by means of the screws in the pedestal of the glass.

A planet, or star of the first magnitude, will be found a very proper object for this purpose.

IV. To make the line of collimation parallel to the plane of the instrument.—Lay the instrument horizontally on a table; place the two adjusting tools on the limb, towards the extremities of one of the diameters of the instrument; and at about fifteen or twenty feet distant let a well-defined mark be placed, so as to be in the same straight line with the tops of the tools; then raise or lower the telescope till the plane, passing through its axis and the tops of the tools, is parallel to the plane of the instrument, and direct it to the fixed object; turn either or both of the screws of the telescope till the mark is apparently in the middle between the wires; then is the telescope adjusted; and the difference, if any, between the divisions pointed out by the indices of the screws will be the error of the indices. Hence this adjustment may in future be easily made.

In this process, the eye-tube must be so placed as to obtain distinct vision.

V. To find that division to which the second index being placed, the mirrors will be parallel, the central index being at zero.—Having placed the first index exactly to 0, direct the telescope to the horizon-mirror, so that its field may be bisected by the line of separation of the silvered and transparent parts of that mirror; hold the instrument vertically, and move the second index until the direct and reflected horizons agree; and the division shown by the index will be that required.

This adjustment may be performed by measuring the sun's diameter in contrary directions, or by making the reflected and direct images of a star or planet to coincide.

Use of the Circular Instrument.

To observe the Distance between the Sun and Moon.

I. The sun being to the right of the moon.

Set a proper coloured glass before the central mirror if the distance between the objects is less than 35°; but if above that quantity, place a coloured glass before the horizon-mirror; make the mirrors parallel, the first index being at 0, and hold the instrument so that its plane may be directed to the objects, with its face downwards, or from the observer; direct the sight through the telescope to the moon; move the second index, according to the order of the divisions on the limb, till the nearest limbs of the sun and moon are almost in contact; fasten that index, and make the coincidence of the limbs perfect by the adjusting screw belonging thereto; then invert the instrument, and move the central index towards the second by a quantity equal to twice the arch passed over by that index; direct the plane of the instrument to the objects; look directly to the moon, and the sun will be seen in the field of the telescope; fasten the central index, and make the contact of the same two limbs exact by means of the adjusting screw: Then half the angle shown by the central index will be the distance between the nearest limbs of the sun and moon.

II. The sun being to the left of the moon.

Hold the instrument with its face upwards, so that its plane may pass through both objects; direct the telescope to the moon, and make its limb coincide with the nearest limb of the sun's reflected image, by moving the second index; now put the instrument in an opposite position; direct its plane to the objects, and the sight to the moon, the central index being previously moved towards the second by a quantity equal to twice the measured distance; and make the same two limbs that were before observed coincide exactly, by turning the adjusting screw of the first index; then half the angle shown by the first index will be the angular distance between the observed limbs of the sun and moon.

To observe the Angular Distance between the Moon and a Fixed Star or Planet.

I. The star being to the right of the moon.

In this case the star is to be considered as the direct object; and the enlightened limb of the moon's reflected image is to be brought in contact with the star or planet, both by a direct and inverted position of the instrument, exactly in the same manner as described in the last article. If the moon's image is very bright, the lightest tinged glass is to be used.

II. The star being to the left of the moon.

Proceed in the same manner as directed for observing the distance between the sun and moon, the sun being to the right of the moon, using the lightest tinged glass if necessary.

Sect. IV.—Of the Method of determining the Longitude from Observation.

Prob. I. To convert degrees or parts of the equator into time.

Rule. Multiply the degrees and parts of a degree by 4, beginning at the lowest denomination, and the product will be the corresponding time; observing that minutes multiplied by 4 produce seconds of time, and degrees multiplied by 4 give minutes.

Ex. Let $26^\circ 45'$ be reduced to time.

$$\begin{array}{c} 26^\circ 45' \\ \times \quad 4 \\ \hline 104^\circ 60' = \text{time required}. \end{array}$$

Prob. II. To convert time into degrees.

Rule. Multiply the given time by 10, to which add the half of the product. The sum will be the corresponding degrees.

Ex. Let $3^h 4^m 28^s$ be reduced to degrees.

$$\begin{array}{c} 3^h 4^m 28^s \\ \times \quad 10 \\ \hline 30^h 44^m 28^s \\ + \quad 15^h 22^m 20^s \\ \hline 46^h 7^m 0^s \end{array}$$

Corresponding deg. = $46^\circ 7' 0''$

Prob. III. Given the time under any known meridian, to find the corresponding time at Greenwich.

Rule. Let the given time be reckoned from the preceding noon, to which the longitude of the place in time is to be applied by addition or subtraction, according as it NAVIGATION.

Of finding its east or west; and the sum or difference will be the corresponding time at Greenwich.

Ex. What time at Greenwich answers to 6h 15m at a ship in longitude 76° 45' W.?

Observations: - Time at ship: 6h 15m - Longitude in time: 5° 7W. - Time at Greenwich: 11 22

Prob. IV. To reduce the time at Greenwich to that under any given meridian.

Rule. Reckon the given time from the preceding noon, to which add the longitude in time if east, but subtract it if west; and the sum or remainder will be the corresponding time under the given meridian.

Ex. What is the expected time of the beginning of the lunar eclipse of February 25, 1793, at a ship in longitude 109° 48' E.?

Begin. of eclipse at Greenwich per Naut. Alm. 9h 23m 45s Ship's longitude in time: 7 19 12 Time of beginning eclipse at ship: 16 42 57

Prob. V. Given the latitude of a place, the altitude and declination of the sun, to find the apparent time, and the error of the watch.

Rule. If the latitude and declination are of different names, let their sum be taken; otherwise, their difference. From the natural cosine of this sum or difference, subtract the natural sine of the corrected altitude, and find the logarithm of the remainder; to which add the log. secants of the latitude and declination: the sum will be the log. rising of the horary distance of the object from the meridian, and hence the apparent time will be known. The equation of time being then applied as directed in the Nautical Almanac, the mean time of observation is obtained.

Example. September 15, 1792, in latitude 33° 56' S. and longitude 18° 22' E., the mean of the times per watch was 9h 12m 10s A.M., and that of the altitudes of the sun's lower limb 24° 48'; height of the eye 24 feet. Required the error of the watch.

Sun's declin. at noon, per Nautical Almanac, 2° 40' 5" N. Equation to 3° 48' A.M.: + 0 3 7 Equation to 18° 22' E.: + 0 1 2 Reduced declination: 2 45' 4" N.

Obs. alt. sun's lower limb: 24° 48' Semidiameter: + 0 16 0 Dip: - 0 4 7 Correction: - 0 1 9 True altitude sun's centre: 24 57' 4 Latitude: 33 56 Declination: 2 45' 4 Sum: 36 41' 4

Secant lat.: 0.08109 Secant dec.: 0.00050 Nat. cosine sum: 80188 Nat. sine alt.: 42193

Difference: 37995 Log. 4.57973 Rising 3 48 51 4.66132 Sun's meridian distance: 3° 48' 51" Apparent time: 8 11 9 A.M. Equation of true, subtract: 0 5 10 Mean time: 8 5 59 A.M. Time per watch: 8 12 10 Watch fast of mean time: 0 6 11

Prob. VI. Given the latitude of a place, and the altitude of a known fixed star, to find the mean time of observation and error of the watch.

Rule. Correct the observed altitude of the star, and reduce its right ascension and declination to the time of observation.

With the latitude of the place, the true altitude and declination of the star, compute its horary distance from the meridian by last problem; which being added to or subtracted from its right ascension, according as it was observed in the western or eastern hemisphere, the sum or remainder will be the right ascension of the meridian.

From the right ascension of the meridian subtract the sidereal time of mean noon, reduced to the meridian of the place of observation, and obtained from the Nautical Almanac; the remainder will be the approximate time of observation; from which subtract the reduction of sidereal to mean time, answering thereto, the result will be the mean time of observation; and hence the error of the watch will be known.

Ex. December 12, 1836, in latitude 37° 46' N., and longitude 21° 15' E., the altitude of Arcturus east of the meridian was 34° 6' 4", the height of the eye 10 feet. Required the apparent time of observation.

Observed alt. of Arcturus: 34° 6' 4" Dip and refraction: - 0 4 4 True altitude: 34 2 0 Latitude: 37° 46' 0" N. Sec. 0-10209 Declination: 20 2 0 N. Sec. 0-02711

Difference: 17 44' 0 N. Co. 95248 Altitude of Arcturus, 34 2 0 N. Sine, 55968

Difference: 39280 4-59417

Arcturus's merid. dist. 4h 7m 35s Rising, 4-72337 Arcturus's right asc. 14 8 12 Right asc. of merid. 10 0 37 Sidereal time of mean noon, 17 24 37

Approximate time: 16 36 0 Reduction of sid. to mean time: - 0 2 43

Mean time of obs: 16 33 17

Prob. VII. Given the apparent altitude of the sun, moon, a planet or star, to find the true altitude.

The altitude obtained from observation corrected for dip and for the semidiameter of the body, if the limb has been observed, is the apparent altitude of the centre.

From the apparent altitude subtract the refraction (Table vol. IV. p. 100, art. Astronomy); and to the remainder add the parallax in altitude; the sum is the true altitude required.

The parallax in altitude is found by adding to the log. cosine of the apparent altitude corrected for refraction, the logarithm of the horizontal parallax; the sum is the logarithm of the parallax in altitude.

The horizontal parallax of the object observed will be found in the Nautical Almanac. The horizontal parallax of the sun may always be supposed 9°. The parallax of a star is insensible.

Ex. The observed altitude of the moon's upper limb was observed 32° 17', the height of the eye being eighteen feet, the moon's horizontal parallax being 57' 58", and semidiameter 16'. Required the apparent and true altitudes. **NAVIGATION**

| Apparent altitude of centre | 32° 57' 0" | |-----------------------------|------------| | Refraction | 0 1 33 | | Cosine | 9-9678 | | Log. hor. parall. S478 | 3-54133 | | Log. parallax in altitude | 3-47011 | | Parallax in altitude | + 0 49 12 | | True altitude of centre | 32 44 39 |

**Prob. VIII.** Given the apparent distance between the moon and the sun or a fixed star or planet, and the apparent and true altitudes of these bodies, to find the true distance.

**Example.** Let the apparent altitude of the moon's centre be 48° 22', that of the sun's 27° 43', the apparent central distance 81° 23' 40", and the moon's horizontal parallax 58' 45". Required the true distance.

| Apparent altitude sun's centre | 27° 43' 0" | |-------------------------------|------------| | Correction | - 0 1 40 | | Sun's true altitude | 27 41 20 | | Sun's apparent altitude | 27 43 0 | | Moon's apparent altitude | 48 22 0 | | Difference | 20 39 0 | | Apparent distance | 81 23 40 | | Sum | 102 24 0 | | Difference | 60 44 40 | | Half difference true altitudes| 10 39 33 | | Arch | 51 27 29 | | Sum | 62 7 2 | | Difference | 40 47 56 | | True distance | 81 4 32 |

This is nearly the method of Borda, similar to Problem IV. and XX.

**Prob. IX.** To find the time at Greenwich answering to a given distance between the moon and the sun, or one of the stars or planets used in the Nautical Almanac.

**Rule.** If the given distance is found in the Nautical Almanac opposite to the given day of the month, or to that which immediately precedes or follows it, the time is found at the top of the page. But if this distance is not found exactly in the ephemeris, subtract the prop. log. of the difference between the distances which immediately precede and follow the given distance (which prop. log. is given in the Almanac) from the prop. log. of the difference between the given and preceding distances; the remainder will be the prop. log. of the excess of the time corresponding to the given distance, above that answering to the preceding distance; and hence the mean time at Greenwich is known.

**Example.** September 19, 1836, the true distance between the centres of the sun and moon was 110° 3' 5". Required the mean time at Greenwich.

**XIX. Practical Astronomy,** which may be consulted, as well as Problems IV. and XX.

**Given distance,** 110° 3' 5"

| Dist. at 3 hours, 109° 3' 58" | |-------------------------------| | Difference | 0 59 7 Prop. log. 4835 | | Prop.log.from Almanac | 2578 | | Excess | 1h 47m 2s Prop. log. 2257 | | Preceding time | 3 0 0 | | Mean time at Green. 4 47 2 |

**Prob. X.** The latitude of a place and its longitude by account being given, together with the distance between, and the altitude of the moon and the sun, or one of the stars or planets in the Nautical Almanac; to find the true longitude of the place of observation.

**Rule.** Reduce the estimate time of observation to the meridian of Greenwich by Problem III., and to this time take, from the Nautical Almanac, the moon's horizontal Of finding parallax and semidiameter. Increase the semidiameter by the Longitude at Sea by Lunar Observations.

Find the apparent and true altitudes of each object's centre, and the apparent central distance; with which compute the true distance by Problem VIII., and find the mean time at Greenwich answering thereto by the last problem.

If the sun or star be at a proper distance from the meridian at the time of observation of the distance, compute the mean time at the ship. If not, the error of the longitude may be found from observations taken either before or after that of the distance.

The difference between the mean times of observation at the ship and Greenwich will be the longitude of the ship in time, which is east or west according as the time at the ship is later or earlier than the Greenwich time.

Ex. 1. On the 8th of April 1835, at 2° 21' 28" P.M., mean time, in latitude 21° 33' N. longitude 155° 44' E. by account, the distance between the sun and moon's nearest limbs was observed to be 111° 6' 7", the observed altitude of the sun's lower limb was 53° 8' 53", that of the moon's lower limb was 14° 38' 32"; the height of the eye 20 feet; the barometer being at 29-20 inches, and Fahrenheit's thermometer at 75°. Required the true longitude.

Mean time at ship, 8th, 2° 21' 28" Longitude in time E. 10 22 56 Est. Greenwich time on 7th, 15 58 32

To est. time sun's semidiameter, 15° 59' Moon's semidiam. + aug. 15 4 Moon's eq. hor. par. 57 20 Red. for lat. 0 2 Red. hor. par. 57 18

Moon's Right Ascension.

April 7th, at 15h, 8° 46' 59-02" Prop. part for 58° 32' + 0 2 16-01 Red. R. A. 8 49 15-03 Sun's obs. alt. lower limb, 53° 8' 53" Dip to 20 feet, - 0 4 26 App. alt. lower limb, 53 4 27 Sun's semidiameter, + 0 15 59 App. cent. alt. 53 20 26 Sun's parallax, + 0 0 5 Refraction, - 0 0 40 Sun's true alt. 53 19 51

Moon's Declination.

22° 28' 46-7" N. - 0 7 10-0 Red. dec. 22 21 36-7 Pol. dist. 67 38 23-3 Moon's obs. alt. lower limb, 14° 38' 32" Dip to 20 feet, - 0 4 26 App. alt. lower limb, 14 34 6 Moon's semidiameter, + 0 15 42 App. cent. alt. 14 49 48 Moon's parallax, + 0 55 31 Refraction, - 0 3 24 Moon's true altitude, 15 41 55

Sidereal Time.

1h 0m 9-48 + 0 2 37-46 Red. S. T. 1 2 46-94 24h - S. T. 22 57 13-06 Obs. dist. n. l. 111° 16' 7" Sun's semid. + 0 15 59 Moon's semid. + 0 15 42 App. central dist. 111 47 48

Apparent central distance, 53 20 26 Secant, 0-223984 Sun's apparent altitude, 14 49 48 Secant, 0-014713 Moon's apparent altitude, Sum, 179 58 2 Half, 89 59 1 Cosine, 6-456427 Difference of half and distance, 21 48 47 Cosine, 9-967736 Sun's true altitude, 53 19 51 Cosine, 9-776115 Moon's true altitude, 15 41 55 Cosine, 9-983490 Sum, 69 1 46 Half, or arc 1st, 34 30 53 (Sum) 16-422465 Half, or arc 2d, 89 4 5 Cosine, (Half) 8-211232 Sum of arcs 1st and 2d, 123 34 58 Sine, 9-920691 Difference, 54 33 12 Sine, 9-910974 Half true distance, 55 28 8 Sine, (Sum) 19-831665 True distance, 110 56 16 Difference, (Half) 9-915832 True distance at 15h, 110 28 27 Prop. log. 0-81097 True distance at 18h, 112 0 10 Prop. log. 0-292982 Proportional par. 0h 54m 35-5 Prop. log. 0-51815 Equation to mean second difference, + 0 0 45 Preceding hour, 15 0 0-0 Greenwich mean time on 7th, 15 54 40-0 To find Moon's true altitude, 15° 41' 55" the Longi-Moon's polar dist. 67 38 23 Cosecant, 0-033947 tude by a Ship's latitude, 21 33 0 Secant, 0-031472 Chronometer. Sum, 104 53 18 Half sum, 52 26 39 Cosine, 9-784996 Difference of half sum and alt. 36 44 44 Sine, 9-778892 Meridian dist. west, 18° 35" 0 Reduced versine, 9-627307 Moon's reduced R. A. 8 49 15 24h — sidereal time, 22 57 13 Mean time at ship, 8th, 2 21 28 Mean time at Greenwich, 7th, 15 54 40

Longitude, 10 26 48 = 150° 42' east.

Remark. In the preceding example, the true distance and mean time at the ship have been computed by methods different from those before given, which would, however, have given the same results.

CHAP. III.—TO FIND THE LONGITUDE BY MEANS OF A CHRONOMETER.

In order to find the longitude at sea by means of a chronometer, its daily rate on mean solar or sidereal time must be established by observations made at some particular place, and its error ascertained for the meridian of that or any other known place.

An observatory is the most proper and convenient place for this purpose, as there the rate and error may be both determined with the utmost accuracy by equal altitudes, or transits over the meridian of the sun or stars. But if an observatory is not adjacent, the rate and error of the chronometer may be found by altitudes taken daily for several days from the horizon of the sea, or by the method of reflection from an artificial horizon.

If by these observations the daily rate is found to be nearly the same; that is, if the chronometer gains or loses nearly the same portion of absolute time daily, it may be depended on for finding the longitude; but if its rate is unequal, it must be rejected, as the longitude inferred from it cannot be expected to be accurate.

It would be proper to have two chronometers, and that they should be wound up at different stated times of the day, so that if one should be found stopped, either through neglect in winding up, or otherwise, it may be set by the other, observing to apply the former interval of time between them, and the change in their rates of going in that interval.

Prob. To find the longitude of a ship at sea by a chronometer.

Let several altitudes of the sun, or of any fixed star or planet, be observed, and find the true mean altitude; with which, the ship's latitude, and object's declination, compute the mean time of observation.

To the mean of the times of observation, as shown by the chronometer, apply its error and accumulated rate. Hence the mean time under the meridian of the place where the error and rate were established will be known; to which apply the difference of longitude in time between that place and Greenwich, and the mean time of observation under the meridian of Greenwich will be obtained. The difference between the time at the place of observation and that at Greenwich will be the longitude of the ship in time; and it is east or west, according as the time, by observation, is later or earlier than the Greenwich time.

Ex. May 19, 1804, in latitude 42° 15' N., five altitudes of the sun's lower limb were observed in the afternoon, the mean being 43° 45', and the mean of the times of observation, as given by a chronometer, 7h 0m 56s, the chronometer's error having been settled at the Royal Observatory at Greenwich, March 16, at noon, 1° 18' fast for mean time, and daily gain 7° 83', height of the eye twenty-six feet. Required the longitude of the place of observation.

The true altitude of the sun's centre is found to be 43° 55', with which, the latitude, and sun's declination 19° 51' N., the sun's meridian distance is found to be 3° 12m 34s, and the equator of time being 3° 51' subtractive, the mean time at the place of observation is 3h 8m 43s.

Time by chronometer, 7h 0m 56s Error, March 16, -0 1 18 Accumulated gain (7° 83' × 644), -0 8 23 Mean time at Greenwich, 6 51 15 Mean time at place of observation, 3 8 43

Longitude in time, 3 42 32 = 55° 38' W.

For various other methods of determining the longitude of a place, the reader is referred to Mackay's Treatise on the Longitude, Inman's, Riddle's, and Norie's Treatises on Navigation; Mendoza Rios's and Thomson's Tables, &c.

CHAP. IV.—OF THE VARIATION OF THE COMPASS.

The variation of the compass is the deviation of the points of the mariner's compass from the corresponding points of the horizon, and is denominated east or west variation, according as the north point of the compass is to the east or west of the true north point of the horizon.

A particular account of the variation, and of the several instruments used for determining it from observation, may be seen under the articles ARMUTH, COMPASS, and VARIATION; and for the method of communicating magnetism to compass needles, see MAGNETISM.

Prob. I. Given the latitude of a place, and the sun's magnetic amplitude, to find the variation of the compass.

Rule. To the log. secant of the latitude add the log. sine of the sun's declination, the sun will be the log. cosine of the true amplitude; to be reckoned from the north or south, according as the declination is north or south.

The difference between the true and observed amplitudes, reckoned from the same point, and if of the same name, is the variation; but if of a different name, their sum is the variation.

If the observation be made before noon, the variation will be east or west, according as the observed amplitude is nearer to or more remote from the north than the true amplitude. The contrary rule holds good in observations taken after noon. Variation of the Compass.

Ex. 1. May 15, 1836, in latitude 33° 10' N. longitude 18° W. about 5h A.M., the sun was observed to rise E. by N. Required the variation.

Sun's decl. May 15, at noon, 18° 58' N. Equation to 7h from noon, -0 4 Equation to 18° W., +0 1

Reduced declination, 18° 55' Sine, 9-51080 Latitude, 33° 10' Secant, 0-07723

True amplitude, N. 67° 13' E. Cosine, 9-58803 Observed amplitude, N. 78° 45' E.

Variation, 11° 32'; which is west, because the observed amplitude is more distant from the north than the true amplitude, the observation being made before noon.

It may be remarked, that the sun's amplitude ought to be observed at the instant the altitude of its lower limb is equal to the sum of 15 minutes and the dip of the horizon. Thus, if an observer be elevated 18 feet above the surface of the sea, the amplitude should be taken at the instant the altitude of the sun's lower limb is 19 minutes.

Prob. II. Given the magnetic azimuth, the altitude and declination of the sun, together with the latitude of the place of observation; to find the variation of the compass.

Rule. Reduce the sun's declination to the time and place of observation, and compute the true altitude of the sun's centre.

Find the sum of the sun's polar distance and altitude and the latitude of the place, take the difference between the half of this sum and the polar distance.

To the log, secant of the altitude add the log, secant of the latitude, the log, cosine of the half sum, and the log, cosine of the difference; half the sum of these will be the log, sine of half the sun's true azimuth, to be reckoned from the south in north latitude, but from the north in south latitude.

The difference between the true and observed azimuths will be the variation as formerly.

Ex. 1. November 18, 1836, in latitude 50° 22' N. longitude 24° 30' W. about three quarters past eight A.M. the altitude of the sun's lower limb was 8° 10', and bearing per compass S. 23° 18' E.; height of the eye twenty feet. Required the variation of the compass.

Observed altitude of sun's lower limb, = 0° 10' Semidiameter, +0 16 Dip and refraction, -0 10

True altitude, 8° 16'

Sun's declin. 18th November, at noon, 19° 25' S. Equation to 3h from noon, -0 2 Equation to 24° 30' W., +0 1

Reduced declination, 19° 24'

Polar distance, 109° 24' Altitude, 8° 16' Secant, 0-00454 Latitude, 50° 22' Secant, 0-19527

Sum, 168° 2 Half, 84° 1 Cosine, 9-01803 Difference, 25° 23' Cosine, 9-55591

Half true azimuth, 22° 43' Sine, 9-58687

True azimuth, S. 45° 26' E. Observed azimuth, S. 23° 18' E.

Variation, 22° 8' W.

CHAP. V.—OF A SHIP'S JOURNAL.

A journal is a regular and exact register of all the various transactions that happen aboard a ship, whether at sea or land, and more particularly that which concerns a ship's way, from whence her place at noon or any other time may be justly ascertained.

That part of the account which is kept at sea is called sea work; and the remarks taken down while the ship is in port are called harbour work.

At sea, the day begins at noon, and ends at the noon of the following day; the first twelve hours, or those contained between noon and midnight, are denoted by P.M., signifying after mid-day; and the other twelve hours, or those from midnight to noon, are denoted by A.M., signifying before mid-day. A day's work marked Wednesday, March 6, began on Tuesday at noon, and ended on Wednesday at noon. The days of the week are usually represented by astronomical characters. Thus O represents Sunday; P, Monday; T, Tuesday; W, Wednesday; Th, Thursday; F, Friday; and S, Saturday.

When a ship is bound to a port so situated that she will be out of sight of land, the bearing and distance of the port must be found. This may be done by Mercator's or Middle Latitude Sailing; but the most expeditious method is by a chart. If islands, capes, or headlands intervene, it will be necessary to find the several courses and distances between each successively. The true course between the places must be reduced to the course per compass, by allowing the variation to the right or left of the true course, according as it is west or east.

At the time of leaving the land, the bearing of some known place is to be observed, and its distance is usually found by estimation. As perhaps the distance thus found will be liable to some error, particularly in hazy or foggy weather, or when that distance is considerable, it will therefore be proper to use the following method for this purpose.

Let the bearing be observed of the place from which the departure is to be taken; and the ship having run a certain distance on a direct course, the bearing of the same place is to be again observed. Now, having one side of a plain triangle, namely, the distance sailed, and all the angles, the other distances may be found by Prob. I. of Oblique Sailing.

The method of finding the course and distance sailed in a given time is by the compass, the log-line, and half-minute glass. These have been already described. In the royal navy, and in ships in the service of the East India Company, the log is hove once every hour; but in most other trading vessels only every two hours.

The several courses and distances sailed in the course of twenty-four hours, or between noon and noon, and whatever remarks are thought worthy of notice, are set down with chalk on a board painted black, called the log-board, which is usually divided into six columns; the first column on the left hand contains the hours from noon to noon; the second and third the knots and parts of a knot sailed every hour, or every two hours, according as the log is marked; the fourth column contains the courses steered; the fifth, the winds; and in the sixth the various remarks and phenomena are written. The log-board is transcribed every day at noon into the log-book, which is ruled and divided after the same manner.

The courses steered must be corrected by the variation of the compass and leeway. If the variation is west, it must be allowed to the left hand of the course steered; but if east, to the right hand, in order to obtain the true course. The leeway is to be allowed to the right or left of the course steered, according as the ship is on the larboard or starboard tack. The method of finding the va- variation, which should be determined daily if possible, is given in the preceding chapter; and the leeway may be understood from what follows.

When a ship is close hauled, that part of the wind which acts upon the hull and rigging, together with a considerable part of the force which is exerted on the sails, tends to drive her to the leeward. But since the bow of a ship exposes less surface to the water than her side, the resistance will be less in the first case than in the second; the velocity in the direction of her head will therefore in most cases be greater than the velocity in the direction of her side; and the ship's real course will be between the two directions. The angle formed between the line of her apparent course and the line she really describes through the water is called the angle of leeway, or simply the leeway.

There are many circumstances which prevent the laying down rules for the allowance of leeway. The construction of different vessels, their trim with regard to the nature and quantity of their cargo, the position and magnitude of the sail set, and the velocity of the ship, together with the swell of the sea, are all susceptible of great variation, and very much affect the leeway. The following rules, are, however, usually given for this purpose.

1. When a ship is close hauled, has all her sails set, the water smooth, with a light breeze of wind, she is then supposed to make little or no leeway.

2. Allow one point when the top-gallant sails are handed.

3. Allow two points when under close reefed top-sails.

4. Allow two points and a half when one top-sail is handed.

5. Allow three points and a half when both top-sails are handed.

6. Allow four points when the fore course is handed.

7. Allow five points when under the main-sail only.

8. Allow six points when under balanced mizen.

9. Allow seven points when under bare poles.

These allowances may be of some use to work up the day's work of a journal which has been neglected; but a prudent navigator will never be guilty of this neglect. A very good method of estimating the leeway is to observe the bearing of the ship's wake as frequently as may be judged necessary; which may be conveniently enough done by drawing a small semicircle on the taffrail, with its diameter at right angles to the ship's length, and dividing its circumference into points and quarters. The angle contained between the semidiameter which points right aft, and that which points in the direction of the wake, is the leeway. But the best and most rational way of bringing the leeway into the day's log is to have a compass or semicircle on the taffrail, as before described, with a low crutch or swivel in its centre; after heaving the log, the line may be slipped into the crutch just before it is drawn in, and the angle it makes on the limb with the line drawn right aft will show the leeway very accurately, which, as a necessary article, ought to be entered into a separate column against the hourly distance on the log-board.

In hard blowing weather, with a contrary wind and a high sea, it is impossible to gain any advantage by sailing. In such cases, therefore, the object is to avoid as much as possible being driven back. With this intention it is usual to lie under no more sail than is sufficient to prevent the violent rolling which the vessel would otherwise acquire, to the endangering her masts, and straining her timbers, &c. When a ship is brought to, the tiller is put close over to the leeward, which brings her head round to the wind. The wind having then very little power on the sails, the ship loses her way through the water; which ceasing to act on the rudder, her head falls off from the wind, the sail which she has set fills, and gives her fresh way through the water, which, acting on the rudder, brings her head again to the wind. Thus the ship has a kind of vibratory motion, coming up to the wind and falling off from it again alternately. Now the middle point between those upon which she comes up and falls off is taken for her apparent course; and the leeway and variation is to be allowed from thence, to find the true course.

The setting and drift of currents, and the heave of the sea, are to be marked down. These are to be corrected by variation only.

The computation made from the several courses, corrected as above, and their corresponding distances, is called a day's work; and the ship's place, as deduced therefrom, is called her place by account, or dead reckoning.

It is almost constantly found that the latitude by account does not agree with that by observation. From an attentive consideration of the nature and form of the common log, that its place is alterable by the weight of the line, by currents, and other causes, and also the errors to which the course is liable, from the very often wrong position of the compass in the binnacle, the variation not being well ascertained, an exact agreement of the latitudes cannot be expected.

When the difference of longitude is to be found by dead reckoning, if then the latitudes by account and observation disagree, several writers on navigation have proposed to apply a conjectural correction to the departure or difference of longitude. Thus, if the course be near the meridian, the error is wholly attributed to the distance, and the departure is to be increased or diminished accordingly; if near the parallel, the course only is supposed to be erroneous; and if the course is towards the middle of the quadrant, the course and distance are both assumed wrong. This last correction will, according to different authors, place the ship upon opposite sides of her meridian by account. As these corrections are, therefore, no better than guessing, they should be absolutely rejected.

If the latitudes are not found to agree, the navigator ought to examine his log-line and half-minute glass, and correct the distance accordingly. He is then to consider if the variation and leeway have been properly ascertained; if not, the courses are to be again corrected, and no other alteration whatever is to be made on them. He is next to observe if the ship's place has been affected by a current or heave of the sea, and to allow for them according to the best of his judgment. By applying these corrections, the latitudes will generally be found to agree tolerably well; and the longitude is not to receive any further alteration.

It will be proper, however, for the navigator to determine the longitude of the ship from observation as often as possible; and the reckoning is to be carried forward in the usual manner from the last good observation; yet it will perhaps be very satisfactory to keep a separate account of the longitude by dead reckoning.

General Rules for working a Day's Work.

Correct the several courses for variation and leeway; place them, and the corresponding distances, in a table prepared for that purpose. From whence, by Traverse Sailing, find the difference of latitude and departure made good; hence the corresponding course and distance, and the ship's present latitude, will be known.

Find the middle latitude at the top or bottom of the traverse table, and the distance, answering to the departure found in a latitude column, will be the difference of longitude; or, the departure answering to the course made good, and the meridional difference of latitude in a latitude column, is the difference of longitude; the sum or difference of which, and the longitude left, according as they are of the same or of a contrary name, will be the ship's present longitude of the same name with the greater. Compute the difference of latitude between the ship and the intended port, or any other place whose bearing and distance may be required: find also the meridional difference of latitude and the difference of longitude. Now the course answering the meridional difference of latitude found in a latitude column, and the difference of longitude in a departure column, will be the bearing of the place; and the distance answering to the difference of latitude will be the distance of the ship from the proposed place. If these numbers exceed the limits of the table, it will be necessary to take aliquot parts of them; and the distance is to be multiplied by the number by which the difference of latitude is divided.

It will sometimes be necessary to keep an account of the meridian distance, especially in the Baltic or Mediterranean trade, where charts are used in which the longitude is not marked. The meridian distance on the first day is that day's departure; and any other day it is equal to the sum or difference of the preceding day's meridian distance and the day's departure, according as they are of the same or of a contrary denomination.

It will be found very satisfactory to lay down the ship's place on a chart at the noon of each day, and her situation with respect to the place bound to and the nearest land will be obvious. The bearing and distance of the intended or any other port, and other requisites, may be easily found by the chart, as already explained; and indeed every day's work may be performed on the chart, and thus the use of tables superseded.

Specimens of a ship's journal may be found in the works on navigation already mentioned. The reader is also referred to these works for the traverse table, and tables of meridional parts, log. rising, middle time, half elapsed time, logarithmic difference, &c. which have been employed in the solution of the various problems.

END OF VOLUME FIFTEENTH. NAVIGATION.

It will be found very satisfactory to lay down the ship's place on a chart at the noon of each day, and her situation with respect to the place bound to or the nearest land will be obvious. The bearing and distance of the intended or any other port, and other requisite, may be easily found by the chart, as already explained; and indeed every day's work may be performed on the chart, and thus the use of tables suspended.

Specimens of a navigator's journal may be found in the works mentioned. The reader is also referred to the traverse table, and tables of logarithmic differences which have been employed in the solution of the problems.

END OF VOLUME FIFTEENTH.