ENCYCLOPÆDIA BRITANNICA.

NAVIGATION.

History. 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 Æginetes, the Phœnicians, 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.

The earliest record of the practice of the art of navigation is that of the Egyptians, who at a very remote period are said to have established commercial relations with India. This traffic was carried on between the Arabian Gulf and the western coast of India, across the Indian Ocean. It would appear, however, that this intercourse was of no long duration, and that the Egyptians soon confined themselves to overland traffic with their neighbours, even excluding from all access to their country those foreigners who would have traded with them by the Mediterranean Sea.

The Phœnicians were the most distinguished of the early navigators; their commercial relations with other nations were the most widely spread; and their capital, Tyre, was for ages the centre of ancient commerce and the "mart of nations." The narrowness and poverty of the little slip of ground they possessed along the coast, and the convenience of two or three good ports, naturally drove an enterprising and industrious people, stimulated by a genius for commerce, to seek by sea those riches which were denied them by land. 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 Phœnician 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.

At an early period of their history, although long subsequently to the rise of the Phœnician navigators, the Greeks learned and practised the art of navigating the adjacent seas, although they seem to have trusted almost entirely to the air as the instrument of propulsion. The celebrated voyage of the Argonauts belongs to a very early period; and in later times the Corinthians and Corcyreans disputed with Athens the empire of the Greek seas. At length 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 great navigation of the Egyptians, which was afterwards so much cultivated by the Ptolemies, that Tyre and Carthage were quite forgotten.

Egypt being reduced to 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 coast of Africa. By them the merchants were expelled, and Alexandria was, until lately, in a languishing state; though it always had a considerable share of the commerce of the Christian merchants trading to the Levant. A fresh impulse, however, has been given of late years to the trade of Alexandria by its having become an important post in the overland route to India.

The fall of Rome and its empire drew along with it not only the overthrow of learning and the polite arts, but also 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 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, recharges, &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 fairest 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 merit 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. This 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 broke out; and the two republics coming to an open rupture, there was almost continual war for three centuries. Towards the end of the fourteenth century, the battle of Chioza ended the strife. The Genoese, who till then had usually the advantage, now lost all; and the Venetians, reduced almost to despair, secured to themselves, by one happy and unexpected blow, the foremost place in commerce and the sole empire of the sea.

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 HANSEATIC LEAGUE, and COMMERCE. For the present state of navigation in the various countries of the world, see under the name of each.)

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 tutelary 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 progress of political and commercial science, which gradually opened men's eyes to the great principle that all trade is most healthy and prosperous when subjected to the fewest possible, and those only the most necessary, restrictions, resulted in the total repeal of these famous navigation laws in 1846, since which period trade with England has been free and open to all the world; but so far from British shipping and commerce being injured or diminished, they have been more prosperous since that repeal than they were at any former period, when most carefully fostered by protective laws.

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; and of late years the extensive and still growing employment of steam as the propelling power, whether applied to paddle-wheels or screws, has still further placed the mariner beyond the adverse retarding influence of calm and contrary winds, and has introduced an element of certainty and punctuality in commercial intercourse unknown at any previous period, and invaluable as it affects the interests of commerce. It is also very certain 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 Judæa were ignorant of this instrument; and

History. it is very improbable that such a 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 1545,—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 farther prosecuted by others, and is now generally accepted as true in the scientific world. Medina's book was soon translated into Italian, French, and Flemish, and for a long time served as a guide to foreign navigators. Cortes, however, 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 discovery 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 leagues were run in each rhumb, together with the departure from the meridian. Some other instruments were also 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 Nonius, published a book in the Portuguese language, to explain a difficulty in navigation proposed to him by the commander Don Martin Alphonso de Susa. 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 nonus.

In the year 1577, 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 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 Edward Wright,

History. 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 inventor of this contrivance is not known; but it was first described in an account of an East India voyage published by Purchas in 1607, from which time it became famous, and was much taken notice of by almost all writers on navigation in every country. 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 Nonius were scarcely practicable, and therefore in a great measure useless. In treating of the sun's declination, he took notice of the gradual decrease in the obliquity of the ecliptic; he also described the cross staff with three transverse pieces, 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 book 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 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 magnetic needle, by 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. He represented the degrees both of latitude and longitude by parallel straight lines, but gradually augmented the space between 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 a 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 Edward Wright, an Englishman.

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 widening between 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 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 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

History. 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 treated of sailing on a great circle, and showed how to draw the rhumbs on a globe mechanically; he also set down Wright's two tables of latitudes and of rhumbs, in order to describe these lines more accurately; and even pretended to have discovered an error in Wright's table. But 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 Willebrordus Snellius, professor of mathematics at Leyden, published a treatise of navigation on Wright's plan, but somewhat obscurely; and as he did not particularly mention all the discoveries of Wright, the latter was thought by some to have taken the hint of all his discoveries from Snellius. But this supposition has been long ago refuted; and Wright's title to the honour of those discoveries remains unchallenged.

Having shown how to find the place of the ship upon his chart, Wright 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, Raphe Handson, amongst the nautical questions which he subjoined to a translation of Pitiscus's Trigonometry, solved very distinctly every case of navigation, by applying arithmetical calculations to Wright's Tables 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, Handson 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 compends 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 operose; yet in high latitudes it is more eligible to use that of the arithmetical mean between the logarithmic cosines, equivalent to the geometrical mean between the cosines themselves—a method since proposed by 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°, 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 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,¹ 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 more readily finding the latitude from the observation. The discoveries concerning logarithms were carried into France in 1624 by 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 scale was projected into a circular arch by the Reverend 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 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. 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 Burrough, 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 Magnes, first printed at Rome in the year 1641, informs us that he had been told of it by John Greaves, and then gives a letter of the famous Marinus Mersennus, 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 did not write an express treatise on navigation, but only for correcting such errors as prevailed in general practice, the log-line did not fall under his notice. Richard Norwood, however, put in execution the method recommended by 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 57,300 French

¹ See GUNTER'S SCALE.

History. fathoms or toises; which is very exact, as appears from many measurements that have been made since that time. Of all this Norwood gave a full account in his treatise called the Seaman's Practice, published in 1657. 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 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 taking into consideration the variation of the compass; 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 work 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, 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 $45^\backslash circ$ 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 $45^\backslash circ$ 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 $45^\backslash circ$; divide the difference of those numbers by the logarithmic tangent of $45^\backslash circ\; 30\text{'}$, 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 stand for the meridional parts of one degree) the same proportion as the logarithmic tangent of half any latitude augmented by $45^\backslash circ$, and the radius neglected, to the like tangent of half a degree augmented by $45^\backslash circ$, with the radius likewise rejected. But here there was still wanting the demonstration of this general theorem, which was at length supplied by James Gregory of Aberdeen, in his Exercitationes Geometricæ, 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 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 $\backslash frac\{1\}\{2\}$th of this division, which does not sensibly differ from the logarithmic tangent of $45^\backslash circ\; 1\text{'}\; 30\text{'}\text{'}$, 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 $45^\backslash circ\; 1\text{'}\; 30\text{'}\text{'}$, diminished by the radius, be taken; that is, the number used by others being 12633, when the logarithmic tables consist of eight places of figures besides the index.

In an edition of a book called the Seaman's Kalender, Bond declared that he had discovered the longitude by having found out the true theory of the magnetic variation; and to gain credit to his assertion, he foretold, that at London in 1657 there would be no variation of the compass, and from that time it would gradually increase the other way; which happened accordingly. Again, in the Philosophical Transactions for 1668 (No. 40), he published a table of the variation for forty-nine years to come. Thus he acquired such reputation, that his treatise entitled The Longitude Found, was, in the year 1676, published by the special command of Charles II., and approved by many celebrated mathematicians. It was not long, however, before it met with opposition; and in the year 1678 another treatise, entitled The Longitude not Found, made its appearance; and as Bond's hypothesis did not answer its author's sanguine expectations, the solution of the difficulty was undertaken by Dr Halley. The result of his speculation was, that the magnetic needle is influenced by four poles; but this wonderful phenomenon seems hitherto to have eluded all our researches. (See MAGNETISM.) In 1700, however, Dr Halley published a general map, with curve lines expressing the paths where the magnetic needle had the same variation; which was received with universal applause. But as the positions of these curves vary from time to time, they should frequently be corrected by skilful persons, as was done in 1644 and 1756, by Mountain and Dodson. In the Philosophical Transactions for 1690, Dr Halley also gave a dissertation on the monsoons, containing many very useful observations for such as sail to places subject to these winds.

After the true principles of the art were settled by Wright, Bond, and Norwood, new improvements were daily made, and everything relative to it was settled with an accuracy not only unknown to former ages, but which would have been reckoned utterly impossible. The earth being found to be, not a perfect sphere, but a spheroid, with the shortest diameter passing through the poles, a tract was published in 1741 by the Reverend Dr Patrick Murdoch, wherein he accommodated Wright's sailing to such a figure; and the same year Colin Maclaurin, in the Philosophical Transactions (No. 461), gave a rule for determining the meridional parts of a spheroid; which speculation is farther treated of in his book of Fluxions, printed at Edinburgh in 1742, and in Delambre's Astronomia (t. iii., ch. xxxvi.).

Amongst the later discoveries in navigation, that of finding the longitude, both by lunar observations and by time-keepers, is the principal. It is owing chiefly to the rewards offered by the British Parliament that this has attained the present degree of perfection. We are indebted to Dr Maskelyne for putting the first of these methods in practice, and for other important improvements in navigation. The time-keepers constructed by Harrison for this express

Practice of purpose were found to answer so well that he obtained the parliamentary reward. These have been improved by Arnold, Earnshaw, and many others, so as now to be almost in common use.

The works which have latterly appeared on navigation are those on the longitude and navigation by Mackay, Inman, Riddle, Norie, Jeans, and others; and these contain every necessary requisite to form the practical navigator.

PRACTICE OF NAVIGATION.

BOOK I.

CONTAINING THE VARIOUS METHODS OF SAILING.

The art of navigation depends upon mathematical and astronomical principles. The problems in the various modes of sailing are resolved either by trigonometrical calculations, or by tables or rules formed by the assistance of plane and spherical trigonometry. By mathematics the necessary tables are constructed and rules investigated for performing the more difficult parts of navigation.

The places of the sun, moon, and 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 investigation of the rules required for this purpose belongs properly to the science of ASTRONOMY, to which the reader is referred. 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 this subject, it seems unnecessary to insert them in this place. The subject naturally divides itself under two heads:—First, The methods of conducting a ship from one port to another by help of rules, in which the log-line and compass are alone required, which is Navigation properly so called. Second, The method of ascertaining the ship's latitude and longitude, and variation of compass, by means of observations on the heavenly bodies; and the rules for that purpose deduced from astronomy, in order to correct the ship's place, and the courses derived from the former method, to which the name of Nautical Astronomy is generally applied. Although the reader is referred to the respective articles on the sciences on which navigation is founded in this work for complete information, we shall, nevertheless, endeavour to make our explanation of the several rules as complete as possible, even at the risk of repeating somewhat the substance of portions of our other articles.

CHAP. I.—PRELIMINARY PRINCIPLES.

SECT. I.—ON LATITUDE AND LONGITUDE; DEFINITION OF TERMS USED IN NAVIGATION; AND GENERAL EXPLANATIONS.

1. Latitude and Longitude.

The situation of a place, or any object on the earth's surface, is estimated by its distance from two imaginary lines on that surface 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 assume that to be the first meridian which passes through the Royal Observatory at Greenwich.

The equator is a great circle on the earth's surface, every point of which is equally distant from the two poles or the extremities of the imaginary axis about which the earth makes her diurnal rotation. It therefore divides the earth into two equal parts, called the Northern and the Southern

Hemispheres, according as the North or the South Pole lies within them.

The latitude of a place is its distance from the equator, reckoned on a meridian in degrees, minutes, and seconds, and decimal parts of seconds (if necessary), being either north or south, according as it is the Northern or Southern Hemisphere. Hence it appears that the latitudes of all places are comprised within the limits $0^\{\backslash circ\}$ and $90^\{\backslash circ\}$ N., and $0^\{\backslash circ\}$ and $90^\{\backslash circ\}$ S.

The first meridian, which is a great circle passing through the poles, also divides the earth into two equal portions, called the Eastern and Western Hemispheres, according as they lie to the right or left of the first meridian; the spectator being supposed to be looking towards the north.

The longitude of a place is the arc of the equator intercepted between the first meridian and the meridian of the given place reckoned in degrees, minutes, and seconds; and is either east or west as the place lies in the Eastern or Western Hemisphere respectively to the first meridian. The longitude of all places on the earth's surface is comprised within the limits of $0^\{\backslash circ\}$ and $180^\{\backslash circ\}$ E., and $0^\{\backslash circ\}$ and $180^\{\backslash circ\}$ W.

On the supposition that the earth is a sphere, the length of all arcs of great circles upon it subtending an angle of $1^\{\backslash circ\}$ at the centre are equal; hence $1^\{\backslash circ\}$ of latitude or longitude is equal to one geographical or nautical mile, of which a degree contains 60. Hence intervals of latitude and longitude, reduced to minutes and parts of minutes, also represent the same number of nautical miles and parts of a nautical mile.

In the practice of navigation, the latitude and longitude of the place which a ship leaves, are called the latitude and longitude from; and the latitude and longitude of the place at which it has arrived, are called the latitude and longitude in.

Fig. 1.

2. Definitions of Terms used in Navigation, and Explanations.

Let QR...V be a portion of the equator, P the pole, and PAQ, PBR, PCS.....PFV be meridians supposed very near to one another, passing through points A, B, C, D, E, F, the line AF being the path traced out by a vessel in passing from A to F, such that it makes equal angles with every meridian over which it passes. From B, C, D, &c., let BH, CI, DK, EL, &c., be drawn perpendicular to the

two meridians between which they respectively lie; or, in other words, the arcs of small circles or parallels of latitude through the points B, C, &c. These are consequently all parallel to one another, and to FG the whole arc of the parallel at F included between the extreme meridians PAQ and PFV.

The constant angle at which the line AF is inclined to the successive meridians, viz., BAP, CBP, DCP, &c., is called the course. Also, if the small circles or parallels at B, C, &c., be continued to the meridian PAQ, the portion of this meridian intercepted between any two consecutive parallels, as cd, will be equal to CK, the distance between the parallels through C and D; and so on for all. Hence the sum of these distances, $AH\; +\; BI\; +\; CK\; +\; DL\; +\; EM\; =\; AG$; which is called the true difference of latitude, or true diff. lat. from A to F.

The corresponding arcs of parallels at different latitudes intercepted between the same meridians are not equal, but gradually decrease from the equator to the poles. Hence the sum of the arcs $BH\; +\; CI\; +\; DK\; +\; EL\; +\; FM$ is less than QV, the intercepted arc of the equator, but greater than FG, the arc of the highest parallel intercepted between PAQ and PFV.

$BH\; +\; CI\; +\; DK\; +\; EL\; +\; FM$ is called the departure; the arc QV of the equator is the difference of longitude; and AF, the curve described by the vessel in passing from A to F, is called the distance.

In navigation, each of the triangles ABH, BCI, &c., is considered as a plane triangle; and as each of them is right-angled, and contains, besides, one constant angle, viz., the course, the other angle in each must also be constant; and all the triangles will be equiangular and similar. Hence we have

And since, when any number of quantities are in continued proportion, as the first consequent is to its antecedent, so are all the consequents to all the antecedents; we have

Hence the true difference of latitude, departure, and distance, may be considered as the sides of a right-angled triangle, similar to each of the small triangles; the angle of which, therefore, between the true difference of latitude and distance, is the course.

Take AB (fig. 2) = the true diff. lat. Draw BC at right angles to it = departure. Join AC. Then AC is the distance, and BAC is the course.

From this it appears, that when any two of the four quantities, true difference of latitude, departure, distance, and course are given, the remaining two can be found by solving the right-angled triangle ABC.

1. Given course (BAC) and distance (AC), to find true difference of latitude (AB), and departure (BC).

2. Given course (BAC) and true difference of latitude (AB), to find distance and departure.

3. Given course and departure, to find distance and true difference of latitude.

4. Given distance and true difference of latitude, to find course and departure.

And the course having been found, we get

5. Given the distance and departure, to find the course and true difference of latitude.

And then we have

6. Given the true difference of latitude and departure, to find the course and distance.

And having found the course, we have

Length of Arc of 1° of Parallels of Latitude.

We have already stated that the lengths of the parallels of latitude diminish as the latitude increases. In fact, it decreases in the ratio of cosine latitude to unity.

Let EQ be the equator, PCP' the polar diameter of the earth passing through C, and LML' any parallel of latitude. Let the angle ECL, or the latitude, be $l$, and let LM and EF be the arcs of the parallel and of the equator intercepted between the meridians PEP' and PFP'. Then angle ECF = angle LOM, because they both measure the angle between the planes of the two meridians. Hence

Hence if FE be the length of an arc 1° of longitude at the equator, or 60 miles, LM the length of an arc 1° of longitude in latitude $l\; =\; 60\; \backslash times\; \backslash cos\; l$.

The departure is less than VQ (fig. 1), the intercepted arc of the equator, or than the intercepted arc of the parallel through A; but it is greater than FG. But since the arc of the parallel gradually decreases from A to F, there is some point intermediate in position between A and F ($y$), the intercepted arc of the parallel of which will be exactly equal to the departure. The exact determination of this point is not very easy. Various methods have been proposed to determine this latitude nearly, with as little trouble as possible: first, by taking the arithmetical mean of the two latitudes for that of the mean latitude; secondly, by using the arithmetical mean of the cosines of the latitudes; thirdly, by using the geometrical mean of these cosines; and, lastly, by employing the latitude deduced from the mean of the meridional parts of the two latitudes. The first of these methods is the one usually employed. It has the merit of great simplicity; and as all the rules in navigation are approximate only, it may perhaps be depended on as much as any of these. Hence $y$ may be considered as the middle point between A and F; and $\backslash text\{dep.\}\; =\; xyz$.

Whence it appears that if the middle latitude be considered as a course, and the departure as a true difference of latitude, the corresponding distance will be the difference of longitude. And conversely, treating the middle latitude as a course, and the difference of longitude as a distance, the corresponding true difference of latitude will be the departure.

The chart used at sea for tracing the ship's track exhibits the surface of the earth on a plane, in which the meridians are parallel, and consequently the distance between them throughout their length equal to the equatorial distance, instead of gradually decreasing as the latitude increases. In other words, FG is increased so as to become equal to VQ. Now, in order that on this chart all points may occupy the same relative position with respect to each other that the points corresponding to them do on the surface of the globe, the distance AF and the distance AG must be increased in the same proportion that BH + CI + DK + &c., i.e., the departure has been increased.

The distance AG so increased is called the meridional difference of latitude, or mer. diff. lat.; and the chart constructed on this principle is called Mercator's Chart. If for any latitude the meridional difference of latitude between this point and the equator be expressed in miles or minutes, the number of miles so expressed is called the meridional parts for that latitude. A table of meridional parts for every minute of latitude from $0^\backslash circ$ up to $90^\backslash circ$, is given in every collection of nautical tables.

This table may be constructed approximately, by dividing the whole meridian from $0^\backslash circ$ to $90^\backslash circ$ into intervals of $1\text{'}$, and supposing the increase of the arc of the parallel of latitude, and consequently that of the arc of latitude, to take place at the end of the successive minutes.

Now we have seen that an arc of any parallel = corresponding arc of equator $\backslash times\; \backslash cos\; \backslash text\{lat.\}$; and since the arcs of the successive parallels have all become equal to the correspond-

ing arc of the equator, they have all been increased in the ratio of $\backslash sec\; \backslash text\{lat.\}$ to 1.

Hence, if the length of $1\text{'}$ of the meridian be 1, and $a,\; b,\; c,\; d$, the corresponding increased lengths between 0 and $1\text{'}$, between $1\text{'}$ and $2\text{'}$, $2\text{'}$ and $3\text{'}$, &c.,

And ; or the meridional parts in any arc of the meridian is equal to the sum of the secants of all the successive angles, differing by $1\text{'}$, from $1\text{'}$ up to the given latitude.

The true investigation is as follows:—Let $m$ = the circular measure of the angle subtended at the centre by the meridional parts in the arc between the latitudes 0 and $l$; and let $l$ become $l+\backslash delta\; l$, and let $\backslash delta\; m$ be the corresponding increase of $m$. Then $\backslash delta\; m$ is proportional to the secant of $l+\backslash delta\; l$;

Diminish the tabular logarithm of the cotangent of one-half the colatitude by 10. Find the logarithm of the remainder, and add to it the constant logarithm 3.8934895. The result is the logarithm of the meridional parts for the given latitude.

Preliminary principles. Ex.—Required the meridional parts for latitude $65^\{\backslash circ\}$ and $65^\{\backslash circ\}\; 20\text{'}$.

Relations between Meridional Difference of Latitude and Difference of Longitude, Course, &c.

It appears that on Mercator's chart the difference of longitude, meridional difference of latitude, and increased distance, form the sides of a right-angled triangle, and are proportional to the departure, true difference of latitude, and distance, in the triangle ABC. The two triangles are therefore similar.

If, then, in AB produced (fig. 4) AB' be taken equal to mer. difference of latitude, and B'C' be drawn parallel to BC, meeting AC produced in C'; the sides of the triangle ABC' will be the meridional difference of latitude, difference of longitude, and increased distance.

Hence, in equations (1.), (2.), (3.), (4.), (5.), (6.), (7.), and (8.), we may substitute meridional difference of latitude for true difference of latitude, difference of longitude for departure, and increased distance for distance; and the equations will still hold.

The relation principally required is that which connects the meridional difference of latitude, difference of longitude, and course;

Parallel Sailing.

When the course is $90^\{\backslash circ\}$, or the ship sails in a parallel of latitude, the equations (1.), &c., give no result when applied to find the distance.

In this case we must apply the formula, arc of parallel = corresponding arc of equator $\backslash times\; \backslash cos\; \backslash text\{\; lat.\}$ Here the arc of the parallel corresponds to the distance, and the arc of equator is difference of longitude, and we have for parallel sailing

from which, any two of the quantities being given, the third may be found. Also, if $d$ and $d\text{'}$ be distances corresponding to the same difference of longitude in parallels $l$ and $l\text{'}$,

which enables us to find the distance on one parallel corresponding to a given distance on another, the difference of longitude being the same.

Middle-Latitude Sailing.

Since departure = diff. of longitude $\backslash times\; \backslash cos\; \backslash text\{\; mid.\; latitude\}$, if in equations (2.), (4.), (6.), (8.), we substitute this value for departure, we shall obtain the following equations:—

from which all the rules for middle-latitude sailing may be derived.

Traverse Tables.

A table in which the true difference of latitude and departure, corresponding to certain distances for every course, expressed in points and degrees, are laid down, is called a traverse table. It is very useful to enable the seaman to solve the several problems which occur in navigation by simple inspection. It must of course be calculated on some of the principles laid down in this chapter.

It is evident that as this table contains the relations of the sides and angles of a right-angled triangle, the solution of any right-angled triangle, whose sides represent any other quantities, will be given by it, by making the requisite changes.

Thus, the course remaining the same, if, for difference of latitude we look out in the table the meridional difference of latitude, the corresponding departure will be the difference of longitude; also, the difference of longitude can be found from the departure by these tables, by looking out the middle latitude as a course, and the departure as a true difference of latitude; then the corresponding distance is the difference of longitude.

SECT. II.—LONGITUDE AND LATITUDE.

PROB. I.—Given latitude from, and latitude in; to find the true difference of latitude.

Rule.—Under latitude in, with its proper name, i.e., N. or S., place latitude from. Subtract the greater from the less, if of the same name, and reduce to minutes. The result is the true difference of latitude required, and is of the same or different name with latitude in, according as latitude in is greater or less than latitude from. If they are of different names, add and affect with the name of latitude in.

Ex. 1.—A vessel sails from the Lizard, Lat. $49^\{\backslash circ\}\; 58\text{'}\; N.$, to Cape St Vincent, Lat. $37^\{\backslash circ\}\; 3\text{'}\; N.$; what is the true diff. of latitude?

Ex. 2.—A vessel sails from New York, Lat. $40^\{\backslash circ\}\; 42\text{'}\; N.$, to Liverpool, Lat. $53^\{\backslash circ\}\; 25\text{'}\; N.$; find the true diff. of latitude.

Ex. 3.—A ship sailed from Funchal, Lat. $32^\{\backslash circ\}\; 38\text{'}\; N.$, to the Cape of Good Hope, Lat. $34^\{\backslash circ\}\; 29\text{'}\; S.$; what is the true diff. of latitude?

PROB. 2.—Given the latitude from, and true difference of latitude; to find latitude in.

Rule.—If the latitude from and true difference be of the same name, add them (the true difference of latitude being turned, if necessary, into degrees and minutes); the sum is the latitude in of the same name. If of unlike names, under latitude from place the true difference of latitude, subtract the less from the greater; the result, with the name of the greater, is the latitude in.

Ex. 1.—A ship sailed from the Lizard, $49^\{\backslash circ\}\; 58\text{'}\; N.$, and made good, in a northerly direction, 207 miles; what is the latitude in?

Ex. 2.—A ship in Lat. $57^\{\backslash circ\}\; 18\text{'}\; N.$ sailed due S. 3789 miles; what is the latitude in?

PROB. 3.—To find the meridional difference of latitude; having given latitude from, and latitude in. Take the meridional parts for the two latitudes from the table for meridional parts; subtract if the names be alike, and add if the names be unlike. The result is the meridional difference of latitude, and is N. or S. according as the latitude in is N. or S. of latitude from.

Ex. 1.—Required the meridional diff. of latitude in sailing from Cape Finisterre, Lat. 42° 52' N., to Port Praya, in the island of Santiago, Lat. 14° 54' N.
Latitude in..... 14° 54' N. Mer. parts..... 904 N.
Latitude from..... 42 52 N. Mer. parts..... 2852 N.
Mer. diff. lat.... 1948 S.

Ex. 2.—To find the meridional diff. of latitude in sailing from Lat. 5° 35' N. to 8° 17' S.
Latitude in..... 8° 17' S. Mer. parts..... 499 S.
Latitude from..... 5 35 N. Mer. parts..... 335 N.
Mer. diff. lat.... 834 S.

PROB. 4.—To find the difference of longitude; having given longitude in, and longitude from.

Rule.—Under longitude in place longitude from; subtract, if of like name; reduce the result to minutes, and call it E. or W., according as longitude in is E. or W. of longitude from. Add, if of unlike names, and attach the name E. or W., according as the longitude in is E. or W. of longitude from. If the longitude found by this rule exceed 180° it must be subtracted from 360°, and affected with the contrary name.

Ex. 1.—A ship sails from Liverpool, Long. 2° 59' W., to New York, Long. 73° 59' W.; required the diff. of longitude.

Longitude in..... 73° 59' W.
Longitude from..... 2 59 W.
Diff. Longitude..... 71 0 W. = 4260 miles W.

Ex. 2.—A ship sails from Maskelyne's Isles, in Long. 167° 59' E., to Olinda, in Long. 35° 54' W.; find the diff. of longitude.

Longitude in..... 35° 54' W.
Longitude from..... 167 59 E.
203 53 W.
360 0
Diff. longitude..... 155 7 E. = 9367 miles E.

PROB. 5.—To find the longitude in; having given longitude from, and difference of longitude.

Rule.—Under longitude from place difference of longitude. If of like names, add, and the result is the longitude in of the same name as the longitude from; if of unlike names, subtract the less from the greater. The result, with the name of the greater, is the longitude in.

Ex. 1.—A ship from Long. 9° 54' E. sailed westerly till the difference of longitude was 1398 miles; what is the longitude in?

Longitude from..... 9° 54' E.
Diff. longitude..... 23 18 W.
Longitude in..... 13 24 W.

Ex. 2.—The longitude sailed from is 25° 9' W., and diff. of longitude 112° 6' W.; find longitude in.

Longitude from..... 25° 9' W.
Diff. longitude..... 18 46 W.
Longitude in..... 43 55 W.

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.)

The interval between two consecutive knots on the line—also technically called a knot—is supposed to be the same fraction of a nautical mile (6080 feet) that half a minute is

of an hour. Hence the proper length of a knot is $\backslash frac\{6080\}\{120\}\; =\; 51$ feet nearly. 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 on the glass, it will be necessary to examine them from time to time; and the distance given by them must be corrected accordingly. The distance sailed, therefore, may 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. 1.—The distance sailed by the log, and the seconds run by the glass, being given; to find the true distance run, the line being supposed right.

Let the number of seconds in which the glass runs out be $n$, and let $d$ and $d\text{'}$ be the true distance and distance by log respectively. Then evidently the longer the time the glass is running, the less is the distance by log compared with the true distance, and conversely. Hence we have the proportion—

Rule.—Multiply the distance given by log by 30, and divide by the number of seconds the glass is running; the result is the true distance run.

Ex. 1.—The hourly rate of sailing by the log is 9 knots, and the glass is found to run out in 35"; required the true rate of sailing.

Ex. 2.—The distance sailed by the log is 73 miles, and the glass runs out in 26"; required the true distance.

PROB. 2.—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".

Here evidently the true distance is greater or less than the distance by log, as the measured interval is greater or less than 51 feet.

Let $a$ be the measured interval between the knots in feet, and $d$ and $d\text{'}$ the true, and measured distances as before.

Then $51\; :\; a\; ::\; d\text{'}\; :\; d$,
or 51 feet : measured distance in feet :: distance by log : true distance

Rule.—Multiply the distance given by log by the measured length of a knot, and divide by 51; the quotient is the true distance.

Ex. 1.—The hourly rate of sailing by the log is 5 knots, and the interval between knot and knot measures 53 feet; required the true rate of sailing.

Ex. 2.—The distance sailed is 85 miles, by a log line which measures 42 feet to a knot; required the true distance run.

PROB. 3.—Given the length of a knot, the number of seconds run by the glass in half a minute, and the distance

Preliminary principles. sailed by the log. To find the true distance we must evidently compound the ratios given in problems 1 and 2 and we have

Rule.—Multiply the distance given by the log by 10 times the measured distance between the knots, and divide by 17 times the number of seconds the glass is running.

Ex.—The distance sailed by the log is 159 miles, the measured length of a knot is 42 feet, and the glass runs out in 33"; required the true distance.

SECT. IV.—ON COURSES AND CORRECTIONS OF COURSES.

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 (fig. 5) gives a general view of

Fig. 5.

the compass. (For a full explanation of its magnetic properties, see MAGNETISM.) The names of the points, and the angles which they form with the meridian, are given in fig. 6, and as thus represented the instrument is called the steering compass.

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.

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

Fig. 6.

effects of local attraction, arising from the effects of the iron, guns, &c., in the vessel itself.

The compass course generally differs from the true course on account of three causes:—1. The variation of the compass; 2. The deviation of the compass; 3. The leeway. We shall now explain how these errors are to be applied.

1. The Variation of the Compass.—This is fully explained under the article MAGNETISM, which see. The mode of ascertaining its amount will be given hereafter.

PROB. I.—To find the true course, having given compass course.

Rule 1.—Allow easterly variation to the right, and allow westerly variation to the left.

Ex. 1.—The compass course is W.N.W., and variation 3½ pts. W.; find the true course.

Ex. 2.—The compass course is S.W.¼W., and variation 2½ E.; find the course.

Ex. 3.—The compass course is N.W., the variation is 3½ E.; required the true course.

PROB. II.—Given the true course, to find the compass course.

Rule 2.—Allow easterly variation to the left, and westerly to the right.

Ex. 1.—The true course is N.N.E.¼E., and variation 1½ W.; find the compass course.

Ex. 2.—The true course is N.¼E., the variation 3½ E.; required the compass course.

2. The Deviation of the Compass.—This error arises from the effects of local attraction, and varies with every different position of the ship's head. Several methods are employed in order to ascertain its amount. That most commonly adopted is to place a compass on shore, out of reach of the ship's attraction, and to take the bearing of the ship's compass, or some other object in the same direction with it; while at the same time the bearing of the compass on shore is taken on board. If now 180° be added to the bearing of the shore compass, so as to bring it round to the opposite point, the difference between this augmented bearing and the bearing at the ship's compass will be the amount of deviation for that position of the ship's head.

Suppose the ship's head is N., and that the reading off at the shore compass is S. 17° 15' W., and that the reading off at the ship's compass is N. 20° E. Adding 180° to the bearing of the shore compass, we get S. 197° 15' W., or N. 17° 15' E.; and subtracting this from the bearing of the ship's compass, N. 20° E., we get the deviation equal to 2° 45' E., when the ship's head is N. The ship is now turned round, so that the head points successively to every point of the compass, and the deviation for each position found as before.

A table is then made, showing the deviation for every point of the compass. The deviation so found is treated exactly as the variation,—i.e., in correcting the compass course to find the true course, easterly deviation is allowed to the right, and westerly deviation to the left; and conversely, to find the compass course from the true course, easterly deviation is allowed to the left, and westerly deviation to the right. Hence it appears, that when both variation and deviation are given, we may consider the latter as a correction of the former—to be added to it if of the same name, and to be subtracted from it if of the opposite name.

Ex. 1.—The compass course is S.W. ¼ W., variation 1¼ E., and deviation ¼ W.; what is the true course?

Ex. 2.—The compass course is W. ¼ N., the variation 2½ W., and deviation ¼ W.; what is the true course?

Ex. 3.—The true course is N.N.W. ¼ W., variation 1¼ E., and deviation ¼ W.; required the compass course.

The following table is taken from the monthly examination-papers at the Royal Naval College, Portsmouth, and will serve as a specimen of the tables which ought to be made for all ships:—

Deviation of the Compass of H.M.S. Vesuvius for different Preliminary positions of the Ship's Head.

3. Leeway.—The effect of the action of the wind upon the sails and hull of a ship is sometimes to produce a motion of the ship in a direction at right angles to that of the head or apparent course, as well as in this latter direction. The true course, therefore, is not that given by the compass, but that which is due to the composition of the two velocities of the ship,—viz., that in the direction of its head, and that at right angles to this direction. To obtain the true course from the compass course, therefore, we must add or subtract the angle of leeway, which is the angle between the compass course and the true course.

If the wind be on the right of a person on board ship who is looking towards the head, the real course is then evidently to the left of the direction of the ship's head,—i.e., of the apparent course. If the wind be on his left hand, the true course is to the right. In the former case the ship is said to be on the starboard tack, and in the latter on the port tack. Whence is derived the rule for obtaining the true course from the compass course.

Rule.—If the ship is on the starboard tack, allow leeway to the left; if on the port tack, allow leeway to the right. Conversely, to obtain the compass course from the true course on the starboard tack, allow leeway to the right; if on the port tack, allow it to the left.

There are many circumstances which prevent laying down accurate rules for the allowance of leeway. The construction of different vessels, their trim with regard to the nature and quantity of cargo, the position and area of sail set, the velocity of the ship and the swell of the sea, are all susceptible of great variation, and very much affect the leeway.

The following rules are usually given for the purpose:—

1. 1. When a ship is close-hauled under all sail, the water smooth, and with a light breeze, allow no leeway.
2. 2. When the top-gallant sails are hauled, allow one point.
3. 3. Under close-reefed topsails allow two points.
4. 4. When one topsail is hauled, allow two points and a half.
5. 5. When both topsails are hauled, allow three points.
6. 6. When the fore-course is hauled, allow four points.
7. 7. When under mainsail only, allow five points.
8. 8. Under balanced mizen, allow six points.
9. 9. Under bare poles, allow seven points.

These rules, however, are not much to be depended upon. 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 taffarel, with its diameter at right angles to the ship's length, and dividing its circumference into points and quarters. The angle contained between the semi-diameter which points right aft,

Plane Sailing. and that which points in the direction of the wake, is the leeway. But the best and most rational way of finding the leeway is to have a compass or semicircle on the taffel, 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.

Ex. 1.—A ship's apparent course is S.S.W. $\backslash frac\{1\}\{2\}$ W., leeway $\backslash frac\{1\}\{2\}$ points, the wind being S.E. $\backslash frac\{1\}\{2\}$ E.; required the true course. In this case the wind is on the left of the vessel, or it is on the port tack, and leeway must be allowed to the right.

Ex. 2.—The apparent course is N.N.W., leeway $\backslash frac\{1\}\{2\}$ points, and wind E.N.E. Here the vessel is on the starboard tack.

Ex. 3.—The true course of a ship is S.E. $\backslash frac\{1\}\{2\}$ S., leeway $\backslash frac\{1\}\{2\}$ points, and wind N.N.E.; required the compass course. Here the ship is on the port tack.

CHAP. II.—ON 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 considered parallel 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; and consequently the departure and difference of longitude are equal. In fact, in the right-angled triangle ABC (fig. 7), where AB is the true difference of latitude, BAC the course, AC the distance, and BC the departure, which is assumed equal to the difference of longitude; all the problems in sailing are solved by the relations of the sides and angles of the single right-angled triangle ABC. Except, however, for a small portion of the earth's surface near the equator, the departure cannot be assumed equal to the difference of longitude without very considerable error; and the longitude in cannot be at all depended on when found by this method. If, however, the departure be an element, this method is correct.

In fig. 7, A is the place from which the ship sails; AB the meridian, and equal to the true difference of latitude; BC perpendicular to the meridian, and equal to the departure. It is always possible and easy to construct a right-angled triangle when two parts, of which one is a side, beside the right angle, are given. Consequently problems in navigation may always be solved by construction, with the aid of the rule and compasses. In making constructions for this purpose, it is only necessary to attend to the following convention:—Let the upper part of the paper or plan on which the drawing is to be made represent the north; then the lower part will be south, the right-hand side east, and the

left-hand side west. This convention we have already tacitly assumed in treating of the corrections of the courses.

To make a Construction.

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 end of this line is to be marked as the position of the place, according as the course is southerly or northerly. From this point as centre, with the chord of $60^\backslash circ$ (on the rule), an arc of a circle 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 on this arc, beginning from the meridian. A straight line drawn through this point and the point sailed from is the direction of the distance, which, if given, must be laid down on this line, beginning at the point sailed from. A straight line is to be drawn from the extremity of the distance perpendicular to the meridian; and hence the true difference of latitude and the departure will be found.

If the true difference of latitude be given, it is to be laid down on the meridian, beginning at the point from which the ship sailed; and a straight line drawn through the extremity of the difference of latitude, perpendicular to the meridian, to meet the distance produced, will limit the figure, and enable us to find the parts required.

If the departure be given, it is to be laid off on a parallel, and the line drawn through its extremity will limit the distance.

If the distance and true difference of latitude be given, through the extremity of the true difference of latitude draw a straight line perpendicular to the meridian; extend a pair of compasses to the given distance, place one of its points at the place from which the ship sailed, and let the other point be in the perpendicular line first drawn; join this point with the point from, and the triangle is determined, and the course and departure found.

If the departure and distance are given, with the point from as centre and the distance as radius, describe an arc of a circle. Let the departure be laid off on a parallel, so that one point is in the meridian and the other in the circle just described. Join this latter point with the point from, and the triangle is formed. The general mode of solving problems in plane sailing has already been given in chap. i., sect. 2. The following examples will show how the formulae are to be applied:—

Obs. It is to be distinctly understood that the above method cannot be applied to obtain the difference of longitude without very sensible error.

Ex. 1.—A ship from St Helena, in Lat. $15^\backslash circ\; 55\text{'}\; S.$ sailed S.W. by S. 158 miles; required the latitude in, and departure.

By Construction.—Draw the meridian AB (fig. 8), and with the chord of $60^\backslash circ$ describe the arc $mn$, 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 true difference of latitude.

To find departure.

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.¹

Ex. 2.—A ship from St George's, in Lat. 35° 45' N., sailed S.E. 1/4 S., and the latitude by observation was 35° 7' N.; required the distance run, and departure.

By Construction.—Draw the portion of the meridian AB (fig. 9) equal to 218 miles; from the centre A, with the chord of 60°, describe the arc mn, which make equal to the rhumb of 3 1/4 points; through A 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 miles, and the departure BC 179 miles.

Fig. 9.

By Calculation.—To find the distance.

To find departure.

By Inspection.—Find the given difference of latitude 218 miles in latitude column, under the course of 3 1/4 points; opposite to which, in distance column, is 282 miles; in departure column 178.9 m.; the distance and departure required.

By Gunter's Scale.—Extend the compass from 4 1/4 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 3 1/4 points on the line of tangent rhumbs (marked T. R.) will reach from 218 miles to 178.9, the departure on numbers.

Ex. 3.—A ship from Palma, in Lat. 28° 37' N., sailed N.W. by W., and made 192 miles of departure; required the distance run, and the latitude come to.

By Construction.—Make the departure BC (fig. 10) equal to 192 miles, draw BA perpendicular to BC, and from the centre C, with the chord of 60°, describe the arc 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.

Fig. 10.

By Calculation.—To find the distance.

To find the true difference of latitude.

By Inspection.—Find the departure 192 miles 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.

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.

Ex. 4.—A ship from a place in Lat. 43° 13' N., sails between the north and east 285 miles, and is then by observation found to be in Lat. 46° 31' N.; required the course and departure.

By Construction.—Draw the portion of the meridian AB (fig. 11) equal to 198 miles; from B draw BC perpendicular to AB; then take the distance 285 miles from the scale, and with one foot of the compass in A describe an arc intersecting BC in C, and join AC. With the chord of 60° describe the arc mn, the portion of which contained between the distance and difference of latitude, applied to the line of chords, will measure 46°, the course; and the departure BC being measured on the line of equal parts, will be found equal to 205 miles.

Fig. 11.

By Calculation.—To find the course.

To find the departure.

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

By Gunter's Scale.—The extent from the distance 285 to the difference of latitude 198 on numbers, will reach from 90° to 44°, the complement of the course on sines; and the extent from 90° to the course 46° on the line of sines being laid from the distance 285, will reach to the departure 205 on the line of numbers.

Ex. 5.—A ship from Fort-Royal, in the island of Grenada, in Lat. 12° 9' N., sailed 260 miles between the south and west, and made 190 miles of departure; required the course and latitude come to.

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 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 arc intersecting AB in A, and join AC. Now describe an arc from the centre A with the chord of 60°, and the portion mn of this arc, 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.

To find the true difference of latitude.

By Inspection.—Seek in the traverse table until the nearest to the given departure is found in the same line with the given distance 200. This is found to be in the page marked 47° at the bottom, which is the course; and the corresponding difference of latitude is 177.3.

By Gunter's Scale.—The extent of the compass, from the distance 200 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 200 to the difference of latitude 177½ on the line of numbers.

Ex. 6.—A ship from a port in Lat. 7° 56' S. sailed between the south and east till her departure was 132 miles, and was then, by observation, found to be in Lat. 12° 3' S.; required the course and distance.

By Construction.—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 arc from the centre A; and the portion mn of this arc, 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.

To find the distance.

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

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

The six problems whose solutions are illustrated above

are all that occur in the solution of the right-angled triangle Mercator's whose sides represent the true difference of latitude, de- Sailing. parture, and distance, and one of whose angles is the course.

CHAP. III.—ON MERCATOR'S SAILING.

We have already explained the principle of Mercator's Chart, and have shown that in every problem of navigation there is a second right-angled triangle similar to that whose solutions formed the subject of investigation in chapter ii.; and the sides of which, corresponding to the true difference of latitude and departure, are the meridional difference of latitude and difference of longitude.

We have already given a rule for finding the meridional parts for any given latitude, on the supposition that the earth is a perfect sphere. If the earth's oblateness be taken into account, and the compression, i.e., the ratio of the difference of the equatorial and polar semi-diameters to the equatorial semi-diameter, be taken as $\backslash frac\{1\}\{345\}$, which it is very nearly, the meridional parts for latitude $l$ will be given by the formula—

Let AD (fig. 14) be the meridian, BAC the course, AB the true difference of latitude, AD the meridional difference of latitude; then

DE is the diff. long., and
$ED\; =\; DA\; \backslash times\; \backslash tan\; BAC$; or,
diff. long. = mer. diff. lat. $\backslash times\; \backslash tan$ course;
or log. diff. long. = log. mer. diff. lat. +
$L\; \backslash tan$ course = 10.

Also, by similar triangles, ADE and ABC, we have

$\backslash therefore$ diff. long. = dep. $\backslash times$ mer. diff. lat.; or,
true diff. lat.
log. diff. long. = log. dep. + log. mer. diff. lat. - log. true diff. lat.
Whence, from the departure and true and meridional differences of latitude, the difference of longitude may be found. The following examples will illustrate the mode of using these formulae:—

Ex. 1. A ship sails from Cape Finisterre, Lat. 42° 52' N., Long. 9° 17' W., to Port Praya, in the island of Santiago, Lat. 14° 54' N., and Long. 23° 29' W.; required the course and distance.

By Construction.—Draw the straight line AD (fig. 15), to represent the meridian of Cape Finisterre, upon which lay off AB, AD, equal to 1678 and 1948, the true 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 distance AC will measure 1831 miles, and the course BAC 23° 37'.

By Calculation.—To find the course.

To find the distance.

Ex. 2.—A ship from Cape Henlopen in Virginia, in Lat. 38° 47' N., Long. 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. 16), and the difference of latitude AB being measured, is 222 miles; hence the latitude in 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; longitude in is therefore 71° 48' W.

Fig. 16.

By Calculation.—To find the true difference of latitude.

To find the difference of longitude.

Ex. 3.—A ship from Port Canso in Nova Scotia, in Lat. 45° 20' N., Long. 60° 55' W., sailed S.E. 1/2 S., and, by observation, was found to be in Lat. 41° 14' N.; required the distance sailed, and longitude come to.

By Construction.—Make AB (fig. 17) equal to 246, and AD equal to 338; draw AE, making an angle with AD equal to 3 1/2 points, and draw BC, DE perpendicular to AD. Now AC being applied to the scale, will measure 332, and DE, 306.

Fig. 17.

By Calculation.—To find the distance.

To find the difference of longitude.

Ex. 4.—A ship sailed from Saltee, in Lat. 33° 58' N., Long. 6° 20' W., the corrected course was N.W. by W., and departure 420 miles; required the distance run, and the latitude and longitude in.

By Construction.—With the course and departure construct the triangle ABC (fig. 18); now AC and AB being measured, will be found to be equal to 476 and 224 respectively; hence the latitude in 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.

Fig. 18.

By Calculation.—To find the distance.

To find true difference of latitude.

To find the difference of longitude.

Or—

Ex. 5.—A ship from St Mary's, in Lat. 36° 57' N., Long. 25° 9' W., sailed on a direct course between the north and east 1162 miles, and was then, by observation, in Lat. 49° 57' N.; required the course steered, and longitude come to.

By Construction.—Make AB equal to 780, and AD equal to 1081; draw BC, DE, perpendicular to AD; make AC equal to 1162, and through A and C draw ACE. Then the course or angle A being measured, will be found equal 47° 50', and the difference of longitude DE will be 1194.

Fig. 19.

By Calculation.—To find the course.

To find difference of longitude.

Longitude from .....

Diff. longitude .....

Longitude in .....

Ex. 6.—From Aberdeen, in Lat. 57° 9' N., Long. 2° 8' W., a ship sailed between the south and east till her departure was 146 miles, and Lat. in 53° 32' N.; required the course and distance run, and longitude in.

By Construction.—With the difference of latitude 217 miles, and departure 146 miles, 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.

To find the distance.

To find the difference of longitude.

Ex. 7.—A ship from Naples, in Lat. 40° 51' N., Long. 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 in.

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 chords, will be found equal to 43° 21', and the difference of latitude applied to the scale of equal parts will measure 183; hence the latitude in 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° 39' E.

Fig. 21.

By Calculation.—To find the course.

To find the true difference of latitude.

To find the difference of longitude.

Ex. 8.—A ship from Terceira, in Lat. 38° 45' N., Long. 27° 6' W., sailed on a direct course, which, when corrected, was N. 32° E.,

and is found, by observation, to be in Long. 18° 24' W.; required the latitude come to, and distance sailed.

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.

Fig. 22.

By Calculation.—To find meridional difference of latitude.

To find the distance.

CHAP. IV.—ON TRAVERSE SAILING, OR COMPOUND COURSES.

It is the first business of the navigator, when he is about to conduct a ship from one port to another, to calculate beforehand the course on which the vessel is to be steered, and the distance she must run on that course. If the sea is perfectly free from obstruction between the two ports, one course and one distance will suffice for this purpose. It very seldom happens, however, that the sea is free from obstruction; but rocks or shoals, islands or some part of a mainland, intervenes, and a change of course is thus rendered necessary. In this case, the course and distance of the vessel, supposing the navigation unobstructed, having been taken from the chart, the mariner will determine how many changes of course are necessary, and will proceed to calculate the several courses and distances which shall be equivalent to the one course and distance on which the vessel would sail if unobstructed. This calculation, it must be remarked, is very different from that of the course and distance actually made good on a given day, when, by reason of variation of winds and other causes, the course requires to be altered; although naturally the modes of making these calculations are similar. In the former case, however, the distances to be dealt with are very much greater, and the changes of course less frequent, than in the latter.

The investigations of this chapter are intended to guide the navigator in making his preliminary calculation; the mode of correcting the course and calculating the distance run in each day will form the subject of a subsequent investigation.

If a ship sail on two or more courses in a given time, the irregular track she describes is called a traverse; and to resolve a traverse is the method of reducing these several courses and distances run into a single course and distance.

RULE 1.—Make a table sufficiently large to contain the several courses, &c. Divide this table 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.

The several courses and their corresponding distances being properly arranged in the table, find the true difference of latitude and departure answering to each in the traverse table, remembering that the true difference of latitude is to be put into a N. or S. column according as the course is in a northern or southern direction, and that the departure is to be put in E. or W. column according as the course is easterly or westerly. Add together these several quantities in each of the columns, and set the sum down at the bottom. The difference between the sums in the N. and S. columns will be the true difference of latitude made good, of the same name with the greater; and the difference between the sums of the E. and W. columns is the departure made good, of the same name with the greater sum.

Look in the traverse table for a true difference of latitude and departure agreeing 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 true difference of latitude is greater or less than the departure, since in the former case the course is less than 45° or 4 points, and in the latter case greater.

Having found the latitude, find also the meridional difference of latitude; and 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 from, will give the longitude in.

It is also easy to resolve a traverse by construction; and we now show how this may be done, although it is scarcely ever practised at sea.

Describe a circle with the chord of 60° as radius, and in it draw two diameters at right angles to each other, at whose extremities are to be marked the initials of the cardinal points, N. 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 the proper length; and so on 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.

To construct for the difference of longitude we must find by the table the meridional difference of latitude, and lay it off on the meridian, and then complete the triangle similar to that whose sides represent the true difference of latitude; distance and departure as usual.

Ex. 1.—A ship from Fayal, in Lat. 35° 32' N., and Long. 28° 35' W., sailed as follows:—E.S.E., 163 miles; S.W. 1/2 W., 110 miles; S.E. 1/2 E., 180 miles; and N. by E. 68 miles: required the latitude and longitude in, the course, and distance made good.

Latitude from ..... 35° 32' N. Mer. parts..... 2569
True diff. latitude.... 3 30 S.
Latitude in ..... 35 2 N. Mer. parts..... 2247
Mer. diff. lat... 262

Now to course 41 1/2°, and opposite 131, half the meridional difference of latitude in latitude column, stands 115 in a departure column, which doubled gives 230 for difference of longitude.

Longitude from..... 28° 36' W.
Diff. longitude ..... 3 50 E.
Longitude in..... 24 46 W.

By Construction.—With chord of 60° describe the circle NESW (fig. 23), 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; through B draw BD equal to 180 miles, and parallel to C3; and draw DE parallel to C4, and equal to 68 miles. Now CE being joined, will represent the distance made good, which, applied to the scale, will measure 281 miles. The arc Sn, which represents the course, being measured on the line of chords, will be found equal to 41 1/2°. From E draw EF perpendicular to CS produced; then CF will be the difference of latitude, and FE the departure made good, which, applied to the scale, will be found to measure 210 and 186 miles respectively. On CF produced lay off to the scale CG equal to 262, the meridional difference of latitude; and through G draw GH parallel to FE, meeting CE produced in H. Then GH is the difference of longitude; and, when applied to the scale, will be found to measure 230 miles.

Fig. 23.

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 2.—Complete the traverse table as before, to which annex five columns. Now, with the latitude from, 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 are 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 Ex. 4, chap. iii., which place in the fourth or fifth columns, according as the course is easterly or westerly; and the difference between the sums of these columns will be the difference of longitude made good upon the whole, of the same name with the greater.

Remarks.

1. When the course is north or south, there is no difference of longitude.

2. When the course is east or west, the difference of longitude cannot be found by Mercator's Sailing; in this case the following rule is to be used:—

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

Ex. 2.—A ship from Lat. 78° 15' N., Long. 28° 14' E., sailed the following courses and distances, viz.:—W.N.W. 154 miles, S.W. 90, N.W. 1/2 W. 89, N. by E. 110, N.W. 1/2 N. 56, S. by E. 78. The latitude in is required, and the longitude, by both methods; the bearing and distance of Huelvit's headland, in Lat. 79° 55' N., Long. 11° 55' E., is also required.

When the course is 8 points or 90° from the meridian, —i.e., due E. or W.,—the true difference of latitude becomes = 0, and the rules we have investigated in chaps. iii. and iv. fail to give any result. In this case the ship sails on a parallel of latitude. We have already proved in chap. i, that, neglecting the earth's oblateness, the arc of a parallel, in any given latitude, intercepted between two meridians, is equal to the corresponding arc of the equator, in other words, the difference of longitude, multiplied by the cosine of the latitude.

Whence we derive these three formulae for parallel sailing:—

Problems in parallel sailing may be solved by construction; for it is evident that we have only to construct a right-angled triangle whose hypotenuse is the difference of longitude, one of the sides the distance, and the angle between this side and the hypotenuse the latitude. Also it is evident, that in a traverse table, if we consider the latitude a course, and the difference of longitude a distance, the distance will be a true difference of latitude.

Ex. 1.—Required the number of miles contained in a degree of longitude in latitude 55° 58'.

By Construction.—Draw the indefinite right line AB (fig. 24); make the angle BAC equal to the given latitude 55° 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—

By Inspection.—To 56°, 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° to 34°, 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 longitude, 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 compass course and distance from A to B.

Variation 1½ E., and deviation as in the table on p. 13. The true course is due E.

To find compass course.

Ex. 3.—A ship sails from Treguer in France, Long. 3° 14' W., to Gaspey Bay, Long. 64° 27' W., the common Lat. being 48° 47' N.; required the distance run.

Ex. 4.—A ship from Cape Finisterre, Lat. 42° 52' N., Long. 9° 17' W., sailed due W. 342 miles; required the longitude in.

By Construction.—Draw the straight line AB (fig. 25), equal to the given distance 342 miles, and make the angle BAC equal to 42° 52', the given latitude; from B draw BC perpendicular to AB, meeting AC in C; then AC applied to the scale will measure 466½, the difference of longitude required.

Fig. 25.

Ex. 5.—A ship sailed due E. 358 miles, and was found by observation to have differed her longitude 8° 42'; required the parallel of latitude.

By Construction.—Make the line AB (fig. 26) equal to the given distance; to which let BC be drawn perpendicular, with an extent equal to 52½, the difference of longitude; describe an arc from the centre A, cutting BC in C; then the angle BAC, being measured by means of the line of chords, will be found equal to 48½, the required latitude.

Fig. 26.

Ex. 6.—From two ports in Lat. 33° 58' N., distance 348 miles, two ships sail directly N. till they are in Lat. 48° 23' N.; required their distance.

By Construction.—Draw the lines CB, CE (fig. 27), making angles with CP equal to the complements of the given latitudes, namely, 56° 2' and 41° 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 arc intersecting CE in E; then EF drawn from the point E, perpendicular to CP, will represent the distance required; which being applied to the scale, will measure 278½ miles.

By Calculation, as under:—

Fig. 27.

Ex. 7.—Two ships, in Lat. 56° 0' N., distant 180 miles, sail due S.; and having come to the same parallel, are now 232 miles distant. The latitude of that parallel is required.

By Construction.—Make DB (fig. 28) 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 arc BE; and through M draw ME parallel to CD, intersecting the arc 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'.

Fig. 28.

By Calculation, as under:—

CHAP. VI.—OF MIDDLE-LATITUDE SAILING.

It has been already explained in chap. ii. that the departure is greater than the intercepted arc of the parallel of the higher latitude, and less than that of the parallel of the lower of two places between which a ship sails; but that there is an intermediate parallel, the arc of which is exactly equal to the departure. This parallel is supposed to pass through the middle point between the extreme latitudes; and hence the latitude of this point is called the middle latitude. The relations between course, distance, departure, and true difference of latitude are to be found as in chap. iii.; and the relation between the departure and difference of longitude is given by the above considerations, viz.—

Hence we get

Whence also

If ABC (fig. 29) be the triangle for plane sailing, where AB is the true difference of latitude, AC the distance, BAC the course, and BC the departure; at C make BCD equal to the middle latitude, and produce CD to meet AB produced in D; then CD is evidently the true difference of longitude, and all the problems may be resolved and constructed by the two triangles which have a common side, viz., the departure BC.

Fig. 29.

Also problems in middle-latitude sailing may be solved by the traverse table; for the relations between middle latitude, difference of longitude, and departure, are the same as those between course, distance, and true difference of latitude, and may therefore be found at once by inspection from the table.

Ex. 1.—Required the compass course and distance from the Island of May, in Lat. 56° 12' N. and Long. 2° 37' W., to the Naze of Norway, in Lat. 57° 50' N., and Long. 7° 27' E.; variation 2½ W.

By Construction.—Draw the right line AD (fig. 30) to represent the meridian of the May; with the chord of 60° describe the arc 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 604', the difference of longitude; and from C draw CB perpendicular to AD; make BA, equal to 96°, 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.

Fig. 30.

Ex. 2.—A ship from Brest, in Lat. $48^\backslash circ\; 23\text{'}\; \backslash text\{\; N.\}$, and Long. $4^\backslash circ\; 30\text{'}\; \backslash text\{\; W.\}$, sailed S.W. $\backslash frac\{1\}\{2\}$ W. 238 miles; required the latitude and longitude in.

By Construction.—With the course and distance construct the triangle ABC (fig. 31), and the difference of latitude AB being measured will be found equal to 142 miles; hence the latitude in is $46^\backslash circ\; 1\text{'}\; \backslash text\{\; N.\}$, and the middle latitude $47^\backslash circ\; 12\text{'}$. Now make the angle DCB equal to $47^\backslash circ\; 12\text{'}$; and DC being measured will be 281, the difference of longitude; hence the longitude in is $5^\backslash circ\; 11\text{'}\; \backslash text\{\; W.\}$

Ex. 3.—A ship from St Antonio, in Lat. $17^\backslash circ\; 0\text{'}\; \backslash text\{\; N.\}$ and Long. $24^\backslash circ\; 25\text{'}\; \backslash text\{\; W.\}$, sailed N.W. $\backslash frac\{1\}\{2\}$ N., till, by observation, her latitude was found to be $28^\backslash circ\; 34\text{'}\; \backslash text\{\; N.\}$; required the distance sailed, and longitude come to.

By Construction.—Construct the triangle ABC (fig. 32), 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.

Ex. 4.—A ship from Lat. $26^\backslash circ\; 30\text{'}\; \backslash text\{\; N.\}$, and Long. $45^\backslash circ\; 30\text{'}\; \backslash text\{\; W.\}$, sailed N.E. $\backslash frac\{1\}\{2\}$ 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. 33), and the distance and difference of latitude being measured will be found equal to 340 and 263 respectively. Hence the latitude in is $30^\backslash circ\; 53\text{'}$, and middle latitude $28^\backslash circ\; 42\text{'}$. Now make the angle BCD equal to the middle latitude, and the difference of longitude DC applied to the scale will measure 246'.

Ex. 5.—From Cape Sable, in Lat. $43^\backslash circ\; 24\text{'}\; \backslash text\{\; N.\}$, and Long. $65^\backslash circ\; 39\text{'}\; \backslash text\{\; W.\}$, a ship sailed 246 miles on a direct course between the S. and E., and was then by observation in Lat. $40^\backslash circ\; 48\text{'}\; \backslash text\{\; N.\}$; required the course, and longitude in.

By Construction.—Make AB (fig. 34) equal to 156 miles, draw BC perpendicular to AB, and make AC equal to 246 miles; draw CD, making with CB an angle of $42^\backslash circ\; 6\text{'}$, the middle latitude. Now DC will be found to measure 246, and the course or angle A will measure $50^\backslash circ\; 39\text{'}$.

Longitude from ..... 65° 39' W.
Diff. longitude ..... 4 16 E.
Longitude in ..... 61 23 W.

Ex. 6.—A ship from Cape St Vincent, in Lat. 37° 2' N., Long. 9° 2' W., sails between the S. and W.; the latitude in is 18° 16' N., and departure 838 miles; required the course and distance run, and longitude in.

Latitude Cape St Vincent 37° 2' N. 37° 2' N.
Latitude in ..... 18 16 N. 18 16 N.
Difference of latitude 18 46 = 1126 S. Sum 55 18 N.
Middle latitude... 27 39 N.

By Construction.—Make AB (fig. 35) 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° 39'. Then the course being measured on chords is about 364°, and the distance and difference of longitude, measured on the line of equal parts, will be found to be 1403 and 946 respectively.

Fig. 35.

By Calculation.—To find the course.
Log. departure ..... 838 + 10 = 12.92324
Log. diff. latitude 1126 = 3.05154
L tan course ..... 36° 39' = 9.87170

To find the distance.
L sec course ..... 36° 39' = 10.09566
Log. diff. latitude ..... 1126 = 3.05154
-10
Log. dist. .... 1403 = 3.14720

To find the difference of longitude.
Log. departure ..... 838 = 2.92324
L sec mid. latitude ..... 27° 39' = 10.05266
-10
Log. diff. longitude ..... 946 = 2.97590
Longitude from ..... 9° 2' W.
Diff. longitude ..... 15 46 W.
Longitude in ..... 24 48 W.

Ex. 7.—A ship from Bordeaux, in Lat. 44° 50' N., and Long. 6° 35' W., sailed between the N. and W. 374 miles, and made 210 miles of westing; required the course, and the latitude and longitude in.

By Construction.—With the given distance and departure make the triangle ABC (fig. 36). Now the course, being measured on the line of chords, is about 341°, and the difference of latitude on the line of numbers is 309 miles; hence the latitude in is 49° 59' N., and middle latitude 47° 25'. Then make the angle BCD equal to 47° 25', and DC being measured, will be 310 miles, the difference of longitude.

Fig. 36.

By Calculation.—To find the course.
Log. departure ..... 210 + 10 = 12.32222
Log. distance ..... 374 = 2.57287
L sin course ..... 34° 10' = 9.74935
To find the true difference of latitude.
L cos course ..... 34° 10' = 9.91772
Log. distance ..... 374 = 2.57287
-10
Log. diff. latitude ..... 309.4 = 2.49059
Latitude from ..... 44° 50' N. 44° 50' N.
True diff. latitude ..... 5 9 N. Half ..... 2 35 N.
Latitude in ..... 49 59 N. Mid. lat. 47 25 N.

To find the difference of longitude.
L sec mid. latitude ..... 47° 25' = 10.16963
Log. departure ..... 210 = 2.32222
-10
Log. diff. longitude ..... 310.3 = 2.49185

Longitude from ..... 6° 35' W.
Diff. longitude ..... 5 10 W.
Longitude in ..... 5 45 W.

Ex. 8.—A ship from Lat. 54° 55' N., Long. 1° 10' W., sailed between the N. and E. till, by observation, she was found to be in Long. 5° 28' E., and has made 220 miles of easting; required the latitude in, course, and distance run.

Longitude from ..... 1° 10' W.
Longitude in ..... 5 28 E.
Difference of longitude 6 38 = 396 E.

By Construction.—Make BC (fig. 37) 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° 15'; hence the latitude come to is 57° 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°.

Fig. 37.

By Calculation.—To find the middle latitude.
Log. diff. of longitude ..... 396 + 10 = 12.59769
Log. departure ..... 220 = 2.34242
L sec mid. latitude ..... 56° 15' = 10.25527
Double middle latitude ..... 112° 30' N.
Latitude from ..... 54 55 N.
Latitude in ..... 57 34 N.
True diff. latitude ..... 2 38 = 158 miles N.

To find the course.
Log. departure ..... 220 + 10 = 12.34242
Log. diff. latitude ..... 158 = 2.19866
L tan course ..... 54° 19' = 10.14376

To find distance.
L sec course ..... 54° 19' = 10.23410
Log. diff. latitude ..... 158 = 2.19866
-10
Log. distance ..... 270.9 = 2.43276

Ex. 9.—A ship from a port in N. Lat., sailed S.E. 18. 438 miles, and differed her Long. 7° 28'; required the latitudes from and in.

By Construction.—With the course and distance construct the triangle ABC (fig. 38), and make DC equal to 448, the given difference of longitude. Now the middle latitude BCD will measure 48° 58', and the difference of latitude AB 324 miles; hence the latitude from is 51° 40', and latitude in 46° 16'.

Fig. 38.

By Calculation.—To find the true difference of latitude.
L cos course ..... 31 pts. = 9.86979
Log. distance ..... 438 miles = 2.64147
-10
Log. true diff. latitude ..... 324.5 = 2.51125

To find middle latitude.
Log. distance ..... 438 ms. = 2.64147
L sin course ..... 31 pts. = 9.82708
Log. diff. longitude ..... 448 = 2.65128
L cos mid. latitude ..... 48° 58' = 9.81727
Mid. latitude ..... 48° 58' N.
Half diff. latitude ..... 2 42 S.
Latitude from ..... 51 40 N.
Latitude in ..... 46 16 N.

CHAP. VII.—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.

Ex. 1.—At 11 a.m. the Girdle Ness bore W.N.W., and at 2 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 NESW (fig. 39), 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 Girdle Ness. Then 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.

Ex. 2.—Running up Channel E. by S. per compass at the rate of 5 knots an hour. At 11 a.m. the Eddystone Lighthouse bore N. by E., and the Start Point N.E. by E.; and at 4 p.m. the Eddystone bore N.W. by N., and the Start N.W. by E.; required the distance and bearing of the Start from the Eddystone, the variation being 2½ points W.

By Construction.—Let the point C (fig. 40) 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.W. 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 22.9, and the bearing per compass BAF will measure 73½°.

Hence BAF = BAC-CAF
= 87° 26' - 1½ pts.
= 87° 26' - 14' 4"
= 73° 22' = 73½° nearly.

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. VIII.—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 of each tack.

By Construction.—Draw the N.W. line CA (fig. 41) 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=2CB=72½ miles.

CHAP. IX.—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 ebbings 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 ahead 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.

It is evident from the above observations that we may consider the direction of the current in the light of a separate course; and by multiplying the rate of the current per hour by the number of hours it has been running, and treating this as a distance, we may estimate the ship's real place by any of the rules for compound courses.

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. 42) 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'.

Fig. 42.

• The angle CAB..... = 9 pts.
• CA..... = 144 m.
• AB..... = 45 m.
• CA + AB..... = 189 m.
• CA - AB..... = 99 m.
• Log. (CA - AB)..... 99 m. = 1.995635
• L cot ¼ CAB..... 4½ pts. = 9.914173
• 11.909808
• Log. CA + CB..... 185 m. = 2.276462
• L tan ¼ (ABC - ACB)..... 23° 15' = 9.633346
• ¼ (ABC + ACB)..... = 39 22
• ∴ ACB = 16° 7' and ABC..... = 62 37
• NCA..... = 22 30
• ∴ NCB the course..... = 6 23
• Log. AB..... 45 m. = 1.653212
• L sin CAB..... 9 pts. = 9.991574
• 11.644786
• L sin ACB..... 16° 7' = 9.443410
• Log. BC..... 159 m. = 2.201376

For first course we have course N.N.E., or 2 points.
And distance..... 144 Whence, from traverse table,
True diff. latitude..... 133.0 N. Departure..... 55.1 E.

For second course we have course N.W. by W., or 5 points.
And distance..... 45 And from traverse table,
True diff. latitude..... 25 N. Departure..... 37.4 W.

• True diff. latitude..... 158 N.
• Departure..... 17.7 W.

• Log. departure..... 17.7 + 10 = 11.24797
• Log. true diff. latitude..... 158 = 2.19866
• L tan course..... N. 6° 23' E. = 9.04931
• L sec course..... 6° 23' - 10 = .00271
• Log true diff. latitude..... 158 m. = 2.19866
• Log. distance..... 159 = 2.20137

Ex. 2.—A ship from Lat. 35° 20' N. sailed 24 hours in a

current setting N.W. by N., and by account is in latitude 35° 42' N., having made 44 miles of easting; but the latitude by observation is 35° 58' N.; required the course and distance made good, and the drift of the current.

Fig. 43.

By Construction.—Make CE (fig. 43) 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 41½°.

By Calculation—

• ABF..... = 3 pts.
• BF = CD - CE..... = 16 miles.
• To find AB.
• Log. BF..... 16 m. = 1.20412
• L sec ABF..... 3 pts. - 10 = 0.08015
• Log. AB..... 19.2 m. = 1.28427
• Or drift of current..... 19.2 miles.
• To find AF.
• Log. BF..... 16 m. = 1.20412
• L tan ABF..... 3 pts. - 10 = 9.82489
• Log. AF..... 10.7 m. = 1.02901
• Hence BD = AE - AF = 44 - 10.7 = 33.4.
• To find the course.
• Log. BD..... 33.4 + 10 = 11.52244
• Log. CD..... 38 = 1.57978
• L tan course..... N. 41° 14' E. = 9.94266
• To find the distance.
• Log. sec course..... 41° 14' E. - 10 = 0.12376
• Log. CD..... 38 = 1.57978
• Log. distance..... 50.5 m. = 7.0354

By Traverse Table.—Taking the current course first, true difference of latitude 16, and course N.W. by N., we find in the traverse table the corresponding distance 19.3, and departure 10.7.

Again, for second course, we have true difference of latitude 38, and departure 44 - 10.7 = 33.4 E.

Whence the course and distance are found as above.

Or, from the traverse table to nearest degree and minute, we find in the columns of distance and angle, opposite to difference of latitude 38.5, and departure 35.5—distance 51, and angle 41°.

CHAP. X.—OF THE DAY'S WORK AND SHIP'S JOURNAL.

The most usual application of the principles laid down in the preceding chapters, is to ascertain from the several courses and distances run by a ship in the interval between the noons of two successive days, the ship's place at the noon of the latter day, i.e., its latitude and longitude; its latitude and longitude being given for the noon of the preceding day. This constitutes a day's work; and the ship's place deduced therefrom is called her place by account or dead reckoning. The day aboard ship, like the astronomical day, commences at noon; and the ship's position is always calculated at every noon. In the Royal Navy, the log is hove once in every hour; but in most trading-vessels only once in every two hours. A record of the knots, and tenths of knots, run every hour or every two hours, the course, the direction of the wind, the leeway, and everything which affects the ship's place, is kept in the journal, which, for this purpose, is usually divided into six or seven columns. The first column on the left hand contains the hours

from noon to noon; the second and third, the knots and tenths of knots sailed every hour, or every two hours; the fourth contains the courses steered; the fifth, the direction of the wind; and the sixth, when there are seven columns, contains the leeway; and the last contains general remarks, including phenomena, variation, &c., &c.

The mode of forming a table showing the deviation of the compass for the several positions of the ship's head, has already been given.

The courses steered, as entered in the log-book, must be corrected for variation, deviation, and leeway. The setting and drift of current, and the heave of the sea, are to be marked down. These are to be corrected for variation only. In the day's work, it is usual to treat a current as an independent course and distance. If the ship does not sail from a place whose latitude and longitude are known (which rarely happens), the bearing of some known place is to be observed, and its distance found, which is usually done by estimation. The ship is then supposed to have taken her departure from this place, in a course exactly opposite to the observed bearing, and to have run the estimated distance on it. If there be any reason to suspect the correctness of the estimated distance, it will be easy to obtain the true distance as follows:—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 again to be observed. We shall then have a triangle, all of whose angles are known from the observed bearings, and one of its sides, viz., the distance the ship has sailed. The other two sides, viz., the distance of the ship from the place of departure at each of the observations, can be immediately found, as in problem 1 on "Oblique Sailing." The distances for each course may be obtained by adding together the hourly distances. The courses being thus corrected, and the distances found, the latitude and longitude may be found by any of the methods explained in chap. iv. As the differences of latitude are not usually great, the traverse table may generally be made use of for finding the latitude in; and having found the middle latitude, the longitude may be obtained by the middle latitude method.

The following example will enable the reader to apply the directions we have just given:—

Ex.—September 12, 1857, at noon, a point of land in Lat. $64^\{\backslash circ\}\; 20\text{'}\; S.$, and Long. $59^\{\backslash circ\}\; 40\text{'}\; E.$, bore by compass S.E., distant 15 miles (ship's head being E.), afterwards sailed as by the following log-account; find the latitude and longitude in, on September 13, at noon.

Enter these in a table as under:—

In this example the true differences of latitude and departures are taken by inspection from the traverse table.

When a ship is bound for a distant port, the bearing and distance of the port must be found. This may be done by calculation or 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 compass course

Sea Charts. by making the requisite allowances for variation and deviation, as already explained.

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 to under no more sail than is sufficient to prevent the violent rolling to which the vessel would be otherwise subjected, to the endangering of her masts and straining her timbers, &c. When a ship is brought to, the tiller or wheel is put down over to the leeward, which brings her head round to the wind. The wind having then little power over the sails, the ship loses her way through the water; and the action of the water on the rudder ceasing, 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 oscillating motion, coming up to the wind and falling off from it again alternately. The middle point between those upon which she comes up and falls off is taken for her apparent course; and the leeway, variation, and deviation are to be allowed from this to find the true course.

It is generally found that the latitude by account does not agree with that by observation. On considering the imperfections of the common log-line, and the uncertainty with regard to variation, an exact agreement of latitudes cannot be expected. When the difference of longitude is to be found by dead reckoning, and 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 is 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 to be in error. This last correction will, according to different authors, place the ship upon opposite sides of her meridian by account. As these corrections, therefore, are no better than guessing, they should be absolutely rejected.

If the latitudes do not agree, the navigator should 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 in 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 may be corrected in the same way.

It will be proper for the navigator to determine the longitude of the ship by 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. The modes of finding the latitude and longitude of a ship by observation, and the variation of the compass, will be given in the next book.

CHAP. XI.—OF SEA CHARTS.

The charts usually employed in the practice of navigation are the Plane and Mercator's charts. The former of these is adapted to represent a portion of the earth's surface near the equator, where the change in the lengths of corresponding arcs of the parallel is very small; and the other for all portions of the earth's surface. (For a particular description of these, see the articles CHART and GEOGRAPHY.) We shall here only describe their use.

Use of the Plane Chart.

PROB. I.—To find the latitude and longitude of a place on the chart.

Rule.—Take the least distance of the given place from the nearest parallel of latitude; this distance applied to the graduated meridian from the extremity of the parallel will give the latitude of the place. In the same way the longitude is found by taking the least distance from the nearest meridian, and applying it to the graduated parallel.

Thus the distance between Bonavista and the parallel of $15^\{\backslash circ\}$ being laid from that parallel on the graduated meridian, will reach to $16^\{\backslash circ\}\; 5\text{'}$, 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; if a parallel ruler be used, keeping the edge of one ruler passing through the places fixed, move the other until it passes through the centre of one of the compasses on the chart; the point of the compass through which this edge passes will show the course.

Or, generally, let a line on the edge of another ruler be placed so as to be parallel to the first ruler, and to pass through the centre of a compass; it will cut the circumference in a point which will determine the course.

The interval between the places being applied to the scale will give the distance.

Thus the course from Palmas to St Vincent will be found to be about S.S.W. $\backslash frac\{1\}\{2\}$ W., and the distance $13\backslash frac\{1\}\{2\}^\{\backslash circ\}$ 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 given place parallel to another ruler laid over one of the compasses, with one edge passing through the centre, and the other the point on the circumference which shows the course, and lay off on it the distance taken from the scale; it will give the point representing the ship's present place.

Thus, supposing a ship has sailed S.W. by W. 160 miles from Cape Palmas; then by proceeding as above, it will be found that she is in Lat. $2^\{\backslash circ\}\; 57\text{'}\; N.$

The reader will have no difficulty in solving various other problems by means of this chart, being, in fact, only the construction of the various problems in plane sailing on this 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, is the same as in the plane chart, which see.

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

CASE 1.—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 2.—When the given places are under the same parallel.

Rule.—If that parallel be the equator, the difference or sum of their longitudes, according as they are on the same or on opposite sides of the first meridian, is the distance; otherwise take the distance between the places, lay it off upwards and downwards from the given parallel, and the intercepted degrees will be the distance between the places.

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

CASE 3.—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 places, and move one point of the compass opened to the difference of latitude just found along the edge of the ruler till the other just touches a parallel; then the distance from the point of the compass on the ruler to the point of intersection of the ruler and the parallel, applied to the equator, will give the distance between the places in degrees and parts of a degree, which, multiplied by 60, will give it in miles.

PROB. II.—Given the latitude and longitude in; to find the ship's place by 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 the given 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; its intersection with the parallel of latitude in, will give the ship's present place.

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

Rule.—Lay a ruler over the given place, in the direction of the given course, take the distance sailed from the equator, and put one point of the compass opened to this distance at the intersection of the ruler with any parallel, and the other point will reach to a certain place by the edge of the ruler. This point being kept fixed, 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 is known; and the intersection of the edge of the ruler with the parallel of this latitude will give the ship's present place.

The above problems sufficiently illustrate the use of Mercator's Chart. The reader will have no difficulty in solving other problems by means of it.

BOOK II.

CONTAINING THE METHODS OF FINDING THE LATITUDE AND LONGITUDE OF THE SHIP AT SEA, THE VARIATION OF THE COMPASS, AND TIME OF HIGH WATER.

CHAP. I.—DESCRIPTION AND USE OF INSTRUMENTS USED IN OBSERVATIONS.

SECT. I.—OF HADLEY'S SEXTANT AND QUADRANT.

The principal difference between these instruments is in the extent of the angle which can be observed by them; and in the more elaborate and careful workmanship of the latter of the two. Indeed the quadrant is only available for taking observations which determine the latitude. The distances of the moon from the sun or other heavenly body, which are frequently used for the determination of the longitude, can only be observed by the help of the sextant.

Allowing for these differences, the principle on which the quadrant and sextant are constructed is the same. In the Royal Navy sextants are almost exclusively in use, although quadrants are still employed for the observation of altitudes in many trading vessels. The sextant, therefore, will first be described, and afterwards those points in which the quadrant differs from the sextant will be explained.

The reader is supposed to be aware of the ordinary laws with regard to the propagation and reflection of light, viz.,—

that in the same medium, light is propagated in straight lines, the smallest conceivable quantity of which that can be stopped or propagated alone is called a ray; and that when a ray of light is incident on a plane reflecting surface, it is bent or reflected after incidence in such manner, that the incident and reflected rays and the straight line perpendicular to the mirror at the point of incidence (called the normal to the surface) lie all in one plane; and that the incident and reflected rays make equal angles with the normal or the surface.

Let MO (fig. 44) be an arc of a circle, CO and CM two radii, and CI be a moveable radius carrying a plane mirror, silvered through its whole extent, firmly fixed to it; EFG another mirror, the lower part of which FG only is silvered, while the upper part EF is unsilvered, so that a ray reflected from the lower portion FG in direction FH, and a direct ray PFH through the unsilvered part EF, may be seen together by an eye at K. This mirror is fixed to the radius CM in such a manner that when the moveable radius occupies the position ACO, the two mirrors ACB and EFG are both perpendicular to the plane of the instrument, and parallel to one another.

Let now S and P be two distant objects whose angular distance is required to be found. Let the instrument be placed so that its plane passes through S and P, and that a ray from P, passing through the unsilvered glass EF, may be seen directly by an eye at K; and while in this position let the bar be moved round C, CI carrying the mirror with it until a ray from S, falling on ACB, is reflected in the direction CF, and again reflected by FG in the direction FH; so that to the eye at K the images of the two objects S and P are seen together, or coincide.

Produce SA to meet PFH in H; then SHP is the angle through which the ray SA has been deflected, and is also the angular distance between S and P. Let A'B' be the new position of the mirror AB; then ACA' is the angle through which the mirror has turned, and consequently also the angle through which CI has moved.

Now angle of deflection

because by law of reflection,

and $\backslash text\{EFC\}\; =\; \backslash text\{GFH\}$; and therefore

But $\backslash text\{EFC\}\; =\; \backslash text\{FCB\}$, because EFG is parallel to ACB;

= twice the angle through which CI has moved.

Hence if the arc OM be divided into degrees, and each degree marked as two degrees, the reading off of the arc OI will be the angle between the distant objects S and P.

Observation Instruments. An instrument constructed on this principle, whose circular arc or limb is a sixth part of a circle, and therefore capable of measuring angles up to $120^\backslash circ$, is called a sextant; if the limb contain only an eighth part of a circle, it is a quadrant, and can only measure angles up to $90^\backslash circ$.

1. (1.) PLM (fig. 45) is the frame of the sextant.
2. (2.) AA the graduated arc or limb.
3. (3.) N the index, carrying the vernier OQ.
4. (4.) I the index-glass.
5. (5.) F the horizon-glass.
6. (6.) D the coloured or dark glasses between the index-glass and horizon-glass.
7. (7.) E the coloured glasses behind the horizon-glass.
8. (8.) K the tube or collar in which the telescope is inserted.

The frame of the sextant. The frame of the sextant consists of an arc AA, firmly attached to the two radii LP, MP, which are bound together by braces, as shown in the figure, to prevent warping and liability to bend.

The index. The index N is a flat bar of brass, and turns on the centre of the sextant; at the lower end of the index there is an oblong opening; to one side of this opening the vernier scale is attached to subdivide the divisions of the arc; at the end of the index there is a piece of brass which bends under the arc, carrying a spring to make the vernier scale lie close to the divisions. It is furnished with a finger-screw C, by which the index is fixed in any position to the limb of the instrument. There is also an adjusting-screw B attached to the index, capable of moving it with greater accuracy than the hand; this screw does not act until the index is fixed by the finger-screw C. Care must be taken not to force the adjusting-screw when it arrives at either extremity of its adjustment. When any considerable movement is required to be given to the index, the screw C at the back of the sextant must be set free; but where the index is brought nearly to the divisions required, this back screw should be tightened, and then the index gradually moved by the adjusting-screw.

The index-glass. Upon the index, and near its axis of motion, is fixed a plane speculum or mirror of glass I, quicksilvered. It is set in a brass frame, which is firmly fixed by a strong cock to the centre plate of the index, with its face 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. As has already been observed, this glass

is to receive the rays from the sun or other object, and reflect them upon the horizon-glass. It is furnished with screws at its back, the object of which is to replace it in a perpendicular position, if by any accident it has been deranged.

To the radius PL is attached a small speculum F, whose surface is parallel to the index-glass when zero on the index coincides with zero on the limb. The under part only of this speculum is silvered, the upper half being left transparent, and the back part of the frame cut away, that nothing may impede the sight through the unsilvered 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. It is set in a brass frame, which turns on axes and 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 is on one side of the socket through which the stem of the finger-screw passes; this screw G is in some instruments under the glass, in others behind it, and in others at the side.

There are four coloured glasses at D, tinged red and green, each of which is set on a separate frame that the brightness of the solar image and the glare of the moon, and may be used separately or together as occasion may require. There are three more such glasses placed behind the horizon-glass at E, to weaken the rays of the sun or moon when 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 sextant is furnished with a plane tube K; and in the order to render objects distinct, it has two telescopes—one a Galileo's telescope, representing the objects erect in their natural position; the longer one, an astronomical telescope, shows them inverted. It has a large field of view; and has parallel wires placed in the principal focus, where a true image of the object viewed by it is seen; thus rendering the position of the image more exact and more easy to be read off, and is that which should be used in taking observations at sea when great accuracy is required. A little use will soon accustom the observer to the inverted position, and to manage the instrument with ease. By a telescope the contact of the images is more perfectly distinguished; and by the place of the images in the field of view, it is easy to perceive whether the sextant is held in the proper position 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 to it by two screws; by turning one 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 which slides in a socket, and, by means of a screw at the back of the sextant, may be raised or lowered so as to move the centre of the telescope to that part of the horizon-glass which shall be deemed most fit for observation. Tinged glasses are provided to screw on the eye-end of either of the telescopes or the plane tube.

The limb of the sextant is divided from right to left into 120 primary divisions, which are to be considered as degrees; the degree is subdivided in some cases into two equal parts, each of which is $30\text{'}$; in others into three equal parts, each of which is $20\text{'}$; and in others again into six equal parts, each of which is $10\text{'}$. If the zero of the index stand exactly at one of the divisions of the limb, the

Observation Instruments. reading off in that case is immediately known. If, however, the zero of the index do not stand exactly at one of the divisions, but distant from it by a small space, the value of this space is known by means of the divisions of the vernier-plate to the left of 0.

Reading off. The vernier contains a space equal to nineteen divisions on the limb, and is divided into twenty equal parts; hence the difference between a division on the vernier and a division on the limb is one-twentieth of a division of the limb, or 1', if the interval between divisions on the limb is equal to 20'. Or supposing the limb divided into intervals of 10', and that fifty-nine divisions of the limb correspond to sixty divisions of the vernier; it is then evident that the difference between a division of the instrument and of the vernier is $\backslash frac\{1\}\{60\}$th part of 10', i.e., 10". This is the most usual kind of division.

To find the actual reading off in any particular case, we must observe which division of the vernier coincides with a division of the limb; the number denoting this, multiplied by the value of the difference between a division of the limb and of the vernier, will give the additional reading. Suppose, for instance, the nearest division of the limb to the zero of the vernier to be 25° 30', and the eighth division of the vernier to be coincident with a division of the limb, the additional angle will be 80" or 1' 20", and the reading off will be 25° 31' 20".

Adjustments of the sextant. The adjustments of the sextant are to set the mirrors perpendicular to the plane of the instrument, and parallel to one another when the index is at zero; and to set the axis of the telescope parallel to the plane of the instrument.

Adjustment 1.—To set the index-glass perpendicular to the plane of the sextant.

Set the index towards the middle of the limb, and hold the sextant so that its plane is 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 the limb seen directly, the speculum is perpendicular to the plane of the instrument. If they do not appear in the same plane, i.e., if the image be seen above or below the arc itself, its position must be gradually and carefully changed by means of the screws at its back until the error is rectified.

Adjustment 2.—To set the horizon-glass perpendicular to the plane of the instrument.

Place the instrument horizontal, and direct the sight to a distant well-defined object, as the sun, so as to view it directly; then move the index until the image of the object seen by reflection is on the field of view, and move the index backwards and forwards so as to make the image pass over the object. If it pass exactly over the object, the fixed mirror is perpendicular to the plane of the instrument; if not, move the screw G until their exact coincidence takes place.

Adjustment 3.—To set the horizon-glass parallel to the index-glass when the zero of the index or vernier-plate coincides with the zero of the graduations of the limb.

Set 0 on the index exactly to 0 on the limb, and fix it in that position by the screw on the under side of it; hold the sextant with its plane vertical, and direct the sight to a well-defined part of the horizon; then if the horizon seen on the silvered part coincides with that seen through the transparent part, the horizon-glass is adjusted; but if the horizons do not coincide, the position of the glass must be altered by moving a screw placed near the fixed reflector, which gives it a motion about an axis perpendicular to the plane of the instrument.

This adjustment is seldom made, as turning the adjusting-screw too often renders this part of the instrument very apt to get out of order. It is usual, therefore, to determine the error in the reading called the Index Error.

To do this, direct the sight to the horizon, and move the index until the reflected horizon coincides with that seen

by direct vision; then the difference between 0 on the limb and 0 on the vernier-plate will be the index error, which is to be added when 0 of the vernier is to the right of 0 on the limb; otherwise subtracted.

A more accurate method than the above is to measure the sun's apparent diameter twice with the index placed alternately on the right and on the left of the zero point of the graduated limb. 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. Care must be taken to measure the sun's horizontal diameter, as the vertical diameter is often affected with refraction. This must be done by keeping the plane of the instrument at right-angles to the vertical diameter of the sun.

For example, on January 2, 1857, the sun's diameter, measured with the index first to the right and secondly to the left of the zero point of division, was 33' and 32' 20" respectively, and the index error obtained by taking the semidifference is 20".

Adjustment 4.—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, or two stars of the first magnitude, be selected, whose distance is not less than 90° or 100°; make the contact of these as perfect as possible at the wire nearest the plane of the instrument; fix the index in this position; move the sextant until the objects are seen at the other wire, and if the same points are in contact, the axis of the telescope is parallel to the plane of the sextant. If, however, the objects are apparently separated, or overlap one another, correct half the error by the screws in the circular part of the supporter, one of which is above, and the other between the telescope and sextant; turn the adjusting-screw at the end of the index till the limbs are in contact; then bring the objects to the wire next the instrument, and if the limbs are in contact, the axis of the telescope is adjusted; if not, proceed as at the other wire, and continue till no error remains. In practice, this adjustment is usually made by means of the sun and moon. The mode of bringing the limbs of the sun and moon into contact will be explained when the use of the sextant is treated of. 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 difference of readings off in these two positions will be the angular distance between the wires. Other and better methods will readily occur to the observer on land.

The Quadrant.

It has been already observed, that this instrument differs from a sextant in the extent of the divided limb and in its rougher manufacture. It is only calculated for observing altitudes. Fig. 46 represents a quadrant of the common construction.

The frame, index, index-glass, and F the fore horizon-glass, are much the same as in the sextant. There is, besides, another horizon-glass G, called the back horizon-glass attached to the same radius as F. Instead of a tube or telescope, the quadrant is furnished with vanes or sights H and I. There are but three coloured glasses, two of which are red and the other green. They are fixed at K, as shown in the figure, when the fore horizon-glass is used.

Observation Instruments. If the back horizon-glass be used, they are transferred to N. The back horizon-glass is silvered at both ends, but has a

transparent slit in the middle through which the horizon may be seen. Each of the horizon-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 or perpendicular, according as it is the fore or back horizon-glass, 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 the other 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.

The sight-vanes H and I are perforated pieces of brass, designed to direct the sight parallel to the plane of the quadrant. 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.

The limb is divided into ninety primary divisions, which are considered as degrees, and each degree subdivided into three equal parts, which are therefore of 20' each. The vernier-plate is generally so divided as to enable the observer to read off accurately to minutes.

These consist in setting the mirrors perpendicular to the plane of the instrument, and the fore horizon-glass parallel, and the back horizon-glass perpendicular to the index-glass, when the zero of the index or vernier-plate coincides with zero of the graduations on the limb.

The adjustments for the index-glass and fore horizon-glass are performed nearly in the same way as for the sextant. The index error, however, must be ascertained by bringing the horizon by reflection into the same line with the horizon seen directly. The method by taking the distance of two stars of the first magnitude, or the sun and moon, is inapplicable here.

The back horizon-glass is so seldom used, that for its adjustments and the mode of taking observations with it, the reader is referred to Norie's Navigation, and other works in which this subject is treated.

The altitude of an object may be determined by either instrument, and is the reading off on the limb, with the proper index error applied, when by reflection that object appears to be in contact with the horizon. The distance between the sun and moon, or other heavenly bodies, may

be observed by the sextant when the limbs of the bodies whose distance is required appear to be in contact. If the quadrant be used for taking the altitude of the sun, when it is 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 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 on the silvered part of the glass adjacent to the line of separation of the two parts.

With the quadrant two different methods of taking observations may be employed. In the first, the observer faces the sun, and looks to that part of the horizon which is immediately under the sun, and the observation is therefore called the fore observation. In the other method, the observer's back is towards the sun, and he looks to the part of the horizon opposite to that which is under the sun; and this is consequently called the back observation. It is not to be employed except when the horizon under the sun is obscured, or rendered indistinct by fog or other impediment.

In all cases of taking altitudes, it must be considered that it is necessary to be quite sure that the distance of the sun or other body from the horizon is the least possible, otherwise it would not be the altitude that is observed. Consequently, after the instrument has been placed as nearly as possible in a vertical position, and a contact made, a motion about the line of sight of the sun must be communicated to the instrument, so as to keep the image always in the same part of the silvered mirror, the plane of the instrument being inclined. In this way we keep the angular distance of the sun from the line through the eye by which it is viewed the same, and the sun's image describes a small circle, whose angular radius is this distance. The horizon being fixed and viewed directly, will always occupy the same position. If, then, on giving this vibratory motion to the instrument, the arc described by the sun touches the horizon, the angular distance observed is the altitude. If it should cut the horizon, so that a portion of the sun's image goes below it, the index must be moved back until this arc simply touches the horizon. In the back observation with the quadrant, and in observing with the sextant furnished with the inverting telescope, the images are inverted, and the arc described by the sun's image lies below the horizon, to which line it is convex.

The motion must be given round the axis passing through the observer's eye and the sun. To do this, a motion about the axis of vision must be given to the instrument, and at the same time the observer must turn himself about upon his heel; for the motion about the line of sight of the sun may be resolved into these two motions; and the observer has no means of giving the requisite motion directly by one movement. When the sun is near the horizon, the line from the eye to the sun will not be far removed from the axis of vision, and the principal motion of the instrument will be performed on this axis; while that part of the motion made about the vertical axis will be small. On the contrary, if the sun be near the zenith, the line from the eye to the sun is nearly vertical and perpendicular to the axis of vision; hence the motion about the vertical axis is the greatest, and that about the axis of vision very trifling. In intermediate positions of the sun the motions of the instrument about these two axes will be more equally divided. When the distance between the moon and sun, a planet or a star, is to be observed, the sextant must be so held that its plane may pass through the eye of the observer and both objects; and the reflected image of the brighter of the two is to be brought into contact with the other seen directly. To effect this, therefore, it is evident that when the brighter object is to the right of the other, the face of the sextant

must be held upwards, and 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 with the right hand. Sometimes a sitting posture will be found 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 be 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 observer's body, the greater part of the weight of the instrument will be supported by his body. In all cases where the sextant is used, when the contact is nearly made, the index should be fixed by the under screw, and the remaining small motion given by the adjusting screw.

Error may arise from two kinds of causes: one inherent in the construction of the instrument—as defect of parallelism or perfect planeness in the fore and back surfaces of the mirrors, as also of the coloured glasses, and of the true circular form of the arc, and true centring, for which no remedy can be provided; and the other arising from the bending and elasticity of the index or moveable radius.

The parallelism of the two surfaces of the mirrors may be tested by viewing through them obliquely a distant distinct object. If the image is perfect and well defined, the surfaces are parallel; otherwise not.

To ascertain whether the surfaces of the mirrors are plane, observe the angle between two distant objects which are nearly of the same altitude, the image of the left-hand object being brought into contact with the right-hand object viewed directly; then move the instrument in its own plane so as to bring the image of the right-hand object into contact with the left-hand object viewed directly. If they continue in contact, the surfaces are plane; otherwise not.

To test the form of the dark glasses, measure the sun's diameter to the right and to the left of the zero point with different combinations of the glasses. If the sum of the diameters so measured be nearly equal to four times the semidiameter given in the Nautical Almanac, the form of the glasses is satisfactory. For the true centering of the arc, and its truly circular form and correct graduation, the navigator must trust entirely to the skill of the maker.

By reason of the bending and elasticity of the index, and the resistance it meets with in turning round the centre, its extremity, on being pushed round the arc, 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, in order to remedy this defect, which he seems to have apprehended, gave special directions that the index be made broad at the end next the centre, and 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 as to be nearly insensible, when the index is made strong, and the proper medium between the two extremes of a shake at the centre on the 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.

Of the reality of the error to which he is liable from this cause, the observer, if he is provided with an instrument furnished with an adjusting screw for the index, may thus satisfy himself:—After finishing the observation, lay the instrument on a table, and note the angle; then cautiously

loosen the screw which fastens the index, and it will immediately, if the instrument is not remarkably well constructed, be seen to start from its former situation, more or less according to the perfection of the joint and strength of the index. This starting, which is due to the index recoiling after being released from the confined state it was in during the observation, will sometimes amount to several minutes; and its direction will be opposite to that in which the index was moved by the screw at the time of finishing the observation. But how far it affects the truth of the observation depends on the manner in which the index was moved in setting it to 0, for adjusting the instrument, or in finishing the observations necessary for finding the index error.

The easiest and best rule to avoid these errors seems to be this:—In all observations made by Hadley's quadrant or sextant, let the observer take notice constantly to finish his observations by moving the index in the same direction which was used in setting it to 0 for adjusting, or in the observations necessary for finding the index error. If this rule is observed, the error arising from the spring of the index will be obviated. For as the index was bent the same way, and in the same degree, in adjusting as in observing, the truth of the observations will not be affected by this bending.

To Observe the Sun's Altitude at Sea.

Turn down one of the dark glasses before the horizon-glass (if the instrument be the quadrant, the fore horizon-glass is to be used) according to the sun's brightness; 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 appears in the horizon-glass. Then give the instrument a slow vibratory motion about the axis of vision, as already described; move the index until the upper or lower limb of the sun is nearly in contact with the horizon at the lowest or highest part of the arc (according as the image is seen erect or inverted) described by this motion; and complete the contact by the tangent-screw, if the sextant be used—if not, by moving the index. The reading off of the limb will be the altitude of the sun.

To Observe the Moon's Altitude at Sea.

Turn down the green glass, and observe the moon in the silvered part of the horizon-glass, the eye being directed towards the horizon; move the index gradually, and proceed as already described in the case of the sun, until the enlightened limb is in contact with the horizon at the lowest or highest point of the arc described by the vibratory motion. The reading off will be the altitude of the moon's observed limb. If the lower limb be observed, the moon's semidiameter must be added; and if the upper limb be observed, it must be subtracted from the observed altitude, in order to obtain the altitude of the moon's centre. If the observation is made in the day-time, the coloured glass is not to be used.

To Observe the Altitude of a Star or Planet.

Put the index to zero; then direct the sight to the star so as to see it through the unsilvered part of the horizon-glass; turn the instrument a 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 will be seen to descend; continue to move the index gradually, until the star is in contact with the horizon at the lowest point of the arc described by the vibratory motion; in the case of the sextant, clamping the index when the contact is nearly made, and completing it with the adjusting or tangent screw.

Observation Instruments. To find the Altitude of the Sun on Shore with an Artificial Horizon.

A flat dish, containing a small quantity of mercury, is generally used for this purpose. The surface of the mercury is horizontal, and is a good reflector of the sun's rays. Let the observer so stand that he may receive on his eye rays from the sun which have been reflected from the surface of the mercury. He will then, according to the principles of optics, see the reflected image of the sun as much below the surface of the mercury as the sun is above it. If, then, he looks at the image through the unsilvered part of the horizon-glass, instead of at the horizon, and brings the image of the sun reflected at the index-glass and horizon-glass into contact with this, it is evident that the angle observed will be double of the sun's altitude. The details of this process are the same as have been already explained. Having read off the angle when the contact has been made, it must be corrected for the index error; and the result, divided by 2, will be the apparent altitude of that limb of the sun which has been observed.

To Observe the Distance between the Moon and any Celestial Object with the Sextant.

1. 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 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 illumined limb of the moon; now clamp the index, and, by a gentle motion of the instrument, make the image of the sun move alternately to one side and the other of the moon; and when in that position, where the limbs are nearest each other, make the contact of the limbs perfect by the tangent-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. Distance 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 downwards if it be to the left. 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; clamp the index, and turn the adjusting or tangent screw till the coincidence of the star and the enlightened limb of the moon is perfect; and the reading off of the limb at 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. II.—OF THE CORRECTIONS TO BE APPLIED TO OBSERVED ALTITUDES AND DISTANCES.

1. Parallax.

In order that the place of a heavenly body may be fixed in space, it is necessary to suppose that all observations are taken from one point. This point is the centre of the earth. Consequently the places of the sun, moon, and planets, whose distances from the earth are measurable, as observed, must be reduced to what they would be if seen from the centre. The correction for this object is called parallax. The fixed stars are at so great a distance from the earth that they have no sensible parallax, and this correction is not to be applied to them.

Let C (fig. 47) be the centre of the earth, P the place of the observer on its surface, and Z his zenith; and let S be a heavenly body whose position is observed. Then ZPS is the observed zenith distance, or complement of the observed altitude, and ZCS the true zenith distance, —i.e., the zenith distance as observed at the earth's centre.

Then clearly the angle ZPS is greater than the angle ZCS by the angle PSC, which is also in the plane of the vertical circle through S. It is evident from the figure that a heavenly body is depressed by parallax; and the observed altitude is less than the true altitude by a certain amount depending on the altitude, which is called the correction of parallax. The correction of parallax, therefore, must always be applied with the positive sign. Its amount may be easily found; for, let $\backslash angle\; ZPS\; =\; z\text{'}$, and $ZCS\; =\; z$, and PSC the parallax = $p$; then, in triangle PSC,

Let $PC\; =\; r$, the radius of the earth, and $SC\; =\; R$, the distance of the heavenly body from C; also PSC is always very small, and $\backslash sin\; PSC\; =\; PSC\; =\; p$ very nearly.

$r$ and $R$ being invariable, $p$ is greatest when $z\text{'}\; =\; 90^\backslash circ$, or when S is in the horizon. Let $P$ be the value of $p$ in this position; then

if $a\text{'}$ be the observed altitude. This will illustrate the principle on which tables of correction for parallax for different heavenly bodies, as sun, moon, &c., for different angles of altitude, are calculated.

2. Refraction.

Refraction is a correction to be applied in consequence of the rays from every heavenly body being bent or refracted as they pass through the successive layers of the earth's atmosphere, in consequence of which they describe curvilinear paths, having their convexity turned towards the zenith of the observer. The tangent to this curve at the eye of the observer is the direction in which he sees the object, and is evidently, from what has been said, bent towards the zenith. Hence the effect of refraction is to raise the heavenly bodies in the heavens above their true

places; and the correction must therefore be applied to the observed altitudes of all bodies with a negative sign.

The law which this correction follows is very complex; it is very great when the body is near the horizon, and vanishes when it is in the zenith. The best position of heavenly bodies for observation, so far as this correction is concerned, is near the zenith.

A table of this correction for every 5' of altitude is calculated, and is to be found in all nautical tables.

3. Correction for Semidiameter.

When the sun or moon, or a planet, is observed, the altitude of one of the limbs is the observed altitude, the altitude of the centre must be obtained by adding or subtracting the semidiameter of the observed body according as the lower or upper limb is observed. The semidiameters of the sun and moon are continually changing; and tables are given in the Nautical Almanac of their values at noon of every day at Greenwich, and in the case of the moon at midnight also. Their values at any intermediate hour may be calculated from these, as will be more fully shown hereafter.

This correction is not to be applied when a fixed star is the object observed.

4. Correction for Dip.

The altitude is supposed to be observed from the true horizon—i.e., from a horizontal plane through the place of the observer. This, however, is never the case, for the observer's eye must then be on the earth's surface. It is, in fact, always elevated above it, and the apparent horizon is depressed in consequence below the true horizon.

The nature of this correction will be easily seen from

fig. 48. B the place of observation at a distance $AB=d$ feet above the surface, $BH\text{'}$ touching the surface at $H\text{'}$; then $BH\text{'}$ is the plane of the sensible horizon; S a heavenly body to be observed. Draw $Ba$ parallel to $AH$, the true horizon of A, and let $BH\text{'}$ and $AH$ intersect in J. Then, since AB is very small compared with BS and AS, the $\backslash angle\; ABS$ may be considered as equal to $\backslash angle\; HAS$. Also let $R$ = earth's radius, $CA$ or $CH\text{'}$. Hence observed altitude of S

Fig. 48.

Or true altitude = observed altitude - $ACH\text{'}$.

Now $R$ in feet = $6120\; \backslash times\; 3958$.

Hence $ACH\text{'}$ in seconds

Hence, putting , feet respectively, we can find the dip when the eye is 1, 2, 3, &c., feet respectively above the surface.

Whence tables for the dip at different elevations may be calculated.

5. Index Error.

This error has already been explained.

It will be observed that all observed altitudes are affected with refraction and dip, of which the first is always subtractive, and the second is subtractive in all observations, except when the back horizon-glass of the quadrant is made use of, when it is additive.

Parallax and semidiameter affect only those bodies whose distance from the earth is not very great, and which have a sensible diameter, as the sun and moon; but no fixed stars. Parallax is always additive; and the semidiameter is to be added or subtracted according as the lower or upper limb of the heavenly body is observed.

CHAP. II.—PRELIMINARY PROBLEMS IN NAUTICAL ASTRONOMY, AND USE OF NAUTICAL ALMANAC AND TABLES.

SECT. I.—OF TIME.

A day is the interval between two successive transits of a heavenly body over the meridian, and derives its name from the body whose motion is observed; and of whatever denomination it be, it is divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. The interval between two successive transits of the sun is a solar day; of the moon, a lunar day; of a fixed star, a sidereal day. The earth's revolution about her axis is performed always in the same time; hence if all the heavenly bodies retained the same position with regard to one another, all days of whatever name would be of the same length. The sun (or, more strictly speaking, the earth), the moon, and planets are always varying their position with respect to one another and the fixed stars; and they move with velocities not only different from each other, but variable in different parts of their own orbits. The length of the day, therefore, determined by these bodies is variable. In order to obtain a definite and uniform measure of time, the mean solar day, which is the average of all the apparent solar days in the year, is employed. An imaginary body, called the mean sun, is supposed to describe the equator uniformly with the true sun's average or mean daily motion, and the interval between two successive transits of this imaginary sun is a mean solar day.

Clocks and chronometers are adapted to mean solar time; so that a complete revolution through twenty-four hours of the hour-hand of one of these instruments would exactly correspond with the revolution of the earth about her axis with regard to the mean sun. Time reckoned in mean solar days and parts of mean solar days is called mean solar time.

The true sun sometimes passes the meridian before and sometimes after the mean sun; the difference in time of the transits of the true and mean suns is called the equation of time, and is sometimes to be added and sometimes to be subtracted from the one of these times to obtain the other, as is pointed out in the Nautical Almanac.

As the earth revolves about her axis from E. to W., different meridians successively come under the mean sun; and since, after the lapse of twenty-four mean solar hours,

Nautical Astronomy the same meridian again comes under the sun; it follows that for every 15° of longitude between the places, there is a difference of 1^h of mean time; and that mean noon is 1^h earlier for every 15° of E. longitude, and 1^h later for every 15° of W. longitude.

A sidereal day is the interval between two successive transits over the same meridian of the vernal equinox, or the first point of Aries. This is not strictly a uniform measure because of the change of the position of this point of Aries in consequence of precession and nutation. Time, therefore, so reckoned, ought strictly to be called apparent sidereal time; and mean sidereal time to be reckoned from the transit not of the true but of the mean equinoctial point. The smallness of the fluctuations to which a clock regulated to apparent sidereal time, compared with one regulated to mean sidereal time, is subject, amounting at the utmost to 2^m 3^s in 19 years, has prevented the practical inconvenience of this being felt; no clock being sufficiently perfect to go for so long a period without requiring frequently to be re-adjusted.

The sun's apparent revolution in his orbit round the earth takes place in 365.242218 mean solar days; the mean sun, therefore, describes an angle of 59° 8' 33" in a mean solar day; hence, in the interval between two successive transits of the mean sun over the same meridian, the earth revolves through 360° 59' 8' 33"; but in a sidereal day the earth revolves through 360°. Whence we obtain the proportion—

A sidereal day : a mean solar day :: 360° : 360° 59' 8" 33".

Whence also it is evident that, if the same interval of absolute time be expressed in mean solar and also in sidereal time, it will be greater in the latter denomination than in the former.

Tables are calculated for the acceleration of sidereal on mean solar time, and, conversely, for the retardation of mean solar on sidereal time, which are useful for converting intervals of sidereal into mean solar time, and conversely. These are to be found at page 516 of the Nautical Almanac.

Astronomical mean time always begins at noon, and goes on to the succeeding noon through 24 hours; so that a clock which shows astronomical mean time, set for any place (as Greenwich), shows 0^h 0^m 0^s when the mean sun is on the meridian, and shows 24 hours, or 0^h 0^m 0^s, when the mean sun is again on the meridian.

A clock set to show sidereal time shows 0^h 0^m 0^s when the mean equinoctial point is on the meridian, and shows 24 hours, or 0^h 0^m 0^s, again when this point is next on the meridian.

Sidereal Time, in the Nautical Almanac, is the sidereal time at mean noon for the meridian of Greenwich; or, in other words, the hour angle of the equinoctial point when the mean sun is on this meridian; and is very useful in all cases where mean time is to be deduced from observations of heavenly bodies.

The civil day is reckoned from midnight to midnight, and is divided into 12 hours, from midnight to noon, called A.M.; and into 12 hours, from noon to midnight, called P.M. Evidently time P.M., and astronomical mean time up to midnight, are the same. But for time A.M., add 12 hours, and date the day one back.

Thus, June 3, 7^h 5^m 27^s P.M. is, June 3, 7^h 5^m 27^s astronomical time; but, June 3, 5^h 12^m 37^s A.M. is, June 2, 17^h 12^m 37^s A.M.

Conversely, if the astronomical time is given, and the corresponding civil time is required, if under 12 hours, the given time is the same time P.M. for the same date; if above 12 hours, subtract 12 hours from it, and date the day one forward, and the result is civil time A.M.

Thus, September 18, 7^h 10^m 15^s is, September 18,

7^h 10^m 15^s P.M.; but September 18, 22^h 13^m 45^s astronomical is, September 19, 10^h 13^m 45^s A.M.

PROB. 1.—To convert degrees, or parts of the equator, into time.

Rule.—Divide the degrees by 15; the result gives hours. Multiply the remaining degrees, if any, by 4; the result is minutes. Divide the number of minutes by 15 for seconds, and then multiply the remaining minutes by 4 for seconds. Divide the seconds by 15 for seconds, and decimals of seconds, if necessary.

Ex. 1.—Reduce 25° 45' to time. Divide 25 by 15, the quotient is 1, with remainder 11. Hence, by the rule—

Ex. 2.—Reduce 135° 48' 18" to time.

PROB. 2.—To convert time into degrees, minutes, and seconds.

Rule.—Multiply the given time by 10, and add to the result half of the product. The result will be the corresponding degrees, minutes, and seconds.

Ex.—Convert 3^h 4^m 28^s into degrees, &c.

PROB. 3.—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 turn it into astronomical time; convert the longitude of the known meridian into time, which add to the given time if the place be W. of Greenwich, and subtract from the given time if it be E.; and, if necessary, increase the time by 24 hours, reckoning the day one back. If the resulting time exceed 24 hours, put the date one day forward, and subtract 24 hours.

Ex. 1.—It is 6^h 15^m P.M., June 23, in a ship whose longitude is 76° 45' W.

Ex. 2.—The time at a ship in longitude 139° 48' 15" E. is, July 24, 5^h 25^m 20^s P.M.; required the Greenwich time.

This being greater than the given time, we next add 24^h to the latter, and we have—

Ex. 3.—The time at a ship in longitude 175° 45' W. is, June 28, 4^h 18^m 12^s A.M.; what is the Greenwich time?

Ex. 4.—The time at a ship in longitude 178° E. is, October 14, 3^h 15^m A.M.; what is the time at Greenwich?
3^h 15^m A.M. on Oct. 14... = Oct. 13, 15^h 15^m 0^s
Longitude 178° ..... = 11 52 0 E.
Time at Greenwich. = Oct. 13, 3 23 0

PROB. 4.—To find the Greenwich time by a chronometer whose error on Greenwich mean time is known.

Rule.—Add or subtract the error of the chronometer to the time shown by it, according as it is slow or fast; the result is Greenwich time. Sometimes 12 hours must be added to this result, and the day reckoned one back. The best way to obviate this error, is to obtain Greenwich time approximately, by Problem 3, by help of the ship's mean time and longitude by account; if the difference between the two (Greenwich) dates, so found in these two ways, is nearly 12 hours, then the date by chronometer must be increased by 12 hours, and the day reckoned one back if necessary, so as to make the two dates agree in the day and hour nearly.

Ex. 1.—June 15, 1857, at 7^h 45^m P.M. mean time nearly, in longitude 45° W., a chronometer showed 10^h 53^m 12^s, being 5^h 15^m fast; required the Greenwich time.

Ex. 2.—October 12, 1857, at 11^h 45^m P.M., in longitude 110° 35' W., a chronometer showed 7^h 0^m 40^s, being 9^h 5^m slow; required the Greenwich date.

PROB. 5.—Given the Greenwich date, to find the date under a given meridian.

Rule.—The Greenwich date being reduced to the preceding noon, reduce the longitude to time, and subtract it from Greenwich date, increased by 24 hours, and put one day back if necessary, if the longitude be W.; and add if the longitude be E.

Ex. 1.—An eclipse of the moon commenced at Greenwich, December 25, 1852, at 22^h 14^m 36^s; at what time did it begin at a place in longitude 175° E.?

Ex. 2.—On the 6th of December 1857, at 9^h at Greenwich, the distance of the moon from the sun was 106° 44' 31"; at what time will the distance be the same in longitude 168° 15' 43" W.?
Long. = 11^h 13^m 2^s 86 W. Greenwich, Dec. 5..... 33^h 0^m 0^s
Add 24 to Greenwich time. Longitude..... 11 13 2 86 W.
Date required, Dec. 5..... 21 46 57 14.

SECT. II.—OF THE SUN'S DECLINATION, RIGHT ASCENSION, EQUATION OF TIME, AND SEMIDIAMETER AND SIDEREAL TIME.

All of these quantities are given in the Nautical Almanac for the noon of every day. They all vary, and consequently their value at any time intermediate between two successive noons is different from that at either noon. For ordinary purposes, it is sufficient to suppose that the change of these quantities is proportional to the time from the preceding noon. Hence, if the difference of the value between the preceding and succeeding noon is taken from the Nautical Almanac, a simple proportion will give the quan-

tity to be applied to the value at the preceding noon, which must be added if the quantity is increasing, and subtracted if it be decreasing. As the values of the quantities are given for Greenwich noon, it is necessary to reduce the ship's time to Greenwich time, in order to obtain their proper value.

This process may be materially shortened by the use of proportional logarithms, which are given in nautical tables, and to which the reader is referred for an explanation. For example, to take out the sun's declination at a given time in a given place, get a Greenwich date, as already explained; take out from the Nautical Almanac the declination at two successive noons between which the date lies; take their difference; take out from the tables the logarithm of the Greenwich date of the sun; add to it the proportional logarithm for the change in 24 hours;—the result is the proportional logarithm of the change of declination for the given time. This will be found from the table, and added or subtracted, according as the declination is increasing or decreasing. Sometimes the declination at two successive noons will be of different names; the difference must then be found by adding them.

The very same mode applies to finding the equation of time, and also the sun's apparent right ascension.

The sun's semidiameter may always be taken to be that corresponding to the nearest noon.

The right ascension of the mean sun, called in the Nautical Almanac the sidereal time, may be found for any other meridian than Greenwich in the same way. Since, however, the motion of the mean sun is uniform throughout the year, the change in any given number of hours, minutes, and seconds is the same for all days, and can be found at once from the "Table of Time Equivalents" given in the Nautical Almanac.

Opposite to each of these quantities in the Nautical Almanac is given the difference for one hour, which enables us to find the quantity for any other hour than noon, by multiplying this hourly difference by the number of hours in the date from noon, and adding to it the proportional parts for the additional minutes and seconds, and applying this difference to the value for the preceding noon, with a positive or negative sign, according as the quantity is increasing or decreasing.

The following examples will illustrate these directions:—

Ex. 1.—October 11, 1857, in Long. 36° 45' E., at 7^h 45^m A.M.; required the sun's declination, right ascension, equation of time, and sidereal time, i.e., the right ascension of the mean sun when crossing the meridian of observer.

To find Declination.—(1.) By proportional logarithms, from Nautical Almanac.

(2.) By hourly differences.

To find right ascension.

Or, from Nautical Almanac—

To find equation of time.

Or, from Nautical Almanac—

which, added to the equation of time at mean noon of October 10, gives 13^m 10^s 81 for equation of time required.

Or, by table of time equivalents,

Ex. 2.—To find the sun's declination at a place in Long. 97° 45' W., at 3^h 46^m p.m., on 20th March 1857,—Greenwich date, March 20, 10^h 17^m.

Or, by hourly difference,

SECT. III.—OF TAKING OUT MOON'S DECLINATION, RIGHT ASCENSION, SEMIDIAMETER, AND HORIZONTAL PARALLAX.

The moon's declination and right ascension are given for every hour in the Nautical Almanac; and the difference for 10^m at the beginning of every hour is recorded. From these data, the moon's declination and right ascension at any hour can be readily obtained; for by multiplying the difference for 10^m by the number of intervals of 10^m in the given time since the last hour, and taking parts for the additional minutes and seconds, and adding the result if the declination be increasing, and subtracting if decreasing, to the declination at the last preceding hour, we shall find the declination. The difference must always be added to the right ascension. Or they may be found by a table of logistic logarithms: thus, add together the logistic logarithms of the minutes and seconds in the Greenwich date, and the proportional logarithms of the difference; the result will be the proportional logarithm of the correction to be applied.

The moon's semidiameter and horizontal parallax are given in the Nautical Almanac for every 12^h,—viz., from mean noon to mean midnight, and mean midnight to mean noon, of every day at Greenwich. Hence we may proceed as for the sun's right ascension and declination, except that we must always take out of the table the declination or right ascension for the mean noon or mean midnight, according as the time lies between noon and midnight or midnight and noon; and if the Greenwich date exceed 12^h, we must subtract 12^h from it, and reckon the difference from the preceding midnight.

Ex. 1.—August 10, 1857, in Long. 84° 30' E., at 7^h 44^m 15^s p.m.; required the moon's right ascension and declination.

Or correction is 1^m 41^s, as before.

Ex. 2.—September 12, 1857, in Long. 149° 18' W., at 5^h 36^m p.m.; required the moon's semidiameter and horizontal parallax.

Rule.—Lunar distances are found for every third hour of Greenwich mean time, and are recorded in the Nautical Almanac, as also the proportional logarithms of the differences of the distances at intervals of every three hours. To find, then, the Greenwich date corresponding to a given distance, find in the table the nearest distance preceding in order of time the given distance, and take the difference between it and the given distance; from the proportional logarithm of this difference subtract the proportional logarithm of the said nearest distance; the remainder will be the proportional logarithm of the correction to be applied to the Greenwich date answering to the nearest distance in order to obtain the approximate Greenwich time.

The distances are here supposed to increase uniformly, which is not the case. It is therefore necessary to apply a correction to the time above obtained, if great accuracy is required. This may be found by help of a "Table showing the Corrections required on account of Second Differences," in the Nautical Almanac, as follows:—(1.) Find the approximate interval by the preceding rule. (2.) Take the difference between the proportional logarithms which stand opposite to the distances which include the given distance. (3.) Look down the column in the table at the head of which is this difference of proportional logarithms, and along the column at the left hand side of which stands the approximate interval; the number so found is the correction, which is to be added to the approximate time if the proportional logarithms are decreasing, and subtracted if they are increasing.

Ex.—Required the time at Greenwich at which the reduced distance of the moon from Jupiter will be 42° 55' 18" on the 23d December 1857.

From the table, it appears this will be between noon and 3^h. The nearest distance preceding therefore is,—

The difference of proportional logarithms at noon and 3^h is 12; and on looking in the table for second differences down the column headed (2), and along the column on the left of which is 2^h, the

nearest to the approximate interval, we find 3^h the correction, which is to be added, as the proportional logarithms are decreasing. Hence the Greenwich time is 1^h 54^m 27^s.

PROB. 1.—Given mean solar time at any given place, to find apparent solar time; and, conversely, given apparent, to find mean solar time.

Find the time at Greenwich, and correct the equation of time for this date. The equation of time so corrected applied to the given time, with the proper sign (as shown in the Nautical Almanac), will give the time required.

Ex. 1.—June 16, 1857, mean solar time 8^h 15^m P.M., in longitude 25° W.; to find the apparent time.

Ex. 2.—Oct. 15, 1857, at 6^h 51^m P.M., apparent time, in Long. 113° 45' E.; find mean time.

Rule.—If the given time be mean solar, find the Greenwich date; correct the right ascension of mean sun for this date, and add to the mean time at ship; the result is sidereal time.

If the given time is apparent solar, convert it into mean solar time, as in Problem 1, and proceed as before.

Ex.—Nov. 18, 1857, at 9^h 35^m 30^s A.M., mean time, in Long. 18° 7' W.; required the sidereal time.

Nautical Astronomy By help of the last problem, we may find what bright stars in the catalogue of the Nautical Almanac will be on the meridian of a given place next after any time, mean or apparent.

For, having found the sidereal time at a ship, it is evident that the star whose right ascension is next greater, is the first that will pass the meridian in question.

It is sometimes required to find all the bright stars that will pass a given meridian between two specified hours. We must in that case find the sidereal time for both hours, and all the bright stars in the catalogue whose right ascensions lie between these limits will pass the given meridian between the specified hours.

Ex. 1.—To find what bright star will pass the meridian in Long. $15^\{\backslash circ\}\; 7\text{'}\; W.$, on November 18, 1857; next after $9^\{\backslash text\{h\}\}\; 35^\{\backslash text\{m\}\}\; 30^\{\backslash text\{s\}\}$ A.M. (See Example to Problem 2.)

Looking in the catalogue of bright stars in the Nautical Almanac, we find that the bright star whose right ascension is the next greater than $13^\{\backslash text\{h\}\}\; 25^\{\backslash text\{m\}\}\; 12^\{\backslash text\{s\}\}\; 44^\{\backslash text\{s\}\}$, is $\backslash beta$ Corvi, which is the star required.

Ex. 2.—What bright stars in the Nautical Almanac passed the meridian of a place in Long. $64^\{\backslash circ\}\; E.$, between the hours of 6 and 9, on March 10, 1857?

On looking at the catalogue, we find that the stars whose right ascensions lie between these limits are from $\backslash beta$ Tauri to $15^\{\backslash circ\}$ Argus, which are the stars required.

PROB. 3.—Given sidereal, to find mean solar time.

Rule.—Take out of the Nautical Almanac the right ascension of the mean sun for noon of the given day. Subtract this from the sidereal time, increased, if necessary, by 24 hours; the remainder is mean time nearly. By help of the table which gives the relation of mean solar and sidereal time already referred to, correct this approximate time, and the result will be the mean time required.

Ex.—July 21, 1857, when a sidereal clock showed $6^\{\backslash text\{h\}\}\; 45^\{\backslash text\{m\}\}\; 35^\{\backslash text\{s\}\}$ find the mean time.

By means of the above we can find the error of a clock or chronometer by comparing the time shown by it with that of a sidereal clock in an observatory. Thus, Greenwich, November 4, 1857, a sidereal clock showed $18^\{\backslash text\{h\}\}\; 15^\{\backslash text\{m\}\}\; 30^\{\backslash text\{s\}\}$, and a chronometer showed $3^\{\backslash text\{h\}\}\; 28^\{\backslash text\{m\}\}\; 15^\{\backslash text\{s\}\}$, the sidereal clock being $3^\{\backslash text\{m\}\}\; 13^\{\backslash text\{s\}\}\; 5$ slow; required the error of the chronometer.

PROB. 4.—To find at what time any heavenly body will pass the meridian of a given place on a given day.

Rule.—From the right ascension of the given star taken out of the Nautical Almanac, increased, if necessary, by 24 hours, subtract the right ascension of the mean sun at mean noon (or sidereal time, as it is called in the Nautical Almanac). This gives the mean time at the ship approximately. Apply the longitude to this, and thus get a Greenwich date, and by this date correct the right ascension of the mean sun, and subtract the quantity so found from the star's right ascension; this will be the time required.

Ex.—At what time will $\backslash beta$ Corvi pass the meridian of a place in Long. $62^\{\backslash circ\}\; 20\text{'}\; E.$, on July 24, 1857?

SECT. VI.—OF THE MOON'S AUGMENTATION—CORRECTION OF SEMIDIAMETER ON ACCOUNT OF REFRACTION—THE MOON'S MERIDIAN PASSAGE, &c., &c.

1. Of the Moon's Augmentation.—The moon's semidiameter given in the Nautical Almanac is calculated as if seen from the earth's centre. Moreover, on account of the comparatively moderate distance of the moon from the earth, the moon's semidiameter subtends an angle at the earth's surface which sensibly varies with the altitude, being greatest when the moon is in the zenith, and least when in the horizon of the observer. The correction to be applied to the semidiameter from this cause is called the augmentation. It is computed for every degree of altitude, and is to be found in all nautical tables.

2. Contraction of Moon's Semidiameter on account of Refraction.—The correction for refraction varies very rapidly near the horizon, and consequently the moon's lower limb is sensibly more raised than the upper; and the moon's apparent form is that of an ellipse instead of a circle. Hence, in taking a lunar distance when the moon is near the horizon, the moon's semidiameter to be added to the observed distance, as taken from the tables, is greater than it ought to be; and a correction must be applied in consequence of the moon's diameter being an oblique diameter of the elliptic face, and not the horizontal or greatest diameter. This correction is always subtractive, and is given in nautical tables.

3. Of the Moon's Meridian Passage.—In west longitude the moon crosses the meridian later with regard to the sun than at Greenwich, and in east longitude earlier; because the moon's distance from the sun in right ascension is constantly changing. Hence to the Greenwich meridian passage of the moon must be applied a correction for every other place, depending on the longitude. All nautical tables contain this correction.

The corrections to be applied to the observed altitudes and distances of the heavenly bodies have been now sufficiently explained, and the reader will, with a little experience, easily understand how the several corrections are to be taken out of the tables. Index error, dip, and refraction are common to all observed altitudes, and must be applied in the order in which they are here set down. If the body observed be a star, this is all that is required.

If a planet, the parallax for the given altitude, taken from the proper table, must be added. If the observed body be the sun, after the dip apply the sun's semidiameter, and then correct for refraction and parallax. These two corrections are frequently given together, under the name of correction in altitude, which is always subtractive, because the refraction is always greater than the parallax. If the observed body be the moon, the semidiameter and horizontal parallax cannot be considered as invariable for 24 hours; they must therefore be corrected for the proper Greenwich date, as already pointed out. To the semidiameter thus corrected must be added the augmentation, from the proper table. The rest of the process is exactly the same as for the sun.

Ex. 1.—Feb. 1, 1857, at 2^h 46^m P.M., in Lat. 53° 20' N., and Long. 13° 20' E., the observed altitude of the sun's lower limb was 13° 35', the index error was +1° 20', and the height of the eye above the sea 18 feet; to find the sun's true altitude.

Ex. 2.—August 24, 1857, in Long. 18° 20' W., the observed meridian altitude of the moon's lower limb was 54° 30' 20", the index correction was +2° 20", and height of the eye above the sea 20 feet; required the moon's true altitude.

CHAP. III.—ON FINDING THE LATITUDE.

SECT. I.—OF FINDING THE LATITUDE BY MERIDIAN ALTITUDES.

1. By a single Meridian Altitude above the Pole.

Subtract the true meridian altitude (corrected from observed meridian altitude) from 90°; the result is the zenith distance—which mark N. or S., according as the zenith is north or south of the observed heavenly body: find also the declination, which must be marked N. or S. If the zenith distance and the declination have the same name, their sum will be the latitude, with the name of either. If they be of different names, take their difference, which will be the latitude, with the name of the greater.

The process will differ in detail, according to the nature of the body observed.

(1.) If it be a fixed star, the altitude must be corrected only for index correction, dip, and refraction; and the declination is given at once by the tables.

(2.) If it be a planet, the declination must be corrected by getting a Greenwich date, applying the proportional increase or decrease for this date. If great accuracy is required, the semidiameter and horizontal parallax must also be found, and the altitude corrected for these.

(3.) If it be the sun, the Greenwich date must be got, and the declination corrected for this; the altitude must be corrected for index, dip, semidiameter, refraction, and parallax.

(4.) If it be the moon, a Greenwich date must be got. If it be only known that the observed altitude is a meridian altitude, the meridian passage at Greenwich must be corrected for the longitude, and the Greenwich date obtained by applying the longitude in time as already shown. The moon's semidiameter, horizontal parallax, and declination, must be corrected for the Greenwich date, and the true altitude and declination thus found. When the true altitude and declination are found, the remainder of the process is according to the rule given above, and is the same for all.

Ex. 1.—May 20, 1857, the observed meridian altitude of Castor was 49° 20' 30" (zenith N. of star), the index correction was -3° 40", and the height of the eye above the sea was 18 feet; required the latitude.

Ex. 2.—October 10, 1857, the observed meridian altitude of α Ophiuchi was 54° 20' 30" (zenith, N. of the star), the index correction was -3° 20", and the height of the eye above the sea was 16 feet; required the latitude.

Ex. 3.—January 6, in Long. 59° 30' E., the observed meridian altitude of the sun's lower limb was 60° 20' 30" (zenith N. of the sun), the index correction was +4° 10", and the height of the eye above the sea was 10 feet; required the latitude.

Ex. 4.—April 4, in Long. 55° 20' E., the observed meridian altitude of the moon's lower limb was 58° 40' 10" (zenith S. of the moon), the index correction was +3' 20", and height of the eye above the sea was 9 feet; required the latitude.

2. By a single Observed Altitude below the Pole.

Rule.—Correct the altitude, and to it add 90°, and from this sum subtract the declination; the remainder is the latitude.

Ex. 1.—June 4, 1857, the observed meridian altitude of $\backslash beta$ Chamaeleontis, under the S. pole, was 29° 56' 40", the index correction was -2° 40", and the height of the eye above the sea was 15 feet; required the latitude.

Ex. 2.—Nov. 20, 1857, the observed meridian altitude of $\backslash alpha$ Draconis under the N. pole, was 31° 40' 20", the index correction was +1' 20" and the height of the eye above the sea 20 feet; required the latitude.

tion was +1' 20" and the height of the eye above the sea 20 feet; required the latitude.

In the following example, the altitude is observed by means of an artificial horizon.

June 8, 1857, in Long. 1° 6' W., the observed altitude of the sun's lower limb (in quicksilver horizon) was 121° 9' 4", index correction 1' 20"; required the latitude.

SECT. II.—BY OBSERVED ALTITUDES OUT OF THE MERIDIAN.

1. By Observations of the Altitude of the Pole-Star (a Polaris).

The tables for this purpose are given in the Nautical Almanac, pp. 513 to 515.

Rule.—From the observed altitude, corrected for the error of the instrument, refraction, and dip of the horizon, subtract 2°.

Reduce the mean time of observation at the place to the corresponding sidereal time.

With the sidereal time found, take out the first correction with its proper sign. If the sign be +, the correction must be added to the reduced altitude; but if it be -, it must be subtracted; in either case the result will give an approximate latitude.

With the altitude and sidereal time of observation, take out the second correction; and with the day of the month and the same sidereal time, take out the third correction. These two corrections, added to the approximate latitude, will give the latitude of the place.

Ex.—January 20, 1857, at 9^h 40^m P.M. (mean time nearly), in Long. 30° 20' E., the observed altitude of Polaris was 31° 40' 20", the index correction was -2° 20", and height of the eye above the sea, 10 feet; required the latitude.

3. By Observed Altitudes of a Heavenly Body very near the Meridian.

The sun is the body most usually observed for the purpose. The apparent time from noon or hour angle is supposed to be known; as also the latitude by account, which is supposed to differ from the true latitude only by a small quantity.

Rule.—Add together the tabular logs. of cosines of the latitude by account, and of the declination of the observed body. To these add log. rising, if a table of log. rising be used; or, 6.301030 + log. haversine hour angle, if a table of haversines be used; or twice the log. sine of half the hour angle with the same constant log., if a table of log. sines only be used.

In all cases cast out the tens from the result, and the remainder is the logarithm of a natural number, which, added to the natural sine of the true altitude, will give the natural cosine of the meridian zenith distance. The declination applied to this—as already pointed out in preceding problems—will give the latitude. The hour angle in this rule must be found by applying to the mean time at the ship the equation of time corrected for the Greenwich date.

If the latitude thus found differ much from the latitude by account, the work must be repeated, using the latitude found instead of latitude by account.

Ex.—October 11, 1857, in Long. 114° E., and Lat. by account 46° 10' N., the chronometer showed 4^h 38^m 10^s, being slow on Greenwich 14^m 10^s. The altitude of the sun's lower limb was observed to be 35° 35' 20", the index error was -2° 20', and the height of the eye above the sea 16 feet; required the latitude.

Nat. number by table of log. rising, 1124.

4. By two Observed Altitudes of the Sun, and the Time between; having given also the Latitude by Account (Douce's Method).

Rule.—To find the hour angle corresponding to the greater meridian altitude—i.e., the hour from apparent noon, at which the greater altitude is observed.

To the log. secant of the latitude by account, add the log. secant of the sun's declination (the mean between the declinations at the first and second observation); this, rejecting 20, is called the logarithm ratio. To this add the log. of the difference of the natural sines of the two altitudes, and the logarithm of the half-elapsed time from the proper column (taken from nautical tables, which are furnished with special tables for this purpose).

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

Or from any table:—

Find the logarithm ratio as before.

Subtract each of the true altitudes from 90°, to get the true zenith distances, and take half their sum and half their difference; and to the logarithm ratio add the sum of the log. of the sines of this semi-sum and semi-difference, and from this sum subtract the log. sine of the half-elapsed time; the remainder is the log. sine of middle time, which find from a table of log. sines, and the difference of the middle time and half-elapsed time is the time from noon, at which the greater altitude is observed.

We have now an observed altitude and the time from noon or hour angle corresponding; the remainder of the solution is therefore the same as in the last case, except that the log. ratio, being the sum of the logarithms of the secants, may be used instead of the sum of the logarithms of the cosines, with the negative sign. If the latitude so found differs much from the latitude by account, the work must be repeated, using the computed latitude for the latitude by account.

This rule is only approximate, and must be used under the following restrictions:—1. The observations must be taken between nine in the forenoon and three in the afternoon. 2. If both the observations be made 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. 3. If one observation be in the forenoon and the other in the afternoon, the interval must not exceed 4^h; and in all instances the nearer to noon the greater altitude is observed the better. 4. If the sun's meridian zenith distance be less than the latitude, the limitations are still more con-

Finding
the Lat-
tude. tracted. 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 3½^h. The observations must be taken still nearer to noon if the latitude exceed the zenith distance in a greater proportion.¹

As the ship is generally in motion between the two observations, it is necessary to apply to the first altitude a correction for its run in the interval, so as to obtain what the altitude would have been if observed at the same time as the second.

If $\backslash theta$ be the angle between the course of the ship and the bearing of the sun, the correction for run is $y\; =\; \backslash pm\; m\; \backslash cos\; \backslash theta$, where $m$ is the number of minutes in the angle subtended by the ship's run, i.e., number of miles run, and $\backslash theta$ the angle between the bearing of the sun and the course; and the sign + or - is to be used according as the ship is moving towards or from the heavenly body: in the former case $\backslash theta$ is less than 90°, and in the latter greater. It is evident, from the form of the expression, that this correction may be found in a traverse table by looking out the difference between the course and bearing, or what it wants of 8 points for a course, and the ship's run as a distance; the corresponding difference of latitude, multiplied by 60, will then be the number of seconds in the correction for run, which is to be added or subtracted according as the angle is less or greater than 8 points or 90°. Thus, the bearing of the sun was S.E., the run was S.S.E. 18 miles; required the correction for run. Here the angle between the bearing and the course is 2 points, and the distance 18 miles; on entering a traverse table with these, we find the true difference of latitude 16.6. Hence the correction for run is +16.12".

Ex.—April 20, 1857, in Lat. 44° 20' N., Long. 35° 50' E., the following double altitude of the sun was observed:—

The run of the ship in the interval was S.S.E. 25 miles, the index correction was -4' 40", and the height of the eye above the sea 15 feet; required the true latitude at the last observation.

The angle between the true bearing at first observation and ship's course is 1 point; the true difference of latitude, corresponding to distance 25 and course 1 point in traverse table, is 24.5. Hence correction for run to be applied to the greater altitude is +24' 30".