INTRODUCTION.

The term Geography, derived from two Greek words, γῆ, the earth, and γράφω, I write, signifies a description of the earth. The description to which this title is applied may be more or less general; either embracing such truths as belong to the earth considered as one whole, or tending to particulars which belong to and distinguish several countries spread over its surface. In whichever of these two aspects the subject be regarded, a vast field opens to the view of the observer. In order to give a full and accurate description of the earth, it would be requisite to consider it in reference to its motion, figure, and magnitude; in reference to its relation to the other bodies of the universe, and more especially to the planetary system of which it forms a part; in reference to its surface, diversified by land and sea, mountains and valleys, lakes and rivers; in reference to the materials which compose its crust, and to its internal structure; in reference to the constitution of the atmosphere with which it is surrounded, and the effects arising from the variations in atmospheric pressure, temperature, and humidity. Nor would it be enough to consider the earth only as a mass of inert and organized matter; it would be necessary to regard it in its relations to vegetable and animal life; and to trace the phenomena which these, in their endless variety, present in various divisions and provinces. It would still further be necessary to view it as the abode of man himself, as modified by his existence; divided into states and kingdoms; adorned with cities, and all the noble monuments of civilized life.

Such is an outline of the picture which geography, in its most unlimited meaning of the term, should exhibit of the globe. To fill up this picture in all its parts, it would evidently be necessary to call in the aid of the whole circle of the sciences. But the description is usually of a less extended character, being confined chiefly to the more obvious and striking features of the various regions and countries of the earth.

In the wide range which the subject presents, several divisions and subdivisions are suggested by the different views in which the earth may be considered. The three following divisions are the most important:

1. Mathematical Geography, which illustrates, on astronomical principles, the figure, magnitude, and motion of the earth; teaches how to determine the positions of places on its surface; explains the construction of globes, with their application to the solution of problems; and shows how the whole or any portion of the earth's surface may, on the principles of projection, be delineated on a map or chart.

2. Physical Geography, which treats of the mutual relations of the diversified objects found on the surface of the earth, including the atmosphere by which it is surrounded; and explains the causes, whether of a chemical or mechanical description, that produce the modifications and changes which they continually undergo.

3. Political or Historical Geography, which describes the earth as divided into countries, occupied by various nations, and improved by human art and industry. It traces the circumstances and character of the different races and tribes of mankind, explaining their social institutions, and ascertaining the place which each occupies in the scale of civilization.

From this general arrangement of the subject, it is evident that geography depends for its rank as a science on its intimate connection with various branches of knowledge, which, taking their rise from investigations instituted in reference to the nature and mutual relations of the objects on the earth, or connected with it, furnish those accurate views which must be obtained before any thing like a precise description can be given of the globe we inhabit, or of any portion of it. With regard to what belongs to Physical Geography, we must refer the reader to the articles Physical Geography, Mineralogy, Meteorology, &c., in this work. What belongs to Political or Historical Geography will be found under the names of the respective countries. The following article will be limited to a view of the progress of Geographical Discovery, and to a brief explanation of the principles of Mathematical Geography.

I.—VIEW OF THE PROGRESS OF GEOGRAPHICAL DISCOVERY.

There are many circumstances in the condition of man which connect him so closely with the globe which he inhabits, as to render absolutely necessary to his existence a knowledge of at least the neighbourhood of the spot where his lot is cast. It is from the earth that he must derive the means of subsistence and accommodation, the materials on which his industry is to be exerted, and those objects in the exchange of which commerce consists. In every stage of his progress, therefore, from barbarism to civilization, he must employ some attention and observation, in order to discover in what respects the objects with which he is surrounded are qualified to contribute to the supply of his wants, and to his comfort and convenience. Even while he roams the forest in the savage state, he must make himself acquainted with many circumstances, a knowledge of which is necessary either to give him success in the chase, or to direct him in retracing his steps to the place where he has fixed his dwelling. But it is not until men have united in society, and that neighbouring communities have begun to hold mutual intercourse, that those feelings and passions are effectually aroused which stimulate to the arduous pursuits of geographical discovery. Commerce and war, with the spirit of adventure which usually accompany them, have without doubt been among the first causes of geographical research. In the train of these have followed the workings of avarice and the aims of ambition. As the human mind has advanced in its career of improvement, curiosity, with an enlargement of views and desires, have been called into action; and voyages have been undertaken for the express purpose of discovering new countries and exploring unknown seas.

In tracing the effects which these causes have produced in the gradual increase of geographical knowledge, it will contribute to distinctness to keep in view a threefold division, which the subject naturally assumes, namely, ancient geography, extending from the earliest period of history down to the time when, the Roman empire having been overrun by barbarous nations from several quarters, Europe was overwhelmed in the darkness which preceded the revival of learning; the geography of the middle ages, extending from the revival of letters to the fifteenth century, when the discoveries of the Portuguese began to lay a The Phoenicians are the earliest commercial people of whose discoveries we have any correct accounts. This people seem first to have explored the coasts of the Mediterranean. Their navigators at length extending their voyages through the Straits of Gades, now called the Straits of Gibraltar, entered the Atlantic Ocean, and visited the western coasts of Spain and Africa. In many places to which they resorted they planted colonies; and sought, by instructing the inhabitants, in some measure, in their arts and improvements, to open a wider sphere for their commerce. The learned Bochart, led by the analogy between the Phoenician tongue and the oriental languages, has endeavoured to follow the tracks of the Phoenicians, both along the shores of the Mediterranean and those of the Atlantic. These analogies are not always sure guides; but there seems no reason to doubt that Cadiz was originally a Phoenician colony, and it is not likely that this was the only one formed by that enterprising people.

The Arabian Gulf, or Red Sea, offered to the Phoenicians another field of naval and commercial exertion, to the improvement of which the distance of Tyre, the emporium of their trade, was the only obstacle. This induced them to make themselves masters of Rhinocurra or Rhinocolura, the port in the Mediterranean nearest to the Red Sea. Commodities purchased in Arabia, Ethiopia, and India were landed at Elah, the safest harbour in the Red Sea towards the north; thence they were conveyed over land to Rhinocolura; and being there reshipped, they were carried to Tyre, whence they were distributed over the world.

The wealth and power which accrued to the Phoenicians from their being in the sole possession of the lucrative trade of the East, incited the Jews, their neighbours, under the prosperous reigns of David and Solomon, to desire a participation in its advantages. Their conquest of Idumea, which stretches along the Red Sea, put it in the power of Solomon to fit out a fleet; while his alliance with Hiram, king of Tyre, enabled him to command the skill of the Phoenicians for the conducting of the voyage. Passing through the Straits of Babelmandel, they carried on commerce in the Indian Ocean; and so distant were the countries to which they traded, that the voyage occupied no less than three years. But though the Jews thus for a time engaged in the pursuits of trade, yet the tendency of their institutions, which were expressly designed to preserve them a separate people, was unfavourable to the development of the commercial spirit which their monarchs wished to foster among them. This joined with the division of the kingdom on the death of Solomon, proved fatal to their rising greatness as a commercial people, and excluded them from ranking among the nations who have contributed to the advancement of geographical knowledge.

It is perhaps impossible to fix with certainty the limits which bounded the geographical researches of the Phoenicians, on account of the difficulty there is of assigning the precise places marked out by the names then given to the countries to which they traded. The length of time occupied in the voyage, and the nature of the cargoes brought home, with a few other circumstances of the same vague kind, are the only particulars afforded to direct us in the determination. Thus the country of Ophir, to which the Phoenicians navigated the ships of Solomon, must be ascertained by the facts that the voyage thither and homeward occupied three years, and that the cargo consisted of "gold and silver, ivory, and apes, and peacocks." Among the various opinions which have been entertained respecting the position of this distant country, the most probable appears to be that it was situated on the eastern coast of Africa, as far south as Sofala. To this quarter every indication seems clearly to point; and whatever objections may appear to stand in the way, in consideration of the remoteness of the region, and the difficulties to be encountered, these admit of being answered by a reference to the length of time required for the voyage, and to the wealth, naval skill, and ample resources, at the command of the monarchs engaged in the traffic.

The Carthaginians, a Phoenician colony, retained in full vigour the commercial spirit of the parent state. They did not, however, attempt to divide with Tyre the wealth and power which she derived from the monopoly of the trade carried on in the Arabian Gulf. They directed their efforts to the opposite quarter, and sailing through the Straits of Gades, pushed their researches far beyond the bounds which had been reached by the mother country in this part of the globe. They visited not only all the coast of Spain, but likewise that of Gaul, and penetrated at length as far as the south-western coast of Britain, where they obtained tin from the mines of Cornwall or in traffic with the natives. Nor was it only towards the northward that they directed their efforts; they explored also the regions southward of the straits, and sailing along the western coast of Africa almost as far as the northern tropic, they planted colonies, as the Phoenicians of Tyre had formerly done, with a view to prepare the natives for carrying on commercial intercourse. The Atlantic Ocean was destined to conceal for ages from the inhabitants of the old world the immense regions which lie beyond it. But the Carthaginians extended the boundary of navigation westward by the discovery of the Fortunate Islands, now known by the name of the Canaries.

The enlargement of views gradually generated by this spirit of commercial enterprise led at length to voyages of which discovery was the special object. The circumnavigation of Africa was one of the earliest attempts of this

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1 The regions always spoken of in Scripture as the most remote with which the Hebrews and Phoenicians were acquainted are, Tarshish, Ophir, the Isles, Sheba, and Dedan, the River, Gog, Magog, and the North. Without entering into any discussion, we may give what appear to be the most probable conclusions with regard to the positions of the countries to which these names were applied. Tarshish is a country from which two voyages are spoken of in Scripture as being made; one by the Mediterranean, bringing iron, silver, lead, and tin, the produce of Spain and Britain; the other by the Red Sea, bringing gold, ivory, and other productions of tropical Africa. These two voyages, though at first sight they appear incongruous if supposed to be made to the same country, may be reconciled by supposing that Tarshish is fundamentally Carthage, which monopolized almost entirely the commerce of Spain and Britain, and was the medium through which the commodities of the west were distributed; and that the name of this great African metropolis was extended to the whole of the continent of Africa. The Isles are the whole southern coast of Europe, consisting either of real islands or peninsular tracts. Sheba is the southern portion of that part of the coast of Arabia which borders on the Red Sea; while Dedan lies upon the opposite coast, that borders on the Persian Gulf. These countries rose to commercial importance in consequence of the valuable commodities which were imported into the former from the African coast, and into the latter from India. Thence arose the traffic carried on by "the companies of Sheba," or caravans, and by "the travelling companies of Dedanim." The River was the name always applied to the Euphrates. Gog, Magog, and the North, appear to be the high table-land in the interior and north of Asia Minor, Phrygia, Galatia, Cappadocia, and Paphlagonia, regions in which may be recognised the peculiarly rude and formidable aspect which belonged to the countries to which in ancient times the names in question were applied. (See Encyclopedia of Geography, by H. Murray, Esq.) The direction which the coast takes beyond the Mediterranean on the one hand, and the Red Sea on the other, suggested the idea of a peninsula which it might be possible to sail round. This voyage was first undertaken by the Egyptians; a people exceedinglyverse in naval affairs, but who at this time were led over by Necho, a monarch whose active spirit prompted him to engage some Phoenicians to descend the Arabian Gulf, and, coasting along Africa, to endeavour to return by the Straits of Gades. Herodotus narrates in a few words the result of this enterprise, which was undertaken about a hundred and four years before the Christian era. He says, "the Phoenicians, setting sail from the Red Sea, made their way into the southern sea; and when autumn approached they drew their vessels to land, sowed a crop, and waited till it was grown; when they reaped it, and put to sea. Having spent two years in this manner, the third year they reached the pillars of Hercules, and turned to Egypt, reporting what does not find belief with us, but may perhaps with some other person; for they said that in passing Africa they had the sun on their right hand. In this manner Libya was first known."

This passage has given rise to much controversy among the learned. But the voyage here so briefly described does not seem to involve any impossibility, notwithstanding its infant state of navigation; and the circumstance which the historian objects to as incredible is the very point which, from its coincidence with what we know should have happened, renders the story more worthy of belief.

Xerxes king of Persia, according to Herodotus, gave a solar commission, about four hundred and eighty years before the Christian era, to one of his satraps, named Sataspes, who, for a heinous offence, had been condemned to death. If successful in the accomplishment of this voyage, Sataspes was to escape a cruel death; but the difficulties were too great to be surmounted by a navigator brought up amidst the luxury and indulgence of the Persian court. Having procured from Egypt a vessel and crew, he passed through the Straits of Gades, entered the Atlantic Ocean, sailed, bending his course towards the south, coasted the continent of Africa, until, after several months, he probably reached the coast of Sahara. The frightful and desolate shores along which he sailed, and the tempestuous ocean which beat against them, combined to fill his mind with terror, and to shake his resolution. He retraced his course through the straits; and hoping perhaps that the labours he had undergone in the partial accomplishment of the task imposed on him would be accepted by his royal master as a sufficient atonement for his offence, or that the offence itself might in a great measure be forgotten, he returned home and presented himself before Xerxes. The cause which he assigned for the failure of the ultimate object of his mission was, that he had encountered a sea so full of herbage that his passage was completely obstructed. This reason (the grounds of which have never been satisfactorily explained, though it has been alleged that obstacles of this description occur in that part of the sea which lies between Cape Verde Islands, the Canaries, and the coast of Africa) appeared so ridiculous to Xerxes, that he ordered the sentence of death by crucifixion, which had been pronounced upon Sataspes, to be immediately executed.

But the most celebrated voyage of antiquity undertaken for the purpose of discovery was the expedition under Hanbro fitted out by the authority of the senate of Carthage, and at the public expense, and that with the view of attempting a complete survey of the western coast of Africa. Of all the voyages performed by the Phoenicians and Carthaginians, this is the only one of which we have an authentic narrative. Mercantile jealousy prevented these two great commercial states from communicating to other nations the knowledge which they acquired of the remote regions of the earth; and from this cause, when the maritime power of Tyre, and the empire of the latter was overthrown by the Roman arms, all monuments of their great skill in naval affairs appear in a great measure to have perished. Even the account of the voyage of Hanno (Periplus Hannonis) has been considered by its learned editor Mr Dodwell as a spurious work. But the arguments of M. de Montesquieu and of M. de Bougainville appear fully to establish its authenticity, which the learned world now generally admits.

Hanno set sail with a fleet of sixty vessels, so constructed that, according to the mode of ancient navigation, he could keep close in with the coast. We are told that, in twelve days after leaving the Straits of Gades, he reached the island of Cerne; that proceeding thence, and following the direction of the coast, he arrived, in seventeen days, at a bay, which he called The West Horn. From this he advanced to another bay, which he named The South Horn. The objects which are described as having been seen by Hanno in his progress belong to tropical Africa. But in attempting to ascertain the places which he visited, or the utmost distance which he sailed southward, much difficulty and uncertainty are experienced. Bougainville supposes Hanno to have reached the Gulf of Benin, and contends that this limit, distant as it is, cannot be regarded as beyond what may be conceived to have been accomplished by the most skilful navigator of antiquity. Major Rennell shortens the distance considerably by conceiving the voyage to have been extended no further southward than Sherbro Sound, a little beyond Sierra Leone. He thus obtains the advantage of avoiding a difficulty involved in the hypothesis of M. de Bougainville; namely, the supposition of ancient ships having sailed upwards of seventy geographical miles in a day. At the same time, the arguments which support the one hypothesis are equally applicable to the other.

According to the views of M. Gosselin, however, the voyage must be confined to much narrower limits southward than even those assigned by Major Rennell. He supposes it to have terminated about the river Nun; an opinion which he supports by alleging that, in such a voyage, the progress must necessarily have been slow. The Carthaginian navigator had to encounter all the obstacles and dangers incident to a course held along a shore, and in a sea, which were equally unknown. He must have found himself impeded by the requisite examination of every part of the coast, as well as by the many precautions which the safety of the fleet under his command must have rendered constantly necessary. With regard to the circumstances given in the narrative which appear to point to tropical Africa, M. Gosselin supposes that the same aspect of life and nature may, at that distant period, have belonged to Morocco, then thinly peopled by the rude native tribes, which is now specially characteristic of more southern regions.

Amidst such diversity of opinion among the learned, it is not easy to decide in reference to a subject beset with so many difficulties. If we assume either of the more remote distances assigned for the termination of the voyage, Cerne must be identified with the isle of Arguin; and, on Major Rennell's hypothesis, the Gulfs of Bissago and Sherbro present those numerous islands described by Hanno, to which there are no islands corresponding on any other part of the coast. On the whole, however, the most limited distance seems preferable, if we admit that part of M. Gosselin's hypothesis which assigns to Morocco features of man and of nature that are usually held to be characteristic of tropical Africa.

The circumnavigation of Africa was an enterprise which in ancient times not only called forth the naval efforts of the most powerful maritime states, but which also awakened... ed the ambition of private adventurers. Eudoxus, a native of Cyzicus, being sent on a mission to Alexandria, at that time the seat of naval enterprise and geographical knowledge, his ardent mind, naturally biased to these pursuits, was aroused to action by the spirit which prevailed in that city. He began his career under the auspices of Ptolemy Euergetes, the reigning Egyptian monarch, who fitted out a fleet, and placed it under his command. According to the destination assigned him, Eudoxus descended the Arabian Gulf, and proceeded probably as far as the southern shore of Arabia. Thence he appears to have returned, after a prosperous voyage, with a valuable cargo of aromatics and precious stones. But of this wealth he appears to have been deprived by Euergetes. After the death of this monarch, which in a short time took place, his widow Cleopatra sent Eudoxus on another voyage, in the course of which he was driven by unfavourable winds on the coast of Ethiopia, where he was kindly received by the inhabitants, and carried on with them an advantageous traffic. After other vicissitudes of fortune, he was induced by circumstances which occurred in his adventurous life to leave the court of Egypt, and repair to the commercial city of Cadiz, in Spain, and there to fit out an expedition for the purpose of African discovery. At Massilia (Marseilles), and other maritime places through which he passed on his way to Cadiz, he took care to make known his views and hopes of success, and to invite all who were actuated by any spirit of enterprise to accompany him. He succeeded in fitting out a ship and two large boats; on board of which he carried not only goods and provisions, but artisans, medical men, and even players on musical instruments. This was no doubt proceeding on a magnificent scale. But his crew was ill calculated to second his bold undertaking. To avoid the danger of stranding, Eudoxus was anxious to keep the open sea. His companions, however, alarmed at the swell, forced him to adopt the usual mode then followed of sailing along the shore; a measure which led to the disaster which he had anticipated. With one vessel of a lighter construction, on board of which was put the more valuable part of the cargo, Eudoxus pursued his voyage until he reached a part of the coast inhabited by a race of people that appeared to him to speak the same language with those whom he had found on the opposite side of the continent. Judging from this circumstance that he had ascertained the main object of his voyage, he returned and endeavoured to obtain the assistance of Bocchus, king of Mauritania. Suspecting, however, treachery on the part of that monarch, he again had recourse to Spain. Here he was again successful in equipping another expedition, consisting of one large vessel fitted for the open sea, and another of smaller size for the examination of the coast. This was a judicious preparation for the accomplishment of the object in view; but with regard to the issue of the voyage no accounts of any authority have been preserved.

Such are the leading circumstances connected with the voyages of Eudoxus, which are narrated by Strabo; and, notwithstanding the scepticism and severe criticisms of that geographer, there is really nothing to which the candid reader can reasonably refuse his belief. Prejudices, founded, for the most part, on his own want of information, led Strabo to treat likewise as fabulous the relation of the only ancient voyage having Europe, and more particularly the British isles, for its object, of which we have any detailed account.

Pytheas, a Massilian navigator, undertook an expedition about three hundred and twenty years before the Christian era. He steered northward; and after examining the coasts of Spain and Gaul, he sailed round the island of Albion; and, stretching still farther to the north, he discovered an island, the *Ultima Thule* of the ancients. What island this was the learned are not agreed. It has been supposed to be the modern Iceland; but this implies too great an extent of open sea for an ancient navigator to traverse; and besides, six days, the period during which he is said to have navigated to the northward of Albion before he made his discovery, are too short a time to admit of his reaching Iceland. Others, amongst whom is Malte-Brun, have considered Jutland as *Ultima Thule*. But it should be kept in view that Pytheas uniformly regarded Thule as British, a character which he could scarcely conceive to belong to Jutland, seeing he could have reached that peninsula only by a long course along the coasts of Germany, which must have impressed on his mind the idea that he had left far behind him everything belonging to Britain. On the whole, Shetland seems best entitled to be considered as the ancient Thule, and suits well with the appellation which Pytheas gives it, when he expressly calls it the "furthest of the Britains."

Strabo endeavours to throw discredit on the statements of Pytheas, by starting objections long known to be of the most groundless description; and it is an advantage which the traveller and navigator possess who describe faithfully the grand features of nature, that, however prejudice may dim their reputation for a time, yet will their accuracy as well as veracity at length, in the progress of knowledge, appear, and secure for them the respect and applause of mankind. At the same time, it must be admitted that, in describing what he saw beyond his *Ultima Thule*, the statement given by Pytheas, as reported by Strabo, assumes a somewhat fabulous character. He asserted, it seems, that beyond Thule there commenced what was neither earth, sea, nor air, but a confused blending of all the three. But even here some allowance is to be made for the workings of imagination under very peculiar circumstances, and a readiness, not unnatural, to believe reports which represented him to have reached the extremity of the habitable globe. If his language is not too literally interpreted, it will be found to convey a strongly figurative, but not altogether imperfect, description of the state of the sea and sky in these climes, which have been so little favoured by nature.

The conquests of Alexander the Great, by making known the East, enlarged the bounds of geographical knowledge. Though the course of his expedition was for the most part by land, his mind was equally intent on commerce and maritime discovery. Checked as he had so long been in the career of his victories by the opposition and efforts of the republic of Tyre, he had an opportunity afforded him of observing the vast resources of a maritime power, and at the same time of forming a judgment respecting the immense wealth to be derived from commerce, especially from that carried on with India, which he found to be wholly in the hands of the Tyrians. With a view to secure this commerce, as soon as he had completed the conquest of Egypt, he founded the city of Alexandria, and thus established for it a station preferable in many respects to Tyre. After his final victory over the Persians, his march in pursuit of Bessus, who had carried off Darius into Bactriana, often led him near to India, and among people accustomed to much intercourse with it, from whom he learned many things concerning the state of the country, that served so to confirm and inflame a desire which he had long cherished of extending his dominion over those regions, that he was induced to conduct his army from Bactria, for the purpose of invasion, across that ridge of mountains which form the northern barrier of India. After passing the Indus, Alexander directed his march to the Ganges, which, from the accounts he heard of it, and of the countries through which it flows, he was eager to reach. The route which he found it necessary to follow, in consequence of being successively engaged in hostilities with various native princes, led him through one of the richest and best peopled countries in India, now called Punjab. In his ultimate object, however, he failed. Is march being performed during the rainy season, his tops had already suffered so much, that notwithstanding the high degree in which he possessed all those qualities that secure an ascendency over the minds of soldiers, he was unable to persuade them to advance beyond the banks of the Hyphasis, the modern Beyah, which was according- ly the utmost limit of Alexander's progress in India.

By this expedition, Alexander first opened the knowl- edge of India to the people of Europe; and as he was ac- companied, wherever he went, by skilful surveyors, Diog- nes and Baeton, who measured the length and determin- ed the direction of every route taken by the army, he fur- nished a survey of an extensive district of it, more accurate than could have been expected from the short time he re- mained in that country. The memoirs drawn up by his officers likewise afforded to Europeans their first authentic information respecting the climate, the soil, the produc- tions, and the inhabitants of India.

Though Alexander did not penetrate to the Ganges, his expedition prepared the way to the knowledge of that mag- nificent stream. For soon after, Seleucus, one of his suc- cessors, sent Megasthenes as his ambassador to Palibothra, capital of a powerful nation on the banks of the Gan- ges. The site of Palibothra was probably the same as that of the modern city of Allahabad, at the junction of the river Jumna with the Ganges. This embassy brought new opulent provinces of India into view, an acquaintance which served to raise still higher the idea generally entertained of the value and importance of the country.

The island Taprobane, so celebrated among the ancients, which appears, notwithstanding some great mistakes with respect both to extent and position, to be the modern Ceylon, seems not to have been known in Europe even by name before the age of Alexander. In consequence, how- ever, of the enlightened and active curiosity with which he explored every country which he subdued or visited, the knowledge of it was at length obtained; and, after some time, it is mentioned by almost every ancient geogra- pher.

Whilst Alexander was attempting to penetrate into In- dia, a numerous fleet was assembled by officers whom he had left on the banks of the Hydaspes, the modern Behat or Chelum, with orders to build and collect as many ships as they could. The destination of this fleet was to sail down the Indus to the ocean, and from its mouth to pro- ceed to the Persian Gulf, with a view of opening a com- munication between India and the centre of his dominions.

When Alexander reached the banks of the Hydaspes on his return, he committed the conduct of this expedi- tion to Nearchus. The voyage down the Indus derived its honour from the greatness and magnificence of the ar- mament, which consisted of an army of a hundred and twenty thousand men, and two hundred elephants, and of a fleet of nearly two thousand vessels. Alexander him- self accompanied Nearchus in his navigation down the river, with one third of the troops on board; whilst the remainder, in two divisions, one on the right and the other on the left of the river, accompanied them in their pro- gress. Having reached the ocean after the lapse of nine months, Alexander left Nearchus and his crew to pursue their voyage, and conducted his army back by land to Persia. A coasting voyage of seven months brought Nearchus, with the fleet, in safety, up the Persian Gulf to the Euphrates. It was at the mouth of the Indus that the Greeks witnessed for the first time, and that with amazement and terror, the ebb and flow of the sea; a phenomenon scarcely perceptible in the Mediterranean, to which their navigation had formerly been confined. In the progress of the voyage they were also struck with sur- prise on observing phenomena belonging to the midsum- mer of the tropics. At noon objects were observed to project no shadows, or to project small shadows declining History. to the south. Their attention was still further attracted by the new appearance of the sky. Stars which they had been accustomed to see high in the heavens were now seen near the horizon. Some stars to the north dis- appeared, while other stars formerly invisible were seen in the south.

The opening of a communication between the Red Sea and the Persian Gulf was with Alexander another great object of ambition. But though with this view he seems to have sent expeditions down both seas, he failed in his attempts to accomplish this project.

Among the Romans, navigation and commerce, the handmaids of geographical science, were never made ob- jects of pursuit, except in so far as they were found to be necessary to forward their schemes of universal dominion. Their conquests opened indeed the west, as those of Alexander had made known the east; and it might be truly said of that great people, that as they were the con- querors, so they were the surveyors of the world. Every new war produced a new survey and itinerary of the coun- tries which were the scenes of action; so that the mate- rials of geography were accumulated by every additional conquest. Some fragments of the itineraries thus com- posed still remain. The most memorable is that which bears the name of Antoninus, and which may be described as a mere skeleton road-book, exhibiting nothing more than the names of places, and their distances from each other. The Jerusalem Itinerary, which details minutely the route from Bordeaux to that holy city, is of the same description.

A more remarkable monument, however, is the Peu- tingerian Table, which forms a map of the world, con- structed on the most singular principles. The map is twenty feet long and only one foot broad, so that we can easily conceive how incorrectly the proportion of the dif- ferent parts is exhibited. Along the high road which traverses the Roman empire in the general direction of east and west, objects are minutely and accurately repre- sented; but of those objects which lie to the north and south of it, only some general notion is conveyed. The Peutingerian Table serves as a specimen of what were called Itineraria Picta, the "painted roads" of the ancients, intended for the clearer direction of the march of their armies.

While the Romans by their surveys contributed much to increase the mass of materials out of which the struc- ture of geographical science was to be reared, they never attempted themselves to combine these materials into one harmonious system. They imbibed in no degree the commercial spirit of the great maritime states of the an- cient world, Carthage, Greece, and Egypt, which their valour and discipline obliged to submit to their dominion. But whilst the trade of the conquered countries continued to be carried on through nearly the former channels after they were reduced to the form of Roman provinces, the wealth accumulated in the capital of the world gave rise to a demand for luxuries of every description. This, com- bined with the comparative peace and security which for a long time prevailed after the complete establishment of the Roman dominion, gave new vigour to commercial en- terprise. Alexandria continued the great centre of na- val affairs. Obstacles which in the time of Alexander were deemed insurmountable, were completely overcome. Trade with India through Egypt acquired new energy, and was carried on to a greater extent. Continued in- tercourse with the shores of India at length made known to navigators the periodical winds which prevail in the Indian Ocean; and taking advantage of these, pilots were emboldened to abandon the slow and dangerous course along the coasts, and to make the open sea their high- Their course was from Ocelis, at the mouth of the Arabian Gulf, to Nelkunda (Nelisuram), on the western shores of the Indian continent (the coast of Malabar), which seems to have been the utmost limit of the ancient navigation in that quarter of the globe. The extensive regions which stretch beyond this to the east were very imperfectly known by the reports obtained from a few adventurers who visited them by land.

If we now bring into one view the amount of information possessed by the ancients respecting the habitable globe, we shall find that it was extremely limited. It was at those places on the earth where the human mind displayed greatest activity and enterprise that this knowledge was naturally accumulated. Proceeding from these stations, the boundary which separated the known from the unknown part of the world was gradually enlarged; but the regions comprehended within it constituted still but a small portion of the whole. In Europe the extensive provinces in the eastern part of Germany were but little known, while the whole of that vast territory which now forms the countries of Denmark, Sweden, Prussia, Poland, and Russia, was buried in the deepest obscurity. The inhospitable and dreary climes within the arctic circle were yet unexplored. In Africa little was known beyond the countries stretching along the Mediterranean Sea, and those bordering on the western shore of the Arabian Gulf. In Asia the rich and fertile countries beyond the Ganges, whence the commerce of modern times has drawn the most valuable commodities for the comfort and embellishment of civilized society, were known, if known at all, only by the most vague and uncertain reports. The immense regions on the north occupied by the wandering tribes called in ancient times by the general names of Sarmatians or Scythians, and which are now inhabited by various tribes of Tartars, and by the Asiatic subjects of Russia, seem never to have been penetrated. Add to this, that the fertile and populous regions within the torrid zone were imagined to be uninhabitable; and we have ample proof that the geography of the ancients was very imperfect.

Having thus far given a succinct view of the progressive steps by which the earth's surface, considered merely as tracts of territory inhabited by men, gradually became known, it will be proper next to trace briefly the advances made towards arranging into a systematic form the materials accumulated. Science required that the relative positions of places, with their distances from each other, should be ascertained in such a manner as to furnish fixed principles on which the whole, or any portion, of the surface of the earth might be represented or delineated with due regard to its figure and dimensions.

The first rude attempt made by the early geographers to determine the position of places appears to have depended on the division of the earth into climates, distinguished by the species of animals and plants produced in each. Thus the appearance of the negro, the rhinoceros, and the elephant, suggested to them the line of division where the torrid zone began towards the north, and ended towards the south. But instead of this very vague method, another was soon adopted, which consisted in observing at places the length of the longest and shortest day. This was determined with some accuracy by means of a gnomon, a method of observation much used by the ancients. An upright pillar of a known height being erected on a level pavement, by observing the lengths of the meridian shadows, they were enabled to trace the progress of the sun from tropic to tropic. The most ancient observation with the gnomon which we meet with is that of Pytheas, in the days of Alexander the Great. Pytheas observed at the summer solstice at Marseilles, that the length of the meridian shadow was to the height of the gnomon as 213\(\frac{1}{4}\) to 600; an observation which makes the meridian altitude of the sun at Marseilles on that day 70° 27'. The merit of the invention of the gnomon in Greece is ascribed to the astronomical school of Miletus, and particularly to Anaximander and Anaximenes. There is reason, however, to believe that this method of observation was originally invented by the Egyptians; and that Thales, who travelled into Egypt, carried thence the knowledge of it into Greece. It has even been conjectured that the Egyptian pyramids and obelisks were intended for the same purpose with the gnomon; and though it would be extravagant to imagine that this was their sole use, this opinion appears to be countenanced by the fact of their being placed in the direction of the four cardinal points.

The determination of the length of the meridian shadow at the solstices for different parts of the earth, by observations made with the gnomon, is important as being the first step towards connecting geography with astronomy; and, when combined with just conceptions of the globular figure of the earth, leads, by a simple train of thought, to the notion of latitude by which the position of a place is fixed relatively to north and south. The position with regard to east and west is the only other element necessary for fixing the absolute situation of the place on the surface of the earth. It might have been supposed not to be more than a reflecting mind could easily accomplish to reach the conception of both these elements, and to apply them to use. Yet so slow was the progress towards the apprehension of the principles on which an accurate system of geography might be founded, that from the days of Thales and his immediate successors, who flourished in the sixth century before the Christian era, there appears to have been little done for the improvement of geography, as a science, until the establishment of the famous astronomical school of Alexandria. Pythagoras had indeed maintained the true system of the world, by placing the sun in the centre, and giving the earth both a diurnal and annual revolution; but this doctrine was so much in advance of the age in which he promulgated it, that it was soon lost sight of.

Eratosthenes was the first who reduced geography to a regular system, and laid its foundations on clear and solid principles. Under the patronage of the Ptolemies, he had access to the materials collected by Alexander, his generals and successors, as well as to the immense mass of documents accumulated in the Alexandrian library. At an early period of the history of astronomical science, the vulgar opinion that the earth is a flat surface, with the heavens resting upon it as a canopy, was rejected; but it was not at once that distinct conceptions of its globular figure were acquired. It was only as astronomical observations increased that the doctrine of its sphericity was fully established. This point had been gained when Eratosthenes began his labours; and what he endeavoured to accomplish was to delineate, in strict conformity with this principle, the known parts of the earth's surface.

With this view, founding his system on the use of the gnomon, he supposed a line to be traced through certain places, in all of which the longest day was known to be exactly of the same length. This line would evidently be a parallel to the equator. But though his method was correct in principle, the want of accurate observations rendered it uncertain in practice. The line was supposed to comprise all the leading positions which lay near it, though they did not actually come within its range. Its western extremity was the Sacred Promontory of Ibera (Cape St Vincent); thence it passed through the Straits of Gades. Proceeding eastward, it passed through the Sicilian Sea, and near the southern extremity of the Peloponnesus, and was continued through the island of... induced M. Gosselin to conclude that they were to be attributed, not to the imperfection of independent observations, but to some general cause, which he endeavoured to assign by imagining that Eratosthenes had access to some early map, found probably by Alexander or his generals in some country in the East, where astronomy had been successfully cultivated; and that misapprehensions respecting the principle of delineation employed, which M. Gosselin supposes to have been that on which the plane chart is constructed, had led him into a regular system of errors. In the plane map the length of a degree of longitude is supposed to be the same at all distances from the equator. By taking for granted that Eratosthenes took his distances from a map of this kind, on which the parts of the globe had been accurately laid down, but that he divided the stadia expressing these distances, not by 700, the number of stadia in a degree at the equator, as he ought to have done, but by 555, the number corresponding to the parallel of Rhodes, M. Gosselin obtains results which have a wonderful coincidence with the positions actually given by Eratosthenes. These results, however, are deduced from a hypothesis which is unsupported by any evidence, except what may be supposed to arise from this coincidence. A more probable solution seems to be that Eratosthenes determined his longitudes from the itinerary measures, which he reduced to degrees at the rate of 700 stadia to a degree at the equator, and of 555 to a degree at the parallel of Rhodes; and that the errors are the consequences of the exaggerated accounts which merchants and travellers of that age gave of the distances over which they passed—-their exaggerations, of course, bearing some proportion to the length and hardships of the journeys undertaken.

The knowledge as yet possessed by geographers with regard to the outline of the habitable globe was far from being such as to enable them to delineate it with any degree of precision. This circumstance, combined with the unavoidable errors in latitude and longitude, produced very great distortions in the representations given of the countries on the surface of the globe. Under the guidance of sound principles of science, however, it was now certain that these imperfections would gradually disappear.

The improvements introduced into geography by Eratosthenes were perfected in principle by Hipparchus. This celebrated astronomer, who flourished between a hundred and sixty and a hundred and thirty-five years before the Christian era, was the first who undertook the arduous task of forming a catalogue of the stars, and fixing their relative positions. His object was to transmit to posterity a knowledge of the state of the heavens at the period of his observations. The extremities of the imaginary axis round which the heavens perform their diurnal revolution suggest two fixed points by which the position of the great circle of the celestial sphere called the equator is determined. If a great circle be supposed to pass through these points and any star, the position of the star will be ascertained if we measure in degrees and parts of a degree the arch of the meridian circle intercepted between the star and the equator, and also the arch of the equator intercepted between a given point in it, and the meridian circle passing through the star. Upon this principle did Hipparchus arrange the stars according to their places in the heavens, a work in which he appears, however, to have been in some measure anticipated by Timocharis and Aristillus, who began to observe about two hundred and ninety-five years before the Christian era. The great improvement which he introduced into geography consisted in this, that he applied to the determining of the position of any point on the surface of the earth the same artifice which he had already so happily introduced in the arrangement of the constel-

Vol. X. lations; and thus furnished the means of ascertaining the relative situations of places with a precision which no itinerary measurements could possibly attain. If we suppose the earth to be a globe concentric with the celestial sphere, and intersected by the planes of the celestial equator and meridian, the principle on which the application of this artifice to the terrestrial sphere depends becomes at once obvious. Hipparchus made a considerable number of observations of latitude, and pointed out how longitudes might be determined by observing the eclipses of the sun and moon. Great as this improvement was, its importance seems not to have been duly estimated until the days of Ptolemy; for none of the intermediate authors, such as Strabo, Vettius, and Pliny, have given the least hint of the latitude and longitude of any one place in degrees and minutes, though all of them have given minutely the geographical position of places according to the length and shadows of the gnomon. Strabo, indeed, even justifies his neglect of the astronomical principles introduced by Hipparchus. "A geographer," says he, "is to pay no attention to what is out of the earth; nor will men engaged in conducting the affairs of that part of the earth which is inhabited, deem the distinction and divisions of Hipparchus worthy of notice."

The true principles of geography being pointed out by the application of latitude and longitude to places on the earth, the way was opened for the improvement of maps, which, with the single exception of the map drawn by Eratosthenes, had hitherto been little more than rude outlines and topographical sketches of the different countries.

No maps more ancient than those formed to illustrate Ptolemy's geography have reached modern times; but the earliest of which there is any account are those of Sesostris, of whom it is said, that having traversed great part of the earth, he caused his marches to be recorded in maps; and that he gave copies of these maps not only to the Egyptians, but to the Scythians, whose astonishment he thus greatly excited.

Some have imagined that the Jews made a map of the Holy Land when they gave the different portions to the nine tribes at Shiloh. For on that occasion, as we are informed by the sacred historian, men were sent "to walk through the land, and to describe it;" and when they had accomplished the object of their mission, by describing "it by cities into seven parts, in a book," they returned unto Joshua. What is here said, however, does not fully determine whether their mensuration of the land was only recorded in numbers, or regularly projected and digested into the form of a map.

The first Grecian map on record is that of Anaximander, mentioned by Strabo, which some have conjectured to have been a general map of the then known world. It has further been imagined to be the same with that referred to by Hipparchus under the designation of The Ancient Map, and which in some few particulars he preferred to that of Eratosthenes.

But some idea of the nature of the maps of those early days will be best obtained from the map of Aristagoras, king of Miletus, which is minutely described by Herodotus. The historian tells us that this map, which was traced on brass or copper, Aristagoras showed to Cleomenes, king of Sparta, in order to induce him to attack the king of Persia, even in his palace at Susa, for the purpose of restoring the Ionians to their ancient freedom. It contained the intermediate countries to be traversed in that march. We must not interpret, however, the words of Herodotus too literally, when he describes it as containing "the whole circumference of the earth, the whole sea or ocean, and all the rivers." Keeping in view the state of geography at that period, it may justly be concluded, that notwithstanding this pompous form of expression, the sea meant only the Mediterranean, and therefore the earth or land the coasts of that sea, and more particularly Asia Minor, extended towards the middle of Persia: by the rivers must be meant the Haly, the Euphrates, and the Tigris, which Herodotus mentions as necessary to be crossed in the expedition in question. The map contained one straight line, called the royal highway, embracing all the stations or places of encampment between Sardis and Susa, so that it was properly an itinerary.

The principle on which Eratosthenes constructed his map we have already considered. With regard to its extent, it seems to have contained little more than the states of Greece, and the dominions of the successors of Alexander, digested from the surveys of the marches of that great general. He had some faint idea respecting the western parts of Europe, which he had acquired from the voyage of Pytheas; but not such a conception as to enable him to delineate their outline on a chart. According to the report of Strabo, he was quite unacquainted with Spain, Gaul, Germany, and Britain, he was equally ignorant of Italy, the coasts of the Adriatic, Pontus, and of all the countries toward the north. His errors with regard to the distances of places were in some instances enormous. The distance of Carthage from Alexandria he represents at 15,000 stadia, instead of 9000.

It was not until Ptolemy commenced his labours that the improvements pointed out by Hipparchus were actually applied to perfect the system which Eratosthenes had so happily begun. Ptolemy composed his system of geography, which escaped amidst the general wreck which has consumed so many other ancient books of science, in the reign of Antoninus Pius, about 150 years after the opening of the Christian era. At this period the Roman empire had reached its utmost extent, and all the provinces had been surveyed, and were well known. The materials then in existence, and in the possession of Ptolemy for completing his great work, were the proportions of the height of the gnomon and its shadow, at the time of the equinoxes and solstices, taken by different astronomers; calculations founded on the length of the longest day; the measures or computed distances of the principal roads contained in the surveys and itineraries; and the various reports of travellers and navigators, whose determinations of the distances of places often rested, however, on no better foundation than hearsay and conjecture. Among these various particulars, there evidently existed considerable differences in point of authority. But Ptolemy undertook the difficult and laborious task of comparing, and reducing into one system, which should possess the order and beauty of science, this apparently incongruous mass. He converted and translated the whole into a new mathematical language expressing in degrees and minutes the latitude and longitude of each place, according to the principles laid down by Hipparchus, but which had been allowed by geographers to lie useless for upwards of two hundred and fifty years. It is in Ptolemy's work, which consists of eight books, that we find for the first time the mathematical principles of the construction of maps, both general and particular, as well as of several projections of the sphere.

Notwithstanding that the light of accurate science thus directed the steps of the first geographer of antiquity, he was far from reaching the precision at which he aimed. This arose from the imperfection of the original materials upon which his work is based in reference to its details. With regard to the remoter boundaries of the then known world, in all its quarters, a wonderful advancement in knowledge had been made since the days of Eratosthenes and of Strabo. But still that additional information was not fitted to make up for the want of astronomical observations, by which alone accuracy could be secured. Besides, in relation to places situated beyond the limits of the Roman The great obstacle with which the ancients had to contend was the finding of the longitude with accuracy, a problem for the solution of which it was long before there was discovered any method sufficiently exact. This accounts for the erroneous longitudes of Ptolemy, and more especially for the length of time, even many centuries, during which the remarkable error with regard to the length of the Mediterranean remained undiscovered and uncorrected.

We have now traced the history of geography from the earliest period of which we have any information, to the time when it assumes a scientific character. We shall conclude our account of ancient geography by shortly noticing the principal geographers of antiquity, some of whom have not yet been mentioned, while others have only been quoted in tracing the rise and progress of the science.

The intimate connection between geography and the sciences of geometry and astronomy, rendered the former an object of attention to many who anciently cultivated the latter. We have already mentioned Anaximander and Anaximenes, of the school of Miletus. Democritus, Eudoxus of Cnidus, and Parmenides, are also reported to have improved geography; and to the last is attributed the division of the earth into zones. These were followed by Eratosthenes, who lived about two hundred and forty years before the Christian era; by Hipparchus about eighty years afterwards; by Polybius, Geminos, and Posidionius. Eratosthenes wrote three books on geography, some passages of which Strabo criticises, though he frequently defends him against Hipparchus, who appears to oppose his opinions with some degree of affectation. Polybius also wrote on geography; as did likewise Geminos and Posidionius, who are frequently quoted by Strabo. Polybius, according to Geminos, argued with considerable acuteness for the possibility of the torrid zone being inhabited; and he even adduced plausible arguments to prove that the countries under the equator enjoy a more temperate climate than the countries that are situated near the tropics.

We must not here omit a geographer and geometer who lived about the time of Alexander the Great. This was Dicearchus of Messina, a disciple of Theophrastus, who wrote a description of Greece in iambic verses, of which some fragments yet remain. But what chiefly renders him remarkable is, that he measured geometrically several mountains, to which an excessive height had been before assigned.

With Dicearchus we may notice another geometer, Xenagoras, a disciple of Aristotle, mentioned by Plutarch in his life of Paulus Emilius, who occupied himself in the measurement of mountains. He found the height of Mount Olympus to be fifteen stadia.

In some of the latter periods which preceded the Christian era there were several writers on geography, as Artemidorus of Ephesus, who wrote a geographical work of eleven books, of which nothing remains; Scymnus of Chio, author of a description of the earth in iambic verses, which remain in a very mutilated state; Isidorus of Charax, who gave a description of the Parthian empire; and Seylax of Caryades, author of a voyage round the Mediterranean, which is still extant.

The works of all these geographers are, however, but small in comparison with the Geography of Strabo; a work in seventeen books, which has come down to us entire. This is one of the most valuable works of antiquity, both from the spirit of discussion which runs through it, and the number of curious particulars which the author has collected from different geographers and navigators who preceded him, and of whose works nothing remains except these extracts. Strabo lived in the reigns of Augustus and Tiberius, and was nearly contemporary with Pomponius Mela. This latter geographer wrote a work De Situ Orbis, which, though little more than a bare summary, is valuable as it gives us a sketch of what was known in his time respecting the state of the habitable globe. Besides Mela, Rome produced in the most flourishing era of its literature another eminent geographer, Pliny. He devoted two books of his extensive work on natural history to a system of geography. His intimate connection with the imperial family, and with many of the most eminent commanders of the time, appears to have given him access to all the military measurements, as well as to the general survey of the Roman empire. Thus furnished with a greater store of authentic materials than any former writer, he has introduced a great number of itinerary details, which are for the most part accurate and valuable. Julius Solenus has also treated of geography in his Polyhistor, a compilation sufficiently valuable from the number of curious particulars which are there collected. Marinus of Tyre was another geographer who appears to have been distinguished, though his works have perished. Even under the Roman empire Tyre continued to be the seat of an extensive commerce; indeed the commercial relations of her citizens appear to have extended over a wider portion of the earth's surface than ever. The enlarged materials furnished by the lengthened journeys of his countrymen, which brought them even to the confines of China, Marinus collected, and sought to apply to them the astronomical principles of Hipparchus, so that he might give to geography a new and more accurate form.

Ptolemy, whom Marinus preceded by a short time, employs a great part of his first book in discussing the means employed by the Tyrian geographer for fixing the relative position of places; and from the references and extracts it appears that the system of Marinus partook largely of the imperfections of a first effort.

The enlarged and scientific views of Ptolemy we have already considered. Some time after Ptolemy lived Dioclesius, commonly called the Periegetic, from the title of a work in verse composed by him, namely, his Periegesis, or Survey of the World. This work was translated into Latin verse by Priscianus, and afterwards by Avienus. There is, besides, a description by Avienus, of the maritime coasts, in iambic verses, of which there remain about seven hundred.

The difficulty of procuring the small and scattered pieces of most of these authors, with those of a few others not here enumerated, induced the learned Hudson to collect them into one work, consisting of four volumes octavo, which were published in the years 1698, 1702, 1712, under the title of Geographiae veteris Scriptores Graeci minores. The originals are accompanied with Latin translations, and notes and dissertations on each by Dodwell. This is a very valuable collection.

We now proceed to consider the progress of geography during the middle ages. From the days of Ptolemy to the revival of letters in Europe, little was done for its solid improvement. The calamities that ere long overwhelmed the Roman empire, involved in the general progress of the intellectual darkness which settled on the world even the imperfect knowledge possessed by the ancient geographers. While barbarous nations poured in from several quarters, art and science ceased to be cultivated. The union by which the Roman power had bound together mankind being now dissolved, Europe was divided into small and independent, and for the most part hostile communities, which had but vague conceptions respecting the situation of each other, while no intercourse subsisted between their members. With regard to remote regions all knowledge was lost; their situations, their commodities, and almost their names, were unknown.

Amidst this ignorance there were but few channels open through which knowledge could be obtained. One circumstance, however, prevented commercial intercourse with foreign nations from being altogether suspended. The opulence and luxury of imperial Rome had long given life and energy to commercial enterprise; that stimulus was now withdrawn; but Constantinople still remained, the last refuge of ancient arts, and taste, and elegance, when the rest of Europe was overspread with barbarism. Fortunately that city had escaped the destructive rage of the fierce invaders; and there, under the cherishing influence of a demand for foreign productions and luxuries, commerce continued to flourish. Alexandria continued to be the emporium whence they imported the commodities of the East Indies, until Egypt, falling under the power of the Arabians, ceased to be a province of the Roman empire. After this event the industry of the Greeks succeeded in discovering a new channel by which Constantinople might still be supplied with the productions of India. These were first conveyed up the Indus as far as that river is navigable, thence by land-carrage they were brought to the Oxus, and were carried down that river to the Caspian Sea. Entering there the Volga, they were conveyed up it, and thence were again transported by land until they reached the Tanais, down which they were conveyed to the Euxine Sea, where vessels from Constantinople awaited their arrival. By this circuitous route was a channel of intercourse kept open with the most distant countries of the East; and an extensive knowledge of remote regions was still preserved in the capital of the Greek empire, while the rest of Europe was sunk in the grossest ignorance.

The missions sent for the conversion of the northern pagans to Christianity served somewhat to illustrate the geography of Europe; though there is sufficient proof that the monks employed were, in many instances, themselves grossly ignorant, some not even knowing the capital of their own country, or the cities nearest to their own. Something was also done by the great sovereigns of Europe towards dispelling the prevailing ignorance of the age on matters connected with geography. Nor did the practical exploits of the Danes and Norwegians under their great sea-kings fail to make them acquainted with the seas and maritime coasts where they carried on their devastations. But it was in the East that a gleam of light and knowledge began to appear, which seemed to be the harbinger of the noon-day splendour of science, that should dispel the darkness of ignorance which had so long oppressed the human mind. Under the influence of a fanaticism which prompted them to own no law but the Koran and the sword, the followers of Mahomet had rushed from the heart of Arabia, and had carried their conquests over half the world. At length, however, under a race of humane and polished princes, having contracted a relish for the sciences of the people whose empire they had contributed to overturn, they stood for some time distinguished as the most learned of nations. They translated into their own language the books of several of the Greek philosophers. The valuable work of Ptolemy was one of the first; and hence the study of geography became an early object of their attention. But the advancement which the science made in their hands towards precision was slow; for they copied and retailed all Ptolemy's principal errors. Still, in all the countries that were under Mahommedan dominion numerous observations were made, which, though not always strictly correct, were entitled to be considered as a step beyond the calculations made merely from the itineraries by the Alexandrian geographers. In the beginning of the ninth century, under their caliph Almanon, who may rank among the most distinguished patrons of science that ever filled a throne, they measured a degree of latitude on the plains of Sinjar, or Shinar, near Babylon, with a view to determine the cir- The tables of Abulfeda and Iug Beg, and of Nazir Eddin, edited by Gravius, and published by Hudson, furnish materials that are still of use in the construction of the maps of the interior of Asia.

The progress and success of the Moslem arms removed the obscurity in which many countries had until then been concealed, as well as the barbarism in which they had been sunk. And even beyond the limits of the Mahommedan world they pushed their researches, by sending missions both to the east and to the west, which they explored to their remotest limits. At that time Europe remained ignorant of the improvements made by the Arabians, though she was destined in subsequent ages to perfect her discoveries.

At length the long period of barbarism which accompanied and followed the fall of the Roman empire, during which the traces of whatever had embellished society, or contributed to the comfort and convenience of life, were almost entirely effaced, drew to a close. Industry began again to shed its blessings on mankind, and Italy was the country where its benign influences were first perceptible. Having from the operation of various causes again obtained liberty and independence, the Italians soon began to feel the impulse of those passions which serve most powerfully to arouse men to activity and enterprise. The reviving demand for the comforts and luxuries of life led to the revival of foreign commerce. The valuable commodities of the East were at first obtained at Constantinople. But the exorbitant price demanded at that mart, in consequence of the circuitous route by which they were conveyed thither, induced the Italians to resort to other ports, as Aleppo and Tripoli, on the Syrian coast, and at length to Egypt itself. After the Soldans had revived the commerce with India in its ancient channel by the Arabian Gulf, Venice, Genoa, and Pisa, rose from inconsiderable towns to wealthy and populous cities. Their trade extended to all the ports in the Mediterranean, and even beyond the straits to the towns on the coasts of Spain, France, the Low Countries, and England; and from these points they diffused through Europe a taste for the luxuries and enjoyments of civilized life, which they at the same time furnished the means of gratifying.

It was not long ere an event occurred, the most extraordinary perhaps in the history of human society, which gave a new impulse to the European mind, and forcibly directed its view eastward to the regions of Asia. Under the influence of a high-wrought enthusiasm, the martial spirit of the Europeans was aroused, and vast armies, composed of all the nations of Christendom, marched towards Asia on the wild enterprise of delivering the Holy Land from the dominion of Infidels. The crusades, however blind the zeal from which they took their rise, had a very favourable influence on the intellectual state of Europe, and prepared it for receiving the light of science which was soon to dawn upon it. Interesting regions, known hitherto only by the scanty reports of ignorant and ridiculous pilgrims, were now made the object of attention and research. Not only was the way opened for the European nations acquiring a correct knowledge of the Holy Land, with the kingdoms of Jerusalem and Edessa, founded by the victorious crusaders; but the extensive regions over which the Saracens and the Turks had extended their empire began to be explored. Search was now made in the writings of the ancient geographers; nor was it improbable that some light was derived even from Arabian writers. Religious zeal, the hope of gain, combined with motives of mere curiosity, induced several persons to travel by land into remote regions of the East, far beyond the countries to which the operations of the crusaders extended. Prompted by superstitious veneration for the law of Moses, and by a desire of visiting his countrymen in the East, whom he hoped to find possessed of wealth and power, Benjamin, a Jew of Tudela, in the kingdom of Navarre, set out from Spain in the year 1160, and travelling by land to Constantinople, proceeded through the countries to the north of the Euxine and Caspian Seas. He then journeyed towards the south, and traversed various provinces of the further India, until, having reached the Indian Ocean, he embarked and visited several of its islands; and at length, after thirteen years, returned by the way of Egypt to Europe. In his progress he had acquired much information respecting a large portion of the globe, then altogether unknown to Europeans.

Various missions were sent by the pope and by Christian princes, for purposes which led them to traverse the remote provinces of Asia. Father John de Plano Carpini, at the head of a mission of Franciscan monks, and Father Ascolino, at the head of another mission of Dominicans, were in the year 1246 sent by Innocent IV. to join Kayuk Khan, the grandson of Zengis, who was then at the head of the Tartar empire, to embrace Christianity, and to cease from desolating the world by his arms. In fulfilling the commands laid upon them by the head of the Christian church, the mendicants had an opportunity of visiting a great part of Asia. Carpini, having taken his route through Poland and Russia, travelled through the northern provinces as far as the extremities of Thibet; whilst Ascolino, who appears to have landed somewhere in Syria, advanced through the southern provinces into the interior parts of Persia.

Father William de Rubruquis, a Franciscan monk, having been sent in the year 1258 on a mission by St Louis of France, in search of an imaginary personage, a powerful khan of the Tartars, who was reported to have embraced the Christian faith, made a circuit through the interior parts of Asia more extensive than that of any European who had hitherto explored them. He had the merit of being the first modern traveller that gave a true account of the Caspian, which had been correctly described by the early Greeks as an inland separate sea; but a notion afterwards prevailed that it was connected with the Northern Ocean. Rubruquis ascertained that it had no connection with the ocean or any other sea. The account of his journey was so little read, however, that the old error was repeated in books of geography long after his time.

While the republics of Italy, and, above all, the state of Venice, were engaged in distributing the jewels, the spices, and the fine cloths of India over the western world, it was impossible that motives of curiosity, as well as a desire of commercial advantage, should not be awakened to such a degree as to impel some to brave all the obstacles and dangers to be encountered in visiting those remote countries where these precious and profitable commodities were produced. A considerable number accordingly are recorded as having penetrated a greater or less depth into the interior of Asia. But of all the old travellers the fame is eclipsed by that of Marco Polo, who has always ranked among the greatest discoverers of any age. This extraordinary man was a noble Venetian, whose family, according to the custom of his country, engaged in extensive commerce. Nicolo Polo, and Maffeo Polo, the father and uncle of Marco, were merchants, who, in partnership, traded chiefly with the East; and in pursuit of their mercantile speculations had already visited Tartary. The recital of their travels on their return fired the youthful imagination of Marco, then between seventeen and eighteen years old. Having, when in the East, gained the confidence of Kublai Khan, the great conqueror of China, at whose court they had resided for a long time, he had sent them back to Italy, accompanied by an officer of his court, that they might repair to Rome as his ambassadors to the pope, of whom, and the potentates of the western world, they had given him an ample account. After many delays they were now, about the year 1265, to set out on their return to the court of Kublai, bearing the papal letters and benediction; and it was resolved that young Marco should accompany them. After a journey that occupied no less than three and a half years, and in the progress of which they passed through the chief cities in the more cultivated parts of Asia, they reached Yen-king, near the spot where Pekin now stands, where they were honourably and graciously received by the grand khan. Struck with the appearance of young Marco, the khan condescended to take him under his protection, and caused him immediately to be enrolled among his attendants of honour. By prudence and fidelity Marco gained so high a place in the esteem and confidence of his protector, that for seventeen years, during which he remained in his service, he was employed in confidential missions to every part of the empire and its dependencies. He made more than one voyage on the Indian Ocean, and traded with many of the islands. Besides what he learned from his own observation, he collected from others many things concerning countries which he did not visit. Considering the very favourable circumstances in which he was placed for geographical research, as well as his passion for travelling, which seems to have increased with his opportunities of gratifying it, it is not surprising that, after the long period of his wanderings in Asia, he should have returned to Europe possessed of the knowledge of many particulars, until his time unknown, respecting the eastern parts of the world. Marco, being afterwards made a prisoner by the Genoese, was induced, with a view to beguile the tediousness of his confinement, to dictate a narrative of his travels. His information was so far in advance of the age, that his veracity was exposed to the most injurious suspicions. But, if we make allowance for some tincture of credulity, characteristic of the times in which he lived, his narrative is supported in all its essential points by modern information.

While great accessions were thus made to the stock of knowledge possessed by the nations of Europe respecting the habitable globe, their ideas were at the same time gradually enlarged; and an adventurous spirit was generated, which prepared them for attempting further discoveries. Still their efforts were limited by certain bounds, in consequence of the imperfect state of navigation. Whatever conceptions a daring mind might venture to form respecting the existence of unknown regions, separated from the known continents by the mighty expanse of the Atlantic Ocean, mankind had not yet so obtained the dominion of the sea as to be able to bring such conceptions to the test of experiment. It was not until the fortunate discovery of the polarity of the magnetic needle, and the consequent construction of the mariner's compass, that man was enabled to visit every part of the globe which he inhabits. This important discovery was made by Flavio Gioia, a citizen of Amalfi, a town of considerable trade in the kingdom of Naples, about the year 1302. Encouraged by the possession of this sure guide, by which at all times and in all places he could with certainty steer his course, the navigator gradually abandoned the timid and slow method of sailing along the shore, and boldly committed his bark to the open sea. At the commencement of the fifteenth century, however, navigation appears to have advanced very little beyond the state which it had reached before the downfall of the Roman empire. But it was now destined to make rapid progress. The growing spirit of enterprise, combined with the increasing light of science, had prepared the states of Europe for entering upon that great career of discovery, of which the details constitute the materials for the history of modern geography.

Portugal took the lead in this new and brilliant path. Her first attempt was to discover the unknown countries situated along the western coast of Africa. Notwithstanding the vicinity of that great continent, and the strong inducement afforded, in the fertility of the countries already known in it, to its further exploration, Cape Non had hitherto limited the researches of the Portuguese, and had been regarded as an impassable barrier. In the year 1412, however, ships sent out for discovery doubled this formidable promontory, and reached Cape Bojador, a hundred and sixty miles to the southward, which became in its turn the boundary of Portuguese navigation; and it continued to be so for upwards of twenty years. Under the coasting system, which still continued to be practised, it was not likely that the obstacles presented by its rocky cliffs, which stretch a considerable way into the Atlantic, would soon have been overcome. But a sudden squall of wind having driven out to sea the vessel next dispatched, this event fortunately led to the discovery of Porto Santo; whence in a little time Madeira was discovered, being first mistaken for a small black cloud in the horizon; and at length, when the Portuguese by their voyages thither had gradually become accustomed to a bolder navigation, Cape Bojador was doubled. Thus, by repeated efforts, the Portuguese navigators gradually approached the northern boundary of the torrid zone. Here their progress was for some time arrested, not by any physical difficulties, but in consequence of the influence which the opinion of the ancient mathematicians and geographers, whom they had hitherto followed as their guides, that excessive heat rendered the middle regions of the earth uninhabitable, had upon their minds. Experience, however, at length enabled them to triumph over ignorance and prejudice. A powerful fleet, fitted out in 1484, after discovering the kingdoms of Benin and Congo, advanced above fifteen hundred miles beyond the equator. Their intercourse with the natives enabled them to obtain information concerning those parts of the country which they had not visited. Not only had they detected the error of the ancients in reference to the torrid zone, but they found also that the direction of the coast was very different from what the description given by Ptolemy had led them to expect. They saw reason to conclude that the continent gradually became narrower as they proceeded southward; so that there was room to believe that the ancient accounts respecting the circumnavigation of Africa were really founded in truth. New and more extensive prospects were thus opened to them; and the finding of a passage to India by sailing round the southern extremity of Africa became a favourite project. In the year 1488 the lofty promontory which terminates that continent was described by Bartholomew Diaz; but it was not until about ten years after that it was doubled, and the coast of Malabar reached, by Vasco de Gama.

Meanwhile, the Cape de Verd Islands, which are said to have been known to the ancients, but afterwards lost sight of, had been discovered in 1446; and soon after the Azores Isles. When we consider the distances at which these two groups of islands lie from the land, the former being upwards of 300 miles from the coast of Africa, and the latter distant 900 miles from any continent, it may be concluded that the Portuguese, when they entered so boldly into the open seas, had made no inconsiderable progress in the art of navigation.

But brilliant as is the lustre which those discoveries shed around the Portuguese name, their glory would have been still more dazzling had they seconded the profound views of Columbus, which led him to the discovery of the New World. That illustrious man and skilful navigator, by revolving in his mind the principles on which the Portuguese had founded their schemes of discovery, and carried them into execution, was led to conceive that he could improve on their plan, and accomplish discoveries which they had found the island of Trinidad, with the neighbouring land; he encountered, before he was aware of danger, the adverse currents and tumultuous waves occasioned by the resistance which the waters of the Orinoco oppose to the tides in the ocean. His attention was thus forcibly called to the immense body of water which is here poured into the Atlantic. This he was convinced was vastly too great to be supplied by any island; and hence he concluded that he had now reached the continent which he had sought through so many dangers.

The tenacity with which an ingenious and enterprising mind adheres to a scheme which it has once proposed to itself as an object of pursuit, was strikingly evinced by Columbus, whose thoughts still dwelt with eagerness on his original and favourite plan of opening a new passage to India. It was not enough that he had astonished mankind by finding a new continent; he conceived the idea that beyond it there might lie a sea extending to the coasts of Asia, and that by diligent search some strait might be found which would conduct him into this sea, or some narrow neck of land, by crossing which it might be reached. To determine this important point, though hitherto his services had met with the most unworthy returns, though years crept upon him, though worn out by fatigue and broken with infirmities, he still undertook with alacrity another voyage. By a lucky conjecture he directed his efforts towards the east of the Gulf of Darien; but he searched in vain for a strait; and though he frequently went on shore and advanced into the country, he never penetrated so far as to enable him to descry the great Southern Ocean.

After the first steps had been taken, the progress of discovery over the globe was astonishingly rapid. No expense or danger deterred even private adventurers from fitting out fleets, crossing oceans, and encountering the rage of savage nations in the most distant parts of the earth. Before Columbus had reached the continent at the mouth of the Orinoco, Newfoundland had been discovered by Cabot, a Venetian by descent, but sailing under the auspices of England. He had also coasted along the present territory of the United States, perhaps as far as Virginia. In the next two or three years, the Cortereals, a daring family of Portuguese navigators, began the long and unavailing search of a passage round the northern extremity of America. They sailed along the coast of Labrador, and entered the spacious inlet of Hudson's Bay. Two of them unfortunately perished in this enterprise. In the year 1501 Alvarez Cabral, a Portuguese navigator, destined for India, having stood out to sea in order to avoid the variable breezes and frequent calms which he was sure to meet with on the African coast, to his surprise, came upon the shores of an unknown country, the coast of Brazil, which he claimed for Portugal. Amerigo Vespucci, a Florentine gentleman, who had already sailed along a great part of Terra Firme and Guyana, now made two extensive voyages along the Brazilian coast. Soon after his return he drew up and transmitted to one of his countrymen an account of his adventures and discoveries, in which he insinuated that to him belonged the honour of having first discovered the continent in the New World. His performance, which was the first description published of the newly-discovered countries, circulated rapidly, was read with admiration, and became the means of procuring for its author the high honour of giving his name to the whole continent. Not many years elapsed before the conjecture of Columbus respecting the existence of an ocean beyond the continent which he had discovered was found to be true; and his favourite project of opening a passage to India by steering westward was actually accomplished. By crossing the narrow isthmus of Panama, Nuñez Balboa reached the Pacific Ocean in the year 1513; and in 1521 Magellan discovered and sailed through the famous straits which bear... After twenty days occupied in navigating this dangerous channel, he beheld spread out before him the boundless expanse of the great Southern Ocean. Directing his course to the north-west, he continued his voyage for nearly four months without discovering land. From want of provisions, and from sickness, he and his crew suffered dreadful distress. But when about to sink under their sufferings, they fell in with the Ladrones Islands, where they found refreshments in abundance. From these isles he proceeded on his voyage, and was not long of discovering the Philippines. Here, in an unfortunate quarrel with the natives, he was slain, with several of his principal officers. But his surviving companions, pursuing their voyage, and returning to Europe by the Cape of Good Hope, solved the great problem of the circumnavigation of the earth.

After the discovery of the Pacific Ocean by Balboa, the investigation of the western coasts of America went speedily forward. Expeditions were soon sent out both northward and southward; so that nearly a full view was obtained of the immense range of coast which the American continent presents to the Pacific Ocean, and at the same time of its great interior breadth.

On the other hand, discovery in the eastern world was no less rapid. Within twenty years from the time that Gama reached India by the way of the Cape of Good Hope, all the coasts of Hindustan, those of Eastern Africa, of Arabia, and Persia, had been explored. Navigators had penetrated to Malacca and the Spice Islands. They had learned the existence of Siam and Pegu; and it was only the characteristic jealousy of the rulers of the Celestial Empire that prevented them from entering the ports of China.

The scientific geographer had now abundance of materials to arrange and digest into one systematic whole. He was now called upon to give such a delineation of the earth's surface, as should connect together the ranges of eastern and western discovery, and should exhibit the true outline and relative positions of countries, as these had been demonstrated by the astronomer and navigator. The ancient system of geography, to which the Arabs seem closely to have adhered, was founded on the idea of the whole earth being surrounded by an ocean as by a great zone. This the Arabians characterized as the "Sea of Darkness," an appellation most usually given to the Atlantic; while the northern sea of Europe and Asia, as inspiring still more gloomy and mysterious ideas, was styled the "Sea of Pitchy Darkness." Such notions could not now keep possession of the human mind, though it was only by degrees that mankind could be expected to be enlightened by doctrines which were not only new, but seemed likewise to be contradicted by the evidence of sense. The fundamental principles of a systematic arrangement had, as we have already seen, been known from the time of Hipparchus, and had been reduced to practice by Ptolemy. But the want of astronomical observations, or even of accurate surveys, which navigators seldom furnished, and for which science had not indeed yet provided suitable instruments, placed it still beyond the resources of modern geography to give any thing like a just representation of the two hemispheres. The Venetian geographers were the first who attempted a systematic arrangement of the immense regions recently discovered, adjusting them to each other, and to the mass of information previously possessed. But a series of Venetian maps, preserved in the king's library, show how much their skill was counteracted by the difficulties with which they had to contend. Instead of exhibiting the vast ocean which separates the east coast of Asia from the west coast of America, the two continents are represented either as meeting, or as separated only by a narrow strait.

The voyage of Magellan across the Southern Ocean had not shown with sufficient distinctness the presentation of the opposite coasts, to enable the geographers of the time to avoid this error. When maps of different dates are compared, we find, as we descend towards modern times, a gradual progress towards accuracy in the representations given of the earth's surface. This is what might be expected; for all maps should be considered as unfinished works, in which there will always be something to be corrected, or something new to be inserted.

At the period of the revival of letters in Europe, the latitudes and longitudes, as given by Ptolemy, were universally received with implicit confidence. When checked, however, by actual observation, they were found to differ materially from the truth. The latitudes in many instances were found very erroneous; that of Byzantium, for example, exceeded the truth by two degrees. As nearly the same excess was found to exist in some other cases, many geographers, unwilling to renounce the authority of Ptolemy, concluded that this difference had arisen from a change having taken place in the position of the earth's axis, in consequence of which the latitudes of all the places in Europe were increased. The progress of observation showed that this opinion was untenable, and that before geography could rest on a sure basis, a general revision of ancient graduation was indispensably necessary. The only observations employed by the ancients for determining longitudes were those of the eclipses of the moon; but it was found that the results derived from this source could not be depended on. In the year 1610, Galileo, having discovered three of Jupiter's satellites, pointed out the use which might be made of their eclipses for finding longitudes. But this method, which gives the greatest degree of accuracy, was turned to little account, until 1668, when Cassini published his tables of the revolutions and eclipses of these satellites. Three years afterwards, by means of simultaneous observations made by him and Picard at Paris, and in the observatories of Tycho Brahe at Copenhagen, the difference of longitude of these two important points, which had been long a matter of dispute, was finally determined. Since that time, other accurate methods of finding the longitude have been discovered; and the instruments employed in observation have been brought to a high degree of perfection. The refinements and improvements of modern science have been brought to bear upon the great problem of determining the figure of the earth, which, though nearly, is not exactly spherical. (See Figure of the Earth.) The labours of scientific men to obtain accurate results on this subject have contributed much to the improvement of geography. The expeditions sent out under Maupertuis to the arctic circle, and under Condorcine to the equator, afforded an opportunity of making various observations of latitude and longitude in regions of which no delineation resting upon proper data had hitherto been given. Within the last thirty years, trigonometrical surveys of France and England have been executed, which have nearly completed the delineation of these countries.

Much advantage has accrued to geographical science, in point of accuracy and precision, from the application in modern times of a sound and judicious criticism to the immense mass of materials which had been accumulating for ages. The labours of M. d'Anville, in the eighteenth century, were employed with great success in this department. He undertook the revision of the whole system on which the delineation of the world, and of the countries into which it is divided, had hitherto been made; and by unhesitatingly rejecting every particular that did not rest on positive authority, he removed many false or uncertain features, and clearly distinguished the known from the unknown parts of the globe. Major Rennell has skilfully arranged and illustrated the important materials collected respecting India. Various authors have in mo- who had been dispatched by the Hudson's Bay Company History. to explore the limits of the coast in this direction, sailed down the Copper-Mine River, and discovered its entrance into the sea; and again, when, in 1780, Sir Alexander MacKenzie traced also to the sea another river about twenty degrees farther to the west. Thus were there furnished strong grounds to believe that the pole was surrounded by an ocean which separated the northern coasts of Asia and America, making these two continents altogether distinct from each other; and that through this ocean lay the long-sought course which would certainly conduct the navigator who should succeed in forcing his way through the ice and storms of the polar regions, from the Atlantic into the North Pacific Ocean.

The determination of this great geographical question, so long agitated, has recently called forth the utmost efforts of the British government. In 1818 an expedition was sent out to Baffin's Bay under the command of Captain Ross, without leading to any important result; as he was led to conclude that no opening existed. Lancaster Sound had, however, forcibly attracted the attention of Lieutenant Parry, second in command; and on returning with a new expedition under his immediate command, he succeeded in penetrating through Lancaster Sound, which he found gradually to widen till it opened into the Polar Sea. He found a chain of large islands to lie parallel to the American coast; and among these he continued his navigation until the accumulation of ice in the straits and channels through which he had to pass stopped his further progress. This circumstance induced him to make his next attempt through Hudson's Bay, by the channel of the Welcome, which had as yet been but imperfectly explored. Here he succeeded in reaching a point considerably beyond that at which Middleton had represented the bay as terminating. He found at length a narrow strait communicating with the Polar Sea, but so encumbered with ice as to preclude the hope of its ever affording an open passage. He was therefore again sent out to renew his efforts in the first direction, where he had already obtained partial success. But the obstacles which he had formerly been unable to overcome still continued, and prevented him from making any material addition to his former discoveries. Whilst these skilfully-conducted voyages were in progress, Captain Franklin was sent out at the head of two successive expeditions by land, and, by actual survey, ascertained three fourths of the boundary coast; his operations terminating at a point beyond the 149th degree of west longitude. On the other hand, an expedition under Captain Beechey, sent to meet Captain Franklin on his second toilsome journey, passed the Icy Cape of Cook, and penetrated nearly as far as the 156th degree of west longitude, leaving only seven degrees, or 160 miles, between the farthest point thus reached and the utmost limit reached by Captain Franklin. The results of this investigation appeared to prove that the whole of the northern coast of America extends in a line not varying much from the parallel of the seventieth degree of latitude. But Captain Ross, in the late expedition fitted out by himself, has found a large peninsula stretching as far north as seventy-four degrees of latitude. The existence of a naval passage farther to the north, in the line of Captain Parry's first voyage, is, however, still probable.

The discovery of a new continent greatly enlarged, as we have seen, the views of mankind respecting the constitution of the globe. But imagination, no longer limited in its range by the notion of a circumambient ocean that could not be passed, soon gave rise to the belief of a southern continent, which was supposed necessary to balance the land in the northern regions of the earth. The immense body of water that was found to occupy so large a portion of the known regions of the southern hemisphere gave ample room for supposing this unknown continent to be of vast dimensions. It was imagined that it might equal in extent as well as in wealth the American continent. Nor was it considered necessary to exclude it from the map of the world till its existence should be proved. It appears in all the early maps as an immense mass of land surrounding the south pole, and presenting to the ocean one unbroken coast. The discovery of certain great insular tracts in the South Seas, which, from ignorance of their true nature, navigators might mistake for continental promontories or portions of coast, no doubt at first gave some countenance to the belief of the existence of antarctic land. But the delusion was gradually dispelled before the light afforded by further discovery. The Portuguese, in less than twenty years after their passage of the Cape of Good Hope, pushed their researches to the most remote islands in the Indian Ocean, including Java and the Moluccas. They appear also to have observed some part of the coast of New Guinea. The Spaniards during their early and adventurous career put forth strenuous exertions to explore the Southern Ocean, and several of the groups of islands scattered over its surface were discovered by their navigators. In 1607, the Dutch having wrested from the Portuguese Java and the Spice Islands, established in them the centre of their Indian dominion. A great maritime power being thus placed so near to the northern shores of the largest portion of land on the globe that is regarded as an island, it became almost impossible that New Holland could long remain unknown. It was discovered early in the seventeenth century, and was long supposed to form a part of the great southern continent. Van Diemen, the Dutch governor of India, sent out several vessels successively to explore its coasts. Hertog, Carpenter, Nuytz, and Ulaming, made very extensive observations on the northern and western shores, but found them so dreary and uninventing that no settlement was attempted. In the year 1644 the eastern coasts were visited by Tasman, who, proceeding beyond his predecessors, reached the southern extremity of Van Diemen's Land, to which he gave its name; but he failed to discover that it is a separate island. Pursuing his course eastward, he surveyed next the western coast of New Zealand, and returned home by the Friendly Islands. He thus proved that New Holland was no part of the southern continent, even should such a continent exist. Cook, by ranging along the borders of the southern pole, where he found only isles and firm fields of ice, showed that the hypothesis of an austral continent had no foundation. The eastern side of New Holland was discovered by Captain Cook in the year 1770, who by his extensive operations in that quarter left little to be done towards completing the full circuit of it. The same great navigator likewise secured glory to his country and to himself by completing the survey of the Great Pacific Ocean. Some of the interesting groups of islands scattered over its vast surface had already been made known by the previous voyages of Byron, Wallis, and Carteret. Cook fully traced the great chain of the Society Islands and the Friendly Islands. He determined also the form and relations of New Zealand, New Caledonia, and other great lands and islands in that region of the globe.

The extensive island of New Holland has recently become doubly interesting from the important relations which now subsist between it and Europe. In the year 1788 the establishment of a British colony on the east coast paved the way for a more complete survey. By the different expeditions undertaken from 1795 to 1799, chiefly under the direction of Bass and Flinders, the east coast, together with Van Diemen's Land and Bass's Strait, which separates that island from New Holland, were accurately explored. In 1801 an expedition was sent out by the British government under the command of Captain Flinders, for the purpose of surveying a large portion of the coast. These surveys were prosecuted with unremitting ardour and perseverance. At the same time that Captain Flinders was carrying on his survey, the French captain Baudin was employed on the same service, and in some parts the discoveries of these navigators intermingle. Some additional observations have more recently been made; and by these various expeditions the whole coast of New Holland and Van Diemen's Land has been accurately surveyed, the position of every point has been ascertained, and every inlet and bay has been traced to its termination.

In tracing the history of geographical discovery, it cannot fail to be observed, that while discovery by sea has been pursued with great advantage, on account of the rapidity of its progress and the extent of its range, the slower and more confined operations of the discoverer by land are no less necessary to make known the interior features and circumstances of the different countries.

The British dominion in India has led to much additional information respecting the interior of Asia; information which is, however, in many respects, only the revival of ancient knowledge. The great mountainous chain which forms the northern boundary of India has been traced and found in many places to tower to such heights as to exceed the Andes, long supposed to be the highest mountains in the world. The source of the Ganges, and that of the Indus, with the early courses of these great rivers, have been found to be situated quite differently from what had been supposed to be their position by modern geographers. The mountainous territories of Cabul and Cashmere, the high interior table-land of Thibet, and the vast sandy plains of Meckran, have all been more or less explored. Information of an authentic character has also been recently obtained respecting the formerly celebrated capitals, Bochara and Samarkand. But a wide field still remains for future research.

The continent of Africa, however, is the quarter of the globe which, more than any other, has baffled the efforts of those who would explore its interior. The vast sandy deserts, high mountains, and impenetrable forests which occur on its surface, joined with the unremitting wars carried on between the petty tribes, and the deeply-rooted antipathy of the African Mahommedans towards the Franks, have presented obstacles of the most formidable kind. The ancients, whose knowledge of the African coasts was very imperfect, except where they border on the Mediterranean and the Red Sea, were accustomed to penetrate into the inland provinces, and are said to have been acquainted with many parts of it which are now altogether unknown. At an early period of modern history reports that Prester John, the Christian prince, who had been sought for in vain in the East, was to be found in the interior of Africa, induced the Portuguese to explore Abyssinia; but the accounts which they gave of the extent of that country were greatly exaggerated. From the western coast they dispatched embassies into the interior in quest of the object of their search; and on one occasion they appear to have reached the city of Tombuctoo, and to have obtained at Benin some information concerning the great interior kingdom of Ghana. Tombuctoo has been for many centuries the grand emporium of the central trade of Africa; and on this account an eager desire has been felt in Europe, ever since the rise of discovery and commercial enterprise, to visit it and establish an intercourse with it. All attempts, however, to reach it made by European merchants and travellers have until recently been entirely baffled. Its actual condition, and even magnitude, are still involved in very considerable uncertainty. Major Laing resided there a considerable time, and made the most diligent inquiries; but the fatal catastrophe which terminated his career prevented the result from ever reaching Europe. But the grand object connected with the interior of Africa by which the attention of Europeans has been from the first chiefly excited, is its great central river, the Niger. The interest with which this stream has been regarded has arisen from the remarkable nature of the regions through which it flows, and still more from our ignorance, combined with the various and contradictory rumours which were so long abroad respecting its course and termination. Herodotus is the earliest author who affords any ideas applicable to this subject. He mentions an expedition into the interior of Africa, undertaken by some Nasamonian youths, who, being taken prisoners, were carried to a great city inhabited by negroes, and situated on the banks of a river which flowed from west to east. This stream he conjectures to be the remote source of the Nile; but the particulars given appear to leave little doubt that it was the Niger. A similar hypothesis was adopted by Strabo, Mela, and Pliny, identifying the waters of these two great rivers.

Ptolemy, whose residence in Alexandria afforded him ample means of information, rejects altogether the idea of communication between them. He describes the Niger as terminating on the west by Mount Mandrus (Mandingo), as giving rise to several extensive lakes as it proceeds in its course. His statements do not, however, involve any dogmatical positive as to the direction in which it flows. The Saccas or Arabians are the next great source of information; for in the course of the dissensions which took place among their dynasties in Northern Africa, large bodies crossed the desert, and founded kingdoms on the eastern part of the shore of the Niger, of which the kingdom of Ghana was the most splendid. According to their testimony, the Niger flows from east to west, and discharges itself into the sea, by which they understood the Atlantic, or great circumambient ocean. With regard to its source, they generally regarded it as the same with that of the Egyptian Nile, identifying the two rivers in the early part of their course. Some were of opinion that the waters of the Niger did not reach the sea; so that they must have supposed them to be discharged into a lake. The system adopted by modern Europeans was derived from Leo Africanus, who retained the delineation of the Niger as flowing from east to west, and falling into the ocean, but instead of deriving it from the Nile, supposed it to rise from the lake of Bornou, lying deep in the interior of Africa. Following this hypothesis, all the early European navigators, who saw the two broad estuaries of the Senegal and Gambia, concluded that one or both gave egress to the water of the Niger. In the beginning of the seventeenth century, the French and English having each formed a settlement, the one on the Senegal and the other on the Gambia, were induced by the hope of gain to seek a route up the rivers to the city of Tombuctoo; and in this enterprise they proved the falsity of the opinion which had been so long held. The streams were traced so near to their source as to become little more than rivulets; whilst the explorers were still far from the great central emporium of Africa, and from the great plain through which the Niger was understood to flow. This result led the two great French geographers Delisle and D'Anville to construct maps in which the Niger, after the lapse of so many ages, was again represented as flowing to the eastward. Instead of a single stream pursuing a course across the whole breadth of Africa, and falling into the Atlantic, D'Anville distinguished three rivers; the Senegal flowing westward, the Niger flowing eastward into a lake in Wangara, and another river still further east, and flowing in the opposite direction. The data on which this scheme rests were never fully made public.

Still new doubt was thrown around this subject by the reports collected by Mr Lucas, who travelled under the auspices of the African Association, and who was assured at Tripoli, by a native merchant, that the river flowed with rapidity in a westerly direction. The time, however, at last arrived when these conflicting opinions were to be silenced, and new light thrown on the subject, by the labours of our illustrious modern traveller Mr Park. In his first expedition, on reaching Segou, he beheld "the long-sought majestic Niger, glittering in the morning sun, as broad as the Thames at Windsor, and flowing slowly to the eastward," and directing its course into the depths of interior Africa. Here, then, was a new point from which theory and conjecture might start in endeavouring to ascertain the termination of the Niger. By the persevering exertions of the British government, an expedition has, however, at length actually made its way into the unknown interior of Africa, and has finally led to the solution of the grand problem which had so long puzzled geographers, as well as exercised their ingenuity. Without entering further into details, it will be sufficient to observe, that instead of only one river being found flowing through the great African plain in one direction, there were several rivers discovered flowing in different directions; all of which, it appears, have been considered at different times, and under different circumstances, the Niger. Hence had arisen the various statements which for ages involved the subject in such deep mystery. The great central stream of Africa has now been very completely traced, whilst much light has been obtained respecting the regions through which it flows. It was reserved for Lander finally to trace the course of the Niger to the Gulf of Benin, where its waters are discharged through a succession of estuaries, extending along the coast a space of about 300 miles. Thus, by means of this river and its numerous tributaries, the prospect is now opened to British commerce of being able to penetrate into the most inland and finest regions of the African continent.

In all parts of the habitable globe the spirit of research, which has already done so much, is still active; nor is it directed only to the determining of the outlines of continents and countries, or to the marking of the leading features of mountains, rivers, and cities, with their relative positions and distances. These are regarded by the geographical inquirer merely as affording a proper basis on which to rest the description of the earth considered as the habitation of man, and as affording him amply the means of improvement and happiness. The picture can be completed only by the continued labours of the scientific observer, who makes the earth, with its various productions, whether natural or artificial, the treasures hid in its bosom, the animals found upon its surface, and, above all, the human beings who people its different regions (and these in all their mutual bearings and relations), the objects of attentive examination and study.

II.—MATHEMATICAL GEOGRAPHY.

CHAP. I.—PRINCIPLES AND DEFINITIONS OF THE TERRESTRIAL SPHERE.

We have already stated that mathematical geography treats of the figure, magnitude, and motion of the earth; its relations to the other bodies of the system to which it belongs; the relative positions of places on its surface and the methods of delineating the whole or any part of its surface. Several of these topics belong as much to astronomy as to geography; for the former science regards the earth, on the one hand, as the grand observatory whence the phenomena of the heavens are contem- Mathematical Geography.

Plated by man; and, on the other hand, as itself constituting a portion of the planetary system, the laws of which it is the object of the astronomer to investigate and explain. It is only, indeed, by celestial observation that the position of a point on the surface of the earth can be accurately determined; so that the first principles of geographical science must necessarily be drawn from astronomy.

We take for granted, then, the doctrine of the celestial sphere; the globular form, the magnitude, and motion of the earth, with the phenomena arising out of and depending upon its motion. These points are fully discussed in the article Astronomy, part ii. It will be necessary, however, for the sake of distinctness, to state briefly such of the results derived from the reasoning there employed as belong specially to the subject of this article.

The doctrine of the earth's rotundity is that with which the student of geography must first make his mind familiar. To the eye the earth appears a circular plane, at the centre of which the spectator imagines himself placed; whilst the heavens, spread over his head like a magnificent canopy, seem to meet the earth all round in a circle which bounds the view. The space comprehended within this circle is found to present often a very irregular surface, rising into mountains, or sinking into cavities, so as apparently to exclude the idea of its bearing any resemblance to a portion of a globe or sphere. This is, however, nearly its true figure. Were we placed at such a distance from the earth that the eye would be able to take in at one glance the vast mass in its full dimensions, it would present the appearance of a circular disk, of greater or less diameter according to our distance. If viewed on all sides, its circular form would remain unchanged; a property characteristic of no other body but a globe or sphere. The highest mountains would be found to bear so little proportion to the whole bulk, as to cause no perceptible deviation from the globular form; and the same thing may be said of the difference which exists between the polar and equatorial diameters, though that quantity amounts to about twenty-eight miles.

The portion of the earth, then, seen from any point above its surface, is to be regarded as the segment of a sphere, which increases in extent according as the eye is more elevated. Knowing the semidiameter of the earth, and the height of the eye above the surface, it is easy to find the diameter of the circle which bounds our view on the earth; supposing, as is the case at sea, that no object should intervene to obstruct our vision. Thus, let \( AB \) (Plate CCLIX. fig. 1) be the height of a mountain on the earth's surface. From \( A \) draw the line \( AD \), a tangent to the surface in any direction; \( D \) will be the most distant point seen by a spectator whose eye is placed at \( A \), the summit of the mountain. If the line \( AD \) be supposed to be carried round the point \( A \), while it at the same time continues to be a tangent to the surface, until, having passed through every possible position, it returns to that from which it set out, the point \( D \) will trace the circumference of the circle which bounds the terrestrial view, and of which the radius will not differ much from \( AD \). Now \( AD \) being a tangent to the arch \( BD \), we have \( AD^2 = (2BC + AB) \times AB = 2BC \times AB \) nearly, since \( AB \) must always be very small compared with \( 2BC \), the diameter of the earth. Assuming the diameter equal to 7912 English miles, we have \( AD^2 = 7912 \times AB \); or, if \( AB \) is expressed in feet,

\[ AD = \frac{7912}{5280} \times AB. \]

Hence, extracting the square root of the numerical part, we obtain

\[ AD = 1.224126 \sqrt{AB}. \]

The result obtained from this formula will deviate from the truth in consequence of the effect of refraction. The error arising from this cause is found to vary from \( \frac{1}{3} \) to \( \frac{1}{5} \) of the whole distance. At a medium, refraction may be considered as increasing the distance by about \( \frac{1}{50} \) or \( \frac{1}{71} \) of the whole. Applying this correction to the above formula, it becomes

\[ AD = 1.3115 \sqrt{AB}. \]

For example, if the Peak of Tenerife is 12,358 English feet in height, then \( AD = 1.3115 \sqrt{12358} = 145.8 \text{ miles} \) is the radius of the circle which bounds the view from that elevation.

The circle bounding terrestrial vision, which we have now been considering, does not, except when the eye is close to the surface of the earth, coincide with the sensible horizon as determined by a plane passing through the eye along the line \( HO \), and perpendicular to the vertical line \( CZ \). (See Astronomy, p. 754.) The angle contained between the visual ray \( AD \) and the horizontal plane, is called the dip of the horizon. For ordinary purposes, however, it is sufficiently accurate to say that the sensible horizon is the circle which bounds the view, where the heavens and the earth appear to meet. To this circle the rational horizon is parallel, being determined by a plane passing through the centre of the earth perpendicular to the vertical line \( CZ \), and continued in every direction to the sphere of the heavens. Let NESQ (Plate CCLIX. fig. 2) be the terrestrial sphere, of which \( C \) is the centre; \( HO \) is the rational horizon of the point \( M \).

The rotatory motion of the earth determines the position of the line NS (Plate CCLIX. fig. 2), on which it revolves, and which is called its axis. The two points \( N, S \), in which the axis meets the surface, are called the poles of the earth. The circle EQ upon the earth, the plane of which passes through \( C \) the centre, and is perpendicular to NS the axis, is called the equator or equinoctial line. This circle is equidistant from the poles of rotation, and its plane coincides with the plane of the celestial equator. The rational horizon and equator being two great circles of the terrestrial sphere, that is, the plane of each passing through the centre, they are mutually bisected. This is a property which belongs to all great circles. Any great circle NFS passing through the two poles is called a meridian. Any given point \( P \) may have a meridian passing through it; and the plane of that circle will coincide with the plane of the celestial meridian which passes through the zenith of that point. Every meridian cuts the equator at right angles.

The horizon HO corresponding to any point \( M \) upon the globe, is divided into four equal parts by the equator and the meridian passing through that point. The points of division are called the four cardinal points. Those through which the meridian passes are called the north and south points of the horizon. Those points, again, through which the equator passes are called the east and west points.

The earth is divided by the horizon into two hemispheres, called the upper, and lower or under hemispheres; by the equator it is divided into the northern and southern hemispheres; and by the meridian, into the eastern and western hemispheres.

The position of any point \( P \) on the surface of the earth is determined by referring it to the co-ordinate circles, the equator and meridian. The distance \( PA \) of the place \( P \) from the equator, measured in degrees and parts of a degree on the meridian passing through the place, is called its latitude, which is said to be north or south, according as the place is situated in the northern or southern hemisphere. The arch EA of the equator, intercepted between the meridian passing through the place \( P \) and some particular meridian NMS, called the first meridian, as being that fixed on from which the reckoning is to commence, is called the longitude of the place in question; and is east or west, according as the place is situated in the eastern or western hemisphere in reference to the first meridian. The difference of longitude between two places the arch of the equator intercepted between their meridians; and is found by subtracting or adding the longitudes, according as the places are on the same or on opposite sides of the first meridian. The ancients assumed for their first meridian that of the Fortunate Islands, which they conceived to be the western limit of the habitable world. In later times, the meridian passing through Ferro, one of the Canary Islands, and nearly coinciding with that of the ancients, was used as the first meridian by the geographers of many countries. But in modern systems of geography the first meridian is that which passes through the capital of that country, or through an observatory, if there is one near the capital, from which the position of other meridians is determined. By British geographers the meridian of the observatory at Greenwich is reckoned as the first meridian. It is usual to call the semicircle NPS passing through the two poles and a given place, the meridian of that place; and the other semicircle NRS passing through the opposite point on the globe, the opposite meridian. All places lying under the same meridian have the same longitude, and at all of them noon or any other hour occurs at the same instant of absolute time. With regard to places that lie towards the east and west of the meridians of each other, this is not the case. The sun in its apparent diurnal revolution moves over fifteen degrees an hour; so that at a place in fifteen degrees east longitude, noon occurs an hour earlier than at places under the first meridian, and at a place in fifteen degrees west longitude, an hour later; and each fifteen degrees by which the longitude east or west is increased, makes an additional hour of difference in the time at which noon occurs. Hence longitude, and difference of longitude, may be estimated in time, allowing fifteen degrees to an hour, or one degree to four minutes.

If a small circle MPK, or mpk, be drawn parallel to the equator at any distance from it, all the places through which that small circle passes will have the same latitude. Such small circles are accordingly called parallels of latitude. The difference of latitude between two places is the arch of the meridian intercepted between their parallels of latitude, and is found by subtracting or adding the latitudes, according as the places lie on the same or on opposite sides of the equator.

On the hypothesis of the earth being a perfect sphere, the degree of latitude must be assumed as everywhere of the same length. Actual observation shows that this is not exactly the case, but that the length of the degree increases slightly as we proceed from the equator towards either pole (see Figure of the Earth), so as to indicate a flattening of the surface in the latter regions, or that the earth is of a spheroidal figure. This, however, we do not at present take into view. With regard to the degree of longitude, it is at the equator of the same length with the degree of latitude. But on both sides of the equator it decreases gradually on account of the meridians approaching each other until they intersect at the poles. At the equator a degree of longitude is the 360th part of the circumference of a great circle of the terrestrial sphere; at every point to the north or south of the equator it is the 360th part of the circumference of a small circle, of which the radius is equal to the cosine of the latitude. Now, since similar portions of the circumferences of circles are to one another as their radii, if we put L for the length of the degree of longitude at the equator, and L' for its length in a latitude denoted by l, we have rad.: cos.l:: L:L'. Hence, radius being unity, we obtain

\[ L' = L \times \cos l. \]

From this formula the following table is calculated; the length of the degree of longitude at the equator being assumed equal to 69-06 English miles.

| Lat. | Geo. Miles | Eng. Miles | Lat. | Geo. Miles | Eng. Miles | Lat. | Geo. Miles | Eng. Miles | |------|------------|------------|------|------------|------------|------|------------|------------| | 1 | 59-99 | 69-05 | 31 | 51-43 | 59-20 | 61 | 29-09 | 33-48 | | 2 | 59-96 | 69-02 | 32 | 50-88 | 58-57 | 62 | 28-17 | 32-42 | | 3 | 59-92 | 68-96 | 33 | 50-32 | 57-92 | 63 | 27-24 | 31-35 | | 4 | 59-85 | 68-89 | 34 | 49-74 | 57-25 | 64 | 26-30 | 30-27 | | 5 | 59-77 | 68-80 | 35 | 49-15 | 56-57 | 65 | 25-36 | 29-19 | | 6 | 59-67 | 68-68 | 36 | 48-54 | 55-87 | 66 | 24-40 | 28-09 | | 7 | 59-55 | 68-55 | 37 | 47-92 | 55-15 | 67 | 23-44 | 26-98 | | 8 | 59-42 | 68-39 | 38 | 47-28 | 54-42 | 68 | 22-48 | 25-87 | | 9 | 59-26 | 68-21 | 39 | 46-63 | 53-67 | 69 | 21-50 | 24-75 | | 10 | 59-09 | 68-01 | 40 | 45-96 | 52-90 | 70 | 20-52 | 23-62 | | 11 | 58-90 | 67-79 | 41 | 45-28 | 52-12 | 71 | 19-53 | 22-48 | | 12 | 58-69 | 67-55 | 42 | 44-59 | 51-32 | 72 | 18-54 | 21-34 | | 13 | 58-46 | 67-29 | 43 | 43-88 | 50-51 | 73 | 17-54 | 20-19 | | 14 | 58-22 | 67-01 | 44 | 43-16 | 49-68 | 74 | 16-54 | 19-04 | | 15 | 57-96 | 66-71 | 45 | 42-43 | 48-83 | 75 | 15-53 | 17-97 | | 16 | 57-68 | 66-38 | 46 | 41-68 | 47-97 | 76 | 14-52 | 16-71 | | 17 | 57-38 | 66-04 | 47 | 40-92 | 47-10 | 77 | 13-50 | 15-54 | | 18 | 57-06 | 65-68 | 48 | 40-15 | 46-21 | 78 | 12-47 | 14-36 | | 19 | 56-73 | 65-30 | 49 | 39-36 | 45-31 | 79 | 11-45 | 13-18 | | 20 | 56-38 | 64-90 | 50 | 38-57 | 44-39 | 80 | 10-42 | 11-99 | | 21 | 56-01 | 64-47 | 51 | 37-76 | 43-46 | 81 | 9-39 | 10-80 | | 22 | 55-63 | 64-03 | 52 | 36-94 | 42-52 | 82 | 8-35 | 9-61 | | 23 | 55-23 | 63-60 | 53 | 36-11 | 41-56 | 83 | 7-31 | 8-42 | | 24 | 54-81 | 62-09 | 54 | 35-27 | 40-59 | 84 | 6-27 | 7-22 | | 25 | 54-38 | 62-59 | 55 | 34-41 | 39-61 | 85 | 5-23 | 6-02 | | 26 | 53-93 | 62-07 | 56 | 33-55 | 38-62 | 86 | 4-19 | 4-83 | | 27 | 53-46 | 61-53 | 57 | 32-68 | 37-61 | 87 | 3-14 | 3-61 | | 28 | 52-98 | 60-98 | 58 | 31-79 | 36-60 | 88 | 2-09 | 2-41 | | 29 | 52-48 | 60-40 | 59 | 30-90 | 35-57 | 89 | 1-05 | 1-21 | | 30 | 51-96 | 59-81 | 60 | 30-00 | 34-53 | 90 | 0-00 | 0-00 | If a great circle $Bd$ be supposed to be drawn on the terrestrial sphere, cutting the equator obliquely at an angle of about twenty-three and a half degrees, this circle will mark out the course of the sun through the year, and is called the ecliptic. It corresponds with the celestial ecliptic, and is divided in the same manner into signs and degrees. The parallel of latitude which passes through the point of the ecliptic in which the sun is placed on any particular day, shows to what points of the earth's surface the sun is vertical on that day. The two parallels BLD, bld, which touch the ecliptic at the points where it recedes farthest from the equator to the north and south, are called the tropics; the one, BLD, the northern tropic, or the tropic of cancer; and the other, bld, the southern tropic, or the tropic of capricorn; because they touch the ecliptic in the first points of these signs. These circles lie in the planes of the corresponding circles in the celestial sphere.

Of the two poles, that which lies in the northern hemisphere is called the north or arctic pole; and the opposite, lying in the southern hemisphere, is called the south or antarctic pole. The two parallels of latitude FG, fg, which encircle these poles respectively at an angular distance equal to the obliquity of the ecliptic, are called the polar circles, the one the north or arctic, and the other the south or antarctic.

Suppose a great circle to be drawn on the terrestrial sphere everywhere equally distant from that point on the surface to which the sun is vertical at any given time; this circle is called the circle of illumination, because it separates the enlightened from the dark hemisphere of the earth. It is upon the position of the circle of illumination that the equal or unequal lengths of the days and nights throughout the year depend over the face of the earth. At every season of the year this circle bisects the equator, so that under the equator the days and nights are always equal. When the sun is over either of the equinoctial points E, Q (Plate CCLIX, fig. 6), the circle of illumination bisects not only the equator, but likewise all the parallels to the equator, in consequence of cutting them at right angles; and hence, as the parallel of latitude passing through any point on the earth's surface may be regarded as the path along which that point is carried by the diurnal motion of the earth, at that season of the year the days and nights are equal over all the earth. When the sun is over any other point of the ecliptic, the parallels of latitude are cut obliquely by the circle of illumination, so that they are divided by it into two unequal parts (Plate CCLIX, fig. 7, 8); and hence the days and nights are unequal all over the earth, except at the equator. If the sun is in north declination (fig. 7) at all places to the north of the equator, the days are longer than the nights; but at all places to the south of the equator the nights are longer than the days; because with regard to places to the north, the larger portion of the parallel of latitude lies within the enlightened hemisphere, while with regard to places to the south the larger portion lies within the dark hemisphere. The reverse has place when the sun is in south declination (fig. 8). When the sun is over the northern tropic BD (fig. 7), or over the southern tropic bd (fig. 8), the day is then the longest or shortest of the year; and the adjacent polar circle is wholly in the light, and the opposite one wholly in darkness. Thus, within the arctic and antarctic circles the inhabitants have their light and darkness extended to a great length; the sun sometimes skirting round a little above the horizon for many days together, and at another season never rising above the horizon at all, but making continual night for a considerable length of time.

Such are the circles supposed to be drawn on the terrestrial sphere. Though these circles are altogether imaginary, and have really nothing corresponding to them on the earth's surface, yet as the positions of the corresponding circles in the heavens can be accurately determined by observation on the celestial bodies, they become actually as effective for fixing the relative positions of places on the globe of the earth, and for other purposes of geography, as if that globe were reduced to such a magnitude as to admit of its being grasped by the hand of man, and surveyed by a single glance of his eye.

The height NO of the pole (Plate CCLIX, fig. 2) adjacent to any place M above HO the horizon of that place, is equal to the latitude of the place; for the arches MO and EN are equal, each being a quadrant; and taking away from each the common arch MN, the remaining arch NO, the elevation of the pole N, is equal to the remaining arch ME, the latitude of the place M.

Hence there are, with regard to the horizon, three positions of the terrestrial sphere, as there are of the celestial, depending on the latitude of the place. If an observer were placed on the pole, the latitude being 90°, his horizon would coincide with the equator, and all the parallels of latitude would be parallel to the horizon. This is the parallel position of the sphere, which is represented in Plate CCLIX, fig. 3. Again, if an observer were placed on any point of the equator, the latitude being 0°, the poles of the earth must be in the horizon; and from the properties of the sphere, the equator, and all the small circles parallel to the equator, must be at right angles to the horizon. This is the right position of the sphere which is represented in fig. 4. With regard to an observer placed at any point between the equator and either pole, the axis of the earth lies obliquely to the plain of his horizon, which is therefore cut obliquely by the equator and all the small circles parallel to the equator: this is the oblique position of the sphere, and is that represented in fig. 2.

The method of determining the latitude of a place, and the difference of longitude between two points on the earth's surface, is given in the article ASTRONOMY, part iv. chap. i. prop. ix. and x. The latitudes and longitudes of two points on the surface of the earth being given, the angular distance may be found; and hence the length of a degree on the earth's surface being known, the distance between the points may be expressed in miles. Thus, let A and B be two points on the earth's surface (Plate CCLIX, fig. 5); let P be the adjacent pole, and PA, PB the meridians passing through A and B; then PA and PB will be the complements of the latitudes, and therefore given; and the spherical angle APB will be the difference of the longitudes, and therefore also given. In the spherical triangle PAB, the base AB of which is the distance required, we have therefore two sides and the included angle, so that the base AB can be found. Put D and D' for the complements of the given latitudes, and P for the difference of the given longitudes; then, by spherical trigonometry, we have

$$\cos AB = \cos D \cos D' + \sin D \sin D' \cos P.$$

By assuming $\tan \phi = \tan D \cos P$, this formula may be reduced to the more convenient form

$$\cos AB = \frac{\cos D \cos (D' - \phi)}{\cos \phi}.$$

From the above expression the number of degrees and parts of a degree contained in the arch intercepted between the points A and B are to be found; and allowing for each degree 69-06 English miles, the distance will be obtained, expressed in English miles, which will necessarily differ a little from the truth, in consequence of the earth not being truly spherical. To determine the distance between two points on the earth's surface with precision, requires the application of some of the most refined improvements in modern mathematics. See FIGURE OF THE EARTH. As an example of the above formula, let it be required to find the distance between Edinburgh and Constantinople.

Colat. of Edinburgh ...34° 3' = D; its long. 3° 12' W. Colat. of Constantinople 48° 59' = D'; its long. 28° 55' E.

Difference of longitude .....= 32° 7' = P.

\[ \tan D = \tan 34° 3' = 9.829805 \\ \cos P = \cos 32° 7' = 9.927867 \]

\[ \tan \phi = 20° 47' = 9.757672 \\ D' = 48° 59' \]

\[ D' - \phi = 19° 12' \quad \text{Its cosine} = 9.975145 \\ \cos D = 9.918318 \]

Arith. compl. cosine \( \phi = 0.061525 \)

\[ \cos AB = 25° 38' = 9.954988. \]

Hence the arch of a great circle of the sphere intercepted between Edinburgh and Constantinople is 25° 38', 1770 English miles nearly.

Each of the four quadrants into which the horizon is divided by the equator and the meridian is supposed to be divided into eight equal parts; so that the whole circumference is divided into thirty-two equal parts, which are called points of the compass; and to each point of vision a name is given indicating its position with regard to the four cardinal points.

The position of one place with regard to another, as estimated by the points of the compass, is called the bearing of the former from the latter. Thus, when a place is said to bear N.E. (north-east), N.N.E. (north-north-east), etc., the meaning is, that it lies in the direction of those points in the horizon from the present position of the observer. If a series of points be assumed on the earth's surface, so situated that all of them, when taken in regular succession, lie towards any the same point of the compass, except either of the four cardinal points, the assumed points lie, not in the circumference of a circle of the sphere, but in a sort of spiral line, the characteristic property of which is, that it cuts all the meridians at the same angle. This line is called a rhumb line; and, whilst it continually approaches the pole, it can never arrive at it, except after an infinite number of revolutions. In passing, therefore, from one point of the surface of the globe to another, by pursuing the direction in which the latter lies from the former, we do not take the shortest way, which is an arch of a great circle, but move over a portion of a rhumb line, passing through the two points. This is the line described by a ship, whilst her course is continually directed towards one and the same point of the compass.

**MAP II.—OF THE SURFACE OF THE EARTH, AND ITS GENERAL DIVISIONS: DEFINITIONS.**

The surface of the earth contains about 196,663,400 square miles. By much the larger portion of this space is water, which is, indeed, more than twice the extent of land. The surface of the land is exceedingly diversified, almost everywhere rising into hills and mountains, sinking into valleys, and sometimes stretching out into plains of great extent. Amongst the most extensive plains are the sandy deserts of Arabia and Africa, the internal parts of European Russia, and a tract of considerable extent in Prussian Poland. But the most remarkable extent of level ground is the vast table-plain of Thibet in Asia, which is the most elevated tract of level ground on the globe. The principal mountain ridges are the Alps and Pyrenees in Europe, the Altai and Himalaya Mountains in Asia, the mountains of Atlas in Africa, and the Andes Cordilleras in South America. The greatest concavities of the globe are those which are occupied by the waters of the ocean; and of these by far the largest forms the bed of the Pacific Ocean, which, stretching from the northeastern shores of Asia and of New Holland to the western coast of America, covers nearly half the globe. The concavity next in extent is that which forms the bed of the Atlantic Ocean, extending between the new and the old worlds; and a third concavity is occupied by the Indian Ocean. The Arctic and Antarctic Oceans fill up the remaining concavities.

Smaller collections of water which communicate freely with the oceans are called seas (Plate CCLIX. fig. 9, 1); and of these, the principal are the Mediterranean, the Baltic, the Euxine or Black Sea, and the White Sea. Seas sometimes take their names from the countries near which they flow; as the Irish Sea, the German Ocean. Some large collections of water, though they have no immediate connection with the great body of waters, being on all sides surrounded by land, are yet called seas; as the Caspian Sea.

A part of the sea running up into the land, so as to form a large hollow, is called a bay or gulf (fig. 9, 2), as the Bay of Biscay, the Gulf of Mexico; but if the hollow be small, it is called a creek, a road, a haven.

When two large bodies of water communicate by a narrow pass between two adjacent lands, the pass is called a strait or straits (fig. 9, 3), as the Straits of Gibraltar, the Straits of Dover, the Straits of Babelmandel. A channel is a wider kind of strait. The water usually flows through a strait with considerable force and velocity, forming what is called a current; and frequently this current, as in the case of the Straits of Gibraltar, flows continually in the same direction.

A body of fresh water entirely surrounded by land is called a lake (fig. 9, 4), as the Lake of Geneva, Lake Champlain.

A considerable stream of water rising inland, and draining a portion of country more or less extensive, discharging its waters into the sea, is called a river. A smaller stream of the same kind is called a rieulet, or brook (fig. 9, 5).

The great extent of land which forms the rest of the surface of the globe is divided into innumerable portions; two of vast extent, but the rest of comparatively small dimensions. The portions of the land which are of great extent are called continents, the one the eastern continent, or the old world, comprehending Europe, Asia, and Africa; the other the western continent, or new world, comprehending North and South America. New Holland is a third portion of land, however, which has by some been also reckoned a continent on account of its great extent.

A portion of land entirely surrounded by water is called an island (fig. 9, 6), as Britain, Ireland, Jamaica, Madagascar. According to the strict meaning of this definition, the largest portions of land on the surface of the globe may be said to be islands; for there is every probability that each of the continents is everywhere bounded by the sea. But this is an extent of signification in which the term island is not employed; and New Holland is the largest portion of land which is called an island. When a number of smaller islands lie near each other they are said to form a group of islands (fig. 9, 7).

A portion of land which is almost entirely surrounded by water is called a peninsula (fig. 9, 8), as the peninsula of Malacca, the Morea or Grecian Peloponnesus, etc. The term peninsula is often applied to a large extent of country. Thus we speak of Spain as a peninsula. Indeed Africa itself may be considered as a vast peninsula, being connected with Asia only by the narrow isthmus of Suez.

The narrow neck of land which joins a peninsula to the main land, or which connects two tracts of country together, is called an isthmus (fig. 9, 9). The most remarkable isthmuses in the world are, the Isthmus of Suez, which joins Africa and Asia, and the Isthmus of Darien, which connects the continents of North and South America. A narrow tract of land stretching out into the sea, and appearing to terminate in a point, is called a cape (fig. 9, e). The most remarkable capes are, the Cape of Good Hope, at the southern extremity of Africa; Cape Horn, at the southern extremity of South America; and the North Cape, at the northern extremity of Europe. A large portion of land jutting out into the sea is called a promontory (fig. 9, f).

It may assist the memory to keep in view the analogy which subsists between the denominations of land and those of water. The large portions of land called continents correspond to the extensive tracts of water called oceans; islands are analogous to lakes; peninsulas to seas or gulfs; isthmuses to straits; capes to creeks; and so with the others.

Until of late, in systems of geography, the earth used to be considered as divided into four quarters; Europe, Asia, Africa, and America. A classification in which the whole world is arranged under seven divisions has now however been very generally adopted: These divisions are, Europe, Asia, Africa, North America, South America, Australasia, and Polynesia. With regard to the last two, the one, Australasia, or South Asia, comprehends certain of the great islands, particularly New Holland, which are usually considered as belonging to Asia; and the other, Polynesia, signifying many islands, comprehends all the smaller islands which are scattered over the great expanse of the Pacific Ocean.

This arrangement and classification of the parts of the earth's surface are founded on the most obvious points of distinction. We shall now explain two divisions employed by the ancients, which are founded upon different principles: that into zones, and that into climates.

The division into zones is suggested by the different degrees of temperature which prevail in different regions of the earth. The temperature of a country depends on a variety of circumstances (see Physical Geography); but of these, one of the most obvious is the position of the sun with regard to the zenith. The more nearly his rays are received vertically, the higher will be the temperature; and, on the contrary, the more obliquely they fall, the less effect will they produce in raising the temperature. Now to every point of the earth's surface between the tropics the sun is vertical twice in the year. It is in this region, then, that the highest temperature will prevail. Again, within the polar circles the sun's rays at all times fall very obliquely; and for a length of time they do not reach these two regions of the globe at all. Here, then, the temperature must be lower than anywhere else, as all other places enjoy more of the sun's genial influence. In the two regions between the tropics and the polar circles, a medium temperature is found, increasing as we approach the former, and diminishing as we approach the latter. Thus is the earth's surface divided, by the two tropics and two polar circles, into five zones, distinguished from one another by the prevailing temperature in each. That between the tropics is called the torrid zone, because there the heat is understood to be extreme. This region, which has the equator passing through the middle of it, the ancients, indeed, considered as uninhabitable. The two regions comprehended within the arctic and antarctic polar circles are called the northern and southern frigid zones, on account of the severity of the cold which there prevails. The two regions situated between the tropics and the polar circles, the one in the northern hemisphere, bounded by the tropic of cancer and the arctic circle, the other in the southern hemisphere, bounded by the tropic of capricorn and the antarctic circle, are called the northern and southern temperate zones, because there neither the heat nor cold is excessive; but the heat reaches the highest temperature of summer, and the cold sinks to the lowest temperature of winter, without either becoming extreme.

As each tropic lies about $23\frac{1}{2}$° from the equator, the breadth of the torrid zone is about $47°$, or nearly 3240 English miles. The breadth of each of the frigid zones, that is, from the circumference of the polar circle to the pole, is $23\frac{1}{2}°$, or nearly 1620 miles; so that there remains for the breadth of each of the temperate zones about $48°$, or nearly 2970 miles. The superficial content of each zone is easily calculated by the ordinary rules of mensuration. Let $s$ be the surface of a segment of the sphere, $d$ the diameter of the sphere, and $h$ the height of the segment; then

$$ s = \frac{3}{4} \pi d h $$

To apply this formula: let NESQ (Plate CCLIX, fig. 3) be the sphere divided into five zones. It is evident that, for the torrid zone, $h$ must be equal to $LR$, or to twice the sine of BE to the radius EC; that for each of the temperate zones $h$ must be equal to LR or $lr$, the difference of the sines FE and BE to the same radius CE; and that for the space comprehended within each polar circle, $h$ must be equal to NR, the excess of the radius NC above CR, the sine of FE corresponding to the radius. Hence, if we put $m$ for twice the natural sine of the arch BE; for the difference of the natural sines of the arches BE and FE; and for the excess of radius above the natural sine of FE, successively;—since in the case of the terrestrial sphere CE or CN is equal to 3956 miles, we have $h = 3956 m$. Putting, therefore, instead of $d$, its value, 7912, and giving the above formula a logarithmic form, we obtain

$$ \log_s = \log_m + 7.9926935 $$

Now, for the torrid zone, $m = 2 \text{ nat. sin. } 23° 30' = 7915$. Hence,

$$ \log_m = \frac{1}{7} \log_9 017307 - \frac{1}{7} \log_9 9926935 $$

$$ s = 78419520 $$

The torrid zone contains, therefore, about 78,419,500 square miles.

Again, for each of the temperate zones, we have $m = \text{ nat. sin. } 66° 30' - \text{ nat. sin. } 23° 30' = 51831$. Hence,

$$ \log_m = \frac{1}{7} \log_9 7145986 - \frac{1}{7} \log_9 9926935 $$

$$ s = 50966300 $$

So that each of the temperate zones contains about 50,966,300 English square miles.

Lastly, for the space contained within each of the polar circles we have $m = \text{ rad. } - \sin 66° 30' = 1 - 0.91706 = 0.08294$: Hence,

$$ \log_m = \frac{1}{7} \log_9 8155630 - \frac{1}{7} \log_9 9926935 $$

The portion of the surface of the globe comprehended within each polar circle is therefore nearly equal to 8,155,600 square miles.

It is evident that the superficial content of any other zone of the terrestrial sphere may be found by the above formula, by putting $m$ equal to the difference of the natural sines corresponding to the latitudes of the parallels by which the zone is bounded.

If it is required to find the area of a segment of a zone bounded at both extremities by meridians, it is only necessary to find first the area of the whole zone, and then to multiply the result by the number of degrees and parts of a degree in the length of the segment; and to divide by 360. Thus, the number of square miles contained in the portion of the torrid zone terminated by two meridians which are separated from each other by $8° 45'$ is equal to $78419520 \times \frac{8}{4} \div 360$, which gives 272,290 square miles. By divid- Besides dividing the surface of the globe into zones and climates, the ancients likewise distinguished the inhabitants of the different regions of the earth by the particular direction in which the shadows of bodies are projected in each region, and by some other circumstances depending on the position of the sun relatively to the zenith or to the meridian. The inhabitants of the torrid zone have their shadows at noon projected, sometimes towards the south and sometimes towards the north, according to the position of the sun in the ecliptic. They were therefore called by the ancients *Amphiscii*, a term derived from *auxi*, about, and *exa*, a shadow. In the temperate zones, the sun is at noon always on the same side of the zenith. Hence the shadows of objects at noon always fall in the same direction; in the northern temperate zone towards the north, and in the southern temperate zone towards the south. The inhabitants of these regions were accordingly called *Heterocii*, from *irrig*, different, and *exa*. Within the polar circles the sun does not always rise and set every twenty-four hours, as in the other zones; but for a certain number of days in our summer he never sets to places within the arctic polar circle, nor rises to places within the antarctic; and the contrary takes place for a certain number of days during our winter. The number of days during which the sun is present or absent increases as we advance from the polar circle towards the pole. When the sun continues above the horizon for twenty-four hours or upwards, the shadows will make a complete circuit round the objects from which they are projected; hence the inhabitants of the frigid zones were called *Periscii*, from *epy*, about, and *exa*.

Again, the inhabitants of two places which lie under the same meridian, and have the same latitude, but are situated on opposite sides of the equator, were called, relatively to each other, *Antecii*, from *arr*, opposite to, and *exa*, a habitation. At such places the hours of the day will always correspond, but the seasons of the year will be opposite. The inhabitants of two places which have the same latitude, and are situated on the same side of the equator, but under opposite meridians, were called *Periciei*, from *epy*, about, and *exa*. They have always the same season of the year at the same time; but any hour of the day at the one place corresponds to the same hour of the night at the other. The inhabitants of two places which have the same latitude, but are situated on opposite sides of the equator, and under opposite meridians, are called *Antipodes* to each other, from *arr*, opposite to, and *evoz*, a foot. They have always opposite seasons and opposite hours.

**CHAP. III.—DESCRIPTION AND USE OF THE GLOBES.**

When geographers became familiar with the doctrine of the sphericity and motion of the earth, it was an obvious step to have recourse to an artificial sphere for illustrating that doctrine. From a very early period, accordingly, the instruments called the *terrestrial* and *celestial* globes have been employed for this purpose. A sphere made of metal, ivory, plaster, paper, pasteboard, or some other convenient substance, is suspended in a brass ring NES (Plate CCLX. fig. 1), of somewhat greater diameter, on two pins N, S, upon which it can be made to revolve. The sphere, thus suspended, is placed in a frame BMC, which may be in many respects variously constructed, according to the taste of the workman; but its upper part BC always consists of a broad ring made of metal or wood, and supported in a horizontal position. The inner circumference of this ring is equal to that of the brass circle in which the globe is suspended; and two notches in its inner edge, diametrically opposite to each other, receive the brass ring, which also rests in a groove below, in such a posi-

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**Table of Climates.**

| Climates | Latitude of the Higher Parallel | Breadth of the Climate | Longest Day under the Higher Parallel | |----------|---------------------------------|------------------------|--------------------------------------| | I | 8° 34' | 8° 34' | 12° 30' | | II | 16° 43' | 8° 9' | 13° 0' | | III | 24° 10' | 7° 27' | 13° 30' | | IV | 30° 46' | 6° 36' | 14° 0' | | V | 36° 28' | 5° 42' | 14° 30' | | VI | 41° 21' | 4° 53' | 15° 0' | | VII | 45° 29' | 4° 8' | 15° 30' | | VIII | 48° 59' | 3° 30' | 16° 0' | | IX | 51° 57' | 2° 58' | 16° 30' | | X | 54° 28' | 2° 31' | 17° 0' | | XI | 56° 36' | 2° 8' | 17° 30' | | XII | 58° 25' | 1° 49' | 18° 0' | | XIII | 59° 57' | 1° 32' | 18° 30' | | XIV | 61° 16' | 1° 19' | 19° 0' | | XV | 62° 24' | 1° 8' | 19° 30' | | XVI | 63° 20' | 0° 56' | 20° 0' | | XVII | 64° 8' | 0° 48' | 20° 30' | | XVIII | 64° 48' | 0° 40' | 21° 0' | | XIX | 65° 20' | 0° 32' | 21° 30' | | XX | 65° 46' | 0° 26' | 22° 0' | | XXI | 66° 6' | 0° 20' | 22° 30' | | XXII | 66° 20' | 0° 14' | 23° 0' | | XXIII | 66° 28' | 0° 8' | 23° 30' | | XXIV | 66° 32' | 0° 4' | 24° 0' |

| Months | | | | |----------|---------------------------------|------------------------|--------------------------------------| | I | 67° 23' | 0° 51' | 1 | | II | 69° 50' | 2° 27' | 2 | | III | 73° 39' | 3° 49' | 3 | | IV | 78° 31' | 4° 52' | 4 | | V | 84° 5' | 5° 34' | 5 | | VI | 90° 0' | 5° 55' | 6 | Mathematical arrangement that the plane of the horizontal ring bisects the spherical Geo- sphere. By this arrangement the globe can be made to revolve on its axis, and the brass circle can be made to slide round in its own plane. On the surface of the globe are delineated the equator or equinoctial line, situated exactly in the middle between the two points on which the globe is suspended, and divided into 360 degrees; the ecliptic, divided into twelve signs, and each of these subdivided into thirty degrees; the two tropics; the two polar circles, with as many more parallels to the equator as are found convenient; and generally twenty-four meridians passing through the points of suspension, which represent the poles. The first meridian is usually made to pass through the intersections of the equator and ecliptic, the points of the vernal and autumnal equinoxes; and from the former of these points the reckoning of the degrees on the equator and ecliptic begins. The brass circle in which the globe hangs may be made to represent the meridian of any given point on the surface of the globe, by simply bringing the given point under it by turning the globe round on its axis. Hence the brass circle is called the universal meridian. It is divided, by the equator and two poles, into four quadrants, each of which is graduated; and on the semicircle NES (Plate CCLX. fig. 1) the degrees are reckoned from the equator towards either pole, while on the opposite semicircle they are reckoned from either pole towards the equator. On the broad horizontal circle of the frame in which the globe stands are drawn several concentric circles, the outer of which is divided into 365 equal parts, answering to the number of days in the year; whilst the other circles are graduated, the innermost (or rather another circle concentric with it, but larger) being, besides, divided into thirty-two equal parts corresponding to the points of the compass. The circle next the outer edge forms the calendar, and it has the names of the months arranged in order around it, whilst the divisions are distributed so as to mark the number of days in each. The adjacent circle contains the signs and degrees of the ecliptic, so arranged that against each day of the year is found the point of the ecliptic in which the sun is situated on that day. The innermost circle represents the horizon; and the two notches in which the brazen meridian rests pass through the north and south points. It is divided into four quadrants by the cardinal points; and the degrees of the two quadrants, which form the northern semicircle, are reckoned from the east and west points towards the north; while the degrees of the other two quadrants are reckoned from the east and west towards the south.

Such is a general view of the parts which belong alike to both globes, the terrestrial and the celestial. A more minute description seems unnecessary, as a careful inspection of the globes themselves, with their appendages, will convey a much more distinct conception of them than can be given either by description or by drawings. It will be necessary, however, to describe shortly the horary or hour-circle, the quadrant of altitude, and the compass.

The hour-circle is a small circle of brass, divided into twenty-four equal parts corresponding to the hours of the day, the divisions being reckoned in two twelves to suit the hours before and after noon. This circle is fixed on the axis of the globe, having its centre coinciding with the north pole. It can be adjusted with the hand to any meridian, but is at the same time tight enough to be moved along with the globe. Some globes have the hour circle fixed on the meridian, with an index that admits of being adjusted with the hand, but is carried round with the globe.

The quadrant of altitude is a thin flexible slip of brass, equal in length to one fourth part of the circumference of a great circle on the globe. It is graduated on one side, and is furnished with a nut and screw at one end, for the purpose of making it fast to the brazen meridian. Its use is to measure degrees on the surface of the globe in any direction.

The compass is simply a magnetic needle suspended over the centre of a circle, on the circumference of which are marked the thirty-two points of the compass. It is fixed to the under part of the frame in which the globe is suspended, and is used for the purpose of placing the meridian due north and south.

On the terrestrial globe the land and water which compose the surface of the earth are delineated, with the various divisions belonging to each. If we suppose the globe to be six or seven feet in diameter, the true height of the mountains on the earth's surface must be reduced, in order to be represented on the globe in due proportion to its bulk, in the ratio of 7912 miles to six or seven feet; that is, the height of the protuberance on the globe which shall represent any particular mountain must be somewhat about the six millionth part of the actual elevation of the mountain above the general surface of the earth. Apply this to the highest mountain in the world, which does not exceed five and a half miles in height; the elevation of the protuberance representing it on a globe of the supposed dimensions would be about one seventeenth part of an inch above the general surface. It is seldom, however, that globes are made of this size. One third of the supposed diameter is more near to the ordinary dimensions. Hence we see with what propriety the earth is represented by a globe having a smooth surface.

But in order to render the representation more complete, it may be supposed necessary that the terrestrial globe should be enclosed in a hollow sphere which would represent the heavens surrounding the earth on all sides. In conformity with this idea the armillary sphere was contrived, in which the several circles of the system of the world, put together in their natural order, are represented, with a small globe in the centre of the sphere to represent the earth. Fig. 3, Plate CCLX., represents a sphere of this description, which will be sufficiently understood by inspection. The ordinary way, however, of representing the heavens, proceeds on the supposition that the eye of the observer is placed, not in the centre of the celestial sphere, but beyond its bounds, so as to look down on a convex surface. Thus are the stars and constellations represented in their relative positions on the celestial globe (fig. 2), with which the concave surface of the visible heavens is easily compared.

Besides answering the general purposes of illustration, the globes furnish also the means of resolving with facility, and with a degree of accuracy sufficient for ordinary purposes, many problems in geography and practical astronomy. It is only requisite to consider the circumstances on which the solution of a problem depends, and to arrange on the globe these circumstances, according to their natural order and dependence, and the result is at once obtained. We proceed to give a few of the more useful problems, with their solutions by the globes.

I. Solution of Problems by the Terrestrial Globe.

Prob. 1. To find the latitude and longitude of a given place.

Bring the place under the graduated edge of the brass meridian, then the degree of the meridian immediately over it is its latitude north or south, and the degree of the equator cut by the meridian is its longitude east or west.

Prob. 2. The latitude and longitude of a place being given, to find the place itself on the globe. Bring the point of the equator corresponding to the given longitude to the brazen meridian; under the degree of latitude on the meridian the place is found.

**Prob. 3.** To find the distance between any two places in the globe.

Lay the quadrant of altitude over the two places, and mark the number of degrees between them. The degrees may then be converted into English miles, if required, by multiplying by 69°06'.

**Prob. 4.** The hour at any one place being given, to find what hour it is at any other place; or to find the difference of longitude between the places in time.

Bring the place at which the hour is given to the brazen meridian, and set the index, or the hour-circle, to that hour; then turn the globe till the other place comes under the meridian, and the index, or the hour-circle, will show the hour required. The difference between the time found and the time given is the difference of the longitudes of the places in time.

**Prob. 5.** To rectify the globe for a given place.

Elevate the pole that is adjacent to the place as many degrees above the wooden horizon as are equal to the latitude.

**Prob. 6.** To find at what hour the sun rises and sets at a given place, for any given day.

Rectify the globe for the latitude of the place; find from the wooden horizon the sun's place in the ecliptic for the given day, and bring it to the meridian. Set the index to XII., and turn the globe till the sun's place comes to the eastern edge of the horizon, the index will show the hour of rising; then turn the globe till the sun's place comes to the western edge of the horizon, and the index will show the time of setting.

By doubling the hour of sunrise, we obtain the length of the night; and by doubling the hour of sunset, we obtain the length of the day. It is evident also that the same arrangement of the globe will give the point of the compass on which the sun rises and sets, by simply observing what point of the circle of rhumbs, on the wooden horizon, is cut by the sun's place in the ecliptic at the time of rising and setting. Further, by observing when the sun is about eighteen degrees below the eastern and western parts of the horizon, the time of the beginning of the morning twilight, and of the ending of the evening twilight, is found.

**Prob. 7.** To find the sun's declination for a given day of the month, and to what places the sun will be vertical that day.

Find, on the wooden horizon, the sun's place in the ecliptic for the given day; bring that point of the ecliptic to the meridian, the degree immediately over it on the meridian is the declination north or south. Turn round the globe till it has made a complete revolution; and to every place which passes under that degree of the meridian, the sun will be vertical on that day.

**Prob. 8.** The hour and day being given at a particular place, to find where the sun is then vertical.

Find the sun's declination by the preceding problem; bring the given place under the meridian, and set the index to the given hour; then turn the globe till the index points to XII. noon; and all the places under the meridian will have noon at the given time; and that place which is under the degree of the meridian that corresponds with the sun's declination will have the sun in the zenith.

**Prob. 9.** The hour and day being given at a particular place, to find where the sun is then rising or setting, and where it is noon or midnight.

Rectify the globe for the given place; and having previously found by last problem the place to which the sun is vertical at the given time, bring that place to the meridian. Then, to all the places under the western edge of the horizon the sun is rising, and to those under the Mediterranean he is setting; to those under the upper half of the meridian it is noon, and to those under the lower half it is midnight.

**Prob. 10.** A place in the torrid zone being given, to find on what two days of the year the sun will be vertical to that place.

Find the latitude of the place; turn round the globe, and note the two points of the ecliptic which pass below the degree of latitude on the meridian. Find in the calendar circle of the wooden horizon the days corresponding to these two points of the ecliptic, and these are the days required.

**Prob. 11.** To find the sun's meridian altitude at a given place on a given day.

Rectify the globe for the latitude of the place; bring the sun's place for the given day to the meridian; count the number of degrees between that place and the horizon for the altitude required.

**Prob. 12.** To find the altitude of the sun at any given place and hour.

Rectify the globe for the latitude; bring the sun's place to the meridian, and set the index to XII. noon; turn the globe till the index point at the given hour; fix the quadrant of altitude on the meridian at the degree of latitude, and lay it over the sun's place; count the number of degrees on the quadrant between that point and the horizon for the altitude required.

**Prob. 13.** To find all the places to which a lunar eclipse is visible at any instant.

Find the place to which the sun is vertical at the given time; rectify the globe for the latitude of that place, keeping the place under the meridian; set the index to XII. noon; then turn the globe till the index point to XII. midnight; the eclipse will be visible to all those places which are above the horizon.

**Prob. 14.** Any place in the north frigid zone being given, to find how long the sun shines there without setting, and how long he is totally absent.

Rectify the globe for the latitude of the place; bring the ascending signs of the ecliptic to the north point of the horizon, and note at what degree the ecliptic is intersected by that point; find on the wooden horizon the day and month corresponding to that degree; from that day the sun begins to shine continually. Next, bring the descending signs to the north point of the horizon, and by observing at what degree the ecliptic is now cut, and referring to the horizon, we find the time when the sun ceases to shine without setting, which is the termination of the longest day. By proceeding in the same manner with the southern point of the horizon, we will find the beginning and end of the longest night.

**Prob. 15.** Two places being given, to find the angle which a great circle passing through them makes with the meridian of each.

Rectify the globe for both places successively, bringing in each case the place for which the globe is rectified to the meridian; fix the quadrant of altitude in each operation on the meridian over the place for which the globe is rectified, and lay it over the other place; the two arches intercepted successively on the horizon between the quadrant and the meridian measure the angles required. It is evident, that if both places lie on the same meridian, the angle is 0°; and that if both lie on the equator, the angles will be each 90°.

If a ship be supposed to sail from the one place to the other, on a great circle of the terrestrial sphere (which supposition implies that the ship's course is altered every instant), then the one angle found by this problem will be the course with which the ship left the one place, and the other angle the course with which she arrived at the Mathema-other place. The arch of the great circle intercepted between the places would be the distance sailed.

**Prob. 16.** To construct a horizontal dial by the globe for a given latitude.

Rectify the globe for the latitude; bring the first meridian under the brazen meridian, and note the arches of the horizon intercepted between the southern point and the several meridians in the eastern and western hemispheres. If the number of meridians drawn on the globe be twenty-four, which is usually the case, the arches intercepted on the horizon will measure the angles which the hour-lines make with the meridian. To find the angles corresponding to half hours and quarters, turn the globe gradually from its position in which the first meridian was under the brazen meridian, noting for each arch of $3^\circ 15'$ of the equator that passes under the brazen meridian, the arch of the horizon intercepted between the southern point and the first meridian; the arches thus found give the positions of the lines corresponding to the half hours and quarters. The style of the dial represents the axis of the earth, and must therefore make with the plane of the horizon an angle equal to the latitude of the place.

A direct north or south dial for any latitude may be constructed in the same manner, by considering it as a horizontal dial for a latitude which is the complement of the given latitude.

**II. Solution of Problems by the Celestial Globe.**

**Prob. 1.** To find the latitude and longitude of any star.

Bring the pole of the ecliptic to the meridian. Then, having fixed the quadrant of altitude over the pole, place it over the given star; the number of degrees between the ecliptic and the given star is the latitude; and the number of degrees between the edge of the quadrant and the first point of Aries indicates the longitude.

**Prob. 2.** To find a star's place in the heavens, its latitude and longitude being given.

Place the extremity of the quadrant of altitude on the pole of the ecliptic, and make its graduated edge cut the ecliptic in the longitude of the star; then the star will be found under the degree of the quadrant that denotes its latitude.

**Prob. 3.** To find the right ascension and declination of the sun or of a star.

Bring the sun's place, or the star, to the meridian; the degree of the equator cut by the meridian gives the right ascension, and the degree of the meridian over the sun's place, or the star, shows the declination north or south.

**Prob. 4.** The latitude of a place, the day and hour being given, to arrange the celestial globe so as to exhibit the appearance of the heavens at that place and time.

Rectify the globe for the latitude of the place; bring the sun's place for the given day to the meridian; set the index to XII.; then turn the globe till the index point to the given hour. In this position, the globe will represent the actual appearance of the heavens.

**Prob. 5.** To find the time when any of the heavenly bodies rises, sets, or comes to the meridian, on a particular day at a given place.

Rectify the globe for the latitude of the place; bring the sun's place to the meridian, and set the index to XII.; then turn the globe till the star comes to the eastern edge of the horizon; the index will show the time of rising. Next, turn the globe till the given star comes to the western edge of the horizon; the index will show the time of setting. Lastly, bring it to the meridian, and the index will show the time of its culmination or southing.

**Prob. 6.** To find on what day of the year any given star comes to the meridian at a given hour.

Bring the given star to the meridian, and set the index to the given hour; turn the globe till the index points to XII. noon, and note the degree of the ecliptic cut by the meridian; the day of the month which corresponds to that degree is the day required.

**Prob. 7.** The latitude of a place, the altitude of a star, and the day of the month, being given; to find the hour of the night.

Rectify the globe for the latitude; bring the sun's place to the meridian, and set the index to XII.; fix the quadrant in the zenith, then move the globe and the quadrant till the star comes under the degree of the quadrant which denotes the given altitude, and the index will show the hour required.

**Prob. 8.** The year and day being given, to find the place of a planet.

Find the sun's place for the given day, and bring it to the meridian; set the index to XII.; then find in the Nautical Almanack the time when the planet passes the meridian on the given day, and turn the globe till the index points to the hour thus found; find in the almanack the declination of the planet for the same day, and under it on the globe is the place of the planet.

These are a few of the more important problems that can be resolved by the globes, and will be sufficient to illustrate the principle of solution. The solution of problems by the armillary sphere depends upon the very same principles.

---

**CHAP. IV.—OF THE CONSTRUCTION AND USE OF MAPS.**

In representing the geographical divisions of the earth's surface, two objects are to be kept in view; on the one hand, to exhibit accurately to the eye the relative position of the different countries; and, on the other hand, to give a delineation sufficiently minute to furnish a distinct knowledge of the necessary details. As the globe has nearly the exact figure of the earth, the representation which it affords of the surface fulfils the first of these objects in the most perfect manner; but to attain the second it would be requisite to enlarge the globe beyond all convenient size. A globe of the ordinary dimensions serves almost no other purpose in this respect, but to convey a clear conception of the earth's surface as a whole; exhibiting the figure, extent, position, and general features of the great continents and islands, with the intervening oceans and seas. To obtain a detailed representation of any part of the earth's surface, geographers have therefore found it necessary to have recourse to Maps, in which countries are delineated on a plane, while the mutual proportions of the distances of places are preserved as nearly as possible the same as on the globe.

For the construction of maps different mathematical hypotheses have been adopted.

By one method of construction, that of projection, the boundaries of countries, and their more remarkable features, are represented according to the rules of perspective, on the supposition of the eye being placed on some point of the sphere, or at some given distance from it, which may be increased indefinitely. Wherever the eye is supposed to be situated, the representation thus obtained answers very well, provided the surface to be represented is of small extent, and the point of view, or projecting point, is nearly over the centre; but when the surface is of great extent, for example, a whole hemisphere, those places which are situated near the border of the projection are in all of them much distorted.

Another method, that of development, is founded on the supposition that the spherical surface to be represented is a portion of a cone, of which the vertex is situated some- An accurate idea of the orthographic projection of any Mathema- line or figure may be obtained by holding it up in the light Geo- light of the sun, and observing the shadow formed on a plane perpendicular to the direction of the sun's rays. The rays which pass close to the figure are the perpendiculars to the plane of projection, and the shadow is the ortho- graphic projection of the figure.

From the nature of this projection, the orthographic representation of half the surface of the globe shows nearly the true figure and proportions of countries about the middle of the map, that is, directly opposite to the supposed position of the eye; but towards the extremi- ties the true figure and position of the countries are im- perfectly exhibited. For this reason this method of pro- jection is seldom employed in geography, but in astro- nomy it is frequently used. We shall give, however, the orthographic projection of the sphere on the plane of the equator and on the plane of the meridian.

To project the Sphere orthographically on the Plane of the Equator. From any point C as a centre, with any radius CA (Plate CCLX. fig. 4), describe the circle ABD. Let this circle represent the equator, upon the plane of which it is required to project the sphere orthographically. It is evident that the centre C will be the projection of the poles of the equator, and that since the planes of the me- ridian circles are perpendicular to the plane of the equa- tor, these circles will be projected into diameters, making with each other the same angles as do the planes of the meridians. Let A 90, B 180, be two perpendicular dia- meters; they will represent two meridian circles at right angles to each other: divide the semicircle B 90 D into twelve equal parts at the points 15, 30, 45, &c., and let diameters be drawn through the points of division; then the twenty-four radii CB, C 15, C 30, &c. will be the pro- jections of the twenty-four meridians usually drawn upon the globe, any one of which, as BC, may be considered as the first meridian.

Next, it is evident that the parallels of latitude will be projected into circles, which have C for their common centre, and of each of which the radius will be equal to that of the corresponding parallel of latitude, or to the cosine of the latitude of the parallel. Let us suppose that a parallel is drawn on the globe for every tenth degree of latitude; then divide the quadrant AB into nine equal parts, at the points 80, 70, 60, &c.; from these points of division let fall perpendiculars upon BC, meeting it in the points 80, 70, 60, &c.; the lines C 80, C 70, C 60, &c. are equal to the cosines of the arches B 80, B 70, B 60, &c. to the radius BC. From the centre C, therefore, de- scribe circles with these lines as radii; and these circles will be the projections of the parallels corresponding to the 80th, 70th, 60th, &c. degrees of latitude.

Lastly, the projections of the polar circles and of the tropics may be found by setting off from the point A to- wards B, and from the point B towards A, twenty-three and a half degrees, drawing perpendiculars to BC through the points of division, and describing circles from C as a centre through the points in which these perpendiculars cut BC.

Thus will the projection of the sphere upon the plane of the equator be completed. The representation given of the polar regions in a map of this description is toler- ably correct, but the countries towards the equator are very much distorted.

To project the Sphere orthographically on the Plane of the Meridian. From any point C as a centre, with any distance CA (Plate CCLX. fig. 5), describe a circle. PASB, to represent the meridian circle, on the plane of which it is required to project the sphere orthographica- lly. Draw the diameters PS and AB at right angles to each other: then may PS be assumed as the projection of Mathematical Geography.

The meridian which is at right angles to the plane of projection; and AB will be the projection of the equator, since that circle cuts all the meridians at right angles. The other meridians cutting the projecting plane obliquely are projected into ellipses, having PS for their common transverse axis, and the cosines of their inclinations to the projecting plane for their several semiconjugate axes. Let the quadrant AS be therefore divided into six equal parts in the points 15°, 30°, &c., from which let 15m, 30m, &c. be drawn perpendicular to AC, and meeting it in m, n, &c. Describe the ellipses PmS, PnS, &c., having a common transverse axis PS, and Cm, Cn, &c., for their semiconjugate axes; these ellipses are the projections of the meridians passing through every fifteenth degree of the equator.

Again, to project the parallels of latitude: Divide either of the quadrants AP or BP into nine equal parts, in the points 10°, 20°, 30°, &c.; draw through the points of division straight lines parallel to AB, the projection of the equator; these lines are the projections of the parallels on the one side of the equator to every tenth degree of latitude. The parallels on the other side of the equator are to be drawn in the same manner; as are also the tropics and polar circles, the former at 23½° on each side of the equator, and the latter at 23½° from the poles.

In this projection there is great distortion in the appearance of the regions about the poles, and of all the countries near the meridian PASB. It is as we approach the centre of the map that this distortion begins so far to disappear as to allow a projected portion of the earth's surface to acquire any considerable resemblance to its delineation on the globe.

2. Stereographic Projection of the Sphere.

In the stereographic projection of the sphere the eye is supposed to be situated at one of the points where the surface of the sphere is intersected by a straight line passing through the centre, and perpendicular to the plane on which the projection is to be made. This plane, as we have already remarked, is that of a great circle of the sphere; so that the point on the sphere at which the eye is supposed situated will be everywhere 90° distant from the circumference of that circle whose plane is assumed as the plane of projection. In order that the lines and circles on the sphere may be visible to the eye in this position, it will be necessary to suppose the sphere to be transparent. Let, then, ABEC (Plate CCLX. fig. 6) be a great circle drawn on a sphere of this description, and passing through the eye at E. Let FG be a plane passing through a, the centre of the sphere, in such a manner as to cut the plane of the circle ABEC at right angles in the diameter HK, and to make each of the arches EH, EK, equal to 90°; then FG will be the plane of projection. Draw EA the diameter of the sphere passing through the eye; also draw EB, EC, to B and C, any two points in the circumference of the circle ABEC; and let the lines EA, EB, EC, intersect the plane FG in the points a, b, c. By the fundamental principle of the stereographic projection, namely, that the representation of any point is where the straight line drawn from it to the eye intersects the plane of projection, the points a, b, c are the projections of the points A, B, C. It is evident also that the line bc, in which the planes EBC and FG intersect each other, is the projection of the line BC, or of the arch BAC; and that the lines ab, ac, are the projections of the arches AB, AC.

Since Eab is a right angle, ab is the tangent of the angle aEb to the radius Ea. But the angle aEb is measured by half the arch AB; hence ab is the tangent of half the arch AB, or its semitangent. Thus it appears, that if a great circle pass through the projecting point E, any arch of it reckoned from the opposite point of the sphere is projected into a straight line passing through the centre, and equal to the semitangent of that arch.

Let BDC be any circle drawn on the sphere, and having BC for a diameter. Take D any point in the circumference of that circle, and draw ED intersecting the plane FG in the point d. If the point D be carried round the circumference BCD, the line ED will trace the surface of a cone of which BDC is the base; and the line bdc, (the intersection of the plane FG with the conical surface), which is traced by the point d, is the projection of the circle BCD.

Now, it can be demonstrated that the angles EBC, ECB, are equal to Ecb, Ebc, each to each; so that the cones EBC, Ebc are similar. Hence the projection bdc is a circle, whose diameter bc is found by taking cd and ac equal to the semitangents of the arches AB, AC.

Thus it appears that every circle of the sphere is, according to the stereographic projection, represented by a circle; but a circle can be described when three points in the circumference are given, or when two points in the circumference and the radius are given. Hence this property renders the projection of the sphere by this method very easy.

Another very elegant and important geometrical property of this projection is, that any two straight lines touching the sphere at one and the same point are represented by two straight lines which make with each other on the plane of projection an angle equal to that contained by the touching lines themselves. Hence also the angle formed by any two circles of the sphere is equal to the angle formed by their projections.

When it is further considered that the stereographic projection gives a representation of a hemisphere, in which the parts about the extremity of the map are less distorted than in the representation obtained by the orthographic projection, it must be concluded that the former method is preferable to the latter.

To project the Sphere stereographically on the Plane of the Equator. Upon C as a centre (Plate CCLX. fig. 7) describe a circle ABGE to represent the equator, on the plane of which it is required to project the sphere; or rather that hemisphere which lies remote from the point of sight. As the eye must be supposed to be situated at the pole, it is evident that the centre C will be the projection of the opposite pole, and that the meridians will be projected into straight lines passing through C, and dividing the circumference of the circle ABGE into as many equal parts as there are meridians supposed to be drawn on the globe. If there are twenty-four meridians, then draw AG any diameter, and divide the semicircle AEG into twelve equal parts. Through the points of division draw diameters, as BCE, FCK, DCH, &c., and the radii thus found will be the projections of the meridians corresponding to every fifteenth degree of the equator, any one of which may be assumed for the first meridian.

Again, to project the parallels of latitude; let AG and BE be perpendicular diameters. Divide the quadrant AB into nine equal parts in the points 10°, 20°, 30°, &c. From E to the several points of division draw straight lines intersecting AC in the points 10°, 20°, 30°, &c. The lines C 10°, C 20°, C 30°, &c., are the semitangents of the distances of the several parallels of latitude from the pole. Hence the points 10°, 20°, 30°, &c., are the points of intersection of the projected parallels with the projected meridian AG. Upon C, then, as a centre, at the distances C 10°, C 20°, C 30°, &c., describe concentric circles, and these will be the projections of the parallels corresponding to every tenth degree of latitude. The tropics and polar circles are found in the same manner.

To project the Sphere stereographically on the Plane of From the centre C (Plate CCLX. fig. 8), with radius CA, describe the circle ASBP to represent the meridian on the plane of which it is required to project the sphere stereographically. As the eye, in this case, is supposed to be situated at a point in the equator, that circle will be projected into a straight line passing through the centre C; let it be represented by the diameter AB. Draw the diameter PS perpendicular to AB; then will PS represent the meridian passing through the eye, and P and S will be the poles. To project the other meridians, divide the quadrant AP into nine equal parts (supposing a meridian to pass through every tenth degree of the equator) at the points 10, 20, 30, &c., and from S draw straight lines to these points of division intersecting AC in the points 1, 2, 3, &c. The lines C 1, C 2, C 3, &c. are the semitangents of the arcs of the equator which measure the distances of the points in which the several meridians cut the equator from the point of the sphere opposite to the eye. Hence the points 1, 2, 3, &c. are the points in which the projected meridians will intersect the projecting equator AB. But the projected meridians also pass through the points P and S. In each, therefore, there are three points given. If we describe, then, the arches P S, P S, P S, &c.; these arches will be the projections of the meridians on one side of that passing through the point opposite to the projecting point, and those on the other side are to be found in like manner.

Again, by dividing each of the quadrants PA, PB into nine equal parts, we find two points in each of the parallels of latitude. A third point in each will be found by drawing straight lines from B to each of the points of division 80; 70, 60, &c., so as to intersect PC in the points 8, 6, &c.; for the lines C 8, C 7, C 6, &c. are the semitangents of the arcs intercepted upon that meridian of which P is the projection between the several parallels and the pole of the sphere opposite to the projecting point. Describe, then, the several arches through the three points found in each; and in this manner the parallels on one side of the equator are found. Proceed in the same manner with regard to the parallels on the other side of the equator, and with regard to the tropics and polar circles, as the projection will be completed. It is evident that the centres of all the projected meridians will lie in the line AB, or in that line produced; and from the construction it is easy to show that the distance of the centre of each projected meridian from the point C is equal to the tangent of the inclination of the meridian to the plane of projection; while the radius of the projected meridian is equal to the secant of the same angle.

Further, the centres of all the projected parallels of latitude lie in the line PS, or in PS produced; and the centre of each projected parallel is distant from C by the semiarch which measures the distance of the parallel from the pole; while the radius of the projected parallel is equal to the tangent of the same arch.

To project the Sphere stereographically on the Plane of the Horizon for a given Latitude. We must here suppose the eye to be situated at the point of the sphere opposite to the place of which the latitude is given. From the centre C (Plate CCLX. fig. 9), with any radius CN, describe a circle NWSE to represent the horizon upon which it is required to project the sphere. To avoid intricacy among the lines necessary to be drawn for the construction, let the subsidiary circle NW'SE' (Plate CCLXI. fig. 1) be described with the same radius. Through C, the centre of such circle, draw two diameters WE, NS (Plate CCLX. fig. 9), and WE', NS' (Plate CCLXI. fig. 1), intersecting each other at right angles. Let the diameters NS, WE (Plate CCLX. fig. 9) be the projections of the meridian of the given place, and of a semicircle at right angles to that meridian, and passing also through the given place. It is evident that the points N, W, S, E are the four cardinal points of the horizon. On the circumference of the sub-tic Geodetic circle NW'SE' (Plate CCLXI. fig. 1) take NP equal to the given latitude; and draw the straight line WP intersecting CN' in P; then will CP be the semitangent of the distance of the pole from the point on the sphere opposite to the projecting point. Make therefore CP (Plate CCLX. fig. 9) equal to CP (Plate CCLXI. fig. 1), and the point P is the projection of the pole. Draw the diameters PP', FA' (Plate CCLXI. fig. 1) at right angles to each other; then will EA' be equal to the latitude; and by drawing WA' to intersect CS' in A, we obtain CA the semitangent of the distance of the point in which the equator intersects the meridian from the point of the sphere opposite to the projecting point. Make CA (Plate CCLX. fig. 9) equal to CA (Plate CCLXI. fig. 1); and through the three points W, A, E, describe an arch of a circle: the arch WAE will be the projection of the equator. Next, we shall show the manner of drawing the parallels of latitude, by taking as an example the two parallels which are twenty and forty degrees distant from the pole. In the subsidiary circle (Plate CCLXI. fig. 1) take P' 20 and P' 40, equal to 20° and 40° on each side of P. Draw from W' to the points thus found, straight lines intersecting CN' and CS' in the points a, b, and c, d. The lines ab and cd are the diameters of the projected parallels corresponding to 50° and 70° of latitude. Make Ca, Cb, Cc, Cd (Plate CCLX. fig. 9) equal to Ca, Cb, Cc, Cd (Plate CCLXI. fig. 1) respectively, and upon ab and cd as diameters describe circles; these circles are the projections of the parallels required. In the same manner are the other parallels of latitude, the tropics, and the polar circles to be drawn.

Again, to project the meridians; the straight lines NP, PS, are the projections of the opposite meridians which pass through the north and south points of the horizon. The meridian circle, which is at right angles to that represented by NS, passes through the east and west points of the horizon, so that three points in its projection are given, namely, the points W, P, E; its projection may therefore be drawn. But it may be more conveniently found by considering that the meridian in question makes with the plane of projection an angle equal to the given latitude: the centre of its projection, which is in CS produced, must therefore be distant from the point C by the tangent of the given latitude, and its radius will be equal to the secant of the same. From the point S' (Plate CCLXI. fig. 1) draw SD, touching the circle in S; and meeting PP' produced in D; then DS' and DC are the tangent and secant of the given latitude. In CS produced (Plate CCLX. fig. 9) take CB equal to DS'; and on B as a centre describe an arch through the point P, which will also pass through W and E. The arches PW, PE will represent the opposite meridians, which are at right angles to the meridian of the place for which the projection is made. With regard to the other meridians, it is not difficult to see that their centres will lie in a straight line GH (Plate CCLX. fig. 9) drawn through B at right angles to CB; and that the distance of the centre of each projection from the point B will be equal to the tangent of the inclination of the corresponding meridian circle to that meridian circle which passes through the east and west points; while the radius of the projection will be equal to the secant of the same inclination. Let us suppose, then, that the meridians are to make with each other angles each equal to 15°: at the point P make the angles BP 15, BP 30, &c. equal to 15°, 30°, &c. respectively, and let P 15, P 30, &c. meet the line GH in the points 15, 30, &c. Upon these points, as centres, describe through the point P the arches mPm', nPn', &c.; then will Pm, Pn, &c. Pm', Pn', &c. be the projected meridians required. In this manner the projection may be completed. 3. Globular Projection of the Sphere.

In the globular projection of the sphere, a point is assumed for the position of the eye, at a finite distance from the centre greater than the radius, and so situated that the degrees in the representation shall be nearly equal to each other, and the deviations from equality in the representations of equal portions of the spherical surface thus in some measure corrected. To determine the position of the point of view so as to answer these conditions, let ADBF (Plate CCLXI. fig. 2) be a section of the sphere made by a plane passing through C the centre, and through the point E which is assumed as the projecting point required. Through C draw EB, meeting the circumference ADBF in A and B, and draw the diameter DF at right angles to EB. Bisect the quadrant DB in I, and draw EI meeting DC in G; the point G will be the projection of the point I. But by hypothesis the projections of equal arches are nearly equal: let then DG be assumed equal to GC. Join DB, and draw CI meeting DB in H. Join GH, and draw IK parallel to DC, or at right angles to AB. Then it is evident that DH is equal to HB; and therefore DH : HB :: DG : GC. Hence the line GH is parallel to CB; and we have IH : HC :: (IG : GE : ) KC : CE. But IH is equal to BK, and HC to KC; therefore BK : KC :: KC : CE, and we have BK · CE = KC² = IK² = BR · KA. Hence CE is equal to KA; and taking away the common part AC, there remains EA equal to KC or to IK; that is, EA, the distance of the point of view above the surface, is equal to the sine of 45°. If the radius CA be divided into 100 equal parts, EA is therefore nearly equal to 71 of these parts.

This projection was first suggested by M. Delahire, and the approximation which it gives to equality in the projection of equal arches of a circle perpendicular to the plane of projection is considerable. The circles of the sphere are, according to this method of projection, represented by ellipses. An approximation to this method is, however, generally all that is aimed at. The circles of the sphere are represented by circles; and, without any regard to the distance of a projecting point, the degrees of longitude on the equator and of latitude on the meridian are made all equal to one another; the plane of the meridian being assumed as the projecting plane. The following is the construction by which such a representation of a hemisphere of the earth's surface is obtained.

From the centre C (Plate CCLXI. fig. 3), with any radius CA, describe a circle PASB to represent the meridian on which it is required to project the hemisphere. Draw the diameters AB, PS, at right angles to each other; and let PS represent the meridian of which the plane is at right angles to the plane of projection: then AB will be the projection of the equator, and the points P and S, the projections of the poles.

To project the parallels on the north side of the equator, divide each of the quadrants PA, PB, into nine equal parts: also divide the radius CP into the same number of equal parts. Let 80, 70, 60, &c., d, e, f, &c. be the points of division: the parallel corresponding to 80° of latitude will pass through the three points 80, d, 80; that corresponding to 70° of latitude through the three points 70, e, 70, and so on. Describe circles, accordingly, through these points, and the parallels on the north side of the equator will be drawn. In the same manner are the parallels on the south side of the equator to be projected, as are also the tropics and polar circles.

Again, to project the meridians, divide the radius AC into six equal parts in the points a, b, c, &c., and through the points P and S, and these points of division, describe circles PaS, PbS, &c. Proceed in the same manner on the other side of PS, and the circles thus described will represent the meridians passing through every fifteenth degree of the equator, any one of which may, in laying down places by their latitudes and longitudes, be assumed for the first meridian. Plate CCLXII. is a planisphere, or map of the world, projected in this manner.

II. Construction of Maps by Development.

The practical application of the three methods of projection which we have now explained, to the construction of maps, is usually confined to the representation of a hemisphere; whilst for the delineation of the geographical features of a single country, the method of development is commonly employed. The particular purpose for which a map is to be used may make it more or less important that it should exhibit, with all the precision that can be attained, some particular features of the country represented. The purposes of civil government require maps that give the true figure and dimensions of territory. For military affairs, maps that give correct distances are chiefly useful; whilst for the purposes of the navigator, the bearings of places, one from another, must be correctly, and, at the same time, simply exhibited. The first two objects are nearly gained in ordinary maps. But, for the attainment of the last, a map of a peculiar construction, called Mercator's Chart, has been invented; which, while it answers completely the purpose of the navigator, is not immediately applicable to any other.

It is an obvious property of a cone and of a cylinder, that the surface of each admits of being spread out on a plane. If a cone be laid with its slant side on a plane, the former will coincide with the latter along a line stretching from the point of contact in the base to the apex of the cone. Hence, if the cone is rolled round, whilst the apex continues at the same point, every point upon the surface of the cone will come in contact with a corresponding point in the plane surface; so that a sector of a circle will be described, with which the surface of the cone, if expanded, would exactly coincide. A cylinder admits of being rolled along a plane surface in a similar manner. But this is not the case with respect to a sphere. For, since a sphere touches a plane only in a point, if the former be rolled along the latter in one direction, the successive points of contact will mark out a straight line. A narrow zone of the sphere may, however, be supposed, without great error, to coincide with the surface of a cone or cylinder; and this hypothesis gives rise to a twofold construction by development, that by the development of a conical surface, and that by the development of a cylindrical surface.

1. Development of the Curve Surface of a Cone.

Let NAMB (Plate CCLXI. fig. 4) be a section of the sphere by the plane of the meridian. Let NM represent the axis; then AB drawn through the centre C, at right angles to NM, will represent the diameter of the equator. Take EF, any arch of the meridian, and bisect it in the point G. Through G draw LD a tangent to the circle NAMB, and meeting the axis MN produced in the point L. Let the plane figure LGMB be now supposed to revolve about the axis LM; then will the circle NAMB generate a sphere, and the line LD will generate a conical surface, which will touch the sphere. Further, the points E, G, F will describe circles, which will be parallels of latitude. If EF be an arch of not many degrees, the zone comprised between the parallel circles EG, IF will nearly coincide with the corresponding portion of the conical surface intercepted between the planes of the same circles. Take any point H in the circumference To find $L'G'$, $L'E'$, $L'F'$.

| 61° | 61° | |-----|-----| | 50° | 50° |

$2)111$

$2)11 = \text{diff. of lat.} = 660'$

$55°30' = \text{mid. lat.}$

$5°30' = 330° = G'E'$ or $G'F'$

Log. cot. mid. lat. $55°30'$,

$9^{\circ}371343$

$34377$...........log. $3^{\circ}362680$

$L'G' = 2362°7$........log. $3^{\circ}3734023$

$G'E'$ or $G'T' = 330°0$

$L'E' = 2692°7$

$L'F' = 2032°7$

To find the angle $G'L'H'$.

Log. sine mid. lat. $55°30'$.............$9^{\circ}9159937$

Diff. of long. $= 13° = 780'$........log. $2^{\circ}8920946$

Angle $G'L'H' = 643° = 10°43'$........log. $2^{\circ}8080883$

To find the chords of the arcs $E'r$, $Fs$.

Chord of arc $E'r = \left\{ \begin{array}{l} 2L'E' \\ \times \sin \frac{1}{2} L' \end{array} \right.$

Chord of arc $E'r = 502°1$........log. $2^{\circ}7008179$

Chord of arc $Fs = \left\{ \begin{array}{l} 2L'F' \\ \times \sin \frac{1}{2} L' \end{array} \right.$

Chord of arc $Fs = 379°1$........log. $2^{\circ}5787032$

Having now determined the four sides of the trapezoid formed by the meridians $E'F'$ and $rs$, and the chords of the arches $E'r$, $Fs$, it is easy to describe that trapezoid, the length of the sides being measured on any convenient scale of equal parts, which is to be considered as a scale of minutes of a degree on the meridian. Let it be described accordingly (Plate CCLXI. fig. 5), and let the sides $E'F'$, $rs$, be produced to meet in a point, which will be the point $L'$. Then, from that point as a centre describe the arches $E'r$, $Fs$, and divide these arches each into thirteen equal parts, since the difference of longitude is $13°$. Also, divide $E'P$ and $rs$ each into eleven equal parts corresponding to the given difference of latitude $11°$. Having fixed upon the number of meridians and parallels of latitude that are to be drawn, describe from the point where the lines $E'F'$, $rs$, intersect when produced, as a centre, the parallels through the proper points of division in $E'F'$ or $rs$; and draw straight lines joining the proper corresponding points of division in $E'r$, $Fs$ for the meridians: we shall here suppose the meridians and parallels to be drawn for every second degree. Number the degrees of latitude and longitude as in the figure, and the map is prepared for having traced upon it the outline of the coasts of the British isles, and places laid down according to their latitudes and longitudes.

It is evident that if the point in which $E'F'$ and $rs$ intersect each other becomes very distant, it may be exceedingly troublesome, or practically impossible, to describe from it as a centre the parallels of latitude. An obvious remedy for this inconvenience is, to join together two rulers, as $AB$, $AC$ (Plate CCLXI. fig. 6), at the point $A$, in such a manner as that they may contain an angle equal to the angle in the segment which has for its arch the parallel to be drawn. If the edges of the rulers be made, by means of two pins, to slide over the extremities $D$, $E$ of the parallel, a pencil fixed at the angular point $A$ will trace the parallel on the map. In the present instance the rulers must be placed so as to form an angle of $\left( \frac{360° - 10°43'}{2} \right) = 174°38'30''$. The conical development has been variously modified, so as to remove as much as possible its defects. Thus, one modification was given by the Rev. Patrick Murdoch in the London Philosophical Transactions, 1758. He supposed the cone to pass through points of the meridian between the middle latitude and the extremities of the arch to be projected, its side being parallel to the tangent at the middle latitude, while the points where the cone intersects the meridian are so situated as that the conic surface is exactly equal to the spherical surface which it represents. Let M denote the arch of the meridian to be represented in the map; then, according to this method of development, \( L'G' \), the radius of the middle parallel on the map, is equal to

\[ \frac{\text{chord of arch } M}{\text{arch } M} \times \cot \text{ mid. lat.} \]

the cotangent being supposed to be expressed by the radius of the sphere. In other respects the construction is the same as in the ordinary conical projection.

But the simplest and most successful method of remedying the defects of the conical development is that known by the name of Flamsteed's projection. The English astronomer by whom this method was invented, and whose name it bears, made use of it in constructing his celestial atlas. He developed all the parallels of latitude on the sphere into straight lines, and also one of the meridians, namely, that which passes through the middle of the chart. To this meridian the lines representing the parallels are perpendicular, and the length of each is the same with that of the parallel on the sphere which it represents. Dividing the parallels in the projection into equal parts, in like manner as the parallels on the sphere are divided, he supposed curved lines to be drawn through the corresponding points of division, and these curve lines represent the meridians. Flamsteed employed this projection in representing the position of the stars; but it is also made use of in geography, particularly for the delineation of countries which extend on both sides of the equator.

To suit it more effectually to this purpose, it has undergone, however, a modification of very considerable importance, as it corrects in some measure the distortion in the figures of countries lying near the extremities of the map, which arises from the obliquity of the curve lines representing the meridians to the straight lines representing the parallels of latitude, which increases as the former recede from the centre of the map. On the globe the meridians cut all the parallels of latitude at right angles; and by employing concentric circles instead of straight lines, as in Flamsteed's projection, to represent the parallels of latitude, the curves representing the meridians on the map may be made more nearly to fulfil this condition. For this purpose the common centre of the circles, which is situated in a straight line drawn through the middle of the map as an axis, is so assumed that the radius of the middle parallel of latitude is equal to the cotangent of the middle latitude; an assumption which diminishes as much as possible the obliquity of the angles made at the intersections of the curves which represent the meridians, with the circles which represent the parallels. In the position of the common centre of the circles representing the parallels of latitude, this modified projection of Flamsteed coincides with the ordinary conical projection.

We shall exemplify this construction by showing how to describe, according to it, the parallels and meridians for a map of Europe.

Let the map be supposed to extend from 35° to 70° north latitude. Hence the middle latitude will be 52° 30', and the radius of the middle parallel of the map (being equal to 3437.7 × cot. 52° 30') will be equal to 3637.8.

Draw, then, any line LE (Plate CCLXI. fig. 7) for the axis of the map; and assuming any point C for the point of intersection of the axis with the parallel of middle latitude, set off the length of the radius of that parallel from C to L, taken from any convenient scale of equal parts, which is to be considered as a scale of minutes of a degree. Thus the point L, the common centre of the circles representing the parallels, is determined.

Let us now suppose that a parallel is to be drawn through every tenth degree of latitude; that is, through 40°, 50°, 60°, 70°. As the middle latitude exceeds 50° by 150 minutes, take 150' from the scale of equal parts, and set it off from C to 50. Again, take 600, which are equal to 10°, from the scale of equal parts, and set off that distance from 50 to 40, from 50 to 60, and from 60 to 70. The point E, corresponding to 35°, is found by setting off 300' from the point 40. If we were to proceed according to Flamsteed's method, as originally employed by him, it would be necessary to draw straight lines through the points E, 40, 50, &c., perpendicular to DE, and therefore parallel to each other, and to make these lines equal respectively to the portions of the parallels on the sphere which they represent; then, dividing each line on both sides of the axis into as many equal parts as are indicated by the number of meridians intended to be drawn on each side of the axis, through the extremities and corresponding points of division to trace curve lines to represent the meridians. In this case the point L would not be required. But in the modification of the method which we are now exemplifying, from the point L as a centre an arch of a circle is to be described through each of the points E, 40, 50, &c., and each of these arches of longitude is to be made equal to the arch of the parallel on the sphere which it represents. Let us then fix upon some meridian of a given number of degrees of longitude to the eastward and westward of that represented by the line DE, which is to be so drawn as to fulfil this condition. Let AB and ab be the meridians assumed, the former 30° to the westward, and the latter 30° to the eastward, of DE.

The arch BE or bE is equal in length to an arch of 30° on the parallel of latitude 35°. Now, the latter arch may be determined in geographical miles or minutes of the meridian by means of the table (page 405), showing the length of a degree of longitude for every degree of latitude in geographical miles. Thus, 30° on the parallel of 35° latitude is found equal to 1474.5. But the arch BE is of the same length. Therefore we have BE or bE = 1474.5.

We might next find the angle which the arch BE subtends at the centre L, and thus determine the points B and b. But it is more convenient to find the chord of the arch, which may be easily done as follows:

Put \( a \) for any arch of a circle whose radius is \( r \). Then

\[ \sin a = a - \frac{a^3}{6r^2} \quad \text{nearly, if the arch is not very great.} \]

In this expression put \( \frac{1}{2}a \) instead of \( a \), and it becomes

\[ \sin \frac{1}{2}a = \frac{1}{2}a - \frac{a^3}{48r^2} \quad \text{nearly.} \]

Doubling both sides of this equation, and observing that \( 2 \sin \frac{1}{2}a = \text{chord of } a \), we obtain

\[ \text{chord of } a = a - \frac{a^3}{24r^2} \quad \text{nearly.} \]

For arches not exceeding 30°, this formula will give the length of the chord with sufficient exactness. Let us apply it to find the chord of the arch BE.

Here \( a = 1474.5 \), and \( r = LE = 3637.8 \). Log. \(a = 3.168637\) Log. \(r = 3.566767\)

\[ \begin{align*} \text{Log.} & \quad a^2 = 9.505911 \\ \text{Log.} & \quad r^2 = 7.133534 \\ \text{Log.} & \quad 24^2 = 8.513745 \\ \text{Log.} & \quad 24 = 1.380211 \end{align*} \]

Diff. of arch and chord 9°-8° = 0.992166 Log. 24 = 8.513745

Hence the chord of the arch \(BE\) is 1464.7 of the meridian. Taking this number from the scale of equal parts, and setting it off from \(E\) towards \(B\) and \(b\), the points \(B\) and \(b\) are determined.

Proceeding in the same manner, we may find the arches of longitude, with their chords, on the other parallels of latitude; and thence determine the remaining parts through which each of the projected meridians \(AB\) passes. The curves drawn through these points will be the representations of the two meridians which have 3° of longitude to east and west of the meridian represented by \(DE\).

The points in which the projections of the intermediate meridians intersect the projected parallels, may be found by dividing each parallel into thirty equal parts from the arc, both towards the right and left; then, by tracing curves through the proper corresponding points of division, as many meridians may be represented as are judged necessary. If the map is to extend farther than 30° on each side of its middle meridian, the division of the parallels may be extended to the necessary distance beyond the meridians \(AB\) and \(ab\).

It is a consequence of the properties of this projection, that distances on the map may be readily and accurately measured by a scale of equal parts. This scale may be constructed as follows:

Draw a straight line \(AB\) (Plate CCLXI. fig. 8) equal to an assumed number of degrees of latitude. If we assume 33 degrees, \(AB\) will be equal to \(DE\). From \(B\) draw an indefinite line \(BC\), making with \(AB\) any angle. Then, since \(A\) is equal to 33°, or to 2417 English miles nearly, take 2417 from any convenient scale of equal parts, and set that distance off from \(B\) towards \(C\), making \(BD\) equal to 2417. From the same scale of equal parts take 100, or 200, &c. (the present instance we shall take 500), according to the number of miles which each of the divisions of the scale is intended to represent; and with this distance in the compasses set off from \(B\) towards \(C\) the divisions \(BM, MN, NP\); the remaining part \(qD\), being only 417, will not again contain it. Join \(AD\); and through the points of division \(m, n, p, q\), draw straight lines parallel to \(AD\), and intersecting \(AB\). Each of the divisions of \(AB\) thus found will represent 500 miles, with the exception of that adjacent to \(A\), which will correspond to 417 miles. The distance of two places of the map applied to this scale will give the distance in English miles.

Development of the Curve Surface of a Cylinder.

The principle of this development is analogous to that of the conical, and may be illustrated in a similar manner. Let the arch \(AB\) (Plate CCLXI. fig. 9) be a fourth part of a meridian, and draw the lines \(AC, BC\) to the centre; these lines will be at right angles to each other, so that if we suppose \(BC\) to represent the semiaxis of the sphere, then \(AC\) will be the radius of the equator. Let \(EF\) be any arch of the meridian, and let it be bisected in \(G\). Through \(G\) draw \(DF\) perpendicular to \(AC\), or parallel to \(BC\). If the plane figure \(BDGAC\) be supposed to revolve round \(BC\) as an axis, the arch \(AB\) will describe one half the surface of the sphere, the line \(DH\) will describe the surface of a ring cylinder, the point \(A\) will describe the equator, and the points \(E, F, G\) will describe parallels of latitude. If \(EF\) be a small arch, the zone of the sphere which it describes will nearly coincide with the corresponding zone of the cylinder. Any tract of country delineated upon the former may, therefore, be nearly represented upon the latter; and the cylinder being developed, the meridians will be represented by parallel and equidistant straight lines, as will also the parallels of latitude; the lines representing the former being at the same time at right angles to the lines representing the latter. Let \(EF\) be a portion of the zone described by \(EF\) comprehended between two meridians of which the difference of longitude is equal to the difference of latitude of the parallels described by the points \(E\) and \(F\). The breadth of the map representing this portion will be equal to the arch \(Gg\) of the middle parallel of latitude. But since the difference of latitude and difference of longitude are supposed equal, the arch \(Gg\) is the same part of the middle parallel that \(EF\) is of the meridian. Hence we have rad.: cos. mid. lat. :: \(EF : Gg\); so that \(Gg\) may be found in minutes of the meridian. Upon this principle depends the construction of the plane chart, which is said to have been invented by Henry, son of John king of Portugal.

As an example, let it be required to construct a plane chart extending from 30° to 50° north latitude, and from 5° to 25° west longitude. Here the difference of latitude and difference of longitude are each equal to 20°, and the middle latitude is 40°. Hence we have \(EF\), the length of the chart, equal to 1200', and the breadth of it equal to 919'25'. Construct a parallelogram \(EFgc\) (Plate CCLXI. fig. 10), of which the sides \(EF, Ff\) are respectively equal to 1200 and 919'25' taken from any the same scale of equal parts, which is to be considered as a scale of minutes of the meridian. Divide \(EF\) and \(Ff\) each into four equal parts, and through the points of division draw straight lines parallel to \(Ff\) and \(EF\), and these lines will represent the parallels of latitude and meridians for every fifth degree of latitude and longitude. If it is required that the chart should extend to a greater number of degrees of longitude east or west, the parallels of latitude may be produced, and additional meridians drawn on the left of \(EF\), or on the right of \(gf\).

The plane chart has nothing to recommend it but its simplicity. The degree of longitude bears to the degree of latitude its proper proportion only at the middle parallel of latitude, where the sphere and the cylinder are supposed to intersect each other. Hence the rhumbs, as shown by this chart, are altogether erroneous, except when the places between which the rhumb is drawn are very near the equator.

The utter inadequacy of the plane chart to serve the purposes of navigation, induced ingenious men to turn their thoughts to the problem of constructing a chart in which the rhumbs should be straight lines, and at the same time the bearings of places accurately represented. The first condition of the problem is satisfied in the cylindrical development which we have already explained, since in it the meridians are represented by parallel straight lines; so that the line that makes equal angles with them all must be a straight line. We are now to consider by what modification of this development the other condition may be satisfied. This can be accomplished in no other way than by preserving at every point on the map the ratio of the degree of longitude to the degree of latitude, the same as it is at the corresponding point of the globe. Let us suppose then that a cylinder is constituted on the equator as a base, and has its axis coincident with that of the globe. Let us suppose further that the globe, with its usual delineations upon it, is expanded as if it were inflated, so as to meet the cylinder, all its parts stretching in such a manner as that the meridians lengthen in the same proportions as the parallels do, until every part of its surface touches the concave surface of the cylinder; then will each parallel have the same diameter as the equator, and the meridians will... Mathematical straight lines parallel to each other; but from the equator towards each pole they will be gradually increased in length at each point, so as to preserve the exact proportion between the degree of the meridian and the degree of longitude on the parallel which passes through that point.

If the cylinder be now spread out on a plane surface, it will present a chart having the important properties required; the rhumbs will be represented by straight lines, and the bearings of places will be accurately shown.

Such is the principle of Wright's, or, as it is usually called, Mercator's chart. A chart of this description was first published by Gerard Mercator in the year 1566. It is not known on what principle he proceeded in constructing it; but it is certain that the true principles of the construction were unknown to him, for these were first found by Edward Wright of Caius College, in Cambridge.

It remains to inquire how the lengthened meridians are to be determined so that the chart may be constructed. This problem can be accurately resolved only by the assistance of the fluxional calculus (see Fluxions). But we may approximate to the solution by reasoning as follows:

Let AB (Plate CCLXI. fig. 11) be a quadrant of the meridian; draw AC, BC, to the centre; and let BC represent the semi-axis; then AC will be the radius of the equator; and if any point D be taken in the quadrant AB, the arch AD will be the latitude of the point D. Draw AG, CG, the tangent and secant of the arch AD, and DE, DF, its sine and cosine. The increase which the arch AD acquires in being transferred from the sphere to the cylinder is made at every individual point of the arch, so that the whole augmentation is the sum of these indefinitely small increments. Let us suppose, then, that AD is divided into a number of small portions, for example, into minutes. Since similar arches of circles are to one another as their radii, and the radius of any parallel is the cosine of the latitude of that parallel to the radius of the sphere, if we put L for any latitude, we have rad. : cos. L :: 1' of the meridian : 1' on the parallel whose latitude is L. But from the similar triangles GAC, CED, it is evident that CD : CE :: CG : CA; that is, for any latitude L, rad. : cos. L :: sec. L :

\[ \text{sec. L : rad. :: d : 1' of equator, or 1' of meridian;} \]

hence \(d = \frac{\text{sec. L} \times 1' \text{ of meridian}}{\text{rad.}}\).

Let 1' of the meridian be now assumed as the radius; then \(d = \text{sec. L}\), to the radius 1' of the meridian; or, since secants to different radii are proportional to one another, and we only require proportional quantities, we have simply

\[d = \text{sec. L}.\]

Thus the increased minute on the chart for each successive minute of which the arch AD on the sphere is made up, is proportional to the secant of the latitude of the higher of the two parallels which pass through the extremities of that minute. Let \(m\) denote the number of minutes in the arch AD, and take L = 1', 2', 3', &c., successively; and we obtain for the lengthened meridian corresponding to AD,

\[\text{sec. L'} + \text{sec. 2L'} + \text{sec. 3L'} + \ldots + \text{sec. mL'}.\]

If, instead of dividing AD into minutes, we had divided it into smaller portions, the approximation would have been so much the nearer to the truth. Upon this principle a table of the lengthened meridians corresponding to every degree and minute of latitude, called a table of meridional parts, is calculated; and by means of such a table Mercator's chart is easily constructed. The following table shows the length of the enlarged meridian for every fifth degree of latitude:

| Lat. | Merid. Parts. | Lat. | Merid. Parts. | Lat. | Merid. Parts. | |------|---------------|------|---------------|------|---------------| | 0° | 0-00 | 25 | 1549-99 | 50 | 3474-47 | | 5 | 300-38 | 30 | 1888-38 | 55 | 3967-97 | | 10 | 603-07 | 35 | 2244-29 | 60 | 4527-37 | | 15 | 910-46 | 40 | 2622-69 | 65 | 5178-81 | | 20 | 1225-14 | 45 | 3029-94 | 70 | 5965-92 |

The approximate numbers obtained in the manner now pointed out are sufficiently correct for all nautical purposes, and indeed might be carried to any degree of accuracy. From the more rigorous investigation afforded by the fluxional calculus, it is found that the enlarged meridian is proportionally equal to the logarithmic tangent of an arch found by adding to 45° half the arch of latitude reckoned from the equator. Thus, the meridional parts corresponding to 40° of latitude are equal to the logarithmic tangent of \((45° + 20°) = \log. \tan. 65° = 33133\); and the meridional parts corresponding to 50° are equal to log. tan. \((45° + 25°) = \log. \tan. 70° = 43389\). These two numbers will be found to be nearly proportional to the numbers set down in the above table as denoting the meridional parts corresponding to 40° and 50°.

To construct Mercator's chart: Draw two straight lines WE, NS (Plate CCLXI. fig. 12), cutting each other at right angles in the point C. Of these lines, WE is to represent the equator, and NS the meridian passing through the middle of the chart. From any convenient scale of equal parts set off equal parts on the equator both ways from C. These divisions are to represent degrees of longitude; and if the size of the chart will admit, each should be subdivided into minutes. Assuming the equator as a scale of minutes, set off from C towards the north and south on the middle meridian the number of minutes in the enlarged meridian, corresponding to each degree of latitude, as shown by a table of meridional parts. Draw straight lines through every fifth or every tenth degree of the equator and divided meridian, and at right angles to them. The lines at right angles to the equator will be the meridians, and those at right angles to the divided meridian, and therefore parallel to the equator, will be parallels of latitude. Any one of the meridians may be assumed as the first meridian.

To find the bearing of any one place from another, it is only necessary to draw a straight line from the one to the other, and to observe the angle which that line makes with the meridians: that angle is the bearing required, or the course on which a ship must be steered in sailing from