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Law of Cosines c 2 = a 2 + b 2 – 2abcos(?) Use to find third side of a triangle Or to solve for unknown angles When c is the hypotenuse of a right triangle. Law of Cosines c 2 = a 2 + b 2 – 2abcos(?) Use to find third side of a triangle Or to solve for unknown angles When c is the hypotenuse of a right triangle we get Pythagoras theorem! Problems 1.Use the law of cosines to find the measure of the largest angle in a 4-5-6 triangle. Best Practice #1 AusVELS Level 9.0 Students will identify similar triangles if the corresponding sides are in ratio and all corresponding angles equal.
Warm Up Find the unknown length for each right triangle with legs a and b and hypotenuse c. Trigonometry 18 April 2015 Learning Objective: To be able to describe the sides of right- angled triangle for use in trigonometry. Course 3 4-8 The Pythagorean Theorem G2.a How Do I Apply Properties of Right Triangles including The Pythagorean Theorem? While containing a less detailed Europe, both the WolfenbA?ttel and Paris manuscripts possess a complete mappamundi, together with a special and interesting addition. To the south of this temperate a€?Australiaa€™, Lambert places a zone of extreme cold, uninhabitable by living creatures.
The ideas expressed here are supplemented by the suggestion of two more unknown continents or a€?earth-islandsa€™, one in the Northern and the other in the Southern [Western] Hemispheres, lying in the expanse of an all-encircling and dividing great ocean. If Lamberta€™s a€?universala€™ conceptions are so narrowly dependent upon classical antecedents, it may be expected that the detailed material of the maps will also display a markedly antique character; and indeed the relationship between the medieval geographers and those of the later Imperial time is seldom found in more complete expression. Discernible on some of the Lambert examples shown here, the seas and rivers are usually colored green, the mountains red, but each of the three copies of the manuscript world map offers peculiarities of its own.
Besides the world map, the Paris manuscript contains (with certain differences) several of the smaller designs which are also found in the Ghent copy of the Lambertian encyclopedia.
In addition to the previously mentioned sources, Lamberta€™s Liber Floridus also drew from such medieval authorities as St.
The fourth or southern continent appears on the right-hand side of the map, covered with a lengthy descriptive text from Martianus Capella. Names on the main map are largely of provinces, with only a few mountains, rivers, or cities included, while peoples of the world are presented in the a€?list mapa€™. The inscription on the island of Paradise tells us that the Garden of Eden is the resting place of Enoch and Elijah, who were waiting in the place where human history had begun for the coming of the Antichrist at the very end of time. Though the early Church Fathers were inclined to reject the idea of a globular earth, there were not a few among who found the theory of a circular earth an acceptable one.
In illustration of the doctrine of a circular earth, terrestrial globes certainly could not have been thought of as having any practical value.
The rejecting of a€?classicala€™ geography and the impetus and rationale for this theocratic trend, while not originating with Cosmas, was synthesized and exaggerated in his works. Unfortunately, the book which he devoted to a description of countries, and which would have revealed his fine powers of observation, has not survived, like all of his other works - his Astronomical Tables, Commentaries on the Psalms, on the Song of Songs, and on the Gospels. The Christian Topography contains references to nearly seventy authorities selected from among philosophers, historians, travelers, doctors of the Church, soldiers, and statesmen. In subsequent Books (II-XII) he fulfills his secondary objective, that of revealing the a€?true doctrinea€? of the universe and the eartha€™s place in it, as defined by Cosmasa€™ interpretation of the Scriptures, confirmed by the Church Fathers (Book X) and even non-Christian sources (Book XII). The heavens come downward to us in four walls, which, at their lower sides, are welded to the four sides of the earth beyond ocean, each to each. This great dome is divided into two strata by the firmament; from the earth to the firmament is the present dispensation of angels and men containing the land, the sea and the inhabitants of the world, with the angels hovering close to the a€?roofa€? holding the sun, moon and stars which they controlled. The sun, said Cosmas via Solomon, in rising, turns first toward the north, where it went down, and thence hastened to the place in which it arosea€?.
In all this Cosmas passed beyond the position of most of the theologians such as Lactantius (the a€?Christian Ciceroa€? of the third century) who preceded him. To illustrate this interpretive description of the earth and the universe, the Christian Topography contains, in all probability, the oldest Christian maps to have survived. The world, as expressed by Cosmas on one of his diagrammatic maps shown here, is of course rectangular and flat, and is divided into two parts: present and antediluvian.
Concerning the dimensions of the world Cosmas writes: a€?for if, on account of a miserable trade, men now try to go to the Seres, would they not much rather go far beyond, for the sake of Paradise, if there were any hope of reaching it?a€? The Seric or Silk Land, indeed, lay in the most distant recesses of India, far past the Persian Gulf, and even past the island of Ceylon.
Cosmas, like all good Christian geographers, shrank from the idea of an inhabited part of the world in the Antipodes, separated from Christianity by an ocean belt near the equator.
In support of the same truth, Cosmas quotes the added testimony of Abraham, David, Hosea, Isaiah, Zachariah and Melchizedek, who clenched the case against the Antipodes - a€?For how, indeed, could even rain be described as a€?fallinga€™ or a€?descendinga€™ in regions where it could only be said to a€?come upa€™?a€? Over against these disproofs of folly and error stands the countless array of evidences for the true tabernacle theory, for the flatness and immutability of earth, founded upon Goda€™s stability, and for the shape of heaven, stretched like a skin-covering over our world, and glued to the edges of it at the horizon. Yet, after all, the Christian Topography will always be remarkable for other than the intended purposes. Unknown artist, A sketch of Cosmasa€™ pattern of the universe, Codex Sinaiticus graecus 1186, fol. Unknown artist, A more detailed sketch of Cosmasa€™ pattern of the universe, Codex Sinaiticus graecus 1186, fol.
The rejecting of a€?classicala€™ geography and the impetus and rationale for this theocratic trend, while not originating with Cosmas, was synthesized and exaggerated in his works.A  Both philosophically and cartographically Cosmasa€™ ideas were strictly dictated by his literal interpretation of the Bible. Of these the PheisA?n [Pison] is the river of India, which some call the Indus or Ganges.A  It flows down from regions in the interior, and falls by many mouths into the Indian Sea, enjoying all of the same products as the Nile, from crocodiles to lotus flowers .
Although there is no evidence to suggest that Herodotus included maps with his history, his ideas influenced the development of Greek cartography. Like Homer, Herodotus includes cardinal directions and topographical landmarks: bounding Egypt beyond Heliopolis, for example, are the Mountains of Arabia, oriented north to south and the site of quarries for the building of pyramids.
Scattered throughout his text is so much information about countries and rivers and seas and their relative size and position that many have tried to draw maps of Herodotusa€™ world from it.
The Persians inhabit Asia extending to the Southern Sea, which is called the ErythrA¦an [i.e. And Asia is inhabited as far as the Indian land; but from this onwards towards the east it becomes desert, nor can anyone say what manner of land it is . For the Nile flows from Libya and cuts Libya through the midst, and as I conjecture, judging of what is not known by that which is evident to the view, it starts at a distance from its mouth equal to that of the Ister [here Herodotus means that the source of the Nile is as far west as that of the Ister-Danube]; for the River Ister begins from the Celti and the city of Pyrene [the Pyrenees?] and so runs that it divides Europe in the midst (now the Celti are outside the Pillars of Hercules and border upon the Cynetes, who dwell furthest towards the sunset of all those who have their dwelling in Europe and the Ister ends, having its course through the whole of Europe, by flowing into the Euxine Sea at the place where the Milesians have their settlement at Istria. For all that sea which the Hellenes navigate, and the sea beyond the Pillars, which is called Atlantis, and the ErythrA¦an Sea are in fact all one, but the Caspian is separate and lies apart by itself. And taking his seat at the temple he [Darius] gazed upon the Pontus [the Euxine], which is a sight well worth seeing.
And I laugh when I see that, though many before this have drawn maps of the Earth, yet no one has set the matter forth in an intelligent way; seeing that they draw Oceanus flowing around the Earth, which is circular exactly as if drawn with compasses, and they make Asia equal in size to Europe . As mentioned above, Herodotus scoffed at the popular belief that Europe, Asia, and Africa (which he called Libya) were all the same size, and made up a circular world. Herodotus tells of five youths from the country of the Nasamones on the Gulf of Sidra, who pushed down through the desert to the south-west until they came to a great river which flowed east.
It can be assumed that maps, charts and plans accompanied even very early examples of these geographical works. Apparently the rectangular grid (a coordinate system consisting of lines intersecting at equal intervals to form squares which were intended to facilitate the measurement of distance, in li), which was basic to scientific cartography in China, was formally introduced in the first century A.D.
Pa€™eia€™s methods are comparable to the contributions made in this regard by Ptolemy in the west; but whereas Ptolemya€™s methods passed to the Arabs and were not known again in Europe until the Renaissance, Chinaa€™s tradition of scientific cartography, and in particular that of the use of the rectangular grid, is unbroken from Pa€™ei Hsiua€™s time to the present.
As the Imperial territories of China increased through the intervening centuries, maps of various scales were made of part or all of the expanding realm. Under the Ta€™angs China had attained a high degree of civilization, perhaps the highest it has ever reached, and cartography then made remarkable progress. As a probable adaptation of the older work, the Hua I Ta€™u is oriented to the North and concentrates primarily on the portion which covers China proper, but also includes part of Korea in the north and part of the Pamir plateau in the west - more than just a map of China, though by no means a world map. Curiously, the Hua I Ta€™u lacks the distinctive rectangular grid system that superimposes most of the other Chinese maps.
Adding to the modern appearance of both of the Xian maps is the lack of the fabulous creatures, the religious themes or adornment of superfluous material so common to many of the European maps of the same period. The assumption of power by the great Chin dynasty has unified space in all the six directions. These three principles are used according to the nature of the terrain, and are the means by which one reduces what are really plains and hills (literally cliffs) to distances on a plane surface.
If one draws a map without having graduated divisions, there is no means of distinguishing between what is near and what is far. But if we examine a map that has been prepared by the combination of all these principles, we find that a true scale representation of the distances is fixed by the graduated divisions. A brief history of the Chinese cartographic tradition will serve to reference the sources upon which these two maps have drawn, and demonstrate the remarkable antiquity and show the continuity that prevailed in Chinese cartography. As the Imperial territories of China increased through the intervening centuries, maps of various scales were made of part or all of the expanding realm.A  Pa€™ei was followed by a number of cartographers, whose work is recorded in the dynastic histories, but of which no example survives.
DESCRIPTION: One of the most interesting and problematical maps of the Middle Ages is the Portolano Laurenziano Gaddiano currently in the Biblioteca Medicea Laurenziana in Florence, Italy. Among the most startling features is its depiction of the recognizable shape of the continent of Africa with remarkable prescience. Nearly a century before the Portuguese age of discovery, the Medici Atlas draws the bend of the Gulf of Guinea and shows that Africa has a southern end, i.e. Now it is well known that the province of the normal portolan chart of the late 14th and early 15th centuries did not extend farther south than Sierra Leone (= Montes Lunae), a region about which the merchants of Genoa and other Mediterranean seaports were beginning to hear as a result of their growing trade with the Sudan. Assuming then that the Laurentian a€?worlda€? map was originally a portolan chart, a resume of the six regional charts that follow in the Atlas, how can we regard the twin African outlines? It is largely on the ground of these and other characteristics common to the two maps that Wiesera€™s arguments are based. Now both the Dulcerto and Pizzigani portolan charts are of the normal type and with one or two exceptions (e.
The first of these arguments assumes, quite arbitrarily, that the de Virga map is the parent map, but it is equally permissible to assume that the Medicean Atlas is the parent map, so that this is really no argument. South of Cape Bojador to the latitude of Sierra Leone the coast is depicted as a rhythmic succession of headlands and bays - an obvious confession of ignorance. In the heart of the Sahara, there is a mountain complex separating the Nile from what presumably the Nile of the Negroes. We see from these illustrations that the world map shares the limitations and defects of all 14th century maps. The last of Wiesera€™s arguments hinges on the association of Ptolemya€™s Geographia in the production of the Medicean Atlas, and in particular, upon the parentage by the gulf which is portrayed in the ink outline. The river systems of the Sahara conform to the views commonly expressed in medieval geographies, while those of the southern portion of the continent recall the traditional ideas held concerning the Terrestrial Paradise. It would appear, therefore, that the problems raised by Wiesera€™s thesis are greater than those that it solves.
To summarize, there would appear to be good reason for retaining 1351 as the approximate date of the original drawing and for regarding the African outlines as the work of later editors: the first dating possibly from the period of the de Virga world map and the second from the period of Portuguese activity along the west African coasts.
Triangles -the basics In any triangle the three angles add up to 180° One of the angles in a right angle. Basic Ratios Find the missing Law of Sines Law of Cosines Special right triangles 100 200 300 400 500.
Trigonometry Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.
Sides and Angles Basic trigonometry is about things to do with the different ANGLES in a right angle triangle Look at the angle labeled. Geometric Mean of 2 #’s If you are given two numbers a and b you can find the geometric mean.
Isidore, Orosius, Julius Honorius, Pomponius Mela, Solinus, Venerable Bede, Raban Maur, the Pseudo-Callisthenes and the Bible.
East is at the top with Paradise a small sunburst to the left of top center, with rivers (Tigris, Euphrates, Nile Ganges) flowing from it into Asia. In addition, Lambert puts a small island in the far west, which he calls the Antipodes, observing that the inhabitants here have night when we have day and vice versa.
Asia, she says, represents the past, the rich golden age of mankind, but also the future, as Enoch and Elijah are waiting in the earthly Paradise for the last day.
The historical content of Lamberta€™s maps is thus fully understood only when one looks at the written portions of the manuscript.

The map also shows Gog and Magog, confined here to an island in a corner of northeast Asia, surrounded by a semicircular ring of water, called mare caspium, is an island on which are the words gog magog, another reference to the Alexander legend.
The surrounding ocean is filled with many more islands, the westernmost of which is Thyle [Iceland], while there are two Hybernias [Ireland] and one Anglia, situated in near correct position on the map.
Isidore, Orosius, Julius Honorius, Pomponius Mela, Solinus, Venerable Bede, Raban Maur, the Pseudo-Callisthenes and the Bible.A  There are at least eight manuscripts of the text preserved in the libraries of Europe, and it was referred to with high praise by writers of the 13th century. With a rejection of the spherical theory of the ancients very naturally went the rejection of their globes. Some of his geographic descriptions are to be found as part of the Topographia, and a few fragments of the above writings do exist. Comasa€™ primary objective and motivation in writing the treatise was to discredit the a€?false and heathen doctrine of a spherical eartha€?. Some of his fellow Christian writers openly declared that it did not matter so far as faith was concerned whether the earth was a sphere, a cylinder or a disc. There is little doubt among scholars that the numerous sketches - of the world, of the northern mountains, of the Antipodes in derision and the rest - which are to be found in the 10th century Florentine manuscript copy were really drawn by Cosmas himself (or under his direction) during the sixth century; and are thus contemporary with the Madaba mosaic map (#121, Book I) and at least two centuries earlier than the map of Albi (#206), or the original sketch of the Spanish monk Beatus (#207).
The theory of such a region, found in some of the pagan writings of the early Greeks and later by the likes of Macrobius (#201), Isidore (#205) and other perpetuators of pagan thought, was impossible, according to Cosmas, on two counts. For many centuries following the fall of the Western Roman Empire, there appears to have been in Christian Europe but little interest in the fundamental principles of geographical or astronomical science. Some of his fellow Christian writers openly declared that it did not matter so far as faith was concerned whether the earth was a sphere, a cylinder or a disc.A  But this sort of rationalizing was not good enough for Cosmas. Cosmas undertakes, with much else, to explain the symbolism of that Tabernacle in detail.A  In calling it worldly, Cosmas explained, St. Predicated upon the concept of a a€?flat eartha€? and oriented with North at the top and Paradise in the East (right) where the human race dwelt until the Flood when they were transported across the now impassable Ocean.
And so the Brahmin philosophers declared that if you stretched a cord from Sina, through Persia, to the Roman Empire, you would exactly cut the world in half. Through his writings of travels Herodotus did much to particularly enlarge contemporary knowledge of Asia. He particularly ridiculed circular maps that displayed landmasses symmetrically divided by the Mediterranean Sea, and he doubted the existence of the Eridanus [Po] River, from which amber was thought to originate. For example, he accepts that the continent of Libya [Africa] is almost entirely surrounded by water, excepting the Isthmus of Suez, as proved by pharaoh Necoa€™s circumnavigation of Africa (ca. He declares, a€?In a few words I will make clear the size [of Asia and Europe] and in what manner each should be depicted.a€? He starts with Persia, delimited by the Persian Gulf and Arabia to the south. His book was intended first and foremost as the story of the Greeksa€™ long struggle with the Persian Empire, but Herodotus also included everything he has been able to find out about the geography, history, and peoples of the world.
His researches for his book took him from his home in Halicarnassus in Asia Minor - the peninsula of western Asia between the Black Sea (Herodotusa€™ Pontus Euxinus) and the Mediterranean - through most of the known world. He cannot guess why three different names, Europa, Libya, Asia, have been given to the earth, which is one physical landmass unit. Now the Ister, since it flows through land which is inhabited, is known by the reports of many; but of the sources of the Nile no one can give an account, for the part of Libya through which it flows is uninhabited and desert . They had seen crocodiles there, and so Herodotus was convinced that they had reached the Nile, which he believed to rise in West Africa.
In length it is a voyage of fifteen days if one uses oars, and in breadth, where it is broadest, a voyage of eight days.
He realized that the Caspian was an inland sea and not, as many geographers thought, a gulf connected to the ocean that was supposed to encircle the earth.
Also their orientation to the North (true of all Sung Dynasty maps that have survived) contributes to the allusion.
So also the reality of the relative positions is attained by the use of paced sides of right-angled triangles; and the true scale of degrees and figures is reproduced by the determinations of high and low, angular dimensions, and curved or straight lines. Soothill, actually argues that the Hua I Ta€™u was Chia Tana€™s map, from which the sheets for the barbarian lands had been lost, and that the text was later substituted for them.A  Sung bibliographies do show, however, that versions of Chia Tana€™s map were still extant at this time, though kept secret at the imperial court and not easily accessible. This world map forms part of an atlas, commonly called the Medicean Atlas, consisting of eight sheets. Most of the maps that I have chosen in these monographs were primarily selected because of their personal aesthetic appeal. First, all details, physical features, pictorial embellishments and legends, stop short of the latitude of the Mos Lune.
Is it possible, therefore, that the draughtsman of the Laurentian chart only drew the part of Africa north of the latitude of Sierra Leone and that the southern portion of the continent was drawn subsequently?
He believes that there are several points of evidence for a later appearance of the Atlas, especially its manifold and often close correspondence with the Catalan Atlas of 1375 (#235).
First the verbal similarity between the text of the Lunar Calendar in the Medicean Atlas and the de Virga map.
The second, namely that the configuration of the southern part of Africa warrants the attribution of a later date, is seen to be poorly grounded when we dismiss, as Wieser does and no doubt rightly, the colored version as a subsequent emendation, for where does the ink outline provide such evidence? The isle of Meroe is placed in close proximity to the headwaters that are supplemented by another river coming down from the northeast extremity of the Atlas range. So it can only be the east coast that furnishes evidence of a late date and here Wieser confesses that the southerly trend may be due to the influence of the narratives of Polo and Odoric.
Now if this is a€?none other than the Sinus Hespericus by Ptolemya€? we should expect to find the map bearing other indications of Ptolemaic influence, but such indications are absent. In the writera€™s opinion the unknown author of the outline map and Albertin de Virga had in mind the gulf of which Western Europe was then beginning to hear a€“ the Gulf of Gold. The Law of Cosines and its Derivation The Law of Cosines is used to solve triangles in which two sides.
The Jordan River, with its double source in the mountains of Lebanon, flowing through the Sea of Galilee to the Dead Sea, is also plainly visible. Some expand on the subject of place, with lists and descriptions of places, while others fill in he historical narrative that gives meaning to the bare schema of the maps.
In Europe the names of the provinces depicted are Germania (twice), Gallia (France, 4 times), Hispania, Venezia, Italia, Magna Grecia [Greater Greece - also referring to Italy], Roma and others - 49 toponyms in all. What was contained in the Scriptures found a more ready acceptance than what was to be found in a€?pagan writersa€?.
In the first place, the region, if indeed there was land there, would be uninhabitable because of the withering heat. The theories of the Greeks and the Romans respecting a spherical earth and a spherical firmament encompassing it, in illustration of which they had constructed globes, were not entirely forgotten, but such theories in general were considered to be valueless hindrances rather than helps to the theological beliefs of the new Christian era. This extensive travel can be substantiated through examination of his detailed description of these areas.A  As a climax to this unusually broad and worldly experience Cosmas embraced Christianity, going so far as to become a religious monk to demonstrate the depth of his conversion. God had once explained to Moses on Mount Sinai exactly how the Tabernacle was to be built, and when it was found in the writings of Saint Paul that there was a passage which could be interpreted to mean that the Tabernacle was a picture of the world, it was quite natural for the Church Fathers to envision the world as a vast tabernacle: a tent with a rectangular base, twice as long as it was broad, and with an arched roof supported by four pillars.
As he discusses lands increasingly distant from the Mediterranean, the details become scanty, and his geography of the Indus is minimal. His work, with the map that can be reconstructed from his descriptions, provides our most detailed picture of the world known to the Greeks of the fifth century B.C.
His geographical descriptions are based on the observations that he made on this journey, combined with what he learned from the people he met. It has been suggested that the Nasamones came upon the Niger near Timbuktu, but it is more probable that they got no further than to the Fezzan, where dried-up river beds bear witness of large prehistoric rivers, and where carvings of crocodiles have been found on rock faces. On the side towards the west of this sea the Caucasus runs along by it, which is of all mountain ranges both the greatest in extent and the loftiest . The length of it is seven thousand one hundred furlongs and the breadth, where it is broadest three thousand three hundred . Also, he stated that Africa was surrounded by sea, and cited the Phoenician voyage commissioned by Necho in 600 B.C. Professors Needham and Ling, writing in their excellent multi-volume Science and Civilisation in China, assert, with ample justification, that this latter map is a€?the most remarkable cartographic work of its age in any culturea€?. If one has a rectangular grid, but has not worked upon the tao li principle, then when it is a case of places in difficult country, among mountains, lakes or seas (which cannot be traversed directly by the surveyor), one cannot ascertain how they are related to one another. Thus even if there are great obstacles in the shape of high mountains or vast lakes, huge distances or strange places, necessitating climbs and descents, retracing of steps or detoursa€”everything can be taken into account and determined. Because the maps are placed in opposite directions on the stone, this particular stele was most likely to have been used to produce rubbings and was not for public display.
The first sheet, an astronomical chart, is of value mainly in the dating of the overall work. The second sheet of the atlas, the world map, is the one that has attracted most attention.
While the remarkable shape of Africa has given rise to speculative theories about ancient sailing and secret voyages, the explanation could be more mundane.
Second, there is a palpable difference of brushwork and of color between the areas to the north and south of this latitude.
A comparison of the European and Asiatic portions of the map with the African suggests that such an explanation is at all events feasible. We can be sure that the Genoese merchants of the latter half of the 14th century were deeply interested in the question of Africa nondum cognita, if only because the Levantine land routes to the East were then occupied by the Turks.
Among other common features he points out is the delineation of the Caspian Sea, which here and they are nearly identical.
Whether or not the Nile of the Negroes was intended, as commonly believed, to connect by some subterranean course with the Egyptian Nile cannot be derived from the map. Prototypes of this may be seen in the Carignano, Sanudo and Vesconte maps, none of which, incidentally, can claim Ptolemaic parentage.
In addition there are biblical names (Judea, Galilea, Philistea, Palestina, Ydumea) used by Martianus, who was not a Christian, but not in the context of the Bible. In Lamberta€™s own area, Flanders, the modern name is put alongside the a€?Morinia€™, the tribe which settled there in Roman times. Thus, while he may have copied the main outline of his map from Martianus, the context as well as his additions, definitely Christianize his map. The far east is still psychologically very far off indeed in the 12th century, the original context of this map. Africa, which is to the right of the Mediterranean, includes the names of Mauritania, Numidia, Libya, Ethiopia and others.
Isaiaha€™s statement, a€?It is He that sitteth upon the circle of the earth,a€? was regarded as one altogether adequate on which to found a theory of the form of the earth, and it was accepted by such biblical interpreters as Lactantius, Cosmas Indicopleustes, Diodorus of Tarsus, Chrysostom, Severian of Gabala, by those who were known as the Syrians, by Procopius and Decuil. Both prophets and apostles, says Cosmas, agree that the Tabernacle was a true copy of the universe, the express image of the visible world. The GeA?n [Gihon or Nile] again, which rises somewhere in Ethiopia and Egypt, and discharges its waters into our gulf by several mouths, while the Tigris and Euphrates, which have their sources in the regions of Parsarmenia, flow down to the Persian Gulf .
Isidore of Seville in the sixth century.A  Two hundred years later Virgil of Salzburg with Basil and Ambrose agreed that even though it was a delicate subject, it was not necessarily closed to the Church. More generally, he raised several geographic questions: Why were three names (Europe, Libya, Asia) given to the earth, which is a single entity? However, given the lack of empirical evidence that the Ocean surrounds the contiguous landmasses of Europe, Libya, and Asia, he rejects this theory. The Caspian Sea and the Araxes River delimit the extreme northeast, but east of India there is an uninhabitable desert whose topography is unknown. Nonetheless, despite his interest in geography and his unequivocal opinions regarding cartography, Herodotus utilized geography primarily to reinforce his presentation of history. Herodotus saw his surroundings far more realistically than did most of his contemporaries; sometimes he even goes to the extent of doubting the truth of a story he reports second-hand. Herodotusa€™ view of the earth was closer to our own, although, because his knowledge was limited, he described Europe as being as long as both Asia and Africa put together. The dromedary camel was not yet in use in Africa in Herodotusa€™ days, and it is difficult to believe that the youths could have crossed the sands of the desert as far as the Niger on horseback.
If one has adopted the tao li principle, but has not taken account of the high and the low the right angles and acute angles, and the curves and straight lines, then the figures for distances indicated on the paths and roads will be far from the truth, and one will lose the accuracy of the rectangular grid. When the principle of the rectangular grid is properly applied, then the straight and the curved, the near and the far, can conceal nothing of their form from us. Based upon Jia Dan's Map of Chinese and Non-Chinese Territories within the Seas of 801, it shows the main natural and administrative features of the Chinese Empire up to the 1120s.

They are now housed among a very large collection of engraved stones in the Pei Lin [the Forest of Steles or Tablets] in the Shensi Provincial Museum at Xian. If the original date of 1351 is true, that would make it the first European extant map to incorporate the travel reports of Marco Polo and Ibn Batuta.
At one extreme comes the view of Alexander von Humboldt that this work, in conjunction with others, offers clear proof of a medieval acquaintance with the southern part of Africa.
The probable source of the a€?Guinea benda€? is the legend of the Sinus Aethiopicus, the rumor of a gulf that lay somewhere south of Cape Bojador that was said to penetrate deeply into the African continent. In Europe the draughtsman contented himself with depicting the region lying to the south of the latitude of central Sweden, that is the region known tolerably well to the medieval geographer. This date, Wieser says, rests solely on the fact that the year 1351 is referred to in the explanatory rubrics of the associated Lunar Calendar. This concordance he deems to be all the more significant because the form of the Caspian Sea on the Dulcerto chart of 1339 and on the Pizzigani chart of 1367 is not entirely closed. The Caspian to most of the cartographers of this period was not a region on which first hand knowledge was generally available for evidence we have only to look at the maps of the 14th and 15th centuries and, therefore, it may easily be that Dulcerto and Pizzigani preferred not to commit themselves and to leave the eastern and northern portions unmapped.
It appears to have its source both in the Atlas Mountains near the region described Hic sunt omines Magni Xll Pedes and in the Mos Lune [The Fouta Djallon Mountains?]. On this view there is no necessity to postulate a 15th century origin for the Medicean Atlas for there is ample evidence of medieval acquaintance with the African east coast as far as Sofala.
The Equatorial Sea [Mediterranean] which here divided the [great land masses or continents of the] world, was not visible to the human eye; for the full strength of the sun always heated it, and permitted no passage to, or from, this southern zone. On the right a text describes the a€?temperate southern continent, unknown to the sons of Adama€™.
Returning to Martianus Capella from which he draws his abbreviated inscription, Lambert seems to be trying to indicate a body of land on the opposite side of the world. The position of Gog and Magog, just beyond a€?Babilona€?, in or at the edge of the Caspian Sea, bespeaks a view of a much smaller world than the one later maps (such as those of Ebstorf and Hereford, #224 and #226) would represent. Philip of Cleves (1456-1528); purchased in 1531 from his estate by Henri III, Count of Nassau (d. Men, however, such as Basil, Gregory of Nyssa, and Philoponos inclined strongly toward the Aristotelian doctrine of a spherical earth.
Cosmas was most emphatic on the subject.A  Pagans, he said, a€?do not blush to affirm that there are people who live on the under surface of the earth .
Giving preeminence to data gleaned from exploration and travel, Herodotus attacks cartographers who utilized only geometry.
We shall have to believe that he was familiar with theories about the sphericity of the earth, but even though he was often critical of other geographers, he nevertheless seems to have accepted the old belief of the world as a flat disc. Of the areas to the north and east he knew little, mentioning neither Britain nor Scandinavia, and confessing ignorance of eastern Asia. As one passes beyond the place of the midday, the Ethiopian land is that which extends furthest of all inhabited lands towards the sunset [i.e. Some 500 years after Herodotus wrote, the geographer Claudius Ptolemy, whose knowledge was more detailed that Herodotusa€™, mistakenly pined southern Africa to Asia, making the Indian Ocean into an inland sea. The texts arranged around the edges of the graphic part of the map provide quotations from historical and other sources and briefly explain the meaning and history of essential markers such as the Great Wall, the size of the empire, and the states to the west. It shows Asia up to India, marking places like the Delhi Sultanate and others with reasonable accuracy. This gulf is described in the fantastical travelogue of the Libro del Conoscimiento (possibly as early as 1350) and finds itself again in the 1459 Fra Mauro map (#249), well before it was discovered by Portuguese explorers. It is difficult, from mere inspection, to decide which rendering is prior in point of execution.
The other may have been drawn by a cartographer at the Portuguese court in the following century after Don Pedro the King of Portugala€™s eldest son had returned from his grand tour of Europe.
As the drawing of the Caspian basin in the Atlas is better documented and more accurate he concludes that the date of the Medicean Atlas cannot be placed earlier than the last quarter of the 14th century. In passing it can be pointed out that as early as 1321 Marino Sanudo represents the Caspian as a closed sea, showing that, because Pizzigani left it open, there is no reason to suppose that the Laurentian map must be later than 1367. Third, the fact that a€?the world map of Albertin de Virga and the Medicean Atlas both testify to the influence of the Ptolemaic theory on the medieval conception of geography. The coast is portrayed with even comparative accuracy only as far as Cape Non and the prominence given to that headland demonstrates quite clearly that close acquaintance with the coast ended at that point.
The Mons Attalas [Atlas Mountains] have a greatly exaggerated length and reach almost to Egypt. 4.A plane which is 100 miles due West of you moves in a roughly Northerly direction at 400 mph. His most important work is an encyclopedia of biblical, theological, geographical, natural historical and musical themes entitled Liber Floridus [Book of Flowers], completed around 1120 and which was composed of extracts from approximately 192 different works. In the latter, however, was a race of Antipodes (as some philosophers believed), wholly different from man, through the difference of regions and climates. He shows Paradise as an island in the Far East with Enoch and Elijah, both of whom were believed in the Middle Ages to have been transported to Paradise without the painful expedient of dying. The problems of representing a sphere on a flat piece of paper has always bedeviled mapmakers. Isidore of Seville (#205) appears to have been a supporter of the spherical doctrine, as was also the Venerable Bede, who, in his De atura rerum, upholds the doctrine of a spherical earth on practically the same grounds as those advanced by Aristotle.
Who fixed the boundary of Asia and Africa at the Nile, and that of Asia and Europe at the Phasis River? He does not know whether Europe is surrounded by water to the west and north, nor the location of the Cassiterides Islands [Great Britain?], from which tin is obtained.
This then [Asia Minor] is one of the peninsulas, and the other beginning from the land of the Persians stretches along to the ErythrA¦an Sea, including Persia and next after it Assyria, and Arabic after Assyria; and this ends, or rather is commonly supposed to end, at the Arabian Gulf [the Red Sea], into which Darius conducted a canal from the Nile . This Pontus also has a lake which has its outlet into it which lake is not much less in size than the Pontus itself, and it is called MA¦otis [the Sea of Azov] and a€?Mother of the Pontusa€™. Herodotusa€™ knowledge of the course of the Nile was, however, as hazy as of that of the Indus.
This document was highly significant as a primary source for any cartographic effort during this and subsequent periods, which also can only be dated with certainty at least as early as the Chou Dynasty. The YA? Kung became and has remained a source of challenge and inspiration to all later generations of Chinese geographers and cartographers.
The notion that the West African coast did not extend straight south but took a sharp eastward bend, could be a hazy reference to the actual Gulf of Guinea. In Asia there is a north-south break (masked, as in the case of Africa, by a color wash) occurring approximately in the longitude of the Indus delta, that is, somewhat to the east of the Caspian Sea, which constituted the eastern frontier of the normal portolan chart. 1351 acceptable, but the fact that the reference to that year is in the past tense and also the fact that the de Virga Calendar starts at the year 1301 suggest that Lunar Calendars started from the first year of a century or half-century and that they do not necessarily bear a definite relation with the date of the map.
Only a few years before the construction of de Virgaa€™s map the geographical work of the great Alexandrine had been rendered accessible to scholars of the West by the Latin translation of Jacobus Angelus finished in 1409.a€? Because of this Wieser concludes that the Medicean Atlas cannot be dated earlier than de Virgaa€™s world map. An unnamed and imaginary river having its source in septem montium Pegio et Civitas Tochorum is made to debouch into the Atlantic on the south side of the cape. On their Saharan front they shelter at least five lakes of considerable size, each being the headwaters of a trans-Saharan river. This river is entirely independent of a third Saharan river-unnamed and taking its source in the Mos Aeris (a westward extension of the Mos Lune). Furthermore the de Virga map, which Wieser considers to be the parent map, places the Kingdom of Organa on the far side of the gulf near the India of Pres. If after 10 minutes the new distance to the plane is 130 miles, determine the exact heading of the aircraft. For when we are scorched with heat, they are chilled with cold; and the northern stars, which we are permitted to discern, are entirely hidden from them . Lamberta€™s solution a€“ to stick it in the margin of his map, like Alaska and Hawaii on a map of the USA. Africa below the familiar northern coast is a land of deserts, a€?full of dragons, serpents and cruel beastsa€™.
Even if he did not use maps himself, his text can still be employed to produce an outline of the oikoumene [the known inhabited world]. Thus Asia also, excepting the parts of it which are towards the rising sun, has been found to be similar to Libya [i.e. According to him, it rose south of the Atlas Mountains, and flowed across Africa before turning north to flow through Egypt toward the Mediterranean Sea.
Nothing is known of the authorship or the raison da€™etre of the work and, beyond the fact that it is of Ligurian provenance, nothing is known of its origin.
The historian Peter Russell notes that the Portuguese Prince Henry the Navigator was entranced by the legend of the Sinus Aethiopicus, as it held out the prospect of a direct sea route around West Africa to the Christian kingdom of Prester Johna€™s Ethiopian Empire, avoiding the complications of travelling through the Muslim lands of Egypt to reach it. In this the Medicean Atlas recalls the Pizzigani Chart of 1367 alone of all mid- and late 14th century maps, suggesting that the date is early rather than late.
Joanes - a tacit acknowledgment that that region was not intended to represent terra incognita of Ptolemy. His reference to days and nights and seasons can refer only to a continent in the southern half of the western hemisphere. Its cursed inheritance from Ham is compounded by its current occupation by the children of Ishmael. But should one wish to examine more elaborately the question of the Antipodes, he would easily find them to be old wivesa€™ fables.A  For if two men on opposite sides placed the soles of their feet each against each, whether they chose to stand on earth or water, on air or fire, or any other kind of body, how could both be found standing upright?A  The one would assuredly be found in the natural upright position, and the other, contrary to nature, head downward. The framework is in place: there are limits to the extent of the world and boundaries between landmasses.
Although he knew that a Greek mariner called Scylax (dates unknown) had sailed down the Indus River and around Arabia into the Red Sea, Herodotus maintained that the Indus River flowed southeast. The author of the Medici-Laurentian atlas is known that he comes from the Liguria region of Italy (probably Genoese), and might have composed it for a Florentine owner.
In the Medici Atlas, the depth of the penetration of the Sinus indeed almost reaches Ethiopia. Among the decipherable words in this hand the following are significant: Mangi, Cipangala(?), Kinsai and Gugut(?). On this theory it is possible that the colored and more plausible rendering dates from the time when the true conformation of the African coast was being disclosed.
Curiously enough the only inscription relating to the traffic of the region is unassociated with any of these rivers, being situated near the western extremity of the continent in Provincia Ganuya. As to Europe, however, it is clearly not known by any, either as regards the parts which are towards the rising sun or those towards the north, whether it be surrounded by sea . The atlas is explicitly dated 1351 (as per its astronomical calendar), but scholars believed it was more likely composed around 1370, possibly from earlier material, and probably amended further later, with emendations as late as 1425-50. Secondly there are two versions of the southeastern extremity of the continent, one in color and the other in ink: the latter appears to have been superimposed upon the former.
In deciding about the ink outline we are helped by the world map of Albertin de Virga, dated 1415 (#240).
Days and nights they have one length; but the haste of the sun in the ending of the winter solstice causes them to suffer winter twice over. Elsewhere, in the far northeast, are the 32 savage nations confined to an enclosure (identified with Gog and Magog confined by Alexander the Great). Herodotusa€™ Arabian Gulf], one who set out from the inner most point to sail out through it into the open sea, would spend forty days upon the voyage, using oars; and with respect to breadth, where the gulf is broadest it is half a daya€™s sail across . A comparison of the two maps suggests that the author (or editor) of the Laurentian map was acquainted with this work, for while the correspondence between the two is only general, it is distinctive and can hardly have been fortuitous as no other medieval maps introduce the Terrestrial Paradise and its river system under exactly the same guise. These maps, which are based upon Capellaa€™s design, contain an equatorial ocean but are quite different than the Macrobian zone-maps (#201). De la Ronciere, for instance, speaks of the map with its a€?prescience de la forme reelle de la€™Afrique avant le periple Portuguaisa€? [foreknowledge of the actual shape of Africa before the Portuguese exploration].

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