Mercator projection

“ Transverse Mercator Projections Constructing a chart using Mercator principles, but with the cylinder tangent along a meridian, results in a transverse Mercator or transverse orthomorphic projection. The word “inverse” is used interchangeably with “transverse.” These projections use a fictitious graticule similar to, but offset from, the familiar network of meridians and parallels. The tangent great circle is the fictitious equator. ”

“ But the renewal of the study of map-projection was mainly owing to German mathematicians, such as Werner of Nuremberg, and Apianus, in the first two decades of the century. In Mercator’s work there are to be observed various tendencies towards modern practice, such as the abolition of the old small sketches or miniatures representing towns and divers other subjects, and the introduction of symbols. On the other hand, the period of the application of criticism by the cartographer to the data before him was not yet come. Mercator was content to supplement data, where imperfect, by imagination ”

“ When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel—that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r = a loge cot 1⁄2z, which is the characteristic equation of Mercator’s projection. ”

“ This projection is available as far as 20° on either side of the Equator; beyond the parallel of 20° and as far as 60° Mercator’s projection is preferable. Between 60° and the pole the distortion of both the plane and Mercator’s projection is so apparent, that a polar or circular projection must be adopted. Sheets of paper, ruled into squares by strong lines, and subdivided by finer ones, afford great assistance in map work. For out-door work the scale of 1in. to one mile is amply large enough to register every particular of one day’s journey on a sheet of 12in. ”

“ Mercator’s Projection.—From the manner in which we have arrived at this projection it is clear that it retains the characteristic property of orthomorphic projections—namely, similarity of representation of small parts of the surface. In Mercator’s chart the equator is represented by a straight line, which is crossed at right angles by a system of parallel and equidistant straight lines representing the meridians. The parallels are straight lines parallel to the equator, and the distance of the parallel of latitude φ from the equator is, as we have seen above, r = a loge tan (45° + 1⁄2φ) . ”

“ When he returned to Louvain, the doctors of the university, shocked at his boldness in questioning what was almost universally received, were ready to attack his new doctrines, anticipating their immediate publication. But he kept his own counsel, and held his cosmogony in his portfolio till such time as he should judge best to make it known. Being obliged to make his living by manual labor, Mercator selected the making of mathematical instruments, with the designing, engraving, and illumination of maps, as his business. ”

“ If the magnitudes in the equations of exchange in other countries with which the United States is connected by trade are constant, the ultimate effect on M is to make it less than what it would otherwise have been, by increasing the exports of gold from the United States or reducing the imports. ”

“ VI.13 Our present interest in international trade, however, is mainly directed to its effects on international price levels. Except for the export or import of money to adjust the price levels, international trade is at bottom merely an interchange of goods. Where the price level is not concerned, the money value of the goods sold by a country will exactly equal the value of those bought. ”

“ The profits on exports, as on all other trade, exceptional cases apart, which cannot impeach the general rule, are measured to a great extent by the distance of the country to which the exports take place, and therefore the length of period, besides the extra risk, before which capital can be replaced and profits realized. Within the compass of a two months' distance from England, we may include the Gulf of Mexico west, the Baltic and White Seas north, the Black Sea south-east, the west coast of Africa to the Gulf of Guinea, and the east coast of South America to Rio Janeiro. ”

“ VI.8 But, although international and interlocal trade will never bring about exact uniformity of price levels, it will, to the extent that it exists, produce an adjustment of these levels toward uniformity by regulating in the manner already described the distribution of money. If one commodity enters into international trade, it alone will suffice, though slowly, to act as a regulator of money distribution; for in return for that commodity, money may flow and, as the price level rises or falls, the quantity of that commodity sold may be correspondingly adjusted. ”

“ In addition to these items, which by enquiry could possibly be ascertained, there are the transactions of private firms who may transfer capital from one country to another, and about these no information can ever be obtained. The only outward indication we can have whether there is an undue excess of imports must be in the foreign exchanges, and if these are constantly adverse, without special reasons, and the value of money were to become permanently higher here than in other centres, then there would be reason to fear that our invisible exports are not sufficient. ”

“ No extension of foreign trade will immediately increase the amount of value in a country, although it will very powerfully contribute to increase the mass of commodities, and therefore the sum of enjoyments. As the value of all foreign goods is measured by the quantity of the produce of our land and labour, which is given in exchange for them, we should have no greater value, if by the discovery of new markets, we obtained double the quantity of foreign goods in exchange for a given quantity of ours. ”

“ As prices rise, the purchasing power of their incomes correspondingly falls.
At length prices begin to go up less rapidly, and the question arises in many minds whether the movement can continue, and if not, when it will cease. Men wish to hold on for the last profits, and are willing to risk something to gain them. When prices rise not only as compared with former domestic prices, but as compared with current foreign prices, foreign imports are stimulated and exports fall. This calls for a new equilibrium of money and requires at length large and continued exportation of specie. ”

“ But whatever may be the source of the difference in price levels, equilibrium will eventually be restored through an international or interlocal redistribution of money and goods brought about by international and interlocal trade. Other elements in the equation of exchange than money and commodities cannot be transported from one place to another. ”

“ Those who are determined, at all hazards, to cry up the advantages of Free Trade, will point to a balance-sheet of extended exports as a sure index of the prosperity of the nation. Is it, after all, a sure index? The whole amount of our national exports is but an infinitesimal portion of the annual creation of wealth in the country; it consists of the products of only a few branches of industry, and represents the employment, not of the masses of the population, but merely of a small section. ”

“ Political economists, seeing the folly of these views, and clearly perceiving that the advantage of commerce consists and must consist solely of the imports, have occasionally suffered themselves to employ language evincing inattention to the fact, that exports, though unimportant in themselves, are important by their influence on imports. So real and extensive is this influence, that every new market which is opened for any of our goods, and every increase in the demand for our commodities in foreign countries, enables us to supply ourselves with foreign commodities at a smaller cost. ”

“ In fact no such extreme is even remotely approached, for a slight movement of money in either direction at once influences prices and sets in motion counteracting forces. Decade after decade the circulating medium of leading countries changes only slightly in amount, and the fluctuations during periods of so-called "favorable balance of trade" and of "unfavorable balance of trade" represent only the smallest fraction of the value of goods passing through the ports of the country. ”

“ The first is where the change in volume of trade corresponds to a change in population, as when there is an increase in trade from the settling of new lands, without any greater concentration in previously settled areas, and without any change in the per capita trade or in the distribution of trade among the elements of the population. Under such conditions no reason has been assigned, nor apparently can be assigned, to show why the velocity of circulation of money should be other for a condition in which the volume of trade is large than for a condition in which it is small. ”

“ Any cause which alters the exports or imports from one country into another, alters the division of the advantage of interchange between those two countries. Suppose the discovery of a new process, by which some article of export, or some article not previously exported, can be produced so cheap as to occasion a great demand for it in other countries. This of course produces a great influx of money from other countries, and lowers the prices of all articles imported from them, until the increase of importation produced by this cause has restored the equilibrium. ”

“ But let trade decay, let domestic industry fail, it will not be the foreign price which will then support the rate of our markets. What a multitude of circumstances contribute to the exportation of our superfluity! I can send from Charing-cross, any parcel whatever, to the distance of fifty miles, cheaper than from my own country habitation, where I can have a man, with a cart, for two shillings a day. ”

“ Professor De Morgan, while assenting to the proposition that a change to the metric system might prove of benefit to foreign commerce, believed that it would create such an immense “amount of confusion throughout the country that the inconvenience would far more than counterbalance the advantages we should derive in our foreign commercial relations.” The home trade of all countries is of far greater volume, and consequently of more importance, than the traffic carried across their borders. ”

“ Yet even here this consequence is by no means certain; for this reason, that what turns the balance of trade in favour of a nation is the demand which foreign markets make for her commodities: now this demand, as it raises the value of her coin above her bullion, so it raises the price of her commodities, by increasing foreign competition to acquire them. ”

“ Trade is at present carried on between Britain and the rest of Europe, at a profit of 15 or 20 per cent. Were that profit raised to 400 per cent., would not their home market, in case of such a fall of prices, be so exhausted by exportation, and in case of such a rise of prices, be so overstocked with foreign commodities, as immediately to restore the general equilibrium? ”

“ It is evident that the exportation of a plentiful year could not raise the price above that of a medium year; because it is the high price of a medium year, and the great demand at home, which prevents any part of that produce from going abroad. The importation in a scanty year would bring the price upon a level with the general free market, common to all the nations of the world, which would always be the same, or nearly the same, with the medium price at home. ”

“ Not only, therefore, does the trade between nations tend to the same equilibrium between exports and imports, whether money is employed or not, but the means by which this equilibrium is established are essentially the same. The country whose exports are not sufficient to pay for her imports, offers them on cheaper terms, until she succeeds in forcing the necessary demand: in other words, the Equation of International Demand, under a money system as well as under a barter system, is the law of international trade. ”

“ So long as our money market remains the cheapest in the world—I look upon the present abnormal position as merely temporary—so long shall we be able to take from foreign nations and our own Colonies the loans they may have to place for the development of their own countries; and the increase of our export trade to these countries will follow as a natural result. When the railways of the United States and of Argentina were built, mostly with British capital, we also sent the material to build them with. ”

“ It gets its imports cheaper, the greater the intensity of the demand in foreign countries for its exports. It also gets its imports cheaper, the less the extent and intensity of its own demand for them. The market is cheapest to those whose demand is small. A country which desires few foreign productions, and only a limited quantity of them, while its own commodities are in great request in foreign countries, will obtain its limited imports at extremely small cost, that is, in exchange for the produce of a very small quantity of its labour and capital. ”

“ Under the first, the country which wants more imports than its exports will pay for, must offer its exports at a cheaper rate, as the sole means of creating a demand for them sufficient to re-establish the equilibrium. When money is used, the country seems to do a thing totally different. She takes the additional imports at the same price as before, and as she exports no equivalent, the balance of payments turns against her; the exchange becomes unfavourable, and the difference has to be paid in money. ”