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remembered. Omitting such modifying words as " nearly,'

nearly,” “ about,” more than,” we have a few countries of Asia of a very convenient size, thus

1. Independent Tartary contains 500,000 square miles.
2. Persia.......

500,000
3. Asiatic Turkey

500,000 4. Arabia

1,000,000 The

eye tells us at once that the first three of these are nearly equal in size; and to the learner it might be made still simpler by telling him that any of the three great divisions was about equal to Austria and France united.

In Africa several countries are related in a similar way. For example :

1. Nubia...........contains 300,000 square miles, and is equal to Austria.
2. Abyssinia......

300,000
3. Egypt
150,000

half
4. Madagascar...
150,000

half The reason why Egypt is so much smaller than the others, though it does not appear so on an inaccurate map, is, that it is a narrow strip of country, -as the sands of the desert encroach upon it, except in the immediate neighbourhood of the banks of the river. The “quarters

The great divisions of the land on the earth's surface are

more easily remembered than appears at first sight. We have seen, for example, that the areas of Asia and Africa may be inferred to be respectively fifteen and eleven millions of square miles. Europe and New Holland are about equal in size, each of them containing four millions of square miles.

In like manner,

North and South America are equal in extent, each containing seven millions of square miles.

The subdivisions of the two great parts of the American continent are also very distinctly marked ::

NORTH AMERICA, SEVEN MILLIONS. British Possessions

2 millions square miles. United States

21 Mexico. ....

le Various (Greenland, Texas, Guatimala, Russian America) i

of the world."

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...

SOUTH AMERICA, SEVEN MILLIONS.
Brazil

3 millions square miles.
Colombia (New Granada, &c.)...... 1
Peru (including Bolivia)

1
Chili and La Plata....

1 Various (Patagonia, Guiana, &c.)... 1

3. Mountains. The principal mountains and mountain chains in the world have their height expressed by an odd number of thousands of feet. This singular fact at once diminishes the chances of error or forgetfulness fifty per cent. ; indeed it does more than this, for the judgment, reasoning from some substantial or collateral fact, will often bring us to the actual number, instead of leaving us to doubt between such numbers as 11,000

and 13,000. In Europe there are seven great chains, which are ranged thus 1. Alps, 15,000 feet high. 5. Ural,

7,000 feet high. } 2. Pyrenees, 11,000

6. Dofrine, 3. Apennines, 9,000

7. Carpathian, 8,000

"} 4. Balkan, 9,000 In Asia there are two important mountain chains, and two single mountains. The chains are the Himalaya, on the south side of the table land, the highest in the world, being 27,000 feet high; and the Altai mountains, on the northern side of this elevation, 11,000 ft. high. The single mountains are Ararat, 17,000, and Lebanon, 9,000.

In Africa there are four chains—the mountains of the Moon, 15,000 feet high; the mountains of Atlas and Abyssinia, 11,000; and the Table Mountain, 3,500.

The chains in North America correspond with those in South America in number, there being two of each, and in position, there being in each case a continuous one on the west, and a short separate one on the east. As before, the Andes and the Rocky Mountains are both expressed by odd numbers, the former being 21,000 feet, and the latter 17,000. Of the other two chains, the Brazilian mountains are just twice the height of the Alleghanies, the former being expressed by 7,000, and the latter by 3,500.

4. Rivers. There may be enumerated no fewer than thirty-eight of the principal rivers in the world, and only seven numbers are required to express their various lengths. Take for example the rivers of EUROPE. The Thames and the Shannon are each about 200 miles long; and these may be taken as our standard of measurement. In France there are four principal rivers, the Rhone, the Loire, the Seine, and the Garonne, three of which are each 400 miles long, and the Loire is 500. Again, in Spain, there is a singular coincidence with this. There are four principal rivers, the Guadiana, the Douro, the Ebro, and the Tagus, three of which are each 400 miles long, and the Tagus is 500. The Po, in Italy, is 400. There are several rivers in the central parts of Europe, of which it is sufficient to say that they vary from 500 to 1,000 miles in length. Three rivers of Europe are of the length of 1,000 miles, the Don, the Dnieper, and the Dwina ; and two, the Danube and the Volga, are 2,000 miles long.

In Asia there are twelve important rivers, none of which flow westward, but they admit of an interesting classification thus-three on the north, three on the south, three on the east, and three on the west. Of the three on the north, the Lena is 2,000 miles long, the Yenisei and the Obi are 3,000 each. On the east, the Segalien is 2,000 miles long; and the Chinese rivers Hoang Ho and Yang-tse-Kiang are 3,000 each. On the south, each of the three rivers, Ganges, Brahmapootra, and Irawaddy, measures 2,000 miles. The rivers not flowing westward, but on the west side, are the Indus, the Euphrates, and the Tigris. The first two are each 2,000 miles long, and the third, which is a tributary, 1,000.

In AFRICA there are only four rivers of much importance—the Nile, the Niger, the Senegal, and the Gambia. Of these, the Nile flows through three countries, and is three thousand miles long; the Niger rises two hundred miles from Sierra Leone, and is two thousand miles long : the others are about 1,500 each,

In North America there are three large rivers—the Mississipi, the Mackenzie, and the St. Lawrence—the lengths of which are respectively 3,000, 2,000, and 1,000 miles. In South AMERICA, in like manner, there are three great rivers—the Amazon, the La Plata, and the Orinoco, and these again are respectively represented by 3,000, 2,000, and 1,000 miles.

5. Population. General considerations. The subject of population is unlike all the preceding, for while the grand features are permanent through ages and centuries, it is constantly changing. It is well to remember, therefore, in teaching, that when two text-books, or two editions of the same book, differ slightly, the greater population is probably the more correct one. comparatively new countries, such as various parts of British America, or New Zealand, the increase of population is extremely rapid ; in countries which have been the seat of war, as Greece, it sometimes diminishes ; where emigration is so regular as to be a national characteristic, (as in Scotland,) the numbers increase slowly. It is only in the “old countries” of the world, or in ordinary circumstances, that any general remarks are applicable. But there are general ideas, sufficiently accurate to serve the purposes

of almost

any
individual, it

may be, during his life :—for the data being given at any point, and the modifying circumstances being supposed known, he cannot fail to make a sufficient approximation to the results. The following are a few which may serve as specimens :

In

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POPULATION OF THE WORLD, EIGHT HUNDRED MILLIONS.
Europe....... 200 millions. Africa......

.100 millions.
Asia.
..400

America, &c. .........100 Some of these again may be subdivided, and stated in a tabular form; or the same figures may be analysed variously according to the different relations which they may be said to represent. The population of Europe, for example, may be divided nationally as well as ecclesiastically, thus

EUROPE, TWO HUNDRED MILLIONS.
Russia....

50 millions (properly about 55.)
Austria

30 Turkey

20 (properly about 15.) Various...

100 N.B.—Half the population is embraced by the three empires.

RELIGIONS OF EUROPE.

Roman Catholics

100 millions. Greek Church .....

50 Protestants......

50 N.B.-There are about eight millions of Mahometans, and two or three of Jews.

In Asia, the analysis is somewhat like the following :

ASIA, FOUR HUNDRED MILLIONS.
Chinese Empire

200 millions.
India (generally)

...... 100 Various....

.....100 Starting point In noticing more minutely the population of the various in calculating. countries in Europe, it is assumed that the pupil is already acquainted with the population of the various sections of the British Islands. He will thus have, as before, a standard of comparison, instead of guessing as irrationally as if he were drawing numbers at a lottery. The same remark applies with additional force to the population of cities and great towns : for that is frequently the only means we have by which to judge of their magnitude or importance; and the ideas which it conveys in those cases, are, of course, much nearer the truth than those derived from comparing the population returns of particular countries or provinces. Let the pupil, then, be made acquainted with the population of his own town or village, or with that of some town which he knows intimately; and he will have no difficulty in realising any similar place to his mind, its population being known. Thus, Liverpool contains 300,000 inhabitants, London 1,500,000; it is clear, therefore, that London is about five times as large as Liverpool. The allowance is readily made for modifying circumstances; thus, Dublin and Edinburgh, being metropolitan cities, are not to be estimated by their population merely : Manchester, Birmingham, and Paisley, being manufacturing towns, would disappoint one slightly in an opposite way :-and so would Limerick, from its poverty, or Washington, from local causes. Particular Examples. The population of Austria is about equal to that of France, each being represented by thirty millions. In like manner, Spain, Prussia, and England (proper) coincide; the number being fifteen millions. The numbers for Norway, Denmark, and Sweden, (which are associated locally), are respectively one, two, and three millions: and these being known with their countries, it is next to impossible to misapply them. The population of Ireland, (eight millions,) is equal to that of the kingdom of the Two Sicilies; it is twice as great as that of Bavaria or Belgium ; four times as great as that of Hanover, Wirtemberg, or Switzerland; and eight times as great as that of the kingdom of Greece.

6. Density of Population. Method of finding Just as the area of a country may, in general, be in| it. Examples. ferred from its dimensions, so the density of the population may always be ascertained from the elements given above. If we are told, for example, that England with its forty shires contains upwards of 40,000 square miles, and supports a population of 15,000,000; while the Two Sicilies with 40,000 square miles, support only 8,000,000, it will be evident that in England the population is nearly twice as dense as in Naples or Sicily. But here, too, a standard of comparison is necessary, and the simplest seems to be—the number of individuals to a

square mile. Asia, for example, contains 15 millions of square miles, and 400 millions of inhabitants; the average, therefore, is about 27 to the square mile. Again, in Europe, the area is only 4 millions, and the population is 200 millions ; hence we have in it 50 inhabitants to the

square mile. Proceeding in a similar way, we should find the density represented by 9 in Africa, and 4 in the whole of America. In Ireland, it is 266; in Bavaria, Hanover, and Switzerland, 133; in Greece, 66; in France and Prussia, 150; in Austria, 100; and in Turkey, 75.

A. HUME. (To be concluded in our next.)

INSTRUCTIONS IN ARITHMETIC.

( Continued from page 333.)

CHAPTER IV.-ON THE RIGHT METHOD OF STUDYING ARITHMETIC.

65. In the preceding chapters of this work we have learned to form, represent, combine, and compare numbers; we have become acquainted with certain properties that belong to them, and have been enabled usefully to apply the knowledge thus acquired.

Hence arithmetic may be defined to be the science which teaches us to form, represent, combine, and compare numbers, to study their properties, and to answer useful questions which relate to them.

66. If we carefully observe the mode in which we have proceeded thus far, we shall then perceive, also, certain principles, which will hereafter be of much more important use, inasmuch as it is by these means we shall be enabled to answer questions much more complex and difficult; and it will be seen, at the same time, that it greatly conduces to the order, clearness, and correctness of our expressions, if we learn to understand and use the language which those best acquainted with the subject have adopted, and even fix in our memory, as far as may be, the several definitions, axioms, and enunciations, upon which the science of number depends.

67. We are instinctively led to inquire “What is the reason?” in any process in which we are engaged, and this natural desire increases as our knowledge increases.

The reason why a thing takes place is called “cause,” and that which the cause produces is called "effect”; ex.gr., the knowledge which any one acquires is an effect, and the cause of this effect is the attention and perseverance given,—the exercise of the natural ability of the student, which ability is the gift of God, and can, as we know, be developed and increased by exercise. An effect may become itself a cause; thus knowledge, which was the effect of study, is a cause relatively to the good which a man of knowledge can produce.

68. In order to facilitate the study of arithmetic, the principles, or truths and definitions, on which it depends, are so laid down and presented to the student, that each thing shall be always explained by those things which precede it. It is this which is meant, when it is said that in arithmetic we proceed from the known to the unknown.

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