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do not begin at the true beginning. Pupils learn to practise mechanically rules for multiplying, dividing, working the rule of three, and so forth; but the reasons of the rules are so imperfectly communicated, that many misconceive them, and, in consequence, soon forget and misapply them.

§ 2. A prime fact not sufficiently impressed on the learner's mind, is, that all ordinary computations are the comparing of numbers in general with some one convenient number chosen as a standard. That number is TEN, most convenient because it is the number of the human fingers, always present whenever men have to count, and as naturally adopted for numbers as the length of the human foot has been for lengths.

$ 3. The crossed-square exhibited and explained above at page 703, presents to the eye, and thereby very clearly to the mind, many important relations among themselves of this number ten with its multiples or powers, and a learner profits much, who by frequent review and reflection makes it familiar to the thoughts. Some of the individuals who at an early age have become remarkable for their readiness in mental arithmetic, have owed their ability to the accident which led to their acquiring familiarity with some such standard arrangement of units.

§ 4. Another arrangement of kindred nature and utility is sketched in the adjoining page, which pictures clearly to the eye with appearance of solid bulk, what the crossed-plane shows in merely extended surface. It exhibits a succession of columns increasing in magnitude, consisting each of ten equal parts. In the first column, marked A, the parts are of very small size. Those of the second column, B, are ten times larger, one of its unit-parts being equal to ten units of the column

Those of the third column, C, are in the same proportion larger than those of B, and so on in succession, a part in any column being equal to ten of those in the preceding column, and being only a tenth of one in the following column on the left. The number of primary units in a cube of each row is inscribed on the bottom one, viz., 1, 10, 100, 1000, 10,000, &c. The cubes of the seventh column, G, not shown here, would contain each a million of the primary units.

§ 5. The drawing represents the smaller columns entire ; but of the larger, only the bottom cubes find room in the page, and the very largest do not appear there at all. The parts are represented as cubes nearly of the comparative sizes which the realities

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would have. In page 712, it is explained why many solid bulks heaped closely together seem to occupy little space.

. To express any particular number of units by this apparatus, the required number of cubes would be in some way marked in the different columns; for instance, the year of the present era, 1866, is signified by one cube in the site D, eight in the site C, and six in each of the sites B and A, as indicated in the figure by diagonal lines drawn across the cubes.

$ 6. If beneath the arrangement of cubes shown, the lines which divide the spaces for the different columns, are carried down to form corresponding spaces for the common arithmetical figures below, the cubes and the figures explain one another. This is seen in the engraving where the figures 1, 8, 6, 6, appear. The figures may be considered as miniature pictures or signs of the sets of the cubes marked above.

§ 7. It is clearly seen how the difference of value is given to an arithmetical digit, according to the column in which it stands. The figure 4, for instance, when alone, means simply four, but when several are placed thus, 4444 in a line, as belonging to the several columns, the extreme figure on the right hand means four units, the next to it four tens or forty, the next four hundreds, the last, four thousands; and the whole together signify four thousand four hundred and forty-four.

$ 8. Near the bottom of the diagram in page 3, is represented still another mode of representing decimal sets of numbers. It is that used all over the Chinese empire, and the apparatus is called the Swan-pan. The present writer had in early life the opportunity of seeing how readily and accurately persons of all classes in China made calculations by means of this swan-pan. It consists, as here shown between the letters X and Y, of a frame having a number of parallel wires crossing it, on which wires balls of wood or ivory slide like beads on a cord. A bar, X Y, runs from end to end of the frame, dividing the wires and their balls unequally, two balls being on one side of the bar and four balls on the other. The wires and their balls correspond, as here shown, to the columns A, B, C, D, E, F, of the cubes above, and to the arithmetical figures or digits in the middle part of the diagram, between the letters V and W. The four balls are used to signify each a single unit of the different orders, the two balls signify each five such units. The balls are significant only when moved into contact with the bar. Here the balls represented as in contact with the bar indicate the present year

of our era, 1866; the one ball near the bar, of the rank D, marks 1000; the three balls of single units, with the one ball of five on the wire C, mark eight hundreds; and the one ball of unit value with one of five-fold value on the wires B and A mark six for each ; the whole expressing 1866.

$ 9. To facilitate reference to the succession of columns or ranks, which determine the different values of the figures placed in them, the adjoining table is given, showing the relation of the figures and the letters to the names written at length.

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§ 10. The explanations here offered show, that any number of units which the mind can conceive may be expressed with absolute precision by any of the modes described, and much better than was done formerly by the Greeks and Romans, who used the letters of their alphabets ; but the ten Indian figures or symbols 1, 2, 3, 4, &c., far surpass the others for the purposes to be served. The aid given by them to profound computations is singularly great, and the compendious form of the written expression is equally remarkable. As exemplifying the last quality, one may see on a single page of a small book, clearly set forth, such a mass of statistical information as the following titles indicate, many of the facts being results of wide observation and great labour of calculation :

The amount of population of a great city.
The amount of population of a kingdom.
The amount of population of the world.

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The number of persons in a community alive at different ages.

The number of persons employed in the special kinds of employment exercised in a society.

The comparative fatality of different diseases.
The comparative duration of life in different countries.

The rate of increase of populations. For instance, the fact is now ascertained, that the mixed race called Anglo-Saxon in North America has doubled in number every twenty-five years for at least four periods, and, if the laws of nature continue unchanged, the increase at the same rate will go on for at least four periods more, before wider diffusion becomes a necessity. At present there are more than thirty millions of the race

30 millions In the first period there will be

60 In the second

120 In the third

240 In the fourth (within a century bence) 480 which is about a third of the present population of this globe. Then North America shows but one of the European colonies that are now similarly progressing, chiefly because of men’s advanced knowledge of the universe, and the many new arts sprung from such knowledge during the last two centuries. It is calculated that the portion of the earth's surface on which the rain falls that fills the channels of the rivers Mississippi and St. Lawrence, could, if completely cultivated and managed according to modern skill, supply the necessaries and even luxuries of life to the whole present population of the world. Such are among the astounding facts which men of the present day have to contemplate; and they may see that the other arts which have worked the change, as that of navigation founded on astronomy, are mainly due to improvements in the means of measuring and computing.

$ 11. It is known that Charles Babbage, who is still living in health amongst us, after leaving the university, invented a machine, now known as Babbage's calculating machine, which, when moved by the turning of a crank or otherwise, is made to calculate and even to set the types and to print, without possibility of error, mathematical and other tables of the highest importance. In former times such tables had to be produced under the superintendence of governments, with enormous labour and cost, and without insuring accuracy.

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