Calculating Machines. Plato, in the 4th century B. C, invented a sliding square to solve the problem of two mean proportionals, and Nicomedes, three centuries afterward, invented his celebrated conchoid curve for solving the same problem and trisecting an angle. Some mechanical devices for assisting in arithmetical computation were also in use at a very-early age; but these were exceedingly limited in their operations, and therefore of little practical advantage. The same may be said of the more ingenious contrivances devised in the beginning of the 17th century, Gunter's scale and Napier's bones. John Napier, who was probably the first man to suggest the modern notation of decimal fractions, and whose invention of logarithms was well called canon mirificus, devised two modes of mechanical computation, one by means of square rods engraved with the Arabic figures, the other by means of circular plates. Napier's discovery of logarithms was made by Edmund Gunter the basis of a very simple machine, consisting merely of a straight line graduated to logarithms, but marked with the corresponding numbers. Addition and subtraction can be performed upon this line by means of a pair of dividers, and the corresponding number by the side of the line will be products, quotients, and factors.

But Pascal, in 1642, at the age of 19, invented the first arithmetical machine properly so called. This machine was improved by L'Eipine and Boitissen-deau about 80 years afterward, but it never came into practical use. It consisted essentially of short barrels, upon whose circumference the 10 figures were inscribed, covered by a box, one figure alone of each barrel being visible through a row of little windows on the upper surface of the box. These barrels were so connected that 10 revolutions in one produced one revolution in the next, the revolutions of the first barrel being performed by hand to correspond with the numbers to be added. Subtraction was performed by the device of having each figure on the wheels accompanied by a smaller figure, such that the sum of the two was equal to 9. Whatever number was added to the large figures was of course subtracted from the smaller. In 1673 Leibnitz published a description of a machine which was much superior to that of Pascal, but complicated in construction and too expensive for the work it was capable of performing, which was only that of arithmetical addition, subtraction, multiplication, and division.

But the glory of Pascal and Leibnitz, as inventors of calculating machinery, has been eclipsed by Charles Babbage and by G. and Ei Scheutz. The British government began in 1822 to build a machine under Mr. Bab-bage's direction. Early in 1833 a small portion of the machine was put together, and was found to perform its work with the utmost precision. In 1834 Mr. Babbage commenced the design of a far more powerful engine, but •nothing has been done toward its construction. These machines of Babbage are enormously expensive, $80,000 having been spent in the partial construction of the first. They are designed for the calculation of tables or series of numbers, such as tables of logarithms, of sines, etc, and are based upon the fact that if we make a new table consisting of the differences between the successive numbers of the first table, then a third consisting of the differences of the successive numbers of the second, then a fourth in like manner from the third, and so on, we shall at length generally obtain a table in which the numbers are all alike. If we had then given to us the first number in each of these tables, we might, beginning with the table in which all the numbers were alike, get back to the original table by a simple process of addition.

Thus, by this principle of differences, the computation of all tables is, in general, reduced to a process of addition. The machine prepares a stereotype plate of the table as fast as calculated, so that no errors of the press can occur in publishing the result of its labors. Many incidental benefits arose from the invention, the most curious and valuable of which was the contrivance of a scheme of mechanical notation by which the connection of all parts of a machine, and the precise action of each part, at each instant of time, may be rendered visible on a diagram, thus enabling the contriver of machinery to devise modes of economizing space and time by a proper arrangement of the parts of his invention. This mechanical notation of Babbage (" Philosophical Transactions," 1826) is for an inventor of machinery what the notation of algebra is to the student of geometry. - The machine in the Dudley observatory, Albany, N. Y., was invented by G. and E. Scheutz of Stockholm, and finished in 1853. The Swedish government paid $20,000 as a gratuity toward its construction. The inventors sought to attain the same ends that Mr. Babbage had attained, but with simpler means.

Their engine proceeds by the method of differences, calculating to the 15th place of decimals, and stamping the eight left-hand places in lead, so as to make a stereotype mould from which plates can be taken by either a stereotype or electrotype process, ready for the printing press. It can express numbers either decimally or sexa-gesimally, and prints by the side of the table the corresponding series of numbers or arguments for which the table is calculated. It has been employed at Albany in calculating a table of the true anomaly of Mars for each tenth of a day.