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often so great as to endanger life. Frequent shifts of workmen are required. In the Brooklyn bridge caissons pressures of ten atmospheres were used. Formerly caissons were used only for foundations in water, but within recent years the foundations for many of the tall buildings in New York city have been laid by means of pneumatic caissons.

^ Calais (kd'ld'), a seaport of France, is situated on the Strait of Dover. A ring of forts and regions of marshy ground on the south and east which can be easily flooded make the city a secure fortress. Calais has a good harbor and is connected with Dover by steamer and by a submarine telegraph. Calais has been important in history. In 1347 it was captured after a siege of eleven months by Edward III of England. Calais was held until the time of Queen Mary (1558), when it was taken back by the French. It had been called the brightest jewel in the English crown, and the boast had been written over one of its own gates,

"Then shall Frenchmen Calais win,

When iron and lead like cork shall swim." Population, 72,322.

Calcium (kdl's%-ifan) is a white, or as usually prepared in the laboratory, yellow metal. The metal is of no practical importance, but its compounds are very common. Limestone, marble and chalk as well as coral, shells, etc. are compounds of calcium carbonate; quick lime is the oxide; gypsum and plaster of paris consist of the sulphate; and calcium carbide is a manufactured compound used for making acetylene. The earthy part of bones is largely made up of calcium phosphate. It is chiefly calcium compounds that make water hard.

Calculating Machine, a machine for adding and subtracting by mechanical

CALCULATING  MACHINE

means. There are various forms of such machines, some of them being very complicated. One constructed by Mr. Babbage for the English government to be used in

preparing logarithmical and trigonometrical tables is said to have cost $100,000. Practically all these machines are constructed upon one principle. There are a number of dial faces on wheels alongside each other, each with the first ten figures on them. These wheels correspond to units, tens, hundreds, etc., and are so interlocked that ten steps on the units dial move the tens dial one step and so on. By some mechanism any number that can be set up on the dials can be added to itself any number of times, and recorded on the same or another set of dials. Multiplication is taken as a successive addition and division as a successive subtraction. The best known of these machines are the Grant and the Thomas. They can be made to multiply or divide any number of figures by any other number, but with increased number of figures the machinery becomes more complicated. In a common form of Grant machine any number up to five figures can be multipled by any other number up to five figures. Thirteen figures are the limit of a common form of the Thomas machine. Forms of calculating machines are now used in banking work as well as in engineering and statistical work.

Calculus (kaVku-lus) means any method of making mathematical investigations by means of algebraic symbols. There are several sorts of calculus, but the term commonly means the infinitesimal calculus, that is, the principles of mathematical reasoning by the use of symbols that represent and express the infinitesimal increases (variations) of quantities. Arithmetic and algebra consider numbers to be finite and discontinuous, but calculus deals with them as capable of growth continuous and infinite. Hence calculus investigates quantities whose values constantly change. Such quantities, for example, are the motions of bodies, as of planets in their orbits, or the amounts of force in the performance of work, neither of which are identically the same at any two instants. Before calculus was invented modern science made but slight progress, but since Newton in 1665 and Leibniz, in 1675, both independently discovered it, science has progressed rapidly. (Leibniz published his discovery in 1684, Newton his in 1687, but Newton was the prior discoverer.) Differential calculus investigates the infinitesimal changes of quantities when the relations between the quantities are given. Integral calculus deduces relations between quantities from those between their infinitesimal variations. The influence of the calculus on nearly all branches of mathematics has been extensive, rendering possible most important advances in astronomy, mechanics and physics.

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