CALCULATING MACHINES (from Lat. calculare, to reckon, compute; see CALCULUS). Mechanical contrivances designed to facilitate computations, to relieve the calculator from the mental strain of his work, and to insure greater accuracy in results. Calculating machines exist in various forms, and are now made in such perfection that large business houses and banks regard them as a necessity, while many scientific computations would have been abandoned but for their help. An instrument which is used for the purpose of illustration or instruction in number work is called a reckoning apparatus, but one which automatically produces the results of number combinations involving the union of different orders is called a calculating machine.

The earliest known instrument of calculation of any importance is the abacus. The Chinese lay claim to its invention. Its use by the Egyptians as early as B.C. 460 is definitely asserted by

A notable example of this type is the set of rods invented by Napier and known as virgulæ; or, popularly, as Napier's rods or bones. These consist of flat pieces of bone or ivory, divided into squares, which (on ten of the rods) are subdivided by diagonals into triangles, except the squares at the upper ends of the rods, which spaces are numbered from 1 to 9.

To illustrate the process of multiplication, consider the product of 5978 by 937. Arrange the proper rods, as in the figure, so that the numbers at the top indicate the multiplicand, and on the left place the rod headed 1. In this rod find the right-hand figure of the multiplier, which in this case is 7. Passing across this horizontal row, add obliquely the two rows of corresponding digits, writing the results in each case as the digits of the first partial product. For example, the first figure on the right is 6; this is written in the units place in the first partial product. Next add the 5 and 9 in the adjoining oblique row, which gives 4 in the tens place, with 1 to carry. This makes 8 in the hundreds column. Proceed in the same way with the other figures of the multiplier, and add the partial products as in ordinary multiplication.

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ARRANGEMENT OF NAPIER'S RODS.

Herodotus. It was probably used by the Babylonians, and certainly by the Greeks and Romans, from whom it spread to all Europe. It has existed in various forms the knotted strings, the sand-board, the pebble-tray, the counters, and the frame of beads. The last form is still in use, known as the Chinese swanpan, the Russian Stchoty, or the Japanese SoroBan. The ordinary swan-pan consists of a frame divided into two sections, holding several parallel rods, each containing several movable beads. In the Chinese swan-pan, each bead on the bottom row in the right division represents one unit, and each on the bottom row in the left division represents five units. In the next higher row the value of each bead is ten times as great, and so on.

The first improvement over the ancient abacus consisted in the use of counters, on a plan attributed, probably erroneously, to Boethius. Later these counters bore numbers, and were attached to rods, disks, or cylinders, which could be moved so as to indicate the desired results.

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The chief point of improvement over the primitive abacus consists in supplying the instrument with moving scales, which enable the calculator to form number combinations without actually counting together the different addends. Kummer (1847) accomplished this by running parallel rods in grooves; Lagrous (1828) by concentric rings; Djakoff and Webb by bands on rollers.

Another form of the calculating machine is the slide rule, which is more generally employed than any other class of calculating instruments, particularly by engineers and statisticians. In its simplest form it consists of two rules, arranged to slide on each other, and so divided into scales that by sliding the rules backward or forward until a selected number on one scale is made to coincide with a selected number on the other, the desired result is read off directly on a third scale. By means of a duplex slide rule, where the rule may be set for four factors instead of two, more complicated problems may be solved. Revolving slide rules are employed to increase the virtual length of the scales and the number of decimal places to which results may be read. In the Thacher calculating instrument, a cylinder 4 inches in diameter and 18 inches long revolves within a framework of triangular bars, each of which contains a scale on two sides. The scales contain 33,000 divisions and 17,000 engraved figures, executed on a dividing machine made expressly for the purpose. Fuller's spiral slide rule consists of a wooden cylinder containing a spiral scale 42 feet long.

Circular slide rules, resembling watches, are also made. The slide-rule principle is also employed in instruments used to work out specific problems, such as the flow of water in pipes, or the strength of beams. Such computers may be either like the ordinary slide rule, with scales in terms of the factors involved, or, as in the various Cox computers, there may be a founda

tion plate, revolving disk, revolving segment, and index or pointer, with proper scales. The various slide rules proper all depend on the mechanical use of logarithms, and the scales are graduated on a logarithmic basis. By referring to the article LOGARITHMS, the operation of a simple slide rule will readily be understood, as the various graduations correspond to the logarithmic functions, and the appropriate length of each is determined from a table of logarithms. The figures inscribed on the scales, however, are those of the numbers corresponding to the logarithms. For example, to multiply 2 by 2, the number 2 on the scale is brought opposite the number 2 on the second scale, and, as a result, the zero of the latter is distant from the zero of the first by an amount equivalent to the sum of the two logarithmic graduations. The number corresponding to the point at which the zero or indicator stands is, of course, the product, which in this case is 4. The complexity of the problems which may be solved with the aid of the slide rule varies with the different rules; but, in general, it may be said that all problems involving multiplication and division may be solved by any of them, including powers, roots, and proportions, simply by setting the rule and reading off the indicated result. By providing scales with trigonometrical instead of arithmetical functions, the uses of the slide rule may be increased greatly, and often the two classes are engraved on reverse sides of the smaller slide rules. The rule is particularly valuable where the same operation is to be repeated many times, as in computing percentages, or where many long and wearisome calculations are to be made.

The improved calculating instruments of Slonimsky (1844) and Lucas (1885) effect multiplication without the supplementary addition required by Napier's rods. Quotients and remainders, in the case of division, are likewise fully determined by Genaille's instrument. Instruments in which mechanisms are combined for both addition and multiplication are sometimes called arithmographs. Rous (1869) constructed an apparatus of this kind, combining a set of Napier's rods with the abacus. More perfect forms are those of Th. von Esersky (1872), Troncet (1891), and Bollée (1895). These form the border line between the elementary reckoning apparatus and the more elaborate calculating machine. As numbers are essential to reckon ing, so number mechanism is the basis of calculating machinery. This mechanism is arranged for the decimal system, and combines elements for the various powers of 10. The elements are usually cylindrical disks, on whose plane or curved surfaces are placed the figures 0, 1, 2, 9, once or several times. Whatever the arrangement of these number disks, their axes of rotation may be parallel and lie in a plane, or may form the elements of a cylindrical surface, or may coincide so that the numbers are beside one another on a common cylinder. This last arrangement, which seems to have appeared for the first time in the machine of Périere (1750), is preferred, because it requires the least space and brings the figures into close proximity. In every calculating machine the mechanism automatically carries over from any order to the next higher. Whenever a number disk is rotated so that it points to the figure 9, any further movement also moves the disk of

.

the next order; that is, for every ten-place rotation of any desired number disk, the next disk rotates one place. For addition, it is only neces sary that each element of the number mechanism admit of being moved forward independently one or more figures. For subtraction, the older machines generally contain rows of red figures arranged in reverse order, so that the motion of the disk may still take place in the same sense. It is immaterial whether this motion be produced directly by the hand or indirectly by a lever; but it makes a difference in the rapidity of the work whether different figures of the same rank are added by the movement of one and the same element, or by the motion of different elements. To the first group belongs the oldest of all calculating machines, the machine arithmétique of Pascal (1642), designed for adding and subtracting. The modern machines of Roth (1843) and Webb (1868), and Orlin's automatische Schraubenrechenmaschine (1893), are modifications of the machine arithmétique. The necessary speed and accuracy of movement have been gained by the introduction of keys, as in the machines of Stettner (1882) and Mayer (1887). A key being provided for the numbers from 1 to 9, in the various orders, one has to fix the eye upon the numbers of one figure only. The latest improvement is a contrivance for automatically printing both the addends and their sum, thus leaving little to be desired in the form of an addition instrument. This is a feature of Burrough's registering ac countant (1888) and Carney's cash register. Goldman's 'arithmachine' (1898) is one of the latest of the simple and practical machines.

In order mechanically to effect repeated addition—that is, multiplication-a rack or special carrying apparatus is necessary. This device makes it possible by a single motion of the hand, as the rotation of a crank, to carry simultaneously the set of number disks over a desired number of places. Four methods have been devised for this, but the most common is the stepped reckoner of Leibnitz, a cylinder with nine teeth of different lengths, corresponding to units, tens, etc. Another means also known to Leibnitz, and lately coming into favor, is the use of toothed wheels, whose teeth may be shoved in at will, thus rendering the wheels inoperative. Among the instruments of this type, with slight modifications, are the arithmometer of Thomas (1820), the machine of Maurel and Jayet (1849), and the arithmometers of Odhner (1878) and Küttner (1894). The machine à calculer of Bollée (1888), designed especially for multiplication, operates on a new principle. The products of numbers from 1 to 9 are represented by pairs of pegs, whose lengths correspond to the units and tens of the products. The pegs limit the freedom of the rack, which can be so moved that the product of the multiplicand by each figure of the multiplier is carried over to the addition machinery. In the calculating machine of Steiger (1892) partial products are expressed by pairs of disks, and in Selling's elektrische Rechenmaschine (1894) by electro-magnets. These machines are defective in that the multiplication must be performed step by step, using a multiplier of one figure only. They are made to perform division by moving a lever, which reverses the motion of the number disks. Much care has also been given to perfecting in

struments having for their object the computing of mathematical and astronomical tables and the tabulation of functions. These are, in fact, the only means of producing thoroughly accurate tables. The idea practically originated with Müller (1786), but Babbage (1823) was the first to obtain valuable results with a machine of this kind. The machines of Wiberg (1863) and Grant (1871) are improved forms of this type. Babbage (1834) also invented an 'analytic engine,' designed to perform various analytic and arithmetical operations, but it was never completed.

The following machines of recent mention are extensively used; the first three are of German make and the last three American, the latter being the more practical: Beher's addition machine (1892), of keyboard type, limited to sums under 500; Illgen's calculator (1888), limited to sums under 1000; Runge's addition machine, Berlin (1896), adding numbers of several figures; Felt's comptometer, Chicago (1887). keyboard type, performing all four operations; Burrough's registering accountant, Saint Louis (1888), an addition machine of 81 keys, with a capacity of 2000 entries per hour, and automatically printing both the addenda and the total sum; Carney's cash register, Dayton (1890), an adding and printing machine of great perfec

tion.

CASH-REGISTERS are a form of calculating machine in general use in retail stores, whose chief functions are to make a record of money received from sales of merchandise in a retail store, as the money is placed in the cash-drawer, and to add automatically this sum to the total previously placed in the drawer; it also indicates to the customers the record which has been made. The more complex cash-registers have been further developed so that it is possible to include an automatic record of other transactions which take place in a retail store, including credit sales and the separate sales of individual clerks or of particular lines of goods, so that they may be referred to at the close of the day's business.

The first practical cash-register was invented by James Ritty, of Dayton, Ohio, who secured his patent in 1879. In this first register the record was made on adding wheels and displayed by hands on a dial, but in later inventions the record is sometimes made by puncturing printed rolls of paper and is shown by indicators which rise and fall as the mechanism is operated, a number equal to the amount of the purchase rising as the cash paid is deposited in the drawer, the same operation causing the number which records the previous purchase to fall. In the 'detail adders,' manufactured by the National Cash Register Company, the mechanism is operated by pressing the proper registering key. A single pressure of the finger unlocks and throws open the cash-drawer, rings a bell, drops the indicator showing the last transaction, raises an indicator showing the amount of the new transaction, and at the same time records it on the adding wheels inside the register. Each regis tering key is connected with a corresponding adding wheel inside the register, which shows the total amount of registrations made on that key. For example, if the '5-cent' key be pressed five times its corresponding adding wheel shows a total of 25 cents. Thus the total amount of the day's sales can be ascertained at any time by

adding together the total amounts shown by the adding wheels. These registers can be arranged to keep separate record of 'charge,' 'received on account,' and 'paid out' transactions, or to show separately the receipts from different classes of goods. A drawer cannot be opened without making both an indication to the customer and an inside record under lock and key.

ELECTRIC TABULATING MACHINES, such as the one devised by Hollerith for recording and summarizing the United States census returns, may be classed under calculating machines. This apparatus is in three parts. The first operation is to punch holes in a card, corresponding to the facts to be recorded for each individual, the punches being operated from a keyboard of 240 characters. After the cards are punched they are fed into a machine, which, by means of the holes and certain electric devices, adds one to the total record for the fact indicated by each hole, such as sex, color, or age. Next the cards are placed in sorting boxes, in order to secure a combination of facts, such as the number of black persons who are married, and by means of electric connections which are acted upon only by cards having holes corresponding to the facts to be tabulated, the record is made.

For descriptions of calculating machines, consult: Mehmke, "Numerisches Rechnen," in Encyklopädie der mathematischen Wissenschaften, Vol. I. (Leipzig, 1901), containing numerous figures; Unger, "Einige Additionsmaschinen," Abhandlungen zur Geschichte der Mathematik, Vol. IX. (Leipzig, 1899); Shaw, "Theory of Continuous Calculating Machines," in Phil. Transactions of Royal Society, Vol. CLXXVI. (London, 1885).

CALCULATORS (Lat. calculator, computer; see CALCULUS), REMARKABLE. Arithmetical prodigies, often spoken of as 'lightning calculators,' having an unusual capacity for combining numbers. The wonderful feats of these prodigies have been pronounced genuine by competent judges, although their psychological peculiarities have not been fully explained. Two peculiarities, however, seem characteristic of most of the known cases: an extraordinary memory for numerical combinations, and unusual methods of grouping numbers. That their ability is not entirely the result of special training is attested by the early age at which the power is manifested. Thus, at the age of 6, T. H. Safford computed mentally the number (617,760) of barleycorns in 1040 rods, and could extract the cube roots of numbers of 9 and 10 figures. Buxton solved the problem, to find the product of doubling a farthing 139 times, the result, expressed in pounds, being a number of 39 figures. Zerah Colburn, at 9 years of age, gave at sight the factors of 294,967,297, and in 20 seconds found

mentally the number of hours in 1811 years. Raising 991 to the fifth power in 13 operations, and giving the product of any pair of two-figure numbers in 12 seconds, are feats accomplished by Arthur Griffith, who also memorized the squares of all numbers up to 130 and the cubes Other noted prodigies are Annich, and Inaudi. Bidder, Vinckler, Pughiesi, Mondeux, Magimelle,

up to 100.

CALCULUS (Lat., a small stone, or pebble, which was used in reckoning, or calculations, by the Romans). A term applied in mathematics

to any method of treating problems by means of a system of algebraic notation. Thus, the Calculus of Forms (see FORMS) is a symbolic treatment of the properties of invariants; Imaginary Calculus is the method of calculating by the use of the imaginary unit (see COMPLEX NUMBER), and the Calculus of Quaternions (see QUATERNIONS) is the method of treating certain problems with the aid of the quaternion symbolism. Usually, however, the term is employed to designate the Differential and Integral Calculus, a branch of mathematical science affording, by one general method, a solution for many of the most difficult problems of pure and applied mathematics.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. This is one of the most useful branches of mathematics. While elementary algebra and geometry deal with quantities whose value is fixed, the calculus investigates quantities whose value is continually changing. Considering that all nature in all its aspects varies continually, the importance of a mathematical method of dealing with variables is evident; and it is easy to see why science had made so little progress before the invention of the calculus, and why progress has been so rapid since.

Three simple examples may serve to show the kind of problems usually attacked by the calculus, and the manner in which it solves them. The first two of these examples can also, on account of their simplicity, be solved by means of elementary algebra, without resorting to the calculus. Nevertheless, they are typical calculus problems, and furnish as good examples of the calculus method as would be furnished by similar but much more complicated problems lying really beyond the power of elementary mathematics.

Problem I. Suppose the sum of two adjoining sides of a rectangle known. What must be the length of each side so that the rectangle may have the greatest possible area?

Problem II. A person in a boat 3 miles from the nearest point on a straight shore wishes to reach a place 5 miles away from that point. He can row 4 miles an hour and walk 5 miles an hour. Where should he land in order to reach his point in minimum time?

Problem III. To determine the work performed when a gas is compressed at constant temperature is one of the fundamental problems of theoretical engineering. Work is generally defined as the force required to move a body, multiplied by the distance traversed. In the case of a gas compressed in a cylindrical vessel, the body moved is the piston. If at the beginning of the experiment the pressure exercised on the pis

ton is, say, p pounds per square inch of surface, and the area of the piston is a; then p X a is evidently the force acting on the piston. This force, however, multiplied by the distance traversed by the piston during compression will not by any means give the work performed. For during compression the force will, of course, have to be continually increased; in other words, it will not retain its original value ap fixed, but will be a variable. In this case algebra and geometry fail to give a method of direct computation and the calculus has to be resorted to.

In order to understand how the calculus deals with problems of this nature, it is necessary to grasp clearly some fundamental ideas, which usually appear somewhat difficult to the beginner in calculus, just as the idea of any fixed number being represented by the letters a, b, c, appears difficult to the child first taking up the study of elementary algebra.

are

Fundamental Ideas: Function, Differential, Differential Coefficient, Limit.-Variables represented in calculus by the Latin letters x, y, etc., or by the Greek letters, 5, etc., just as unknown quantities are represented in algebra. If the value of one variable continually depends on that of another variable, the first variable is said to be a function of the second, and the fact is denoted by writing: y = f(x). Thus, the variable area y of a square is said to be a function of the variable length x of its side, and in this case the expression y = f(x) stands for the equation y = x2. In investigating the functions and their variables, the calculus catches them at a given moment for the purpose of determining the relative rate of their variation at that moment. Consider the motion of a ball thrown up in the air. Its velocity changes from instant to instant. We might get a rough idea of its motion by measuring the distance traversed during the first second, during the second second, during the third second, etc. But our results would be far from precise; for, however small an interval of time a second is, the velocity of our ball, changing continually, must be different at the end of that interval from what it is at its beginning. Our results would be even rougher if instead of the second we employed as a unit of time the minute. To render the results mathematically precise, we would have to take for our unit not a finite, but an infinitely small interval of time, an instant. The distance traversed dur ing such an interval would be called the differ ential of distance and would be denoted in calculus by the symbol dl, if I stand for distance. Similarly, our infinitely small interval of time would be called the differential of time and would be denoted by the symbol dt, if t stand for time. But as this idea of what a differential is is somewhat vague, owing to the difficulty of actually conceiving something that is infinitely small,' the following considerations may be resorted to. Studying the motion of a ball thrown up in the air, we consider infinitely small intervals of time dt merely in order to be able to think of the motion as uniform; for within any finite interval the motion is variable. But if at a given instant the motion should actually become uniform, and continue so, we might think of our differential dt as representing any finite length of time, be it 5 minutes, or 10 minutes, or 500 minutes. For when a body moves with perfectly uniform speed, that speed may be readily determined by ascertaining the distance traversed during any interval of time whatever; the result is the same whether we divide the distance traversed in 5 minutes by 5, or that traversed in 10 minutes by 10. We may, accordingly, define the differential of distance dl as the distance that would be traversed by the ball in an arbitrary, finite interval of time, dt, beginning at a given instant, if at that instant the motion became uniform. In this manner we may avoid thinking of infinitely small quantities. velocity would then be dl÷dt, no matter how

The

great or small dt is supposed to be. The ratio

d'

dt

is called the differential coefficient of l with respect to t-the distance being of course 'a function' of the time t. This ratio represents a limit. For, considering again the ball thrown up in the air, the error introduced by choosing a finite instead of an infinitesimal interval of time is the less the smaller an interval is chosen, and dl

finally the true velocity is approached as a dt limit, when the interval of time becomes infinitely small. All this is concisely represented by a few symbols, as follows:

limit At=0

(笠)

dl

dt

Maxima and Minima.-Since represents the velocity of the ball at any moment of the flight, it is evidently itself a variable quantity. For when, say, a rubber ball is thrown up in the air, the velocity of its motion becomes smaller and smaller until the highest point in its flight is reached; at that point the ball pauses for an instant and then begins to descend with increasing speed until it reaches the ground. Here it pauses again for an instant and then again goes up in the air. At the instant the ball is at the highest point, as well as at the instant it touches the ground, the velocity is therefore zero; i.e.

di =0. But as the two points reached by the

dt

ball are respectively the highest and the lowest, it may be said that when the function 7 has its maximum or minimum value, its differential dl coefficient with respect to its variable (i.e. -)

is zero. This must be carefully remembered.

dt

Bearing in mind the ideas explained in the preceding paragraphs, the problems cited at the beginning of the article may now be analyzed without any difficulty.

I. Solution of the First Problem.-In the problem of the maximum rectangle, let a be the known sum of two adjacent sides, let a be one of the sides, and let y be the area of the rectangle. Then y = x (ax), or y = ax — - x2. Seizing the rectangle at some point in its variation, let us lengthen the side a by some finite amount, Ar, and suppose that this causes the area to increase by a finite amount, Ay. Our

equation then becomes

y+▲ y=a(x + ▲ x ) − ( x + Ax)2=
ax+a▲ x-x2 - 2 x ▲ x − ( ▲ x )2.

Subtracting the original equation, y=ax-x2, we get

Aya ▲x-2x Ar — ( ▲ x)2, and, dividing throughout by Ax,

Ay Δε

a-2x-Ax.

But this tells us that each side must be onehalf of the known sum, i.e. that the two adjoining sides must be made equal in order that the rectangle may have its maximum area.

The process just employed in solving the problem may be described as 'differentiating with the aid of the theory of limits.' Indeed, we started with the law that the area of a rectangle equals the product of two adjoining sides, a law expressed in our case by the equation y = x(a - x). We then ascertained the ratio of the finite increment of area to an actual finite increment of the variable side x. Next we ascertained the limiting value of that ratio corresponding to an infinitely small increase of the side. This gave dy of our

the value of the differential coefficient dx

function as a 2x. And as it had been shown before that the differential coefficient is zero at the point where a function has its maximum value, we wrote a 2x0, which gave the value of the side x for that point.

-

By analogous processes of reasoning we may 'differentiate' any function whatever, and thus determine the form of its differential coefficient. In practical work, however, it is not necessary to go through the whole process every time a function is differentiated, and the differential coefficient of a function is usually obtained directly by the use of a few general formulas, the demonstrations of which are given in all text-books of calculus. In solving our other problems we will make direct use of two such formulas.

II. Solution of the Second Problem.-In the problem of the person in a boat, call A the point