Circubtr slide rules, resembling aro also made. The slide-rule principle is also em ployed in instruments used to work out specific problems, such as the (low of water in pipes. or the strength of beams. Such computers may he either like the ordinary slide rule, with scales in terms of tlw factors Involved, or, as in the carious Cox computers, there may he a founda Lion plate, revolving disk. revolving segment. and index or pointer, with proper scale.. The vari ous slide rides proper all depend on the meelum ieal use of logarithms. and the scales are gradu ated on it logarithmic basis. By referring to the article LooAttrritxis. the operation of a simple slide rule will readily be understood. as the vari ous graduations correspond to the logarithmic functions, a.nd 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 log arithms. For 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 graduations. The number corresponding to the point at which the zero or indicator stands is, of course, the product, which in this ease is 4. The complexity of the problems which may he solved with the aid of the slide rule varies with the different rifles: 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 arithmet ical 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 Slo nimsky (1844) and Lucas (1885) effect multi plication without the supplementary addition required by Napier's rods. Quotients and re mainders, in the case of division, are likewise fully determined by Genaille's instrument. In struments in which mechanisms are combined for both addition and multiplication are sonic times called a ri t Inoy ea phs. IZotts (1869) eon struettql an apparatus of this kind, combining a set of Napier's rods with the abacus. More per fect forms are those of Th. von Esersky (1872). Troncet (18911. and ( 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 meehanism is the basis of cal culating machinery. This mechanism is ar ranged 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 ire placed the figures 0, 1. 2 .... it, once or several times. What ever the arrangement of these number disks, their a Xl•A of rotation may be parallel and lie in pia Ile, or may form the element, of a surface, or may coincide so that the num bers are beside one ;mother on a common cylin der. This last arrangement, which seems to have appeared for the first tune in the machine of (1750), is preferred. beeause it require; 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 point- to the figure 9. any further movement also moves the disk of the next order: that is, for every ten-place rota tion of any desired number disk. the next disk rotates one place. For addition, it is only neces sary that each clement of the number mechan ism admit. of being moved forward independently one or more figures. Fur subtractiim, the older machines' generally eontain rows of red figures •ranged in reverse order. so that the motion of the disk may still take place in the saute sense. It is immaterial whether this motion iK produced directly by the hand or indirectly by a lever; but it makes a difference in the rapid ity of the work whether different figures of the same rank are added by the movement of one and the same eleme.nt, or by the motion of dif
ferent elements. To the first group belongs the oldest of all ca leula Ong machines, the machine of Pascal ( 1(142 ). designed for adding and subtracting. The modern machines of Roth (1843) and Webb (1868), and (zit tom a t ische ‘4e111.(1 ',ben rrch e schine ( 1893 ), are modifications of the machin• a rit h atetique. 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 I to 9. in the various orders, one has to fix the eye upon the numbers of one figure only. The latest improvement is a con trivance for automatically printing both the addends and their sum. thus leaving little to lie 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'; ritlunachine' (1898) is one of the latest of the simple and practical machines.
In order mechanically to effect repeated addi tion—that is, multiplkation—a rack or special earrying apparatus is necessary. This device makes it possible by it single motion of the hand, as the rotation of a crank, to carry simul taneously the set of number disks over a desired number of places. Emir methods have been de vised for this, but the most common is the stepped reckoner of Leibnitz. a cylinder with nine teeth of different lengths. eorrespondmg to units. tens, etc. Another means also known to Leibnitz, and lately eoinin1.4 into favor, is the use of toothed wheels. whose teeth may he shoved in at will, thus rendering the Wheels in operative. A In (mg the instruments of this type, with slight modifications. are the arithmonicter of Thomas (1820). the of .laurel and dayet (1849), and the a ritimomieters of Odh ner (1378) and Kiittner (1894). The machine a (wielder of Bollce (1888), designed especially for multiplication. operates On It principle. The products of numbers from 1 to 9 are repre sented 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 lie so moved that the product of the multiplicand by each ligu rc of the multiplier is carried over to the addition machinery. In the caleulating ma chine of Steiger (1892) partial products are expressed by pairs of disks, and in Selling's clek frisehe Vcchrn inosch in( ( 1894) by electro-mag nets. These machines are defective in that the multiplication must be performed step by step. using a multiplier of one figure only. They aro made to perform division by moving a lever, which reverses the motion of the number disks.
•u•li care has also been given to perfecting in strnments having for their object the c(imput mg 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 :Milner (1786), but Babbage (1823) was the first to obtain valuable results with a machine of this kind. The machines of \Viberg (1863) and Grant (1871) are improved forms of this type. Badinage (1834) also invented an 'analytic engine,' designed to perform various analytic and arithmetical operations, but it was never com pleted.
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 (1SSS), lim ited to sums under 1000: Runge's addition ma chine, Berlin (1896), adding numbers of several figures; Felt's comptometer. Chicago (1887). keyboard type. performing all four operations: Burrongh's registering accountant, Saint Louis (BS'S I. an addition machine of Si keys, with a capacity of 2000 entries per hour, and automat ically printing both the addenda and the total sum; C'arney's cash register, Dayton (1890), an adding and printing machine of great perfec tion.