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Modern electronic calculators are small (often pocket-sized), digital, and usually inexpensive devices to perform the basic operations of arithmetic. In addition to general purpose calculators, there are those designed for specific markets; for example, there are scientific calculators which focus on operations slightly more complex than those specific to arithmetic - for instance, trigonometric and statistical calculations. Some calculators even have the ability to do computer algebra. Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space. They often serve other purposes, however. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.
In the past, mechanical clerical aids such as abaci, comptometers, Napier's bones, books of mathematical tables, slide rules, or mechanical adding machines were used for numeric work. This semi-manual process of calculation was tedious and error-prone.
Modern calculators are electrically powered (usually by battery and/or solar cell) and vary from cheap, give-away, credit-card sized models to sturdy adding machine-like models with built-in printers. They first became popular in the late 1960s as decreasing size and cost of electronics made possible devices for calculations, avoiding the use of scarce and expensive computer resources. By the 1980s, calculator prices had reduced to a point where a basic calculator was affordable to most. By the 1990s they had become common in math classes in schools, with the idea that students could be freed from basic calculations and focus on the concepts.
Computer operating systems as far back as early Unix have included interactive calculator programs such as dc and hoc, and calculator functions are included in almost all PDA-type devices (save a few dedicated address book and dictionary devices).
Most calculators contain the following buttons: 1,2,3,4,5,6,7,8,9,0,+,-,×,÷ (/),.,=,%, and ± (+/-). Some even contain 00 and 000 buttons to make larger calculations easier to compute.
In most countries, students use calculators for schoolwork[citation needed]. There was some initial resistance to the idea out of fear that basic arithmetic skills would suffer. There remains disagreement about the importance of the ability to perform calculations "in the head", with some curricula restricting calculator use until a certain level of proficiency has been obtained, while others concentrate more on teaching estimation techniques and problem-solving. Research suggests that inadequate guidance in the use of calculating tools can restrict the kind of mathematical thinking that students engage in.[1] Others have argued that calculator use can even cause core mathematical skills to atrophy, or that such use can prevent understanding of advanced algebraic concepts.
There are other concerns - for example, that a pupil could use the calculator in the wrong fashion but believe the answer because that was the result given. Teachers try to combat this by encouraging the student to make an estimate of the result manually and ensuring it roughly agrees with the calculated result. Also, it is possible for a child to type in −1 × −1 and obtain the correct answer '1' without realizing the principle involved. In this sense, the calculator becomes a crutch rather than a learning tool, and it can slow down students in exam conditions as they check even the most trivial result on a calculator.
Errors are not restricted to school pupils. Any user could carelessly rely on the calculator's output without double-checking the magnitude of the result — i.e., where the decimal point is positioned. This problem was all but nonexistent in the era of slide rules and pencil-and-paper calculations, when the task of establishing the magnitudes of results had to be done by the user. In addition, algorithmic flaws and rounding techniques can sometimes lead to minor precision errors.[2]
Some fractions such as 2⁄3 are awkward to display on a calculator display as they are usually rounded to 0.66666667. Also, some fractions such as 1⁄7 which is 0.14285714285714 (to fourteen significant figures) can be difficult to recognize in decimal form; as a result, many scientific calculators are able to work in vulgar fractions and/or mixed numbers.
Personal computers and personal digital assistants can perform general calculations in a variety of ways:
Searchjavascript:alert(1337*7331)" in a web browser's address bar (as opposed to "http://website name"). Such calculations can be embedded in a separate Javascript or HTML file as well.The fundamental difference between calculators and computers is that computers can be programmed to perform different tasks while calculators are pre-designed with specific functions built in, for example addition, multiplication, logarithms, etc. While computers may be used to handle numbers, they can also manipulate words, images or sounds and other tasks they have been programmed to handle. However, the distinction between the two is quite blurred; some calculators have built-in programming functions, ranging from simple formula entry to full programming languages such as RPL or TI-BASIC. Graphing calculators in particular can, along with PDAs, be viewed as direct descendants of the 1980s pocket computers, essentially calculators with full keyboards and programming capability.
The market for calculators is extremely price-sensitive, to an even greater extent than the personal computer market; typically the user desires the least expensive model having a specific feature set, but does not care much about speed (since speed is constrained by how fast the user can press the buttons). Thus designers of calculators strive to minimize the number of logic elements on the chip, not the number of clock cycles needed to do a computation.
For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point. Bit serial logic designs are more common in calculators whereas bit parallel designs dominate general-purpose computers, because a bit serial design minimizes the languages chip complexity, but takes many more clock cycles. (Again, the line blurs with high-end calculators, which use processor chips associated with computer and embedded systems design, particularly the Z80, MC68000, and ARM architectures, as well as some custom designs specifically made for the calculator market.)
The first calculators were abathia, and were often constructed as a wooden frame with beads sliding on wires. Abathias were in use centuries before the adoption of the written Arabic numerals system and are still used by some merchants, fishermen and clerks in Africa, Asia, and elsewhere.
Devices have been used to aid computation for thousands of years, using one-to-one correspondence with our fingers.[3] The earliest counting device was probably a form of tally stick. Later record keeping aids throughout the Fertile Crescent included clay shapes, which represented counts of items, probably livestock or grains, sealed in containers.[4]
The counter abacus was devised by Egyptian mathematicians in Egypt in 2000 BC. It was used for arithmetic tasks. The Roman abacus was used in Babylonia as early as 2400 BC. Since then, many other forms of reckoning boards or tables have been invented. In a medieval counting house, a checkered cloth would be placed on a table, and markers moved around on it according to certain rules, as an aid to calculating sums of money (this is the origin of "Exchequer" as a term for a nation's treasury).
A number of analog computers were constructed in ancient and medieval times to perform astronomical calculations. These include the Antikythera mechanism and the astrolabe from ancient Greece (c. 150-100 BC), which are generally regarded as the first mechanical analog computers.[5] Other early versions of mechanical devices used to perform some type of calculations include the planisphere and other mechanical computing devices invented by Abū Rayhān al-Bīrūnī (c. AD 1000); the equatorium and universal latitude-independent astrolabe by Abū Ishāq Ibrāhīm al-Zarqālī (c. AD 1015); the astronomical analog computers of other medieval Muslim astronomers and engineers; and the astronomical clock tower of Su Song (c. AD 1090) during the Song Dynasty. The "castle clock", an astronomical clock invented by Al-Jazari in 1206, is considered to be the earliest programmable analog computer.[6]
Scottish mathematician and physicist John Napier noted multiplication and division of numbers could be performed by addition and subtraction, respectively, of logarithms of those numbers. While producing the first logarithmic tables Napier needed to perform many multiplications, and it was at this point that he designed Napier's bones, an abacus-like device used for multiplication and division.[7]
In 1622 William Oughtred invented the slide rule, which was revealed by his student Richard Delamain in 1630.[8] Since real numbers can be represented as distances or intervals on a line, the slide rule allows multiplication and division operations to be carried out significantly faster than was previously possible.[9] The devices were used by generations of engineers and other mathematically inclined professional workers, until the invention of the pocket calculator. The engineers in the Apollo program that sent a man to the moon made many of their calculations on slide rules, which were accurate to three or four significant figures.[10]
German polymath Wilhelm Schickard built the first digital mechanical calculator in 1623, and thus became the father of the computing era.[11] Since his calculator used techniques such as cogs and gears first developed for clocks, it was also called a 'calculating clock'. It was put to practical use by his friend Johannes Kepler, who revolutionized astronomy when he condensed decades of astronomical observations into algebraic expressions. Some 20 years later, in 1642, French philosopher Blaise Pascal invented the calculation device later known as the Pascaline, which was used for taxes in France until 1799. An original Pascaline is preserved in the Zwinger Museum. A machine by Gottfried Wilhelm von Leibniz (1671) followed. Leibniz once said "It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used."[12]
The first half of the 20th century saw the gradual development of the mechanical calculator mechanism.
The Dalton adding-listing machine introduced in 1902 was the first of its type to use only ten keys, and became the first of many different models of "10-key add-listers" manufactured by many companies.
In 1948 the miniature Curta calculator, which was held in one hand for operation, was introduced after being developed by Curt Herzstark in 1938. This was an extreme development of the stepped-gear calculating mechanism.
From the early 1900s through the 1960s, mechanical calculators dominated the desktop computing market (see History of computing hardware). Major suppliers in the USA included Friden, Monroe, and SCM/Marchant. (Some comments about European calculators follow below.) These devices were motor-driven, and had movable carriages where results of calculations were displayed by dials. Nearly all keyboards were full — each digit that could be entered had its own column of nine keys, 1..9, plus a column-clear key, permitting entry of several digits at once. (See the illustration of a 1914 mechanical calculator.) One could call this parallel entry, by way of contrast with ten-key serial entry that was commonplace in mechanical adding machines, and is now universal in electronic calculators. (Nearly all Friden calculators had a ten-key auxiliary keyboard for entering the multiplier when doing multiplication.) Full keyboards generally had ten columns, although some lower-cost machines had eight. Most machines made by the three companies mentioned did not print their results, although other companies, such as Olivetti, did make printing calculators.
In these machines, addition and subtraction were performed in a single operation, as on a conventional adding machine, but multiplication and division were accomplished by repeated mechanical additions and subtractions. Friden made a calculator that also provided square roots, basically by doing division, but with added mechanism that automatically incremented the number in the keyboard in a systematic fashion. Friden and Marchant (Model SKA) made calculators with square root. Handheld mechanical calculators such as the 1948 Curta continued to be used until they were displaced by electronic calculators in the 1970s.
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Typical European four-operations machines use the Odhner mechanism, or variations of it. This kind of machines included the Original Odhner, Brunsviga and several following imitators, starting from Triumphator, Thales, Walther, Facit up to Toshiba. Although most of these was operated by handcranks, there were motor-driven versions.
Although Dalton introduced in 1902 first ten-keys printing adding(two operations) machine, these features was not present in computing (four operations) machines for many decades. Facit-T (1932) was the first 10-keys computing machine having a large commercial diffusion. Olivetti Divisumma-14 (1948) was the first computing machine with both printer and 10-keys keyboard. Full-keyboard machines, including motor-driven ones, were also built until 60ties. Some machines had as many as 20 columns in their full keyboards. The monster in this field was the Duodecillion made by Burroughs for exhibit purposes.
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The first mainframe computers, using firstly vacuum tubes and later transistors in the logic circuits, appeared in the late 1940s and 1950s. This technology was to provide a stepping stone to the development of electronic calculators.
In 1954, IBM, in the U.S., demonstrated a large all-transistor calculator and, in 1957, the company released the first commercial all-transistor calculator, the IBM 608, though it was housed in several cabinets and cost about $80,000.[18]
The Casio Computer Co., in Japan, released the Model 14-A calculator in 1957, which was the world's first all-electric "compact" calculator. It did not use electronic logic but was based on relay technology, and was built into a desk.
In October 1961, the world's first all-electronic desktop calculator, the Bell Punch/Sumlock Comptometer ANITA (A New Inspiration To Arithmetic/Accounting) was announced.[19][20] This British designed-and-built machine used vacuum tubes, cold-cathode tubes and Dekatrons in its circuits, with 12 cold-cathode "Nixie"-type tubes for its display. Two models were displayed, The Mk VII for continental Europe and the Mk VIII for Britain and the rest of the world, both for delivery from early 1962. The Mk VII was a slightly earlier design with a more complicated mode of multiplication and was soon dropped in favour of the simpler Mark VIII version. The ANITA had a full keyboard, similar to mechanical Comptometers of the time, a feature that was unique to it and the later Sharp CS-10A among electronic calculators. Bell Punch had been producing key-driven mechanical calculators of the Comptometer type under the names "Plus" and "Sumlock", and had realised in the mid-1950s that the future of calculators lay in electronics. They employed the young graduate Norbert Kitz, who had worked on the early British Pilot ACE computer project, to lead the development. The ANITA sold well since it was the only electronic desktop calculator available, and was silent and quick.
The tube technology of the ANITA was superseded in June 1963, by the U.S. manufactured Friden EC-130, which had an all-transistor design, 13-digit capacity on a 5-inch CRT, and introduced reverse Polish notation (RPN) to the calculator market for a price of $2200, which was about triple the cost of an electromechanical calculator of the time. Like Bell Punch, Friden was a manufacturer of mechanical calculators that had decided that the future lay in electronics. In 1964 more all-transistor electronic calculators were introduced: Sharp introduced the CS-10A, which weighed 25 kg (55 lb) and cost 500,000 yen (~US$2500), and Industria Macchine Elettroniche of Italy introduced the IME 84, to which several extra keyboard and display units could be connected so that several people could make use of it (but apparently not at the same time).
There followed a series of electronic calculator models from these and other manufacturers, including Canon, Mathatronics, Olivetti, SCM (Smith-Corona-Marchant), Sony, Toshiba, and Wang. The early calculators used hundreds of Germanium transistors, since these were then cheaper than Silicon transistors, on multiple circuit boards. Display types used were CRT, cold-cathode Nixie tubes, and filament lamps. Memory technology was usually based on the delay line memory or the magnetic core memory, though the Toshiba "Toscal" BC-1411 appears to use an early form of dynamic RAM built from discrete components. Already there was a desire for smaller and less power-hungry machines.
The Olivetti Programma 101 was introduced in late 1965; it was a stored program machine which could read and write magnetic cards and displayed results on its built-in printer. Memory, implemented by an acoustic delay line, could be partitioned between program steps, constants, and data registers. Programming allowed conditional testing and programs could also be overlaid by reading from magnetic cards. It is regarded as the first personal computer produced by a company (that is, a desktop electronic calculating machine programmable by non-specialists for personal use). The Olivetti Programma 101 won many industrial design awards.
The Monroe Epic programmable calculator came on the market in 1967. A large, printing, desk-top unit, with an attached floor-standing logic tower, it was capable of being programmed to perform many computer-like functions. However, the only branch instruction was an implied unconditional branch (GOTO) at the end of the operation stack, returning the program to its starting instruction. Thus, it was not possible to include any conditional branch (IF-THEN-ELSE) logic. During this era, the absence of the conditional branch was sometimes used to distinguish a programmable calculator from a computer.
The first handheld calculator was developed by Texas Instruments in 1967. It could add, multiply, subtract, and divide, and its output device was a paper tape.[21][22]
The electronic calculators of the mid-1960s were large and heavy desktop machines due to their use of hundreds of transistors on several circuit boards with a large power consumption that required an AC power supply. There were great efforts to put the logic required for a calculator into fewer and fewer integrated circuits (chips) and calculator electronics was one of the leading edges of semiconductor development. U.S. semiconductor manufacturers led the world in Large Scale Integration (LSI) semiconductor development, squeezing more and more functions into individual integrated circuits. This led to alliances between Japanese calculator manufacturers and U.S. semiconductor companies: Canon Inc. with Texas Instruments, Hayakawa Electric (later known as Sharp Corporation) with North-American Rockwell Microelectronics, Busicom with Mostek and Intel, and General Instrument with Sanyo.
By 1970, a calculator could be made using just a few chips of low power consumption, allowing portable models powered from rechargeable batteries. The first portable calculators appeared in Japan in 1970, and were soon marketed around the world. These included the Sanyo ICC-0081 "Mini Calculator", the Canon Pocketronic, and the Sharp QT-8B "micro Compet". The Canon Pocketronic was a development of the "Cal-Tech" project which had been started at Texas Instruments in 1965 as a research project to produce a portable calculator. The Pocketronic has no traditional display; numerical output is on thermal paper tape. As a result of the "Cal-Tech" project, Texas Instruments was granted master patents on portable calculators.
Sharp put in great efforts in size and power reduction and introduced in January 1971 the Sharp EL-8, also marketed as the Facit 1111, which was close to being a pocket calculator. It weighed about one pound, had a vacuum fluorescent display, rechargeable NiCad batteries, and initially sold for $395.
However, the efforts in integrated circuit development culminated in the introduction in early 1971 of the first "calculator on a chip", the MK6010 by Mostek,[23] followed by Texas Instruments later in the year. Although these early hand-held calculators were very expensive, these advances in electronics, together with developments in display technology (such as the vacuum fluorescent display, LED, and LCD), lead within a few years to the cheap pocket calculator available to all.
In early 1971 Pico Electronics.[24] and General Instrument also introduced their first collaboration in ICs, a complete single chip calculator IC for the Monroe Royal Digital III calculator. Pico was a spinout by five GI design engineers whose vision was to create single chip calculator ICs. Pico and GI went on to have significant success in the burgeoning handheld calculator market.
The first truly pocket-sized electronic calculator was the Busicom LE-120A "HANDY", which was marketed early in 1971[25]. Made in Japan, this was also the first calculator to use an LED display, the first hand-held calculator to use a single integrated circuit (then proclaimed as a "calculator on a chip"), the Mostek MK6010, and the first electronic calculator to run off replaceable batteries. Using four AA-size cells the LE-120A measures 4.9x2.8x0.9 in (124x72x24 mm).
The first American-made pocket-sized calculator, the Bowmar 901B (popularly referred to as The Bowmar Brain), measuring 5.2×3.0×1.5 in (131×77×37 mm), came out in the fall of 1971, with four functions and an eight-digit red LED display, for $240, while in August 1972 the four-function Sinclair Executive became the first slimline pocket calculator measuring 5.4×2.2×0.35 in (138×56×9 mm) and weighing 2.5 oz (70g). It retailed for around $150 (GB£79). By the end of the decade, similar calculators were priced less than $10 (GB£5).
The first Soviet-made pocket-sized calculator, the "Elektronika B3-04" was developed by the end of 1973 and sold at the beginning of 1974.
One of the first low-cost calculators was the Sinclair Cambridge, launched in August 1973. It retailed for £29.95, or some £5 less in kit form. The Sinclair calculators were successful because they were far cheaper than the competition; however, their design was flawed and their accuracy in some functions was questionable. The scientific programmable models were particularly poor in this respect, with the programmability coming at a heavy price in transcendental accuracy.
Meanwhile Hewlett Packard (HP) had been developing its own pocket calculator. Launched in early 1972 it was unlike the other basic four-function pocket calculators then available in that it was the first pocket calculator with scientific functions that could replace a slide rule. The $395 HP-35, along with all later HP engineering calculators, used reverse Polish notation (RPN), also called postfix notation. A calculation like "8 plus 5" is, using RPN, performed by pressing "8", "Enter↑", "5", and "+"; instead of the algebraic infix notation: "8", "+", "5", "=").
The first Soviet scientific pocket-sized calculator the "B3-18" was completed by the end of 1975.
In 1973, Texas Instruments(TI) introduced the SR-10, (SR signifying slide rule) an algebraic entry pocket calculator for $150. It was followed the next year by the SR-50 which added log and trig functions to compete with the HP-35, and in 1977 the mass-marketed TI-30 line which is still produced.
The first desktop programmable calculators were produced in the mid-1960s by Mathatronics and Casio (AL-1000). These machines were, however, very heavy and expensive. The first programmable pocket calculator was the HP-65, in 1974; it had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader. A year later the HP-25C introduced continuous memory, i.e. programs and data were retained in CMOS memory during power-off. In 1979, HP released the first alphanumeric, programmable, expandable calculator, the HP-41C. It could be expanded with RAM (memory) and ROM (software) modules, as well as peripherals like bar code readers, microcassette and floppy disk drives, paper-roll thermal printers, and miscellaneous communication interfaces (RS-232, HP-IL, HP-IB).
The first Soviet programmable desktop calculator ISKRA 123, powered by the power grid, was released at he beginning of the 1970s. The first Soviet pocket battery-powered programmable calculator, Elektronika "B3-21", was developed by the end of 1977 and released at the beginning of 1978. The successor of B3-21, the Elektronika B3-34 wasn't backward compatible with B3-21, even if it kept the reverse Polish notation (RPN). Thus B3-34 defined a new command set, which later was used in a series of later programmable soviet calculators. Despite very limited capabilities (98 bytes of instruction memory and about 19 stack and addressable registers), people managed to write all kinds of programs for them, including adventure games and libraries of calculus-related functions for engineers. Hundreds, perhaps thousands, of programs were written for these machines, from practical scientific and business software, which were used in real-life offices and labs, to fun games for children. The Elektronika MK-52 calculator (using the extended B3-34 command set, and featuring internal EEPROM memory for storing programs and external interface for EEPROM cards and other periphery) was used in soviet spacecraft program (for Soyuz TM-7 flight) as a backup of the board computer.
This series of calculators was also noted for a large number of highly counter-intuitive mysterious undocumented features, somewhat similar to "synthetic programming" of the American HP-41, which were exploited by applying normal arithmetic operations to error messages, jumping to non-existent addresses and other techniques. A number of respected monthly publications, including the popular science magazine "Наука и жизнь" ("Science and Life"), featured special columns, dedicated to optimization techniques for calculator programmers and updates on undocumented features for hackers, which grew into a whole esoteric science with many branches, known as "eggogology" ("еггогология"). The error messages on those calculators appear as a meaningless Russian word "EGGOG" ("ЕГГОГ").
A similar hacker culture in the USA was centered around the HP-41, which was also noted for a large number of undocumented features and was much more powerful than B3-34.
Mechanical calculators continued to be sold, though in rapidly decreasing numbers, into the early 1970s, with many of the manufacturers closing down or being taken over. Comptometer type calculators were often retained for much longer to be used for adding and listing duties, especially in accounting, since a trained and skilled operator could enter all the digits of a number in one movement of the hands on a Comptometer quicker than was possible serially with a 10-key electronic calculator. The spread of the computer rather than the simple electronic calculator put an end to the Comptometer. Also, by the end of the 1970s, the slide rule had become obsolete.
Through the 1970s the hand-held electronic calculator underwent rapid development. The red LED and blue/green vacuum fluorescent displays consumed a lot of power and the calculators either had a short battery life (often measured in hours, so rechargeable nickel-cadmium batteries were common) or were large so that they could take larger, higher capacity batteries. In the early 1970s liquid crystal displays (LCDs) were in their infancy and there was a great deal of concern that they only had a short operating lifetime. Busicom introduced the Busicom LE-120A "HANDY" calculator, the first pocket-sized calculator and the first with an LED display, and announced the Busicom LC with LCD display. However, there were problems with this display and the calculator never went on sale. The first successful calculators with LCDs were manufactured by Rockwell International and sold from 1972 by other companies under such names as: Dataking LC-800, Harden DT/12, Ibico 086, Lloyds 40, Lloyds 100, Prismatic 500 (aka P500), Rapid Data Rapidman 1208LC. The LCDs were an early form with the numbers appearing as silver against a dark background. To present a high-contrast display these models illuminated the LCD using a filament lamp and solid plastic light guide, which negated the low power consumption of the display. These models appear to have been sold only for a year or two.
A more successful series of calculators using the reflective LCD display was launched in 1972 by Sharp Inc with the Sharp EL-805, which was a slim pocket calculator. This, and another few similar models, used Sharp's "COS" (Crystal on Substrate) technology. This used a glass-like circuit board which was also an integral part of the LCD. In operation the user looked through this "circuit board" at the numbers being displayed. The "COS" technology may have been too expensive since it was only used in a few models before Sharp reverted to conventional circuit boards, though all the models with the reflective LCD displays are often referred to as "COS".
In the mid-1970s the first calculators appeared with the now "normal" LCDs with dark numerals against a grey background, though the early ones often had a yellow filter over them to cut out damaging UV rays. The advantage of the LCD is that it is passive and reflects light, which requires much less power than generating light. This led the way to the first credit-card-sized calculators, such as the Casio Mini Card LC-78 of 1978, which could run for months of normal use on button cells.
There were also improvements to the electronics inside the calculators. All of the logic functions of a calculator had been squeezed into the first "Calculator on a chip" integrated circuits in 1971, but this was leading edge technology of the time and yields were low and costs were high. Many calculators continued to use two or more integrated circuits (ICs), especially the scientific and the programmable ones, into the late 1970s.
The power consumption of the integrated circuits was also reduced, especially with the introduction of CMOS technology. Appearing in the Sharp "EL-801" in 1972, the transistors in the logic cells of CMOS ICs only used any apreciable power when they changed state. The LED and VFD displays had often required additional driver transistors or ICs, whereas the LCD displays were more amenable to being driven directly by the calculator IC itself.
With this low power consumption came the possibility of using solar cells as the power source, realised around 1978 by such calculators as the Royal Solar 1, Sharp EL-8026, and Teal Photon.
At the beginning of the 1970s hand-held electronic calculators were very expensive, costing two or three weeks' wages, and so were a luxury item. The high price was due to their construction requiring many mechanical and electronic components which were expensive to produce, and production runs were not very large. Many companies saw that there were good profits to be made in the calculator business with the margin on these high prices. However, the cost of calculators fell as components and their production techniques improved, and the effect of economies of scale were felt.
By 1976 the cost of the cheapest 4-function pocket calculator had dropped to a few dollars, about one twentieth of the cost five years earlier. The consequences of this were that the pocket calculator was affordable, and that it was now difficult for the manufacturers to make a profit out of calculators, leading to many companies dropping out of the business or closing down altogether. The companies that survived making calculators tended to be those with high outputs of higher quality calculators, or producing high-specification scientific and programmable calculators.
The first calculator capable of symbolic computation was the HP-28, released in 1987. It was able to, for example, solve quadratic equations symbolically. The first graphing calculator was the Casio FX-7000G released in 1985.
The two leading manufacturers, HP and TI, released increasingly feature-laden calculators during the 1980s and 1990s. At the turn of the millennium, the line between a graphing calculator and a handheld computer was not always clear, as some very advanced calculators such as the TI-89, the Voyage 200 and HP-49G could differentiate and integrate functions, solve differential equations, run word processing and PIM software, and connect by wire or IR to other calculators/computers.
The HP 12c financial calculator is still produced. It was introduced in 1981 and is still being made with few changes. The HP 12c featured the reverse Polish notation mode of data entry. In 2003 several new models were released, including an improved version of the HP 12c, the "HP 12c platinum edition" which added more memory, more built-in functions, and the addition of the algebraic mode of data entry.
Online calculators are programs designed to work just like a normal calculator does. Usually the keyboard (or the mouse clicking a virtual numpad) is used, but other means of input (e.g. slide bars) are possible.
Thanks to the Internet, many new types of calculators are possible for calculations that would otherwise be much more difficult or impossible, such as for real time currency exchange rates, loan rates and statistics.
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CALCULATING MACHINES. Instruments for the mechanical performance of numerical calculations, have in modern times come into ever-increasing use, not merely for dealing with large masses of figures in banks, insurance offices, &c., but also, as cash registers, for use on the counters of retail shops. They may be classified as follows:-(i.) Addition machines; the first invented by Blaise Pascal (1642). (ii.) Addition machines modified to facilitate multiplication; the first by G. W. Leibnitz (1671). (iii.) True multiplication machines; Leon Bolles (1888), Steiger (1894). (iv.) Difference machines; Johann Helfrich von Muller (1786), Charles Babbage (1822). (v.)Analytical machines; Babbage (1834). The number of distinct machines of the first three kinds is remarkable and is being constantly added to, old machines being improved and new ones invented; Professor R. Mehmke has counted over eighty distinct machines of this type. The fullest published account of the subject is given by Mehmke in the Encyclopadie der mathematischen Wissenschaften, article " Numerisches Rechnen," vol. i., Heft 6 (1901). It contains historical notes and full references. Walther von Dyck's Catalogue also contains descriptions of various machines. We shall confine ourselves to explaining the principles of some leading types, without giving an exact description of any particular one. Practically all calculating machines contain a " counting work," a series of " figure disks " consisting in the original form of horizontal circular disks (fig. 1), on which the figures o, 1, 2, to 9 are marked. Each disk can turn about its vertical axis, and is covered by a fixed plate with a hole or " window " in it through which one figure can be seen. On turning the disk through onetenth of a revolution this figure will be changed into the next higher or lower. Such turning may be called a " step," positive if the next higher and negative if the next lower figure appears. Each positive step therefore adds one unit A;tion maddchr?es. to the figure under the window, while two steps add two, and so on. If a series, say six, of such figure disks be placed side by side, their windows lying in a row, then any number of six places can be made to appear, for instance 000 373. In order to add 6425 to this number, the disks, counting from right to left, have to be turned 5, 2, 4 and 6 steps respectively. If this is done the sum 006798 will appear. In case the sum of the two figures at any disk is greater than 9, if for instance the last figure to be added is 8 instead of 5, the sum for this disk is II and the 1 only will appear. Hence an arrangement for " carrying " has to be introduced. This may be done as follows. The axis of a figure disk contains a wheel with ten teeth. Each figure disk has, besides, one long tooth which when its o passes the window turns the next wheel to the left, one tooth forward, and hence the figure disk one step. The actual mechanism is not quite so simple, because the long teeth as described would gear also into the wheel to the right, and besides would interfere with each other. They must therefore be replaced by a somewhat more complicated arrangement, which has been done in various ways not necessary to describe more fully. On the way in which this is done, however, depends to a great extent the durability and trustworthiness of any arithmometer; in fact, it is often its weakest point. If to the series of figure disks arrangements are added for turning each disk through a required number of steps, FIG. I.
we have an addition machine, essentially of Pascal's type. In it each disk had to be turned by hand. This operation has been simplified in various ways by mechanical means. For pure addition machines key-boards have been added, say for each disk nine keys marked i to 9. On pressing the key marked 6 the disk turns six steps and so on. These have been introduced by Stettner (1882), Max Mayer (1887), and in the comptometer by Dorr Z. Felt of Chicago. In the comptograph by Felt and also in " Burrough's Registering Accountant " the result is printed. These machines can be used for multiplication, as repeated addition, but the process is laborious, depending for rapid execution essentially on the skill of the operator.' To adapt an addition machine, as described, to rapid multi 11 cation the turnings of the separate figure disks are replaced by one motion, commonly the turning of a handle. As, however, the different disks have to be turned through different steps, a contrivance has to be inserted which can be " set " in such a way that by.one turn of the handle each disk is moved through a number of steps equal to the number of units which is to be added on that disk. This may be done by making each of the figure disks receive on its axis a ten-toothed wheel, called hereafter the A-wheel, which is acted on either directly or indirectly by another wheel (called the B-wheel) in which the number of teeth can be varied from o to 9. This variation of the teeth has been effected in different ways. Theoretically the simplest seems to be to have on the B-wheel nine teeth which can be drawn back into the body of the wheel, so that at will any number from o to 9 can be made to project. This idea, previously mentioned by Leibnitz, has been realized by Bohdner in the " Brunsviga." Another way, also due to Leibnitz, consists in inserting between the axis of the handle bar and the A-wheel a " stepped " cylinder. This may be considered as being made up of ten wheels large enough to contain about twenty teeth each; but most of these teeth are cut away so that these wheels retain in succession 9, 8, ... 1, o teeth. If these are made as one piece they form a cylinder with teeth of lengths from 9, 8 ... times the length of a tooth on a single wheel.
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In the diagrammatic vertical section of such a machine (fig. 2) FF is a figure disk with a conical wheel A on its axis. In the covering plate HK is the window W. A stepped cylinder is shown at B. The axis Z, which runs along the whole machine, is turned by a handle, and itself turns the cylinder B by aid of conical wheels. Above this cylinder lies an antis EE with square section along which a wheel D can be moved. The same axis carries at E' a pair of conical ?,r?i '&Or ??,???? ' FIG. 2.
wheels C and C', which can also slide on the axis so that either can be made to drive the A-wheel. The covering plate MK has a slot above the axis EE allowing a rod LL' to be moved by aid of a button L, carrying the wheel D with it. Along the slot is a scale of numbers o 12. .. 9 corresponding with the number of teeth on the cylinder B, with which the wheel D will gear in any given position. A series of such slots is shown in the top middle part of Steiger's machine (fig. 3). Let now the handle driving the axis Z be turned once round, the button being set to 4. Then four teeth of the Bwheel will turn D and with it the A-wheel, and consequently the figure disk will be moved four steps. These steps will be positive or forward if the wheel C gears in A, and consequently four will be added to the figure showing at the window W. But if the wheels CC' are moved to the right, C' will gear with A moving backwards, with 1 For a fuller description of the manner in which a mere addition machine can be used for multiplication and division, and even for the extraction of square roots, see an article by C. V. Boys in Nature, 11th July 1901.
the result that four is subtracted at the window. This motion of all the wheels C is done simultaneously by the push of a lever which appears at the top plate of the machine, its two positions being marked " addition " and " subtraction." The B-wheels are in fixed positions below the plate MK. Level with this, but separate, is the plate KH with the window. On it the figure disks are mounted.
This plate is hinged at the back at H and can be lifted up, thereby throwing the A-wheels out of gear. When thus raised the figure disks can be set to any figures; at the same time it can slide to and fro so that an A-wheel can be put in gear with any C-wheel forming with it one " element." The number of these varies with the size of the machine. Suppose there are six B-wheels and twelve figure disks. Let these be all set to zero with the exception of the last four to the right, these showing I 4 3 2, and let these be placed opposite the last B-wheels to the right. If now the buttons belonging to the latter be set to 3256, then on turning the B-wheels all once round the latter figures will be added to the former, thus showing 4688 at the windows. By aid of the axis Z, this turning of the B-wheels is performed simultaneously by the movement of one handle. We have thus an addition machine. If it be required to multiply a number, say 725, by any number up to six figures, say 357, the buttons are set to the figures 725, the windows all showing zero. The handle is then turned, 725 appears at the windows, and successive turns add this number to the first. Hence seven turns show the product seven times 725. Now the plate with the A-wheels is lifted and moved one step to the right, then lowered and the handle turned five times, thus adding fifty times 725 to the product obtained. Finally, by moving the plate again, and turning the handle three times, the required product is obtained. If the machine has six B-wheels and twelve disks the product of two six-figure numbers can be obtained. Division is performed by repeated subtraction. The lever regulating the C-wheel is set to subtraction, producing negative steps at the disks. The dividend is set up at the windows and the divisor at the buttons. Each turn of the handle subtracts the divisor once. To count the number of turns of the handle a second set of windows is arranged with number disks below. These have no carrying arrangement, but one is turned one step for each turn of the handle. The machine described is essentially that of Thomas of Colmar, which was the first that came into practical use. Of earlier machines those of Leibnitz, Muller (1782), and Hahn (1809) deserve to be mentioned (see Dyck, Catalogue). Thomas's machine has had many imitations, both in England and on the Continent, with more or less important alterations. Joseph Edmondson of Halifax has given it a circular form, which has many advantages.
The accuracy and durability of any machine depend to a great extent on the manner in which the carrying mechanism is constructed. Besides, no wheel must be capable of moving in any other way than that required; hence every part must be locked and be released only when required to move. Further, any disk must carry to the next only after the carrying to itself has been completed. If all were to carry at the same time a considerable force would be required to turn the handle, and serious strains would be introduced. It is for this reason that the B-wheels or cylinders have the greater part of the circumference free from teeth. Again, the carrying acts generally as in the machine described, in one sense only, and this involves that the handle be turned always in the same direction. Subtraction therefore cannot be done by turning it in the opposite way, hence the two wheels C and C' are introduced. These are moved all at once by one lever acting on a bar shown at R in section (fig. 2).
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In The Brunsviga, The Figure Disks Are All Mounted On A Common Horizontal Axis, The Figures Being Placed On The Rim. On The Side Of Each Disk And Rigidly Connected With It Lies Its A Wheel With Which It Can Turn Independent Of The Others. The B Wheels, All Fixed On Another Horizontal Axis, Gear Directly On The A Wheels. By An Ingenious Contrivance The Teeth Are Made To Appear From Out Of The Rim To Any Desired Number. The Carrying Mechanism, Too, Is Different, And So Arranged That The Handle Can Be Turned Either Way, No Special Setting Being Required For Subtraction Or Division. It Is Extremely Handy, Taking Up Much Less Room Than The Others. Professor Eduard Selling Of Wiirzburg Has Invented An Altogether Different Machine, Which Has Been Made By Max Ott, Of Munich. The B Wheels Are Replaced By Lazy Tongs. To The Joints Of These The Ends Of Racks Are Pinned; And As They Are Stretched Out The Racks Are Moved Forward O To 9 Steps, According To The Joints They Are Pinned To. The Racks Gear Directly In The A Wheels, And The Figures Are Placed On Cylinders As In The Brunsviga. The Carrying Is Done Continuously By A Train Of Epicycloidal Wheels. The Working Is Thus Rendered Very Smooth, Without The Jerks Which The Ordinary Carrying Tooth Produces; But The Arrangement Has The Disadvantage That The Resulting Figures Do Not Appear In A Straight Line, A Figure Followed By A 5, For Instance, Being Already Carried Half A Step Forward. This Is Not A Serious Matter In The Hands Of A Mathematician Or An Operator Using The Machine Constantly, But It Is Serious For Casual Work. Anyhow, It Has Prevented The Machine From Being A Commercial Success, And It Is Not Any Longer Made. For Ease And Rapidity Of Working It Surpasses All Others. Since The Lazy Tongs Allow Of An Extension Equivalent To Five Turnings Of The Handle, If The Multiplier Is 5 Or Under, One Push Forward Will Do The Same As Five (Or Less) Turns Of The Handle, And More Than Two Pushes Are Never Required.
The Steiger Egli Machine Is A Multiplication Machine, Of Which Fig. 3 Gives A Picture As It Appears To The Manipulator. The Lower Part Of The Figure Contains, Under The Covering Plate, A Carriage With Two Rows Of Windows For The Figures Marked If And Gg. On Pressing Down The Button W The Carriage Can Be Moved To Right Or Left. Under Each Window Is A Figure Disk, As In The Thomas Machine. The Upper Part Has Three Fig. 3.
Sections. The One To The Right Contains The Handle K For Working The Machine, And A Button U For Setting The Machine For Addition, Multiplication, Division, Or Subtraction. In The Middle Section A Number Of Parallel Slots Are Seen, With Indices Which Can Each Be Set To One Of The Numbers O To 9. Below Each Slot, And Parallel To It, Lies A Shaft Of Square Section On Which A Toothed Wheel, The A Wheel, Slides To And Fro With The Index In The Slot. Below These Wheels Again Lie 9 Toothed Racks At Right Angles To The Slots. By Setting The Index In Any Slot The Wheel Below It Comes Into Gear With One Of These Racks. On Moving The Rack, The Wheels Turn Their Shafts And The Figure Disks Gg Opposite To Them. The Dimensions Are Such That A Motion Of A Rack Through 1 Cm. Turns The Figure Disk Through One " Step " Or Adds 1 To The Figure Under The Window. The Racks Are Moved By An Arrangement Contained In The Section To The Left Of The Slots. There Is A Vertical Plate Called The Multiplication Table Block, Or More Shortly, The Block. From It Project Rows Of Horizontal Rods Of Lengths Varying From 0 To 9 Centimetres. If One Of These Rows Is Brought Opposite The Row Of Racks And Then Pushed Forward To The Right Through 9 Cm., Each Rack Will Move And Add To Its Figure Disk A Number Of Units Equal To The Number Of Centimetres Of The Rod Which Operates On It. The Block Has A Square Face Divided Into A Hundred Squares. Looking At Its Face From The Right I.E. From The Side Where The Racks Lie Suppose The Horizontal Rows Of These Squares Numbered From O To 9, Beginning At The Top, And The Columns Numbered Similarly, The O Being To The Right; Then The Multiplication Table For Numbers O To 9 Can Be Placed On These Squares. The Row 7 Will Therefore Contain The Numbers 63, 56,. 7, O. Instead Of These Numbers, Each Square Receives Two " Rods " Perpendicular To The Plate, Which May Be Called The Units Rod And The Tens Rod. Instead Of The Number 63 We Have Thus A Tens Rod 6 Cm. And A Unitsrod 3 Cm. Long. By Aid Of A Lever H The Block Can Be Raised Or Lowered So That Any Row Of The Block Comes To The Level Of The Racks, The Units Rods Being Opposite The Ends Of The Racks.
The Action Of The Machine Will Be Understood By Considering An Example. Let It Be Required To Form The Product 7 Times 385. The Indices Of Three Consecutive Slots Are Set To The Numbers 3, 8, 5 Respectively. Let The Windows Gg Opposite These Slots Be Called A, B, C. Then To The Figures Shown At These Windows We Have To Add 21, 56, 35 Respectively. This Is The Same Thing As Adding First The Number 165, Formed By The Units Of Each Place, And Next 2530 Corresponding To The Tens; Or Again, As Adding First 165, And Then Moving The Carriage One Step To The Right, And Adding 253. The First Is Done By Moving The Block With The Units Rods Opposite The Racks Forward. The Racks Are Then Put Out Of Gear, And Together With The Block Brought Back To Their Normal Position; The Block Is Moved Sideways To Bring The Tens Rods Opposite The Racks, And Again Moved Forward, Adding The Tens, The Carriage Having Also Been Moved Forward As Required. This Complicated Movement, Together With The Necessary Carrying, Is Actually Performed By One Turn Of The Handle. During The First Quarter Turn The Block Moves Forward, The Units Rods Coming Into Operation. During The Second Quarter Turn The Carriage Is Put Out Of Gear, And Moved One Step To The Right While The Necessary Carrying Is Performed; At The Same Time The Block And The Racks Are Moved Back, And The Block Is Shifted So As To Bring The Tens Rods Opposite The Racks. During The Next Two Quarter Turns The Process Is Repeated, The Block Ultimately Returning To Its Original Position. Multiplication By A Number With More Places Is Performed As In The Thomas. The Advantage Of This Machine Over The Thomas In Saving Time Is Obvious. Multiplying By 817 Requires In The Thomas 16 Turns Of The Handle, But In The Steiger Egli Only 3 Turns, With 3 Settings Of The Lever H. If The Lever H Is Set To I We Have A Simple Addition Machine Like The Thomas Or The Brunsviga. The Inventors State That The Product Of Two 8 Figure Numbers Can Be Got In 6 7 Seconds, The Quotient Of A 6 Figure Number By One Of 3 Figures In The Same Time, While The Square Root To 5 Places Of A 9 Figure Number Requires 18 Seconds.
} Machines Of Far Greater Powers Than The Arithmometers Mentioned Have Been Invented By Babbage And By Scheutz. A Description Is Impossible Without Elaborate Drawings. The Following Account Will Afford Some Idea Of The Working Of Babbage'S Difference Machine. Imagine A Number Of Striking Clocks Placed In A Row, Each With Only An Hour Hand, And With Only The Striking Apparatus Retained. Let The Hand Of The First Clock Be Turned. As It Comes Opposite A Number On The Dial The Clock Strikes That Number Of Times. Let This Clock Be Connected With The Second In Such A Manner That By Each Stroke Of The First The Hand Of The Second Is Moved From One Number To The Next, But Can Only Strike When The First Comes To Rest. If The Second Hand Stands At 5 And The First Strikes 3, Then When This Is Done The Second Will Strike 8; The Second Will Act Similarly On The Third, And So On. Let There Be Four Such Clocks With Hands Set To The Numbers 6, 6, I, O Respectively. Now Set The Third Clock Striking 1, This Sets The Hand Of The Fourth Clock To 1; Strike The Second (6), This Puts The Third To 7 And The Fourth To 8. Next Strike The First (6); This Moves The Other Hands To 12, 19, 27 Respectively, And Now Repeat The Striking Of The First. The Hand Of The Fourth Clock Will Then Give In Succession The Numbers 1, 8, 27, 64, &C., Being The Cubes Of The Natural Numbers. The Numbers Thus Obtained On The Last Dial Will Have The Differences Given By Those Shown In Succession On The Dial Before It, Their Differences By The Next, And So On Till We Come To The Constant Difference On The First Dial. A Function.
Y = A 4 Bx Fcx2 Dx3 Ex4 Gives, On Increasing X Always By Unity, A Set Of Values For Which The Fourth Difference Is Constant. We Can, By An Arrangement Like The Above, With Five Clocks Calculate Y For X= I, 2, 3,. To Any Extent. This Is The Principle Of Babbage'S Difference Machine. The Clock Dials Have To Be Replaced By A Series Of Dials As In The Arithmometers Described, And An Arrangement Has To Be Made To Drive The Whole By Turning One Handle By Hand Or Some Other Power. Imagine Further That With The Last Clock Is Connected A Kind Of Typewriter Which Prints The Number, Or, Better, Impresses The Number In A Soft Substance From Which A Stereotype Casting Can Be Taken, And We Have A Machine Which, When Once Set For A Given Formula Like The Above, Will Automatically Print, Or Prepare Stereotype Plates For The Printing Of, Tables Of The Function Without Any Copying Or Typesetting, Thus Excluding All Possibility Of Errors. Of This " Difference Engine," As Babbage Called It, A Part Was Finished In 1834, The Government Having Contributed 17,000 Towards The Cost. This Great Expense Was Chiefly Due To The Want Of Proper Machine Tools.
Meanwhile Babbage Had Conceived The Idea Of A Much More Powerful Machine, The " Analytical Engine," Intended To Perform Any Series Of Possible Arithmetical Operations. Each Of These Was To Be Communicated To The Machine By Aid Of Cards With Holes Punched In Them Into Which Levers Could Drop. It Was Long Taken For Granted That Babbage Left Complete Plans; The Committee Of The British Association Appointed To Consider This Question Came, However, To The Conclusion (Brit. Assoc. Report, 1878, Pp. 92 102) That No Detailed Working Drawings Existed At All; That The Drawings Left Were Only Diagrammatic And Not Nearly Sufficient To Put Into The Hands Of A Draughtsman For Making Working Plans; And " That In The Present State Of The Design It Is Not More Than A Theoretical Possibility." A Full Account Of The Work Done By Babbage In Connexion With Calculating Machines, Acrd Much Else Published By Others In Connexion Therewith, Is Contained In A Work Published By His Son, General Babbage.
Slide Rules Are Instruments For Performing Logarithmic Calculations Mechanically, And Are Extensively Used, Especially Where Only Rough Approximations Are Required. They Are Slide Almost As Old As Logarithms Themselves. Edmund Gunter Drew A " Logarithmic Line " On His " Scales " As Follows (Fig. 4): On A Line Ab Lengths Are Set Off To Scale To Represent The Common Logarithms Of The Numbers 1 2 3 ... Io, And The Points Thus Obtained Are Marked With These Numbers.
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As log i = o, the beginning A has the number 1 and B the number io, hence the unit of length is AB, as log ro = i. The same division is repeated from B to C. The distance 1,2 thus represents log 2, 1,3 gives log 3, the distance between 4 and 5 gives log 5 - log 4= log, and so for others. In order to multiply two numbers, say 2 and 3, we have log 2 X3 = log 2 +log 3. Hence, setting off the distance 1,2 from 3 forward by the aid of a pair A g be performed. It is then convenient to make the scales circular. A number of rings or disks are mounted side by side on a cylinder, each having on its rim a log-scale.
The " Callendar Cable Calculator," invented by Harold Hastings and manufactured by Robert W. Paul, is of this kind. In it a number of disks are mounted on a common shaft, on which each turns freely unless a button is pressed down whereby c FIG.
of compasses will give the distance log 2+log 3, and will bring us to 6 as the required product. Again, if it is required to find of 7, set off the distance between 4 and 5 from 7 backwards, and the required number will be obtained. In the actual scales the spaces between the numbers are subdivided into io or even more parts, so that from two to three figures may be read. The numbers 2, 3... in the interval BC give the logarithms of r o times the same numbers in the interval AB; hence, if the 2 in the latter means 2 or 2, then the 2 in the former means 20 or 2.
Soon after Gunter's publication (1620) of these " logarithmic lines," Edmund Wingate (1672) constructed the slide rule by repeating the logarithmic scale on a tongue or " slide," which could be moved along the first scale, thus avoiding the use of a pair of compasses. A clear idea of this device can be formed if the scale in fig. 4 be copied on the edge of a strip of paper placed against the line A C. If this is now moved to the right till its r comes opposite the 2 on the first scale, then the 3 of the second will be opposite 6 on the top scale, this being the product of 2 and 3; and in this position every number on the top scale will be twice that on the lower. For every position of the lower scale the ratio of the numbers on the two scales which coincide will be the same. Therefore multiplications, divisions, and simple proportions can be solved at once.
Dr John Perry added log log scales to the ordinary slide rule in order to facilitate the calculation of a x or e' according to the formula log loga z = log loga+logx. These rules are manufactured by A. G. Thornton of Manchester.
Many different forms of slide rules are now on the market. The handiest for general use is the Gravet rule made by TavernierGravet in Paris, according to instructions of the mathematician V. M. A. Mannheim of the E. Polytechnique in Paris. It contains at the back of the slide scales for the logarithms of sines and tangents so arranged that they can be worked with the scale on the front. An improved form is now made by Davis and Son of Derby, who engrave the scales on white celluloid instead of on box-wood, thus greatly facilitating the readings. These scales have the distance from one to ten about twice that in fig. 4. Tavernier-Gravet makes them of that size and longer, even 2 metre long. But they then become somewhat unwieldy, though they allow of reading to more figures. To get a handy long scale Professor G. Fuller has constructed a spiral slide rule drawn on a cylinder, which admits of reading to three and four figures. The handiest of all is perhaps the " Calculating Circle " by Boucher, made in the form of a watch. For various purposes special adaptations of the slide rules are met with - for instance, in various exposure meters for photographic purposes. General Strachey introduced slide rules into the Meteorological Office for performing special calculations. At some blast furnaces a slide rule has been used for determining the amount of coke and flux required for any weight of ore. Near the balance a large logarithmic scale is fixed with a slide which has three indices only. A load of ore is put on the scales, and the first index of the slide is put to the number giving the weight, when the second and third point to the weights of coke and flux required.
By placing a number of slides side by side, drawn if need be to different scales of length, more complicated calculations may 4.
the disk is clamped to the shaft. Another disk is fixed to the shaft. In front of the disks lies a fixed zero line. Let all disks be set to zero and the shaft be turned, with the first disk clamped, till a desired number appears on the zero line; let then the first disk be released and the second clamped and so on; then the fixed disk will add up all the turnings and thus give the product of the numbers shown on the several disks. If the division on the disks is drawn to different scales, more or less complicated calculations may be rapidly performed. Thus if for some purpose the value of say ab 3 VC is required for many different values of a, b, c, three movable disks would be needed with divisions drawn to scales of lengths in the proportion i: 3: 2. The instrument now on sale contains six movable disks.
In order to measure the length of a curve, such as the road on a map, a wheel is rolled along it. For one revolution of the wheel the path described by its point of contact is equal to the circumference of the wheel. Thus, if a cyclist counts the number of revolutions of his front wheel he can calculate the distance ridden by multiplying that number by the circumference of the wheel. An ordinary cyclometer is nothing but an arrangement for counting these revolutions, but it is graduated in such a manner that it gives at once the distance in miles. On the same principle depend a number of instruments which, under various fancy names, serve to measure the length of any curve; they are in the shape of a small meter chiefly for the use of cyclists. They all have a small wheel which is rolled along the curve to be measured, and this sets a hand in motion which gives the reading on a dial. Their accuracy is not very great, because it is difficult to place the wheel so on the paper that the point of contact lies exactly over a given point; the beginning and end of the readings are therefore badly defined. Besides, it is not easy to guide the wheel along the curve to which it should always lie tangentially. To obviate this defect more complicated curvometers or kartometers have been devised. The handiest seems to be that of G. Coradi. He uses two wheels; the tracing-point, halfway between them, is guided along the curve, the line joining the wheels being kept normal to the curve. This is pretty easily done by eye; a constant deviation of 8° from this direction produces an error of only i %. The sum of the two readings gives the length. E. Fleischhauer uses three, five or more wheels arranged symmetrically round a tracer whose point is guided along the curve; the planes of the wheels all pass through the tracer, and the wheels can only turn in one direction. The sum of the readings of all the wheels gives approximately the length of the curve, the approximation increasing with the number of the wheels used. It is stated that with three wheels practically useful results can be obtained, although in this case the error, if the instrument is consistently handled so as always to produce the greatest inaccuracy, may be as much as 5%.
Planimeters are instruments for the determination by mechanical means of the area of any figure. A pointer, generally called the " tracer," is guided round the boundary of the figure, and then the area is read off on the recording apparatus metetr-s. of the instrument. The simplest and most useful is Amsler's (fig. 5). It consists of two bars of metal OQ and QT, which are hinged together at Q. At 0 is a needle-point which is driven into the drawing-board, and at T is the tracer. As this is guided round the boundary of the figure a wheel W mounted on QT rolls on the paper, and the turning of this wheel measures, to some known scale, the area. We shall give the theory of this instrument fully in an elementary manner by aid of geometry. The theory of other plani meters can then be easily FIG. 5. understood.
Consider the rod QT with the wheel W, without the arm OQ. Let it be placed with the wheel on the paper, and now moved perpendicular to itself from AC to BD (fig. 6). The rod sweeps over, or generates, the area of the rectangle Acdb =lp, where 1 denotes the length C 0 of the rod and p the distance AB through which it has been moved. This distance, as measured by the rolling of the wheel, which acts as a curvometer, will be called the " roll " of the wheel and be denoted by w. In this case p = w, and the area P is given by P=wl. Let the circumferA ence of the wheel be divided into say a hundred equal parts u; then w registers the number of u's rolled over, and w therefore gives the number of areas lu contained in the rectangle. By suitably selecting the radius of the wheel and the length 1, this area lu may be any convenient unit, say a square inch or square centimetre. By changing 1 the unit will be changed.
Again, suppose the rod to turn (fig. 7) about the end Q, then T will describe an arc of a circle, and the rod will generate an area 11 2 9, where 0 is the angle AQB through which the rod has turned. The wheel will roll over an arc cO, where c is the distance of the wheel from Q. The " roll " is now w =c0; hence the area generated is 2 P II 1 w, and is again determined by w.
B Next let the rod be moved FIG. 7. parallel to itself, but in a direction not perpendicular to itself (fig. 8). The wheel will now not simply roll. Consider a small motion of the rod from QT to Q'T'. This may be resolved into the motion to RR' perpendicular to the rod, whereby the rectangle QTR'R is generated, and the sliding of the rod along itself from RR' to Q'T'. During this second step no area will be generated. During the first step the roll of the wheel will be QR, whilst during the second step there will be no roll at all. The roll of the wheel will therefore measure the area of the rectangle which equals the parallelogram QTT'Q'. If the whole motion of the rod be considered as made up of a very great number of small steps, each resolved as stated, it will be seen that the roll again measures the area generated. But it has to be noticed that now the wheel does not only roll, but also slips, over the paper. This, as will be pointed out later, may introduce an error in the reading.
We can now investigate the most general motion of the rod. We again resolve the motion into a number of small steps. Let (fig. 9) AB be one position, CD the next after a step so small that the arcs AC and BD over which the ends have passed may be considered as straight lines. The area generated is AB DC. This motion we resolve into a step from AB to CB', parallel to AB and a turning about C from CB' to CD, steps such as have been investigated. During the first, the " roll " will be p the altitude of the parallelogram; during the second it will be cO. Therefore w=p+c9. The area generated is 1p+11 2 9, or, expressing p in terms of w, lw+(1l 2 - lc)9. For a finite motion we get the area equal to the sum of the areas generated during the different steps. But the wheel will continue rolling, and give the whole roll as the sum of the rolls for the successive steps. Let then w denote the whole roll (in fig. Io), and let a denote the sum of all the small turnings 0; then the area is P =lw+(11 2 -1c)a. (I) Here a is the angle which the last position of the rod makes with the first. In all applica tions of the planimeter the rod is brought back to its original position. Then the angle a is either zero, or it is if the rod has been once turned quite round.
Hence in the first case we have P =lw. (2a) and w gives the area as in case of a rectangle.
In the other case P =lw+lC. (2b) where C = (1l - c)27r, if the rod has once turned round. The number C will be seen to be always the same, as it depends only on the dimensions of the instrument. Hence now again the area is determined by if C is known.
Thus it is seen that the area generated by the motion of the rod can be measured by the roll of the wheel; it remains to show how any given area can be generated by the rod. Let the rod move in any manner but return to its original position. Q and T then describe closed curves. Such motion may be called cyclical. Here the theorem holds : - If a rod QT performs a cyclical motion, then the area generated equals the difference of the areas enclosed by the paths of T and Q respectively. The truth of this proposition will be seen from a figure. In fig. II different posit r ons of the moving rod QT have been marked, and its motion can be easily followed. It will be seen that every part of the area TT'BB' will be passed over once and always by a forward motion of the rod, whereby the wheel will increase its roll. The area AA'QQ' will also be swept over once, but with a backward roll; it must therefore be counted as negative. The area between the curves is passed over twice, once with a forward and once with a backward roll; it therefore counts once positive and once negative; hence not at all.. In more complicated figures it may happen that the area within one of the curves, say TT'BB', is passed over several times, but then it will be passed over once more in the forward direction than in the backward one, and thus the theorem will still hold.
To use Amsler's planimeter, place the pole 0 on the paper outside the figure to be measured. Then the area generated by QT is that of FIG. 12. the figure, because the point Q moves on an arc of a circle to and fro enclosing no area. At the same time the rod comes back without making a complete rotation. We have therefore in formula (I), a = o, and hence P = 1w, 9 T r' T w 0 FIG. 6.
A FIG. 9.
D FIG. IQ.
FIG II.
which is read off. But if the area is too large the pole 0 may be placed within the area. The rod describes the area between the boundary of the figure and the circle with radius r =OQ, whilst the rod turns once completely round, making a=21r. The area measured by the wheel is by formula (I), lw+ (2l 2 -1c)27. To this the area of the circle irr 2 must be added, so that now P =lw {-(2l2 - lc)2 ?+ ?r2, P =lw +C, where C = (1l 2 -lc)27r+1rr 2 is a constant, as it depends on the dimensions of the instrument alone. This constant is given with each instrument.
Amsler's planimeters are made either with a rod QT of fixed length, which gives the area therefore in terms of a fixed unit, say in square inches, or else the rod can be moved in a sleeve to which the arm OQ is hinged (fig. 13). This makes it possible to change the unit lu, which is proportional to 1. In the planimeters described the recording or integrating apparatus is a smooth wheel rolling on the paper or on some other surface. Amsler has described another recorder, viz. a wheel with a sharp edge. This will roll on the paper but not slip. Let the rod QT carry with it an arm CD perpendicular to it. Let there be mounted FIG. 14.
on it a wheel W, which can slip along and turn about it. If now QT is moved parallel to itself to Q'T', then W will roll without slipping parallel to QT, and slip along CD. This amount of slipping will equal the perpendicular distance between QT and Q'T', and therefore serve to measure the area swept over like the wheel in the machine already described. The turning of the rod will also produce slipping of the wheel, but it will be seen without difficulty that this will cancel during a cyclical motion of the rod, provided the rod does not perform a whole rotation.
The first planimeter was made on the following principles: A frame FF (fig. 15) can move parallel to OX. It carries a rod TT movable along its own length, hence the tracer T can be guided along any curve ATB.
A When the rod has been pushed back to Q'Q, the tracer moves along the axis OX. On the frame a cone VCC' is mounted with its axis sloping so that its top edge is horizontal and 0 A' o ?' X parallel to TT', whilst its vertex V is opposite Q'. As the frame moves it turns the cone. A wheel W is mounted on the rod at T', or on an axis parallel to and rigidly connected with it. This wheel rests on the top edge of the cone. If now the tracer T, when pulled out through a distance y above Q, be moved parallel to OX through a distance dx, the frame moves through an equal distance, and the cone turns through FIG. 15. an angle do proportional to dx. The wheel W rolls on the cone to an amount again proportional to dx, and also proportional to y, its distance from V. Hence the roll of the wheel is proportional to the area ydx described by the rod QT. As T is moved from A to B along the curve the roll of the wheel will therefore be proportional to the area AA'B'B. If the curve is closed, and the tracer moved round it, the roll will measure the area independent of the position of the axis OX, as will be seen by drawing a figure. The cone may with advantage be replaced by a horizontal disk, with its centre at V this allows of y being negative. It may be noticed at once that the roll of the wheel gives at every moment the area Aatq. It.
will therefore allow of registering a set of values off ydx for any values of x, and thus of tabulating the values of any indefinite. integral. In this it differs from Amsler's planimeter. Planimeters. of this type were first invented in 1814 by the Bavarian engineer Hermann, who, however, published nothing. They were reinvented by Prof. Tito Gonnella of Florence in 1824, and by the Swiss engineer Oppikofer, and improved by Ernst in Paris, the astronomer Hansen in Gotha, and others (see Henrici, British Association Report, 1894). But all were driven out of the field by Amsler's simpler planimeter.
Altogether different from the planimeters described is the hatchet planimeter, invented by Captain Prytz, a Dane, and made by Herr Cornelius Knudson in Copenhagen. It consists of a single rigid piece like fig.
16. The one end T is the tracer, the other Q has a sharp hatchet-like edge. If this is placed with QT on the paper and T is moved along any curve, Q will follow, describ ing a " curve of pursuit." In consequence of the sharp edge, Q can only move in the direction of QT, but the whole can turn about Q. Any small step forward can therefore be considered as made up of a motion along QT, together with a turning about Q. The latter motion alone generates an area. If therefore a line OA = QT is turning about a fixed point 0, always keeping. parallel to QT, it will sweep over an area equal to that generated by the more general motion of QT. Let now (fig. 17) QT be placed on OA, and T be guided round the closed curve in the sense of the arrow_ Q will describe a curve OSB. It may be made visible by putting a piece of " copying paper " under the hatchet. When T has returned to A the hatchet has the position BA. A line turning from OA about 0 kept parallel to QT will describe the circular sector OAC, which is equal in magnitude and sense to AOB. This therefore measures the area generated by the motion of QT. To make this motion cyclical, suppose the hatchet turned about A till Q comes from B to O. Hereby the sector AOB is again described, and again in the positive sense, if it is remembered that it turns about the tracer T fixed at A. The whole area now generated is therefore twice the area of this sector, or equal to OA. OB, where OB is measured along the arc. According to the theorem given above, this area also equals the area of the given curve less the area Osbo. To make this area disappear, a slight modification of the motion of QT is required. Let the tracer T be moved, both from the first position OA and the last BA of the rod, along some straight line AX. Q describes curves OF and BH respectively. Now begin the motion with T at some point R on AX, and move it along this line to A, round the curve and back to R. Q will describe the curve Dosbed, if the motion is again made cyclical by turning QT with T fixed at A. If R is properly selected, the path of Q will cut itself, and parts of the area will be positive, parts negative, as marked in the figure, and may therefore be made to vanish. When this is done the area of the curve will equal twice the area of the sector RDE. It is therefore equal to the arc DE multiplied by the length QT; if the latter equals 10 in., then 10 times the number of inches contained in the arc DE gives the number of square inches contained within the given figure. If the area is not too large, the arc DE may be replaced by the straight line DE.
To use this simple instrument as a planimeter requires the possibility of selecting the point R. The geometrical theory here given has so far failed to give any rule. In fact, every line through any point in the curve contains such a point. The analytical theory of the inventor, which is very similar to that given by F. W. Hill (Phil. Mag. 1894), is too complicated to repeat here. The integrals expressing the area generated by QT have to be expanded in a series. By retaining only the most important terms a result is obtained which comes to this, that if the mass-centre of the area be taken as R, then A may be any point on the curve. This is only approximate. Captain Prytz gives the following instructions: - Take a point R as near as you can guess to the mass-centre, put the tracer T on it, the knife-edge Q outside; make a mark on the paper by pressing the knife-edge into it; guide the tracer from R along a straight line to a point A on the boundary, round the boundary, IG. 13.
or Q' FIG. 16.
FIG. 17.
and back from A to R; lastly, make again a mark with the knifeedge, and measure the distance c between the marks; then the area is nearly cl, where l = QT. A nearer approximation is obtained by repeating the operation after turning QT through 180° from the original position, and using the mean of the two values of c thus obtained. The greatest dimension of the area should not exceed 21, otherwise the area must be divided into parts which are determined separately. This condition being fulfilled, the instrument gives very satisfactory results, especially if the figures to be measured, as in the case of indicator diagrams, are much of the same shape, for in this case the operator soon learns where to put the point R.
Integrators serve to evaluate a de finite integral fJ(x)dx. If we plot out the curve whose equation is y= f(x), the integral f ydx between the proper limits represents the area of a figure bounded by the curve, the axis of x, and the ordinates at x= a, x= b. Hence if the curve is drawn, any planimeter may be used for finding the value of the integral. In this sense planimeters are integrators. In fact, a planimeter may often be used with advantage to solve problems more complicated than the determination of a mere area, by converting the one problem graphically into the other. We give an example: Let the problem be to determine for the figure ABG (fig. 18), not only the area, but also the first and second moment with regard to the axis XX. At a distance a draw a line, C'D', parallel to XX. In the figure draw a number of lines parallel to AB. Let CD be one of them. Draw C and D vertically upwards to C'D', join these points to some point 0 in XX, and mark the points C 1 D 1 where OC' and OD' cut CD. Do this for a sufficient number of lines, and join the points C 1 D 1 thus obtained. This gives a new curve, which may be called the first derived curve. By the same process get a new curve from this, the second derived curve. By aid of a planimeter determine the areas P, P 1, P2, of these three curves. Then, if X is the distance of the mass-centre of the given area from XX; X I the same quantity for the first derived figure, and I =Ak 2 the moment of inertia of the first figure, k its radius of gyration, with regard to XX as axis, the following relations are easily proved: Px = aP 1; P1 1 = aP 2; I = aP 1 x 1 = a 2 P 1 P 2; k2=, which determine P, `x and I or k. Amsler has constructed an integrator which serves to determine these quantities by guiding a C' D 0 FIG. 18.
tracer once round the boundary of the given figure (see below). Again, it may be required to find the value of an integral yc(x)dx between given limits where 4(x) is a simple function like sin nx, and where y is given as the ordinate of a curve. The harmonic analysers described below are examples of instruments for evaluating such integrals.
Amsler has modified his planimeter in such a manner that instead of the area it gives the first or second moment of a figure about an axis in its plane. An instrument giving all three quantities simultaneously is known as Amsler's integrator or moment-planimeter. It has one tracer, but three recording wheels. It is mounted on a zontal disk A, movable about a vertical axis Q.
carriage which runs on a straight rail (fig. 19). This carries a horiSlightly more than half the circumference is circular with radius 2a, the other part with FIG. 19.
radius 3a. Against these gear two disks, B and C, with radii a; their axes are fixed in the carriage. From the disk A extends to the left a rod OT of length 1, on which a record- in` ing wheel W is mounted. The disks B and C have also r recording wheels, W 1 and W2, the axis of W 1 being per pendicular, that of W2 parallel to OT. If now T is guided round a figure F, 0 will move to and fro in a straight line. This part is therefore a simple planimeter, in which the one end of the arm moves in a straight line instead of in a circular arc. Consequently, the "roll" of W will record the area of the figure. Imagine now that the disks B and C also receive arms of length l from the centres of the disks to points T1 FIG. 20.
and T2, and in the direction of the axes of the wheels. Then these arms with their wheels will again be planimeters. As T is guided round the given figure F, these points T 1 and T2 will describe closed curves, F 1 and F2, and the " rolls " of W 1 and W2 will give their areas A 1 and A2. Let XX (fig. 20) denote the line, parallel to the rail, on which 0 moves; then when T lies on this line, the arm BT1 is perpendicular to XX, and CT 2 parallel to it. If OT is turned through an angle 0, clockwise, BT 1 will turn counter-clockwise through an angle 20, and CT 2 through an angle 30, also counterclockwise. If in this position T is moved through a distance x parallel to the axis XX, the points T 1 and T2 will move parallel to it through an equal distance. If now the first arm is turned through a small angle do, moved back through a distance x, and lastly turned back through the angle do, the tracer T will have described the boundary of a small strip of area. We divide the given figure into X such strips. Then to every such strip will correspond a strip of equal length x of the figures described by T 1 and T2.
The distances of the points, T, T1, T2, from the axis XX may be called y, y l , y2. They have the values y =1 sin 0, y1=1 cos 20, y 2 = - 1 sin 30, from which dy =l cos 0. dB, dy1= - 2l sin 20. dB, dye= - 3l cos 30. d9. The areas of the three strips are respectively dA=xdy, dA i =xdy i, dA2=xdy2.
Now dy 1 can be written dy i = - 4l sin 0 cos OdO= - 4 sin Ody; therefore dA1= - 4 sin 0.dA= - lydA; whence A l = - Z f ydA= - l Ay, where A is the area of the given figure, and y the distance of its mass-centre from the axis XX. But A 1 is the area of the second figure F 1, which is proportional to the reading of W 1. Hence we may say Ay = C1w1, where C l is a constant depending on the dimensions of the instrument. The negative sign in the expression for A 1 is got rid of by numbering the wheel W 1 the other way round.
Again dye=-31 cos B { 4 cost 0 - 3 } dB= -3{4 cos t 19-3 }dy = - 3 y2 S dy, dA 2 = - l2 y2dA+9dA, 1 A2 = - d2 y2dA+9A.
2 But the integral gives the moment of inertia I of the area A about the axis XX. As A2 is proportional to the roll of w 2, A to that of W, we can write I = Cw - C2w2, Ay = Clwl, A = Cox.
If a line be drawn parallel to the axis XX at the distance y, it will pass through the mass-centre of the given figure. If this represents the section of a beam subject to bending, this line gives for a proper choice of XX the neutral fibre. The moment of inertia for it will be I +A5 72 Thus the instrument gives at once all those quantities which are required for calculating the strength of the beam under bending. One chief use of this integrator is for the calculation of the displacement and stability of a ship from the drawings of a number of sections. It will be noticed that the length of the figure in the direction of XX is only limited by the length of the rail.
This integrator is also made in a simplified form without the wheel W2. It then gives the area and first moment of any figure.
While an integrator determines the value of a definite integral, hence a irate= mere constant, an integraph. gives the value of an indefinite integral, which is a function of x. Analytically if y is a given function f(x) of x and Y =f ydx or Y =f +const. the function Y has to be determined from the condition dY ax = y. Graphically y =f(x) is either given by a curve, or the graph of the equation is drawn: y, therefore, and similarly Y, is a length. But is in this case a mere number, and cannot equal a length y. Hence we introduce an arbitrary constant length a, the unit to which the integraph draws the curve, and write dY a = and aY= fydx. Now for the Y-curve d = tan 0, where is the angle between the tangent to the curve, and the axis of x. Our condition therefore becomes P; a variety of mechanisms to p a B obtain the object in question are described. Some years FIG. 21.
later G. Coradi, in Zurich, carried out his ideas. Before this was done, C. V. Boys, without knowing of Abdank-Abakanowicz's work, actually made an integraph which was exhibited at the Physical Society in 1881.. Both make use of a sharp edge wheel. Such a wheel will not slip sideways; it will roll forwards along the line in which its plane intersects the plane of the paper, and while rolling will be able to turn gradually about its point of contact. If then the angle between its direction of rolling and the x-axis be always equal to 4), the wheel. will roll along the Y-curve required. The axis of x is fixed only in direction; shifting it parallel to itself adds a constant to Y, and. this gives the arbitrary constant of integration.
In fact, if Y shall vanish for x=c, or if x Y = ydx, then the axis of x has to be drawn through that point on the y-curve which corresponds to x = c. In Coradi's integraph a rectangular frame_F 1 F 2 F 3 F 4 (fig. 22)/ b -'rY FIG. 22.
rests with four rollers R on the drawing board, and can roll freely in the direction OX, which will be called the axis of the instrument.. On the front edge F 1 F 2 travels a carriage AA' supported at A' on another rail. A bar DB can turn about D, fixed to the frame in its axis, and slide through a point B fixed in the carriage AA'.. Along it a block K can slide. On the back edge F 3 F 4 of the frame another carriage C travels. It holds a vertical spindle with the knife-edge wheel at the bottom. At right angles to the plane of the wheel, the spindle has an arm GH, which is kept parallel to a.
P F which gives and tan 4= y a This 4) is e tsily constructed for any given point on the y-curve :- From the foot B' (fig. 21) of the ordinate y = B'B set off, as in the figure, B'D =a, then angle BDB'= 4). Let now DB' with a perpendicular B'B move along the axis of x, ---; whilst B follows the y-curve, '`- then a pen P on B'B will E describe the Y-curve provided it moves at every moment in a direction parallel to BD. The object of the integraph is to draw this new curve when the tracer of the instrument is guided along the y-curve.
The first to describe such instruments was AbdankAbakanowicz, who in 1889 published a book in which x; similar arm attached to K perpendicular to DB. The plane of the knife-edge wheel r is therefore always parallel to DB. If now the point B is made to follow a curve whose y is measured from OX, we have in the triangle BDB', with the angle cp at D, tan 4) = y/a, where a= DB' is the constant base to which the instrument works. The point of contact of the wheel r or any point of the carriage C will therefore always move in a direction making an angle 4) with the axis of x, whilst it moves in the x-direction through the same distance as the point B on the y-curve - that is to say, it will trace out the integral curve required, and so will any point rigidly connected with the carriage C. A pen P attached to this carriage will therefore draw the integral curve. Instead of moving B along the y-curve, a tracer T fixed to the carriage A is guided along it. For using the instrument the carriage is placed on the drawing-board with the front edge parallel to the axis of y, the carriage A being clamped in the central position with A at E and B at B' on the axis of x. The tracer is then placed on the x-axis of the y-curve and clamped to the carriage, and the instrument is ready for use. As it is convenient to have the integral curve placed directly opposite to the y-curve so that corresponding values of y or Y are drawn on the same line, a pen P' is fixed to C in a line with the tracer.
Boys' integraph was invented during a sleepless night, and during the following days carried out as a working model, which gives highly satisfactory results. It is ingenious in its simplicity, and a direct realization as a mechanism of the principles explained in connexion with fig. The line B'B is represented by the edge of an ordinary T-square sliding against the edge of a drawing-board. The points B and P are connected by two rods BE and EP, jointed at E. At B, E and P are small pulleys of equal diameters. Over these an endless string runs, ensuring that the pulleys at B and P always turn through equal angles. The pulley at B is fixed to a rod which passes through the point D, which itself is fixed in the T-square. The pulley at P carries the knife-edge wheel. If then B and P are kept on the edge of the T-square, and B is guided along the curve, the wheel at P will roll along the Y-curve, it having been originally set parallel to BD. To give the wheel at P sufficient grip on the paper, a small loaded three-wheeled carriage, the knife-edge wheel P being one of its wheels, is added. If a piece of copying paper is inserted between the wheel P and the drawing paper the Y-curve is drawn very sharply.
Integraphs have also been constructed, by aid of which ordinary differential equations, especially linear ones, can be solved, the solution being given as a curve. The first suggestion in this direction was made by Lord Kelvin. So far no really useful instrument has been made, although the ideas seem sufficiently developed to enable a skilful instrument-maker to produce one should there be sufficient demand for it. Sometimes a combination of graphical work with an integraph will serve the purpose. This is the case if the variables are separated, hence if the equation Xdx +Ydy = 0 has to be integrated where X =p(x), Y = 4)(y) are given as curves. If we write au= f Xdx, av = f Ydy, then u as a function of x, and v as a function of y can be graphically found by the integraph. The general solution is then u+v=c with the condition, for the determination for c, that y=yo, for x = xo This determines c = u,,+vo, where uo and v 0 are known from the graphs of u and v. From this the solution as a curve giving y a function of x can be drawn: - For any x take u from its graph, and find the y for which v = c - u, plotting these y against their x gives the curve required.
If a periodic function y of x is given by its graph for one period c, it can, according to the theory of Fourier's Series, be Harmonic expanded in a series.
analysers. y= Ao+A l tos B+A 2 cos2 B+. where 8=e The absolute term Ao equals the mean ordinate of the curve, and can therefore be determined by any planimeter. The other coefficients are An = J y cos nO.dO; = y sin nO.d6. A harmonic analyser is an instrument which determines these integrals, and is therefore an integrator. The first instrument of this kind is due to Lord Kelvin (Proc. Roy. Soc., vol. xxiv., 1876). Since then several others have been invented (see Dyck's Catalogue; Henrici, Phil. Mag., July 1894; Phys. Soc., 9th March; Sharp, Phil. Mag., July 1894; Phys. Soc., 13th April). In Lord Kelvin's instrument the curve to be analysed is drawn on a cylinder whose circumference equals the period c, and the sine and cosine terms of the integral are introduced by aid of simple harmonic motion. Sommerfeld and Wiechert, of Konigsberg, avoid this motion by turning the cylinder about an axis perpendicular to that of the cylinder. Both these machines are large, and practically fixtures in the room where they are used. The first has done good work in the Meteorological Office in London in the analysis of meteorological curves. Quite different and simpler constructions can be used, if the integrals determining A n and B,, be integrated by parts. This gives nA n sin n8.dy; nB n = 1 cos n6.dy. An analyser presently to be described, based on these forms, has been constructed by Coradi in Zurich (1894). Lastly, a most powerful analyser has been invented by Michelson and Stratton (U.S.A.) (Phil. Mag., 1898), which will also be described.
The Henrici - Coradi analyser has to add up the values of dy. sin nO and dy. cos nO. But these are the components of dy in two directions perpendicular to each other, of which one makes an angle nO with the axis of x or of 8. This decomposition can be performed by Amsler's registering wheels. Let two of these be mounted, perpendicular to each other, in one horizontal frame which can be 23.
turned about a vertical axis, the wheels resting on the paper on which the curve is drawn. When the tracer is placed on the curve at the point 0=o the one axis is parallel to the axis of 0. As the tracer follows the curve the frame is made to turn through an angle nO. At the same time the frame moves with the tracer in the direction of y. For a small motion the two wheels will then register just the components required, and during the continued motion of the tracer along the curve the wheels will add these components, and thus give the values of nA n and nB n. The factors I/ir and - Or are taken account of in the graduation of the wheels. The readings have then to be divided by n to give the coefficients required. Coradi's realization of this idea will be understood from fig. 23. The frame PP' of the instrument rests on three rollers E, E', and D. The first two drive an axis with a disk C on it. It is placed parallel to the axis of x of the curve. The tracer is attached to a carriage WW which runs on the rail P. As it follows the curve this carriage moves through a distance x whilst the whole instrument runs forward through a distance y. The wheel C turns through an angle proportional, during each small motion, to dy. On it rests a glass sphere which will therefore also turn about its horizontal axis proportionally, to dy. The registering frame is suspended by aid of a spindle S, having a disk H. It is turned by aid of a wire connected with the carriage WW, and turns n times round as the tracer describes the whole length of the curve. The registering wheels R, R' rest against the glass sphere and give the values nA n and nB n. The value of can be altered by changing the disk H into one of different diameter. It is also possible to mount on the same frame a number of spindles with registering wheels and glass spheres, each of the latter resting on a separate disk C. As many as five have been introduced. One guiding of the tracer over the curve gives then at once the ten coefficients A n and B r, for n = I to 5.
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Frame
All The Calculating Machines And Integrators Considered So Far Have Been Kinematic. We Have Now To Describe A Most Remarkable Instrument Based On The Equilibrium Of A Rigid Body Under The Action Of Springs. The Body Itself For Rigidity'S Sake Is Made A Hollow C ')?
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C?: _ A N Cos Nb }.. .
B I Sin 0 B 2 Sin2 0. .. B N Sin No. .. Cylinder H, Shown In Fig. 24 In End View. It Can Turn About Its Axis, Being Supported On Knife Edges O. To It Springs Are Attached At The Prolongation Of A Horizontal Diameter; To The Left A Series Of N Small Springs S, All Alike, Side By Side At Equal Intervals At A Distance A From The Axis Of The Knife Edges; To The Right A Single Spring S At Distance B. These Springs Are Supposed To Follow Hooke'S Law. If The Elongation Beyond The Natural Length Of A Spring Is A, The Force Asserted By It Is P = Kx. Let For The Position Of Equilibrium 1, L Be Respectively The Elongation Of A Small And The Large Spring, K, K Their Constants, Then Nkla = Klb.
The Position Now Obtained Will Be Called The Normal One. Now Let The Top Ends C Of The Small Springs Be Raised Through Distances Yl, Y2,...Ym. Then The Body H Will Turn; B Will Move Down Through A Distance Z And A Up Through A Distance Bz. The New Forces Thus Introduced Will Be In Equilibrium If Or N B K N (B } L) This Shows That The Displacement Z Of B Is Proportional To The Sum Of The Displacements Y Of The Tops Of The Small Springs. The Arrangement Can Therefore Be Used For The Addition Of A Number Of Displacements. The Instrument Made Has Eighty Small Springs, And The Authors State That From The Experience Gained There Is No Impossibility E Of Increasing Their Number F C Even To A Thousand. The Displacement Z, Which Necessarily Must Be Small, Can Be Enlarged By Aid Of A Lever Ot'. To Regulate The Displacements Y Of The Points C (Fig. 24) Each Spring Is Attached To A Lever Ec, Fulcrum E. To This Again A Long Rod Fg Is Fixed By Aid Of A Joint At F. The Lower End Of This Rod Rests On Another Lever Gp, Fulcrum N, At A Changeable Distance Y = Ng From N. The Elongation Y Of Any Spring S Can Thus Be Produced By A Motion Of P. If P Be Raised Through A Distance Y', Then The Displacement Y Of C Will Be Proportional To Y'Y"; It Is, Say, Equal To µY'Y" Where µ Is The Same For All Springs. Now Let The Points C, And With It The Springs S, The Levers, &C., Be Numbered Co, C I, C 2. There Will Be A Zero Position For The Points P All In A Straight Horizontal Line. When In This Position The Points C Will Also Be In A Line, And This We Take As Axis Of X. On It The Points Co, C1, C 2. .. Follow At Equal Distances, Say Each Equal To H. The Point Ck Lies At The Distance Kh Which Gives The X Of This Point. Suppose Now That The Rods Fg Are All Set At Unit Distance Ng From N, And That The Points P Be Raised So As To Form Points In A Continuous Curve Y' _ 0(X), Then The Points C Will Lie In A Curve Y = µ)(X). The Area Of This Curve Is µ F C(X)Dx. Approximately This Equals Ehy = H2Y. Hence We Have ?(X)Dx=µ?Y = ?L Where Z Is The Displacement Of The Point B Which Can Be Measured. The Curve Y' = Cb(X) May Be Supposed Cut Out As A Templet. By Putting This Under The Points P The Area Of The Curve Is Thus Determined The Instrument Is A Simple Integrator.
The Integral Can Be Made More General By Varying The Distances Ng = Y". These Can Be Set To Form Another Curve Y" =F(X). We Have Now Y= A Y'Y"= T Cf(X)0(X), And Get As Before ' F(X)Cb(X)Dz = Z. These Integrals Are Obtained By The Addition Of Ordinates, And Therefore By An Approximate Method. But The Ordinates Are Numerous, There Being 79 Of Them, And The Results Are In Consequence Very Accurate. The Displacement Z Of B Is Small, But It Can Be Magnified By Taking The Reading Of A Point T' On The Lever Ab. The Actual Reading Is Done At Point T Connected With T' By A Long Vertical Rod. At T Either A Scale Can Be Placed Or A Drawing Board, On Which A Pen At T Marks The Displacement.
If The Points G Are Set So That The Distances Ng On The Different Levers Are Proportional To The Terms Of A Numerical Series Uo Ul U2. .
And If All P Be Moved Through The Same Distance, Then Z Will Be Proportional To The Sum Of This Series Up To 80 Terms. We Get An Addition Machine. The Use Of The Machine Can, However, Be Still Further Extended. Let A Templet With A Curve Y' _ O() Be Set Under Each Point P At Right Angles To The Axis Of X Hence Parallel To The Plane Of The Figure. Let These Templets Form Sections Of A Continuous Surface, Then Each Section Parallel To The Axis Of X Will Form A Curve Like The Old Y' =4)(X), But With A Variable Parameter, Or Y' = (1)(E, X). For Each Value Of E The Displacement Of T Will Give The Integral Z' = F ° Where. .. (I) Where Y Equals The Displacement Of T To Some Scale Dependent On The Constants Of The Instrument.
If The Whole Block Of Templets Be Now Pushed Under The Points P And If The Drawing Board Be Moved At The Same Rate, Then The Pen T Will Draw The Curve Y=F(). The Instrument Now Is An Integraph Giving The Value Of A Definite Integral As Function Of A Variable Parameter. Having Thus Shown How The Lever With Its Springs Can Be Made To Serve A Variety Of Purposes, We Return To The Description Of The Actual Instrument Constructed. The Machine Serves First Of All To Sum Up A Series Of Harmonic Motions Or To Draw The Curve Y=A L Cos X A 2 Cos 2X A 3 Cos 3X. . (2) The Motion Of The Points P I P 2. .. Is Here Made Harmonic By Aid Of A Series Of Excentric Disks Arranged So That For One Revolution Of The First The Other Disks Complete 2, 3,.. . Revolutions. They Are All Driven By One Handle. These Disks Take The Place Of The Templets Described Before. The Distances Ng Are Made Equal To The Amplitudes A L, A 2, A3, . The Drawing Board, Moved Forward By The Turning Of The Handle, Now Receives A Curve Of Which (2) Is The Equation. If All Excentrics Are Turned Through A Right Angle A Sine Series Can Be Added Up.
It
Is A Remarkable Fact That The Same Machine Can Be Used As A
Harmonic Analyser Of A Given Curve. Let The Curve To Be Analysed Be
Set Off Along The Levers Ng So That In The Old Notation It Is Y
" =F(X), Whilst The Curves Y' _ 4(X0 Are Replaced By
The Excentrics, Hence By The Angle 0 Through Which The
First Excentric Is Turned, So That Y'K = Cos Ko.
But Kh = X And Nh =R,N Being The Number Of
Springs S, And 7R Taking The Place Of C. This
Makes Ko = No.X. Hence Our Instrument Draws A Curve Which
Gives The Integral (I) In The Form Y = ? F O F
(X) Cos (Ox) Dx As A Function Of O. But This Integral Becomes
The Coefficient A M In The Cosine Expansion If We Make
On/Ir =M Or O = Mir/N. The Ordinates Of The Curve
At The Values 0=7R/N, 27R/N. .. Give Therefore All
Coefficients Up To M =80. The Curve Shows At A Glance Which And How
Many Of The Coefficients Are Of Importance.
The Instrument Is Described In Phil. Mag., Vol. Xlv., 1898. A Number Of Curves Drawn By It Are Given, And Also Examples Of The Analysis Of Curves For Which The Coefficients A M Are Known. These Indicate That A Remarkable Accuracy Is Obtained. (0. H.)
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