...

Jettons. During the Renaissance, Europeans counted with Roman numerals. But even for people accustomed to using them, Roman numerals are not easy to calculate mentally, and paper and pens were hard to come by. Merchants used copper tokens called "jettons" to calculate prices.

The jettons were moved about on lines. Merchants could draw the lines on the ground or scratch them on a table. The lines represented different values of ten; ones, tens, hundreds, and so on. For intermediate values, like five or fifty, a space was left between the lines. In the same way that you know immediately what is meant by $1.98, the Renaissance buyer and seller immediately recognized the price by the position of the jettons on the lines.

Until 1700, calculating tokens were common in Europe. The tokens usually derived their name from moving them about the lines while calculating; the word "jettons" comes from the French verb "jeter" meaning "to throw." Adept calculators must have made their jettons fly across their counting boards! By the mid- 18th century both the Roman numeral system and jettons had disappeared from everyday use.

A competition between the Hindu-Arabic form of mental and paper arithmetic that we use today and Roman numeral figuring using jettons. The dismay of the jetton user shows graphically who is winning. The mid-18th century saw the widespread availability of paper, of printing, and the use of the Hindu-Arabic number system with the simultaneous decline of the use of Roman numerals and their computation with jettons.

These copper jettons are from 14th-century Italy. The designs on jettons were often symbols of different trades or coats of arms. Depending upon the wealth of the owner, jettons were pro- duced in metals varying from copper to gold but they were not coins. In fact, a new set of jettons was a customary New Year's gift. The old set would then be thrown in a river, symbolically clearing last year's accounts. Remnants of jettons remain with us today. Merchants had boards in their shops on which to toss their jettons to calculate bills; stores today still have "counters." From the collection of Gwen and Gordon Bell.

Abacus. Most eastern countries used an abacus of some sort. It emerged from the Middle East sometime after 500 A.D. and was based on a system in which pebbles were moved around on the ground to represent numbers and perform calculations. (The word abacus is from the Semitic word "abaq," meaning dust.) The Chinese developed a version that they called a suapan. The Japanese modified the suapan and called it a soroban. An abacus is rather similar to jettons. The difference is that instead of scratching lines and carrying loose jettons, the tokens were strung on wires, and then framed. The abacus became the indispensible calculator for eastern merchants. It was easy to carry. And for the skilled user, it is very fast. People who use an abacus learn to recognize numbers simply by looking at the position of the beads.

The Japanese "soroban" has sharp edges to its beads and only one bead in heaven and four beads in earth to make operations faster. From the collection of the Peabody Museum of Salem. The soroban has not declined in use since the advent of its rivals the calculator and computer. In some banks, the daily computerized totals are double checked with a soroban. In learning the soroban, students learn to visualize the position of numbers. The soroban champion, Ms. Nishida, can add eight ten-digit numbers in less than ten seconds simply by visualizing the position of the beads in her head. In fact, the Japanese claim that learning the use of the soroban can increase a students I.Q.

The Chinese "suapan," or "counting tray," has round beads and is divided into two sections. The top section is called heaven and contains two beads, each worth five units. The bottom section is called earth and contains five beads, each worth one unit. The suapan was used in China by the 1300s, and it became widely popular in 1593 when the mathematician Chen Ta-wei published a book on abacus computation. The abacus is still such an important part of Chinese culture that May 10 is celebrated as National Abacus Day. From the collection of Gwen and Gordon Bell.

Scientists' Instruments The year 1670 marked the first recorded appearance of Halley's Comet. A seventeenth- century astronomer bent on knowing the heavens and such predictions as the recurrence of the comet, faced very complicated calculations. He often needed to multiply vast numbers to describe the motions of the planets and the stars.

Astronomers were greatly aided by the Hindu-Arabic numeral system introduced to Europe around the fifteenth century. This number system made the sophisticated arithmetic of science possible. In addition, a wide variety of calculating tools were developed that stored information including the development of printing and the production of books of tables. These tools saved the scientist time and increased the accuracy of his calculations.

Napier's Bones were invented in 1617 , when John Napier, a Scottish baron, published a book describing the device. Within a few years, it had spread throughout Europe and as far as China. Napier's Bones (so-called because they were often made of bone) were rods with multiplications tables on them. At the time, educated people often knew their multiplication tables only as far as 5 x 5.

The concept of the pocket book of tables started with the development of printing itself. (1) The 1683 table of trigonometric values was useful to navigators, surveyors, astronomers, mathematicians and architects. Such tables eliminated the need to constantly calculate the trigonometric values of numbers. However, few tables were free from mistakes, and corrections were often put in by hand. From the collection of Gwen and Gordon Bell. (2) This set of logarithms tables was compiled in 1839 in England by The Society for the Diffusion of Useful Knowledge. To multiply two large numbers an astronomer would look up the numbers in the table, and add the listed logarithm values. The number in the tables that corresponded to the sum of the logarithms was the answer to the original multiplication problem. From the collection of the IBM Corporation. (3) Easily carried in a shirt pocket, Thompson and Thomas's Electrical Tables and Memoranda, published in London in 1898, was a handy reference for the electrical engineer wherever he went. From the collection of Gwen and Gordon Bell.

These calculations were performed by Johannes Kepher for his Ephermerides, dedicated to Napier for his invention of Logarithms. These were typical of the long hand arithmetic used to numerically describe the movements in the heavens.

The 18th century scientist/scholar/gentleman had a number of elegant devices that could be carried in the pocket and be the mark of a learned man. These include such items as (1) A pocket set of drawing instruments in an elegant shagreen and silver case would have been useful in producing a map of the heavens. From the collection of Gwen and Gordon Bell. (2) A portable sun dial, made in Augsburg, Germany, during the middle of the 18th century, was a precursor to the pocket watch, for telling the time of day. From the David Eugene Smith Collection, Rare Book and Manuscript Library, Columbia University. (3) An Arab astrolabe that could be used to determine the position of the stars and sun on any day of the year. The spikes on the top piece of brass represent the major stars and could be turned about the brass plate below it. The etching on the plate is a map of the heavens. Different plates are used depending upon the user's latitude. From the David Eugene Smith Collection, Rare Book and Manuscript Library, Columbia University. (4) Napier's Bones in its secure case. From the collection of Gwen and Gordon Bell.

Napier's Bones were used for multiplication, division, and square and cube root problems. It was simple to arrange the rods to solve complicated problems. In the Museum's exhibit visitors utilize a super-sized model of the device.

Digital Adders. Most mechanical adders use gears to "count." As you enter a number the machine "counts" the number of gear teeth that you advance. Calculating by counting is called digital calculation. The idea of using a stylus to advance gears to perform addition dates to 1642, when the French mathematician Blaise Pascal (1623- 1662) invented a calculator called the Pascaline. Many mechanical calculators built for the pocket operated on similar principles. The gears only went in one direction and the machines only held one register. Subtraction was carried out by 9's complement arithmetic, and multiplication by repeated addition. These were able to be widely produced for a very low cost and became the mechanical helper for many people.

The Webb Adder was patented in the United States in 1869. Any two-digit number could be directly entered on the large gear with a stylus. When the large gear had made one complete revolution it advanced the smaller gear one place, thus "carrying" to the hundreds place. Gift of Gwen and Gordon Bell.

The oldest mechanical pocket calculator, designed by Englishman Samuel Morland (1625-1695) in 1666, avoided some of the mechanical problems that plagued the Pascaline. Morland did not link together the gears for different digits. Instead, each time a digit gear completed a full turn it advanced the small gear above it one place. At the end of a problem the small gearsindicated how much to add (carry) to the next digit places. From the collection of the IBM Corporation.

The Addiator was a very inexpensive and widely-sold pocket calculator. Introduced in 1920, over 100,000 were sold the first year. The Addiator was not truly mechanical in operation. The user added by sliding either up or down strips of metal with numbers marked on them. No gears or inter-linked parts were involved. The basic idea was first invented in 1889 by a Frenchman named Troncet. Troncet called his calculator the Arithmographe. From the collection of Gwen and Gordon Bell.

Slide Rules. Edmund Gunter (1581-1626) was the first to construct a scale rule that could be used to multiply. He divided his scale according to Napier's principle of logarithms, so that multiplication could be done by measuring and adding lengths on the scale. In about 1630 William Oughtred (c.1574-1660) improved upon Gunter's idea by fixing two rules together so they could slide against one another.

Slide rules were not the only widely used analog calculators. Quadrants evolved from instruments used for measuring angles between stars in ancient Babylonia. In the 16th century scales were etched on these devices which made them more useful to laymen as calculators. Gunter was one of those most responsible for the quadrant's improvement and use. Other popular analog calculators were the proportional compass and the sector.

In the seventeenth century, the English government devised an efficient system for taxing ale and wine by producing a slide rule to help the assessors calculate the tax, right at the barrels. The alcohol tax was levied only on the amount that had been sold, and the slide rule allowed the assessor simply to determine the liquor that remained in the barrel. He used a gauging rod, like a dip stick in a car.

The first such slide rule was described in 1683 by Thomas Everard. In 1739 Charles Leadbetter improved upon the design by adding scales that could calculate the contents of a keg whether it was standing on end or lying on its side. The rule was used with a folding gauging rod to measure the depth of liquor in a keg. Sometimes tax assesors had their rule and gauging rode fit into a cane; not quite a pocket calculator but certainly of the same notion.

Slide rules were the work horse of scientific calculation for many decades, They were fast and reliable, and an experienced user could perform a long and complex calculation with ease.

The "Unique Log-Log" slide rule, and the later Dietzgen Redirule are slide rules designed for the shirt pocket. In general, the shorter the slide rule, the less accurate it is. From the collection of Gwen and Gordon Bell. Gift of I. Bernard Cohen.

Circular slide rules operated on the same principle as straight slide rules, but they took up less space. Both William Oughtred and his student Richard Delamain claimm to have first thought of the circular slide rule. (1) A general circular slide rule. Gift of Stanton Vanderbilt. (2) This slide rule was used by pilots to estimate arrival times, and to calculate other aspects of their flight according to changing conditions. From the collection of Steve Kallis. (3) A bombardier would have used this slide rule to calculate the chances of destroying his target under various conditions. Gift of David Martz. (4) Harvard Project Physics circular slide rule could be slipped into a student's textbook. Gift of I. Bernard Cohen. (5) This homemade version made by Charles Bachman helped the family compare the price of goods at the grocery before the days of unit pricing. Gift of Charles Bachman.

Slide rules had two drawbacks. First, slide rules were only as accurate as the fineness of their scales, because they were analog calculators and measured quantities. If the scale were any finer you could not read it. Notice how you have to estimate the position of a three-digit number on the scale. This degree of accuracy, however, was generally enough to estimate the answer to most scientific and engineering problems. Second, slide rules have no decimal points. The same mark can be read as 0.125, 1.25, 12.5, or 125. The user had either to keep track of the decimal point or to place it wherever it was reasonable when the problem was finished.

Until the 1970s, when an engineer wanted a quick answer to a problem, he usually reached for his slide rule. The slide rule was designed to simplify complicated calculations. Leather cases that could be clipped to the belt were often used by scientists to carry large slide rules with them wherever they went. The slide rule (or "slip stick" as it was nick-named) was the constant companion of engineers and scientists. Slung from the belt or stuck in the pocket, it was the mark of the serious scientist.

Mechanical Multiplying Calculators. In 1671, Leibniz conceived the idea of a multiplying machine by repeated addition, and constructed his earliest model in 1694. Since 1879 it has been preserved in the Royal Library at Hanover, where at one time Leibniz was the librarian. An important feature of the machine was the stepped wheel which is the basis for many subsequent mechanical calculators. Most of these are quite large and heavy, couldn't even think about being portable - not to mention fit into the pocket.

The Curta was the only multiplying, mechanical pocket calculator. Built with the precision of a fine watch, it took mechanical calculation to its finest development. However, like a fine watch, this degree of mechanical precision was not cheap. The Curta sold for close to $150 in the early 1960s. Like many of the fine pocket instruments of earlier days it became a symbol for the need for precise calculations and was closely associated with car rallying. Its manufacturing costs only increased and by the mid- 1970s electronic calculators were faster, smaller, lighter, more powerful and less expensive than the Curta.

The Curta was invented by Curt Herzstark and manufactured in Liechtenstein starting about 1950. Each part was manufactured to a tolerance of .001 millimeter. Gift of Robert Brickford.

Electronic Calculators. By the mid-1970s electronic calculators reached the masses. The development of very small and cheap electronic circuits for computers during the late 1960s and early 1970s allowed small electronic calculators to be constructed and sold inexpensively. Over time, calculators became even more sophisticated, cheaper, and smaller.

Today electronic calculators can:

• store numbers and information like early pebble calculating systems and wax tablet records,
• add large numbers quickly like the abacus or Webb Adder,
• multiply rapidly like Napier's Bones, and the Curta,
• quickly find the values of mathematical functions like mathematical tables or slide rules,
• be programmed to perform complicated and lengthy calculations at the push of a button,
• perform whole new tasks such as translating languages and dialing phones.

The Hewlett-Packard HP-35 was the first scientific pocket calculator. It could very quickly and accurately perform many of the slide rule functions that were too complicated for simple fourfunction calculators. It was nicknamed the "electronic slide rule." Thanks to the speed of electronic circuits, the HP-35 could calculate the logarithm of a number at the push of a button. When introduced on February 1, 1972, the HP-35 cost $395. Prior to the first scientific calculator, the HP-35, finding the value of a function (such as the sine of an angle) meant looking it up in a table, or being satisfied with the limited accuracy of a slide rule. The HP-35 could instantly calculate the sine of an angle to ten decimal places. The same is true of other trigonometric functions and logarithms. Gift of the HewlettPackard Company

...