Stepped Reckoner
Encyclopedia
The Step Reckoner was a digital mechanical calculator
invented by German mathematician Gottfried Wilhelm Leibniz around 1672 and completed in 1694. The name comes from the translation of the German term for its operating mechanism; staffelwalze meaning 'stepped drum'. It was the first calculator that could perform all four arithmetic operations: addition, subtraction, multiplication and division.
Its intricate precision gearwork, however, was somewhat beyond the fabrication technology of the time; mechanical problems, in addition to a design flaw in the carry mechanism, prevented the machines from working reliably.
Two prototypes were built; today only one survives in the National Library of Lower Saxony
(Niedersächsische Landesbibliothek) in Hannover, Germany. Several later replicas are on display, such as the one at the Deutsches Museum
, Munich
. Despite the mechanical flaws of the Stepped Reckoner, it gave future calculator builders new possibilities. The operating mechanism, invented by Leibniz, called the stepped cylinder or Leibniz wheel
, was used in many calculating machines for 200 years, and into the 1970s with the Curta hand calculator.
It is unclear how many different variants of the calculator were made. Some sources, such as the drawing to the right, show a 12 digit version. This section describes the surviving 16 digit prototype in Hannover.
The machine is about 67 cm (26 inches) long, made of polished brass and steel, mounted in an oak case. It consists of two attached parallel parts; an accumulator
section to the rear, which can hold 16 decimal digits, and an 8 digit input section to the front. The input section has 8 dials with knobs to set the operand
number, a telephone-like dial to the right to set the multiplier digit, and a crank on the front to perform the calculation. The result appears in the 16 windows on the rear accumulator section. The input section is mounted on rails and can be moved along the accumulator section with a crank on the left end that turns a worm gear, to change the alignment of operand digits with accumulator digits. There is also a tens-carry indicator and a control to set the machine to zero. The machine can:
Addition or subtraction is performed in a single step, with a turn of the crank. Multiplication and division are performed digit by digit on the multiplier or divisor digits, in a procedure equivalent to the familiar long multiplication and long division
procedures taught in school. Sequences of these operations can be performed on the number in the accumulator; for example it can calculate roots
by a series of divisions and additions.
. Later he learned about Pascal's machine when he read Pascal's Pensees
. He concentrated on expanding Pascal's mechanism so it could multiply and divide. He presented a wooden model to the Royal Society of London on 1 February 1673 and received much encouragement. In a letter of 26 March 1673 to Johann Friedrich
, where he mentioned the presentation in London, Leibniz described the purpose of the "arithmetic machine" as making calculations "leicht, geschwind, gewiß" [sic
], i.e. easy, fast, and reliable. Leibniz also added that theoretically the numbers calculated might be as large as desired, if the size of the machine was adjusted; quote: "eine zahl von einer ganzen Reihe Ziphern, sie sey so lang sie wolle (nach proportion der größe der Machine)" [sic]. In English: "a number consisting of a series of figures, as long as it may be (in proportion to the size of the machine)". His first preliminary brass machine was built 1674 - 1685. His so-called 'older machine' was built 1686 - 1694. The 'younger machine', the surviving machine, was built from 1690 to 1720.
In 1775 the 'younger machine' was sent to the University of Göttingen for repair, and was forgotten. In 1876 a crew of workmen found it in an attic room of a university building in Göttingen
. It was returned to Hannover in 1880. In 1894-1896 Artur Burkhardt, founder of a major German calculator company restored it, and it has been kept at the Niedersächsische Landesbibliothek ever since.
number to the accumulator
register, as many times as desired (to subtract, the operating crank is turned in the opposite direction). The number of additions (or subtractions) is controlled by the multiplier dial. It operates like a telephone dial
, with ten holes in its circumference numbered 0 - 9. To multiply by a single digit, 0 - 9, a knob-shaped stylus is inserted in the appropriate hole in the dial, and the crank is turned. The multiplier dial turns clockwise, the machine performing one addition for each hole, until the stylus strikes a stop at the top of the dial. The result appears in the accumulator windows. Repeated subtractions are done similarly except the multiplier dial turns in the opposite direction, so a second set of digits, in red, are used. To perform a single addition or subtraction, the multiplier is simply set at one.
To multiply by numbers over 9:
In this way, the operand can be multiplied by as large a number as desired, although the result is limited by the capacity of the accumulator.
To divide by a multidigit divisor, this process is used:
It can be seen that these procedures are just mechanized versions of long division
and multiplication.
Mechanical calculator
A mechanical calculator is a device used to perform the basic operations of arithmetic. Mechanical calculators are comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator....
invented by German mathematician Gottfried Wilhelm Leibniz around 1672 and completed in 1694. The name comes from the translation of the German term for its operating mechanism; staffelwalze meaning 'stepped drum'. It was the first calculator that could perform all four arithmetic operations: addition, subtraction, multiplication and division.
Its intricate precision gearwork, however, was somewhat beyond the fabrication technology of the time; mechanical problems, in addition to a design flaw in the carry mechanism, prevented the machines from working reliably.
Two prototypes were built; today only one survives in the National Library of Lower Saxony
Lower Saxony
Lower Saxony is a German state situated in north-western Germany and is second in area and fourth in population among the sixteen states of Germany...
(Niedersächsische Landesbibliothek) in Hannover, Germany. Several later replicas are on display, such as the one at the Deutsches Museum
Deutsches Museum
The Deutsches Museum in Munich, Germany, is the world's largest museum of technology and science, with approximately 1.5 million visitors per year and about 28,000 exhibited objects from 50 fields of science and technology. The museum was founded on June 28, 1903, at a meeting of the Association...
, Munich
Munich
Munich The city's motto is "" . Before 2006, it was "Weltstadt mit Herz" . Its native name, , is derived from the Old High German Munichen, meaning "by the monks' place". The city's name derives from the monks of the Benedictine order who founded the city; hence the monk depicted on the city's coat...
. Despite the mechanical flaws of the Stepped Reckoner, it gave future calculator builders new possibilities. The operating mechanism, invented by Leibniz, called the stepped cylinder or Leibniz wheel
Leibniz wheel
A Leibniz wheel or stepped drum was a cylinder with a set of teeth of incremental length which, when coupled to a counting wheel, was used in the calculating engine of a class of mechanical calculators...
, was used in many calculating machines for 200 years, and into the 1970s with the Curta hand calculator.
Description
The Stepped Reckoner was based on a counting device that he invented and that is now called a Leibniz wheel.It is unclear how many different variants of the calculator were made. Some sources, such as the drawing to the right, show a 12 digit version. This section describes the surviving 16 digit prototype in Hannover.
The machine is about 67 cm (26 inches) long, made of polished brass and steel, mounted in an oak case. It consists of two attached parallel parts; an accumulator
Accumulator (computing)
In a computer's central processing unit , an accumulator is a register in which intermediate arithmetic and logic results are stored. Without a register like an accumulator, it would be necessary to write the result of each calculation to main memory, perhaps only to be read right back again for...
section to the rear, which can hold 16 decimal digits, and an 8 digit input section to the front. The input section has 8 dials with knobs to set the operand
Operand
In mathematics, an operand is the object of a mathematical operation, a quantity on which an operation is performed.-Example :The following arithmetic expression shows an example of operators and operands:3 + 6 = 9\;...
number, a telephone-like dial to the right to set the multiplier digit, and a crank on the front to perform the calculation. The result appears in the 16 windows on the rear accumulator section. The input section is mounted on rails and can be moved along the accumulator section with a crank on the left end that turns a worm gear, to change the alignment of operand digits with accumulator digits. There is also a tens-carry indicator and a control to set the machine to zero. The machine can:
- add or subtract an 8 digit number to / from a 16 digit number
- multiply two 8 digit numbers to get a 16 digit result
- divide a 16 digit number by an 8 digit divisor
Addition or subtraction is performed in a single step, with a turn of the crank. Multiplication and division are performed digit by digit on the multiplier or divisor digits, in a procedure equivalent to the familiar long multiplication and long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...
procedures taught in school. Sequences of these operations can be performed on the number in the accumulator; for example it can calculate roots
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
by a series of divisions and additions.
History
Leibniz got the idea for a calculating machine in 1672 in Paris, from a pedometerPedometer
A pedometer is a device, usually portable and electronic or electromechanical, that counts each step a person takes by detecting the motion of the person's hips...
. Later he learned about Pascal's machine when he read Pascal's Pensees
Pensées
The Pensées represented a defense of the Christian religion by Blaise Pascal, the renowned 17th century philosopher and mathematician. Pascal's religious conversion led him into a life of asceticism, and the Pensées was in many ways his life's work. "Pascal's Wager" is found here...
. He concentrated on expanding Pascal's mechanism so it could multiply and divide. He presented a wooden model to the Royal Society of London on 1 February 1673 and received much encouragement. In a letter of 26 March 1673 to Johann Friedrich
John Frederick, Duke of Brunswick-Lüneburg
John Frederick was duke of Brunswick-Lüneburg and ruled over the Principality of Calenberg, a subdivision of the duchy, from 1665 until his death....
, where he mentioned the presentation in London, Leibniz described the purpose of the "arithmetic machine" as making calculations "leicht, geschwind, gewiß" [sic
Sic
Sic—generally inside square brackets, [sic], and occasionally parentheses, —when added just after a quote or reprinted text, indicates the passage appears exactly as in the original source...
], i.e. easy, fast, and reliable. Leibniz also added that theoretically the numbers calculated might be as large as desired, if the size of the machine was adjusted; quote: "eine zahl von einer ganzen Reihe Ziphern, sie sey so lang sie wolle (nach proportion der größe der Machine)" [sic]. In English: "a number consisting of a series of figures, as long as it may be (in proportion to the size of the machine)". His first preliminary brass machine was built 1674 - 1685. His so-called 'older machine' was built 1686 - 1694. The 'younger machine', the surviving machine, was built from 1690 to 1720.
In 1775 the 'younger machine' was sent to the University of Göttingen for repair, and was forgotten. In 1876 a crew of workmen found it in an attic room of a university building in Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
. It was returned to Hannover in 1880. In 1894-1896 Artur Burkhardt, founder of a major German calculator company restored it, and it has been kept at the Niedersächsische Landesbibliothek ever since.
Operation
The machine performs multiplication by repeated addition, and division by repeated subtraction. The basic operation performed is to add (or subtract) the operandOperand
In mathematics, an operand is the object of a mathematical operation, a quantity on which an operation is performed.-Example :The following arithmetic expression shows an example of operators and operands:3 + 6 = 9\;...
number to the accumulator
Accumulator (computing)
In a computer's central processing unit , an accumulator is a register in which intermediate arithmetic and logic results are stored. Without a register like an accumulator, it would be necessary to write the result of each calculation to main memory, perhaps only to be read right back again for...
register, as many times as desired (to subtract, the operating crank is turned in the opposite direction). The number of additions (or subtractions) is controlled by the multiplier dial. It operates like a telephone dial
Rotary dial
The rotary dial is a device mounted on or in a telephone or switchboard that is designed to send electrical pulses, known as pulse dialing, corresponding to the number dialed. The early form of the rotary dial used lugs on a finger plate instead of holes. Almon Brown Strowger filed the first patent...
, with ten holes in its circumference numbered 0 - 9. To multiply by a single digit, 0 - 9, a knob-shaped stylus is inserted in the appropriate hole in the dial, and the crank is turned. The multiplier dial turns clockwise, the machine performing one addition for each hole, until the stylus strikes a stop at the top of the dial. The result appears in the accumulator windows. Repeated subtractions are done similarly except the multiplier dial turns in the opposite direction, so a second set of digits, in red, are used. To perform a single addition or subtraction, the multiplier is simply set at one.
To multiply by numbers over 9:
- The multiplicand is set into the operand dials.
- The first (least significant) digit of the multiplierMultiplierThe term multiplier may refer to:In electrical engineering:* Binary multiplier, a digital circuit to perform rapid multiplication of two numbers in binary representation* Analog multiplier, a device that multiplies two analog signals...
is set into the multiplier dial as above, and the crank is turned, multiplying the operand by that digit and putting the result in the accumulator. - The input section is shifted one digit to the left with the end crank.
- The next digit of the multiplier is set into the multiplier dial, and the crank is turned again, multiplying the operand by that digit and adding the result to the accumulator.
- The above 2 steps are repeated for each multiplier digit. At the end, the result appears in the accumulator windows.
In this way, the operand can be multiplied by as large a number as desired, although the result is limited by the capacity of the accumulator.
To divide by a multidigit divisor, this process is used:
- The dividendDivision (mathematics)right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
is set into the accumulator, and the divisorDivisorIn mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
is set into the operand dials. - The input section is moved with the end crank until the lefthand digits of the two numbers line up.
- The operation crank is turned and the divisor is subtracted from the accumulator repeatedly until the lefthand (most significant) digit of the result is 0. The number showing on the multiplier dial is then the first digit of the quotient.
- The input section is shifted right one digit.
- The above two steps are repeated to get each digit of the quotient, until the input carriage reaches the right end of the accumulator.
It can be seen that these procedures are just mechanized versions of long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...
and multiplication.