|
Engineering has always been seen as the application of mathematics. Since the time of Archimedes to Galileo this relationship appears to be a one way street. But as they say, necessity is the mother of invention, and as mathematics became more and more complex, we require better tools to handle the more mundane aspects of it.
This is when application turns back to help formulate theories. |
"The arithmetical machine produces effects which approach nearer to thought than all the actions of animals. But it does nothing which would enable us to attribute will to it, as to the animals." - Blaise Pascal, Pensées (1669)
Thus far we’ve always talked about mathematics’ contributions to science, whether its basic science (e.g. theoretical physics) or applied (e.g. engineering, navigation), but how about contributions flowing from the other direction? Certainly we would refer to the computer. From solving simple arithmetic on a calculator, to confirming mathematical models on an Excel spreadsheet, to crunching out the hundred billionth decimal point of pi on a supercomputer; it certainly did its job.
But that would be in modern times. What about before the age of microprocessors and electronics? Before transistors and vacuum tubes? Before we take these things for granted, and simply accept a few pushes of a button would give us reliable results?
But that would be in modern times. What about before the age of microprocessors and electronics? Before transistors and vacuum tubes? Before we take these things for granted, and simply accept a few pushes of a button would give us reliable results?
The 17th Century: Origins
Blaise Pascal (1623 – 1662) was a French philosopher, mathematician, scientist, inventor, and theologian. In mathematics, he was an early pioneer in the fields of game theory and probability theory. In philosophy he was an early pioneer in existentialism. Despite chronic ill health, Pascal made historic contributions to mathematics and to physical science, including both experimental and theoretical work on hydraulics, atmospheric pressure, and the existence and nature of the vacuum. As a scientist and philosopher of science, Pascal championed strict empirical observation and the use of controlled experiments.
In 1642, after seeing his father struggling with menial calculation due to his job as a tax collector, Pascal designed a device that was inspired by the mechanical clocks of the day. Just as clock gears step up every 60 seconds or 60 minutes, Pascal’s ‘calculating clock’ steps up depending on what use it’s for.
The Pascaline was initially developed for accounting (i.e. counting money) but at the time the French state uses a non-decimal monetary system. The basic monetary unit is the Livre (Pound in English), each Livre is divided into 20 Sols, and each Sol is divided into 12 Deniers. So we can see why Pascal got his solution from clocks. BTW, decimal Pascalines came later.
This carry mechanism is at the heart of the Pascaline, unlike in the minute and hour gears of a clock, they are independent (the display of a clock is a dial, while the Pascaline have digital indicators).
Blaise Pascal (1623 – 1662) was a French philosopher, mathematician, scientist, inventor, and theologian. In mathematics, he was an early pioneer in the fields of game theory and probability theory. In philosophy he was an early pioneer in existentialism. Despite chronic ill health, Pascal made historic contributions to mathematics and to physical science, including both experimental and theoretical work on hydraulics, atmospheric pressure, and the existence and nature of the vacuum. As a scientist and philosopher of science, Pascal championed strict empirical observation and the use of controlled experiments.
In 1642, after seeing his father struggling with menial calculation due to his job as a tax collector, Pascal designed a device that was inspired by the mechanical clocks of the day. Just as clock gears step up every 60 seconds or 60 minutes, Pascal’s ‘calculating clock’ steps up depending on what use it’s for.
The Pascaline was initially developed for accounting (i.e. counting money) but at the time the French state uses a non-decimal monetary system. The basic monetary unit is the Livre (Pound in English), each Livre is divided into 20 Sols, and each Sol is divided into 12 Deniers. So we can see why Pascal got his solution from clocks. BTW, decimal Pascalines came later.
This carry mechanism is at the heart of the Pascaline, unlike in the minute and hour gears of a clock, they are independent (the display of a clock is a dial, while the Pascaline have digital indicators).
An accounting Pascaline
|
Its internal mechanism
|
Just a few years after Pascal, the German Gottfried Wilhelm Leibniz (co-inventor with Newton of calculus) managed to build a four-function (addition, subtraction, multiplication, and division) calculator that he called the Stepped Reckoner because, instead of gears, it employed fluted drums having ten flutes arranged around their circumference in a stair-step fashion. Although the stepped reckoner employed the decimal number system (each drum had 10 flutes), Leibniz was the first to advocate use of the binary number system which is fundamental to the operation of modern computers.
The 19th Century: Boom Years
By the 19th century, miniaturization became a norm in the watchmaking industry. This is coupled with advances in metallurgy, providing strong, yet light, gears that can better resist the wear and tear from repetitive use.
The first amongst these new generation of mechanical calculators was the Arithmometer (in 1851), the brainchild of Charles Xavier de Colmar of France. The Arithmometer, a direct descendent of the Stepped Reckoner, became the first commercially successful mechanical calculator due to the technical advances mentioned previously, as well as a clever construction that makes complex calculation less labour intensive.
By the 19th century, miniaturization became a norm in the watchmaking industry. This is coupled with advances in metallurgy, providing strong, yet light, gears that can better resist the wear and tear from repetitive use.
The first amongst these new generation of mechanical calculators was the Arithmometer (in 1851), the brainchild of Charles Xavier de Colmar of France. The Arithmometer, a direct descendent of the Stepped Reckoner, became the first commercially successful mechanical calculator due to the technical advances mentioned previously, as well as a clever construction that makes complex calculation less labour intensive.
Colmar's Arithmometer
|
Odhner's Arithmometer, a later development based on Colmar's
|
As shown in the diagram, the Arithmometer allows for rapid long multiplication and divisions using a movable accumulator that allows for multiplication by digits. To illustrate if one were to multiply a figure by 53, rather than cranking the handle 53 times, one would only need to turn it three times (for the ones digit), shift the accumulator and turn another five times (for the tens digits).
De Colmar’s invention was widely copied (patent laws aren’t as enforceable then as it is today), but it also spurred innovation for original designs, motivated by high demands for calculators and inspired by the Industrial Revolution’s obsession for efficiency. This led to the next step, the Comptometer.
The Comptometer was a design patented by the American industrialist Dorr Eugene Felt and was intended to rapidly enter and calculate long figures. This makes sense as the Arithmometer proves to be too cumbersome for its main users in academia and business. This was achieved by allowing for the entry of all the digits of an intended figure to be done simultaneously using a key (rather than a dial or slider) driven mechanism, with each digit is attached to a column of nine keys (1 to 9, with 0 being the default status).
For much of the 19th century, variations and improvements on these two basic designs led to the reduction in costs and wide availability of calculating machines. Some are integrated with printers, leading to the invention of the cash register. Others are programmable, which eventually led to the birth of the computer. But even with the developments in electrical and electronic technologies in the 20th century, the mechanical calculator has yet to reach its potential.
De Colmar’s invention was widely copied (patent laws aren’t as enforceable then as it is today), but it also spurred innovation for original designs, motivated by high demands for calculators and inspired by the Industrial Revolution’s obsession for efficiency. This led to the next step, the Comptometer.
The Comptometer was a design patented by the American industrialist Dorr Eugene Felt and was intended to rapidly enter and calculate long figures. This makes sense as the Arithmometer proves to be too cumbersome for its main users in academia and business. This was achieved by allowing for the entry of all the digits of an intended figure to be done simultaneously using a key (rather than a dial or slider) driven mechanism, with each digit is attached to a column of nine keys (1 to 9, with 0 being the default status).
For much of the 19th century, variations and improvements on these two basic designs led to the reduction in costs and wide availability of calculating machines. Some are integrated with printers, leading to the invention of the cash register. Others are programmable, which eventually led to the birth of the computer. But even with the developments in electrical and electronic technologies in the 20th century, the mechanical calculator has yet to reach its potential.
The 20th Century: Zenith
As mentioned, rather than replacing the mechanical calculator, electrical and electronic technologies apparently complements its development. The electrical motor allows for strings of calculation to be done instantaneously, much like a factory machine, but churning out numbers instead.
Electrification also allows for further utilization. With more power, complicated gearing and transmission controls became possible, making the mechanism less complex in terms of user operations, but more functional in capabilities. For example, before electrification, multiplications are done by successive additions (e.g. 2 x 4 = 2 + 2 + 2 + 2), now multiple grindings of the crank happen at a single push of a button.
This is all well and good, but in terms of functionality smaller is better, and miniaturization is the key. The most elegant and collectable of this type of mechanical calculators is the Curta. This hand held device is a beautifully designed four function mechanical calculator. The Curta is about 11cm high and 4cm in diameter and is a masterpiece of mechanical engineering. It was designed by an Austrian called Curt Herzstark. Herzstark completed the drawings for the Curta in 1938; then World War 2 intervened and prevented the machine being put into production.
In 1943 Herzstark was sent to the Buchenwald concentration camp for “helping Jews and subversive elements”. The camp commandant knew of Herzstark’s work on the Curta and he was set to work redrawing the calculator blueprints from memory. The commandant’s idea was to build a calculators and present it to Hitler after the war ended. After the Allies bombed the camp Herzstark was transferred to a deserted salt mine nearby and continued work on the Curta 600 metres below the surface. After the war Herztark moved to Liechtenstein and a factory was built to produce the Curta. The calculator sold well because there was nothing like it on the market and production continued until 1972. By then 140,000 units had been produced.
As mentioned, rather than replacing the mechanical calculator, electrical and electronic technologies apparently complements its development. The electrical motor allows for strings of calculation to be done instantaneously, much like a factory machine, but churning out numbers instead.
Electrification also allows for further utilization. With more power, complicated gearing and transmission controls became possible, making the mechanism less complex in terms of user operations, but more functional in capabilities. For example, before electrification, multiplications are done by successive additions (e.g. 2 x 4 = 2 + 2 + 2 + 2), now multiple grindings of the crank happen at a single push of a button.
This is all well and good, but in terms of functionality smaller is better, and miniaturization is the key. The most elegant and collectable of this type of mechanical calculators is the Curta. This hand held device is a beautifully designed four function mechanical calculator. The Curta is about 11cm high and 4cm in diameter and is a masterpiece of mechanical engineering. It was designed by an Austrian called Curt Herzstark. Herzstark completed the drawings for the Curta in 1938; then World War 2 intervened and prevented the machine being put into production.
In 1943 Herzstark was sent to the Buchenwald concentration camp for “helping Jews and subversive elements”. The camp commandant knew of Herzstark’s work on the Curta and he was set to work redrawing the calculator blueprints from memory. The commandant’s idea was to build a calculators and present it to Hitler after the war ended. After the Allies bombed the camp Herzstark was transferred to a deserted salt mine nearby and continued work on the Curta 600 metres below the surface. After the war Herztark moved to Liechtenstein and a factory was built to produce the Curta. The calculator sold well because there was nothing like it on the market and production continued until 1972. By then 140,000 units had been produced.
The Curta, disassembled. A mistake many owners made, as it is so complex, it's almost impossible to reassemble.
This success was however, the swansong of the era of mechanical calculator, as the invention of the transistor allows for solid state calculations based on logic gates (i.e. no mechanical movements needed).
Ponder this
Why wasn't mechanical calculators became commercially successful prior to the 19th century? Governments still need to reckon revenues, commercial enterprises still need to keep accounts, and yet it took almost 200 years for it to be widely adopted.
What about functions beyond arithmetic? Logarithmic or trigonometric functions? Powers and roots? Why were these functions not explored by the engineers of the age, or were they?
Discuss
Discuss the mechanical principles in doing additions, subtractions, multiplications and division in a mechanical calculator. Try to design one that can execute these four functions.
Further readings
The Pascaline, the first mechanical calculator, invented by Blaise Pascal in the 17th century.
The Stepped Reckoner, by the co-founder of calculus Gottfried Leibniz, came a close second in the race.
The Arithmometer, the first commercially successful mechanical calculator.
The Curta, the zenith of mechanical calculation.
"Why Dividing by Zero Makes a Mechanical Calculator Go Berserk", somebody tried to end the world using a mechanical calculator.
