calcolatoria


Download the paper slide rule with the instructions


A brief history of the slide rule

The slide rule made it possible to design the world as we can still see it today.

Here we see a brief history of the slide rule, the full version can be freely downloaded.

"Houston, Tranquility Base here. The Eagle has landed"

with these words Armstrong announced the landing on the Moon. One of the on-board computers was a pocket slide rule, supplied to all the Apollo missions. Invented in 1622 this tool came in space. An history long forgotten, overtaken by a digital age that seems to exist forever.

Pickett N600 ES slide rule

Pickett N600 ES slide rule, in use on the Apollo's missions


Galileo's compass

In the sixteenth century for the development of science was necessary to calculate with big numbers and the old systems were no longer sufficients. Many searched for a solution. At the end of '500 Galileo was perhaps the first to develop a tool that help to solve mathematical operations. Multiplication, division, roots, calculation of areas and volumes, measuring gauges of guns. His "Compasso geometrico et militare", based on the proportionality of corresponding sides of two triangles, was very convenient for aiming artillery pieces. Galileo advertised it with modern methods, selling throughout Europe accompanied by a comprehensive instruction's booklet entitled "Le operazioni del Compasso".

Riproduzione del Compasso di Galileo

Reproduction of the Galileo's Compass (@ Museo Galileo - Firenze)

Around 1620 Edmund Gunter added a logarithmic scale to make a more precise multiplications and divisions. This model, known as Sector, was used in the Royal Navy until the Second World War.

Sector rule o Compasso di proporzione

Sector rule, ca. 1850


The slide rule

In 1654, just few years after the invention of Oughtred, Robert Bissaker made the "Gauging Rule", with 4 slides, specialized in measuring the contents of the barrels of wine, beer or spirits and calculate the tax burden. A very succesfull instrument that was marketed for over 300 years.

Everard slide rule

The Gauging Rule of Thomas Everard, half of the eighteenth century

In 1677 Henry Coggeshall created the "Carpenter's Slide Rule", mounted on two wooden rulers with the gradation in inches, the central sliding scale in bronze and several other scales to solve various problems. It is a combined instrument that has allowed the common people to measure and calculate, remained in use until the beginning of 1900 especially in shipyards and workshops.

Carpenter's Rule

Carpenter's Rule, ca. 1840

At the beginning of 1700 there were slide rules specific to all the needs of the time. The Carpenter's Slide Rule was used to find the volume and weight of shipments o timber, the Gauging Rule to calculate the taxation of beer barrel. The Gunter's Scale allowed a great work: the mapping of the United States.

Gauging slide rule

Gauging Rule, ca. 1820

Towards mid-800, however, there was a pressing need for computational tools not only specialized in tax or workshop use. Essentials for the design of steam engines and the development of railroads, the generic slide rule began to appears. Soon became the secret weapons of the industrial revolution.


The golden age

In 1859 the French artillery lieutenant Amedee Mannheim perfected the scales introducing the movable cursor. the modern slide rule was born.

Tavernier slide rule

With the Mannheim slide rule appears the cursor, ca. 1860

Around 1920 the slide rule had assumed its final form. Einstein used it to develop the theory of relativity, Marconi for the radio. Fermi for the atomic bomb, Korolev for the Sputnik program and Von Braun for the engines of the Saturn V, the Apollo vector.

Einstein Nestler slide rule

the model preferred by Einstein, ca. 1930

In order to improve accuracy, proportional to the length of the scales, where produced models very large, also circular or cylindrical.

Fowler slide rule

Fowler, ca. 1910

Fuller slide rule

Fuller, 1921


The twilight of the analog era

The first computers appeared around 1946, but they were huge and expensive, the same IBM planned to sell up to four a year, and the slide rule seemed irreplaceable.

Nobody imagined a world without the slide rule: they served to housewives in the kitchen, to tracing the routes on the ship "Star Trek". Appeared on the cover of Playboy, were also proposed in the form of cufflinks and tie clips. Existing in all types, Walt Disney had a simplified model for the children, was built in Braille for the blind, with scales dedicated to solving statistical problems and also in hexadecimal, octal or binary for computer programmers. It was the laptop of the era, always sticking out of engineers' pocket. A true sign to identify the category.

Hexadecimal slide rule

Hexadecimal model for computer programmers

In 1969 the slide rule was used on the Apollo 11 landing on the Moon: a very long career which began more than 350 years before. However these tools are only accurate to three decimal places and the engineer needed to make continuous estimates with the help of their experience.

The slide rule on the moon

From the seventeenth century to the Moon, a really long career

Approximating calculations for excess created the myth of the "Olde Good Things", but the modern structural analysis required now exact results, thus promoting the development of small electronic calculators. These were of course designed using the slide rule: Robert Ragen said to have literally consumed two to realize in 1963 its revolutionary "Friden 130".

Electronic slide rule on board the shuttle

From 1972 only electronic in space

Finally in 1972 the Helwett Packard, advertising it as "Innovative electronic slide rule", put on sale the first economic scientific calculator, 50 times smaller than the competitors and so modern that it is still on the market. The capabilities of the new HP 35 were indispensable. Forbes cites it among the 20 objects that have changed the world, and analog computers disappeared from the market in a flash.

IBM ad

The first computers were bulky and not very powerful, but they were able
to replace 150 engineers equipped with slide rule: their time had come


Logarithms and the basis of the slide rule

The mathematician John Napier discovered in 1614 the logarithms, published in "Mirifici logarithmorum canonis descriptio", capable of expressing any positive number via powers. Since the product of two powers with the same base is a power with the same base and exponent given by the sum of the exponents, with logarithms multiplications and divisions can be made as simple additions and subtractions.

To multiply two numbers just look out for their logarithms and add them together: the result is the number whose logarithm correspond to the sum. In practice the logarithm of a number in a certain base is the exponent to which the base must be raised to obtain the number. The logarithm of 10,000 in base 10 is 4 (104 = 10,000) and 10,000 x 1,000 become 104x103 = 104+3 = 107 = 10,000,000. Multiplication and division of exponents allow to find squares, cubes and roots. Things get complicated when dealing numbers other than 10: we need a volume with more than a million values. The tables had a very long life as they were cheap and their precision made them indispensable for astronomers and navigators ap to ca 1975.

With logarithms we are unable to work quickly as the consultation of the tables is very laborious and in 1620 Edmund Gunter, to expedite the proceedings, designed the logarithmic scale by placing numbers on a ruler at a distance from the origin proportional to the value of their logarithm. Here is the table:

Regolo di Gunter

Now we can try to construct the scale: the 1 is the starting point, the 2 is located at 3.01 cm, the 3 to 4.77 and so on up to 10. We can therefore represent each number as we can read, for example, the number 3 as 30, 300, 0.003, 0.3, etc.

Logbase

How far it is possible, for reasons of space, we now add the minor divisions (logs between 1 and 99). Instead of search the logarithms in the tables we can simply add them with the help of a compass. Calculating with a compass was however laborious, slow and difficult. In 1622 William Oughtred marked the logarithmic scales on two sliding parallel rulers. An innovation that allows the direct reading of the result. The slide rule was born.

The slide rule, being an analog instrument, replace the mathematical functions with linear measurements. To show how it works let's start to see how we can execute an addition using two common metric rules. To add 2 and 4, align first the 0 of the rule B with the 2 of the rule A. We have set 2+ and the sum can be read on the mark of slide A corresponding to the second addendum.

Slide rule addition

To add 2+6 we don't need to move again the rule (set on 2+), but just read the sum directly on the figure 6 of the B rule. To subtract, we use the opposite proceeding.

From the accuracy of the construction depends the precision of the results but, also dividing further the scales, it is not possible to operate with numbers greater than 100. It 'is therefore clear that, as regards the addition and subtraction, the slide rule is much less practical than the abacus and to any other type of calculator. This system, however, becomes very powerful if the scales are drawn using the logarithmic succession that we have seen previously.

To perform 2 x 4 we align the 1 of scale B in correspondence of 2 in scale A and the result can be read on the same scale above the 4 of scale B.

Slide rule multiplication 1

We now have a tool that can perform multiplication (2x with this setting); the previous picture also shows how to perform 8/4. Just put the 4 of scale B under the 8 of scale A and read the result on the same scale above the 1 in scale B.

There are also some disadvantages: if we want process 4x3 the slides are positioned as follows:

Slide rule multiplication 2

The total is now located out of the scale. To solve this problem, we need to use the 10 of the rule B, instead of the previous 1:

Slide rule multiplication 3

So we obtain 1.2, but the right total is 12: the slide rule gives only the numbers and how to locate the dot or how to add ten or hundreds we must find by ourselves.

This was just a brief outlook on how the system work, but the slide rule has many other scales and can reach the computing power of a modern calculator. His only flaw is the poor readability. The secret is: practice, practice ...

Slide rule joke


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