The Slide Rule 

However the more complex operations: multiplication and division, remained as an annoying process. It was until 1617 that a Scottish amateur mathematician:  John Napier, invented the first device to perform multiplications and divisions: the "Virgulas" (Rods), known in present times as "Napier's Bones". 

These were a series of rods (by that time carved from bones) that had squares and numbers inscribed in them. Using the rods, one could perform multiplication by looking up partial products and summing them. Division was a more complex process but could be performed similarly as a series of look-ups and subtractions.  (https://en.wikipedia.org/wiki/Napier%27s_bones) 

Other invention to ease multiplications-divisions, roots and powers computations, also  surged from John Napier's bright mind: The Logarithms.

 After 20 years of tedious calculations, Napier published in 1614 his " Canon of Logarithms" containing tables of numbers that allowed compute multiplications and divisions as simple additions and subtractions. Napier set forth his motivation writing at the beginning of his book:

Seeing there is nothing that is so troublesome to mathematical practice, nor doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances... 

Logarithms were so useful that the illustrious French mathematician Pierre Simon Laplace wrote that logarithms were: "an admirable artifice which, by reducing to a few days the labor of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations".  A very interesting story of the invention of logarithms, can be found in the lecture: John Napier and the Invention of Logarithm. 1619, given by E.W. Hobson, Math Professor at Cambridge University, in 1914, to commemorate the 300th  anniversary of the invention of the Logarithms.

But even though the great help the tables of logarithms were to make computational work easier, they were not fully efficient  because additions or subtractions had to be performed "by hand", and the conversions "numbers - to - logs - to - number" were annoying.

However logarithms were quickly adopted and used by the scientific community, and in 1620, a English clergy and mathematician: Edmund Gunter, engraved a logarithmic scale (a number line where logarithms of numbers 1 to 10, and some fractions between them are represented) on a wood ruler. With this device he was able to perform basic calculations with the aid of compasses.

In the figure below, the number 1 on the lower scale is aligned with the number 2 on the upper scale, then it is easy to see that above the number 2 on the lower scale is the number 4, and above the number 2.5 is the number 5, and above the number 3 is the number six and so on... the results to multiply the numbers in the lower scale by two can be read directly above on the upper scale! With this procedure William Oughtred invented the Slide Rule. 

In 1657, Seth Partridge, an English surveyor and mathematician, made up the first slide rule with three strips of wood, with the two outer strips held together by bridging cleats, and the strip at the middle sliding freely. Partridge included scales on both faces... the duplex slide rule was born

In 1675, Sir Isaac Newton, devised a method to solve cubic equations using three movable slide rule scales, and a separated edge or marking line laid across them to bring together three numbers on the scales.

The use of the slide rule quickly spread to other countries, and during the next centuries modifications and additions were made to the original designs, Additional to the basic logarithmic scales to perform multiplications and divisions, other scales to compute squares and square roots, cubes and cubic roots, values for trigonometric functions were added, and the slide rule's power grew when scales to compute any power or any root for a number, the Log-Log scales ,were created by the English physician Peter Roget (Yes!, the creator of the Thesaurus!) in 1845. The slide rule was an important tool for scientist and engineers during the Industrial Revolution.

The slide rule had also the possibility to be adapted to perform calculations for specific disciplines, so it was not only a tool for engineers, scientists and mathematicians, but creating specific logarithmic scales, was very useful for accountants, bankers, tax men, or merchant's. 

But the modern general purpose slide rule was created by the French Artillery Lieutenant Amedee Mannheim in 1851.  He created a simple arrangement with one pair of two-cycle logarithmic scales labeled A-B, and one pair of one-cycle logarithmic scales labeled C-D. The positioning of the A-B scales above the C-D scales, with B and C scales engraved on the rule's slider, allowed to calculate, multiplications-divisions, squares and square roots, and mixed multiplications-divisions of numbers with squares-square roots, in a very simple way. At the back of the slide sine and tangent scales were engraved. The array (A/B, C/D) remained the foundation for all further slide rule models, but Mannheim's greatest contribution was the reintroduction and refinement of a functional cursor. Mannheim described his slide rule design in a note in the Nouvelles Annales de Mathematiques in 1853 .

The technological advance of the next years and the development of new materials, made possible to produce slide rules of better quality and precision. The first line cursor was produced by the French Company Tavernier-Gravet in 1905, and this added to the invention of the plastic celluloid, gave to the slide rule its modern face.

With only the use of logarithmic scales, the slide rule became a powerful aide to solve problems. But scientists and engineers required to perform more complex operations to solve also more complex problems, so more sophisticated scales were created and slide rules became also more complex. Next are shown some masterpieces created during 1959-1975 (Click on the images to see the ISRM panoramic picture):

The Slide Rule has been also the most efficient and versatile mechanical calculating device ever, able to be adapted to specific tasks. Next are some examples for chemical engineering, science and military:

For about 350 years, the slide rule was the most efficient and effective calculating device, until it was displaced by the pocket electronic calculators, these, by the way, were designed with the help of the slide rule. The Empire State Building, the Eiffel Tower, The Golden Gate Bridge, the Panama Channel, Airplanes, Automobiles, Electrification, Radio and TV, Nuclear Power, Space Rockets, and all the great scientific and engineering achievements done from the 1700's to the early 1970's, were calculated with a slide rule. 

Below is the famous picture of Buzz Aldrin using a Pickett N-600 in the Apollo 11. It is known that when the Gemini XII computer docking system failed, Buzz Aldrin calculated the docking trajectory using a sextant and a slide rule. 

But the inexorable advance of science and technology during the 1960's and early 1970's quickly changed this panoramic,  In 1961, with the help of their slide rules,  IBM scientist and engineers produced the first transistor  computer for engineering applications: the 7030 Stretch System, with much more computer power and precision than the best slide rule, but even though these computers were compact compared with those built with vacuum tubes, they were accessible only to a reduced number of privileged users (Original cost: $7,780,000,  equivalent to $68,000,000 in 2021), so for the common people, or for working on your desk or at home, the slide rule was still necessary and useful, so, during the 1960's the slide rule market steadily expanded, and excellent and new fancy models were produced (see pics of the rules above). 

By the same time, Casio Electronics produced in 1957 the first desktop electronic calculator: the Model 14-A, In the early 1960's were produced the first solid state calculators, but these devices were quite expensive for the standard user.  

However, technological developments continued and in 1972 Hewlett Packard introduced the first pocket scientific calculator: the HP-35, with capabilities that made the slide rule look a very primitive gadget, but the not very affordable price of the HP-35, $395.00 USD (about $2,600.00 in 2021), compared with the lower cost of an  "engineering" slide rule, (about $25 USD, ~$160 in 2021) gave to the slide rule market a few more years. 

Unfortunately for the slide rule manufacturers, a new road was already open and nothing stopped the advance in that direction. In 1973, Texas Instruments produced its pocket scientific calculator TI-SR10,  with a selling price of $150.00 USD (~ $850.00 USD in 2021).  This price was already affordable for professionals and small companies, but still not very much for the common people.  

Some slide rule manufacturers tried to ride on the new wave. In 1974, Faber-Castell introduced  the TR1, TR2 and TR3 Electronic Slide Rules,  handheld electronic calculators with slide rules attached on the back, but unfortunately for them, that same year, Texas Instruments launched the TI-30 at a cost of only $24.95 USD (~$112.00 USD in 2021), cheaper than some high end slide rules!  The slide rule market had no way to compete now. During the next  year, all manufactures stopped production, and slide rules were put in drawers, or simply throwed away.  There were even plans from Pickett Co, manufacturer of the aluminum slide rules, (Yes, the ones used in the Mission Apollo), to sell the unsold production as scrap metal in Mexico. 

Here is  a scan of the central pages of the November 1972 Life magazine, showing the prices of the new fast-selling electronic gadgets for that Christmas:

Something  like this have never seen before, or after.  A flourishing market with a production around the 40 million of pieces, collapses to zero in less than two years. As Cliff Stoll wrote in his classic paper When Slide Rules Ruled: ...the slide rule helped to design the very machines that would render it obsolete...

Same as automobiles, that do not require you make an effort to go large distances,  electronic calculators do not require users to make the effort to think  to get a result, only a few keystrokes and... voila!...  the answer is there! For many people the results obtained with an electronic calculator are unquestionable, undoubtedly correct, they ignore or don't know the Computer Sciences GIGO rule: garbage in, garbage out... flawed, or nonsense input data produces nonsense output...

Additionally, slide rules are very ingenuous devices. They are the materialization of the use of logarithms to solve problems, and a hands-on example that logarithms were invented to make complex operations simpler, and not to make math students' lives miserable... 

Mastering of the slide rule was not only a need, but also was a passion that was taken into competitions, and participant students spent several hours per day learning tricks and practicing methods, but progress is at the end unstoppable, and those slide rule contests became calculator  contests...

Finally, the slide rules versatility to be adapted to different disciplines and problems, and their intrinsic charm and beauty, make them very interesting collectibles. I personally declare myself a slide rule lover, and of course an irremediable collector. Click on the link below to see my little collection.