# The Slide Rule

Perhaps, in the far distant back of times, man in his need to somewhat quantify what he had collected to survive (animals or plants or fruits), needed to count, and his own fingers and toes became a good first instrument always available, so that easily got to 20 units. And his system was digital, quite advanced, in contrast with the slide rule we try to deal with herein, which is analogical, so that the figure above has nothing to do with the slide rule. An apology is kindly offered to the reader for the unintended mixture we got in.

## How counting Started to be aSsisted

Obviously a true digital devise, the abacus (plural abaci or abacuses) is a counting frame tool that was used in Europe, China, and Russia, centuries before the adoption of the written numeral system. Its exact origin is unknown. Today, it is more of a toy or a curiosity adorning perhaps a furniture in a living room.

## Who were involved in developing counting instruments? [1]

Several eminent figures in science contributed to the creation or invention of the slide rule, including Galileo Galilei (1564–1642), unjustly condemned by the dogmatic and ignorant church of those times. John Napier de Merchiston (1550–1617), a Scottish mathematician recognized for being the first in defining logarithms, produced the essential concept backing up the slide rule. Sir Isaac Newton (1643–1727) and other prominent scientists, such as Johann Carl Friedrich Gauss (1777–1885), mathematician, physicist, and astronomer; James Watt (1736–1819), Scottish engineer; and Joseph Priestley (1733–1804), were part of the group, too. Other more recent scientists played a role in the development and improvement of the device, although perhaps in a lesser degree. As we advance in the text, different, more recent contributors’ names will be brought over.

## Early Attempts

Galileo popularized the sector at the very end of the 16th century. The sector is a graduated ruler that uses trigonometric formulae and a caliper to calculate squares, cubes, reciprocals, and tangents of numbers. Galileo’s design of the sector as a mathematical tool can be seen as the moment when calculation aids cease to be based upon counting and instead exploited the deeper relationships among numbers (Theory of Numbers).¹ His invention was still in use as a navigation aid in the 20th century… 300 years later! Napier dramatically advanced the understanding of number relationships in 1614 with his invention of logarithms: They are the foundation on which the slide rule is built; its history rightly begins with Napier. His early concept of simplifying mathematical calculations through logarithms makes possible the slide rule as we know it today. Napier came up with the so-called Napier’s Bones in 1617, a set of sticks based on the *gelosia*, or lattice multiplication method.² In 1620, Edmund Gunter (1581–1626) of London made a straight logarithmic scale able to perform multiplication and division with the use of a set of dividers, or calipers. Let us underline again that the slide rule cannot be called a counting device for it is an analog device. About 1622, William Oughtred (*Figure 1, right*), an Anglican Minister, today recognized as the inventor of the slide rule in its actual form, by placing two such scales side by side and sliding them to read the distance relationships, thus multiplying and dividing directly. He also developed a circular slide rule.

Later on in 1675, Newton solved cubic equations using three parallel logarithmic scales and made the first suggestion toward the use of the cursor. Two years after, Henry Coggeshall perfected the timber and carpenter’s rule. Incredibly, the Coggeshall rule remained in common use 200 years later. His design and its standardization made the slide rule from a tool of mathematical inquiry to specialized applications. Beginning in 1683, Thomas Everard (1560–1633) popularized the gaging rule to determine the content of ale, wine, and spirits barrels and to calculate the excise tax. This design, first created by William Oughtred in 1633, saw widespread use well into the 19th century. In 1722, John Warner, a London instrument dealer, applied square and cube scales.^{3} By 1790, after trying it first, Watt,^{4} in Birmingham, U.K., commissioned the design of a slide rule with a set of scales suitable for his work. This became known as the Soho Slide Rule (Watt lived in Soho). By 1799, their device had greatly influenced the Industrial Revolution development for it helped the design and manufacture of his seminal machine, the steam engine.^{5}

In 1815, Peter Roget, an English physician, invented a log–log scale, which he used to calculate roots and powers to any number or fraction thereof.^{6} It was regarded at the time as a mathematical curiosity. Fifty years later, advances in electrical engineering, thermodynamics, dynamics, and industrial chemistry made these scales so necessary that they were rediscovered! In the next 50 years, they increased from three, to six, to eight scales on the slide rule, as engineering extended its grip on modern computations.

In 1851, a French artillery officer named Amedee Mannheim^{7} standardized a set of four scales for the most common calculation problems. The four scales included two double-length, named A and B, for squares and square roots, and two single- length, C and D, for multiplication and division. This scale set became the basis of slide rule design for the next 100 years and bears his name today. His design and use of a cursor hastened the eventual widespread acceptance of this feature.

Early in the 19th century, the first slide rules came into use in the United States. By 1870, Germany produced two giants of the slide rule world: Dennert and Pape (makers of Aristo) and Faber (later Faber-Castell). The slide rule’s importance to the Industrial Revolution, and the impact of the Industrial Revolution upon the slide rule, are demonstrated by the proliferation of designs. From 1625 to 1800, the first 175 years after its invention, a total of 40 slide rule types, including circular and spiral designs, are recorded. The next 100 years, from 1800 to 1899, saw the creation of 250 slide rule types and manufacturers. Over 90 designs are recorded in the first 10 years of the 20th century. Cylindrical calculators with extra-long logarithmic scales were invented by George Fuller of Ireland in 1878^{8} and Edwin Thacher of New York in 1881.^{9} Production of Thacher’s calculator was soon taken over by a Hoboken, NJ, USA, instruments company, named Keuffel and Esser (K&E), which had previously imported slide rules for sale.

A revolutionary linear slide rule construction with scales on both front and back and with a cursor referring to all scales simultaneously was patented in 1891 by William Cox.^{10} Folded scales CF, DF, and CIF were put on slide rules in about 1900 to reduce the amount of movement and resetting of the slide. Log–log scales in three sections appeared about 1901, enabling very accurate calculation of powers and roots to any number or fraction. Our last century could not have been built without the slide rule, yet its direct evidence is almost totally missing to the uninformed eye.

In the United States, this indispensability is seized upon most successfully by K&E. This firm moved from importing rules in the 1870s, to building a complex calculating instrument (the Thacher Calculator) in the 1880s, to manufacturing their own slide rules in the 1890s. Their contributions are legion [2].

A different path to market dominance was adopted in Japan by Jiro Hemmi. Hemmi systematically experimented with both natural base materials and celluloid, settling upon bamboo as the core, and combining this with very modern manufacturing (including celluloid surface lamination), resulting in both high quality and quantity. In the 1960s, Hemmi was producing 1 million slide rules annually. Hemmi was known for using a large-diameter variety of bamboo grown in Kagoshima Prefecture on the island of Kyushu. Company founder Jiro Hemmi (1878–1953) patented this innovation in several nations, including the United States in 1920.

The philosophy of engineering moved on, and the triumphs of 20th century design became limitations for the slide rule. Einstein favored a Nestler slide rule in his work. The approaches to the Golden Gate Bridge and the thrust profile of the Redstone Rocket were designed with simple slide rules. Slide rules in different models provided emergency computational power aboard Apollo missions.

But structural design and first principles in physics both seeked answers to how structures would react to changing loads and forces. Wind speeds, tidal friction, and interstellar collision all require dynamic computational models, rather than the answer to static structural problems. Large, slow, obese computing engines designed for these questions gave birth to the sleek four-function calculator. K&E’s travails with its Analon slide rule model were an excellent example of the change in engineering demands that surpassed the slide rule. The four-function electronic calculator is a symptom as much as a cause of this change.

Slide rule researchers have estimated that possibly 40 million slide rules were produced in the world in the 20th century alone. Among these are many types of specialty slide rules developed and made for specific applications in chemistry, surveying, electricity and electronics, artillery ranging, hydraulics, steam and internal combustion engines, concrete and steel structures, radio, and other special fields.

In summary, the slide rule has a long and distinguished ancestry, from Oughtred in 1622 to the Apollo missions to the moon, a span of three and a half centuries. It was used to perform design calculations for virtually all the major structures built on this earth during that long period of our history, an amazing legacy for something so mechanically simple. I still dearly keep an Albert Nestler unit, bought in Buenos Aires during my engineering student years, and a K&E one bought in the United States when I was there back in the 1960s. Deeply embarrassed, I confess I cannot operate them anymore! Let us bow humbly before this device!

## An old and unknown book worth remembering

The author of this article will adopt from here onward the personal way of writing because he was, when still a young engineering student, an active part as an assistant to Dr. Manuel Sadosky, along with a former also student mate, Ricardo Gayoso, in the generation of the book, mainly by proof correction, recalculation of all its exercises and examples, and even checking punctuation and other minor editorial details. I am referring to *Cálculo Numérico y Gráfico (Numerical and Graphic Calculus)*, in Spanish, by Sadosky (Ediciones Librería del Colegio, Buenos Aires, distributed by Editorial Sudamericana, also in Buenos Aires, 1952, 347 pages), (Figure 2). Chapter 4 deals fully with the Slide Rule (pp. 50–72). I do not know how many, but perhaps relatively few, copies went into the market, and by now, it has disappeared. A real pity. I will briefly describe the contributions to the field of calculations found in its content, which reads:

Chapter 1 – Numerical Approximations (pp. 9–29);

Chapter 2 – Scales (pp. 30–37);

Chapter 3 – Logarithmic Graphics (pp. 38–49);

**Chapter 4 – Slide Rule (pp. 50–72);**

Chapter 5 – Nomography (pp. 73–98);

Chapter 6 – Linear Systems (pp. 99–158);

Chapter 7 – Numerical Solution of Equations (pp. 159–201);

Chapter 8 – Interpolation (pp. 202–232);

Chapter 9 – Numerical Differentiation and Integration (pp. 233–262);

Chapter 10 – Graphic and Mechanic Integration (pp. 263–291);

Chapter 11 – Approximate Integration of Ordinary Differential Equations (pp. 292–320);

Appendix – Evolution of the Mechanic and Automatic Calculus (pp. 321–333).

Let us give a brief and incomplete reference to Sadosky’s life; more can be easily found in the Internet. Manuel Sadosky (1914–2005) was a mathematician and a physicist recognized as the father of computer science in Argentina. He perfected studies at the Poincaré Paris Institute in 1946–1947 with a fellowship from the French government. He brought *Clementina*, the first computer (a Ferranti-Mercury) in the country. Sadosky, very cleverly and efficiently, used the machine as the seed for the creation of the studies of Scientific Computer Specialist. During the military dictatorship, persecuted, he left the country as exiled, returning when Raúl Ricardo Alfonsín won the election as the president for the period 1983–1989. Then, Sadosky was appointed Secretary of Science and Technology. From 1955 onwards, he was a professor at the University of Buenos Aires, Buenos Aires, Argentina, reaching the position of Vice-Dean of the Faculty of Exact and Natural Sciences between 1957 and 1966.

## Footnotes

- Regarding this theory, there is an abundant bibliography easily found on the Internet for the interested reader, but unnecessary to go deeper herein.
- Lattice multiplication, also known as the Italian method, Chinese method, Chinese lattice, gelosia multiplication, sieve multiplication, shabakh, or Venetian squares, is a method of multiplication that uses a lattice to multiply two multidigit numbers. See more: Wikipedia, Youtube
- No information could be found regarding this man.
- Scottish engineer who improved the Newcomen machine that led to the water vapor machine; the latter was an essential piece in the development of the First Industrial Revolution. See more.
- Boulton & Watt was located in Birmingham, while Gravet Lenoir in Paris, France, made many rules of the same kind. See also Soho engineers’ slide rule.
- Peter Mark Roget (1779–1869) studied medicine at Edinburgh University. In 1814, he invented a slide rule to calculate the roots and powers of numbers. It became the basis of slide rules that were common currency in schools and universities until the age of the calculator. Later in his life, he attempted to construct a calculating machine.
- Victor Mayer Amédée Mannheim (1831–1906) brought up the modern version of the slide rule, around 1850, with a new scale using a runner. It became known as the Mannheim.
- No information could be found regarding this man.
- Letters Patent No. 249117, dated November 1, 1881.
- No information could be found regarding this man.

## References

- The Oughtred Society. Slide rule history. (Dec. 2013). [Online].
- R. Paselk, “Thacher calculator,” Feb. 2012. [Online].