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# Slide Ruler – a simple invention that changed the history of science in the world

In the mid-16th century, the scientific revolution began thanks to Copernicus’ astonishing discovery that the Earth and other planets orbit the Sun in perfectly circular orbits. Johannes Kepler believed that Copernicus was wrong about the shapes of these orbits and suggested that they were not perfect circles.

Kepler already had some relatively accurate measurements of Mars’ motion in the sky, made by his teacher, astronomer Tycho Brahe. Using these measurements, Kepler manually calculated the orbit of Mars, but the addition, subtraction, multiplication and division calculations dragged on and he repeatedly miscalculated without being able to find an answer.

Finally, after four years, Kepler completed his 900-page calculation – and he had to repeat it over the course of 70 times. Fortunately, Kepler didn’t know that, there were errors in those calculations, but miraculously, the errors negated each other and made the calculation still correct.

Kepler’s case is not unique. Throughout history, the difficulty of performing complex calculations has troubled scholars and hindered the development of human knowledge. That difficulty paved the way for the invention of a calculation tool that revolutionized human history: the sliding ruler.

# The birth of a slide ruler

As the scientific revolution continued to develop, calculations became more and more difficult to perform. Researchers and technicians need to make precise scientific measurements, but math has evolved and scientists need to do longer and more complex calculations. And then the miracle happened.

Scottish mathematician and astronomer John Napier discovered a function he called the “logarithm” that turned long and complex multiplication and division into simple addition and subtraction.

For example, if 103 equals 1,000 then the base 10 logarithm of 1,000 is 3 or Log(1000)=3. Similarly, Log(100)=2. Adding them together, we get 5, which is equivalent to the power of 5 of 10 = 100,000. More generally, this calculation can be written as adding two real numbers: Log(ab)= Log(a) + Log(b).

In 1614, after nearly 20 years of hard work, Napier published his discovery in a 90-page spreadsheet – a spreadsheet that lists the logarithms of about 10 million numbers. It allows users to find the product of 2 numbers in the following way: find the logarithm of each number in the table, add these two logarithms together and find the number whose logarithm matches the result of this calculation, thereby obtaining the result for multiplication. .

For example, when an astronomer needs to perform the multiplication of two values ​​of a trigonometric function: 0.57357 by 0.42261, using the Logarithmic worksheet, he can find the two Logarithmic values ​​that are their best approximations. are -0.24141 and -0.37406. Add these two Logarithms together to get -0.61547 and the corresponding number for this Logarithm in the spreadsheet is 0.242399. This number approximates the result of the above multiplication. A page from John Napier’s Logarithmic. The leftmost column is the angles from 30 to 60 degrees (the right is from 0 to 30 degrees), adjacent to it is the column recording the sine of those angles, and then the Natural Logarithm of the sine of that angle. The middle column is the difference between these two Logarithms, or the Natural Logarithm of the Tan function.

At a time when computers weren’t around, this was the easiest way to perform complex multiplication and division calculations with relative precision. But mathematicians still have to grope the numbers they need in a 90-page spreadsheet, and that’s still a time-consuming process.

In 1620, the English astronomer and mathematician Edmund Gunter developed a special ruler that could calculate the product of two numbers by measuring the length on the ruler, instead of groping it in a 90-long alphanumeric table. Page. The ruler is made so that its length, from its tip, to any number x corresponds to its logarithm.

This ruler starts from 1, since Log(1)=0 and so the distance to 1 is 0. The distance from 1 to 3 is half the distance from 1 to 9 because Log(3) = ½ Log( 9). To calculate the product of a certain number a and b, the user will measure the distance from the tip of the ruler to that number to find the Logarithm of this number and then add them together, and then compare this distance on ruler to find the result of the calculation.

Then, in 1622, William Oughtred created the first sliding ruler. It is a simple and easy to use calculation tool, with 2 Logarithmic rulers that can be matched and slid parallel to each other. To calculate the product of two numbers a and b, all you have to do is align these two rulers so that the number 1 on the upper ruler is aligned with the number a in the lower ruler and compare which number is on the lower ruler. line up with the number b in the upper ruler. Oughtred’s slide rule example. To do a 2×3 calculation. Aligning the number 1 on the upper ruler with the number 2 on the lower ruler and the number on the bottom ruler aligning with the number 3 on the upper ruler is the result of the calculation: the number 6.

Since the distance of Log(ab) = Log(a)+Log(b) is calculated from the top of the ruler below to this number, this is the result of the product of a and b. Doing the opposite will result in the division of two numbers. For example, to divide ab by b, put the number b on the upper ruler in line with ab on the lower ruler, then find the number that aligns with the number 1 on the upper ruler to get the result of the division.

# 350 years of computing service for mankind

This simple ruler revolutionized the way multiplication and division were performed. It became a popular tool for mathematicians, scientists, engineers, doctors, geographers, military personnel, pilots, tax officers and many more. This tool has appeared alongside nearly every invention, every design of historic structure, and in every scientific advance for nearly 350 years now.

As it became more commonly used, it also grew more complex. Along with the “regular” Logarithmic scale, the slider is also supplemented with many other calculations, such as sine, square root and exponent. The front and back of the Keuffel & Esser 4081-3 slider, in addition to calculating more calculations, it also has a slider above to make it easier for users to see the numbers.

In addition to the calculation of common calculations, some special types of sliders with their own scales are designed for specific calculations. For example, they are used to convert between different units of measure, calculate bank loans, engineering calculations and even complete flight gauges used for calculations involving cargo. are not. There are also circular and also cylindrical rulers. A technician on a British bomber aircraft uses a slide to calculate the plane’s fuel consumption, October 1941.

Parachute was considered an advanced tool for hundreds of years and was even used in the invention of a variety of other mechanical calculation tools. However, the rapid development of digital computers with outstanding computing power has made these sliders increasingly difficult to find a foothold.

Even in 1972, an article lamented the fatal blow of electronic computers that had caused the rulers to almost completely disappear: “When an engineer or scientist needs a quick answer to a problem that requires a lot of multiplication, division, or complex functions, he often turns to his sliding ruler. Soon, however, that faithful cane will have to be retired. Now, a pocket electronic calculator can also give answers much easier, faster and more accurate..”