Who invented the slide rule. Slide rule: the history of the world's best computer

The ruler looks very much like a mechanical stopwatch, only it does not have a clock mechanism, and instead of buttons there are rotating heads, with the help of one we turn the hands, with the help of the other - a movable dial.

Unlike ordinary slide rulers, it does not allow you to count logarithms and cubes, the accuracy is one bit lower, well, you won’t use it like a regular ruler (and you won’t scratch your back), but it is very compact, you can carry it in your pocket.

Fast calculations

The attached (below) instruction suggests multiplying and dividing in three movements: rotating the movable scale on the pointer, rotating the arrow to the desired value, and rotating the dial to another value. However, it is much more interesting to use both dials, movable and fixed on the back of the ruler, and do the calculations in two movements. In this case, it is possible to obtain at once the entire range of values, simply by rotating the dial, and immediately reading the values.

To do this, on the stationary dial, you need to set the arrow with either a multiplier (in the case of multiplication) or the dividend (in the case of division), and, turning the ruler, turn the moving dial to set the second multiplier on the arrow, or the divider on the pointer, and immediately read the result. Continuing to rotate the dial, we immediately read other values \u200b\u200bof the function. An ordinary calculator cannot do this.

Inches to centimeters

For example, we need to convert centimeters to inches, or vice versa. To do this, by rotating the crown with a red dot, set the value of 2.54 on the stationary dial with an arrow. After that, we will see how many centimeters there are in our 24 "monitor - by rotating the head with a black dot of the moving dial, set the value 24 on the arrow, and read the value 61 cm from the fixed pointer (2.54 * 24 \u003d 60.96). In this case, you can easily find out the reverse values , for example, we find out how many inches are in our 81 cm TV, for this, by rotating the head with a black dot of the movable dial, set the value 81 on the stationary pointer, and read the value 32 "on the arrow (81 ⁄2 .54 \u003d 31.8898).

Faringate degrees to Celsius degrees

On a stationary dial we set the value 1.8, subtract 32 from degrees Fahrenheit in our mind and set the resulting value opposite the stationary pointer, read the degrees Celsius on the arrow. For the reverse calculation, set the value on the arrow, and mentally add 32 to the value on the pointer.

20*1.8+32 = 36+32 = 68

(100-32)/1.8 = 68 ⁄ 1 .8 = 37.8 (37.7778)

Miles to kilometers

We set the value 1.6 on a fixed scale, by rotating the movable scale we get miles in kilometers or kilometers in miles.

Let's calculate the speed of acceleration of the time machine in the movie "Back to the Future": 88 * 1.6 \u003d 141 km / h (140.8)

Time and distance from speed

To find out how long it will take to drive 400 kilometers at a speed of 60 km / h, set the value 6 on the fixed dial, and turn the movable dial to 4, we get 6.66 hours (6 hours 40 minutes).

Ruler instruction

The instructions I have on the ruler are very shabby, because they are already produced in 1966. Therefore, I decided to digitize it for safekeeping in electronic form.

Complete instructions for slide rule "KL-1":

Circular slide rule "KL-1"

1. Housing.
2. Black dot head.
3. Red dot head.
4. Movable dial.
5. Fixed pointer.
6. Main scale (counting).
7. Scale of squares of numbers.
8. Arrow.
9. Fixed dial.
10. Counting scale.

ATTENTION! Pulling the heads out of the housing is not allowed.

The circular slide rule "KL-1" is designed to perform the most common mathematical operations in practice: multiplication, division, combined actions, raising to a cladrate, extracting the square root, finding the trigonometric functions of the sine and tangent, as well as the corresponding inverse trigonometric functions, calculating the area circle.

The slide rule consists of a case with two heads, 2 dials, one of which rotates with a crown with a black dot and 2 hands that rotate with a crown with a red dot. There is a fixed pointer opposite the black dotted crown above the movable dial.

On the movable dial there are 2 scales: internal - main - counting and external - scale of squares of numbers.

There are 3 scales on the fixed dial: the outer scale is a counting one, similar to the inner scale on the moving dial, the middle scale “S” -values \u200b\u200bof the angles for reading their sines and the inner scale “T” -values \u200b\u200bof the angles for reading their tangents.

Performing mathematical operations on the "KL-1" ruler is as follows:

I. Multiplication

1. Turn the head with the red dot to align the arrow with the “1” mark.
2. Against the pointer on the counting scale, count the desired value of the product.

II. Division

1. Rotate the crown with a black dot to turn the movable dial until the dividend on the counting scale aligns with the pointer.
2. Against the pointer on the counting scale, count the sought value of the quotient.

III. Combined actions

1. Rotate the head with a black dot to turn the movable dial until the first factor on the counting scale aligns with the pointer.
2. By rotating the head with the red dot, align the arrow with the divider on the counting scale.
3. Rotate the head with a black dot to turn the movable dial until the second factor is aligned with the arrow on the counting scale.
4. Count the final result against the index on the counting scale.

Example: (2x12) / 6 \u003d 4

IV. Squaring

1. Rotate the head with a black dot to turn the movable dial until the square value of the number is aligned with the pointer on the counting scale.
2. Against the same pointer on the scale of squares, read the required value of the square of this number.

V. Extraction of the square root

1. Rotate the crown with a black dot to turn the movable dial until the value of the radical number on the scale of the squares is aligned with the pointer.
2. Against the same pointer on the internal (counting) scale, read the required value of the square root.

Vi. Finding trigonometric functions of an angle

1. By rotating the crown with a red dot, align the arrow above the stationary dial with the value of the specified angle on the sine scale (“S” scale) or on the tangent scale (“T” scale).
2. Against the same arrow on the same dial on the outer (counting) scale, read the corresponding value of the sine or tangent of this angle.

Vii. Finding Inverse Trigonometric Functions

1. By rotating the head with a red dot, align the arrow above the fixed dial on the external (counting) scale with the specified value of the trigonometric function.
2. Against the same arrow on the sine or tangent scale, read the value of the corresponding inverse trigonometric function.

VIII. Calculating the area of \u200b\u200ba circle

1. Rotate the crown with a black dot to turn the movable dial until the circle diameter on the counting scale is aligned with the pointer.
2. Turn the head with the red dot to align the arrow with the “C” mark.
3. Rotate the crown with a black dot to turn the movable dial until the mark “1” is aligned with the arrow.
4. Against the pointer on the scale of squares, count the desired value of the area of \u200b\u200bthe circle.

Technical and sales organization "Rassvet" Moscow, A-57, st. Ostryakov, house number 8.
STU 36-16-64-64
Article B-46
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Ruler size:

Now slide rule are produced only in wristwatches. Humanity has lost something by completely switching from analog computers to purely digital ones.

PS: The photos are not mine, taken from the Internet. In the last picture on the dial, the MLTZKP factory markings, if anyone knows what this abbreviation means, please let me know. I was able to decipher only part of it: “Moscow L? T? Plant of Control Devices ", produced this line of" Moscow Experimental Plant of Control Devices "Controlpribor" ".

In the age of computer technology, most calculations in the design of equipment are fully automated, engineers can only enter the required parameters through a convenient interface.

The XX century was called by different names. It was atomic, space and informational. Aircraft designers improved airplanes, and they turned from clumsy biplanes into fast-moving supersonic MiGs, Mirages and Phantoms. Giant aircraft carriers and submarines began to plow the seas and oceans at all latitudes. In Los Alamos (New Mexico), the first nuclear power plant was tested and in Obninsk near Moscow, the first nuclear power plant began to give energy. Rockets soared up ...

How missiles were calculated and

Historical chronicles show the process of working towards these achievements. Scientists and engineers in white coats, standing at drawers and sitting at tables piled up with drawings, perform the most complex technical and scientific calculations on adding machines. Sometimes in the hands of Tupolev, Kurchatov or Teller suddenly appeared a thing unfamiliar to a modern young man - a slide rule. Photos of those whose youth passed in the post-war decades, up to the 80s, also recorded this simple object, which successfully replaced a calculator with it while studying at the institute or graduate school. Yes, and dissertations were also considered on her, on a dear one.

What is the principle of a logarithmic ruler?

The basic principle of operation of this wooden object, carefully pasted over with celluloid white scales, is based on the logarithmic calculus, as the name suggests. More precisely, on After all, everyone who taught knows that their sum is equal to the logarithm of the product, and, therefore, by correctly applying division into the moving parts, you can achieve that multiplication (and therefore division), squaring (and extraction of the root) will become easy.

The logarithmic ruler became popular in the 19th century, when ordinary abacus was the main means for performing calculations. This invention is a real find for the then scientists and engineers. Not at once they all figured out how to use this device. To learn all the intricacies and reveal its capabilities to the fullest, fans of the new calculating mechanism had to read special manuals, quite voluminous. But it was worth it.

There are different rulers, even round

Nevertheless, the main advantage of the logarithmic ruler is its simplicity and, consequently, reliability. Compared to other calculation methods (there were no calculators yet), operations were performed much faster. But there are also points that should not be forgotten. Calculations can only be made with mantissa, that is, the whole (up to nine) and fractional parts of the number, with an accuracy of two (three, who have very good eyesight) decimal places. The order of the numbers had to be kept in mind. There was one more drawback. The logarithmic ruler, although small, can hardly be called a pocket device - 30 centimeters after all.

However, the size did not become an obstacle for inquisitive minds. For those who, by the nature of their activity, must have a calculating device always with them, a compact slide rule was invented. The dial with hands gave it a watch-like appearance, and some models of expensive chronometers contained it on their dials. Of course, the capabilities of this device and its accuracy were somewhat inferior to the corresponding parameters of the classical line, but it could always be carried in a pocket. And it looked more aesthetically pleasing!

Most have seen the slide rule (or counting rule) only in pictures or in films such as Titanic (1997), This Island Earth (1955) and Apollo 13 (1995). If you are a Star Trek fan, you should be aware that Mr. Spock uses the Jeppesen CSG-1 and B-1 slide rules for several episodes. However, there was a time when engineers did not walk around with calculators or mobile phones, but with slide rules on their belts. The Pickett slide rule flew to the moon with the cosmonauts, and the K&E ruler made the atomic bomb possible.

Slide rules are part of mathematics and history. They are not affected by electromagnetic impulses, which means they are able to survive the Apocalypse, which everyone predicts to us. In the case of slide rules, like many other things in this life, the rule applies: the more the better.

Slide rule history

The slide rule was developed by the English mathematician William Oughtred in the 17th century. It retained its popularity among people who were serious about mathematics until the early 1970s. In fact, the idea of \u200b\u200bperforming various calculations with a ruler was not new at the time. Previously, Edmund Gunther developed a sector with the same division as a slide rule, but in order to solve any problem with it, you needed a separate set of dividing compasses. Otred's device was a circular slide rule. One of his students, Richard Delamaine, claimed to have also invented the slide rule. Both men accused each other of stealing ideas.

Modern scientists believe that they simultaneously created a circular slide rule. Delamaine was the first to publicly announce his invention, but Outred appears to have completed the development of the slide rule earlier than his student.

The usual slide rule was created by Otred in about 1650.

Slide rule theory

Slide rules are associated with Napier's discovery of logarithms. Logarithms played an important role in the world of pre-computer mathematics. Let's take a decimal logarithm as an example. If you square 10, you get 100. Therefore, the logarithm of 100 is 2. If you raise 10 to the fifth power, you get 100000. Hence, the logarithm of 100000 is 5. The resulting digits do not have to be whole numbers. So, for example, the logarithm of 200 is 2.3.

Logarithm table

If you spent a lot of time on calculations, you would certainly create a table of numbers and their logarithms. The question is: why? The answer is simple. Suppose you wanted to multiply two numbers - 200 and 100. This is easy enough to do without resorting to any tricks. You write down “200x100” on a piece of paper and multiply each number. Logarithms make this much easier. The logarithm of 200 is 2.301, and the logarithm of 100 is 2. The sum of the logs of 200 and 100 is 4.301 (2.301 + 2). If you raise 10 to the power of 4.3, you will get a not quite accurate answer (19998.6), since we rounded the logarithm of 200. Obviously, the more digits in your table, the better.

This is not a very good example. But if you need to multiply 7329 by 8115, then knowing the logarithms of these numbers (3.8650 and 3.9093, respectively), it will be very easy for you to perform this calculation. Raise 10 to the power of 7.7743, and you will find out the correct answer - 59470282 (actually 59474835, but, again, very close).

Movable tables

How does this relate to a slide rule? A slide rule is an efficient logarithm table made of wood, plastic or metal. The marks are plotted on the surface based on the logarithm of the number, however, they are indicated with real numbers, that is, the distance between 0 and 1, for example, is much greater than the distance between 8 and 9.

Let's look at the principle of using a slide rule with a simple example: 2x3. Move the C scale so that the one is above the number 2 on the fixed D scale. Then set the slider to 3 on the C scale. Now you just need to look at the number on the fixed D scale to get the answer (6). The principle of using a slide rule is very easy to understand if you hold it in your hands. You can also use the web simulator available on link... You can see a screenshot of the calculation below.

If you are dealing with large numbers, first reduce them by the n-th number of tens of times, and then mentally increase the result obtained by the same amount. For example, to calculate the product of the numbers 20 and 30, you must first reduce them by 10 times, and then increase the result by 100 times.

Division and other operations

Division works in much the same way, but is based on subtraction. If you shift the C scale so that the number 3 is above 6 on the fixed D scale, you can see the answer 2 (D scale) below 1 on the C scale. A transparent plastic slider with a thin line in the middle will help you not to get confused in the numbers. Some rulers even have a small magnifying glass to help you see the marks on the scale better.

Getting the right answer

Unlike a calculator, a slide rule usually requires you to have some idea of \u200b\u200bthe answer in order to interpret the results. You should also be able to tell the difference between, say, 7.3, 7.35 and 7.351. That's why the more the better.

A typical slide rule is about 25 centimeters long. Pocket rulers were short but impractical. There were also huge slide rule designed for classroom use (some of them were up to 2 meters 15 centimeters long). For more accurate calculations, engineers used cylinder-shaped rulers. They were the equivalent of slide rules up to 10 meters long.

Above is Otis King's slide rule, which corresponded to a 170 centimeter long ruler, but easily fit in a pocket. It looks very much like a telescope. In fact, it is a slide rule with a scale drawn in a spiral around the instrument. Otis King's ruler had more digits than a regular slide rule, but the calculations performed with it were often not entirely accurate.

How to start collecting slide rule and where to get them?

Many people think that slide rule is difficult to collect, but in fact it is quite easy and inexpensive. At one time they were widespread, but after the invention of a calculator and a computer, they instantly became of no use to anyone. If you try, you can find people who have preserved used or completely new slide rules.

EBay is where you search results show you can find over 3,000 slide rule lines. They can also be purchased for cheap at local stores. Often people do not understand what slide rules are for, so they are only happy to get rid of them. In addition, if people find out that you are a collector, they can just give you slide rules that once belonged to their distant relatives. They will be pleased to know that you will keep them.

If you decide to buy a slide rule, make sure that the scale C works and the transparent slider does not fog up. Their repair or replacement is very painstaking work. Also, avoid rulers with corrosion or faded marks. They can be restored, but it takes a lot of time and effort. On the Internet, you can find tips on how to properly clean the various rulers.

If you have purchased a slide rule, you should remember that it, like any other thing, requires special care. To make its moving parts work well, wipe them with furniture polish (if the ruler is wooden). Previously, people smeared iron slide rule with Vaseline. It is also important to keep the slide rule clean at all times and to ensure that no dirt gets under the engine.

Also, do not leave the ruler in direct sunlight. Also, try to avoid using soap, water, and other substances that could damage your ruler.

Slide rule was once a kind of computers and will probably replace modern PCs when the Apocalypse comes.

The slide rule (see photo below) was invented as a device to save the mental costs and time associated with mathematical calculations. It was especially widespread in the practice of engineers in research-oriented institutes and in statistical offices before the introduction of electronic computing technology.

Logarithmic ruler: history

The scale for calculations of the English mathematician E. Gunter was the prototype of the calculating device. He invented it in 1623, shortly after the discovery of logarithms, to make it easier to work with them. The scale was used in conjunction with a compass. The necessary graduated segments were measured with them, which were then added or subtracted. Operations with numbers were replaced by operations with logarithms. Using their basic properties, multiplying, dividing, raising to a power or calculating the root of a number turned out to be much easier.

In 1623 the slide rule was improved by W. Otred. He added a second movable scale. She moved along the main ruler. Measuring line segments and reading the results of calculations has become easier. To improve the accuracy of the device, an attempt was made in 1650 to increase the length of the scale by arranging it in a spiral on a rotating cylinder.

The addition of a slider to the design (1850) made the calculation process even more convenient. Further improvement of the mechanism and method of applying logarithmic scales to the standard ruler did not add accuracy to the instrument.

Device

The logarithmic ruler (standard) was made of solid wood, resistant to abrasion. For this, a pear tree was used on an industrial scale. The body and engine were made from it - a smaller bar mounted in an internal groove. It can be moved parallel to the base. The runner was made of aluminum or steel with a glass or plastic viewing window. A thin vertical line (sight) is drawn on it. The slider moves along side guides and is spring-loaded by a steel plate. The body and engine are lined with light celluloid with embossed scales. Their divisions are filled with printing ink.

On the front side of the ruler there are seven scales: four on the body and three on the slider. On the side faces, there is a simple measurement marking (25 cm) with 1 mm divisions. The scales (C) on the slider below and (D) on the body immediately below it are considered main. On top of the base there is a cubic marking (K), below it - a quadratic (A). Below (on top of the slider) there is exactly the same symmetrical auxiliary scale (B). At the bottom of the case there is still a markup for the values \u200b\u200bof the logarithms (L). In the very center of the front of the ruler, between markers (B) and (C), there is an inverse number scale (R). On the other side of the engine (the bar can be removed from the grooves and turned over) there are three more scales for calculating trigonometric functions. The upper (Sin) is for sines, the lower (Tg) is for tangents, and the middle (Sin and Tg) is common.

Varieties

The standard logarithmic ruler has a measuring scale length of 25 cm. A pocket version with a length of 12.5 cm and a device of increased accuracy of 50 cm were also produced. There was a division of the rulers into first and second grades, depending on the quality of execution. Attention was paid to the clarity of the strokes, symbols and auxiliary lines. The engine and body had to be flat and perfectly matched to each other. Products of the second grade could have minor scratches and dots on the celluloid, but they did not distort the designations. There could also be a slight backlash in the grooves and deflection.

There were other pocket (similar to a watch with a diameter of 5 cm) variants of the device - a logarithmic disk ("Sputnik" type) and circular (KL-1) rulers. They differed both in design and in lower measurement accuracy. In the first case, a transparent cover with a line-sight was used to set numbers on closed circular logarithmic scales. In the second, the control mechanism (two rotating knobs) was mounted on the body: one controlled the disk engine, the other controlled the arrow-sight.

Opportunities

A general-purpose slide rule could be used to divide and multiply numbers, square and cube them, extract the root, and solve equations. In addition, trigonometric calculations (sine and tangent) were performed on the scales at given angles, the mantissa of logarithms and inverse actions were determined - the numbers were found by their values.

The correctness of the calculations largely depended on the quality of the ruler (the length of its scales). Ideally, one could hope for accuracy to the third decimal place. Such indicators were quite sufficient for technical calculations in the 19th century.

The question arises: how to use a slide rule? Knowledge of the purpose of the scales and the methods of finding numbers on them is not enough for making calculations. To use all the possibilities of the ruler, you need to understand what a logarithm is, know its characteristics and properties, as well as the principles of construction and dependence of scales.

Certain skills were required to operate the device confidently. Comparatively simple calculations with one slider. For convenience, the engine (so as not to distract) can be removed. By setting the line to the values \u200b\u200bof any number on the main (D) scale, you can immediately get the result of squaring it on the scale above (A) and cube on the uppermost (K) scale using the viewer. The bottom (L) will be the value of its logarithm.

Division and multiplication of numbers is done using the engine. The properties of logarithms are applied. According to them, the result of multiplying two numbers is equal to the result of adding their logarithms (similarly: division and difference). Knowing this, you can quickly make calculations using graphic scales.

Why is slide rule complicated? Instructions for its correct use were included with each copy. In addition to knowing the properties and characteristics of logarithms, it was necessary to be able to correctly find the initial numbers on the scales and be able to read the results in the right place, including independently determining the exact location of the comma.

Relevance

How to use a slide rule, in our time, few know and remember, and we can confidently say that the number of such people will decrease.

Slide rule from the category of pocket calculators has long become a rarity. To work confidently with it, you need constant practice. The calculation methodology with examples and explanations is enough for a brochure of 50 sheets.

For an average person who is far from higher mathematics, a slide rule can be of some value only with reference materials placed on the back of the case (density of some substances, melting point, etc.). Teachers do not even bother to impose a ban on its presence when passing exams and tests, realizing that it is very difficult for a modern student to understand the intricacies of its use.

In informatics lessons, studying the topic "History of Computing", the slide rule device is mentioned. What it is? How she looks like? How to use it? Consider the history of the creation of this device and the principle of operation.

Is a calculating device that was used before the advent of calculators and personal computers. It was a fairly versatile device that could multiply, divide, square and cube, calculate square and cube roots, sines, tangents and other values. These mathematical operations were performed with a sufficiently high accuracy - up to 3-4 decimal places.

History of the slide rule

In 1622 William Oughtred (William Oughtred March 5, 1575 - June 30, 1660) creates perhaps one of the most successful analog computing mechanisms - the slide rule. Otred is one of the founders of modern mathematical symbolism - the author of several notation and operation signs, standard in modern mathematics:

• The multiplication sign is an oblique cross: ×
• The division sign is a forward slash: /
• Parallelism symbol: ||
• Short notation of functions sin and cos (previously written in full: Sinus, Cosinus)
• The term "cubic equation".

"All his thoughts were focused on mathematics, and he was constantly pondering or drawing lines and shapes on the ground ... His house was full of young gentlemen who came from everywhere to learn from him.".

Unknown contemporary of Otred

Otred made a decisive contribution to the invention of a user-friendly slide rule by proposing the use of two identical scales sliding along one another. The very idea of \u200b\u200ba logarithmic scale was previously published by the Welshman Edmund Gunther, but in order to perform calculations, this scale had to be carefully measured with two compasses.

Gunther also introduced the now generally accepted notation log and the terms cosine and cotangent. In 1620, Gunther's book was published, where a description of his logarithmic scale was given, as well as tables of logarithms, sines and cotangents. As for the logarithm itself, it was invented, as you know, by the Scotsman John Napier. Seeing the bewilderment of Forster, who highly appreciated this invention, Otred showed his student two calculating tools he had made - two slide rule.

Gunther's slide scale was the ancestor of the slide rule and has undergone multiple revisions. So in 1624, Edmund Wingate published a book in which he described a modification of the Gunther scale, which makes it easy to square and cube numbers, as well as to extract square and cubic roots.

Further improvements led to the creation of a slide rule, however, the authorship of this invention is disputed by two scientists, William Oughtred and Richard Delamain.

Otred's first ruler had two logarithmic scales, one of which could shift relative to the other, stationary. The second tool was a ring, inside which a circle rotated on an axis. On the circle (outside) and inside the ring, logarithmic scales “rolled into a circle” were depicted. Both rulers made it possible to do without compasses.

In 1632, Otred and Forster's book "Circles of Proportions" was published in London with a description of a circular slide rule (of a different design), and a description of Otred's rectangular slide rule is given in Forster's book "An Addendum to the Use of a Tool Called" Circles of Proportions ", published in the following year.

The ruler of Richard Delamaine (who at one time was Otred's assistant), described by him in the pamphlet "Grammelogy, or Mathematical Ring", which appeared in 1630, was also a ring, inside which a circle revolved. Then this brochure with changes and additions was published several times. Delamaine described several variants of such rulers (containing up to 13 scales). In a special recess, Delamain placed a flat pointer that could move along a radius, making it easier to use a ruler. Other designs have also been proposed. Delamain not only provided descriptions of the rulers, but also gave a calibration methodology, suggested ways to check accuracy, and gave examples of using his devices.

And in 1654 the Englishman Robert Bissaker proposed the construction of a rectangular slide rule, the general view of which has been preserved to our time ...

In 1850, a nineteen-year-old French officer, Amedeus Mannheim, created a rectangular slide rule, which became the prototype of modern rulers and provides accuracy to three decimal places. He described this tool in the book "Modified Ruler", published in 1851. For 20-30 years, this model was produced only in France, and then it began to be produced in England, Germany and the USA. Soon, Mannheim's lineup became popular around the world.

For many years the slide rule remained the most popular and affordable device for individual computing, despite the rapid development of computers. Naturally, it had a low accuracy and speed of solution in comparison with computers, however, in practice, most of the initial data were not accurate, but approximate values \u200b\u200bdetermined with varying degrees of accuracy. And, as you know, the results of calculations with approximate numbers will always be approximate. This fact and the high cost of computing technology allowed the slide rule to exist almost until the end of the 20th century.

Addition

2 + 4 = 6

Subtraction

8 – 3 = 5

Multiplication

a ∙ b = from at a = 2 , b = 3

Taking the logarithm of both sides of the equality, we have: Lg(a ) + lg(b )= lg(from ) .

Taking two rulers with logarithmic scales, we see that the addition of values lg2 and lg3 results in lg6 , that is, the product 2 on the 3 .

On the main scale of the ruler body (second from the bottom), the first factor is selected and the beginning of the main, lower, scale of the engine is set on it (it is on the front side of the latter and is exactly the same as the main scale of the body).

On the main scale of the slider, the hair of the slider is set on the second factor.

The answer is on the main scale of the ruler body under the hair. If, in this case, the hair goes beyond the scale, then not the beginning, but the end of the engine (with the number 10) is set on the first factor.

Division

a / b = from at a = 8 , b = 4

Taking the logarithm of both sides of the equality, we get: Lg(a ) – lg(b ) = lg(from ) .

The difference between the logarithms of the dividend and the divisor gives the logarithm of the quotient, in our case - 2 .

On the main scale of the ruler body, the dividend is selected, on which the slider hair is set.

A divider found on the main scale of the engine is placed under the hair. The result is determined on the main body scale opposite the start or end of the slider.

Exponentiation and root extraction

The scale of squares of numbers is the second from the top, cubes are the first from the top.

The hair is set to the raised number on the main scale of the body, and the result is read under the hair on the corresponding scale.

When extracting square and cube roots, on the contrary, the result is on the main scale.

Carrying for calculations with comma

If, for example, one of the factors is 126 , then the ruler uses the value 1,26 , and the found work is enlarged 100 times. When cubing the number 0,375 result found for number 3,75 , decreases 1000 times, etc.