Slide rule (history of invention). Slide Rule Inventor of the Slide Rule

Well adapted to perform addition and subtraction operations, the abacus turned out to be an insufficiently efficient device for performing multiplication and division operations. Therefore, the discovery of logarithms and logarithmic tables by J. Napier at the beginning of the 17th century, which made it possible to replace multiplication and division with addition and subtraction, respectively, was the next major step in the development computing systems manual stage. His “Canon of Logarithms” began: “Realizing that in mathematics there is nothing more boring and tedious than multiplication, division, square and cubic roots, and that these operations are a useless waste of time and an inexhaustible source of elusive errors, I decided to find a simple and reliable means to get rid of them.” In his work “Description of the Amazing Table of Logarithms” (1614), he outlined the properties of logarithms, gave a description of the tables, rules for using them and examples of applications. The basis of Napier's logarithm table is an irrational number, to which numbers of the form (1 + 1/n) n approach without limit as n increases indefinitely. This number is called the Neper number and is denoted by the letter e:

e=lim (1+1/n) n=2.71828…

Subsequently appears whole line modifications of logarithmic tables. However, in practical work their use has a number of inconveniences, so J. Napier, as an alternative method, proposed special counting sticks (later called Napier sticks), which made it possible to perform multiplication and division operations directly on the original numbers. The basis this method Napier laid down the lattice method of multiplication.

Along with sticks, Napier proposed a counting board for performing multiplication, division, squaring, and square root operations in the binary number system, thereby anticipating the benefits of such a number system for automating calculations.

So how do Napier logarithms work? A word from the inventor: “Discard the numbers, product, quotient or root of which you need to find, and take instead those that will give the same result after addition, subtraction and division by two and three.” In other words, using logarithms, multiplication can be simplified to addition, division can be reduced to subtraction, and square and cube roots can be reduced to division by two and three, respectively. For example, to multiply the numbers 3.8 and 6.61, we determine using a table and add their logarithms: 0.58+0.82=1.4. Now let’s find in the table a number whose logarithm is equal to the resulting sum, and we’ll get an almost exact value of the desired product: 25.12. And no mistakes!

Logarithms served as the basis for the creation of a wonderful computing tool - the slide rule, which has served engineers and technicians around the world for more than 360 years. The prototype of the modern slide rule is considered to be the logarithmic scale of E. Gunther, used by W. Oughtred and R. Delamaine when creating the first slide rules. Through the efforts of a number of researchers, the slide rule was constantly improved and the appearance closest to the modern one is due to the 19-year-old French officer A. Manheim.

A slide rule is an analog computing device that allows you to perform several mathematical operations, including multiplying and dividing numbers, exponentiation (most often squaring and cube), calculating logarithms, trigonometric functions and other operations

In order to calculate the product of two numbers, the beginning of the moving scale is combined with the first factor on the fixed scale, and the second factor is found on the moving scale. Opposite it on a fixed scale is the result of multiplying these numbers:

log(x) + log(y) = log(xy)

To divide numbers, find the divisor on the moving scale and combine it with the dividend on the fixed scale. The beginning of the moving scale indicates the result:

log(x) - log(y) = log(x/y)

Using a slide rule, only the mantissa of a number is found; its order is calculated in the mind. The calculation accuracy of ordinary rulers is two to three decimal places. To perform other operations, use a slider and additional scales.

It should be noted that, despite its simplicity, quite complex calculations can be performed on a slide rule. Previously, quite voluminous manuals on their use were published.

The principle of operation of a slide rule is based on the fact that the multiplication and division of numbers is replaced, respectively, by the addition and subtraction of their logarithms.

Up until the 1970s. slide rules were as common as typewriters and mimeographs. With a deft movement of his hands, the engineer easily multiplied and divided any numbers and extracted square and cube roots. A little more effort was required to calculate proportions, sines and tangents.

Decorated with a dozen functional scales, the slide rule symbolized the innermost secrets of science. In fact, only two scales did the main work, since almost all technical calculations came down to multiplication and division.

The ruler is very similar in appearance to 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 rules, it does not allow you to count logarithms and cubes, the accuracy is one digit lower, and you can’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 instructions (below) suggest multiplying and dividing in three movements: by rotating the moving scale to 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 stationary with reverse side rulers, and do calculations in two movements. In this case, it is possible to obtain the entire range of values ​​​​at once, simply by rotating the dial and immediately reading the values.

To do this, on a fixed dial you need to set either the multiplier (in the case of multiplication) or the dividend (in the case of division) with the arrow, and, turning the ruler over, by rotating the movable dial, set the second multiplier on the arrow, or the divisor on the pointer, and immediately read the result. Continuing to rotate the dial, we immediately read other function values. A regular 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 head with the red dot, we set the arrow to 2.54 on the stationary dial. After this, we will look at how many centimeters there are in our 24" monitor - by rotating the head with the black dot of the movable dial, we set the value 24 on the arrow, and read the value 61 cm from the fixed pointer (2.54 * 24 = 60.96). In this case, you can easily find out and reciprocal values, for example, we find out how many inches there are in our 81 cm TV, to do this, by rotating the head with the black dot of the movable dial, we set the value 81 on the fixed pointer, and read the value 32" on the arrow (81 ⁄ 2 .54 = 31.8898).

Fahrenheit to Celsius

On the fixed dial we set the value to 1.8, subtract 32 from degrees Fahrenheit in our minds and set the resulting value opposite the fixed pointer, read the degrees Celsius on the hand. To do the reverse calculation, set the value on the arrow, and add 32 in your head 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 to 1.6 on the fixed scale, and by rotating the moving scale we get miles in kilometers or kilometers in miles.

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

Time and distance from speed

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

Instructions for the ruler

The instructions for the line I have are very tattered, because it was produced in 1966. Therefore, I decided to digitize it for safekeeping in electronic form.

Complete instructions for the slide rule “KL-1”:

Circular slide rule “KL-1”

1. Frame.
2. Head with black dot.
3. Head with a red dot.
4. Movable dial.
5. Fixed pointer.
6. Main scale (counting).
7. Number square scale.
8. Arrow.
9. Fixed dial.
10. Counting scale.

ATTENTION! Pulling 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 operations, raising to cladrarate, extracting square roots, finding trigonometric functions of sine and tangent, as well as the corresponding inverse trigonometric functions, calculating area circle.

A slide rule consists of a body with two heads, 2 dials, one of which rotates using a head with a black dot, and 2 hands, which rotate using a head with a red dot. Opposite the crown with a black dot above the movable dial there is a fixed pointer.

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

There are 3 scales on the fixed dial: the outer scale is counting, similar to the inner scale on the movable dial, the middle scale is “S”-values ​​of angles for counting their sines, and the inner scale is “T”-values ​​of angles for counting their tangents.

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

I. Multiplication

1. Rotate 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. By rotating the head with the black dot, 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 desired value of the quotient.

III. Combined actions

1. By rotating the head with the black dot, 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. By rotating the head with the black dot, turn the movable dial until the second factor on the counting scale aligns with the arrow.
4. Count the final result against the pointer on the counting scale.

Example: (2x12)/6=4

IV. Squaring

1. By rotating the head with the black dot, turn the movable dial until the value of the squared number on the counting scale aligns with the pointer.
2. Against the same pointer on the square scale, read the desired value of the square of this number.

V. Extracting the square root

1. By rotating the head with the black dot, turn the movable dial until the value of the radical number on the square scale aligns with the pointer.
2. Against the same pointer on the internal (counting) scale, read the desired value of the square root.

VI. Finding Trigonometric Angle Functions

1. By rotating the head with the 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 the red dot, align the arrow above the stationary dial on the outer (counting) scale with the given 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 ​​a circle

1. By rotating the head with the black dot, turn the movable dial until the value of the diameter of the circle on the counting scale aligns with the pointer.
2. Rotate the head with the red dot to align the arrow with the “C” mark.
3. By rotating the head with the black dot, turn the movable dial until the “1” mark aligns with the arrow.
4. Against the pointer on the square scale, count the desired value of the area of ​​the circle.

Technical and sales organization “Rassvet” Moscow, A-57, st. Ostryakova, house No. 8.
STU 36-16-64-64
Article B-46
Quality Control Department stamp<1>
Price 3 rub. 10 kopecks

Ruler size:

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

P.S.: the photos are not mine, taken from the Internet. On last picture On the dial there is a factory marking MLTZKP, 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 “Moscow Experimental Plant of Control Devices “Kontrolpribor”“.

In computer science lessons, while studying the topic “History of Computer Science,” the slide rule device is mentioned. What it is? How does she look? How to use it? Let's consider the history of the creation of this device and the principle of operation.

is a calculating device used before the advent of calculators and personal computers. It was a fairly universal device on which you could multiply, divide, square and cube, calculate square and cube roots, sines, tangents and other values. These mathematical operations were performed with fairly 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. Oughtred is one of the creators of modern mathematical symbolism - the author of several standard notations and operation signs in modern mathematics:

• Multiplication sign - oblique cross: ×
• The division sign is a slash: /
• Concurrency symbol: ||
• Brief designations functions sin and cos (previously written in full: Sinus, Cosinus)
• The term "cubic equation".

“All his thoughts were concentrated on mathematics, and he was always meditating or drawing lines and figures on the ground... His house was full of young gentlemen who came from all over to learn from him.”.

Unknown contemporary of Oughtred

Oughtred made a decisive contribution to the invention of the easy-to-use slide rule by proposing the use of two identical scales, sliding one along the other. The very idea of ​​a logarithmic scale was previously published by the Welshman Edmund Gunther, but to perform calculations this scale had to be carefully measured with two compasses.

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

The Gunter logarithmic scale was the progenitor of the slide rule and was subject to numerous modifications. 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 extract square and cube roots.

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

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

In 1632, Oughtred 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 Oughtred’s rectangular slide rule is given in Forster’s book “Addition to the use of an instrument called “Circles of Proportions”, published in the following year.

The ruler of Richard Delamain (who was at one time Oughtred's assistant), described by him in the brochure “Grammeology, or the Mathematical Ring,” which appeared in 1630, was also a ring with a circle rotating inside it. Then this brochure with changes and additions was published several more times. Delamain 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, which made it easier to use the ruler. Other designs have also been proposed. Delamaine not only presented descriptions of the rulers, but also gave a calibration technique, suggested methods for checking accuracy, and gave examples of using his devices.

And in 1654, the Englishman Robert Bissacker proposed the design of a rectangular slide rule, general form which has survived to this day...

In 1850, nineteen-year-old French officer Amédée Mannheim created the 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 Calculating Ruler,” published in 1851. For 20-30 years, this model was produced only in France, and then it began to be manufactured in England, Germany and the USA. Soon the Mannheim line gained popularity all over the world.

For many years, the slide rule remained the most widespread and accessible device for individual calculations, despite the rapid development of computers. Naturally, it had low accuracy and speed of solution compared to computers However, in practice, most of the initial data were not exact, but approximate values ​​determined 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 computer 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 = With at a = 2 , b = 3

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

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 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 slider is set to 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 engine, the slider hair is installed on the second factor.

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

Division

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

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

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, onto which the slider hair is installed.

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

Exponentiation and root extraction

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

The hair is set on the number being erected on the main scale of the case, 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.

Carry in calculations with commas

If, for example, one of the factors is equal to 126 , then the value is used on the ruler 1,26 , and the found product increases 100 times. When cubed a number 0,375 result found for number 3,75 , decreases by 1000 times, etc.

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

The 20th century has been called differently. It was atomic, and cosmic, and informational. Aircraft designers improved the aircraft, and they turned from clumsy biplanes into fast supersonic MiGs, Mirages and Phantoms. Giant aircraft carriers and submarines began to plow the seas and oceans at all latitudes. The first nuclear power plant was tested in Los Alamos (New Mexico), and the first nuclear power plant began producing energy in Obninsk near Moscow. Rockets soared up...

How the rockets were calculated and

Historical chronicles demonstrate the process of work on these achievements. Scientists and engineers in white coats, standing at drawing boards and sitting at tables littered with drawings, perform the most complex technical and scientific calculations on adding machines. Sometimes, in the hands of Tupolev, Kurchatov or Teller, a thing suddenly appeared that was unfamiliar to modern young man- logarithmic ruler. Photos of those whose youth passed in the post-war decades, right up to the 80s, also recorded this simple object, which successfully replaced a calculator during their studies at the institute or graduate school. Yes, and dissertations were also considered on it, on my dear one.

On what principle is a slide rule constructed?

The main operating principle of this wooden object, neatly covered with celluloid white scales, is based on logarithmic calculus, as the name suggests. More precisely, after all, everyone who taught knows that their sum is equal to the logarithm of the product, and, therefore, by correctly applying the divisions to the moving parts, you can achieve that multiplication (and therefore division), squaring (and root extraction) will become a simple matter.

The slide rule became popular back in the 19th century, when the main means for carrying out calculations was ordinary abacus. This invention was a real find for the scientists and engineers of that time. It took them some time to figure out how to use this device. In order to learn all the intricacies and fully reveal its capabilities, fans of the new counting mechanism had to read special manuals, quite voluminous. But it was worth it.

There are different rulers, even round ones

However, the main advantage that a slide rule has is its simplicity, and therefore reliability. Compared to other methods of calculation (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 mantissas, that is, the integer (up to nine) and fractional parts of a number, accurate to two (three, for those with very good eyesight) decimal places. The order of the numbers had to be kept in mind. There was one more drawback. A slide rule, although small, can hardly be called a pocket device - it’s still 30 centimeters.

However, size did not become an obstacle for inquisitive minds. For those who, due to their line of work, must always have a counting device with them, a compact slide rule was invented. The circular scale with arrows made it look like a watch, and some models of expensive chronometers contained it on their dial. Of course, the capabilities of this device and its accuracy were somewhat inferior to the corresponding parameters of the classic line, but it could always be carried in your pocket. And it looked more aesthetically pleasing!

Slide rule or tally rule- a computing device that allows you to perform several mathematical operations, including multiplication and division of numbers, exponentiation (most often squaring and cube) and calculation of square and cube roots, calculation of logarithms, potentiation, calculation of trigonometric and hyperbolic functions and other operations. Also, if you divide the calculation into three steps, then using a slide rule you can raise numbers to any real power and extract the root of any real degree.

Don't be scared! You don't need to calculate bases and logarithms, cosines and arctangents every day. In most cases, slide rules built into watches are not equipped with scales for calculating the values ​​of trigonometric functions.

A number of watches are equipped with computing lines, the functions of which are close to everyday life.

By the way, it was Mark Carson, the head of the theoretical department at the nuclear center, USA, who was the first to come up with the idea of ​​putting the logarithmic school into a clock.

So, the clock Citizen Promaster Sky– just from the designations on the graduated scale it is clear that they are perfectly suited for calculating fuel consumption when traveling by car or traveling on a motor boat.

Let's start with the simplest. The circular slide rule consists of a ruler on the bezel and a ruler on the dial. Rotate the bezel until the value on the bezel ruler aligns with the desired mark on the dial.

In order to divide 150 by 3, the number 15 (=150) on the outer scale should be set against the number 30 (3) on the inner scale. The result is counted on the internal scale opposite “10” and is equal to 50.

You can find an example on the Internet Triple rule, or calculating the rate of descent using the circular ruler on the clock.

A pilot in a glider at an altitude of 3300 meters determines that he is losing altitude at a rate of one meter per second, i.e. 60 m per minute. How much time does he have until the end of the flight? In order to know the answer, you should set the number 33 (=3300) on the outer scale against the number 60 on the inner scale. The result is opposite the "10" sign on the internal scale and is 55 minutes.

But let’s leave aviation problems alone and apply this rule for calculations in a closer area. How far will 40 liters of gasoline last for you with a fuel consumption of 8 liters per 100 kilometers? We set the number 40 opposite the number 8. We get 50, taking into account the scale of 1 to 10 - for 500 km.

There are many symbols on various watches to make it easier to recalculate length measures.

STAT means English mile, NAUT- nautical mile, M- American mile, and on the clock Citizen Promaster Sky - KM– which in both Latin and Russian transliteration means kilometers.