Logarithmic line how to use. History of logarithmic ruler

The logarithmic line (photo see below) was invented as a device for saving mental costs and time associated with mathematical calculations. She received special distribution in the practice of engineers in institutions focused on research activities, and in statistical bureaus until the introduction of electronic computing equipment.

Logarithmic line: history

The prototype of the counting device was a scale for calculating the English mathematics E. Ganter. He came up with it in 1623, shortly after the opening of logarithms, to simplify work with them. The scale was used in combination with a circular. They were measured by the necessary graded segments, which then folded or subtracted. Operations with numbers were replaced by actions with logarithms. Using their basic properties, multiply, divide, to raise the degree or calculate the root of the number turned out to be much easier.

In 1623, the logarithmic line was improved by W. End. He added the second moving scale. It moved along the main line. Measure the segments and read the results of the calculus became easier. To increase the accuracy of the device in 1650, an attempt was implemented to increase the length of the scale due to its disposal on the rotating cylinder.

Adding a runner to the design (1850) made the calculus process even more convenient. Further improvement of the mechanism and method of applying logarithmic scales on the standard line did not add accuracy to the device.

Device

The logarithmic line (standard) was made of dense wood, resistant to abrasion. For this, a pear tree was used on an industrial scale. From it, the hull and the engine is smaller, mounted in the inner groove. It can be moved parallel to the base. The slider was made of aluminum or steel with an observation window of glass or plastic. A thin vertical line (Vizier) is applied to it. The slider moves along the side guides and spring plans. The housing and the engine are lined with a light celluloid, on which the scales are embossed. Their divisions are filled with typographic paint.

On the front side of the line there are seven scales: four- on the case and three - on the engine. A simple measuring markup (25 cm) with divisions of 1 mm is applied on the side faces. Scale (C) on the engine at the bottom and (d) on the housing immediately under it are considered the main. Based on top there is a cubic markup (K), under it - quadratic (A). Below (on top on the engine) there is exactly the same symmetrical auxiliary scale (B). Below on the case there is still a markup for the values \u200b\u200bof logarithms (L). In the heart of the front part of the line between the markups (b) and (c), the reverse scale of numbers (R) is applied. On the other hand, the engine (bar can be removed from the grooves and flip) there are three more scales for calculation trigonometric functions. Top (SIN) - designed for sinuses, lower (TG) - tangents, average (SIN and TG) - total.

Varieties

The standard logarithmic line has a length of the measuring scale 25 cm. A further pocket version of 12.5 cm long and a 50 cm device was existed. There was a division of lines to the first and second grade depending on the quality of execution. Attention was paid to the clarity of the strokes applied, designations and auxiliary lines. The engine and the case should have been smooth and perfectly adjusted to each other. The products of the second grade could have minor scratches and points on celluloid, but they did not distort the designations. There was also a slight backlash in the grooves and deflection.

There were other pocket (similar to the clock with a diameter of 5 cm) options for the device - the logarithmic disc (satellite type) and the circular (CL-1) ruler. They differ in both the design and less measurement accuracy. In the first case, for the installation of numbers on closed circular logarithmic scales, a transparent lid with a linus-vizir was used. In the second, the control mechanism (two rotating handles) was mounted on the housing: one was controlled by a disk engine, the other was driving an arrow-vizir.

Capabilities

Logarithmic ruler general purpose It was possible to divide and multiplying numbers, erect them into a square and cube, extract the root, solve equations. In addition, the scales were carried out trigonometric calculations (sinus and tangent) at the given angles, the logarithms and reverse actions were determined - there were numbers by their values.

The correctness of the calculations in many respects depended on the quality of the line (its length is long). Ideally, it was necessary to hope for accuracy to the third decimal sign. Such indicators were quite sufficient for technical calculations in the XIX century.

The question arises: how to use the logarithmic ruler? One knowledge of the appointment of the scales and ways to find numbers is not enough for the work of calculations. To use all the capabilities of the line, you need to understand what logarithm is, to know its characteristics and properties, as well as the principles of construction and the dependence of the scales.

For confident work with the device, certain skills were required. Comparatively simple calculations with one slider. For the convenience of the engine (so as not to distract) you can delete. By setting the line to the values \u200b\u200bof any number on the main (D) scale, you can immediately get the result of the construction of it into the square on the scale above (A) and in the cubic meter on the top (K). Below (L) there will be the value of its logarithm.

The division and multiplication of numbers is performed using the engine. Properties of logarithms apply. According to them, the result of multiplying two numbers is the result of the addition of their logarithms (similarly: division and difference). Knowing it, you can quickly make calculations using graphic scales.

What is the complex logarithmic ruler? The instruction for its correct use was bundled with each instance. In addition to knowledge of the properties and characteristics of logarithms, it was necessary to be able to properly find initial numbers on the scales and be able to right place Receive results, including independently determine the exact location of the comma.

Relevance

How to use the logarithmic ruler, in our time they know and remember few, and confidently can argue that the number of such people will decline.

The logarithmic line from the discharge of pocket countable devices has long become a rarity. For confident work with it you need a permanent practice. The method of calculations with examples and explanations pulls on a brochure of 50 sheets.

For an average man, far from higher mathematics, the logarithmic line can represent some value except with reference materials placed on the reverse side of the housing (the density of some substances, the melting point, etc.). The teachers do not even bother to bring a ban on its presence when passing exams and tests, realizing that it is very difficult to deal with the intricacies of its use.

Inventor: William Otred and Richard Delaminian
Country: England
Time of the invention: 1630

Inventors of the first logarithmic are the British - Mathematics and Teacher William Outfilled (William Oughtred) and Mathematics Teacher Richard Delamaine.

The Son of the Priest, William Otred studied first in Iton, and then in the Cambridge Royal College, specialized in the field of mathematics. In 1595, the final received the first scientific degree and entered the college council. He was then a little over 20 years old. Later, it was concluded to combine classes in mathematics with the study of theology and in 1603 became a priest. Soon he received a parish in Albury, near London, where he lived most of his life. However, this man was the teaching of mathematics.

In the summer of 1630, his student and friend, London Mathematics Teacher, William Forster, came down. Colleagues talking about mathematics ke and, as it were, they said today, about the method of her teaching. In one of the conversations, the Otred critically responded about the Günther scale, noting that the manipulation of two takes a lot of time and gives low accuracy.

Wallen Edmund Günther built a logarithmic scale that was used together with two circulators. The gunter scale was a segment with divisions corresponding to the logarithms of numbers or trigonometric values. With the help of circulators, the amount or difference between the length segments of the scale was determined, which, in accordance with the properties of logarithms, it allowed to find a product or private.

Günther also entered the generally accepted Log designation and the term cosine and Kotangenes.

First Lie neuka had two logarithmic scales, one of which could shift relative to the other, stationary. The second tool was a ring, inside of which rotated on the axis circle. On the circle (outside) and inside the ring were depicted "rolled into the circle" logarithmic scales. Both rules allowed to do without circulas.

In 1632 in London, the Book of Credential and Forster "Circles of Proportions" was published with a description of a circular logarithmic (already different design), and the description of the rectangular logarithmic ruler was given in the Forster book "Supplement to the use of a tool called" circles of proportions ", released next year. The rights to manufacture their linecas subferee transferred to the famous London mechanics Elias Allen.

Richard Delaminian line (which was at one time an assistant asked), described by him in the brochure "Grammotegia, or the Mathematical Ring", which appeared in 1630, also represented a ring inside of which the circle rotated. Then this brochure with changes and additions was published several times. Delaminine described several variants of such lines (containing up to 13 scales). IN special deepened Dealer placed a flat pointer capable of moving along the radius, which facilitated the use of the ruler. Other designs were offered. Delaminine not only presented the descriptions of Linek, but also gave the method of graduation, offered ways to check accuracy and led examples of using their devices.

In informatics lessons, studying the topic "The history of computing equipment", the device is mentioned a logarithmic ruler. What it is? How she looks like? How to use it? Consider the history of creating this device and the principle of operation.

- This is a countable device applied before the appearance of calculators and personal computers. It was a fairly universal device on which it was possible to multiply, divide, erect into a square and cube, calculate square and cubic roots, sines, tangents and other meanings. These mathematical operations were performed with a sufficiently large accuracy - up to 3-4 decimal places.

History of logarithmic ruler

In 1622. William Otred (William Oughtred March 5, 1575-30 June 1660) creates, perhaps, one of the most successful analog computing mechanisms is a logarithmic ruler. Credential is one of the creators of modern mathematical symbolism - the author of several standard in modern mathematics designations and signs of operations:

• Multiplication sign - oblique cross: ×
• Sign of division - oblique trait: /
• Parallelism symbol: ||
• Brief symbols functions sin. and COS (previously wrote fully: Sinus, Cosinus)
• The term "cubic equation".

"All his thoughts focused on mathematics, and he all the time pondered or hello lines and figures on Earth ... his house was full of young gentlemen who came from everywhere to learn from him".

Unknown contemporary expandering

Conducting a decisive contribution to the invention convenient to use the logarithmic line with the fact that he suggested using two identical scales, sliding one along the other. The idea of \u200b\u200bthe logarithmic scale itself was previously published Wallen Edmund Günther, but to fulfill the calculations, this scale had to be thoroughly measured by two circulas.

Günther also entered the generally accepted Log designation and the term cosine and Kotangenes. In 1620, the book of Günther, where the description of its logarithmic scale was given, and the tables of logarithms, sinuses and catangers were placed. As for the logarithm itself, it was invented, as it is known, Scotlandz John Never. Seeing the bewilderment of the Forster, highly appreciated by this invention, the subdon showed his student two manufactured computing tools - two logarithmic rules.

The Logarithmic scale of Günther was the progenitor of the logarithmic line and was subjected to multiple refinement. So in 1624, Edmund Wengate issued a book that described the modification of the Günther's scale, which makes it easy to erect the numbers into the square and to the cube, and extract square and cubic roots.

Further improvements led to the creation of a logarithmic line, however, the authorship of this invention challenges two scientists by William Otred and Richard Delaminian.

The first ruler of the final had two logarithmic scales, one of which could shift relative to the other, fixed. The second tool was a ring, inside of which rotated on the axis circle. On the circle (outside) and inside the ring were depicted "rolled into the circle" logarithmic scales. Both rules allowed to do without circulas.

In 1632, in London, the Book of Crane and Forster "Circles of Proportions" was published with a description of a circular logarithmic line (already different design), and the description of the rectangular logarithmic line was given in the Forster's book "Supplement to the use of a tool called" circles of proportions ", released in the following year.

Richard Delaminian line (which was at one time an assistant asked), described by him in the brochure "Grammotegia, or the Mathematical Ring", which appeared in 1630, also represented a ring inside of which the circle rotated. Then this brochure with changes and additions was published several times. Delaminine described several variants of such lines (containing up to 13 scales). In a special deepening, the Delaminian placed a flat pointer capable of moving along the radius, which facilitated the use of the ruler. Other designs were offered. Delaminine not only presented the descriptions of Linek, but also gave the method of graduation, offered ways to check accuracy and led examples of using their devices.

And in 1654, the Englishman Robert Bissaker proposed the design of the rectangular logarithmic line, the general view of which was preserved to our time ...

In 1850, a nineteen-year-old French officer Amedea Mannheim created a rectangular logarithmic ruler, which became a prototype of modern lines and ensuring accuracy to three decimal signs. It described this tool in the "Modified Computing Line" book, published in 1851. For 20-30 years, this model was produced only in France, and then it began to make it in England, Germany and the United States. Soon the Mannheim ruler won popularity around the world.

The logarithmic line for many years remained the most massive and affordable instrument of individual calculation, despite the rapid development of computing machines. Naturally, she had a small accuracy and speed of the solution compared to computing machines, however, in practice, most of the source data were not accurate, but approximate values \u200b\u200bdefined with one degree 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 equipment allowed the logarithmic line to exist almost until the end of the 20th century.

Addition

2 + 4 = 6

Subtraction

8 – 3 = 5

Multiplication

a. ∙ b. = from for a. = 2 , b. = 3

Logarithming both parts of equality, we have: LG(a. ) + lG(b. )= lG(from ) .

Taking two rules with logarithmic scales, we see that the addition of values lG2 and lG3 as a result lG6 , that is, the work 2 on the 3 .

On the main scale of the line of the line (second below), the first factory is selected and the beginning of the main, lower, engine scale is set to it (it is on the front side of the latter and exactly the same as the main body of the case).

On the main scale, the mechanic runner is installed on the second beggar.

The answer is on the main scale of the line of the lineup. If at the same time the hairs goes beyond the scale, then the first factor is not started, but the end of the engine (with a number 10).

Division

a. / b. = from for a. = 8 , b. = 4

Logarithming both parts of equality, we get: LG(a. ) – lG(b. ) = lG(from ) .

The difference between the logarithms of the divide and divider gives the logarithm of the private, in our case - 2 .

On the main scale of the line of the line is chosen divide, which is installed by the Runner's hairs.

Under the hairs is supplied by a divider found on the main scale of the engine. The result is determined on the main scale of the case opposite the beginning or end of the engine.

Erend the degree and extraction of the root

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

The hairs are installed on an erection of the number on the main scale of the case, and the result is read on the scarc on the corresponding scale.

When removing square and cubic roots, on the contrary, the result is on the main scale.

Transfer when calculating a comma

If, for example, one of the factors is equal 126 then the line is used on the line 1,26 , and the work found is 100 times. When erected into a list of numbers 0,375 number 3,75 , decreases 1000 times, etc.

A person who is not familiar with the use of the logarithmic ruler, it will seem work Picasso. It has at least three different scales, almost every of which numbers are not even at the same distance from each other. But having understood what, what, you will understand why the logarithmic ruler was so comfortable during the invention of pocket calculators. Correcting the desired numbers on the scale correctly, you can multiply two any numbers much faster than performing calculations on paper.

Steps

Part 1

general information

Pay attention to the gaps between the numbers. Unlike the usual line, the distance between them is not the same. On the contrary, it is determined by a special "logarithmic" formula, less on one side and more on the other. Thanks to this, you can combine two scales in the desired way and get an answer to the multiplication task as described below.

Tags on scale. Each logarithmic line scale has a letter or symbolic designation on the left or right side. The generally accepted designations on the logarithmic rules are described below:

• Scale C and D are similar to a single-digit extended line, the labels on which are located left to right. Such a scale is called a "single-digit decimal" scale.
• Scale A and B - "Two-digit decimal" scales. Each consists of two small elongated lines located in principle.
• K is a three-digit decimal scale or three elongated rules located in principle. Such a scale is not available on all logarithmic rules.
• Scale C | and D | Similar to C and D, but read right to left. Often they have red color. They are not present on all logarithmic rules.
• The logarithmic rules are different, therefore the designation of the scales may be different. On some rules, the scales for multiplication can be labeled as a and b and be on top. Regardless alphabetic designations, on many rules next to the scales there is a symbol π, noted in a suitable place; In most, the scales are opposite each other or in the upper or in the lower gap. We recommend solving a few simple tasks to multiply so that you can understand whether you use the scale correctly. If the product 2 and 4 does not equals 8, try using the scales on the other side of the line.

1. Learn to understand the division of the scale. Look at the vertical lines on the C or D scale and get acquainted with how they are read:

   • The main numbers on the scale begin with 1 from the left edge and continue up to 9, and then completed another one on the right. Usually all of them are applied to the ruler.
   • Secondary divisions, designated slightly smaller vertical lines, share each main figure by 0.1. You should not confuse if they are indicated as "1, 2, 3"; All the same, they correspond to "1,1; 1.2; 1.3 "and so on.
   • Smaller divisions may also be present, which usually correspond to step 0.02. Watch for them carefully, as they can disappear at the top of the scale, where the numbers are closer to each other.

2. Do not expect to get accurate answers. When reading the scale, you will often have to come to the "most likely assumption" when the answer will not fall into the baillet. The logarithmic line is used for quick counts, and not for maximum accuracy.

   • For example, if the answer is between the marks 6.51 and 6.52, write down the value that you seem closer. If it is completely incomprehensible, write the answer as 6,515.

   Part 2

   Multiplication

   1. Write down the numbers that you will multiply. Write down the numbers that are subject to multiplication.

      • In Example 1 of this section, we will calculate how much it will be 260 x 0.3.
      • In Example 2, we will calculate how much it will be 410 x 9. It is a bit more complicated than an example 1, so first consider a simpler task.

   2. Move the decimal points for each number. The logarithmic ruler has numbers from 1 to 10. Move the decimal point of each multiplying number so that they match their values. After solving the problem, we will move the decimal point in response to the desired position, which will be described at the end of the section.

      • Example 1: To calculate 260 x 0.3, start instead from 2.6 x 3.
      • Example 2: To calculate 410 x 9, start instead from 4.1 x 9.

   3. Find smaller numbers on the D, then move the scale to it. Find a lower digit on the scale D. Slide the scale C so that the "1" left (left index) was located on the same line with this number.

      • Example 1: Slide the scale C so that the left index coincides with 2.6 on the D.
      • Example 2: Slide the scale C so that the left index coincides with 4.1 on the D. scale

   4. Move the metal pointer to the second digit on the C. The pointer is a metal object that moves throughout the line. Align the pointer with the second digit of your task on the C scale. The pointer will indicate the response to the task on the scale D. If it does not move so far, go to the next step.

   5. If the pointer is not moving to the answer, use the right index. If the pointer is blocked by the partition in the center of the line or the answer is located outside the scale, then use a slightly different approach. Slide the scale C so that right index Or 1 on the right was located over the large coefficient of your task. Move the pointer to another coefficient on the C scale and read the answer on the D.

      • Example 2: Move the scale C so that 1 to the right coincided with 9 on the scale D. Move the pointer to 4.1 on the C scale. The pointer shows on the scale D at a point between 3.68 and 3.7, so the most likely answer will be 3.69.

   6. Puck the right decimal point. Regardless of the multiplication produced, your answer will always be read on the scale D, which contains only numbers from one to ten. You can not do without assumptions and mental counting to determine the location of the decimal point in the actual response.

      • Example 1: Our initial task was 260 x 0.3, and the ruler gave an answer 7.8. Round the initial task to convenient numbers and decide it in my head: 250 x 0.5 \u003d 125. Such an answer is much closer to 78 than by 780 or 7.8, so the correct answer will be 78 .
      • Example 2: Our initial task was 410 x 9, and the ruler gave the answer 3.69. Count the initial task as 400 x 10 \u003d 4000. The nearest number will be 3690 which will become the actual answer.

   Part 3.

   Construction of the Square and Cube

   Part 4.

   Extraction of a square and cubic root

   1. Record the number in the exponential representation to extract the square root. As always, there are only values \u200b\u200bfrom 1 to 10 on the line, so you will need to write a number in the exponential representation to extract the square root.

      • Example 3: To solve √ (390), write down the task as √ (3.9 x 10 2).
      • Example 4: To solve √ (7100), write down the task as √ (7.1 x 10 3).

   2. Determine what way the scale A must be used. To remove the square root of the number, to start, move the pointer to this number on the scale A. But since the scale A is applied twice, it is necessary to decide what to use.

      Find a response on the scale D. Read the value of the D, to which the pointer is hovering. Add to it "x10 n". For counting N, take the starting degree 10, round down to the nearest even number and divide by 2.

      • Example 3: The corresponding value of the scale d at a \u003d 3.9 will be 1.975. The initial digit in the exponential representation had 10 2. 2 already even, so just divide into 2 to get 1. The final answer will be 1.975 x 10 1 \u003d 19,75 .
      • Example 4: The corresponding scale value d at a \u003d 7.1 will be 8.45. The initial figure in the exponential representation had 10 3, so rounded 3 to the nearest even number, 2, and then divide to 2 to obtain 1. The final answer will be 8.45 x 10 1 \u003d 84,5 .

   3. Similar way to remove cubic roots on the K. scale The process of extracting the cubic root is very similar. The most important thing is to determine which of the three scales K should be used. To do this, divide the number of numbers of your number to three and find out the residue. If the residue 1, use the first scale. If 2, use the second scale. If 3, use the third scale (another way - to be repeatedly considered from the first scale to the third until you reach the number of numbers in your response).

      • Example 5: To remove the cubic root of 74,000, it is necessary to calculate the number of numbers (5), divide it to 3 and find out the residue (1, residue 2). Since the remainder 2, we use the second scale (you can also count on the scales five times: 1-2-3-1- 2 ).
      • Move the cursor to 7.4 by the second scale K. The corresponding value on the scale D will be approximately 4.2.
      • Since 10 3 less than 74,000, but 100 3 more than 74,000, the answer must be within 10 to 100. Move the decimal point to get 42 .
   • The logarithmic line allows you to calculate other functions, especially if it has a scale of logarithms, a trigonometric calculation scale or other specialized scales. Try to deal with them yourself or read the information on the Internet.
   • You can use multiplication method for conversion between two units of measurement. For example, since 1 inch \u003d 2.54 centimeters, the task "convert 5 inches to centimeters" can be interpreted as an example of multiplication of 5 x 2.54.
   • The accuracy of the logarithmic ruler depends on the number of distinguishable scale marks. The larger the length of the line, the higher its accuracy.

Logarithmic ruler or counting ruler - a computing device that allows you to perform several mathematical operations, including multiplication and division of numbers, the construction of a degree (most often in the square and cube) and the calculation of square and cubic roots, the calculation of logarithms, potentiation, the calculation of trigonometric and hyperbolic functions and other operations. Also, if you split the calculation into three actions, then with the help of the logarithmic line you can build the numbers to any actual degree and extract the root of any actual degree.

Do not scare! You do not need to calculate the foundations and logarithms, cosine and arctanges daily. In most cases, logarithmic rules embedded in hours are not equipped with scales to calculate the values \u200b\u200bof trigonometric functions.

A number of watches are equipped with computational rules whose functions are close to everyday life.

By the way, the first came up with putting a logarithmic school in Mark Carson's watch - the head of the theoretical department in the Nuclear Center, USA.

So hours Citizen Promaster Sky - Already on the designations on a separated scale, it is clear that they are well adapted to calculate fuel consumption during car travel or travel on a motor boat.

Let's start with the simplest. The circular logarithmic ruler consists of a ruler on the nearest and ruler on the dial. Turn the bezel before combining the value on the Berli line with the desired mark on the dial.

In order to divide 150 by 3, follows the number 15 (\u003d 150) on the outer scale to establish a number 30 (3) on the inner scale. The result is counted on the internal scale opposite "10" and is 50.

On the Internet you can find an example Triple Rules, or calculating the rate of reduction using a circular computing ruler on the clock.

The pillar in the glider, which is at an altitude of 3300 meters, determines that it loses height at a speed of one meter per second, i.e. 60 m per minute. How long does he have time to end the flight? In order to know the answer, you should set the number 33 (\u003d 3300) on the outer scale against Numbers 60 on the inner scale. The result is against the sign "10" on the internal scale and is 55 minutes.

But we will leave in the aforemention of aviation tasks and apply this rule to calculate in a closer area. What distance do you have enough 40 liters of gasoline at a fuel consumption of 8 liters per 100 kilometers? We establish the number 40 opposite the number 8. We obtain 50, taking into account the scale of 1 to 10 - by 500 km.

At various hours there are many designations that facilitate the recalculation of length of length.

Stat means english mile Naut. - marine mile, M. - American mile, and on the clock CITIZEN PROMASTER SKY - KM - As in Latin, and Russian transliteration means kilometers.