Slide rule

A slide or slide rule is an analog computational tools for the mechanical- optical implementation of basic arithmetic, preferably of multiplication and division. Depending on the design more complex arithmetic operations (including root, square, logarithmic and trigonometric functions or parameterized conversions ) can be performed. The principle of a slide-rule consists in the addition or subtraction of distances, which are as a logarithmic scale on the fixed and the movable part of the slide rule. Until the widespread use of the calculator slide rule were indispensable for many calculations in School, Science and Technology.

• 3.1 designs
• 3.2 The scale systems 3.2.1 System Mannheim
• 3.2.2 System Rietz
• 3.2.3 System Darmstadt
• 3.2.4 System Duplex

• 3.3.1 Slide Rule for surveying calculations
• 3.3.2 Electric Slide
• 3.3.3 Other

• 4.1 The scale
• 4.2 The rotor
• 4.3 Accuracy and decimals
• 4.4 multiplication
• 4.5 Division
• 4.6 proportions
• 4.7 magnitude 4.7.1 squares
• 4.7.2 cubes

• 4.8.1 square Root
• 4.8.2 cube root

• 4.11.1 sine
• 4.11.2 cosine
• 4.11.3 tangent
• 4.11.4 radians

History

Development of logarithms

The history of the slide rule is based on the development of logarithms. Although there are Indian sources from the 2nd century BC, in which already logarithms to the base 2 were mentioned, it was the Swiss watchmaker Jost Bürgi ( 1558-1632 ) and the Scottish mathematician John Napier ( 1550-1617 ), the at the beginning of the 17th century the first known system of logarithms developed independently.

The Greek word " logarithm " means in German ratio and comes from Napier. Were first published logarithms of this in 1614 under the title Mirifici logarithmorum canonis descriptio, what can be translated description of the wonderful canon of logarithms.

After the Oxford Professor Henry Briggs (1561-1630) was intensely occupied with this work, he took the author to contact and suggested for the logarithms of the base 10 to use ( " briggssche " or " decadal " logarithms ). This spread quickly and were particularly appreciated in astronomy.

With the logarithms of the mathematical basis for the development of the mechanical slide-rule was set, since the operation of the slide-rule is based on the multiplication and division, on the principle of addition or subtraction of logarithms.

Development of the slide rule

Even in 1624, ten years after the recognition of the existence of logarithms by John Napier of Merchiston, gave the English theologian and mathematician Edmund Gunter ( 1581-1626 ) for the first time his basic thoughts about the logarithmic numbers known. With his development of the " Gunter scale ", a bar with logarithmically arranged scale, one could initially only be carried out with the help of compasses addition and subtraction calculations by thumbed the logarithmic distances. However, the calculation with the compass was very complicated and labor intensive. Groundbreaking was the idea of the Englishman William Oughtred (1574-1660) in 1632, to be used in place of compasses two congruent logarithmic scales straight or circular, so the latter than the actual inventor of the slide rule applies.

Even Robert Bissaker (1654 ), who built a slide rule with movable tongue between two scales, and in 1657 Seth Patridge ( 1603-1686 ) contributed to the development of the slide rule in. From Patridge the idea of ​​also logarithmically scaled tongue which can be moved against the rod body, which calculations could be carried out much easier derived.

In 1722 Warner used first square and cube scales. The Isaac Newton (1643-1727) invented runners - also called indicator - was implemented in 1775 by John Robertson ( 1712-1776 ). He remained, however, over a hundred years in oblivion. This extremely practical development made ​​possible by their cross the line mark the union of two non-contacting scales, thus increasing the accuracy of the tongue setting or reading.

The - usually located on the back of the tongue - double logarithmic Exponentialskalen were invented in 1815 by the English physician and lexicographer Peter Mark Roget ( 1779-1869 ). They are useful for simplifying Exponentialaufgaben any kind

Of great importance in the history of the slide rule comes to the French Amédée Mannheim (1831-1906), who first suggested a common structure for the slide rule in 1850. With this new standard slide rule named " Mannheim " also since this version transparent runner was established. By the end of production of the slide rule, numerous other improvements were introduced.

The end of the slide rule

The invention of the electronic calculator in 1969 sparked a boom in the development of this new product in high demand computing instrument. In 1972 came with the HP -35 from Hewlett -Packard, the first technical-scientific calculator with trigonometric, exponential and logarithmic functions to the market. Calculators are more versatile, more accurate and more convenient to handle than slide rule. In addition, the calculator could always be purchased cheaper by the increased production due to the immense demand. The schools started around ten years later in the Federal Republic in 1975, in the GDR, use the electronic calculator instead of the mechanical slide rule, which ultimately meant the end for the once considered crucial slide rule and thus also for its manufacturer. Therefore, the slide rule is now only of historical significance - only a few young people know him, hardly anyone knows more to deal with it.

However, some prefer the slide rule before the calculator, since they are already counting since school time and can perform calculations quickly and so well versed. But there are also individuals, which retain slide rule for nostalgic reasons, and collectors whose passion it is to swap rare models or to acquire it - mostly through contacts or Internet auction houses. In Germany, but also at the international level, such as through the existing since 1991 " Oughtred Society ", meeting of slide rule collectors are held regularly in which these their range and their knowledge can expand on the slide rule. Also very rare specimens are occasionally exhibited there that achieve prices of over $ 100,000 among collectors. The most sought-after slide rule, the four original slide rules by William Oughtred, each of which about $ 250,000 worth, are all located in museums.

Manufacturer of slide rules

In the Federal Republic of Germany slide rules, for instance by Aristo ( Dennert & Pape ) in Hamburg, A. W. Faber Castell in Stein near Nuremberg, Nestler in Lahr, IWA produced at Esslingen. In the GDR, it was the Company Reiss (later VEB measuring and drawing equipment manufacturing ) in Bad Liebenswerda and the Meissner KG in Dresden. Both companies were merged in 1974 to VEB Mantissa in Dresden. Well-known foreign manufacturers of slide rules were the U.S. has Keuffel & Esser (New York), Pickett and mail. In Japan, slide rules were produced at Sun Hemmi, who produced numerous slide rule for the American company mail; in France and in the UK at Graphoplex Bludell - Harling. In addition there exist numerous other lesser-known companies at home and abroad.

Applications

In the first two hundred years after its invention of the slide rule was used very little. It was not until the late 18th century its meaning from James Watt was newly detected.

Because of technical progress in the time of the Industrial Revolution of the slide rule was a much-used tool for engineering and scientific calculations. In the 1950/1960er years he was regarded as the symbol par excellence of the engineers, similar to the stethoscope at the doctors.

In addition to the school slide rules, which in everyday life found their use in the classroom and with simple calculations, also many special slide rules were produced, often in very specific areas, such as in aerospace, electrical and mechanical engineering, chemistry, the military or the commercial were used. Aluminum slide the brand Pickett were carried on the Apollo space missions, including on flights to the moon.

Construction

Designs

A slide rule consists of a body on which usually several parallel scales are attached, a movable tongue with similar own scales and a sliding body on the runner with a slash mark. By displacement of the scale relative to one another, the computing operation is carried out and read from the corresponding position value. The runners marking allows you to set values ​​between scale ticks and also reading the spaced parallel scales that do not touch directly on the edges of body and tongue.

The standard scale length - measured from the mark "1" up to " 10 " - the slide rule models is 25 cm; small versions (eg pocket models ) have a scale length of 12.5 cm, office or table-top models of 50 cm.

The first slide rules were made of wood, bamboo, brass and ivory, later there were further developments with the use of aluminum and plastic.

Variants of the slide rule are

Since you can not directly add and subtract with the slide rule, there are also designs that have on the back of a number cruncher ( Griffeladdierer ).

The scale systems

System Mannheim

The French professor of mathematics Amédée Mannheim (1831-1906) developed in 1850 a scale selection and arrangement of slide rules, the learned first a large and vendor-neutral distribution. With this new standard slide rule, which consisted of the basic scales C and D, the square scales A and B and the sine and tangent scale, since this version also the transparent runner was established. The use of the sine and tangent scale on the back of the tongue, the tongue must be turned. Slide rule with this system have been described by many manufacturers as a school computer.

System Rietz

The German engineer Max Rietz (1872-1956) added in 1902, the system Mannheim to the cubic scale K and L. Partial Mantissenskala was added a reciprocal scale. Turning over the tongue for use of the sine and tangent scales could be some models avoided by the introduction of index lines on the back of the body. At some later models the sine and tangent scales have been moved to the front side of the rod body. Occasionally, the translated fundamental scales CF and DF were used. The Rietz system was until the end of the slide rule production one of the most widely used scales arrangements.

System Darmstadt

The German mathematics professor Alwin Walther (1898-1967), developed in 1934 at the TH Darmstadt a new scale selection and arrangement. The front was supplemented by a Pythagorean scale P. The sine and tangent scales have been moved to the front side of the rod body. In order that the tongue back for three Exponentialskalen LLn was free. These improvements came to meet the needs of engineers. The system remained until the end of the Rechenschieberära next to the System Rietz in the offer of the manufacturer.

Duplex system

The so-called duplex slide rule was invented by William Cox in 1891. This designation is not a standalone system, but the name for a slide rule with two-sided arrangement of the already known scales. Especially in the 1950s brought several manufacturers duplex models on the market. This also further reciprocal of scale and root scales were introduced. There are bars with over 30 scales.

Special slide rule

Special functions can be scaled to any slide, making it suitable for special technical applications for which you would otherwise need table books.

Examples are:

Slide rule for surveying calculations

In surveying calculations, trigonometric functions play an important role. Corresponding slide rule not only scales for the elementary functions sin?, Cos, tanα, but also for more complex functions such as cos ² α, cos sin?, 1/tan ( α / 2). There are often counted in surveying angle not in degree but in Gon with decimal division (90 ° = 100 gon ), there is such a slide rule (eg " Aristo - geodesic " ) in Gon - design, so that here the argument of all trigonometric functions in Gon is set.

Electric Slide

The so-called electric slide rule represent a further development of the system Rietz, which specifically meets the requirements in electrical and mechanical engineering. However, the arrangement of the scales may differ from the system Rietz. In some models, the sine scale S does not correspond to the basic scale C, but with the square ruler B. In addition, two LL2 and LL3 Exponentialskalen are on the electric grid bars attached. Especially for applications in electrical and mechanical engineering have such a slide rule on special scales for the calculation of efficiencies and the voltage drop in copper pipes as well as special brands that facilitate the conversion of PS in kW and the calculation of the resistance of copper lines.

Other

There are also slide rule for other special applications, eg for selecting bearings or belts in mechanical engineering.

• Examples of special slide rule

Machine tool operating parameters

Humidity indicators

Operation

The scales

On a bar or slide are several (usually logarithmic ) scales, each with a specific function. The capital letters used in the table correspond to the usual designation on most modern slide rule. Each slide has usually only a selection of the enumerated scales. Since there are a plurality of scale systems, the indication of the position of the scale is not always valid in general. In addition, the individual systems are not always clear; just the additional scales were arranged differently by each vendor.

In general, the scales from the left point to the right ascending values ​​. Scales, decreasing from left to right, are usually labeled in red.

The rotor

The sliding runners not only serves for exact reading and setting the various scales. He often has additional runners strokes that allow a simplified direct calculation. The short runner dashes at the top left or bottom right are used in conjunction with the main line for the calculation of circular areas. Some models have additional markings for converting KW into HP or for direct calculation by a factor of 360 (eg for calculation of interest ).

Precision and decimal places

The accuracy with which can be set or read a number that depends on the size of the slide rule. In a 30 -cm-long slide rule can be set or read the numbers on the bottom scale C and D with an accuracy of two to three decimal places directly. Another decimal point can be estimated with some practice. Larger slide rules have a finer subdivision of the scale and thus enable a more accurate statement.

Since the actual distances numerically equidistant scale marks vary according to the logarithmic scale, you can set larger numbers absolutely less accurate or read than smaller numbers. However, the relative inaccuracy, ie the ratio of the inaccuracy of a number to the number itself is the same for all numbers. Therefore, in several successive multiplications is not only the result but also the accuracy is independent of the order in which the individual steps of multiplication are performed.

The slide rule, however, does not indicate the magnitude of a number. Thus, e.g. the reading is 6, both 6; 60; 600, as well as 0.6; 0.06; 0,006 so mean. The position of the decimal point is determined by a rough calculation. This is indispensable for the correct application of the slide rule.

Multiplication

The multiplication is the simplest and most primitive of calculation of the slide rule. It is based on the calculation rule that the logarithm of the product can be determined by the sum of the logarithms of the individual factors.

Since the scale C, and D are separated logarithmically on the slide, obtained by the addition of two geometric paths to these scales, a total of two logarithms. First, the initial flag is "1" of the movable scale C is pushed ( in the mouth ) by the first factor to the fixed scale D. The rotor is then slid over the second factor on the C scale. The result is read at this point on the scale D.

However, where the product is 10, it can not be read in the manner described this. You imagine now that you attach a second virtual D scale at the end of the first. This corresponds to a shift of the C- 10 on the first scale factor of the D scale. The product can then be read in the second factor of the C scale to D by means of the rotor. This procedure is called " pushing through " or " setback" of the tongue.

By the same method can be used for the multiplication of the scales A and B. This is very handy when one of the factors is a square number or if you want to pull a root of the product. For the simple multiplication, this approach is rather unusual, since one obtains a lower accuracy due to the greater division of scales A and B.

Division

The division is the inverse of the multiplication. It is based on the calculation rule that the logarithm of a quotient (numerator divided by the denominator) equals the difference between the logarithm of the dividend (numerator) and the logarithm of the divisor is ( denominator).

First, the divisor is on the movable scale C pushed ( on the tongue ) on the dividends on the fixed scale D. The rotor is then pushed onto the top marked " 1" on the C scale. The result is read at this point on the scale D.

Falls below the ratio is 1, you can read the result alternatively at the end mark " 10" on the movable scale C.

Same method can also be used for the scales A and B, the division; However, in this case, the accuracy is lower.

Another possibility consists in that multiplying the dividend by the reciprocal of the divisor. It is to proceed exactly as for the multiplication, with the only difference that you can use instead of the tongue scale C, the inverse scale CI.

Proportions

The ratio between the values ​​on the scales C and D or A and B is always the same setting with unchanged tongue.

Thus, the slide rule is very well suited for proportional invoices or for three-set tasks. It is helpful, before multiplication to perform the division, since the task can then usually be calculated with a single tongue adjustment.

A major advantage of the slide rule is three calculations is that not only the result for the second factor, but at the same tongue setting for any number of other factors can be read.

An example of the table learning: It will convert English yards to meters and vice versa. It is the ratio of 82 yards, 75 yards. For this purpose, it sets the value of the movable scale 75 on the value of C 82 on the fixed scale D. Now, there is any value of yards on the D scale reading the appropriate meter number on the C scale. Conversely, one can read off the corresponding yardage on the scale D for any values ​​of m on the C scale.

Potencies

Square numbers

For the square scales A and B, the following relationship applies, ie they have two decades ( from 1 to 10 and 10 to 100 ) in the area of ​​basic scales decade ( 1-10 ).

The squaring is performed by the transition from the scale C or D on the scale B, or A, where advantageous, the average runner bar is used. It provides, for example, the rotor bar about the value on scale D and reads on scale A from the square.

Some slide rules exist on the runner a short add-on bar on the square scales A and B, which is offset π / 4 around the track. With the help of this additional stroke the circular area can be read directly on the scale A or B, if the pitch diameter is set to the middle line of the rotor on the scales C or D.

Cubes

For the cube or cubes scale K, the relation holds, ie she has three decades ( 1 to 10, 10 to 100 and from 100 to 1000) in the field of basic scales decade.

Determining the cubic number is determined by the transition from the D scale on the scale K, where advantageous, the average runner bar is used.

Root

Square root

To find the square root of a number whose value is between 1 and 100, provides this figure with the rotor on the scale A or B and reads the result from the basic scale D and C respectively.

Is the square root to determine a number whose value is not between 1 and 100, the radicand is split into two factors, one factor is a power of the base 100 and the second factor is in the range between 1 and 100. According to the formula one can separately take the square root and multiply the results of each factor.

One can also apply the following rule of thumb: All numbers greater than 1 with an odd number of digits before the decimal point and all numbers less than 1 with an odd number of zeros after the decimal point of the scale A are in the left- decade ( 1 to 10 ) is set. All numbers greater than 1 with an even number of digits before the decimal point and all numbers less than 1 with an even number of zeros ( 0 is also an even number) after the decimal point are in the right decade ( 10 to 100 ) set the scale A.

Cube root

To find the cube root of a number whose value is between 1 and 1000, represents this figure with the rotor on the scale K and reads the result on the basic scale from D.

The cubic root of a number to identify the value is not between 1 and 1000, the radicand is split into two factors, one factor is a power to base 1000 and the second factor is in the range between 1 and 1000. According to the formula one can separately take the square root and multiply the results of each factor.

Reciprocals

The inverse scale CI or DI match in appearance the basic scales C and D, but run in the opposite direction. Therefore, they are usually colored red. These scales may be used for different processing options.

Is the reciprocal to identify a number of times, this figure with the rotor on the basic scale and reads the return value directly on the reciprocal scale from.

Using the reciprocal value of scale can be replaced by the calculation of a multiplication, a division, and vice versa. The following applies: A number is multiplied by dividing by the reciprocal. Thus one can identify products of several factors with less tongue settings.

Compound multiplications and divisions can be calculated with the inverse of the scale efficient.

For other uses of the reciprocal of scale arise in the trigonometric functions and exponential calculations.

Logarithmenbestimmung

The linear split logarithms or Mantissenskala L contains values ​​for the mantissa ( decimal ) of the logarithm.

To find the logarithm of a number, you provide them with the runners on the base scale D and reads the mantissa on the Mantissenskala L from. The reference mark ( decimal point ) of the logarithm is obtained, as well as in the application of logarithms, in numbers greater than or equal to 1 from the number of digits before the decimal point minus 1 For numbers less than 1, the number of zeros after the decimal point is determined. This number is set to negative and decreases by 1. As a rule of thumb: This corresponds to the number of places by which the comma must be postponed until exactly one different from the zero digit appears before the decimal point. A left shift is considered positive, a right shift is negative.

The Logarithmenbestimmung is mainly used for the calculation of powers and roots of arbitrary exponent. However, since the final accuracy is significantly affected even by small inaccuracies in the determination of the logarithm, this method is used only for rough calculations.

Trigonometric values

For all trigonometric functions applies: If an angle given is greater than 90 °, so it must only be attributed to an acute angle, which provides the same function value.

Sine

The sinusoidal scale S is divided decimal and, in conjunction with the base scale D is the angle function, or in the reverse reading of the angle.

The sine value for angles between 5.7 ° and 90 ° can be determined by adjusting the rotor the number of degrees on the sine scale S and the function value is reading on scale D.

Sine values ​​for angles smaller than 5.7 ° can be personalized with the Bogenmaßskala ST determine (see below).

Cosine

The sine scale S is usually additionally labeled with red angle values ​​increase from right to left. These values ​​, in the interval from 0 ° to 84.3 °, allow the complementary angle is and allow the reading of the cosine on the basic scale D. Conversely, determine the corresponding angle.

Tangent

For determining the tangent values ​​to use the scale T1 and T2, wherein T1 is for angular values ​​between 5.7 ° and 45 ° and T2 used for the angular values ​​between 45 ° and 84.3 °. The reading of the tangent value is indicated on the basic scale D. Conversely, determine the corresponding angle.

The cotangent can be read on the reciprocal scale DI.

Tangent values ​​for angles smaller than 5.7 ° can be personalized with the Bogenmaßskala ST determine (see below).

For angle values ​​between 84.3 ° and 90 ° to determine the complementary angle and sets it on the Bogenmaßskala ST. After the relationship you can read the tangent value of the reciprocal scale DI.

Radians

The ST is divided Bogenmaßskala decimal and, in conjunction with the base scale D radians or in the reverse reading of the angle.

The radian measure for angles between 0.57 ° and 5.7 ° can be determined by adjusting the rotor the number of degrees on the Bogenmaßskala ST and reads the radians on the basic scale D.

For angles less than 5.7 °, the relation holds. That radians is approximately equal to the sine function and the tangent function. The deviation is less than 1.5 ‰ here. Therefore, this scale is used to determine the sine and tangent values ​​for small angles.

General power calculations

The three Exponentialskalen LL1, LL2, LL3 strung together provide the natural logarithm of the function values ​​from 1.01 to 50,000 dar. With the help of these scales to any powers, roots and logarithms can be calculated. The Exponentialskalen are important economies of scale, ie its decimal equivalent of the endorsed label and is not, as in the basic scale variable.

Addition and subtraction

An addition or subtraction is not directly possible with conventional slide rules. However you came to reach your goal by the transformation of the addition problem into a multiplication problem.

A suitable formula is derived by excluded. It is

This task can be solved by the application of division and multiplication with the slide rule. The necessary as an intermediate result adding or subtracting the number 1 can be done in your head. This type of calculation is certainly not trivial and hardly plays a role in the use of the slide rule.

When slide rules with linear scales, the addition and subtraction can be executed directly.

Watches

There are still watches that are equipped with a slide rule, about Breitling, Sinn, Casio or Citizen.