Table of logarithms#Tables of logarithms
Logarithms is called a tabular representation of the mantissas of logarithms. Logarithms were for centuries an important computational tool, especially in the natural and engineering sciences. Many calculations in the school mathematics, such as the drawing of difficult roots could only be carried out with their help. The invention and widespread use of calculators and computers has made the use of logarithmic tables, similar to the slide rules, practically completely useless within a few years.
Most tables were the common logarithm ( base 10) in resolution from 1.00 to 9.99.
History
On the history of logarithms, see the main article logarithm: History.
John Napier was with his work Mirifici Logarithmorum Canonis Descriptio 1614 was the first one out of logarithms and is considered the inventor. Napier was already on trigonometric functions. In an appendix Constructio Napier remembered to take a firm basis what his friend Briggs did soon.
Jost Bürgi, was involved in the introduction and development of the decimal numbers that were necessary for the practical calculation, and calculated independent of the first logarithmic tables Napier 1603-11. Kepler urged him several times to make it public, but not until 1620 Arithmetic and geometric Progresstabuln, to Napier, happened. As an employee of Johannes Kepler he used the logarithm tables created for astronomical calculations. These panels were numerically.
Henry Briggs in 1624 led a unified basis, the ten, one, and could his board, here are the logarithms of the numbers from 1 to 20,000 and 90000-100000 to 14 sites were listed exactly not finish itself. It was 1627/28 fully published by Dutch publishers Adriaan Vlacq and Ezechiel de Decker in the Netherlands. The Vlacqschen panels contained relatively low 603 error. They displaced the Napier completely and left for Kepler Chilias logarithmorum 1624 no interest to pay.
Tables were calculated by exponentiation. Only after the invention of calculus, more and more convergent series offered for calculation.
William Gardiner 1773 brought a board with exact values out.
Man with Nicholas Mercator had the opportunity to series (1668 for ln (1 x)) to be used to calculate, but it still took over 100 years to Jurij Vega 1783 his Thesaurus logarithmourum completus flawlessly brought out, which was the most well-known board and for almost all niederstelligeren the foundation was formed.
Carl Bremiker improved the Vegaschen panels.
With the panels was expected mainly to the 1970s.
Use of logarithms
Logarithms allow it to return the multiplication and division of numbers on the simple addition and subtraction. Before there were mechanical or electrical calculators, logarithmic tables facilitate the computing immensely. So logarithmic tables were indispensable companion in school, among other things in mathematics and physics senior classes.
The product of two numbers a and b is due to the Logarithmengesetzes
Computed by the logarithm of the number a to the base and that of the number x b x looked up in the table to the base. The sum of the two logarithms is formed and searched in the table. The resulting sum as the logarithm of this number is then the product of a and b.
With the help of a table of logarithms, arithmetical operations can be traced back to the next simpler operation: multiplication to addition, division to subtraction, exponentiation to multiplication and square root ( square root ) on Division. These returns are based on the following Logarithmengesetzen:
Build a table of logarithms
The most common were three -, four - and five-figure logarithms. The greater the accuracy of a panel is to be, the greater is their scope. In school were up in the 1970s usually four digit logarithms in use.
Simple three-digit logarithmic tables are constructed so that the first two digits ( ie 10 to 99) form the left of the table, while the third digit (0 to 9) is used as the column heading.
The number range from 1.00 to 9.99 is sufficient when using logarithms to base 10 Indeed, it can calculate the logarithm of 10 times, 100 times, etc. a number calculated by multiplying the integer part is modified according to the number of points ( number of integer digits minus 1). See the Logarithmengesetz of multiplication:. Example: The logarithm of the 1- digit number 2 is about 0.30103; that of the 2- digit number 20 is 1.30103; the logarithm of the 3- digit number 200 is 2.30103, etc. For numbers smaller than 1 applies accordingly: and.
Logarithms to numbers with four valid Sections can be determined by linear interpolation.
Since logarithms were considered daily used tools, they have often been enriched by additional information. Were recorded formularies for example from geometry and trigonometry, data collections, for example on the body that make up our solar system, as well as mortality tables and many others as examples of demographic data collections
Generating a table of logarithms
Logarithms were determined from a list of values of the inverse function, the exponentiation, by interpolation.
P.P. tablet
The panels are Interpolationstafeln added to a linear interpolation. P.P. represents partes proportional and linear interpolation.
Panel cut-out from the decadic ( base ( b ) is 10) logarithm, humerus ( the numerical value ( a) ) to the left and above, mantissa (meaning here the decimal (x ) are ) right for five-figure logarithms. The decimal places are divided into groups of two and threes, right are the last three digits. In other panels are like here, for example, 82 is not repeated but written down only once in the column, and only when it increases to 83, including written in the column:
P.P panel:
If you want here determine an interpolated mantissa for the number 66108, you have eight times the tens part of the panel difference 7 (horizontal difference between the table values ) add, so 5.6 or 0.000056 and would then rounded m = 4.82026.
If you want to add one more point, you take parts of the table difference divided by 100 instead of 10, where only the last digit should be rounded. For the six-digit number N = 6613.78 in the first step 4.2 in the second 0.48 and then receives five digits m = 82040 4.2 0.48 = 82045, ie 3.82045.
If you have a four-digit number M = 82116 ( 3.82116 ) between M = 82112 and M = 82119 N must be between N = 6624 and N = 6625. The Tafeldiffenz is 7, the additional 4 of the mantissa is found most likely in the table, at 3.5 so the number is 6624.5, 4.2 to round them off would mean 6624.6. 3.5 one can again enlarge 0.49 which meant in the table 0.07, ie, the number N is called after all 6624 0.5 0.07 = 6624.57 what rounding up to 6624.6. As one easily checks using the calculator.
As you can see, the differences are boards for 7 and 6 indicated, as both occur in the panel, 027-033, six, and then come back seven, 033 to 040
Trivia
Logarithms played a role in the discovery of the Benford law. The side with the 1 as the leading digit is often needed as the other digits and therefore wore out faster.
Known issues
- Vega - Bremiker, 7-digit logarithms and trigonometric functions, 1795
- Wilhelm Jordan ( surveyor ), logarithms and tables