Panel interval
In addition to the number of decimal places, the table interval of tabulated calculation aids is the most important key figure in tables, for example angle function or logarithm tables, astronomical ephemeris or auxiliary tables for the earth ellipsoid .
The step size of the argument with which one enters the table is called the table interval . For example, this is the angle itself for trigonometric functions and the date for precalculated planets .
Efficiency of use
A balanced relationship between the interval and the number of digits in a table is essential for the efficiency and speed of looking up, because you have to interpolate between the tabulated values for greater accuracy . The only exception are three- or four-digit tables from which you can take each value directly .
If there are more than 5 decimals, the table interval needs to be carefully considered if the numbers are not to fill several volumes. If the interval is too large, the linear interpolation between the columns is no longer sufficient , so that the user has to switch to the time-consuming quadratic interpolation . If, on the other hand, the table interval is too small, the size of the table or the volume of the book grows rapidly - up to the point of unusability or rapid wear and tear of a book that is too thick .
The 7-digit Vega-Bremiker may serve as an example of a very efficient and balanced table . This logarithmic table , which was published in over 100 editions from 1795 to around 1960, logarithmic - trigonometric tables , along with other tables and formulas set up for use in mathematics, was calculated for military technology by the Slovenian-Austrian officer Baron von Vega in 1793-97 and quickly spread to the various countries Subjects and applications.
Numerical example from the "Vega-Bremiker"
The trigonometric part of the "Vega-Bremiker" (parts II and III) contains the trigonometric functions sine and tangent , namely in 2 steps:
for angles from 0 ° to 5 ° in the table interval 1 " ( arcsecond ) and from 0 ° to 45 ° (as a result of the co-functions de facto up to 90 °) in the interval of 10 " . How wisely this choice was made over 200 years ago is shown by some values of the logarithms (increased by 10):
log sin Tafeldiff. log tan Intervall 1" und 2°00'00" 8,542 8192 (603 604) 8,543 0838 Tafeldifferenzen von 600: 2 00 01 8,542 8795 603 603 8,543 1442 zur Interpolation auf 0,01" 2 00 02 8,542 9397 602 603 8,543 2045 genügt Rechenschieber, 2 00 03 8,543 0000 603 604 8,543 2649 für 0,1" kurze Kopfrechnung.
4°00'00" 8,843 5845 (301 302) 8,844 6437 Die Tafeldifferenz ist 4 00 01 8,843 6146 301 303 8,844 6740 nur mehr halb so groß, 4 00 02 8,843 6447 301 302 8,844 7042 deshalb bei 5° Übergang 4 00 03 8,843 6748 301 303 8,844 7345 auf 10" Tafelintervall:
6°00'00" 9,019 2346 (2004 2026) 9,021 6202 Intervall 10": 6 00 10 9,019 4348 2002 2025 9,021 8227 Bis 45° sinken die Tafel- 6 00 20 9,019 6350 2002 2024 9,022 0251 differenzen auf 210 u. 420, 6 00 30 9,019 8351 2001 2023 9,022 2274 sind also noch sinnvoll.
Theoretically, one could make several gradations and thus reduce the volume of the book (which is 4.5 cm thick) somewhat - for example
0 - 5° Tafelintervall 1" (wie oben), 5 - 10° Tafelintervall 5" (statt 10" wie oben) 10 - 25° Tafelintervall 10" 25 - 45° Tafelintervall 20".
The table differences between which each has to be interpolated would thereby become more uniform - e.g. B. for the sine (2-45 °) in the range 250-900 (instead of Vega-Bremiker 210-2400) .. and the book about 15% thinner. However, this small advantage would result in a large increase in calculation errors, because the manageable gradation ( 1:10 ) would be replaced by several non-round steps (1: 5: 10: 20).
Today's meaning of table works
Since the advent of the electronic pocket calculator in the 1970s, the above logarithm tables have lost much of their importance, but similar tables are important for various functions such as harmonic spherical surface functions , elliptic integrals or for solving transcendent equations . They are also of lasting importance as auxiliary tables for complex technical and structural tasks, etc.
In the history of mathematics and the history of technology , the optimal choice of panel intervals has been an important task in preparing various calculations. Even today, every Astronomical Yearbook shows the experienced user whether this choice has been given sufficient attention.
Variable speed of the planets
The ephemeris (projections) of the 5 bright planets Mercury to Saturn for the year 2008 serve as an example . These freely visible planets have orbital times between 0.24 and 30 years. In the usual table interval of 10 days, Mercury moves up to 20 ° further in the starry sky, Venus and Mars by around 10 °, Jupiter and Saturn only by a maximum of 2.5 ° and 1 ° respectively. Therefore it makes sense to adapt the table intervals to this speed.
The German calendar for Sternfreunde only does this for Mercury (5-day intervals), while the other four planets are tabulated at 10-day intervals. A very practical yearbook, the Austrian sky calendar , on the other hand, uses time quite useful, because you usually only have to interpolate in quarters of an interval and this is easier in your head than with tenths.
swell
- Vega-Bremiker , logarithmic-trigonometric manual . 100th edition, Weidmannsche Verlagsbuchhandlung, Berlin 1959.
- Th.Neckel, O.Montenbruck, Ahnerts Astronomisches Jahrbuch 2008 . Stars and Space Publishing House, Heidelberg 2007.
- H.Mucke, Himmelskalender 2008 . Austrian Astro Association, Vienna 2007.