Chord (math)

Definition of the chord function on the unit circle, for comparison sine function

Chord (from chorda , lat. Tendon ) is a mathematical function that is uncommon today, which maps an angle to the corresponding chord length on the unit arc .

calculation

Like the trigonometric functions in use today, chord also relates a length ratio on a circle with any radius to an angle, namely the ratio of chord length and circle radius to the angle that the center of this circle includes with the end points of the chord. The following applies: ${\ displaystyle s}$ ${\ displaystyle r}$ ${\ displaystyle \ alpha}$

{\ displaystyle s / r = \ operatorname {chord} (\ alpha) = 2 \ cdot \ sin (\ alpha / 2)}

Nowadays it is used very rarely, corresponding places are expressed with the sine function according to the above equation . ${\ displaystyle \ operatorname {chord}}$

As for the angle functions commonly used today, tables were previously used for the chord function , in which values of the chord function value that were precalculated in degrees for certain angles could be looked up and vice versa. ${\ displaystyle \ sin, \ cos, \ tan}$

A chord board can be put together with the above relationship between and : ${\ displaystyle \ operatorname {chord}}$ ${\ displaystyle \ sin}$

α	0 °	10 °	20 °	30 °	40 °	50 °	60 °	70 °	80 °	90 °	...	180 °
chord (α)	0.0000	0.1743	0.3473	0.5176	0.6840	0.8452	1.0000	1.1472	1.2856	1.4142	...	2.0000

Such chordal tablets were already known in ancient times. The first is attributed to the Greek astronomer Hipparchus of Nicaea , its division was 7.5 °. In addition, proportional circles were provided with chord scales , which allowed simple geometric construction. Chord scales were used in land surveying until the 19th century to improve the measuring accuracy of the catoptric compass , which works on the same principle as the mirror sextant .

Individual evidence

^ Hans-Joachim Vollrath: Historical angle measuring devices in projects of mathematics lessons . In: Mathematics Lessons . tape 45 , no. 4 , 1999, p. 42-58 ( PDF ).

^ Nicholas Bion: Traité de la construction et des principaux usages des instrumens de mathématique . In: Jombert . Paris 1709. Translated reprint: N. Bion: The Construction and Principal Uses of Mathematical Instruments: Including Thirty Folio Illustrations of Several Instruments . Astragal Press, 1995, ISBN 1-879335-60-3 .

↑ Dr. Klöffler, Martin: Surveying in the training and practice of Prussian officers in the early 19th century , in: Brohl, Elmar (ed.), Military threat and structural reactions - Festschrift for Volker Schmidtchen, German Society for Fortress Research e. V., Marburg (2000), ISBN 3-87707-553-3

↑ Alto Brachner (ed.): GF Brander, 1713-1783: scientific instruments from his workshop . Deutsches Museum, Munich 1983, p. 131 ff .

[Vollrath-1] Hans-Joachim Vollrath: Historical angle measuring devices in projects of mathematics lessons . In: Mathematics Lessons . tape 45 , no. 4 , 1999, p. 42-58 ( PDF ).