The Cord Table in the Ancient Greek Times

Transcription

1 The od Table in the ncient Geek Times 1 The od Table in the ncient Geek Times Septembe 8, 000 Ryoji Tatsukawa 1. Intoduction They say that Hippacus systematically used tigonomety, calculated chods of cicle and made a chod table, which coesponds to today's tigonometic atio table, fo the fist time in about Unfotunately, it is not existing today, but they say, Ptolemy Ptolemaios, laudios often cited it in his witings, lmagest; howeve, Geald Toome estoed it in Let us look at what kind of table they used. Note: The lmagest is a compilation of all the study about astonomy in the ancient Geek times. It had been liteally the geatest astonomical study, till heliocentic theoy was advocated in the 17th centuy.. Ptolemy's cod table 1 Fist of all, he divided a quadant into 180 equal angles aound its cente and its adius is divided into 10 equal pats at evey 1/ Secondly, he calculated the length of chod ', cd. at evey cental angle. eing shown in the Table1, he epesented the factions of chod using sexagesimal system. cd.a veage =60 fo evey incease of choad 1' incease 1 0;31,5 0;1,,50 M ' 1 1;,50 0;1,, ;3,15 0;1,,50 ; 5,0 0;1,,50 1 ;37, 0;1,,8 3 3; 8,8 0;1,, ;39,5 0;1,,8 Fig.1 Tab.1

2 The chod Table in ncient Geek Times Note: The half length of the chod ' coesponds to today's sine M. This means that they used the length of the chod instead of a atio with two sides of a ight tiangle. It was the 16th centuy when tigonometic atio was dealt as a ight tiangle. Rheticus, Geog Joahim epesented it as length of side of a tiangle, not a atio. chod table had been made at evey segment of adius, till the decimal notation was established by Stevin, Simon The tigonometic atio was defined in the fou quadants, as the atio with two sides of the ight tiangle, by ule, Leonhad in the 18th centuy, and sine notation was also established by him. onstuction of egula inscibed polygons in a cicle The following lengths of chods ae able to be calculated by using inscibed egula tiangle, squae and hexagon. cd.10= 3 = ;55'3" cd.60= = 60; cd.90= = 60 1;'51" nd we can also calculate the lengths of chods fo 7 and 36 by using well-known constuction of egula hexagon. aw a cicle centeed at the Point, of which diamete is, and a midpependicula on. With a adius, daw a cicle centeed at the point which is a midpoint of. This cicle meets the diamete at a point. The line segments, ae the sides of a egula decagon and a egula pentagon, espectively. Poof: + = ( + )() + = = = + Fig.

3 The od Table in the ncient Geek Times 3 = = This means that the segment is divided in extenal and intenal atio at the point. ecause, the segment is a side of the inscibed egula pentagon, then we find that the segment is a side of inscibed hexagon in the cicle. Poposition 9 in the books 13 of uclid's lement = + The segment is a side of egula hexagon and is a side of egula decagon, then we find that is a side of egula pentagon. Poposition 10 in the ooks 13 of uclid's lement Note: Setting = in the Fig., we find =. = + = + = 5 = We can calculate cd.36 and cd.7 by using and = = = 5 1 = + = + = 10 5 cd.36= ;'55", cd.7= ;3'3" (3) (cd.a ) + cd.(180 a) = (cd.180) Poof: Setting =1, we get =cd., =cd. 180 '. + = (cd.) + = cd.(180 (cd.180) Fig.3

4 The chod Table in ncient Geek Times Note: Using this fomula, we can calculate cd.108 and cd.. (cd.108) + cd.( ) = (cd.180) (cd.108) = (cd.180) (cd.7) =() ( 10 5 ) = cd.108= = 60 =30 ( 5 + 1) 97;'55" (cd.1) + cd.(180 1) = (cd.180) (cd.1) = (cd.180) (cd.36) =() ( 5 1 ) = cd.1= 11;7'36" = Lemma: Fo the inscibing quadangle in a cicle, we have + =. Ptolemy's theoem Poof: Take a point on the segment so that =. = : = : = =, = : = : = + : + = ( + ) = Fig.

5 The od Table in the ncient Geek Times 5 5 Theoem Theoem 1: When chod and meet at an end point of diamete, if = 1, = and =, the following addition theoem is established: cd.180 cd.( ) = cd. cd.(180 ) cd.(180 ) cd. Poof: y using Ptolemy's theoem, we have =. = cd.( ), = cd.180, =cd.,=cd.(180 ), = cd. Then, we get cd.180 cd.( ) = cd. cd.(180 ) cd.(180 ) cd.. Note: This chod addition theoem coesponds to the following sine addition theoem: sin(ab) = sina cosb cosa sinb Fig.5 Poof: = cosa, = cosb, = sina = sinb y using Ptolemy's theoem, we have =. = sina cosbcosa sinb In, = = = 1 R = sin (ab) y using and, we get sin(ab) = sina cosb cosa sinb. Fig.6

6 6 The chod Table in ncient Geek Times Note: If we take the point on ac, we get the following fomula: sin(a + b) = sina cosb + cosa sinb. Note: We can calculate cd.1. cd.180cd.(7 60) = cd.7cd.(180 60)cd.(180 7)cd.60 cd.1= cd.7cd.10 cd.108cd = cd.1= = = ( )15 1;3'36" Theoem : Setting = and = in the Fig.7, we have the following fomula: cd.180 cd.(180 ) = cd.(180 ) cd.(180 ) cd.cd.. Poof: =, = cd. 180 (+), = cd. (180 ), = cd.(180 ), = cd., = cd. y using Ptolemy's theoem, we get = +. cd.180cd. 180 (a + b) = cd.acd.(180 b)cd.bcd.a Note: cos(a + b= cosa cosbsina sinb Fig.7

7 The od Table in the ncient Geek Times 7 Poof: Setting = and = in the Fig.8, we get this fomula. = = cosa, = sina = cosb, = sinb = 1, = cos(a + b) y using Ptolemy's theoem, we get =. cos(a + b= cosa cosbsina sinb Theoem 3: (cd. a ) (Fig.8) = 1 (cd.180 )cd.180 cd.(180 a) Poof: Setting = in the Fig.9, we get this fomula. The bisecto of meetsthecicleatapointandapointsatisfies=. So we find that = =. Let F be a foot of the pependicula thatwedoptofomthepoint. F = F = = F = y using and we have = 1 ( ) = cd. a, = cd.180, = cd.(180 a) Fig.9

8 8 The chod Table in ncient Geek Times (cd. a ) = 1 (cd.180 )cd.180 cd.(180 a) Note: When =, theoem 3 coesponds to the following fomula fo half angle. sin a = 1 (1 cosa ) We can calculate the following chods by using this fomula. (cd.6) cd.168= cd.6= = ;16'9" 6( ) 86( ) Similaly, we can calculate the following chods. cd.3= ;8'8" cd.1 1 = ( ) 1;3'1" = (cd.180)(cd.180 cd.1) = (cd.168) (cd.168) = (cd.180) (cd.1) cd. 3 = + 0;7'7" ( ) 6 Intepolation fomula cd.b cd.a < b fo 0<a < b < p a Poof: Let be the point at which the bisecto of the angle meets the side,

9 The od Table in the ncient Geek Times 9 we find =. : = : < Let F be a pependicula to, then we find, > > F. The cicle which has a point as a cente and a adius, meets the line at the point G and the extension of F at the point H. F : = F : (sectoh) : sectog = H : G F : < F : G H F : < F : componendo Fig.10 : < : (multiplying both sides by ) : < : (dividendo) : < : = : cd.b : cd.a < b : a Note: We can calculate cd.1 as follows: 3 <1< 3 cd. 3 cd.1 1 <, cd. 3 3 cd.1 < cd. 3 < cd.1< 3 cd. 3 1;'50"< cd.1<1;'50 3 " cd.1 1;'50"

10 10 The chod Table in ncient Geek Times Refeence ooks 1. T. L Manual of Geek Mathematics Thomas L. Heath xfod I.L.Heibeg et H.Menge, uclidis lementa opea omnia, Lipsiae