Date: Sep 18, 2010 2:04 AM
Author: shyamal kumar das

To:Peter Scales
From: Shyamal Kumar Das
Thanks for your quick reply posted sept.6, to my last letter on the subject.
I thank you once again for allowing me and encouraging me to think over further regarding Arc length vs Chord length to minimize the difference and to get acceptable accuracy.
Here are some outcomes:
1 ) when the arc length is equal to the radius of the arc ,the angle subtended at the center = 1 Radian = 57.3 degree again, when the chord length is equal to the radius of the arc, the angle subtended at the center= 60 degree Therefore, when angle subtended at the center = 57.3 degree, arc length minus chord length corresponds to the difference of ( 60?57.3)= 2.7 degree So, for 22.5 degree subtended angle, the difference = (2.7/57.3) *22.5 =1.06degre = 4.7% Hence, error accrued = 1.06 degree = 4.7%
2 )for 60 degree subtended angle and R being the bigger radius = nr, where n = no. of parts and r = smaller arc radius, Arc length minus Chord length = R*pi/3 -R =R (pi/3 - 1) = R (22/21 - 1)= R/21 so for22.5degree subtended angle, the difference =R/21 * 22.5/60 =1/56 of R = =1.8% Hence, error accrued = 1.8% of R. If we take n=no of division =5and r =small arc =0.5 cm, then, R =n.r = 5*0.5 =2.5 cm, error accrued = 1.8% of 2.5 cm = .045cm
Regarding the no of operations involved, let us take the example of 180 degree to be divided into 5 no of divisions .
a) 3 bisections involve 3operations
b) One sector of 22.5degree to be divided into 5 parts to get 5 divisions ? requires 1 operation
c) Taking one small div., or one graduation i,e, 1/5 of 22.5 degree , another 3 graduations to be added to get 8 graduations, means 1 operation
d) Taking 8 graduations at a time, whole arc of 180 degree is to be divided into 5 equal parts is the sixth operation
Hence, dividing into 5 equal parts for an obtuse angle involves 6 operations, while dividing any straight line into 5 equal parts requires at least 5 operations.
Please correct me if I am wrong in my above observations. Otherwise, please let me know if the errors accrued and no of operations involved are within the acceptable accuracies.
I am now 65 years old . I don?t know how do you like to teach an old man like me. But, believe you me ,pleasure is mine. Regards .Shyamal Kumar Das.