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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Bowditch 2002**

**From:**Michael Wescott

**Date:**2002 Oct 5, 12:03 -0400

MBurrill@LEESBURGVA.ORG said: > I told my HP48 to leave default mode, gave it 7 decimal places, asked > it for 5, and it disagreed with Bowditch. How many places should I > give it ? That question piqued my interest and the answer is 9, at least for sines. First I downoaded the pdf file for Table 2 and extracted its contents. I managed to reconstruct all the significant data except the difference columns. Then I made use of a calculation program on Linux/Unix systems (bc) that does it's calculations all in decimal and permits selection of the number of decimal places it keeps. It doesn't round, it truncates calculations to the specified number of places. I programmed bc to calculated sines using the classic series x - x^3/3! + x^5/5! - ... calculation terminated when the term became 0 at the specified scale. I compared these results against the data from Bowditch. With all the calculations done with 5 decimal places there were, as you might expect, many disagreements in the data sets. About half the values disagreed in the last decimal place, but the differences were all no more than +/-0.00002. Carried to 6 decimal places, the calculated values differed from Bowditch by at least 0.000005 in 619 values (out of 5401) but never more than +/-0.000007. At 7 digits there were 38 discrepancies of 0.0000050 or more. The greatest being 0.0000052. At 8 digits, there were five of precisely 0.00000500 dd:mm Bowditch bc ===== ======== ======== 21:37 0.36839 .36839500 23:40 0.40141 .40141500 28:8 0.47153 .47152500 35:19 0.57810 .57809500 56:47 0.83660 .83660500 At 9 digits, all differences were less than 0.000005000. -- Mike Wescott Wescott_Mike@EMC.COM