Bürgi's Kunstweg to Calculate Sines

05072022, 11:31 AM
Post: #1




Bürgi's Kunstweg to Calculate Sines
Reference
From: Jost Bürgi’s Method for Calculating Sines Abstract Quote:From various sources we know that the Swiss instrument maker and mathematician Jost Bürgi (15521632) found a new way of calculating any sine value. Division of a right angle into nine parts Program to initialize the registers: Code: 00 { 39Byte Prgm } Quote:The other nine cells are filled with an arbitrary series of natural numbers, for which Bürgi selects 2, 4, 6, 7, 8, 9, 10, 11, 12. Code: 00 { 42Byte Prgm } After the initialisation we perform 4 steps to get the values of column 5: XEQ INIT XEQ STEP R/S R/S R/S RCL 04 8273441 RCL 09 12871192 ÷ 0.64278747 Compare this to the correct value: 40 SIN 0.64278761 Division of a right angle into 90 parts It's easy to adjust the program to do the same for \(n = 90\) to calculate the sines of all degrees from sin 1° to sin 90°. To initialize the registers I use an approximation of the sines similar to before: \(\text{round}(12 \sin(x))\): Code: 00 { 34Byte Prgm } Code: 00 { 46Byte Prgm } SIZE 91 XEQ INIT90 XEQ STEP90 R/S R/S RCL 47 310796277956 RCL 90 424964111250 ÷ 0.73134712 Compare this to the correct value: 47 SIN 0.73135370 Here we performed only 3 steps instead of 4. That's why the accuracy is not as good. But we can also notice that the numbers get bigger than before. 

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Messages In This Thread 
Bürgi's Kunstweg to Calculate Sines  Thomas Klemm  05072022 11:31 AM
RE: Bürgi's Kunstweg to Calculate Sines  Thomas Klemm  05082022, 01:19 PM
RE: Bürgi's Kunstweg to Calculate Sines  Albert Chan  05092022, 01:48 AM
RE: Bürgi's Kunstweg to Calculate Sines  Albert Chan  05092022, 04:55 PM

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