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Sexagesimal math with the "40 method"
From: Paul Hirose
Date: 2003 Feb 14, 12:21 -0800
From: Paul Hirose
Date: 2003 Feb 14, 12:21 -0800
In an old surveying publication I saw a nice trick for adding sexagesimal numbers on an electromechanical calculating machine. If the minutes or seconds summed to 60 or more you added 40 to the part that "rolled over". Here's a simple example: by hand calculator equivalent 3 14' 314 + 1 48' + 148 ------- ----- 5 02' 462 + 40 ----- 502 In the following example, seconds do not roll over, so 40 is added to minutes part only: 281 45' 32" 2814532 + 68 59' 14" + 685914 ------------ ------- 350 44' 46" 3500446 + 4000 ------- 3504446 Here's an example with minutes and seconds rolling over, so 40 is added to both parts: 316 38' 13" 3163813 + 12 43' 54" + 124354 ------------ -------- 329 22' 07" 3288167 + 4040 -------- 3292207 Finally, in the following example seconds roll over, and the carry will make minutes roll over, so 40 must be added to both parts: 82 34' 16" 823416 + 34 25' 53" + 342553 ------------ -------- 117 00' 09" 1165969 + 4040 -------- 1170009 The algorithm works for subtraction too; you just subtract 40 instead of adding. Here's an example where seconds roll over. This time I'll use a decimal point between degrees and minutes for easier reading: 142 52' 11" 142.5211 - 21 10' 56" - 21.1056 ------------ ------- 121 41' 15" 121.4155 - .0040 ------- 121.4115 In general, I don't see much use for this old technique if you have a scientific calculator. It's easier to use the sexagesimal and decimal conversion functions and do the math in decimal. On the other hand, the 40 method is usable on the simplest calculator. Also, it easily handles degrees and decimal minutes, a format which has no direct conversion on the calculators I've used. For paper and pencil computation, I think I'll continue to do the math in sexagesimal. A variation on the 40 method is the 76 method for dealing with 24 hour days. This can be combined with the 40 method: 2 days 17 hours 27 minutes 2.1727 +1 day 20 hours 35 minutes +1.2035 --------------------------- ------- 4 days 14 hours 2 minutes 3.3762 + .7640 ------- 4.1402 or for subtraction: 2 days 17 hours 27 minutes 2.1727 -1 day 20 hours 35 minutes -1.2035 --------------------------- ------- 0 days 20 hours 52 minutes .9692 - .7640 ------- .2052 I may try that the next time such a calculation comes along. Carrying or borrowing 24 hours always makes me think hard, and if I'm tired it's easy to mess up.