Freestetter’s world of formulas: Sometimes feelings are more important than logic
Angles are a wonderful thing. They are essential to scientific calculations – and at the same time they show the deeply human aspects of mathematics.
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Rationality plays an important role in science. But sometimes researchers also make a decision on their gut feeling, for example when defining units.
If you want to be precise, an angle cannot be described that easily: An angle is a portion of a plane bounded by two straight lines starting from a common point. On the other hand, what we usually mean when we say angle is the measure of the angle or the width of the angle – that is, the size of the angle. If you rotate a line around its starting point by a certain amount, you get a corresponding large angle. The specific numbers we use to represent the angle depend on how we divide the so-called perfect angle.
A perfect angle is obtained if the straight line is allowed to move in a complete circle. We usually say that a full angle is 360 degrees – but that’s just an imitation. There is another way, as this formula shows:
The Gon unit is defined here as the 400th part of a perfect angle. So one gon is 0.9 degrees, which seems unusual. But upon closer inspection, it’s actually a clear idea. The right angle has exactly 100 johns – and we also define our remainder using decimal, i.e. as multiples of ten. Surprisingly, this system has become almost universally adopted in terms of height or weight; But not when measuring angles.
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When it comes to angles, we still use the sexagesimal system that the Sumerians used 5,000 years ago. On the one hand, astronomy may have been responsible for this, as it found it practical to represent a complete angle in multiples of 60: a number that can be easily divided into subsections. On the other hand, 360 degrees also fits well with the 365-day length of the year, making calendar calculations easier.
A circle of 6 x 60 degrees, each divided into 60 minutes of 60 seconds each: Despite all attempts at measurement, this system is still in use almost everywhere today. When units were converted to decimal during the French Revolution, the meter was defined as a ten-millionth of the distance from the North Pole to the equator (along the Paris meridian). Hence a gon would correspond to a 100 km arc, a gon would correspond to a 1 km arc, and 10 microgon would be exactly 1 meter along the meridian arc.
A day consists of 10 hours, 100 minutes each
Everything is very clear. But just like trying to divide the calendar and time into decimal, the Gon unit never worked. Apart from the individual areas of surveying and robotics, 360 degrees of the circuit is still calculated. When it comes to time, we’ll likely stick to the usual 24 hours (or 12 hours) a day; Another remnant of the old sexagesimal system (5 times 12 equals 60).
Mathematics itself does not care what the actual numbers it deals with look like. Whether we divide a perfect angle into 360 degrees, 400 gon, 24 hours, or 2π radians – or come up with some other measurement – will have no effect on our knowledge of geometry and the results derived from it. But we humans cannot be as abstract and objective as the laws of mathematics. In some areas, tradition seems more important to us than systematic theology. A day consisting of 10 hours of 100 minutes each would last just as long as a day of 24 hours. A circle with 400 sides looks the same as a circle with a length of 360 degrees. But it is wrong to measure this way – and in this case, feelings are clearly more important than mathematical rankings.
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