What’s Your Angle on Angular Measurements?

CNC Machines and Angular Measurements

Everyone’s Got an Angle, and it’s All About Your Angle

You know, when I started researching this article I really thought it was going to be an in depth intellectual undertaking.  In reality, it’s much more like trying to roll a square wheel!

The Angle: It Started a Long Time Ago and Far Away

Degrees, hours, days, years, and seconds all have their roots in Sumerian and Babylonian civilizations. These are the people that lived in Mesopotamia, modern Southern Iraq, about 7,000 years ago. This was so long ago that Egypt and the Pyramids weren’t even a twinkle in Ramses’ eye yet. As with most things of value, the concepts were passed down and improved upon, first by the Egyptians, then the Greeks, then the Romans, and then on to the world (including the machine world!).

The Angle: It Needs a “Base” Number to Begin

The use of base 12 numbering and sexagesimal (or base 60) numbering systems also had great influence on the evolution of our current designations for angular measurements and time.

The Babylonians used the base numbering system of 60. It is hypothesized that the base 60 was used because this base made it possible to take the radius of a circle to which a cord line could be drawn inside of the circle with the result that a perfect hexagonal figure could be obtained. Divide this by 60 and you get 360. Others say this is just coincidence.

The Angle: Development from Days, Hours, Minutes, Seconds … Degrees!

Another thought is that since 7,000 years ago you couldn’t go down to the local department store and buy a watch, some people stabbed a stick in the ground and noticed the shadow traveled in an arc. If they put a small stone at the top of the shadow projected every day by the sun against the stick, at the point the sun was directly overhead; it would form an elongated circle. Every 365 days the rock would start the circular path again. Now we know it’s actually 365 ¼ days but the rock told them it was 360 by all evidence found in ancient writings. Ah what the heck, move the rock over a little bit every few years and it’ll be OK!

A further twist on this is observed when a small hole is drilled in a cave by which the light shines on a flat surface being tilted just right, which results in a perfect circle in a years’ time. This is then divided into 360 or so days, hence 1 day equaling 1 degree. This is a system that has been around for 7,000 years!

This is really interesting up to this point but now it gets really high tech. If there are 360 days in a year by dividing up the sun’s path, then why not divide the day into 360 something’s?

Well, along come the Egyptians. They have this calendar and geometry thing down quite well, but adding and subtracting in the base 60 numbering system doesn’t work out easily when you’re down at the local market. They decide to use the base 12 system, because if you take your thumb on one hand and count, the easily reached joints on each of your four fingers, starting with the tip, you get 12 joints. They probably would have had a base 16 system, but if only their hands would have been more flexible … Pounding and chiseling pyramid stones all day takes a toll on flexibility! The Egyptians also realized that light and dark occurred for about the same amount of time each day. Therefore, one hand represented day-time, the other hand night-time; hence a 24 hour day.

The Angle: The Formal Derivation

An article from Princeton Press (PDF) has this to say about formal deviations and degrees:

The word degree originated with the Greeks. According to mathematics historian, David Eugene Smith, they used the word μοίρα (moira), which the Arabs translated into daraja (akin to the Hebrew dar’ggah, a step on a ladder or scale); this in turn became the Latin de gradus, from which came the word degree. The Greeks called the sixtieth part of a degree the “first part,” the sixtieth part of that the “second part,” and so on. In Latin the former was called pars minuta prima (“first small part”) and the latter pars minuta secunda (“second small part”), from which came our minute and second.

The Angle: Next Step – Into the World of Metrology

There we have it, the history of years, days, hours, minutes, and seconds. It probably all started when that guy with the hammer put it down after knocking off the corners of that “square wheel” and noticed the arc made by the shadow of the handle pointing skyward.

Next month we’ll get into how these measurements and angles get into the world of metrology and some of the history of being able to measure them.