Introduction: The Sector
The sector was invented or at least first deployed by Thomas Hood or Galileo Galilei at the turn of the 17th century. Although capable of many functions, you can use the sector to find proportions -- thus its other name, the proportional compass.
The phenomenon of the sector is a dramatic paradox. The sector was created to eliminate the need for tedious arithmetic, but its use accelerated the mathematics of natural science. According to Wikipedia, the sector advanced science itself.
The sector was a very useful instrument at a time when artisans and military men were poorly educated in mathematics and, often, were unable to perform even elementary arithmetical operations. The inaccuracy induced by the analog scales of the sector were usually of no concern to those attempting to find a rapid solution to an approximate problem. It is striking, however, that the disciplines to which these instruments were applied, particularly perspective, music, architecture and fortification, traditionally classed as mechanical sciences, soon emerged as mathematical sciences in the seventeenth century. Indeed there is evidence that the universality of these practical applications helped to make possible the universality of science at a theoretical level. Hence this technology was not simply a consequence of advances in science. Rather, the technology helped make possible the mathematical sciences that led to modern science.
As you can see from the fancy version, the sector serves many functions. You might find yours useful if not for fortification and gunnery practice, but as a good way to explore proportional equations. If you make a sector with 13 sections, you'll be able to approximate the golden ratio (1.618:1). Or in my case, as an American in London, I can use it as a quick currency or temperature conversion tool.
Step 1: Tools and Materials
I've made these with index cards and standard fasteners, but this version is made with coffee stirrers and 1/8 inch brads. Both aren't so common, but they are available and cheap.
One of the tools I am using is non-standard; it is the twist gimlet. I almost bought the twist gimlet for the name alone. I wonder if the gimlet drink was named for its potent and stabbing effect. They are very good -- the tool, that is -- for working with soft wood. You can probably find a twist gimlet in your local hardware store for less than five dollars. I bought this one at the local iron mongers for less than three pounds.
Step 2: Make the Hinge
Use the twist gimlet to make a small hole at the top of the stirrer. Work the screw back and forth so as not to crack the wood. This is the gimlet's advantage over an awl. Once you pierce through the other side, flip the stirrer over and open the hole a bit more. If the tip has gone through the other side, you probably can fit the 1/8th inch brad through. Repeat with the other stirrer and connect the two with a brad.
Step 3: Set the Angle
The proportional scale of the vector works on the principle of similar triangles. Similar triangles have sides that are proportional. The angle for the sector will be set at 60 degrees. When you measure from leg to leg, you are essentially creating an equilateral triangle.
Step 4: Mark Off Increments
Starting at zero, mark off your increments. The coffee stirrer is slightly longer than 13 centimeters, so I used centimeters as the unit of measure.
Step 5: Check Your Measurements
Even though we're not using this sector for navigational purposes, it's best to check each of the increments to see if they create equilateral triangles. The goal here is practical precision (good enough for who it's for). With patience and a steady hand you should be able to mark off millimeters.
Step 6: Start Using Your Sector
In our similar triangles from Step Four, you can see with the linear scales how the different sized triangles are proportional.
Suppose you want to divide a line six inches long into five equal parts. Measure off six inches with dividers, and open the sector until the distance between the 5 and 5 on the linear scales is six inches apart. By the principle of similar triangles, the distance between 1 and 1 on the linear scales is then one fifth of the distance of the distance between five and five. Measure the distance 1 to 1 with dividers and the five equal lengths can be measured off the original six inch line.
Please know that technicians usually took measurements with dividers, just like the half-naked Newton statue at the British Library.
Step 7: Variations, Improvisations, and Meditations
I made a sector using an index card. I'd probably use a wider piece of wood for my next sector. I would definitely use a thumbscrew. That way I could lock an angle when needed.
Galileo and others had marked their sectors with additional scales besides the arithmetic scale. These were used for calculating areas and volumes, and in some instances the weight of cannon balls. By adding an adjustable strut between the legs, you can measure angle. Sectors were also used in surveying by laying them flat and supporting them with a pole.
What happens when you change the angle and measure different increments? Can you predict the lengths using trigomometry?