Analyzing a Sector

In March I saw an article from the Lost Art Press weblog advertising a free plan on the First Light Works web site, for making a paper Sector. Just to pass time during the quarantine. I’d heard of the Sector but never looked into it as a woodworking tool so I download and printed the design and instructions. I think I’m hooked, but how does it work? Down the rabbit hole I go.

Sector from Wikipedia
Sector from Wikipedia

A Sector is a very old tool to graphically lay out proportions with dividers. Based on the principles of similar triangles, it’s mainly used for scaling drawings but can be directly applied to a project if necessary. The tool opens to a wide V where you can, on the Line of Lines, see that the distance between the “6” markings is half the distance between the “12” markings, and the “4” marking separation is a third of the “12” separation. That much is intuitive, useful for scaling a drawing up by say, a third. Quick – how tall is a 7 1/2” drawer front if scaled up by a third? Set your dividers to the height of the front, open the sector so the divider just spans the “9” markings, then open your dividers to span the “12”. No math involved, and mathematicians were scarce in the 17th century.

There is also a scale labeled “Line of Circles”. It has markings for “Radius”, Diameter”, and “Circumference”. These are also intuitive proportions. Diameter = 2 X Radius and Circumference = Pi X Diameter. Linear relationships.

Similar models of the Sector are freely available on Jim Tolpin’s By Hand And Eye web site. He will sell you an 11 inch assembled model. Also Brenden Gaffney’s web site has information and a link to a useful YouTube video on the subject. Brenden appears to have an obsession with ancient measuring techniques, you can buy a ruler calibrated in Cubits from him. Brendon also teaches occasional classes at Lost Art Press on making your own Sector. The description file that accompanies the Tolpin, Gaffney, or FirstLightWorks models give several examples of a Sector in use. Gaffney’s YouTube video above, is also filled with information.

Sector Models
Top: byhandandeye.com Middle: Firstlightworks.co.uk Bottom: modified folding ruler

Galileo is often credited with the invention of the Sector in the 1690s, but there are versions known from at least a century earlier. Galileo perfected and expanded the idea of a proportioning tool as an aid to military operations and his design includes scales dealing with solid objects, like cannonballs. He made a hundred copies in brass of his design, which he called a “Proportional Compass”, then wrote a document on using the tool, but printed 80 copies only for those who bought one of his instruments. There were no pictures in the tutorial so it was useless unless you had the purchased tool in hand. That’s how you copyrighted things in the 17th century.

Galileo's Proportional Compass
Galilean compass: front and back

The front of Galileo’s compass carries four pairs of scales:

arithmetic lines, 245 mm in length, divided into 260 equal parts, used for a variety of proportional calculations;
geometric lines, for solving the following problem: given a regular polygon, find the side of another polygon with the same number of sides, but with area n times the area of the first polygon; these lines can also be used to extract of the square root of a number;
stereometric lines, for finding the solid with a volume n times that of a given solid;
metal lines, used, as Galileo says, to give “proportions and differences in weight between the materials indicated on them”—in other words, to determine the specific weights of metals.

On the back are engraved:

polygraphic lines, for finding, from a side of given length, the circle circumscribed around a regular polygon with any number of sides;
tetragonic lines, for finding the side of a square, pentagon, hexagon or other polygon with the same area as a circle of given radius, and vice versa;
adjunct lines, “added” to the tetragonic lines for finding the square of the same area as a circular segment of given chord and radius.

The FirstLightWorks and Tolpin Sectors have a third scale: “Line of Polygons”. You draw a circle, transfer the radius to the Sector Polygon scale at the “6” division, then with the dividers pick off the correct side length to inscribe a Polygon with 4 to 12 sides inside the circle. This is very useful, but not as intuitive as the Line or Circle scales.

Circle Inscribed With 4, 5, and 6 Side Polygons
Circle Inscribed With 4, 5, and 6 Side Polygons

IF I can find a suitable hinge, I’d like to make my own copy of the Tolpin Sector. The “Line of Circles” and “Line of Lines” are linear and easy to lay out but “Line of Polygons” is decidedly not-linear. There are web sites that calculate the length of a polygon side (some giving wrong answers) but Google could not find a description that showed how the Polygon line worked. After many pages of crossed out equations and a less than satisfactory prototype, I found a geometric hint that led to a solution to the problem: How far from the pivot point is the mark for the various polygons?

Skipping over all the details, the solution is x = l \sqrt{ 1 - cos (\frac{360}{n})} where n is number of polygon sides, l the total length of the Polygon Line, x is the distance from the pivot point. These calculated distances agree with the Tolpin rule I have.

Sector Open To Radius R on"Line of Polygons"
Sector Open To Radius R on”Line of Polygons”

This is the formula from the University of Regina web site for the side length of an nth degree polygon inscribed in a circle with radius R. The formula was developed from the familiar “Law of Cosines” c{^2} = a{^2} + b{^2} -  2ab \ cos (C) where C is the angle opposite side c.

1.0 c = \sqrt 2 R \sqrt{ 1 - cos (\frac{360}{n})}

The farthest mark on the Polygon line is n = 4, a square. At this point the side length formula reduces to c = \sqrt 2 R because cos (90) is zero. Drawing a line perpendicular to side c through the center of the circle divides the Sector triangle into a pair of right triangles with far end c' = \frac {c}{2}, the apex angle is C' = \frac {C}{2}. Since the 4 mark is at the far end of the Polygon line, the sine of the apex angle is
2.0 sin C' = \frac {\sqrt 2 R}{2 l}
or
2.1 sin C' = \frac {R}{\sqrt 2 l}.

For all the other Line of Polygon markings, the general formula 1.0 above applies. The c side of those similar triangles will be half of eq. 1.0
3.0 c' = \frac { R \sqrt {1 - cos (\frac {360}{n})}}{\sqrt 2}
Dividing that opposite side by x, the hypotenuse which is the quantity we are trying to derive gives again the sine of the angle C’
3.0 sin C' = \frac { R \sqrt {1 - cos (\frac {360}{n})}}{\sqrt 2 x}

Now we can equate the two sine formulas.
4.0 sin C' = \frac { R \sqrt {1 - cos (\frac {360}{n})}}{\sqrt 2 x}  = \frac {R}{\sqrt 2 l}
Cancelling like terms gives:
4.0 \frac { \sqrt {1 - cos (\frac {360}{n})}}{ x} = \frac {1}{l}
and re-arranging finally shows:
4.1 l  \sqrt {1 - cos (\frac {360}{n})} = x
Which gives the distance x from the Sector pivot to any nth degree polygon mark, for a given length of scale l.

So of course I made a spread sheet. What ancient tool is complete without it’s own spread sheet? Input the length of your Sector’s scale and the sheet gives distance from the pivot for each graduation. You can also change the number of graduations on the Line of Lines scale.

Sheet to Calculate Sector Rulings
Sheet to Calculate Sector Rulings

You can download a copy of the spread sheet from Dropbox
https://www.dropbox.com/s/8q7elf674e1omqe/Sector.zip?dl=0

    • Ralph J Boumenot
    • April 17th, 2020

    I made Brendan’s sector and got a few free copies. After playing with them in the shop I found it easier to use dividers sans the sector. Cell phones apps do the math so much quicker and with better results.

    • danny
    • January 18th, 2021

    I believe a simpler formula for the P line is to normalize a length for a hexagon (n=6) and then the distance for the other marks is

    d = 2*sin(‘pi’/n)

    This takes advantage of sin(‘pi’/6) = 1/2 (the denominator of the normalization)

    Danny

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