The Sector: 500 Years Old and Still Useful

The Sector: 500 Years Old and Still Useful

Galileo

Proportioning and accurate layout are central to design and building.  The Sector, almost overnight, has become one of my favorite tools for these tasks.  It is a simple device consisting of two legs attached at one end by a hinge.   There is one scale on each leg and they are identical to one another.  (For other purposes, sectors may have multiple scales. See Note 1.)

The sector was invented by Galileo (and a few other mathematicians) in the late 1500’s as a kind of analog calculator (see Note 1).  In the shop, however, it provides a wonderful way to accurately proportion and divide parts without the use of rulers or calculators or formulae.  In my hands, rulers are not foolproof: Mistakes occur from not lining up the end perfectly, not seeing appropriate divisions or misreading the division, or errors in marking.  I am also prone to miscalculation:  I sometimes make mistakes in arithmetic but rescaling is especially annoying, i.e., converting measures in decimal to fractional equivalents or from English to metric or vice versa.  The sector eliminates many of these sources of error.  All one needs is a pair of dividers and the sector. 

 
Below is the first sector I built. It consists of 2 pieces of wood about 24” long joined at one end with a butt hinge.  Starting at the center of the hinge pin, there are 13 equal segments numbered from 1 to 13 down each leg. The units of the scale are arbitrary but they must be equally spaced, a task that is made simple with a pair of dividers.  Its construction was not difficult, didn’t take very long and required only some lumber scraps and a butt hinge.  Believe it or not, it is a fairly accurate instrument!  And, it takes nothing more than a pair of dividers to build in and to capture that accuracy. (See Note 2 for sources of construction instruction.)

Sector with two similar triangles. 

How the Sector works in theory.  The sector works on a simple principle of Euclidian geometry.   Even if they differ in size, two triangles are similar if they have the same angles.  The nice thing about similar triangles is that the ratio of the sides is identical, i.e., the ratio of side a to side b (or a to c, or b to c) in triangle 1 is the same as the ratio of side a to side b (or a to c, or b to c) in triangle 2.  Let’s see how this plays itself out with the sector.  ABC is one triangle and ADE is another (see photo). ABC and ADE are similar triangles, i.e., all the corresponding angles are identical. We know the length of two sides of each triangle.  Two sides in ABC are 6 units and two sides in ADE are 12 units.  We do not know the length of line BC or the length of line DE.  However, because the triangles are similar we know that the ratio of the length of line AB to line BC is equal to the ratio of the length of line AD to the line DE.  And since line AB (6 units) is ½ the length of line AD (12 units) then the line BC must be exactly half the length of the line DE.   If line BC cut the sector at 4 units on each leg, instead of 6 units as pictured, then AB would be 1/3 of the line AD;  so, the line BC would be 1/3 length DE.  (By the same reasoning, since line AD (12 units) is 3 times the length of line AB (4 units) if follows that line DE is 3 times the length of Line AB.)  In short, we need not have a ruler measured edge length to find its center or several other whole number fractions (or multiples) of its length.  These geometric relationships are very useful in the studio/shop.

Some practical examples.  Let’s suppose that the piece of plywood in the illustration below is intended as a drawer front and we wish to put a pull in the middle of that drawer front.  One can of course, measure the width of the front divide by two and measure to the middle.  It can be done more simply by setting the sector to the width of the drawer at any even number on the sector scale.   In the illustration below I have set the width of the drawer front to the number 12.  To get half that distance all I have to do is set the divider to 6 and transfer that setting to the panel.

Set drawer front to 12 on both legs.

Use divider to mark center of panel

Divider set to 6.

 Again, any even number can be used.  If the sector touches the drawer front width at 10 on each leg then setting the divider to 5 will give half the distance.  If the drawer front width is at 8 on the sector legs then the dividers set at 4 will yield half the distance.   My sector put me within less than a hair’s breadth of the middle. And since the distance is laid off with dividers, a tiny depression is left to mark the spot; the depression provides a positive stop for a pencil and square.  Note that I never used a ruler to get the precise width of the drawer front and it is not necessary for me to know that measurement.

 Now, suppose I still want to place the pull in the middle of the width but a bit higher than the middle with respect to the height of the drawer front.  In many iconic pieces, the pull is set at 4/7 of the height.  I can use a ruler to measure the height of the drawer front, divide by 7 and multiply by 4 to give me the distance from the bottom.  However, with the sector all I have to do is set the height of the drawer front to 7 and then set my dividers to 4.  The distance between the legs of the dividers will give me the distance I need. Now, with one leg of the divider at the bottom of the drawer front, the other leg marks the place for the pull at exactly 4/7 up from the bottom. 

Mark 4/7 of height

Set height of panel to 7 on both legs

Set dividers to 4

Now let us consider 2 pulls on the drawer.  Let’s space the pulls an equal distance from each side edge of the drawer front.  Further, let’s set the space between the pulls to be 3 times the space each pull is from its respective side.  Of course, if there is 1 unit of space from the left edge to the first pull, and 3 units of space between the first and second pull, and 1 unit of space from the second pull to the right edge then there is a total of 5 spaces.  Laying out the appropriate distances is quick and easy.  I simply set the width of the sector to 10 across the width of the drawer front; and, then set the dividers to 2 on the sector.  (Two is one fifth of ten.  I could have set the sector to 5 across the width of the drawer and set my dividers to 1 across the sector with exactly the same result.)  Now use the dividers to step off one unit from each edge and the pulls will be spaced precisely as specified: Each pull is one unit from the side and there are three units between them!
            
Thirteen divisions and the Golden Ratio.  It is easy to see why there are at 12 divisions on the sector legs.  Twelve is an even multiple of 1, 2, 3, 4, and 6.  So, with 12 division one can easily find 1/12, 1/6, 1/4, 1/3, ½.   But, why 13?  Thirteen is only divisible by 1 and by itself so finding nice “round” divisions is not facilitated.  But, having 13 units is crucial to classic design proportioning through the golden ratio, 1.618 to 1.

As you probably know this ratio turns up in many design classics, in biology including various human proportions and in the unfolding of many botanical phenomena such as sunflowers.  The golden ratio is based on an arithmetic series known as the Fibonacci series.  Each number in the Fibonacci series is found by summing the preceding two numbers.   Starting with 0, 1, here is what a short series looks like: 0, 1, 1, 2, 3, 5, 8, 13, 21...  The 3rd number in the series was found by summing the first two, 0+1=1; the 4th number in the series is 1+1=2; the 5th is 2+1=3; the 6th is 3+2=5; then 5+3=8; 8+5=13 and so on. As the series progresses the ratio of each number in the series to the next smaller number converges on the Golden Ratio.  Happily, the convergence is relatively rapid and the ratio of 13 to 8 is a close approximation (less than 1/100th of a unit) to the Golden ratio.

Use the sector to lay out a golden rectangle.  Mark or set the width of rectangle to anything you wish.  Capture that width with the sector at scale value 13 on each leg.  Maintain that setting on the sector and set your dividers to 8 on each leg of the sector.   Use the dividers set at 8 to mark the height of the rectangle.  The ratio of 13 to 8 approximates a golden rectangle, i.e., 1 to 1.62.  in short,  including 13 divisions on the sector facilitates the use of the golden ratio.
          
It is easy to see why the sector is a welcome addition to my shop.  It makes life a lot easier for designing, proportioning and dividing.  Will it replace a ruler and computation?  No.  The smallest division of the sector is 1/13th of a unit and we often need smaller units.  And, we often need fractions that are not immediately available using the sector. However, the sector is convenient and useful for laying out proportions that are possible with the limited scale that it has.  Fortunately, from a design perspective, that leaves a lot (see note 3).  Since the time of Pythagoras, many design schemes utilize a harmonic approach.  Just as consonant musical harmonies in which whole number ratios such as 1/3, ¼, 1/5 produce attractive sounds the history of design shows that spatial proportions using such whole number ratios also produce attractive visual elements.  In addition, the golden ratio, 13/8, also available on the Sector, is particularly useful for slicing and dicing space.

Notes
1.     To learn more about the sector as a scientific instrument see The Sector: its history, scales, and uses, by Erwin Tomash and Micheal R. Williams.  Click here for the PDF.
2.    Want to build your own sector?  The sector shown in this article is based on instructions given by Jim Tolpin and slightly modified by me (click here).  Other instructions are given by Chris Schwartz.
3.     I was introduced to the sector by a book titled By Hand & Eye  by George Walker and Jim Tolpin, (Lost art Press, Fort Mitchell, 2013).    The book is about methods for designing, proportioning and laying out furniture.  It does a good job of increasing our awareness of the importance and ubiquity of proportioning schemes in well-designed furniture.  One of the most useful tidbits that I got from the book is its introduction to the sector.