Thursday, 23 April 2015

Merewen's Pentathlon Documentation - Part 4 of 5 (Scientific Instrument)

The Sector of Thomas Hood

Scientific Instrument (Group III) Beginner

Merewen de Sweynesheie

 
Photo by Lord Alexander Gladstone.
Goal:  To recreate the sector of Thomas Hood as an introduction to scientific instruments of the Middle Ages.  This is my first scientific instrument project.

Contents:
1.      Study of Extant Material and Examples
2.      Construction Steps
3.      Process, Choices, and Materials
4.      Further Information about the Markings and Marking Guides
5.      Considerations
6.      Use of the Sector
7.      Resources


1. Study of Extant Material and Examples

I decided to replicate a sector created by Thomas Hood, as per his book, "The Makinge and Use of the Geometricall Instrument called a Sector," from 1598.  This was a good first choice for a sector I found out as I went along, because the book is written in English and Thomas Hood shows diagrams and explains how to create the markings on the instrument and which pieces were included.  My research found that Galileo, who invented a similar instrument concurrently but separately from Hood, did not do this. Galileo, who published his manuscript on his sector in 1606, was afraid of imposters taking credit for his work, which did actually happen less than 10 years later.  The sector originally invented by Thomas Hood was designed to be used to solve trigonometric problems, inscribe regular polygons inside or around the outside of circles, and create regular polygons proportional to other regular polygons.  It could have been used by architects, mathematicians, or anyone else who needed to create proportional drawing, although there are also other tools available for this.

Pieces of the sector - Hood's sector was comprised of four essential parts and a number of "accidental parts."  The essential parts were the minimum number of parts required to effectively use the sector, and the accidental parts were those which would aid further in the sector's use or allow additional uses but which were not required for basic use of the sector.  The four essential parts are the two feet, the hinge, the points of the sector, and the circumferential limb.  The accidental parts include an index, three sights and their mounting hardware, and a socket.
  
Use of Hood's book - Hood provided a very good guide for creating the markings on the sector, although he did not provide illustrations showing how the markings looked on both sides on the instrument's parts.  He did not provide information on how to construct the sector's parts, including the very important hinge piece.  He provided a detailed list of the uses of the sector, and expounded upon them by explaining what they were, how they worked, and providing examples of this use, but some of the language no longer matches today's language.  For example, we no longer use the term "enneagon;" we instead use the term "nonagon."

Hinge - I found one modern article containing a drawing of one kind of hinge, but the hinge was slightly too complicated for me to make.  I found a number of extant examples of sectors from mostly the mid 1650s through 1700s, and tried to analyze the hinges as best I could, from limited views.



Feet - Hood's book called for points on the feet of the sector, centered on each foot.  Most extant examples I found either had sheet brass points (such as Hood's sector) or no points, but I did find at least one example of points set into wooden carved feet.

Sector materials - Extant examples showed sectors made of boxwood, ivory, brass, or silver, with the majority being made out of brass.

Sector parts - Most extant sectors I found only appeared to have feet; no circumferential limb, points, or other pieces.  There is very little description in Hood's book of what the pieces look like, and none about how to construct any part of them.  It is here where I looked at as many extant examples as possible (mostly photographed from only one view, unfortunately) to better understand how the pieces might fit together.

2. Construction Steps

 
Photo by Master Eirik Andersen.
1)      Cut feet
2)      Cut hinge pieces
3)      Cut slots in feet for hinges
4)      Drill setting holes in hinge plates
5)      Drill central hinge pin hole
6)      Drill setting holes in first foot
7)      Adjust hinge plate to fit first foot
8)      Attach hinge plates to first foot
9)      Trim curved section out of both feet for central hinge pin placement
10)  Adjust hinge plate for second foot to allow good hinge alignment
11)  Drill setting holes in second foot
12)  Attach second set of hinge pieces to second foot
13)  Cut feet bottoms to shape
14)  Carve feet bottoms smooth
15)  Prepare feet for marking
16)  Prepare marking guides for feet
17)  Mark feet (both sides)
18)  Wax and re-sand feet
19)  Set central hinge pin
20)  Drill holes for points
21)  Insert and shape points
22)  Prepare marking guides for circumferential arc
23)  Create and mark circumferential arc

3. Process, Choices, and Materials


a)      What parts to recreate - Essential parts only.
As this was my first time, I wanted to choose something that would not be too challenging.  As such, I decided to stick with creating only the essential parts of Hood's sector.
b)      Foot material - Hickory. 
The easiest material for me to work with, having no experience, and create was of wood.  Although extant examples were made of boxwood, boxwood is a very slow-growing shrub, and not readily available in this day and age in the size required.  The greatest benefits of boxwood are its hardness, density, easy workability, colour, lack of prominent grain, and its flexibility.  For this reason, I selected a local hardwood with properties as similar as possible to boxwood - hickory.  Hickory is slightly less dense, but almost as hard as boxwood.  It is not quite as easily worked due to its stringiness, but it not horrible to work with either.  It is lightly-coloured, but has a much more prominent grain.  Finally, its strength is similar to that of boxwood.  The feet were cut to their rough shape with a table saw (including cutting the thickness of the wood in half), and I hand-sanded them.
c)      Hinge - Brass, my own design.
Photo by Merewen de Sweynesheie.
The most important part of the sector is the hinge.  The hinge must allow the sector feet to fold together side-by-side and also open to 180 degrees.  It must be tight enough to hold the sector in any position so that measurements can be easily transferred.  For this reason, I addressed this most important part early in the design process.  I had no detailed drawings or clear extant examples or descriptions from which to work.  Although the instrument would be made mainly of wood, it was obvious that a metal (brass) hinge would be required to fulfill the working and durability requirements of the instrument.  I ended up designing a plausible hinge that could be set into a slot at the top of the wood.  Because a filler brass piece is used on the opposite side to the hinge piece, the slots in both feet can be cut identically.  The slots were cut with a table saw because that was the only way, using the tools available, that they could be cleanly and evenly cut.  The brass used was the only sheet brass I had available.  I believe it's 12 or 14 gauge.
d)      Cutting the brass hinge pieces - Dremel and file. 
Photo by Merewen de Sweynesheie.
I did research online and asked questions about how to cut the brass using period methods.  The most period method would have been to use cold chisels to trim the pieces to size, then file down.  Unfortunately, I searched all over town and could not find a cold chisel smaller than a straight 1/2 inch width.  This would not do to cut the curves.  The next best method was a jeweller's saw.  I searched all over town and could not find one.  I went to the local jeweller, but he had left town 6 years ago.  I called a goldsmith 1/2 hour from my home, but he was unwilling to let me use his saw, even if I paid for blades.  As a last resort, I used a dremel to cut the rough shape of the pieces.  I then used a file to file down the shape to match what I had planned.  Unfortunately,  the hinges did require additional adjustment in shape after the initial cut and shape.  I was out of my depth in metalworking at this point, and received some help from someone more experienced in metalworking to achieve the final shape.  This could have been predicted based on my research - I read that mostly only instrument makers made sectors in period because the hinge is so difficult to create accurately.  There was some evidence of hingeless sectors being made by the layman, but these were noted as being very difficult to use.
e)      Drilling holes - powered drill. 
This was the only option I had available to me.  For drilling the metal hinge plates, I used sewing machine oil to lubricate and expanded the holes slowly using progressively larger bits.
f)       Attaching hinge plates to feet - brass wire pins. 
This was accurate to extant wooden examples that I had found - three metal pins cut from wire, set into the holes, and trimmed flush.  A dremel was used to further smooth out the faces after trimming with wire cutters.
g)      Making marking guides - I followed Hood's instructions in creating the various marking guides, using a modern protractor, compass, divider, scales, rulers, pencil, and paper.
h)      Making the markings on the wood - oiling, inscribing, writing with ink, waxing, and sanding. 
It was pointed out to me early on that simply marking the feet with ink would result in the ink spreading and crawling through the grain.  To avoid this, I created a test piece of hickory where I tested various combinations of marking and masking media in order to find the best way to cleanly mark the sector.
    inscribed and inked: not bad
    beeswax, inscribed, and inked: pretty good
    just ink: bled quite a bit
    inscribing through masking tape then inked: line barely made it through the tape
    masking tape set on either side of line and inked: worst bleed of all
    masking tape set on either side of line, inscribed, and inked: bled
    tung oil then inscribed and inked: pretty good
    tung oil, beeswax, then inscribed and inked: very good

So I decided to make my markings by first laying down a coat of tung oil, then a coat of beeswax, then inscribing the line with an X-acto knife, then carefully creating the ink line.  With a fine hand sand after marking, most of the beeswax is removed, leaving a beautiful, soft finish.  Oiling the wood removes most of a pencil mark, so I would have to oil the wood before marking my lines.  In practice, oiling the wood, making the marks, and then applying solid beeswax did not work.  The beeswax smeared the lines until they were completely unreadable when used over oil.  I decided to skip the beeswax, since it did not significantly improve the line marking.  After applying ink, I then applied the beeswax and rubbed and sanded off the excess.  This filled in some of the incisions and made a smoother finish.
i)        Cut away the appropriate amounts of the top corners of the feet for the hinge pin - Exacto knife and metal washer. 
I marked the appropriate curves lightly using the metal point of a compass and then an exacto knife was used to cut and a metal washer was used as a guide to prevent accidental inaccurate cuts.
j)        Setting the central hinge pin - hammer, rivet set, steel bench block, and brass rivet.
I used these tools to set and peen down the central hinge pin until the sector's movement was stiff but acceptably mobile.
k)      Creating the feet and points - saw, Flexcut Cutting Knife carved, drill, wire cutter, hammer, and file. 
First, I chose an extant design from the 1700s as a basis for my foot design, as I could not find anything closer to the time period that had points that I could emulate.  I did not feel comfortable enough with my metalworking ability to create points from sheet brass, so brass wire would have to do.  I created a cardboard template for the foot bottoms, traced it onto the feet with a pencil, cut the lines with a saw, and then used a carving knife to carve the feet to a smoother and more even shape.   I then drilled holes in the bottom of the feet, keeping the drill and the sector both level to create straight holes.  After drilling, the wire (craft brass wire, unknown gauge, but the same as the hinge setting pins) was inserted into the holes.  I cut the wire to length with a pair of wire cutters, and then shaped the wire to a point using a hammer and a steel bench block.  To smooth the points further, I filed them down.

Photo by Merewen de Sweynesheie.

l)        Create circumferential arc - cardboard, compass, protractor, ruler, pencil, and ink.
I did not have any wood or brass material that could be cut thin enough and be in a large enough size to make the circumferential arc.  I also did not have the means (tool-wise) to join two pieces together effectively.  In addition, it was very unclear whether, and if so, how, the circumferential arc should attach to the sector's feet.  I decided that in this case it was best to make this portion of the sector out of cardboard, as doing so would not affect the use of the piece in any way.  I marked out the requisite arc and markings using a compass, protractor, and pencil, and then used ink to mark my lines.  Although Hood's manuscript states that 100 degrees is the minimum size needed, and that no more is needed than that, but more (up to 180 degrees) was acceptable, I decided to mark 120 degrees, to coincide with 120 parts of inches on the feet.  The ink used for my markings was the same as was used to mark the feet of the sector.  It was a Schaeffer calligraphy pen, fine nib, with brown ink.  When I got to the back side of the arc, I found that Hood had not made it clear how the markings should be laid out on the arc.  While marking the 6-parts-per-inch section (which I had not had a reliable way of creating a marking guide for anyway), I made some choices that did not result in a clearly-marked scale.  I remedied this for the remaining two scales.  Although period sectors did not have words telling what the different scales were, in this case I felt it prudent to add the words, because viewing all three scales and trying to figure out how to read the markings would have been very confusing otherwise.

4. Further Information about the Markings and Marking Guides

Photo by Master Eirik Andersen.

Front side of feet:  On the front side of the feet, I used a drafting scale to mark out measurements in tenths of an inch, starting from the center of the sector's hinge pin and extending out to the ends of the tips on the feet, following Hood's instructions.


Back side of feet:  The internal measurements are created to replicate the length of the sides of various polygons that would fit perfectly into a circle the diameter of the length of the sector's feet.  This would typically have been used by architects/masons for building arches and decorative features as are often seen in churches.  Through proofs which I will be happy to demonstrate but which I choose not to detail here, these lengths are calculated. The external measurements on the backside of the sector represent lines which are of different proportions to one another, relative to the diameter of a circle which has the same diameter as the length of the sector's feet.  Again, I will decline to mathematically show the proof here, but will perform this proof if anyone reviewing this work so desires.

For the external measurements on the backside, I again created a semicircle.  This time, I divided the length of the diameter in two, three, four, etc, and drew a line perpendicularly up to the arc from each of these points.  From this, I found the length of the arc formed from one side of the semicircle to each of these points along the arc.  These lines were marked 1/2, 1/3, 1/4, etc. on the feet of the sector, and represent proportions to one another.

Front side of the circumferential arc:  For the front side, I used a protractor, ruler, compass, and pencil to mark 120 degrees of a circle.  I chose 120 degrees because the sector feet were 12 inches long divided into tenths, so I wanted to be consistent.

Back side of the circumferential arc:  On the backside of the circumferential arc, there are three 15-inch scales.  The scales are not true to size, but when the sector is opened to the marking on the circumferential arc the points of the feet will be spaced at the correct measurement.  Each of the 15-inch scales contains inches, divided into 6ths, 8ths, and 10ths, respectively, and each of these divisions is further divided in half.  To create these markings on the circumferential arc, a line of fifteen inches' length is drawn on a piece of paper, and the inches divided up accordingly.  The sector is then set to each measurement in turn, and the circumferential arc is set against this and the required marking made on the circumferential arc.  This is a very time-consuming process.  I began by marking the sixths scale.  With no reference for how these scales were best marked, the choice I made for the sixths of an inch scale was easily seen as not the best choice, but once I had started using that method, I had to continue on that scale.  I changed to a much easier to read method of marking for the eighths and tenths of an inch scales.  I was not able to transfer to the arc the first inch of each scale, as the closest that the points can be is approximately 1 1/16 inches apart, or the width of a foot of the sector.  I also did not have any fully accurate way to measure out sixths of an inch, so the sixths scale I used to transfer markings to the circumferential arc was not as accurate as the eighths and tenths scales.  In the end, the markings of the sixths scale were not much different in accuracy from those of the eighths and tenths scales.

5. Considerations

a)      Descriptions given in Hood's book - I began by transcribing the chapter of the book regarding the making of the instrument.  This was important because the book was written in gothic script and in a medieval manner of speaking.  I found that while first starting out it was very difficult to use both the language and mathematical parts of my brain at once in both interpreting the language and figuring out how to use it.  After transcribing the first chapter, I became more adept at directly assimilating the knowledge, and transcription was no longer required. 
b)      Reading the front of the circumferential arc - In order to provide divisions into tens of arc-minutes, Hood's design calls for six arcs radiating out from the measurement at the inside arc, and isosceles triangles connecting each even degree on the inside arc with each odd degree on the outside arc.  In this way, reading the first line after 0 degrees at the first arc from the inside yields a measurement of 0 degrees and 10 arc-minutes.
c)      Reading the back side of the circumferential arc - On the backside of the circumferential arc, there are three 15-inch scales.  The scales are not true to size, but when the sector is opened to the marking on the circumferential arc the points of the feet will be spaced at the correct measurement.  Each of the 15-inch scales contains inches, divided into 6ths, 8ths, and 10ths, respectively, and each of these divisions is further divided in half.  I was not able to transfer to the arc the first inch of each scale, as the closest that the points can be is approximately 1 1/16 inches apart, or the width of a foot of the sector.
d)      Accuracy of the sector - It was clear to me when marking these scales onto the circumferential arc the truth of certain things I had read concerning the sector.  Although the sector I have made is very stiff in most of its movement, it was not very accurate in its transfer of the measurements.  For most use, this may be fine, but to inscribe the parts of an inch on the circumferential arc, it was not quite adequate to the task.  Markings ended up slightly unevenly spaced despite my best efforts.  This corroborates the claim I had read that the sector, although popular, eventually fell out of popularity because it was not accurate enough.
e)      Foot design - While Hood specifically called out the importance of precise points on the feet of the sector, a future instrument maker, Gunter, who based his sector on Hood's design, did not use feet, rather creating a decorative woodwork at the bottom.  I chose the design for the feet from an extant sector found in "Drawing Instruments," by Maya Hambly.  Although the sector in question was created in the 1700s, the foot design was plausible given Hood's requirements, and I could not find any other valid extant examples. 

6. Use of the Sector

 
Photo by Master Eirik Andersen.
As the points are set at the center of the feet, all measurements should be taken from the center (angled) line of the sector, unless stated otherwise.

Using the front side of the feet of the sector:
1.                  Finding proportional lines:
a)      If given a proportion only - (5/8, 1:2, etc.)  If the proportion does not fit between 0 and 120, multiply or divide both sides of the proportion to allow the proportion to fit between 0 and 120.  Open the sector any amount.  Given an example:  if the proportion is 5:8, measure with a divider the line between the two 50 marks on the sector and the two 80 marks on the sector.  These lines will have proportions to each other as 5 is to 8.
b)      If given a line and a proportion, looking for a line smaller than the line given - If doing so will result in a whole number, multiply 120 (the length of the feet of the sector) by the proportion to find a whole number.  Measure the given line with the points of the feet of the sector.  Use dividers to measure the distance between the feet of the sector at the number found.  If multiplying will not result in a whole number, place the length of the line between the feet of the sector at the larger number of the proportion, and use dividers to measure the distance between the feet of the sector at the smaller number of the proportion given.
c)      If given a line and a proportion, looking for a line larger than the line given -  If doing so will result in a whole number, multiply 120 (the length of the feet of the sector) by the inverse of the proportion to find a whole number.  Measure the given line with the feet of the sector at the number found (use dividers).  The length of the larger proportional line will be marked by the points of the sector.  If multiplying will not result in a whole number, place the length of the line between the feet of the sector at the smaller number of the proportion, and use dividers to measure the distance between the feet of the sector at the larger number of the proportion given.
d)      If given two lines - the same methods can be used to find the proportions of the lines to one another.

2.                  To find a proportional figure, find the length of the line upon which the original figure is based (for a circle, it could be the diameter), use 1 to find its proportional equivalent line, and then create the proportional figure based on this proportional line.

3.                  To find a proportional triangle given a triangle:
a)      To find a smaller triangle inside the given triangle, with one identical angle - take the length of one side and find the length of a line proportionally smaller than that line.  Starting at a point of the triangle at one end of the line measured, mark the calculated length parallel to and on the same line as the line measured.  From the end of this line that does not fall on the original triangle's point, draw a line parallel to the matching line on the original triangle.  If the given triangle is larger than the triangle sought, terminate the new parallel line when it reaches the existing line of the given triangle.
b)      To find a triangle or outside the given triangle, equidistant to all sides of the given triangle - draw lines perpendicular to the given triangle sides, toward the center of the triangle.  The intersection of these lines is the center of the triangle.  Measure the distance from the center to one corner.  Find its proportional equivalent and mark this length from the center.  Do the same with the remaining two distances to find the proportionally smaller or larger triangle.
c)      To find a larger triangle outside the given triangle, with one identical angle - take the length of one side and find the length of a line proportionally larger than that line.  Starting at a corner of the given triangle at one end of the line measured, draw the length of the new proportional line.  Do the same with the other line of the given triangle extending out from the original corner.  Join the two ends of the new proportional lines to create the new proportional triangle.
d)      Outside the given triangle - measure one line and create a proportional line where you wish.  Measure the second line, find the proportional line, and draw an arc of that length at one end of the first proportional line.  Measure the final line, find the proportional line, and draw an arc of that length at the other end of the first proportional line.  Where the two arcs cross is the final point of the triangle.  Draw the two lines from the two ends of the first proportional line to the found point.
e)      Similarly, the proportion of one like figure to another can be found.

4.                  Given a polygon or geometric figure (regular or irregular) that can be made to be built of triangles of various sizes (a triangulate) and a proportion, to create a proportional figure:  Divide the figure into triangles and use the same methods as were used for triangles in (3) to create the proportional figure.

5.                  To find a proportional figure if the figure is not geometric but curved, inscribe on the original figure a square grid (the more squares the better).  Label the lines of the grid.  Measure the length of one side of the square and create a proportionally larger or smaller grid, labelling it the same way.  Mark all points on the proportional grid that cross the figure in the first grid, then connect the points.

6.                  To divide a geometric figure proportionally so that one part will be of an area the given proportion to the other part's area:
a)      Triangle - reduce one side by the given proportion and draw a line from that mark to the opposite point of the triangle.
b)      Square, rectangle, or quadrilateral - reduce two parallel sides proportionally (from the same side) and draw a line between the marks.

Using the back side of the feet of the sector:
1.                               To find the length of a chord of a given circle that will inscribe a polygon of a given number of sides within that circle, measure the diameter of the circle with the feet of the sector.  Find the given number of sides for the polygon on the feet of the sector and measure the distance between them to find the length of the chord.  The reverse of this procedure will find the diameter of the circle that a given polygon can be inscribed within, given a chord length.
a)      If you draw a line perpendicularly from the center of one of the chords creating the polygon to the circle arc and then draw a line from one end of that chord to the point on the circle arc, continuing creating equal chords around the circle, you will create a polygon of twice as many sides as the polygon from which you started.
b)      If you reduce or increase the length of one side of the polygon proportionally using the front of the sector's feet and then inscribe the proportionally altered length of chord around the circle, you will create a polygon with a proportionally different number of sides.
c)      If a chord that would make the side of a polygon with number of sides x and the chord that would make the side of a polygon of number of sides y are inscribed side-by-side within a circle, originating from the same point, the arc distance between the other ends of the two chords will be one x*y portion of the whole circumference.  Example: If the two chords are that of a hexagon (6-sided) and heptagon (7-sided), the distance along the arc between the ends of these chords will be one forty-second (6*7) of the circumference.

2.                              Using the external markings on the backside of the feet, lengths can be found that, when squared, are a fraction (1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8) or multiple (2, 3, 4, 5, 6, 7, 8) of a given line.
a)      To find a smaller line than the one given, set the given line in the points of the sector and measure the distance between the markings for the specified proportion on the external scales.
b)      To find a larger line than the one given, set the length of the given line between the scales at the specified proportion, and measure the resulting line at the points of the feet.
c)      The same can be done with areas of squares, triangles, polygons, and circles/semicircles/ovals, given one line in the figure.

Using the feet with the circumferential limb:
1.      Using the arc, the feet can be set 90 degrees apart, allowing the feet of the sector to be used to create right angles to draw perpendicular lines.
2.      The sector may be used as a protractor to create or measure angles up to and including the maximum angle measurement given on the circumferential limb.
3.      Creating the right angle, the sector may be used to draw a sundial, providing the locations of the hour lines.
4.      To measure the distance between two bodies in space, the left foot should be held steadily pointing to the first object.  Move the right foot until it points to the second object.  The degrees of difference between the two points can be read on the circumferential limb.

In addition to the uses given above, there are additional uses which make use of the accidental parts of a sector, which I have not reproduced.  Many of these functions can be reproduced to varying degrees without the other parts of the sector, but may require other, separate instruments, such as a ruler, plumb line, or a moveable sight.

7. Resources

 
Photo by Lord Alexander Gladstone.
My main resource when completing this project was Thomas Hood's own manuscript, "The Makinge and Use of the Geometricall Instrument Called the Sector," of 1598:

Hood, Thomas, The Makinge and Use of the Geometricall Instrument Called the Sector, 1598.  Printed by John Windet at the North Dore of Paules Church by Samuel Shorter.

Other Resources:

Bennett, JA, The Divided Circle - A History of Instruments for Astronomy, Navigation and Surveying, 1987.  Phaidon Christie's Limited, Oxford, UK.

Drake, Stillman, Galileo at Work His Scientific Biography, 1978, Dover Phoenix Editions, Mineola, NY.

Fahie, JJ, Galileo his Life and Work, unknown date.

Galilei, Galileo, Le Operazioni del Compasso Geometrico et Militaire, 1606.  Padoua, Italy.  Accessed through the Internet Archive in January 2015, https://archive.org/stream/leoperazionidelc17paol#page/80/mode/2up

The Galileo Project - The Sector, accessed January 2015.  http://galileo.rice.edu/sci/instruments/sector.html

Hambly, Maya, Drawing Instruments 1580-1980, 1988.  Sotheby's Publications, London, UK.

Hutton, Charles, A Mathematical and Philosophical Dictionary - The Sector, accessed January 2015, http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/archim/dict/hw?lemma=SECTOR&step=entry&id=d006

Instruments of Science - An Historical Encyclopedia, 1998.  The Science Museum, London, and The National Museum of American History, Smithsonian Institution, in Association with Garland Publishing Inc, New York, USA, and London, UK.

Sangwin, CJ, Edmund Gunter and the Sector, 2003.  School of Mathematics and Statistics, University of Birmingham, Birmingham, UK.  Accessed January 2015, http://chsi.harvard.edu/docs/gunter_sector.pdf

Tomash, Erwin, and Williams, Michael R, The Sector:  Its History, scales, and uses, unknown date.  Accessed January 2015 from http://wenku.baidu.com/view/01d34d48e45c3b3567ec8bba.html


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