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Thread: Using a sector

  1. #1
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    Using a sector

    The latest issue of Popular Woodworking has an article by Jim Tolpin on using a sector, a hinged tool for scaling and proportions. It highlights enough cool tricks that I began thinking of making a version for my own use. There is a picture of an old sector attached to the article. That version has several scales marked on the blades including some marked on an angle. This is hard to imagine without a picture.

    Here is my question. Any ideas about the calculation functions of those "angled" scales?" Scales that run parallel with the blades are obviously useful for proportions but I can't think far enough to see the use of the angled scales.

    Peace,
    Harlan B

  2. #2
    Oh my, there's so much. I think I'll try to get Don to explain it because he's worked with it a lot more than me.

    sector.jpg

  3. #3
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    Thanks for the image Larry. That helps. My head tells me there are some good secrets there but I need a little explaining.

  4. #4
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    Here is a link that may explain more:

    http://collectingme.com/Measuring/French_Sector.aspx

    It does not mention anything about angles, but the markings at the hinge on the instrument shown looks like some angles could be measured.

    So often there are uses for such items that were never published which then become lost to history.

    jtk
    "A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
    - Sir Winston Churchill (1874-1965)

  5. #5

  6. http://www.popularwoodworking.com/ar...of-the-sector’
    In Chris Schwarz'z blog a video of explanation.

  7. #7
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    Jim, thats a beautiful version. I'm looking to make more of a "user" variety.

    Caspar, that link is so informative I printed it out. It may have the answer to my original question if I understood it...

    Richard, the way Chris uses a sector in that clip are all proportional. I'm interested in non-proportional uses.

    Thanks all,
    Harlan Barnhart

  8. #8
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    Harlan et al,

    Unfortunately, I think the only way to really understand how the sector works is to obtain one and sit down with one of the more detailed descriptions of its use and work through the various functions. There are several of those descriptions which I can supply references to if you'd like. But the one I would recommend at the outset, because it includes information on its use in conjunction with the classical orders, is _A Treatise of Mathematical Instruments_ by John Robertson, © 1775 (third edition). Reprint by Flower-de-Luce Books.

    But, maybe I can at least describe the primary purpose of some of the "double scales" usually found on sectors. They are the ones which appear as diagonal scales which you are asking about. They are called double scales because each such scale has exactly the same divisions and markings along each leg of the rule. And, they are diagonal so that the operative line of each scale can accurately radiate from the center/pivot-point of the hinge. They can be at an angle to the edges of the legs of the sector because the distance between the relevant points of each part of the scale is set to your dividers by opening the legs of the sector the required distance.

    In fact, the line of lines (marked "L"), by which one can divide a line of a given length into any number of equal parts, is one of those diagonal "double scales." In this sense, the shop-made sectors shown by Jim Tolpin and Chris Schwarz misrepresent the typical sector's line of lines. It is easier to set the dividers on an actual sector because one isn't trying to set the tip of ones dividers on the arris of the sector's leg. Also, I'm a little puzzled why they are using divisions up to 13. All of the sectors, and older texts describing them, that I've seen, have ten equal divisions along the line of lines, each of which, in turn, is divided into ten sub-divisions. So I'm curious as to the rationale for going up to thirteen.

    In any event, the other double scales usually found on the same face of the sector as the line of lines are the polygon scales (usually marked "POL"), the scale of chords (marked "C"), and the line of secants (marked "S").

    The polygon scale can be used to determine the length of the side of any polygon, having between four and twelve sides, which can be constructed in a circle of a given diameter. For example, say you wish to construct a seven-sided polygon in a circle having a diameter of 4 inches. Set your dividers to the radius, 2 inches, then open the legs of the sector so that the distance between the "6" division of each leg of the polygon scale are equal to the span of your dividers. (This, because the radius of any given circle is equal to one-sixth of the circumference of that circle.) Then reset your dividers to correspond to the distance between the "7" divisions on the polygon scale and you have the length of one side of the seven-sided figure which can be constructed in that circle. You can reverse this procedure to determine the radius/diameter of the circle required for constructing a polygon if you already know the length of one side.

    The line of chords operates similarly. With it, you can determine the length of a chord of any number of degrees of a circle whose diameter/radius you know. Again, set your dividers to the radius and open the legs of the sector so that the 60 degree division of each leg of the line of chords correspond to that distance, then, reset your dividers between the appropriate angle divisions on each leg and you have the length of that chord. It becomes more complicated for chords of more than 60 degrees, but the principle remains the same.

    I haven't spent any real time on the secant scale, but it also operates similarly. With your dividers set to the radius of a given circle, open the legs of the sector so that the zeros, or starting points, of the secant scales correspond to that distance. Then reset your dividers to the angle divisions corresponding to the secant you seek.

    The diagonal, or double, scales on the other face of the sector are for sines (marked "S") and tangents (marked "T"). There are two tangent double scales, one for less than 45 degrees and the other for more than 45 degrees. Again, I haven't spent much time on these, but they are used similarly to the secant scale. Except that the legs of sector are opened so that the ends of these scales, rather than the beginning, are set at the distance of the known radius. I think there are other ways to use these scales, too, but I haven't looked into that.

    Hope this hasn't been too confusing.

    Don McConnell
    Eureka Springs, AR

  9. #9
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    Don,
    Thanks for the explanation. That helps me get started. I see now that the fact that the scales are on an angle does not mean they operate differently from the scales parallel to the edges because they all radiate from a common pinion. After thinking about it, the functions I find most usable would be the dividing lines or circles into equal segments which seem to be the line of lines scale and the polygon scale or perhaps the line of chords. I think my next step is to watch ebay for a sector and a book to play with.

  10. #10
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    Quote Originally Posted by Don McConnell View Post
    Harlan et al,

    In fact, the line of lines (marked "L"), by which one can divide a line of a given length into any number of equal parts, is one of those diagonal "double scales." In this sense, the shop-made sectors shown by Jim Tolpin and Chris Schwarz misrepresent the typical sector's line of lines. It is easier to set the dividers on an actual sector because one isn't trying to set the tip of ones dividers on the arris of the sector's leg.
    I was wondering that as well when I saw the video - about the only rationale I could come up with was that it makes it easier to set direct measurements from a workpiece by butting the edges of the sector against it? Still not sure why they ended up with 13, but I've only seen the video and not the article.

    But as Harlan said, thanks again for the great explanation, Don! I knew a bit about sectors before this, (well, that they existed, and the general principle) but you really shed a lot more light on this.

    I had seen a sector before in a museum that also had some sort of weird proportional scales on it, something for figuring out the weights different metals of the same size, or something? It was interesting.

    Harlan - you mention cruising the 'bay for a sector - be careful with materials - a lot of the ones I've seen pictures of are ivory, and if they're anything like the folding rules I've seen (I have a co-worker who collects them) they change size over the years, but not uniformly! Seems like this could be an issue...

    EDIT: Just started looking at the previous links; one of them mentions the ivory shrinkage thing. Sorry for restating the obvious...
    Last edited by Joshua Pierce; 04-26-2011 at 9:43 PM.

  11. #11
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    Don, thank you for reminding me of how intellectually lazy I am. I read "polygons" and my mind started to wander. All I could think was "dang, I wish I had some 'sectors' to put on ebay right about now...."

    BTW, if you are the same Don McConnel that has written articles about wooden planes, I think they're great articles.
    Last edited by john brenton; 04-26-2011 at 11:31 PM.

  12. #12
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    Quote Originally Posted by Don McConnell View Post
    Also, I'm a little puzzled why they are using divisions up to 13. All of the sectors, and older texts describing them, that I've seen, have ten equal divisions along the line of lines, each of which, in turn, is divided into ten sub-divisions. So I'm curious as to the rationale for going up to thirteen.

    Don McConnell
    Eureka Springs, AR
    According to the blog:
    Jim responds:

    The reason I went to 13 divisions is close to what Bear discovered: you can proportion to 8 to 13 which creates the most pleasing (according to actual research!) rectangle. (Regardless of whether that’s a “golden” rectangle or not – turns out that whole Golden Ratio thing is largely a Victorian-era fabrication!)

    Also, you want to go to at least 12 so you can easily scale in thirds, fourths, sixths. There is rarely any need to go beyond 13 as all the proportioning you might want happens below 13!

    — Jim Tolpin

  13. #13
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    Kris et al,

    This will have to be brief as I need to get back down to the shop.

    While I think the sector is a quite ingenious instrument, I'm not really an advocate of everyone rushing out to buy one or make one up for use in the shop. I'm aware of a couple of sectors which have shown up in pattern maker's kits of tools, but, beyond that, I'm not aware of any evidence that they were historically used in woodworking shops. There was an effort to introduce a similar instrument, known as Scamozzi's Rule, into the carpentry trade as early as the 17th century. And while there are a few passing references to it in later sources, there is no real evidence that it was widely adopted.

    Rather, it appears to have been mostly used in conjunction with navigation, gunnery and by architects or designers. And, regarding the latter, Sheraton has a nice little section on the sector in the complete version of _The Cabinet-Maker and Upholsterer's Drawing-Book_, so we know that at least one furniture designer was familiar with its use. In this regard, I feel the primary value of the sector, today, is the light it sheds on architectural and furniture design process in the eighteenth and nineteenth centuries. (There is also a brief account of the sector in Blackie and Son's _The Victorian Cabinet-Maker's Assistant, © 1853.)

    For this reason, I find Jim Tolpin's "redesign" of the line of lines on the sector confusing at least. One of the things which becomes clear if one studies the sector, is that there was no accommodation to a supposed "special" ratio in the instrument. (And, this is consistent with other architecture/design sources from the period.) Given that he seems aware of the a-historical nature of this alteration, I think the least Jim could have done is first accurately describe the instrument as it was used historically.

    Toward that end, there is a brief and readable article on the history and use of the sector authored by mathematicians Erwin Tomash and Michael Williams, which people might be interested in. The following url does not lead to a web-page, rather it should (hopefully) begin an automatic download of the article. At least it does for me:

    ftp://ftp.cpsc.ucalgary.ca/pub/users...%20version.doc

    Don McConnell
    Eureka Springs, AR

  14. #14
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    One comment that supports Don's thoughts about sector use being primarily confined to applied mathematics and architects in the past - I collect rules and measuring instruments; I have never seen a sector "in the wild" that was more than about 12" long. Most are about 7" long. That doesn't mean longer ones don't exist, but it does suggest that their use was uncommon in the woodshop, at least as direct-measurement devices as Chris demonstrates in the article.

    Based on Don's description of the lines on a typical sector, their uncommonness in a size that would be appropriate for cabinet makers also fits. In a typical cabinet-maker's shop, there would be little, if any, call for making items of a polygon nature other than 4-sided. And their use as a proportionment of linear length could be fairly easily accomplished with dividers a spare board, and some chalk. And the world is full of antique dividers, and there are examples in several museums of case pieces where the back boards or the inside of the case sides have chalk or graphite drawings of the draw layout of the piece.

  15. #15
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    Dave, et al,
    While I agree that a sector was not likely to have been used by 18th century woodworkers (I do not believe that a sector was found in the Dominny workshop, for instance; and period plans for furniture are few and far between), taking the sector as described by Tolpin as a useful tool is interesting. If you remove the "antiquity" of it, and perhaps the name (a real period sector has several scales, not just one) and think of it that way, it has some application.
    For that, I think it is worth a try. Certainly for a couple pieces of scrap and a cheap hinge it is one of the least expensive tools promoted.
    Mike
    From the workshop under the staircase, Clinton Township, MI
    Semper Audere!

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