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  • Circle Division

    I downloaded a copy of "Foundations of Mechanical Accuracy" that you guys recommended in my other threat. Wow! What a fascinating book and a wealth of information.

    In Part 3, Moore describes dividing the circle as one of the four pillars of mechanical accuracy. Is circle division used as a basis for establishing consistent linear measurements, or is it only used to create angle standards, (or both)?

  • #2
    I have not seen any fundamental connection between circle division and linear measure. But in practice there are some obvious ones. For instance, a micrometer relies on a well made, accurate screw thread and a thimble that is accurately divided into a number of equal, circular divisions. This is more of a practical matter and not fundamental to the idea of measurement. There are other ways to accurately measure things that do not use either a screw or divisions of a circle. A vernier caliper is one.

    Another old trick for producing circular divisions is to wrap a linear scale around a cylinder. When the linear scale fits the circumference of the cylinder exactly, it then divides that circle. Again, this is one of usage, not any fundamental principle.
    Paul A.
    SE Texas

    And if you look REAL close at an analog signal,
    You will find that it has discrete steps.

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    • #3
      Originally posted by Paul Alciatore View Post
      Another old trick for producing circular divisions is to wrap a linear scale around a cylinder. When the linear scale fits the circumference of the cylinder exactly, it then divides that circle. Again, this is one of usage, not any fundamental principle.
      It might not be fundamental, but it's still pretty cool and practical.

      What is the main purpose of circle division if not for measurement?

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      • #4
        "Foundations" is one of the most excellent books, I have re-read it several times. Dividing the circle has had applications since ancient times. The Babylonian astronomers had a protractor with a 40-meter radius, dividing the circle into minute parts.

        Being able to divide the circle is very useful for navigation among other things -- since ancient times, traders and sailors have used this.
        In more modern times, it is used for gear teeth, and all kinds of locating and positioning. The hard disks in a computer, or a DVD, depend on positioning very accurately, for example -- all of that accuracy began somewhere. (Probably at IBM Research in the 60's -- they pioneered the hard drive and had fantastic machine shops)

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        • #5
          Originally posted by Stargazer View Post
          What is the main purpose of circle division if not for measurement?
          anything you need to divide a circle for....e.g. dividing plate, a gear, graduated collar, etc
          in Toronto Ontario - where are you?

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          • #6
            Circle division is interesting. Divide by 2 is easy- draw a line across the diameter, crossing the center. Divide by 6- first divide by 2. Then set a compass to the radius. Pivot at a point where the diameter line crosses the circle. Swing around and mark two points on the circle. Pivot where the diameter crosses the other side of the circle, and mark out two points on the circle. That gives 6 equidistant points. Divide by 7 is interesting- start with divide by 6. Then draw a line through two of those points perpendicular to the diameter line. Where that line crosses the diameter, set your compass from that point to where that line crosses the circle. That's your compass spacing to divide the circle into 7 parts. I'm sure this can go on and on. There is a way to divide by 5, but I don't remember it.

            The compass is a pretty simple device, but you can obviously use it to mark out some interesting geometry. If you're marking out bolt hole locations for instance, the accuracy you get is determined by how well you can center your center punch, and not anything ambiguous about the method of layout.
            Last edited by darryl; 02-13-2021, 12:35 AM.
            I seldom do anything within the scope of logical reason and calculated cost/benefit, etc- I'm following my passion-

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            • #7
              Originally posted by Stargazer View Post
              Is circle division used as a basis for establishing consistent linear measurements, or is it only used to create angle standards, (or both)?
              You got that right, it is both.

              Problem being, you are trying to mix circles with angles. Wont work.

              The circle is perfect, the angle is not.

              Simple math here right? Divide (Subtract if you need) a circle into its self, Zero. Try to do that mathematically with an angle. No can do, too many angles LOL JR

              My old yahoo group. Bridgeport Mill Group

              https://groups.yahoo.com/neo/groups/...port_mill/info

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              • #8
                Originally posted by JRouche View Post

                You got that right, it is both.

                Problem being, you are trying to mix circles with angles. Wont work.

                The circle is perfect, the angle is not.

                Simple math here right? Divide (Subtract if you need) a circle into its self, Zero. Try to do that mathematically with an angle. No can do, too many angles LOL JR
                If you were "starting from scratch" (conceptually, not necessarly in real life) how would you relate circle division to making more precise linear measurements?

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                • #9
                  Originally posted by nickel-city-fab View Post
                  "Foundations" is one of the most excellent books, I have re-read it several times. Dividing the circle has had applications since ancient times. The Babylonian astronomers had a protractor with a 40-meter radius, dividing the circle into minute parts.

                  Being able to divide the circle is very useful for navigation among other things -- since ancient times, traders and sailors have used this.
                  In more modern times, it is used for gear teeth, and all kinds of locating and positioning. The hard disks in a computer, or a DVD, depend on positioning very accurately, for example -- all of that accuracy began somewhere. (Probably at IBM Research in the 60's -- they pioneered the hard drive and had fantastic machine shops)

                  It masterful book. Thanks again for recommending it. I spent the entire weekend on the circle division section. When I googled the subject, I came across a post from last October on some Internet message board topic on the Whitworth Three Plate method where one poster wrote:

                  "A similar process can also get you a master square, although there are other ways to do it (such as via circle division)." https://news.ycombinator.com/item?id=24684365

                  That I can't see to understand how you go from a layout trick (circle division) and obtain a master square. What could he be referring to?

                  Comment


                  • #10
                    I'm sure this is redundant now, but laying out a perfect 90 is pretty easy. Draw one line, mark a point in the middle of it, roughly. Use a compass from that mark to draw two other marks along that line. Then from those marks, increase the spacing of the compass and draw two marks above the line. Where those marks intersect, draw a line through the first mark you put on the line. That's a perfect perpendicular.

                    In our woodwork shop, I had to repair the saw a few days ago. In order to do that I had to crank an angle to the blade so I could access what I needed to. Then I had to return it to 90 degrees. The standard way is to use the pointer and adjust to the mark. That is very close to 90, but maybe not 100%- so I make a cut across a piece of scrap. Then turn one piece over and butt the edges together. If the pieces are laying flat, but the edges don't touch all the way across, you don't have a 90. On our saw, the pointer has to align with one edge of the marking, not down the center of the mark, to give a perfect 90. I know this is only a fraction of a degree, but I like my corners to fit perfectly, and this is a way to check it.

                    There are many way of 'mcgyvering' in metalworking. Some are quite simple, and it's mainly some rational thinking that will lead you to ways of achieving an accuracy. I've read some of that book on foundations of accuracy- or at least the same kind of thing in another book. It is interesting how accuracy was brought about over time.
                    I seldom do anything within the scope of logical reason and calculated cost/benefit, etc- I'm following my passion-

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                    • #11
                      Darryl in Post # 6 nailed it !
                      Remember fellows, the Protractor is a "New Tool"
                      In the 1800's and before , every angle was done using circles and chords , or circles and dividers
                      Since the circle could be divided , and the cord gave an absolute angle . Drawings made in those old days would state the circle diameter and the chord length and dividers were the common tool of the trade Or for example the drawing says " 6 bolts on a 20" circle " and the machinist would mark off the distance with his dividers ....
                      The chord for a right angle (90) with two equal legs is 1.414141414 ( x the leg)
                      Rich
                      Green Bay, WI

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                      • #12
                        I have a 24” rule of chords rule, very handy thing
                        mark

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                        • #13
                          Originally posted by Stargazer View Post


                          It masterful book. Thanks again for recommending it. I spent the entire weekend on the circle division section. When I googled the subject, I came across a post from last October on some Internet message board topic on the Whitworth Three Plate method where one poster wrote:

                          "A similar process can also get you a master square, although there are other ways to do it (such as via circle division)." https://news.ycombinator.com/item?id=24684365

                          That I can't see to understand how you go from a layout trick (circle division) and obtain a master square. What could he be referring to?
                          In the "Foundations" book, they used the master plates to create their master straightedges, which also happened to be master squares -- they were square or box in cross-section. Darryl and Rich both mention one of the ancient methods of using just dividers. Either way is perfectly valid. It's amazing how much high school geometry applies to mechanical stuff -- using just a dividers and a ruler or other straight edge, you can create whatever you need. Notice that either method still requires the use of a straight edge --
                          Last edited by nickel-city-fab; 02-14-2021, 04:49 PM.

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                          • #14
                            Originally posted by nickel-city-fab View Post

                            In the "Foundations" book, they used the master plates to create their master straightedges, which also happened to be master squares -- they were square or box in cross-section. Darryl and Rich both mention one of the ancient methods of using just dividers. Either way is perfectly valid. It's amazing how much high school geometry applies to mechanical stuff -- using just a dividers and a ruler or other straight edge, you can create whatever you need. Notice that either method still requires the use of a straight edge --
                            I gotta tell you, I find the whole thing captivating. I'm studying math at UB and it's a big topic in mathematical philosophy how math is a real thing - not just a tool someone came up with, but that the objects and relationships are surrounding everything we do. And what's academic and philosophical in that discussion is just clear as a bell reality when you see geometry and mathematical objects come to life in the shop. It's like you could never get bored doing this kind of work.

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                            • #15
                              You would love DaVinci's "virtruvian man". How the proportions of the human body are all in the golden ratio, and how they became the foundation of the Imperial system like 8,000 yrs ago. Some of which we still use today. Unfortunately, some of my favorite websites on metrology have "gone dark" especially about relating ancient units of measures and math that they did way back when. I'll have to go looking again.

                              EDIT: This is the guy I was remembering... Awesome writing, kinda like "Foundations" but it goes back to ancient times. Bummer their website is gone: https://en.wikipedia.org/wiki/Livio_Catullo_Stecchini
                              Last edited by nickel-city-fab; 02-14-2021, 05:48 PM.

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