Creating a Master Square With Circle Division

# Thread: Creating a Master Square With Circle Division

1. Aluminum
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## Creating a Master Square With Circle Division

A comment entered on this page The Whitworth Three Plates Method (2017) | Hacker News about the Whitworth Three Plate Method said that a master square could be arrived at using circle division. ("A similar process can also get you a master square, although there are other ways to do it (such as via circle division).")

Is this a real thing? How would that even work?

2. Very precise indexing table used to grind two sides of a square at 90° should work.

3. Aluminum
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Originally Posted by eKretz
Very precise indexing table used to grind two sides of a square at 90° should work.
Interesting. What are the practical limits to accuracy with that setup?

4. Diamond
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Originally Posted by eKretz
Very precise indexing table used to grind two sides of a square at 90° should work.
Originally Posted by Euclidean
Interesting. What are the practical limits to accuracy with that setup?
It's a shame we lost RJ Newbold.....

5. Diamond
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I would depend on the RT and squareness required of course. With one of the really fancy RT's with the discrete steeps (Moore?), you could probably get a really really high level of accuracy. I guess that assumes perfect linear motion as well.

As for practical? This is likely a dancing on pin heads type topic, you want practical, grind squareness to a tenth or phone up Mcmaster Carr and get the Starrett sq of the size you need. Or maybe you do have a practical requirement? otoh, nothing wrong with just being curious.....if so, read Moore's Foundations of Mechanical accuracy...there's a whole chapter devoted to dividing circles. That, and a bit of trig, should give you your answer

6. If you need perfection, make three on the indexer and scrape them to each other. Otherwise, some of the most precise rotary indexers can get pretty damn close. RJ's were able to directly index one arc-second, though I don't remember what his specified accuracy was. But they are pretty small. I know that Moore made some larger ones in a similar vicinity accuracy-wise.

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Originally Posted by Mcgyver
I would depend on the RT and squareness required of course. With one of the really fancy RT's with the discrete steeps (Moore?), you could probably get a really really high level of accuracy. I guess that assumes perfect linear motion as well.

As for practical? This is likely a dancing on pin heads type topic, you want practical, grind squareness to a tenth or phone up Mcmaster Carr and get the Starrett sq of the size you need. Or maybe you do have a practical requirement? otoh, nothing wrong with just being curious.....if so, read Moore's Foundations of Mechanical accuracy...there's a whole chapter devoted to dividing circles. That, and a bit of trig, should give you your answer
Just curious here. I'm very new to all of this and just trying to figure out how you could calculate the precision on it. When I came across the thread on the other website, I was just having difficulty wrapping my mind around how you could go from circle division to square. Your reply really helped me out and I appreciate it.

8. Hot Rolled
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I suppose that to grind up a cylinder square,you arrive at perfect squareness in one fell swoop. Although it will only register squareness along a line contact. I have a matched pair of lapped 9" cast iron cubes,made by Pitter Tool and Guage,,which I put on the surface plate when I want to check anything for squareness.

9. The simplest and most accurate way I know of to generate a "square" (or a right angle) is on a lathe. It should be noted that many precision master right angles are produced this way so it is not a new idea. I had a commercial made one in my shop that was 12" tall by 4" diameter. One can generate a pretty good right angle standard even on a fairly clapped out lathe. It just takes more work. Simply turn a reasonably straight cylinder of suitable length and undercut the inner 9/10 of the diameter of the right end and face the remaining outer 1/10. The level of achievable precision is limited only by patience and the level of precision of your gages. It can be done with only a micrometer but a dial test indicator helps.

Next thing (or first thing!) of course is how you measure the precision of any right angle.
One method is to use a master precision square and look for light between the beam edge and the test piece or feeler gages or cigarette papers or whatever.
Another method is to setup a surface gage with a DTI with the probe end of the DTI at the height you want and the ball end of the rod or ball bearing in the V "crotch" of the surface gage. Zero it out on your standard and then measure the test piece. You will get a precise number of "X thousandths (or tenths) at Y height/".
Some companies make purpose built gages for measuring "squareness". Hermann-Schmidt used to make one. I made my own from the plans in James Harvey's "Machine Shop Trade Secrets."

If you do a search you will find that this very subject has been discussed many times on PM.

-DU-

10. Titanium
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I don't positively recall the maximum permissible error of RJ's Newbould Indexer, but it was several times that of Moore's offerings. Don't misunderstand, RJ designed and manufactured a precise machine that sold at an excellent Precision : Price ratio.

Moore's queen of the ball was the 1440 Indexer, a Hirth-coupling divider that divided the circle into 1/4 degree increments having 1/10 arcsecond maximum permissible error. Moore also offered a small (10 inch, IIRC), purely mechanical, rotary table having a maximum permissible error of 2 arcsecond.

11. Grind the sides of a small monolith on the surface grinder. If it (and you) are any good, it will be near dead parallel. Check it on the surface plate and scrape if necessary. Then, use the ball in crotch method to see if it sits vertical. Just look at one side, then the other. Scrape until there's no difference. Or just go the McMaster or other tool site.

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Whatever the method used to make things square, you must be able to inspect for squareness. We typically used a squareness gage similar to this one:

The squareness gage is used on a surface plate. It is zeroed on a known good square (typically a cylindrical square) then the part squareness is compared using the gage. Both cylindrical squares and squareness gages can be shop made (excepting the indicator).

Hope this helps.

Best Regards,
Bob

13. Just for reference since O.P. is asking, by my calculation an arc-second equates to about 60 (58) millionths of an inch across a foot. That's 0.000058".

And while yes cylindrical squares are an easy method to create a good square reference, OP was specifically asking about creating a master square using circle division.

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Is there any chance the reference was suggesting the use of geometric construction like shown in this link?

How to construct a square inscribed in a circle with compass and straightedge or ruler - Math Open Reference

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Originally Posted by eKretz
Just for reference since O.P. is asking, by my calculation an arc-second equates to about 60 (58) millionths of an inch across a foot. That's 0.000058".
That's insanely precise.

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Originally Posted by NRDock
Is there any chance the reference was suggesting the use of geometric construction like shown in this link?

How to construct a square inscribed in a circle with compass and straightedge or ruler - Math Open Reference
I thought that at first, but the other thread referred to a master square, so I didn't think it was referring to merely a geometric construction but something that could be used to test other squares against.

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I do not know what this circle division thing means, any easy to understand help for rookies?
Bob

18. Titanium
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My understanding is that "dividing the circle" was done in the 1700s by first carefully scribing a circle, and then using dividers to step all the way around the circle. If the final divider marking did not fall precisely on the first divider marking after the desired number of steps, the dividers were adjusted slightly, and the entire stepping-off process, error evaluation, divider adjustment, and re-stepping is repeated until the final point is precisely superimposed on the starting point.

This same fundamental process can be applied to division of a cylindrical surface.

Bottom line, though, is that Euclidian construction is a much more direct way of constructing a perpendicular.

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"Circle division" does not necessarily mean the Euclidean method using compass and a straight edge. Another more practical method involves cutting a number of disks from a round rod of uniform diameter. Then, using a bit of math you can calculate the diameter of a circle which will exactly fit inside a circular ring of a given number of those discs. So, if you want to divide a circle into six parts (I am making the math simple) then the central circle will be the same diameter as the six discs. And when the seven discs are arranged in a hexagonal pattern with the center one touching all six of the others, then you have divided the circle into six equal parts. With a means of holding these discs in place on a shaft and with an external arm with a finger on it that can be brought into each of the six valleys between the discs, you have a very accurate indexing device.

Other numbers of the discs can be used. For instance if you want to cut a 40 tooth worm gear to make an indexing head, then you would use 40 of the discs and the diameter of the inside circle or hub would be given by the formula:

D = 2 (( r / sin(180/n)) - r)
where
D = diameter of inner hub/circle
r = radius of individual discs
n = number of divisions

so for 40 divisions D = 2 (( r / sin(22.5)) - r)

The accuracy of this method depends on the accuracy of machining the two circles: the inner hub of diameter D and the individual discs of diameter 2r. And on the accuracy of centering the affair on the shaft. Four discs or any multiple thereof will provide you with four 90 degree indexing points.

Of course, once you have an accurate worm gear installed in an indexing device, you can divide the circle by any number of divisions via other, well known methods with that indexing head.

Originally Posted by Euclidean

A comment entered on this page The Whitworth Three Plates Method (2017) | Hacker News about the Whitworth Three Plate Method said that a master square could be arrived at using circle division. ("A similar process can also get you a master square, although there are other ways to do it (such as via circle division).")

Is this a real thing? How would that even work?

20. Originally Posted by eKretz
RJ's were able to directly index one arc-second, though I don't remember what his specified accuracy was.
His website says "Top Side Shotpin: Allows for lock-in indexing every 15 degrees, repeatable within 10 arc seconds." So about a half thou over 10". There are other relevant specs here: The Newbould Indexer | Meadville, PA | (814) 337-8155

-DU-