Squaring Work With Dividers and a Rule

I read that if you can measure accurately enough, you can square up a work piece without using a square based on the Pythagorean Theorem. This got me to thinking: If calipers/dividers can make precise measurements, could you accurately square up work with dividers and a rule/vernier/micrometer? Just wondering.

Hi,

Get a High School Geometry book or look for Geometric Constructions on line and check out constructing perpendicular and parallel lines. All you need is a straight edge and a set of dividers. Eculid came up with some pretty useful stuff.

rwilliams said:

Hi,

Get a High School Geometry book or look for Geometric Constructions on line and check out constructing perpendicular and parallel lines. All you need is a straight edge and a set of dividers. Eculid came up with some pretty useful stuff.

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That I understand. What I'm wondering is whether it is too hard or impossible to use the dividers and a scale to determine square of if that's something people do to calculate error.

You could wind up with a parallelogram using a rule or dividers. Two long sides same length, two short sides same length. Not always 90° .

Yeah, but that's why you compare the diagonals, which the old a^2 + b^2 = c^2 is going to encourage you to do anyway.

Impossible? No!

Easy? No!

Accurate? That depends.

Done to check other methods? Not really.

The geometric construction of a line that is perpendicular to another line is done with a line on, in the interior of a plane; that usually is in the middle of a sheet of paper. It assumes that one point of the dividers can be placed at a point that has been marked on that plane. The first problem that I see with using it to square a piece of material is that you usually want to work from AN EDGE. And it is hard to stick the point of a divider into an edge. You either will fall off the edge is it is placed only half way on it or you will be inside of that edge if you can make it stick.

If I were forced, kicking and screaming, to do it this way, the first thing I would do would be to draw another line on the surface and parallel to the edge. But then you still need to put one of the points of the dividers on that edge and you are back to the original problem.

Euclid's geometry was not intended to be a set of practical procedures for precision work. Yes, people like carpenters and stone masons (pyramid builders, for instance) may have used some of these procedures, but they were not working to thousandths of an inch or better. Euclid was a philosopher, a "thinker" not a machinist or even a carpenter. Philosopher in his day would be equivalent to a mathematician in today's world.

In a machine shop and even in modern carpentry shops there are methods that are far better in that they are far more practical. They can be carried out a lot more easily and will produce more accurate results. Ask any carpenter who knows his stuff how he knows his square is accurate and he will slap it down on the edge of a board and draw a line with it. Then he will flip it over and draw a second line either on top of that first one or just a small distant from it. If the two lines are parallel, he assumes the square is accurate. This is usually good enough for carpentry.

A good machinist will use other means. One such is the use of a block that is ground with parallel faces on a surface plate and a device called a squareness comparator. Another is the use of a cylindrical square, which can be made with a good lathe.

Taft-Peirce COMPARATOR SQUARES by Suburban Tool, Inc.

Euclidean said:

That I understand. What I'm wondering is whether it is too hard or impossible to use the dividers and a scale to determine square of if that's something people do to calculate error.

Click to expand...

Euclidean said:

I read that if you can measure accurately enough, you can square up a work piece without using a square based on the Pythagorean Theorem. This got me to thinking: If calipers/dividers can make precise measurements, could you accurately square up work with dividers and a rule/vernier/micrometer? Just wondering.

Click to expand...

I am glad to see someone use the correct terminology for "rule".

Traditional Greek geometry uses a compass and straight edge, not a rule. There is no direct measurement only comparison or ratio. Analytic geometry came much later.

An ancient quest, squaring the circle, finding the geometric equivalent of a circle in square form, don’t think it was ever done but I may be completely wrong!, keep filing flat, rotate 90 degrees, repeat, do that 4 times, when the flats meet you’ve won, vee block and clamp required
Mark

you could find zero with the tip of your dick, but it's much easier with a $10 chinese dial indicator?

if you want to go the hard row to hoe, that's your business. just don't count me in. i don't want to waste
my eyeballs with some lame 1/100 scale when my MTI digimatic will do.

don't be a hero . use modern tools to find the zero.

i have read that you can also find square .....with a freakin' square......much easier than postulating
and pontificating about grease.

EPAIII said:

The geometric construction of a line that is perpendicular to another line is done with a line on, in the interior of a plane; that usually is in the middle of a sheet of paper. It assumes that one point of the dividers can be placed at a point that has been marked on that plane. The first problem that I see with using it to square a piece of material is that you usually want to work from AN EDGE. And it is hard to stick the point of a divider into an edge. You either will fall off the edge is it is placed only half way on it or you will be inside of that edge if you can make it stick.

If I were forced, kicking and screaming, to do it this way, the first thing I would do would be to draw another line on the surface and parallel to the edge. But then you still need to put one of the points of the dividers on that edge and you are back to the original problem.

Click to expand...

I think that's what I was getting at. If you had to do it, or just wanted to try it as an experiemnt, what would you do to put one of the divider points accurately at the end of the line? Would you use a loupe?

While he may be using correct terminology "rule" when grouping it with Vernier calipers and micrometers, he is not correctly stating the rules of Euclid. The classic statement is "a straight edge and compass".

Rules or rulers have divisions marked on them. Straight edges do not and in the Euclidean sense, they can not be marked with any. They are used without any marks on them.

There is even some disagreement about how the compass should operate. Some think that it should keep it's setting when lifted off the paper (plane). But others think that it should not which would mean that both points of the compass must stay on the paper while transferring a distance to another place. That makes things considerably more difficult but I am told it can be done.

CORONA VIRUS said:

I am glad to see someone use the correct terminology for "rule".

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I'd build the equivalent of a "chair" for edge/corner finding, use the physical edge of the work to locate the chair, and a physical edge of the chair to locate one end of the dividers.

I still do a fair amount of manual layout, and I'm very lucky to be within 0.002" of true position most of the time. Even with a recently ground centerpunch or divider tip "clicking" positively into the intersection of two lightly scribed lines. The width of a heavy scribed line can easily be 0.002 or 0.003" wide. You are not getting to tenths with dividers and a straightedge, even with the benefit of a loupe.

Again, Euclid was a theoretician, a thinker, not a practical, shop guy. He ASSUMED that the point of a compass could be simply and accurately placed on a point or on the intersection of two lines. He did not address any of the practical problems associated with doing that.

People, like the builders of pyramids or carpenters who built houses or stone masons who built the cathedrals and other stone structures of antiquity had to work out those "details" on their own when they used or tried to use the methods spelled out by Euclid and the other theoretical mathematicians.

More on that "point" or "the intersection of two lines": an actual point would have NO real dimensions. It would have zero width and zero height. In the real world, we could not ever, EVER see it. It is a mathematical construct, an ideal idea, not a real thing. Likewise, a "line" has zero width. It does have length, but with zero width, again we could never see one. Lines are also mathematical constructs. Euclid WAS aware of this. He was not trying to invent ways to make things. He was trying to describe how those things worked in IDEALISTIC terms.

"...as an experiemnt, what would you do to put one of the divider points accurately at the end of the line?" As a robot in a sifi movie or program would say, "INSUFFICIENT DATA". As I stated above, an actual line would be totally invisible, no matter how much magnification you employed. So, you must be talking about something that we poor humans commonly call a line. But that line will have actual, real, physical dimensions. It will have a width and it will even have a height. Different, physical psudolines will have different characteristics so you need to state just exactly what kind of "line" you are talking about. Pencil line? Ink line? Scribed line? Etc. And it would also be nice to state IN ADVANCE just how accurate you desire that location of the point of the compass to be.

BTW, the point of a compass is not a real, zero dimension point. It will have a radius. It will have some surface roughness. And the traditional compass (bow compass) has it's legs at an angle to the surface while it is in use. So do we use the actual "point", whatever that is, or some, admittedly small distance above it which the instrument actually rotates around.

There are so many questions here and so few answers. Your very question is so poorly stated that it is meaningless.

A real world device that attempts to work around some of these difficulties is the "tooling ball". It replaces an actual point with a sphere of a "known" diameter and that is installed at a "known" distance above the surface where that theoretical point is. In theory you can measure from a side of that tooling ball and get an accurate measurement. But tooling balls are not perfect. And they can not be located exactly above a given point, whatever that is. So we work withing the limits, the tolerances that this device and it's installation has. We could probably fill a page or more with the factors that come into play when using one or more tooling balls.

Here is a kick. I tried to look up tooling balls. I did find them, but out of three sources, only one of them gave any tolerances for them. And neither of those two tolerances was on the ball's diameter or on the distance the ball's center was above the surface it was installed on. Those are the first two numbers I would be interested in when using them. But none of the sources seemed to think that providing them was a good idea.

Euclidean said:

I think that's what I was getting at. If you had to do it, or just wanted to try it as an experiemnt, what would you do to put one of the divider points accurately at the end of the line? Would you use a loupe?

Click to expand...

EPAIII said:

While he may be using correct terminology "rule" when grouping it with Vernier calipers and micrometers, he is not correctly stating the rules of Euclid. The classic statement is "a straight edge and compass".

Rules or rulers have divisions marked on them. Straight edges do not and in the Euclidean sense, they can not be marked with any. They are used without any marks on them.

Click to expand...

I think I may have confused some people. I was referring to checking square with a modern rule and a set of dividers, not to an unmarked straightedge.

Every school boy in my day knew 90d=3,4,5, new math??...Phil

EPAIII said:

"...as an experiemnt, what would you do to put one of the divider points accurately at the end of the line?" As a robot in a sifi movie or program would say, "INSUFFICIENT DATA". As I stated above, an actual line would be totally invisible, no matter how much magnification you employed. So, you must be talking about something that we poor humans commonly call a line. But that line will have actual, real, physical dimensions. It will have a width and it will even have a height. Different, physical psudolines will have different characteristics so you need to state just exactly what kind of "line" you are talking about. Pencil line? Ink line? Scribed line? Etc. And it would also be nice to state IN ADVANCE just how accurate you desire that location of the point of the compass to be.

Click to expand...

Well, that was the question. Could you use dividers (or even calipers) to measure three lengths of a part to determine whether the part was in or out of square, and if so, could that be done with .1, .01, .001 accuracy?

When I was laying out the position for my current shop building, I used a surveyor's tape and the 3-4-5 principle to make a rectangle with square corners. Because I wanted the shop right up against the setback limits, I had the lot line surveyed and took distances from the stakes. I quickly found that the corner of my lost was slightly non-square: when I aligned one side of my rectangle with the lot line, the perpendicular side was noticeably not parallel to the adjacent side of the lot.

I wasted a lot of time redoing my layout multiple times (and getting consistent results) thinking there was an error in my technique, maybe cosine error due to ground slope, whatever. Finally, I checked the detailed plot map and found the lot corner was indeed non-square, by about 1/2 of one degree.

My point is that simple, stupid 3-4-5 is precise enough to detect small angular discrepancies, and if you were doing this with machinists tools instead of fiberglass tapes and wooden stakes, you could do a lot better.

Building larger patterns it was and is in non cnc shops pretty common practice to use trammels and simple geometry for laying out square or angles and over large distances. Far more accurate than trying to use undersized squares or protractors. Just like a sine bar is more accurate than a protractor over small distances.

I mentioned Euclid's idea of a straight edge for historical accuracy. In practical terms and with today's marked rulers, it makes no difference in theory; only more or less work in practice.

My comments about the use of such instruments to make or check the square of a block or of a square are still 100% applicable. I would not choose to use them for these purposes and I have not seen any references to any shops or calibration services which do. If any of you know of such usage, please post the specific details.

Yes, laying out a 3, 4, 5 triangle can detect a half degree error when we step up to the dimensions of a piece of land suitable for a building. And an individual may have success using them. But here again, the pros would not use that method. Modern survey instruments will produce results that are much more accurate in a much smaller time.

survey transit - Google Search

Euclidean said:

I think I may have confused some people. I was referring to checking square with a modern rule and a set of dividers, not to an unmarked straightedge.

Click to expand...