# Extending the range of calipers

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• Jun 11, 2021, 08:46 PM
WmRMeyers
Quote:

Originally Posted by mklotz
Tony Foale has described a radius gauge...

based on the principle of measuring a chord and its related sagitta and then using these values to compute the radius of the measured curve.

The same principle can be applied to measure the diameter of circular objects whose diameter exceeds the capacity of your calipers. (Capacity here can refer to: calipers whose jaws do not open far enough to encompass the diameter to be measured or calipers whose jaws are too short to reach the diameter of the object.)

The first photo shows a too small caliper attempting to measure the diameter of an object. The caliper's jaws are too short to reach the diameter so it ends up measuring the length of some chord of the circle formed by the object. Note how the object is touching the beam of the caliper; it's pushed as far into the gape of the caliper as it will go.

This diagram will help to understand the mathematical notation...

The tips of the caliper jaws are at locations B and D. The line BD is the chord and that is what the calipers are measuring. The sagitta, EC (marked S in the diagram) is the distance from the chord to the circle's circumference. It should be obvious that that is the length of the caliper's jaws. Just to be clear, the following picture shows what you need to measure to find the length of your caliper's jaws...

(but use calipers not a scale; the picture is meant to show what to measure, not how to measure it.)

Now, if you know the length of the chord and the sagitta, you can compute the radius of the curve from a simple formula...

r = (c^2 + 4 * s^2) / (8 * s)

d = 2 * r = (c^2 + 4 * s^2) / (4 * s)

c = chord
s = sagitta
d = diameter

How well does it work ? Well, of course, that depends on your instruments and how carefully you measure.

For the bench block pictured, I made the following measurements...
c = 2.859
s = 1.159

and computed a diameter of 2.922. I then used my height gauge to measure the diameter (the jaws of my larger calipers are too short) and got a diameter of 2.924. YMMV.

Stuff like this makes me reconsider not finding my HS algebra teacher's grave and desecrating it. He told me the only reason I needed to learn algebra was to graduate from high school. It took me most of a decade to realize he was wrong. I hope he found out that I wound up teaching math for a short time. That would, I'm sure, have helped him shuffle off this mortal coil.

OK, it probably didn't help that I was a smart-aleck, and I do lack all talent for mathematics, but I did finally learn that it was useful in so very many ways, and I've spent far more time trying to make up for the deficits I allowed to remain back when my brain was younger and might have benefited more from the exercise. Though thank God for digital calculators. :)

Bill
• Jun 12, 2021, 08:56 AM
mklotz
Quote:

Originally Posted by old_toolmaker
...
Back in my early years before the electronic calculator, we solved these trigonometric issues via long hand with pencil and paper.

And you used a sundial to find the time :-)

Thankfully, now we have electronic, self-setting timepieces and inexpensive scientific calculators.

One learns nothing from doing the arithmetic, other than one's inability to do arithmetic. The real, generalized understanding comes from manipulating symbols, not numbers, and implies using algebra.
• Jun 12, 2021, 09:54 AM
WmRMeyers
Quote:

Originally Posted by mklotz
And you used a sundial to find the time :-)

Thankfully, now we have electronic, self-setting timepieces and inexpensive scientific calculators.

One learns nothing from doing the arithmetic, other than one's inability to do arithmetic. The real, generalized understanding comes from manipulating symbols, not numbers, and implies using algebra.

Bill
• Jun 12, 2021, 11:20 AM
mklotz
Quote:

Originally Posted by WmRMeyers
Everything you need to do with arithmetic can be done with a mechanical adding machine. It's just slow, and error-prone.

Ever tried extracting a square root with a mechanical, four function (no square root key) calculator? :-)

In grade school they taught us an alogorithm that was as complex as it was impossible to remember; I made a point of forgetting it as soon as the class was over.

The easiest way is to use Heron's iterative algorithm. Say 'N' is the number whose root is to be found. Make a guess of the root, 'x'. Then if x is smaller than the true root, N/x will be larger; If x is larger than N/x is smaller. Thus a better estimate is the average of these two...

z = (x + N/x) / 2

Now, keep iterating. Use z in place of x and compute a new value of z. As these iterations proceed, z will converge to the true value of the root.

A good initial guess for x is the largest integer whose square is less than N. The better the guess, the faster the convergence.

In the example below, I deliberately chose a poor initial guess; six would have been better. Despite this poor guess, I've got a damn accurate value in only three iterations.

N = 49
x = 5

z = (5 + 49/5) / 2 = 7.4

z = (7.4 + 49/7.4) / 2 = 7.0108...

z = (7.0108 +49/7.0108) / 2 = 7.00000833525

Close enough for government work.

Choosing a first guess larger than the correct value doesn't matter...

N = 49
x = 9

z = (9 + 49/9) / 2 = 7.222...

z = (7.222 + 49/7.222) / 2 = 7.0034

etc.
• Jun 12, 2021, 11:49 AM
WmRMeyers
Quote:

Originally Posted by mklotz
Ever tried extracting a square root with a mechanical, four function (no square root key) calculator? :-)

In grade school they taught us an alogorithm that was as complex as it was impossible to remember; I made a point of forgetting it as soon as the class was over.

The easiest way is to use Heron's iterative algorithm. Say 'N' is the number whose root is to be found. Make a guess of the root, 'x'. Then if x is smaller than the true root, N/x will be larger; If x is larger than N/x is smaller. Thus a better estimate is the average of these two...

z = (x + N/x) / 2

Now, keep iterating. Use z in place of x and compute a new value of z. As these iterations proceed, z will converge to the true value of the root.

A good initial guess for x is the largest integer whose square is less than N. The better the guess, the faster the convergence.

In the example below, I deliberately chose a poor initial guess; six would have been better. Despite this poor guess, I've got a damn accurate value in only three iterations.

N = 49
x = 5

z = (5 + 49/5) / 2 = 7.4

z = (7.4 + 49/7.4) / 2 = 7.0108...

z = (7.0108 +49/7.0108) / 2 = 7.00000833525

Close enough for government work.

Either I missed that class, or they weren't teaching it when I was going to elementary school. I was getting much of my info from books. I do have problems remembering complex written processes, like the kind of algorithm you described. This is one reason why I like checklists. Check off the steps as you go and you're less likely to goof it. I was getting my early math instruction in the days of the New Math that Tom Lehrer described so neatly. https://www.youtube.com/watch?v=W6OaYPVueW4

I was also anemic as a baby, and required blood transfusions directly into the veins in the temple since they couldn't hit anything else. Quite possibly, I'm dain brammaged. Though the family joke is that my mom dropped me on my head. Repeatedly. ;)

I've actually had some corroboration on that. The middle child, my eldest daughter, enlisted as a Navy Nuc. Went to the tech school for that career field in 2014. In her class of 38 students, she was the only one who didn't have calculus-based math and physics in high school. There were 8 graduates. She was the lowest ranked grad, but she learned enough calculus and physics in her copious spare time to pass the class, when 30 of her classmates who'd already had that work did not. She is the most like me of my children, so it's possible she got the unimpaired brain I should have had. OTH, her mom is also a math wiz, so she could have just gotten Mom's math brain. I know which one I want to believe. :lol:

She says if you know someone who needs that sort of education and can't get it in their school, that Khan Academy is the place to go. https://www.khanacademy.org/ & https://www.youtube.com/results?sear...y=khan+academy

Bill
• Jun 12, 2021, 06:49 PM
old_toolmaker
Calculator
When electronic calculators first came out we didn’t trust them, so we cross checked the calculator with pencil and paper! When both agreed we were satisfied. I’m sure this sounds strange to the younger crowd, but “true story”.
• Jun 12, 2021, 07:08 PM
WmRMeyers
Quote:

Originally Posted by old_toolmaker
When electronic calculators first came out we didn’t trust them, so we cross checked the calculator with pencil and paper! When both agreed we were satisfied. I’m sure this sounds strange to the younger crowd, but “true story”.

There was a check for the digital calculators that involved multiplying the number 12345679 by a whole number multiple of 9 that was greater than 0 and less than 10. If the calculator gave a result that was the multiplier, supposedly the calculator was working correctly. 12345679 x 81 gave you 999999999, 12345679 x 18 gave you 222222222, etc. I doubt it checks all the functions even on a 4-function calculator, but I never ran into a calculator that didn't get the correct results. I was told that one of the spreadsheet programs, I think an early version of Excel didn't get it right, but my brain has fried a number of times since then. I barely remembered how to do this check myself. And the 2007 version does get it right. ;)

Trust but verify!

Bill
• Jun 12, 2021, 11:23 PM
Toolmaker51
The arithmetic/ mathematics-education system generates all kinds of hoohah, apparently most everywhere. My initial years were ok, then something happened and it went kablooey. Some things like positive and negative numbers, no issue, but other students had trouble. Enjoyed geometry then and now, but I'm no Marv Klotz.
Trig gave me trouble, due some variety of dyslexia, I suppose. Soon, but by accident, I turned a worksheet around to wherever the most recognizable intersection or right angle 'popped out', it no longer bothers me.
I'm sure now, that some math works as a visual function, others not so well, abstract. It does seem trying to teach math should demonstrate more 'how' instead of 'why', in application, ignore algebraic types until the others are solid. I base this on personal experience, once exposed to how machining couples math to action, bingo, purely tangible. It worked again in auto shop, flight school, ocean navigation, some gunnery, and just now learning survey of plots.

Afterall, how hard would language or music be, native or otherwise, if it was all on paper?

Got so wound on my thoughts, completely fizzed on Marv's.......the calculator originally banned from education, I remember it.
That never made sense. If you miss order operations, it will be wrong; burning up 6 sheets of paper won't make it apparent halfway through. Hate to think what state things like space program or major construction would be dependent on manual calculation. At same time, engineering being fully digitized is not perfect. We enjoy so many old products that were papered or slide-ruled, with that extra 10%~, while the newer comparatively shoddy from paring thickness, material shear strength, etc.
• Jun 13, 2021, 06:03 AM
WmRMeyers
Quote:

Originally Posted by Toolmaker51
The arithmetic/ mathematics-education system generates all kinds of hoohah, apparently most everywhere. My initial years were ok, then something happened and it went kablooey. Some things like positive and negative numbers, no issue, but other students had trouble. Enjoyed geometry then and now, but I'm no Marv Klotz.
Trig gave me trouble, due some variety of dyslexia, I suppose. Soon, but by accident, I turned a worksheet around to wherever the most recognizable intersection or right angle 'popped out', it no longer bothers me.
I'm sure now, that some math works as a visual function, others not so well, abstract. It does seem trying to teach math should demonstrate more 'how' instead of 'why', in application, ignore algebraic types until the others are solid. I base this on personal experience, once exposed to how machining couples math to action, bingo, purely tangible. It worked again in auto shop, flight school, ocean navigation, some gunnery, and just now learning survey of plots.

Afterall, how hard would language or music be, native or otherwise, if it was all on paper?

Got so wound on my thoughts, completely fizzed on Marv's.......the calculator originally banned from education, I remember it.
That never made sense. If you miss order operations, it will be wrong; burning up 6 sheets of paper won't make it apparent halfway through. Hate to think what state things like space program or major construction would be dependent on manual calculation. At same time, engineering being fully digitized is not perfect. We enjoy so many old products that were papered or slide-ruled, with that extra 10%~, while the newer comparatively shoddy from paring thickness, material shear strength, etc.

I spent a short time as a teacher, and a good part of that, I was teaching math. Mostly arithmetic, but some of my teaching time I was subbing for a higher maths HS teacher undergoing chemo. For most of that, all I could do was tell them that I didn't know how to do that math myself, but I did know folks who used it daily, and told them some of the fields where that math was used. My wife had done physical chemistry in college, working for her BS in Chemistry, and that is a calculus-based course. I'd worked with communications and aeronautics engineers myself, and knew that to become an astronaut when I was a kid you had to be an engineer of some sort, and a test pilot. Had one kid in that class that wanted to be a machinist. I'd been bringing in books from various fields that showed the maths they needed. Mostly electronics, but I had a copy of the USAF's Space Handbook, my wife's P-Chem text book, and some others. Brought my Machinery's Handbook in for the kid who wanted to be a machinist, and after it passed around the classroom, he spent the rest of that class thumbing through the book finding all the places where it described math useful to a machinist. Funny thing, after that, he started applying himself to the classwork. When I was in HS, math was all theory, and they thought you'd get the practical aspects in college. When I was teaching, they were talking some of the practical aspects, but mostly they were doing theoretical environmental stuff. I guess they were getting ready for "Climate Change."

And I've since learned that for practical aspects of math, you need to be in the field for a while, whatever the field is. Much of my wife's P-Chem course was teaching the future chemists how to use the technology, both lab equipment, and software to do analysis. A lot of the students were having to thumb through books on using Excel, for example.

For the classes I was teaching, I tried for the more practical maths. Cooking, finance and book keeping. Had the kids in the government class I was teaching one semester calculate their share of our National Debt. Had my special ed kids figuring out what it cost their families to live, and what they'd have to make to be able to live the way they wanted. Of course, one of the kids wanted a Lamborghini. So I set him to finding jobs that paid enough he could get one. He didn't much like the answers, but he did figure them out!

A lot of kids don't get abstract. That is apparently something that develops after the teen years, for most folks. And for decades, that is all they taught in school. I hope they someday learn to actually follow the research on what when and how to teach. I wasn't seeing a great deal of it when I was teaching, and that ended in 2012, finally, though I should have realized it was really done in 2008.

Bill
• Jun 13, 2021, 06:48 AM
old_toolmaker
Geometry
I did very well with geometry in high school. I was an A+ student. I struggled with Trigonometry though.
It wasn’t until I began my first year in Trade school the geometry really began to click for me. Suddenly I saw how it could be applied and used in the real world! I became an ace with geometry, so much so that I became the go to person if anyone had a geometry problem they deemed unsolvable. My ego became inflated! Then came CAD and now we were all on equal footing again. Everything seems to go full circle in the end. New technology replaces old an si goes evolution. Nothing is forever. Enjoy today!
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