Accuracy of Hand Drilled holes

[SillyOldDufferModerator @sillyoldduffer]

Posted by Neil Wyatt on 10/10/2021 13:06:27:

…

They needed a circle of 'x' holes to step the mechanism around in 'x' steps per rotation.. Micron accuracy is not required, so there is no reason for them to go chasing it.

To me the obvious solution is scribe a ring, use a marked strip wrapped around a circumference as an index. Scribe or punch each hole, then drill by hand.

…

I agree, and I wonder if the practicalities of hole placing determined the size of the whole mechanism, said to be about the size of a shoebox. Could be all the mechanism's dimensions are driven by these holes deciding the size of the ring.

Shoebox width suggests a ring diameter of about 150mm. Using mathematics unknown to the Antikythera makers, we can easily calculate the angle and distance between holes. Assuming the lowest possibility (354), the angle is 0.983333333° and the spacing about 1.3314mm.

The Greeks didn't have our advantages. Their maths were restricted to whatever could be done with a plain divider and a straight-edge, which is a lot. But the workman's tools where behind greek mathematical technique – he had no graduated rules, screw-adjustable dividers or loupes.

Spacing graduations at 1.33mm by hand is quite challenging, and it may represent the smallest distance the Greek maker could see well enough to do the job.

To make the ring, I think he worked out a representative space between 8 or 10 holes (say 13.3mm), and used that to determine the approximate circumference of the ring by laying out a tape. The actual ring was turned with a somewhat larger diameter to accommodate a more carefully graduated tape.

A tape could be accurately graduated by rolling it around the newly made ring to mark the ends and then laying it out flat and dividing it into parts as described by Euclid, Proposition 9, Book VI I wouldn't attempt to mark all the holes by Euclid, perhaps marking every fourth hole would be done accurately and the others pricked in by eye.

Might also have been done by laying out a polygon, but this cramps the work because the diameter of the ring is divided into parts rather than the circumference, reducing the working length by over a third. I think Neil's way is easier and more accurate.

Dave

[pgk pgkParticipant @pgkpgk17461]

These guys were bright enough to build nice boats, fancy buildings and design cities and irrigation. It's hardly stretching things to accept they had ways to scale stuff up and down.
Lots of ways to accurately measure out the divisions without math. Stick your 6" disc down on some wet plaster and use string to scrobe a 10' circle round it.. you can then walk round the circle with a short stick to see how many times it fits and nibble the stick until you get it right.
You could even get more creative and attach your disc to the middle of a very slowly turning water wheel (adjust the sluice so it turns once every few minutes and adjust a water drip for the drips per rev you want and scribe as it passes the datum.
They understood gears to make the thing so a simple rotary table is hardly stretching imagination either.
Not understanding or publishing math rules doesn't mean you can't use them intuitively. they obviously had a way of creating right angles for building projects and must have been aware that a circle's radius gives a nice hexagon of chords.
I do'lt believe that marking out would have been any issue even if slow. As or drilling the holes the rotating your disc under a drill guide is simple enough.

pgk

[Michael GilliganParticipant @michaelgilligan61133]

pgk

Although Neil, for good reasons, separated the two current Antikythera threads … may I suggest that you have a look at p4 of the companion thread; where [amongst other things] I wrote :

“ … Although Dave has moved this to a problem of placing 352 holes … The real [Antikythera] issue is estimating the number of holes in a circle, based on a relatively small sample thereof.

The level of accuracy required to do that prediction successfully, to the nearest integer value, is astonishing.

From my table … if 352 is the right answer: we can only get it if we know the angle per step is greater than 1.020° and less than 1.026° … “

MichaelG.

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Ref. __ https://www.model-engineer.co.uk/forums/postings.asp?th=174972&p=4

Edited By Michael Gilligan on 15/10/2021 14:09:24

[pgk pgkParticipant @pgkpgk17461]

MG,

I had read that thread. Not sure of your point..if it's the smallness of the angles then as you realise extending those to a larger circle and using measurements around the large circumference would make it practicable.
And as Neil has suggested the absolute placement of the holes isn't quite that important if being used as a diary marker…They just have to be good enough to avoid confusion as to which day is which.

I’d guess it was easier to count holes when the disc was one piece (unless the guy got fed up, broke it apart and tossed it into the ocean)

pgk

[SillyOldDufferModerator @sillyoldduffer]

Posted by pgk pgk on 15/10/2021 13:31:07:

…
Lots of ways to accurately measure out the divisions without math. Stick your 6" disc down on some wet plaster and use string to scrobe a 10' circle round it.. you can then walk round the circle with a short stick to see how many times it fits and nibble the stick until you get it right.

… they obviously had a way of creating right angles for building projects…

pgk

I think the Antikythera mechanism is remarkable considering how limited the technology of the day was. It's more challenging than I first thought. Quite easy to get a feel for the problem by trying to transfer the entire millimetre scale of a 30cm steel rule to paper. Try it and see!

The ancients didn't have graduated steel rules and had to make their own with a straight-edge and pair of dividers. Making an accurate straight-edge from scratch is a tricky challenge in itself.

The stick method in basic form is out because it accumulates error. And it's not very practical when the requirement is to drill 354 holes on a 150mm diameter – allowing for the diameter of the hole, the stick is 1.131166mm wide.

Re right angles, the ancients knew how to create them: the method is in Euclid's Elements.

Dave

[pgk pgkParticipant @pgkpgk17461]

SOD

354 dots on a 10 foot circumference has them just over an inch apart so i submit sticks and thread could work.
As for a straight edge – their ancestors were capable of knocking up pyramids with pretty straight stone edges..apart from the tedium and time involved making a wooden or metal straight edge isn't that hard – reference against a string line perhaps or somewhere cold enough for a bucket of ice – or just 'have a borrow of someone’s'.
sword'

I have a pal who is an ex toolmaker and very OCD (cleans his skirting board corners with cotton buds). When he decided to make a clock he stated that he couldn't file more than 2 gear teeth an evening and keep them up to his standards. Once again it's all a matter of patience. The Antikythera may have been the result of several years work and lots of trial and error.

The Antikythera mechanism post-dates such things as the Parthenon with it's truncated fluted columns – dividing circles was a known skill. As must have been the idea of drafting and lofting and making small models of finished designs.

pgk

Edited By pgk pgk on 15/10/2021 17:55:07

[Nigel Graham 2Participant @nigelgraham2]

I wonder how advanced the Ancient Greek craftsmen were in maths, compared to the philosophers of the time. The philsosophers with all their Theorems and things tended to believe their knowledge mystical so kept it hermetic.

Perhaps the artisans worked out the practical geometry themselves but the Euclids and Platos stole the glory so 20C school-children could enjoy having to prove some random shape is a Cycling Quadri-thingummybob!

[Michael GilliganParticipant @michaelgilligan61133]

Posted by pgk pgk on 15/10/2021 16:10:02:

MG,

I had read that thread. Not sure of your point..if it's the smallness of the angles then as you realise extending those to a larger circle and using measurements around the large circumference would make it practicable.
And as Neil has suggested […]

.

Have you ever checked what an angle of six thousandths of a degree actually looks like ?

Note: I am NOT suggesting that the Greeks needed to make it that accurately

… There are two different issues.

MichaelG.

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Edit: __ Not sure when you read that other thread : If not recently, then may I suggest you note Neil’s interpretation of the logic behind the question.

Ref. 10/10/2021 12:59:11

Edited By Michael Gilligan on 15/10/2021 20:12:23

[pgk pgkParticipant @pgkpgk17461]

6/1000deg = not a lot
For the mechanism to work at all then gear teeth numbers have to be formed with an accuracy greater than the apparent accuracy of these holes? I have trouble getting my head around this mechanism anyway and assumed that the holes have an intention to be used as a setting point for part of the mechanism – either as a sighting guide or perhaps a pin point?
For the amount of work involved in building it one has to assume (or at least I do) that trial parts were made for sub assemblies and perhaps some were re-used or remanufactured. It’s hard to believe that whatever hole number there is is accidental unless the holes are redundant to operation from a re-purposed scrap part

pgk.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by pgk pgk on 16/10/2021 07:05:48:

.

6/1000deg = not a lot
[…]

.

cos (0.006) = 0.999999994517
tan (0.006) = 0.000104719756
sin (0.006) = 0.000104719755

Ponder those numbers, please, whilst I get some Coffee

I will then try to explain the fundamental difference between the two issues:

A. How accurately did the Greeks need to place the holes, to achieve fitness for purpose ?

B. How accurately would we need the Greeks to have placed the holes, to do the forensic calculation ?

I too have trouble getting my head around this mechanism … but I’m pretty sure I understand why there is a big difference between the two answers.

MichaelG.

[Neil WyattModerator @neilwyatt]

Posted by Nigel Graham 2 on 15/10/2021 19:16:04:

I wonder how advanced the Ancient Greek craftsmen were in maths, compared to the philosophers of the time. The philsosophers with all their Theorems and things tended to believe their knowledge mystical so kept it hermetic.

Perhaps the artisans worked out the practical geometry themselves but the Euclids and Platos stole the glory so 20C school-children could enjoy having to prove some random shape is a Cycling Quadri-thingummybob!

I think it's a mistake to assume a split between philosophers and artisans. Even if an artisan made the Antikythera mechanism, someone with far greater skill in geometry and mathematics than I was involved very closely – playing the role that Excel and CAD play for me when I make a mechanism.

Neil

[Michael GilliganParticipant @michaelgilligan61133]

O.K. … here goes

A. We obviously don’t know for certain, but; it seems reasonable to assume that the holes are on a setting-ring which serves some calendric purpose. To fit that purpose, I would suggest that the Greeks only needed to be able to drill ‘n’ holes on a circle, without them breaking-into one another and forming slots. … This seems realistically achievable with hand-tools.

B. If, however, we are attempting to ‘reverse-engineer’ a complex and incomplete mechanism; we really need to know [or be very confident in our estimate of] the ‘count’ of any gears and setting-rings in the remains.

The available fragment appears to be approximately one fifth of the full ring.

The BHI paper makes a good case for wanting it to have 354 holes …

… but I am questioning the confidence level of that number.

My own working [from published images rather than the originals] made a reasonable case for the number being 352, but that would not fit the research team’s model.

I have invited forum members to repeat my exercise, and compare our results; but no-one has yet done so.

Accepting that my ‘personal equation’ may have skewed my estimate [*], I would be grateful if someone could check.

I do believe, however, that I have demonstrated how difficult [near impossible] it is to extrapolate from that fragment to the full circle.

MichaelG.

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[*] **LINK**

https://en.wikipedia.org/wiki/Personal_equation

[Michael GilliganParticipant @michaelgilligan61133]

For convenient reference, I have copied this table of ‘other estimates’ from the BHI document:

.

.

Obviously … the text continues, and the whole document is worth reading.

MichaelG.

[FarmboyParticipant @farmboy]

Lacking even an O-level in maths I have played around with the photo in TurboCad, measuring angles by visually centreing the cross-hairs on the holes ( the harder I try, the harder it seems to get ) and using the centre of arc marked on the photo since I didn't feel capable of re-calculating it. Most of my somewhat cruder findings are at least in the same 'ball park' as MichaelG's results. However, by selecting different groups of 20 holes I found it just possible to get the hole count up to 354. The variation in apparent hole size and placement is a limiting factor.

My thoughts, for what they're worth, pretty much agree with MichaelG. There are just too many variables to allow accurate extrapolation from the sector to the full circle. For that to happen we need to know that there is no image distortion at any of the stages between the original x-ray and the viewing device we are using, and that the actual mechanism was not distorted by physical damage or corrosion. We also need to be sure that the hole spacing was actually intended to be regular in the first place, although it is hard to imagine why it wouldn't be.

As for making the original, I think it would be fair to assume that there were many simpler earlier mechanisms on which they developed their skills. By the time of the Antikythera machine there were possibly many skilled craftsmen capable of achieving the required accuracy. I would imagine they had also developed some fairly sophisticated measuring devices, even if they were not widely available. A couple of thousand years earlier, the Egyptians were obviously capable of some pretty accurate design and marking out, albeit on a larger scale!

Mike.

[Michael GilliganParticipant @michaelgilligan61133]

Many thanks for the input, Mike

[ and not just because you pretty much agree with me ]

Your comment that: “( the harder I try, the harder it seems to get )” is spot-on

MichaelG.

[pgk pgkParticipant @pgkpgk17461]

Accepting my ignorance I've done a little digging into the tech of the time. This post-dates the screw water lift , Pythagoras and Archimedes, pulleys, the steam powered pigeon, anecdotally some automata and even a chain drive and apparently some of the lettering on this mechanism is 1.6mm high characters – technical and fine work was available (whether used or not). And if one is to believe the even older chinese account some sophisticated automata indeed:
https://en.wikipedia.org/wiki/Automaton

**LINK**

Out of curiosity, I note that 353 is a prime number – not that i see any significance in that

pgk

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