# How do you make more precise instruments while only using less precise instruments?

I'm not sure where this question should go, but I think this site is as good as any.

When humankind started out, all we had was sticks and stones. Today we have electron microscopes, gigapixel cameras and atomic clocks. These instruments are many orders of magnitude more precise than what we started out with and they required other precision instruments in their making. But how did we get here? The way I understand it, errors only accumulate. The more you measure things and add or multiply those measurements, the greater your errors will become. And if you have a novel precision tool and it's the first one of its kind - then there's nothing to calibrate it against.

So how it is possible that the precision of humanity's tools keeps increasing?

• Hi everybody -- I have removed comments that were attempting to provide answers to the question. Please use comments to suggest improvements or seek clarifications only. Any answers should be added below. Thanks! Feb 17, 2021 at 18:24
• From your interest in this topic, I think you'd enjoy browsing a few telescope mirror making sites. Eg, stellafane.org/tm/atm/general/myths.html Feb 17, 2021 at 18:28
• From a computational perspective, you may be interested to read up on bootstrapping, where compilers are a great example. Feb 17, 2021 at 19:03
• Hi again everyone. I've left a couple of related-topic comments, but I've removed some back-and-forth about how related they are. If you're after back-and-forth you are invited to Physics Chat.
– rob
Feb 18, 2021 at 5:40
• Book recommendation for further reading on this topic: How precision engineers created the modern world - Simon Winchester. (Title may vary across regions.) Feb 18, 2021 at 10:34

I work with an old toolmaker who also worked as a metrologist who goes on about this all day.

It seems to boil down to exploiting symmetries since the only way you can really check something is against itself.

Squareness: For example, you can check a square by aligning one edge to the center of straight edge and tracing a right angle, then flip it over, re-align to the straight edge while also trying to align to the traced edge as best you can. Then trace it out again. They should overlap if the square is truly square. If it's not, there will be an angular deviation. The longer the arms, the more evident smaller errors will be and you can measure the linear deviation at the ends relative to the length of the arms to quantify squareness.

Other Angles: A lot of other angles can be treated as integer divisions of 90 degree angle which you obtained via symmetry. For example, you know two 45 degrees should perfectly fill 90 degrees so you can trace out a 45 degree angle and move it around to make it sure it perfectly fills the remaining half. Or split 90 degrees into two and compare the two halves to make sure they match. You can also use knowledge of geometry and form a triangle using fixed lengths with particular ratios to obtain angles, such as the 3-4-5 triangle.

Flat Surfaces: Similarly, you can produce flat surfaces by lapping two surfaces against each other and if you do it properly (it actually requires three surfaces and is known as the 3-plate method), the high points wear away first leaving two surfaces which must be symmetrical, aka flat. In this way, flat-surfaces have a self-referencing method of manufacture. This is supremely important because, as far as I know, they are the only things that do.

I started talking about squares first since the symmetry is easier to describe for them, but it is the flatness of surface plates and their self-referencing manufacture that allow you to begin making the physical tools to actually apply the concept of symmetries to make the other measurements. You need straight edges to make squares and you can't make (or at least, check) straight edges without flat surface plates, nor can you check if something is round...

"Roundness": After you've produced your surface plate, straight edges,and squares using the methods above, then you can check how round something is by rolling it along a surface plate and using a gauge block or indicator to check how much the height varies as it rolls.

EDIT: As mentioned by a commenter, this only checks diameter and you can have non-circular lobed shapes (such as those made in centerless grinding and can be nearly imperceptibly non-circular) where the diameter is constant but the radius is not. Checking roundness via radius requires a lot more parts. Basically enough to make a lathe and indicators so you can mount the centers and turn it while directly measuring the radius. You can also place it in V-blocks on a surface-plate and measure but the V-block needs to be the correct angle relative to the number of lobes so they seat properly or the measurement will miss them. Fortunately lathes are rather basic and simple machinery and make circular shapes to begin with. You don't encounter lobed shapes until you have more advanced machinery like centerless grinders.

I suppose you could also place it vertically on a turntable if it has a flat squared end and indicate it and slide it around as you turn it to see if you can't find a location where the radius measures constant all the way around.

Parallel: You might have asked yourself "Why do you need a square to measure roundness above?" The answer is that squares don't just let you check if something is square. They also let you check indirectly check the opposite: whether something is parallel. You need the square to make sure the the gauge block's top and bottom surfaces are parallel to each other so that you can place the gauge block onto the surface plate, then place a straight edge onto the gauge block such that the straight edge runs parallel to the surface plate. Only then can you measure the height of the workpiece as it, hopefully, rolls. Incidentally, this also requires the straight edge to be square which you can't know without having a square.

More On Squareness: You can also now measure squareness of a physical object by placing it on a surface plate, and fixing a straight edge with square sides to the side of the workpiece such that the straight edge extends horizontally away from the workpiece and cantilevers over the surface plate. You then measure the difference in height for which the straight edge sits above the surface plate at both ends. The longer the straight edge, the more resolution you have, so long as sagging doesn't become an issue.

From these basic measurements (square, round, flat/straight), you get all the other mechanical measurements. The inherent symmetries which enable self-checking are what makes "straight", "flat", "round", and "square" special. It's why we use these properties and not random arcs, polygons, or angles as references when calibrating stuff.

Actually making stuff rather than just measuring: Up until now I mainly talked about measurement. The only manufacturing I spoke about was the surface plate and its very important self-referencing nature which allows it to make itself. That's because so long as you have a way to make that first reference from which other references derive, you can very painstakingly free-hand workpieces and keep measuring until you get it straight, round or square. After which you can use the result to more easily make other things.

Just think about free-hand filing a round AND straight hole in a wood wagon wheel, and then free-hand filing a round AND straight axle. It makes my brain glaze over too. It'd also be a waste since you would be much better off doing that for parts of a lathe which could be used to make more lathes and wagon wheels.

It's tough enough to file a piece of steel into a square cube with file that is actually straight, let alone a not-so-straight file which they probably didn't always have in the past. But so long as you have a square to check it with, you just keep correcting it until you get it. It is apparently a common apprentice toolmaker task to teach one how to use a file.

Spheres: To make a sphere you can start with a stick fixed at one end to draw an arc. Then you put some stock onto a lathe and then lathe out that arc. Then you take that work piece and turn it 90 degrees and put it back in the lathe using a special fixture and then lathe out another arc. That gives you a sphere-like thing.

I don't know how sphericity is measured especially when lobed shapes exist (maybe you seat them in a ring like the end of a hollow tube and measure?). Or how really accurate spheres, especially gauge spheres, are made. It's secret, apparently.

EDIT: Someone mentioned putting molten material into freefall and allow surface tension to pull it into a sphere and have it cool on the way down. Would work for low tech production of smaller spheres production and if you could control material volume as it was dropped you can control size. Still not sure how precisely manufactured spheres are made though or how they are ground. There doesn't seem to be an obvious way to use spheres to make more spheres unlike the other things.

• Comments are not for extended discussion; this conversation has been moved to chat.
– Chris
Feb 19, 2021 at 2:28
• Unfortunately we can’t move comments to chat except once. The ones I just cleaned out were super-interesting, but they weren’t about improving this answer.
– rob
Feb 20, 2021 at 22:27

The more you measure things and add or multiply those measurements, the greater your errors will become.

Not necessarily. If the errors in a series of measurements are independent and there is no systemic bias in the measurements then taking the average of all the measurements will give you a more accurate value than any individual measurement.

• As an example, if you make a thousand metersticks using tooling that produces random errors in length, putting them end-to-end will give you a very good idea of how long a kilometer is. Any single meterstick is probably wrong, but averaged over a thousand of them, the errors mostly cancel out.
– Mark
Feb 18, 2021 at 2:10
• @Mark Except that there will be systematic errors in putting them end to end; even if all metre sticks are perfect, to use them in the way you describe they need to be perfectly straight and parallel. Any deviation will cause an error, and this error is always negative, so you will systematically underestimate the length of a kilometre. Metrology is fun, and also hard :). Feb 18, 2021 at 9:43
• I once learned that sometimes tiny errors necessarily accumulate, whic was clear in hindsight but first surprised me: I wanted to make about 100 cookies from a big portion of dough. So I thought I'd go for 128 cookies because I suppose I can quite well divide an arbitrary portion of dough roughly into halves. Let's say my eye-balling error is so small that one part is about 48% and the other 52% (and I cannot tell which). Errors should cancel and all cookies should be roughly the same size, shouldn't they? Actually, with the above numbers, the biggest cookie will be almost twice the smallest! Feb 18, 2021 at 20:13
• @gerrit: You're getting lost in the example. The takeaway was that things average out in large numbers. Feb 19, 2021 at 9:06
• @HagenvonEitzen That would only happen if each division is necessarily 48 vs 52, in practice if the limiting factor is that you cannot tell an approx 8% difference between 2 portions (but can determine differences greater than that), you could quite possibly end up with something like 50.1 vs 49.9 on a particular division, so even though errors will still propagate, a max of .52^7 and a min of .48^7 like you describe is just the very worst case scenario. Feb 19, 2021 at 9:14

One thing I haven't seen mentioned is amplification.

Amplification: Imagine you have a lever that is 10 cm on one side of the pivot and 1 m on the other. Then any change in position on the short side is amplified 10 times on the long side. This new precision can be used to make new even more precise tools. Rinse and repeat.

Even just by iterating through this crude method over and over you will eventually get to levels of precision far beyond what the human eye can detect.

• Whoa! Good one! And so obvious in hindsight! :) Although the precision of the lever length, pivot placement and pivot hole matter too. Feb 17, 2021 at 22:10
• Yes but you can use the lever to make more precise tools with which you can make a more accurate lever. This hits a limit once things like thermal expansion or the flex of the lever itself starts to overtake your margin of error. But by that point you are already approaching industrial era precision. Feb 17, 2021 at 22:45
• The increase in precision is from the long side to the short side. (Similarly with gearing in another answer). Feb 18, 2021 at 8:49
• Isn't that more or less how a vernier caliper works? Feb 18, 2021 at 9:48
• @gerrit Maybe? But it would be worth describing since it is not straightforward. It is how a pantomiming machine (mimograph? I can't remember its name) works though. The thing that uses levers in a four-bar linkage to trace out a large letter at the long end of the lever and reproduce it smaller at the short end. Feb 18, 2021 at 15:32

That's a really nice one! I'm not an expert on experiments and measurements but this is how I see it:

The ultimate calibration tool is always nature. We pick special phenomena which rely on certain parameters.
Take temperature, for example. The phenomenon here is the phase transition of water. You set boiling water to a $$100^\circ$$ C and freezing to $$0^\circ$$ C and calibrate all measurement instruments (for example a mercury thermometer) around it.

But sometimes, one does not have such a perfect phenomenon in nature. Length is such an example. In these cases, the most precise instrument sets the standard.
At some point in France, someone forged a rod to define the metre. Later, someone else measured the speed of light based on the definition of the metre. When it became apparent that measurements of the speed of light are actually very precise (since time can be quantified precisely through the decay of atoms and we are very good at optics), the speed of light was set to a fixed value ($$299.792.458 ~m/s$$), now in reverse defining the metre. And this is how we calibrated the now most precise measurement of length.

• Hmm, good one! That's true, this would do nicely for calibration - just tie the various units to things found in nature. The more fundamental the better. But how do you go up in precision? Say, if all you have a rulers that have 1mm as their smallest unit, how do you create a ruler that has 1/10th of a milimeter unit? Etc. I mean, in the beginning we didn't even have a ruler. Someone took a length, said "this is an absolute unit" and divided it and divided it... how was this division done precisely enough? Feb 17, 2021 at 12:24
• You do not need to be precise to create something that is precise. Time, for example: You really only need any process that repeats itself, taking roughly the same time. Then, you calibrate a number of repititions of this process against your measurement unit. Take a pendulum and you can measure seconds by tweaking the pendulum length to what you think it should be. Use a spring with many repititions in one second and you can increase accuracy. Take a quartz clock with a frequency of 32768 Hz and you have great accuary! Feb 17, 2021 at 13:19
• @jamesqf Example: Draw a straight line, get a compass and and make circles of identical diameters at each end of the line. The line that intersects the two points where the two circles intersect each other bissect the original line. The old toolmaker showed me this when he was showing me his old tools and pointed to the compass and called it a divider and I asked why he called it a divider and not a compass. Feb 17, 2021 at 19:54
• Water isn't precise enough unless you also specify the isotopes of hydrogen and oxygen Feb 18, 2021 at 2:10
• @Nayuki Actually, first you would need specify the pressure. Since 1948, the triple point was used to increase precision. The isotope composition is part of the definition of Celsius only since 2005 and from 2019 on, it is completely decoupled form water and defined via Boltzmann's constant (see Wikipedia on Celsius). But the point remains: You would want to calibrate using a special phenomenon (e.g. a phase transition) in nature. Feb 18, 2021 at 12:25

Measurement errors can accumulate, yes.

But we are not talking about measurements here, we are talking about processes and tooling. That's another deal.

If you fling off a piece of flint from a flintstone by banging another stone into it, then the "precision" of that other stone doesn't matter. What matters is that you supply enough force - not precise force but enough force - to interfere with the structure of the flint in line with its internal structure (such as along grain boundaries, along fibres etc depending on material) .

The flint piece might come off sharp as a knife due to the way it fractures. The "precision" of this new tool you just created (the knife-like flint piece) depended on the disruption of its internal structure, not on the tool used to make it.

• Your story reminds me of the wire tip used in a scanning tunneling microscope, which can be made from ordinary wire and a pair of wire cutters. It has been a long time since the device was demonstrated to me, but as far as I remember we were able to see individual atoms using the piece of wire as the detector tip. Apparently if you cut the wire while also pulling on the cutters in the right way, the tip of the wire does have the few-atoms diameter required for this machine (apparently the proper tips sold with the machine were so expensive that some grad student figured out an alternative). Feb 17, 2021 at 20:02

The way I understand it, errors only accumulate.

That is not always the case. Human ingenuity found and systematically selected processes which improved a particular quality. Two examples:

• You can hand polish a spherical mirror or even a plane mirror down below 100 nm surface roughness with easily available raw materials.
• Zone melting of semiconductor or other crystalline materials can yield a final product with less than 1:1,000,000,000 impurity. You only need an oven with the right design.

I expect that worm drives are part of the story.

For every rotation of the driven axis the driving axis goes through multiple rotations. I expect that it is mechanically possible to capitalize on that ratio.

Other than that, forms of very high level consistency can be achieved with very low level means. A necessary tool for precision measurement is a large flat surface.

Flat surfaces like that are fabricated by grinding stone slabs against each other. If you would use just two slabs then you cannot prevent the possibility of one slab ending up slightly concave and the other slightly convex. So to fabricate flat surfaces a set of three slabs is used, changing out the pairing from time to time. So that way you can create very high precision flat surfaces, that are subsequently used as reference.

I assume similar procedures are used to create high precision instruments that check for squareness.

I recommend you look up articles and videos about the process of 'scraping'. The process of scraping involves steps where a high quality flat stone is used as reference to achieve a high quality of flatness of a machine bed.

The lead screw that moves the table must be manufactured with precision but I expect that a milling machine can create parts to a higher level of precision than the level of precision of its own parts. The lead screw has a small pitch compared to its diameter; a complete turn of the table feed wheel results in a relatively small displacement of the table.

youtube video How to Measure to a MILLIONTH of an Inch (The Dawn of Precision)

• When I saw the question, those two videos are what immeadiately came to mind. They are excellent answers in and of themselves, Feb 17, 2021 at 18:28
• I have always wondered about lead screws on lathes. I suspect (but I have not seen a convincing argument) that if you have a lathe with a not-so-good lead screw, you can use it to turn a better lead screw. So the very first lathe had a really crappy lead screw (perhaps hacked out by hand using some imprecise manual technique), but then it didn't take long to bootstrap a much better one. Feb 17, 2021 at 18:57
• @SteveSummit I did a quick internet search. It appears the threading of a lead screw is often accomplished with thread rolling. The rolls do two things in one process: they create the threading, and the setup is such that it is the rolls that are advancing the material, ensuring the same thread pitch along the entire length. Let's assume the rolls are manufactured on a milling machine. I expect that thread rolls can be manufactured to a higher level of precision than the lead screw of the machine that mills them, because the table of the milling machine moves slower than the feed wheel. Feb 17, 2021 at 19:46

In addition to excellent answer of Cream. Let us not confuse precision and accuracy.

Accuracy is the ability to obtain correct measurement in absolute scale. Following Cream's example, if you use a metal slab to define length of a meter, then indeed all your measurements are limited by how well you can produce and measure such slab. E.g. the reference platinum-iridium meter rod used at the beginning of 20th century was manufactured with 200 nm uncertainty, which limits relative accuracy of all measurements based on it to $$2\cdot 10^{-7}$$. This is why at some point we begun to use a more reliable measurement speed of light to define the meter.

Precision is the ability to distinguish small differences in the quantity you measure. Again, read the history of measurements of light speed, it's a truly amazing lecture.

Let's look at the example of Fizeau-Foucault experiment with rotating mirror (https://en.wikipedia.org/wiki/Fizeau%E2%80%93Foucault_apparatus). Notice how imperfections of the device propagate to your final result. If you measure the mirror rotation speed with e.g. 1% precision (I think they actually did better), this propagates to only 1% in your time measurement, even if it's a time period way too short to measure directly.

If instead you conducted a Galileo-style experiment where you attempt to measure time of light propagation over a distance of e.g. 1km (today we know it's ~3µs) using a 0.01s precision stopwatch, your uncertainty would be 0.01s/3µs = 300'000%, far less useful (though Galileo's work had significance at his time).

This example demonstrates how it is both important and possible to design your measurement device in a way that is either insensitive to its own imperfection, or at least minimizes their impact on the final measurement.

I would also like to emphasize the importance of periodic waves which can magnify tiny errors. The most precise length measurements ever, by LIGO, are done with interferometry. A similar principle was used in the Michelson Morley experiment.

This also allows us to measure the speed of light for waves of different frequencies over long distances (gamma ray bursts over the distance of the observable universe).

Edit: So, if we imagine that light waves are sort of uber-precise ruler, it's not as though we've contructed this more precise ruler out of a less precise ruler, but it's more like we "found" a more precise ruler in nature (which we are able to use because understand the laws of physics) and decided to use that one instead.

• This answer could be improved if it contained a bit more detail. Feb 18, 2021 at 9:46

## If an instrument involves counting, just count more

Some measurements are done by counting things. For example, the definition of the second is n = 9,192,631,770 cycles produced by Cs-133. The number of whole cycles that have elapsed on an atomic clock are exactly known, even if there is some uncertainty in the additional fraction of a cycle. No matter how many cycles have elapsed, the uncertainty is always one cycle.

So the trick to improve precision is to just count more cycles. Counting n cycles gives you one second, with an uncertainty of one cycle. Counting 1000n gives you 1000 seconds, but the uncertainty is still one cycle. Counting one billion n gives you one billion seconds, still with an uncertainty of one cycle. The absolute precision is always one cycle, but the relative precision improves by increasing how much you count.

The only limit is how high you are willing and able to count.

• you don't even need to count the cycles, the knowledge that it's an integer is enough. You start with a very rough measure of the period, then time for an unknown integer number of cycles to obtain a better measure of the period. Then iterate this procedure. Provided you don't increase the number of cycles slowly enough that the uncertainty is always less than one cycle, you can obtain results with arbitrarily good precision (obviously other sources of error will still cause inaccuracies though) Feb 18, 2021 at 11:51
• one of my labs in first year was precisely measuring g with this method of exact fractions using a torsion pendulum, a standard wooden desktop metre ruler, and an ordinary sports stopwatch. In a couple of hours it was fairly easy to obtain a measurement more precise than one part in one thousand (most pairs managed nearer one part in ten thousand). Easily enough to see that the city we were in has noticeably stronger gravity than the standard value of g Feb 18, 2021 at 11:52

I'll point out that you can use multiple low-precision measuring tools in conjunction to get a measurement that is more precise than any of the tools individually. This is the idea behind a Vernier scale, which essentially uses two rulers of different spacing to interpolate distances between the markings. A Vernier scale can use two rulers that are marked at no less than millimeter spacing to precisely measure distances to the hundredth of a millimeter - this is possible despite the fact that none of your tools can measure that precisely individually! Basically, by comparing measurements among low-precision tools, you can get a high-precision measurement.

• I just looked up what a Vernier scale is, but... those two rulers themselves need to be made with 1/100mm precision. Sure, the marks are no less than 1mm apart, but they are VERY precisely placed. So how do you make them if none of your tools have this precision to begin with? Feb 17, 2021 at 16:25
• @Vilx- To be honest, this is kind of like asking "how do you turn a rock into a microchip?" There are many steps involved in between and you may be expecting too succint an answer. Feb 17, 2021 at 18:07
• @DKNguyen - Well, sort of. In fact, that's almost exactly the meme I saw which inspired me to post this question. :D And yes, I do of course understand that there are a bazillion steps inbetween, but somewhere along that line the precision problem must have been solved, and more than once, in various fields and for various types of instruments. Feb 17, 2021 at 18:24
• @Vilx- Well, at the very least, you can always compare a ruler against a bunch of other supposedly identical rulers all along their lengths. The spacing between each increment should be the same as each increment at every other position on every other ruler. Or, more practically, use a physical gauge the same width as the spacing of an increment and compare that against every increment (which was probably your reference to begin with when you first started making the machine to make the rulers). Feb 17, 2021 at 19:35
• @Vilx A similar idea to a Vernier scale is a transversal en.wikipedia.org/wiki/Transversal_(instrument_making). I believe this really does allow you to get greater precision than you had before Feb 18, 2021 at 0:27

You exploit the relationships between theory and the thing you want to measure. You come up with devices that are, in a sense, independent from the error. Let's focus on time.

Imagine you wanted to build a clock. You go to your theory toolbox and look around. Immediately you notice things like pendulums and springs. Theory tells us that 'ideal' versions of these things completely fit the bill. But theory also tells us that in real life objects are not 'ideal'. There's all sorts of things that go wrong when you try and build one, both internally (try and build a pendulum with a rope of zero mass) and externally (air resistance, etc). In other words, the theory is simple but trying to implement a real world version of it is complex. That's where the idea of independence comes in. You don't need a more accurate clock to know that lessening air resistance will improve the regularity of the oscillations, that comes from your knowledge of the theory. You can greatly improve the performance of the device simply by making the device closer to its idealized version -- you lengthen the rope of the pendulum, you make the thing swinging at the end extremely heavy and dense vs the weight of the rope and so on.

After a while you start to hit a wall where your improvements to the design become increasingly marginal vs the effort. Luckily it's now the early 1900s -- some other people have been playing around with this electricity thing for about a century and have developed a lot of theory around it. You exploit this theory and make a quartz watch. But this too ultimately has implementation limitations. And then atomic theory is developed and you realize you can exploit the vibrations of atoms and build a device that is even closer to the 'ideal' version of itself. Interestingly, as this happens the theories become significantly more complex, but the implementations in a way become simpler. In other words, a quartz watch is much closer to its theoretical equivalent than any pendulum is to an ideal pendulum. This is why you can buy a quartz watch for a couple dollars that outperforms basically any physical watch.

So the story of time measurement is really the story of people using theory to come up with devices whose real world versions can more closely match the theoretical versions of themselves.

From another angle: consider predictiing future systems behavior. If you have tools that measure an observable effect (maybe latitude of sunrise vs. day-of-year) to a crude accuracy, then design and build a new tool, see if the new tool fits observed data better, with more precision ,.. and predicts future events better, then you know you've improved the state of the art.

Why can't you use imprecise tools to make more precise tools? Consider how humans created hammers and anvils, they must have used lumps of metal to make a vague outline of a hammer, and then used that to create a hammer with a handle, then used that to create other tools such as molds to standardize production. It seems to me that a better comparison that an imprecise tool takes LONGER to measure or do a certain work, and a more precise tool can do that same work quicker and better. There's some cut off point, a hammer can't work at the atomic level, but it seems like a ladder that each technology builds off the previous.

There is a nice answer by Steeven, I feel like I would like to add some interesting notes.

The first ever particle accelerator was built in the 1930s and it was just a 100mm in diameter. Today, we are using particle accelerators that are giant (The Large Hadron Collider (LHC) is the world's largest and highest-energy particle collider and the largest machine in the world. It lies in a tunnel 27 kilometres (17 mi) in circumference and as deep as 175 metres (574 ft) beneath the France–Switzerland border near Geneva.) in size and can reach ever bigger energy levels.

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

We are using ever newer (in this case bigger) tools as you say, to make ever more precise measurements (scattering experiments) to test the quantum realm.

In quantum mechanics, if you want to probe physics at short distances, you can scatter particles at high energies. (You can think of this as being due to Heisenberg's uncertainty principle, if you like, or just about properties of Fourier transforms where making localized wave packets requires the use of high frequencies.) By doing ever-higher-energy scattering experiments, you learn about physics at ever-shorter-length scales. (This is why we build the LHC to study physics at the attometer length scale.)

A list of inconveniences between quantum mechanics and (general) relativity?

So I believe this is a beautiful example of how as you say we use newer (bigger and bigger) tools and the "precision" of these new tools depends on how powerful (what energy levels they can reach) they can be built, and ultimately we are using these giant tools to probe the smallest possible world, the quantum realm.

I think this has a place here, but seems to be asked in a manner where more engineering concepts can be utilized.

Take electron microscopes. All's you really need is a vacuum, any manner in producing free electrons into the vacuum, and then a focusing magnetic field. You can experiment with the strength of the electron beam, and degree of focusing. Then two orthogonal fields are used of varying strengths to deflect the beam an x and y direction. Any conductor and capacitor will absorb the reflected electrons and store a voltage, that voltage is read, and the beam is offset slightly again. The stored voltages are used to represent brightness of an image pixel, and you look at the image. You play around with all these voltages and get higher magnifications.

Here nobody ever had to build small parts. The vacuum can be produced in a variety of different ways, the electrons can be stripped from ionized gas, or ejected from metals.

Similar is the process of many nan technologies. They use silk screens, chemical emulsion, or image focusing lenses.

The limiting factors end up being the physical theories, which are probably approximated well enough by basically approaching the atom scale. And often this can be reached. But to go to precisions beyond the atom, they go to cern which can be viewed analogously as a large scale electron microscope, which is also much much slower version of the same process. And today they are presumably still assembling this image.

Then there are different statistical concepts used in chemistry, to isolate chemical compounds via stirring or agitating. Alot of these things use classical particles to model things, and are long complicated process. This is how they can extract dna and other things. Which originally exist in tiny proportions. They also can engineer reactions which magnify the small presence of a substance.

On the large scale, it is different yet again. Watch makers can build highly precise and efficient watches, useful for celestial observation, navigation, with reduced error propagations. In some space vehicle applications such large scale engineering can still be the only solutions.

As you can see with these three examples, the mechanisms which dictate the degree at which precision can be obtained, are wildly unrelated mostly. Instead dictated by the predictable aspects of nature in the regime in question. There does seem to be reduction of applicability on larger scales. But this could be due to the broad range coupled with our expectations for smaller scales. Also many times large equipment is unfeasible, requiring a lot of resource.

There does appear to be empirical observation of a common phenomena were increased precision is obtained by increasing the size of some parameter. However as we look at the examples, typically this gain is small and limited ounce the proportions are stretched beyond what the process had been developed for. This is were we see that a sound theoretical understanding yields the greatest gains, by aiding the intuition in the development of an entirely new process optimal for the regime of phenomena in question. An example of this would be when scientist use celestial observations on large scales to validate theories where such devices would be impossible to achieve merely by expanding some parameter' size.

• I don't completely understand this line of reasoning, but it seems to the point Feb 18, 2021 at 12:47

Your intuition that you can only get precision from precision is correct, but your conclusion that you need precise instruments to get precision is incorrect. The trick is always to use precision found in nature to get precision. You should actually look at how people make things to get an idea of where they are getting their desired precision from. For example, one can obtain a nearly perfect parabolic mirror by using the physics of a rotating liquid under uniform gravity. Another is to get a very precise measurement of time using the physics of atomic state transition, which is a clear example of using the central limit theorem to obtain a statistical guarantee on the precision of the output of the atomic clock.

Related to other answers, UW physics professor John Sidles presented this as a puzzle about a decade ago:

https://rjlipton.wordpress.com/2012/04/25/cutting-a-graph-by-the-numbers/#comment-20036

You are marooned on a desert island whose sole resources are sand, sea, and coconut trees. Construct a surface flat to atomic dimensions.

The following answer beautifully blends engineering, physics, and mathematics.

• Construct a fire-bow from bark and branches (engineering) • Start a fire, and melt sand into three irregular lumps of glass (physics) • Using sea-water as a lubricant, grind each shape against the other two (mathematics) Continue this grinding process, varying the pairing and direction of the grinding strokes, until three flat surfaces are obtained. Then it is an elegant theorem of geometry that three surfaces that are mutually congruent under translation and rotation, must be exactly flat. Essentially in consequence of this geometric theorem (which was known to Renaissance mirror-grinders), all modern high-precision artifacts are constructed by (seemingly) low-precision grinding processes... the quartz-sphere test-masses of Gravity Probe B are perhaps the most celebrated examples.

• Nice one! One thing though that makes me suspicious - you'll need about 1400°C-1600°C to melt sand, but wood fire doesn't burn in excess of 1000°C. I didn't find any information about coconut trees specifically, but if it did burn hot enough, it would be waaay outside the average. That said, it's probably possible to get a high enough temperature through some other low-tech means (like making charcoal and primitive furnaces). Feb 20, 2021 at 19:16

Greeks used a compass and straight edge. A compass is as precise as an A-frame anchored with a needle, and a straight edge is some wood that is sanded flat by a carpenter. One wooden straight edge can be used to create a straighter straight edge because the sanding grit is small and it evens out the bumps.

Distances were an issue until the micrometer which gave birth to nano precise engineering. There's very good documentaries about early micrometers. Later micrometers and straight edges become laser precise, as reliable as photons, i.e. relatively accurate, not perfectly.

In error analysis, the errors multiply/add up (basically effect) only if all these sources of error are independent. That need not be the case always. And further using different instruments like moving form Cloud Chamber detector to Semiconductor detectors have increased the resolution my 100s of magnitudes, the reason being some materials are more sensitive than other, and you can always create new materials!

Measure many input variables. Compute the resultant measurement upon all of them.

An aggregate measurement on more than 1 estimator with performance better than random chance is better than any one of them.