How Machines Learned Precision

This is a screw-cutting lathe of the kind Henry Maudslay built in London around 1800. A labourer turns the large wheel to spin the bar in the middle, and a cutting tool moves along beside it on a carriage that slides on straight iron rails. A long screw moves the carriage the same distance on every turn of the bar, cutting a thread into its surface. Lathes like this made shafts and screws for steam engines, looms, locomotives and other lathes.

But the lathe's own parts have to be accurate first. If a rail is bent, the tool follows the bend. If the turns of the screw are unevenly spaced, the thread it cuts will be uneven too. These errors leave pistons leaking, shafts wobbling in their bearings and nuts jamming on their bolts. In the 1770s, workshops were still struggling to make large cylinders round, long rails straight and screw threads evenly spaced. Even checking their work was difficult: the smallest mark on an ordinary workshop rule was a sixteenth of an inch, about a millimetre and a half.

By the 1850s, Joseph Whitworth had built a machine that could detect a difference of a millionth of an inch. In this article we'll follow how workshops learned to make flat surfaces, cut accurate screws and measure their parts, using interactive figures to explore each step (try rotating the lathe above with two fingers, or pinching to zoom indragging the lathe above, or zooming with ⌘/Ctrl + scroll). Let's start with the cylinder of Watt's steam engine, which we built in the previous article.

The cylinder

Newcomen's engine filled a cylinder with steam, then condensed it with a spray of cold water. The atmosphere pushed the piston down into the partial vacuum, but the spray also chilled the cylinder. Watt moved the condensation into a separate cold vessel so that the cylinder could stay hot, saving about two thirds of the coal. This also meant he had to change the way he sealed the piston. Below, we can compare the two engines.

Newcomen kept a pool of water on his piston to fill the gaps against the uneven cylinder wall. Water would have cooled Watt's cylinder, so he used dry packing instead. The tightly rammed packing couldn't give way enough to fill the gaps, so the hole through the cylinder, called the bore, had to be round and straight along the whole stroke.

The cylinders of the 1760s were far from that. The eighteen-inch cylinder of Watt's test engine at Kinneil, in 1769, was about ⅜ of an inch wider one way than the other at its worst point. That is nearly a centimetre of error in a cylinder less than half a metre wide.

The traditional way to make two parts fit was to press them together and file away the bright spots where they rubbed. The workman doing this was called a fitter. He could make the small surfaces of a valve or a gun lock fit closely, but a piston had to fit at every point along its stroke. Filing one part of the bore wouldn't make the rest match, and most of it was deep inside the cylinder, where a file couldn't reach.

To see why this was such a challenge, let's go through how the cylinders were built. The thick iron walls were made in one piece by pouring molten iron into a mould, a process called casting. The mould formed the outside, while a core in the middle left a hole through the iron.¹ Below, we can watch a cylinder being cast in a sand mould.

Molten iron is far denser than sand, so it pushes the core upward against whatever holds it down. If it lifts even slightly, the casting is ruined. Even a core that stays in place leaves a rough hole that isn't straight or round. A boring mill cuts away the inside of the casting to bring it to the right shape and size.

The boring mills of the time held the cutter on the end of a long bar supported from one side, like a broomstick held at arm's length. The bar sagged under its own weight,² and the uneven wall pushed it from side to side as it cut. Instead of making a straight hole, the cutter followed much of the crooked one. Large cylinders were bored four times, turned a quarter turn between passes, and still did not always come out round.³

Around 1775, the ironmaster John Wilkinson found an ingenious solution. He built a boring mill with a much heavier bar that ran right through the cylinder and turned in bearings at both ends.⁴ The casting was clamped in place, and the cutter head slid along the bar as it turned. The cutter now followed the bar the way a pencil follows a ruler. Supporting both ends greatly reduced the bending, so the cutter could make a straight bore through a crooked casting.

In 1776, Watt's partner Matthew Boulton wrote that a fifty-inch cylinder Wilkinson had bored for them “does not err the thickness of an old shilling in any part.” A worn shilling was a little under a millimetre thick, which puts the error at less than one part in a thousand across a cylinder wider than a metre. Relative to its size, that was about thirty times more accurate than Watt's test cylinder from seven years earlier!

The coin was thinner than the smallest mark on a workshop rule. Even a common modern tolerance for a precision bore that large is only about seven times tighter.⁵ Wilkinson had made a cylinder accurate enough for Watt's dry packing to seal. For the next twenty years, nearly every engine Boulton & Watt built had one of his cylinders.⁶

The slide rest

Most other round parts, like shafts and piston rods, were made on a lathe. The metal spinning in a lathe is called the work, and the person running the lathe is the turner. In the 1770s, the turner rested the cutting tool on a T-shaped support and guided it by hand, much as a woodturner steadies a chisel. But cutting iron takes a lot of force, and a turner's arms couldn't always hold the tool steady.

A heavy cut could push the tool away from the work, leaving a raised patch on the spinning bar. One revolution later, that patch struck the tool again. The uneven surface made the tool bounce, leaving more unevenness for the next turn. This growing vibration is called chatter. Below, we can watch the same cut in slow motion, first with the tool braced by hand and then clamped to cast iron.

A slide rest clamps the tool in a carriage of cast iron and advances it with screws. The carriage slides along two rails called the ways, which run the length of the lathe's bed. The work is held between the headstock and tailstock, which sit on the same ways. The headstock carries the spindle, the shaft that turns the work. If the ways are straight and the two stocks line up, the tool travels parallel to the work's axis.

With a slide rest, the forces of the cut go into iron instead of the turner's arms. The spinning surface drags down across the cutting edge, pressing the tool and carriage onto the bed, while the ways resist the push away from the work. In our model, the iron support bends about a hundred times less than the turner's braced arms,⁷ so a cut that set the hand-held tool chattering runs smoothly.

Slide rests weren't new. Instrument makers and French mechanics had used them for decades.⁸ Beginning in the 1790s, Henry Maudslay made them rigid enough for heavy cuts in iron and combined them with the screws and gears of a practical screw-cutting lathe.

The tool now followed the ways, so any bend in them would be copied into the work. A workshop could check the ways against a flat reference surface, but first it had to make one.

Flat

A workshop that needed a flat surface filed it by hand. A file is a hardened steel bar whose fine teeth shave metal as it's pushed across a surface. Filing takes off the largest bumps quickly, but the result depends on how steadily the file is held. Rock it slightly from heel to toe and it leaves a shallow slope behind.

As the surface gets flatter, each stroke starts to add about as much unevenness as it removes. The fitter can work more carefully, but can't hold the file perfectly steady or feel where the smallest high spots remain.

Fitters got past this limit with a scraper, a short hard blade that shaves off flakes a few thousandths of a millimetre thick, and with a way of finding errors far too small to feel. They spread a thin film of colour over a flat reference plate and rubbed the work against it. Colour transferred only where the work touched, marking the high spots that needed cutting. The fitter scraped those spots and rubbed the work against the plate again.⁹

The fitter doesn't have to hold the scraper level across the whole surface. The colour shows where to cut, and the short edge removes only a little metal with each stroke. If a spot is cut too deep, it stops touching the plate and picks up no colour. The fitter leaves it alone while scraping down the higher metal around it. Each new set of marks shows where more metal needs to come off, allowing the fitter to make a surface far flatter than filing alone could produce.

This method needs a flat reference plate to start with. Rubbing two plates together only shows whether they fit each other. A slightly domed plate can nest against a slightly hollow one and print colour evenly across both, even though neither is flat, and scraping them only makes them fit each other more closely. Below, two roughly filed plates are printed and scraped until they fit, one domed and the other hollow.

Here A is slightly domed and B slightly hollow, so they fit perfectly. Now take a third plate, C, and scrape it to fit A. C becomes hollow too. When we rub B and C together, the two hollows touch only around their edges, leaving a gap in the middle. The colour marks the edges, so the fitter scrapes them down and then compares all three pairs again: A against B, B against C and C against A. A dome can fit a hollow, but two domes or two hollows can't fit each other. As the fitter keeps comparing and scraping, the curves become shallower. All three plates can fit in every pairing and orientation only when they're flat.

This is one of the most beautiful ideas in engineering. The fitter could start with three rough castings and make a flat reference without having a flat surface to copy! The colour and the comparisons showed where to remove metal. Toolmakers still use the three-plate method to make a flat reference from scratch.¹⁰ By 1829, Maudslay's workshop had a scraped plate on every bench. Most workshops had none, and in 1840 Whitworth published the method and urged them to adopt it.

A workshop could use its flat plate to check a straightedge, then use the straightedge to find and scrape the high spots on a machine's ways. Straight ways could also guide a cutting tool. In a planing machine, which appeared around 1817, a casting travelled back and forth beneath a fixed tool, quickly removing most of the metal before the final scraping. Below, a later planer takes the top off a rough casting, one strip on each stroke. Like the lathe at the top, it can be turned and zoomed.

Whitworth reckoned in 1856 that truing a square foot of cast iron by chipping and filing had cost twelve shillings in labour thirty years earlier. A planing machine now did the work for less than a penny, more than a hundred times cheaper¹¹!

We can now make the straight ways that guide our lathe's tool. To cut a thread, the carriage also has to move the right distance for every turn of the work. The long screw that moves it has to be accurate too.

The screw

Before Maudslay, each workshop cut screw threads its own way, by hand or with dies, hardened tools that cut a thread around a bolt. The dies were copied from older screws, carrying their errors into new threads. A nut made for one bolt rarely fitted another, so each bolt and its nut had to be marked as a pair. Threads were often uneven as well. Workmen called a thread whose turns weren't evenly spaced drunken.

On a screw-cutting lathe, a long screw called the lead screw moves the carriage, copying its spacing into every thread the lathe cuts. An uneven lead screw makes uneven threads, even if the ways are perfectly straight. Instrument makers already had fine screws for dividing scales, but these were small and only had to move a point across brass. Maudslay needed a long, accurate screw strong enough to push a cutting tool through iron. How could he make the first one without an accurate screw to guide the tool?

The distance between neighbouring turns of a thread is its pitch. On the screws in this article, it's also how far the screw advances through its nut in one turn.¹² If we unwrap the surface of a screw, the thread becomes a straight sloping line that rises one pitch for every circumference it crosses. You can try this at your desk: cut a right triangle out of paper and wrap it around a pencil, and its sloped edge winds itself into a helix.

The pitch depends on the diameter of the cylinder and the slope of that line. Maudslay used this geometry to make his first screws. He turned a soft cylinder of wood, tin or brass between two pivots and pressed a hardened knife against it at a precisely set angle. As the cylinder turned, the knife cut a shallow groove, and the groove drew the knife along the way a thread draws a nut.¹³ Turning the crank faster made the knife travel faster, but didn't change how far it moved on each revolution. That distance was set by the cylinder's circumference and the knife's angle.

Maudslay had made a screw without copying one! This is a remarkable piece of bootstrapping. A chasing tool, with a row of teeth filed to the pitch, followed the shallow groove and deepened it into a working thread. Small errors in the knife, the cylinder and the pivots still left some turns farther apart than others. Those errors would be copied into any thread the new screw guided, so Maudslay set about improving it.

Improving the screw

Maudslay also devised an ingenious method that used two imperfect screws to guide the tool cutting a third. Equal gears turned both guide screws at the same speed. A bar joined their two nuts, with the tool fixed at its midpoint, so the tool's position was the average of the two nut positions. Where one guide ran ahead and the other lagged, their errors partly cancelled. Each guide contributed only half its error to the new screw.¹⁴

Averaging only helps where the errors differ. If both guides run ahead together, so does the tool. To check whether the average spacing was right, Maudslay measured the nut's travel over many turns against an ordinary rule. A screw of fifty threads to the inch should carry its nut exactly one inch in fifty turns. Dividing that measured distance by fifty gives the average pitch, with fifty times less uncertainty than measuring a single turn. It doesn't tell us whether every turn is evenly spaced, but it can catch a screw whose pitch is too large or too small overall.¹⁵

After repeated copying and correction, Maudslay made a master screw five feet long and two inches across, with fifty threads to the inch and a foot-long nut that engaged six hundred of them at once. It was used to divide the scales of astronomical instruments.

Many pitches from one screw

Maudslay's first screw-cutting lathe, built around 1797, coupled its lead screw directly to the spindle. Every thread it cut had the lead screw's own pitch, and a different pitch meant fitting a different lead screw. Within a few years he had joined the two through gears that could be exchanged, so that one lead screw could cut many pitches.¹⁶

The lead screw in our model moves the carriage six millimetres per turn, so coupled directly to the spindle, it cuts a six-millimetre thread. To cut a three-millimetre thread, the lead screw has to make exactly half a turn for every turn of the work, and a pair of gears can do that. Below, a 24-tooth wheel drives a 48-tooth wheel through one full turn of the small wheel.

In one full turn, all 24 teeth on the small wheel pass through the mesh, moving 24 of the large wheel's 48 teeth past the same point. The tooth counts set the ratio at exactly two to one. But if we put these two wheels directly on our lathe, they turn the spindle and lead screw in opposite directions, sending the carriage the wrong way. They may not reach each other across the gap between the two shafts, either.

A third wheel, called an idler, fits between them. It reverses the drive once more, so the spindle and lead screw turn in the same direction, and bridges the gap between their shafts.

The idler doesn't change the ratio. However many teeth it has, every tooth moved by the first wheel moves one tooth on the last, so only the end wheels set the pitch. Equal end wheels give six millimetres. Put 24 teeth on the spindle and 48 on the lead screw and the pitch becomes three millimetres; change the last wheel to 72 and it becomes two. The turner kept the wheels in a drawer beside the lathe and swung their mounting plate until the teeth met. Below, one lead screw cuts all three pitches on the same bar.

The tooth counts fix the ratio over complete turns, but poorly made teeth can still make the lead screw speed up and slow down within a turn.¹⁷ With well-made gears, one accurate lead screw could cut many accurate pitches.

A deep thread takes several passes, since cutting it all at once would overload the tool. Each pass has to follow the groove left by the one before. The turner doesn't have to find that groove by eye. The gears return the tool to the same thread on every pass! As long as they stay connected, they keep the carriage's movement in step with the work's rotation. Running the machine backwards brings the tool home along the same thread. The turner then sets the tool a little deeper and runs forwards again, and it follows the groove on the next pass. If the gears are separated and rejoined out of step, the tool cuts across the thread instead.

We can now cut an even thread at several pitches and deepen it over repeated passes. But the turner still needs to know when the screw has reached the right diameter. A workshop rule can't distinguish two parts that differ by a thousandth of an inch.

Measuring with a screw

Maudslay checked sizes with a bench micrometer. A fine screw, turned by a wheel divided around its edge, moved one of two flat jaws along a bed. It could show a difference of a thousandth of an inch, about a quarter of the thickness of a sheet of paper and more than sixty times finer than the smallest mark on a workshop rule. When two workmen disagreed about a measurement, the part went between the jaws and the instrument settled it. Maudslay called it the Lord Chancellor, after England's highest judge.¹⁸

The screw works as a lever for length. In our model of the Chancellor, one turn moves the rim of the wheel about 251 millimetres while the jaw advances only half a millimetre. A small movement at the jaw appears five hundred times larger at the rim, where we can read it against engraved marks. The smallest change those marks can show is the instrument's resolution.¹⁹

The two thin metal plates below, called shims, both look about three and a half millimetres thick against a rule. Between the jaws, they read 3.615 and 3.640 millimetres. The difference is 0.025 millimetres, about a third of the width of a hair. The screw magnifies it five hundred times, so on the rim of the wheel the two readings are more than a centimetre apart.

But the reading also depends on how hard the operator turns the wheel. Once the jaws touch the shim, extra force bends the frame slightly and lets the wheel turn a little further. The shim hasn't changed thickness, but the reading has. To compare two parts, the jaws have to close with the same force each time.

Reversing the wheel causes a subtler problem. A screw needs a little clearance in its nut so that it can turn. While it moves forward, one face of each thread carries the load. Turn it back, and the screw has to cross the clearance before the opposite faces touch, so for a small part of a turn the wheel moves while the jaw doesn't. This lost motion is called backlash, and we can see it most clearly on a plain screw and carriage.

On the micrometer, backlash means the same shim can give two readings, depending on which way the wheel last turned. Below, the jaws close on a shim, then the operator eases the wheel back a little to reduce the pressure. The wheel changes its reading before the jaw starts to move.

Machinists deal with this by always closing onto the part from the same direction. If they overshoot, they open the jaws well past the clearance and close again, so the screw always bears on the same thread faces. Using the same approach and the same force makes a reading repeatable: measuring the same shim again gives the same number. That still doesn't make it accurate. If the screw's pitch is slightly off, repeated readings can agree and still be wrong. To check the instrument's accuracy, we need a part whose length is already known.

The millionth of an inch

Whitworth wanted workshops to stop measuring with rules. He argued that it was easier to work to a ten-thousandth of an inch by comparing a part with a steel bar of known length than to a hundredth by reading the lines on a rule. He called these bars standards of end measure, because the length was the distance between their ends. To make them, he needed a measuring instrument much finer than the Chancellor.

Whitworth's machine used a screw-driven jaw like the Chancellor's, but with a nut made in two adjustable halves. He first fitted the nut to the measuring screw, then cut it in two. Small screws drew the halves together until one pressed on the forward faces of the thread and the other on the rear faces. This took up the clearance that caused backlash, much as two doorstops on opposite sides keep a door from moving either way. As the threads wore, the halves could be drawn a little closer.

To reduce the effect of an unsteady hand, Whitworth put a worm between the handwheel and the measuring screw. A worm is a short screw whose thread meshes with the teeth of a wheel. One turn of the worm moves the wheel by a single tooth, so with a 200-tooth wheel, the measuring screw turns only a two-hundredth as far as the handwheel. A small unwanted turn of the handwheel is reduced by the same ratio.

The measuring screw had twenty threads to the inch, the worm wheel two hundred teeth, and the handwheel was divided into 250 parts. Twenty times two hundred times two hundred and fifty is a million, so one division on the wheel moved the measuring faces one millionth of an inch, about 25 nanometres.²⁰ A fingernail grows that much in about twenty seconds.

A movement that small is far too fine to judge by eye. It's only about a twentieth of a wavelength of green light, much narrower than an engraved line. Whitworth used a small, flat piece of steel, the gravity piece, to find when the faces had closed far enough. It sat between one measuring face and the end of the bar. The operator closed the faces a division at a time, letting go of the piece after each step. At first it fell, but when the faces pressed closely enough, friction held it. Closing them just one millionth of an inch further could make the difference between a piece that fell and one that stayed put!

The machine compared two nearly equal lengths. The operator first measured a standard, closing the faces until they held the gravity piece, then exchanged the standard for the work and repeated the same test. The change in the wheel's reading gave the difference in length. The measuring screw only travelled through that small difference, and a fixed error shared by both readings cancelled when one was subtracted from the other.²¹ It didn't have to measure the whole bar from zero.

At the Great Exhibition of 1851, Whitworth had someone else adjust the machine a millionth of an inch at a time without telling him, and detected every change. Below, we can see the machine compare a bar with a standard, then try a similar test ourselves.

Even the operator's body heat mattered. A yard of steel grows about 230 millionths of an inch for every degree Fahrenheit, and Whitworth found that the slightest touch of a finger on a yard bar in the machine warmed it enough to stop the gravity piece from falling. A standard of length is only exact at one stated temperature, which for the British standard yard was 62 °F.

The measuring machines were used to make and check standard gauges, steel plugs and rings of known diameter. A workshop could check its parts against these gauges without having a millionth machine of its own. Whitworth showed how much a small change in size affected the fit: in a ring of 0.5770 inch, a plug of 0.5770 inch “fits tightly … when both are clean and dry.” A plug of 0.5769 inch, one ten-thousandth smaller, “is so loose in it as to appear not to fit at all.” The difference was about a thirtieth of the width of a hair. By 1857, the workmen in Whitworth's own works measured to a twenty-thousandth of an inch.²²

In about fifty years, the smallest difference a workshop instrument could reveal had shrunk from the Chancellor's thousandth of an inch to Whitworth's millionth, a thousand times finer. The gauges carried the benefit into everyday work. Now, let's return to the lathe.

The machine that made machines

Here is the complete lathe from the beginning of the article. We've now seen how each of its main parts could be made accurate. The ways can be checked against a flat reference and scraped straight. The slide rest holds the tool steady under the force of the cut. A lead screw, made by copying and correcting earlier screws, moves the carriage evenly along the bed. Change wheels set how far it moves for each turn of the work. Together, these parts let the turner cut a straight shaft or an even thread, then check its size with a micrometer or gauge.

Steam engines needed true cylinders, straight piston rods and flat valve faces, and machine tools made them more quickly and consistently than hand fitting could. A lathe could also make screws and spindles for another lathe, while a planing machine cut its bed. The new machine would follow the geometry of those parts each time it cut. A skilled turner still had to set it up and choose the cut, but no longer had to guide the tool's whole path by hand.

Epilogue

Workshops used machine tools like Maudslay's lathe to make parts for the railways, steamships and factories of the nineteenth century, and standards like Whitworth's to check their sizes. Machine tools also made parts for new machine tools, copying their accuracy and improving it as workshops learned better ways to guide a cut or check a size. Small gauge blocks, the standards of length that workshops use to check their instruments, can now be made within a millionth of an inch of their stated size.²³ A millionth of an inch, the smallest difference Whitworth's machine could detect, is now a manufacturing tolerance.