How, and with what, did man make the first straight lines? How did he layout the first square or rectangular building? How did he (or she) make the first machine without the benefit of a machine-making machine? These questions have intrigued me for decades. Did you know that even today people grind their own mirror lenses for telescopes without the benefit of machines? How is such a thing possible? Have you ever gazed at the mirror-like surface of a hydraulic piston, ground and polished to a precision measured in ten-thousandths of an inch, and wondered how we progressed from chipping flint tools to manufacturing such seemingly perfect shapes?

We live in a world of irregular shapes. Rivers don't flow straight to the sea, tree limbs don't radiate out in straight lines, and rocks aren't square like bricks. We humans seem to like straight lines and predictable, smooth curves. We have imposed an irregular regularity on our environment. Skyscrapers both rectangular and curved, reach skyward with precise, regular profiles.

It's not as if there are no straight lines, smooth surfaces, or regular curves in nature, it's just that she seems to prefer the irregular, at least at scales visible to our naked eye. When we do see straight lines, regular polygons, or smooth predictable curves in nature, they demand our attention. The apparent roundness of the the sun and moon, the exquisitely smooth and predictable curves of a Robin's egg, the startling squareness of the stones in the Devil's Postpile, or the spectacularly smooth surfaces of a piece of quartz almost insist we take note. Who has not marveled at the astounding hexagonal regularity of a honeycomb. And yet a beehive observed in the wild is often just a large lumpy mass hanging from a tree, despite its inner regularity.

Photo: The Devil's Postpile

A curious thing about straight lines is there are none, at least not any truly straight lines. But there are plenty of lines that are more than straight enough to satisfy our aesthetic or practical desire for straightness. It is curious that without any actual example of a truly straight line, we should be able to conceive the idea of perfect, or ideal straightness. G.K. Chesterton, in 'Orthodoxy', explained his need to believe in a Divine being partly in terms of his need to believe in the existence of ideals.

 It's not too hard to imagine a curious neanderthal stretching a piece of hair, or sinew between his outstretched hands and noting that it made a straight line. The ability to create straightness at will may have been deeply satisfying to primitive man. Could his delight in this ability have influenced his highly stylized early art? Imagine his delight when he discovered that holding one end of the taut sinew fixed, while swinging the other about, created wonderfully smooth arcs and circles.

 Visit any construction job site today, and during the process of setting up the concrete formwork, you will find construction lines of strong, yellow string stretched over and between nails hammered into batter boards, defining in our real world the ideal grid lines of the blueprints. If a circular arc needs to be created on that job site, it will be done by a method nearly identical to that we have just imagined our ancient ancestor using.

The states of being level, or plumb, i.e. being horizontal or vertical, must certainly have occurred to man in prehistoric times. As a teen-ager I watched my father quite accurately level the formwork for a house foundation using nothing but a garden hose—he simply observed the water level at each end, and adjusted the formwork to match the water level. A technique ancient, yet still useful.

 How did our ancient man construct right angles? And why did he conceive of, and prefer right angles? We don't know. We can't know with any certainty the first method used to construct right angles, but long before Thales—the father of Greek philosophers—came along, the Egyptians were pulling ropes to construct right angles for surveying and construction. Their precise technique is not known, but it seems likely that 'magic' right triangles such as (3,4,5) and (5,12,13) were known, and that ropes knotted at regular intervals were stretched taut to create such triangles.

 We do know that Thales (624 BC - 546 BC) discovered and proved that any triangle inscribed in a semi-circle was a right angle. I still experience actual delight every time I'm reminded of this simple and beautiful principle. The best ever precalculus book, the tiny 'Precalculus Mathematics in a Nutshell', by George F. Simmons, gives a clear, simple proof of Thales' theorem. So given a rudimentary compass and a straight-edge, the construction of right triangles is trivial.

 The great builders of early civilization displayed a decided preference for square corners... but why? I'll hazard a guess. They liked squareness because it helped ensure that furniture, casework, and other interior features would fit, and made for good use of interior space. Consider that in a circular room, curved furniture made to fit at the perimeter will not fit the perimeter of a room with a different radius. Rectangular furniture and rooms don't suffer from this shortcoming.

Groma: Transit-like device used by Romans.Credit: self

Sketch: Groma

 So now our ancient builders have the ability to create right triangles at will. How do they use them to layout building foundations? Using a plumb bob in conjunction with the right triangles created by the Egyptian rope pullers would be one way. Later on the Romans used a device known as the Groma (see sketch). They constructed a wooden cross having only right angles, with a plumb bob at each end of the each crosspiece, and another bob at the intersection of the two crosspieces.  This center plumb bob was placed directly over one corner of the building, one pair of strings was lined up with an established edge of the future building (say A and B) and then they sighted across the remaining pair of strings (C and D) to establish the other edge of the building. Presumably a helper in the distance, holding a staff, or perhaps another plumb bob, moved it to the location indicated by the user of the groma. The groma was used for centuries before the first transit-like device made was invented. The groma was usually suspended from a portable frame.

Credit: self

Figure: A Plumb Bob 

How does one go about creating a flat surface? The technique is clever. Imagine a mostly flat but slightly uneven stone plate (no, not a dinner plate, just a flat plank like surface). Given a tool hard enough to scrape or grind the stone, a good craftsman can eyeball a fairly flat surface, but then he’ll need a bit of help. It turns out that he can get from fairly flat to extremely flat, using three stone plates, and perhaps some grinding grit. None of the three plates need be truly flat, just close. If he were to use just two stone plates, rubbing them against each other, there is the risk that he might create a subtle concave depression or bowl in one plate, and a slight convex bulge in the other. Of course the fit would get better the more he rubbed the plates together, but he would have no guarantee they were getting flatter.

Here's how he can ensure increasing flatness as he rubs. He needs three stone plates the same size. Let's call them A, B, and C. He begins rubbing A against B, and unbeknownst to him, A begins to develop a very slight depression, and B a very slight convex bulge—but our craftsman doesn't detect these tiny flaws. No matter. He wisely begins to rub C against B, and doggone it if B doesn't begin to create a depression in C, but C in return begins to wear down B's projecting bulge—because unlike A, C doesn't have a depression that perfectly matches B. Our craftsman proceeds. He next begins to rub C against A, and now the two slightly concave plates begin to flatten out (this is easy to visualize). And on, and on it goes until our craftsman reaches the desired level of flatness. Expert craftsmen of today can hand lap plates to incredible degrees of flatness. Combined with hand-scraping techniques, such lapping techniques can produce steel plates so flat that one can be suspended from the other, without clamps, due to molecular attraction. Such plates are said to be 'wrung', from the slight twist given to them when pressing them together.

How do you make a machine without a machine? You could use simple tools, knives, hammers—stone hammers if you insist—along with readily available materials, tree limbs, animal hides, etc, to manufacture simple machines. Wheels, gears, and levers could certainly be carved by hand. An early example is the screw thread used by Archimedes to lift water from one elevation to another. The yet to be threaded blank could be held between lathe like supports. Then a knife could be held against the blank, at the helix angle of the thread. When the work was rotated, the knife would trace out the helix of the thread along the length of the blank. But to finish cutting the thread you would have to resort to hand carving in the absence of a ‘lead screw’, or some similar device that could reliably advance the cutter x inches to the right, for every y revolutions of the blank (the blank being the eventual Archimedes screw).

 We should not sell the ancients short and assume that they could not come up with a reliable substitute for a lead screw—one that could maintain a constant  number of revolutions for every inch the cutter advanced. Threads are described in terms of tpi—threads per inch—a single thread results from a single revolution of the blank. For example, if the blank revolves 12 times for every inch the cutter travels to the right, we would end up with a 12-tpi screw thread. Of course threads can be measured in terms of any system of units—perhaps turns per cubit for an Archimedes screw. To give you a better idea of what a lead screw is, see the two items marked as ‘feed screws’ in the image below—feed screw is another word for lead screw.

 Take some time to study the engine lathe photograph below. The blank would be mounted between the headstock and the tailstock for rotation at the desired speed. Gears to the left of the headstock determine the speed at which the blank rotates—the headstock powers the rotation of the blank. One feed screw moves the carriage (or saddle) to the right—when engaged by the half nuts on the carriage—the other moves the carriage to the left. The carriage carries the cutter (held in the tool holder) along with it. If the speed of the lead screw is held constant, then when the speed at which the blank rotates is changed, the number of threads cut per inch will change proportionately.

Credit: self

Photo: An Engine Lathe

 Why so much time discussing lead screws, and thread making? Because once the difficulties in manufacturing accurate lead screws was overcome, accurate, uniform threads could be made. With uniform threads, two great benefits accrue: Interchangeable parts, and precision measuring instruments. Astoundingly accurate vernier scales could be created, along with adjusting screws that reliably moved them tiny fractions of an inch at a time. Micrometers that could measure in fractions of 1/10,000 of an inch, and affordable theodolites (surveyor’s transits) capable of measuring just a few seconds of arc, followed soon after. Precision gears, bolts, and screws could now be made in incredible variety with perfect repeatability and consequent interchangeability.

 The prevailing opinion is that lathes equipped with hand filed lead screws were used to turn ever more accurate lead screws in a process of iteration. It is also widely believed that longer nuts were used to transfer the motion of the lead screw to the carriage and cutter. The thinking being that a nut that engages 20 threads on the lead screw, instead of just a few, would tend to average out the motion of the carriage and cutter—by resisting the tendency of any single out-of-place thread to speed up, or retard, the motion of the carriage and cutter.

I mentioned at the beginning of this article that people still hand grind telescope lenses. To be precise, what is usually ground is the mirror for a reflecting telescope, which is nothing but a heavy—one or two inches thick—circular glass blank. One side of this glass blank will be ground convex. That  such a precise convexity can be ground by hand (no electric powered tools) and then polished by hand to the most perfectly clear, glistening surface is, to me, amazing. The hand tools used are astoundingly simple, perhaps a 2" 2" 1" square of mild or cast steel, along with some 80 grit carborundum to begin the process. A central depression is ground out, then widened with a variety of ad hoc grinding implements and grit. At all times the lens grinder strives to maintain a random pattern, turning his grinding tool in one direction, and his glass mirror blank in the opposite direction, at short and regular intervals. The lens grinder will periodically test his lens for curvature and regularity, often using the simplest of home built tools to assure the desired mirror profile is achieved. Hand grinding telescope lenses and mirrors deserves an article of its own. The techniques employed are both fascinating and clever.