The Endless Screw

Jesse Ramsden, the Dividing Engine, and the Sub-Arcsecond Circle

One arcsecond of angle—$1/3,600$th of a single degree, or $1/1,296,000$th of a full circle—subtends the width of a human hair held twenty meters from the eye. On the surface of the Earth, one arcsecond of latitude corresponds to thirty-one meters of ground; in astronomy, measuring the annual stellar parallax of even the nearest stars requires resolving angles well below one arcsecond. Yet throughout the first three quarters of the eighteenth century, every quadrant, sextant, and astronomical transit circle in Europe was trapped inside a mechanical circularity problem: to measure an angle to one arcsecond, an optician first had to engrave thousands of degree and arcminute marks around a brass circle with sub-micron linear spacing.

Before 1770, dividing a brass scale was an exhausting hand craft guarded by a handful of London instrument makers such as George Graham and John Bird. Working in a room held at constant temperature, Bird used beam compasses fitted with magnifying lenses to bisect circular arcs repeatedly—stepping off sixty degrees with a radius chord, bisecting to thirty, fifteen, and down through prime-number subdivisions using an auxiliary scale of equal parts. Engraving a single eight-foot mural quadrant for the Royal Observatory took weeks of crouched manual labor. Worse, every time the craftsman breathed on the brass arc or leaned his warm torso over the bench, the metal expanded locally by several microns, locking permanent thermal stepping errors into the graduations. Small portable marine octants and sextants needed by Royal Navy officers could not be hand-divided economically at all.

In Piccadilly, London, the Yorkshire-born former clothworker and optician Jesse Ramsden set out in 1766 to replace the human hand and eye with a self-correcting mechanical indexing machine. After an initial 1768 prototype proved slightly imprecise, Ramsden completed his second circular dividing engine in 1775, winning a £615 reward from the Board of Longitude in exchange for publishing his blueprints and training rival London makers to divide naval sextants at three shillings apiece.

Instead of stepping off chords with compasses, Ramsden's dividing engine transferred the geometry of a circle into the continuous rotation of an endless tangent screw meshing with a large toothed wheel. To eliminate the periodic errors that normally plague gear teeth, Ramsden engineered a five-stage mechanical architecture:

• * The Forty-Five-Inch Gunmetal Wheel: A rigid horizontal bell-metal wheel forty-five inches in diameter cast with ten reinforced spokes and turned completely flat on its rim, braced against thermal warping by double conical bearings over a central steel spindle.
• * The Two-Thousand-One-Hundred-Sixty Ratchet Teeth: The outer circumference of the wheel was cut with exactly 2,160 fine teeth—six teeth for every degree of the $360^\circ$ circle—so that one full turn of the driving tangent screw advanced the wheel by exactly ten arcminutes (600 arcseconds).
• * Self-Lapping Screw Homogenization: To carve those 2,160 teeth without cumulative gap errors, Ramsden first cut a master hardened-steel tangent screw on a specialized lead-screw lathe using a diamond-tipped tool, then gashed longitudinal flutes into a second steel worm screw so it acted as a rotary cutting hob. By forcing the hardened hob screw to turn the gunmetal wheel continuously for hundreds of revolutions under oil and diamond paste, high spots on any individual tooth were ground away against the average pitch of all 2,160 teeth simultaneously.
• * The Micrometer Brass Drum and Ratchet Treadle: Mounted on the tangent screw shaft was a graduated brass micrometer disk divided into sixty parts, where each division advanced the giant wheel by exactly ten arcseconds. A spring-loaded foot treadle connected to a spiral catgut cord allowed an operator to step the screw forward by an exact, preset fraction of a turn with each press of the foot while keeping both hands on the engraving carriage.
• * The Kinematic Radial Scriber: Above the wheel, a diamond-pointed cutting tool slid purely along a radial steel guide frame hung on frictionless gimbal pivots, impossible to deflect sideways while scoring hair-thin radial lines into the brass sextant or theodolite ring clamped to the master wheel.

With the dividing engine, an apprentice pressing the foot treadle could graduate a naval sextant arc in thirty minutes with higher accuracy than John Bird could achieve in a week of hand division. A decade later, in 1787, Ramsden applied his engine to construct his masterpiece: the Great Three-Foot Theodolite built for General William Roy's triangulation linking the Royal Observatory at Greenwich with the Paris Observatory. Looking through high-magnification micrometer microscopes focused directly onto the engine-divided thirty-six-inch horizontal brass circle, surveyors could read horizontal angles to a single arcsecond—representing a linear displacement of just 2.2 microns at the brass rim. When British and French surveyors met at the White Cliffs of Dover in autumn 1787 to connect the Royal Observatory at Greenwich with the Paris Observatory across the Channel, Ramsden's three-foot brass theodolite closed the twenty-six-mile optical triangle to within two arcseconds.