Simple, Compound, Differential and Block Indexing

§1The 40:1 heart

Inside every classic dividing head a single-start worm drives a 40-tooth wheel on the work spindle — the other-threads page’s worm, employed as pure arithmetic: forty crank turns rotate the work exactly once.

Everything follows from that ratio. One crank turn moves the work 360/40 = exactly 9.000°; a fraction of a crank turn moves it that fraction of nine degrees; and the index plate — the hero’s disc of concentric hole circles, with a spring plunger on the crank dropping into whichever circle is chosen — makes fractions of a turn as exact as the plate’s drilling. Two sector arms, set once to embrace the required hole count, then bracket every subsequent move so the operator counts spaces between the arms instead of holes around a circle: swing crank to the far arm, drop the pin, flip the arms round — a rhythm that survives hundreds of divisions without a miscount. The worm itself contributes the mechanism’s integrity: self-locking (the systems page’s test, safely passed at this ratio), so cutting forces at the work cannot stir the crank, and fitted with backlash control so the count is real. On this one instrument the era cut its gear teeth, fluted its taps and reamers, indexed its thread starts and graduated its dials; the four sections that follow are the four grades of arithmetic it runs on.

§2Simple indexing

For N divisions, turn the crank 40/N turns between cuts — whole turns counted, the leftover fraction expressed as holes on whichever circle makes it exact.

§3Compound indexing

The first answer to an impossible fraction was to build it from two possible ones: move the crank on one circle, then move the plate itself on another, and let the sum land where neither could alone.

The plate on a compound-capable head can be unlocked and rotated against a second stationary pin, so each division becomes two counted motions: the crank through its holes in the plate, then the plate — carrying the pinned crank with it — through holes counted against the frame, added or subtracted as the identity requires. Take N = 69, hopeless for §2 since no circle carries its factors: the standard circles offer 10/15 − 2/23, which a line of fraction arithmetic confirms equals 40/69 exactly — crank forward ten holes on the 15, plate back two on the 23, every division. It works, and it was hated: two counts per move, one of them often backwards, doubled chances for the single miscount that scraps a gear at tooth fifty-nine, and the search for a valid a, b, C₁, C₂ was an evening with tables. Compound indexing is best understood as the pressure that produced §4 — the same two-motion idea, with the second motion taken away from the fallible hand and given to a gear train.

§4Differential indexing

Gear the work spindle back to the index plate and the plate creeps automatically as the crank turns — the compound correction, made continuous, exact and impossible to miscount.

§5Block (direct) indexing

For the small, friendly numbers there is a faster gear entirely: disengage the worm and locate the spindle straight off a slotted plate on its nose — one motion, no crank, no arithmetic.

The direct plate is a disc of slots or holes fixed to the work spindle itself, engaged by a plunger in the frame; with the worm dropped out of mesh, indexing is pull-turn-click, as fast as the hand moves. The divisions available are simply the divisors of the plate’s count: the common 24-slot plate yields 2, 3, 4, 6, 8, 12 and 24 — precisely the hexagons, squares, flats, keyway pairs and cross-drillings that constitute most of a jobbing shop’s indexing, done in a tenth of the crank method’s time. The trade is exactness of a different kind: accuracy now rests wholly on the plate’s slot spacing with no 40:1 worm dividing its errors down, and rigidity rests on the plunger, so direct work is light work. The same idea, detached from the dividing head altogether, becomes the collet indexer and spacer block of every toolroom — and, scaled up with hardened face-tooth couplings, the turret and rotary-station indexing inside production machines. Block indexing is the reminder that the section’s cleverness is optional: when 24 divides your problem, the answer is a click.

§6Angular indexing and thread starts

Because one crank turn is exactly nine degrees, the head divides angles as readily as counts — down to twenty minutes of arc on the standard plates — and it is the tool that spaces a multi-start thread.

The conversion is a single division by 9. One hole on the 27-circle is 9/27° = 20 minutes of arc, the finest the common plates offer; an angle like 37° becomes 37/9 = 4 1/9 turns — 4 turns + 2 holes on the 18-circle, since 1/9 = 2/18, and the check multiplies back to 37.000° exactly. Graduating a dial, drilling holes on an angular pattern, or setting a cam’s events all reduce to this arithmetic. The section closes where it began: the multi-start threads of the systems, Acme and closure pages exist only because something can rotate a blank by exactly 360/n between passes — 180.000° for two starts (20 crank turns), 120° for three — with an error small enough that the starts share load equally, which is precisely a 40:1 head’s gift of dividing its own plate errors by forty. Cut one start on the lathe, index, cut the next: the dividing head is the quiet co-author of every fast-lead thread in the section. And its modern descendant — the servo rotary axis reading a 360° encoder — changes the actuator, not the arithmetic: 40/N merely became a number typed instead of cranked.

§7Quick reference

The working core of the page on one card rack.