Dividing head: indexing tables

November 3, 2026  ·  Indumor

Seven divisions: 5 turns of the crank and 15 holes on the 21 circle. Seventeen: 2 turns and 6 holes on the 17. Fifty-one cannot be done with the Brown & Sharpe plates: you set up the differential train, or switch to the Cincinnati plate, which has a 51 circle. That is the table in this chapter, computed for the two sets of plates found in most shops and for the 40:1 ratio of nearly every head. What follows explains where the numbers come from, for the day the job asks for a count that is not in the table or the head is not 40:1.

What is inside the head

A universal dividing head is a single-start worm meshing with a 40-tooth worm wheel on the spindle. Forty turns of the crank are one turn of the work; one turn of the crank is one fortieth, which is 9°. Fractions of a turn are counted on the index plate, a disc with concentric circles of different hole counts where the crank pin drops in, and with the sector arms, two blades you open to mark how many holes to move so you never count them twice. There are 60:1, 90:1 and 120:1 heads, mostly on large machines and optical dividers; the arithmetic is the same with the right ratio.

Direct indexing

The direct index plate sits on the spindle nose, ahead of the worm, and has 24 holes; some have 30 or 36. You disengage the worm, turn the work by hand and lock it with a pin. It covers the numbers that divide the plate: hexagons, squares, double keyways. It is fast and has no backlash, but it never leaves that list.

Simple indexing: the table from 2 to 100

With the worm engaged, the crank turns for N divisions are 40/N. If the division comes out whole, done; if it leaves a fraction, the fraction becomes holes on a circle whose count is a multiple of the denominator. For 7: 40/7 = 5 turns and 5/7, and since 21 is a multiple of 7, 5/7 of 21 is 15 holes. The three Brown & Sharpe plates carry 15 to 20, 21 to 33 and 37 to 49; the two-sided Cincinnati plate carries 24 to 43 and 46 to 66, which is why it reaches counts like 51, 53, 57 and 59 that need the differential with the B&S set.

The table gives the smallest circle that works; any multiple is the same move (15 on 21 = 35 on 49). "diff." marks a count that cannot be indexed directly with that set of plates: it goes by differential indexing.

With the B&S plates, 21 counts between 2 and 100 fail (51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 96, 97, 99); with the Cincinnati plate, 16. The next table covers them. Gear teeth are indexed like any other N; the cutter numbers are in chapter 13 and the tooth formulas in chapter 15.

When it will not go: differential indexing

The fix is to gear the head spindle to the index plate shaft with a train of change gears and unlock the plate. You pick a number A close to N that does index directly and index as if there were A divisions; meanwhile the plate creeps round, driven by the spindle, and makes up the difference. If A is larger than N the plate turns the same way as the crank; if smaller, the opposite way, and the direction is set with one or two idlers depending on the head. The B&S set has 24, 24, 28, 32, 40, 44, 48, 56, 64, 72, 86 and 100 teeth, and with it every count up to 200 comes out. Two limits: the sector arms are set for A, not for N, and it cannot be combined with helical milling, because the train is already in use.

Computed from train = 40·(A − N)/A, taking the nearest A that indexes directly and can be built from those gears. "Same" means the plate turns in the same direction as the crank; the direction is set with the idlers according to the head. The sector arms are set for A, not for N. Compound trains come from the ratio alone: if two gears foul on the quadrant, swap the drivers or the driven gears, which gives the same ratio.

Angular indexing

One turn of the crank moves the work 9°, so the turns for an angle are the angle divided by 9. For 30°: 3 turns and 1/3, that is 3 turns and 5 holes on the 15. The resolution depends on the circle: on the 18 each hole is half a degree, on the 27 a third, and on the Cincinnati 54 a sixth, 10 minutes. When the angle does not land on a hole you take the nearest one and note the error; the calculator does that. For bolt circles laid out by coordinates, the table in chapter 23 is the better tool.

Check before you cut

Most mistakes at the dividing head are not in the arithmetic but in the counting. The sector arms are opened by counting spaces, not holes: to move 15 holes there are 16 holes between the arms, the starting one included. The drawing shows it on the 21 circle, and the routine below takes care of the rest.

• Open the arms to the 15 spaces, tighten them and leave them alone. After each move, swing them round until the trailing arm rests against the pin.
• Always turn the crank the same way. If you overshoot a hole, back up half a turn and come in again, to take the backlash out of the worm.
• Do a dry run: seven moves of 5 turns and 15 holes must bring the pin back to the starting hole with the work one full turn round. If it does not close, do not cut.
• Lock the spindle before every pass and unlock it before indexing; the worm holds the division, not the shock of the cutter.

Frequently asked questions

How many crank turns for 7 divisions on a dividing head?

On a 40:1 head, 5 turns and 15 holes on the 21 circle (or 20 on the 28 with the Cincinnati plate). It comes from 40/7 = 5 turns and 5/7.

Which divisions cannot be indexed directly on a dividing head?

With the three Brown & Sharpe plates and a 40:1 ratio, between 2 and 100 these fail: 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 96, 97, 99. They are done by differential indexing, or several of them with the Cincinnati plate.

How do you index 51 divisions?

By differential indexing: index as for 50 (12 holes on the 15) with the 32/40 train between the spindle and the plate, which turns against the crank. With the Cincinnati plate it is direct: 40 holes on the 51 circle.

Sources: Brown & Sharpe and Cincinnati universal dividing head manuals (historical names of the plates and change gears); Machinery's Handbook, Indexing section (simple, differential and angular indexing); A. L. Casillas, Máquinas: cálculos de taller (historical reference); every table in this chapter was computed from the formulas given in it.