Angles can be generated without any angle standard, because a circle divided into N equal steps must close exactly after N steps, and any leftover gap shows the accumulated error and how to correct it. A dividing plate drilled with hole circles handles the divisions a compass can construct, such as 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 and 24, while a worm-driven dividing head reaches the rest.
Part of Precision Tool Making
Splitting a circle or a length into equal parts for gear teeth, hole circles and graduations, and checking the result without a reference.
A Circle Checks Itself Because It Must Close
Flatness needs three plates to check it, and length needs a physical standard. Angle needs neither. A full turn is 360 degrees whatever you do, so if you step round a circle N times with the same setting, the last step must land on the first mark. If it overshoots or falls short, the shortfall divided by N is the error in each step, and you correct the setting by that much and go round again (Bootstrapping Precision).
That self-closing property is what makes angle the cheapest quantity to generate and the reason divided circles can be trusted before any instrument exists. The error that remains is not the angle itself but the physical positions of the holes or marks.
What Can And Cannot Be Divided With A Compass
The carrier lists the divisions made by geometric construction with a compass alone: 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 and 24. Others, such as 7, 9, 11 and 13, cannot be constructed with compass and straightedge alone (a mathematical fact, not a workshop limit), so they need the dividing head. A dividing plate is a disc of 200-300 mm with rows of holes at such divisions; the carrier names rows of 24, 30, 36, 48 and 60 holes as covering most common requirements (Dividing Plate).
What You Need
- A disc of steel or brass, 200-300 mm diameter, faced flat on a lathe (Lathe Fundamentals)
- Dividers, a scribe and a centre punch
- A drill for 2 mm holes and a way to hold the disc (Workholding)
- A spring-loaded indexing pin that fits the holes without play
Method: Divide And Close
Step 1: Scribe a precise circle with a pointed tool in the lathe.
Step 2: Divide by construction into 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 or 24, then centre-punch each point.
Step 3: Step round with dividers to check: set the dividers to the chord, step N times, and see where the last step lands.
Step 4: Correct the setting by the shortfall divided by N and repeat until the steps close.
Step 5: Drill the 2 mm holes at the points; for finer divisions add rows at other radii.
Step 6: Check the pin fit. Any looseness in the pin is an error on every division.
Worked Example
A circle of radius 100 mm divided into 30 parts has a chord of 2 x 100 x sin(6 degrees) = 200 x 0.10453 = 20.906 mm. Suppose the dividers are set to 20.90 mm, short by 0.006 mm. After 30 steps the shortfall is about 30 x 0.006 = 0.18 mm. Dividing by 30 returns the 0.006 mm per step, and you widen the dividers by that amount.
The error grows with N: Gear Cutting notes that a 0.5% error in the divider setting on a 30-tooth gear puts the last tooth off by 0.15 of a pitch, which is 30 x 0.5% = 15% of a pitch, and enough to bind. Closure is therefore the check that matters more than any one tooth.
Common Mistakes
- Trusting each division and never closing the circle. Small errors add.
- Using a loose indexing pin. The play is an error on every step.
- Assuming every division can be constructed. Seven cannot.
- Punching marks too deep and too large. The mark itself is a source of position error.
Safety
Drills, lathes and sharp points
Dividing plates are drilled and turned: clamp the disc so it cannot spin or lift (Workholding), keep sleeves clear of the chuck, and know where the stop is. Dividers and scribes are sharp points: carry them point-down in a cork.
What’s Next
- Learn next: Dividing Head - reaching the divisions a compass cannot
- Combine with: Gear Cutting - using the divisions to cut teeth