Compass-Arm Gear
Part of Gear Making
Making and using compass-arm (trammel) methods to lay out gear tooth spacing with precision.
Why This Matters
The single greatest challenge in hand-cutting a gear is spacing the teeth exactly. A gear with 36 teeth must have all 36 tooth spaces divided precisely around the circumference — if any space is too wide or too narrow, the gear will not mesh smoothly with its partner, and the variation will repeat as a cyclic irregularity that causes vibration, noise, and accelerated wear.
The compass-arm method — sometimes called the trammel method or beam compass approach — is the traditional pre-industrial technique for accurately dividing a circle into equal parts for gear tooth layout. It requires only a compass (or beam compass for large diameters), a sharp scriber, and patience. Combined with a dividing plate for fine divisions, it gives the craftsman sufficient precision to make functional gears without any specialized indexing equipment.
Understanding the geometry behind compass-based division explains why it works and where its limits are. This understanding lets you judge when your layout is good enough and when to invest more time in verification.
The Geometry of Equal Division
To divide a circle of radius R into N equal parts, the chord length between adjacent division points is:
c = 2R × sin(π/N) = 2R × sin(180°/N)
For example, to divide a 200 mm pitch circle (R = 100 mm) into 24 equal parts: c = 2 × 100 × sin(180°/24) = 200 × sin(7.5°) = 200 × 0.1305 = 26.1 mm
Set a compass to exactly 26.1 mm (this is the critical step — the accuracy of the division depends on this setting), place the point on one reference mark on the pitch circle, and step the compass around the circle. After 24 steps, you should arrive back at the start mark. The accumulated error tells you how accurate your compass setting was.
Setting the Compass Precisely
The compass setting is the critical precision step. Methods:
Method 1: Direct chord measurement. Calculate the chord length and set the compass using a steel rule or vernier caliper. The limiting factor is the precision of the chord calculation and the sharpness of the compass points. Achievable accuracy: ±0.1–0.2 mm on compass setting, which translates to ±0.1–0.2 mm × N cumulative error after going around the full circle. For 24 teeth, this is potentially ±2.4–4.8 mm total accumulated error — significant.
Method 2: Trial-and-adjust (iteration). Set the compass approximately, step around the circle, and measure the remaining gap or overlap at the end. If the gap is too large, the compass was set too small. Adjust by half the remaining gap / N. Repeat until the gap closes completely. This iterative approach can achieve very good results — ±0.1 mm or less around the full circle — with patience.
Method 3: Geometric construction. For divisions that correspond to regular polygon constructions (3, 4, 5, 6, 8, 10, 12, etc.) geometric methods without calculation give exact results. For 6 divisions, set the compass to the radius and step around — this is exact by Euclidean geometry. For 12 divisions, bisect the 6. For 8 divisions, bisect the 4. For prime numbers of teeth (7, 11, 13, etc.) geometry alone is insufficient and calculation plus iteration are needed.
The Beam Compass for Large Gears
A standard draftsman’s compass is limited to circles of about 300 mm diameter. For larger gears, a beam compass is used: a long straight beam with two adjustable points, one fixed, one sliding. The beam compass can be set to any chord length regardless of the circle diameter.
Building a beam compass:
- A straight hardwood or iron bar, square section, at least 1 m long
- Two adjustable sliding clamps with sharp steel points (scribed hardened steel works well)
- A locking screw on each clamp
The beam compass can be used for initial rough division and also for checking — after marking all the tooth spaces, step the compass around and verify that each consecutive step lands within acceptable tolerance of the previous mark.
Cumulative Error and Correction
When stepping a compass around a large circle, cumulative error from small individual setting errors builds up. After N steps, the compass point may land ahead of or behind the starting mark by a visible amount.
If the error is small (less than one tooth pitch): Redistribute it by adjusting the position of every mark by a small equal correction. Divide the total error by N and shift each mark by that fraction multiplied by its sequential position (mark 1 shifts 1/N of total error, mark 2 shifts 2/N, etc.). Use a second pass of the compass to apply corrections.
If the error is large: Re-check the compass setting and start over. A large cumulative error means the chord setting was significantly wrong.
The target for gear tooth layout: cumulative error after going fully around the circle should be less than 0.5 mm, ideally less than 0.2 mm, for gears of 100–300 mm pitch diameter.
Combining Compass and Dividing Plate
For high-accuracy work, use the compass to establish approximate tooth positions and then refine them using a simple dividing plate. The dividing plate has holes drilled at the correct angular positions; each tooth space is indexed by moving from hole to hole.
The compass method is valuable for gears with unusual tooth counts that may not be available on a standard dividing plate — the compass is completely flexible in tooth count, requiring only the correct chord calculation.
Verifying the Layout
Before cutting any teeth, verify the layout:
- Check that the number of marked points is correct by counting
- Use the compass to check several random pairs of adjacent points — each should measure the same chord length
- Check points directly opposite each other (N/2 apart for even tooth counts) — they should be diametrically opposite to within ±0.1 mm
- Use a straight edge to check that points 180° apart are truly diametrically opposed
Any systematic error in the verification (all marks on one side seem too close, all on the other too far) indicates a center point error — the circle was drawn from an inaccurate center. Re-establish the center and redo the pitch circle before marking tooth positions.