Proportional Compasses are a draughting instrument made of two crossed legs joined by a sliding pivot, with points at both ends, used to transfer a length at a fixed ratio between the two pairs of points. Typical scaling range runs from 1:1 up to about 1:10 with positional accuracy around ±0.1 mm on a quality brass instrument. They let a draughtsman scale dimensions off a drawing without arithmetic — a single setting handles every transfer. You see them used today by model-makers, marine architects, and conservators copying historical maps at Stanfords in London.
Proportional Compasses Interactive Calculator
Vary the pivot ratio, leg length, pickup opening, and clamp slip to see the transferred short-point distance and scaling error.
Equation Used
The proportional compass is governed by similar triangles: the long-point opening divided by the short-point opening equals the long pivot arm divided by the short pivot arm. This calculator also shows how a small pivot clamp slip changes the effective arm lengths and creates transfer error.
- The two compass legs share the same included angle, so openings scale directly with pivot arm length.
- Leg length is the total point-to-point distance along one leg, split into long and short pivot arms.
- Positive clamp slip moves the effective pivot toward the long points, reducing the ratio.
- The slip term is applied as an effective two-sided relative pivot shift for the crossed slotted legs.
Operating Principle of the Proportional Compasses
Take two flat legs of equal length, drill a slot down the centre of each, and join them face-to-face with a clamp screw that runs in both slots. Sharpen all four ends to fine points. That's the whole instrument. When you slide the pivot toward one end, the legs become a lever — the short ends open by a small amount while the long ends open by a larger amount, and the ratio between those two openings equals the ratio of the lever arms. Set the pivot at the mark labelled '3' on the LINES scale, and any distance you pick up with the long points reduces to one-third when you flip to the short points.
The geometry is pure similar triangles. If the long arm is L<sub>long</sub> and the short arm is L<sub>short</sub>, the opening at each end scales as the arm length, because both pairs of legs share the same included angle at the pivot. This is why the scaling compass works for any opening within the leg's range — the ratio is locked by pivot position, not by how far you spread the legs.
Where it goes wrong is the pivot clamp. If the screw isn't bottomed hard against the index mark, the pivot creeps under load and your ratio drifts. A 0.3 mm slip on a 150 mm leg shifts a nominal 1:3 ratio to roughly 1:2.94 — small, but on a 90 mm pickup that's a 0.6 mm error at the short points, enough to ruin a scale model joint. The other failure mode is point wear: blunted tips don't sit in a pricked hole the same way, and you'll see scatter of ±0.2 mm even with the pivot perfectly set. Quality instruments from Stanley, Riefler, or Haff used hardened steel inserts in brass legs to keep the points sharp through years of use.
Key Components
- Legs: Two matched flat bars, typically 150-250 mm long in brass or German silver, each milled with a central slot running most of the length. The slots must be parallel to the leg centreline within 0.05 mm or the ratio varies as the legs spread.
- Sliding pivot screw: A knurled clamp screw passing through both slots, locking the legs at the chosen ratio. The screw shoulder must seat firmly without crushing the legs — a slip of 0.1 mm at the pivot translates directly into ratio error at the points.
- Engraved scales: Graduated markings along one leg labelled LINES (linear ratios 1:1.5 up to 1:10), CIRCLES (for dividing a circle into equal arcs), PLANES (area scaling) and SOLIDS (volume scaling). Quality instruments engrave the scales to ±0.05 mm of true position.
- Steel points: Four hardened steel needles pressed into the leg ends, ground to a 30° included cone. Tip radius below 0.05 mm is required for clean pricking on cartridge paper — anything blunter wanders in the fibres and adds scatter to every transfer.
- Index line: A fine engraved line across the pivot screw that registers against the scale on the leg. Reading parallax matters here — a 0.2 mm misread at the index gives a visible ratio shift on the 1:8 setting.
Real-World Applications of the Proportional Compasses
Proportional Compasses earned their place on the draughting board because they remove arithmetic from repetitive scaling. Architects copying details from a 1:50 site plan to a 1:20 detail sheet, cartographers reducing survey work, model-makers scaling shipyard drawings down to 1:48 — anywhere you transfer a dimension at a fixed ratio many times in a row, the instrument is faster than picking up a scale rule and reading both ends. The sector compass, a close cousin developed by Galileo around 1597, extends the same idea with engraved trigonometric scales for gunnery and surveying.
- Architecture & Heritage Drawing: Conservation architects at the SPAB (Society for the Protection of Ancient Buildings) use proportional dividers to transfer dimensions off measured survey drawings of timber-framed barns onto detail sheets at 1:5.
- Marine Model-Making: Model shipwrights at the Rogers Ship Model Collection at the US Naval Academy scale Admiralty draughts from 1:48 down to 1:96 using LINES-scale proportional compasses.
- Cartography: Stanfords in London still uses proportional dividers in-house when transferring features from one historical map projection to another, where digital scaling would lose register marks.
- Numismatics & Engraving: Die engravers at the Royal Mint set proportional compasses on the PLANES scale to lay out reduced-area portrait studies before cutting reduction master dies.
- Education: Engineering drawing courses at TU Delft keep Riefler-pattern proportional compasses in the teaching collection so students see how the sector compass and proportional compass relate to modern CAD scaling.
- Restoration Draughting: Restoration draughtsmen at the Bluebell Railway carriage shop use proportional dividers to scale fragmentary original carriage drawings up to full-size templates on plywood.
The Formula Behind the Proportional Compasses
The ratio a Proportional Compass produces is set entirely by pivot position. The formula tells you what scaling ratio you get when you place the pivot screw a distance d from one end of a total leg length L. At the low end of the practical range, with the pivot near the centre, you get ratios close to 1:1 — useful for slight reductions but the instrument adds nothing over a plain divider here. At the high end, pushing the pivot close to one end gives ratios up to about 1:10, but the short arm becomes so short that point flex and pivot slop dominate the error budget. The sweet spot is between 1:2 and 1:6, where the scaling is meaningful and the geometry stays stiff.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| R | Scaling ratio between long-point opening and short-point opening | dimensionless | dimensionless |
| L | Total leg length from point to point | mm | in |
| d | Distance from pivot centre to the short-end points | mm | in |
| Llong | Distance from pivot centre to the long-end points (= L − d) | mm | in |
| Lshort | Distance from pivot centre to the short-end points (= d) | mm | in |
Worked Example: Proportional Compasses in a model-shipwright scaling a frigate draught
A model-shipwright at a workshop in Brest is reducing an Admiralty draught of an 18th-century frigate from its original 1:48 scale down to a display model at 1:96. They are using a 200 mm Riefler-pattern proportional compass and need to set the pivot for a clean 1:2 reduction so they can transfer hull station spacings directly from the original draught onto the new building board.
Given
- L = 200 mm
- Rtarget = 2 dimensionless
Solution
Step 1 — solve the ratio formula for d at the nominal target ratio of 1:2:
So the pivot index line must sit 66.7 mm from the short-point ends. The long arm is then 200 − 66.7 = 133.3 mm. Pick up any station spacing with the long points and the short points reproduce it at half size.
Step 2 — check the low end of the practical range, a gentle 1:1.5 reduction the same shipwright might use later for a detail study:
At 1:1.5 the short arm is 80 mm — almost half the leg — and the instrument is rock solid. Ratio error from a 0.1 mm pivot slip is tiny, around 0.2% on the output. Honestly though, a plain divider plus a scale rule does this just as well; the proportional compass really shines once the ratio gets larger.
Step 3 — now the high end, a 1:8 reduction for laying out fitting details from a full hull plan onto a small inset:
The short arm is now only 22.2 mm. A 0.1 mm pivot slip here pushes the actual ratio from 1:8.00 to roughly 1:8.05 — a 0.6% error, and on a 100 mm pickup that's 0.75 mm at the short points instead of 0.625 mm. Visible to the eye on a precise model build. Above 1:8 the geometry gets twitchy and most experienced draughtsmen split the job into two passes (1:3 then 1:3 again gives 1:9, with both passes done in the stiff middle of the range).
Result
The pivot index line sets at d = 66. 7 mm from the short-point end for a 1:2 reduction on a 200 mm instrument. Practically, that means picking up a 90 mm hull-station spacing from the original draught will prick a 45.0 mm spacing on the new building board, repeatable to about ±0.1 mm with sharp points. Compared across the range, the 1:1.5 setting (d = 80 mm) is bulletproof but barely worth the tool, the nominal 1:2 setting hits the sweet spot, and the 1:8 setting (d = 22.2 mm) is at the edge of usable — pivot slop and short-arm flex start to show. If your reduced spacings come out wrong by more than 0.2 mm, suspect three things in this order: (1) the pivot screw not seated against the index mark so the ratio drifts mid-job, (2) one or more of the four steel points blunted past 0.1 mm tip radius and walking in the paper fibres, or (3) the slot edges burred from a previous overtightening, letting the pivot rock perpendicular to the legs.
When to Use a Proportional Compasses and When Not To
Proportional Compasses sit between plain dividers and the engraved sector compass. The choice depends on how often you scale, how big the ratio is, and how much arithmetic you want to do at the board. Modern CAD has displaced these instruments for production drawing, but they remain in active use anywhere paper originals must be copied without digitising.
| Property | Proportional Compass | Plain Dividers + Scale Rule | Sector Compass (Galileo type) |
|---|---|---|---|
| Typical scaling accuracy | ±0.1 mm at 1:2, ±0.3 mm at 1:8 | ±0.05 mm (limited by rule reading) | ±0.2 mm (depends on engraved scale quality) |
| Useful ratio range | 1:1.5 to 1:10 | Any ratio, but each transfer needs arithmetic | 1:1 to about 1:6 plus trig functions |
| Speed for repeated transfers | Fast — set once, transfer many | Slow — read, calculate, set, transfer for each | Moderate — scales engraved but reading is fiddly |
| Cost (quality instrument, 2024) | £80-£300 (Haff, vintage Riefler) | £15-£40 plus a scale rule | £200-£600 (collectible, rare new production) |
| Failure modes | Pivot slip, point wear, slot burr | Reading error, point wear | Scale wear, hinge slop |
| Best application fit | Repeated fixed-ratio scaling on paper | One-off transfers, any ratio | Trig calculations and gunnery layout |
| Skill floor | Low — set the pivot and go | Moderate — arithmetic per transfer | High — must understand sector geometry |
Frequently Asked Questions About Proportional Compasses
The ratio is geometrically constant — if it varies with opening, your slots aren't parallel to the leg centreline. On cheap instruments the milled slots wander by 0.1-0.2 mm along their length, so the effective pivot position shifts as the legs spread. Check by setting 1:2 and pricking a 20 mm pickup, then a 100 mm pickup — both reduced lengths should be exactly half. If one is off, the slots are the cause and no amount of pivot adjustment will fix it.
Quality instruments from Riefler, Haff, or pre-war Stanley hold slot parallelism to better than 0.05 mm and don't show this fault.
LINES every time. PLANES is for area scaling — if you pick up the side of a square and transfer with PLANES set at 2, the short points give you the side of a square with half the area, not half the side length. That's useful for engravers laying out portrait reductions where the visual weight of the image matters, but for a scale model you want every linear dimension reduced by the same factor, which is exactly what LINES does.
SOLIDS works the same way for volume scaling and is even more specialised — sculpture reduction work, mostly.
Walking error. Each prick has a finite tip footprint, and if you pick up the next pickup distance from the previous prick rather than from a master reference, errors stack. On a 0.05 mm tip radius point, 30 transfers can accumulate 1-2 mm of drift along a hull centreline.
Fix it by stepping off all stations from a single fixed origin on the original drawing each time, not from the previous prick. The proportional compass itself isn't drifting — your method is.
Whenever the short arm drops below about 30 mm. Below that the leg flex and pivot slop dominate, and a single pass at 1:9 typically gives ±0.5 mm scatter on a 100 mm pickup. Two passes at 1:3 each give 1:9 overall but each pass runs in the stiff middle of the range — short arm around 50 mm — and the scatter drops to ±0.15 mm even after compounding.
The downside is doubled handling time and the need for an intermediate sheet of paper to hold the first-pass marks. Worth it for precision model work, overkill for sketch-level reduction.
Almost always pivot bias. The pivot screw has a finite shoulder diameter — typically 4-5 mm — and the index line is engraved on one face. If you read the index against the leg scale from the wrong side, you offset the apparent pivot position by half the shoulder diameter, around 2 mm. On a 1:4 setting that pushes the actual ratio to roughly 1:4.1, so reduced lengths come out 2-3% small.
Quality instruments engrave a fine index line on both faces of the screw to eliminate this. If yours has only one, always read from the same side and calibrate by pricking a known length first.
Yes — flip the instrument. Pick up the source dimension with the short points and transfer with the long points. The same pivot setting that gives a 1:3 reduction one way gives a 3:1 enlargement the other. The geometry doesn't care which end is input and which is output.
The practical limit is paper size and point reach. A 200 mm instrument set at 1:5 only spreads the long points to about 80 mm absolute, so you can't enlarge anything bigger than that in one bite. Enlarging a 300 mm feature at 5:1 needs a 600 mm instrument or a stepped approach with intermediate reference marks.
References & Further Reading
- Wikipedia contributors. Sector (instrument). Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
