A Parallel Ruler form 2 is a four-bar parallelogram linkage that links two straight rulers with a pair of equal-length swing arms, so one rule translates relative to the other while staying perfectly parallel. Built right, it holds parallelism within ±0.1° across a 300 mm sweep — tight enough for marine chart work and drafting board layouts. The linkage exists to transfer a heading line across a chart or drawing without rotation. Generations of navigators used Captain Field's pattern Parallel Rule to step bearings between a compass rose and a destination on Admiralty charts.
Parallel Ruler Form 2 Interactive Calculator
Vary arm length, arm mismatch, pivot clearance, and chart transfer distance to see parallelism error and plotted drift.
Equation Used
The calculator follows the worked example: parallelism error is the arm length mismatch divided by nominal arm length, plus the effect of radial pivot clearance at two pivots. The angular result is converted from radians to degrees, then projected over the selected chart transfer distance as plot drift.
- Small-angle linkage error model.
- Both swing arms are nominally parallel and nearly equal length.
- Pivot clearance contributes twice because two arm pivots affect blade alignment.
- Plot drift is evaluated over the selected bearing transfer distance.
How the Parallel Ruler (form 2) Actually Works
The form 2 Parallel Ruler is a parallelogram four-bar linkage in its purest form. You have two straight rules — call them the upper blade and the lower blade — connected by two identical swing arms pinned at four points. Because opposite sides of the parallelogram stay equal in length, the two blades remain parallel through every position of the sweep. Open the rule, lift one blade, walk it across the chart, and the bearing line you scribed at the start transfers untouched.
The geometry only works if the four pivot pin centres form a true parallelogram. That means the two swing arms must be matched in length to within about 0.05 mm over a typical 100 mm arm — any longer on one side and the upper blade fans out as you walk it, introducing a parallelism error you can read off the chart as a fake course deviation. The pivot bores must run a sliding fit on the pins, typically H7/g6, around 3 mm diameter on a brass shipwright's rule. Slop in the pivots is the dominant failure mode in cheap rules: a 0.1 mm radial clearance at each pin compounds across the linkage and you'll see the upper blade rock by half a degree when you press down on one end.
Why two arms and not one? A single swing arm gives you a circular sweep, not parallel translation. You need the second arm at the same length and same angle to constrain the rotation out of the system. Skew one arm even 0.5° at assembly and the rule will visibly racks as it opens — the kind of fault you'll only catch by closing the rule flat and checking the blade edges line up along their full length.
Key Components
- Upper blade: The straight rule the navigator or draughtsman lifts and walks across the chart. Typically 250-450 mm long in brass-bound boxwood or solid brass for ship-model work, with one bevelled edge ground straight to within 0.05 mm over its length.
- Lower blade: The reference rule held flat against the chart. Identical in length and edge geometry to the upper blade. The two blades must seat perfectly edge-to-edge when fully closed — any visible gap means an arm-length mismatch.
- Swing arms (pair): Two identical link bars, usually 80-120 mm centre-to-centre between pin holes. Length match between the pair must be within ±0.05 mm or the parallelogram closes into a trapezium and parallelism drifts across the sweep.
- Pivot pins: Four shouldered brass or steel pins, typically 3 mm diameter, riveted or threaded into the blades. Pin-to-bore clearance sits at H7/g6 — a sliding fit with no perceptible rock. Excess clearance is the number-one cause of measured parallelism error in field use.
- Friction stops or detent washers: Thin fibre or felt washers at each pivot that add controlled drag, so the rule holds its position when you release it. Without them the rule collapses under its own weight as soon as you let go.
Real-World Applications of the Parallel Ruler (form 2)
The Parallel Ruler form 2 lives wherever someone needs to transfer a line across a flat surface without rotating it. Chart navigation is the classic use, but the same linkage shows up in drafting boards, ship-model layout jigs, optical bench alignment, and any benchtop process where a parallel reference edge has to walk across a workpiece. Anywhere you'd otherwise reach for a rolling parallel rule, a T-square, or a parallel-motion drafting machine, the form 2 linkage is the simpler, cheaper, more compact option — at the cost of limited sweep range.
- Marine navigation: Captain Field's Improved Parallel Rule on Admiralty charts — stepping a bearing from the compass rose to a fix point during coastal pilotage.
- Ship modelling: Brass parallel rules used at Bristol model-shipwright benches to lay parallel plank lines on 1:48 plank-on-frame Napoleonic-era hulls.
- Architectural drafting: Vintage Kuhlmann and Mutoh drafting boards using a parallel-motion arm to keep the horizontal rule perpendicular to the page edge across an A0 sheet.
- Optical bench alignment: Lab-built parallel rules in Thorlabs cage-system layouts, transferring beam-path centrelines between mounted optics on a 600 mm breadboard.
- Cabinet making: Festool MFT-style benchtop layout where a parallel rule walks a reference fence across a workpiece for matched dado cuts.
- Cartography teaching: Royal Navy bridge simulator training rooms at HMS Collingwood, where cadets still learn manual chart work with traditional parallel rules before moving to ECDIS.
The Formula Behind the Parallel Ruler (form 2)
The number that matters on a Parallel Ruler form 2 is the parallelism error — how far off parallel the two blades drift as a function of arm-length mismatch and pivot clearance. At the low end of the typical operating range, a well-machined brass rule with matched arms and tight pins holds error under 0.05° and you'll never see it on a chart. At the high end, a worn or sloppy rule pushes past 0.5° and a single bearing transfer across a 600 mm chart introduces a fix error large enough to put a vessel on the wrong side of a shoal. The sweet spot is around 0.1° — invisible to the eye, well inside the precision a navigator can plot anyway.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Δθ | Parallelism error between upper and lower blade, in radians (multiply by 57.3 for degrees) | rad | rad |
| ΔL | Length mismatch between the two swing arms (centre-to-centre) | mm | in |
| Larm | Nominal swing-arm length, centre-to-centre between pivot pins | mm | in |
| c | Radial clearance at each pivot pin (bore diameter minus pin diameter, divided by 2) | mm | in |
Worked Example: Parallel Ruler (form 2) in a brass parallel rule for marine chart work
A marine instrument restorer in Falmouth is rebuilding a 1950s Captain Field's pattern Parallel Rule for use on Admiralty paper charts. The swing arms are nominally 100 mm centre-to-centre. The restorer needs to know whether their machined-arm tolerance and pivot fit will hold parallelism tight enough for chart work, where a 0.2° error across a 600 mm bearing transfer translates to roughly 2 mm of plot drift — about the width of a sharp pencil line.
Given
- Larm = 100 mm
- ΔL (nominal build) = 0.05 mm
- c (nominal pivot clearance) = 0.02 mm
Solution
Step 1 — at nominal build tolerance, ΔL = 0.05 mm and c = 0.02 mm, compute the parallelism error in radians:
Convert to degrees:
Across a 600 mm bearing transfer, that's about 0.5 mm of drift — well under a pencil-line width and invisible on the chart.
Step 2 — at the low end of a precision build (matched arms ground in pairs, ΔL ≈ 0.01 mm, lapped pins at c ≈ 0.005 mm):
This is the territory of a high-grade Stanley or Captain Field's original — error so small you'd need a 1-metre baseline to see it on a drafting board. Overkill for chart work, appropriate for surveying instruments.
Step 3 — at the high end, a worn shipboard rule with arms drifted to ΔL ≈ 0.3 mm and pivots loosened to c ≈ 0.15 mm:
Across a 600 mm transfer that's roughly 3.5 mm of plot error — wider than a pencil line, enough to misplace a fix by a noticeable fraction of a nautical mile. This is the rule that gets quietly retired from the chart table.
Result
Nominal parallelism error works out to about 0. 052° — well inside the 0.2° threshold for chart work and indistinguishable from a pencil line on a 600 mm bearing transfer. The low-end precision build at 0.011° is overkill for navigation but useful for surveying jigs, while the worn high-end case at 0.34° is the threshold where you'll catch yourself plotting a course on the wrong side of a navigation hazard. If you measure parallelism worse than predicted on a freshly built rule, the usual culprits are: (1) one swing arm machined a few hundredths long because the operator zeroed off the wrong shoulder, (2) pivot bores reamed oversize on a worn 3 mm reamer giving you 0.05+ mm of clearance per pin, or (3) the rivets at the pivots not pulled tight enough so the swing arm rocks axially as well as radially.
Choosing the Parallel Ruler (form 2): Pros and Cons
The Parallel Ruler form 2 isn't the only way to walk a line across a flat surface. The two main competitors are the rolling parallel rule (a single rule with two knurled rollers that grip the chart) and a full parallel-motion drafting machine (rule mounted on a pantograph arm). Each picks a different point on the precision-versus-range-versus-cost curve.
| Property | Parallel Ruler form 2 (linkage) | Rolling parallel rule | Drafting-machine parallel motion |
|---|---|---|---|
| Parallelism accuracy across full sweep | ±0.05° to ±0.1° well-built | ±0.5° typical (depends on chart surface friction) | ±0.02° on a quality Mutoh or Kuhlmann |
| Effective sweep range | Limited to ~2 × arm length, typically 200 mm | Unlimited — rolls as far as the chart allows | Full board, typically 1200 × 1800 mm |
| Cost (typical workshop quality) | £30-£150 for a brass-bound boxwood rule | £40-£90 for a machined acrylic rolling rule | £400-£2000 for a full drafting machine |
| Setup and bench footprint | Pocketable — 250-450 mm closed | Pocketable — 300 mm long | Permanent fixture on a drafting board |
| Failure mode with wear | Pivot slop → parallelism drift | Roller skid on rough chart paper → angular drift | Slide-rail wear → unsquare reference edges |
| Best application fit | Coastal pilotage, ship modelling, optical alignment | Open-ocean chart work, large bearing transfers | Architectural and engineering drafting |
Frequently Asked Questions About Parallel Ruler (form 2)
This is a classic signature of unequal swing-arm lengths. When the rule is closed, both arms are folded flat against the blades and any length mismatch hides because the blade edges seat against each other. As you open the rule, the longer arm forces its pivot to swing on a larger radius than the shorter one, and the upper blade fans out into a trapezium instead of staying parallel.
Quick diagnostic — open the rule to maximum sweep and measure the gap between blade edges at both ends with a feeler gauge. If one end is tight and the other end is open by more than 0.1 mm over a 300 mm blade, your arms are mismatched and need re-machining as a matched pair off the same setup.
Different jobs. The form 2 linkage rule excels in coastal pilotage where bearings are short — under 200 mm — and you need precision and confidence the rule isn't slipping. A rolling rule wins when you're plotting long ocean transits across a 1200 mm chart, because the linkage rule simply runs out of sweep range and you'd have to walk it in multiple steps, accumulating error each time.
Rule of thumb on a paper chart larger than A1: rolling rule. Smaller harbour or coastal chart with a need for repeatable bearing transfers: form 2 linkage. Most navigators carry both.
No — and this is a common rebuild mistake. The pins must be a sliding fit, not press-fit, because the linkage relies on free rotation at all four pivots. Press-fit pins seize the linkage solid or introduce stick-slip friction that makes the rule jump rather than walk smoothly.
Target H7/g6 — about 0.005 to 0.015 mm radial clearance on a 3 mm pin. Below 0.005 mm and the linkage binds on dust and oil residue. Above 0.025 mm and you start seeing the parallelism error from clearance dominate the error budget. The fix for a sloppy rule is reaming the bores oversize and fitting fresh pins, not pressing harder.
Side-to-side rock under load is almost always axial slop at the pivots, not radial clearance. The rivets or shoulder pins aren't pulled tight enough against the swing arms, so the arm can tilt out of plane. You'll see this as a slight twist of the upper blade rather than a parallel offset.
Pull each pivot apart and check the shoulder against the blade face — there should be zero detectable axial play but the swing arm should still rotate freely. A 0.05 mm fibre washer between the arm and the blade is the traditional fix because it adds friction without binding and absorbs minor manufacturing variation in the shoulder length.
Because clearance acts at both pivots on the same arm — the pin can sit anywhere within its bore at the upper blade and anywhere within its bore at the lower blade independently. Worst-case alignment puts the upper pin at one extreme of clearance and the lower pin at the opposite extreme, doubling the effective offset of that arm relative to the other.
Arm-length mismatch, by contrast, is a fixed dimensional error that only counts once. So when you're budgeting tolerance for a precision build, spend your machining effort on lapping the pivot fits before you worry about matching arm lengths to the last 0.01 mm.
Geometrically yes, practically not without redesign. The parallelism error formula scales with the ratio ΔL / Larm, so longer arms actually relax your tolerance — a 300 mm arm with 0.15 mm mismatch hits the same 0.05° error as a 100 mm arm with 0.05 mm mismatch. The problem is rigidity. A 1 metre boxwood blade flexes under its own weight enough to introduce more parallelism error from sag than from the linkage itself.
Boat lofters historically solved this with chalk lines and battens rather than oversized parallel rules. If you need parallel reference at metre scale, a parallel-motion drafting arm or a sliding T-square against a planed reference edge will outperform any scaled-up form 2 rule.
References & Further Reading
- Wikipedia contributors. Parallel rulers. Wikipedia
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