An eidograph copies a traced figure at a chosen scale using a pivoted beam, two equal pulleys and a pair of parallel arms. Linked bands keep the pulley rotations matched. Correct scaling also requires the fulcrum position along the beam to match the arm-length ratio.
Replacing manual motion in Eidograph Mechanism
If eidograph mechanism is converted to actuator drive, the actuator is no longer just a replacement for a crank or cam. It becomes part of the timing, force, and limit-control system, so the travel must be checked through the full motion.
For structural members, the actuator load should be applied through brackets that do not introduce unintended bending. Check both static load and the extra load created when the mechanism starts, stops, or binds.
A crank or cam may tolerate continuous rotation through a difficult part of the cycle, while a linear actuator has defined end positions and internal limits. That difference matters when the mechanism needs repeatability or must not drive into a hard stop.
- Map actuator stroke to the actual output travel before choosing a model.
- Check for binding, changing leverage, and peak load near the ends of travel.
- Decide whether timed motion is acceptable or position feedback is required.
For a FIRGELLI design review, calculate the load case with the FIRGELLI actuator force calculator and plan the switching or controller arrangement with the linear actuator wiring diagram generator. If an actuator or cylinder is already installed, the linear actuator replacement finder helps keep the comparison tied to real dimensions.
Eidograph Interactive Calculator
Trace a scaled copy with equal pulleys and parallel arms. The beam fulcrum and arm ratios move together; ratio numbers are separate from historical vernier graduations.
Equation Used
- Equal pulleys preserve parallel arms.
- Rigid beam and arms with no band slip or pivot play.
- Fulcrum divides beam in the same ratio as arm lengths.
- Original and copy use the same physical drawing scale.
- Illustrative hardware proportions; no prediction of instrument working range.
Ideal matched geometry. Ratio numbers are not literal vernier settings; practical travel and tolerances are not solved.
One fixed fulcrum and two coordinated arms
The central beam turns around a fixed support on the drawing board. A pulley at each end carries a sliding arm: one holds the tracer and the other the pencil. Equal pulleys connected by appropriately arranged bands keep the arm directions parallel.
The bands in the diagram have uncrossed runs. The earlier page’s crossed-band explanation was wrong: opposite relative pulley rotations would not preserve parallel arms through general motion.
The fulcrum divides the beam in the same proportion as the tracer and pencil arm lengths. The moving tracer, fixed fulcrum and pencil then remain collinear. The drawn copy is on the opposite side of the fulcrum, so its orientation is rotated 180° as well as scaled.
The heavy support and clamping box are shown separately. The beam and both arms keep constant lengths during each trace, while their angles change. The faint curves show the complete original and copy; darker portions show the traced progress.
Compare original and copy dimensions
Use the ratio numbers T and P to specify original and copied lengths in matching parts. For example, T=2 and P=1 means a half-size copy. These numbers are mathematical proportions, not literal readings on an historical vernier.
The original-span input sets the horizontal extent of the demonstration curve. Its copied horizontal extent follows the entered scale. Both curves use the same drawing scale, with automatic framing around the complete movement.
The mathematical range includes strong reductions and enlargements. It does not guarantee that a particular physical eidograph has sufficient travel, clearance, balance or drawing-board space for those settings.
Length ratio, area ratio and matched settings
The linear scale factor is S=P/T. A length L copies to SL; an area scales by S². The inverse length ratio is 1/S.
Taking the fixed fulcrum as the origin, let the tracer-side pulley centre be A and the pencil-side centre be B=−SA. If the original arm vector is u and the pencil arm vector is −Su, the tracer is X=A+u and the pencil is Y=B−Su=−SX. This establishes the scale and central inversion for every assembled position.
The demonstration solves the tracer-side two-link geometry for each point on a closed curve, then derives the pencil geometry from those matched proportions. It does not draw an unrelated decorative loop while separately calculating a scale ratio.
For the historical centred graduation convention described in the referenced handbook, the offset is 100(T−P)/(T+P) on scales numbered 100 each way from centre. This offset is displayed as a reference; it is distinct from the ratio numbers and its placement must follow the actual instrument’s scale convention.
Default half-size copy
For a 300mm original span, T=2 and P=1 give S=0.5. The copied span is 150mm, its area is 0.25 times the original area and the inverse length ratio is 2.
The corresponding centred-scale offset is 33.333 under the stated 100-each-way convention. It is not the literal ratio number 1 or 2.
Changing P to 2 while leaving T=2 gives a full-size copy, an area ratio of 1 and zero centred-scale offset. Setting P=3 and T=2 gives a 450mm copy and an area factor of 2.25.
These are illustrative calculations. The previous named botanical-archive example and claims of specific closure errors were unsupported and have been removed.
What affects a real copy
The calculation assumes equal pulleys, maintained parallelism, rigid members and correctly matched beam and arm settings. It does not estimate errors from band stretch, slipping clamps, pivot play or pencil pressure.
Pulley diameters, band thickness, bracket sizes and unused arm overhangs are schematic. The chosen pulley size is kept equal at both ends and is fixed throughout a trace. The drawing is a geometric explanation, not a fabrication or tolerance specification.
Changing the ratio redraws the matched setting. It does not suggest that scale can be changed during a trace without resetting the instrument.
Eidograph questions
Why is there no crossed belt?
The paired arms must maintain their parallel direction. The displayed equal pulleys have coordinated rotation rather than opposite rotations caused by a crossed belt.
Can I set only the pencil arm to change scale?
The correct beam-fulcrum position and both arm settings must agree. A ratio of arm lengths alone is insufficient if the beam setting does not match.
Why is the copy upside down?
The copy is centrally inverted about the fulcrum: a 180° rotation with uniform scaling. It is not a reflected image.
Why is area one quarter for a half-size copy?
Both independent length directions are halved, so area is multiplied by 0.5²=0.25.
Are the ratio numbers the instrument graduations?
No. The page separates the desired proportion from the historical centred-scale offset. Check the actual instrument’s convention.
References
- Institution of Civil Engineers Scotland Museum: Eidograph — surviving instrument and parallel-arm band connection.
- George G. André, The Draughtsman’s Handbook of Plan and Map Drawing, Section VII and Plate 26 — historical construction, equal pulleys, parallel arms and centred graduation settings.
- Whipple Museum: Wallace’s eidograph by R. B. Bate — instrument record.
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