This document discusses different ways of classifying planimeters, which are instruments used to determine the area of shapes. It describes planimeters that were developed before and after 1851, focusing on their variable gear mechanisms. These include wheels rolling on discs, spheres, cones, and cylinders. The document also examines Henrici's 1894 classification of planimeters into three types based on their tracing mechanisms: orthogonal, polar-coordinate, and Amsler-type polar planimeters. In the end, it notes there are several valid ways to categorize planimeters based on their components, geometry, or whether they record area by slipping or rolling.

1. A Taxonomy of Planimeters
Charles Care
September 2004

2. What is a Planimeter?
• “Planimeters are instruments for the determination by mechanical
means of the area of any figure.”
(Encyclopedia Britannica 11th Ed.)
• An integrating machine.
• A component of the differential analyser.

3. Where Planimeters Fit In
Engineering
Machines
Electronic
Analogue
Moment
Napier’s Bones Planimeters
Watt−hour
Integraphs meters
Differential
Analysers
Astrolabes Planimeters Mechanical
Integrators
Mathematical Instruments

4. Pre-1851 Planimeters
Various isolated∗ developments
• Hermann (1814)
• Gonella (1824)
• Oppikofer (1836)
• Wetli (1850)
• Sang (1850)
∗ Some sources indicate a link between Gonella and Oppikofer

5. Post-1851 Planimeters
An explosion in manufacture
• Amsler (1854) – 12,000 sold by 1892
• Maxwell (1855)
• Thompson (1861-1864)
• Coradi & Ott (Late 19th Century)
• Developments in Uk and US

6. Variable Gears
• All mechanical planimeters use some kind of variable gear.
• This is archetypally a wheel rolling on a disc but can be many
other things
Wheel Sphere
Cone Hermann Gonnella Oppikofer
Disc Wetli Coradi Thomson
Sphere Boys Maxwell
Cylinder Boys
Surface Amsler Coradi etc.

7. Variable Gear: Cone and Wheel

8. Variable Gear: Disc and Wheel

9. Variable Gear: Sphere and Sphere

10. Variable Gear: Disc, Ball and Cylinder

11. Variable Gear: Wheel and Cylinder

12. Variable Gear: Wheel and Sphere

13. But what about Polar Planimeters?

14. Or this one?

15. Difficulties
Wheel Sphere
Cone Hermann Gonnella Oppikofer
Disc Wetli Coradi Thomson
Sphere Boys Maxwell
Cylinder Boys
Surface Amsler Coradi etc.

16. Difficulties
Wheel Sphere
Cone Hermann Gonnella Oppikofer
Disc Wetli Coradi Thomson
Sphere Boys Maxwell
Cylinder Boys
Surface Amsler Coradi etc.

17. Classification by the tracing mechanism
• System used by Henrici in 1894
• Three basic types
I Orthogonal
II Polar-Coordinate (not Amsler-type)
III Amsler-type (commonly called Polar Planimeters)

18. Type I Orthogonal
• Position generated by varying
two lengths
• Type of early planimeters
– Wheel & Cone
– Wheel & Disc
– Sphere & Sphere

19. Type II Polar-Coordinate
• Position generated by varying
one lengths and one rotation
• Very few constructed
– Wheel & Cylinder
– Wheel & Sphere
– Sphere & Sphere

20. Type III Amsler (Polar Planimeter)
• Position generated by varying
two rotations
• All large-scale productions
belong to this class
– Wheel & Disc (Coradi)
– Slipping Wheel

21. Another classification
Is the area recorded by slipping or rolling∗.
∗ Used by Hele Shaw

22. Summing Up
Several ‘characteristics’ used to classify
planimeters
• Classification by components
– Difficult to separate polar and orthoganal
• Henrici’s classification
– Useful to understand the geometry of the various planimeters
• Hele Shaw’s classification
– But need to consider pure rolling systems as well

23. A Taxonomy of Planimeters
Charles Care
September 2004