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Embodying the Calculus: Planimeters and Analogue Computing

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Charles Care
Charles Care

Paper given at BSHM Postgraduate Research in Progress, Oxford, February 2005.

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Embodying the Calculus: Planimeters and Analogue Computing Charles Care Feburary 2005
• Planimeters invented and re-invented throughout the 19th Century. – Hermann (1814) – Gonnella (1824) – Oppikofer (1827) – Wetli (1849) – Sang (1851) – Amsler (1854) • Principle uses in: – Land area calculation – Indicator diagrams
Using a planimeter to calculate the area of an indicator diagram
A roller can be used to calculate the length S ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         h ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         S
Calculating the area under a two-state function ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     h2 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     h1 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     S2 S1
Scaling the rotation according to the function’s height
The cone provides a different gear for every possible value in the y-axis
An Oppikofer planimeter
A model of an Oppikofer planimeter
The Hermann Planimeter
The Wetli Planimeter
A differential analyser∗ ∗ Crank, J. The Differential Analyser. Longmans, London, 1947.
Embodying the Calculus: Planimeters and Analogue Computing Charles Care Feburary 2005

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Embodying the Calculus: Planimeters and Analogue Computing

  • 1. Embodying the Calculus: Planimeters and Analogue Computing Charles Care Feburary 2005
  • 2. • Planimeters invented and re-invented throughout the 19th Century. – Hermann (1814) – Gonnella (1824) – Oppikofer (1827) – Wetli (1849) – Sang (1851) – Amsler (1854) • Principle uses in: – Land area calculation – Indicator diagrams
  • 3. Using a planimeter to calculate the area of an indicator diagram
  • 4. A roller can be used to calculate the length S ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         h ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         S
  • 5. Calculating the area under a two-state function ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     h2 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     h1 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     S2 S1
  • 6. Scaling the rotation according to the function’s height
  • 7. The cone provides a different gear for every possible value in the y-axis
  • 9. A model of an Oppikofer planimeter
  • 12. A differential analyser∗ ∗ Crank, J. The Differential Analyser. Longmans, London, 1947.
  • 13. Embodying the Calculus: Planimeters and Analogue Computing Charles Care Feburary 2005