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# Embodying the Calculus: Planimeters and Analogue Computing

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

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### Embodying the Calculus: Planimeters and Analogue Computing

1. 1. Embodying the Calculus:Planimeters and Analogue Computing Charles Care Feburary 2005
2. 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. 3. Using a planimeter to calculate the area of an indicator diagram
4. 4. A roller can be used to calculate the length S ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                        h ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ £ £ £                         S
5. 5. Calculating the area under a two-state function ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                    h2 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     h1 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £ £ £ £ £ £ £ £ £ £ £                     S2 S1
6. 6. Scaling the rotation according to the function’s height
7. 7. The cone provides a diﬀerent gear for every possible value in the y-axis
8. 8. An Oppikofer planimeter
9. 9. A model of an Oppikofer planimeter
10. 10. The Hermann Planimeter
11. 11. The Wetli Planimeter
12. 12. A diﬀerential analyser∗∗ Crank, J. The Diﬀerential Analyser. Longmans, London, 1947.
13. 13. Embodying the Calculus:Planimeters and Analogue Computing Charles Care Feburary 2005