1. The earliest solution of the inverse-square orbit problem was evidently constructed by John Keill and published in the 1708 volume of thePhilosophical Transactions. The latest published solution, so far as I am aware, can be found in Robert Weinstock, “Inverse-square orbits: Three little-known solutions and a novel integration technique,”Am. J. Phys. 60 (1992) 615–619.

   Article MATH MathSciNet  Google Scholar 

2. If, instead of erecting the perpendicular toOp at its midpoint in Fig. 1 (as does J/G²), we constructed the perpendicular toOp at, say, 0.75 of its length from O, and considered the pointP′ at which this perpendicular intersectsCp, then, as θ increases from 0 to 2π, the intersection P′ would still trace a closed smooth curve. However, you will find the tangent to this curve at each P′ is not (except for θ = 0 and θ= π perpendicular to the segmentOp. Only the choice of the halfway pointP works.

3. J. Clerk Maxwell,Matter and Motion, D. Van Nostrand, New York (1878), Chap. VIII; reprinted from Van Nostrand’s Magazine.

4. T.L. Hankins,Sir William Rowan Hamilton, Johns Hopkins University Press, Baltimore (1980), pp. 326–333.

   MATH  Google Scholar 

References

1. H. M. Collins,Artificial Experts (Cambridge, Massachusetts: MIT Press, 1990).

   Google Scholar 

2. John Searle,Minds, Brains and Science (Cambridge, Massachusetts: Harvard University Press, 1984).

   Google Scholar 

3. John Searle,The Rediscovery of the Mind (Cambridge, Massachusetts: MIT Press, 1992).

   Google Scholar 

4. Alan Turing, “Computing Machinery and Intelligence,” inThe Mind’s I, edited by Douglas Hofstader and Daniel Dennett (New York: Basic Books, 1981).

   Google Scholar 

5. Ludwig Wittgenstein,Philosophical Investigations, translated by G. E. M. Anscombe (New York: Macmillan Publishing, 1958).

   Google Scholar 

6. Ludwig Wittgenstein,Wittgenstein’s Lectures on the Foundations of Mathematics, edited by Cora Diamond (Chicago: University of Chicago Press, 1976).

   Google Scholar 

Download references