- David J. Furley, Two Studies in the Greek Atomists, (Princeton: Princeton University Press, 1967), p. 7. text
- Furley, p. 7. text
- Furley, p. 8. text
- After one bisection the parts are half the size of the original and are both extended. If a part after N bisections is extended, then half that is still extended. The parts are extended after one bisection, and if the parts are extended after N bisections, then they are also extended after N+1 bisections. Mathematical induction concludes that the parts are extended after K bisections for all K. text
- Furley, p. 10. text
- Furley, p. 13. text
- Furley, p. 13. text
- Furley, p. 13. text
- Furley, p. 14. text
- Furley, p. 16. text
- Furley, p. 17. text
- Furley, p. 17. text
- Furley, p. 19. text
- Furley, pp. 17-18. text
- Furley, p. 18. text
- Furley, p. 18. text
- Furley, p. 22. text
- Furley, p. 25. text
- Furley, pp. 36-37. text
- Furley, p. 36. text
- Arnauld, Antonie, The Art of Thinking: Port-Royal Logic, trans. James Dickoff and Patricia James, (New York: Bobbs-Merrill, 1964), p. 299. text
- 22. Such a proof might go as follows. Suppose
is rational. Then there exists numbers P and Q such that = P/Q (1). We may suppose that the fraction P/Q is expressed in lowest terms. This would require that P and Q are relatively prime, that is, that they have no greatest common divisor (GCD) (2). Now then, squaring both sides gives us the equation
2 = P
^{2}/Q^{2}or that Q^{2}= 2感^{2}(3). Since the right hand side of the equation is divisible by 2, so must the left hand side be. But for that to be possible, Q must itself be divisible by 2, and Q must be of the form 2愛 (4). Consequently Q^{2}(= 2感^{2}) must equal (2愛)^{2}(5). This allows us to conclude that P must also be divisible by 2 (8,9) contradicting the assumption that could be expressed as a rational number in lowest terms.(1) = P/Q. Assume is rational. (2) GCD(P,Q)=1 P/Q is expressed in lowest terms (3) Q ^{2}= 2感^{2}Square both sides & multiply by Q ^{2}(4) Q = 2愛 Both sides must be divisible by 2 (5) (2愛) ^{2}= 2感^{2}Substituting (6) 4愛 ^{2}= 2感^{2}Expanding (7) 2愛 ^{2}= P^{2}Divide both sides by 2. (8) P = 2惹 Both sides must be divisible by 2. (9) GCD(P,Q)=2 Both P and Q divisible by 2. (10) P/Q By reductio cannot be rational. - See note 8 of chapter VI and page 174. text
- Arnauld, 1964. p. 299. text
- N. Kretzmann, ed., Infinity and Continuity in Ancient and Medieval Thought, (Ithica, NY: Cornell University Press, 1982). text
- J. D. North, "Finite and otherwise: Aristotle And Some Seventeenth Century Views", in Nature Mathematized vol. 1., University of Western Ontario Series in Philosophy of Science, no. 20. William R. Shea, ed., (Dordrecht, Holland and Boston, U. S. A.: D. Reidel, 1983), 113-148. text
- Donald L. M. Baxter, "Hume on Infinite Divisibility", History of Philosophical Quarterly 5 (April 1988): 133. text
- Baxter, pp. 134-5. text
- D. E. Thomsen, "Atomic nuclei: Quarks in leaky bags", Science News Magazine Vol. 125, No. 18, May 5, 1984, p. 297. text
- Dietrick E. Thomsen, "Experimenting With 40 Trillion Electron-Volts", Science News Magazine Vol. 132, No. 20, November 14, 1987, p. 315. text