Let ε < 0.

03.1.09

The mathematical theory of big game hunting I

The following is the famous, seminal paper by H. Petard*, which appeared in the American Mathematical Monthly in 1938 and introduced to the academic community at large the mathematical theory of big game hunting. As is evident from the other articles in this section, Petard’s work prompted a good many others to add to this literature. This, together the subsequent articles we shall reproduce this week, do not form a complete compendium, but they do provide the interested reader a solid introduction into this exciting branch of mathematics.

A contribution to the mathematical theory of big game hunting

– H. Petard
Princeton, New Jersey

This little known mathematical discipline has not, of recent years, received in the literature the attention which, in our opinion, it deserves. In the present paper we present some algorithms which, it is hoped, may be of interest to other workers in the field. Neglecting the more obviously trivial methods, we shall confine our attention to those which involve significant applications of ideas familiar to mathematicians and physicists.

The present time is particularly fitting for the preparation of an account of the subject, since recent advanaces both in pure mathematics and theorectical physics have made available powerful tools whose very existence was unsuspected by earlier investigators. At the same time, some of the more elegant classical methods acquire new significance in the light of modern discoveries. Like many other branches of knowledge to which mathematical techniques have been applied in recent years, the Mathematical Theory of Big Game Hunting has a singularly unifying effect on the most diverse branches of the exact sciences.

For the sake of simplicity of statement, we shall confine our attention to Lions (Felis leo) whose habitat is the Sahara Desert. The methods which we shall enumerate will easily be seen to be applicable, with obvious formal modifications, to other carnivores and to other portions of the globe. The paper is divided into three parts, which draw their material respectively from mathematics, theoretical physics, and experimental physics.

The author desires to acknowledge his indebtedness to the Trivial Club of St. John’s College, Cambridge, England; to the MIT chapter of the Society for Useless Research; to the F o P, of Princeton University; and to numerous individual contributors, known and unknown, conscious and unconscious.

1. Mathematical methods

1. The Hilbert, or axiomatic, method. We place a locked cage onto a given point in the desert. After that we introduce the following logical system:

  • Axiom I. The set of lions in the Sahara is not empty.
  • Axiom II. If there exists a lion in the Sahara, then there exists a lion in the cage.
  • Rule of procedure. If P is a theorem, and if the following is holds: “P implies Q”, then Q is a theorem.
  • Theorem 1. There exists a lion in the cage.

2. The method of inversive geometry. We place a spherical cage in the desert, enter it and lock it from inside. We then perform an inversion with respect to the cage. Then the lion is inside the cage, and we are outside.

3. The method of projective geometry. Without loss of generality, we can view the desert as a plane. We project the surface onto a line, and then project the line onto an interior point of the cage. Thereby the lion is projected onto that same point.

4. The Bolzano-Weierstrass method. Divide the desert by a line running from N-S. The lion is then either in the E portion or in the W portion; let us assume him to be in the W portion. Bisect this portion by a line running from E-W. The lion is either in the N portion or in the S portion; let us assume him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion ultimately surrounded by a fence of arbitrarily small perimeter.

5. The “Mengentheoretisch” method. We observe that the desert is a separable space. It therefore contains an enumerable dense set of points, from which can be extracted a sequence having the lion as limit. We then approach the lion stealthily along this sequence, bearing with us suitable equipment.

6. The Peano method. Construct, by standard methods, a continuous curve passing through every point of the desert. It has been remarked [1]that it is possible to traverse such a curve in an arbitrarily short time. Armed with a spear, we traverse the curve in a time shorter than that in which a lion to move a distance equal to its own length.

7. A topological method. We observe that a lion has at least the connectivity of a torus. We transport the desert into four-space. Then it is possible [2] to carry out such a deformation that the lion can be returned to three-space in a knotted condition. He is then completely helpless.

8. The Cauchy, for function theoretical, method. We examine a lion-valued function f(z). Let ζ be the cage. Consider the integral

where C represents the boundary of the desert. Its value is f(ζ), i.e. there is a lion in the cage [3].

9. The Wiener-Tauberian method. We obtain a tame lion, L0, from the class L(-¥, ¥), whose Fourier transform vanishes nowhere, and release it in the desert. L0 then converges toward our cage. By Wiener’s General Tauberian Theorem [4], any other lion, L (say), will converge to the same cage. Alternatively we can approximate arbitrarily closely to L by translating L0 through the desert [5].)

10. The Eratosthenian method. Enumerate all the objects in the desert. Examine them one by one, and discard all those that are not lions. A refinement will capture only prime lions.

2. Methods from theoretical physics

11. The Dirac method. We observe that wild lions are, ipso facto, not be observable in the Sahara desert. Consequently, if there are any lions at all in the Sahara, they are tame. We leave catching a tame lion as an exercise to the reader.

12. The Schroedinger method. At any given moment there is a positive probability that there is a lion in the cage. Sit down and wait.

13. The nuclear physics method. Place a tame lion into the cage, and apply a Majorana exchange operator [6] on it and a wild lion.

As a variant, let us suppose, to fix ideas, that we require a male lion. We place a tame lioness into the cage, and apply the Heisenberg exchange operator [7] which exchanges spins.

14. A relativistic method. We distribute about the deser lion bait containing large portions of the Companion of Sirius. When enough bait has been taken, we project a beam of light across the desert. This will bend right around the lion, who will hen become so dizzy that he can be approahced with impunity.

3. Experimental physics methods

15. The thermodynamics method. We construct a semi-permeable membrane, permeable to everything except lions, and sweep it across the desert.

16. The atom-splitting method. We irradiate the desert with slow neutrons. The lion becomes radioactive, and a process of disintegration set in. When the decay has proceeded sufficiently far, he will become incapable of showing fight.

17. The magneto-optical method. We plant a large lenticular bed of catnip (Nepeta cataria), whose axis lies along the direction of the horizontal component of the earth’s magnetic field, and place a cage at one of its foci. We distribute over the desert large quantities of magnetized spinach (Spinacia oleracea), which, as is well known, has a high ferric content. The spinach is eaten by herbivorous denizens of the desert, which in turn are eaten by lions. The lions are then oriented parallel to the earth’s magnetic field, and the resulting beam of lions is focus by the catnip upon the cage.

References

[1] After Hilbert, cf. E. W. Hobson, “The Theory of Functions of a Real Variable and the Theory of Fourier’s Series” (1927), vol. 1, pp 456-457

[2] H. Seifert and W. Threlfall, “Lehrbuch der Topologie” (1934), pp 2-3

[3] According to the Picard theorem (W. F. Osgood, Lehrbuch der Funktionentheorie, vol 1 (1928), p 178) it is possible to catch every lion except for at most one.

[4] N. Wiener, “The Fourier Integral and Certain of its Applications” (1933), pp 73-74

[5] N. Wiener, ibid, p 89

[6] cf e.g. H. A. Bethe and R. F. Bacher, “Reviews of Modern Physics”, 8 (1936), pp 82-229, esp. pp 106-107

[7] ibid

This first apperared in Amer. Math. Monthly 45 (1938) p446-447. In fact, the Method 10 included here did not actually appear in the original Monthly version, but in a slightly expanded version of that appeared in Eureka.

* It is perhaps generally not known that Petard’s** full initials are H. W. O., standing for “Hoist With Own.”

** Actually, H. Petard is the pen-name for the mathematician E. S. Pondiczery***, who preferred to publish the paper pseudonymously.

*** In fact, Pondiczery himself was a fictious mathematician invented by Ralph P. Boas and Frank Smithies.****

**** This was published in The American Mathematical Monthly, 1938, with one editorial alternation: a “footnote to a footnote was ruthlessly removed.” Consider this last footnote to a footnote to a footnote of a footnote as our nod to the masters.

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