# A Theory with Zero Probability

Pannell and Campbell’s work received no mention when, in February 1918, their colleagues at the National Physical Laboratory, W. L. Cowley and Hy­man Levy, published their authoritative Aeronautics in Theory and Experi­ment.74 (This book is the one I used in chap. 2 to introduce some of the ba­sic ideas of classical hydrodynamics.) Cowley and Levy’s chapter titled “The Mathematical Theory of Fluid Motion” was, for the most part, devoted to the discontinuity theory. The weaknesses of the approach were candidly ac­knowledged, but it was still deemed “remarkable” that its predictions agreed as well as they did with the experimental results. By contrast, their treatment of Lanchester’s theory was brief. It consisted of just two paragraphs. Cowley and Levy clearly believed it was beset by a fundamental hopelessness. They began by noting that, apart from the discontinuity theory, “the only other serious attempt to originate a reaction between a body and a perfect fluid is that strongly advocated by Mr F. W. Lanchester, among others, in which he supposes a cyclic motion, about the aerofoil say, superposed on the ordinary steady streaming” (65). Note the word “suppose.” Lanchester did not explain the cyclic motion; he merely supposed that it was present. Cowley and Levy then showed how, granted the supposition, there would be a resultant pres­sure. This would have been the moment to mention that Pannell and Camp­bell had actually detected the presumed circulation. The authors did not take this opportunity. Instead, the exposition was followed by two terse sentences giving objections to the cyclic theory: “It can be shown that this force is pro­portional to the intensity of the cyclic motion and is thus apparently quite arbitrary. The method moreover gives rise to a lift on the aerofoil and no drag” (66). Both points are correct. First, consider the no-drag problem. The force generated by combining a circulation and a uniform horizontal flow is directed vertically upward. There is no horizontal component, so there will be a lift without a drag. But no wing can move through the air without experiencing some drag force, however small. On factual grounds, therefore, this theoretical analysis is doomed from the outset to give empirically false results.

Second, why was the amount of circulation said to be arbitrary? Here Cowley and Levy were pointing to a structural feature of the mathematics. Their point can be illustrated by going back to Rayleigh’s analysis of the ten­nis ball. Rayleigh’s formula for the stream function of the flow had two parts to it. Call them y and y2. The first part dealt with the uniform flow of speed V, which went toward and over the ball, while the second dealt with the cir­culating flow of strength Г, which went around it. Rayleigh’s formula thus looked like this:

W = Wl +¥2-

The two parts of the formula are independent of one another. The speed of flow in the first part can be varied without altering the strength of the circula­tion, and the strength of the circulation can be altered without affecting the speed of the oncoming flow. There is no grand, overarching stream function y* from which these two features of the flow can be deduced. They are not values to be deduced but merely parameters to be specified. And what was true of the analysis that Rayleigh gave of the flow with circulation around a tennis ball applied to the flow with circulation around a wing. This is what Cowley and Levy meant by “arbitrary.” Applied to Lanchester’s theory it meant that there was nothing in the theory that actually predicted the amount of circula­tion, and therefore nothing that predicted the amount of lift.

Isn’t the intensity of the circulation determined by the shape of the aero­foil, for example, its curvature or thickness? Plausible though this is, Cow­ley and Levy could point out that nothing in the mathematics indicated any such connection. The theory simply implied that lift was proportional to the density, the speed of the free stream, and the circulation. These were the only variables, and shape does not feature in the list.75 The formula applies to any shape of cylinder and not just to those whose cross section looks like that of a typical wing. Lanchester, as we have seen, was aware of this. He ex­pressed the point succinctly in his Aerodynamics when he said that all bodies, of whatever shape, must count as being “streamlined” in a perfect fluid (22). Lanchester then drew the obvious, but striking, conclusion: “From the hy­drodynamic standpoint irregularity of contour is no detriment, as obstruct­ing neither the cyclic motion nor that of translation. The consequence is that peripteroid motion [that is, motion of a kind that generates lift] is theoreti­cally possible in the case of a cylinder of infinite extent, no matter what its cross-section. This conclusion applies naturally only in the case of the invis­cid fluid” (163).

The circulation theory, when it is based on the behavior of a perfect fluid in irrotational motion, thus has some disconcerting features, both factually and formally. First, the circulation is independent of the shape of the wing and, second, it cannot be created by the movement of the wing through the air. Both problems derived from the same source and appeared to be in – eradicably connected with the mathematics of a perfect fluid. This was the root of all the trouble. Both circulation theory and discontinuity theory were doomed because they were built on the same unreal foundation. Until a way was found to overcome the limitations of classical hydrodynamics, progress in this branch of aerodynamics would be impossible. As Cowley and Levy put it, “the failure of the various treatments of the problem of the motion of a body and the forces experienced, to approximate to that of practice is evidently due to the supposition that the fluid dealt with is perfect” (66). The need was for a general theory of viscous flow around a wing. As yet, said Cowley and Levy, no such theory existed, but if and when it did, it would “clarify at one stroke the whole problem of aerodynamics” (75).

Why did Cowley and Levy make no mention of Pannell and Campbell’s results, given that the results came from their own laboratory? Could the ex­planation be that their book was written under wartime restrictions? Certainly it would not have been prudent in such circumstances to advertise problems with a prototype aircraft, but such restrictions hardly explain the negative as­sessment of Lanchester. Even if the details of the evidence could not be given, it is difficult to see why the mere existence of experimental support could not have been admitted. This neglect suggests that the evidence was deemed inadmissible on scientific grounds rather than for reasons of security.

What scientific grounds could ever justify the neglect of evidence? The answer calls for a brief look at the principles of scientific inference. Philoso­phers sometimes analyze science in terms of what they call Bayesian confir­mation theory.76 The analysis depends on a mathematical theorem associated with the name of Thomas Bayes. The idea is that new experimental evidence that confirms the predictions of a theory increases the scientist’s assessment of the probability that the theory is true. The size of the increase is given by a simple formula derived from the calculus of probabilities. The degree of belief in a theory h, given the new evidence e, depends on the initial or a priori probability of the evidence p(e) and the initial or a priori probability of the theory p(h). If the theory h entails, that is, predicts, the evidence, so that h ^ e, then the posterior probability, that is, the probability of h given e, written p(h/e), is given by Bayes’ theorem as

p(h / e) = .

p(e)

The initial probability of the theory is divided by the initial probability of the evidence to give the new, increased, probability that the theory is true. If the predicted evidence is itself surprising and improbable, then the value of p(e) will be smaller than if the prediction is less surprising. A successful but sur­prising prediction will thus increase the posterior probability of the theory to a greater extent than a less surprising prediction.

Suppose that Lanchester’s theory is symbolized by h. Pannell and Camp­bell knew about the theory and did not dismiss it, but they had no indepen­dent grounds for expecting the flow around the wingtips. Accordingly, they called that piece of evidence “remarkable.” They also knew that Lanchester had predicted it. If they were behaving like Bayesians, the probability of Lanchester’s theory p( h) would have been enhanced by the evidence e so that p(h/e) > p(h). Their subjective degree of belief would have been increased.77 What about Cowley and Levy? They too must have known about the result, so why were they unmoved by it? Their response makes sense in terms of Bayes’ theorem provided that one, simple, further condition is satisfied. The a priori probability they accorded to the theory must have been zero. For Cow­ley and Levy, p(h) = 0. This has the result that p(h/e) = 0, whatever the value of p(e). The a posteriori probability will always be zero if the a priori prob­ability is zero. Mathematically this follows because any number multiplied by zero again yields zero. Psychologically, it means that if a scientist starts with a zero degree of belief in a theory, then the subsequent course of belief will be wholly unresponsive to new evidence in its favor.

Scientists do not behave exactly like Bayesian calculating machines, but the model dramatizes the logic of the situation. The association between Lanchester’s circulation theory and perfect fluid theory was sufficient, in the minds of some scientists, to render his account of lift irredeemably false. It represented an essential failure, and the failure was fatal. As Cowley and Levy put it: “The absence of reaction between body and fluid is extremely unfor­tunate for it implies an essential failure in the application of results obtained for a perfect fluid to a real case. Mathematical physicists have striven for years to introduce some new assumption into the nature of the flow that will avoid this fatal result, but it is clear that no matter how ingenious the suggestions may be, they must of necessity be artificial since they attempt to simulate the action of viscosity without actually assuming its existence” (53). The argu­ment was that perfect fluids are mathematical fictions. A theory built upon such a foundation cannot possibly offer a true account of the world. It fol­lowed that Lanchester could not possibly be right.