Category The Enigma of. the Aerofoil

The Sociological Perspective

It is important to appreciate the difference between the professional perspec­tive of the sociologist and the perspective that prevails, and perhaps comes naturally, to social actors themselves in the course of everyday life. I refer to these as the “analyst’s perspective” and the “actor’s perspective.” The concerns of those engaged in sociological analysis are usually not identical to those of the social actors they study, though, of course, analysts themselves will some­times occupy the very roles that they investigate professionally. Conversely, sociological perspectives are sometimes invoked in the course of everyday interaction. (Major Low adopted such a stance when he speculated on what would have happened if Rayleigh had backed the circulation theory. He was reflecting on the role played in British aerodynamics by Rayleigh’s authority.) Despite this overlap and interweaving, it is the differences in the perspectives of the analyst and the actor that I want to emphasize.

In everyday life much of our curiosity centers on deviations from what normally happens or from (what we feel to be) our justified expectations. We want to know why things go wrong more than we want to know why they went right. Going right tends to be taken for granted. It is the failure of the airline to keep to its schedule that makes irate travelers demand to know the causes of the delay. They do not demand to know how and why a punctual departure was achieved. If they were to pose such a question, it would be heard as a hostile comment rather than a disinterested inquiry. The structure of everyday curiosity can be remarkably one-sided. Using a terminology that has become current in the sociology of science, such everyday curiosity may be described as “asymmetrical.” For the sociologist, however, the atypical is not the only thing that needs explaining. The typical is as interesting as the atypical, and the normal or the expected course of events is at least as im­portant as the deviations. The professional curiosity of the sociologist may therefore be called “symmetrical,” in contrast to the “asymmetry” of much commonsense curiosity.4

If an “asymmetrical” curiosity prompts us to ask for causes for half the story, then a “symmetrical” curiosity must lead us to demand causes for the whole story. If the commuters only want to know the causes of delay, then sociologists must risk the resentment prompted by their wanting to know the causes of nondelay. They must ask the questions others don’t ask or don’t want to answer. If sociologists were to study the workings of an airline, they would try to grasp its organizational features and see how its various parts related to one another. There are pilots and crew to be trained, maintenance schedules to be established, fuel supplies to be arranged, safety standards to be adhered to, duty rosters and wages to be negotiated, and shareholders to be satisfied. These dimensions of the organization would be common to all or most airlines, so the sociologist could construct a general model of such an organization and note the difference of practice between different instances of the model. This airline might devote twice as much time to safety training as that one; this one might repeatedly demand more flying hours between checks and repairs than that one; this one might meet its schedules by taking more risks. Such an investigative procedure would bring both the successful and the unsuccessful, the efficient and the inefficient, the safety conscious and the risk takers under the scope of the same model. By casting both sides of the story in the same terms, it is possible to use the different performances to probe the working of the general model, and hence to explore more deeply what it is to have a social organization capable of producing the range of ob­served outcomes.5

These considerations, drawn from the practice of general sociology, also apply to the sociology of knowledge. The central thrust of the Strong Pro­gram is that explanations in the sociology and history of science should be both “causal” and “symmetrical” in the sense that I have just explained. The same type of cause should explain the attractions of both true and false the­ories, and both successful and unsuccessful lines of work (where the judg­ments of truth and success derive from hindsight or are the analyst’s own). I have said that this approach has informed my case study, but when I asked “Why did the British resist the circulation theory?” I may seem to have ad­opted the asymmetrical stance of common sense. It is true that the question could be posed in a purely commonsense way. This, I suspect, is how Major Low meant it when he asked why Lanchester had been ignored. Despite his sociological insight, he primarily wanted someone to take the blame. Stated in isolation Low’s question is worded in a way that is consistent with either a symmetrical or an asymmetrical stance. What differentiates the two stances is the purpose behind the question and the distribution of curiosity informing the answer. The evidence I have presented indicates that it was local cultures, and the institutions sustaining them, that explain the reactions to the theory of circulation exhibited by both the British and the German experts. These

were the causes of the phenomena that I set out to explain, and the causes were of the same kind in the two cases. Cambridge was not Gottingen, but both were influential and brilliant research institutions. Mathematical phys­ics was not technical mechanics, but both were based on rich, mathemati­cal traditions. Lamb’s Hydrodynamics was not Foppl’s Vorlesungen, but both were much-used textbooks that, respectively, encouraged and transmitted their own characteristic, mental orientation.

The scientific study of a complex phenomenon, or the development of a complex technology, typically calls for the cooperation of specialists from a number of fields. The creation of the atomic bomb in Los Alamos involved physicists, chemists, metallurgists, engineers, and experts in fluid dynamics. An episode in the history of science and technology of the kind I have ana­lyzed likewise counts as a complex, real-world phenomenon, and its study will involve specialists from different fields, for example, historians, sociolo­gists, and psychologists. The psychologist studies the mental capacities nec­essary for learning about the world and becoming a competent member of society, perhaps even a member of a specialist subgroup—for example, a subgroup whose members are able to read Foppl’s textbook or sit the Tri­pos examination. The sociologist studies the social processes without which, ultimately, there would be no professional identities such as “the engineer” or “the physicist” and no institutions such as “the textbook,” “the examina­tion,” or “the university.”

It is evident that the episode I have described in my case study cannot be called “purely” sociological any more than it is “purely” psychological or “purely” a matter of grappling with the world. Likewise, the desire for a “complete” description or a “complete” explanation of the episode can be dismissed as utopian. But it is not unrealistic to hope for insights into parts of the problem, and some aspects of the episode may call for psychological study, while other aspects may call for sociological study. The one does not exclude the other. My emphasis on the sociological dimension is not a denial of the psychological dimension or any other naturalistic dimension. Rather, the emphasis on society arises because sociological variables are the ones most relevant to the question I am asking. British and German experts did not diverge because their basic cognitive faculties differed or because their personalities were different or because one group engaged with the material world while the other turned its back on it. As far as the present episode is concerned, they differed primarily because their education and professional lives were different. They worked in different disciplines and institutions whose traditions and reward structures diverged from one another.

One final feature of the sociological approach must be emphasized. It is central to my account that the actors involved were not detached intel­ligences moving in an abstract world of thoughts, theorems, and deductions. Nor did they move exclusively in a world of committee meetings, personal confrontations, status conflicts, and power struggles. These things were part of their world but not the whole of it. The experts in my story experimented in wind tunnels, built models, observed and measured the forces on them, flew airplanes, and sometimes died in them. The sociological variables to which I have drawn attention are not to be conceived in a way that excludes these practical, experimental activities or diminishes their importance. The sociological processes I have identified do not stand between people and the material environment with which they are engaged. Contrary to the claims repeatedly made by their critics, those who follow the Strong Program do not treat the social world as something to which scientists respond instead of re­sponding to the natural world. The cultures, institutions, and interests that I have identified did not block the active involvement with material reality but were the vehicle of that involvement and gave a specific meaning to it.6

The Euler Equations

The second principle states that fluid elements obey Newton’s second law of motion, F = MX A, that is, force equals mass times acceleration. Figure 2.2 is a picture of a fluid element whose sides are parallel with the x – and y-axes. As in the previous discussion, the flow is assumed to be “two-dimensional,” so the z-axis plays no role in the analysis. The element is again assumed to be of unit depth and can be represented by a rectangle ABCD (whose “depth” bz = 1). Thus the mass of a small, rectangular fluid element with sides bx and by is pbxby. In aerodynamics the effect on the air of external forces, such as gravity, can be neglected. The resultant force on the fluid element in the x-direction comes from difference between the pressures on the two faces AD

The Euler Equations

figure 2.2. Small fluid element showing pressure on the faces AD and BC. The pressure difference is the force producing the acceleration along the x-axis. From Cowley and Levy 1918, 37.

and BC. Let the pressure on AD be p. The pressure on BC is then p+d-Sx,


that is, the original pressure plus the change in pressure over the length of the element. The resultant thrust on the element is

pSy—| p+dp Sx lay=—dp Sx8y.

dx ) dx

This is the force causing the acceleration. Newton’s second law then takes the form

What is the acceleration in the x-direction? The desired term is du/dt, the rate of change of the velocity u with time t. The velocity u is a function of three variables: x, y, and t. The acceleration is given by a process of differentiation involving all three variables which is called “differentiation following the mo­tion of the fluid.”16 Thus,

du du dx + du dy + du dt dt dx dt dy dt dt dt

du dx + du dy + du dx dt dy dt dt

This expression can be simplified. If the restriction is introduced that the flow is steady, then d. = 0. Also, by definition, – А = u and – А = v. Substituting these terms in the expression for Newton’s law yields

dp I du du |

— = 4 U + V ) •

Similar reasoning gives the equation for the y-direction:

dp I dv dv

-ay=plu dX+v dy

These are the Euler equations for the two-dimensional steady flow of an ideal fluid. A mathematician will identify them as nonlinear, partial differen­tial equations that relate the pressure p and the velocity components u and v. Along with the equation of continuity they constitute the fundamental equa­tions of motion of an ideal fluid. They completely determine the motion. The integration, or solution, of the equations will, however, involve arbitrary functions and constants, and these require knowledge of (1) the initial condi­tions of the motion and (2) the position of any fixed boundaries. These two specifications are called the boundary conditions.

An Old Anomaly or a New Crisis?

How did the leading British aerodynamicists react? The theory of lift derived from the picture of discontinuous flow could not be right, at least, not as it stood. Was this situation to be treated (in Kuhn’s terms) as a “crisis”? Did it call for a radical response involving the use of new models and the creation of a wholly new approach to lift? Or was it no more than an “anomaly”? Was there still the possibility that some refinement of the old, discontinuity ap­proach would eventually allow the problem to be met? The answer is that the problem was seen as both an anomaly and a crisis, but it was seen in these different ways by different groups. The split was roughly along disciplinary lines. For the more purely mathematical contributors, the emerging results represented, or were collectively treated, as an anomaly rather than a crisis; for the experimentalists and physicists, even the mathematical physicists, the findings were more than merely anomalous: they were taken to herald a full­blown crisis.34

Greenhill, Bryan, and a number of other research mathematicians con­tinued to work on the problems of discontinuous flow. They aimed to refine the analysis by taking account of the curvature that was characteristic of the cross section of a wing. This involved complicating the Schwarz-Christoffel transformation, which applied to straight-sided figures. In 1914 G. H. Bryan and Robert Jones published a paper called “Discontinuous Fluid Motion Past a Bent Plane, with Special Reference to Aeroplane Problems.”35 They were able to align the analysis with some of the qualitative facts, for example, that a moderate degree of camber can increase lift without increasing drag. In 1915 J. G. Leathem (the sixth wrangler in 1894 and a fellow of St. John’s) published “Some Applications of Conformal Transformations to Problems of Hydrodynamics,” which was meant to put the introduction of “curve – factors” on a systematic basis.36 Also in 1915, Hyman Levy published “On the Resistance Experienced by a Body Moving in a Fluid,” in which he set out to link a discontinuity analysis to recent work on vortices by von Karman.37 In 1916 Greenhill published a substantial appendix to his R&M 19 which was titled “Theory of a Stream Line Past a Curved Wing.” Greenhill noted that a curved surface could be approximated by a large number of short, straight surfaces so that the way was open to work with a more realistic model of a wing. He added to his previous discussion by surveying the contributions to discontinuity theory of French and Italian mathematicians such as Bril – louin, Villat, Cissotti, and Levi-Civita.38 The same year, 1916, saw Levy’s “Dis­continuous Fluid Motion Past a Curved Boundary.”39 The investigation was again justified by its relevance to aerodynamics. The author asserted that “In aeronautics alone recent developments have shown the practical necessity for an effective discussion of the case where the plane is cambered” (285).

What kept the mathematicians at work?40 Partly, they hoped to bring dis­continuity theory into closer contact with experimental results, as indeed they did; it would be wrong, however, to overstate the optimism associated with this project. The main factor seems to have been the lack of any per­ceived alternative. The question they faced was whether there was any chance of digging beneath the equations of perfect fluid theory and making progress with the full, governing equations of viscous flow. If there was no chance, or very little chance, then it was reasonable to carry on as before. Consider the stance of Cowley and Levy. They concluded that the fatal flaw in the theory of discontinuous flow was that it depended on the assumptions of the theory of a perfect fluid, although, they argued, “it is remarkable. . . that the results ob­tained for the resistance are comparable at all with those derived from experi­ment” (65), and they speculated that perfect fluid flows with vortices might be used to simulate the flows that could be photographed in a turbulent, viscous fluid.41

While Cowley and Levy spoke of the mathematics of viscous flow as “not yet sufficiently developed” (75)—thus holding out hope—a more pessimistic induction is hinted at by G. H. Bryan’s remarks in the Mathematical Gazette of 1912.42 With a nod to Greenhill’s article on hydrodynamics in the Encyclo­paedia Britannica, Bryan said that the subject really consists in the study of certain partial differential equations and not “town water supply, resistance of ships, screw propellers and aeroplanes” (379). There was not much induce­ment for the mathematician to adapt his work to the needs of engineers. If he were going to do that, he might as well become an engineer “and give up most of his mathematics, relying on the introduction of constants or coefficients to save him from running his head against insoluble differential equations” (379). As for the hypothetical conditions that make hydrodynamics so unreal, these have “pretty well done their duty when they have been made use of to write down differential equations” (379). So it was not the empirical status of these conditions (that is, their falsity) that counted, but their power to help the mathematician frame tractable equations. The real issue was “remarkably simple”: if you give up ideal fluid theory, you get equations for which nobody can find the integrals, “at least, mathematicians have tried over and over in vain to find them” (379). And the same argument applied to the simplifying assumption of steady motion, which involved, and justified, ignoring “for example, eddy formation in the rear of planes” (380). Bryan’s position was the same as the one he adopted in his main field of research into stability. There the equations turned out to be accurate, but for Bryan, this was a bo­nus rather than something that was necessary for justifying the work. Even if inaccurate, he said, the analysis might still furnish a useful basis for the interpretation of experimental data.

Such was the reasoning by which a small number of high-status math­ematicians justified their continued elaboration of discontinuity theory and sustained a pessimistic form of “normal science.” For Bairstow, and others at

the NPL who were more experimentally inclined, the emerging problems in­dicated the end of the road for discontinuity theory. The theory was artificial and doomed to failure because it was grounded in the unreal conception of a perfect, frictionless fluid. What was needed was a return to the full equations of viscous flow and the attempt to develop new methods of approximation. For the moment, however, Bairstow accepted that the complex flow around a wing was beyond the comprehension of the mathematician.

In a lecture at the Aeronautical Society, on February 12, 1913, Bairstow asked what shape of aerofoil or strut would give the most lift or the least resistance.43 “A true theory of aerodynamics,” he said, “would answer these questions for us completely, but unfortunately for us the answers to such questions are beyond the reach of our present mathematical knowledge.” To reinforce the point Bairstow showed his audience a photograph that, he said, “illustrates a motion which has defied the mathematician” (117). The photo­graph was one of the NPL water-channel pictures showing a wing with the characteristic turbulent wake associated with a stall, that is, a wing exhibiting Kirchhoff-Rayleigh flow. Greenhill was in the audience—but he did not de­fend his mathematical model of discontinuous flow around a wing, nor did he challenge Bairstow’s conclusion. It is difficult to know how to interpret this disregard. Greenhill initiated the discussion that followed the lecture, but only to make jocular comments on the law of mechanical similarity. Ac­cording to Greenhill’s calculations, if angels existed in the form in which they are usually depicted, then they would have to be about the size of a bee.44 This lack of an explicit response to the shortcomings of the discontinuity theory was wholly characteristic. There had been no discussion of the de­mise of discontinuity theory at the meetings of the Advisory Committee (or none at which minutes were taken). Greenhill, though in regular attendance, appears to have made few contributions to the business of the committee. Anecdotal evidence reveals that Sir George was actually prone to fall asleep during meetings. On one such occasion Mervin O’Gorman, a talented artist, drew a sketch of the slumbering mathematician and left it on the table. The caption was from the well-known hymn: “There is a green hill far away.”45 But sleep patterns are not really very illuminating. Perhaps the reason for the reticence was simply that Greenhill, and his fellow wranglers on the commit­tee, shared something of Bryan’s pessimism. Without an analysis of viscous flow, the choice was between inviscid theory and empiricism—and inviscid theory had failed.

Bairstow’s slightly more optimistic view, that progress of some kind was possible with the equations of viscous flow, was shared by Geoffrey Ingram Taylor. Taylor came up to Trinity in 1905, took part I of the Mathematical

Tripos in 1907 and part II of the Natural Sciences Tripos in 1908. He was given a major scholarship at Trinity and in 1910 was elected to a fellowship. After war was declared on August 4, 1914, Taylor hurried to submit his dis­sertation for the Adams Prize, unsure whether it would be awarded because of the uncertainty of the international situation. He volunteered his services to the military on August 5, hoping to work in meteorology, but was immedi­ately drafted to Farnborough and later co-opted onto the Aerodynamics Sub­Committee of the ACA. Soon after his arrival at Farnborough, Taylor took his first flight. He flew with Edward Busk, on the day before Busk’s death. Later, after gaining his “wings” with the Royal Flying Corps, Taylor carried out numerous pieces of research, including measurements of the pressure distribution across the wing of a BE2. His work enabled comparisons to be made between wind-channel results on models and full scale data.46 In the early months of the war, however, he used every spare moment to bring his thesis to completion. It was titled “Turbulent Motion in Fluids.”47

The preface and introduction of the thesis were used to set out the current situation in fluid dynamics. Taylor’s position contrasted starkly with that of Bryan, who placed the emphasis on getting hold of some differential equa­tions rather than worrying about what had to be assumed or discarded en route. In Taylor’s view: “In no other branch of applied mathematics is the danger of neglecting the physical basis of the subject greater than it is in hy­drodynamics” (5). It has been possible, he said, to get “rigorous mathemati­cal solutions” in hydrodynamics, but they have no relation to what is found experimentally. Unfortunately, mathematicians adhere to these physically unrealistic assumptions simply because it makes the mathematics easier. As an example Taylor cited the theory of discontinuous flow. The theory is based on the assumption that the pressure in the dead-water region is the same as that of the undisturbed flow, when measurements show it is less than this. Referring to measurements made by Melvill Jones at the NPL, Taylor drew an unequivocal conclusion: “This, I think, finally disposes of the discontinuity theory, which . . . must now be placed among the curiosities of mathematics” (4). The rigor of the old methods, argued Taylor, was well worth sacrificing if there was a chance of explaining the turbulent behavior of real fluids—and this is what he set out to do in the remaining sections of the bulky thesis.48 His aim was to develop a new (statistical) theory of turbulence.

For Taylor, like Bairstow, discontinuity theory and Rayleigh flow were things of the past. The same judgment can be read into what was said, and what was not said, in an important lecture given by Glazebrook in June 1914. His title was “The Development of the Aeroplane,” and the aim was to de­scribe the achievements of the National Physical Laboratory. He mentioned the work of Stanton and Bairstow and explained how pressure measurements reveal the important role of the upper surface of the wing (from the outset, the weak point of the discontinuity theory). Glazebrook mentioned no names and did not make it explicit that the one candidate for the explanation of lift that had been taken seriously in British aerodynamics was now being quietly abandoned. All the attention was directed toward the successful work on sta­bility. The occasion of Glazebrook’s assessment was the second of the Wright Memorial Lectures. It was an important event so the assessment would have been carefully considered.49 The unspoken message was that, notwithstand­ing the opinion of a few of the older mathematicians, the discontinuity theory of lift was dead.

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) = .


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.

Section i. introduction

Kutta began with a nod not only to Lilienthal but to more recent develop­ments in aviation. These, he said, gave a great practical significance to the old, but difficult hydrodynamic problem of calculating the forces on a body immersed in a moving fluid such as air. Calculating the lift forces on a wing was particularly important. It should be possible to do this because the rel­evant flow can be understood (“aufgefafit werden kann”) as the superimpo­sition of a circulation and a steady stream. For an explanation of the basic ideas of the circulatory theory, Kutta directed the reader to the first volume of Lanchester’s Aerial Flight and a 1909 article by Finsterwalder. Finsterwalder’s article was titled “Die Aerodynamik als Grundlage der Luftschiffahrt” (Hy­drodynamics as the basis of airship flight), but it also dealt with the basics of the circulation theory of lift.29 Kutta’s choice of words, in saying that the flow can be so understood, simply indicates that if the flow is interpreted in this way, then the lift force becomes intelligible. There is, however, no reason to suppose that he doubted the reality of circulation. He was probably stepping carefully because of the highly artificial character of the concepts he was ap­plying to the problem: friction was being ignored, the flow was to be treated as two dimensional, and the fluid was taken to be free of vorticity (that is, the flow was irrotational).

Kutta then made three observations about his own earlier work. First, he said that in 1902 he had discovered a general theorem about lift which was re­discovered by Joukowsky in 1906. He thus made a priority claim for the result that lift is proportional to circulation and is given by the product of density, velocity, and circulation.30 Second, he said that the 1902 predictions about lift were supported by Lilienthal’s data but acknowledged that the discussion had been confined to wings at zero angle of incidence. This restriction would not apply to the more general analysis he was about to offer. Third, in his earlier account, the magnitude of the circulation could be fixed by specifying that the flow was to be smooth at both the leading and trailing edge. This was pos­sible because of the symmetry of the arc-like wing at zero angle of incidence. In the more general treatment, with an arc whose chord was at an angle to the flow, adjusting the circulation could only make the flow smooth at one edge, for example, the trailing edge. At the leading edge the fluid would divide, generally at a point on the lower surface, and some of it would be forced to flow around the leading edge. Because Kutta represented the wing by a geo­metrically thin line, this meant the fluid at the leading edge would achieve infinite speeds and pose a significant problem for the analysis.

Kutta indicated to the reader that the lift force on the wing would have to be broken down into two parts. Leaving the explanation until later, he stated that one part of the lift would be produced by pressure on the surface of the arc, while the other part could be represented as a tangential, suction force at the leading edge. He also signaled his intention to make his abstract, geo­metrical model of the wing more realistic by studying the effects of rounding off the leading edge.

The Horseshoe Vortex and the Biot-Savart Law

Prandtl’s earliest publications on aeronautics did not deal with the circula­tion theory of lift in either its two – or three-dimensional form. He wrote mainly on airships, the general mechanical problems associated with build­ing an airplane, or the engineering that went into the construction of wind channels for testing models. Thus in June 1909 he lectured to the annual gen­eral meeting of the association of German engineers at Mainz on the signifi­cance of the model experiments to be done at Gottingen and the equipment that had been developed.47 September of the same year saw him speaking at the International Aeronautical Exhibition (ILA) in Frankfurt. Here Prandtl concentrated on the principles and preconditions of flight and the practical problems of achieving adequate lift and stability.48 In a series of articles in the new Zeitschrift fur Flugtechnik he laid out for the reader’s benefit those parts of mechanics of special relevance to aeronautics.49 His topics included gyroscopes, stability, and air resistance, with a mention of his own boundary – layer theory in connection with the phenomenon of separation and vortex formation. One of Prandtl’s concerns was to remove false ideas that contin­ued to cause trouble in the field. Many “inventors,” he explained, tried to achieve automatic stability in aircraft by some sort of pendulum device whose rationale was based on a misunderstanding. Prandtl also stressed that resis­tance depends as much on the suction effects behind a body as it does on what happens at its front face. From the outset he had no time for Rayleigh – Kirchhoff flow as an account of lift. He viewed it as wrong at the level of general principle.

In these articles for the ZFM, Prandtl recommended some of the relevant literature (64). Significantly he included both Lanchester’s book on aerody­namics and Lamb’s book on hydrodynamics. He thus symbolically conjoined what, in the homeland of those two authors, was proving so resistant to uni­fication. On the first page of the initial article in the ZFM series, Prandtl also nailed his colors to the mast regarding the principle of the unity of theory and practice. Not for him the slogans of the antimathematical movement, which treated theory, and especially mathematically based theory, as something in fundamental opposition to practice. Using words that closely echoed the for­

mulations adopted a decade earlier by August Foppl, in the preface to his Vorlesungen uber technische Mechanik, 50 Prandtl asserted:

Um nun meine Absichten naher zu kennzeichnen, mochte ich vorweg be – tonen, dafi ich das viel ausgesprochene Schlagwort vom Gegensatz zwischen Theorie und Praxis nicht gelten lassen will; der Gegensatz liegt fur mein Emp – finden zwischen guter und schlechter, richtig und unrichtig angewandter Theorie; eine gute Theorie is in Ubereinstimmung mit den Ergebnissen der praktischen Erfahrung, oder sie gibt zum mindesten wesentliche Zuge der Er – fahrungstatsachen wieder. (3)

In order to characterize my intentions more precisely, I should emphasize at the outset that I do not want to endorse the much used slogans about the oppo­sition of theory and practice. In my experience the contrast lies between theo­ries that are good or bad for application and between correctly or incorrectly applied theories. A good theory is in agreement with the results of practical experience or, at least, captures the essential thrust of the facts of experience.

Prandtl was not just striking attitudes. His work on the three-dimensional wing would meet the demanding requirement of applicability to engineering practice.

Prandtl tells us that it was in the winter semester of 1910-11 that he began to develop his own mathematical theory of the finite wing.51 It was treated in a course of lectures that he gave in Gottingen on the theme of aeronau­tics. The lectures were attended by Otto Foppl and, though the text of the lectures appears to have been lost, hints and mentions of their content are to be found in the early papers coming from the Gottingen group. The basic picture with which Prandtl worked was similar to the one already proposed by Lanchester, namely, a finite wing with circulation around it and with two vortices emerging from the wingtips. Prandtl’s aim was to make this picture amenable to a mathematical analysis, but to do so he had to resort to extreme simplification. He treated this complex three-dimensional, physical system as equivalent to three connected-line vortices. One vortex, called the bound vortex, represented the wing and was assumed to run along the span of the wing. The other two vortices, called free vortices or trailing vortices, extended back from the wingtips and were at right angles to the wing. The arrange­ment, as shown in figure 7.5, first appeared in a brief, qualitative account of the theory that Prandtl published in 1912.52 The two diagrams, which show the progressive abstraction of the picture, reappeared in Prandtl’s published lectures in the form shown here.

Physically the trailing vortices are assumed to extend right back from the wing in flight to the point where the aircraft left the ground. Mathematically

The Horseshoe Vortex and the Biot-Savart Law

The Horseshoe Vortex and the Biot-Savart Law

figure 7.5. Trailing vortices from the wingtips represented diagrammatically (a) and by the even more simplified “horseshoe” vortex system (b). Tietjens 1931, 197. (By permission of Springer Science and Business Media)

they are said to extend back “to infinity.” The three connected vortices thus form a single bent line of vorticity. A vortex line in an ideal fluid (whether straight or bent) has the same strength everywhere along its length, so the circulation Г around the wing, that is, around the bound vortex, also gives the vortex strength along the trailing vortices. For reasons examined later, this model was called, rather implausibly, the Hufeisen Schema, or “horseshoe schema.”

Prandtl’s task was to see how the presence of the two trailing vortices modified the flow in the vicinity of the wing. If the trailing vortices are ig­nored (as they are in the two-dimensional case), the law of lift is (using the usual notation) Lift = p V Г. Is this law preserved in the three-dimensional case? Surely some significant modification of the flow must occur, and this must have some consequences for the behavior of the wing—but what modi­fications and what consequences? Before I go into mathematical details, it may be useful to sketch the outcome of Prandtl’s analysis in qualitative terms. Using his simplified model, Prandtl was able to predict two unexpected and important effects.

First, Prandtl found that the effect of the trailing vortices ought to be the creation of a downwash at, and behind, the wing. The swirling air of the trail­ing vortices would (according to his mathematical analysis) influence the air in the vicinity of the wing in such a way as to give it a downward velocity component. The downward component would have the effect of tilting the airflow. The way that the tilt arises from the introduction of the downwash component is shown in figure 7.6a. The presence of the tilt means that the wing operates at a slightly lower angle of incidence relative to this new local flow. The effective angle of incidence is therefore reduced. Because (over the working range of the aerofoil) lift is proportional to the angle of incidence, the lift should be reduced. The same point may be expressed the other way round. In order to get the same lift per unit length in a finite aerofoil as in an infinite aerofoil of the same cross section, the angle of incidence relative to the main flow has to be increased.

The second prediction followed from the first. If the resultant lift force on the wing is at right angles to the local flow, and the local flow is tilted in the manner shown in figure 7.6b, then the resultant aerodynamic force will be tilted back relative to the main flow and hence to the direction of motion of the wing. The effect will be that the resultant force will now have a compo­nent that opposes the motion. There will be a drag. It is important to recall that this analysis was all done using ideal-fluid theory. The two-dimensional analysis showed how a flow involving a vortex and circulation could yield a lift but no drag, the three-dimensional analysis now predicts that it can pro­duce both a lift and a drag. For reasons that will become clear in a moment, Prandtl and his co-workers came to call this novel drag an “induced drag,” and the tilt of the local flow due to the downwash was called the “induced angle of incidence.” Induced drag was a form of drag that did not result from viscosity or skin friction or turbulence. It was produced by the very same inviscid mechanisms that generated the lift.

Having sketched these two initial predictions, I now look at their math­ematical derivation. How did the idealization of the “horseshoe” vortex help Prandtl to develop a mathematical description of the flow which led to these results? The answer lies in an analogy that exists between the hydrodynam­ics of a perfect fluid and electromagnetic phenomena. A line vortex is like a wire carrying an electric current which sets up, or “induces,” a magnetic field around it. The vortex in an ideal fluid does not, of course, induce a mag­netic field, it induces a velocity field. The velocity was exactly what Prandtl wanted to understand because once he had a mathematical expression for the velocities, he could deduce the pressures, and thus the forces on the wing. Prandtl’s use of the analogy was explicit. “Fur die Verteilung der Geschwindig – keit in der Umgebung irgendeines Wirbelgebildes besteht eine vollkommne

The Horseshoe Vortex and the Biot-Savart Law

figure 7.6. In (a), the trailing vortices induce a downward component win the flow behind the wing. This tilts the airflow and effectively lowers the angle of incidence of the wing. The new “effective” angle of incidence is the original geometrical angle of the wing to the free stream minus the angle of the downwash produced by the tailing vortices. In (b), the tilt in the airflow means that the resultant aerodynamic force, which is at right angles to the local flow, is no longer at right angles to the free stream, that is, the direction of motion. This produces a drag component called the “induced drag.”

Analogie mit dem Magnetfeld eines stromdurchflossenen Leiters”53 (For the velocity distribution in the neighborhood of any such vortex formation there exists a complete analogy with the magnetic field around a current carrying conductor).

The analogy to which Prandtl referred was well known before the work on

aerodynamics and was discussed in standard textbooks in both electromag­netic theory and hydrodynamics. It was to be found in the books on Max­well’s theory and on technical mechanics written by Prandtl’s own teacher and father-in-law, August Foppl.54 The analogy is not a superficial one but exists at the level of the fundamental equations that can be used to describe the two different areas. Experts in electromagnetism, such as August Foppl, could easily describe the magnetic effects of a current-carrying wire shaped in the form of the idealized “horseshoe,” and Prandtl carried these results over to the corresponding system of line vortices.

The law of induction common to the hydrodynamic and electromagnetic cases is called the Biot-Savart law.55 The law may be explained by considering figure 7.7, which shows an infinite, straight-line vortex of strength Г. A small segment of the vortex is identified as ds. The point P lies at a distance r from the vortex element, and the line joining ds and P makes an angle 0 (theta) with the vortex. Following the electromagnetic analogy, there is a small com­ponent of velocity dw “induced” at the point P by the small element ds. Ac­cording to the Biot-Savart law, the component is

, rds. sind

dw = -—.


The velocity component is perpendicular to the plane determined by ds and the line r. This formula links infinitesimal quantities, and the causal relation between such infinitesimals that seems to be implied by the law occasioned some puzzlement. August Foppl had no time for such subtleties. Taken by itself, said Foppl, the formula has absolutely no meaning: “Es hat uberhaupt keinen Sinn.”56 Foppl’s point was that the real significance of the law only

The Horseshoe Vortex and the Biot-Savart Law

figure 7.7. According to the Biot-Savart law, the infinitesimal amount of induced velocity dw at a point P due to the infinitesimal vortex element ds of a vortex of strength Г is given by the formula dw = (Г/4ТСГ2) ds sin0.

emerges when it is integrated in order to give the finite effect of a finite length of vortex or, by extrapolation, the finite effect of a very long (and effectively infinite) vortex. This is what Prandtl needed in order to compute the effect of the, effectively infinite, trailing vortices.

As Prandtl first employed it, the Biot-Savart law was used to give the fi­nite velocity component w at an arbitrary point P in the vicinity of a finite, straight-line segment AB of a vortex (see fig. 7.8). If the strength of the vortex is Г, and the point P is at a perpendicular distance h from the line AB, then, after integration, the law now reveals that the contribution to the velocity of the finite segment is


w = (cosa + cos в).


The angles a and в are the angles made by the lines joining the point P with the ends of the finite vortex segment under consideration. The direction of the velocity w at P is at right angles to the plane that passes through the points A, P, and B. Whether the velocity vector faces downward, into the page, or upward, depends on the sense of the circulation around AB. (If an observer who looks from B to A is confronted with a clockwise circulation, then the induced velocity vector points into the page and vice versa.) Notice that if the finite line segment AB is extended to infinity in both directions (so the angles a and в get smaller and smaller as the line gets bigger and bigger), then the velocity at point P should correspond to the velocity at a point situated a distance h from the center of a two-dimensional vortex, that is, a point vortex, of circulation Г. This is exactly what the formula provides. The expression (cos a + cos в) assumes the value 2 for a = в = 0, so that w = r/2nh.

The Horseshoe Vortex and the Biot-Savart Law

figure 7.8. The induced velocity w at a point P due to a finite vortex segment AB of a vortex of strength Г is, according to the Biot-Savart law, w = (Г/4-nh) (cosa + cosp).

Prandtl’s aim was to apply the Biot-Savart law to the “horseshoe” vortex because he was interested in the effect of the trailing vortices. The trailing vor­tices count as “semi-infinite” lines because they start from the wingtip and go to infinity in one direction. To understand Prandtl’s reasoning when he ap­plied the law to his horseshoe system, think of the arrangement in figure 7.8 modified in two ways until it turns into that in figure 7.9. First, A is moved to infinity so that a = 0 and cos a = 1. Second, the point B is moved inward until it coincides with the base of the perpendicular from P. This makes в = 90° so that cosP = 0. The formula then gives the value for the induced velocity w:


w = .


Interpreted in terms of the horseshoe vortex model of the wing, this formula gives the contribution of one of the trailing vortices to the flow at a point on the wing that is distance h from the wingtip generating the vortex. The full downwash at any given point on the wing needs the contribution of both trailing vortices to be added together, but the formula reveals the mechanics of the process that generates the downwash. Prandtl spoke of a zusatzliche Abwartsgeschwindigkeit, an additional downward speed. Max Munk called the quantity w the “induced velocity,” and this name was taken over by the Prandtl school.

Подпись: FIGURE 7.9. Application of the Biot-Savart law to the horseshoe vortex. The point P is now a point on the wing. B is now the wingtip. A is at infinity, and AB is the semi-infinite trailing vortex from one wingtip. The Biot-Savart formula gives the induced velocity created by the trailing vortex at point P of the wing.

Sufficient has now been said to show how Prandtl was able to reach his predictions about the general effect of the trailing vortices, that is, (1) the

creation of a downwash, (2) the tilt that the downwash creates in the local flow, and (3) the resulting (induced) drag. One final point to notice about the formula for the induced velocity w of the downwash is that it takes the form of a fraction with h in the denominator. It contains a singularity at the point h = 0. For this value of h, the formula requires that the velocity w be infinite, which is physically impossible. Recall that h refers to the distance from the wingtip. This means the application of the Biot-Savart law to the horseshoe model of the three-dimensional wing breaks down for points close to the wingtip. Prandtl had made novel and important predictions, but, because of the singularity, the predictions carried with them a problem. Even if in many cases they were proven correct, they were based on a physically impossible model.

The Albatross Wing

Major Low’s paper, the third Congress paper of the morning, was titled “The Circulation Theory of Lift, with an Example Worked out for an Albatross Wing-Profile.”99 The “albatross” of his title was not the bird but the name of a German aircraft company that had played a prominent role in the war.100 Low’s aim was to apply the circulation theory to an actual aircraft wing and to show the relation of the theory to drawing-office practice. He also wanted to straighten out one or two points of recent history. He began by reminding his listeners that the origin of the circulatory theory was grounded in the work of British physicists. Fifty years ago, said Low, Rayleigh had published his paper on the spin of the tennis ball and explained the force that made it veer by reference to the circulation around the ball. A similar idea, he said, was to be found in the work of P. G. Tait of Edinburgh (where Low had himself been a student, graduating with an honors degree in mathematics and natural philosophy in 1903).101 Low regaled his audience with a story about experi­ments, done in the dark cellars of the old Edinburgh University buildings, on the spin of golf balls. Tait was helped by his son, “the lamented Freddie Tait.” Freddie had been a professional golfer and, in the name of science, was required by his father to shoot golf balls through screens in order to trace their trajectory. On one occasion, in the gloom, he missed the screens. This resulted in the experimenters dodging around as the ball “ricocheted inter­minably off the walls of the cellar” (255).

This story was merely the disarming prelude to a point that was not in­tended to be amusing. Lanchester, Low went on, had boldly applied the idea of circulation to the wings of an aircraft and had given a thorough, descriptive account of the mechanism of flight. That was nearly twenty years ago. Why was it only now that the circulation theory was being taken seriously in the land of its origin? Low had an answer, and it was not a flattering one: “Had Rayleigh put forward the theory, how we should have vied with each other in the will to believe it, if not in power to understand it! But when it was offered by a man outside the circle of recognised physicists it was ignored” (255).

Leaving his audience to ponder this sociological point, Low went on to expound some of the basic techniques associated with the theory, confin­ing himself to “strictly graphical and descriptive” methods. First, he gave a graphical method for transforming a circle into Joukowsky profiles and then tackled the more difficult, inverse task of going from a given aerofoil back to a circle or a close approximation to a circle. As before, Low was conveying to his audience the content of recent German material, this time using a postwar publication by Geckeler.102 Low showed how to start from the Albatross wing and, using drawing-office methods, map it back to an approximate circle by a series of trial-and-error steps. “There now remains only a routine of laying off and measuring straight lines on the drawing board to determine the ve­locity and hence the pressure at every point of the field” (273). Low assumed a velocity U = 10 m/sec and an air density of p = 1.2 kg/m3. Using the formula L = p U Г, he derived the data to construct a theoretical curve for the Albatross wing relating lift to angle of incidence. Because the formula was based on the assumption of an infinite span, the curve could not be compared directly with wind-tunnel data derived from a finite wing. Low then appealed to the Gottingen transformation formulas relating wings of the same section but different aspect ratio. This allowed him to recast known experimental data on the Albatross wing into its equivalent for an infinite wing. Low now had two curves that linked lift and incidence for the Albatross wing, one curve coming from wind-tunnel tests, the other derived from the circulation theory. For the range of -5° to +10° the two graphs were close together. The theory was sup­ported by experiment. Given an arbitrary wing, a designer could now predict from the circulation theory the curve relating lift to angle of incidence at least up to the point of stall.

Having achieved his main goal, Low then returned to the theme with which he had begun. “In conclusion,” he said, “it is desired to call attention to the fact that this fundamental physical theory was first stated by an English writer, and then allowed to fall into complete neglect in the country of its origin, largely owing to the attitude taken up by some of Lanchester’s fellow members of the Advisory Committee for Aeronautics” (275). On this note the talk ended.

Calling the circulation account of lift a “fundamental physical theory” can only have been meant as a thrust at Bairstow, who had just explicitly denied it the status of being a fundamental theory. But the remarks blaming the Advisory Committee for Aeronautics for the neglect of Lanchester were even more pointed. The austere figure of Professor Sir Richard Glazebrook, who had been the chairman of the Advisory Committee, and who was thus the main focus of Low’s complaint, was present at the talk. In fact, he was more than present. He was presiding over the session at which Low had just delivered his paper.103

Reasons and Causes

One of the objections that critics have repeatedly directed against the Strong Program is that the commitment to causal, sociological explanation entails neglecting the role of reasons. The critics say that a Strong Program analysis involves disregarding the reasons that social actors themselves offer for their behavior. According to the critics these reasons can, on occasion, provide a sufficient explanation of the behavior and thus render redundant any attempt to construct a causal, sociological explanation. A recent example of this criti­cism is to be found in a 2006 paper by Sturm and Gigerenzer.7 The authors say: “Even after a strong sociological explanation has been given for the be­liefs of a scientist, it remains sensible to ask: Very fine, but how are these beliefs connected to the scientist’s justificatory reasons? Can these reasons perhaps explain better why the scientist acquired the relevant beliefs?” (144). As the wording indicates, these critics assume that the candidate sociological explanation will have been constructed without any significant reference to the agent’s reasons. The critics wish to make good this alleged lack, and they put their money on finding cases where nothing more than the scientist’s own reasoning is needed to explain some pattern of scientific belief. The agent’s reasons, they say, can be the cause of their beliefs, and a proper explanation of these beliefs should be in terms of these reasons (141).

Does the causal explanation that I have put forward to account for my findings in the history of aerodynamics depend on, or result in, a lack of

serious attention to the agent’s reasons? I hope it will be evident that such a complaint is groundless. The problem I have posed, the central problem of the book, is a problem about the reasons that were used to justify a certain scientific judgment. I have attended closely to the reasons given by the actors in the story and subjected them to a close analysis. I fear, however, that critics will dismiss my discussion of scientific reasoning as mere window dressing. Thus Sturm and Gigerenzer say, “Mentioning that scientists claim to employ certain reasons or reasoning standards, and mentioning which ones these are, is not the same as taking these standards seriously—seriously in the sense that they are acknowledged as causes of the scientist’s acceptance or rejection of claims” (142). For these critics, nothing short of treating reasons as self­sufficient explanations will count as taking them seriously. This I certainly have not done and will not do because it would corrupt the analysis. Despite this, attending to the actor’s reasons has played a central role, even though the analysis culminates in a causal explanation. The actor’s reasons are not merely mentioned; they are given a substantial role. Consider the stated reasons of­fered by the British to justify their rejection of the ideal fluid approach, for example, their complaint that the origin of circulation must forever remain a mystery. These reasons certainly illuminate the British rejection. When the reasons are examined, however, it becomes clear they do not adequately ex­plain the behavior of the British experts. This is not because the British had other “real” reasons. The inadequacy of the explanation is because their Ger­man counterparts understood the reasons that moved the British as well as the British did, but responded differently. Kelvin’s theorem was as familiar to Prandtl as it was to Taylor or Jeffreys. For me, therefore, taking reasons seri­ously, and assessing their causal role, requires asking why the same facts and the same theorems, that is, the same reasons, caused such divergent responses in the two groups of professionals.

Faced with a situation of this kind, where stated reasons underdetermine the response, should the historian look for a difference in the intellectual presuppositions behind the reasoning of the two groups? Perhaps this ap­proach could uncover hidden premises and lead to an explanation of the dif­ferent reactions. An appeal to presuppositions might keep the analysis within the realm of self-sufficient reasons in the way that the critics want. A search for presuppositions is certainly important in any historical analysis, and it has been a feature of my own procedure. As I have shown, such a search uncovers subtly different conceptions of an ideal fluid. The British experts treated an ideal fluid as a fluid of zero viscosity (p = 0), while their German counterparts treated it as the limit of a fluid of small viscosity (p ^ 0). The British drew a strong boundary between inviscid theory and viscous theory.

German-language work involved a weaker and differently positioned bound­ary. Thus von Mises was inclined to treat the objects of both the Euler and the Stokes equations as abstractions, while Prandtl was inclined to treat them both as realities. Despite their differences, both of these German-language thinkers placed viscous and inviscid fluids on a par with one another. This aligned the two against the more literal-minded realism of the British, who treated viscous fluids as real and inviscid fluids as unreal.

There can be no doubt, then, that identifying presuppositions of this kind deepens the analysis, but it still cannot furnish an explanation of the diver­gent responses of the British and the Germans. Presuppositions are simply reasons for reasons, so the real problem is postponed rather than solved. It merely leads to further questions: Where do the presuppositions come from? Why did the British and Germans have different presuppositions?

The point may be made in another way. I identified a sequence of judg­ments that informed the technical content of the aerodynamic knowledge of lift. At each point in the sequence the British experts jumped in one direction while the German experts jumped in the other. Such was the case regard­ing (1) the significance attached to the “arbitrary” value of the circulation,

(2) the meaning of the zero-drag result, (3) the importance of explaining the critical angle at which a wing stalled, (4) the reaction to the overoptimistic lift predictions derived from the theory of circulation, and (5) the problem of explaining the origin of the circulation around a wing within the confines of the theory of ideal fluids. The divergence of judgment on these questions was systematic, fundamental, and constitutive of the rival understandings of the two groups. It cannot be dismissed as a coincidence, but nor can it be ex­plained by the divergent reasons themselves. The deployment of reason is the problem, not the solution. The phenomenon calls for a causal explanation, and that is what I have given.

So far I have described my procedure in terms of an apparent transi­tion from reasons to causes. I have said, in effect, that my analysis may have started with reasons but it finished by my making an appeal to causes because reasons became equivocal. I justify the claim that the analysis is causal (and conforms to the Strong Program) by saying that an appeal to causes is, in the end, unavoidable. This argument is correct, but as a way of speaking it can generate problems. At best it is a provisional way to state the methodology behind the analysis.

The problem is that two modes of speech and two perspectives are in play: those of the actors and those of the analyst. Keeping both modes of speech in play may suggest that there are two different sorts of cause at work, namely, rational causes and sociological causes. Are we to conclude that reasons cause some of the behavior of the scientists under study but not all of it, so that the remainder has to be explained by sociological and nonrational causes? Does rationality provide a partial cause alongside other kinds of cause furnished, say, by the social context? Some such view may seem to be underwritten by the historian’s own investigative procedure or, at least, by the way the procedure is sometimes presented. First, it seems, reasons are examined and then, and only then, are sociological causes to be invoked (as if they were a residual cat­egory). But granted that the work of the historical analyst sometimes exhibits such a pattern, it would be wrong for the analyst to project this expository sequence into the picture of the historical actor and imagine that actors are, or may be, subject to a corresponding sequence of influences. I do not believe that a dualism of rational and sociological causes, which allegedly compete or alternate with one another, or supplement one another, can be the basis of a satisfactory perspective. It is eclectic and merely encourages baldly posed questions. Something more unified, and hence reductive, is called for. If the Strong Program is correct, then “rational causes,” which have so often been treated as sui generis, are really nothing more than a species of social causes.

Think of the confrontation between Lanchester and Bairstow when they clashed in public in 1915. Bairstow said that Lanchester could not explain why aircraft stalled, so his theory could not be taken seriously; Lanchester said that he did not need to explain this stalling because a theory of narrow scope could still be valuable within its limited domain. My claim is that, in tracing the arguments that Bairstow and Lanchester used against one another, a good historical analyst will, at the same time, be tracing the causal texture of their interaction. There will be no duality of rational and social causes and no tran­sition from one to the other. A properly historical account of the interaction between Lanchester and Bairstow will be in terms of social causation from the outset. A unified, social-causal perspective of this kind can be sustained if the analyst focuses relentlessly on the credibility that the participants and their audience attach to the arguments that are being advanced. Why did the failure to explain the onset of a stall worry Bairstow in a way that it did not worry Lanchester? Why was Bairstow’s concern shared by other British ex­perts but not, to the same degree, by Kutta, Prandtl, and other supporters of the circulatory theory in Germany? These are the questions that will expose the sociological basis of the power of reason, and these are the questions to which I have given answers.

The importance of credibility as a causal category, with its variable and distributed character, is at its most striking when the overall scene is brought into view, for example, the systematic divergence of the German and British responses to circulation. But actual or possible divergences of this kind are not confined to the large scale. They are a feature of every act of reason giving and every act of responding to reasons, whether interpersonal or intraper­sonal, whether public or private. This is because, on its own, invoking and formulating reasons can never be sufficient to render a belief causally intel­ligible or a course of action causally explicable. Things may not look this way from the actor’s point of view. Sometimes the reasons that are advances in the course of an interaction are accepted by other actors as sufficient justifica­tion or explanation. But I am giving the analyst’s perspective. I am speaking here from the point of view of a historian or sociologist who is committed to giving a causal analysis of a passage of interaction and behavior. Of course, critics say the claim that reasons are never sufficient is mere dogmatism or the result of an irresponsible generalization. Like others before them, Sturm and Gigerenzer see here nothing but a lack of prudence on the part of sup­porters of the Strong Program.8 They think it is more judicious to allow that reasons sometimes explain rather than never explain. In fact my claim is not dogmatic; it is made on the basis of a general and principled argument. The argument comes from Wittgenstein’s analysis of rule following and has ex­plicitly informed the Strong Program from the outset.9 Because the argument is so important, and so often misunderstood or ignored, I rehearse it here and then connect it to my overall analysis.

Stream Functions and Streamlines

To apply and solve the Euler equations, mathematicians had to introduce various techniques to relate them to specific flow problems. “As they stand,” said Cowley and Levy, “these equations are not very suitable for solution” (39). They need to be fleshed out. This was done by means of a variety of auxiliary concepts such as source, sink, vortex, and stream function. The gen­eral connotations of the labels “source,” “sink,” and “vortex” will be evident, and their mathematical idealization refines, but does not essentially alter, the everyday meaning of the word. A vortex is like a whirlwind around a central point. A source is a geometrical point at which fluid is created at a certain rate, and a sink is a geometrical point at which it disappears and is destroyed. The words “stream function,” however, do not have any obvious counterpart in common usage. In this section I describe briefly what they mean.

Imagine a coordinate system of x – and y-axes that is to be used to describe a flow of fluid. The value of the stream function at some point P is given by the amount of fluid that flows in unit time across a line drawn from the origin to P. To specify this quantity is to specify the value of the stream function. In hydrodynamics this value is usually designated by the Greek letter psi, y. Altering the position of the origin only alters the value of y by the same con­stant amount at all points in the flow. It follows from the definition that such a function has a simple relation to the velocity components of the flow, and this is the utility of the stream function. If u is the speed along the x-axis at P and v is the speed of flow along the y-axis, it can be shown that

u = -—, and


v ~~dX ■

Given the stream function, a process of differentiation gives the velocity com­ponents. Here is a simple example. The stream function for a uniform flow of speed U along the x-axis is

W = —Uy = —Ur sinft

The first expression is in Cartesian coordinates and the second is in polar co­ordinates, giving the value of у at the point (r, 0). In Cartesian coordinates, differentiating у with respect to x gives the correct answer v = 0, meaning that the flow has zero velocity along the y-axis. Differentiating with respect to y gives the speed u = U along the x-axis. Notice that putting у = c, a constant, gives a straight line parallel with the x-axis. Such a line can be called a stream­line of the flow. Later in the discussion it will become evident that, for all its simplicity, this flow plays a basic role in hydrodynamic reasoning. Logically, it provides the foundation of the edifice.

I have referred to a streamline of this basic flow, but what is a stream­line? In everyday language the words connote speed. Modern aircraft are “streamlined,” whereas aircraft in the period of the old Advisory Committee for Aeronautics, with their struts and protruding engines and undercarriage, were certainly not. This usage, and the idea of low-resistance, streamlined bodies, was already well established in early aerodynamics, even if it could not be realized in the construction of flying machines.17 The technical mean­ing of the term “streamline” in hydrodynamics, though related to this popu­lar meaning, is more specific. A streamline drawn through a point in a fluid flow is a line that conforms to the direction of motion of the fluid element that is located at that point at that moment in time. A moment later the point may be occupied by another fluid element with a different velocity. The pic­ture becomes much clearer if the flow is steady so that the speed and direction of the flow at a given point are constant over time. When the flow is steady, then streamlines will coincide with the path taken by the fluid element. Look­ing at the streamlines will give a picture of what the fluid elements are doing. Streamlines also indicate something about the speed of the flow. For steady incompressible flow they come closer together as the flow speeds up and be­come wider apart as the flow slows down.18

How does the mathematician identify streamlines in order to draw a dia­gram of a flow? The answer is by reference to the stream function. Once in possession of an expression у for the steam function of the flow, the math­ematician generates a series of curves by putting у = c, a constant, and giving

the constant a sequence of values q, c2, c3, etc. The curves are convention­ally plotted at equal intervals. These are the streamlines. As a simple example, recall the stream function for the uniform flow parallel to the x-axis—the basic flow. The formula for the stream function was у = – Uy. Putting у = (say) 0, 1, 2, 3, etc. gives the straight, horizontal lines y = 0, y = -1/U, y = -2/U, y = -3/U, etc. Notice that the greater the speed U, the smaller the gap between the lines. Because, by definition, a fluid element will not cross over a stream­line, then any streamline can be selected and interpreted as a solid boundary without this in any way changing the picture of the flow. (It is sometimes said that the fluid bounded by a streamline can be suddenly “frozen” or “solidi­fied” without altering the rest of the flow.) In the present case the line у = 0 can be selected for this treatment. The flow then becomes (that is, can now be regarded as) the uniform flow of an infinite ideal fluid along a flat, smooth wall located on the x-axis.

Other, more complicated, flows call for other, more complicated, formu­las for the stream function. For example, there are stream functions for the flow around point sources and point sinks and for vortices. The streamlines of sources and sinks radiate away from, or toward, their center point while the streamlines of a simple vortex are concentric circles. By the expedient of adding the stream functions, the flow can be found for combinations of sources, sinks, and vortices. Shortly I shall give the stream function and the streamlines for another, particularly important flow; for the moment, how­ever, the point to retain is that a streamline is specified by setting the stream function equal to a constant у = c.

Intuitive and Holistic Aerodynamics

The practical men did not like “scientific” aerodynamics.50 So what sort of aerodynamics did they like? I begin to answer this question by identifying what might be called their “practical epistemology.” Then I look in more de­tail at the accounts of lift that are to be found in books written for the design­ers of airplanes and in articles that appeared in the Aeroplane, Flight, and Aeronautics.

The epistemology of the practical man was intuitive and qualitative. It was formulated in conscious opposition to the pedantic concern with ac­curacy and irrelevant detail attributed to the despised figure of the mathema­tician.51 Reality must be grasped in all its complexity rather than simplified and broken down into imagined elements. In this sense their epistemology was holistic. It was also artistic. A good designer could rely on his eye, his experience, and his judgment. In a literature review in Aeronautics the editor said: “I don’t deny the infinitely valuable role of pure science, still less that of theory, but science should have some relation to practice, since it is its foster-mother. There is more than one aeroplane designer who knows just enough mathematics to make twice two work out at four, but he will turn out machines equal in performance to the best. We in this country know, as they do in the United States, of eminent designers who see a new type of machine rather than design it.”52

Grey made the point more bluntly with no genuflection in the direction of science: “Never mind what the scientists calculate. Trust the man who guesses, and guesses right.”53 The claim was that some designers have a track record of guessing rightly, and these are the people to trust. We may not be able to see how they do it, but we should not let this put us off. Trust rather than understanding lies at the root of things. This was indeed Grey’s view: there were not only unknown factors involved in the design of aircraft but there were actually unknowable factors, and this was something the “slide – rule scientists” could not grasp.54

The implication was that the reasons behind practical success will remain mysterious. This notion implied a species of intellectual pessimism or even nihilism. Such pessimism was not unusual among practical men and was sometimes echoed by those in the other camp. For example, writing as J. C., a reviewer of G. P. Thomson’s Applied Aerodynamics recommended the book to practical designers (even though it was the product of Farnborough) and said, “One of the first ideas that arises in the reading is the state of ignorance that still exists in aerodynamics; it is safe to say that we know practically noth­ing of the reasons for the experimental results that we find. The amazing thing is that we are able to make aeroplanes as well as we can.”55

At least two of these statements come from spokesmen of the practi­cal men rather than from designers themselves, but they seem to articulate a widely held view. Grey’s characteristic denunciations were repeated in a foreword he wrote in 1917 for the book Aeroplane Design by F. S. Barnwell, who was the chief designer at the British and Colonial Aeroplane Company. This firm, usually known as the Bristol Company, became famous during the Great War for the Bristol fighter, which was designed by Barnwell.56 Much harm had been done, said Grey, “both to the development of aeroplanes and to the good repute of genuine aeroplane designers by people who pose as ‘aeronautical experts’ on the strength of being able to turn out strings of incomprehensible calculations resulting from empirical formulae based on debatable figures acquired from inconclusive experiments carried out by persons of doubtful reliability on instruments of problematic accuracy.”57 If one asks what is left when all the hated calculations, experiments, and instru­ments have been swept away, the answer is intuition. This was Grey speaking, not Barnwell, so we cannot be sure that Barnwell endorsed it. Authors do not necessarily agree with what others say in the forewords of their books, but it is reasonable to expect general agreement.

W. H. Sayers, a strong critic of the National Physical Laboratory, was in­volved with the development of seaplanes during World War I. In an article written after the war, in 1922, called “The Arrest of Aerodynamic Develop­ment,” Sayers described the current conception and form of the airplane.58 It was, he said, “the hybrid product of two utterly different and independent methods of development.” From 1908 to 1914, its evolution was “the result almost entirely of individual adventure.” There were, he insisted, no wind – tunnel results worth mentioning, the mathematics of stability had no appar­ent connection with the facts, and even engineers regarded the airplane as a mechanical curiosity. “Individual designers worked, as artists worked, by a sort of inspiration as to what an aeroplane ought to be like, and built as nearly to their inspiration as the limited means, appliances and increasing knowledge they possessed would allow them” (138). Sayers went on to de­plore the degree of standardization that had set in with regard to design. This, he said, gave a spurious sense of understanding and control. In reality we did not know how to predict what would happen outside the limited range with which we had become familiar. Similarly, the laboratory workers had been in error in concentrating on simple bodies, especially “such simple bodies as might be used as components of the standard type of aeroplane” (138). The result, he said, was a bias toward an additive conception of the different aspects of design and a tendency to overlook large, qualitative effects such as the interference of different components.

Like many other practical men, Sayers was skeptical about model work.59 In his view, aerodynamicists did not yet know what dynamic “similarity” re­ally was, so that inferences from models remained doubtful. Full-scale ex­perimentation was the real basis of knowledge. Grey could be relied upon to give the relatively measured prose of Sayers, his frequent contributor, a more colorful rendering: “I would back any one of a dozen men I know to find out more about streamlines in a month at Brooklands, with the help of a borrowed racing car, a jobbing carpenter, and a spring-balance, than the combined efforts of the National Physical Laboratory, Chalais-Meuden, the Eiffel Tower, the laboratory at Kouchino, and the University of Gottingen have discovered since flying first attracted the attention of that section of hu­manity which the Americans expressively call ‘the high-brow.’”60

This cavalier dismissal of all the major aerodynamic laboratories of Eu­rope dramatizes the anti-intellectual strand in the epistemology. Not all of its expressions were so markedly of this character, but there is no denying a tendency in this direction. Nor can one deny a certain justice in the stance. If scientists have a tendency to simplify the complex and decompose it into its elements, where does this leave the designer who has to reassemble the ele­ments in novel ways? Even if simple principles can be discovered, it can still be unclear how these principles interact when they work together. Design is still a matter of judgment about their combination and compromise in their balance.

Grey’s dismissive attitude toward Gustav Eiffel’s work was not shared by all practical men. The impression created by articles and reviews in the tech­nical journals is that Eiffel was seen as an engineer who could be relied upon to operate in a practical way. If Eiffel’s large, empirical monograph, replete with tables of data, graphs and diagrams of airplanes, is laid side by side with

Greenhill’s mathematical report, there can be no more striking visual proof of the extremes of style that can be represented in aeronautical work. What is more, Eiffel’s work was frequently compared favorably with the experimental work of the NPL. Where the two laboratories diverged, the practical men backed Eiffel.

The reviewer of Eiffel’s La resistance de Fair et Faviation, for the Aero, in March 1911, was enthusiastic: “One is hardly going too far in describing this book as the most authoritative work on the subject that has yet appeared, and it is especially valuable in as much as the experiments have been evolved with an eye specially inclined toward their value in practical aeronautics. . . while experiments of a more purely academic interest have. . . been relegated to the background.” This, the reviewer continued, was strikingly different from the situation that “obtains in more than one experimental laboratory.”61

Writing in July 1916, the editor of Aeronautics invited readers to compare Eiffel, “working almost single handed,” with the National Physical Labora­tory: “It would not be unjust to say that Eiffel attains practical results, ne­glecting a slight margin of error, accounting probably 2 per cent. in extreme cases, which for the time being and for practical purposes is inappreciable. On the other hand, the N. P.L., in its beautiful work, seems rather to strive for the meticulous elimination of this negligible margin of error and passes by the major facts.” Ask Eiffel for the air resistance of, say, an airship hull and the job is done “in a couple of days,” while it would last “heaven knows how many weeks” at the N. P.L.62

The report of the Advisory Committee for 1911-12 noted that, between Eiffel’s laboratory and the NPL, there were differences of some 15 percent between the values of the lift coefficient for certain wings. The probable rea­son, it was said, was observational errors. The ACA resolved to investigate the matter and to ensure that a high degree of accuracy was maintained at Teddington. The “Editorial View” in the Aero was that to the “lay mind” such differences are “disquieting,” and the writer of the editorial chose to read the ACA’s response “almost as an acknowledgement of error on the part of Teddington.”63