Category The Enigma of. the Aerofoil

Southwell entered Trinity in 1907 to read mechanical sciences. He was an engineer, but an engineer with impressive mathematical skills.38 In 1909 he was placed in the first class of part I of the Mathematical Tripos and in 1910 graduated with first-class honors in the Mechanical Sciences Tripos. He was coached by Pye and Webb, two of the best mathematical coaches of the time. On graduation he began research on elasticity theory and the strength of ma­terials and in 1912 became a fellow of Trinity. In 1914 Trinity offered Southwell the post of college lecturer in mathematics but he did not take up the offer because of the outbreak of war. He volunteered for the army and was sent to France. In 1915, however, he was brought back to work on airships for the

navy. In 1918 he was transferred to the newly created Royal Air Force, with the rank of major, and was sent to Farnborough in charge of the aerodynamic and structural department. After demobilization, and a brief return to Trin­ity, in 1920 he went to the National Physical Laboratory as superintendent of the Aerodynamics Department. He stayed at the NPL for five years and then returned again to Cambridge, where (unusual for an engineer) he was a fac­ulty lecturer in mathematics.

It was in the field of applied mathematics, rather than practical engineer­ing, that Southwell made his outstanding contribution. He developed novel mathematical techniques for the analysis of complex structures of the kind used in the building of airships. The technique was called “the relaxation of constraints” and depended on replacing the derivatives in the equations and boundary conditions by finite differences.39 Though the technique was initially developed to deal with engineering problems, Southwell later dem­onstrated its power as a general method of solving differential equations. Referring to the unavoidable complexities of practice, and the uncertainties

in data of whatever kind, he called his own Relaxation Method “an attempt to construct a ‘mathematics with a fringe.’”40 He was not only interested in elasticity and the strength of materials but also worked on viscous flow. Like Bairstow, Southwell started from Oseen’s approximation to the full equations of viscous flow and the developments provided by Lamb.41 In 1929 Southwell was offered the chair in engineering at Oxford, which he accepted after some hesitation but where he stayed until his retirement. Southwell had a lively sense of the different demands confronted by engineers and mathematical physicists, but it may be revealing that Glazebrook said of him that, although he was an Oxford professor, he was still a Cambridge man.42

As superintendent of the Aerodynamics Department at the NPL, South­well played a prominent role in the discussions that took place in the Aero­nautical Research Committee after the war when plans for future work were thrashed out. Southwell always placed great emphasis on fundamental scien­tific research. It was the long term, not the short term, that counted. Though an engineer by training, he defended the value of academic research of the kind so often attacked by the practical men. This came out clearly in the pol­icy discussions that took place in February 1921, devoted to the topic of “The Aeroplane of 1930.” The participants were invited to anticipate the character and needs of aviation in ten years’ time. Southwell wittily subverted the discussion by posing the question If we could know where we would be in ten years’ time, why wait? His point was that fundamental advances could not be predicted. He suspected that, whatever we said, we would be wrong.43 The most we can do is to be conscious of the gaps in existing knowledge and try to fill them. Consider, he said to the committee, the fundamental cause of the lift and drag on an aircraft wing: “We have much empirical data in regard to aerofoils, but our ignorance of the mechanisms by which their lift and drag are obtained has hitherto been almost absolute.” Here was a worthy focus for research: the true mechanisms of lift and drag must be identified.44

One might assume that Bryant and Williams’ experiments, as well as those of Fage and Simmons, were performed to identify the mechanisms that Southwell had in mind. But if this were so, we would expect that the results of the work (give or take Taylor’s reservations) would have been seen by South­well as furnishing the desired account of lift and drag. This was not how he saw them. The same sense of ignorance about fundamental causes still per­vaded Southwell’s thinking after this experimental work had been completed and after Glauert had begun to provide his superbly clear exposition and de­velopment of the circulation theory. The same pessimism that was expressed privately in committee in 1921 was expressed again, and publicly, some four years later in two lectures that Southwell gave in 1925. One of these lectures, on January 22, was to the Royal Aeronautical Society; the other, on August 28, was to the British Association meeting in Southampton.

The lecture to the RAeS was titled “Some Recent Work of the Aerodynam­ics Department” and was meant as a summary of the achievements of the department during the years of Southwell’s superintendence.45 His return to Trinity was an opportunity to take stock. Southwell began by welcoming the change from ad hoc wartime experimentation to programs of research guided by theory. Two main lines of theoretical concern were identified. First, there was the classical theory of stability, and Southwell described in detail the re­cent work of Relf and others. This had taken the experimental determination of the damping coefficients for roll, yaw, and pitch to new levels of sophistica­tion. The second set of theoretical concerns dealt with the fundamentals of fluid flow. For aerodynamics, said Southwell,

I suppose no problem is so fundamental as the question—why does an aero­foil lift? We can hardly rest satisfied with the present position—which is, that we have next to no idea. To answer the question completely would involve no less than the solution of the general equations of motion for a viscous fluid, and attacks on these equations have been made from all angles. Considering the energy expended, the results have been very small; but then, these are about the most intractable equations in the whole of mathematical physics. (154)

Southwell mentioned the role played in this (so far fruitless) endeavor by Bairstow, Cowley, and Levy and then moved on to the approach adopted by Prandtl, namely, using the inviscid theory of the “hydrodynamic textbooks” informally conjoined with the idea of a viscous boundary layer. In this way the “once discounted” classical theory of the perfect fluid had been “rein­stated” and could provide a close approximation to the truth when used “un­der proper control, and aided by assumptions based on physical intuition” (156). At the NPL, said Southwell, every opportunity had been taken to check the validity of Prandtl’s theory, and “in the main one must say, I think, that it has passed the ordeal with flying colours” (156). The most important tests “are those which Messrs. Fage, Bryant, Simmons and Williams have made” (156). Southwell explained that at the time of his lecture this work had not yet been published but it had confirmed the most important result, namely, “the theoretical relation between lift-coefficient and the circulation” (156).

At this point Southwell’s audience might have been puzzled. They were being told that Prandtl’s theory had passed the tests to which it had been subject with “flying colours,” and yet a moment before, Southwell had de­clared that experts had “next to no idea” how a wing produced lift. Didn’t these claims contradict one another? The answer is that Southwell’s argument was consistent but depended on a suppressed premise. For Southwell, the experiments of Fage and Simmons only justified the use of inviscid theory as a way of representing the real flow. They did not show that it truly described the flow. As far as Southwell was concerned, Fage and Simmons were not tracing footprints in the snow. In their experiments the imprint of reality had not been made in some familiar and reliable medium. Their analysis had used ideal fluid theory. The nature of the beast that left the footprints was still under discussion. The inviscid approach left it an open question whether the “actual flow” corresponded to the representation, and the most plausible answer was that it did not. The no-slip condition was violated by the inviscid representation, and Prandtl had assumed that the flow was steady. The eddies in the wake were neglected. The place to look to resolve these issues, Southwell concluded, was the boundary layer. It was this aspect of Prandtl’s work that really engaged Southwell. As he put it, “the conditions in this layer are the ultimate mystery of aerodynamics: somehow or other, in a film of air whose thickness is measured in thousandths of an inch, that circulation is generated which we have just seen to be the essential ingredient of ‘lift’” (158). Research should concentrate on the boundary layer. Theoreti­cally this required a deeper understanding of the equations of viscous flow; experimentally it called for the development of special instruments such as microscopic Pitot tubes to probe the boundary layer. Southwell mentioned that Muriel Glauert was working mathematically and experimentally on the calibration of such an instrument.46

Here was the explanation of Southwell’s apparently conflicting claims. Prandtl’s theory of the finite wing “worked,” but it could not be true because the mathematical analysis depended on false boundary conditions. This was the suppressed premise, which rendered the argument consistent. Although Prandtl’s wing theory could pass many tests, and even pass them with flying colors, it could not, by its very nature, answer the question that Southwell wanted to answer. In a very British way, he wanted to know how a viscous fluid generates lift. In the discussion after the lecture, in response to Major Low, Southwell said: “The really interesting part of Prandtl’s work was the work he had been doing subsequently in his study of the ‘boundary layer,’ because that work might ultimately explain why the assumptions which could not be correct could make such amazingly true predictions” (166).

In a lecture titled “Aeronautical Problems of the Past and of the Future,” delivered later in the same year, Southwell insisted that the aim of research was “not so much to achieve, as to understand.”41 Scientists should not be content with “achievement,” “unless it be the result of understanding’—something of which the “practical man” would never be persuaded (410). Understanding meant understanding based on a sound theory. Southwell identified three triumphs of British aeronautics that, in his opinion, met this condition. They were (1) the ability to build stable aircraft, (2) the analysis of the dangerous maneuver of spinning and its avoidance, and (3) the achievement of control in low-speed flight even after the aircraft had stalled. In all three cases, he argued, the end result had enormous practical value but the driving force had been the aim to understand. And it was mathematical analysis that had furnished the understanding.

The theory of lift was conspicuous by its absence from this list of tri­umphs. For Southwell, Prandtl’s wing theory was an achievement that was not yet informed by an adequate theoretical understanding. Bryant, Wil­liams, Fage, and Simmons were mentioned by name, and Southwell used diagrams taken from their papers. The role that he accorded the work, how­ever, was that of showing that the effects of viscosity can be ignored as far as the sliding of air on air is concerned but cannot be ignored very close to the surface of a wing or in the wake behind the wing. It is what happens in these regions that constitutes “the ultimate problem of hydrodynamics” (417). It was this “ultimate” problem that Southwell had in mind when he asked: Why does a wing generate lift? He was not denying the role of circulation, nor was he belittling the insights of Lanchester, Prandtl, or Glauert as they continued to develop the inviscid theory of lift. His point was that no one, following this route, could hope to explain the origin of circulation.48 Within inviscid theory, circulation had to be a postulate not a deduction.

Southwell’s skeptical position was endorsed by H. E. Wimperis, the quiet but influential director of scientific research at the Air Ministry.49 Wimperis had trained as an engineer in London and Cambridge and had sat the Me­chanical Sciences Tripos in 1890. During the Great War he had served as a scientist with the Royal Naval Air Service and had designed a bomb sight that carried his name. After the war he worked at Imperial College in a labora­tory financed by the Air Ministry. Along with Tizard, he was later to play an important role in the development of Britain’s radar defense system. In 1926 Wimperis, in his role as director of research, published a survey article in the Journal of the Royal Aeronautical Society called “The Relationship of Physics to Aeronautical Research.”50 One of Wimperis’ aims was to send the message that the Air Ministry and government were aware of the need for fundamental research. What, he asked, was engineering but applied physics? Government scientists at the National Physical Laboratory and Farnborough must have the freedom to pursue basic, physical problems. A second aim was to argue that this policy had already produced significant results. Here Wimperis cited, among other examples, the mathematical work that had been done on fluid flow and, in particular, the flow around a wing. It rapidly became clear, how­ever, that in Wimperis’ view, the approach based on inviscid theory was not an exercise in real physics but a mere preliminary to a genuine understanding of lift. On a classical hydrodynamic approach, he noted, the circulation must be added in an arbitrary way to the flow, and this only provides an “analogy with the lift force experienced by an aerofoil” (670). Admittedly there have been some successful predictions made “by the employment of this conven­tion” (670), but the theory becomes “somewhat far-fetched” in its account of what is happening on the surface of the wing. “Circulation,” said Wimperis, “must have a physical existence since velocity is greater above the wing than below; though this real circulation is a circulation with no slip, whereas the mathematical circulation has slip. Hence the rather amusing situation arises of adding to the mathematical study of streamlines a conventional motion which could not really arise in an inviscid fluid!” (670). Southwell was right, said Wimperis, in insisting that the real problem lay in discovering what was actually happening in the very thin, viscous layer close to the wing. This was a problem in physics rather than something that could be evaded by the use of mathematical conventions and unreal boundary conditions.

The continuity between Frank’s ideas, developed in the 1930s, and the more recent work in the sociology of scientific knowledge was noted by the phi­losopher Thomas Uebel in his paper “Logical Empiricism and the Sociology of Knowledge: The Case of Neurath and Frank.”71 Uebel concluded (I think rightly) that Frank had anticipated all the methodological tenets of the Strong Program (147), but he insists that there is an important difference: the advo­cates of the Strong Program are “relativists,” whereas Frank “did not accept the relativism for which the Strong Programme is famous” (149). This state­ment is incorrect. The similarity does not break down at this point. Frank was also a relativist. I first want to establish this fact and then I shall use Frank’s relativism to illuminate some examples of aerodynamic knowledge.

Frank’s relativism was implicit, but clearly present, in his paper on the acceptance of theories, for example, in his assertion that there was no such thing as “perfect” simplicity. He meant that there is no absolute measure of simplicity that could exist in isolation from the circumstances and perspec­tives of the persons constructing and using the theory. If there is no absolute measure, then all measures must be relative, that is, relative to the contingen­cies and interests that structure the situation. Recall also the trade-off be­tween simplicity and predictive power. Frank said this meant there was no such thing as “the truth” because there was no absolute, final, or perfect com­promise. The relativist stance is epitomized by Frank’s comparison between assessing a theory and assessing an airplane. Talk about an “absolute aircraft” would be nonsense. All the virtues of an aircraft are relative to the aims and circumstances of the user. If the process of scientific thinking has an instru­mental character, and theories are technologies of thought, then talk about an absolute theory, or the absolute truth of a theory, is no less nonsensical.

Frank made his relativism explicit in a book called Relativity: A Richer Truth.72 Einstein wrote the introduction, and the book contains a number of examples drawn from Einstein’s work, but the book is not primarily about relativity theory. It is a discussion of the general status of scientific knowl­edge and its relation to broader cultural concerns. Frank’s purpose is much clearer in the title of the German edition, Wahrheit—Relativ oder absolut? (Truth—relative or absolute?),73 which poses the central question of the book. Does science have any place within it for absolutist claims? Frank said no. No theory, no formula, no observation report is final, perfect, beyond revision or fully understood. The world will always be too complicated to permit any knowledge claim to be treated as absolutely definitive. In devel­oping this argument Frank draws out the similarities between relativism in the theory of knowledge and relativism in the theory of ethics. Are there any moral principles that must be understood as having an absolute character? The claim is often made, but Frank argues that if close attention is paid to the actual employment of a moral principle, it always transpires that qualifica­tions and complications enter into their use. “For this so-called doctrine of the ‘relativity of truth’ is nothing more and nothing less than the admission that a complex state of affairs cannot be described in an oversimplified lan­guage. This plain fact cannot be denied by any creed. It cannot be altered or weakened by any plea or admonition on behalf of ‘absolute truth.’ The most ardent advocates of ‘absolute truth’ avail themselves of the doctrine of the ‘relativists’ whenever they have to face a real human issue” (52).

The book on relativism was written during the 1940s after Frank had left Prague. It was a response to a systematic attack on science by theological writ­ers in the United States. They blamed science for the ills of the time, such as the rise of fascism, the threat of communism, the decline in religious belief, and the loss of traditional values. The critics said that science encouraged relativism and relativism was inimical to responsible thinking. Frank con­fronted the attack head on. He did not seek to evade the charge by arguing that scientists were not relativists (and therefore not guilty); indeed, he said that scientists were relativists (and should be proud of the fact). The danger to rational thought and moral conduct came, he said, not from relativism but from absolutism. If we try to defend either science or society by making absolutist claims, we will merely find ourselves confronted by rival creeds making rival, absolute claims. If we take the issue outside the realm of reason, we must not be surprised if it is settled by the forces of unreason (21). Relativ­ism, he argued, is the only effective weapon against totalitarianism and has long been instrumental in the progress of knowledge. It has been made “a scapegoat for the failures in the fight for democratic values” (20).

Frank alluded to the many caustic things that critics said about relativism and then added, “this crusade has remained mostly on the surface of scientific discourse. In the depths, where the real battle for the progress of knowledge has been fought, this battle has proceeded under the very guidance of the doctrine of the ‘relativity of truth.’ The battle has not been influenced by the claim of an ‘absolute truth,’ since the legitimate place of this term in scientific discourse has yet to be found” (20-21). Notice that Frank placed the words “absolute truth” in quotation marks because, as a positivist, he would have been inclined to dismiss the words as meaningless. For him they had no real content and no real place in meaningful discourse. The claims of the absolut­ists were to be seen as similar to the claims of, say, the theologian. But if the best definition of relativism is simply the denial that there are any absolute truths, and if relativism is essentially the negation of absolutism, then relativ­ism is meaningless as well. The negation of a meaningless pseudoproposi­tion is also a meaningless pseudoproposition. Relativism would, likewise, be revealed as an attempt to say what cannot be said. This may explain why Frank also placed the words “relativity of truth” in quotation marks. There is much to recommend this analysis. It might be called the Tractatus view of relativism.74 Where, however, does this analysis leave Frank’s book? Does it not render the book meaningless and pointless? The answer is no. The reason is that absolutism, like theology, has practical consequences, and whatever the status of its propositions, the language is woven into the fabric of life. It provides an idiom in which things are done or not done. Even for the strictest positivist this penumbra of practical action has significance.

What is done, or not done, in the name of absolutism? The answer that Frank gave is clear. Absolutism inhibits the honest examination of the real practices of life and science. It is inimical to clear thinking about the human condition. The meaningful task of the relativist is grounded in this sphere. It is to be expressed by combating obscurantism and fantasy and by replacing them with opinions informed by empirical investigation. That is the “richer truth” referred to in the title of the English-language edition of Frank’s book. This down-to-earth orientation also provides the answer to another prob­lem that may appear to beset Frank’s relativist position. What is scientific knowledge supposed to be relative to? The answer is that it is relative to what­ever causes determine it. There are as many “relativities” as there are causes. That is the point: knowledge is part of the causal nexus, not something that transcends it. Knowledge is not a supernatural phenomenon, as it would have to be if it were to earn the title of “absolute.” Knowledge is a natural phenomenon and must be studied as such by historians, sociologists, and psychologists.

Frank’s relativism, and the relativist thrust of the positivist tradition, seems to have been forgotten.75 A number of prominent philosophers paid a moving tribute to Frank after his death in 1966, but they did not mention his relativism.76 In the course of this forgetting, a strange transformation has taken place. In his Kleines Lehrbuch des Positivismus, von Mises spoke of “the phantom of absolute cognition.”77 That phantom still stalks the intellectual landscape, but in Frank’s day it was scientists who were accused of relativism, whereas today it is scientists, or a vocal minority of scientists, who accuse others of relativism. From being the natural home of relativism, science has been polemically transformed into the abode of antirelativism and hence of absolutism. A significant role in this transformation has been played by phi­losophers of science who are today overwhelmingly, and often aggressively, antirelativist in their stance. The involvement of analytic philosophers should have ensured that the arguments for and against relativism were studied with clarity and precision. This has not happened. The philosophical discussion of relativism is markedly less precise today than when Frank addressed it fifty years ago and provided his simple and cogent formulation of what was at stake.

Before we look at the early Reports and Memoranda generated by this re­search program, something should be said about the public and professional reception that was given to Haldane’s new committee. If there was a cautious welcome in some quarters, elsewhere bitter disappointment was expressed that the commercial manufacturers of aircraft and the pioneers of flight (who were usually the same men) were not represented. To these critics the ACA was just a committee of professors, not of producers. Even the inclusion of Lanchester did not satisfy the critics. He had written books on airplanes, but these were dismissed as theoretical works. He had not built airplanes, only motorcars (and some of the critics didn’t even like his cars). Where were the names of Britain’s aeronautical pioneers, such as Handley Page, Fairey, Roe, Rolls, Short, or Grahame-White?48

To prepare the ground for the prime minister’s announcement of the Ad­visory Committee, Haldane had written on May 4, 1909, to the newspaper magnate Lord Northcliffe, who had been agitating for government action. “We have,” Haldane said, “at last elaborated our plans for the foundation of a system of Aerial Navigation.” The government had created “a real scientific Department of State” for its study. In his reply of May 9, Northcliffe was dismissive. He gracelessly declared that the composition of the committee “is one of the most lamentable things I have read in connection with our national organisation.” He conceded that Rayleigh was a good choice as chairman, but “the Committee should certainly include the names of some of the now numerous English practical exponents.” As for Lanchester, he was known to be critical of the Wright machine,49 about which Northcliffe was enthu­siastic, and was “the same Mr Lanchester, I understand, who is responsible for. . . one of the most complicated motor cars we have ever had.” This was Northcliffe writing to a member of the Cabinet; when corresponding with his political cronies he simply referred to the Advisory Committee as Haldane’s “collection of primeval men.”50

In reply to Northcliffe (on May 18) Haldane said that the advice he had been given had convinced him “of what I was very ready to be convinced, that here as in other things we English are far behind in scientific knowledge. The men you mention are not scientific men nor are they competent to work out great principles: they are very able constructors and men of business. But in this big affair much more than that is needed.”51 Predictably, this response failed to mollify Northcliffe and his friends, such as J. L. Garvin of the Ob­server (“too many theorists”; May 7). Nor were they alone in their negative response. The Tory Morning Post of May 7 had declared that “too much value has been attached to the purely theoretical side, while no evidence is forth­coming that the practical side will be advanced at all.” In an interview for the Post, the aviation and motoring enthusiast Lord Montague said that “the Commission is composed of theoretical and official people as distinct from practical men. . . . I do not recognize the name of any man on it of actual practical experience.”52 The journal Flight joined in and got its revenge for its failed prediction over the Wright brothers’ contract: “It is a bad system to encumber enterprise by establishing ‘Boards of Opinion.’ The opinions of the practical men who are doing the work are worth more to the nation than those of a miscellaneous collection of scientists.”53

A more positive response to the committee was to be found in a short article in the pages of Nature on May 13, 1909.54 It was by the brilliant and opinionated mathematician George Hartley Bryan, himself a wrangler and a former fellow of Peterhouse College.55 Bryan welcomed the creation of the Advisory Committee, saying: “It is clear. . . that mathematical and physi­cal investigations are to receive a large share of attention, and that the mere building of aeroplanes and experience in manipulating them are not to in­terfere with the less enticing and no less important work of finding out the fundamental principles underlying their construction” (313). The problem of stability, he noted with satisfaction, had been singled out for attention, though the “mathematics of this problem are pretty complicated” (313). Bryan was not surprised that newspapers were complaining that the committee was too theoretical in its orientation and that the “practical man” was not properly represented. The real problem, said Bryan, was not too much theory but too many publications that contained equations and algebraic symbols written by people who did not understand mathematics. “Indeed, in many cases it is the ‘practical man’ who revels in the excessive use and abuse of formulae, and the mathematician and physicist who would like to bring themselves in touch with practical problems are consequently deterred from reading such litera­ture” (314). There was an urgent need, Bryan concluded, “for a clear division of labour between the practical man and the physicist” (314). The failure to create such a division, he argued, had already cost England the loss of its chemical and optical industries, and France had a long head start in automo­biles. Now at last there was a chance to make up the ground in aeronautics.

Bryan’s reaction was just the kind that Haldane would have been hoping for, though the reference to the “mere building of aeroplanes” was hardly politic. These two initial responses—that of Northcliffe and his allies and that of Bryan—indicate the tension surrounding the ACA. They also serve to introduce some of the labels that were used at the time to signalize the differ­ent and opposed parties. The term “practical man” does not refer to a cloth – capped artisan but primarily to engineers and entrepreneurs, and included, for example, the Hon. Charles Stewart Rolls. The label was a badge of honor intended to mark the contrast with university academics, civil servants, and others with no direct involvement in market processes.56 I follow out some of the further expressions and consequences of this social divide.

Consider the idealized model of the postcard experiment, that is, the stream­lines around a flat plate normal to the flow. The flow could take the form shown in figure 2.7 as well as that already shown in figure 2.5. Instead of curl­ing around the edges of the plate and moving down the back of the plate, the flow of ideal fluid can break away at the edges. Behind the plate the flow is not a mirror image of the flow in front of it but consists of a body of “dead air,” or dead fluid, bounded by the moving fluid which has met the plate, moved along the front face of the plate, and separated at the edges. The pressure in the “dead air” will be the same as that at a great distance from the plate and can be equated with the atmospheric pressure. In a real, viscous fluid, the moving fluid and the dead or stationary fluid would interact. There would be a transition layer, with a speed gradient created by the stationary fluid retarding the moving fluid while the moving fluid sought to drag the station­ary fluid along with it. In an ideal fluid there will be no such transition layer because there will be no traction between the two bodies of fluid. The free

stream will pass smoothly over the dead fluid so there will be a sudden transi­tion from fluid with zero velocity to fluid with a nonzero velocity. Mathema­ticians call this sudden transition a discontinuity in the velocity because there are no intermediate values. This term gives rise to the general label for flows of this kind, which are called discontinuous flows. The streamline that marks the mathematically sharp discontinuity between the moving and stationary bodies of fluid is called a free streamline. It is a line of intense vorticity along which the flow possesses rotation in the technical sense defined earlier in the chapter.

This attempt to make mathematical hydrodynamics more realistic was introduced by Helmholtz in 1868 in a paper titled “Uber discontinuirliche Flussigkeits-Bewegungen” (On discontinuous fluid motions).40 Helmholtz argued that all the flows that had produced d’Alembert’s paradox had de­pended not only on the assumption that the flow was inviscid but also on the assumption that the velocity distribution was continuous. Helmholtz explored flows involving surfaces of separation (Trennungsflache) or (what is mathematically equivalent) sheets of vorticity (Wirbelflache).41 Of course, said Helmholtz,

Die Existenz solcher Wirbelfaden ist fur eine ideale nicht reibende Flussigkeit eine mathematische Fiction, welche die Integration erleichtert. (220-21)

The existence of such a vortex sheet for an ideal inviscid fluid is a mathemati­cal fiction to make the integration [of the equations] easier.

But fiction or no fiction, Helmholtz had raised the hope that the glaringly false consequences of the standard picture of ideal-fluid flow could be avoided.

If a steady, discontinuous flow is to be possible, certain conditions must be satisfied. It must be the case that the static pressure on either side of the free streamline is the same, otherwise the flow pattern would not be in equi­librium and would modify itself. Since the flow at a great distance in front of the plate is assumed to have a constant speed V and to be at atmospheric pressure pa, while the dead air is also at atmospheric pressure, then the speed of the flow along the free streamlines that bound the dead air must also be V. This conclusion follows from Bernoulli’s equation relating speed and pres­sure. Bernoulli’s law also leads to the conclusion that a flow of this kind will generate a greater pressure on the front of the plate than on the back.

Consider the streamline that terminates at the stagnation point at the front of the plate. What is the pressure on the front of the plate at the stagna­tion point? Call the pressure p. Everywhere along the streamline that goes to the stagnation point, the static and dynamic pressure will sum to the same constant value, that of the Bernoulli constant or the total pressure head. The value of the constant, or the total pressure, at a distance from the plate is H = pa+ Уг p V2. On the plate, at the stagnation point, the speed is zero. There will be no dynamic pressure but only a static pressure that will equal the total pressure, therefore ps = H = pa + 4 p V2. The pressure produced by bringing the air to a standstill at the stagnation point thus exceeds the atmospheric pressure pa by the quantity Уі p V2. But the pressure on the back of the plate is also pa, so at this point there is an excess pressure on the front of the plate.

This argument only applies to the stagnation point, which is the point of maximum pressure on the front of the plate. What happens at other points on the front of the plate as the fluid moves away from the stagnation point and moves toward the edges? The fluid will speed up so its pressure will drop. But the pressure exerted by the moving fluid only drops to atmospheric pres­sure as it reaches the free stream velocity at the edges. It follows that, at all points on the front of the plate, there will be a higher pressure than the at­mospheric pressure on the rear of the plate. On this account, therefore, the forces on the plate do not cancel out, except at the very edges, and there is an overall resultant aerodynamic force on the plate.42

Discontinuous flows of this kind thus avoid the paradoxical-seeming zero-resultant outcome found by D’Alembert, but it is still necessary to ask whether the predicted forces are the right size. It is one thing to avoid a bla­tantly false outcome and another thing to do so by giving the right answer in quantitative terms. The question still remains: Do the forces predicted on the basis of discontinuous flow fully correspond to the observed forces? Quantitative knowledge of the forces on the plate calls for a quantitative knowledge of the speed and pressure of the flow along the front of the plate, not just at the stagnation point and the edges. Until this information could be provided, the picture was merely qualitative. Working independently of one another, Rayleigh and Kirchhoff provided testable answers.43

The quantitative analysis of discontinuous flows was not an easy task, but by the use of ingenious transformations, it proved possible to connect the discontinuous flow around a flat plate to the simple, uniform, horizon­tal flow. There was no guaranteed way to find the required steps leading to the simple flow. It called for a high order of puzzle-solving ingenuity. The character of the thinking required can be glimpsed from the first few steps of the process. Rayleigh and Kirchhoff noticed that in the original flow, the direction of the boundary streamline along the plate was known but not the velocity. For the free streamline, the reverse held: the velocity was known but not the direction. If the flow could be redrawn on a diagram where one axis was proportional to speed while the other axis was proportional to direction, then both parts of the streamline would be transformed into straight lines. This was a step toward the desired simplicity because the straight lines could be interpreted as “polygons” of the kind to which the Schwarz-Christoffel theorem could be applied. Neither Kirchhoff nor Rayleigh explicitly used the Schwarz-Christoffel theorem but used a number of ad hoc transformations to achieve the same goal.44 But once the formula describing the flow had been found, pressures and velocities could be calculated and quantitative predic­tions made.

Rayleigh’s achievement was to generalize Kirchhoff’s analysis, which dealt with plates that were normal to the flow, and give the analysis required for plates that were oblique to the flow. This classic result in hydrodynamics was published in 1878 and provided the starting point for the work of the Advi­sory Committee for Aeronautics when its members tried to explain the lift generated by an aircraft wing. The work was officially overseen by Rayleigh himself as president of the ACA. It was monitored on a day-to-day basis by Glazebrook and other mathematical physicists who were closely associated with Rayleigh. In the next chapter I describe this early British work on the lift and drag of a wing, which was based on the idea of discontinuous flow.

Frederick William Lanchester (fig. 4.7) was born in 1868, the son of an archi­tect.10 He was educated at the Royal College of Science and Finsbury Techni­cal College. He began work in 1889 with a company making gas engines, and in 1895 he began to develop his own motorcar. Until 1919 he was managing director and then the consultant engineer of the Lanchester Motor Company Ltd. He was responsible for some important patents for devices that suc­cessfully reduced the vibration that plagued early engines. During this time Lanchester had also been working on the problems of artificial flight and experimenting with model gliders. In 1907 he assembled the ideas about lift that he had been developing since the mid-1890s and published them in the form of a bulky volume called Aerodynamics, the first of a two-volume Trea­tise on Aerial Flight.11 This work is now recognized as the locus classicus of the circulation theory of lift, though it does not read like a modern textbook on aerodynamics. The circulation theory is only one strand in the argument that had evolved over some dozen years and had been changed to bring it more into line with the concepts used in, for example, Lamb’s Hydrodynamics.12 The precise character of the changes and the form in which the theory was

first conceived are not known, though they may, to some extent, be guessed from the variations in the uneven text.

Aerodynamics shows the traces of at least five interwoven lines of argu­ment: (1) an evolutionary perspective, (2) the concept of a wing being carried on a wavelike airflow, (3) a quasi-Newtonian idea of the “sweep” of a wing, (4) examples of the theory of discontinuous flow, and (5) versions of the theory of circulation or the vortex theory. Although Lanchester devoted all of chapter 3 of his book to an exposition of basic hydrodynamic ideas, the assimilation was incomplete. He did not avail himself of the mathematical formula expressing the circulation as an integral, though he did accept the ideas behind it. Furthermore, his use of the word “circulation” was not con­sistent. It was often used informally to refer to fluid that was displaced by a body and pushed from the front to the rear.13 Tracking the word “circula­tion” in Lanchester’s text does not necessarily reveal those places where the circulation theory of lift was being developed. Terminologically, Lanchester preferred to speak of “cyclic flow.”

The theoretical centerpiece of the 1907 book was chapter 4, called “Wing Form and Motion in the Periptery.” The word “periptery” was coined by Lanchester to refer to the characteristic form of airflow in the vicinity of a lift­ing surface. The chapter began with an evolutionary argument and a criticism of an existing theory of lift. In order to perform its biological function, argued Lanchester, the wing of a bird must have evolved into a shape that conforms to the pattern of airflow necessary to provide lift. It should therefore be pos­sible to read off this pattern of flow from the shape of the wing. All such natu­rally occurring wings show a similar “design and construction” that involves an arched profile and a slight downward inclination of the front edge. From this Lanchester inferred that the air must be moving upward as it approaches the leading edge of the wing and downward as it leaves the trailing edge.

The advantages of the dipping front edge was first recognized by Horatio Phillips, who made it the subject of two patents in 1884 and 1891, but, said Lanchester, Phillips gave an incorrect account of it. According to Phillips the air impinged on the sloping, upper surface of the leading edge and was deflected upward, off the surface of the wing, leaving a partial vacuum on the upper surface. Lanchester rejected this in favor of an explanation based on principles drawn from Newton’s mechanics. The central point, he said, was the exchange of momentum. The air, which was rising at the front of the wing, had to have the vertical component of its motion reduced to zero. The air then had to be given a downward direction, and thus supplied with another vertical component of motion, but this time in the opposite direc­tion. It was important, said Lanchester, that during this process the flow of air remained conformable to the shape of the wing and that no surfaces of discontinuity were created.

These ideas were developed by means of a thought experiment involv­ing the fall of a flat plate. The plate was to fall so that it presented its full surface-area to the direction of its descent. During the fall the air would be pushed around the edges from the lower to the upper surface. “There is at first a circulation of air round the edge of the plane from the under to the upper surface, forming a kind of vortex fringe” (145). (Notice that here the word “circulation” refers to the air being literally displaced from the front to the back of the object.) Lanchester then supposed the falling plate was given a horizontal velocity. This, he said, made the case equivalent to an inclined plane moving horizontally. If, following Newton, air is treated as if it is made up of independent particles, the analysis gives the wrong answers. Lanchester concluded that it was necessary to take account of the continuous nature of the fluid medium but knew of no way to do this for the flow under consid­eration. This led him to introduce a (more or less) arbitrary “sweep” factor to define the amount of air that was involved in the momentum exchange. It was Lanchester’s references to “sweep,” rather than circulation, that were picked up by the practical men.

Lanchester then explored a number of different approaches. The reader was told that “the peripteroid system may be regarded as a fixed wave” (156), though this idea was nowhere adequately explained. It seems to have been part of an early version of the theory. A few pages later Lanchester explained that because the disturbances in the neighborhood of the aerofoil possess an­gular momentum, it can be inferred that the flow comprises a cyclic motion. Lanchester went on: “The problem, then, from the hydrodynamic stand­point, resolves itself into the study of cyclic motion superposed on a transla­tion” (162). He then used the formulas of mathematical hydrodynamics to plot the streamlines and potential lines for the flow around a circular cylinder on the assumption that the flow contained a circulation or cyclic component. Graphical methods were used to establish that the imbalance of pressures furnished a lift force. The flow depicted in the plot did not look like a picture of the flow round a wing, because there was a circular cylinder, rather than a winglike profile, at the center of the action, but, said Lanchester, we may look upon this figure “as representing in section a theoretical wing-form, or aerofoil, appropriate to an inviscid fluid” (163). He justified this statement by observing that from “the hydrodynamic standpoint,” that is, with a perfect fluid, the shape of the aerofoil section is irrelevant.

Lanchester then moved from perfect fluids to real fluids and from the in­finite wing to the wing of finite length where the behavior of the air at the tips had to be considered. Finite wings could be understood by supposing that the cyclic flow extends beyond the wingtips in the form of two vortices issuing from the ends of the wing and trailing behind it. The trailing vortices could be assumed to extend back to the point on the ground from which the aircraft took off. Such a picture meets the requirement, first identified by Helmholtz, that a vortex can only end on a surface of the fluid. Lanchester acknowledged that, because of Kelvin’s theorem, the creation of such vortices in a perfect fluid presented a problem. He argued that viscosity had to be invoked to start the process, but inviscid theory could be used for the subsequent description. He also noted that the two trailing vortices would interact with one another. As Lanchester put it, “We have seen. . . that the lateral terminations of the aerofoil give rise to vortex cylinders. . . . Such a supposition presents no dif­ficulties in a viscous fluid. . . . Now we know that two parallel vortices, such as we have here, possessed of opposite rotation. . . will precess downwards as fast as they are formed” (173). Lanchester then referred his readers to the diagram that is reproduced here as figure 4.8.

Lanchester now had a qualitative account of lift that fulfilled the following conditions: (1) it was based on “cyclic” flow, that is, a flow around a vortex with circulation; (2) it was applicable to a finite wing; (3) it identified the role of trailing vortices; and (4) it made appeal to the viscosity of air as well as to conceptions derived from ideal fluid theory.14

At 9.30 a. m., on August 24, 1912, Lamb took the chair of section III of the Fifth International Congress of Mathematicians that was being held in Cam­bridge.38 Section III was devoted to mechanics, physical mathematics, and astronomy. Lamb wanted to say a few words before getting down to business. He noted that, in spite of the subdivision of the field, the scope of the section was still a wide one. He then went on to offer a classification of the different styles of work that were to be represented. He also identified the predomi­nant style of what he called the “Cambridge school” within this typology. His words are revealing.

It has been said that there are two distinct classes of applied mathematicians; viz. those whose interest lies mainly in the purely mathematical aspect of the problems suggested by experience, and those to whom on the other hand analysis is only a means to an end, the interpretation and coordination of the phenomena of the world. May I suggest that there is at least one other and an intermediate class, of which the Cambridge school has furnished many examples, who find a kind of aesthetic interest in the reciprocal play of theory and experience, who delight to see the results of analysis verified in the flash of ripples over a pool, as well as in the stately evolutions of the planetary bodies, and who find a satisfaction, again, in the continual improvement and refine­ment of the analytical methods which physical problems have suggested and evoke? All these classes are represented in force here today; and we trust that by mutual intercourse, and by the discussions in this section, this Congress may contribute something to the advancement of that Science of Mechanics, in its widest sense, which we all have at heart. (1:51)

The tone may have been lofty but Lamb had a purpose. He was making a plea for the representatives of the different tendencies in the discipline to communicate and cooperate. The remarks suggest a background anxiety that there might be problems on this score, and Lamb may have known about the convoluted and acrimonious arguments between pure and applied math­ematicians that had been taking place in Germany. We must also remember that Lamb was addressing a gathering of men of powerful intellect, many with significant achievements behind them and reputations to make or break. Larmor, Levi-Civita, Darwin, Moulton, and Abraham were all in the audience, while the Gottingen laboratory was represented by the presence of two of Prandtl’s colleagues and former assistants, Theodore von Karman and Ludwig Foppl. All of these men played an active part in the session that fol­lowed. Given his unifying purpose, Lamb could not have risked caricaturing the different classes of mathematician.

The care with which Lamb would have chosen his words lends a particular interest to his description of the Cambridge school. Lamb saw the character­istic concern of its practitioners as lying between pure mathematics, on the one hand and, on the other, a purely instrumental view of mathematics, one in which its role was simply the interpretation and coordination of data. The point on which he placed the emphasis was that mathematical results should be verified by the interplay of theory and experience. Lamb obviously saw this process as more than mere success in the ordering of data. Truth and correspondence with reality were the central aims. He described this con­cern as “aesthetic”—a word chosen, surely, to portray an intellectual involve­ment that was dignified rather than merely useful. The emphasis on truth was certainly consistent with what Lamb had said elsewhere, for example, in the discussion of Stokes’ equations in his Hydrodynamics. Applied mathematics, as practiced at Cambridge, was to be justified by its capacity to portray the nature of physical reality, not by its employment of useful fictions.

Other Cambridge luminaries expressed themselves somewhat differently but conveyed a similar orientation. At a different session of the same con­ference the Cambridge mathematical physicist Joseph Larmor voiced senti­ments that reinforced Lamb’s message. Larmor asserted that the role of the mathematician and the physicist were essentially identical.39 A. E. H. Love had spoken out in support of Larmor at the conference.40 Love was reiterating a position already developed in his authoritative Treatise on the Mathemati­cal Theory of Elasticity.41 This volume contained a historical introduction in which tendencies and distinctions similar to those identified by Lamb were rehearsed and evaluated. Love declared, as one of his aims, that he wanted to make his book useful to engineers and this had led him “to undertake some rather laborious arithmetical computations” (v). But he also wanted to “em­phasise the bearing of the theory on general questions of Natural Philosophy” (v), and it was clear that this was where his heart lay. His historical comments were judicious, but he went out of his way to emphasize the non-utilitarian origins of the subject matter he was about to expound. Thus,

The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Philoso­phy than in material progress, in trying to understand the world than in try­ing to make it more comfortable. From this attitude of mind it may possibly have resulted that the theory has contributed less to the material advance of mankind than it might otherwise have done. Be this as it may, the intellectual gain which has accrued from the work of these men must be estimated very highly. (30)

Technical mechanics is to be distinguished from natural philosophy, and he, Love, was doing a species of natural philosophy. Any resulting failure to contribute to material progress did not seem to distress him unduly. He was more interested in the link with fundamental physics and in recounting the detailed discussions that had taken place over the number and meaning of the elastic constants. These had thrown light on “the nature of molecules and the mode of their interaction” (30). The wave theory of optics and the theory of the ether had benefited from advances in the theory of elasticity, as had, even, certain branches of pure mathematics. Though Love and Lamb expressed themselves differently, we see a similar distancing of applied math­ematics from issues of utility and an affirmation of the fundamental char­acter of the relation between mathematics and physical reality. G. I. Taylor’s demand, made a few years later in his Adams Prize essay, that applied math­ematics should have a firm basis in physics was the expression of a stance already endorsed by figures of authority on the Cambridge scene and already characteristic of the Cambridge school.42

The demand for a firm basis in physics had not always characterized what had passed as “mathematical physics” or “mixed mathematics” at Cambridge. Mathematicians of earlier generations had often been happy to see mathemat­ics arise from physical problems but had then developed it independently of experimental data or with only a loose or analogical link to physical reality. An example of this earlier phase, which was still evident as late as the 1870s, was James Challis’ Essay on the Mathematical Principles of Physics in which he offered a speculative, hydrodynamic cosmology.43 The closer connection between mathematics and real physics that Lamb and, later, Taylor were tak­ing for granted had originally been forged in the work of Stokes, Thomson, and Maxwell, who were critical of the earlier style.44 Lamb, however, still felt the need to express himself carefully when he said that the Cambridge school provided “many examples” of the intermediate path between an overly ab­stract and an overly utilitarian approach. He thus acknowledged a continuing diversity in Cambridge work. This should come as no surprise since tradi­tions, even vigorous traditions, will always encompass a range of positions as they change and develop. Rayleigh, like Lamb, spoke of “the Cambridge school,” and he too noted a certain inner complexity and development. In connection with Routh’s textbook on dynamics, Rayleigh took the view that the earlier editions had been overly abstract, whereas later editions evinced a closer engagement with genuine scientific problems.45 In other words Routh had shifted toward the position that Lamb, like Rayleigh himself, saw as the strong point of the Cambridge school.46

Otto Blumenthal had been Hilbert’s first doctoral student at Gottingen and continued to help Hilbert edit the distinguished journal Mathematische An – nalen.55 In the winter semester of 1911-12 Blumenthal, now at the TH in Aachen, gave a course of lectures on the hydrodynamic basis of flight. He de­scribed, mathematically, the irrotational flow of an ideal fluid over a range of different Joukowsky profiles. Along with his colleagues at Aachen, Karl Toep- fer and Erich Trefftz, he drew up diagrams of the precise shape of the profiles. The result of the joint work was published in two papers in the ZFM for 1913. The main paper, by Blumenthal, was titled “Uber die Druckverteilung langs Joukowskischer Tragflachen” (On the pressure distribution along Joukowsky wings).56 It was followed by a short note by Trefftz giving a simplified geo­metrical method for drawing Joukowsky profiles and a graphical technique for rapidly computing the predicted air velocities, and hence pressures, on the surface of the wing.57

Blumenthal began by drawing attention to a unifying principle that had not emerged in Joukowsky’s original paper. Joukowsky had used two geo­metrical constructions. The first, which was the more complicated, gener­ated the wing profile, while the second, which was simpler, generated the symmetrical rudder. Blumenthal pointed out that only the second of the two constructions need be used. What is more, the process could be represented by a simple mathematical formula. This formula was the version of the Jou – kowsky transformation that was to achieve such fame.58 The formula can yield wing shapes and curved, Kutta-like arcs as well as rudder shapes and flat plates. Only one transformation, not two, was needed. It was all a matter of the position of the circle on the coordinate system of the plane that was to be transformed. The totality of Joukowsky contours, said Blumenthal, could be generated by the set of all circles that can be drawn on the Z = + i П plane

that pass through = – I/2, provided they either pass through, or contain, the point = +I/2. All that is required is that the circles are then subject to the transformation:

Those circles that pass through both = – I/2 and = +I/2 will have their cen­ters on the n-axis and will generate arcs similar to Kutta’s wing. The one circle in this family that has its center precisely at the origin, and hence has the line from = – I/2 to = +I/2 as its diameter, will be transformed into the straight line that is the limiting case of the arc. Wing shapes will be generated by all of the (off-center) circles whose circumference passes through = – I/2 but contains = +I/2, that is, which are sufficiently large that the circumference goes around the point = +I/2. The sharp trailing edge of the wing will be the transformation of the point = —/2, and the curved leading edge will go round the transformation of the point = +I/2. As a point moves around the circumference of such an off-center circle, the transformation will trace out the curve of an aerofoil shape with a rounded nose and an elongated tail.59 These, said Blumenthal, are “the Joukowsky figures in the proper sense” (“die Joukowskischen Figuren im eigentlichen Sinne”; 125).

It was Blumenthal who provided the unity lacking in Joukowsky’s original paper but which, today, is so often taken for granted. But Blumenthal’s aim was not merely to achieve a formal unity. He was bringing the generation of Joukowsky figures under intuitive control in order to facilitate their practical use. He isolated the features of the construction process that had an aero­dynamically significant effect on the overall geometry of the wing. Where Joukowsky had merely said that the geometrical construction of the wing de­pended on an angle and two lengths, Blumenthal identified the results of the choices that are to be made.

Blumenthal referred his readers to the diagram reproduced here as fig­ure 6.8. The circle in the figure has center M and passes through the point H, which is at a distance I/2 from the origin O. (Notice that I/2 featured in the formula that Blumenthal chose to specify the transformation.) The off-center circle in the diagram is to be transformed by means of the Joukowsky formula and turned into a wing profile. The point H (sometimes called the “pole” of the transformation) is to be transformed into the all-important trailing edge. The radial line from M to H cuts the vertical axis at a point labeled M’. Then, explained Blumenthal, the distance OM’ (labeled f/2) controls the height of curvature of the wing, while the distance M’M (labeled 6) controls the thick­ness of the wing. In general, if the center of the circle to be transformed is on
the positive vertical axis, the result is one of Kutta’s arcs; if the center is on the positive horizontal axis, the result is a symmetrical rudderlike figure; if it is somewhere in between (as in fig. 6.8), the result will be a curved profile of the characteristic Joukowsky type. How curved and how rounded will depend on the factors that Blumenthal had just identified.

Blumenthal gave four examples of Joukowsky profiles to show the effects of modifying these parameters, that is, the effect of moving the center of the circle while ensuring that its circumference still passed through H. Thus the curvature parameter (expressed as the ratio f/l) was given the value of 0, 1/10, 1/5, and 1/5 (again), while the thickness parameter (expressed as the ratio 8/l) was set at 1/10, 1/20, 1/20 (again), and 1/50. The effect of these choices was clearly visible as the Joukowsky profiles that he illustrated went from a sym­metrical shape to a markedly curved shape and from fat to thin.

From the velocity q of the flow (provided by Trefftz’s speedy method of graphical calculation), the pressure on the surface of the aerofoil could be

computed, and this led to the most striking feature of Blumenthal’s paper. Each of the four Joukowsky profiles that he had constructed was accompa­nied by a graph showing the theoretical pressure distribution on the upper and lower surfaces. (In all cases Blumenthal assumed that the profile was at an angle of incidence of 6°.) One of his profiles and its accompanying pressure graph is shown in figure 6.9. Summing the areas enclosed by the graphs gave a quantity proportional to the resultant lift. Blumenthal discussed each aero­foil in turn, pointing out the significance of the predicted pressure distribu­tion and its dependence on the parameters of the profile. Some features were common and stood out very clearly in the graphs, for example, the greater contribution of the suction effect on the upper surface of the wing compared to the pressure effect on the lower surface. Others were special to one shape, for example, the presence of a small suction effect even on the lower surface of the symmetrical (rudderlike) aerofoil and the very high speeds at the lead­ing edge of the thinnest profile.

Looking back to the period of the Great War, after some sixty years, Max Munk expressed the belief that the aeronautical work he had carried out in Gottingen had rapidly fallen into the hands of the Allies. According to Munk, the secret Technische Berichte “were translated in England a week after appear­ance and distributed there and in the U. S.”3 Exactly how this feat of espionage was performed Munk did not say. Similar stories have been related about the flow of sensitive information in the other direction, from the Allies to the Germans. I have already mentioned the secret testing of the Dunne biplane in the Scottish Highlands before the war. This was said to have attracted the at­tention of numerous German “spies,” though these stories surely owed more

to John Buchan than to reality.4 A more sober counterpart to Munk’s beliefs is provided by J. L. Nayler, one of the secretaries to the Advisory Committee. Also speaking retrospectively, he said that the wartime Reports and Memo­randa produced in Farnborough and Teddington eventually found their way into German hands. Nayler, though, suggested that this took months rather than weeks.5 Perhaps British spies were just superior to German spies.6

The truth was almost certainly more pedestrian than these claims sug­gest. There is no evidence that agents acting on behalf of the British gov­ernment got their hands on any information about the wartime Gottingen work and passed it on to their masters in Whitehall or their allies in Paris and Washington. There appears to have been no successful espionage activ­ity. It is not the speed with which information traveled that is striking but its slowness. When information did travel, the channels were overt and obvious rather than mysterious.7 The war had the predictable effect of attenuating the flow of technical information between different national groups, but even during the prewar years, with no military or diplomatic impediments, the flow was surprisingly limited. It is important to identify where the restriction lay. It did not arise because of what might be called material or external fac­tors, such as censorship, but because of more subtle, cultural constraints. It was not the physical inaccessibility of reports, journals, or books that caused the problem. What counted was the response, on the intellectual level, even when they were accessible. For example, both Sir George Greenhill and G. H. Bryan were present at the congress in Heidelberg, in 1904, when Prandtl presented his revolutionary, boundary-layer paper.8 Bryan explicitly men­tioned Prandtl’s contribution in his postconference report for Nature, but he ignored its mathematical content entirely and confined his comments to the experiments and photographs.9 It is difficult to resist the conclusion that if such important matters can be passed over in these circumstances, then even if there had been “spies” reporting back to the British Advisory Committee, their efforts would have been wasted.

To reinforce this claim I start with some other prewar events and look at the information that members of the Advisory Committee had available to them about their German counterparts. From the outset the committee, and the Whitehall apparatus that supported it, accepted the principle that it was important to monitor the work of foreign experts. Haldane stressed the point in Parliament, and the theme was picked up by the aeronautical press.10 The commitment to gathering intelligence was made apparent in three ways. First, the preliminary documentation of the committee, when it was estab­lished in 1909, included what was, in effect, a reading list for the committee members. The list cited some twenty-two works by French, German, Ital-

ian, and American writers. The German authors included Ahlborn, Finster – walder, and Lilienthal.11 Second, the sequence of Reports and Memoranda issued by the committee began with a description of the program of German airship research. It was presented by Rear Admiral Bacon at the very first meeting of the Advisory Committee on May 12, 1909.12 R&M 1 consisted of translated extracts from the publications of the German Society for the Study of Airships and included a lengthy quotation from Prandtl.13 There was men­tion of Prandtl’s wind channel, his experiments on model airships, and, in­triguingly, a passing reference was made to his “hydraulic machine” (shown earlier in fig. 7.1). This was the apparatus used to take his boundary-layer photographs. There was, however, no mention of the mathematical theory. Third, and most important of all, the committee was provided with a series of summaries of foreign papers from leading journals such as the Zeitschrift fur Flugtechnik. A steady stream of these summaries was published in the period between the founding of the committee and the outbreak of the Great War, when such material was immediately withdrawn from public circulation.14

A measure of the size of the intelligence initiative can be gathered by count­ing the number of such abstracts published yearly in the annual report of the Advisory Committee. Such a procedure can only provide an approximate measure of the potential flow of information because it does not take account of the different scope of the individual publications, but it gives some guide. Figure 8.1 charts the year-by-year production of summaries and abstracts of foreign-language publications that were made available to the committee.15 Two things stand out. First, the size of the effort put into tracking foreign work was clearly considerable. Second, there was a consistently high level of attention given to German work, amounting on average to identifying and abstracting some eighteen items per year for a period of six years.

Moving from the quantitative to the qualitative character of the informa­tion, it is important to know which authors the committee deemed interest­ing. The answer is that Prandtl and his collaborators were prominent among them. In December 1910, Glazebrook, as chairman, explicitly drew the Got­tingen work to the attention of the members of the Advisory Committee.16 In August 1913, in preparation for a forthcoming visit to the laboratory in Teddington, Prandtl sent a number of his papers to the National Physical Laboratory (NPL) and received acknowledgment from Selby, the secretary.17 Thus, by one route or another, all of the major prewar work of the Gottin­gen school had been made available, including accounts of the wind channel and the airship work but also material directly concerned with the circulation theory of lift. In addition there were abstracts of papers of indirect interest

because of their significance for fluid dynamics in general. More specifically, among the papers summarized, sometimes at length, were those of Foppl on the resistance of flat and curved plates (abstracts 93, 94, 97, 98, 118, and 131), Fuhrmann on the resistance of different airship models (abstracts 95, 96, and 127), and Prandtl’s classic study of the flow of air over a sphere in which he in­troduced turbulence into the boundary layer by means of a trip wire (abstract 234). Of those explicitly related to the idea of circulation and Prandtl’s wing theory, accounts were given of Foppl’s 1911 study of the downwash behind a wing (abstract 128, but incorrectly attributed to Fuhrmann); Wieselsberger’s 1914 study of formation flying in birds (abstract 276); the 1914 paper by Betz on the interaction of biplane wings (abstract 279); Joukowsky’s pioneering 1910 article (abstract 299); Blumenthal’s 1913 paper on the pressure distribu­tion along a Joukowsky aerofoil (abstract 301); and Trefftz’s 1913 graphical construction of a Joukowsky aerofoil (abstract 302).

The principal mathematical formulas associated with the circulation theory in both its two – and three-dimensional forms were also to be found in the abstracts. Thus the basic law of lift, linking density, velocity, and cir­culation, L = pvr, was stated, as was the law of Biot-Savart, which was the basis of the three-dimensional development of the theory. The abstracts pro­vided everything that was needed to show that the circulation theory was capable of mathematical development and was more than a mere collection of impressionistic ideas. The abstracts gave clear, documentary evidence of the progress that the German engineers were making. It would appear that the circulation theory was there for the taking. Nevertheless, the availability of the abstracts generated no more enthusiasm for the theory of circulation in its mathematical form than did Lanchester’s original publication with its more intuitive treatment of the subject.

Why might this be? Obviously, the abstracts had no power to force them­selves on anyone’s attention. They were things to be used selectively and were subject to the filtering effects of interpretation, both in their composition and their evaluation. Thus Glazebrook’s act in drawing attention to the Gottingen work was probably indicative of his enduring concern with discrepancies be­tween the results of different wind channels and the fundamental problems shared by the NPL and Gottingen in the interpretation of their findings. Gla – zebrook was acutely aware that such problems would be grist to the mill of the “practical men” and was anxious lest they be used to persuade the govern­ment to cut the budget of the NPL.18 Furthermore, the precise content of the abstracts reveals the way that reported work may be glossed so that certain as­pects of it are given salience at the expense of other readings. Take, for exam­ple, the account given in abstract 131, which was devoted to Foppl’s 1910 paper in the Jahrbuch der Motorluftschiff-Studiengesellschaft.19 This paper contained a comparison of Rayleigh flow with Kutta’s theory of circulatory flow. After summarizing the contents of the paper, the abstract writer drew the conclu­sion that neither approach to the flow over an inclined plate was satisfactory. What was needed was an understanding of certain subtle, viscous effects. “It is suggested that Kutta’s theory throws some light on the experimental results, and in some respects, qualitatively, is in fair agreement with the experiments. At present, however, no entirely satisfactory theory seems to be possible until more is known of the nature of the air flow, the main differences being due to the difficulty of including the frictional effects” (257). The need to include frictional effects was, of course, an abiding theme in the British work. The abstract writer then went on to single out, as the “most striking result” of Foppl’s investigation, “the discontinuity in lift and drift coefficients within the region from 38° to 42°” (258). All the attention was thus directed toward extremely difficult, fundamental, and unstable features of the flow that lay far outside the typical working range of an aerofoil. Once again, the British were drawn to the phenomenon of stalling. The focus was on all the things that could not be understood on the basis of inviscid flow at small angles of inci­dence rather than on what could be achieved using perfect fluid theory over a limited range. Thus the abstract and summary itself prefigured the selective tendencies and implicit evaluations that worked against the circulatory theory.

The prewar information about German thinking on aerodynamics was rich but unexploited, whereas during the war, the pressure of short-term work added to the tendency to pass over the significance of the German theoretical approach.20 What of the pattern of information flow, and the reaction to it, immediately after the cessation of hostilities? In some quarters in Britain, the outcome of the war produced a jingoistic complacency. Such sentiments were exemplified by C. G. Grey, editor of the Aeroplane, when he said in 1918: “We have nothing to learn from the Hun in aerodynamics.”21 This boast was a continuation of a commonplace theme in the aeronautical press, which, throughout the war, dismissed German inventiveness, originality, and skill.22 Such vulgarity was largely absent from the writing of the more technically so­phisticated members of the British aerodynamic community, who had, if not an admiration, at least a healthy respect for German achievements.23 Among the members and associates of the Advisory Committee there was an under­standable degree of self-congratulation as the war drew to a close, but it was modest in tone.24

The first reaction to the outbreak of peace by the scientists at the Royal Aircraft Establishment (formerly the Royal Aircraft Factory) was to poke fun at themselves and their critics. The period immediately after the cease-fire, between November and December 1918, saw the production of a light-hearted work titled “The Book of Aeron: Revelations of Abah the Experimenter.”25 This undergraduate-style spoof was composed by the remarkable group of young men who had been recruited by Mervin O’Gorman. Many of them were billeted in a large house in Farnborough called Chudleigh (see fig. 8.2).

The house acted as a mess for the RAE and had originally been organized by Major F. M. Green, who, until 1916, had been the engineer in charge of designs.26

The “Book of Aeron,” circulated as an internal report, had been written by a committee that included R. V. Southwell and Hermann Glauert. It was couched in mock, Old Testament terms, with ancient Egyptian overtones, and told the story of the Land of Rae (the RAE), its ruler Bah Sto (Bairstow), and the wicked scribe Grae (C. G. Grey). The Abah of the subtitle appears to be a reference to the designation “Department BA” in which the experi­mental work was conducted at Farnborough. Naturally, the clash between the aircraft manufacturers (“the merchants”) and the Farnborough scientists (“the men of Rae”) played a significant role in the story. The following pas­sage conveys the spirit of the enterprise:

2. And the men of Rae built air chariots for their king, and brought forth new chariots of diverse sorts; and to each chariot did they give a letter and a num­ber, that the wise men might learn their habits:

3. but the multitude comprehended it not.

4. Then murmured the merchants one to another saying, Why strive the men of Rae so furiously against us? For the king goeth to war with their chariots, and behold our chariots are cast into the pit. (3)

After the relief of the armistice, and the lessening of tension, came the serious business of taking stock. Just what had been achieved during the war? What had been sacrificed, scientifically, because of the demands of the war effort? What, if anything, was to be learned from newly accessible German literature? Bairstow and other committee members rapidly let it be known that they deplored the cutback in basic research that had been caused by the war. In terms of fundamentals, they argued, the period of rapid achievement in aerodynamics had been before the war. It was now time to get back to deeper questions and that meant solving Stokes’ equations of viscous flow. Pure mathematicians may have given up on this task, but new techniques, perhaps using graphical methods, might be developed for this purpose. In a confidential report of November 1918,27 mapping out a program of work for 1919-20, the argument was put like this: “General research in fluid motion has been discontinued during the war and it is very desirable that it should be resumed at the earliest possible moment. It is proposed as soon as the opportunity occurs to continue the study of the motion of viscous fluids to which considerable attention has already been given” (9-10). These senti­ments represented the beginnings of a campaign by Bairstow and Glazebrook to channel more resources from short-term to basic research. They argued, in letters to the Times, that government figures for the aeronautics budget con­flated the spending on development with that on research proper. This made the expenditure on fundamental work, which was vital for future technology, appear larger than it really was.28

What had the Germans being doing during the war? At first, informa­tion filtered through to the British via the French and Americans. In 1919 the American National Advisory Committee for Aeronautics, the NACA, es­tablished an office at 10, rue Victorien Sardou in Paris. Their representative, William Knight, actively pursued a policy of information gathering. To the ir­ritation of military and diplomatic circles in Paris and Washington, he made contact with Prandtl and suggested that information on recent developments be shared. Knight contacted Prandtl on November 15, 1919, by letter and man­aged to get official agreement to visit him in April 1920. He made a second visit in the autumn of 1920.29 It appears to be due to the efforts made in Paris that the British Advisory Committee was furnished with translations of reports by A. Toussant and by Colonel Rene Dorand. Toussant was an engineer at the Aerotechnical Institute of the University of Paris at St. Cyr. In November 1919 he had produced a resume of the theoretical work done at Gottingen based on the Technische Berichte. Extracts of Toussant’s work were translated by the NACA, and these surfaced in February 1920 as a report designated Ae. Tech. 48 for consideration by Glazebrook and his colleagues.30 There was, however, little that Toussant could add to the information that was already at hand from the prewar abstracts. He gave mathematical formulas but none of the background reasoning. Dorand’s report was titled “The New Aerodynamical Laboratory at Gottingen.” It was translated as report T. 1516 in October 1920 and reached the Advisory Committee via the Inter-Allied Aeronautical Com­mission of Control.31 In Dorand’s report Prandtl was referred to through­out as “Proudet,” but it included three pages of blueprints of the Gottingen wind channels and provided technical details of the automatic speed con­trol and the measuring apparatus. Bairstow felt that there was nothing new in Dorand’s report.32 Others, however, noticed that Prandtl seemed to have solved certain problems that plagued the NPL channels and had “managed to achieve good velocity distribution in the working sections.”33

Both the British and the Americans were keen to locate and translate cop­ies of the Technische Berichte. At a meeting that took place on July 13, 1920, at the Royal Society, minute 28 records Treasury authority for the “employment of abstractors to make abstracts of German technical reports.”34 Across the Atlantic, Joseph Ames, who was the chairman of the Committee on Publica­tions and Intelligence, and one of the founding members of the American National Advisory Committee for Aeronautics, wrote from the NACA head­quarters to J. C. Hunsaker at the Navy Department in Washington, D. C., on October 15, 1920:

It is with great pleasure that I am informing you that the National Advisory Committee for Aeronautics has been successful in obtaining a number of sets of “Technische Berichte” and we are mailing you under separate cover vol­umes No.1, 2 and 3. The Committee is also forwarding a carefully prepared translation of the index of the first three volumes with a list of the symbols used. . . . The importance of the information contained in the “Technische Berichte” cannot be over-estimated and it is the desire of the Committee that all research laboratories and individuals interested in aeronautical research should become familiar with the results of the aeronautical research carried on in Germany during the War.35

The success of this search appears to have been due to Knight’s persistence. He used Prandtl’s good offices to approach the German publishers but had to overcome the numerous obstacles arising from the immediate postwar currency and customs restrictions.36 Meanwhile the U. S. National Advisory Committee continued negotiations with the British about the translation and abstraction of the reports.37 The enthusiastic terms of Ames’ letter, and the continuing efforts by both the British and the Americans, provide a sufficient basis for rejecting Max Munk’s claim that the reports were in the possession of the allies soon after their completion. Had this intelligence already been gathered, all the postwar concern would have been unnecessary.38

In the German-speaking world the circulation theory of lift and Prandtl’s wing theory found their way into the textbooks during the Great War. Rich­ard Grammel, who taught mechanics at the technische Hochschule in Danzig, led the way in 1917 with his Die hydrodynamischen Grundlagen des Fluges5 In 1919, immediately after the war, Arthur Proll of the TH in Hannover, pub­lished Flugtechnik: Grundlagen des Kunstfluges.52 H. G. Bader published his Grundlagen der Flugtechnik in 1920, and in 1922 Richard Fuchs of Berlin and Ludwig Hopf of Aachen produced their comprehensive Aerodynamik.53 The content and level of these books contrasted markedly with what was available on the British textbook scene. As described earlier, both Cowley and Levy’s Aeronautics in Theory and Experiment of 1918 and Bairstow’s Applied Aero­dynamics of 1920 dismissed the circulatory theory, whereas G. P. Thomson’s Applied Aerodynamics of 1919 was almost purely empirical. Thomson spoke of the “complete failure” (26) of mathematical hydrodynamics to account for lift. He concluded that an account of lift required an understanding of eddies and turbulent motion: “This is the solution we want for aerodynamics, and not that found by the ordinary mathematical method” (32). In 1926 the situa­tion changed radically when Cambridge University Press published Hermann

Glauert’s Elements of Aerofoil and Airscrew Theory.54 This work showed the power of the “ordinary mathematical method” of which Thomson, like most of his Cambridge companions, had despaired. Glauert’s Elements proved to be an outstanding work of exposition which, even today, some eighty years later, is still confidently recommended to students.55

The book consisted of seventeen brief chapters and surveyed all the main themes of modern aerofoil theory for a reader with no previous knowledge of fluid mechanics (though a significant degree of mathematical competence was presupposed). The first chapter described the main facts to be explained, while chapters 2-5 outlined the theory of perfect fluids. Chapters 6 and 7 in­troduced the theory of conformal transformation and the specific properties of the Joukowsky transformation. Chapter 8 dealt with viscosity and drag. Here Glauert introduced the Stokes equation for viscous flow and informally derived Prandtl’s boundary-layer approximation. Chapter 9 was called “The Basis of Aerofoil Theory,” and in it Glauert sought to bring together the ap­parently antithetical ideas of viscous and inviscid flow into a practical synthe­sis. His aim was “to obtain the true conception of a perfect fluid” (127). The form of the desired synthesis was described by Glauert as follows: “The vis­cosity must be retained in the equations of motion and the flow of a perfect fluid must be obtained by making the viscosity indefinitely small” (117).

When analyzing the motion of an object in a fluid, the concept of a per­fect fluid must be deployed in a way that retains the effects of viscosity. If a perfect fluid is defined as a fluid devoid of viscosity, such a requirement is contradictory: how can the viscosity, which is excluded by definition, also be “retained”? Here, once again, Glauert sought to convey his novel, Gottin – gen-style methodology. He argued that the requirement he formulated can be satisfied even though the boundary conditions, the way the fluid behaves at a solid boundary, are wholly different for a perfect fluid and for a viscous fluid. (For a perfect fluid the boundary conditions are that it cannot penetrate the solid boundary but it can slide smoothly along it; for a viscous fluid the conditions are zero penetration of the boundary and zero slip along it.) The trick needed to retain viscosity in the equations of motion is to start with a viscous boundary layer of finite thickness around the object and imagine it to become an infinitely thin sheet of vorticity. The boundary layer is a real phenomenon belonging to viscous fluids, and a vortex sheet is an idealization appropriate to perfect fluids. The connecting link that allows viscosity to be “retained” is that

in the limit the boundary layer becomes a vortex sheet surrounding the sur­face of the body and the vortices of this sheet act as roller bearings between

the surface of the body and the genera! mass of the fluid. The conception of a perfect fluid with a vortex sheet surrounding the surface of the body therefore represents the limiting conditions of a viscous fluid when the viscosity tends to zero, and the existence of the vortex sheet implies that the perfect fluid so­lution need not satisfy the condition of zero slip at the boundary. (117-18)

If the cross section of a wing is drawn in two dimensions, the line that traces its profile is to be thought of as made up of an infinite number of points acting like infinitesimal roller bearings. These rollers are said to be rotating fluid elements, that is, “fluid elements in vortical motion” (119), and they stand in for the boundary layer. The rotating fluid elements pass along the surface of the body and finally leave it to pass downstream in the wake. Glauert linked this picture to von Karman’s work on the so-called vortex street that exists behind a bluff body placed in a fluid flow. Clearly Glauert was introducing a significant degree of idealization, but even this degree would soon be surpassed. Having replaced the boundary layer on the wing surface by a sheet of vorticity, the chord and profile of the wing was then ignored altogether. In two dimensions the wing was reduced to a single point, that is, to the “cross section” of a line of vorticity imagined to be perpendicular to the page on which the figure is drawn.

How did Glauert explain the origin of the circulation around a wing? The discussion of this sensitive and difficult question was located in a short sec­tion of chapter 9. Combining candor with British understatement, Glauert introduced the issue as follows: “The process by which the circulation round an aerofoil develops as the aerofoil starts from rest presents certain theoreti­cal difficulties, since the process would be impossible in a perfect fluid, and it is again necessary to consider the limiting condition as the viscosity tends to zero” (121). Circulation is impossible if the analysis starts with p = 0 and con­fines itself to this condition. To overcome the difficulty it is necessary to start by considering p Ф 0 and then make the transition from viscous to nonvis­cous flow by imagining that p ^ 0. Here within the one sentence we see the British and German conceptions of an ideal fluid directly juxtaposed. Glauert then proceeded to offer a qualitative account of the required transition. The analysis effectively hinged on two diagrams of the flow at the trailing edge of a wing. I reproduce the diagrams in figure 9.11.

At low speeds, as the aerofoil starts from rest, the air behaves like an ideal fluid and curves round the trailing edge as shown in (a) in figure 9.11. There is a stagnation point S on the upper surface not far from the trailing edge. Glauert argued that, as the velocity increases, the streamlines coming from the undersurface are unable to turn round the trailing edge “owing to the large viscous forces brought into action by the high velocity gradient” (121).

The flow thus breaks away from the trailing edge in the manner shown in (b). The result is that “a vortex is formed between the trailing edge and the old stagnation point S” (121). When the vortex has reached a certain stage of development, it breaks away and floats downstream in the wake of the wing.

Although Glauert does not make the point explicitly, it is obviously im­portant for the argument that the vortex that detaches itself and moves down­stream is rotating in the correct direction. Its direction of rotation determines the direction of the circulation in any contour that surrounds it. The infor­mation about the direction in which the vortex rotates is contained in Glau – ert’s diagram rather than his text. The diagram shows that the vortex rotates in a counterclockwise direction. Why counterclockwise? The presumption must be that the flow that initially went round the trailing edge and then progressively failed to navigate the sharp corner is moving more rapidly than the flow that comes away from the stagnation point. The speed difference of the adjacent bodies of fluid would constitute a surface of discontinuity and hence a surface of vorticity—and the differences, in this case, would produce a counterclockwise vortex.

The next step in the argument was equally crucial. In the diagram a vortex has detached itself from the wing, and this, the argument goes, generates an equal and opposite circulation around the wing. The detached vortex had a counterclockwise circulation, so the circulation around the wing will be clockwise, and this produces the speed differential postulated by the circula­tion theory. But why does this process create an equal and opposite circula­tion? The answer given by Prandtl and Glauert was that such an outcome is required in order that Kelvin’s theorem is satisfied. Kelvin’s theorem initially looked as if it would rule out the onset of circulation entirely. If there is zero circulation at the onset of movement, there will be zero circulation at all later times. But the enemy is now converted into an ally. Glauert explained that the circulation around a large contour, enclosing both the wing and the detached vortex, will indeed stay zero. This is necessary to satisfy Kelvin’s theorem, but the theorem can be satisfied by virtue of two opposing vortices whose

respective circulations cancel each other out. The argument is represented diagrammatically in figure 9.12, again taken from Glauert. The large contour is called ABCD. The vortex is called E, and the circulation around the wing is K, while the circulation around the vortex is, accordingly, -K. As Glauert put it: “the circulation round any large contour ABCD which surrounded the aerofoil initially was and must remain zero, and as this contour includes the vortex E there must be a circulation K round the aerofoil which is exactly equal and opposite to the circulation round the vortex E” (121). The detached vortex floats away on the free stream with velocity V, leaving the wing with circulation K and lift given by the Kutta-Joukowsky law L = pKV. Glauert’s argument was moving rapidly at this point, but though potentially puzzling, this sequence of steps is now to be found in all standard textbooks.

Having prepared the ground in the first nine chapters, Glauert moved on to discuss the aerofoil in three dimensions. In chapter 10 he introduced the mathematics of the simple and refined horseshoe vortex system as well as the important concepts of induced velocity and induced drag. In chapters 11-13 he dealt with the effects of varying the aspect ratio of a wing and generalized the results to biplanes. In chapter 14 he discussed wind-tunnel corrections, while in the final chapters, 15 and 16, he applied the circulation theory to the airscrew.

Time and again critics attempt to refute relativism by drawing attention to the objectivity of what is known in both science and daily life.78 Such attempts are misguided. The only kind of counterexample that could refute relativism would be an example of absolute knowledge. Proof or evidence of objectiv­ity will not suffice unless the objectivity in question can be shown to be an absolute objectivity. The demand for objectivity is legitimate, but it is meant to preclude subjectivity, and subjectivism is not the same as relativism. The subjective-objective distinction is one thing, the relative-absolute distinc­tion is another thing, and the two should not be conflated. Frank was ad­mirably clear on this point and knew that his defense of relativism was not an attack on objectivity. He (rightly) believed in both the relativity and the objectivity of scientific knowledge.79

Rather than explore this theme in an abstract way, let me take an example from aerodynamics. The example, which concerns the rolling up of the vor­tex sheet behind a wing, is designed to show the objective (that is, nonsub­jective) character of knowledge at its most dramatic. The question is: Can the example be understood in relativist terms? Here is the example. In the spring of 1944, at a crucial stage of the Second World War, London was at­tacked by V-1 flying bombs. The V-1 was a large bomb fitted with small wings and a ram-jet engine, and it flew some 300 miles per hour. The bombs were launched from sites on the French and Dutch coasts by means of a shallow ramp that pointed in the direction of the target. After the bomb had trav­eled a predetermined distance, its engine was switched off by an onboard device that simultaneously altered the trim of the wings, causing the bomb to fall to the ground and explode. In an effort to stem the attacks, the pi­lots of the Royal Air Force chased after the bombs and tried to bring them down in open country where they would do less harm. It was not possible to close in on the bomb to shoot it down because of the danger that the re­sulting explosion would destroy the attacking aircraft. Some pilots therefore developed the technique of flying close to the bomb, making use of the air­flow behind the wing of their aircraft to flip the missile on its side so that it would drop to the ground.80 This technique did not involve direct, metal-on – metal contact with the V-1 but, it has been argued, exploited the rolling up of the vortex sheet behind the wing of the aircraft. It was the rotating air of the vortex that turned the missile on its side. According to an article in the An­nual Review of Fluid Dynamics in 1998, the rolling up of the trailing vortices behind a wing of high-aspect ratio was, for a long time, considered to be a matter of little practical importance by experts in aerodynamics. The experts acknowledged its existence but not its utility. But, says the author, if the theo­rists ignored the significance of the roll up, “fighter pilots who used their own vortices to topple V-1 flying bombs had another opinion.”81

The example shows knowledge and skill tested by uncompromising, ex­ternal criteria. The pilots’ subjective feelings had to be mastered and their judgments subordinated to the objective demands of the situation. What, then, is “relative” about this episode and the knowledge and skills involved? The brief quotation from the Annual Review of Fluid Dynamics already indi­cates the lines on which an answer can be given. First, the relevant knowledge and beliefs were distributed unevenly across the groups mentioned in the Review article. The experts who worked theoretically, or who experimented with wind tunnels, had one opinion about vortex sheets; the pilots who chased the bombs over the fields of Kent are attributed with another opinion. Second, the character of the knowledge varied. The experts had a mathemati­cally refined understanding; the pilots had a rough-and-ready but practical sense of what they needed to do. What they lacked in rigor they made up in skill. Third, although the experts and the pilots were oriented to the same features of reality, they did not share a common language or common con­cepts. The article makes no mention of communication between pilots and aerodynamic experts on the matter, but it would almost certainly have been problematic. It had never been easy for pilots and aerodynamicists to talk to one another.82 Fourth, the range of circumstances that the members of the two groups took into account differed markedly. The experts operated in a world that was artfully controlled, shielded, and simplified; the pilots functioned in an environment saturated with complexity, interaction, noise, vibration, jolting, turbulence, and distraction.

The conclusion must be that although both groups were actively engaged with a reality that was largely independent of their subjective will, the qual­ity of that engagement was different. In both cases their understanding was objective rather than subjective, but it was also to be seen as relative to their standpoints. In neither case did it have an absolute character. In developing the argument of his book, Frank was therefore right to insist that the doctrine of the relativity of truth “does not imperil by any means the ‘objectivity’ of truth” (21).

It may be objected that the pilots had causal knowledge of reality. Theo­ries may come and go, and verbal accounts may vary, but don’t actions and interventions put an agent into direct contact with reality? This, it may be said, proves that there is a way of grasping reality that is not merely relative. But does it? The critic who takes this line must confront and answer the ques­tion What is supposed to be absolute about the knowledge of causes and the exploitation of this knowledge? The correct answer is that there is nothing absolute about causal knowledge. This conclusion ought to be well known because it was established over two hundred years ago by David Hume in his Treatise of Human Nature. Hume gave a relativist analysis whose essential points remain unchallenged to this day. His argument was that all knowledge about causes, for example, that A causes B, whether expressed in words or ac­tions, is inductive knowledge based on experience. Inductive inferences, said Hume, can never be given an absolute justification. Inductive knowledge is irremediably relative.83

The limited, fallible, and relative character of practical knowledge can be generalized from the example of the flying bombs to my entire case study. The pattern of flow over a wing described by Lanchester, Kutta, Joukowsky, and Prandtl is not the only one that can render mechanical flight possible. Their picture, which is now called “classical aerodynamics,” and whose his­tory I have been describing, rests on the principle that the separation of the flow from the surface of the wing must be minimized. Flow separation, it was assumed, always leads to a breakdown of the lift. It has now been dis­covered that flow separation can be both exploited and controlled in a way that actually generates lift. Leading-edge vortices, and even shockwaves, can be exploited to create lift.84 This was not realized until many years after the events I have described. As one authority put it, “We must realise, however, that Prandtl’s is but one of many possible bases of wing theory and there can be no doubt that more comprehensive assumptions will eventually be

developed for this interesting type of physical flow.” 85 Until the late 1950s, all of the technical knowledge in aerodynamics concerning lift had been de­veloped on what can now be seen as a narrow basis. What the future holds is always unknowable, but the more recently acquired, broader perspective serves to expose the hitherto unappreciated relativities of past achievements. But we should not allow ourselves to think that, as these historical relativi­ties are exposed, knowledge progressively sheds its relative character and moves closer to something absolute. To cherish such a picture is to indulge in metaphysics.