Category The Enigma of. the Aerofoil

Kutta ended with some ideas about extending his mathematical methods to a variety of different wing profiles with rounded leading edges and with flaps attached to the trailing edge. He mentioned the need to develop a more gen­eral form of the Schwarz-Christoffel theorem and indicated the demanding amount of computational effort that would be involved, but Kutta did not feel that the limits of his approach had been reached and hinted at their fur­ther application to biplanes.

Kutta’s next paper, in 1911, utilized the same mathematical techniques as those adopted in 1910 but it dealt with more complicated cases.33 The analysis was generalized in two ways. First, Kutta showed how to apply his conformal transformations to an aerofoil whose cross section was composed of not one but two circular arcs in the form of a crescent or sickle shape. Such a sickle­shaped profile was used in the successful Antoinette monoplane, and von Mises has suggested that its practical use was prompted by Kutta’s analysis.34 Second, Kutta generalized the approach in order to describe the flow over a number of wings that could be arranged to make a biplane or a triplane or even a multiwing arrangement in the form of a “Venetian blind.” Again, all of these forms had actually been used, or experimented with, by those who tried to build flying machines.

Having made these two outstanding contributions to aerodynamics, Kutta fell silent. He never published anything again. The reason for the silence is unknown.

The wartime activities of the Gottingen group, and their colleagues in the technische Hochschulen, proceeded on a much broader front than I have so far described.92 On the theoretical side, building on Betz’s early papers, there were studies by Betz, Munk, and Prandtl on the aerodynamics of biplanes and triplanes. Using the apparatus of the Biot-Savart law they produced some general theorems that helped to guide the aircraft designer through the maze of possible multiplane configurations. Betz had proven that, for an unstag­gered biplane, the induced drag effects of the wings on one another would be equal. For a staggered arrangement it was now shown that the sum of the mutually induced drags was constant and independent of stagger, provided that the lifts, and their distribution, were not changed. (This condition could be satisfied by changing the angle of attack.) In general, it was shown that the best biplane configuration was one with wings of equal length, with the up­per wing ahead of the lower. The biplane work also confirmed the important finding that elliptical lift distributions provided a good approximation for wings with non-elliptical planforms.93

On the empirical side, the war effort called for wind-channel studies of the drag generated by different aircraft components such as undercarriages and machine-gun mountings, engine-cooling systems, and the ubiquitous struts and bracing wires of the period.94 Work was also done on the lift and drag of the fuselage, the interaction between the fuselage and the wing, the effect of dividing the wing, the forces on fins and rudders, and the empirical properties of the triplane configuration.95 Experiments were done to test the resistance of the nose shapes required by different engine types, for example, rotary as compared with in-line engines, and attempts were made to add rotating propellers to the wind-channel models to achieve realism.96 Some wind-channel tests were also done on models of complete aircraft.97 Munk and Cario continued the studies initiated by Foppl of the downwash behind a wing. Whereas Foppl had worked with the overall force exerted by the down – wash on the elevator, Munk and Cario studied the downwash in much more empirical detail, using fine silk threads to trace the local variations. They un­covered significant complexities in the flow and made clear the need for a more extended program of work.98

Numerous studies were carried out to measure the lift and resistance of individual wing profiles. These were mainly overseen by Munk and his col­laborator Erich Huckel.99 Significant efforts were made to ensure that the re­sults were intelligible to those who might use them in practice.100 An attempt was also made to introduce order into the vast amount of data that had accu­mulated for different aerofoils, though the classification remained largely at the empirical level.101 One trend was toward an interest in thicker rather than thinner aerofoils, something that surprised the British when they examined captured German aircraft.102 An example of these thick aerofoils was the Got­tingen 298 used on the famous Fokker triplane. The use of the 298 profile by the designer Anthony Fokker does not, however, appear to have been a con­sequence of Prandtl’s recommendation or scientific knowledge of its good lift and drag characteristics (characteristics that, a priori and wrongly, the British designers doubted). In fact, the aerofoil was introduced into the Fokker pro­duction line by their chief engineer Reinhold Platz on the basis of trial-and – error knowledge. Later, and unknown to the people at Fokker, it was tested in Gottingen, where it was given its designation.103

The Fokker episode indicates that there was a continuing gap between the “practical men” of Germany and those self-consciously developing science- based procedures and working in academic and government institutions. The alienation of the practical men was not purely a British phenomenon, though it seems to have been less acute as a problem for German aviation than for British. Evidence in the technical reports indicates that the members of the Gottingen school were themselves aware of this gap and found it frustrating. Max Munk addressed the issue directly in a brief report of October 15, 1917, titled “Spannweite und Luftwiderstand” (Span and air resistance).104 Refer­ring to the practical conversion formulas linking wings of different aspect ratio, Munk complained:

Die kurzlich von Betz veroffentlichen Prandtlschen Flugelformeln werden wohl, da sie auf theoretischen Grundlagen beruhen, in der Praxis nicht so freundlich aufgenommen werden, wie sie verdienen. Das ist sehr schade, denn die Formeln enthalten mehr und leisten Besseres als der Praktiker geneigt ist, ihnen zuzutrauen. (199)

The formulas of Prandtl’s wing theory that Betz has recently published will probably not be welcomed in the realm of practice as much as they deserve because they rest on theoretical grounds. This is a great shame because the formulas offer more and give better service than the practical man is inclined to believe.

Munk went on to give an explanation of the significance of the formulas for the aircraft designer and some simple, general rules for the rapid calculation of the induced resistance and angle of incidence. Despite this evidence of skepticism in certain quarters, there was no shortage of contract work to be done for individual aircraft firms during the war years. This is attested by the frequency with which such names as AEG, Aviatik, Rumpler, Siemens and Schuckert, and Zeppelin were mentioned in the technical reports. Despite the problems of communication between the representatives of theory and practice, Prandtl’s institute had achieved a central position in what would now be called the military-industrial complex of Wilhelmine Germany. If this development brought frustrations as well as the advantages of government support, it is clear that striking progress had been made in aerodynamics, both empirically and theoretically.

One of the more unusual aerofoils whose properties were reported on by Munk and his colleagues was designated as profile 95. Visually it stood out from the usual run of aerofoil shapes (see fig. 7.16). The aerofoil looked like the back of a cat when the animal stretched, and it was duly given the nick­name Katzenbuckelflache. The Gottingen tests showed that the performance characteristics of the “cat’s-back” profile 95 were notably poor. It was tested by Max Munk and Carl Pohlhausen in the course of a run of work on nearly one hundred aerofoils. The results were listed together in the Technische Ber – ichte of August 1917 and showed that the maximum-lift coefficient for each wing in this sequence was typically in the region of 130 or 140. The maxi­mum lift coefficient for profile 95, by contrast, was given as 95.2. Again, the maximum lift-to-drag ratio was typically 14 or 15, while the ratio for profile 95 was 10.8.105 The designer of the cat’s-back wing was the celebrated physicist Albert Einstein.106 In retrospect Einstein felt that his excursion into aerody­namics had been irresponsible—he used the word Leichtsinn. From 1915 to

1917, Einstein had been a consultant to two aircraft firms, LVG and Merkur, and an aircraft had been equipped with the Einstein wing. The test pilot for LVG, Paul Ehrhardt, barely managed to get the machine off the ground and gave his professional opinion on the wing by saying that the airplane flew like a pregnant duck.107 The Gottingen tests made the same point in more scientific terminology.

No account remains of how Einstein actually designed the wing, but some insight into his thought processes may be gained from an article he published in 1916 in Die Naturwissenschaften. Here he set out to explain, in elemen­tary terms, the basic principles of lift.108 How does a wing support an air­craft and why can birds glide through the air? Einstein declared, “Uber diese Frage herrscht vielfach Unklarheit; ja ich mufi sogar gestehen, dafi ich ihrer einfachsten Beantwortung auch in der Fachliteratur nirgends begegnet bin” (400) (There is a lot of obscurity surrounding these questions. Indeed, I must confess that I have never encountered a simple answer to them even in the specialist literature). This is a striking claim, given that Einstein was writing a number of years after the publications of Kutta, Joukowsky, and Prandtl.

Einstein drew an analogy between the flow of fluid through a pipe of vari­able cross section and its flow around a wing. As fluid passes along a pipe that gets narrower, the fluid speeds up. By Bernoulli’s law the pressure will be lower in the fast, narrow section than in the broader section. Einstein then invited the reader to consider a body of incompressible fluid with no signifi­cant viscosity (that is, a perfect fluid) which flowed horizontally but where the flow was divided by a thin, rigid, dividing wall. The wall was aligned with the flow except that it also had a curved section where the wall bulged up­ward. (See fig. 7.17, which is taken from Einstein’s paper.) The curved section looks a bit like a cat’s back and would appear to have the same shape as the underside of the Einstein wing tested at Gottingen. Einstein argued that the fluid above the dividing wall will behave like the fluid in a pipe when it en­counters a narrowing of the pipe and will speed up and exert a diminished pressure on the wall. The fluid below the wall will behave like the fluid in a pipe when it encounters a widening out of the pipe so it will slow down and increase the pressure on the wall. The fluid pressure pushing upward on the curved section of the dividing wall will thus be in excess of the pressure push­ing downward, so there will be a resultant force upward All that is necessary now is to imagine that most of the dividing wall has been removed, leaving behind just the curved section. This procedure, argued Einstein, will retain the features of the flow that generate the pressure difference and hence will represent a wing with lift.

Einstein’s argument rested on the assumption that the removal of all but

the curved portion of the dividing wall would leave the flow unchanged at the leading and trailing edge of the remaining arc. He appeared to take this as ob­vious: “Um diese Kraft zu erzeugen, braucht offenbar nur ein so grofies Stuck der Wand realisiert zu werden, als zur Erzeugung der wirksamen Ausbiegung der Flussigkeitsstromung erforderlich ist” (510) (To generate this force it is obviously only necessary for part of the wall to be real. It need only be suf­ficiently large to produce the effective curvature of the flow). It is puzzling that Einstein made no mention of circulation. Was he aware of the circulation theory of lift? This remains unclear, but given that he assumed the air to be a perfect fluid, it makes it all the more important to ask how he proposed to circumvent d’Alembert’s paradox. How did Einstein expect to get a lift force rather than a zero resultant?

Although Einstein wrote in a dismissive way about the aerodynamic liter­ature, he had, in effect, taken the discussion back to where Kutta started it in 1902. When, following Einstein’s instructions, all of the wall dividing the flow is removed except for the curved piece, what is left is essentially Kutta’s arc at a zero angle of incidence. Einstein did not specify that the curve he discussed in his article was precisely the arc of a circle, but his argument was offered as a general one. If it were right it would apply to Kutta’s arc. But it does not. The arc is a counterexample to what seems to be Einstein’s argument, that is, to any argument that depends on ideal fluids but does not make provision for circulation. In order to generate lift, and to generate the requisite speed dif­ferential between the upper and lower surfaces of the arc, a circulation must be postulated. In a continuous perfect fluid flow, without an independently specified circulation, such an arc would not produce lift. D’Alembert’s para­dox would come into play. Such an arc would not have its stagnation points on the leading and trailing edge. The formula for the complex velocity has singular points indicating infinite velocities for the ends of the arc.109

In his diagram Einstein put the stagnation points on the leading and trail­ing edges of the arc even though he did not explicitly invoke a circulation. He put them where they would have been, given the appropriate amount of circulation needed to avoid infinite velocities at the edges. He did this by sup­posing that the guiding effect of the dividing wall, smoothly leading the fluid toward, around, and then away from the arc, would still be present even when the wall was removed and only the arc was left. There are no grounds for this assumption. Perhaps the best that can be said is that Einstein had, in fact, made provision for circulation in his analysis, but had done so tacitly and by questionable means. In his published lectures, a few years later, Prandtl gave a diagram that was almost identical to Einstein’s but, in Prandtl’s case, the com­ponent of circulation in the flow was properly identified and made explicit.110 Prandtl’s diagram is reproduced as figure 7.18 for purposes of comparison with Einstein’s figure.

It would seem that Einstein had little knowledge of current developments in the field of aerodynamics. This episode is a salutary reminder of the differ­ence between fundamental physics and technical mechanics. Eminence in the former does not guarantee competence in the latter.

A significant part of the experimental evidence for the circulation theory came from Arthur Fage (1890-1977). Fage was a retiring man who had trained as an engineer at the Royal Dockyard School in Portsmouth. His fa­ther had been a coppersmith in the dockyards. Fage won a scholarship to the Royal College of Science in London and then moved to the National Physical Laboratory in October 1912 as a junior assistant in the aeronautics section. In 1915 he published The Aeroplane: A Concise Scientific Study.29 The book was “written to meet the requirements of engineers” (v). It embodied a wholly empirical approach and made no mention of either the discontinuity theory or the circulation theory. With his “infinite capacity for taking pains,” Fage “became progressively a better and better research scientist” and acquired the reputation of being one of the NPL’s most meticulous experimenters.30

After Bryant and Williams’ work on the infinite wing, the next step in the Research Committee’s plan was to test Prandtl’s account of the flow around a finite wing. This work was undertaken by Fage and L. F. G. Simmons.31 They tested a rectangular model wing with a 3-foot span and a chord of 0.5 foot that was set at an angle of incidence of 6° to a wind of 50 feet per second. The wing had a cross section known as the RAF 6a. This particular wing was chosen because it had been studied in earlier experiments at the NPL, and the lift coefficient and the distribution of lift along the span were already known. Fage and Simmons used a speed-and-direction meter to probe the space round the wing to build up a detailed, quantitative picture of the flow. They measured the flow as it cut a number of transverse planes across the wind channel at various distances behind the wing (see fig. 9.5). If the x-axis is taken as the longitudinal axis of the channel, then positions on these trans­verse planes would be defined in terms of y – and z-coordinates where the y-axis lay parallel to the span of the wing and the z-axis indicated positions above and below the level of the wing. These coordinate axes are shown in the figure. Notice that the origin is located at one of the wingtips. The transverse planes chosen for study were called A, B, C, and D. Distances were expressed in terms of the chord of the wing. Plane A, which featured very little in the subsequent discussion, lay at a distance of about half a chord in front of the wing. Planes B, C, and D, the main interest of the experimenters, lay respec­tively at distances of x = 0.57, x = 2.0, and x = 13.0 chords behind the wing. The aim was to measure the properties of the trailing vortices as they cut through planes B, C, and D. On the basis of previous work (such as Piercy’s), Fage and Simmons were by now confident of the existence of the vortices but their question was: Did these vortices behave quantitatively in the way that Prandtl had assumed?

For convenience the test wing was actually mounted vertically and the speed-and-direction meter was inserted through a hole in the floor of the wind channel. The meter could be moved up and down, parallel to the lead­ing and trailing edges. The tips of the wing were fastened to transverse run­ners on the roof and floor of the wind channel so that the wing could be made to slide from one side of the channel to the other without altering its angle of incidence. These two degrees of freedom (the wing going from side to side and the meter going up and down) allowed measurement to be made within

a given, transverse plane. After taking measurements in plane A, they had to remount the wing at a different distance from the meter to measure the flow in plane B, and similarly for planes C and D. The measurements were made with an instrument of a standard type developed at the NPL (see fig. 9.6). It consisted of four open-ended manometer tubes forming a head that could be tilted on its mounting as well as turned from side to side. The orientation of the head of the meter could be manipulated from outside the wind channel. When the head was directly aligned with the local flow, the pressure in the four tubes was equal, which allowed the direction of the air at that point to be read off. The speed of the air was given by the difference in pressure between one of the four tubes and a fifth open-ended tube pointing in the direction of the free flow.32

Fage and Simmons took hundreds of measurements in order to build up a quantitative picture of the flow, and they were able to draw a number of important conclusions. First, they found that the velocity components (u) in the direction of the main flow (that is, parallel to the x-axis) were very little changed by the air having passed over the wing. The circulation theory implies that the speed immediately above and below the wing was modified, but measurement showed that soon after its passage over the wing the u-component settled down to a steady value that only varied by about 1 percent. This (approximately) constant velocity could therefore be ignored when considering the flow in the transverse planes. Exploiting this fact, Fage and Simmons argued that the flow within each plane could be considered as a two-dimensional flow. This two-dimensional flow was taking place in the (y, z) plane, so the two components of speed in the flow were v (along the y-axis) and w (along the z-axis). Fage and Simmons proceeded to com­pute the stream functions, streamlines, and vorticity of this flow at a large

figure 9.6. Apparatus used by Fage and Simmons for detecting the speed and direction of the flow behind a wing. As shown, the neck of the apparatus is inserted through the floor of the wind tunnel. The head of the apparatus can be raised or lowered, turned from side to side, and also rotated. From Fage and Simmons 1926, 306. (By permission of the Royal Society of London) number of points in each transverse plane by using the values taken from their measurements.

By the term “vorticity” Fage and Simmons meant (dw / dy—dv / dz). This is the quantity previously introduced and defined (in chap. 2) as the measure of the rotation of a fluid element. The two terms “rotation” and “vorticity” were used interchangeably. Each of the two parts of the expression represents a rate of change and hence the slope of a graph. Thus the value of dw/dy at a given point refers to the rate of change of the speed w with distance along the y-axis at that point. A corresponding definition applies to dv/dz. Fage and Simmons had a sufficient number of velocity readings of v and w at a sufficient number of points to find the relevant slopes and rates of change. They were thus in a position to compute the rotation or vorticity at each of these points. Once numerical values of this expression were established for a large number of points on the planes B, C, and D, it was possible for the ex­perimenters to draw curves linking up the points of equal vorticity. The result was a striking picture of the flow.

Fage and Simmons’ figures show the contours of equal vorticity in each of the transverse planes behind one-half of the wing (see fig. 9.7). The wing is not shown on these diagrams but lies horizontally, obscured, as it were, by the vorticity. The wingtip would be on the right, and the center of the wing, which is off the diagram, would be on the left. The top figure refers to plane B immediately behind the wing; the other two refer to planes C and D, which are set farther back. Examination of the figures shows that initially (at about half a chord behind the wing) vorticity is spread in a nar­row sheet along the trailing edge but becomes more concentrated near the tip. At a distance of two chords behind the wing, the vorticity at the tip has becomes less intense and more spread out. These changes, said Fage and Simmons, demonstrate the unstable character of the vortex sheet and show that it rolls up in the way predicted by Prandtl. The third picture shows the rolling up to be almost complete. The vorticity is now confined to the area near the wingtip. Citing Lamb’s Hydrodynamics, Fage and Simmons state that the diagrams show that, in accordance with the classical mechanics of vortex motion, the vortices behave as if they are attracted to one another. Their centers are progressively displaced toward the center and away from the tips.

Fage and Simmons then used their data to test the quantitative aspects of the theory and to construct detailed connections between their observations and the mathematics of Prandtl’s picture. The central questions were whether the flow (outside the trailing vortices) was irrotational and whether the lift and circulation around the wing were connected to the trailing vorticity

according to the laws identified by Prandtl. To address the question of irrota – tional flow, Fage and Simmons computed the circulation around a sequence of rectangular contours of increasing size drawn on their transverse planes. They chose rectangular contours, with sides parallel to the y – and z-axes, be­cause they already had the relevant velocity components needed for the cal­culation. The circulation they sought was given by the quantity (vdy + wdz) calculated around the contour. The chosen rectangles all enclosed the same areas of vorticity in the wake of the wing. According to Stokes’ theorem the circulation around these contours gave a measure of the vortex strength they enclosed. The calculated values for the circulation were close to one another, and this, argued Fage and Simmons, showed that outside the vortex wake the flow was irrotational—that is, as the contours got bigger, no further rotating fluid elements, and hence no further vorticity, can have been enveloped by the contour.

This argument for the irrotational character of the flow outside the vor­tex wake had exactly the same form as Glauert’s inference when he gave his preliminary analysis of Bryant and Williams’ results. A number of contours of increasing size display the same amount of circulation, ergo the contours pass through an irrotational flow. This was the inference to which Taylor had taken exception. Had Fage and Simmons fallen into the same trap as Bry­ant and Williams? The answer is no. As a deductive inference the argument put forward by Fage and Simmons is not compelling (for the reasons that Taylor gave), but as an inductive inference the conclusion is plausible. Given their other computations dealing with vorticity in the flow, and the contours showing its distribution, these results reinforced the conclusion that the flow outside the wake was, indeed, irrotational.

The most important quantitative question was whether the strength of the vortices trailing from the three-dimensional wing had been correctly predicted by the proponents of the circulation theory. Fage and Simmons deduced that, according to Lanchester and Prandtl, the strength of vorticity leaving the semi-span of the wing should equal the circulation around the median section of the wing. If these two quantities could be found and com­pared, then the prediction could be tested. Fage and Simmons developed this argument mathematically, but the intuitive basis of the deduction can be seen immediately by inspecting figure 9.8.

The figure shows a version of the refined horseshoe vortex. The vortices that run along the span of the wing all pass through a contour around the me­dian section of the wing. The value of the circulation around this contour is determined by the vortices that run through it (from Stokes’ theorem). All of these vortices then peel off the trailing edge of the wing. Their total strength can be assessed by the circulation around another contour, namely, any con­tour that captures and includes them. For example it can be measured by the circulation around the rectangular contours drawn in the planes B, C, and D that have been previously discussed. Fage and Simmons compared these empirically based values with the predicted circulation around the median section.

How did Fage and Simmons evaluate the predicted circulation? It is the

quantity that has previously been called Г0. Recall that in chapter 7, it was shown that by making the assumption that the lift distribution was elliptical, Prandtl had given an expression that predicted the value of Г0 from known properties of the wing and the flow. Prandtl had shown that

2VSC.

г = L,

0 bn

where CL is the coefficient of lift, S is the area of the wing, b is the span, and V is the free-stream velocity. All of these were known quantities for the wing used in Fage and Simmons’ experiment. In this way they were able to compare the predicted value of Г0 and the observed results for the circula­tion around the vorticity coming away from the trailing edge. The two values should be the same. The theoretical prediction was corroborated for the flow up to two chord lengths behind the wing, that is, for planes B and C. On plane D, at a distance of thirteen chord lengths, the agreement was not so good. The measured vorticity was around 18 percent too small. Some of the vortic – ity appeared to have dissipated, but, within the limits of experimental error, Fage and Simmons declared that the match between theory and observation was a good one.

———– СглЬтти1 Ur*S of Y

——- – Фспіоиґ Uno of Ф

figure 9.9. The streamlines of a trailing vortex at three distances behind the tip of a finite wing. The position of the wing is shown on the diagrams with the wingtip on the right, behind the center of the vortex. The centerline of the wing is on the left. From Fage and Simmons 1926, 320-22. (By permission of the Royal Society of London)

Fage and Simmons not only drew the lines of equal vorticity (shown in fig. 9.7) but also drew the actual streamlines of the vortices that issued from the wings and passed through the planes B, C, and D. This, too, was an exer­cise that involved considerable computation. First they had to check that the preconditions for the existence of a stream function, y, were fulfilled. Was the continuity equation satisfied? Laboriously, they concluded that it was. They then identified the values of the stream function at a large number of points in the (y, z) plane in order to draw the streamlines y = constant. On grounds of symmetry they took the line y = 3, the center line of the wing, as the streamline y0. The pictures they produced from their measurements and computations were convincing. “The diagrams,” they said, “illustrate clearly the changes in the character of the airflow behind the aerofoil: close to the aerofoil, the contour lines are ovals with the ends pointed inwards; as the distance behind the aerofoil increases, the contour lines become more and more circular—indicating that the vorticity is becoming more concentrated” (320). The results are shown in figure 9.9.

Fage and Simmons had significantly strengthened the case for the cir­culation theory. They had effectively picked up the argument at the point where Pannel and Campbell had left it in 1916 when curiosity seems to have been swamped by collective incredulity. Unlike the conclusions of Bryant and Williams, the work of Fage and Simmons was not challenged by Taylor.33 Others, such as Lanchester and Foppl, had made qualitative observations of trailing vortices and, more recently, photographs had been taken, but now the vortices had been directly measured and linked to the mathematically defined relations of the circulation theory. Could it now be said that the entities that were subject to empirical measurement in the laboratory had been clearly identified as one and the same with the entities referred to in the circulation theory? If so, the work of Fage and Simmons would represent a qualitative change in the status of the circulation account of lift. What had previously been a theory could now, arguably, be counted as a directly verifi­able fact.

How can the Tripos tradition be invoked to explain both failure, in the case of lift, and success, in the case of stability? Should not the same cause have the same effect? The answer is that my causal explanation is not meant to ex­plain success and failure. It explains the preconditions of success and failure, that is, what succeeded or failed, and the response to success and failure. It concerns the path that the Cambridge scientists and their associates chose to follow, or declined to follow, rather than the consequences of the choice and what was found along the path that was selected. Neither the British nor the Germans knew what awaited them. Neither group could select their course of action in the knowledge of what would be successful or unsuccessful. Both had to take a gamble. In the case of the theory of lift, the British gamble failed and the German gamble succeeded.

Lanchester did not approve of the theoretical choices made by what he called the Cambridge school, but he had a clear understanding of the meth­odological gamble involved. He did not use the language of gambling, but as we have seen in his reflections on the role of the engineer, he chose the more cerebral metaphor of playing chess. The scientist and the engineer were like players in a game of chess who were confronting an opponent called Nature. As a player, Nature was subtle and her moves could not be easily predicted. Lanchester expressed the contingency of the outcome by saying that, at the outset, no one could identify which moves in the game were good ones and which were bad. A sound-looking move might turn out to be a mistake, and an apparent mistake might turn out to be the winning move. Only the un­known future would reveal this.

Lanchester’s metaphor can, and should, be taken further. It does not just apply to the opening moves in the game of research or to what is sometimes called “the context of discovery” as distinct from “the context of justifica­tion.” It applies to the entire course of research and development ranging from the origin of ideas to their acceptance and rejection. The game with Nature does not come to an end when results begin to emerge and when the chosen strategy of research starts to generate successes and failures. The uncertainties Lanchester identified at the outset of the game still inform the responses that have to be made to the feedback from experience. If the scien­tist or engineer scores a success, the question remains whether it will prove to be of enduring significance or short-lived. If there is a failure, does it indicate the need for a revision of the strategy or merely call for more resolve? These radical contingencies and choices can never be removed, and in one form or another, whether remarked or unremarked, they are present throughout sci­ence. Indeed, they are present in every single act of concept application. The game never comes to an end.49