Anonymity and Connectivity

Lanchester’s Aerodynamics was reviewed anonymously in Nature on August 18, 1908.46 The overall judgment was ungenerous and negative. No reader was likely to come away with the idea that the book contained striking insights into the nature of flight but instead that Lanchester was proposing a theory that was neither original nor successful. The theory was, perhaps, the product of a lively mind, but not a mind whose powers could be relied upon. The parts of the book that contained Lanchester’s most characteristic opinions were de­scribed by the reviewer as “the more shaky theoretical chapters” (338).

There was qualified praise for some of the more empirical sections, which described Lanchester’s experiments on viscosity and skin friction. The glider experiments, conceded the reviewer, gave results that were “remarkably con­sistent.” Lanchester’s account of the “chief methods and results of hydrody­namics,” which lay at the basis of his theory of lift, were described as “on the whole very clearly written,” but the reader was warned that Lanchester was “not, however, content to follow orthodox theory.” It was in chapter 4 of the book, noted the reviewer, that Lanchester “leaves behind the solid ground of orthodox theory” and “attempts to work out the motion of a curved lamina,” that is, a winglike surface (338). Furthermore, Lanchester’s originality was challenged: “It seems to us that the author is wrong in claiming to be the first to give a theory of the motion of curved surfaces, and [in claiming] that Lilienthal had only practical acquaintance with the curved form, for Lilien – thal clearly realised that the effect of curvature was to diminish eddy motion and to give an increased upward pressure due to the centrifugal force of the air. The theory has been worked out mathematically by Kutta, and his results are in fair agreement with Lilienthal’s experiments” (338).

The reviewer then turned to Lanchester’s own explanation of how a curved plate generates lift. It was introduced and dismissed in one sentence: “The author of the present volume attempts to work out the problem by applying the theory of cyclic motion to the motion of a surface in two dimensions, but it is difficult to see how this can have any application to the case of a lamina moving in free air” (338). Before looking into this expression of doubt I must address two preliminary points that concern the reviewer’s mention of Kutta. First, it looks as if the reviewer did not appreciate that Kutta had put forward a cyclic theory. Second, Kutta’s main contributions were published in 1910 and 1911, two or three years after the review. So what was the reviewer’s source?47

Other than personal contact, there were two possible sources of informa­tion. One was a brief account of his work that Kutta himself published in 1902 in the Illustrirte Aeronautische Mittheilungen.48 He gave his main results in the form of a complicated and opaque formula (not the simple product of density, circulation, and free-stream velocity). Kutta said that to reach the given formula he had used conformal transformations, but the assumptions behind his analysis were not explained. The other source was a footnote ref­erence to this article by Sebastian Finsterwalder, Kutta’s research supervisor at the technische Hochschule in Munich.49 Finsterwalder had contributed the article on aerodynamics to Felix Klein’s multivolume Encyklopadie der Math – ematischen Wissenschaften. The relevant volume had been published before the Nature review appeared. The cyclic character of Kutta’s theory was not apparent in the 1902 paper, though its relation to Lilienthal’s work was explic­it.50 The same holds true of the Finsterwalder reference: there was no men­tion of the role of circulation. If these were the sources used, it could account for the misleading way in which Kutta was invoked in the review.

Why did the reviewer find it “difficult to see” how an account of a two­dimensional, cyclic motion could have any application to the motion of a lamina in free air? The reasons behind the difficulty were not explained, so it is necessary to make a conjecture about the argument that was probably in the reviewer’s mind. The worry was about the move from two dimensions to three dimensions. Why should there be a problem about generalizing an ac­count of cyclic or vortex motion in this way? The answer lies in the properties of the space around the wing that mathematicians call “connectedness”—a topological theme with which all Cambridge-trained mathematicians would be familiar.51

Connectedness refers to the conditions under which a contour in the form of a closed loop can be shrunk into a point or stretched and distorted so that it coincides with another closed loop. A “simply-connected” space is one in which every closed loop can be changed into any other closed loop without going outside the space. A “multiply-connected” space is one that is divided by barriers so that it ceases to be true that any two arbitrary loops can be made to coincide. Now a loop enclosing the infinitely long wing cannot be unhooked from it. It can be transformed into any other loop that is itself already around the wing, but it cannot be transformed into a loop that does not go around the wing. The space around an infinite wing is thus “doubly connected,” while the space around a finite wing is “simply connected.”

The move from a two-dimensional analysis to a three-dimensional analy­sis thus involves a move from a multiply connected space to a simply con­nected space. But why should this matter? A mathematically sophisticated re­viewer will have known that, in a simply connected region, the only possible form of irrotational motion is acyclic.52 In an acyclic motion there is no circu­lation and hence no lift. The reviewer seems to have assumed that Lanchester was exploiting a special, topological feature of two-dimensional flow but was then illegitimately applying the analysis to the three-dimensional case.53 This assumption may explain why it was “difficult to see” how a theory of cyclic motion in a surface of two dimensions could have any application to a lamina moving in free air, that is, in three dimensions.

Was Lanchester’s work really vitiated by these considerations? The answer is no. If this was the reviewer’s argument, it was wrong. Lanchester had at­tended with some care to issues of connectivity. He stated explicitly that “we are consequently confined, in an inviscid atmosphere, strictly to the case where the aerofoil is of infinite extent, for a cyclic motion is only possible in a multiply connected region” (162).

How did Lanchester, having formulated the topological problem for himself, get round it? He needed some way to render the space of the three­dimensional case multiply connected. Lanchester did this by appeal to the trailing vortices issuing from the wingtips and reaching back to the ground. This method divided the space in such a way as to destroy its simple con – nectivity.54 In figure 81 of his book (175), Lanchester gave a clear diagram of the vortices reaching back from the wingtips to the ground. But if Lanchester had anticipated and solved this problem, there was still another issue left. If circulation now makes mathematical sense, there is still the physical problem of how it gets started. Lanchester conceded that, as long as the atmosphere was viewed as an inviscid fluid, his vortices could be neither created nor de­stroyed. Such a system, he said, “in a fluid that is truly inviscid would be un – creatable and indestructible” (174). His response was to appeal to the viscos­ity of real fluids: “In dealing with a real fluid the problem becomes modified; we are no longer under the same rigid conditions as to the connectivity of the region” (175). Lanchester’s remarks were perceptive, but the problem of the creation and destruction of vortices, and thus the problem of how circulation could arise, would continue to haunt the theory.