Sine-squared law

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The sine-squared law was a principle derived from Isaac Newton that air resistance to a plane in motion increased in proportion with the square of the sine of the angle between the plane and the air.

Experimentation revealed that this calculation dramatically overestimated air resistance.


Hallion, 2003 (pp. 102–103):

Interpretation of one of Newton's propositions in his landmark Principia Mathematica indicated that in calculating the resistance of a plate set in a flow, one had to square the sine of the angle formed by the plate and the relative flow. Fortunately this concept, the so-called Newtonian sine-squared law proved incorrect: the resistance force acting on the plate is proportional to the sine of the angle, not the square of the sine of the angle. Squaring the sine implied drastic increases in drag as the angle of attack increased. If true, this would have demanded construction of totally impractical flying machines having enormous wings that could furnish the requisite lift for an airplane only while operating at minimal angles of attack. Hence it would have called into question whether a successful airplane could ever be built. After further examining Newton's work, later researchers fortunately recognized the error, realizing that wings could operate quite well at modest angles of incidence, and Francis Wenham experimentally disproved this bogus "law" in his first wind-tunnel tests. Nevertheless, it continued to haunt aeronautics even into the early twentieth century, used by ill-meaning critics to assert flight's "impossibility."

And Maxim, 1909, Artificial and Natural Flight (pp. ix–x):

Mathematics of the higher order expressed in elaborate formulæ do very well in communications between college professors—that is, if they happen to be agreed. When, however, these calculations are so intricate as to require a clever mathematician a whole day to study out the meaning of a single page, and if when the riddle is solved, we find that these calculations are based on a fallacy, and the results in conflict with facts, it becomes quite evident to the actual experimenter that they are of little value. For many years, Newton's law was implicitly relied upon. Chanute, after going over my experimental work, wrote that Newton's law was out as 20 is to 1—that is, that an aeroplane would lift twenty times as much in the practice as could be shown by the use of Newton's formula. Some recent experiments, which I have made myself, at extremely high velocities and at a very low angle, seem to demonstrate that the error is nearer 100 to 1 than 20 to 1. It will, therefore, be seen how little this subject was understood until quite recently, and even now the mathematicians who write books and use such an immense amount of fomulæ, do not agree by any means, as will be witnessed by the mass of conflicting controversy which has been appearing in Engineering during the last four months.

Samuel Pierpont Langley:

After many years and in mature life, I was brought to think of these things again, and to ask myself whether the problem of artificial flight was as hopeless and as absurd as it was then thought to be. Nature had solved it, and why not man? Perhaps it was because he had begun at the wrong end, and attempted to construct machines to fly before knowing the principles on which flight rested. I turned for these principles to my books and got no help. Sir Isaac Newton had indicated a rule for finding the resistance to advance through the air, which seemed, if correct, to call for enormous mechanical power, and a distinguished French mathematician had given a formula showing how rapidly the power must increase with the velocity of flight, and according to which a swallow, to attain a speed it is known to reach, must be possessed of the strength of a man.[1]


  1. Reprinted ultimately from an article in McClure's Magazine, June 1897. In Langley, 1908, Researches and Experiments in Aerial Navigation, p. 201. Quoted in Maxim 1909, ibid. See also Langley's Law.