Mathematics on Appeal: Hamilton’s Equations in Lapsley v. Xtek, Inc.

Better off Ted is a satiric TV series about wacky scientists and managers at the amoral high-tech company, Veridian Dynamics. In Lapsley v. Xtek, Inc., 689 F.3d 802, 812 (7th Cir. 2012), Meridian Engineering supplied the scientific breakthrough -- about dynamics -- for plaintiff. But the resemblance ends with the company names. There is nothing funny about this case.

In Xtek, a machine at a steel rolling mill accidentally ejected industrial grease with such force that it shot through a worker’s body, permanently disabling him. As the court of appeals explained:

At trial the jury found that the accident was caused by a design defect in a heavy industrial product designed and manufactured by defendant Xtek, and sold and installed in the mill. That equipment contained an internal spring that could exert over ten thousand pounds of force. The jury accepted the theory of plaintiffs' expert witness, Dr. Gary Hutter, that the spring was the culprit mechanism behind the accident and that an alternative design of a thrust plate in the equipment would have prevented the disabling accident. Xtek has appealed, challenging the district court's denial of its Daubert motion that sought to bar Dr. Hutter from offering his expert opinions, which were essential to the plaintiffs' case.

Xtek’s Daubert challenge, as described in the opinion, was that Dr. Hutter, an engineer who founded the consulting firm, was performing “the simulation of science, not science,” when he used a mathematical model to infer the cause of the release and to conclude that an alternative design would have prevented it. He did no empirical testing to verify his theories, and to Xtek’s lawyers, the notes attached to Dr. Hutter’s report were the “equivalent of Sanskrit.”

The U.S. Court of Appeals for the Seventh Circuit squashed this argument. The opinion could have been rather mundane. It need only have stated that (1) Dr. Hutter used equations that were not in dispute; (2) defendant raised no question about the expert’s assumptions; and (3) supplementary physical testing was not feasible.

In addition to making these points, however, the court took the opportunity to lecture counsel about science. The lecture began with the advice that

Lawyers and judges who were not trained in science can benefit from the famous “Two Cultures” lecture given in 1959 by British scientist and novelist C.P. Snow, in which he described the cultural gap between persons schooled in the sciences and those schooled in the humanities:

A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: Have you read a work of Shakespeare's?

Nowadays, the opinion admonished, “[j]udges and lawyers do not have the luxury of functional illiteracy in either of these two cultures.” The trial judge was not guilty of such illiteracy. There was “no indication, either from the district court's Daubert ruling or its later discussions of the expert evidence during trial, of any deficiency in the court's preparation or in its understanding of the proposed evidence.”

This leaves the lawyers. “For the curious” the court supplied a link to a digitalized version of Sir Isaac Newton’s Philosophise Naturalis Principia Mathematica. This classic is not written in Sanksrit, but the original Latin is challenging enough. Consequently, the court reproduced and described the equations for force, kinetic energy, and pressure. After a nod to quantum mechanics (but not relativity), it announced that “Newtonian physics still provides a reliable and workable description for the mechanical systems of a steel mill.” Seeing no problem with the expert’s use of these “basic equations of classical mechanics” and invoking Galileo and Descartes in addition to Newton and two of the Bernouillis (Daniel and Johann, whose animosity was such that they would not have appreciated being mentioned in the same sentence), the court concluded that “physical tests of [the] theories with regard to causation (the effect of the spring releasing) and alternate design (the reduction in pressure from the grease grooves)” were not essential.

Despite its unusual historical and mathematical sweep, Xtek has a narrow holding. It does not approve of all mathematical modeling of complex phenomena in lieu of physical tests. To be sure, the opinion asserts that “[a] mathematical or computer model is a perfectly acceptable form of test,” and “that simulation is one of the most common of scientific and engineering tools.” Moreover, it contains the memorable line,“We do not require experts to drop a proverbial apple each time they wish to use Newton's gravitational constant in an equation.” But that analogy only goes so far. The reason physicists need not remeasure G every time they wish to use the inverse-square law of gravitational force is that the value already is known from many experiments. In Xtek, the question is the value of an unknown quantity.

The holding is simply that when the equations (including the constants in them) are familiar and their applicability is unchallenged, an expert can use them to make relevant computations. Additional physical testing may be required under Daubert in some cases, but not when it would be too expensive or dangerous. “Around the world, computers simulate nuclear explosions, quantum mechanical interactions, atmospheric weather patterns, and innumerable other systems that are difficult or impossible to observe directly” (emphasis added). Thus, Dr. Hutter did not have “to try to recreate the binding up of a ten thousand pound spring to produce a potentially deadly jet of industrial grease ... to testify to the results of his mathematical simulations.”

It is not often that one encounters equations in a judicial opinion. Years ago, in Branion v. Gramly, 855 F.2d 1256 (7th Cir. 1988), Judge Frank Easterbrook tossed out a partial differential equation and a few calculations of his own to chastise a habeas corpus petitioner’s lawyers for “fooling with algebra” in their brief. I, 1991, I described that effort as needlessly opaque but basically correct.

In this case, the author of the opinion, Judge David Hamilton, stuck to simpler equations. And that takes me to another similarity in names. Judge Hamilton's equations also are simpler than the elegant formulation of dynamics known to physicists as Hamilton’s equations (invented by William Rowan Hamilton, 1805-1865). 

References

• David H. Kaye, David E. Bernstein Jennifer L. Mnookin, The New Wigmore: A Treatise on Evidence: Expert Evidence, New York: Aspen Pub. Co., 2d ed., 2011(updated annually)
• David H. Kaye, Statistics for Lawyers and Law for Statistics, Michigan Law Review, Vol. 89, No. 6, May 1991, pp. 1520-1544 (review essay discussing Branion v. Gramly)