by Benjamin Katzenellenbogen
This article was originally published in Law360. Download PDF

I. Introduction

Mathematical formulas and other abstract ideas cannot be patented because the Supreme Court has held that they do not constitute patentable subject matter under 35 U.S.C. § 101.  This issue frequently arises in patents involving business methods and computer software.  Unfortunately, neither the courts nor the patent office has provided a satisfactory explanation for why processes that consist entirely of applying mathematical formulas are not patentable.  Decisions regarding patentable subject matter would be better reasoned, and more consistent, if they were based on meaningful guidance regarding what it is about formulas that makes them unpatentable.  This would focus the patentability analysis directly on whether specific patent claims suffered from these concerns. 

The discussion below addresses conventional arguments for not allowing patents on mathematical formulas, and suggests why they are not satisfactory.  The Pythagorean Theorem has been cited as the archetypal example of an unpatentable formula.  Accordingly, this discussion is presented in the context of the question: Why shouldn’t Pythagoras have been entitled to a patent on the Pythagorean Theorem?

II. The Ground Rules

Evaluating whether mathematical formulas should be eligible for patent protection requires considering a claim that would at least be arguably patentable under the other patent law provisions.  Of course, neither Pythagoras, nor anyone else, would be entitled to a patent on the formula or a process for using it if they applied for it today.  It has been known and used for thousands of years.  This discussion assesses patentability of the formula under current United States patent law, using the hypothetical assumption that Pythagoras has just “discovered” that A² + B² = C². describes the relationship of the lengths of the sides of a right triangle.

III. The Pythagorean Theorem Is Not A “Process”

The first conventional argument against allowing patents on formulas is that they are not a “process, machine, manufacture, or composition of matter,” as required by § 101.  Rather than debate the potentially metaphysical question of whether a formula is a process, this discussion will focus on a claim that expressly recites a process of using the Pythagorean Theorem.  Consider the following claim, which resembles the types of claims that are at issue in many cases regarding software and business method claims:

(1)      A process of calculating the length of a side of a right triangle that comprises using the relationship A² + B² = C².

This claim is literally directed toward a process for accomplishing something useful, but is directed toward the pure application of a mathematical formula.  As discussed below, the case law has not provided a satisfactory explanation for why this type of process is not patentable.

IV. The Pythagorean Theorem Is Not New

One conventional argument is that Pythagoras’ formula was not really “new,” even on the day he came up with it, because Pythagoras did not invent right triangles.  Such triangles have always existed, and have always obeyed the relationship embodied in his formula.  According to this argument, if Pythagoras were able to get a patent on pre-existing triangles, then he would be taking something out of the public domain.  Suddenly, anything that had the shape of a right triangle would infringe a patent.

It is fundamental to our patent laws is that one cannot take something out of the public domain.  However, this concern is addressed by the “novelty” requirement of § 102, and there does not appear to be any reason to engraft a redundant requirement onto § 101.  A claim to pre-existing subject matter is unpatentable because it lacks “novelty,” not because it claims unpatentable subject matter.  For example, the following claim would be unpatentable under § 102:

(2)      A right triangle having sides whose lengths obey the following relationship: A² + B² = C².

All right triangles would infringe this claim.  Therefore all pre-existing right triangles would anticipate this claim.  Such a claim could never be allowed because right triangles pre-dated the Pythagorean Theorem (regardless of when it was discovered).  There is no need for any special rule precluding patents on formulas or other “abstract ideas” to address the problems that would result if one could patent pre-existing subject matter.  The fundamental requirement of novelty should adequately ensure that nobody obtains patent claims on things that are old.

The process claim discussed above does not raise the same novelty concerns.  If nobody before Pythagoras ever performed the calculation expressed in the Pythagorean Theorem, or described how to perform such a calculation, the process of claim (1) would appear to be novel.  Triangles that obeyed this relationship would not infringe, or anticipate.  Nor would this claim prohibit anyone from doing anything they knew how to do before Pythagoras.  Accordingly, it would not remove anything from the public domain.  Allowing process claim (1) would merely give Pythagoras a legal right to exclude others, for a limited time, from performing a calculation that nobody knew how to perform before he disclosed it.  Such a claim would appear to be consistent with the general bargain encouraged by the patent laws.

V. The Pythagorean Theorem Is A Discovery, Not An Invention

Another argument against patenting formulas is that, even if nobody knew about the Pythagorean Theorem, a formula is at most a “discovery” and not an “invention.”  This argument is based on the questionable premise that there is a fundamental distinction between “discoveries” and “inventions,” and that the patent laws are supposed to protect only inventions, not discoveries.  This “discovery” argument points out that the Pythagorean Theorem is an inescapable law of nature that all right triangles obey, and have always obeyed.  Therefore, Pythagoras did not “invent” anything and is not entitled to a patent. 

There are at least two problems with this argument.  First, the same section of the patent laws that has been interpreted as precluding patents on formulas says that, “[w]hoever invents or discovers” any new and useful subject mater, or improvement thereof, may be entitled to a patent.  Thus, the statute does not appear to support a distinction between “inventions” and “discoveries.”

Moreover, even if Pythagoras only “discovered” the mathematical relationship among the sides of a right triangle, a claim to a process of performing that calculation, such as claim (1) above, would appear to encompass a new and useful process that Pythagoras “invented.”  If nobody before Pythagoras knew how to use the lengths of two sides of a right triangle to calculate the length of the third side, it would seem that he “invented” a first process for calculating the length of that third side.  Whether it was based on an underlying “discovery” appears beside the point.

VI. The Pythagorean Theorem Is A Law of Nature

A closely related argument is that patents on formulas are improper attempts to patent fundamental laws of nature that should remain available to all.  This argument appears to overlook the fact that unknown laws of nature are not “available” for exploitation by others in any meaningful sense.  The discovery and disclosure of a previously unknown law of nature, and particularly its use or application, adds to the available pool of human knowledge.

Even if Pythagoras is regarded as having only “discovered” an existing law of nature, prohibiting patents on such “discoveries” would appear to go too far because all inventions can be characterized as mere “discoveries” of laws of nature.  As former Chief Judge Markey of the Federal Circuit once wrote in the context of discussing obviousness:  “Only God works from nothing.  Man must work with old elements.” 

For example, Thomas Edison did not “invent” the law of nature that applying an electrical current to a carbonized filament in a vacuum produces light.  He merely discovered certain applications of that particular law of nature (ignoring whether he really was the first person to do so).  The laws of physics are such that applying an electrical current to a carbonized filament in a vacuum produces a light bulb.  That was as true before Edison as it was after.  Thus, the law of nature argument seems to go too far because it suggests that allowing Edison a patent on the “discovery” of the light bulb would have been taking something out of the public domain.  This would eliminate patents on all sorts of things that are conventionally regarded as properly patented “inventions.”

Arguments that seek to characterize the inevitable results of applying laws of nature as unpatentable “discoveries” rather than patentable “inventions” are particularly problematic for the chemical arts.  No person “invented” any of the laws of nature that result in chemical reactions or that produce new chemical compounds.  For example, no person “invented” the law of nature that combing the chemical toluene with nitric acid under appropriate conditions produces the stable and powerful explosive TNT.  Julius Wilbrand was just the first person to “discover” that law of nature.  As with the light bulb, or virtually any other “invention,” anyone who had added nitric acid to toluene under the right conditions would have produced the same result.  This is because the laws of physics, chemistry and mathematics apply to all people equally.  Inventors are just those who happen to be the first to discover certain interesting applications of those laws.

The patent system distinguishes between those who could have figured out how to do something and those who actually figured out how to do it.  The patent system assumes that providing a legal (or financial) reward for disclosing how to do things benefits society by encouraging people to make such discoveries.  Accordingly, the patent laws reward the first person to discover and publicly disclose how to apply the laws of nature to further some useful end with the legal right to exclude other from doing so for a limited period of time.  These assumptions may not be correct, but precluding a patent on anything that is a law of nature would arguably eliminate all patents.  Accordingly, this does not appear to be a satisfactory basis for precluding only patents on formulas.

VII. The Pythagorean Theorem Is Too Fundamental To Be Patented

Another related argument is that patents on formulas should not be allowed because they preclude further research.  This argument again appears to selectively raise a concern that is not unique to patents on formulas.  There does not appear to be any reason why patents on mathematical formulas are inherently more likely to discourage subsequent inventions than any other type of dominating patent. 

Patents are traditionally assumed to promote innovation, both with respect to broad “pioneering” inventions that create entirely new fields of research, and narrower inventions that merely improve upon known concepts.  Prohibiting patents on formulas because they tend to be “pioneering” inventions would stand conventional wisdom on its head.  If formulas tend be broad pioneering inventions that constitute the necessary foundations for entirely new fields of human endeavor, that would seem to be a strong argument in favor of encouraging such inventions by allowing formulas to be patented, not a reason to preclude such patents.

Concerns that the breadth of claims on mathematical formulas can easily exceed the true scope of the invention should be adequately addressed through other requirements of the patent law.  As with all patents, § 112 would limit the scope and validity of mathematical formula claims based on the breadth of the written description and enabling disclosure provided by the inventor.

VIII. Conclusion

If there is a well-reasoned basis for precluding patents on formulas, making it more explicit will enable courts and patent examiners to make better and more consistent decisions based on whether specific claims implicate those concerns and policy considerations.  On the other hand, if there really is no such explanation, the concerns that have been associated with claims on mathematical formulas can be better addressed through applications of other aspects of the patent laws.  Such recognition would produce better and more consistent outcomes by focusing decisions on more relevant criteria.  Accordingly, if there is a sound reason why Pythagoras would not have been entitled to a patent, the Supreme Court or the Federal Circuit should provide the legal and scientific communities with that analysis.