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Patenting Mathematical Methods in Germany

The German Federal Supreme Court has extended its existing jurisprudence on the patentability of programs for computers under paragraphs 1 (3) and (4) PatG to allow a claim to a mathematical method.

Features lacking “technical character” should not be taken into account in the examination for inventive step – in line with the so-called “Comvik/Hitachi” approach employed by the EPO - but a mathematical method is not necessarily non-technical if employed in conjunction with natural forces, since natural laws are themselves generally defined mathematically.


Paragraph 1 (1)  of the German Patent Law (PatG) corresponds to Article 52 (1) EPC and specifies that patents are granted for inventions in all fields of technology, provided that they are new, involve an inventive step and are susceptible of industrial application.

The established jurisprudence of the German Federal Supreme Court (BGH) is that an invention requires a technical teaching, and that this must involve the application of controllable natural forces to achieve a causal perceivable result (BGH Z 52.74, GRUR 69, 672 - Rote Taube). This is the BGH’s way of expressing that an invention must have technical character.

Paragraph 1 (3) PatG - like Article 52 (2) EPC – contains a list of exclusions from patentability, including “programs for computers”. However paragraph 1 (4) PatG limits these exclusions to the subject matter or activities “as such” in a similar manner to the EPC.
In a recent series of decisions the BGH has developed criteria on interpreting the exclusion of programs for computers: the exclusion from patentability is held not to apply if the claimed teaching solves a concrete technical problem by concrete technical means. It is immaterial whether the claimed teaching additionally includes non-technical features. However, these non-technical features should be excluded from the subsequent examination for inventive step.  To date, this test has not been applied to a mathematical method.

Claim 1 in the Flugzeugzustand case 17W(pat)15/11

A method for determining the status of an aircraft, namely the position, velocity and attitude of the aircraft, comprising the steps of:

•    determining a number n of measured values xi, with i = 1, ..., n, of an inertial system which determines the aircraft status, wherein the measured values xi represent points in k-dimensional space, 
•    processing the measured values xi in a Kalman filter to estimate the aircraft status,
characterized in that
•    for each number n of measured values  xi of the inertial system, a first variable mn, and a second variable rn are derived, and 
•    these derived variables are supplied to the Kalman filter for further processing, wherein 
•    the variable mn is the centre vector and the variable rn is the radius of a k-dimensional sphere Bn, within which lie all points xi with  i = 1, …, n; 
•    wherein the sphere Bn is the smallest possible k –dimensional sphere which contains all the points xi, with  i = 1, …, n, of the number n of measured values.

The Facts

The appellant had filed a German patent application which related to a method for determining the status of an aircraft. It was refused by the German Patent and Trademark Office.

On appeal, the Federal Patent Court noted that the prior art already disclosed a method for determining the status of an aircraft using data from both an inertial system and a satellite-based navigation system supplied to a Kalman filter.  A Kalman filter uses an algorithm to calculate a variable from a plurality of measurements known to contain inaccuracies. In one process known in the prior art the Kalman filter was supplied with a calculated average instead of individual variables and covariance data derived to give the scattering of the measured values.

The subject matter of claim 1 of the application differed from the prior art in using different variables, namely a central vector and a sphere with n measurement values within it, the position and size of the sphere being held as small as possible.

The 17th Senate, which heard the case, did not take a position on excluded subject-matter, but held  that the claimed subject-matter differed from the prior art only in respect of considerations in the fields of data modelling, statistics and geometry. It was held that these did not require technical considerations and were not to be taken into account in assessing inventive step.

However, the Senate approved a judicial review under paragraph 100.2 PatG and noted that the question of whether an inventive step could be based on a purely mathematical method which turned out to give a non-predictable advantage had never been answered. The question arose because the appellant had referred to advantages arising from the differences over the prior art which could not be found in the originally filed application; the Senate was sceptical. 

The Federal Supreme Court’s decision

The appeal was successful and led to a referral back to the Federal Patent Court.

In its reasoning  the Federal Supreme Court affirmed the existing jurisprudence that if technical means were used then there was an invention in a field of technology within the meaning of paragraph 1 (1) PatG  (X ZR 121/09, GRUR 2011, 610 Rn 16). In the case in suit the requirement was satisfied by the use of an inertial system in an aircraft and by the Kalman filter, which required a data processing system.

The Federal Supreme Court also stated that the same criteria as for programs for computers applied to mathematical methods within the meaning of paragraph 1 (3) 1 PatG, namely that the exclusion did not apply if the claimed teaching solved a specific technical problem by specific technical means. The claimed method met these criteria, since it was used to determine the status of an aircraft by evaluation of measurements representing technical parameters.

The Court then referred to its jurisprudence that excluded subject-matter should not be taken into account in assessing inventive step; this also applied to mathematical methods excluded in accordance with paragraph 1 (3) 1 PatG.

The Court however noted that a mathematical method was not necessarily non-technical if employed in conjunction with natural forces, since natural laws are generally defined mathematically. It held that the use of mathematical methods to obtain a particular technical result had technical character and such methods could only be considered as non-technical when the claimed teaching did not apply natural forces.

It also held that features distinguishing the invention from the prior art must be taken into account when assessing inventive step. Although these features concerned arithmetic operations affecting the application of statistical methods, there was sufficient connection to the targeted application of natural forces, given that calculation steps based on the acceleration parameters and the Kalman filter served to obtain more reliable knowledge of the condition of the aircraft and thus to influence the operation of the system.

Regarding the question used by the Federal Patent Court to justify a judicial review, the Federal Supreme Court held that it was irrelevant whether solutions to the problem were known in the art and whether claimed differences gave advantages.  It was stated that technical advantages were not a criterion for patentability and that the claimed alternative method of determining the status of an aircraft could not be dismissed as technically nonsensical.

The decision is notable in that the Court puts on an equal footing its existing jurisprudence and that developed by the European Patent Office. This strengthens the impression that the Court is seeking a convergence between German and European practice on exclusions from patentability, a practice to be welcomed in view of the upcoming UPC.


a) Mathematical methods are only patentable if they serve to solve a specific technical problem by technical means.

b) A mathematical method is only considered non-technical if it does not require the use of controllable natural forces in the context of the claimed teaching.

c) In the case of an aircraft, controllable natural forces are involved when a mathematical method uses available measurements to derive more reliable findings about the status of the aircraft, thereby influencing the operation of the system.

d) An object which involves an inventive step is not unpatentable merely because it offers no recognizable advantage compared to the prior art.


This decision is helpful so far as it goes.  It can be applied to many cases with mathematical methods as long as they have some physical reality applied to them, but it adheres to the controllable natural forces doctrine of technicality and does not necessarily help assess the patentability of many mathematical methods that relate more to the realm of, for example, statistical outcomes, Baysean learning, regression algorithms and the like.

Of interest is the last point.  According to EPO Guideline G-VII 5.4(iii), if subject matter claimed solves no technical problem vis-à-vis the closest prior art, there is no technical contribution and it cannot be inventive.  It is helpful here to have affirmation that subject matter can be inventive and yet offer no recognizable advantage compared to the prior art.