Plaintiff sues the United States for patent infringement, within our jurisdiction under 28 U.S.C. § 1498 (1976). The case is before us on plaintiffs exception to the recommended decision submitted by Trial Judge Browne under Rule 54(a). The trial judge concluded that plaintiffs patent was invalid under 35 U.S.C. § 101 (1976), but denied defendant’s motion for patent invalidity under 35 U.S.C. § 103 (1976). He also held that if the patent had been valid, then defendant’s actions amounted to infringement. Upon consideration of the briefs and the oral argument of the parties, we agree with the trial judge’s conclusion of invalidity under 35 U.S.C. § 101 and to this extent adopt his opinion and conclusion of law as the basis *183for our judgment in this case. Because of our holding of invalidity under 35 U.S.C. § 101, it is not necessary to render a determination of infringement. For the same reason, we also do not render a determination as to the correctness of the denial of defendant’s motion for patent invalidity under 35 U.S.C. § 103. We have modified the trial judge’s opinion, therefore, to delete all references to the 35 U.S.C. § 103 invalidity question and the infringement question. It is, thus, concluded that plaintiff is not entitled to recover, and the petition is dismissed.
The trial judge’s report, modified as above indicated and also by certain minor additions and deletions, follows.
OPINION OF TRIAL JUDGE
BROWNE, Trial Judge:This action is brought by George Arshal (plaintiff) against the United States (defendant) under the provisions of 28 U.S.C. § 1498(a) for recovery of reasonable and entire compensation for the alleged use or manufacture by or for defendant of the invention covered by claims 2, 3, 6, and 71 of his United States Patent No. 3,319,052 (the ’052 patent), issued May 9, 1967, for a "Directional Computer.” The application on which the patent is based was filed January 20, 1966 (Serial No. 533,106) as a continuation of a subsequently abandoned application (Serial No. 206,801) filed July 2,1962.
Defendant moved for summary judgment on March 30, 1978, on the issue of validity, and plaintiff moved for summary judgment on April 7, 1978, on the issues of validity, infringement, and license. The court referred the motions to the trial judge on July 25, 1978, pursuant to Rule 54 for his opinion and recommendation for the conclusion of law.
Upon review of the motions, supporting briefs, and extensive oral arguments of both parties heard on February 7 and 8, 1979, we have concluded that the ’052 patent is invalid; wherefore, plaintiff is not entitled to recover on his claim for reasonable and entire compensation. Accordingly, defendant’s motion for summary judgment is granted and *184plaintiffs motion for summary judgment on the issue of validity is denied. Since holding the patent to be invalid makes moot the issues of infringement and license, these two issues are not considered. The holding of invalidity also being dispositive of the case, the petition is dismissed.
I. The Pleadings
Plaintiff filed his petition, pro se, on December 12, 1975, after having failed to negotiate an administrative settlement of his claim with the Department of the Navy. In his petition, plaintiff alleges that claims 2, 3, 6, and 7 of his patent are infringed by the guidance system manufactured for and used by defendant in its Poseidon missile, and also by the guidance system intended for use in the Trident C4 (Trident) missile. Lockheed Missiles and Space Company, Inc., of Sunnyvale, California (Lockheed) is alleged to be the sole manufacturer of the missiles for defendant.
Defendant filed its answer to the petition on April 9, 1976. In its answer defendant admitted that the Poseidon missiles were supplied by Lockheed and used by defendant, and that certain plans exist with respect to manufacture and use of the trident C4 missile by defendant. Defendant denied, however, that the patent has been infringed or that it was duly and legally issued, or otherwise enforceable. Defendant contends that, in any event, it is licensed under the patent in suit.
II. The Patent in Suit
In order to comprehend the legal issues of this case, it is necessary first to have a basic understanding of the underlying mathematics of the invention, since it involves vectors, vector algebra, and vector calculus. A very brief-discussion of these concepts is presented in Appendix B of this opinion. The discussion which follows presumes both a basic knowledge of elementary algebra and an understanding of the appended discussion of vector mathematics.
The ’052 patent is entitled, "Directional Computer.” It defines the invention as "a vectorial data processing system for extracting or prescribing directions and their rates of *185change from given input vectors.” The patent specification specifically states that the output direction is affected by the input vector in one or both of the following two ways:
(1) So that the output direction rotates around the input vector to describe a conical surface,
(2) So that the output direction rotates directly towards the input vector and seeks coalignment.
The system is described as useful in aircraft flight control, as one example. It is also intended for use to "service demanding applications requiring wide ranging directional controls under a variety of options.” The remaining portion of the specification describes the invention primarily in terms of vector algebra. The drawings illustrate examples of the relationship of the algebraic expressions disclosed in the specification in terms of circuit diagrams and other graphic symbols.
III. Validity of the Claims Under 35 U.S.C. §101 A. The Positions of the Parties
Defendant has alleged in its motion for summary judgment that claims 2, 3, 6, and 7 of the patent are invalid because they are drawn to subject matter which is not patentable under the Patent Act of 1952 (Title 35, United States Code). Specifically, defendant contends that the claims, essentially, are drawn to a mathematical equation, and that a mathematical equation is not patentable subject matter under 35 U.S.C. § 101.2
Claim 2 of the patent is illustrative of all of the claims in suit and states:3
2. A directional computer comprising
(a) a physically defined, stablilized reference frame having means of sensing and suppressing its angular velocity,
(b) means of obtaining a plurality of input signals in representation of any desired values, said input *186signals being referred to said reference frame to express an input vector, and
(c) means to receive the said input signals and generate a plurality of output signals expressing an output vector in relation to said reference frame as the integral of the cross product vector between the said output vector and the said input 'vector, whereby the said output vector rotates around said input vector. [Emphasis added.]
To support its position that claim 2 is essentially directed to a mathematical equation, defendant points to the italicized phrase in part (c) of claim 2. (This language also appears in the remaining claims in suit, namely, claims 3, 6, and 7.) Although plaintiff does not dispute that this language is the verbal equivalent of a mathematical expression,4 plaintiff points out that this language forms only a portion of the claim and that, when considered in its entirety, the claim is not directed to an equation, but to a machine.
B. The Applicable Law
Since mathematical expressions or equations, per se, are not susceptible to patent protection under section 101 of the Patent Act,5 we must determine whether or not the asserted claims define an invention which is merely a mathematical expression or equation. If the mathematical expression or equation is simply an adjunct to the invention sought to be covered, the inclusion thereof in the claims is not fatal to their validity. If, however, the invention itself can be expressed only in terms of the mathematical expression or equation, it is outside the scope of 35 U.S.C. § 101 and, therefore, is invalid under the applicable case law.
*187Tests for determining whether a claim is directed to subject matter which is not patentable have been applied in some cases which predated the 1952 Patent Act, and many others subsequent to that Act. In some early cases, because the "point of novelty” or truly novel element in the claims was determined to be a mathematical expression or equation, the claims were held to be nonstatutory.6 See e.g., In re Abrams, 188 F.2d 165, 89 USPQ 266 (CCPA 1951).
The "point of novelty” test was, at first, followed after the effective date of the 1952 Act, but was formally rejected by the Court of Customs and Patent Appeals (CCPA) in In re Bernhart,7 417 F.2d 1395, 163 USPQ 611 (CCPA 1969).8 In Bernhart, among the claims before that court was an apparatus claim directed to a general purpose computer programmed to calculate the results of a complex mathematical equation. In reversing the Examiner’s rejection of this claim (which was based upon the Examiner’s belief that the novelty of the claim lay in the equations with which the computer was programmed), the court stated:
The principle [behind the Examiner’s rejection] may, we think, be fairly stated as follows: If, in an invention defined by a claim, the novelty is indicated by an expression which does not itself fit in a statutory class (in this case not a machine or a part thereof), then the whole invention is non-statutory since all else in the claim is old. We do not believe this view is correct under the Patent Act and the case law thus far developed. [Id. at 1399,163 USPQ at 615-16.]
Instead, the court concluded that:
[I]f a machine is programmed in a certain new and unobvious way, it is physically different from the machine without that program; its memory elements are *188differently arranged. The fact that these physical changes are invisible to the eye should not tempt us- to conclude that the machine has not been changed. If a new machine has not been invented, certainly a "new and useful improvement” of-the unprogrammed machine has been, and Congress has said in 35 U.S.C. § 101 that such improvements are statutory subject matter for a patent. [Id. at 1400,163 USPQ at 616.]
The CCPA reaffirmed its rejection of the "point of novelty” test in In re Musgrave, 431 F.2d 882, 167 USPQ 280 (CCPA 1970), and pointed out the fallacy in that test:
[I]t is our view that [the point of novelty test is] logically unsound. According to [this test], a process containing both "physical steps” [the statutory portion] and so-called "mental steps” [the nonstatutory portion] constitutes statutory subject matter if the "alleged novelty or advance in the art resides in” steps deemed to be "physical” and nonstatutory if it resides in steps deemed to be "mental.” It should be apparent, however, that novelty and advancement of an art are irrelevant to a determination of whether the nature of a process is such that it is encompassed by the meaning of "process” in 35 U.S.C. § 101. Were that not so, as it would not be if [the point of novelty test] were the law, a given process including both "physical” and "mental” steps could be statutory during the infancy of the field of technology to which it pertained, when the physical steps were new, and non-statutory at some later time after the physical steps became old, acquiring prior art status, which would be an absurd result. Logically, the identical process cannot be first within and later without the categories of statutory subject matter, depending on such extraneous factors. [Id. at 889, 167 USPQ at 286-87.]
Instead, that court stated that all that was needed to make the claimed process statutory under 35 U.S.C. § 101 was "that it be in the technological arts so as to be in consonance with the Constitutional purpose to promote the progress of 'useful arts.’ ” 431 F.2d at 893, 167 USPQ at 289-90. Finding the claims at issue to be within the technological arts, the court reversed the Examiner’s rejection.
The new "technological arts” test was reaffirmed by the CCPA in In re Foster, 438 F.2d 1011, 169 USPQ 99 (CCPA 1971), as it stated that "it is not important whether the *189claims contain mental steps [the nonstatutory portion] or not if the process is within the technological arts.” Id. at 1015, 169 USPQ at 101. And in In re Waldbaum, 457 F.2d 997, 173 USPQ 430 (CCPA 1972), aff’d, 559 F. 2d 611, 194 USPQ 465 (CCPA 1977), the CCPA explained that "[t]he phrase 'technological arts,’ as we have used it, is synonymous with the phrase 'useful arts’ as it appears in Article I, Section 8 of the Constitution.” Id. at 1003, 173 USPQ at 434.
In In re Benson, 441 F.2d 682, 169 USPQ 548 (CCPA 1971), the CCPA again applied their ''technological arts” test to reverse a section 101 rejection of a claim drawn to a method of converting one form of numerical representation into another, utilizing a novel algorithm. After concluding that the process had no practical use except in connection with a digital computer, the court stated:
It seems beyond question that the machines - the computers - are in the technological field, are a part of one of our best-known technologies, and are in the "useful arts” rather than the "liberal arts,” as are all other types of "business machines,” regardless of the uses to which their users may put them. How can it be said that a process having no practical value other than enhancing the internal operation of those machines is not likewise in the technological or useful arts? We conclude that the Patent Office has put forth no sound reason why the claims in this case should be held to be non-statutory. [Id. at 688,169 USPQ at 553.]
Certiorari was granted in Benson, and the Supreme Court, however, reversed. Gottschalk v. Benson, 409 U.S. 63, 175 USPQ 673 (1972).9 The Supreme Court began by pointing out that the case law has well established the principle that one may not patent an idea. Directing its attention to the claims at issue, the Court found two fatal objections to their validity: first, that "[t]he claims were not limited to any particular art or technology, to any particular apparatus or machinery, or to any particular end use,” id. at 64, 175 USPQ at 674, but were "so abstract and sweeping as to cover both known and unknown uses” of the claimed invention, id. at 68, 175 USPQ 675; and second, *190that because the mathematical formula embodied in the claims had "no substantial practical application except in connection with a digital computer, * * * the patent [claims] would wholly pre-empt the mathematical formula and in practical effect would be a patent on the algorithm itself.” Id. at 71-72,175 USPQ at 676.
The determination of whether a claim preempts nonstat-utory subject matter under the rationale of Benson has been determined to require a two-step analysis:
First, it must be determined whether the claim directly or indirectly recites an "algorithm” in the Benson sense of that term, for a claim which fails even to recite an algorithm clearly cannot wholly preempt an algorithm. Second, the claim must be further analyzed to ascertain whether in its entirety it wholly preempts that algorithm. [In re Freeman, 573 F.2d 1237, 1245, 197 USPQ 464, 471 (CCPA 1978)].
We now direct our attention to that analysis.
Plaintiff contends that his claims do not recite an "algorithm” in the Benson sense, but are directed to "physical geometry.” Specifically, plaintiff states that his claims are distinguishable from those in Benson because his are "concerned with creating a circumstance,” while those in Benson are concerned with "determining the value of a variable that already exists in the context of a specific set of circumstances.” We find, however, this distinction, if it is one at all, to be of no consequence. The important point is that the claims both in Benson and in this suit are directed to a particular means for transforming defined input signals into useful output signals. Although Benson did not render all algorithms (as defined by the dictionary) to be nonstatutory, subsequent case law has clearly established that a formula or equation, even if expressed in its prose equivalent, is the type of subject matter to which the Benson rationale must be applied. See In re Freeman, 573 F.2d at 1246, 197 USPQ at 471 (and cases cited therein).
Plaintiff further argues that Benson is inapplicable because the claims in Benson are drafted in the form of a "method,” whereas the claims in this suit are drafted in the *191form of an "apparatus” (using "means plus function” language).10 However, even the language used in Benson itself indicates that the fundamental principles governing the statutory determination of "method” claims are equally applicable to "apparatus” claims:
As we stated in Funk Bros. Seed Co. v. Kalo Co., "He who discovers a hitherto unknown phenomenon of nature has no claim to a monopoly of it which the law recognizes. If there is to be invention from such a discovery, it must come from the application of the law of nature to a new and useful end.” We dealt there with a "product” claim, while the present case deals with a "process” claim. But we think the same principle applies. [409 U.S. at 67-68, 175 USPQ at 675 (citations omitted and emphasis added)].
As the CCPA stated in In re Freeman, it is not the form of the claim which is controlling but its substance:
Though a claim expressed in "means for” (functional) terms is said to be an apparatus claim, the subject matter as a whole of that claim may be indistinguishable from that of a method claim drawn to the steps performed by the "means” * * *. [I]f allowance of a method claim is proscribed by Benson, it would be anomalous to grant a claim to apparatus encompassing any and every "means for” practicing that very method. [573 F.2d at 1247, 197 USPQ at 472 (footnote omitted)].
To do otherwise would make the determination of patentable subject matter depend simply on the draftman’s skill which would ill serve the principles underlying the prohibition against patents for ideas. See Parker v. Flook, 437 U.S. 584, 593, 198 USPQ 193, 198 (1978).
Plaintiff finally urges that Benson is inapplicable because Benson dealt with an algorithm used only in digital computers, whereas plaintiffs patent discloses his equation for use in analog computers. While it is true that the *192Supreme Court in Benson expressly declined to extend its holding to algorithms used in analog computers,11 it is also true that they did not in any way disapprove of such an extension. As we can see no factual reason for making such a legal distinction, we believe that the rationale of Benson is equally applicable to algorithms used in analog computers. This conclusion, however, is unnecessary to our decision as the claims in suit are in no way limited to analog computers. That the specification may disclose analog computers in one embodiment of the claimed invention is irrelevant. Although the specification may be properly used to interpret limitations already present in the claims, Westwood Chemical, Inc. v. United States, 207 Ct. Cl. 791, 525 F.2d 1367, 187 USPQ 656 (1975), the specification may never be used to add additional and otherwise nonexistent limitations to those claims, Siegel v. Watson, 267 F.2d 621, 121 USPQ 119 (D.C. Cir. 1959). To be sure, not even plaintiff seriously contends that his claims do not emcompass digital computers for plaintiff, himself, contends that defendant’s alleged infringing computer, which is a digital computer, is fully encompassed by his patent claims. It is a fundamental principle of patent law that a claim may not be narrowly construed to avoid invalidity and then broadly construed to encompass the accused device. See e.g., Dominion Magnesium Ltd. v. United States, 162 Ct. Cl. 240, 320 F.2d 388, 138 USPQ 306 (1963).
Accordingly, the rationale of Benson is applicable to a determination of validity of the claims in this case. The issue to which we now direct our attention is whether, under the rationale of Benson, these claims "wholly preempt the mathematical formula” which they incorporate.
C. Application of the Law to the Claims 1. Claim 212
Claim 2 recites three elements which can, functionally, be summarized as follows:
*193(a) a physically defined, stabilized reference frame13
(b) means of obtaining input signals defining an input vector in relation to the stabilized reference frame; and
(c) means to calculate
b = f (bxY) dt,
where Y is the input vector, b is the output vector, and the integration is relative to the stabilized reference frame.14
The Supreme Court has indicated in Benson that a claim should be "limited to [a] particular art or technology, to [a] particular apparatus or machinery, [and] to [a] particular end use.” 409 U.S. at 64, 175 USPQ at 674. If we consider only element (c) of claim 2 for the moment, it is clear that it violates each and every requirement of this rule. First, it in no way specifies any use for the calculated output vector and, therefore, is not "limited to any particular end use.” Second, the "means plus function” language encompasses any and every type of apparatus or machine which could be used to implement the claimed equation and, therefore, the element is not "limited to any particular apparatus or machinery.” Finally, the element has application in arts or technologies as diverse as the equation itself and, therefore, is not "limited to any particular art or technology.” The ;element truly does, in the Benson sense, wholly preempt the mathematical formula which it embodies.
Of course, claim 2 recites more than merely element (c). However, the effect of additional elements (a) and (b) in claim 2 is insufficient to factually or legally distinguish the claim from the type held to be unpatentable in Benson. The *194reason for this conclusion becomes clear upon an understanding of the purpose of these elements.
As we have stated above, element (c) merely calls for means to perform a mathematical computation which, by itself, is unpatentable subject matter. Because the equation recited in element (c) is a calculation using an input vector, i.e., Y, apparatus performing that calculation necessarily requires means to generate the value of the input vector, Y. Element (b) is for just that purpose and, therefore, is essential to the performance of the computation. Thus, the "limitation” in the claim effectuated by the addition of element (b) is, for all practical purposes, not a limitation at all because element (c) cannot function in the absence of element (b). Of course, if element (b) recited apparatus which generated a particular type of input signal, such as a specified physical quantity, our conclusion might be different. But it does not, and, in fact, it states that the input signals defining the input vector may be representative of "any desired value.”
The same is true of element (a). Because the equation claimed in element (c) operates exclusively on vectors, as we have pointed out in our background mathematics discussion, Appendix B, infra, all vectors must be expressed in terms of a frame of reference and, therefore, a reference frame is also essential to the implementation of the claimed equation.15 This is the purpose of element (a) and it too, therefore, is essential to the performance of the computation claimed in element (c). While it is true that element (a) is limited to a particular type of reference frame, i.e., a "physically defined, stabilized reference frame,” the very nature of the claimed equation requires this specific type of reference frame to produce a meaningful result.16
*195In In re Christensen, 478 F.2d 1392, 178 USPQ 35 (CCPA 1973), the CCPA was presented with a similar case. The claim at issue contained a mathematical equation and several "data gathering” steps. In rejecting the contention that the addition of "data gathering” steps could convert a nonstatutory claim into statutory subject matter, the court said:
Given that the method of solving a mathematical equation may not be the subject of patent protection, it follows that the addition of the old and necessary antecedent steps of establishing values for the variables in the equation cannot convert the unpatentable method to patentable subject matter. [Id., at 1394, 178 USPQ at 37-38 (emphasis supplied)].
The rationale behind this pronouncement of law was later explained by Chief Judge Markey in In re Sarkar, 588 F.2d 1330, 200 USPQ 132 (CCPA 1978):
No mathematical equation can be used, as a practical matter, without establishing and substituting values for the variables expressed therein. Substitution of values dictated by the formula has thus been viewed as a form of mathematical step. If the steps of gathering and substituting values were alone sufficient, every mathematical equation, formula, or algorithm having any practical use would be per se subject to patenting as a "process” under § 101. [Id. at 1335, 200 USPQ at 139.]
Even where the antecedent "data gathering” steps are novel and unobvious, their recitation in combination with a mathematical computing step is legally insufficient to give rise to a patentable claim under § 101 of the Patent Act. In re Richman, 563 F.2d 1026, 195 USPQ 340 (CCPA 1977); accord, In re Sarkar, supra at 1336 n. 18, 200 USPQ at 139 n. 18.
Of course, these CCPA precedents dealt with claimed processes, not apparatus. However, as we have already pointed out, the controlling principles are the same. Otherwise, the anomaly might result that a claim to apparatus encompassing any and every means for performing a method might be valid while, at the same time, a claim encompassing all methods of obtaining the desired result *196might be proscribed by Benson. In re Freeman, 573 F.2d at 1247, 197 USPQ at 472.
Plaintiff maintains, however, that claim 2 does not preempt a mathematical equation because the preamble "[a] directional computer” adequately limits the scope of the claim to a particular technology, i.e., the directional computer technology. We find as a matter of law, however, that those words do not effectively limit the claim to any particular technology.17
First of all, computers are applicable to all technologies, not to any particular technology. Although the type of computer is specified as a "directional” computer, in the present case, this is a limitation without substance because vector computations (the type of computation recited' in the claim), by definition, include directional computations. Thus, the claimed mathematical formula already intrinsically calls for directional computations and, therefore, use of the word "directional” in the preamble creates no additional limitation.
Second, there is a substantial question whether words in a preamble which do not appear in the body of the claim may even properly be used to limit the scope of the claim. See e.g., Stradar v. Watson, 244 F.2d 737, 113 USPQ 365 (D.C. Cir. 1957). Where, as here, the effect of the words is at best ambiguous (i.e., as pointed out above, there is uncertainty whether the words have any limiting effect on the scope of the claim), a compelling reason must exist before the language can be given weight. Cf. In re de Castelet, 562 F.2d 1236, 1244, n. 6, 195 USPQ 439, 447, n. 6 (CCPA 1977) (Where there is a potential for misconstruction of preamble language, a compelling reason must exist before the language can be given weight.). We can see no such reason in this case, for the language in the body of claim 2, standing alone, is clear and unambiguous.
Even assuming, arguendo, that the preamble language "[a] directional computer” does have the effect of limiting the scope of claim 2 to a particular technology, the fact *197remains that claim 2 still preempts a mathematical equation, albeit in a limited and specific application. As the CCPA established in In re Waldbaum, 559 F.2d 611, 194 USPQ 465 (CCPA 1977), preemption of a mathematical calculation is fatal to the validity of a claim regardless of the scope of that preemption:
[Cjlaims 1, 2, and 19 do not represent "no less of a 'computer processing program’ than did claim 8 in Benson,” because they are limited to the specific application of calculating the number of busy and idle lines in a telephone system. They would not preempt all uses of the algorithm [citing uses of the algorithm outside of the claimed invention], but would preempt only use of the algorithm in calculating the number of busy and idle lines in a telephone system. At the same time, it must be recognized that a patent on these claims would, in practical effect, be a patent on the algorithm itself - albeit in its limited, specific application to calculating the number of busy and idle lines in a telephone system.
In ‘view of the foregoing, we hold that claims 1, 2, and 19 do not define a statutory process within the meaning of 35 U.S.C. §§ 100 and 101. [Id. at 617, 194 USPQ at 469-70 (footnotes omitted)].
Similarly, the Supreme Court recently held in Parker v. Flook, 437 U.S. 584, 595 n. 18, 198 USPQ 193, 199 n. 18 (1978), that "a claim for an improved method of calculation, even when tied to a specific end use, is unpatentable subject matter under § 101.”
It is important to point out that a claim is not invalid merely because it recites a mathematical equation or computation. Parker v. Flook, supra. Rather, a claim reciting a mathematical equation or computation as one of its elements will be invalid only if, when considered in its entirety, the essence of the claim is nothing more than that element. Although the line of demarcation between a patentable and an unpatentable (or non-patentable) claim does not always shimmer with clarity, a persuasive index of a valid mathematical-calculating type of claim is when the claimed computation or equation is used not merely to calculate a numerical value, but is used to effectuate a physical result. For example,,the CCPA recently held a *198series of claims directed to methods for removing unwanted noise from seismic data not to be invalid under § 101 because the purpose of the claimed invention was to "filter out extraneous and erroneous components and to physically record a noiseless seismic trace on a record medium,” rather than to merely "compute a new value.” In re Johnson, 589 F.2d 1070, 200 USPQ 199 (CCPA 1978). As is apparent from a reading of claim 2 of the patent in suit, however, the sole purpose of the invention recited therein is merely to "compute a new value.” Thus, for this reason and all of the reasons discussed above, we conclude that, under the rationale of Benson and its progeny, unlike the claims in Johnson, claim 2 is invalid.
2. Claim 618
We find the additional language placed in the remaining claims in suit insufficient to distinguish them from nonstat-utory claim 2, as a matter of law. With respect to claim 6, the parties agree that it differs from claim 2 in that it calls for the following additional elements:
(d) a second reference frame defined with respect to the said stabilized frame by means to perform a coordinate transformation therebetween,
(e) the input signals being referred to the second reference frame to express an input vector, and
(f) the "means” receiving the input signals and generating the output signals to include the coordinate transformation means in (d) above.
See PRDRFA at |f 21. The essential difference between claim 6 and nonstatutory claim 2, therefore, is that claim 6 calls for an additional data gathering element, the "second reference frame,” and for an additional mathematical calculating element, the "means to perform a coordinate transformation.”19 Certainly, the addition of a second mathematical calculating element (an element which is on its face nonstatutory) cannot change a nonstatutory claim *199into one which is statutory; and we think that, given the principle that the addition of a "data gathering” element cannot change an otherwise nonstatutory claim into a statutory claim, the inclusion of an additional "data gathering” element in the claim will also not have that effect. Accordingly, we conclude that claim 6 does not define an invention which is entitled to patent protection under 35 U.S.C. § 101. We now consider the effect of the differences between claim 3 and claim 2.
3. Claim 320
When the value of the output vector, b, recited in claim 2 is computed, as we have explained in the mathematical background discussion, infra at Appendix B, what is actually computed is the scalar values of the output vector’s three projections on the axes of a cartesian coordinate reference frame, i.e., bi, b2, and b3. Thus, while claim 2 recites only a single mathematical equation, i.e.,
b = f (bXY) dt,
the solution of that equation requires the following three separate computations (a fact not disputed by the parties):21
b, = f (b2Y3 - b3Y2)dt;
b2 = f (b3Y[ - b,Y3)dt; and
b3 = /(b,Y2 - b2Y,)dt
Claim 3, on the other hand, expressly limits its scope by defining only the third axis projection (Y3) with respect to the reference frame. Unlike claim 2, the remaining projections of the input vector (i.e., Y2 and Y2) are not expressly defined. Also, unlike claim 2, only two of the projections of the output vector are calculated (b2 and b2),22 rather than *200the full set (bx, b2, and b3) calculated by the apparatus recited in claim 2. Based upon this analysis (the accuracy of which is not disputed), the only difference between claim 3 and claim 2 is that claim 3 calls for apparatus which performs only two computations, while claim 2 intrinsically calls for apparatus which performs three computations. This difference, however, does not make claim 3 any less "abstract” or "sweeping” in the Benson sense than claim 2. It merely alters the quantity and perhaps the substance23 of the computations claimed, not the fact that a computation has been claimed. We conclude, therefore, that claim 3 is also invalid under the rationale of Benson.
4. Claim 724
Claim 7 is identical to claim 6, except that it incorporates the language which distinguished claim 3 from claim 2. In other words, claim 7 is identical to claim 6, except that it calls for only two computations as opposed to the three computations called for by claim 6. Accordingly, for the same reasons given above for finding claim 3 nonstatutory, claim 7 is also directed to nonstatutory subject matter.
In summary, we have concluded that all of the claims in suit are invalid under the rationale of Benson and its progeny, because they preempt the mathematical formulae which they embody and, in practical effect, claim the mathematical formula itself. This conclusion, we think, is fortified by plaintiffs own admission that his invention is disclosed in a certain report, but that this report does not disclose any hardware or structure, only mathematical equations. See PRDRA at ¶ 17.
*201IV. Summary and Conclusion
While most courts are reluctant to grant summary judgment in patent cases involving technically sophisticated subject matter, see e.g., Xerox Corp. v. Dennison Manufacturing Co., 322 F. Supp. 963, 966-67, 168 USPQ 700, 703 (S.D.N.Y. 1971), it is well established that summary judgment is permissible when the court can understand the relevant technology without the aid of expert opinion, see e.g., Grayson v. McGowan, 543 F.2d 79, 192 USPQ 571 (9th Cir. 1976). In the present case, the voluminous body of instructional materials offered in evidence by both parties and the helpful comments offered at the extensive and thorough oral hearing on the motions have enabled the court to sufficiently comprehend the subject technology to render a technically sound decision without hearing the testimony of technical expert witnesses.
Our analysis of the claims in suit and the accused structures has led us to conclude that all of the claims in suit fail to define an invention which is susceptible to patenting under 35 U.S.C. § 101 and are, therefore, invalid. In view of this finding, we have decided that it is unnecessary to rule on plaintiffs motion for summary judgment on the license and infringement issues, or other issues raised by either party.
For the reasons set forth herein, defendant’s motion for summary judgment on the dispositive issue of invalidity is allowed, and plaintiffs motion for summary judgment on the issue of validity is denied. The petition is dismissed.
APPENDIX A
The asserted claims of the ’052 patent read as follows:
2. A directional computer comprising a physically defined, stabilized reference frame having. means of sensing and suppressing its angular velocity, means of obtaining a plurality of input signals in representation of any desired values, said input signals being referred to said reference frame to express an input vector, and means to receive the said input signals and generate a plurality of output signals expressing an output vector in
*202relation to said reference frame as the integral of the cross product vector between the said output vector and the said input vector, whereby the said output vector rotates around the said input vector.
3. A directional computer comprising a physically defined, stabilized reference frame having means of sensing and suppressing its angular velocity, the axes of said reference frame being denoted by the numerals 1, 2, 3, means of obtaining an input signal in representation of any desired value, said input signal being referred to said axis 3 to express an input vector, and means to receive the said input signal and generate a pair of output signals expressing an output vector in reference to the said axes 1 and 2 as' the integral of the cross product vector between the said output vector and the said input vector, whereby the said output vector is driven by the said input signal to rotate around the said axis 3.
6. A directional computer comprising a physically defined, stabilized reference frame having means of sensing and suppressing its angular velocity, a second reference frame defined with respect to the said stabilized frame by means to perform a coordinate transformation therebetween, means of obtaining a plurality of input signals in representation of any desired values, said input signals being referred to said second reference frame to express an input vector, and means, including said coordinate transformation means, to receive the said input signals and generate a plurality of output signals expressing an output vector in relation to said stabilized reference frame as the integral of the cross product vector between the said output vector and the said input vector, whereby the said output vector rotates around the said input vector.
7. A directional computer comprising a physically defined, stabilized reference frame having means of sensing and suppressing its angular velocity, a second reference frame defined with respect to the said stabilized frame by means to perform a coordinate transformation therebetween, means of obtaining an input signal in representation of any desired value, said input signal being referred to an axis of said second reference frame to express an input vector, and means, including said coordinate transformation means, to receive the said input signal and generate a plurality of output signals expressing an output vector in relation to said stabilized reference frame as the integral of the cross product vector between the said output vector and the said input vector, whereby the said output vector rotates around the said input vector.
*203 APPENDIX B
A. Vectors
Vectors are commonly used to describe certain physical quantities which require a statement of direction, as well as magnitude, to adequately specify the physical quantity. The velocity of a particle is one example of such a type of physical quantity. Although it would not be improper to specify the magnitude of the velocity alone {e.g., 10 ft/sec), a full description of the vector quantity would require a statement of its direction as well (e.g., N.E.). Physical quantities which require only a specification of magnitude to fully specify the physical quantity are known as scalars. Common examples of these are temperature, humidity, and time. When a vector is represented by a mathematical symbol, a horizontal line is drawn over that symbol to indicate that it is representative of a vector quantity so that it may be distinguished from a scalar quantity. Thus the variable "V” is understood to be representative of a scalar quantity, while the variable "V” is understood to be representative of a vector quantity.
The specification of a vector quantity is always dependent on the frame of reference of its observer. For example, the velocity of a ball rolling on the floor of a moving train may appear to be relatively slow when viewed by a passenger seated on that train, but would appear to be relatively fast when viewed by an observer stationed on the ground near the train tracks. Thus, the specification of a vector quantity must always include a specification of the frame of reference with respect to which the vector has been expressed. Otherwise, the expression would be meaningless.
A vector may be graphically illustrated by an arrow having a length proportional to the magnitude of the vector and pointing in the direction specified by the vector. Thus, for the velocity vector example given above, the vector (labeled for convenience as V) may be graphically illustrated as follows:
*204It is frequently useful to specify the vector, not in terms of its magnitude and direction, but in terms of its "projections” on the perpendicular axes of a special type of reference frame called a cartesian coordinate reference frame. Graphically, this may be represented as follows:
With the perpendicular axes of the cartesian coordinate reference frame defined as axes Í and % the projection of the velocity vector, V, on the i axis would be the scalar quantity Vi, while the projection of the velocity vector, V, on the 2 axis would be the scalar quantity V2.
The representation of a vector can be extended into three spatial dimensions. This would simply require the addition of a third axis, 3, which would be normal (perpendicular) to the plane defined by axes i and % and a projection of the vector, V, on that third axis to define the scalar quantity V3. Therefore, when using a cartesian coordinate reference frame, a vector can be completely specified by reference to its three projections on the axes of that reference frame, i.e., the scalar quantities Vi, V2, and V3. These quantities are also known as the "cartesian coordinates” of the vector. B. Vector Algebra
Vectors are subject to many types of mathematical operations. One such operation which is important to an understanding of this case is the "cross-product.” The cross product between two given vectors, e.g., A and B, always results in a_third_ vector, e.g., C, and 4s mathematically written as: C = A X B. The symbol "X” denotes "cross product” and is not to be confused with the use of that *205symbol to indicate mere multiplication of A and B. When the vectors are expressed by their cartesian coordinates {i.e., A = A1( A2, A3; B = B1( B2, B3; C =_C2, C3), the result of the cross product between vectors A and B, i.e., Ci, C2, C3, is defined by the following equations:
Ci = A2B3 — A3B2;
C2 = A3B, — AtB3; and
C3 = a,b2 — a2b, .
Often a vector , is expressed with respect to one frame of reference, but it is desirable to know its expression with respect to still another frame of reference. There is another mathematical operation called "coordinate transformation,” which can be used to calculate this needed information. However, unlike the cross product, it is unnecessary to describe the mechanics of this computation.
For the purposes of this opinion, it is most important to understand that when the value of a vector expressed by three cartesian coordinates is calculated, the actual calculation requires three separate computations; viz., one calculation for determining the value of each of the three cartesian coordinates. Thus, although a vector equation may appear to require only a single calculation for its solution, the calculation in fact requires three separate computations when the result is expressed in cartesian coordinates.
C. Vector Calculus
Another type of mathematical operation to which a vector may be subjected is integration. Integration is a mathematical operation unique to calculus and is denoted by the mathematical symbol " / .” Although the integration operation is very difficult to appreciate conceptually, it may nevertheless be intuitively understood by considering one of the types of problems which it was designed to solve.
Suppose one knew the rate of flow of water (R) coming out of a hose and wanted to know the volume of water (V) expelled after the passage of a certain period of time (T). The answer would be easily calculated by using the well-known equation: volume (V) = rate of flow (R) multiplied by the elapsed time (T) or V =RT. For example, if the rate of flow, R, is 10 gallons per minute (i.e., R=10 gal/min), *206after the passage of 2 minutes (i.e., T=2 min), 20 gallons of water would have been expelled (i.e, RT = V).
Graphically, the rate of flow, R, may be illustrated as follows:
Suppose, instead, that the water pressure was erratic and thus caused the rate of flow, R, to vary from time to time. For example, suppose the rate of flow, R, followed the "profile” shown in the following graph:
On the graph, the rate of flow, R, has been indicated as "R(t)”, rather than merely "R,” to indicate that the rate is no longer constant, but varies as a function of time (the addition of "(t)” to a variable name to indicate that it varies as a function of time (t) is standard mathematical nomenclature). Now, the computation of the volume of water expelled, V(t), after the passage of a known period of time is not so easily calculated for the equation V = RT is not applicable when the rate of flow does not remain constant during the given time interval.
Nevertheless, the volume of water expelled, V(t), after the passage of 2 minutes, can be approximated by calculating the volume of water expelled by an imaginary water *207hose whose rate of flow profile, R’(t) (the apostrophe is used after the symbol "R” to distinguish the imaginary variables from the actual ones), is chosen to closely approximate the actual rate of flow profile, R(t), but has a shape susceptible to the application of the equation, V = RT.
One such imaginary rate of flow profile, R’(t), may be created by sampling the real rate of flow profile, R(t), every half-minute and by maintaining the value of the imaginary rate of flow profile between samples equal to the value of the last rate of flow. Graphically this would appear:
Although the imaginary rate of flow, R’(t), does not remain constant during the entire 2 minute time interval, it does remain constant during each half-minute time interval (designated as ta, t2, t3, and t4). Thus, the volume of water expended by the imaginary water hose, V’(t), during each half-minute segment would simply be the imaginary rate of flow during that time segment, R’(ti), R’(t2), R’(t3), and R’(t4), times its respective flow time, t1; t2, t3, and t4, which in each case is one-half a minute. Mathematically, therefore, the total imaginary volume of water expended after two minutes, V’(2 min), would be:
V'(2 min) = V'(t,) + V'(t2) + V'(t3) + V'(t4)
= R'(tj) 1/2 + R'(tj) 1/2 + R' (t3) 1/2 + R' (t4) 1/2
~ V( 2 min) (“ — ’' means approximately equals.)
*208The error in the approximation of V(t) by equating it to V’(t) is directly related to the amount of difference between the two profiles. This difference can be minimized by increasing the frequency of samples in the imaginary rate of flow profile, R’(t). Thus, the following imaginary rate of flow profile, R”(t), which has twice as many samples per period of time as does R’(t), will yield an even more accurate approximation of V(t) than will V’(t):
As the density of the samples increases to infinity, R”(t) will become identical to R(t) and the accuracy of the approximation for V(t) based on a calculation of V”(t) will approach theoretical perfection. At the limit when the density of samples does reach infinity, the calculation process is defined as integration and is denoted by the following equation (for this particular problem):
V(t) = /R(t)dt
where "dt” signifies an infinitesimal length of time.
Essentially, therefore, integration is a method to calculate and sum an infinite series of mathematical computations. However, because of the infinitesimal (microfinite) nature of the time which passes between calculations, integration is known as a "continuous” method of computation, while the sample method described above, because it involves the calculation of a series of time-separated computations, is known as an individual or "discrete” method of calculation.
Devices called "analog integrators” can actually perform an integration operation on electronic signals. There are *209also mechanical devices which can integrate. An example of a mechanical device which can actually perform the integration operation required to precisely calculate V(t) in the given example is, quite simply, a large bucket! If the water expelled from the hose is channeled into the large bucket, no matter how the rate of flow, R(t), varies with time, the integrator, i.e., the large bucket, will "keep track” of V(t) precisely. One need merely measure the volume of water in the large bucket at the desired time to ascertain the value of V(t). This example of an integrator serves to intuitively illustrate the fundamental difference between an integrator which "continuously calculates” and the sample method which performs "discrete computations.”
When a vector, expressed in terms of its cartesian coordinates, is intergrated, its three cartesian coordinates are separately integrated (which, therefore, requires three integrations). Mathematically stated, to perform
b= /Zdt,
whát really must be performed is:
bi = / Z,dt;
b2=1 Z2dt; and
b3 = /Z3dt
If we let Z = b X Y in the above example, we know from the definition of cross products that:
Z, = b2Y3 - b3Y2;
Z2 = b3Y, - b,Y3; and
Z3 = b,Y2 - b2Y[;
and, therefore, the vector calculus equation
b = / bX Y dt
is solved by solving the following three scalar calculus equations:
bi = /(b2Y3 - b3Y2)dt;
b2= f (b3Y, - b,Y3)dt; and
b3 = f (b[Y2 - b2Y,)dt
We have examined the particular equation
b = f bxYdt
*210because an understanding of this particular equation is essential to an understanding of the claims in suit. With this knowledge in hand, the analysis of the legal issues in this case may be more easily perceived.
The text of the asserted claims is set forth in Appendix A, infra.
35 U.S.C. § 101 states: "Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent therefor, subject to the conditions and requirements of this title.”
The claim has been broken down into three parts for analytical purposes. It does not appear in this format in the patent.
Specifically, the language recites the mathematical expression
b = JjbxY) dt,
where Y represents a desired input vector and 5 represents a directional output vector.
Section 101 of the Patent Act, see n. 2, supra, does not expressly exclude a mathematical expression or equation as patentable subject matter. Case law, however, has established that they are not included. See e.g., Gottschalk v. Benson, 409 U.S. 63, 175 USPQ 673 (1972); Mackay Radio & Telegraph Co. v. Radio Corporation of America, 306 U.S. 86, 40 USPQ 199 (1939).
The word "nonstatutory” is sometimes used to apply to a claim directed to subject matter which is not expressly entitled to patent protection.
As a point of fact, the CCPA first rejected this approach in In re Prater, 415 F.2d 1378, 159 USPQ 583 (CCPA 1968). However, the opinion in Prater was subsequently vacated, 415 F.2d 1378, 160 USPQ 230, and replaced by an opinion which did not discuss the propriety of the "point of novelty” test, see 415 F.2d 1393, 162 USPQ 541 (CCPA 1969).
The decisions of the Court of Customs and Patent Appeals, while being neither controlling nor binding on this court, are nevertheless accorded great weight and respect in view of the manifest expertise of that court in determining whether or not claims presented in pending applications for patents satisfy the requirements of the patent statutes.
All subsequent references to "Benson” are to Gottschalk v. Benson, 409 U. S. 63, 175 USPQ 673 (1972).
It should be noted that plaintiff, who prosecuted his patent application pro se, did not have the benefit of the CCPA and Supreme Court decisions rendered after May 9, 1967, since the patent issued on that date was based on his application which was first filed in 1962 and refiled in 1966. In re Abrams, supra, was the only significant precedent available during the pendency of plaintiffs applications.
Specifically, the Court stated: "It is said that we have before us a program for a digital computer but extend our holding to programs for analog computers. We have, however, made clear from the start that we deal with a program only for digital computers.” 409 U.S. at 71, 175 USPQ at 676.
For the full text of claim 2. see Appendix A, infra.
This element is further described in the claim as "having means of sensing and suppressing its angular velocity.” However, it is clear from the specification that this language merely clarifies the essential nature of a "stabilized reference frame” and thus is .without limiting effect. See col. 5, lines 4-9. To be sure, plaintiff wholly agrees with this conclusion.
This element is further described in the claim by the language "whereby the said output vector rotates around the said input vector.” However, this language has no limiting effect for it is clear that in every implementation of the claimed equation the "output vector rotates around the said input vector” whether it rotates to trace a conical surface around the input vector or seeks coalignment with it.
The statement that "the integration is relative to the stabilized reference frame” is used to mean what the inventor in his specifications has defined it to mean, namely that the vector quantities of the integrand, i.e. 5 and Y are expressed in reference to the stabilized reference frame. See cols. 4 and 5, lines 64-75 and 1-4, respectively.
Plaintiff wholly agrees with this statement.
The claimed equation,
b = f (bxY) dt,
is but one embodiment of the more general equation
b = f ((A/b2-B)tT + by Y + bVw) dtL
(see col. 4, equation (17)) with A/b2-B = w = 0 by definition. To make w=0, however, required a "physically defined, stabilized reference frame.” See col. 5, lines 1-7. Thus, the claimed equation cannot be meaningfully implemented with any other type of reference frame. Accord, see plaintiffs submission to standard pretrial order on liability (PSPOL), f[ 23,24 (filed February 27,1978).
While the specification describes the invention as being a "vectorial data processing system” which is useful in "aircraft flight control,” all of the claims are directed to a "directional computer,” without recitation or limitation as to any field of use.
The full text of claim 6 is recited in the Appendix A, infra.
The specifications disclose that the "means to perform a coordinate transformation” is merely apparatus which performs a series of known mathematical computations. See cols. 6 and 7, lines 15-17 and 1-3, respectively.
The full text of claim 3 is presented in Appendix A, infra.
The derivation of these equations is shown in the background mathematics discussion, infra at Appendix B.
The parties dispute the implication to be drawn from the failure of the claim to expressly define the value of two of the three projections of the input vector (i.e., Y2 and Yj). Plaintiff contends that the inference to be drawn is that they may be of any value. Thus, plaintiff contends, the two claimed computations are
bl = J <b2Y3 - b3Y2)dt and b2 = J (b^ - b^) dt
which are identical to two of the three computations claimed in claim 2. Defendant contends that the only proper interpretation of claim 3 is that the two unnamed projections (Y2 and Y3) must be consistently zero. Thus, defendant contends, the two *200claimedcomputations are
bl = J (b2Y3>dt and b2 = J ( - b^3) dt
which are not identical to any of the three computations claimed in claim 2. However, the resolution of this dispute is irrelevant to the issue of claim validity under § 101 because, regardless of which interpretation is correct, the fact remains that a computation has been claimed.
See n. 22, supra.
The full text of claim 7 is presented in Appendix A, infra.