## Opinion

Patent Appeal No. 77-519.

October 6, 1977.

Melvin P. Williams, attorney of record, Hartford, Conn., for appellant.

Joseph F. Nakamura, Washington, D.C., for the Commissioner of Patents, Thomas E. Lynch, Washington, D.C., of counsel.

Appeal from the Board of Appeals of the United States Patent Office.

Before MARKEY, Chief Judge, and RICH, BALDWIN, LANE and MILLER, Judges.

This appeal is from the decision of the Patent and Trademark Office ("PTO") Board of Appeals ("board"), unchanged on reconsideration, sustaining the rejection of claims 1-4 under 35 U.S.C. § 101 for being directed to nonstatutory subject matter. We affirm.

In application serial No. 177,560, filed September 2, 1971, for "Radar Boresight Calibration and Velocity Vector Determination."

### *The Invention*

The invention involves a method of calculating (according to a mathematical formula) an average boresight correction angle for an airborne, coherent pulse doppler, synthetic aperture, signal processing radar, using actual terrain measurements, and a method of calculating (according to a mathematical formula) the average vertical velocity component of the aircraft carrying the radar, using these same measurements. Appellant describes the invention in terms of the figure below:

The invention is based upon the principle that, although the depression angle ( or ) and the range (R[1] or R[2]), and even the absolute distance of the velocity vector (12) of the aircraft to the map cell (16), vary along the flight path, the product of the range and the sine of the depression angle should be constant for the same map cell along a straight line of flight.

In actual practice, however, this product varies when measured from different points. Appellant utilizes this variation in the product of the range and the sine of the depression angle, from the antenna to a given map cell from two separate points (P[1], P[2]) along a straight line of flight of the aircraft, using a specific mathematical formula, to calculate a boresight correction angle. Summation over many cells is employed to calculate an average boresight correction angle (). The average actual vertical (the vertical direction is the normal direction to a plane below which the depression angle is measured — see the dashed lines in the above-copied figure) velocity component V[z] and the average actual vertical velocity component V[z], which latter value utilizes the aforementioned average boresight correction angle, are also calculated from summation over many cells, utilizing the depression angles and the ranges in specific mathematical formulae.

Claims 1 and 4 are illustrative:

1. In an airborne coherent pulse doppler synthetic aperture depression angle sensing processing radar, the method of calculating a correction factor , for measured values of depression angle, , comprising:

recording a plurality of signal sets from at least two points in a flight path, each of said signal sets relating to a map cell on a radar map of M cells in which each map cell comprises the intersection of one of K slant range slices with one of J doppler cones, said signal sets comprising a depression angle and a slant range R[kj] to each kj[th] map cell in each of the maps, the slant ranges R[kj1] and depression angles [1] relating to the first map and the slant ranges R[kj2] and depression angles [2] relating to the second map, said two points being separated along the flight path and therefore in time by a period of time greater than the processing interval for the making of one of said maps; and

calculating a boresight correction angle from the slant range to each cell in the first and second map R[kj1], R[kj2], respectively, and the measured depression angle to each cell in each map , , respectively as follows:a b — M ab = _________________ M [b2] — ( b)[2] Where a = R[kj2] sin — R[kj1] sin b = R[kj2] cos — R[kj1] cos = summation across K J = M cells and M = the number of map cells is each map.

4. In an airborne coherent pulse doppler synthetic aperture depression angle sensing processing radar, the method of calculating the magnitude of the component of velocity perpendicular to the plane from which depression angle is measured, including in said calculation compensation for boresight-induced error in measured values of depression angle, comprising:

recording a plurality of signal sets from at least two points in a flight path, each of said signal sets relating to a map cell on a radar map of M cells in which each map cell comprises the intersection of one of K slant range slices with one of J doppler cones, said signal sets comprising a depression angle and a slant range R[kj] to each kj[th] map cell in each of the maps, the slant ranges R[kj1] and depression angles [1] relating to the first map and the slant ranges R[kj2] and depression angles relating to the second map, said two points being separated along the flight path and therefore in time by a period of time greater than the processing interval for the making of one of said maps; and

calculating the average value of said velocity component, , from the slant range to each cell in the first and second map R[kj1], R[kj2], respectively, and the measured depression angle to each cell in each map , , respectively, as follows:1 a b[2] — a ab = (---) _______________________ T M b[2] — ( b)[2] where a = R[kj2] sin sign; — R[kj1] sin b = R[kj2] cos — R[kj1] cos = summation across K J = M cells.

### *The Board*

[9] The majority opinion of the board states that, although the claimed invention is limited to a particular art or technology,

we do not find that reciting an algorithmic process in such a manner as to preempt the use of an arithmetic procedure in a limited field as opposed to in general [sic, in a general field], would convert an unpatentable method to patentable subject matter [In re Christensen,478 F.2d 1392, 178 USPQ 35 (Cust. Pat.App. 1973)].

The majority opinion answers appellant's argument that the claims include novel steps in addition to the novel calculating step as follows:

Although we find no evidence before us indicating that the step of repeating the aforementioned steps at a later time in the flight path of the aircraft is old, we note that such a step would be required in order to obtain the necessary values of the variables required for solution of the equation set forth. Hence, the repeating step is dictated by the equations to be calculated. Under these circumstances we fail to find any significant distinction between the nature of that which is claimed here, and that which was condemned in [Gottschalk v. Benson,409 U.S. 63, [ 93 S.Ct. 253, 34 L.Ed.2d 273,] 175 USPQ 673 (1972) (hereinafterBenson)] andChristensen.

A concurring opinion states that *Benson,* as explained in *Dann v. Johnston,* 425 U.S. 219, 96 S.Ct. 1393, 47 L.Ed.2d 692, 189 USPQ 257 (1976), involved no particular art or technology other than pure mathematics; that, therefore, *Benson* neither compels nor authorizes rejection of claims 1-4 under 35 U.S.C. § 101; but that the calculating step is crucial to establishing unobviousness of the claimed method, and thus *Christensen* requires affirmance of the section 101 rejection.

### OPINION

Both the majority and concurring opinions of the board note that the antecedent steps to obtain data are at least obvious; further, the majority opinion notes that the repeating step is "dictated by the equations to be calculated," while the concurring opinion states that this step is "no more than mere duplication of the prior art practice of map making." Such statements ignore the fact that until the formulae are known, they cannot "dictate" the data-acquiring steps or provide motivation for duplicating the prior art practice of map making. By using the formulae to find the antecedent steps obvious, the PTO is improperly used appellant's invention against him. *Cf. In re Hirao,* 535 F.2d 67, 190 USPQ 15 (Cust. Pat.App. 1976); *In re Kuehl,* 475 F.2d 658, 664-65, 177 USPQ 250, 255 (Cust. Pat. App. 1973).

Moreover, by regarding the mathematical formulae as the novel subject matter and determining the section 101 issue solely on that basis, the PTO has dissected each of the claims on appeal (instead of considering each claim as a whole), a procedure rejected in *In re Chatfield,* 545 F.2d 152, 191 USPQ 730 (Cust. Pat.App. 1976), *cert. denied,* U.S., 46 U.S.L.W. 3203 (October 4, 1977):

Our reference inChristensento the mathematical equation as being "at the point of novelty" does not equate to a holding that a claim may be dissected, the claim components searched in the prior art, and, if the only component found novel is outside the statutory classes of invention, the claim may be rejected under 35 U.S.C. § 101. That procedure is neither correct nor within the intent of Congress . . .. [Id.at 158, 191 USPQ at 736.]

Accordingly, we conclude that, properly stated, the issue in this case is whether the claimed methods of calculating (utilizing mathematical formulae) an airborne radar boresight correction angle or a velocity component for the aircraft carrying the radar, which methods include new and unobvious steps for acquiring data for use in the formulae, are statutory subject matter under 35 U.S.C. § 101.

In *In re Christensen, supra* at 1394, 178 USPQ at 37-38, this court stated:

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.

This portion of *Christensen* was explained in *In re Chatfield, supra* at 158, 191 USPQ at 736, as follows:

In affirming the rejection of the claims [inChristensen], we followed the tenet ofBensonthat solving an equation constituted non-statutory subject matter. We determined that the several steps antecedent to the solving of the equation in Christensen's claims merely established the variables for the equation and did not suffice to render the claimed method patentable as a whole.

In the present case too, notwithstanding that the antecedent steps are novel and unobvious, they merely determine values for the variables used in the mathematical formulae used in making the calculations. Thus, such antecedent steps do not suffice to render the claimed methods, considered as a whole, statutory subject matter.

Appellant points out that "there is a tendency, when applying the `issue' portion of *Christensen,* to over-emphasize that the equation is the final step, and to under-emphasize that the equation is the point of novelty." However, the fact that an equation is the final step is not determinative of the section 101 issue. Moreover, as pointed out above, this court has rejected the "procedure" of deciding whether claimed subject matter is nonstatutory on the sole basis of whether a mathematical formula is "the point of novelty." See *In re Chatfield, supra.*

Appellant argues that "[c]onsidering the claim as a whole, the particular order of the steps is not determinative of statutory matter." We agree. But if a claim is directed essentially to a method of calculating, using a mathematical formula, even if the solution is for a specific purpose, the claimed method is nonstatutory. As this court stated in *In re Waldbaum,* 559 F.2d 611, 617, 194 USPQ 465, 470 (Cust. Pat.App. 1977):

[I]nIn re Christensen,478 F.2d 1392, 178 USPQ 35 (Cust. Pat.App. 1973), . . although the claims were limited to a specific application of technological significance, they were, nonetheless, directed to acalculationand would have preempted use of the algorithm in making the calculation. The situation is in marked contrast to that we recently considered inIn re Deutsch,553 F.2d 689, 193 USPQ 645 (Cust. Pat.App. 1977), where the claims were to methods ofoperating. . an entire system of manufacturing plants, employing particular algorithms, and to that inIn re Chatfield, supra,where the claims were to a method ofoperatingthe machines of a computer system more efficiently, some of the claims also employing particular algorithms, [Emphasis in original.]

*Accord, In re Flook,* 559 F.2d 21 (Cust. Pat.App. 1977).

Appellant contends that —

we should not penalize the inventor whose attorney drafts the claims in a simple, formula expression of a mathematical relationship as against the inventor whose attorney simply prolongs the expression with words which mean the same thing.

However, this misses the point of *Christensen.* That a claim includes a mathematical expression is not determinative. The decisive factor is whether a claimed method is essentially a mathematical calculation. If it is, deletion from the claims of the mathematical formula involved and substitution of "words which mean the same thing" would not transform the claimed method into statutory subject matter. See *In re Waldbaum* and *In re Flook,* both *supra.*

In view of the foregoing, we hold that the claimed methods of calculating an airborne radar boresight correction angle or a velocity component for the aircraft carrying the radar, utilizing mathematical formulae, which methods include new and unobvious steps for acquiring data for use in the formulae, are not statutory subject matter under 35 U.S.C. § 101.

The decision of the board is *affirmed.*

*AFFIRMED.*

RICH, J., concurs in the result.