CITE:  562 F.2d 1236
CMON:  October 1977
PLAIN: In re De Castelet, Gaetan De Coye
DEFND: United States Patent and Trademark Office
COURT: United States Court of Customs and Patent Appeals
DATE:  October 6, 1977

Appeal from the decision of the Patent and Trademark Office Board of Appeals
(board), affirming the rejection of claims 5-7, all the claims in application
serial No. 68,507, filed August 31, 1970, entitled "Method of Generating a
Curve, Applicable to Tracing Machines or Machine Tools." We affirm.

  Solution of mathematical equations have to be used to achieve a result
  to be patentable.

JUDGE: MARKEY, Chief Judge.
RICH, J., and LANE, J., concur in result.

The Invention

De Castelet's invention relates to a method of generating a curve or family
of curves, employing a computer in conjunction with drafting and milling
machines. The invention employs these principles: (1) a curve may be
represented by four points defining two straight line tangents to the ends
of the curve, and (2) a curve can be represented by a transformed
base-of-reference curve, by transforming coordinates of points representing
a unit or base curve into a coordinate system of a curve to be generated.

In de Castelet's overall system:

           [Graphic omitted.   See illustration in original.]

Data are inputed to computer 10 through control boards 19 and 20.
Calculations, using the inputed data and pursuant to stored programs,
are performed by computer 10.  Computer 10 also receives milling machine
11 position information from encoder 13 and drafting machine 12 position
information from encoder 14.  Based upon the results of its calculations
and information received from encoders 13 and 14, computer 10 generates
speed-change signals for controlling position motor driving shafts 21-25.
The net result of the operation is a machined surface (milling machine 11)
or a drawing (drafting machine 12) which corresponds to a desired curve

According to de Castelet's method, curves to be generated are first broken
down into a series of successive arcs. Those arcs can be defined by two
straight line tangents, for example, AM and BN in the following illustrations:

              [Graphic omitted.  See illustration in original.]

The data inputed to the computer constitutes the X, Y, and Z Cartesian
coordinates of points A, M, N, and B.  The computer, pursuant to a stored
program, treats each are so defined as though it were the transformed curve
of a base curve inscribed within and between two opposite vertices of a
unit cube.  The coordinates of that base curve are transformed by computer
calculations, according to the following transformation equations, into
coordinates of the curve to be generated:

    X1 = Xa + alpha  x1 + beta  y1 + gamma  z1

    Y1 = Ya + alpha' x1 + beta' y1 + gamma' z1

    Z1 = Za + alpha" x1 + beta" y1 + gamma" z1

    alpha  = Xm - Xa        alpha' = Ym - Ya        alpha" = Zm - Za.
    beta   = Xn - Xm        beta'  = Yn - Ym        beta"  = Zn - Zm.
    gamma  = Xb - Xn        gamma' = Yb - Yn        gamma" = Zb - Zn.

De Castelet claims:

  7.  A machine method of generating a curve from data supplied to a
  computer in the form of coordinates of points defining two given
  segments of tangents to the curve to be generated extending from the
  end of and subtended by said curve for controlling numerical control
  system type model forming means, wherein data, in the form of
  electrical signals representing a table of coordinates of points, of
  characteristics of a base curve inscribed on a unit-cube between two
  opposite vertices of said unit-cube is stored in a memory bank of said
  computer and said computer is programmed (1) to treat electrical signals
  representing a given arc, defined by the coordinates of the ends of
  two segments of tangents extending from and subtended by the ends of
  said given arc, as the transformed curve of said base curve, wherein
  said ends are considered as the transformed points of vertices of said
  unit-cube and (2) to calculate and transmit to the control system of
  said model forming means electrical signals representing the coordinates
  of a sequence of points of successive ones of said given arc defining
  said curve to be generated, said computer thereafter automatically
  performing the steps of:

  (a) transforming the electrical signals representing said coordinates
  of points defining two given segments of tangents to said curve to be
  generated by program (1) to define a corresponding change in reference
  coordinates with respect to the characteristics of the stored base curve;

  (b) calculating the sequence of coordinates of the current points of the
  transformed base curve of program (1) through the change in reference
  coordinates obtained from step (a); and

  (c) transmitting electrical signals representing said sequence of
  coordinates calculated in step (b) from said computer to said model
  forming means by program (2).

  5.  The method according to claim 7, wherein step (b) further comprises
  calculating said sequence of coordinates of the current points according
  to the following equations:

       Xw = Xa + alpha x + beta y + gamma z

       Yw = Ya + alpha' x + beta' y + gamma' z

       Zw = Za + alpha" x + beta" y + gamma" z


   Xw, Yw and Zw are the coordinates of a current point of each arc of the
   curve to be generated;

   Xa, Ya and Za are the coordinates of the point of origin of the arc of
   the curve to be generated; and

   x, y and z are the coordinates of a current point of the base curve, and

   alpha, alpha', alpha"; beta, beta', beta"; gamma, gamma', gamma";

   are the respective projections on the X, Y and Z axes of the two segments
   of tangents to the arc of the curve to be generated and of the segment
   joining the ends of said two segments opposing the point of tangency.

   6.  The method according to claim 7, wherein step (c) comprises the 
   further step of interpolating between adjacent points of said sequence
   of points obtained by program (2) according to a calculation of the
   director cosines of the tangents of each one of said sequence of points.

The Board

The board, citing Gottschalk v. Benson, 409 U.S. 63, 93 S.Ct. 253 (1972),
stated that "where the claimed novelty involves a formula, equation or
algorithmic process and has no substantial practical application except in
connection with a digital computer, a patent to such would in practical
effect be a patent on the algorithm itself, and should not be granted."

Acknowledging that independent claim 7 does not recite a specific algorithm,
equation, or mathematical formula, the board nevertheless found an
algorithmic process contained therein, and concluded that, because
the apparatus was known, any novelty in the claims resulted from that
algorithmic process.  That novelty, the board said: "is of the same nature
as that which is condemned in Benson."

The board was of the opinion that patenting the present claims would, in
effect, preempt the algorithmic process despite the fact that "they recite
a machine environment, a particular art or a particular end use."

The Issue

As in In re Chatfield, 545 F.2d 152, (CCPA 1976), cert. denied, 46 U.S.L.W.
3203 (Oct. 4, 1977), the sole issue before us is whether the particular
claims on appeal define statutory subject matter under 35 USC 101. {1}


The Board's Interpretation of Benson

Though we agree with the board's ultimate conclusion, and with its reference
to Benson as precluding patentability of claims to a mathematical equation,
we disagree with several of the board's expressions of the applicable law.
Initially, the board found:

  [The] thrust of the decision in Benson to be that computer programs or
  program implemented algorithms are not patentable subject matter at
  least until such time as the Congress acts otherwise.

That "computer programs" are not patentable is not the "thrust" of
Benson. As the Court cautioned:

  It is said that the decision precludes a patent for any program servicing
  a computer, We do not so hold. [409 U.S. at 71, 93 S.Ct. at 257]

and as the Court's characterization of Benson in Dann v. Johnston, 425 U.S.
219, 224, 96 S.Ct. 1393, 1396 (1976) confirmed:

  Our limited holding * * * was that respondent's method was not a
  patentable "process" as that term is defined in 35 U.S.C. 100(b).

Absent contrary directions, no basis exists for a moratorium on protection
of inventions embodying or using computer programs.  Such broad prohibition
could subject meritorious statutory inventions to unabatable piracy, and
could forestall invention disclosure, the hallmark of the patent system,
until Congress chooses to act.

We disagree, also, with the breadth of the board's succeeding comment:

  Stated another way, where the claimed novelty involves a formula,
  equation or algorithmic process and has no substantial practical
  application except in connection with a digital computer, a patent
  to such would in practical effect be a patent on the algorithm
  itself, and should not be granted.

That second comment is not a restatement of the board's preceding comment,
insofar as it envisions the question of statutory subject matter as residing
in a "claimed novelty" and its practical application.  As a majority indicated
in Chatfield, supra, claim dissection and comparison against prior art to
determine a limited "claimed novelty" is not proper.  What an applicant
chooses to encompass by the whole of his claims comprises his "claimed
novelty." There is no 35 USC 103 rejection before us, based on the view
that it would have been obvious to employ de Castelet's equations in
apparatus described by the examiner as old. {2}

The board's focus on practical applications for the "claimed novelty" springs
from the "nutshell" holding in Benson:

  It is conceded that one may not patent an idea.  But in practical effect
  that would be the result if the formula for converting BCD numerals to
  pure binary numerals were patented in this case.  The mathematical formula
  involved here has no substantial practical application except in
  connection with a digital computer, which means that if the judgment
  below is affirmed, the patent would wholly pre-empt the mathematical
  formula and in practical effect would be a patent on the algorithm itself.
  [409 U.S. at 71-72, 93 S.Ct. at 257]

That language is interpreted by the solicitor and the board as foreclosing
patentability of the appealed claims because, as de Castelet admitted at oral
argument, the only practical application of the involved algorithm (equations)
is in the claimed method.  But the notion that the "nutshell" language makes
a method nonstatutory whenever it involves mathematical equations having
their only practical use in the method is neither impelled by the rationale
of Benson nor supported by precedent.

The "meat" within the "nutshell" is not that patentability is precluded
whenever an inventor discloses and claims only one use for his formula.
The Supreme Court felt that "Benson's claims would have preempted all
practical use  of both the underlying mathematical formula and the
involved algorithm," [Emphasis added.] Chatfield, 545 F.2d at 156, i.e.,
Benson's claims were considered as in effect drawn to the algorithm itself.

This court, in In re Benson, 58 CCPA 1134, 441 F.2d 682 (1971), recognized
that the only practical use of Benson's method was in conjunction with a
computer, but stated why that fact would not support a "mental steps"

  Realistically, the process of claim 13 has no practical use other than the
  more effective operation and utilization of a machine known as
  a digital computer. 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?  [Emphasis in original.  58 CCPA at 1143-44, 441 F.2d at 688]

Benson's limitation of his claims to a computer machine environment overcame
the rejection bottomed on a "mental steps" theory.  Because operation of a
formula-solving computer constituted the only practical use of the involved
algorithm, however, the Supreme Court deemed unpatentable what it considered
claims to an algo6ithm performed in a computer, i.e., claims to the algorithm

The precedent for the Benson "nutshell" language is found in two basic
propositions.  The first is that "[phenomena] of nature, though just
discovered, mental processes, and abstract intellectual concepts are not
patentable, as they are the basic tools of scientific and technological
work."  409 U.S. at 67, 93 S.Ct. at 255. Support for that proposition
appears in MacKay Co. v. Radio Corp. of America, 306 U.S. 86 (1939);
Rubber-Tip Pencil Co. v. Howard, 87 U.S. (20 Wall.) 498 (1874);
LeRoy v. Tatham, 55 U.S. 167, 14 How. 156 (1852); and Funk Bros. Seed Co.
v. Kalo Inoculant Co., 333 U.S. 127 (1948). A corollary was stated in
MacKay: "While a scientific truth, or the mathematical expression of it,
is not a patentable invention, a novel and useful structure created with
the aid of knowledge of scientific truth may be." {3}  306 U.S. at 94.
To "novel and useful structure," encompassed within "machine", should be
added the other statutorily patentable categories, i.e., processes,
manufactures, compositions of matter, and new and useful improvements

The second proposition is that a statutory claim must define more than a
mere effect. In O'Reilly v. Morse, 56 U.S. 65, 15 How. 62 (1853), cited by
the Court in Benson, it was alleged that a Morse claim was void because
drawn to "a principle or effect, and not a machine, manufacture, or
composition of matter, or an improvement upon either." 56 U.S. at 106,
15 How. at 100. Morse sought to claim:

  Eighth.  I do not propose to limit myself to the specific machinery or
  parts of machinery described in the foregoing specification and claims; 
  the essence of my invention being the use of the motive power of the
  electric or galvanic current, which I call electro-magnetism, however
  developed for marking or printing intelligible characters, signs or
  letters, at any distances, being a new application of that power of
  which I claim to be the first inventor or discoverer.
  [56 U.S. at 119, 15 How. at 112.]

The Supreme Court characterized the claim in Morse:

  It is impossible to misunderstand the extent of this claim.  He claims
  the exclusive right to every improvement where the motive power is the
  electric or galvanic current, and the result is the marking or printing
  intelligible characters, signs, or letters at a distance.

If this claim can be maintained, it matters not by what process or machinery
the result is accomplished.  For aught that we now know some future inventor,
in the onward march of science, may discover a mode of writing or printing at
a distance by means of the electric or galvanic current, without using any
part of the process or combination set forth in the plaintiff's specification.
His invention may be less complicated -- less liable to get out of order --
less expensive in construction, and in its operation.  But yet if it is
covered by this patent the inventor could not use it, nor the public have
the benefit of it without the permission of this patentee.
[56 U.S. at 119-20, 15 How. at 112-13.]

Though principally involving an issue raisable today under section 112, Morse
illustrates the Supreme Court's concern that an "effect" shall not be deemed
patentable subject matter. A claimed process or machine for implementing an
"effect," however, is patentable, The Telephone Cases, 126 U.S. 1, 534 (1887),
and that is so regardless of whether it constitutes the only known means of
accomplishing the "effect":

  We see nothing in Morse's case to defeat Bell's claim; on the contrary,
  it is in all respects sustained by that authority.  It may be
  that electricity cannot be used at all for the transmission of speech
  except in the way Bell has discovered, and that therefore, practically,
  his patent gives him its exclusive use for that purpose, but that does
  not make his claim one for the use of electricity distinct from the 
  particular process with which it is connected in his patent.
  [126 U.S. at 535, emphasis added.]

If such were not the law, the anomalous result would occur that an inventor
who first discovered a practical utility for a principle, law or force
of nature, or their mathematical expression, could not obtain a patent on a
particular process employing that practical utility.  The inventor would have
to wait until (if ever) he discovered at least one other practical utility,
and the public would have to wait for a disclosure of the inventor's initial
discovery. {4}

The type of problem facing the Supreme Court in Morse, The Telephone
Cases, and Benson, was described by one commentator over a hundred years ago,
as "how far a discovery or invention which may first disclose and practically
embody some truth in physics or some law in the operation of the forces of
nature, for a useful purpose, is capable of being carried in the exclusive
privileges secured by the grant of letters-patent," Curtis, A Treatise on the
Law of Patents For Useful Inventions 124 (4th ed. 1873).  Recent authority
is in accord.  See 1 Deller's Walker on Patents 23 (1964).

It is thus clear that the "nutshell" language of Benson, expressed the
ancient rule that practical application remains the key.  Because it did not
consider the performance of an algorithm by a computer as constituting a
practical application of that algorithm under the rule, the Court must have
viewed Benson's claims as effectively claiming the "effect," principle, or
law or force of nature (the algorithm) itself.

Claim Analysis

The distinction may thus be fine indeed between statutory and nonstatutory
subject matter, considering the glorious flexibility and frustrating
limitations of the English language on the one hand, and the ingenuity
of patent draftsmen on the other.  Nonetheless, the line required by
precedent, and which must here be drawn, is clear.  The mathematical
expression of scientific truth or principle is itself not patentable.
"The decisive factor is whether a claimed method is essentially a
mathematical calculation."  In re Richman, 563 F.2d 1026, (CCPA 1977).

In In re Chatfield, supra, the claims were drawn to a method of operating a
computer machine system.  Though it employed solutions from equations, the
method simply used the results of those equations, and the claims were not
drawn to an equation or algorithm per se.  In In re Deutsch, 553 F.2d 689
(CCPA 1977), the claims were drawn to a method of operating a system of
manufacturing plants, the method used the results of an algorithm, and the
algorithm was not the method nor the method the algorithm. In In re Flook,
F.2d, (CCPA 1977), unlike the situation in In re Christensen, 478 F.2d 1392,
(CCPA 1973), the post-algorithm solution activity recited in the claims
established that the claimed method involved simply the use of an algorithm,
and the claim was not in effect a claim to the algorithm per se.

On the other hand, in In re Waldbaum, F.2d, (CCPA 1977), a number of the
claims, drawn to an algorithm with no practical application other than in
a data processing apparatus, were deemed nonstatutory under the tenets
of Benson, and other claims, limiting the algorithm to a particular
environment, were nonetheless directed solely to a process for calculating,
i.e., to an algorithm per se. The same was true of the claims in
In re Richman, supra.

De Castelet's claim 7 begins by describing his method as one for "generating
a curve from data supplied to a computer." According to the claims, the
ultimate objective, curve generation, is achieved through use of a computer
instructed (programmed) to perform certain calculations upon stored and
inputed data.

Specifically, the inputed data are "supplied to a computer in the form of
coordinates of [end] points defining two given segments of tangents to the
curve to be generated." The data stored are "in the form of electrical
signals representing a table of coordinates of points, of characteristics of
a base curve inscribed on a unitcube between two opposite vertices of said
unit-cube." The computer is instructed "(1) to treat electrical signals
representing a given arc, defined [by the inputed data] * * * as the
transformed curve of said base curve, wherein [the inputed data] * * * are
considered as the transformed points of vertices of said unit-cube and
(2) to calculate and transmit to the control system of said model forming
means electrical signals representing the coordinates of a sequence of
points * * * of said given arc defining said curve to be generated."

The recited method steps are computation steps performed by the computer
pursuant to instructions "(1) and "(2)." Instruction "(2)" also includes
the non-computation step of transmitting electrical signals to a "model
forming means."

Overall, therefore, de Castelet's claimed method involves the storing of
certain mathematical data in a computer, inputing additional mathematical
data, causing the computer to perform programmed computations using the
stored and inputed data, and, finally, causing the computer to transmit
the results of those computations to a "model forming means."

We think the instant claims recite a process for solving a set of
mathematical equations per se, the solution being a set of points along
a curve, and not a process which merely uses equation solutions as one step
in achieving some result other than solution of the equations. We hold that
de Castelet's claims are drawn to nonstatutory subject matter. {5}

The claimed method is not one for operating a machine system or plant system
in a particular manner, as in Chatfield, supra, or Deutsch, supra. The
preamble recites that de Castelet's claimed method is for generating
a curve. The method steps claimed, however, are directed to computer
instructions for solving mathematical equations and transmitting
electrical signals representing those solutions.

That the computer is instructed to transmit electrical signals, representing
the results of its calculations, does not constitute the type of "post
solution activity" found in Flook, supra, and does not transform the
claim into one for a process merely using an algorithm. The final
transmitting step constitutes nothing more than reading out the result
of the calculations.  Recitations of specific machine elements, i.e.,
the mere reference in the claims to a computer and a model-former, cannot
alone render statutory the presently claimed subject matter as a whole.
Claims to nonstatutory processes do not automatically and invariably
become patentable upon incorporation of reference to apparatus. {6}

Though the preamble of the appealed claims refers to curve generation, the
recited steps do not result in the formation of a curved drawing or surface.
The "model forming means" is merely recited as receiving signals.

The potential for misconstruction of preamble language requires that
compelling reason exist before that language may be given weight.  No
such reason has been presented in this case.

De Castelet's reliance on In re Bernhardt, 57 CCPA 737, 417 F.2d 1395 (1969)
is misplaced.  The basis for the PTO rejection in Bernhardt being unclear,
we deemed it grounded on a "mental steps" theory:

  In the case now before us, the disclosure shows only machinery for
  carrying out the portrayal process.  In fact it is the chief object of
  the invention to eliminate the drudgery involved in a draftsmans' making
  the desired portrayals.  Accordingly, a statutory process is here
  disclosed.  Looking then to method claim 13, we find that it in no way
  covers any mental steps but requires both a "digital computer" and a
  "planar plotting apparatus" to carry it out.  To find that the claimed
  process could be done mentally would require us to hold that a human mind
  is a digital computer or its equivalent, and that a draftsman is a planar
  plotting apparatus or its equivalent.  On the facts of this case we are
  unwilling so to hold.  We conclude that the method defined by claim 13 
  is statutory, and its patentability must be judged in light of the prior
  art.  [57 CCPA at 745, 417 F.2d at 1401]

We recognize that de Castelet's claims include a computer and a drafting or
machine tool and, accordingly, are not drawn to "mental steps", but there is
no "mental steps" rejection before us.  Similarly, the basis of the present
rejection was not before us in Bernhardt.  That the same section of the 
statute is employed does not imply the same rationale underlying a rejection.

However, in Bernhardt this court cautioned against allowance of claims
preempting equations, and what was there said, regarding 101 in general,
does have relevance here:

  We think it is clear that in enacting section 101 Congress meant to exclude
  principles or laws of nature and mathematics, of which equations are an
  example, from even temporary monopolization by patent. [57 CCPA at 743,
  417 F.2d at 1399.]

The present equations remain "basic tools of scientific and technological
work", 409 U.S. at 67, 93 S.Ct. at 255, when resolved by a computer and when
computer resolution produces corresponding electrical signals. In sum, we are
convinced, as was the Supreme Court in Benson, that the resolution of de
Castelet's equations by a computer, which then transmits the electrical result
to a tool, is not a practical application within the rule, and that a patent
containing the appealed claims would in effect be no more than a patent on
de Castelet's equations.

Hence the present case falls on that side of the statutory-nonstatutory line
occupied, for example, by Morse, Benson, Christensen, Waldbaum, and Richman,
and not on that occupied by The Telephone Cases, Chatfield, Deutsch, and

Accordingly, the decision of the board is AFFIRMED.


{1} 35 USC 101 provides: "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."

Broader questions regarding general patentability of computer programs or
computer-implemented processes must await the action of Congress.  Dealing
only with the particular circumstances of the present appeal, we reiterate
that the analysis under 101 must be directed to the claimed subject matter
as a whole. Chatfield, 545 F.2d at 158. We specifically reject the
solicitor's suggestion that a piecemeal analysis is either useful or

{2} The examiner rejected earlier claims under 35 USC 103, citing substantial
prior art, of which a listing appears in the record, and asserting that the
art disclosed computer controlled drafting and machine tools and methods of
generating continuous curves from predetermined data points.  The record is
silent with respect to the reasons for not applying the same rejection to
claims 5-7.  The application was passed to issue, containing claims 5-7, and
withdrawn from issue after the Supreme Court's decision in Benson.

{3} Thus, it is important to distinguish between situations in which (1) the
scientific truth per se constitutes the sum of the subject matter sought
to be patented, and (2) the scientific truth is merely used in an overall
process or to describe a physical relationship.

In MacKay, RCA owned a patent claiming a particular antenna structure
having a relationship of elements conforming to a mathematical formula:

   15.  An antenna comprising a pair of relatively long conductors disposed
   with respect to each other at an angle substantially equal to twice 50.9
   (1/[*] ) -3.513 degrees, 1 being the length of the wire and the operating
   wave length in like units, and means in circuit with said antenna for
   exciting the conductors in phase opposition whereby standing waves of
   opposite instantaneous polarity are formed on the conductors throughout
   their length.

The formula per se would not have been patentable, but its use in describing
the new and useful antenna device would not render a claim to the device

{4} The anomaly is similar to that observed in In re Tarczy-Hornoch,
55 CCPA 1441, 397 F.2d 856 (1968), attendant the "mere function of a
machine" rejection.

{5} We voice no opinion on the patentability of disclosed but unclaimed
subject matter.

{6} A recitation that a process is performed by a machine may preclude
rejection on a "mental steps" theory.  See In re Prater, 56 CCPA 1381,
415 F.2d 1393 (1969).