India: Mathmatical Equation Per Se Cannot Be Copyrightable

Last Updated: 26 February 2019
Article by Pratibha Ahirwar

There is no legal provision till now which would establish that the mathematical equations are copyrightable or not? The conclusion is being made with the help of idea-expression dichotomy and the doctrine of merger.

The mathematical equations are not copyrightable per se because granting the copyright will create a monopoly and every person would not be able to use the equations (established fact). It is very similar to the saying that the law has been made but no person can use it. As, no one has copyright over the law/rule, the very same way mathematical equation also cannot be copyrighted. The mathematical equations are the law of nature and hence every person has the opportunity to use it. Therefore, copyright is not granted to the mathematical equations.

The aim of the Indian Copyright Law is to promote the progress of the useful arts and science by protecting the exclusive right of authors and inventors to benefit from their works of authorship. Copyright law protects literary, musical, graphical or other artistic forms of the works in which the author expresses the intellectual concepts. While there are some exceptions to this rule, one of them is mathematical equations which cannot be copyrighted. In the normal parlance, if the excerpt is original work of the author it is a subject-matter of the copyright. But this is not the case with mathematical equations as they are the matter of fact and laws of nature so each and every person is authorized to use it without the fear of copyright infringement.

The discovery of such equations cannot create monopoly over the person who describes it, as it will restrict the free-flow of information and can be used by only the proprietor owner of that work or the licensee of the owner. No other person can use it.

In the case,Chancellor Masters and Scholars of The University of Oxford v. Narendra Publishing house &Ors[1], Justice Bhat held that "mathematical questions are expression of laws of nature. The discovery of such laws cannot confer monopoly to those who describe it. The reason is that language is a limited medium, which enables description of such laws of nature – in only a few ways."

For Example – A mathematical equation A2 +B2 = C2. There is no copyright protection for this equation. These all are well-established laws of the nature. This is the point where the doctrine of merger comes into play which is discussed in detail in next section.

While, if the book with same set of equation comprising of many examples and questions in it, than the author of that book has the right of copyright in its original work.

For Example – Mrs. Shakultala Devi, Dr. R.D. Sharma do have the copyright in their respective mathematics book. Both books will have the general concept of the mathematics but in totality the book have different examples, sets of arrangement, questions, etc., so they have different copyright in their own books.

In a legal sense also, you can copyright the content of a math book (such as diagrams, word problems, and illustrations) but the formula, problems, proofs, theorems themselves cannot be copyrighted. You cannot copyright a fact which is established and natural i.e., 1+1=2. This means that if you express your math formula in a book copyright protection will extend to the book as it the original work of authorship. Copyright subsists will be limited to the copying of the mathematical publication but not on the mathematical formulas contained in the publication.


A foundational element of copyright law is that it does not grant the author of a literary work protection on ideas and facts.[2] It is the creative expression of such ideas and facts that is rewarded by law, by conferring a privilege to exclusively exploit such expression for a limited time. Law does not protect every such expression. The law affords protection to expressions that are fixed in a medium and are "original".

In India, Section 13 of the Copyright Act states that only "original" literary, artistic, dramatic and musical works are subject matter of copyright. A literary work, in order to qualify as work in which copyright can subsist, must therefore be original. The important thing to note is that the work must be fixed in some tangible form of expression; you cannot copyright an idea or concept based on the way it is expressed.

In order to balance the interests of society in the free flow of information against the property interests of authors gave rise to the idea of protecting expression but not ideas. Courts have restricted themselves to put forth a straitjacket definition for the term idea. An idea has been described as a thought, as a virtual image, as a conception of a theory. In layman terms, an idea can thus be described as a formulation of thought on a particular subject while expression would constitute implementing the said thought. Needless to state, the issue of copyright arises when the same idea can have numerous expressions. If the same idea can be expressed in a number of different ways, a number of different copyrights may coexist and no infringement will result. However, problem arises when it becomes difficult to trace the difference between the idea and its expression. Herein lays the idea of merger where an idea and the expression cannot be delineated and they are said to have merged. In case of merger, the expression may not be copyrighted, because to do so would in effect be copyrighting the idea and idea is not copyrightable.

However an often quoted policy concern of this doctrine is that, to avoid the monopoly of the idea upon copyright owner, when the expressions are inseparable from the ideas, those expressions are not protected. At the same time, an idea also has certain expressions, without which the idea cannot exist. In other words, there can exist an idea where changing the expression of the same in a particular form would, in effect change the very idea itself. Mostly, courts are in the opinion that these essential ideas not copyrightable, as to copyright them would also, in effect, copyright the idea. This type of merger is called scenes a faire.

Another example of merger is when there are very few ways to express a given idea. This is called 'Idea-expression identity' exception when specific instructions, even though previously copyrighted, are the only and essential means of accomplishing a given task, their later use by another will not amount to an infringement. The idea/expression dichotomy has been miserably fallen for its failure to provide practical guideline instead of the fact that the idea/expression dichotomy is such a time-honored doctrine. The complexity lies in the fact that very few, if any, works contain exclusively either ideas or expressions. Indeed, almost any and every work can be abstracted into a spectrum of various levels of generality, at one extreme of which is the principal goal or theme of the work and the other extreme is the literary expression.

Courts recognize degrees of merger."[3] An expression is considered indispensible or necessary to an idea. In certain circumstances, protection of the expression not the idea would confer a monopoly on the idea which is contrary to the intent of copyright law.[4] In the case of Freedman v. Grolier Enterprises,[5] the court implied that, "In the case of copyright protection is not given "to a form of expression necessarily dictated by the underlying subject matter." Inseparable expression that is copied is permitted.[6] In other circumstances, courts statedthat ideas and expressions are not inseparable, but a limited change in the form of the expression of an idea is possible; here, infringement occurs only with whole appropriation of writing.[7]

Thus, under this doctrine, the scope of protection for the writings depends upon the possibility of existence of a variety of expressions.[8] The merger doctrine is open to criticism. It merely represents an epicycle generated bythe faulty metaphysics of the idea/expression dichotomy: an "expression" would never be taken as adding anything to an "idea" unless an idea is treated as an abstract entity behind the expression. Merger presupposes that normally there is room for substantial variation in the expression of ideas; only when there is little choice in the expression does the expression collapse into the idea.[9] It is also true of expressions in all contexts since the form and content of writing shape each other. Writings at best differ from one another only in degree.

The above rule of doctrine of merger accurately fits in the case of mathematical equations. They are not copyrightable as it is difficult to separate its idea and expression from each other so granting the copyright will indirectly mean copyrighting the idea as well, which is not permissible under copyright law.


From the above discussion, it is clear that mathematical equation per se is not copyrightable the reason behind it is the Doctrine of Merger. If mathematical equations are reduced in compiled form, having original idea to it than it is protectable under copyright law.

In the case of Eastern Book Company & Ors v. D.B. Modak & Anr[10] the court held that if the compiled form of any written excerpt fulfills the two condition/doctrines i.e.,

  1. Sweat of Brow; and
  2. Modicum of Creativity.

Than that work will be considered to be 'original' and will be protectable under the copyright.


[1] 2008(106)DRJ482.

[2] Baker v. Seldon [1879] 101 US 99;,Nichols v. Universal Pictures Corp. 45 F.2d (2d Cir. 1930), RG Anand v. DeluxFilmsMANU/SC/0256/1978 : [1979]1SCR218.

[3] Francione, Facing the Nation: The Standards for Copyright, Infringement, and Fair Use of Factual Works, 134 U. PA. L. REV. 519, 573 n.265 (1986);

[4] Herbert Rosenthal Jewelry Corp. v. Honora Jewelry Co., 509 F.2d 64, 65 (2d Cir. 1974); Kalpakian, 446 F.2d at 742.

[5] 179 U.S.P.Q. (BNA) 476, 478 (S.D.N.Y. 1973).

[6] Johnson Controls v. Phoenix Control Sys., 886 F.2d 1173, 1175 (9th Cir. 1989);

[7] Frybarger v. International Business Mach. Corp., 812 F.2d 525, 530 (9th Cir.1987) (extended protection against virtually identical copying);

[8] Cooling Systems and Flexibles v. Stuart Radiator, 777 F.2d 485, 491 (9th Cir. 1985);

[9] The doctrine in Baker,101 U.S. 99, 101-03 (1879), is open to the same criticism as merger. In the words of Professor Nimmer: "It is factually erroneous to conclude that there is any system or method which can be performed by the use of only one particular form of written expression."

[10] (2008) 1 SCC 1.

The content of this article is intended to provide a general guide to the subject matter. Specialist advice should be sought about your specific circumstances.

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