The Mathematics of Nikola Tesla

In the past I have always tried to ask people to quote Tesla; and with regards to what other people have printed about him... take it with a pound of salt.

It has been remarked to me that “You can’t be a engineer without Math.” This statement led me to ask the question:

What mathematics did Tesla use?

Without a doubt Telsa learned algebra, trigonometry, and--in all probability--calculus. But, in engineering today, little can be done without the use of vectors. And, at the time Tesla was being schooled as an engineer, there was only one system that was being discussed on the continent, and that was a system called Quaternions.

(Here I would like to remind the reader that Tesla was born between the 9th and 10th of July, 1856.)

Quaternions were developed by Sir William Rowan Hamilton on the 16th of October 1843. Actually it took a little longer than a day to develope the system, but on that day in a flash of inspiration, much like Tesla’s inspired alternating current motor, Hamilton solved the problem of using imaginary numbers to represent meaningful geometric relations. A better description of the enlightened discovery of quaternions is best provided in a letter by Hamilton to his son Archibald in 1865:

If I may be allowed to speak of myself in connexion with the subject, I might do so in a way which would bring you in, by referring to an ante-quaternionic time, when you were a mere child, but had caught from me the conception of a “Vector”, as represented by a “Triplet”: and indeed I happen to be able to put the finger of memory upon the year and month - October, 1843--when having recently returned from visits to Cork and Parsonstown, connected with a Meeting of the British Association, the desire to discover the laws of the multiplication referred to regained with me a certain strength and earnestness, which had for years been dormant, but was then on the point of being gratified, and was occasionally talked of with you. Every morning in the early part of the above cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, “Well, Papa, can you multiply triplets”?

Whereto I was always obliged to reply, with a sad shake of the head: “No, I can only add and subtract them.”

But on the 16th day of the same month - which happened to be a Monday, and a Council day of the Royal Irish Academy--I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a “result”, whereof it is not too much to say that I felt “at once” the importance. An “electric” circuit seemed to “close”; and a spark flashed forth, the herald (as I “foresaw, immediately”) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse - unphilosophical as it may have been--to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely

i² = j² = k² = ijk = -1

which contains the “Solution” of the “Problem”, but of course, as an inscription, has long since mouldered away. A more durable notice remains, however, on the Council Books of the Academy for that day (October 16th, 1843), which records the fact, that I then asked for and obtained leave to read a Paper on Quaternions, at the First General Meeting of the Session: which reading took place accordingly, on Monday the 13th of November following.

Hamilton spent the next twenty two years developing quaternions until his death on September 2, 1865. In his letters the first few days after his discovery, he felt his system had applications in such diverse fields as heat theory, electricity, and spherical trigonometry.

Tesla was 9 years old when Hamilton went to his reward.

The banner of quaternions was carried on by Peter Guthrie Tait, who made contributions toward thermodynamics. Tait was a graduate of Cambridge, and later became Professor of Mathematics at Queen’s College. Later he served as Professor of NaturalPhilosophy at Edinburgh from 1860 till his death on July 4th, 1901. Tait was also a prime mover for quaternions during the whole of this period. Tait’s importance to the influence of quaternions on Tesla is outside the scope of this article. However, one of Tait’s good friends was James Clark Maxwell. As a matter of fact Tait and Maxwell corresponded quite frequently.

Some of these letters make for very interesting reading. In one letter dated November 17, 1870, Maxwell wrote to Tait:

With regard to my dabbling in Hamilton I want to leaven my book Treatise on Electricity and Magnetism with Hamiltonian ideas without casting the operations into Hamiltonian form for which neither I nor I think the public are ripe.

Later, in October of 1872, Maxwell wrote to Tait:

I am going to try, as I have already tried, to sow quaternion seed at Cambridge.

This statement was made after Maxwell was chosen as the first professor of Experimental Physics at Cambridge and Director of the prestigious Cavendish Laboratory the year before. In November of 1872 Maxwell wrote to Tait:

But the virtue of the quaternions lies not so much as yet in solving hard questions, as in enabling us to see the meaning of the question and of its solution, instead of setting up the question in x y z, sending it to the analytical engine, and when the solution is sent home translating it back from x y z so that it may appear as A, B, C to the vulgar.

In 1873 Maxwell wrote a review of Kelland’s and Tait’s recently published Introduction to Quaternions. This review was published in Nature and rather appropriately titled Quaternions. In this report, Maxwell had this to say:

There are physical quantities of another kind which are related to directions in space, but which are not vectors. Stresses and strains in solid bodies are examples of these, and so are some of the properties of bodies considered in the theory of elasticity and in the theory of double refraction. Quantities of this class require for their definition “nine” numerical specifications. They are expressed in the language of quaternions by liner and vector functions of a vector.

To say that Maxwell was an influence on Tesla, or at least on Tesla’s teachers, is an understatement. Maxwell’s impact on electricity and magnetism extends even until today and in all probability into the foreseeable future. It is also interesting to note the fact that if Maxwell were to see his own equations today that they would be almost unrecognizable, because the notation they appear in was not developed until about 30 years after his death.

Maxwell died when Tesla was 23 years old.

When Tesla was nine years old, Hamilton died and the second book he had been working on was published posthumously. (The first book, Lectures on Quaternions, was published in 1853. The second, and last, Elements of Quaternions was published in 1866.) At this time Tesla was just starting his education at the Real Gymnasium at Gospic.

John J. O’Neill in his book, Prodigal Genius, reported that Tesla had from an early age the ability to envision objects with great clarity. Additionally, he could envision a blackboard and solve problems of arithmetic or algebra more rapidly than writing the problem out, to the point of rendering a solution almost as quickly as the whole problem was stated. O’Neill also states in his book that at this early age, that mathematics was Tesla’s favorite subject.

Tesla at fifteen attended the Higher Gymnasium at Karlovac in Croatia, completing the four-year course in three years. There Tesla apparently had a rather good physics professor who had impressed him favorably with electrical phenomena.

It was 1875 when Tesla went to Gratz, in Austria, to study electrical engineering. Quaternions had been in existence almost 30 years, and considerable amounts of information had been developed with regard to their application in all branches of engineering. It is almost inconceivable that quaternions were not known at the Polytechnic Institute. Certainly, later when Tesla attended the University of Prague, where he studied mathematics and physics, he was exposed to Hamilton’s ideas. As well, Tait’s Introduction to Quaternions had been in print for two years, Maxwell was at work “sowing quaternion seed” all about Cambridge, and Maxwell’s Treatise on Electricity and Magnetism had been in print almost ten years. The latter must have been required reading.

At this point another statement should be called to the reader’s attention that was made in Maxwell’s treatise:

...Quaternions, will be of great use to us in the study of all parts of our subject, and especially in electrodynamics, where we will have to deal with a number of physical quantities, the relations of which to each other can be expressed far more simply by a few expressions of Hamilton’s than by the ordinary equations.

One particularly interesting remark made by Maxwell in his review of Tate’s “Introduction to Quaternions” is, (and here I am forced to paraphrase for lack of the complete reference),

Now, Quaternions, or the doctrine of Vectors, is a mathematical method, but it is a method of thinking, and not, at least for the present generation, a method of saving thought.... It calls upon us at every step to form a mental image of the geometrical features represented by the symbols, so that in studying geometry by this method we have our minds firmly fixed on the problem before us.

In light of this statement it would seem that Tesla would have been particularly well suited to use the quaternion system. It would also seem that the quaternion system was in place at the proper time for Tesla to have used it. In Michael Crowe’s book, A History of Vector Analysis, he describes Maxwell’s Treatise on Electricity and Magnetism as follows:

In the text of the book quaternion notations were used in a substantial number of cases. These may be classified into three groups: to the first belong those cases in which at the end of a section Maxwell stated that he would give the quaternion form of an equation; to the second group belong those cases in which quaternion notation was used in such an elementary form that little importance can be attributed to it; and to the third class belong the most important cases, those in which Maxwell used quaternion expressions almost in such a way as to integrate them into the development.

Here I would like to quote a statement from my old but trusty 1968 edition of Britannica, in the section: Number: Quaternions and Hypercomplex Numbers.

For many years quaternions were widely used in solving physical problems, but since about 1900 their use in physics has been replaced by that of the ‘Vector Calculus’.

With modern vector calculus there are two cases to consider. The scalar ‘dot’ product and the ‘vector’ cross product. With both cases the associative law is not relevant, division is ambiguous, and the law of the moduli fails. Additionally, with the vector cross product the commutative properties fail as well. It is interesting to note that with quaternions only the commutative property has to be abandoned. (In quaternions: qq’ = -q’q). This was a suitable replacement for quaternions? It would seem, at best, a suitable subset of quaternions, and with a seemingly suitable subset of solutions.

My parting statements are these: In 1900 Tesla was 44 years old. Quaternions were 57 years old. Tesla grew up in the shadow of quaternions. I am convinced Tesla used quaternions.

Afterthoughts:

Currently I am convinced that Tesla used quaternions in his work. I am in the process of obtaining more complete information on quaternions and researching if Tesla did indeed use quaternions in his work. If at such time I am able to determine that Tesla did not use quaternions, I am prepared to retract the assertion that Tesla used quaternions--just to keep the record clear.

References:

• John J. O’Neill, Prodigal Genius: The Life of Nikola Tesla, Inventor Extraordinary, Hollywood, CA:
• Michael Crowe, A History of Vector Analysis: The evolution of the idea of a vectorial system. Notre Dame, IN 1967.
• Richard E. Williamson, Richard H. Crowell, and Hale F. Trotter, Calculus of Vector Functions: 2nd Edition, Englewood Cliffs, NJ: Prentice-Hall, Inc, 1968.