Algebraic Triplets

The standard story of algebraic triplets is that William Rowan Hamilton wanted to generalise the geometric view given by the complex plane (the Argand diagram) to three dimensions so that applications in 3-dimensions could benefit from the system of triplets in an analogous way to how the complex numbers give a powerful way of making applications in 2-dimensions. For example in 1842 Hamilton was so preoccupied with the triplets that even his children were aware of it. Every morning they would inquire:-

Well, Papa can you multiply triplets?
but he had to admit that he could still only add and subtract them. This version certainly does not give credit to Charles Graves and his brother John Thomas Graves who were Hamilton's friends and did a great deal to keep him interested in multiplying triplets. The part of the story which does not often get a mention is that after Hamilton discovered the 4-dimensional quaternions in October 1843, he, the Graves brothers, and other mathematicians continued to make strenuous efforts to find a system of triplets. We can get some indication of this from the four papers in the Proceedings of the Royal Irish Academy from 1845. These four papers, all entitled On Algebraic Triplets, report mainly on work by Charles Graves but also indicate contributions by others. The four papers are:

  1. On Algebraic Triplets, Proceedings of the Royal Irish Academy (1836-1869) 3 (1844-1847), 51-54.

  2. On Algebraic Triplets (Continued), Proceedings of the Royal Irish Academy (1836-1869) 3 (1844-1847), 57-64.

  3. On Algebraic Triplets (Continued), Proceedings of the Royal Irish Academy (1836-1869) 3 (1844-1847), 80-84.

  4. On Algebraic Triplets (Continued), Proceedings of the Royal Irish Academy (1836-1869) 3 (1844-1847), 105-108.

We now give a number of quotes from these papers which indicate the work that was going on. As we will explain, the mathematics in these papers is of little interest, so we make little mention of it:

Quotes from the paper (i):

The Rev Charles Graves read the first part of a paper on Algebraic Triplets. The object which he proposes to himself is to frame, for the geometry of three dimensions, a theory strictly analogous to that by which Mr Warren has succeeded in representing the combined lengths and directions of right lines in a plane. In carrying out this design Mr Charles Graves has necessarily been led to the consideration of new imaginaries ... The problem now proposed by Mr Charles Graves is to assign two distributive symbols, i and k, of such a nature that:

  1. the sum or product of two triplets, x + iy + kz and x'+ iy'+ kz', shall be itself a triplet of the same form:
  2. there shall be theorems concerning the moduli and amplitudes of triplets, similar to those already enunciated for couplets:
  3. the equation x + iy + kz = 0 shall be equivalent to the three, x = 0, y = 0, z = 0:
  4. the symbols i and k shall admit of a geometric interpretation analogous to that which has been provided for the symbol √-1. ...
Comment on the above quote:

Although Charles Graves takes i and k to be abstract symbols, the properties he gives them means that the system he is considering is isomorphic to triplets

x + iy + i2 z
where i is a cube root of 1 not equal to 1. Now it appears that De Morgan had come up with a similar idea but using the cube root of -1.

Quotes from the paper (ii):

Mr Charles Graves mentioned that, since he had obtained per mission to read the present paper, Sir William R Hamilton had kindly communicated to him the abstract and the proof sheets of a memoir by Professor De Morgan, on Triple Algebra. That paper contains the discussion of a system of triplets, which is most closely connected with the one now proposed: the only difference being that Professor De Morgan uses what are in fact new imaginary cube roots of negative unity. Mr Charles Graves thinks that in the interpretation and generalization of his results he has met with greater success; but he fully concedes to Professor De Morgan the prior possession of what must be looked upon as fundamental in this theory, the conception of symbols which act upon each other in the same manner as the imaginary cube roots of unity. Mr Charles Graves also stated that his brother, John T Graves, Esq., had anticipated him in the idea of using cube roots of positive unity in the constitution of algebraic triplets.

Comment on the quotes:

In the next two papers various results are proved for Graves' triplets which are analogous to the results for complex numbers. For example analogues of de Moivre's theorem and Cotes' theorem are proved. It is certainly not surprising that analogous theorems could be proved since the system under consideration is, of course, just the complex numbers in disguise. For this reason the system of triplets given by Charles Graves and that given by De Morgan are of no particular interest.

Quotes from the paper (iii):

Mr Charles Graves stated that Sir Wm Hamilton had been the first to announce that if the real unit line, the factors, and the product line, be projected upon the symmetric axis, the projections will form a proportion in the simple sense of that term ... Having thus interpreted the results of multiplication by means of the existing trigonometry, Mr Charles Graves proceeded to show how the use of a new kind of trigonometry gives in creased symmetry and flexibility to the present theory of algebraic triplets. ... Mr Charles Graves mentioned that his elder brother, Mr John T Graves, had been the first to conceive the notion of employing the functions l, m, and n in the interpretation of this theory of triplets ... The President (Sir William R Hamilton) stated that the remarkable researches respecting algebraic triplets, made lately by Professor De Morgan and John T Graves, Esq. in England, and here by the Rev Charles Graves, had led him to perceive the following theorem ... . ... some of the results of the systems of the two Messrs Graves (Charles Graves and John Thomas Graves) may be reproduced by [Hamilton's results] ... Sir William Hamilton pretends to no farther merit in the matter than to that of having sought to illustrate, by generalizing in one direction, the foregoing points of the theories of his friends.

Comment on the quotes:

In the third of the papers, further geometric interpretations are made. In the fourth and final paper, however, it is reported that Charles Graves had come up with another system of triplets.

Quotes from the paper (iv):

The Rev Charles Graves made a further communication relative to Algebraic Triplets, and their Geometric Interpretation. Besides the system of algebraic triplets developed in former communications to the Academy, Mr Charles Graves has conceived another, which appears to admit of an interpretation in some respects more closely analogous to Mr Warren's geometrical representation of imaginary quantities.

Comment on the quotes:

We now know that all attempts to construct the type of algebraic triplets with the properties that Hamilton, Charles Graves, John Graves and De Morgan were looking for are doomed to failure since none exists.

JOC/EFR February 2016

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