Part I:
In the contemporary urban world, there is no educated person who is ignorant of the meaning of the word 'computer'. Consider now the following sentence, which appears in a brief biographical sketch of L.W. Reid (1867-1961), the author of a well-known book on number theory: "....He then worked for the U.S Bureau of the Census and the U.S. Coast and Geodetic Survey, where his job title was listed as "computer"." This usage of the word 'computer' the modern reader will doubtless find rather surprising. It was in usage in the distant past when electronic computing devices had not become available, and calculational work in professional institutions was carried out by employees specially trained to do so. These men were called 'computers', i.e human computers.
In this article, I intend to discuss the mathematics of Euler and Jacobi, two of the greatest mathematicians of all time, and indeed, the two greatest human computers of all time.
Students of engineering are very familiar with the name of Euler because engineering mathematics is replete with results discovered by him. The name of Jacobi, the greatest exponent of Eulerian mathematics amongst the successors of Euler, is also well-known amongst these students, mainly because of their acquaintance with the determinant of partial derivatives called the Jacobian. Honors students of mathematics have to acquire a deeper knowledge of his wide-ranging contributions than their engineering counterparts.
Both these men were indefatigable in calculations. As Arago said, "Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the air." The same could have been said about Jacobi. Their mathematics was dominated by formulae, and they were consummate masters of symbolic work. A careful study of their original publications reveals that the manner in which their minds operated was essentially the same as that in which our own (electronic) computers run.
E. T. Bell, in his book entitled 'Men of Mathematics', famously named the chapter on Jacobi 'The Great Algorist'. In the chapter on Euler, he writes: "As an algorist Euler had never been surpassed, and probably never even closely approached, unless perhaps by Jacobi. An algorist is a mathematician who devises "algorithms" (or "algorisms") for the solution of problems of special kinds....An algorist is a "formalist" who loves beautiful formulas for their own sake."
The contributions of both Euler and Jacobi abound in feats of spectacular
calculations.. Euler developed the theory of trigonometric functions, deriving many of the numerous identities which generations of students since his time till the present have been taught. Much of what is called higher algebra was developed by him, and his seminal works on calculus constituted the foundation on which modern analysis was constructed by his successors. In number theory, he proved many of the results which were found, but left unproved, by Fermat, and discovered numerous results of his own. In all this, Euler carried out extensive calculations to gather data and discerned patterns amongst them which he thereafter formalized into provable theorems. His mind searched for significant results amongst reams of numerical data which he gathered by computation, and in so doing he foreshadowed the mode of operation of the computers that we use. In his study of Diophantine equations (i.e equations solvable in integer values of the variables), he was driven by his insatiable hunger for computation. I cannot do better than quote Bell: "If a single equation of the second degree in two unknowns proved unexciting, Euler increased the degree to three or four. If this failed to provide an attractive equation, he simultaneously increased the number of unknowns. As a last resort, he increased the number of equations and exercised his unequaled ingenuity on simultaneous systems." ( 'The Development of Mathematics', p.299)
Jacobi continued the Eulerian tradition of algoristic mathematics, enriching it with his own creative ideas and prodigious powers of calculation. His papers on elliptic functions and theta functions abound in numerous formulae and identities derived by computations of amazing ingenuity. His instinct for discerning patterns and symmetries amongst labyrinthine masses of formulae was of the most incredible order conceivable. He, like Euler, appears to have been capable of programming his mind in the manner in which we program our computers, for carrying out exceedingly difficult computations in a systematic and methodical manner. He developed his own algorithms and was able to effectively implement them computational contexts.
In a memoir on theta functions, Jacobi derived a master identity from which a total of 256 identities can be derived. While reading this memoir, I felt that perhaps it should be possible to write a program and run it on a sophisticated computer to derive all of them.
Both Euler and Jacobi were masters of iteration and recursion, the twin pillars on which many of our computer programmes rest.
Part II:
The names of Laplace and Lagrange are well-known to students of engineering. They too relied on extensive calculations in their work, and Jacobi learned his mathematics by reading their publications in addition to the monumental oeuvre of Euler. He wrote in a letter to his uncle Lehmann: "The huge colossus which the works of Euler, Lagrange, and Laplace have raised demands the most prodigious force and exertion of thought if one is to penetrate into its inner nature and not merely rummage about on its surface. To dominate this colossus and not to fear being crushed by it demands a strain which permits neither rest nor peace till one stands on top of it and surveys the work in its entirety. Then only, when one has comprehended its spirit, is it possible to work justly and in peace at the completion of its details." I can only add that those contemporary students of mathematics and engineering who are academically ambitious and desirous of engaging in higher studies have much to learn from this quotation.
Students of engineering mathematics are familiar with determinants. It is known that it is not difficult to evaluate 2×2 and 3×3 determinants by hand calculations. But any student who has ever tried to evaluate a 10×10 determinant knows how prohibitively difficult the calculations become, as the order of a determinant increases. In such cases, a computer has to be used. One of the major contributions of Jacobi was the development of the theory of determinants. Earlier mathematicians had used determinants, but it was Jacobi who first fully realized their immense importance in algebra and various other branches of mathematics.
Jacobi clearly comprehended the importance of the derivation and storage of large masses of numerical data, ready for recall and reference as and when necessary. Accordingly, he published in 1839 a table of indices and powers with respect to primitive roots for prime powers less than 1000, entitled 'Canon Arithmeticus'. A soft copy of it is now available on the internet, and anyone who glances at it will realize the mind-boggling magnitude of the computational labour and ingenuity that had enabled Jacobi to construct it. We have to remember that there were no computing devices in the 1830's - not even the most primitive ones - and Jacobi must have relied entirely on hand calculations.
It is well-known that computers do much else than numerical or algebraic computation. Indeed, for most people, the chief function of a computer is as a word-processor. Recently, computers have begun to be used extensively in the translation of texts, and even oral speech, from one language to the other. The major role of computers in language studies and linguistics is now well established.
Euler and Jacobi were both extraordinary linguists. The minds of both were steeped in classical learning, and both were equally fluent in several languages. Euler wrote most of his books and papers in Latin and wrote his highly influential book entitled 'Elements of Algebra' in German. He must have been fluent in French too and learned Hebrew early in his life. Jacobi wrote with equal facility in Latin, French, and German (and if my memory serves me, he wrote at least one paper in Italian). How his writing mind could have operated almost simultaneously in at least three languages with equal proficiency, and that too in abstruse mathematical contexts, is a redoubtable mystery which contemporary psycholinguistic researchers should endeavor to unravel. Not only in respect of numerical and symbolic computation but also the processing of several languages in their minds, Euler and Jacobi were human computers par excellence. A close study of their contributions reveals that in their minds, language and mathematics were intimately intertwined with each other. Indeed, looking at their works, one feels that language and mathematics are but two aspects of the same central intellectual entity.
It was Leibniz’s dream to construct a universal symbolic language for science, mathematics, and metaphysics. A computing machine can be interpreted to have been a part of this dream. Euler’s contributions to the development of calculus depended heavily on Leibniz’s ideas and his notation for differentiation and integration, and Jacobi was profoundly influenced by Euler. Therefore, the contributions of Euler and Jacobi, and all that has been achieved in the recent past in computing science, together constitute a substantial part of the realization of the great dream of Leibniz.
The ideals of Euler and Jacobi have re-emerged in the computer age of the 21st century in the new mathematics, which is actually the old mathematics of the 18th and 19th centuries in a new incarnation. This phenomenon demonstrates clearly that amongst all disciplines, it is only in mathematics that nothing really is either new or old; for everything in it is eternal.
Regards,
Prof. Somjit Datta,
Department of Mathematics,
Heritage Institute of Technology
Enriching ideas
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