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[Alastair Broderick]

_{(OOC Note: For any interested mathematics buffs (are there any?) I absolve myself of incorrect mathematical thinking; Alastair thinks in keeping with the trends of his time. Do not send me PMs about chaos theory )}

The following paper contains several results in the theory of operative symbols as employed by the differential calculus. To claim them all as new would be, in these days, a somewhat hazardous pretension; the author is, besides, fully sensible of the fragmentary and unfinished shape in which a few detached results are here presented, which evidently form but indications of a great extension of the infinitesimal calculus.

The present paper is confined to the consideration of symbols of the form ƒ(x, D); where D=d/dx. If µ be any such symbol, it is well to remember that it is commutative with any function of itself; in fact, that
ø(µ)π(µ) = π(µ)ø(µ).

In treating of these symbols, an operand will be usually understood when not expressed. A symbol may be isolated; i. e., considered independently of any operand; it is important to remember that this is different from the same symbol operating on unity. To illustrate this distinction, and the notation employed, let us consider the symbol xD, as operated on by D^-1, in the following cases:--

1. D^-1 xD = x - D^-1,
2. D^-1 |xD| = ½ (x²D) + ø(D),
3. D^-1 xD.1 = C,
4. D^-1 |xD|.1 = C.

In (1) and (2), an operand is understood; but in (2) the operation D^-1 is limited by the bars || to the symbol xD.

Alastair rested his pen for a moment, examining the words in his note-book. The article he was writing was one he hoped to publish in Proceedings of the Lindebo Mathematical Society, a small but respectable journal of mathematics in the city. He would have sent the article to London to be published in one of the larger journals, but he was an unknown author and would likely be laughed at, even though his mathematical basis was solid.

He leaned back against the bole of the tree he sat under, near the south shore of Windemere, one of Lindeboshire’s lakes, looking up at the sky and luxuriating in the feel and sound of nature around him. He sighed contentedly. His best thinking was always done out here; the only problem was that he had to persuade himself to concentrate and not simply fritter the day away philosophizing about trivialities or just taking in the beauty of God’s creation. He gazed into the leaves overhead and thought that there must be a way to describe the pattern of the winds; there was order in everything, even something so seemingly random as the movement of the air. Surely if there were devices to measure them accurately enough, a table could be constructed whereby predictions might be made of future movements?

A wing of birds flew overhead, and the mathematician’s thoughts flew off with them, abandoning the patterns of air for the dynamics of a bird’s wing. He had done some experimental mathematics to try and prove why the Wright brother’s machine was the only way that man would ever fly; that is, with a fixed-wing craft. He had yet to be able to prove it soundly, in a mathematically acceptable fashion, but he was certain he was correct. Actually, to be honest with himself, he supposed with strong and light enough metals it could be done. But it would require a wingspan many meters in length, and he could not think of any material that would be light enough for a single man to operate such wings by himself. Adding more men would increase the need for lifting power, and that in turn would increase the wingspan, and the whole problem would cycle itself quickly into improbability.

But how marvelous it would be to fly, he thought. Simply wonderful. Many people knew about the balloons lifted from heated air, and the dirigibles lifted from hydrogen gas. But only the rich (well, he allowed to himself, the middle-class or better, not the extremely poor such as himself) could afford to lark about on them, something that Alastair was most certainly not. The Wright brother’s machine was not practicably useful as yet, and although rumours were circulating amongst the people that paid attention to engineering, such as he, about research being done by Count von Zeppelin, those were nothing but smoke on the wind, and had been making the rounds for years.

His eyes followed the birds as they flew out over the lake, and the pen gradually slumped as both it and his hand came to rest on the grass. The mathematician was lost in his musings.

[Alastair Broderick]

Alastair had determined, upon further reflection, that it was his greatest dream to fly. This, he supposed, was slightly whimsical, given that he would never have the money to spare for a trip on a hot-air balloon or a dirigible. It was possibly slightly more attainable than his last greatest wish, which was previously to find a proof for Fermat’s Last Theorem. The irritating fellow had died without ever proving his conjecture, and Alastair had become intrigued with the problem a year before. His proof had gotten nowhere, however, and he had been obliged to give it up in the interest of research into more currently fashionable (and more marketable) areas of mathematics.

That was why he was now working on the paper on the symbols and theory of the differential calculus. Such a paper was recent, and up-to-date, and in the fashion of mathematicians just now. If he could succeed in getting it published, it would go a long way towards his recognition as a mathematician of note; and as such, his employability would increase and he might be expected to be offered a position to teach at one or two of the universities. Less desirable but also possible would be the offer of a position as some Lord’s private tutor for a son that the Lord wanted educated in the best possible manner.

It wasn’t that tutoring was abhorrent to Alastair, that wasn’t why he didn’t particularly want that position and would rather have a position as professor. After all, he tutored Quartermaine’s son—a bright lad but not one cut out for a life in mathematics to Alastair’s mind—part-time at the Lindeman Theatre. It was just that tutoring took up so much time if one did it full-time and really left little to spare for one’s own pursuits. A professor was expected to publish, and was given the time necessary to research.

He realised that he had not gotten much work done to-day, though, his attention being distracted by fancies of flight. He would have to work on that. But, he forgave himself generously, he was usually a most diligent and hard-working fellow and could be expected to have at least one day where he indulged himself in such relaxing past-times. He pocketed his pen and note-book, and stood to leave the Windemere lake-side. It was not a very long journey into the city, and he quickly made his way to his boardinghouse, in the much lower ends of town.

(OOC: Alastair next posts in Bad news at the boarding-house)