As we have seen, Maxwell took the molecular-vortex model quite seriously – with ontological intent – when he first presented it in 1861–2, and although he later lost confidence in certain aspects of the model and removed it to the periphery of his research program, he continued in his allegiance to the core hypothesis of the model – that is, to the hypothesis of molecular vortices. The centrality of the molecular-vortex model in Maxwell's general thinking about electromagnetic theory and the particular importance of this model in the background of the displacement current and the electromagnetic theory of light together provide motivation for a careful study of this intricate mechanical model of the electromagnetic field.

Maxwell's work on the molecular-vortex model was guided, above all, by his desire – his commitment – to fashion a coherent and comprehensive theory unifying the full range of electromagnetic phenomena from the field-primacy point of view. This was required in order to produce a credible alternative to Wilhelm Weber's unification of electromagnetic theory within the charge-interaction framework; comprehensiveness and coherence were required also in connection with the intended realistic status of the theory – Scottish and Cambridge methodologies converged on this requirement.

Three perennial issues in Maxwell scholarship have woven their way through our study of the origins of the displacement current and the electromagnetic theory of light in the context of the molecular-vortex model: The first concerns the relationship between Maxwell's accomplishments and the mechanical worldview, the second addresses the role of the field-primacy approach in the genesis of Maxwell's innovations, and the third concerns the unity and coherence of Maxwell's mechanical models and mathematical formalisms.

Maxwell and the mechanical worldview

We have seen that Maxwell's stance with respect to mechanical models and his use of them was conditioned by the confluence, in his educational background and scientific training, of Scottish (Edinburgh) and Cambridge traditions, with the former inclining toward an analogical interpretation of mechanical representations, and the latter toward a more ontologically committed approach, in which mechanical hypotheses were viewed as candidates for reality, and evidence of a hypothetico-deductive character was accepted as providing support for their realistic status.

We have examined the movement of Maxwell's own ideas and practices concerning mechanical modeling: from an initial reliance on mechanical models as heuristic physical analogies – in the Scottish, skeptical vein – thence to the installation of the molecular-vortex model as the basis for a realistically intended physical theory – as countenanced in Cambridge methodology – and finally to an attenuated mechanism – representing Maxwell's own, carefully balanced position - in which the physical universe continued to be viewed as ultimately mechanical, but the possibility of coming to know the details of the mechanism receded indefinitely.

‘JUMP-LIKE’ MUTATIONS – THE WORKING-GROUND OF NATURAL SELECTION

The general facts which we have just put forward in evidence of the durability claimed for the gene structure, are perhaps too familiar to us to be striking or to be regarded as convincing. Here, for once, the common saying that exceptions prove the rule is actually true. If there were no exceptions to the likeness between children and parents, we should have been deprived not only of all those beautiful experiments which have revealed to us the detailed mechanism of heredity, but also of that grand, million-fold experiment of Nature, which forges the species by natural selection and survival of the fittest.

Let me take this last important subject as the starting-point for presenting the relevant facts — again with an apology and a reminder that I am not a biologist:

We know definitely, today, that Darwin was mistaken in regarding the small, continuous, accidental variations, that are bound to occur even in the most homogeneous population, as the material on which natural selection works. For it has been proved that they are not inherited. The fact is important enough to be illustrated briefly. If you take a crop of pure-strain barley, and measure, ear by ear, the length of its awns and plot the result of your statistics, you will get a bell-shaped curve as shown in Fig. 7, where the number of ears with a definite length of awn is plotted against the length.

The theory of molecular vortices had constituted the focus of Maxwell's research program in electricity and magnetism in the late 1850s and early 1860s, and his two major innovations of that period – the introduction of the displacement current and the treatment of electromagnetism and optics within a single theoretical framework – grew out of the theory of molecular vortices and reflected that context in their initial formulations. In the course of Maxwell's elaboration of the molecular-vortex model, however, problems had accumulated, to the point that he had serious reservations concerning certain parts of the model. In addition, Maxwell's research program in the theory of heat and gases was, in the years around 1860 and thereafter, developing in such a way as to undermine support for the theory of molecular vortices in that area, which had been its original stronghold. Finally, and relatedly, Maxwell's general views on the use of mechanical models in science were developing in a new direction that involved less emphasis on specific and concrete models. All of these factors converged in encouraging Maxwell to begin a measured retreat from the molecular-vortex model.

As part of this general retreat from the model, Maxwell took steps to free his signal innovations in electromagnetic theory from their original matrix in the theory of molecular vortices. The modification of Ampère's law and, more significantly, the incorporation of optics into electromagnetic theory defined new research programs, based on those innovations.

In this last chapter I wish to demonstrate in a little more detail the very strange state of affairs already noticed in a famous fragment of Democritus of Abdera - the strange fact that on the one hand all our knowledge about the world around us, both that gained in everyday life and that revealed by the most carefully planned and painstaking laboratory experiments, rests entirely on immediate sense perception, while on the other hand this knowledge fails to reveal the relations of the sense perceptions to the outside world, so that in the picture or model we form of the outside world, guided by our scientific discoveries, all sensual qualities are absent. While the first part of this statement is, so I believe, easily granted by everybody, the second half is perhaps not so frequently realized, simply because the non-scientist has, as a rule, a great reverence for science and credits us scientists with being able, by our ‘fabulously refined methods’, to make out what, by its very nature, no human can possibly make out and never will be able to make out.

If you ask a physicist what is his idea of yellow light, he will tell you that it is transversal electro-magnetic waves of wave-length in the neighbourhood of 590 millimicrons!. If you ask him: But where does yellow come in? he will say: In my picture not at all, but these kinds of vibrations, when they hit the retina of a healthy eye, give the person whose eye it is the sensation of yellow.

Figures 4.8 and 4.9a depict schematically the vortex rotations and idle-wheel translations associated with a uniform current density inside of a long, straight wire with uniform circular cross section. Inside the wire, the magnetic field grows linearly with distance from the axis; because of this, neighboring vortices rotate with different angular velocities; this engenders motion of the idle-wheel particles interposed between the vortices, constituting a nonzero current density J; and the inhomogeneity of the magnetic field H is associated with a nonzero value for curl H, which is equal to the nonzero current density J.

Outside of the wire, the H field falls off as 1/r, where r is the distance from the axis [E. R. Peck, Electricity and Magnetism (New York: McGraw-Hill, 1953), 214–17]. One might, at first thought, expect that because of this, neighboring vortices would rotate with different angular velocities, and this would engender motion of the idle-wheel particles, constituting a nonzero current density J. Even though the magnetic field H is inhomogeneous, however, curl H and hence curl ω* are zero outside the wire, and Maxwell's calculation leading to equation (3.7a) shows that in this situation there will be no net flux of the idle-wheel particles, and hence no current ι or J.

As Uraniborg took shape, word of the remarkable new establishment began to circulate among the scholars who perambulated across northern Europe. By the mid-1580s, Tycho was receiving not only numerous applications from aspiring disciples, but also a steady stream of overnight visitors from the summer circuit of touring intellectuals and curiosity seekers. Among all of those who ever appeared on Hven in either category, the one who probably made the greatest impression on Tycho turned up in July 1580. The visitor was a mathematician named Paul Wittich. He and Tycho seem to have crossed paths already at Wittenberg during one of Tycho's stops at the university there. But that first meeting had obviously been very brief, for when Wittich appeared at Uraniborg, he had with him an introduction from Hayek. Of course, even without such a formality, Wittich would have been more than welcome, for Tycho at that time still had only Flemløse for observational help and intellectual company.

Like Flemløse and all the rest of Tycho's later recruits, Wittich came without much experience in the use of astronomical instruments: Tycho was later to report that his smiths were very amused by Wittich's first attempts to observe. The complete theoretician, Wittich not only knew none of the constellations but unrepentantly argued that such knowledge was no more necessary for an astronomer than a knowledge of herbs was for a physician.

Among the mass of detail that constitutes the personal, social, cultural, and intellectual background of Tycho Brahe's scientific achievement, the one indispensable fact is that he was born a Brahe, that is, born not merely into the Danish nobility but also into the small fraction of the noble class that had historically played significant roles in the administration, governance, and defense of the realm. The epitome of this special status was membership in the Rigsraad, or Council of the Realm. Nominally an advisory body for the king but actually an oligarchical institution devoted to defending the interests of the most powerful noble families, the Rigsraad consisted of twenty-odd members who declared war, concluded treaties of peace, appointed regents (among themselves, naturally), seated kings, and participated with kings in virtually every aspect of the daily affairs of state.

All four of Tycho's great-grandfathers and both of his grandfathers had been councillors (see Fig 1.1). His paternal grandfather and namesake, Tyge Brahe, had held that honor only briefly before being killed during the siege of Malmø in 1523, fighting in the cause that put Frederick I on the throne and brought the Reformation to Denmark. But Tyge's widow, Sophie Rud, was descended from the equally powerful Rosenkrantz and Gyldenstierne families and thus had her father and brother on the council to look after the interests of her young family.

Toward the end of the summer of 1581 as the finishing touches were being applied to Uraniborg, Kirsten gave birth to a son. In a family that to that point consisted solely of daughters, it must have been a great occasion, no matter how elevated the rights and status of Danish women may have been relative to those in other contemporary cultures. After having disposed of his father's name (Otte) and used his maternal grandfather's name (Claus), Tycho now baptized his third son Tyge, after his paternal grandfather. When another son was born in 1583, he was named Jørgen, presumably after Tycho's uncle and stepfather but perhaps after Kirsten's father as well.

Tycho's sons were only the vanguard of a considerable increase in the Brahe household. From at least as early as the planning of the eight-room garret on the top floor of Uraniborg, Tycho had envisioned having a significant number of assistants to work with him. As space, instruments, and time became available to Tycho in the early 1580s, therefore, he began to select collaborators until he accumulated a group of eight to twelve members of varying degrees of permanence and competence. Thanks to the curiosity of an anonymous inmate who compiled a fragmentary list of his fellows toward the end of the 1580s, we can obtain a glimpse of at least the upper half of the spectrum of Tycho's assistants.

As the flood of activity associated with the initial planning of Uraniborg began to ebb, Tycho was able to turn some of his attention to astronomical matters. For the most part, that attention was absorbed first by the necessity of exploiting the appearance of the comet to the greatest possible degree and then by the need to put into production instruments worthy of his long-term professional aspirations. But already in 1578 Tycho was able to do enough observing both to obtain some insight into the problems that might arise in doing more extended work on the island and to start accumulating some useful observations. The latter seem to have consisted primarily of meridian altitudes of the sun, for Tycho continued to take about a hundred of them annually. A couple of hundred distances recorded between various stars used as reference points for the nova and the comet seem merely to have convinced Tycho that such work was premature, given the instruments available to him for after March 1578, he made very few observations of either the stars or the planets for the next three years.

As Tycho began to settle into Uraniborg in 1581 and get access to the instruments from his shop, he gradually shifted into being a full-time professional astronomer. The most prominent manifestation of this transition was his expansion into serious nighttime observation.

Although Benatky had been Tycho's choice among three manor houses, it was far from being the new Uraniborg that Rudolph had agreed to provide for Tycho. Its best feature was its location. It was the farthest of the three from Prague; it was situated on high ground, offering (as Tycho wrote Longomontanus) a clear view of the horizon in all directions; and it was splendid and commodious. Compared with Uraniborg – to say nothing of the temporary quarters in which Tycho had been living during the year since his departure from Wandesbeck – all three of the places probably looked commodious. But although Benatky had room for people, it had no obvious place to set up instruments or perform chemical distillations.

Taking Rudolph's promises literally, Tycho began to make repairs, modify rooms, and even plan a completely new wooden building – whether as an alchemical laboratory or some kind of analogue to Stjerneborg is not clear. Within a few weeks the administrator of the estate, Caspar von Mühlstein, complained to Barwitz about the cost of Tycho's projects. By late November, when the original estimates had doubled and Tycho had shown Mühlstein a letter from Barwitz stating that Rudolph had granted Tycho a salary that far exceeded the proceeds of the estate, Mühlstein had had enough. He wrote a letter to the chamber of deputies outlining the problem and formally refusing to authorize further expenditures without an official order – and the money to pay for it – from the treasury.

As of Tycho's day, the island of Hven had played no role in Danish history for a long time. Several folktales associated with it were still in circulation, and the ruins of four forts could still be discerned at strategic points on the island, but nothing important had happened there since the Norwegian king, Eric the Priest Hater, had reportedly destroyed the forts in 1288. Through the years, some forty families had together tilled the land, grazed a few animals on the less tractable areas, and shared their meager yields with the crown through a series of provincial governors living on the mainland.

In 1576, however, the scene changed radically. On May 23 Frederick II signed a document conferring “to our beloved Tyge Brahe … our land of Hven, with all our and the crown's tenants and servants who live thereon, and with all the rent and duty which comes from it … to have, enjoy, use, and hold; free and clear, without any rent, all the days of his life.”

In fact, by that date the new landlord was probably already a notorious figure on the island. By then he must have been almost accustomed to the two–hour boat trip from Landskrona harbor to the landing on the north side of the island and familiar with the 150-foot bluffs that greeted the eye from any other perspective of the island.

At the beginning of July 1579, Tycho wrote letters to Vedel and Dançey announcing that his house was now far enough along to be worth seeing. It was far from complete. Tycho did not move into the house for another eighteen months and was not to pronounce it “finished” until a year after that, in the late fall of 1581. Moreover, he would be adding outbuildings right up to the end of the decade. But with the exterior complete except for ornamentation, the rough framing done inside, and the grounds generally laid out, Uraniborg had assumed enough form to permit Tycho to convey to his friends a good idea of how it would eventually look.

What first greeted the visitor to Uraniborg was Tycho's only concession to the medieval tradition of noble residences as fortified bastions of defense: the wall. Five and a half meters high and nearly five meters thick at its base, this stone-veneered earthen edifice completely enclosed the seventy-eight-meter-square area that constituted the heart of Tycho's island estate. The square was oriented astronomically, with its principal avenue running from the main gate at the east corner (see Figure 4.2) due west to the other portal at D. Through its other diagonal was a north-south path servicing the servants' quarters at the north extremity of the compound (C) and what would become Tycho's printing establishment at the south corner (B).

Although Tycho's decision to publish De Stella nova seems to have been based on a decision to leave Denmark, the publication itself, ironically, was probably largely responsible for keeping him in Denmark. Certainly in the short term it was. The trip abroad that Tycho had conceived as the springboard to emigration was first delayed by ill health – probably the consequence of an overly zealous regimen of daytime writing and winter-night observing – and then postponed for a year, as the numerous details of publication kept Tycho occupied beyond the normal spring departure times.

Starting with the letters to Pratensis concerning the publication of De Stella nova in the spring of 1573, the occasional references to Tycho's “address” are to Knudstrup rather than Herrevad. Whether this means that Tycho was actually occupying the main residence, living in the vicinity, or simply using it as his address because he was now entitled to be addressed as “Tycho Brahe of Knudstrup” is not clear. It is probable, however, that the principal occupant of the manor was Tycho's mother. She was certainly living there with her youngest children in the spring of 1574 and even wrote to a friend that Tycho was living with her. Presumably, she should have said that she was living with Tycho, if he were now legally lord of the manor.

Until its final revision by Longomontanus in 1600, Tycho's lunar theory was fundamentally Copernican: Tycho (or Longomontanus) had been able to represent the Variation by simply putting an extra circle, AB in Figure A.3.1, into the center of Copernicus's theory. By making the center of the deferent (B) revolve around the earth (A) at twice the synodic velocity of the moon, he produced displacements that mounted to ±45' 20′ when the moon was in the octants and yet vanished in syzygy and quadrature. But the Variation was not the only extra circle grafted onto the traditional theory. The center of the large epicycle also moved on a new circle, CD, producing a periodic displacement up to ±11'.

This correction appears without a prior hint of any kind in either Tycho's log notes or his letters and would therefore be unknown if Tycho had not allowed Jöstel to “prepublish” his theory in 1599. What it represented, although somewhat imperfectly, was a phenomenon called the annual equation. When it finally attained recognition as a legitimate lunar phenomenon and was explained by Newton a hundred years later, it was seen as an annual variation of the speed of the moon in its orbit, owing to the annual fluctuation of the sun's distance from (and hence influence on) the earth–moon system.

That Tycho also associated the phenomenon with the sun in some sense is obvious from the fact that his period for it was a year.