Figure 1.17. The title page from Hevelius’s Cometographia (1668). To the left in the image is Aristotle, centrally seated is Hevelius and to the right is Kepler. In the center background a crowd of onlookers, some with arms raised to the sky, contemplate the approaching comet. Meanwhile, lofted above the superstitious plebeians, are seen the “astronomers” at work. Three large sextants (characteristic of the design promoted by Tycho Brahe) can be seen in the image, along with an observer viewing the comet through a telescope. The dividers placed on the table next to Helvelius’s right hand are a sign of authority and exactness.
Kepler, Galileo, and Helvelius (and indeed all astronomers prior to their time) accepted that comets were large ephemeral objects, quasi-planets perhaps, but objects with a limited lifetime. This notion began to change, however, with the realization, through Edmund Halley’s investigations, that at least some comets moved along elliptical paths and were in principle at least long-lived and subject to repeated returns to the inner solar system. Newton’s thoughts on the topic were discussed in his Principia (1687) and in his second magnum opus, Opticks (first published in 1704). In these works Newton suggested that while comets had a solid and durable nucleus, they were additionally surrounded by a tenuous atmosphere. Newton additionally suggested an alchemical link with respect to the utility of the vapors emanating from cometary tails, arguing that planets needed to accrete such matter in order to replenish the fluids vital to the process of vegetation and putrefaction [6]. Indeed, in his Opticks Newton asked, “to what end are comets?” - answering in effect that it was only by the accretion of cometary material that sustained life on Earth was possible. With Newton, the swirling vortices of Descartes were laid to rest. “The motions of the Comets”, Newton writes in his Principia, “are exceedingly regular, are governed by the same laws with the motions of the planets, and can by no means be accounted for by the hypothesis of vortices. For comets are carried with very eccentric motion through all parts of the heavens indifferently, with a freedom that is incompatible with the notion of a vortex”. While Newton freed comets from the random buffetings supplied by Descartes swirling tourbillions, he none-the-less entrapped them within the ever searching web of gravity. By this act, Newton placed comets within the realm of direct calculation; their paths, at least in principle, being described by mathematical formula and direct computation.
With the mathematical techniques put in place by Isaac Newton, the 18th Century saw ever more calculations of cometary orbits being made. With, in principle, their orbits now being understood and numerated, the issue as to their origins and purpose became a matter of greater interest. It was now clear that comets, for at least some of the time, moved in a realm beyond the planets. Indeed, Halley’s Comet, at aphelion, is located three times further from the Sun than Saturn, the outermost of the known planets until the discovery of Uranus, by William Herschel, in 1781. One of the first diagrams to clearly illustrate cometary orbits and contrast them against those of the planets was that constructed by Thomas Wright of Durham. Writing in his text Clavis Cœlestis: being the explanation of a diagram entitled a Synopsis of the Universe (published 1742). Wright notes that, “Besides the Planets, which all move round the Sun from West to East near the Plain of the Ecliptic; there are other surprising Bodies in the System call’d Comets, whose Motions are perform’d in very different Plains, and in all manner of Directions, both direct and retrograde”. Lavishly illustrated, and massive in scale, Wright shows in the supplementary diagrams that accompanied Clavis Cœlestis the paths deduced for Halley’s three ‘periodic’ comets near to the Sun (figure 1.18), but leaves as a mystery their more distant peregrinations. Other diagrams of the time simple stop once the cometary distance extends beyond that of the orbit of Saturn (figure 1.19) – in many ways, rather than invoking the terrestrial map maker’s “here be dragons”, the astronomers responded with the notion that in the realm beyond Saturn, “there be comets”.
Figure 1.18. A small section of Thomas Wright of Durham’s A Synopsis of the Universe, published in 1742, showing the orbits deduced for the comets of 1661, 1680 and 1682 interior to the orbit of Mars. These three comets were highlighted by Edmund Halley as being periodic – that of 1661 being, in fact, Halley’s Comet.
At the time of publishing Clavis Cœlestis Wright was actively engaged in the study of comet C/1742 C1, describing his observations in the February, March, and April issues of The Gentleman’s Magazine. Wright ultimately submitted his derived orbit for the comet to the Royal Society. Employing these same mathematical skills a few years later, Wright speculated upon a possible extension of Kepler’s laws of planetary motion with respect to the comets. Writing in his 1750 work, An Original Theory or New Hypothesis of the Universe, Wright noted (in letter III) that, “I am strongly of the opinion that the comets in general, throughout all their respective orbits, describe one common area”. This idea, that all cometary orbits should encompass the same area, moves beyond Kepler’s laws, and there is no physical reason, other than the direct influence of an omnipotent maker, that such a constant condition should apply to cometary orbits. The area of an ellipse is given by A = π a b, where a and b are the semi-major and semi-minor axes. The major and minor axes are further related through the eccentricity, e, with b = a(1 - e2)1/2. Accordingly, Wright was suggesting that with A being constant, so the semi-major axis and the eccentricity for any comet’s orbit were related as: a2 = (A/π)/(1 - e2)1/2. Substituting modern-day numbers for cometary a and e values, however, reveals that Wright’s idea is entirely fallacious: for 1P/Halley, 2P/Encke and 3D/Biela, we find areas of 253.6, 8.1 and 25.6 astronomical units squared respectively. Even though Wright was wrong in his supposition, and indeed there is no dynamical requirement for the semi-major axis and eccentricity to be linked according to a constant area, the idea is interesting since it builds upon the notion that comets are somehow ‘different’ from the planets; their orbits being constrained not only by Kepler’s three laws of planetary motion but by an additional, dare one say divinely engineered, constant orbital area rule.
Figure 1.19. A scheme of the solar system, by William Whiston, published circa 1720, and engraved by John Senex. The cometary orbits are those deduced by Edmund Halley in his 1705 Synopsis of the Astronomy of Comets (recall figure 1.9). The map was first composed circa 1712 and Whiston writes in the (densely packed) surrounding text that comets, “are the more numerous bodies of the entire Solar system”. The large number of comets that cut across the circular planetary orbits have become a dominant feature of the map, and the sence that collsions might occasionally occur between planets and comets is apparent, albeit geometrically enhanced by the two-dimensional restriction of the page.
Beyond the 2-dimensional page
Human beings see the world in three-dimensions and this trait can be exploited to provide a better minds-eye view of a comet’s path through space. The ecliptic provides a natural plane for orienting the cometary orbit (recall figure 1.13), and the circle of the Earth’s orbit, centered upon the Sun, provides a graphic representation for the advancement of time when divided into appropriate date markers. Cometaria constructed in this manner have no moving parts as such, but when constructed to scale, sky locations, phase angles5 and distances relative to the Earth and/or Sun can be read-off with a straightedge or piece of string.
The 2-dimensional page can be transformed, and literally lifted into the third dimension, by the ingenious use of folded inserts and origami-style constructions. In this way, by turning the page, a 3-dimensional vista can be made to unfold – the scene animating itself before the reader’s eye. The history of the pop-up book stretches back many centuries, and a number of early astronomy texts are known to have used combinations of fold-out pages, fold-up diagrams and diagrams composed of multiple moving layers and/or moving parts – books containing the latter kind of diagrams generally being known as volvelle. Indeed, the addition of rotating discs within books