The Astronomical Heritage of Scientific Method
The Case of the Discovery of the Orbit of Ceres

by Tarrajna Dorsey

PROLOGUE

The history of the determination of the orbit of the minor planet, Ceres Ferdinandea, exemplifies a dilemma in the long arc of the scientific maturation of mankind, which, though only a proverbial `drop in the bucket' to a deluge of scientific achievements, nevertheless presents to us an infinitesimal reflection of the essence of the ongoing struggle over the nature of the mind of man, the Universe, and the Creator. For as long as men have been known to inhabit the Earth, had eyes to see, and a mind with which to reflect, they have taken curious and careful note of the heavens. The motions of these twinkling lights of the firmament have been carefully observed and recorded from the time of the `old men' of Egyptian civilization, to the Pythagoreans of antiquity, through the many succeeding ages of mankind. In each case, contemplation of the nature of the universal order which conducts this magnificent opus, has elicited a strong sense of wonder in the mind of the beholder, accompanied by a compelling desire to make the invisible composition of its movements visible, to the mind. It is the pursuit of this very inclination which has yielded the fruits of civilization to man; both the actual harvests of cultivation of the earth, as well as the ripening of man's knowledge of truth, or what we call science. On the other hand, the denial of this sense of awe and creative hypothesis has only served to stultify and impede man's development. The proof of this crucial point is illustrated in the case of Ceres, as will be expounded upon, presently.

The Pythagoreans, Plato, and Piety

First, we must turn our sights to a much earlier time, to the well-springs from whence this discovery came. This brings us to the `early' investigations of the Pythagoreans, whose ideas have been most clearly captured and unfolded to us through the works of Plato.1 Before elaborating upon the role of the Pythagoreans, let it be emphasized that we are not attributing the origins of investigation of the heavens, mensuration, trigonometry, etc., to the Greek era. By current knowledge, the monitoring of lunar cycles and eclipses occurred long before Greek civilization made its entrance onto the stage of world history, and the Egyptians had developed language for the measurement of pyramids, etc., in their age. Proclus, a Greek scholar who writes commentaries upon the work of the Greeks, tells us of Thales bringing the knowledge of geometry from the Egyptians to the Greeks. Also, references to a far more ancient people and mode of thought appear again and again in the course of Plato's dialogs.

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Star trails over the course of the night: a sight only viewable by the mind in the Pythagoreans' time. (Nasa.gov)

Thus, what is emphasized here as originating uniquely with the Pythagoreans, relates more to the foundations, or epistemology ("what one stands upon"), of their mode of thought and manner of inquiry. For example, the concept of the Earth and universe possessing a spherical nature, which must be investigated from that standpoint, is uniquely `Pythagorean'. This point is illustrated to us by an Alexandrine poet, Hermesianax, in the following manner: "What inspiration laid forceful hold on Pythagoras when he discovered the subtle geometry of (the heavenly) spirals and compressed in a small sphere the whole of the circle which the aether embraces."2 In this way, they sought to uncover the fabric of the world which binds together the pinions of creation, by looking into those things which instigate man's wonder and curiosity, provoking him to seek out their cause, such as the musical harmonies, the five regular solids, the commensurability of certain types of number, etc. The Pythagoreans are well known for the saying, "a figure and a step, but not a figure and three oboli,"3 which succinctly portrays that the steps upon which they climbed were laid out of a love for truth in and of itself, which looked for unity in all investigations into the nature of things, as opposed to a utilitarian search for formulae which could serve as `useful tools'.

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The Analemma: The Sun, photographed at the same time each day, over the course of one year. (antwrp.gsfc.nasa.gov)
In the Epinomis dialog4 between an Athenian, a Cretan, and a Lacedaemonian, on the establishment of laws and what studies will lead a mortal man to wisdom, their discussion arrives at the question of: "what single science there is, of all those we have, such that were it removed from mankind, or had it never made its appearance, man would become the most thoughtless and foolish of creatures?"
Namely, what is the species character of mankind? What is the essence of that which distinguishes man from all other living creatures? This discussion of antiquity lights upon the concept of number, not as learned from the so-called `counting' of objects, but rather from the lessons taught by the heavens: the period of the Sun, distinguished by day and night, the waxing and waning of the Moon, falling within the greater period of a year, the return of a planet to a certain grouping of stars, etc. Or, in the words of the Athenian: "...when God had made the moon in the sky, waxing and waning, as we have said, he combined the months into a year and so all the creatures, by a happy providence, began to have a general insight into the relations of number with number." Thus, we see that the distinguishing characteristic of man, or cognitive beings, is ascribed to the faculty of reflection upon the periodic celestial motions. But this "reflection upon" does not merely consist of simple observation. Let us listen further to what the Athenian has to teach.
...We must do what we can to enumerate the subjects to be studied and explain their nature and the methods to be employed, to the best of the abilities of myself who am to speak and you who are to listen-to say, in fact, how a man should learn piety, and in what it consists. It may seem odd to the ear, but the name we give to the study is one which will surprise a person unfamiliar with the subject-astronomy. Are you unaware that the true astronomer must be a man of great wisdom? I do not mean an astronomer of the type of Hesiod and his like, a man who has just observed settings and risings, but one who has studied seven out of the eight orbits, as each of them completes its circuit in a fashion not easy of comprehension by any capacity not endowed with admirable abilities. I have already touched on this and shall now proceed, as I say, to explain how and on what lines the study is to be pursued. And I may begin the statement thus.
The moon gets round her circuit most rapidly, bringing with her the month, and the full moon as a first period. Next we must observe the Sun, his constant turnings throughout his circuit, and his companions. Not to be perpetually repeating ourselves about the same subjects, the rest of the orbits which we enumerated above are difficult to comprehend, and to train capacities which can deal with them we shall have to spend a great deal of labor on providing preliminary teaching and training in boyhood and youth. Hence there will be a need for several sciences. The first and most important of them is likewise that which treats of pure numbers-not numbers concreted in bodies, but the whole generation of the series of odd and even, and the effects which it contributes to the nature of things. When all this has been mastered, next in order comes what is called by the very ludicrous name [geometry] (gewmetria), but is really a manifest assimilation to one another of numbers which are naturally dissimilar, effected by reference to areas. Now to a man who can comprehend this, it will be plain that this is no mere feat of human skill, but a miracle of God's contrivance. Next, numbers raised to the third power and thus presenting an analogy with three-dimensional things. Here again he assimilates the dissimilar by a second science, which those who hit on the discovery have named stereometry [the gauging of solids], a device of God's contriving which breeds amazement in those who fix their gaze on it and consider how universal nature molds form and type by the constant revolution of potency and its converse about the double in the various progressions. [emphasis in original]
We are now able to clearly see how the Pythagorean's practice of "piety" necessitated undertaking the study of "geometry" (which is mistakenly asserted by some scholars to have originated solely from the measurement of plots of land), stereometry, etc. Already here, there exists the concept of a unity underlying these studies, the pursuit of which promises to ripen the wisdom of man, and in turn, increase his wellbeing. As if issuing a challenge, to those after him, to take up and continue this method of investigation, our Athenian leaves off his speech with the following call:
To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves, and disclose themselves they will, if, as I say, a man pursues his studies aright with his mind's eye fixed on their single end. As such a man reflects, he will receive the revelation of a single bond of natural interconnection between all these problems. If such matters are handled in any other spirit, a man, as I am saying, will need to invoke his luck.
The Harmony of the World
The man who fully heeded the call of the Athenian, nearly two millennia later, was Johannes Kepler (1571-1630). From the beginning of his theological studies at Tübingen as a young man, Kepler committed himself to investigating the true nature of the system of the world, rejecting the widespread belief in the Ptolemaic model, and even defending the Copernican model in debates amongst his peers. However, he was more driven by the desire to understand the reason why the universe is ordered in one particular way rather than any other, as opposed to adopting any particular model which could accurately represent the apparent phenomena of that order. Thus, while maintaining his first job of teaching mathematics in the small town of Graz, Kepler began to work on answering several very prominent questions in his mind: "Why are there six planets? Why are their distances from the Sun exactly such and such? Why do they move more slowly the further they are from the Sun?"5 The result of this first inquiry, "my little book," as he later called it, was the Mysterium Cosmographicum (The Secret of the Universe), published in 1596. In the greeting to the reader, Kepler announces the content of his investigation, thus:
The nature of the universe, God's motive and plan for creating it, God's source for the numbers, the law for such a great mass, the reason why there are six orbits, the spaces which fall between all the spheres, the cause of the great gap separating Jupiter and Mars, though they are not in the first spheres-here Pythagoras reveals all this to you by five figures. Clearly he has revealed by this example that we can be born again after two thousand years of error, until the appearance of Copernicus, in virtue of this name, a better explorer of the universe. But hold back no longer from the fruits found within these rinds.
This was only the very beginning of Kepler's lifelong passion to seek truth. The sheer honesty with which he pursued this end, is demonstrated by his eagerness to throw aside his own failed hypotheses, in order to take up new, more developed ones. This quality was especially reflected by his annotations to the Mysterium Cosmographicum, which he added many years later, after having published the crowning jewel of all of his works, the Harmonices Mundi (Harmony of the World), in 1619.6 However, in spite of this constantly developing process, Kepler's epistemology remained firmly rooted in the understanding that the universe is composed as a unity, and capable of being comprehended by the mind of man. Not only were Kepler's inquiries consciously built upon the foundations which the Pythagoreans had erected, but he himself expounded upon this relationship of the mind of man to the universe, and to the Creator.
...Would that excellent Creator, who has introduced nothing into Nature without thoroughly foreseeing not only its necessity but its beauty and power to delight, have left only the mind of Man, the lord of all Nature, made in his own image, without any delight? Rather, as we do not ask what hope of gain makes a little bird warble, since we know that it takes delight in singing because it is for that very singing that the bird was made, so there is no need to ask why the human mind undertakes such toil in seeking out these secrets of the heavens. For the reason why the mind was joined to the senses by our Maker is not only so that Man should maintain himself, which many species of living things can do far more cleverly with the aid of even an irrational mind, but also so that from those things which we perceive with our eyes to exist we should strive toward the causes of their being and becoming, although we should get nothing else useful from them. And just as other animals, and the human body, are sustained by food and drink, so the very spirit of Man, which is something distinct from Man, is nourished, is increased, and in a sense grows up on this diet of knowledge, and is more like the dead than the living if it is touched by no desire for these things. Therefore as by the providence of Nature nourishment is never lacking for living things, so we can say with justice that the reason why there is such great variety in things, and treasuries so well concealed in the fabric of the heavens, is so that fresh nourishment should never be lacking for the human mind, and it should never disdain it as stale, nor be inactive, but should have in this universe an inexhaustible workshop in which to busy itself.7
Proving the truth of his own musings, Kepler busied himself quite diligently in this workshop of the Creator, hammering away at the pliable sheets of his own hypotheses, heated in the forges of a burning passion for truth, and molded by the strong force of his keen intellect. In the Harmonices Mundi, Kepler utilizes the manifold quality of this workshop to investigate the principle underlying the motions of the heavenly bodies. By looking at the relationships which arise in vision from the division of the circle, the formation of regular solids in physical space, and juxtaposing them with the proportions arising from the audibly perceivable divisions of the string-the musical harmonies-Kepler is able to demonstrate the coherence of the planetary orbits in this way, in their relations to one another, and to the Sun.8
It is no mere lucky coincidence, nor strange outcome of historical probability, that by precisely this method, and no other, Kepler was the very individual to draw the veil from the mysteries of the planetary orbits, discovering the principle of universal gravitation, and making his discovery communicable to mankind. Admittedly, he stood upon the shoulders of many great thinkers who preceded him, but more importantly, again, the crux of the matter rests upon the issue of epistemology; the method of thinking, and the fundamental concepts of the coherence of God, the creation, and the mind of man. This is the very arc of thought which will come into play in the matter of the dilemma of determining the orbit of Ceres.

LET THE PLAY BEGIN

As is the case with all fruitful scientific inquiry, however, Kepler's monumental contributions not only served to open the doorway to an entirely new domain of investigation, but they left unanswered questions concerning the order of the universe as well. Among the harmonies of the world which Kepler demonstrates to exist as the governing cause for the distances and periods of the planets' orbits, an anomaly arises, which Kepler finds himself unable to reconcile with his hypothesis. The anomaly appears in the case of the proportions between the orbits of Mars and Jupiter. Contrary to harmonic proportions found among the relations of the other planets' motions, nothing quite satisfying could be found for those between Mars and Jupiter. The distances between them could be explained, using the inscribing and circumscribing spheres of the tetrahedron, one of the five regular Platonic solids. However, as Kepler found from his investigations into the various harmonic proportions in the Harmonices Mundi, the reason for the particular structure of the solar system did not lie in the proportions of the Platonic solids alone, as he had originally thought, but in the proportions of musical harmony, as expressed by what Kepler calls the converging and diverging motions of the heavenly bodies. In the converging and diverging motions of Mars and Jupiter,9 however, no "pleasing" musical harmony is to be found, and this leaves Kepler, and those after him, to continue to wonder.10
One of the possibilities which Kepler considers, and experiments with, even in the Mysterium Cosmographicum, is the existence of another, yet unknown planet, hiding somewhere in the vast gap between Mars and Jupiter. Although actual distances of the planets are not known exactly in Kepler's time, their relative distances are, and this immense gap is knowable even by comparison of their relative periods. In this gap, lies, so to speak, the `sleeping giant', which will not be awoken until many, many years after Kepler's death, and, when stirred from slumber, will cause great trouble to all of the geometers, mathematicians, and astronomers of Europe.
The Missing Planet
In the period following that of Kepler's breakthroughs in physical scientific method, there endured another long silence for mankind-certainly not of two thousand years-but with Kepler's death came a tremendous battle over the preservation of his life's work.11 Apart from the studies of Gottfried Wilhelm Leibniz and his collaborators, the furtherance of physical scientific method, as developed and applied by Kepler, fell by the wayside.
We now arrive at the core of our narrative, commencing in the middle of the 18th century. Amidst a heated battle over the ideas of Leibniz and Kepler, one of their chief defenders, Abraham Gotthelf Kästner (1718-1800), takes the helm on the matter of scientific inquiry and astronomy. In 1747, he pens a piece entitled "Lob der Sternkunst" (The Praise of Astronomy), in which he cleverly enumerates the uses of astronomy to human society, but then moves to the following, more profound discussion, revealing himself to be an inheritor of the arc of thought of the Pythagoreans and Kepler:
...It may seem superfluous to speak of the delightfulness of astronomy, now that its uses have been demonstrated. However, those who have felt the delectation given us by the knowledge of truth, would not forgive me were I to be silent of this [emotion] in astronomy. Are we satisfied with a secure insight into such laws which seem impossible to the ignorant? Do we take delight in comprehending how, from a very limited understanding, the most hidden truths have been deduced? Does one wish to know, how far the powers of human reason reach? Then, one should study astronomy. [Astronomy] establishes the orbit of the daytime stars; she proclaims the eclipses unto the moons; she orders each planet to move faster or slower; she commands how many seconds the whole army of fixed stars should move in uniform order each year....Actually, she does not command, rather she investigates the edicts which the Creator prescribed for the whole universe, with reverent curiosity."12

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The Comet of 1769, espied by the French astronomer Charles Messier (1730-1817). Here depicted in the Amsterdam sky. (Dionari.de)

Despite the current stall in the advancement of scientific method, the world of observational astronomy is abuzz with activity. New developments in instrumentation are being made, new observatories built across Europe, new comets sighted regularly,13 and even the new phenomena of "comet scares" have become all the rage.14 Star charts, featuring dozens of new stars are constructed, the previous obscurity surrounding various astronomical phenomena such as nebulae and star clusters is cleared, and even a new planet, Uranus, is discovered! However, Kepler's sleeping giant, the question of the gap between Mars and Jupiter, looms in the periphery of these advancements. Some references are made to the idea of a missing planet, here or there, but for the most part, Kepler's mode of inquiry, which had produced the question in the first place, is scorned by `modern' empiricists. Illustrative of the case are the remarks made by a most renowned astronomer of the day, hailed in his time as "the Newton of France," Pierre Simon de La Place (1749-1827), concerning Kepler's method and work. In his Exposition du systéme du monde (1796), he opines:
...In [Kepler's] time, the world had just begun to get a glimpse of the proper method of proceeding in the search of truth, at which genius only arrived at by instinct, frequently connecting errors with its discoveries. Instead of passing slowly by a succession of inductions, from insulated phenomena, to others more extended, and from these to general laws of nature; it was more easy and more agreeable to subject all the phenomena to the relations of convenience and harmony, which the imagination could create and modify at pleasure."15
This tiny glimpse into the thoughts of La Place suffices as a telling insight into the paucity of the entirety of empirical scientific method, a highly dominant fad of the time.
The vacuity of La Place's observations is by no means the only affliction plaguing the currents of scientific thought at this time: a notion, tossed about in seeming answer to Kepler's unexplained anomaly, surfaces in the form of what is referred to as the "Titius-Bode law of harmonic progression." The story begins with a certain Baron von Wolf, a supposed protégé of Leibniz,16 who had published a work on physics in German, in 1741. Therein, he includes a scheme to describe the distances of the planets from one another by dividing the distance from the Sun to the Earth into ten parts, and assigning the other planets numerical values accordingly. Although he does not cite the origin of this mechanism, our researches have uncovered that this scheme was lifted nearly word for word from a passage of Newton follower David Gregory's The Elements of Astronomy, published in 1715.17

Images/ZMCTitlePage.gifThe cover of von Zach's astronomical journal, launched in 1799.

Unfortunately, this scheme was s natched up and developed into an empiricist gimmick to explain the planetary distances by Johannes Daniel Titius, a German professor of mathematics and physics at Württemberg. He is cited by Baron Franz von Zach, astronomer and editor of the main astronomical and scientific journal of the day, the Monatliche Correspondenz zur Beförderung des Erd- und Himmelskunde (Monthly Correspondence for the Furthering of Geography and Astronomy), as the originator of the aforementioned so-called "harmonic progression" in the planetary orbits.18 Johann Elert Bode, an astronomer and mathematician heading up the Berlin observatory, and chief