Functions of Mathematical Description in Astronomy and Optics,
Illustrations from Antiquity —Stephen Boydstun (1999)

Use of mathematics in our descriptions of nature makes our conceptions of natures more definite, sensitive, and probative. By what pathways?

I shall take the process of scientific conception of natures to consist of three interconnected moments: observation, characterization, and explanation. Into these three moments, I shall cast work in early astronomy and optics, attending to mathematical descriptions and the epistemological functions they serve.

I. Mathematical Description in Observations

Writing and drawing can extend memory of past perceptions. They can improve the fidelity of memory to past perceptions. They can make aspects of past perceptions accessible to other people.

Records of Timocharis' observation of the location of the star Spica relative to the autumnal equinox was used by Hipparchus about 170 years later in establishing the precession of the equinox (Neugebauer 1975, 292). Aristarchus' observation of the summer solstice was used 145 years later by Hipparchus, who compared it with an observation of his own to establish an accurate length of the year (Thurston 1994, 126). And essential parameters (period relations) in Hipparchus' lunar theory had originated in Babylonian astronomy (Neugebauer 1975, 321).

In ancient astronomy, ratios between synodic and sidereal periods of planets could be established with fair accuracy from observations over a few decades because the error of individual observations would be distributed over the long intervals of integral period equations (Neugebauer 1975, 14, 386–91; Thurston 1994, 79–81). Similarly, one might establish, to fair precision in the fractional day, the number of days in a sidereal year; by counting the number of days between successive first visible heliacal risings of a certain fixed star over a large number N of such first risings; the imprecision in the fractional day in a sidereal year would be inversely proportional to N.

On a special day each summer, clouds permitting, I watch the sun rise out of Lake Michigan. That is how I perceive it—the sun rising up from the lake horizon—even though my standing belief is that the phenomenon is due to the axial rotation of the earth. I can see the phenomenon as the horizon descending below the sun, but that requires addition of an intellectual construction to the experience. It seems, then, very natural that people would first conceive the sun, moon, and stars (and planets) as moving over the earth; that is just as we experience the circumstance.

Risings, settings, eclipses, and occultations are celestial events we can observe with unaided normal vision. With normal memory over a day, we can observe as well the daily journey of the sun and moon and the journey of the shadows they induce around us. Systematic measurement and recording of locations (angular) and times of recurring events in the celestial dome, and of attendant shadow lengths, directions, and motions on our ground, can reveal patterns of motion not discernible by direct perception and memory alone. We can become cognizant of the variability of the speed of the moon through records of direct observations of the day-by-day progress of the moon with respect to the fixed stars (Neugebauer 1975, 71, 371; 1969, 210; cf. 1975, 85). By gnomon and polos and by marking records, we can determine the dates of the equinoxes, learn that the seasons are unequal (Meton and Euctemon in 432 B.C.), and determine the obliquity of the ecliptic to the equator (assuming one has apprehended those two natural distinct planes; Pedersen 1993, 37–41; Thurston 1994, 41–44; Neugebauer 1975, 371–73, 627–29; Heath 1981, 130–31). This is mathematics, and instruments and records, extending our indirect observations.

As indirect observations, I should also count interpolations, such as the Seleucid Babylonians evidently performed to approximate daily longitudes of planets (between conjunction, opposition, or stationary points) using higher-order difference sequences (Neugebauer 1969, 127; 1975, 397, 412–18). Likewise, I should count as indirect observations inferred distances and angles, such as were attained by Hipparchus' resort to procedures such as tables of chords, analemmata, and stereographic projection (Neugebauer 1975, 299–304, 868–69; 1948, 1017, 1028–37; Goldstein 1983a). I should count as indirect observation Ptolemy's use of stereographic projection to determine zodiac-sign rising times by plane trigonometry (Neugebauer 1969, 185, 220) and his use of Menelaos' theorems to infer solar declinations, thence right ascensions, for a given solar longitude (Neugebauer 1975, 30–32).

It is commonly noticed that mathematics is abstract, precise, and concise (Metaph. 982a25–29; 1061a29–61b2). I expect the latter two ride upon the former, though upon two different aspects of the former. Precision is borne by the prescinding aspect of abstraction. Conciseness is borne by the hypostatizing aspect of abstraction (Ph. 193b31–194a11).

Conciseness would seem to be a matter of cognitive economy, such as we enjoy in counting and elementary arithmetic, all the more with place-value notation (Neugebauer 1969, 18–22), and such as we enjoy by widely applicable relationships made manifest in formal mathematics. I take the following mathematical techniques as effecting cognitive economies, whether applied to observation, to characterization, or to explanation: Babylonian use of the concept of the geometrical relation similarity (Neugebauer 1969, 46) and their use of formulae for obtaining solutions of quadratic equations (ibid., 41, 149–51); use by Autolycus, Euclid, and Theodosius of representative figures of spherical astronomy, with lettered elements, for reference in text (Neugebauer 1975, 752–54); use by Menaechmus of an algebraic relationship as proxy for a geometric object, so that connections among geometric objects "may be deduced by manipulation of their algebraic equivalents" (North 1987, 176).

Though cognitive economy is an important function served by mathematical descriptions, I want to put most of my attention on precision. Precision narrows possibilities of what is in nature. That narrowing is a function of mathematical description in science in general, whether in observations, in characterizations, or in explanations. The special gift of mathematical precision to observation is as we have seen above: expansion of the range of phenomena accessible to indirect perception.

II. Mathematical Description in Characterizations

What I mean by a characterization, in the context of physics and astronomy, is the finding of an essential form of a phenomenon. The form need not characterize the phenomenon exactly, only approximately. Such forms as I have in mind are mathematical or at least antemathematical. The contrast I want to make between what we should, by our present lights, call mere characterization and what we should call explanation is that the latter is the finding of working or constituting causes. For Pythagoreans, of course, and perhaps for many thinkers today, characterization by essential forms suffices for explanation. I think rather not, although a middle course (in which what I am calling mathematical characterization of phenomena be taken as a formal explanation, though not invoking formal causes) does not seem unreasonable (e.g. Gaukroger, chp. 6).

In saying mere, I intend no necessary inferiority of mere characterization in comparison with causal explanation. The latter also is a characterization, and I intend by the qualification mere only to designate a characterization that is not a causal explanation.

One reservation I have with placement of mere characterization by essential form into one hand and causal explanation into the other hand is that the causal buck always stops somewhere. If asked why the earth continues its axial rotation, I should say this is due to conservation of angular momentum. I should also be fairly satisfied, justifying my satisfaction of this causal stopping-place by reminding us how deep are angular momentum and its conservation: their ubiquity in both classical and quantum regimes; the importance of the unit of angular momentum in quantum mechanics; the connections of angular momentum conservation to other deep principles (of physics, including physical geometry) in Hamiltonian mechanics and in general relativity. Still, I am left in the end with angular momentum conservation and its cohorts as brute, unexplained physical facts, just as the Pythagorean is left in the end with brute mathematical forms. Perhaps, ultimately, the two hands and what is in them must be brought together, but for today it seems most sensible to me to keep the two hands apart, that is, to maintain an ontological distinction between mere essential forms and working or constituting causes.

Babylonian astronomers, at least by the Seleucid era, had devised elaborate, purely arithmetical procedures (using periodic step or zigzag functions; Neugebauer 1969, 110–15; 1975, 373–79) from which they could predict dates of characteristic planetary phenomena such as oppositions or onsets of retrogression (Neugebauer 1969, 125–34; 1975, 420–31; Thurston 1994, 79–81) and times of lunar events such as full moons and eclipses of the moon, even the extent of lunar eclipses (Neugebauer 1969, 109, 117; 1975, 474, 549–50, 1094–95, 1124; Thurston 1994, 74–78). The key to Babylonian prediction of lunar eclipses was "the construction of a common period (later known as the Saros) of syzygies and latitudes that made it possible to select those syzygies which would be accompanied by eclipses" (Neugebauer 1975, 664; also 1969, 118–19). Nevertheless, Otto Neugebauer judges that this does not imply that the Babylonians had a geometric model for the sun, moon, earth, and earth shadow; only arithmetic methods were required for predicting times of full moons and for describing the lunar motions in latitude, thence predicting lunar eclipses (Neugebauer 1975, 664; 1948, 1020). Furthermore, we do not know whether Babylonian astronomers explained lunar eclipses—"to introduce the concept of the earth's circular shadow is [tantamount] to postulating the sphericity of the earth, a concept which otherwise is completely lacking in Babylonian astronomy"—nor even lunar phases (Neugebauer 1975, 550; also, 1093–94). What we do know is that they had arrived at more and less effective arithmetical characterizations of lunar, planetary, and solar phenomena. Some of these were characterizations whose precision could be improved by corrective adjustments in response to deviations from observations accumulating over time (Neugebauer 1969, 116; 1975, 484–86, 497–99).

A second example of pure characterization would be the Aristotelian schematic geometric characterization of the shape of rainbows, a pure characterization anyway when we consider it in isolation from Aristotle's full, causality-dressed picture. For a rainbow formed when the sun is at some place on the horizon, Aristotle considers the straight line from the horizon point (say, at the center of the sun's width) to some point on the rainbow, which rainbow is to the side of the observer opposite the sun, and the straight line from that rainbow point to the observer. Whatever be the reasons for color forming that point of the rainbow, the apparently crucial angle between the aforesaid two straight paths (of rays of vision) will be preserved if those two paths are together rotated about the line from the observer to the sun's horizon point. Under such a rotation, the bow point will trace a full semicircle of rainbow above the ground; as we observe. The sun's horizon point, the point of the observer, and the center of the rainbow's (semi)circle will be collinear. For rainbows formed when the sun is above the horizon, the sun's location is treated as if it were the former horizon point elevated. The center of the rainbow's circle is equally lowered beneath the ground. Those two points keep their collinearity with the point of the observer. The portion of the rainbow's circle remaining above the ground will then be less than a semicircle; as we observe (Boyer 1959, 42–44; Meteor. 371b26–29, 375b17–76b21, 376b28–77a11). The Aristotelian geometric characterization of the rainbow is primitive, but promising (Descartes!) for future geometric modeling of rainbow phenomena.

III. Mathematical Descriptions in Explanations

Bernard Goldstein and Alan Bowen have argued that Eudoxus was the first to use the fundamental two-sphere model of Greek astronomy "to account for the risings and settings of stars, to provide a framework for geographical studies, and to justify a more mathematically sophisticated sundial. At the same time, he laid the foundations for the application of geometrical argument to the study of celestial phenomena" (Goldstein 1983b, 234). The first sphere represents the stationary earth. The second, surrounding and concentric sphere rotates daily and uniformly about a fixed axis passing through the common center of the two spheres. The second sphere is just the celestial sphere, the orb of the fixed stars. If all the two-sphere model did was describe the nightly risings and settings of stars as seen from some point on the earth, I should be inclined to class this model as a mere geometric characterization of diurnal stellar motion, really a straight extraction of the essential character of that phenomenon. But the combination of rotating celestial star-studded sphere with sphericity of the earth does explain, by working causes, the changes in the nightly stars passing overhead as one travels north or south on the earth.

For quite a different reason, I should also take Eudoxus' characterization of the motions of moon, sun, and planets by interconnected homocentric uniformly rotating spheres (with axes of rotation variously oriented) to also count as explanatory. For Eudoxus, as for the Pythagoreans and Plato, uniform rotation of an isolated sphere was an elementary motion. I construe Eudoxus' portrayal of the observed motions of moon, sun, or planets each in terms of a trio or quartet of homocentric rotating spheres (nested about the spherical earth) to be a mechanically-minded way of composing as resultant the motion of the moon, sun, or planet from elementary uniform circular motions. Eudoxus' composition of the resultant motion of each the moon, sun, or planet does not have the freedom we have in composing an observed motion, say specifically an angular velocity, from its components along freely chosen basis vectors. Eudoxus' composition of motions does not have that sort of freedom. Inflexibility of the will-he-nil-he sort in our constructions suggests we are encountering autonomous realities. Eudoxus' assemblies of homocentric rotating spheres are as if the kinematics of rotary machines. It seems that each rotating sphere and its characteristics—its order in the nest of three or four and its angular speed (required to be uniform) and orientation of rotation—is evidently uniquely fixed by the required resultant motion of moon, sun, or planet. I am inclined to class Eudoxus' homocentric-sphere models as not only characterizations of the motions of moon, sun, and planets, but as constitutive causal explanations: the resultant motion of the celestial body is constituted by the elementary motions in its assembly of homocentric spheres. The elementary motions cause the motion of the celestial body in that they compose it, or constitute it.

Eudoxus' model specifies the number, order, inclinations, and periods of the homocentric spheres; the celestial body being affixed to the equator of the innermost sphere. The model is specific enough to get itself into trouble, and that is a great virtue. That, I should say, is a broad function served by mathematical description in explanation, as in pure characterization: mathematical precision begets sensitivity of characterization, causal or pure.

The Eudoxan model for the motion of the sun implies equality of the seasons; Callippus tried to remedy this defect by adding two more spheres for composing the solar motion. The Eudoxan model for the motion of the moon captures the Saros period, variations in the latitudes of the moon, and eclipses of the moon. This model of lunar motion implies (other, unknown conditions being constant) that the apparent diameter of the moon will be constant. But, as a student of Eudoxus came to realize, the apparent diameter of the moon does change. In addition, the model for the lunar motion implies that the moon's motion around the ecliptic will be uniform, whereas in actuality the speed of the moon varies in that migration (Pedersen 1993, 69; Thurston 1994, 113–14). Callipus may have known about this variation, from Babyonian astronomy. It was perhaps for the sake of bringing this variation into the homocentric account of the moon's motion, that he added two more spheres for composing the motion of the moon, as he had done for the sun (Neugebauer 1975, 625).

The Eudoxan model of the planetary motions yields motions resembling retrogradations and variations in latitude for some planets, but the model's shortcomings, qualitative and quantitative, are several (Thurston 1994, 116–17; Neugebauer 1975, 679–84). Among them are the noticeable facts of variable brightness of Mars and of Venus over their circuits. This variability, like the variability of the apparent diameter of the moon, could not be accounted for by homocentric models; each celestial body, through all its gyrations, is necessarily conjectured to remain a fixed distance from the earth under such models.

The eccenter geometric model of Apollonius was able to account for evident variations of distances and angular speeds of the sun and moon as they are observed from the earth. The eccenter model retained uniform circular motion as elementary. The moon, sun, or planet orbits at constant speed and distance about some fixed center, the eccenter, which is in the neighborhood of the fixed earth (or, as the phenomena may require, the eccenter moves uniformly in a small circle about the fixed earth).

The epicycle method of modeling motions, discovered some time before Apollonius, also retains uniform circular motion as elementary. Apollonius came to realize that all the results obtainable with an eccenter model could also be obtained by an epicycle model. These two ways of modeling the motion of moon, sun, or planet about the earth are equivalent in the sense that they both yield the same angular motion of the moon, sun, or planet about the earth. Apollonius then realized that an epicycle model can be made to display retrograde motions of planets, and he proceeded to develop both epicycle and eccenter models for the stations of the planets (Pedersen 1993, 70–73; Neugebauer 1975, 263–70).

Hipparchus suited these models with quantitative values taken from observations and from the Babylonian arithmetic characterizations of phenomena. He applied an eccentric model (with fixed eccenter) to the motion of the sun, deriving basic parameters of the model (eccentric-quotient and longitude of solar apogee) from the positions (true longitudes) of the sun at cardinal points and from the lengths of seasons between them. From this model, solar longitudes can be calculated as a function of time to an accuracy of about half a degree (Thurston 1994, 128–31; Pedersen 1993, 73–75; Neugebauer 1975, 306–8.). He applied an epicycle model to the motion of the moon, deriving basic parameters from (two sets of) three positions-at-times of the moon, defined by three lunar eclipses. With his epicycle model for lunar motion, conjoined with his model of solar motion, he could have predicted lunar eclipses with some assurance (Thurston 1994, 131–34; Pedersen 1993, 75–76; Neugebauer 1975, 319–22, 129–31).

Hipparchus' mathematical characterizations had that good quality of being specific enough to get themselves into trouble. Ptolemy's mathematical characterizations of the apparent motions of sun, moon, and planets would be more refined, more capable of accounting for further aspects of those motions. The refinements and extensions of Hipparchus upon the work of Apollonius and the refinements and extensions of Ptolemy upon the work of Hipparchus are further cases of mathematical precision begetting sensitivity of characterization. Along these courses of refinements, lasting discoveries of nature were made, notably the precession of the equinoxes, or distinctness of sidereal and tropical year, and the second lunar inequality.

Firm discoveries and fineness of fit (including predictive power) of mathematical models are indications that we are approaching the independent natures of things. How does that square with the fact that throughout the modeling by Apollonius, Hipparchus, and Ptolemy, there is equivalence between the geometric eccenter model of a particular apparent motion and a geometric epicycle model of that same particular apparent motion?

My perspective on the dual geometric characterization, by eccenter and by epicycle, is as follows: An eccenter model and its equivalent epicycle model are each, after the fashion of characterization by homocentric spheres, prima facie constitutive causal explanations of the apparent motions of sun, moon, or planet. The "resultant" motions of a celestial body are prima facie constituted by the elementary motions in both its eccenter and epicycle renditions. The elementary motions in each of the two renditions are prima facie constituting causes of the celestial body's apparent motion in that each rendition composes that motion.

Unlike the constitutive causation of a resultant physical motion by an assembly of homocentric rotating spheres, there is freedom to choose between an eccenter- or an epicycle- constitutive cause of the resultant physical motion. That is, Apollonius, Hipparchus, and Ptolemy have a freedom in their constitutive causal characterizations that Eudoxus did not have. That freedom seems moderately similar to our freedom to choose between two coordinate systems (e.g., elliptical cylindrical coordinates v. parabolic cylindrical coordinates) for a problem. With the freedom to choose—apparently indifferently to nature—between an eccenter or an epicycle characterization of the apparent motion of sun, moon, or planet, it seems very natural to reclassify these characterizations as not constitutive causes of the motions of sun, moon, or planet, but as mere compositional characterizations of those motions. According to this adjusted stance, what remains in nature (as in nature itself) from these characterizations would be whatever elements of the eccenter and epicycle characterizations are in common between them (Neugebauer 1975, 57 and 1220, fig. 51), including of course, the apparent motion of the celestial body. Invariance under transformation (eccenter-epicycle) demarcates degree of autonomous reality.

What remains most definitely physically given are those resultant (of eccenter and epicycle characterization) apparent motions of the sun, moon, and planets and the working effects (optical) of those bodies among themselves, the earth, and vice versa, such as the cast of the earth's shadow across the moon. Physically, the earth can remain central, if not dispositively at the center; with sun, moon, and planets dancing out orbits around the realm of the earth, still besinging the praises of uniform circular motion in the hymns of eccenter and epicycle characterizations (but with a discord by Ptolemy's use of equants). This would seem close to the overall picture settled upon by Ptolemy (Pedersen 1993, 87–89; Thurston 1994, 171–76).

We have at hand then a further way in which mathematical characterization can make our conceptions of natures sensitive to those natures. By what and what not we are free to choose in our so-far true characterizations, we can sort the likely facts of natures themselves from their accessories.


References

Aristotle 1984 [c. 348–22 B.C.]. The Complete Works of Aristotle. J. Barnes, editor. Princeton: University Press.

Boyer, C.B. 1959. The Rainbow: From Myth to Mathematics. New York: Thomas Yoseloff.

Gaukroger, S. 1978. Explanatory Structures: Concepts of Explanation in Early Physics and Philosophy. Atlantic Highlands, NJ: Humanities Press.

Goldstein, B.R. 1983a. The Obliquity of the Ecliptic in Ancient Greek Astronomy. In Goldstein 1985.
——. 1983b. A New View of Early Greek Astronomy. In Goldstein 1985.
——. 1985. Theory and Observation in Ancient and Medieval Astronomy. London: Variorum Reprints.

Heath, T. 1981 [1913]. Aristarchus of Samos. New York: Dover.

Neugebauer, O. 1948. Mathematical Methods in Ancient Astronomy. In Astronomy and History: Selected Essays. 1983. New York: Springer-Verlag.
——. 1969. The Exact Sciences in Antiquity. 2nd ed. New York: Dover.
——. 1975. A History of Ancient Mathematical Astronomy. Providence: Brown University Press.

North, J.D. 1987. Coordinates and Categories: The Graphical Representation of Functions in Medieval Astronomy. In Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages. E. Grant and J.E. Murdoch, editors. Cambridge: University Press.

Pedersen, O. 1993. Early Physics and Astronomy. 2nd ed. Cambridge: University Press.

Thurston, H. 1994. Early Astronomy. New York: Springer-Verlag.

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For some uses of mathematics in modern science, see:
http://www.solopassion.com/node/2361#comment-61725

Stephen writes;


Still, I am left in the end with angular momentum conservation and its cohorts as brute, unexplained physical facts, just as the Pythagorean is left in the end with brute mathematical forms. Perhaps, ultimately, the two hands and what is in them must be brought together, but for today it seems most sensible to me to keep the two hands apart, that is, to maintain an ontological distinction between mere essential forms and working or constituting causes.

I respond:

angular momentum and its cohorts are not quite as brutally unexplained. By way of Noether's theorem the conservation of these quantities follow from the underlying symmetries in nature. It is the symmetries that are basic and primordial.

As for causes, causes are itch scratchers. I am with Hume on this. When we see an event of type A always preceding and event of type B we are just driven to postulate a cause connecting the two. Causes live up in our heads. Events are Out There.

Our so-called explanations are mental dispositions that follow from our acceptance of and love for our models. Remember when the cause of heat was the flow of caloric? Ah yes. Those were the days.

Ba'al Chatzaf

Right, conservation of angular momentum is the result of space rotation symmetry, i.e. there is no preferred direction in space, all directions are equivalent. This is the simplest assumption we can make about the rotational structure of space, and therefore we should really only need an explanation if angular momentum were not conserved. Assume nothing special and you get conservation of angular momentum (in a similar way conservation of momentum follows from space translation symmetry and conservation of energy from time translation symmetry).


Exactly.

Folks:

One of the most common fallacious "reasoning" events...after this, therefore because of this.

Adam

More correct is: not after this therefore not because of this.

Ba'al Chatzaf

Reading this composition these nine years later, it took me a while to see why I was referring to the principle of conservation of angular momentum as a causal explanation of phenomena at all. Conservation principles are reasons, they fulfill because, but are they causal explanations? I had written “if asked why the earth continues its axial rotation, I should say this is due to conservation of angular momentum.” That’s pretty bad. All it really says is that unchanging continuance of the earth’s axial rotation is a case of the principle of inertia, Newton’s first law. That is a principle that sorts motions into those not requiring an explanation in terms of an external cause (force or torque) and the sorts of motion that do require an external cause. The steady spinning earth (supposing its angular velocity were exactly unvarying) is the sort of motion not requiring an external causal explanation at all.

That each cubic inch of matter composing the earth is rotating about the earth's axis, rather than traveling along some straight line at constant speed, does require causal explanation external to each such part. To begin mathematical characterization to serve causal explanations, we turn from Newton’s first law to his second.

But let us not simply toss conservation principles into the bin of characterization by an essential form. Instead of the whole spinning globe, think of the way a spinning figure skater draws in her arms to spin faster. Appealing to the conservation of angular momentum is a squarely causal explanation for what is going on in this maneuver. The lesson I draw then is that a given conservation principle can be appealed to merely to point out an essential form present in a case at hand (unvarying spin of earth) or appealed to by way of causal explanation (changing spin of skater).

In my 1991 essay “Induction on Identity,” I had written “Out of all the conditions that obtain in a situation, we typically take only one or a few as cause of some distinctive result, only a select portion of the ways in which the law of identity applies to an action or a becoming. We try to discover among antecedent conditions ones that will make a certain result under a wide range of variations in the remaining variable conditions. . . . In a primary sense, causes make things happen” (Objectivity V1N3, pp.25–26). That still goes. I’m pleased to say that this conception of causality has been developed in a really big way by James Woodward in Making Things Happen: A Theory of Causal Explanation (OUP 2003).

Looking to our further, modern understanding of why angular momentum is conserved, Bob and Peter remind us that isotropy of space joined with Hamilton’s principle implies that conservation. I say that causality is quartered in Hamilton’s principle, and the way causality can be brought out of it in classical domains, such as for the extended causal explanation of the changing spin of the skater, is through the relation between the Hamiltonian (a composition of energies in this situation) and force (or torque). With rendition of an application in terms of forces, we could turn to Newton’s second law (and third) to produce a causal analysis, in mathematical form.

As you would expect, I do not agree that causation is in the head and not given by the world. There are true causes, and mathematical characterizations help us find them and their situation in the mind-independent world.

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Following on the lesson I drew above, it appears that in Newton’s derivation of the form of the law of gravity in Book 1 of Principia, he uses the conservation of angular momentum only as an essential form. However, the principle is here doing significant explanatory work (noncausal), unlike in an appeal to conservation of angular momentum to “explain” the continued rotation of the earth on its axis.

In Newton’s dynamical analysis of planetary motion, the conservation of angular momentum was quartered in the law of equal areas. Kepler had discovered that law, you will recall, for the elliptical orbits of the planets. Newton demonstrated that the law holds more generally for any orbit due to any sort of central force.

This generalization is necessary for his grand strategy of demonstrating the various separation-dependencies of the various force laws that would be required for various supposed forms of orbits (mathematically specified of course)—to the end of showing that his demonstration of the form of the actual force for the actual planetary orbits is an instance of a general form of demonstration upon very general dynamical principles ( http://www.objectivistliving.com/forums/in...amp;#entry45789 ). (Instructive: The Key to Newton’s Dynamics by J. B. Brackenridge [U of CA 1995] and, rocking the boat, essay 3 in Reading Natural Philosophy edited by D. B. Malament [Open Court 2002])

Newton’s generalized law of equal areas in equal times is Proposition I, Theorem I of Principia Book 1. The demonstration relies on Newton’s first law alone. (Newton’s use of the second law in his proof of the parallelogram rule is dispensable.) In Proposition II, Theorem II, Newton demonstrates that any body subject to central forces (as in Prop. I) is subject, more specifically, to a centripetal force. This demonstration relies on Newton’s second law, as well as the first. Proposition IV, Theorem IV derives the general mathematical form of centripetal forces, relying on Propositions I and II. (The Scholium here mentions his alternative demonstration, from his student days, of this mathematical form [ http://www.solopassion.com/node/2361#comment-61725 ].) Proposition VI, Theorem V codes centripetal force into a characteristic of the instantaneous arc along the path of an orbiting body. This demonstration relies on Proposition I and its corollaries.

The specific form of the centripetal force (gravity) that sets an orbiting body in an elliptical orbit, where the source of the force is located at one focus of the ellipse, is demonstrated in Proposition XI, Problem VI. The demonstration relies on Proposition VI. All of these demonstrations rely on geometry. That is not a causal element here, unlike in machinery.

Newton’s demonstration of the mathematical form of the law of gravity is an explanation. His demonstration displays explanatory structure, following the causal structure of nature. Forces have sources, and the form of those forces dictate the form of orbits about the source. The form of the orbits indicates the form of their external cause.

Proposition XI relies on Newton’s first law and second law. Conservation of angular momentum enters in the way that the first law enters. Causality enters under the second law. Conservation of angular momentum is part of the Newtonian explanation of the law of gravity, but it is not the causal strand in the explanation.

When we come to general relativity, the gravitational force is seen as a particular type of curvature relation between nearby geodesics in a curved spacetime. It is sensible to say that spacetime curvature dictates that planets shall orbit the sun in the way they do. Certain aspects of spacetime (not mere space) become causes. But, of course, spacetime curvature is caused by source matter/fields. Causality runs from source mass-energy density to spacetime curvature to characteristics of the motion of bodies in that spacetime vicinity. Newton’s law of gravitation is recovered from general relativity in the joint limit of (i) velocities small in comparison to c and (ii) weak gravitational fields.

On the loss of traction from Noether’s theorem when applied in GR so as to include not only mass-energy density but energy of the (nonlocal) gravitational field, see Roger Penrose’s The Road to Reality, pp. 489–90 (Knopf 2004).

Related:
http://www.objectivistliving.com/forums/in...amp;#entry28908

http://www.objectivistliving.com/forums/in...amp;#entry27765

http://rebirthofreason.com/Forum/Objectivi...0242_3.shtml#69

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Using ancient astronomy for examples, I moved from the uses of mathematics in observation to its uses in characterization by essential form to its uses more particularly in characterizations that are causal explanations. In my next note, I want to look at a case, from optics, having those three, but scientific experimentation as well.

Rainbows

Aristotle (384–22 B.C.)

The idea that rainbows are produced by sunlight and its reflection by clouds goes back at least as far as Anaximenes (flourished c. 575 B.C.). Anaxagoras (born c. 500 B.C.) and Democritus (fl. c. 460 B.C.) may have held more advanced views on the rainbow specifically and on the character of reflection in general. Plato (427–347 B.C.) had some sense of the fixed character of reflection angles off mirrors, though not the law of reflection itself (Timaeus 46a–c; Boyer 1987, 34–39).

Aristotle offered an explanation of the rainbow. Some sciences are related to others such that “one is under the other—e.g. optics to geometry and harmonics to arithmetic” (An.Post 75b16–17). “Geometrical demonstrations apply to mechanical or optical demonstrations, and arithmetic to harmonical” (76a 24–25). Considering optics by means of geometry, “it is for the empirical scientist to know the fact and for the mathematical to know the reason why; for the latter have the demonstrations of the explanations, and often they do not know the fact, just as those who consider the universal often do not know some of the particulars through lack of observation” (79a3–7).

“While geometry investigates natural lines but not quo natural, optics investigates mathematical lines, but quo natural, not quo mathematical” (Phys 194a10–11). Sciences considered by means of mathematics “are those which, being something different in substance, make use of forms, for its objects are not said of any underlying subject—for even if geometrical objects are said of some underlying subject, still it is not as being said of an underlying subject that they are studied. / Related to optics as this is related to geometry, there is another science related to it—viz. the study of the rainbow; for it is for the natural scientist to know the fact, and for the student of optics—either simpliciter or mathematical—to know the reason why” (An.Post 79a8–13).

What is the phenomenon of the rainbow and its circumstances for which Aristotle offers an explanation? “The rainbow never forms a full circle, nor any segment [portion of circle] greater than a semicircle. At sunset and sunrise the circle is smallest [radius is smallest] and the segment largest: As the sun rises higher the circle is larger and the segment smaller. . . . There are never more than two rainbows at one time. Each of them is three-colored; the colors are the same in both and their number is the same, but in the outer rainbow they are fainter and their position [order] is reversed. In the inner rainbow the first and largest band is red; in the outer rainbow the band that is nearest to this one and smallest is of the same color: the other bands correspond on the same principle” (Meteor 371b27–372a6). The three colors Aristotle sees in the rainbow are red, green, and purple, “though between the red and the green an orange color is often seen” (372a9–10). http://www.arrowphotos.com/Rainbows.htm

We have seen Aristotle’s geometrical explanation of the shape of the rainbow in the last paragraph of §II of the starter for this thread. I will repeat that explanation now.

For a rainbow formed when the sun is at some place on the horizon, Aristotle considers the straight line from the horizon point (say, at the center of the sun’s width) to some point on the rainbow, which rainbow is to the side of the observer opposite the sun, and the straight line from that rainbow point to the observer. Whatever the reasons for color forming that point of the rainbow, the apparently crucial angle between the aforesaid two straight paths will be preserved if those two paths are together rotated about the line from the observer to the sun’s horizon point. Under such a rotation, the bow point will trace a full semicircle of rainbow above the ground, which is as we observe. The sun’s horizon point, the point of the observer, and the center of the rainbow’s (semi)circle will be collinear. For rainbows formed when the sun is above the horizon, the sun’s location is treated as if it were the former horizon point elevated. The center of the rainbow’s circle is equally lowered beneath the ground. Those two points keep their colinearity with the point of the observer. The portion of the rainbow’s circle remaining above the ground will then be less than a semicircle, which is as we observe (Boyer 1987, 42–44; Meteor 371b26–29, 375b17–76b21, 376b28–77a11).

In the starter for this thread, I had offered Aristotle’s schematic geometrical interpretation of the shapes of rainbows as an example of pure characterization. I am going to have to take back now what I said there: that this geometrical characterization is itself not a causal explanation. This characterization is too similar to the case of the spinning figure skater to not be counted a causal explanation of the shape of the rainbow. In particular the geometry explains how a greater elevation of the sun above the horizon brings it about that the rainbow forms a smaller portion of a semicircle. We now have examples of conservation-of-angular-momentum explanations that are causal (spinning skater changing her spin rate) and not causal (planetary orbits*) as well as geometrical explanations that are causal (elevation of sun reducing rainbow circular portion) and not causal (say, Newton’s use [in Prop. II, Th. II] of Euclid’s Prop. XL).

Aristotle supposed that for a given rainbow, a particular band of color lies at some single angle between the direction of the sunlight falling on the water in the air and the line of sight of the viewer (Meteor 375b17–377a11). He thought that the way the droplets of water produce color is by reflection. Aristotle did not know of the dispersion of light into colors by refraction. He knew of reflection by manmade mirrors and by natural ones, such as the appearance of the Milky Way on a body of still water.

He figured that just as shapes are revealed in mirror reflections under tight angular rules, there is another kind of reflection—by very tiny mirrors—which yield no shapes, only colors. These colors by minute reflectors, it is assumed, also operate under tight angular rules (Meteor 372a17–b8, 373a32–b32; An 435a5–10).

Aristotle may have known the law of reflection (of shapes) from mirrors, which is the rule that the angle of light incidence (angle with the plane it falls upon) equals the angle (with that plane) of line from eye to mirror (Boyer 1987, 39–41). But if he did know that rule and carried it over to “color reflection,” he failed to see an important implication that then follows from his geometrical account of the rainbow: arcs formed under all elevations of the sun should have equal radii, contrary to his statement of rainbow phenomena. The visual observation that rainbows less than semicircular have larger radii than full semicircular rainbows is a visual illusion, which could have been deposed by reasoning that enlisted the law of reflection (43–44).

Ptolemy (c. A.D. 100–180)

Look into a plane mirror. Reflected objects appear in it. In optics we call an object in there an image.

Ptolemy sets out three general principles for reflections in plane mirrors and, by tangent planes, for reflections in spherical mirrors. (1) The image in the mirror lies along a straight line from the location e of the eye to the image. That line intersects the plane M of the mirror at some point r. Let the location of the image in the mirror be called the point i.

Consider the straight line from a point o on the real object to its image i in the mirror. (2) That line is perpendicular to the mirror. Imprecise confirmation of this perpendicularity can be obtained by holding the blunt end of a pencil against the mirror, holding the pencil perpendicular to the mirror. Observe that the line of the pencil continues into the mirror without any bend. Tilt the pencil. Its line in the mirror will bend such that the line between the actual lead point o and its image point i looks to be perpendicular to the mirror plane M.

We now have two lines intersecting the point i, the lines ei and oi. From our Euclidean geometry, we recall that two intersecting lines determine a plane. Because one of our two lines lying in that plane P is perpendicular to the mirror plane M, we know that P is perpendicular to M. Now the point o lies in the plane P, and the point r lies in the plane P (for the lines oi and ei determine and lie in P, and r lies on ei). Therefore the line or lies in the plane P. But the line er also lies in P. The lines er and or determine a plane, and that is the very plane in which oi lies. The real object, the eye, the point of reflection on the mirror, and the image in the mirror all lie on a single plane.

The intersection of two planes determines a line. Call by L the line formed by the intersection of M and P. Only L lies on both M and P, and because r lies on both M and P, r lies on L. With all this understanding, it is proposed: (3) The angle between er and L is the same as between or and L. This we know as the law of reflection. Ptolemy gives a way of imprecise confirmation of this principle, then turns to a more certain demonstration by experiment (Optics Bk. 3, [3]–[6]).

Ptolemy has inscribed on a round bronze plate a circle, two perpendicular diameters through the circle’s center, and fine marks graduated by one angular degree within each quadrant. He has made three mirrors of polished iron. One mirror is plane, one convex cylindrical, and one concave cylindrical. He mounts a diopter along a radius of the circle, at some angle from one of the diameter lines, to be able to view the center of the circle from a definite known angle along the edge of the plate. At the center he puts a mirror aligned with the other diameter line. He slides a little colored object along the inscribed circle until it comes into view in the mirror through the diopter. Lo! The angle between the object-to-center radius and the plane of the mirror at the center point equals the angle between the diopter-viewing radius and the plane of the mirror at the center point (Bk. 3, [8]–[11]). http://brunelleschi.imss.fi.it/museum/esim.asp?c=201201

Ptolemy turns to refraction, the deflection of images seen through transparent media. To begin, make a watertight cylindrical basin. The basin shall be open along its length such that the remaining lengthwise wall has cross section significantly greater than a semicircle. Set the basin horizontal. Rest a coin inside the basin such that it is just out of view when one peers over the open edge of the basin. Gently fill the basin with water. The coin will become visible from one’s same vantage point, peering just over the edge of the basin (Bk. 5, [6]).

Now to quantification. Set that round inscribed disk from the reflection experiment on its edge square in a semicylindrical horizontal basin. Align one of the disk’s perpendicular lines horizontally, and fill the vessel with water to that line. Set a marker at some angle along the circle of the disk above the water. The disk has a nodule at its center, to use for sighting (as with a rifle). Sight from the edge marker such that it and the center marker appear to coincide. At the same time, move another marker along the basin wall under water until the marker appears aligned with the center-sight and the marker on the disk arc above water. The angle between the underwater marker (its radial line) and the disk’s inscribed vertical line will be less than the above-water marker (its radial line) and the disk’s vertical line (Bk. 5, [7]–[8]).

In this manner, Ptolemy obtains the refractive bending of sight-line from air to water for angles of sight at 10, 20, . . . 80 degrees from the vertical line. (Today we would say he was observing the refractive bending of light-line from water to air in this experiment.) The amounts of bending he reports are fairly accurate, except for the one at 80 degrees. It has been argued that Ptolemy was doctoring his results to fit a sequence of numbers that could be generated by an algorithm that had been used for generating sequences in Babylonian astronomy of the Seleucid (Smith 1996, 44–45, 233n9; Neugebauer 1969, 110–14, 135).

Ptolemy learned from his experiments in refraction that the greater the angle between sight-line and the line perpendicular to the water surface, the greater will be the difference between that angle and the angle between the straight extension of the air sight-line on into the water and the actual, bent sight-line continued to objects under the water. But he does not have the sine-formula we call the law of refraction which is able to capture the amount of water-surface bending for all values of air sight-line.

If Ptolemy worked on theory of the rainbow, his work has not come down to us. Ptolemy lived in Alexandria. A point was added to the phenomenology of the rainbow by an Athenian philosopher, Alexander of Aphrodisias. He became head of Aristotle’s school, the Lyceum, between A.D. 198 and 211. He reasoned from Aristotle’s theory of the rainbow (parts of the theory not presented here) that the sky between the primary and secondary bows should be brighter than the rest of the blue sky. He noted, however, that the sky between the bows is in fact darker than the rest of the sky. Today we call this dark arc Alexander’s band, and it is one characteristic to be explained by theories of the rainbow.

(To be continued.)

Note

* I said in #6 of this thread that Kepler’s second law (and, therein, conservation of angular momentum) does not enter as causal in Newton’s dynamical analysis of planetary motion. It does not enter as causal in Kepler’s account either. Recall that a planet travels through the path of its orbit faster while nearer the sun than while farther the sun. That is explained by Kepler’s second law: equal areas are swept out in equal times throughout the (elliptical) orbit of a given planet. Kepler did not see this as a causal explanation of the speed-distance inverse variation. In his conception, the cause of the planet moving faster while nearer the sun was that the strength of the sun’s ability to move a planet along its orbit is stronger the nearer the planet is to the sun (Kozhamthadam 1994, Chapter 8; Voelkel 2001a, 2001b)

References

Aristotle 1984. The Complete Works of Aristotle. J. Barnes, editor. Princeton.

Boyer, C. B. 1987 (1959). The Rainbow: From Myth to Mathematics. Princeton.

Kozhamthadam, J. 1994. The Discovery of Kepler’s Laws. Notre Dame.

Neugebauer, O. 1969. The Exact Sciences in Antiquity. 2nd ed. Dover.

Smith, A. M. 1996. Ptolemy’s Theory of Visual Perception: An English Translation of the Optics. The American Philosophical Society.

Voelkel, J. R. 2001a. Commentary on Ernan McMullin, “The Impact of Newton’s Principia on the Philosophy of Science.” Philosophy of Science 68:319–26.

——. 2001b. The Composition of Kepler’s Astronomia Nova. Princeton.