How Gauss Determined the Orbit of Ceres

8/9/2019 How Gauss Determined the Orbit of Ceres

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The following presentation of Carl Gausss

determination of the orbit of the asteroidCeres, was commissioned by Lyndon H.

LaRouche, Jr., in October 1997, as part of an

ongoing series of Pedagogical Exercises high-lighting the role of metaphor and paradox in

creative reason, through study of the great

discoveries of science. Intended for individual and classroom study, the weekly install-mentsnow chapterswere later serial-

ized in The New Federalist newspaper. Theyare collected here, in their entirety, for the first time, incorporating additions and revi-

sions to both text and diagrams.

Through the course of their presentation, itbecame necessary for the authors to reviewmany crucial questions in the history of math-

ematics, physics, and astronomy. All of these

issues were subsumed in the primary objective,the discovery of the orbit of Ceres. And,

because they were written to challenge a layaudience to master unfamiliar and conceptu-ally dense material at the level of axiomatic assumptions, the installments were often pur-

posefully provocative, proceeding by way ofcontradictions and paradoxes.

Nonetheless, the pace of the argument

moves slowly, building its case by constant ref-

erence to what has gone before. It is,th

a mountaintop you need not fear to climWe begin, by way of a preface, w

following excerpted comments by L

H. LaRouche, Jr. The authors retthem in the concluding stretto.

* * *

From Euclid through Legegeometry depended upon axiic assumptions accepted as if theyself-evident. On more careful intion, it should be evident, that

by Jonathan Tennenbaum

and Bruce Director

How GaussDeterminedThe Orbit of Ceres

PREFACE

4


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assumptions are not necessarily true.Furthermore, the interrelationshipamong those axiomatic assumptions, isleft entirely in obscurity. Most conspicu-

ous, even today, generally acceptedclassroom mathematics relies upon theabsurd doctrine, that extension in spaceand time proceeds in perfect continuity,with no possibility of interruption, evenin the extremely small. Indeed, everyeffort to prove that assumption, such asthe notorious tautological hoax concoct-ed by the celebrated Leonhard Euler,

FIGURE 1.1. Positions of unknownplanet (Ceres), observed byGiuseppe Piazzi on Jan. 2, Jan. 22,and Feb. 11, 1801, moving slowlycounter-clockwise against thesphere of the fixed stars.

5

Carl F.Gauss


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was premised upon a geometry which preassumed perfectcontinuity, axiomatically. Similarly, the assumption thatextension in space and time must be unbounded, was shownto have been arbitrary, and, in fact, false.

Bernhard Riemanns argument, repeated in the con-cluding sentence of his dissertation On the HypothesesWhich Underlie Geometry, is, that, to arrive at a suitabledesign of geometry for physics, we must depart the realm

of mathematics, for the realm of experimental physics.This is the key to solving the crucial problems of represen-tation of both living processes, and all processes which, likephysical economy and Classical musical composition, aredefined by the higher processes of the individual humancognitive processes. Moreover, since living processes, andcognitive processes, are efficient modes of existence withinthe universe as a whole, there could be no universal physicswhose fundamental laws were not coherent with that anti-entropic principle central to human cognition. . . .

By definition, any experimentally validated principleof (for example) physics, can be regarded as a dimen-

sion of an n-dimensional physical-space-time geome-try. This is necessary, since the principle was validatedby measurement; that is to say, it was validated by mea-surement ofextension. This includes experimentallygrounded, axiomatic assumptions respecting space andtime. The question posed, is: How do these n dimen-sions interrelate, to yield an effect which is characteris-tic of that physical space-time? It was Riemannsgenius, to recognize in the experimental applicationswhich Carl Gauss had made in applying his approachto bi-quadratic residues, to crucial measurements inastrophysics, geodesy, and geomagnetism, the key to

crucial implications of the approach to a general theoryof curved surfaces rooted in the generalization fromsuch measurements. . . .

What Art Must Learn from Euclid

The crucial distinction between that science and artwhich was developed by Classical Greece, as distinctfrom the work of the Greeks Egyptian, anti-Mesopotamia, anti-Canaanite sponsors, is expressed mostclearly by Platos notion ofideas. The possibility of mod-ern science depends upon, the relatively perfected formof that Classical Greek notion of ideas, as that notion is

defined by Plato. This is exemplified by Platos Socraticmethod of hypothesis, upon which the possibility ofEuropes development depended absolutely. What ispassed down to modern times as Euclids geometry,embodies a crucial kind of demonstration of that princi-ple; Riemanns accomplishment was, thus, to have cor-rected the errors of Euclid, by the same Socratic methodemployed to produce a geometry which had been, up toRiemanns time, one of the great works of antiquity. This

has crucial importance for rendering transparent underling principle of motivic thorough-compositionClassical polyphony. . . .

The set of definitions, axioms, and postulates dedufrom implicitly underlying assumptions about spacexemplary of the most elementary of the literate usethe termhypothesis. Specifically, this is adeductive hypesis, as distinguished from higher forms, including n

linear hypotheses. Once the hypothesis underlyinknown set of propositions is established, we may anpate a larger number of propositions than those originconsidered, which might also be consistent with tdeductive hypothesis. The implicitly open-ended coltion of theorems which might satisfy that latter requment, may be named atheorem-lattice . . . .

The commonly underlying principle of organizainternal to each such type of deductive lattice, isextensas that principle is integral to the notion of measuremThis notion of extension, is the notion of a type of exsion characteristic of the domain of the relevant choic

theorem-lattice. All scientific knowledge is premiupon matters pertaining to a generalized notion of exsion. Hence, all rational thought, is intrinsically geomrical in character.

In first approximation, all deductively consistent tems may be described in terms of theorem-lattices. Hever, as crucial features of Riemanns discovery illustmost clearly, the essence of human knowledge is chachange of hypothesis, this in the sense in which the prlem of ontological paradox is featured in Platos P

menides. In short, the characteristic of human knowleand existence, is not expressible in the mode of deduc

mathematics, but, rather, must be expressed as chafrom one hypothesis, to another. The standard for chanis to proceed from a relatively inferior, to superior hythesis. The action of scientific-revolutionary change, fra relatively inferior, to relatively superior hypothesis, ischaracteristic of human progress, human knowledge, of the lawful composition of that universe, whose masmankind expresses through increases in potential relapopulation-density of our species.

The process of revolutionary change occurs othrough the medium of metaphor, as the relevant priple of contradiction has been stated, above. Just as Eu

was necessary, that the work of descriptive geometryGaspard Monge et al., the work of Gauss, and so fomight make Riemanns overturning Euclid feasible, sohuman progress, all human knowledge is premised uthat form of revolutionary change which appears as

agapic quality of solution to an ontological paradox.Lyndon H. LaRouche

adapted from Behind the NoFidelio, Summer 1997 (Vol.VI, N


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J

anuary 1, 1801, the first day of a new century. In theearly morning hours of that day, Giuseppe Piazzi,

peering through his telescope in Palermo, discoveredan object which appeared as a small dot of light in thedark night sky. (Figure 1.1) He noted its position withrespect to the other stars in the sky. On a subsequentnight, he saw the same small dot of light, but this time itwas in a slightly different position against the familiarbackground of the stars.

He had not seen this object before, nor were there anyrecorded observations of it. Over the next several days,Piazzi watched this new object, carefully noting its changein position from night to night. Using the methodemployed by astronomers since ancient times, he recorded

its position as the intersection of two circles on an imagi-nary sphere, with himself at the center. (Figure 1.2a)(Astronomers call this the celestial sphere; the circles aresimilar to lines of longitude and latitude on Earth.) One setof circles was thought of as running perpendicular to thecelestial equator, ascending overhead from the observershorizon, and then descending. The other set of circles runsparallel to the celestial equator.

To specify any one of these circles, we require an angu-

lar measurement: the position of a longitudinal circle isspecified by the angle (arc) known as the right ascension,

and that of a circle parallel to the celestial equator, by thedeclination.* (Figure 1.2b). Hence, two angles suffice tospecify the position of any point on the celestial sphere.This, indeed, is how Piazzi communicated his observa-tions to others.

Piazzi was able to record the changing positions of thenew object in a total of 19 observations made over the fol-lowing 42 days. Finally, on February 12, the object disap-peared in the glare of the sun, and could no longer beobserved. During the whole period, the objects totalmotion made an arc of only 9 on the celestial sphere.

What had Piazzi discovered? Was it a planet, a star, a

comet, or something else which didnt have a name? (Atfirst, Piazzi thought he had discovered a small comet withno tail. Later, he and others speculated it was a planetbetween Mars and Jupiter.) And now that it had disap-peared, what was its trajectory? When and where could it

7

R.A.=

0h

rs.

ecliptic

star

SouthCelestialPole

NorthCelestialPole

Earth

equator

R.A. DECL

.

#

60N

30N

30S

60S

SouthCelestialPole

22h

0

2h

6h4h

NorthCelestialPole

##

FIGURE 1.2 The celestial sphere. (a) Since ancient times, astronomers have recorded their observations of heavenly bodies as points

on the inside of an imaginary sphere called the celestial sphere, or sphere of the fixed stars, with the Earth at its center. Arcs of rightascension and parallels of declination are shown. (b) Locating the position of an object on the celestial sphere by measuring rightascension and declination.

(a) (b)

CHAPTER 1

Introduction

__________

* Figure 1.1 shows the celestial sphere as seen by an observer, with agrid for measuring right ascension and declination shown mappedagainst it.


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be seen again? If it were orbiting the sun, how could its tra-

jectory be determined from these few observations madefrom the Earth, which itself was moving around the sun?

Had Piazzi observed the object while it was approach-ing the sun, or was it moving away from the sun? Was itmoving away from the Earth or towards it, when theseobservations were made? Since all the observationsappeared only as changes in position against the back-ground of the stars (celestial sphere), what motion didthese changes in position reflect? What would thesechanges in position be, if Piazzi had observed them fromthe sun? Or, a point outside the solar system itself: aGods eye view? (Figure 1.3)

It was six months before Piazzis observations werepublished in the leading German-language journal ofastronomy, von ZachsMonthly Correspondence for the Pro-

motion of Knowledge of the Earth and the Heavens, but newsof his discovery had already spread to the leadingastronomers of Europe, who searched the sky in vain forthe object. Unless an accurate determination of the objectstrajectory were made, rediscovery would be unpredictable.

There was no direct precedent to draw upon, to solvethis puzzle. The only previous experience that anyonehad had in determining the trajectory of a new object inthe sky, was the 1781 discovery of the planet Uranus by

William Herschel. In that case, astronomers were able toobserve the position of Uranus over a considerable time,recording the changes in the position of the planet withrespect to the Earth.

With these observations, the mathematicians simplyasked, On what curve is this planet traveling, such thatit would produce these particular observations? If onecurve didnt produce the desired mathematical result,another was tried.

As Carl F. Gauss described it in the Preface to his 1

book, Theory of the Motion of the Heavenly Bodies Moabout the Sun in Conic Sections,

As soon as it was ascertained that the motion of the newplanet, discovered in 1781, could not be reconciled with

the parabolic hypothesis, astronomers undertook to adapa circular orbit to it, which is a matter of simple and vereasy calculation. By a happy accident, the orbit of thi

planet had but a small eccentricity, in consequence owhich, the elements resulting from the circular hypothesis sufficed, at least for an approximation, on which th

determination of the elliptic elements could be based.There was a concurrence of several other very favor

able circumstances. For, the slow motion of the planeand the very small inclination of the orbit to the plane othe ecliptic, not only rendered the calculations much morsimple, and allowed the use of special methods not suitedto other cases; but they removed the apprehension, lest th

planet, lost in the rays of the sun, should subsequentlelude the search of observers (an apprehension whichsome astronomers might have felt, especially if its ligh

had been less brilliant); so that the more accurate determination of the orbit might be safely deferred, until a selection could be made from observations more frequent and

more remote, such seemed best fitted for the end in view.

Linearization in the Small

The false belief that we need a large number of obsertions, filling out as large an arc as possible, in ordedetermine the orbit of a heavenly body, is a typical pruct of the Aristotelean assumptions brought into scieby the British-Venetian school of mathematicsschool typified by Paolo Sarpi, Isaac Newton, and Lehard Euler. Sarpiet al. insisted that, if we examine sm

8

FIGURE 1.3.Artistsrendering of a Gods eyeview of the first six

planets of the solar system.(Note that the correct

planetary sizes, andrelative distances from the

sun of outer planetsJupiter and Saturn, arenot preserved.)


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tained, in this case, only if the problem of determining

the orbit of an unknown planet is treated as a purelymathematical one.

For example, think of three dots on a plane. (Figure1.5) On how many different curves could these dots lie?Now add more dots. The more dots, covering a greaterpart of the curve, the more precise determination of thecurve. A small change of the position of the dots, can

er and smaller portions of any curve in nature, we shall

find that those portions look and behave more and morelike straight line segmentsto the point that, for suffi-ciently small intervals, the difference becomes practicallyinsignificant and can be ignored. This idea came to beknown as linearization in the small.

In the mid-Fifteenth century, Nicolaus of Cusa hadalready demonstrated conclusively that linearization inthe small had no place in mathematicsif that mathe-matics were to reflect truth. Cusa demonstrated that thecircle represents a fundamentally different species of curvefrom a straight line, and that this species difference doesnot disappear, or even decrease, when we examine very

small portions of the circle. (Figure 1.4) With respect totheir increasing number of vertices, the polygonsinscribed in and circumscribing the circle become moreand more unlike it.

Extending Cusas discovery to astronomy, JohannesKepler discovered that the solar system was orderedaccording to certain harmonic principles. Each small partof the solar system, such as a small interval of a planetaryorbit, reflected that same harmonic principle completely.Keplers call for the invention of a mathematical conceptto measure this self-similarity, provoked G.W. Leibniz todevelop the infinitesimal calculus. The entirety of the

work of Sarpi, Newton, and Euler, was nothing but afraud, perpetrated by the Venetian-British oligarchyagainst the work of Cusa, Kepler, and Leibniz.

Applying the false mathematics of Sarpi et al. toastronomy, would mean that the physical Universebecame increasingly linear in the small, and that, there-fore, the smaller the arc spanned by the given series ofobservations, the less those observations tell us about theshape of the orbit as a whole. This delusion can be main-

FIGURE 1.4.Nicolaus of Cusa demonstrated, that no matter how many times its sides are multiplied, the polygon can never attainequality with the circle. The polygon and circle are fundamentally different species of figures.

FIGURE 1.5. (a) Here are just a few of the curves that canbe drawn through the same three points. (b) With moreobservation points, we may find that the curve is not asanticipated.

(a)

(b)


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mean a great change in the shape of the curve. The fewerthe dots and the closer together they are, the less precise isthe mathematical determination of the curve.

If this false mathematics were imposed on the Uni-verse, determining the orbit of a planet would hardly bepossible, except by curve-fitting or statistical correlationsfrom as extensive a set of observations as possible. But the

changes of observed positions of an object in the nightsky, are not dots on a piece of paper. These changes ofposition are a reflection of physical action, which is self-similar in every interval of that action, in the sense under-stood by Cusa, Kepler, and Leibniz. The heavenly body isnever moving along a straight line, but diverges from astraight line in every interval, no matter how small, in a

characteristic fashion.In fact, if we focus on the characteristic features of the

non-linearity in the small of any orbit, then the smallerthe interval of action we investigate in this way, the moreprecise the determination of the orbit as a whole! This

key point will become ever clearer as we work throughGauss determination of the orbit of Ceres.

It was only an accident that the problem of the deter-mination of the orbit of Uranus could be solved withoutchallenging the falsehood of linearization in the small.But such accidental success of a wrong method, was shat-tered by the problem presented by Piazzis discovery. TheUniverse was demonstrating Euler was a fool.

(Years later, Gauss would calculate in one hour, thetrajectory of a comet, which had taken Euler three daysto figure, a labor in which Euler lost the sight of one eye.I would probably have become blind also, Gauss said of

Euler, if I had been willing to keep on calculating in this

manner for three days!)It was September of 1801, before Piazzis observati

reached the 24-year-old Gauss, but Gauss had alreanticipated the problem, and ridiculed other mathem

cians for not considering it, since it assuredly comme

10

FIGURE 1.7. Some characteristic properties of the ellipse (a fuller description is presented in the Appendix).

FIGURE 1.6. Generation of the conic sections by cutting acone with a rotating plane. When the plane is parallel tothe base, the section is a circle. As the plane begins to rotatelliptical sections are generated, until the plane parallel tothe side of the cone generates a parabola. Further rotationgenerates hyperbolas.

Construction of a tangent tothe ellipse: Draw a circlearound focus f, with radiusequal to the constantdistance d + d. The tangentat any point q is the lineobtained by folding the

circle such that point q

touches the second focus f.This construction can beinverted to generate

ellipses and other conicsections as envelopes ofstraight lines (see text andFigure 1.9).

Every ellipse has twofoci f, f, such that thesum of distances d and dto any point q on thecircumference of theellipse is a constant.

The ellipse as acontraction of the

circumscribed circle, inthe direction perpendicularto the major axis. Theratio pq: pqremains thesame, no matter where plies on the major axis.

d d

f f

q

q

q

p

d d

d

f f

q

q

(a) (c)

(b)


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ed itself to mathematicians by its difficulty and elegance,even if its great utility in practice were not apparent.Because others assumed this problem was unsolvable,and were deluded by the accidental success of the wrongmethod, they refused to believe that circumstances wouldarise necessitating its solution. Gauss, on the other hand,considered the solution, before the necessity presenteditself, knowing, based on his study of Kepler and Leib-niz, that such a necessity would certainly arise.

Introducing the Conic Sections

Before embarking on our journey to re-discover themethod by which Gauss determined the orbit of Ceres, wesuggest the reader investigate for himself certain simplecharacteristics of curves that are relevant to the followingchapters. As we shall show later, Kepler discovered that

the planets known to him moved around the sun in orbitsin the shape of ellipses. By Gausss time, objects such ascomets had been observed to move in orbits whose shapewas that of other, related curves. All these related curvescan be generated by slicing a cone at different angles, andare therefore called conic sections. (Figure 1.6)

The conic sections can be constructed in a variety ofdifferent ways. (SEE Figure 1.7, as well as the Appendix,The Harmonic Relationships in an Ellipse) The readercan get a preliminary sense of some of the geometricalproperties of the conic sections, by carrying out the fol-

lowing construction.Take a piece of waxed paper and draw a circle on it.

(Figure 1.8) Then put a dot at the center of the circle.Now fold the circumference onto the point at the centerand make a crease. Unfold the paper and make a new fold,bringing another point on the circumference to the point

FIGURE 1.8. Using paperfolding to generate a circle asan envelope of chords.

. . .

FIGURE 1.9. Conic sections generated as envelopes of straight lines, using thewaxed paper folding method. (a) Ellipse. (b) Hyperbola. (c) Parabola.

(a) (b)

(c)


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at the center. Make another crease. Repeat this processaround the entire circumference (approximately 25 times).At the end of this process, you will see a circle enveloped

by the creases in the wax paper.Now take another piece of wax paper and do the

same thing, but this time put the point a little awayfrom the center. At the end of this process, the creaseswill envelop an ellipse, with the dot being one focus.(Figure 1.9a)

Repeat this construction several times, each time mov-ing the point a little farther away from the center of thecircle. Then try it with the point outside the circle; thiswill generate a hyperbola. (Figure 1.9b) Then make thesame construction, using a line and a point, to construct a

parabola. (Figure 1.9c)In this way, you can construct all the conic secti

as envelopes of lines. Now, think of the different cur

tures involved in each conic section, and the relatiship of that curvature to the position of the dot (focu

To see this more clearly, do the following. In eachthe constructions, draw a straight line from the focuthe curve. (Figure 1.10) How does the length of this change, as it rotates around the focus? How is tchange different in each curve?

Over the next several chapters, we will discover hthese geometrical relationships reflect the harmoordering of the Universe.

Bruce Dire

12

FIGURE 1.10. The length of a line drawn from the focus to the curve changes as it movesaround the curve, except in the case of the circle. In the case of a planetary orbit, thatlength is the distance from the sun to the planet . Note that the circle and ellipse are closed

figures, whereas the parabola and two-part hyperbola are unbounded.

Circle Ellipse Parabola Hyperbola

CHAPTER 2

Clues from Kepler

What did Gauss do, which other astronomersand mathematicians of his time did not, andwhich led those others to make wildly erro-

neous forecasts on the path of the new planet? Perhapswe shall have to consult Gausss great teacher, Johannes

Kepler, to give us some clues to this mystery.Gauss first of all adopted Keplers crucial hypothesis,

that themotion of a celestial object is determined solely by itsorbit, according to the intelligible principles Keplerdemonstrated to govern all known motions in the solarsystem. In the Keplerian determination of orbital motion,no information is required concerning mass, velocity, orany other details of the orbiting object itself. Moreover, asGauss demonstrated, and as we shall rediscover for our-

selves, the orbit and the orbital motion in its totality, be adduced from nothing more than the internal curture of any portion of the orbit, however small.

Think this over carefully. Here, the science of KepGauss, and Riemann distinguishes itselfabsolutely fr

that of Galileo, Newton, Laplace, et al. Orbits changes of orbit (which in turn are subsumed by highorder orbits) areontologically primary. The relation ofKeplerian orbit, as a relatively timeless existence, toarray of successive positions of the orbiting body, is that of an hypothesis to its array of theorems. From standpoint, we can say it is the orbit which movesplanet, not the planet which creates the orbit bymotion!


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If we interfere with the motion of an orbiting object,

then we are doing work against the orbit as a whole. Theresult is to change the orbit; and this, in turn, causes thechange in the visible motion of the object, which weascribe to our efforts. That, and not the bestial pushingand pulling of Sarpian-Newtonian point-mass physics,is the way our Universe works. Any competent astro-naut, in order to successfully pilot a rendezvous in space,must have a sensuous grasp of these matters. Gausssentire method rests upon it.

Gauss adopted an additional, secondary hypothesis,likewise derived from Kepler, for which we have beenprepared by Chapter 1: At least to a very high degree of

precision, the orbit of any object which does not passextremely close to some other body in our solar system(moons are excluded, for example), has the form of asimple conic section (a circle, an ellipse, a parabola, or ahyperbola) with focal point at the center of the sun.Under such conditions, the motion of the celestial objectisentirely determined by a set of five parameters, knownamong astronomers as the elements of the orbit,which specify the form and position of the orbit inspace. Once the elements of an orbit are specified, and

for as long as the object remains in the specified orbit, itsmotion is entirely determined for all past, present, and

future times!Gauss demonstrated how the elements of any orbit,

and thereby the orbital motion itself in its totality, can beadduced from nothing more than the curvature of anyarbitrarily small portion of the orbit; and how the lattercan in turn be be adducedin an eminently practicalwayfrom the intervals, defined by only three good,closely spaced observations of apparent positions as seenfrom the Earth!

The Elements of an Orbit

Theelements of a Keplerian elliptical orbit consist of thefollowing:

Two parameters, determining the position of theplane of the objects orbit relative to the plane of the Earthsorbit (called the ecliptic). (Figure 2.1) Since the sun isthe common focal point of both orbits, the two orbitalplanes intersect in a line, called the line of nodes. Therelative position of the two planes is uniquely deter-mined, once we prescribe:

(i) their angle of inclination to each other (i.e., theangle between the planes); and

(ii) the angle made by the line of nodes with somefixed axis in the plane of the Earths orbit.

Two parameters, specifying the shape and overallscale of the objects Keplerian orbit. (Figure 2.2) It is notnecessary to go into this in detail now, but the chieflyemployed parameters are:

(iii) the relative scale of the orbit, as specified (forexample) by its width when cut perpendicular to itsmajor axis through the focus (i.e., center of the sun);

(iv) a measure of shape known as the eccentricity,which we shall examine later, but whose value is 0 for cir-cular orbits, between 0 and 1 for elliptical orbits, exactly 1

for parabolic orbits, and greater than 1 for hyperbolicorbits. Instead of the eccentricity, one can also use the peri-helial distance, i.e., the shortest distance from the orbit tothe center of the sun, or its ratio to the width parameter;

Lastly, we have:(v) one parameter specifying the angle which the

main axis of the objects orbit within its own orbitalplane, makes with the line of intersection with theEarths orbit (line of nodes). For this purpose, we can

ecliptic (planeof Earth's orbit)

line of nodes

inclination of orbitalplane to ecliptic

i

major

axis

FIGURE 2.1.A set of three angles is used tospecify the spatial orientation of a givenKeplerian orbit relative to the orbit of theEarth. (1) Angle of inclination i, whichthe plane of the given orbit makes withthe ecliptic plane (the plane of theEarths orbit). (2) Angle , which theorbits major axis makes with the lineof nodes (the line of intersection ofthe plane of the given orbit and theecliptic plane). (3) Angle , whichthe line of nodes makes with somefixed axis in the ecliptic plane(the latter is generally taken to be

the direction of the vernalequinox ).


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take the angle between the major axis of the objects orbitand the line of nodes. (Figure 2.1)

The entire motion of the orbiting body is determinedby these elements of the orbit alone. If you have masteredKeplers principles, you can compute the objects preciseposition at any future or past time. All that you mustknow, in addition to Keplers laws and the five parame-ters just described, is a single time when the planet was (orwill be) in some particular locus in the orbit, such as theperihelial position. (Sometimes, astronomers include thetime of last perihelion-crossing among the elements.)

Now, let us go back to Fall 1801, as Gauss pondered

over the problem of how to determine the orbit of theunknown object observed by Piazzi, from nothing but ahandful of observations made in the weeks before it dis-appeared in the glare of the morning sun.

The first point to realize, of course, is that the tiny arcof a few degrees, which Piazzis object appeared todescribe against the background of the stars, was not thereal path of the object in space. Rather, the positionsrecorded by Piazzi were the result of a rather complicat-ed combination of motions. Indeed, the observed motionof any celestial object, as seen from the Earth, is com-pounded chiefly from the following three processes, or

degrees of action:

1. The rotation of the Earth on its axis (uniform circularrotation, period one day). (Figure 2.3)

2. The motion of the Earth in its known Keplerian orbitaround the sun (non-uniform motion on an ellipse,period one year). (Figure 2.4)

3. The motion of the planet in an unknown Keplerian

orbit (non-uniform motion, period unknown in case of an elliptical orbit, or nonexistent in case oparabolic or hyperbolic orbit). (Figure 2.5)

Thus, when we observe the planet, what we see kind of blend of all of these motions, mixed or muplied together in a complex manner. Within any inteof time, however short, all three degrees of action operating together to produce the apparent positionthe object. As it turns out, there is no simple way to sarate out the three degrees of motion from the obsertions, because (as we shall see) the exact way the thmotions are combined, depends on the parameters ofunknown orbit, which is exactly what we are tryingdetermine! So, from a deductive standpoint, we woseem to be caught in a hopeless, vicious circle. We sget back to this point later.

Although the main features of the apparent motare produced by the triple product of two elliptmotion and one circular motion, as just mentioned, eral other processes are also operating, which have a coparatively slight, but nevertheless distinctly measuraeffect on the apparent motions. In particular, for his p

cise forecast, Gauss had to take into account the follow

known effects:4. The 25,700-year cycle known as the precession of

equinoxes, which reflects a slow shift in the Earaxis of rotation over the period of observation. (Fig2.6) The angular change of the Earths axis in course of a single year, causes a shift in the apparpositions of observed objects of the order of tens of onds of arc (depending on their inclination to the cetial equator), which is much larger than the margin

sun(focus)

'parameter'

major axis

(line of apsides)

B(semi-minor axis)

csun(focus)

f

A(semi-major axis)

(a) Relative scale (b) Eccentricity

FIGURE 2.2. (a) The relative scale of the orbit can be measured by the line perpendicular to the line of apsides, drawn through thefocus (sun). This line is known as the parameter of the orbit. (b) The eccentricity is measured as the ratio of the distance f from thefocus to the center of the orbit (point c, the midpoint of the major axis) divided by the semi-major axis A. For the circle, in whichcase the focus and center coincide, f = 0; for the ellipse, 0 < f/A < 1.


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ing the time it takes the light to reach him.

7. The apparent positions of stars and planets, as seenfrom the Earth, are also significantly modified by thediffraction of light in the atmosphere, which bends the

rays from the observed object, and shifts its apparentposition to a greater or lesser degree, depending on itsangle above the horizon. Gauss assumed that Piazzi, asan experienced astronomer, had already made the nec-

15

precision which Gauss required. (In Gausss timeastronomers routinely measured the apparent positionsof objects in the sky to an accuracy of one second ofarc, which corresponds to a 1,296,000th part of a fullcircle. Recall the standard angular measure: one fullcircle = 360 degrees; one degree = 60 minutes of arc;one minute of arc = 60 seconds of arc. Gauss is alwaysworking with parts-per-million accuracy, or better.)

5. The nutation, which is a smaller periodic shift in theEarths axis, superimposed on the 25,700-year preces-sion, and chiefly connected with the orbit of the moon.

6. A slight shift of the apparent direction of a distantstar or planet relative to the true one, called aber-ration, due to the compound effect of the finitevelocity of light and the velocity of the observer dur-

FIGURE 2.3.Rotation of Earth (daily). FIGURE 2.4. Orbit of Earth (yearly).

poleof the

ecliptic

ecliptic

celestialequator

tothe

'Pole

Star'

FIGURE 2.5. Unknown orbit of mystery planet (periodunknown).

FIGURE 2.6.Precession of the equinoxes (period 25,700years).The precession appears as a gradual shift in theapparent positions of rising and setting stars on the horizon,as well as a shift in position of the celestial pole. Thisphenomenon arises because Earths axis of rotation is notfixed in direction relative to its orbit and the stars, butrotates (precesses) very slowly around an imaginary axis

called the pole of the ecliptic, the direction perpendicularto the ecliptic plane (the plane of the Earths orbit).


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16

essary corrections for diffraction in the reported obser-vations. Nevertheless, Gauss naturally had to allow fora certain margin of error in Piazzis observations, aris-ing from the imprecision of optical instruments, in thedetermination of time, and other causes.

Finally, in addition to the exact times and observedpositions of the object in the sky, Gauss also had to knowthe exact geographical position of Piazzis observatory on

the surface of the Earth.

What Did Piazzi See?

Let us assume, for the moment, that the complicationsintroduced by effects 4, 5, 6, and 7 above are of a relative-ly technical nature and do not touch upon what Gausscalled the nerve of my method. Focus first on obtainingsome insight into the way the three main degrees ofaction 1, 2, and 3 combine to yield the observed positions.

For exploratory purposes, do something like the follow-ing experiment, which requires merely a large room and

tables. (Figures 2.7 and 2.8) Set up one object to representthe sun, and arrange three other objects to represent threesuccessive positions of the Earth in its orbit around the sun.This can be done in many variations, but a reasonable firstselection of the Earth positions would be to place themon a circle of about two meters(about 6.5 feet) radius aroundthe sun, and about 23 cen-timeters (about 9 inches)apartcorresponding, let ussay, to the positions on the Sun-days of three successive weeks.

Now arrange another threeobjects at a greater distancefrom the sun, for example 5meters (16 feet), and separatedfrom each other by, say 6 and 7centimeters. These positionsneed not be exactly on a circle,but only very roughly so. Theyrepresent hypothetical positionsof Piazzis object on the samethree successive Sundays ofobservation.

For the purpose of the sight-ings we now wish to make, thebest choice of celestial objectsis to use small, bright-coloredspheres or beads of diameter 1cm or less, mounted at the endof thin wooden sticks which arefixed to wooden disks or otherobjects, the latter serving as

bases placed on the table, as shown in the photographFigure 2.7.

Now, sight from each of the Earth positions to the cresponding hypothetical positions of Piazzis object, beyond these to a blackboard or posters hung fromopposing wall. Imagine that wall to represent part ofcelestial sphere, or sphere of fixed stars. Mark the ptions on the wall which lie on the lines of sight betw

the three pairs of positions of the Earth and Piazobject. Those three marks on the wall, represent data of three of Piazzis observations, in terms of objects apparent position relative to the backgroundthe fixed stars, assuming the observations were madesuccessive Sundays. Experimenting with different relapositions of the two in their orbits, we can see how observational phenomenon of apparent retrograde moand looping can come about (in fact, Piazzi observeretrograde motion). (Figure 2.9) Experiment also wdifferent arrangements of the spheres representPiazzis object, as might correspond to different orbits.

From this kind of exploration, we are struck byenormous apparent ambiguity in the observations. WPiazzi saw in his telescope was only a very faint poinlight, hardly distinguishable from a distant star excepits motion with respect to the fixed stars from day to d

FIGURE 2.7.Author Bruce Director demonstrates Piazzis sightings. The models on the table ithe foreground represent the three different positions of the Earth. The models on the table infront of the board represent the corresponding positions of Ceres. Marks 1, 2, and 3 on the boarrepresent the sightings of Ceres, as seen from the corresponding positions of the Earth.


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17

On the face of things, there would seem to be no way toknow exactly how far away the object might be, nor inwhat exact direction it might be moving in space. Indeed,all we really have are three straight lines-of-sight, run-ning from each of the three positions of the Earth to thecorresponding marks on the wall. For all we know, eachof the three positions of Piazzis object might be locatedanywhere along the corresponding line-of-sight! We doknow the time intervals between the positions we arelooking at (in this case a period of one week), but howcan that help us? Those times, in and of themselves, do

not even tell us how fast the object is really moving, sinceit might be closer or farther away, and moving more orless toward us or away from us.

Try as we will, there seems to be no way to determinethe positions in space from the observations in a deduc-tive fashion. But havent we forgotten what Keplertaught us, about the primacy of the orbit, over themotions and positions?

Gauss didnt forget, and we shall discover his solutionin the coming chapters.

Jonathan Tennenbaum

1

2

312.35 in.

Earth

Earthorbit

wall('fixedstars')

3

Piazzis object

1

3

22

sun

9in

.

orbit ofPiazzis object

16 ft.

6 ft.

lines of sight

FIGURE 2.8.Paradoxes of apparent motion.The apparent motion of Ceres as seen from the Earth (here indicated by the

successive positions 1,2,3 on the wall at the right) is very different from the actualmotion of Ceres in its orbit. In the case illustrated here, the order of points 1,2,3 on the

wall is reversed relative to Ceres actual positions; thus, Ceres will appear from Earth to bemoving backwards! The bizarre appearance of retrograde motion and looping is due to the

differential in motion of Earth and Ceres, combined with their relative configuration in space,Earths orbital motion being faster than that of Ceres (see Figure 2.9). In reality, the apparent

motion is further complicated by the circumstance that the two bodies are orbiting in different planes.

FIGURE 2.9. Star charts show apparent retrograde motion for the asteroids (a) Ceres, and (b) Pallas, during 1998.

(a)

(b)


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18

In investigations such as we are now pursuing, it should not be so much asked what has occurred, as what has

occurred that has never occurred before.C. Auguste Dupin,in Edgar Allan Poes

The Murders in the Rue Morgue

With Dupins words in mind, let us return to thedilemma in which we had entangled ourselvesin our discussion in the previous chapter. That

dilemma was connected to the fact, that what Piazziobserved as the motion of the unknown object against thefixed stars, was neither the objects actual path in space,

nor even a simple projection of that path onto the celestialsphere of the observer, but rather, the result of the motionof the object and the motion of the Earth, mixed together.

Thanks to the efforts of Kepler and his followers, thedetermination of the orbit of the Earth, subsuming itsdistance and position relative to the sun on any given dayof the year, was quite precisely known by Gausss time.Accordingly, we can formulate the challenge posed byPiazzis observations in the following way: We candetermine a precise set of positions in space from which

Piazzis observations were made, taking into account

Earths own motion. From each of the positionPalermo, where Piazzis observatory was located, drastraight line-of-sight in the direction in which Piazzi the object at that moment. All we can say with certaabout the actual positions of the unknown object at given times, is that each position liessomewhere alongcorresponding straight line. What shall we do?

In the face of such an apparent degree of ambiguany attempt to curve fit fails. For, there are no wdefined positions on which to fit an orbit! But, dwe knowsomething more, which could help us? AfterKepler taught that the geometricalforms of the orbits

(to within a very high degree of precision, at least) plconic sections, having a common focus at the center ofsun. Kepler also provided a crucial, additional set of cstraints (to be examined in Chapter 7), which determthe precise motion in any given orbit, once the elemenof the orbit discussed last chapter have been determine

Now, unfortunately, Piazzis observations dont etell us whatplane the orbit of Piazzis object lies in. Hdo we find the right one?

Take an arbitrary plane through the sun. The linessight of Piazzis observations will intersect that planas many points, each of which is a candidate for the p

tion of the object at the given time. Next, try to constra conic section, with a focus at the sun, which gthrough those points or at least fits them as closely as psible. (Alas! We are back to curve-fitting!) (Figure 3.1

Finallyand this is the substantial new featurcheck whether the time intervals defined by a Keplemotion along the hypothesized conic section betweengiven points, agree with the actual time intervalPiazzis observations. If they dont fit, which will be nly always the case, then we reject the orbit. For examif the intersection-points are very far away from the sthen Keplers constraints would imply a very s

motion in the corresponding orbit; outside a certain tance, the corresponding time-intervals would becolarger than the times between Piazzis actual obsertions. Conversely, if the points are very close to the sthe motion would be too fast to agree with Piazzis tim

The consideration of time-intervals thus helps to lithe range of trial-and-error search somewhat, but domain of apparent possibilities still remains monstrolarge. With the unique exception of Gauss, astronom

CHAPTER 3

MethodNot Trial-and-Error

E1

E3

E2

O

P3

P2P1

FIGURE 3.1.Piazzis observations define three lines of

sight from three Earth positions E1,E2,E3, but do not tellus where the planet lies on any of those lines. We do know

that the positions lie on some plane through the sun.


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felt themselves forced to make ad hoc assumptions andguesses, in order to radically reduce the range of possibili-ties, and thereby reduce the trial-and-error procedures toa minimum.

For example, the astronomer Wilhelm Olbers andothers decided to start with the working assumption thatthe sought-for orbit was very nearly circular, in whichcase the motion becomes particularly simple. Keplers

third constraint (usually referred to as his Third Law)determines a specific rate of uniform motion along thecircle, as soon as the radius of the circular orbit isknown. According to that third constraint, the square ofperiodic time in any closed orbiti.e., a circular or anelliptical oneas measured in years, is equal to the cubeof the orbits major axis, as measured in units of themajor axis of the Earths orbit. Next, Olbers took two ofPiazzis observations, and calculated the radius which a

circular orbit would have to have, in order to fit thosetwo observations.

It is easy to see how to do that in principle: The twoobservations define two lines of sight, each originatingfrom the position of the Earth at the moment of observa-tion. Imagine a sphere of variable radiusr, centered at thesun. (Figure 3.2) For each choice ofr, that sphere willintersect the lines-of-sight in two points, P and Q.

Assuming the planet were actually moving on a circularorbit of radiusr, the pointsP and Q would be the corre-sponding positions at the times of the two observations,and the orbit would be the great circle on the sphere pass-ing through those two points. On the other hand,Keplers constraints tell us exactly how large is the arcwhich any planet would traverse, during the time inter-val between the two observations, if its orbit were a circleof radius r. Now compare the arc determined from

19

It seems somewhat strange that thegeneral problem to determine theorbit of a heavenly body, without anyhypothetical assumption, from observa-

tions not embracing a great period of time,and not allowing a selection with a viewto the application of special methods

was almost wholly neglected up to thebeginning of the present century; or, atleast, not treated by any one in a man-

ner worthy of its importance; since itassuredly commended itself to mathe-maticians by its difficulty and elegance,

even if its great utility in practice werenot apparent. An opinion had univer-

sally prevailed that a complete determi-nation from observations embracing ashort interval of time was impossible,an ill-founded opinion,for it is now

clearly shown that the orbit of a heav-enly body may be determined quitenearly from good observations embrac-ing only a few days; and this without

any hypothetical assumption.Some ideas occurred to me in the

month of September of the year 1801,

[as I was] engaged at that time on avery different subject, which seemed

to point to the solution of the greatproblem of which I have spoken.

Under such circumstances we notinfrequently, for fear of being too

much led away by an attractive inves-tigation, suffer the associations ofideas, which, more attentively consid-ered, might have proved most fruitful

in results, to be lost from neglect. Andthe same fate might have befallenthese conceptions, had they not happi-ly occurred at the most propitious

moment for their preservation andencouragement that could have beenselected. For just about this time the

report of the new planet, discoveredon the first day of January of that year

with the telescope at Palermo, was thesubject of universal conversation; andsoon afterwards the observations madeby that distinguished astronomer

Piazzi, from the above date to theeleventh of February were published.

Nowhere in the annals of astrono-

my do we meet with so great anopportunity, and a greater one couldhardly be imagined, for showing most

strikingly, the value of this problem,than in this crisis and urgent necessity,when all hope of discovering in the

heavens this planetary atom, amonginnumerable small stars after the lapse

of nearly a year, rested solely upon asufficiently approximate knowledge of

its orbit to be based upon these veryfew observations. Could I ever havefound a more seasonable opportunityto test the practical value of my con-

ceptions, than now in employing themfor the determination of the orbit ofthe planet Ceres, which during theseforty-one days had described a geocen-

tric arc of only three degrees, and afterthe lapse of a year must be looked forin a region of the heavens very remote

from that in which it was last seen?The first application of the method

was made in the month of October1801, and the first clear night (Decem-ber 7, 1801), when the planet wassought for as directed by the numbers

deduced from it, restored the fugitiveto observation. Three other new plan-ets subsequently discovered, furnished

new opportunities for examining andverifying the efficiency and generalityof the method. [emphasis in original]

Excerpted from the Preface to the Eng-lish edition of Gausss Theory of the

Motion of the Heavenly Bodies Movingabout the Sun in Conic Sections.

C.F. Gauss: To determine the orbit of a heavenly body,without any hypothetical assumption


17/85

Keplers constraint, with the actual arc betweenP and Q,

as the length of radius r varies, and locate the value orvalues ofr, for which the two become coincident. Thatdetermination can easily be translated into a mathemati-cal equation whose numerical solution is not difficult towork out. Having found a circular orbit fitting twoobservations in that way, Olbers then used the compari-son with other observations to correct the original orbit.

Toward the end of 1801 astronomers all over Europebegan to search for the object Piazzi had seen in January-February, based on approximations such as Olbers. Thesearch was in vain! In December of that year, Gauss pub-lished his hypothesis for the orbit of Ceres, based on his

own, entirely new method of calculation. According tocalculations based on Gausss elements, the object wouldbe located more than 6 to the south of the positions fore-cast by Olbers, an enormous angle in astronomical terms.Shortly thereafter, the object was found very close to theposition predicted by Gauss.

Characteristically, Gausss method used no trial-and-error at all. Without making any assumptions on the par-ticular form of the orbit, and using only three well-chosen observations, Gauss was able to construct a goodfirst approximation to the orbit immediately, and thenperfect it without further observations to a high precision,

making possible the rediscovery of Piazzis object.To accomplish this, Gauss treated the set of observa-

tions (including the times as well as the apparent posi-tions) as being the equivalent of aset of harmonic intervals.Even though the observations are, as it were, jumbled upby the effects of projection along lines-of-sight andmotion of the Earth, we must start from the standpointthat the underlying curvature, determining an entireorbit from any arbitrarily small segment, is somehow

20

lawfully expressed in such an array of intervals. To de

mine the orbit of Piazzis object, we must be able to idtify the specific, tell-tale characteristics which revealwhole orbit from, so to speak, between the intervalsthe observations, and distinguish it from all other orThis requires that we conceptualize the higher curvatunderlying the entire manifold of Keplerian orbits, taas a whole. Actually, the higher curvature required, cnot be adequately expressed by the sorts of mathematfunctions that existed prior to Gausss work.

We can shed some light on these matters, by the lowing elementary experimental-geometrical investtion. Using the familiar nails-and-thread method, c

FIGURE 3.3. Constructing an ellipse in the shape of the orbof Mars.

sunP

Q

r

r

P

Q

E1

E2

CC

FIGURE 3.2.Method to determine the orbit of Ceres,on the assumption that the orbit is circular. Twosightings of Ceres define two lines of sight comingfrom the Earth positions E

1, E

2(the Earths positions

at the moments of observation). A sphere around thesun, of radius r, intersects the lines of sight in twopoints P,Q, which lie on a unique great circle C onthat sphere. A sphere of some different radius rwoulddefine a different set of points P, Qand a differenthypothetical orbit C. Determine the unique value ofr, for which the size of the arc PQ agrees with the rateof motion a planet would really have, if it weremoving according to Keplers laws on the circularorbit C over the time interval between the givenobservations.


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21

struct an ellipse having the shape of the Mars orbit, as fol-lows. (Figure 3.3) Hammer two nails into a flat boardcovered with white paper, at a distance of 5.6 cm fromeach other. Take a piece of string 60 cm long and tie eachend to one of the nailsor alternatively, make a loop of

string of length 60 + 5.6 = 65.6 cm, and loop it aroundboth nails. Pulling the loop tight with the tip of a pencil asshown, trace an ellipse. The positions of the two nails rep-resent the foci. The resulting curve will be a scaled-downreplica of Mars orbit, with the sun at one of the foci.

Observe that the circumference generated is hardlydistinguishable, by the naked eye, from a circle. Indeed,mark the midpoint of the ellipse (which will be the pointmidway between the foci), and compare the distancesfrom various points on the circumference, to the center.You will find a maximum discrepancy of only about onemillimeter (more precisely, 1.3 mm), between the maxi-

mum distance (the distance between the points on the cir-cumference at the two ends of the major axis connectingthe two foci) and the minimum distance (between theendpoints of the minor axis drawn perpendicular to themajor axis at its mid-point). Thus, this ellipses deviationfrom a perfect circle is only on the order of four parts inone thousand. How was Kepler able to detect anddemonstrate the non-circular shape of the orbit of Mars,given such a minute deviation, and how could he correct-

ly ascertain the precise nature of the non-circular form,on the basis of the technology available at his time?

Observe in Figure 3.4a, that the distances to the sun(the marked focus) change very substantially, as we movealong the ellipse.

Now, choose two pointsP1 andP2 anywhere along thecircumference of the ellipse, two centimeters apart. Theinterval between them would correspond to successivepositions of Mars at times about seven days apart (actual-ly, up to about 10 percent more or less than that, depend-ing on exactly where P

1andP2 lie, relative to theperihe-

lion [closest] andaphelion [farthest] positions). Draw radi-al lines from each ofP

1,P2 to the sun, and label the corre-

sponding lengthsr1,r

2.

Consider what is contained in the curvilinear triangleformed by those two radial line segments and the smallarc of Mars trajectory, from P

1toP2. Compare that arc

with that of analogous arcs at other positions on the orbit,and consider the following propositions: Apart from thesymmetrical positions relative to the two axes of theellipse,no two such arcs are exactly superimposable in any of

their parts. Were we to change the parameters of theellipsefor example, by changing the distance betweenthe foci, by any amount, however smallthen none ofthe arcs on the new ellipse, no matter how small, wouldbe superimposable withany of those on the first, in any of

sun

P6

P5

P4

P3

P2

P1

r6r5

r4

r3

r2

r1

(focus)

FIGURE 3.4. (a) The positions of Mars in its orbit around thesun at equal time intervals of approximately 30 days. Notethat the orbital arcs are longer when Mars is closer to the sun(faster motion), shorter when Mars is farther away (slower

motion), in such a way that the areas of the corresponding

orbital sectors are equal (Keplers Area Law). (b) In aclose-up of Mars orbit, note the small areas separating thechords and the orbital arcs, and reflecting the curvature ofthe orbit in the given interval. These areas change in size andshape from one part of the orbit to the next, reflecting aconstantly changing curvature.

P2

P1

r6 r5r4

r3

r2

r1

P3

P4

P5

P6(b)(a)


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Any successful solution of the problem posed toGauss must pivot on conceptualizing the char-acteristic curvature of Keplerian orbits in the

small. Before turning to Keplers own investigationson this subject, it may be helpful to take a brief look atthe closely related case of families of catenaries on thesurface of the Earththese being more easily accessibleto direct experimentation, than the planetary orbits

themselves.

Catenaries, Monads, andA First Glimpse at Modular FunctionsWhen a flexible chain is suspended from two points, andpermitted to assume its natural form under the action of itsown weight, then, the portion of the chain between thetwo points forms a characteristic species of curve, knownas a catenary. The ideal catenary is generated by a chainconsisting of very small, but strong links made of a rigidmaterial, and having very little friction; such a chain is

practically inelastic (i.e., does not stretch), while at the sametime being nearly perfectly flexible, down to the lower lim-it defined by the diameter of the individual links.

Interestingly, the form of the catenary depends only onthe position of the points of suspension and the length ofthe chain between those points, but not on its mass orweight.

With the help of a suitable, fine-link chain, suspendedparallel to, and not far from, a vertical wall or board (so

that the chains form can easily be seen and traceddesired), carry out the following investigations.

(For some of these experiments, it is most convenienuse two nails or long pins, temporarily fixed into the wor board, as suspension-points; the nails or pins shouldrelatively thin, and with narrow heads, so that the linkthe chain can easily slip over them, in order to be ablvary the length of the suspended portion. In some exp

ments it is better to fix only one suspension-point winail, and to hold the other end in your hand.)

Start by fixing any two suspension-points and an atrary chain-length. (Figure 4.1) Observe the way shape of each part of the catenary, so formed, dependall the other parts. Thus, if we try to modify any porof the catenary, by pushing it sideways or upwards w

22

their parts! Thus, each arc is uniquely characteristic ofthe ellipse of which it is a part. The same is true amongall species of Keplerian orbits.

Consider what means might be devised to reconstructthe whole orbit from any one such arc. For example, bywhat means might one determine, from a small portion

of a planetary trajectory, whether it belongs to a parabol-ic, hyperbolic, or elliptical orbit?Now, compare the orbital arc betweenP

1andP2 with

the straight line joiningP1

andP2. (Figure 3.4b) Togeth-er they bound a tiny, virtually infinitesimal area. Evident-ly, the unique characteristic of the particular elliptical

orbit must be reflected somehow in thespecific mannewhich that arc differs from the line, as reflected in tinfinitesimal area.

Finally, add a third point, P3, and consider the culinear triangles corresponding to each of the three p(P

1,P2), (P2,P3), and (P1,P3), together with the co

sponding rectilinear triangles and infinitesimal awhich compose them. The harmonic mutual relatiamong these and the corresponding time intervals, lithe heart of Gausss method, which is exactly the oppof linearity in the small.

CHAPTER 4

Families of Catenaries(An Interlude Considering Some Unexpected Facts About Curvature

A B

FIGURE 4.1A catenary is formed by suspending a chainbetween points A and B.


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23

FIGURE 4.3. Varying the endpoint position of a fixed lengthof chain generates a second family of catenaries.

ABL

B

BB

B

B

B

the tip of a finger, we see that the entire curve is affected,at least slightly, over its entire length. This behavior ofthe catenary reflects Leibnizs principle of least action,whereby the entire Universe as a whole, including its

most remote parts, reacts to any event anywhere in theUniverse. There is no isolated point-to-point action inthe way the Newtonians claim.

Note that the curvature of each individual catenarychanges constantly along its length, as we go from itslowest point to its highest point.

Next, generate a family of catenaries, by keeping thesuspension-points fixed, but varying the length of thechain between those points. (Figure 4.2) Observe thechanges in the form and curvature, and the changes inthe angles, which the chain makes to the horizontal at thepoints of suspension, as a function of the suspended

length.Generate a second family of catenaries, by keeping the

chain length and one of the suspension-points fixed, whilevarying the other point. (Figure 4.3) IfA is the first sus-pension-point, andL is the length of the suspended chain,then the second suspension-point B (preferably held byhand) can be located anywhere within the circle of radius

L aroundA. ForB on the circumference of the circle, thecatenary degenerates into a straight line. (Or rather, some-thing close to a straight line, since the latter would requirea physically impossible, infinite tension to overcome thegravitational effect.) Observe the changes of form, as B

moves around A in a circle of radius less than L. Also,observe the change in the angles, which the catenarymakes to the horizontal at each of the endpoints, as afunction of the position ofB. Finally, observe the changesin the tension, which the chain exerts at the endpoint B,held by hand, as its position is changed.

Examine this second family of catenaries for the case,where the suspended length is extremely short. Combin-ing the variation of the endpoint with variation of length

A B

C

D

S

A B

C

D

S

A B

(a)

(b)

FIGURE 4.2. Varying the lengths of the chain generates afamily of catenaries of varying curvatures.

FIGURE 4.4.Release catenary AB to points C,D. Every arcof a catenary, is itself a catenary!


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24

whole. In consequence of this, secondly, when we lat different parts of a given catenary, we are in a se

looking at different local expressions on the same gloentity. Although various, small portions of the canary have different curvatures in the sense of visgeometry, in a deeper sense they all share a commhigher curvature, characteristic of the catenarywhich they are parts. Finally, there must be a shigher mode of curvature, which defines the commcharacteristic of the entire family of catenaries. Tlatter entity would be congruent with Gausss concof a modular function for the species of catenaries, special case of his hypergeometric function; the lasubsuming the catenaries together with the analog

crucial features of the Keplerian planetary orbits.the Earth-bound case of elementary catenaries, distinction among different catenaries is, to a vhigh degree of approximation, merely one of self-silar scaling. That is not even approximately the cfor Keplerian orbits.)

In a 1691 paper on the catenary problem, Leibnotes that Galileo had made the error of identifyingcatenary with a parabola. Galileos error, and the discrancy between the two curves, was demonstratedJoachim Jungius (1585-1657) through careful, diexperiments. However, Jungius did not identify the

law underlying the catenary. Leibniz stressed, that catenary cannot be understood in terms of the geomwe associate with Euclid, or, later, Descartes, but isceptible to a higher form of geometrical analysis, whprinciples are embodied in the so-called infinitesicalculus. The latter, in turn, is Leibnizs answer to challenge, which Kepler threw out to the worlds geoters in hisNew Astronomy (Astronomia Nova) of 1609.

C

D

S

B

B

B

S

S

(families one and two) gives us the manifold of all ele-mentary catenaries.

Consider, next, the following remarkable proposition:Every arc of a catenary, is itself a catenary! To wit: On acatenary with fixed suspension-points A,B, examine thearc S bounded by any two points C andD on the curve.(Figure 4.4a) Drive nails through the chain at C andD

into the wall or board behind it. Note that the form of thechain remains unchanged. If we then remove the parts ofthe chain on either side of the arc, or simply release thechain from its original supportsA andB, then the portionof the chain between C and D will be suspended fromthose points as a catenary, while still retaining the origi-nal form of the arc S. (Figure 4.4b)

Consider another remarkable proposition: The entireform of a catenary (up to its suspension-points), is implicitly determined by any of its arcs, however small. Or, to put itanother way: If any arc of one catenary, however small, iscongruent in size and shape to an arc on another cate-

nary, then the two catenaries are superimposable overtheir entire lengths. (Only the endpoints might differ, aswhen we replacedA,B by C,D to obtain a subcatenary ofan originally longer catenary.) To get some insight intothe validity of this proposition, try to beat it by anexperiment, as follows.

Fix one of the endpoints of the arc in question, say C,by a nail, and mark the position of the other endpoint,D,on the wall or board behind the chain. (Figure 4.5) Nowtaking the end of the chain on Ds side, say B, in yourhand (i.e., the right-hand endpoint, ifD is to the right ofC, or vice versa), try to move that endpoint in such a way,

that the corresponding catenary, whose other suspension-point is now C, always passes through the position D asverified by the mark on the adjacent wall or board.Holding to that constraint, we generate a family of cate-naries having the two common points C andD. In doingso, observe that the shape of the arc between C and Dcontinually changes, as the position of the movable end-pointB is changed. This change in shape correlates withthe observation, that the tension exerted by the chain atits endpoints, changes according to their relative posi-tions; according to the higher or lower level of tension,the arc between C and D will be less or more curved.

Only a single, unique position ofB (namely, the originalone) produces exactly the same tension and same curva-ture, as the original arc CD. Our attempt to beat thestated proposition, fails.

While admittedly deserving more careful examina-tion, these considerations suggest three things: Firstly,that all the catenary arcs, which are parts of one andthe same catenary, share a common internal character-istic, which in turn determines the larger catenary as a

FIGURE 4.5. Only one unique position of B produces theexact tension and curvature of catenary CD. Different partsof a given catenary are local expressions of the whole, sharina common internal characteristic.


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Non-linear curvature, exemplified by our explo-

ration of catenaries, stands in the forefront ofJohannes Keplers revolutionary work NewAstronomy. There Kepler bursts through the limitationsof the Copernican heliocentric model, where the plane-tary orbits were assumeda priori to be circular.

The central paradox left by Aristarchus and Coperni-cus was this: Assume the motions of the planets as seenfrom the Earthincluding the bizarre phenomena of ret-rograde motionare due to the fact that the Earth is notstationary, but is itself moving in some orbit around thesun. These apparent motions result from combinations ofthe unknown true motion of the Earth and the unknown

true motion of the heavenly bodies. How can we deter-mine the one, without first knowing the other?

In the New Astronomy, Kepler recounts the excitingstory, of how he was able to solve this paradox by aprocess of nested triangulations, using the orbits ofMars and the Earth. Having finally determined the pre-cise motions ofboth, a new set of anomalies arose, leadingKepler to his astonishing discovery of the elliptical orbitsand the area law for non-uniform motion. Keplersbreakthrough is key to Gausss whole approach to theCeres problem, one hundred fifty years later. It is there-fore fitting that we examine certain of Keplers key steps

in this and the following chapter.As to mere shape, in fact, the orbits of the Earth, Mars,

and most of the other planets (with the exception of Mer-cury and Pluto) are very nearly perfect circles, deviatingfrom a perfect circular form only by a few parts in athousand. The centers of these near-circles, on the otherhand, do not coincide with the sun! Consequently, thereis a constant variation in the distance between the planetand the sun in the course of an orbit, ranging between theextreme values attained at the perihelion (shortest dis-tance) and theaphelion (farthest distance).

As Kepler noted, the perihelion and aphelion are at

the same time the chief singularities of change in theplanets rate of motion along the orbit: the maximum ofvelocity occurs at the perihelion, and the minimum at theaphelion.

In an attempt to account for this fact, while trying tosalvage the hypothesis of simple circular motion as ele-mentary, Ptolemy had devised his theory of the equant.According to that theory, the Earth is no longer the exactcenter of the motion, but rather another point B. (Figure

5.1) The planet is driven around its circular orbit

(called an eccentric because of the displacement of itscenter from the position of the Earth) in such a way, thatitsangular motion is uniform with respect to a third point(the equant), located on the line of apsides opposite theEarth from the center of the eccentric circle.* In otherwords, the planet moves as if it were swept along theorbit by a gigantic arm, pivoted at the equant and turningaround it at a constant rate.

On the basis of his precise data for Earth and Mars,Kepler was able to demolish Ptolemys equant once andfor all. This immediately raised the question: If simplerotational action is excluded as the underlying basis for

planetary motion, then what new principle of actionshould replace it?

Step-by-step, already beginning in theMysterium Cos- mographicum (Cosmographic Mystery), Kepler developedhis electromagnetic conception of the solar system,referring directly to the work of the English scientistWilliam Gilbert, and implicitly to the investigations ofLeonardo da Vinci and others on light, as well as Nico-laus of Cusa. Kepler identifies the sun as the original

CHAPTER 5

Kepler Calls for a New Geometry

__________

* Readers should remember that in Ptolemys astronomical model, thesun and planets are supposed to orbit about the Earth.

25

equantEarth

planet or sun

B

FIGURE 5.1. To account for the differing rates of motion ofthe planet, Ptolemys description placed the Earth at aneccentric (off-center) location, with the planets uniformangular motion centered at a third, equant point.


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source and organizing center of the whole system,which is run on the basis of a harmonically ordered,but otherwiseconstantly changing activity of the sun vis--vis the planets. Keplers conception of that activity, hasnothing to do with the axiomatic assumption of smooth,featureless, linear forms of push-pull displacement inempty space, promoted by Sarpi and Galileo, and revivedonce more in Newtons solar theory, in which the sun is

degraded to a mere attracting center.On the contrary! According to Kepler, the solar activi-ty generates a harmonically ordered, everywhere-densearray ofevents of change, whoseongoing, cumulative resultis reflected inamong other thingsthe visible motionof the planets in their orbits.

The need to elaborate a new species of mathematics,able to account for the integration of dense singularities,emerges ever more urgently in the course of the New

Astronomy, as Kepler investigates the revolutionary impli-cations of his own observation, thatthe rate of motion of a

planet in its orbit is governed by its distance from the sun.

This relationship emerged most clearly, in comparing themotions at the perihelion and aphelion. The ratio of thecorresponding velocities was found to be precisely equalto the inverse ratio of the two extreme radial distances.For good reasons, Kepler chose to express this, not interms ofvelocities, but rather in terms of the timerequired for the planet to traverse a given section of itsorbit.*

Keplers Struggle with Paradox

Let us join Kepler in his train of thought. While still

operating with the approximation of a planetary orbit asan eccentric circle, Kepler formulates this relationshipin a preliminary way as follows: It has been demonstrat-ed,

that the elapsed times of a planet on equal parts of theeccentric circle (or equal distances in the ethereal air) arein the same ratio as the distances of those spaces from the

point whence the eccentricity is reckoned [i.e., thesunJT]; or more simply, to the extent that a planet isfarther from the point which is taken as the center of the

world, it is less strongly urged to move about that point.

Since the distances are constantly changing, the exis-tence of such a relationship immediately raises the ques-tion: How does the temporally extended motionas, forexample, the periodic time corresponding to an entirerevolution of the planetrelate to the magnitudes ofthose constantly varying urges or impulses?

A bit later, Kepler picks up the problem again. To low Keplers discussion, draw the following diagr(Figure 5.2) Construct a circle and its diameter and lthe center B. To the right ofB mark another poinThe circumference of the circle represents the planeorbit, and point A represents the position of the sKepler writes:

Since, therefore, the times of a planet over equal parts o

the eccentric, are to one another, as the radial distanceof those parts [from the sunJT], and since the individ

ual points of the entire . . . eccentric are all at differendistances, it was no easy task I set myself, when I soughto find how one might obtain the sums of the individuaradial distances. For, unless we can find the sum of all o

them (and they are infinite in number) we cannot sahow much time has elapsed for any one of them! Thusthe whole equation will not be known. For, the whol

sum of the radial distances is, to the whole periodic time, aany partial sum of the distances is to its corresponding time[Emphasis added]

I consequently began by dividing the eccentric into 36parts, as if these were least particles, and supposed thawithin one such part the distance does not change . . . .

However, since this procedure is mechanical antedious, and since it is impossible to compute the wholequation, given the value for one individual degree [othe eccentricJT] without the others, I looked aroun

for other means. Considering, that the points of theccentric are infinite in number, and their radial lines arinfinite in number, it struck me, that all the radial line

are contained within the area of the eccentric. I remembered that Archimedes, in seeking the ratio of the cir

26

__________

* Cf. Fermats later work on least-time in the propagation of light.

AB

aphelion perihelion

FIGURE 5.2.Keplers original hypothesis: The planetaryorbits are circles whose centers are somewhat eccentric withregard to the sun. Kepler observed that the planet movesfastest at the perihelion, slowest at the aphelion, in appareninverse proportion to the radial distances.


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cumference to the diameter, once divided a circle thusinto an infinity of trianglesthis being the hidden forceof his reductio ad absurdum. Accordingly, instead of

dividing the circumference, as before, I now cut the areaof the eccentric into 360 parts, by lines drawn from thepoint whence the eccentricity is reckoned [A, the position

of the sunJT] . . ..

This brief passage marks a crucial breakthrough intheNew Astronomy. To see more clearly what Kepler hasdone, on the same diagram as above, mark two positions

P1,P2 of the planet on the orbit, and draw the radial linesfrom the sun to those positionsi.e.,AP

1andAP

2. (Fig-

ure 5.3) Kepler has dropped the idea of using the lengthof the arc between P

1andP

2as the appropriate measure

of the action generating the orbital motion, and turnedinstead to thearea of the curvilinear triangle bounded by

AP1,AP

2and the orbital arc fromP

1toP

2.

We shall later refer to such areas as orbital sectors.Kepler describes that area as the sum of the infinitenumber of radial linesAQ, of varying lengths, obtainedas Q passes through all the positions of the planet fromP

1toP

2! Does he mean this literally? Or, is he not express-

ing, in metaphorical terms, the coherence between themacroscopic process, fromP

1toP

2, and the peculiar cur-

vature, which governs events within any arbitrarilysmall interval of that process?

The result, in any case, is a geometrical principle,which Kepler subsequently demonstrated to be empiri-cally valid for the motion of all known planets in theirorbits: The time, which a planet takes in passing from any

position P1 to another position P

2in its orbit, is proportional

to the area of the sector bounded by the radial lines AP1,AP

2,

and the orbital trajectory P1P

2, or, in other words, the area

swept out by the radial line AP. This is Keplers famousSecond Law, otherwise known as the Area Law. Allthat is needed in addition, to arrive at an extremely pre-cise construction of planetary motion, is to replace theeccentric circle approximation, by a true ellipse, asKepler himself does in the later sections of the New

Astronomy. We shall attend to that in the next chapter.

Time Produced by Orbital Action?Are you not struck by something paradoxical in Keplersformulation? Does he not express himself as if nearly tosay, that time is produced by the orbital action? Or, doesthis only seem paradoxical to us (but not to Kepler!),because we have been indoctrinated by the kinematicconceptions of Sarpi, Descartes, and Newton?

There is another paradox implicit here, which Keplerhimself emphasized. Sticking for a moment to the eccen-tric-circle approximation for the orbit, Kepler found a verysimple way to calculate the areas of the sectors. In our ear-

lier drawing, choose P1 to be the intersection of the cir-cumference and the line of apsides passing through B and

A. (Figure 5.4)P1

now represents the position of the plan-et at the point of perihelion. TakeP

2to be any point on the

circumference in the upper half of the circle. IfA andBwere at the same place (i.e., if the sun were at the geometri-cal center of the orbit), then the sectoral area betweenAP

1andAP

2would simply be proportional to the angle formed

atA between those two lines. Otherwise, we can transform

27

AB

P2

P1

Q

AB

P2

P1

FIGURE 5.3.Assuming the momentary orbital velocitiesare inversely proportional to the radial distances, Keplertries to add up the radii to determine how much time theplanet needs to go from one point of the orbit to another.

FIGURE 5.4.Keplers method for calculating the areaswept out by the radial line from the sun to a planet onthe assumption that the orbit is an eccentric circle, i.e., acircle whose center B is displaced from the position of thesun A.


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the sector in question into a simple, center-based circular

sector, byadding to it the triangular areaABP2.Indeed, as can be seen in Figure 5.5, the sum of the

two areas is the circular sector between BP1

and BP2.

The area of the circular sector, on the other hand, is pro-portional to the angle formed by the radial lines BP

1,

BP2

at the circles center B, as well as to the circular arcfrom P

1to P

2. Turning this around, we can express the

sector AP1P

2, which, according to Kepler, tells us the

time elapsed between the two positions, as the result ofsubtracting the triangle ABP

2from the sector BP

1P

2. In

other words: The time T to go fromP1

toP2, is propor-

tional to the area AP1P

2, which in turn is equal to the

area of the circular sector between BP1 and BP2 minusthe area of triangleABP

2. Of these two areas, the first is

proportional to the angleP1BP

2at the circles center and

to the circular arc P1P

2; while the second is equal to the

product of the base of triangle ABP2, namely the length

AB, times its height. The height is the length of the per-pendicular line P

2Ndrawn from the orbital position P

2to the line of apsides, which (up to a factor of the radius)is just the sine of the angle P

1BP

2. In this wayleaving

aside, for the moment, a certain modification requiredby the non-circularity of the orbitKepler was able tocalculate the elapsed times between any two positions in

an orbit.These simple relationships, which are much easier to

express in geometrical drawings than in words, are crucialto the whole development up to Gauss. They involve thefollowing peculiarity, highlighted by Kepler: The elapsedtime is shown to be a combined function of the indicated

angle orcircular arc on the one side, and the length of theperpendicular straight line drawn from P

2to the line of

apsides, on the other. Now, as Kepler notes, in implicit ref-

28

erence to Nicolaus of Cusa, those two magnitudes are erogeneous; one is essentially a curved magnitude,other a straight, linear one. (That is, they are incommsurable; in fact, as Cusa discovered, the curve is transdental to the straight line.) That heterogeneity seemblock our way, when we try to invert Keplers solutand to determine the position of a planet after any gielapsed time (i.e., rather than determining the time a

relates to any position). In fact, this is one of the problewhich Gauss addressed with his higher transcendenincluding the hypergeometric function.

Let us end this discussion with Keplers own chlenge to the geometers. For the present purposesdering some further dimensionalities of the probuntil Chapter 6you can read Keplers technical tein the following quote in the following way. WKepler calls the mean anomaly, is essentially elapsed time; the term, eccentric anomaly, refers toangle subtended by the planetary positionsP

1,P

2as s

from the center B of the circlei.e., the angle P1B

Here is Kepler:But given the mean anomaly, there is no geometricamethod of proceeding to the eccentric anomaly. For, th

mean anomaly is composed of two areas, a sector and triangle. And while the former is measured by the arc othe eccentric, the latter is measured by the sine of tha

arc. . . . And the ratios between the arcs and their sineare infinite in number [i.e., they are incommensurable afunctional speciesed.]. So, when we begin with th

sum of the two, we cannot say how great the arc is, anhow great its sine, corresponding to the sum, unless wwere previously to investigate the area resulting from

given arc; that is, unless you were to have constructedtables and to have worked from them subsequently.

That is my opinion. And insofar as it is seen to lack

geometrical beauty, I exhort the geometers to solve mthis problem:

Given the area of a part of a semicircle and a point on

the diameter, to find the arc and the angle at that pointthe sides of which angle, and which arc, encloses the given area. Or, to cut the area of a semicircle in a given ratiofrom any given point on the diameter.

It is enough for me to believe that I could not solvthis, a priori, owing to the heterogeneity of the arc andsine. Anyone who shows me my error and points th

way will be for me the great Apollonius.*

A NB

P1

P2

__________

* Apollonius of Perga (c. 262-200 B.C.), Greek geometer, authoOn Conic Sections, the definitive Classical treatise. Drawn byreputation of the astronomer Aristarchus of Samos, he livedworked at Alexandria, the great center of learning of the Helletic world, where he

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