aft - Kerry K. Kuehn 38 CHAPTER 6. THE WORLD OF PTOLEMY teenth and seventeenth centuries. Indeed under-standing the Almagest is essential in understand-ingtheastronomicalworksofCopernicus,

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Chapter 6The world of Ptolemy

only mathematics can provide sure andunshakeable knowledge to its devotees

—Claudius Ptolemy

Introduction

Claudius Ptolemy (ca. 100-165 A.D.) is themost influential astronomer of the ancient world.He lived and worked in the city of Alexandria,carrying out astronomical observations and writ-ing about diverse topics such as optics, harmon-ics, geography, mathematics, philosophy and as-tronomy.Ptolemy’s most famous work, the Almagest,

served as a textbook and a practical referencefor calculating astronomical events for the next1500 years. The name of the book, literally “thegreatest”, derives from the title given it by Arabsscholars and translators from the original greektext during the ninth century.The Almagest itself consists of thirteen books.

In the first book Ptolemy begins with his the-ory of the earth: its size, shape and place in theuniverse. The remainder of Books I and II pro-vide careful demonstrations of how to performnumerical calculations using plane and spheri-cal trigonometry. For instance, he demonstrateshow to determine the right ascension (angular

distance from the vernal equinox) of any pointon the ecliptic (I,16), how to divide the surfaceof the earth into geographical zones using par-allels (II, 6), and how to calculate the length ofshadows, the position and time of sunrise, andthe angle between the ecliptic and the horizon(II, 5, 2, 7 and 11, respectively). These are someof the methods that will prove useful when heis developing his theories of motion of the sun(III), the moon (IV), the fixed stars (VII-VIII),and the planets (IX-XIII).

The reading selection included below is com-prised of the first nine chapters of Book I of theAlmagest. Ptolemy begins in chapter 1 by ex-plaining the di↵erence between theoretical andpractical philosophy. Where does Ptolemy sit-uate the study of astronomy? Next, after ex-plaining the outline of the Almagest in chapter2, Ptolemy o↵ers a summary of his theory ofthe heavens and the earth. Thus, chapters 3-8provide a framework for understanding the Al-magest as a whole. He argues against those whowould claim that the earth itself is in motion.This, after all, was an ancient opinion espousedby the Pythagoreans but later rejected by Aris-totle in his book On the Heavens. The astutereader of the Almagest may recognize many ofAristotle’s arguments and proofs in these chap-ters. Can you identify them?

The Ptolemaic worldview came under renewedand vigorous attack by thinkers during the six-

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38 CHAPTER 6. THE WORLD OF PTOLEMY

teenth and seventeenth centuries. Indeed under-standing the Almagest is essential in understand-ing the astronomical works of Copernicus, Keplerand Galileo, who wrote in reaction to the geo-centric worldview which Ptolemy so clearly andmeticulously laid out in the Almagest.

Reading

Ptolemy, C., Ptolemy’s Almagest, Princeton Uni-versity Press, Princeton, NJ, 1998. Book I,Chapters 1-9.

Chapter 1

Preface

The true philosophers, Syrus, were, I think,quite right to distinguish the theoretical part ofphilosophy from the practical. For even if prac-tical philosophy, before it is practical, turns outto be theoretical, nevertheless one can see thatthere is a great di↵erence between the two: in thefirst place, it is possible for many people to pos-sess some of the moral virtues even without beingtaught, whereas it is impossible to achieve theo-retical understanding of the universe without in-struction; furthermore, one derives most benefitin the first case [practical philosophy] from con-tinuous practice in actual a↵airs, but in the other[theoretical philosophy] from making progress inthe theory. Hence we thought it fitting to guideour actions (under the impulse of our actual ideas[of what is to be done]) in such a way as neverto forget, even in ordinary a↵airs, to strive fora noble and disciplined disposition, but to de-vote most of our time to intellectual matters, inorder to teach theories, which are so many andbeautiful, and especially those to which the epi-thet ‘mathematical’ is particularly applied. ForAristotle divides theoretical philosophy too, veryfittingly, into three primary categories, physics,mathematics and theology. For everything thatexists is composed of matter, form and motion;

none of these [three] can be observed in its sub-stratum by itself, without the others: they canonly be imagined. Now the first cause of the firstmotion of the universe, if one considers it simply,can be thought of as an invisible and motion-less deity; the division[of theoretical philosophy]concerned with investigating this [can be called]‘theology’, since this kind of activity, somewhereup in the highest reaches of the universe, canonly be imagined, and is completely separatedfrom perceptible reality. The division [of the-oretical philosophy] which investigates materialand ever-moving nature, and which concerns it-self with ‘white’, ‘hot’, ‘sweet’, ‘soft’ and suchlikequalities one may call ‘physics’; such an order ofbeing is situated (for the most part) amongst cor-ruptible bodies and below the lunar sphere. Thatdivision [of theoretical philosophy] which deter-mines the nature involved in forms and motionfrom place to place, and which serves to inves-tigate shape, number, size, and place, time andsuchlike, one may define as ‘mathematics’. Itssubject-matter falls as it were in the middle be-tween the other two, since, firstly, it can be con-ceived of both with and without the aid of thesenses, and, secondly, it is an attribute of all ex-isting things without exception, both mortal andimmortal: for those things which are perpetuallychanging in their inseparable form, it changeswith them, while for eternal things which havean aethereal nature, it keeps their unchangingform unchanged.

From all this we concluded: that the first twodivisions of theoretical philosophy should ratherbe called guesswork than knowledge, theologybecause of its completely invisible and ungras-pable nature, physics because of the unstableand unclear nature of matter; hence there is nohope that philosophers will ever be agreed aboutthem; and that only mathematics can providesure and unshakeable knowledge to its devotees,provided one approaches it rigorously. For itskind of proof proceeds by indisputable methods,namely arithmetic and geometry. Hence we weredrawn to the investigation of that part of theo-retical philosophy, as far as we were able to the

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whole of it, but especially to the theory concern-ing divine and heavenly things. For that alone isdevoted to the investigation of the eternally un-changing. For that reason it too can be eternaland unchanging (which is a proper attribute ofknowledge) in its own domain, which is neitherunclear nor disorderly. Furthermore it can workin the domains of the other [two divisions of theo-retical philosophy] no less than they do. For thisis the best science to help theology along its way,since it is the only one which can make a goodguess at [the nature of] that activity which isunmoved and separated; [it can do this because]it is familiar with the attributes of those beingswhich are on the one hand perceptible, movingand being moved, but on the other hand eternaland unchanging, [I mean the attributes] havingto do with motions and the arrangements of mo-tions. As for physics, mathematics can make asignificant contribution. For almost every pe-culiar attribute of material nature becomes ap-parent from the peculiarities of its motion fromplace to place. [Thus one can distinguish] thecorruptible from the incorruptible by [whetherit undergoes] motion in a straight line or in acircle, and heavy from light, and passive fromactive, by [whether it moves] towards the centreor away from the centre. With regard to virtu-ous conduct in practical actions and character,this science above all things, could make mensee clearly; from the constancy, order, symmetryand calm which are associated with the divine, itmakes its followers lovers of this divine beauty,accustoming them and reforming their naturesas it were, to a similar spiritual state.

It is this love of the contemplation of the eter-nal and unchanging which we constantly striveto increase, by studying those parts of these sci-ences which have already been mastered by thosewho approached them in a genuine spirit of en-quiry, and by ourselves attempting to contributeas much advancement as has been made possi-ble by the additional time between those peopleand ourselves. We shall try to note down every-thing which we think we have discovered up tothe present time; we shall do this as concisely as

possible and in a manner which can be followedby those who have already made some progressin the field. For the sake of completeness in ourtreatment we shall set out everything useful forthe theory of the heavens in the proper order,but to avoid undue length we shall merely re-count what has been adequately established bythe ancients. However, those topics which havenot been dealt with [by our predecessors] at all,or not as usefully as they might have been, willbe discussed at length, to the best of our ability.

Chapter 2

On the order of the theorems

In the treatise which we propose, then, thefirst order of business is to grasp the relation-ship of the earth taken as a whole to the heavenstaken as a whole. In the treatment of the individ-ual aspects which follows, we must first discussthe position of the ecliptic and the regions of ourpart of the inhabited world and also the featuresdi↵erentiating each from the others due to the[varying] latitude at each horizon taken in or-der. For if the theory of these matters is treatedfirst it will make examination of the rest eas-ier. Secondly we have to go through the motionof the sun and of the moon, and the phenom-ena accompanying these [motions]; for it wouldbe impossible to examine the theory of the starsthoroughly without first having a grasp of thesematters. Our final task in this way of approachis the theory of the stars. Here too it would beappropriate to deal first with the sphere of theso-called ‘fixed stars’, and follow that by treatingthe five ‘planets’, as they are called. We shall tryto provide proofs in all of these topics by usingas starting-points and foundations, as it were, forour search the obvious phenomena, and those ob-servations made by the ancients and in our owntimes which are reliable. We shall attach the sub-sequent structure of ideas to this [foundation] bymeans of proofs using geometrical methods.

The general preliminary discussion covers thefollowing topics: the heaven is spherical in shape,and moves as a sphere; the earth too is sensibly

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spherical in shape, when taken as a whole; inposition it lies in the middle of the heavens verymuch like its centre; in size and distance it hasthe ratio of a point to the sphere of the fixedstars; and it has no motion from place to place.We shall briefly discuss each of these points forthe sake of reminder.

Chapter 3

That the heavens move like a sphere

It is plausible to suppose that the ancients gottheir first notions on these topics from the follow-ing kind of observations. They saw that the sun,moon and other stars were carried from east towest along circles which were always parallel toeach other, that they began to rise up from be-low the earth itself, as it were, gradually got uphigh, then kept on going round in similar fash-ion and getting lower, until, falling to earth, so tospeak, they vanished completely, then, after re-maining invisible for some time, again rose afreshand set; and [they saw] that the periods of these[motions], and also the places of rising and set-ting, were, on the whole, fixed and the same.What chiefly led them to the concept of a

sphere was the revolution of the ever-visiblestars, which was observed to be circular, and al-ways taking place about one centre, the same[for all]. For by necessity that point became[for them] the pole of the heavenly sphere: thosestars which were closer to it revolved on smallercircles, those that were farther away describedcircles ever greater in proportion to their dis-tance, until one reaches the distance of the starswhich become invisible. In the case of these, too,they saw that those near the ever-visible starsremained invisible for a short time, while thosefarther away remained invisible for a long time,again in proportion [to their distance]. The re-sult was that in the beginning they got to theaforementioned notion solely from such consid-erations; but from then on, in their subsequentinvestigation, they found that everything else ac-corded with it, since absolutely all phenomena

are in contradiction to the alternative notionswhich have been propounded.

For if one were to suppose that the stars’ mo-tion takes place in a straight line towards infin-ity, as some people have thought, what devicecould one conceive of which would cause each ofthem to appear to begin their motion from thesame starting-point every day? How could thestars turn back if their motion is towards infin-ity? Or, if they did turn back, how could thisnot be obvious? [On such a hypothesis], theymust gradually diminish in size until they dis-appear, whereas, on the contrary, they are seento be greater at the very moment of their disap-pearance, at which time they are gradually ob-structed and cut o↵, as it were, by the earth’ssurface.

But to suppose that they are kindled as theyrise out of the earth and are extinguished againas they fall to earth is a completely absurd hy-pothesis. For even if we were to concede thatthe strict order in their size and number, theirintervals, positions and periods could be restoredby such a random and chance process; that onewhole area of the earth has a kindling nature,and another an extinguishing one, or rather thatthe same part [of the earth] kindles for one set ofobservers and extinguishes for another set; andthat the same stars are already kindled or ex-tinguished for some observers while they are notyet for others: even if, I say, we were to concedeall these ridiculous consequences, what could wesay about the ever-visible stars, which neitherrise nor set? Those stars which are kindled andextinguished ought to rise and set for observerseverywhere, while those which are not kindledand extinguished ought always to be visible forobservers everywhere. What cause could we as-sign for the fact that this is not so? We willsurely not say that stars which are kindled andextinguished for some observers never undergothis process for other observers. Yet it is utterlyobvious that the same stars rise and set in certainregions [of the earth] and do neither at others.

To sum up, if one assumes any motion what-ever, except spherical, for the heavenly bodies,

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it necessarily follows that their distances, mea-sured from the earth upwards, must vary, wher-ever and however one supposes the earth itselfto be situated. Hence the sizes and mutual dis-tances of the stars must appear to vary for thesame observers during the course of each revolu-tion, since at one time they must be at a greaterdistance, at another at a lesser. Yet we see thatno such variation occurs. For the apparent in-crease in their sizes at the horizons is caused, notby a decrease in their distances, but by the ex-halations of moisture surrounding the earth be-ing interposed between the place from which weobserve and the heavenly bodies, just as objectsplaced in water appear bigger than they are, andthe lower they sink, the bigger they appear.

The following considerations also lead us tothe concept of the sphericity of the heavens. Noother hypothesis but this can explain how sun-dial constructions produce correct results; fur-thermore, the motion of the heavenly bodies isthe most unhampered and free of all motions,and freest motion belongs among plane figuresto the circle and among solid shapes to thesphere; similarly, since of di↵erent shapes havingan equal boundary those with more angles aregreater [in area or volume], the circle is greaterthan [all other] surfaces, and the sphere greaterthan [all other] solids; [likewise] the heavens aregreater than all other bodies.

Furthermore, one can reach this kind of notionfrom certain physical considerations. E.g., theaether is, of all bodies, the one with constituentparts which are finest and most like each other;now bodies with parts like each other have sur-faces with parts like each other; but the only sur-faces with parts like each other are the circular,among planes, and the spherical, among three-dimensional surfaces. And since the aether isnot plane, but three-dimensional, it follows thatit is spherical in shape. Similarly, nature formedall earthly and corruptible bodies out of shapeswhich are round but of unlike parts, but allaethereal and divine bodies out of shapes whichare of like parts and spherical. For if they wereflat or shaped like a discus they would not al-

ways display a circular shape to all those observ-ing them simultaneously from di↵erent places onearth. For this reason it is plausible that theaether surrounding them, too, being of the samenature, is spherical, and because of the likenessof its parts moves in a circular and uniform fash-ion.

Chapter 4

That the earth too, taken as a whole, issensibly spherical

That the earth, too, taken as a whole, is sen-sibly spherical can best be grasped from the fol-lowing considerations. We can see, again, thatthe sun, moon and other stars do not rise andset simultaneously for everyone on earth, butdo so earlier for those more towards the east,later for those towards the west. For we findthat the phenomena at eclipses, especially lunareclipses, which take place at the same time [forall observers], are nevertheless not recorded asoccurring at the same hour (that is at an equaldistance from noon) by all observers. Rather,the hour recorded by the more easterly observersis always later than that recorded by the morewesterly. We find that the di↵erences in the hourare proportional to the distances between theplaces [of observation]. Hence one can reason-ably conclude that the earth’s surface is spheri-cal, because its evenly curving surface (for so itis when considered as a whole) cuts o↵ [the heav-enly bodies] for each set of observers in turn ina regular fashion.

If the earth’s shape were any other, this wouldnot happen, as one can see from the following ar-guments. If it were concave, the stars would beseen rising first by those more towards the west;if it were plane, they would rise and set simul-taneously for everyone on earth; if it were tri-angular or square or any other polygonal shape,by a similar argument, they would rise and setsimultaneously for all those living on the sameplane surface. Yet it is apparent that nothinglike this takes place. Nor could it be cylindrical,

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with the curved surface in the east-west direc-tion, and the flat sides towards the poles of theuniverse, which some might suppose more plau-sible. This is clear from the following: for thoseliving on the curved surface none of the starswould be ever-visible, but either all stars wouldrise and set for all observers, or the same stars,for an equal [celestial] distance from each of thepoles, would always be invisible for all observers.In fact, the further we travel toward the north,the more of the southern stars disappear and themore of the northern stars appear. Hence it isclear that here too the curvature of the earth cutso↵ [the heavenly bodies] in a regular fashion in anorth-south direction, and proves the sphericity[of the earth] in all directions.There is the further consideration that if we

sail towards mountains or elevated places fromand to any direction whatever, they are observedto increase gradually in size as if rising up fromthe sea itself in which they had previously beensubmerged: this is due to the curvature of thesurface of the water.

Chapter 5

That the earth is in the middle of theheavens

Once one has grasped this, if one next con-siders the position of the earth, one will findthat the phenomena associated with it could takeplace only if we assume that it is in the middleof the heavens, like the centre of a sphere. For ifthis were not the case, the earth would have tobe either

(a) not on the axis [of the universe] but equidis-tant from both poles, or

(b) on the axis but removed towards one of thepoles, or

(c) neither on the axis nor equidistant from bothpoles.

Against the first of these three positions mili-tate the following arguments. If we imagined [the

earth] removed towards the zenith or the nadirof some observer, then, if he were at sphaerarecta, he would never experience equinox, sincethe horizon would always divide the heavens intotwo unequal parts, one above and one below theearth; if he were at sphaera obliqua, either, again,equinox would never occur at all, or, [if it didoccur,] it would not be at a position halfway be-tween summer and winter solstices, since theseintervals would necessarily be unequal, becausethe equator, which is the greatest of all paral-lel circles drawn about the poles of the [daily]motion, would no longer be bisected by the hori-zon; instead [the horizon would bisect] one ofthe circles parallel to the equator, either to thenorth or to the south of it. Yet absolutely ev-eryone agrees that these intervals are equal ev-erywhere on earth, since [everywhere] the incre-ment of the longest day over the equinoctial dayat the summer solstice is equal to the decrementof the shortest day from the equinoctial day atthe winter solstice. But if, on the other hand, weimagined the displacement to be towards the eastor west of some observer, he would find that thesizes and distances of the stars would not remainconstant and unchanged at eastern and westernhorizons, and that the time-interval from risingto culmination would not be equal to the intervalfrom culmination to setting. This is obviouslycompletely in disaccord with the phenomena.

Against the second position, in which theearth is imagined to lie on the axis removed to-wards one of the poles, one can make the fol-lowing objections. If this were so, the plane ofthe horizon would divide the heavens into a partabove the earth and a part below the earth whichare unequal and always di↵erent for di↵erent lat-itudes, whether one considers the relationship ofthe same part at two di↵erent latitudes or thetwo parts at the same latitude. Only at sphaerarecta could the horizon bisect the sphere; at asphaera obliqua situation such that the nearerpole were the ever-visible one, the horizon wouldalways make the part above the earth lesser andthe part below the earth greater; hence anotherphenomenon would be that the great circle of

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the ecliptic would be divided into unequal partsby the plane of the horizon. Yet it is apparentthat this is by no means so. Instead, six zodia-cal signs are visible above the earth at all timesand places, while the remaining six are invisible;then again [at a later time] the latter are visiblein their entirety above the earth, while at thesame time the others are not visible. Hence it isobvious that the horizon bisects the zodiac, sincethe same semi-circles are cut o↵ by it, so as toappear at one time completely above the earth,and at another [completely] below it.And in general, if the earth were not situated

exactly below the [celestial] equator, but wereremoved towards the north or south in the di-rection of one of the poles, the result would bethat at the equinoxes the shadow of the gnomonat sunrise would no longer form a straight linewith its shadow at sunset in a plane parallel tothe horizon, not even sensibly. Yet this is a phe-nomenon which is plainly observed everywhere.It is immediately clear that the third position

enumerated is likewise impossible, since the sortsof objection which we made to the first [two] willboth arise in that case.To sum up, if the earth did not lie in the mid-

dle [of the universe], the order of things whichwe observe in the increase and decrease of thelength of daylight would be fundamentally up-set. Furthermore, eclipses of the moon wouldnot be restricted to situations where the moonis diametrically opposite the sun (whatever partof the heaven [the luminaries are in]), since theearth would often come between them when theywere not diametrically opposite, but at intervalsof less than a semi-circle.

Chapter 6

That the earth has the ratio of a point tothe heavens

Moreover, the earth has, to the senses, the ra-tio of a point to the distance of the sphere of theso-called fixed stars. A strong indication of thisis the fact that the sizes and distances of thestars, at any given time, appear equal and the

same from all parts of the earth everywhere, asobservations of the same [celestial] objects fromdi↵erent latitudes are found to have not the leastdiscrepancy from each other. One must also con-sider the fact that gnomons set up in any partof the earth whatever, and likewise the centresof armillary spheres, operate like the real centreof the earth; that is, the lines of sight [to heav-enly bodies] and the paths of shadows caused bythem agree as closely with the [mathematical]hypotheses explaining the phenomena as if theyactually passed through the real centre-point ofthe earth.

Another clear indication thaI this is so is thatthe planes drawn through the observer’s lines ofsight at any point [on earth], which we call ‘hori-zons’, always bisect the whole heavenly sphere.This would not happen if the earth were of per-ceptible size in relation to the distance of theheavenly bodies; in that case only the planedrawn through the centre of the earth couldbisect the sphere, while a plane through anypoint on the surface of the earth would alwaysmake the section [of the heavens] below the earthgreater than the section above it.

Chapter 7

That the earth does not have any motionfrom place to place, either

One can show by the same arguments as thepreceding that the earth cannot have any mo-tion in the aforementioned directions, or indeedever move at all from its position at the centre.For the same phenomena would result as wouldif it had any position other than the central one.Hence I think it is idle to seek for causes forthe motion of objects towards the centre, once ithas been so clearly established from the actualphenomena that the earth occupies the middleplace in the universe, and that all heavy objectsare carried towards the earth. The following factalone would most readily lead one to this notion[that all objects fall towards the centre]. In ab-solutely all parts of the earth, which, as we said,

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has been shown to be spherical and in the mid-dle of the universe the direction and path of themotion (I mean the proper, [natural] motion) ofall bodies possessing weight is always and every-where at right angles to the rigid plane drawntangent to the point of impact. It is clear fromthis fact that, if [these falling objects] were notarrested by the surface of the earth, they wouldcertainly reach the centre of the earth itself, sincethe straight line to the centre is also always atright angles to the plane tangent to the sphereat the point of intersection [of that radius] andthe tangent.

Those who think it paradoxical that the earth,having such a great weight, is not supported byanything and yet does not move, seem to me tobe making the mistake of judging on the basis oftheir own experience instead of taking into ac-count the peculiar nature of the universe. Theywould not, I think, consider such a thing strangeonce they realised that this great bulk of theearth, when compared with the whole surround-ing mass [of the universe], has the ratio of a pointto it. For when one looks at it in that way, it willseem quite possible that that which is relativelysmallest should be overpowered and pressed inequally from all directions to a position of equi-librium by that which is the greatest of all and ofuniform nature. For there is no up and down inthe universe with respect to itself, any more thanone could imagine such a thing in a sphere: in-stead the proper and natural motion of the com-pound bodies in it is as follows: light and rarefiedbodies drift outwards towards the circumference,but seem to move in the direction which is ‘up’for each observer, since the overhead direction forall of us, which is also called ‘up’, points towardsthe surrounding surface; heavy and dense bodies,on the other hand, are carried towards the mid-dle and the centre, but seem to fall downwards,because, again, the direction which is for all ustowards our feet, called ‘down’, also points to-wards the centre of the earth. These heavy bod-ies, as one would expect, settle about the centrebecause of their mutual pressure and resistancewhich is equal and uniform from all directions.

Hence, too, one can see that it is plausible thatthe earth, since its total mass is so great com-pared with the bodies which fall towards it, canremain motionless under the impact of these verysmall weights (for they strike it from all sides),and receive, as it were, the objects falling on it.If the earth had a single motion in common withother heavy objects, it is obvious that it wouldbe carried down faster than all of them becauseof its much greater size: living things and indi-vidual heavy objects would be left behind, ridingon the air, and the earth itself would very soonhave fallen completely out of the heavens. Butsuch things are utterly ridiculous merely to thinkof.

But certain people, [propounding] what theyconsider a more persuasive view, agree with theabove, since they have no argument to bringagainst it, but think that there could be no evi-dence to oppose their view if, for instance, theysupposed the heavens to remain motionless, andthe earth to revolve from west to east aboutthe same axis [as the heavens], making approxi-mately one revolution each day; or if they madeboth heaven and earth move by any amountwhatever, provided, as we said, it is about thesame axis, and in such a way as to preservethe overtaking of one by the other. However,they do not realise that, although there is per-haps nothing in the celestial phenomena whichwould count against that hypothesis, at leastfrom simpler considerations, nevertheless fromwhat would occur here on earth and in the air,one can see that such a notion is quite ridicu-lous. Let us concede to them [for the sake ofargument] that such an unnatural thing couldhappen as that the most rare and light of mat-ter should either not move at all or should movein a way no di↵erent from thaI of matter withthe opposite nature (although things in the air,which are less rare [than the heavens] so obvi-ously move with a more rapid motion than anyearthy object); [let us concede that] the dens-est and heaviest objects have a proper motion ofthe quick and uniform kind which they suppose(although, again, as all agree, earthy objects are

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sometimes not readily moved even by an externalforce). Nevertheless, they would have to admitthat the revolving motion of the earth must bethe most violent of all motions associated withit, seeing that it makes one revolution in such ashort time; the result would be that all objectsnot actually standing on the earth would appearto have the same motion, opposite to that of theearth: neither clouds nor other flying or thrownobjects would ever be seen moving towards theeast, since the earth’s motion towards the eastwould always outrun and overtake them, so thatall other objects would seem to move in the di-rection of the west and the rear. But if they saidthat the air is carried around in the same direc-tion and with the same speed as the earth, thecompound objects in the air would none the lessalways seem to be left behind by the motion ofboth [earth and air]; or if those objects too werecarried around, fused, as it were, to the air, thenthey would never appear to have any motion ei-ther in advance or rearwards: they would alwaysappear still, neither wandering about nor chang-ing position, whether they were flying or thrownobjects. Yet we quite plainly see that they doundergo all these kinds of motion, in such a waythat they are not even slowed down or speededup at all by any motion of the earth.

Chapter 8

That there are two di↵erent primarymotions in the heavens

It was necessary to treat the above hypothesesfirst as an introduction to the discussion of par-ticular topics and what follows after. The abovesummary outline of them will su�ce, since theywill be completely confirmed and further provenby the agreement with the phenomena of the the-ories which we shall demonstrate in the follow-ing sections. In addition to these hypotheses itis proper, as a further preliminary, to introducethe following general notion, that there are twodi↵erent primary motions in the heavens. Oneof them is that which carries everything from

east to west: it rotates them with an unchang-ing and uniform motion along circles parallel toeach other, described, as is obvious, about thepoles of this sphere which rotates everything uni-formly. The greatest of these circles is called the‘equator’, because it is the only [such parallelcircle] which is always bisected by the horizon(which is a great circle), and because the revo-lution which the sun makes when located on itproduces equinox everywhere, to the senses. Theother motion is that by which the spheres of thestars perform movements in the opposite senseto the first motion, about another pair of poles,which are di↵erent from those of the first rota-tion. We suppose that this is so because of thefollowing considerations. When we observe forthe space of any given single day, all heavenly ob-jects whatever are seen, as far as the senses candetermine, to rise, culminate and set at placeswhich are analogous and lie on circles parallel tothe equator; this is characteristic of the first mo-tion. But when we observe continuously with-out interruption over an interval of time, it isapparent that while the other stars retain theirmutual distances and (for a long time) the par-ticular characteristics arising from the positionsthey occupy as a result of the first motion, thesun, the moon and the planets have certain spe-cial motions which are indeed complicated anddi↵erent from each other, but are all, to char-acterise their general direction, towards the eastand opposite to [the motion of] those stars whichpreserve their mutual distances and are, as itwere, revolving on one sphere.

Now if this motion of the planets too tookplace along circles parallel to the equator, that is,about the poles which produce the first kind ofrevolution, it would be su�cient to assign a sin-gle kind of revolution to all alike, analogous tothe first. For in that case it would have seemedplausible that the movements which they un-dergo are caused by various retardations, andnot by a motion in the opposite direction. Butas it is, in addition to their movement towardsthe east, they are seen to deviate continuously tothe north and south [of the equator]. Moreover

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the amount of this deviation cannot be explainedas the result of a uniformly-acting force pushingthem to the side: from that point of view it isirregular, but it is regular if considered as the re-sult of [motion on] a circle inclined to the equa-tor. Hence we get the concept of such a circle,which is one and the same for all planets, andparticular to them. It is precisely defined and,so to speak, drawn by the motion of the sun, butit is also travelled by the moon and the planets,which always move in its vicinity, and do notrandomly pass outside a zone on either side ofit which is determined for each body. Now sincethis too is shown to be a great circle, since thesun goes to the north and south of the equatorby an equal amount, and since, as we said, theeastward motion of all of the planets takes placeon one and the same circle, it became necessaryto suppose that this second, di↵erent motion ofthe whole takes place about the poles of the in-clined circle we have defined [i.e. the ecliptic], inthe opposite direction to the first motion.

If, then, we imagine a great circle drawnthrough the poles of both the above-mentionedcircles, (which will necessarily bisect each ofthem, that is the equator and the circle inclinedto it [the ecliptic], at right angles), we will havefour points on the ecliptic: two will be producedby [the intersection of] the equator, diametricallyopposite each other; these are called ‘equinoc-tial’ points. The one at which the motion [ofthe planets] is from south to north is called the‘spring’ equinox, the other the ‘autumnal’. Two[other points] will be produced by [the intersec-tion of] the circle drawn through both poles;these too, obviously, will be diametrically op-posite each other; they are called ‘tropical’ [or‘solsticial’] points. The one south of the equatoris called the ‘winter’ [ solstice], the one north,the ‘summer’ [solstice].

We can imagine the first primary motion,which encompasses all the other motions, as de-scribed and as it were defined by the great circledrawn through both poles [of equator and eclip-tic] revolving, and carrying everything else withit, from east to west about the poles of the equa-

tor. These poles are fixed, so to speak, on the‘meridian’ circle, which di↵ers from the afore-mentioned [great] circle in the single respect thatit is not drawn through the poles of the ecliptictoo at all positions of the latter. Moreover, itis called ‘meridian’ because it is considered tobe always orthogonal to the horizon. For a circlein such a position divides both hemispheres, thatabove the earth and that below it, into two equalparts, and defines the midpoint of both day andnight.

The second, multiple-part motion is encom-passed by the first and encompasses the spheresof all the planets. As we said, it is carried aroundby the aforementioned [first motion], but itselfgoes in the opposite direction about the polesof the ecliptic, which are also fixed on the cir-cle which produces the first motion, namely thecircle through both poles [of ecliptic and equa-tor]. Naturally they [the poles of the ecliptic]are carried around with it [the circle throughboth poles], and, throughout the period of thesecond motion in the opposite direction, they al-ways keep the great circle of the ecliptic, whichis described by that [second] motion, in the sameposition with respect to the equator.

Chapter 9

On the individual concepts

Such, then are the necessary preliminary con-cepts which must be summarily set out in ourgeneral introduction. We are now about to be-gin the individual demonstrations, the first ofwhich, we think, should be to determine the sizeof the arc between the aforementioned poles [ofthe ecliptic and equator] along the great circledrawn through them. But we see that it is firstnecessary to explain the method of determiningchords: we shall demonstrate the whole topic ge-ometrically once and for all.

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Study questions

Question 6.1. In what sense is the Almagest awork of theoretical philosophy?

a.) What, according to Ptolemy, is the di↵erencebetween theoretical and practical pursuits?Provide an example of each.

b.) What are three branches into which Ptolemydivides theoretical philosophy? Did he con-ceive of this division? With what is eachbranch concerned? Is his description consis-tent with the contemporary understanding ofthe subjects by the same names?

c.) Which branch of theoretical philosophy doesPtolemy seem to favor? Why? Would youagree?

d.) Is it true, as Ptolemy states, that theology is“completely separated from perceptible real-ity”?

Question 6.2. What is the shape and motionof the heavens?

a.) Upon what ancient observation is Ptolemy’sbelief based?

b.) On what grounds does he reject the notionthat the stars move in straight lines?

c.) On what grounds does he reject the “absurdhypothesis” that the stars are kindled andextinguished daily?

d.) What other evidence does he o↵er in supportof his view? Are these reasonable? Are theycorrect?

Question 6.3. What is the shape of the earth?

a.) What do the di↵erent rising and settingtimes of the sun, moon and stars for di↵er-ent observers tell us about the shape of theearth?

b.) Why does Ptolemy here speak of eclipses?Consider: if the ancients used the motion

of the moon as a standard to measure time,then how could they measure whether themoon rose at a di↵erent time for people atdi↵erent locations on the earth?

c.) How would our observations of the sun bedi↵erent if the earth were, say, concave? pla-nar? polygonal? cylindrical?

d.) What other evidence does he o↵er for his be-lief? Are his arguments convincing?

Question 6.4. What is location of the earthwith respect to the celestial sphere?

a.) If the earth were located at the center of thecelestial sphere, then how much of the zodiacwould appear above the horizon on any givennight?

b.) How would the length of daylight di↵er fora person standing at the equator or stand-ing in Europe? Would the hours of daylightand darkness be equal throughout the year?Does it depend on where the person is stand-ing on the earth?

c.) How would your answers to the previousquestions di↵er if the earth were locatedo↵ the axis of the celestial sphere? If theearth were located on the axis of the celes-tial sphere, but nearer the north pole of thesphere?

d.) What does Ptolemy conclude, based uponthese considerations? Is his argument con-vincing?

Question 6.5. What does Ptolemy believe con-cerning the size and the motion of the earth?

a.) What evidence does he provide for the sizeof the earth?

b.) If the earth were not at the center of theworld, but objects were attracted to the cen-ter of the center of the world, then what con-trafactual observation would follow?

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c.) How does Ptolemy address the apparentparadox that the earth, being heavy, is sup-ported by nothing?

d.) If the earth were indeed falling, then whatcontrafactual observation would follow?

e.) Does his reasoning rely upon Aristotelianphysics?

f.) On what grounds does Ptolemy reject the no-tion that the earth spins on its axis? Are hisarguments reasonable? Are they correct?

Homework exercises

Exercise 6.1 (Ptolemaic geocentrism). InChapter V of the Almagest, Ptolemy argues thatEarth must be located precisely at the center ofthe heavens. In so doing, he carefully consid-ers three distinct possibilities (though not in thefollowing order); we will refer to them as

(i) the earth is located at the center of thesphere of fixed stars,

(ii) the earth is located along the equator of thesphere, but o↵ its axis, and

(iii) the earth is located on the axis of the spherebut nearer one pole than the other.

Let us begin by carefully analyzing case (i) to-gether. Afterwards, you will be asked to sim-ilarly analyze the other two cases, and explainwhy Ptolemy finds them to be erroneous.Case (i) is depicted schematically in Fig. 6.1.

The earth is located at the center of a celes-tial sphere, which rotates about the axis ab onceper day. This westward rotation (clockwise fromabove the axis at a) accounts for the daily risingand setting of the stars, which are fixed to thecelestial sphere.From the perspective of a person standing on

the equator of the stationary earth,1 the sphere

1This is referred to as sphæra recta [K.K.].

�� e

s

w

a

b

c

d

g

h

Figure 6.1: Geocentric diagram depicting thehorizon, hg, for an observer standing on theequator. The earth (&) is at the exact centerof the sphere of fixed stars which rotates aroundthe axis ab daily. The sun (�) appears to driftamong the stars between the tropics labeled sand w throughout the year since it is fixed to adi↵erent sphere which rotates yearly about theaxis cd.

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of fixed stars (and hence the zodiac) is bisectedby the horizon which is depicted by the verticalline hg.2 Polaris, located at a, lies just on (orslightly below) the northern horizon.The inner sphere, to which the sun is fixed,

rotates counterclockwise about the axis cd onceper year. Axis cd is tilted at an angle of approxi-mately 23° with respect to the axis of the sphereof fixed stars.3 The yearly rotation about cd,coupled with the daily rotation about ab, givesrise to both the rising and setting of the sunand also its seasonal eastward drift along theecliptic. The limits of the sun’s northern andsouthern motion is indicated by the horizontaldashed lines which are formed by the intersec-tion of the equator of sphere cd with the sphereof fixed stars.On the day of the winter solstice, the sun will

appear to lie in front of the stars at w from thevantage point of an observer on the earth. Asthe year progresses, the sun’s position will driftnorthward. On the day of the vernal equinox,the sun will appear in front of the stars at e. Onthe day of the summer solstice, it will appearin front of the stars at s. It will then drift backsouthward during the course of the year, past thepoint e on the autumnal equinox, until it reachesthe solsticial point w once again.

Notice that, for an observer standing on theearth’s equator, the sun would spend an equaltime above and below the horizon on every dayof the year. Also, the sun would be directly over-head at noon on only two days of the year, thevernal and autumnal equinoxes.

What about for an observer located in thetemperate region, say in Milwaukee? In sucha case, the horizon hg would again bisect thesphere of fixed stars, but it would do so at anoblique angle, as shown in Fig. 6.2.4 Polaris

2An accurate bisection is only e↵ected insofar as theearth’s radius is negligibly small compared to the sphereof fixed stars. [K.K.]

3For the sake of clarity, I have omitted the severalintervening spheres which, according to Ptolemy, governthe motion of the planets which lie between the sun andthe sphere of fixed stars [K.K.].

4This is referred to as sphæra obliqua [K.K.].

�� e

s

w

a

b

c

d g

h

Figure 6.2: Similar to Fig. 6.1, but depicting thehorizon for an observer in the mid-latitudes.

would reside at an angle above the horizon equalto the observer’s latitude. During the courseof the year, the motion of the sun would againbe confined between the two horizontal dashedlines. But during the days of winter, the sunwould spend far less time above the horizon thanduring the summer months. Only on the vernalequinox and the autumnal equinox would the dayand night be of equal duration.

Finally, for an observer located above the arc-tic circle, the horizon line hg would bisect theheavenly sphere at a very oblique angle, as shownin Fig. 6.3. Again, Polaris would reside at an an-gle above the horizon equal to the observer’s lati-tude. During certain days in the depth of winter,the sun never rises above the horizon, and dur-ing certain days in the height of summer, the sunnever sets below the horizon.

Now, finally, here are the homework exercises.

a.) How would the observations di↵er if theearth were located o↵ of the axis of thesphere of fixed stars, but equidistant fromthe poles? Draw appropriate diagrams, sim-ilar to Figs. 6.1, 6.2 and 6.3, and answer thefollowing questions. (i.) Would the horizonbisect the zodiac? (ii.) Would Polaris be visi-

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��

e

s

w

a

b

c

d

g

h

Figure 6.3: Similar to Fig. 6.1, but depicting thehorizon for an observer in the arctic region.

ble on the horizon? (iii.) Would the hours ofdaylight and darkness be equal throughoutthe year? If not, would they ever be equal?When? (iv.) Would the sun pass directlyoverhead on the equinoxes? And (v.) wouldthe equinoxes lie midway between the sol-stices?

b.) What would be observed if the earth, in-stead, were located on the axis of the sphereof fixed stars, but nearer its northern pole?Draw appropriate diagrams and answer thesame questions as before.

Exercise 6.2 (Shape of the earth). Supposethat there were no satellite photographs of theearth from space. How might you determine theshape and size of the earth? Be as clear, detailedand complete as possible in your explanation.

Vocabulary

1. virtue2. disposition3. epithet4. aethereal

5. arithmetic6. geometry7. corruptible8. passive9. latitude

10. ecliptic11. subsequent12. period13. pole14. exhalation15. interpose16. sphericity17. constituent18. eclipse19. concave20. equidistant21. zenith22. nadir23. equinox24. sphaera obliqua25. solstice26. bisect27. equinoctial28. culmination29. disaccord30. gnomon31. diametrical32. armillary sphere33. tangent34. paradoxical35. plausible

Laboratory exercises

A cross sta↵ is a simple and inexpensive toolthat allows one to consistently measure the an-gular separation of two distant objects. The twoobjects might be celestial objects, such as twostars in a constellation or two craters on themoon, or they might be terrestrial objects, suchas the top and bottom of a mountain or the earson each side of a face. Historically, the cross-sta↵was used extensively in both astronomy and nav-igation, but it has been largely replaced by more

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sophisticated devices and techniques, such as thesextant and global positioning system.

Constructing a cross-sta↵

A cross sta↵ is shown in Fig. 6.4. It consistsof a scale fixed perpendicularly to the end of along straight rod. By holding the rod just belowthe eye and looking down the rod, the transversescale of the cross-sta↵ can be used to measure theangular separation between two objects. Slidingmarkers on the transverse scale can be used tofacilitate such measurements.

Figure 6.4: Using a cross-sta↵ to measure theangle CAB between the sun and the horizon,from John Seller’s Practical Navigation (1672).

In order to construct a cross-sta↵ you will needthe following materials: (1.) a straight woodendowel at least 60 cm long (2.) a flexible plastic orwooden ruler at least 30 cm long, (3.) at least 150cm of string, (4.) two rectangular pieces of card-board from a cereal box which measure about4 ⇥ 7 cm, (5.) a razor blade or precision utilityknife, (6.) a small wood-cutting saw, (7.) a wood

screw or nail, (8.) a small drill to make a starterhole for the screw.

Begin by cutting the wooden dowel to an ap-propriate length. What is an appropriate length?By choosing wisely, you can design your cross-sta↵ so that one centimeter on the transversescale corresponds to an angular separation of onedegree. So you will need to use a bit of geometry:when considering a sector of a circle, what is therelationship between the arc length, the radiusof the circle, and the angle of the sector? Howcan you design your cross-sta↵ such that an arclength of one centimeter corresponds to an angleof one degree?

Once you have your dowel cut to an appropri-ate length, you will need to drill a hole in thefar end and attach your plastic ruler so that itforms a T . Next, cut two lengthwise slits, abouta centimeter apart, in your 4 ⇥ 7 cm pieces ofcardboard. These can be slid over the ends ofthe ruler and will serve as markers.

In order to make your cross-sta↵ more pre-cise, you may wish to bend the ruler into an arc,so that all points on the ruler are equidistantfrom the tip of the dowel near the observer’s eye.To this end, the string can be fed through thecardboard markers, wrapped around the edgesof the ruler, pulled taught and fixed with tapeto the near end of the wooden dowel, oppositethe transverse scale. It may be helpful to cut ashallow groove in the near end of the dowel inwhich to lay the string.

Cross-sta↵ precision

How precise is your cross sta↵? It may dependupon what you are trying to measure. Beginby using your cross-sta↵ to measure the angularwidth of an object across the room, such as achalkboard or a window. Adjust your markersto measure the angular width of the object.

Now, how confident are you in the precisionof your angular measurement? In other words,by how much could you change the positions ofthe markers and still feel that you have done areasonable angular measurement? This is the

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(angular) precision of your cross-sta↵.Note that the precision does not depend solely

upon the spacing of the ticks on the transversescale; it may depend upon the type of objectbeing measured and on your ability to hold thesta↵ steady during your measurement. Thesefactors should be considered when assigning anuncertainty to your angular measurement.Now, suppose that you wished to determine

the linear width (in, say, meters) of your dis-tant object from your measurement of its an-gular width. What additional information ormeasurement(s) are necessary? After collectingthis information, calculate the linear width ofthe object from your measurement of its angularwidth. Finally, measure the linear width of theobject using a tape measure. How accurate wasyour calculation? In other words, by what factordid your calculated linear width di↵er from yourmeasured linear width?

Practicing with the cross-sta↵

Go outside during the daytime and measurethe angular size of various distant objects—towers, trees, mountains—so as to become ac-quainted with using your cross-sta↵. The morecomfortable you are with your device, the easierit will be for you to use it later, in the dark.Now, go outside at night and practice measur-

ing the angular size of distant objects in the dark.You may need a small red-tinted light source toassist in reading your angular scale.

Documents

aft - Kerry K. Kuehn 38 CHAPTER 6. THE WORLD OF PTOLEMY teenth and seventeenth centuries. Indeed under-standing the Almagest is essential in understand-ingtheastronomicalworksofCopernicus,