Aristarco de Samos - GÓMEZ GÓMEZ, A. (2011)

8/10/2019 Aristarco de Samos - GMEZ GMEZ, A. (2011)

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Aristarchos of Samos the Polymath

Alberto Gomez [email protected]

(2011-05-05; revised on 2011-11-20)

Part 1Abstract

The evidence on Aristarchos of Samos is gathered and pieced together, revealing a genius who hasinfluenced history more than history has been aware of.

Contents of Part 1

1. Introduction........................................1 2. The early earth movers.......................2 3. Aristarchos and the human eye ........10 4. Eagle eyes?.......................................11 5. The suns distance............................12

6. Archimedes chiliagon .................... 13 7. The myriad figure............................ 14 8. On Sizes and Distances ................... 17 9. The solar system scale..................... 19 Appendix 1 .......................................... 22

1. Introduction

The ravishes of history have left us few traces of the Greatest Mind of Antiquity. Yet, the remnantsalone are more than enough to show that he was a supernova who once shone through the dust ofprejudice to light up the very workings of the world around us, and to remind us forever of how easilythe truths of science can be hidden by the lies of ignorance. His was a mind capable of seeing truthsthat would remain hidden from most of us for nearly two thousand years. Can this be said of anyoneelse? Men like this are rarely met with, says Vitruvius ( On Architecture 1.1.17) in the 1 st century (1c)BC, echoing the high esteem and admiration Aristarchos was held in by his contemporaries, or ratherby the wisest and best hearted of them who, for good reason, called him The Mathematician. 1

Aristarchos was born on the small but important and wealthy island of Samos. We dont even knowexactly when but, because he had become a prominent astronomer by the year 280 BC, we might ten-tatively assign him an age of about 30 or 40 years by then, which would put his birth date somewherebetween 320 and 310 BC. We might generously allow him to live up to about 230 BC. He is said tohave attended Strato of Lampsacos lectures on the nature of light and human vision; yet, John ofStobi ( Anthology 1.16.1, 1.52.3), who quotes in the 5c the now lost 2c works of Aetios on this detail,

1 In Marcus Vitruvius Pollios 1c BC words (ed. Rose 1899:9), Quibus vero natura tantum tribuit sollertiaeacuminis memoriae ut possint geometriam astrologiam musicen ceterasque disciplinas penitus habere notas,

praetereunt officia architectorum et efficiuntur mathematici. itaque faciliter contra eas disciplinas disputare pos-sunt, quod pluribus telis disciplinarum sunt armati. hi autem inveniuntur raro, ut aliquando fuerunt AristarchusSamius, Philolaus et Archytas Tarentini, Apollonius Pergaeus, Eratosthenes Cyrenaeus, Archimedes et Scopinasab Syracusis, qui multas res organicas et gnomonicas numero naturalibusque rationibus inventas atque explicatasposteris reliquerunt. Meaning (tr. Gwilt 1826:9), Those unto whom nature has been so bountiful that they are atonce geometricians, astronomers, musicians, and skilled in many other arts, go beyond what is required of thearchitect, and may be properly called mathematicians, in the extended sense of that word. Men so gifted, dis-criminate acutely, and are rarely met with. Such, however, was Aristarchus of Samos, Philolaus and Archytas ofTarentum, Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and Scopinas of Syracuse: each of whomwrote on all the sciences.


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A. G. Gomez Aristarchos of Samos the Polymath

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leaves us wondering whether these lectures took place in Alexandria or in Athens, or in both. 1 Weknow from the 3c biographer Diogenes Laertios ( Lives 5.58) that Strato spent the best part of his lifetutoring Ptolemy II in Alexandria, where he might also have taught the young Aristarchos, who wasthen about the same age as the future king, or just a little older. Ptolemy I Soter, who had strategicallychosen Strato to tutor his son, made every effort not only to attract brilliant minds to Alexandria, but

also to move the very grounds of Aristotles Lyceum with all the benefits that this would bring to theCity of the Great Library. The latter was never to happen: in 287 BC Strato moved to Athens instead,succeeding Theophrastos as head of the Lyceum and remaining in this position for eighteen years. 2 Whether Aristarchos followed him or stayed in Alexandria among other prominent astronomers likeTimocharis and Aristyllos we do not know, but the chances are that he would not sympathize with theAristotelian School as much as with that of the more mathematically inclined Pythagoras of Samos. Itwas in the latter school, after all, that the novel idea of a moving earth was first conceived.

2. The early earth movers

According to Laertios ( Lives 8.85), the first to say that the earth moves in a circle were Philolaos ofCroton, one of the main exponents of Pythagorism, and Hiketas of Syracuse. 3 They did not have itmove around the sun though, but around a fanciful heavenly body called Central Fire, which we, mere

earthlings, can never see because we live on the side of the earth that always looks away from it. Noteven from the earths underside could we see it, for Zeus had so disposed matters that his home at thiswondrous hearth be permanently eclipsed from our sight by another fanciful heavenly body calledCounter Earth. The total number of heavenly bodies moving around this central fire is ten, as befits thePythagorean ideal of perfection, and Philolaos was the first to put them in the right order: the sky, thefive planets, the sun, the moon, the earth, and the counter earth. The last two whirl around it in just oneday, while the others take much longer: the moon, for example, takes a month, the sun, one year. Thisexplains the succession of days and nights as the earth gets lit up every time it races past the sun.

1 The 5c writings of John of Stobi (Macedon) have been traditionally divided into two volumes called Eclogae(Extracts) and Florilegium (Anthology) , which it is the current trend to group together under the name Anthol-ogy . Several editions can be consulted for the original words connecting Aristarchos and Strato, like Wachsmuthand Hense (1884:149, 483), Diels (1879:313, 403), or Meineke (1860:98), , , ! . Meaning (tr. Heath 1913:300), Aristar-

chos of Samos the Mathematician, following Strato, says that light is the colour impinging on a substratum.Also, " #$%& ' () . * +, ' . Meaning, Strato says that colours are emanations from bodies,material molecules, which impart to the intervening air the same colour as that possessed by the body, whileAristarchos says that colours are shapes or forms stamping the air with impressions like themselves. Also,- . / ! 0 1 " ' 2. Meaning, Epikouros and Aristarchos[both of Samos] say that colours in darkness have no colouring.2 In Laertios own 3c words (ed. Hicks 1925 1:510), 3+4( + ', 5 6 56 789 , : . ! +; !9< !55#= . !5. , . > ? , !54 +4. 55/ . #; @5> , A5+45 . 25B, >, ', 65 C#+;< 5 +4, 6 559+ !

D , E( F >G . HF . IF J56+ , K 5K #6 2 C>+.Meaning (tr. Hicks 1925 1:511), His [Theophrastos] successor in the school was Strato, the son of Arcesilaus, anative of Lampsacus, whom he mentioned in his will; a distinguished man who is generally known as ThePhysicist, because more than anyone else he devoted himself to the most careful study of nature. Moreover, hetaught Ptolemy Philadelphus and received, it is said, eighty talents from him. According to Apollodorus in hisChronology [of which only fragments now remain] he became head of the school in the hundred and twenty-third Olympiad [288 to 284 BC], and continued to preside over i t for eighteen years.3 In Laertios 3c words (ed. Hicks 1925 2:398), . #K / ?5 H < L + M4 [ ] 99 . Meaning (tr. Hicks 1925 2:399), He [Philolaos] was the first to declare that the earthmoves in a circle [round the central fire], though some say that it was Hicetas of Syracuse.


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All this (see Figure 1) can be pieced together from the scanty accounts of Philolaos theory thathave come down to us, such as John of Stobis ( Anthology 1.22.1) 1 and Ps-Plutarchs ( Placitis 3.9.1), 2 both of which serve to reconstruct the writings of Aetios ( Placita 3.9.1), as published by Diels (1879).

Figure 1 : Philolaos cosmos, conceived in the 5c BC, is made up of ten heavenly bodies whirlinground a central fire that cannot be seen from earth. In this scheme of things, we are closest to thesun at noon (position 1) and furthest from it at midnight (position 2).

1 In Aetios 2c words (ed. Diels 1879:336) as reconstructed from John of Stobi (ed. Wachsmuth and Hense1884:196, Meineke 1860:134), A55 , ! 1 . , N I , 5 . 3 . , B9 . . O< . 5 , P , .@" +& O , . +Q , + " ?, '9 , ) 5*, R S T5, R U 5*, R V #K, R V , R W O , , I . / 6( !4. X Q Y , , ! U H5 , Z5 5 < / +Q , J5O , ! U ) 5* R [5 . 5* 6, . X +R O 459 . # , ! U / K 5B95 # , '9 . \. . Q / #4 = #> >, . +Q ] #4 (> ; , 5> Q !>, 5K +Q ?. Meaning (authors tr.), Philolaos says there is fire around the worlds centre, whichhe calls Zeus hearth and home. Beyond the worlds outer reaches there is another fire surrounding all there is.The centre came first by nature, and around it dance ten heavenly bodies: the sky, the five planets, then the sun,

then the moon, then the earth, then the counter earth and, after all these, the fire around the centre. The regionbeyond the sky is a place where the elements are in their pure state, unmixed, and that place he calls Olympus.All that lies beneath Olympus, namely the part where the five planets, the sun, and the moon lie, he calls cosmos;and the region under the moon and around the earth, he calls heaven. The first two regions are wisely designed tostay in perfect order and nothing ever changes in them. In the last region, it is the virtue of those subject to birthand death to experience imperfection and change.2 In Aetios 2c words (ed. Diels 1879:376) as reconstructed from Pseudo-Plutarch (ed. Dbner 1841:1092),^5K . L & ', > #K. M @# +O, O . . Meaning,Thales and those following him say that there is one earth; Hiketas the Pythagorean, that there are two: this pre-sent one and the counter earth.

Sun

Moon

Central Fire

Counter EarthEarth

109

876

54

32

1

Sun

Moon

Central Fire

EarthCounter Earth


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A variant of this model can be taken from the words that Aetios ( Placita 3.11.3) later adds in hisaccount saying that Philolaos earth and counter earth move in opposite circles, and thus remain per-manently hidden from each other. 1 If we take this to mean that they are on opposite sides of the centralfire, then we have another perfectly working model different from the one first described, but this isnot all! There is yet another dissenting passage by the 1c BC Roman orator Cicero ( Academica priora

2.39.123) saying that Theophrastos portrayed Hiketas as brandishing another novel idea: a spinningearth. According to him, Hiketas thought that nothing in the universe moves except the earth, whichspins about its axis producing the illusion that all moves but us. 2 Sarton (1993:290) says that thesewords by Cicero, who was no astronomer, show a total forgetfulness of planetary motion along thezodiac and are better understood as an oversimplification of Hiketas original thought that it is theearth that turns around its axis every day, not the starry heavens.

If we are to trust Cicero in this, hard though his words are to fit in with Philolaos, then Hiketas is,as far as we can tell and early in the 4c BC, the first westerner ever to have spoken of a spinning earth.Alternatively, Aetios ( Placita 3.13.3) gives two more names in connection with such a novel ideawhen he says that most philosophers believe that the earth remains fixed in space and immobile, thatPhilolaos has earth, sun, and moon whirling around the central hearth, and that Heraclides of Pontosand the Pythagorean Ecphantos have earth spinning on its own axis, but not moving through space or,as Sarton (1993:291) puts it, moving in the sense of rotation, but not in the sense of translation. 3

1 In Aetios 2c words (ed. Diels 1879:337) as reconstructed from Pseudo-Plutarch (ed. Dbner 1841:1093),A595 @#9, Q , 4< , #/ , I>< +4 +Q >< > +Q H4 #K !( !> 4 . 4 F >< R . +_ ! F+ ) ! !>. Meaning (tr. Goodwin 1874 3:155), Philolaus the Pythagorean gives to fire themiddle place, and this is the hearth-fire of the universe; the second place to the Antichthon ; the third to that earthwhich we inhabit, which is seated in opposition unto and whirled about the opposite, which is the reason thatthose which inhabit that earth cannot be seen by us. Also (authors tr.), The Pythagorean Philolaos puts thecentral hearth in first place, for it is the hinge of the world. Second comes the counter earth, and third, the earthwhere we live. The last two move in opposite circles, which is why the inhabitants of both worlds have neverseen each other.2 In Ciceros 1c BC words (ed. Reid 1885:322), Hicetas Syracosius, ut ait Theophrastus, caelum solem lunamstellas, supera denique omnia stare censet neque praeter terram rem ullam in mundo moueri, quae cum circumaxem se summa celeritatee conuertat et torqueat, eadem effici omnia, quae si stante terra caelum moueretur.Meaning (tr. Reid 1880:81), Hicetas the Syracusan, as Theophrastos says, believes that the sky, sun, moon,stars, and in fact all the heavenly bodies stand still, and that nothing at all moves in the universe except the earth;and that because it turns and twists with great speed about its axis, all the same phenomena are produced as if thesky was in motion and the earth standing still.3 In Aetios 2c words (ed. Diels 1879:378) as reconstructed from Pseudo-Plutarch (ed. Dbner 1841:1093),A595 +Q @#9, ?51 4 . , / ?5 5( 9 [5>1 . 5; . `5>+ @ . a @#9 , Q #K, ' ; # B ,55/ , , +> !(4, + !& 5/ . b+ 'K 4. Ps-

Plutarchs version is slightly different, A595 +Q @#9, ?51 4 . , / ?5 5(, , 9 [5>1 . 5; . `5>+ @ . a @#9 ,

Q #K, ' ; # B , , [+Q] +> !%4, + !R 5/ , . b+ 'K 4. Meaning (tr. Fortenbaugh and Pender 2009:158, Goodwin 1874 3:156, Heath 1913:251), Theothers [believe] that the earth is stationary, but Philolaus the Pythagorean [believes] that it moves in a circlearound the [central] fire in an oblique circle in the same way as the sun and the moon. Heraclides of Pontus andEcphantus the Pythagorean make the earth move not by changing its place but by turning from west to eastaround its own centre, while it is fixed on an axis in the manner of a wheel. This passage is quoted by Coperni-cus ( Praefatio 1543: Folio IV recto) as one of his inspirational sources (tr. Africa 1961:405), The rest believethe earth is stationary, but Philolaus the Pythagorean says that it moves around the Fire in an oblique circle likethe sun and moon. Heraclides of Pontus and the Pythagorean Ecphantus also make the earth move, not throughspace, but rotating about its own centre like a wheel on an axis from west to east.


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Heraclides name appears again in connection with the spinning earth idea in a brief passage by the5c Greek writer Proclos ( Timaeum 281E), who says that he made the earth wind round and round incontrast to Plato, who kept it still. 1 Also in the words of Simplikios, who, a century later, mentions himfour times in his comments on Aristotle saying, in one passage ( Caelo 2.14), that Heraclides made theearth spin and the heavens stand still (just as Cicero, Aetios, and Proclos had said of him before); 2 in

another ( Caelo 2.8), that not only Heraclides, but also Aristarchos forms part of this team of earthspinners (and this is the first time in recorded history that both names are put together); 3 in a third(Caelo 2.13), that Heraclides spins the earth and makes it the [worlds] centre; 4 and yet in a fourth pas-

1 In Proclos 5c words (ed. Diehl 1906:138), , + Y c5# [_ H554 ? L54 . 4 ' , d @56 4 54#; `5>+ Q Y @9, ' @56 e ; , ? !4 +9(, ?51 #K @56 +Q > ' f. Meaning (tr.

Heath 1913:255), How can we, when we are told that the earth is wound round, reasonably make it turn roundas well and give this as Platos view? Let Heraclides of Pontus, who was not a disciple of Plato, hold this opinionand move the earth round and round; but Plato made it unmoved.2 In Simplikios 6c words (ed. Heiberg 1894:541, Karsten 1865:242), '+ g L K 5; !5>8 ? 6( , h / [/ ] +6 'K . T5 6 5 ,55/ . ! !56 , [5> +;. , +Q g 4B, . B ! > [ #K< H +Q ?51 . 4, d `5>+ @ >, '> i? , H Q +? ! g !6 / j 455, H +Q 56 , H Q . ) , H, 95, ' g +9 +>% 9 T5 . L j55 56 455, H +Q . ) , %1+,, ' g L 5 ' . 9 455, h ,< b +Q . ) , H, b . , %1+,, g != 54 [ H / I9 %$+ 6B ; '> k. Meaning (tr. Heath 1913:255), This would equally have happened [that is, the stars wouldhave seemed to be at different distances at different times instead of, as now, appearing to be always at the samedistance, whether at rising or at setting or between these times, and the moon would not, when eclipsed, alwayshave been diametrically opposite the sun, but would sometimes have been separated from it by an arc less than asemicircle] if the earth had a motion of translation; but if the earth rotated about its centre while the heavenlybodies were at rest, as Heraclides of Pontus supposed, then (1), on the hypothesis of rotation towards the west,the stars would have been seen to rise from that side, while (2) on the hypothesis of rotation towards the east, (a)if it so rotated about the poles of the equinoctial circle (the equator), the sun and the other planets would not haverisen at different points of the horizon and, (b) if it so rotated about the poles of the zodiac circle, the fixed starswould not always have risen at the same points, as in fact they do; so that, whether it rotated about the poles ofthe equinoctial circle or about the poles of the zodiac, how could the translation of the planets in the direct orderof the signs have been saved on the assumption of the immobility of the heavens?3 In Simplikios 6c words (ed. Heiberg 1894:444, Karsten 1865:200), 4 +Q i(> . 4 i? , > . +, =% 4 ' 6B 4 i? , +/ ##4 6, l `5>+ @ m . >, >% =% / 9 , Q ', . j i? , K +Q #K . ? , H, 95 + 4 I6 n4 > 2## ;< +Q 2## 9 +/ , n5> K _ > !>< d , b# [ #K, N R C5># Q +>(, , +Q d 9 25B,+? , ', . j4 i? K / 9. Meaning (tr. Cohen and Drabkin

1948:106), There have been some, among them Heraclides of Pontus and Aristarchus, who thought that thephenomena could be accounted for by supposing the heaven and stars to be at rest, and the earth to be in motion

about the poles of the equator from west [to east] making approximately one complete rotation each day. Also(tr. Fortenbaugh and Penten 2009:174), There have been some, including Heraclides of Pontus and Aristarchus,who thought that the phenomena were saved if the heavens and the heavenly bodies were at rest, while the Earthperformed approximately one rotation per day around the poles of the equinoxial [circle] from west [to east].4 In Simplikios 6c words (ed. Heiberg 1894:519, Karsten 1865:232), - 0 1 +Q Y #K . O51 , +Q ' i `5+ @ "% o / .Meaning (tr. Heath 1921 1:317), Heraclides of Pontus supposed that the earth is the centre and rotates (lit.moves in a circle) while the heaven is at rest, and he thought by this supposition to save the phenomena. Also(tr. Fortenbaugh and Penten 2009:162), Heraclides of Pontus believed that he was saving the phenomena byhypothesizing that the earth is in the centre [of the All] and moves in a circle, while the heavens are at rest.


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sage ( Physica 2.2), that Heraclides moves the earth and stills the sun in order to explain the apparentmotion of the latter. 1 Stated in this way, this last statement may sound really striking to us and, indeed,anyone after Copernicus might think that by moving the earth and stilling the sun Heraclides meantsome form of heliocentrism. Pre-Copernican minds could not be fooled though, for, on closer inspec-tion, nothing new is really said: The earth moves (that is, rotates); the sky stands. Thomas Aquinas,

for example, gets it right from Simplikios when, on commenting Aristotles Book On the Heavens inthe 13c, says that Heraclides kept earth spinning at the centre of the world ( Caelo 2.21.5), 2 while theheavens were at rest ( Caelo 2.11.2). 3

This last statement by Simplikios, together with yet another passage by the 4c Latin writer Chal-cidius ( Timaeus 110) in which Heraclides is reputed to have said that Venus is sometimes above,sometimes below the sun, and that the circles of both bodies have the same centre, 4 was enough toconvince the French scholar Thomas Henri Martin (1849/1971:119-23, 419-28) that Heraclides wassaying that Venus circled the sun. Schiaparelli (1873/1925 1:401-20) took the man at his word andwent even further to say that Heraclides was proposing that all planets circled the sun and that the suncircled the earth, just as Tycho Brahe proposed at the end of the 16c. Yet, once again, nothing of thelike can be derived from Chalcidius actual words: on closer inspection, we can see that if the circlesof Venus and of the sun have the same centre, then Venus is not orbiting the sun, but both are orbitingthe same centre: presumably, the earth! Neugebauer (1975:694) points out that Mercury is never men-tioned in this whole discussion, and that the words sometimes above, sometimes below the sun donot mean in Chalcidius thinking further or closer to us as would happen in a sun-centred system but,simply, that Venus is sometimes right and sometimes left of the sun.

Therefore, in the light of the evidence before us and as far as the extant original sources can tell, wemust conclude that Heraclides never harboured any form of heliocentrism nor did he ever have anything moving around the sun. So the frequently quoted statement that he had Mercury and Venus or-biting the sun is groundless and goes no further back in time than Martins 19c interpretation. It is tobe noted, though, that there is a certain resemblance between this tale and an extant fragment byVitruvius ( Arch. 9.1.6) clearly stating that Mercury and Venus revolve around the sun, 5 but this frag-

1 In Simplikios 6c words (ed. Diels 1882:292), + . 5= > `5>+ @9 N . 4 K #K, , +Q [5> 49 +? [ . T5 4 5> $%.Meaning (tr. Fortenbaugh and Pender 2009:162),A certain Heraclides of Pontus even came forward to say that[on the hypothesis that] the earth somehow actually moves, and the Sun somehow remains stationary, the appar-ent unsmoothness [or motion] of the sun can be saved.2 In Thomas Aquinas 13c words (ed. Rome 1886:205), Quidam Heraclitus Ponticus posuit terram in mediomoveri, et caelum quiescere. Meaning (authors tr.), A certain Heraclides [Thomas spells it Heraclitus] ofPontus put the earth in motion in the centre, and the heaven at rest.3 In Thomas Aquinas words (ed. Rome 1886:162), Et ideo quidam, ponentes stellas et totum caelum quiescere,posuerunt terram in qua nos habitamus, moveri ab occidente in orientem circa polos aequinoctiales qualibet diesemel; et ita per motum nostrum videtur nobis quod stellae in contrarium moveantur; quod quidem dicitur po-suisse Heraclitus Ponticus et Aristarchus. Meaning (authors tr.), Some put the stars and the whole heaven atrest, while making the earth on which we live move eastwards around its equinoctial poles once a day, and this is

why the stars seem to move against the direction we travel. Such is said to have been the opinion of Heraclitus ofPontus and Aristarchus.4 In Chalcidius 4c words (ed. Wrobel 1876:176), Denique Heraclides Ponticus, cum circulum Luciferi de-scriberet, item solis, et unum punctum atque unam medietatem duobus daret circulis, demonstrauit ut interdumLucifer superior, interdum inferior sole fiat. Meaning (tr. Neugebauer 1975:694, Heath 1913:256), Heraclidesof Pontus, when describing the circle of Venus as well as that of the sun, and giving the two circles one centreand one mean motion, showed how Venus is sometimes above, sometimes below the sun.5 In Vitruvius 1c BC words (ed. Rose 1899:217), Mercurii autem et Veneris stellae circa solis radios utiquecentrum eum itineribus coronantes regressus retrorsus et retardationes faciunt, et ita stationibus propter eam cir-cinationem morantur in spatiis signorum. Meaning (tr. Gwilt 1826:270), The stars of Mercury and Venus near-


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ment has no connection whatever with the name of Heraclides and was written long after heliocen-trism had been developed. Centuries later, Proclos wrote a similar fragment ( Timaeum 259A), againwithout connection to Heraclides. 1 So, the fact that two thousand years after Heraclides later genera-tions of such renowned astronomers made up this connection is nothing but an unfair twist of fate.

This leaves us only one candidate on whom to bestow the high honour of having been the first per-son ever to discover that the earth moves around the sun: Aristarchos of Samos. The ancient testimonyis unanimous on the point, says Heath (1921 2:2), quoting Archimedes ( Sandreckoner 1.4) on the mat-ter, from whom we know that Aristarchos brought out a book explaining that the world is many timesgreater than previously thought, that the stars and the sun remain unmoved, that the earth movesthrough space following a circular path whose centre is the sun, and that the sphere of the fixed stars,whose centre is approximately the centre of the sun, is so great that the circle in which he supposesthe earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the spherebears to its surface. 2

est the rays of the sun, move round the sun as centre, and appear sometimes retrograde and sometimes progres-sive, seeming occasionally, from the nature of their circuit, stationary in the signs [of the zodiac].1 In Proclos 5c words (ed. Diehl 1906:65), Q +Q T5 +> . pK, [5) k . +#, '0 . 5#> N5 '0 5,< + . H9+ #6 k [5>1 . . '9 H d #, '0 K ; . Meaning (tr. Taylor1820:228), Above the sun are Venus and Mercury, these planets being solar, and fabricating in conjunction withthe sun, and also contributing together with him to the perfection of wholes. Hence their course is equally swiftwith that of the sun, and they revolve about him, as communicating with him in the production of things.2 In Archimedes words (ed. Heiberg 1881:244), 4 +4, +9 5> 9 Q 5> 59# q , r ! 4 Q _ #_ 4, q +Q ! , 4 b s '>t s () , 4 , q5> . , 4 _ #_. , #6 ! / #9, d / 59# +6.> +Q 6 > !(4+ #6, ! u ! 4 B> 9 5556 , , H4. 4 #/ / Q 54 j . v5 4 >, / +Q #_ 4 . v5 / ?5 4, N ! ! 41 0 +91 >, / +Q 54 j . ' 4 0 q5>1 4 0 #4 5? , h ?5, R w / #_ 4 4, ? 2 5#> . / 54 >, f 2 4 _ > . / !6. , #R c+5 d +?9 !. !. #/ _ > 4 '+Q 2 4#, '+Q 59# 2 '+4 . / !6 _ > 54 '9 . !+4 +Q > +> 9+< !+ / #_ 5B6 h 4 , 9, w 2 59# q #_ . R q H4 9, , 2 59# / , ! x ! ?5, R w / #_ 4 4, . / 54 j . / #/ +(> 4 y 41 !9%, . 65 4 4# _ >, ! x > / #_ 4, b 4 0 R q H41 91.Meaning (tr. Heath 1897:221, 1913:302), You [King Gelon] are aware that universe is the name given by mostastronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight linebetween the centre of the sun and the centre of the earth. This is the common account, as you have heard fromastronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premiseslead to the result that the universe is many times greater than that now so called. His hypotheses are that the

fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, thesun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as thesun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance ofthe fixed stars as the centre of the sphere bears to its surface. Now it is easy to see that this is impossible; for,since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surfaceof the sphere. We must, however, take Aristarchus to mean this: since we conceive the earth to be, as it were, thecentre of the universe, the ratio which the earth bears to what we describe as the universe is the same as the ratiowhich the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixedstars. For he adapts the proofs of his results to a hypothesis of this kind, and in particular he appears to supposethe size of the sphere in which he makes the earth move to be equal to what we call the universe .


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Taken literally, these last words imply an infinite universe for, as Archimedes points out, the centreof a sphere is but a point that bears no ratio whatever to the surface of any sphere. Archimedes is notwilling to accept this (for his reckoner needs a finite world to fill up with sand) and immediately tellsus what he thinks Aristarchos means: that the earth is to its orbit what its orbit is to the sphere of thefixed stars or, put another

way, that the earths radius isto that of its orbit what theradius of its orbit is to that ofthe sphere of the fixed stars(see Figure 2).

Apparently, this scheme ofthings is as whimsical as anyof Philolaos fancies andmany a good scholar, unableto make head or tail of it, hasbeen misled into thinking thatit is, to say the least, nothingbut idle guesswork. 1 Yet, aswe shall see, there is more toAristarchos than meets theeye.

Destroyed by centuries of vicious persecution, his books and thoughts are only accessible to us nowas a few surviving references by other authors. What is left us, though, is only the tip of a huge icebergof sound scientific research, which we can only hope to reconstruct. To do this, we need to look mostcarefully at every single fragment that has made it down to us, and Archimedes is key! On trying toreckon how many grains of sand it would take to fill the largest world then conceived, he tells us(Sandreckoner 4.12) that Aristarchos places the sphere of the fixed stars almost a myriad (or 10,000)times farther away than the sun, 2 and also that he makes the earth and its orbit bear the same ratio,

placing the sun almost a myriad earth radii away! This is not only to be deduced from Archimedeswords, but also explicitly stated in the Sandreckoner (2.1). 3

Two other sources add detail to this piece of evidence: one is Cleomedes ( Heavens 2.1.79, 1.11.56),who tells us that Poseidonios, in the 1c BC, assumes (either from Archimedes or directly from Aristar-chos) that the earths (lit. suns) orbit is a myriad earths wide, adding the obscurely worded detail thatit might be more rather than less this figure. 4 He also adds a sound description of what the sun and the

1 Neugebauer (1975:643), for one, concludes that Aristarchos treatise on the sizes and distances is a purelymathematical exercise which has as little to do with practical astronomy as Archimedes Sandreckoner in whichhe demonstrates the capability of mathematics of giving numerically definite estimates even for such questionsas the ratio of the volume of the universe to the volume of a grain of sand. He even goes as far as to considersome [extendable to all?] of Aristarchos methods as pure mathematical pedantry of no astronomical interest.2 In Archimedes 3c BC words (ed. Heiberg 1881:288), q +6 _ 54 j > !56 !. z 5> _ +4 , 9. Meaning (authors tr.), The diameter of the sphere

of fixed stars is less than a myriad times the diameter of the cosmos [or sphere then equal to the earths orbit].3 In Archimedes 3c BC words (ed. Heiberg 1881:262), q +6 , 9 _ #_ !56 !. z 5>. Meaning (authors tr.), The diameter of the cosmos [or sphere then equal to the earths orbit] isless than a myriad times the diameter of the earth.4 In Cleomedes words (ed. Ziegler 1891:144-6), @+= 4 5> [5 ?5 , K #K ?5, [...] !+4 +Q . >% ' k z 65 > [_ # . Meaning(authors tr., see also Kidd 1999:171, Bowen and Todd 2004:114, Heath 1913:344, 346), Poseidonios assumesthat the suns circle is as wide as a myriad earths, [] but i t could be more rather than less, for all we know.

Earth Sun

Sphere of thefixed stars

Earthsorbit

Earth

Figure 2: Archimedes thinks Aristarchos means that the earthis to its orbit what its orbit is to the sphere of the fixed stars.

10000 AU 10000 e 1

AU 1e


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earth would look like from the sphere of the fixed stars, making it clear that the sun, for all its bright-ness, could not be told from other stars, and that the earth too would not be seen at all. 1 All this pointsto Poseidonios taking Aristarchos myriad earth-orbit distance to the stars as a lower not upper bound(as Archimedes would prefer in order to count sand grains). But, does this tell us what Aristarchoshimself, with whom all these ideas originate, actually believed? It strongly suggests an endless uni-

verse, but what really confirms this is our second piece of evidence on the matter: Aetios ( Placita 2.24.8) says that Aristarchos thought of the sun as of another star. 2

Even if we didnt have the benefit of Poseidonios or Aetios words to tell us, we could still findwhat Aristarchos really thought by reasoning like this: He proposed heliocentrism and then was facedwith the question of why we cant see any stellar parallax. Indeed, if the earth moves, we should seethe stars shifting their positions from each other. So, why do we not? There are two answers: one isthat they are all at the same distance from us, studding the sphere of the fixed stars; the other is thatthey are exceedingly far away and their shift is too small to be seen. (Actually, this is what happens.)Now, which of these options would Aristarchos favour? The answer is in the myriad figure: had hebelieved that there was such a thing as a sphere of fixed stars, a much shorter distance would havebeen enough to account for null star shifts. Yet, the very fact that he placed the stars a myriad timesfarther away than the sun and that this figure links parallax to human vision, as we shall see, provesthat he had infinity in mind.

1 In Poseidonios 3c BC words as quoted by Cleomedes (ed. Ziegler 1891:102-6), {Y +Q 4# [ #K,[5> +/ 4 !9+ !+4+, ' 9 ? , 9 4# > 59# !4 +/ 55 !54#, 55/ . [5 y8, : / 5) >% !. [ 4 / 5K j . [] | +Q , y+ ) ' g ; 65, 55R '+Q } 5 . ~ Y ) '+4 59# 2 / H4, y '+Q 4# K #K 59# / , 9 4# 2. [...] @ Q #/ T5 / 5) >% e 6 K #K . 56, d +/ I(K !+;, N +> [ > 4, . , 59 . @6 Y ! , d , H , [5, y8 $ H # , E '+R N5 g [ C> z [d ] 4 , B6 4# 2. H +Q !. / 859 [4] , [5> > R 9 . !R ' 5 , '+R N5 g [ C> [ #K,'+R H b $ [5>1 59 2 !>. . ) 4 'K >% # , b # : Q R 'K , +Q , y8 K > 5 ' g ' / 5? # , [5> 4# !56 6< !. 6 . ' T5 ! 0 y8 5 !. 4# ; . Meaning (tr. Bowen and Todd 2004:86-8), While the earth hasthe size demonstrated through the procedures just described, there are several ways of proving that it has theratio of a point not only to the total size of the cosmos, but also to the height of the sun, which the sphere thatencloses the fixed stars far exceeds. [] A [single] pitcher of water would not measure the sea, not even theNile. So just as a pitcher has no [significant] ratio to the [quantities] mentioned, so too the size of the earth hasno [significant] ratio to the size of the cosmos. [] Although the sun is much larger than the earth and sea com-bined, it sends out to us an appearance of being about one foot wide, despite being very bright. We can thus formthe notion that the earth, if we should look toward it from the height of the sun, would either not be seen at all, orbe seen with the size of a minuscule star [under such conditions the earth would appear to be moving in the sameorbit as the sun moves when observed from the earth]; but if by hypothesis we were elevated to a distance far

beyond the sun, and right up to the sphere of the fixed stars, the earth would not be seen by us at all, not even ifimagined as having a brightness equal to [that of] the sun. Hence the stars too must be larger than the earth, inthat they are visible from it, whereas the earth could not be seen from the height of the sphere of the fixed stars.The earth is certainly far smaller in size than the sun, since the sun itself too, if imagined at the height of thefixed stars, will perhaps appear as large as a star.2 In Aetios 2c words (ed. Diels 1879:355) as reconstructed from Ps-Plutarch ( Placitis 2.24.8) and John of Stobi( Anthology 1.25.3 k ), > T5 f / 5 , +Q #K . [5 ?5 . / / ? !#5> 6% +>. Meaning (tr. Heath 1913:305), Aristarchus sets the sunamong the fixed stars and holds that the earth moves around the suns circle [that is, the ecliptic], and is put inshadow according to its [that is, the earths] inclinations.


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3. Aristarchos and the human eye

Before we see how Aristarchos got his myriad figure (which is the Greek word for 10,000 and thepoint from where the Greek number system starts repeating itself), I must say that what follows isbased on the discovery by the American astronomer Dennis Rawlins (2008:13-32), whom I greatlyadmire, of the connection between Aristarchos and human vision. You wont find here a mere recount-ing of his theory, for I found there is room for a little improvement. You will find that I have beenrather critical sometimes. Yet, I must own that, had it not been for him pointing in the right direction, Imost likely never would have thought of it. All started because I find his maths sometimes hard to fol-low; (my own limitation;) but this, together with the fact that it is my job to put things in as plain alanguage as possible (and I hope Ive done so here), has turned into an advantage!

Since none of Aristarchos books on optics has survived, we can only imagine what his studies mayhave been like. If we were to study human vision from scratch, where should we start? The first thingone would think of would be to try to measure the ability of the eye to see fine detail. This is easy: weonly need to draw two black bars on a white background (for maximum contrast) and walk away fromthem until our eyes can no longer tell them apart because they seem to merge into one (see Figure 3). 1 The gap w between these bars divided by the distance d the eye needs to tell them apart yields the an-

gular resolution of the eye in radians:

[ ]d w

rad (1)

or directly in degrees by any of these relations:

[ ]d w

d w

180

2arctan2

= (2)

Now, the typical human eye can make out gaps of about 30 cm from a distance of about 1 km,meaning an angular resolution of almost exactly 1 minute of a degree. With this sort of training onecan easily gauge the angular width of any distant object by viewing it through two bars adjusted to

match the objects apparent width as seen from a known distance (preferably from afar, so that the eyedoes not have to refocus from distant object to closer bars). The gap between these bars (matching theobjects width) and their distance from the viewers eye give the angular width of the distant objectwith impressive precision, such as the one Aristarchos got for the moon and the sun. Archimedes re-ports ( Sandreckoner 1.10) that Aristarchos discovered that the suns apparent size is , which isabout right. 2 The method just described would allow him or anyone else to obtain this degree of preci-sion with ease by gauging the apparent size of the moon first, for safety of viewing, and applying thesame result to the sun, which seems to be the same size as the moon when seen from earth. Ar-chimedes himself attempted to measure just this with the same sort of basic geometry, but by a slightlysimpler method (for he didnt place his sun-occulting bar or disc far enough to improve accuracy) andaccordingly got slightly inaccurate results, which, nevertheless, average the same as Aristarchos.

Now, if the full moon is (or 30 ) wide, then the half moon is (or 15 ) wide. We also knowthat the unaided eye can make out the line exactly halving the moon with a precision of 1 . Anysmaller than this and we wont be able to tell whether the moon is exactly half or not; any bigger andwe shall detect it. Aristarchos tried to find the exact moment when the moon is half lit by the sun asseen from earth because then, and only then, the sun-moon-earth angle would be exactly 90, and this

1 In todays optics, Snellen charts, Landolt rings, and Knig bars are used to determine visual acuity.2 In Archimedes 3c BC words (ed. Heiberg 1881:248), 6 Q 9 , ?5 %1+> v5 9 d H . I9. Meaning (tr. Heath 1897:223, 311), Aristarchus discov-ered that the suns apparent size is about one 720 th part of the zodiac circle.

Figure 3: In todays optics, pairs ofbars as wide as the gap betweenthem are used to determine visualacuity (Grosvenor 2007:9).


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would allow him to apply simple triangle geometry to work out nothing less than the suns distance!He only needed to gauge just one other angle, the sun-earth-moon angle, then he would have all thatwas needed: two angles (one of them right) and the side between them (equal to the moons distance).With these, the suns distance could be expressed in units of that of the moon.

But when he attempted to measure this other angle (that is, the angular distance between the sunand the moon as seen from earth), he saw to his amazement that there was no way to tell it from 90.What sort of triangle is this? No triangle can have two right angles, so he reasoned that the angle heneeded must be imperceptibly close to 90, yielding a huge distance for the sun as compared to that ofthe moon. Now, how far should the sun be for the half moon to subtend an angle that is seen as plainlydistinct from 90 by the human eye? We have all we need to find this: we know that this angle is 1 fora moon disc of 30 (or for a half moon disc of 15 ). Translate this into full blown spherical degreesby imagining that we are not seeing just a flat disc, but a full blown sphere of which only 180 face us,with only 90 of these being lit by the sun. The equivalence is then plain to see: 1 is to 30 what 3 areto 90. So the smallest detectable angle for a half moon is 90 3 = 87, which is exactly what Aris-tarchos is reported to have arrived at by the author of the book bearing his name (see Figure 4). 1

4. Eagle eyes?

I hope the previous maths is simple and plain enough. I got it by trying to understand and simplifyRawlins approach, but this time a perfect hit upon the 87 is achieved through simple maths and real-istic estimates of visual acuity, instead of the approximation Rawlins had to round up in order to reachthe mark because he was working on the assumption that the limit of human vision is one 10,000 th of aradian ( ), which allowed him to link the myriad factor to Aristarchos cosmic size. But, is this true?Can we really see this sharply? Let me disagree. As we have seen, a visual acuity of 1 is considerednormal. (In modern optics this is called 20/20 vision). Yet it is not difficult to find people who see

better than this (I myself can do it), with visual acuities around or (acuity of 20/15 or 20/13).Exceptionally, very few people have been found who see as sharply as (acuity of 20/10), and thats

1 Pseudoaristarchos fourth hypothesis (ed. Heath 1913:352) states that, [ 5; +9 [ >, 9 ' 4 , [5> 25 > 0 , > 0. Meaning (tr.Heath 1913:353), When the moon appears to us halved, its distance from the sun is then less than a quadrant byone-thirtieth of a quadrant [that is, less than 90 by 1/30 th of 90 or 3, and is therefore equal to 87]. Note that Ishare with Rawlins the opinion that the book On Sizes and Distances cannot be authentic for the many reasonsthat will be given later.

Figure 4: If you walk away from this discuntil it is 30 wide (like the moon), then(unless you have better than normal eyesight)you wont be able to tell the two halvinglines apart, because the gap between themwill be 1 wide and this is the best the typicalhuman eye can spot. If instead of thinking ofthis figure as of a 30 wide disc we think of itas of a sphere, then the 1 becomes 6 of the180 facing us. So we can tell the moment ofhalf moon as happening somewhere within 3either side of true middle; any larger than thisand we would confidently tell non-halfness.This is what Aristarchos had in mind whenhe proposed his 87 angle.


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the sharpest that has ever been reported (as far as I know). The reason is that there is a physical limit toour vision: the cone cells responsible for vision cannot be compacted closer than 0.4 at the fovea,making it only theoretically possible to reach a visual acuity of 0.4 (acuity of 20/8), but in practicethis can only be achieved by using a laser interferometer to project images directly onto the retina soas to bypass the eyeballs inner irregularities that prevent the unaided eye from seeing this sharply. So,

Rawlins claim that the finest we can see is the 10,000th

part of a radian or (acuity of less than 20/7)is rather overstated. We shall come back to this later. 1

5. The suns distance

Now let us see what Aristarchos did with his 87 (see Figure 5).

Figure 5: Should the sun be closer than 19 times the distance M to the moon, the angle betweenthem at half moon would then be plainly detectable by the eye as smaller than 87. Since this isnever the case, we must conclude that the sun is at least 19 times farther away than the moon.

Aristarchos triangle shows that the sun must be at least 19 times farther away than the moon. Were

it closer, the sun-earth-moon angle at the moment of half moon would then be plainly detectable bythe naked eye as smaller than 87. Today we have the benefit of trigonometry to work the distance S tothe sun out of Aristarchos triangle by any of the following relations:

M M M

S 193sin87cos

== (3)

But trigonometry was not yet fully developed back then, so Aristarchos gauged the triangle directly,or perhaps he relied on a geometric method that placed the result somewhere between 18 and 20. 2 Thisgeometric method is explained in the seventh proposition of the book On Sizes and Distances : readingit carefully shows that, even though the wording seems to assert that the sun is more than 18 times andless than 20 times as far away as the moon, the whole thing put in context means that if the half moonis taken to be 87 from the sun, then the sun is 18 to 20 times as far away as the moon.

1 In modern optics, the numerator of the 20/20 term refers to the distance in feet from which the subjects eye istested; the denominator refers to the distance from which the thickness of the lines shaping the letters in the Snel-len chart subtend and angle of 1 . Inverted, this fraction gives the eyes angular resolution in minutes of a degree.2 In Pseudoaristarchos words (ed. Heath 1913:376), X 9 w 4 T5 K #K , ; : 4 [ 5; K #K % 4 ! z C+56, 25 +Q z H56. Meaning (tr. Heath 1913:377), The distance of the sun from the earth is greater than eighteentimes, but less than twenty times the distance of the moon from the earth.

SunMoon

Earth

87

390

M M

M 193tan

87tan =

M M M

193sin87cos

=

M


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But Aristarchos knew that the angle actually lay somewhere between 87 and 90, so he knew thatthe 18 to 20 figure was only a lower bound for the suns distance. Even if we allow for Archimedes(Sandreckoner 1.9) telling us that Aristarchos tried to prove that the sun was 18 to 20 times wider (andfarther away) than the moon (which follows from their apparent size), 1 we may confidently add thelittle tag at least, for we also know from Archimedes same book (and we deduce from Poseidonios)

that Aristarchos did indeed place the sun not just 19 times farther away than the moon, but almost amyriad earth radii away, which is 9 times as much. And why should he do this? Why almost a myriad?To answer this question we must see first what Archimedes exactly means by this.

6. Archimedes chiliagon

Archimedes spent a good part of his Sandreckoner (1.17-22) trying to show that the diameter of thesun is larger than the side of a chiliagon (or polygon of a thousand sides) inscribed in its path. We canunderstand this more simply by reasoning that each side of a chiliagon inscribed in a circle (such asthe suns apparent path) subtends an angle equal to the thousandth part of 360 (or 21 36), which iscertainly less than the suns apparent size that Archimedes put between the 164 th and 200 th part of aright angle (that is, between 32 56 and 27 , or about 3 either side of Aristarchos own value).

Then he made use of the old lore that any polygon of more than six sides has a perimeter greaterthan three (and less than pi) times its diameter (in fact, the more sides a polygon has the more its girthbecomes like that of a circle, which is pi times its breadth). Because this number three could be neatlyused as a round figure (instead of pi itself), he surpassed Aristarchos lower bound and convenientlymade the sun thirty times as wide as the moon (which is roughly ten pi), reasoning in the process that,because the earth is wider than the moon, the sun must be less than thirty times as wide as the earth.

Then, because thirty earths are wider than the sun, which in turn is wider than a chiliagons side, itfollows that 30,000 earths are wider than 1000 suns, which in turn are wider than a chiliagons girth.Now, making use of the fact that a chiliagons girth is more than three times its breadth, and that itsbreadth is the same as that of the earths orbit (or cosmos as then understood) within which it is in-scribed, we can say that 30,000 earths are wider than three times the earths orbit. Divide it all by threeand you will find that a myriad earths are wider than the earths orbit or, put another way, that the

cosmos is less than a myriad earths wide.But there is a loophole in Archimedes thinking, for it all hinges on how big the sun is made to be:

take the sun to be as wide as thirty moons, and the cosmos becomes less than a myriad earths wide;take it to be ten times pi as wide as the moon and the cosmos becomes exactly a myriad earths wide;take it to be more than ten times pi and the cosmos then becomes more than a myriad earths wide (andwe are only changing the suns size by a small amount around the pi number)! If you want to work itall out by yourself, here is a formula for a polygons side s, where r is the radius (of 1 AU in this case)and n the number of its sides (you can find the polygons girth as n times s):

( )nr s 180sin2= (4)

Or, if you prefer to get the cosmos size k straight without further ado, here is another one, wherethe sun is m times the moons width, and r and n as before. Archimedes made the sun thirty times aswide as the moon for ease of computation. How big would you make it? Try it and see what happens:

( )nr nm

k 180sin

1000= (5)

1 In Archimedes 3c BC words (ed. Heiberg 1881:248), 6 +Q 4 +?, N !. q +6 , q5> _ +4 _ 5; >% Q z C+5>, !56 +Q z H5>. Meaning (tr. Heath 1897:223), Aristarchus tried to prove that the diameter of the sun is greaterthan 18 times but less than 20 times the diameter of the moon.


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All this proves that Archimedes end product is as arbitrary as his choice of sun size and, therefore,the expression less than a myriad is no better than the expression exactly a myriad or more than amyriad at telling us how far Aristarchos placed the sun (just as Poseidonios had hinted at). Yet, thisfigure is of the utmost importance in trying to unravel how Aristarchos measured the world. We maynever know whether he gave a more precise figure and later rounded it down to the nearest myriad or

whether the rounding down was done by Archimedes, but we can certainly get a good idea of what theoriginal numbers that led to such an outcome were like, as we shall now see.

7. The myriad figure

Moving the earth had raised an interesting question: the stars may be awfully far away but, even so,is there any chance that we may spot any shift in their position? Our best hope is to keep track of closepairs throughout the year. If we find a pair of stars so arranged that they overlap at a certain time of theyear and six months later, when the earth is on the opposite side of its orbit, they can be told apart,then we have found the shift we were after. The smallest gap the typical human eye can see is 1 of adegree, so our chosen pair of stars must split at least 1 at some time along the year. Let us see whethersuch an arrangement is possible (see Figure 6).

In the right triangle above, the distance n from earth to a nearby star and the distance f to a fartheraway star that happens to lie just behind the first one when viewed from one point of the earths orbitare found by the following expressions where h is the width of the earths orbit and is the shift (ortwice the parallax) observed between both stars when seen from the opposite side of the earths orbit:

+

+

+=

hnhn

h f arctan45tan (6)

+

+

+

=

hnhn

h f h

arctan45cos

or 22 h f f h += (7)

+

+=

h f h f hn arctan45tan (8)

+

+

=

h f h f

hnh

arctan45cos

or 22 hnnh += (9)

+

+

=

hnhn

h f h f

arctanarctan (10)

f

n

f h

nh

h

Figure 6: Parallax is (half) the apparent shift or displacement of a nearby ob-ect (such as a star) against a distant background (such as a farther away star),

when viewed from two different positions (like two ends of the earths orbit).


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All these equations simplify greatly if we take the farther of our two stars to be infinitely far away:

tanh

n = (11)

sin

hnh = (12)

Let us consider whether Aristarchos farthest away star can be thought of as being infinitely faraway. Well, the farthest object we can see with the naked eye is the Andromeda Galaxy, which is 2million light years away and looks like a whitish glow with none of its individual stars clearly dis-cernible (unless they explode as supernovas). So we must look for our farthest star within our owngalaxy. One of the farthest away stars that can be seen with the naked eye is x Carin which, at itsimpressive distance of 5930.9 light years (or 375,076,504 AU) from us, shines with magnitude 3.93.Not that Aristarchos could possibly know any of this, of course, but we are considering it here only toassess whether his method is sound. Now let us suppose x Carin to be the farthest away of an idealpair of overlapping stars such as described above. How far should its imaginary companion be from usin order to shift 1 from x Carin once a year? The answer by (8) is 6875 AU and by (11) is 6875AU, which proves that there is virtually no difference between calculating distances using a naked-eyebackground stars actual distance or just using infinity. Aristarchos method is therefore sound andallowed him to assess the worlds size in terms of thousands of AU (or cosmos ) for the first time ever.

So, thinks Aristarchos to himself, let us go then after the big game! Let us keep our eyes peeledfor any shift in virtually any star [since any of them could be suspected of being an overlapping pair]during a year. Oh-oh, none found! So stars must be more than 6875 cosmos away! Is this the best wecan do? Couldnt we pick the most eagle-eyed among us and try again? Well, after a year gazing atclose line-of-sight pairs, no shift was found either, but at least this allowed him to resize the world tothe best of human ability, and just how sharply can humans see? It is not difficult to find people whocan see as sharply as , try a little harder and youll find eyes sharp, while sharp eyes are veryrare indeed. Now, which do you think was Aristarchos choice (if indeed he himself couldnt see thatsharply)? The answer is in Archimedes myriad. How far away should a star be to shift from an

infinitely far away background of stars? Compute it by (11) and you get 10313 AU which, roundeddown, is the myriad we were after. 1

Note that 6875 and 10313 are both multiples of a curious number, 1146 (or rather of 3600/ ),which, as Rawlins (2008:23) discovered, is the number of earth radii spanning the suns distance inAristarchos triangle if we take the moon to be 60 earth radii away (see Figure 7)! He also points outthat al-Battani ( Zij 50) in the 9c chose this very same distance as the farthest the sun can get from us,failing to supply coherent justification for the choice. Though suggestive, this coincidence hardlyproves any connection between both ancients, even though al-Battani was explicitly building upon theremains of Greek solar theory. What really proves that Aristarchos did indeed work out the suns dis-tance as (at least) 1146 earth radii (and, therefore, the moons distance as 60 earth radii) is the fact thatArchimedes ( Sandreckoner 1.19), in the course of his long dissertation on the virtues of the chiliagon,mentions that the suns breadth and distance bear a ratio smaller than 11 to 1148, all while describingthe largest world then conceived! 2 Let us see how this fits with Aristarchos maths.

1 I am indebted to Rawlins (2008:17) for having inspired these maths, though I disagree with him in key pointssuch as his claim that a shift of leads to compute a distance of 10,000 AU for the nearest star. In fact, neitheris a gap of visible to the naked eye nor does it compute a myriad AU, but twice as much because, though heseems to be aware that the actual base line is 2 AU, his maths do not really take this into account: a slight slipthat does not diminish at all his groundbreaking discovery that Aristarchos relied so heavily on optics!2 In Archimedes words (ed. Heiberg 1881:258), !56 Y 59# 2 q . / ^\ , z / . / . Meaning (my tr.), [The suns width] to [the suns distance from us] has a ratio smaller than 11 to 1148.


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Figure 7: If we take the moon to be as far away as 60 e, then thesuns distance in Aristarchos triangle becomes (at least) 1146 e.

In Aristarchos triangle (see Figure 7) we can see that if we take the moon to be 60 e away, then thesun is (at least) 1146 e away. 1 This, together with the fact that both bodies appear to span half a degreeacross the sky, is all we need to find the suns width w expressed in earth radii by rearranging (2):

1802tan2

d d w

= (13)

and putting Aristarchos data in it:

eeSunw 10tan11462 == (14)

So, the suns width in Aristarchos triangle is easy to find by any of the expressions in (13) as 10 e,meaning that the sun is (at least) 5 times wider than the earth and that Aristarchos put the width-to-distance ratio of the sun at 10:1146 which, as Archimedes said, is a bit less than his own 11:1148 ratio.Now, the number 11 in the latter comes straight from Archimedes taking the suns apparent size asthe 164 th part of 90, rather than Aristarchos . Put this in (13), including his own 22/7 value for pi,and the suns width becomes exactly 11 e, which is perhaps why he chose such an awkward looking .The other number in Archimedes ratio, 1148, comes from his long calculation in Sandreckoner 1.19,which essentially boils down to multiplying 164 by 7 (both the denominators of his chosen and ).This calculation is likely to have been tailored to yield a number that is the hypotenuse of a triangle (ordiagonal of a rectangle) whose sides are 60 and 1146: a personal interpretation of Aristarchos triangleby Archimedes the Geometer! Anyway, if we compute back the moons distance that corresponds to asuns distance of 1148 e, we get it as just slightly over 60 e, which Aristarchos would most likely rounddown to 60 e, thus confirming the only possibility left: that of his 10:60:1146 ratio.

eeer Sandreckon M 603sin1148 (15)

1 The symbol e is taken here to mean earth radii. Likewise, m and s will stand for the moons and the suns radiiwhen encountered, and M and S , for the distance to the moon and the sun respectively.

SunMoon

Earth

87

390

ee

11443tan

60

ee

11463sin

60

e60

e


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8. On Sizes and Distances

As stated before, a suns width of (at least) 10 e means that the sun is (at least) 5 times wider than theearth. This is in stark contrast to the fifteenth proposition of the book On Sizes and Distances , whichstates that the sun is 19/3 to 43/6 (a mean of 6) times wider than the earth. 1 This is not the onlystrange thing about this book that has long been ascribed to Aristarchos. The first who noted that itcant be genuinely his was Rawlins (2008:19), who pointed out a number of compelling argumentssuch as the fact that the author of On Sizes wrongly takes the suns and moons apparent size as 2(that is, four times the truth), 2 gainsaying Archimedes report ( Sandreckoner 1.10) that the true Aris-tarchos rightly takes it as . So, according to the author of On Sizes , whoever it was, eclipses can lastmore than half a day (or about four times longer than they really do)!

Now let us draw some conclusions from the data contained in On Sizes . If the sun is 6 times widerthan the earth (that is, if it is 13 e wide), then it must also be about 387 e away in order to subtend anapparent size of 2, as can be found by rearranging (2) thus:

wwd

180)2 / tan(2

= (16)

where d , w, and are the suns distance, width, and apparent size. From the suns distance thus ob-tained, we can derive the moons distance as 20 e by rearranging (3). 3 The same can be deduced fromthe seventeenth proposition of On Sizes , which states that the earth is 108/43 to 60/19 (a mean of2316/817) times wider than the moon. 4 If so (that is, if the moon is 193/270 e wide), then the moonmust again be about 20 e away in order to look 2 wide (see Table 1).

Table 1. Summary of sizes and distances (in terms of the earths radius),as derived from the 15 th and 17 th propositions of the book On Sizes .

Prop. 15 values lower [mean] upper Prop. 17 values lower [mean] upper

Suns width 12 13 14 Moons width 43/54 193/270 19/30Suns distance 363 387 411 Suns distance 344 387 430Moons distance 19 20 21 Moons distance 18 20 22Moons shift 6 5 5 Moons shift 6 5 5

1 The fifteenth proposition of the book On Sizes and Distances states that (ed. Heath 1913:402), ` , [5> +6 K #K +6 >% 59# 2 z w / #, !56 +Q z w / # / .Meaning (tr. Heath 1913:403), The diameter of the sun has to the diameter of the earth a ratio greater than thatwhich 19 has to 3, but less than that which 43 has to 6.2 The sixth hypothesis of the book On Sizes and Distances states that (ed. Heath 1913:352), X 5; > +4 4 %1+>. Meaning (tr. Heath 1913:353), The moon subtends one fif-teenth part of a sign of the zodiac [that is, 1/15

th of 30, or 2].3 Note that Heath (1913:350), quoting Hultsch (1900:199), rightly derives the same conclusion, with the only

difference that, instead of using the mean value of 6 e for the suns width as I have done here, he uses the lower19/3 value given in the fifteenth proposition and correspondingly obtains a distance for the sun and the moon ofabout 360 e and 19 e respectively. Had he used the upper 43/6 value, he would have obtained 411 e and 21 e in-stead, as readers can derive from the equations and method presented here and summarized in Table 1.4 The seventeenth proposition of the book On Sizes and Distances states that (ed. Heath 1913:408), ` +6 K #K +6 K 5; ! >% Q 59#1 !. z w [2] #, ! !56 +Q z w ( . Meaning (tr. Heath 1913:409), The diameter of the earth is to the diameter of the moon in a ratiogreater than that which 108 has to 43, but less than that which 60 has to 19.


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So, according to the 15 th and 17 th propositions of the book On Sizes , the moon is about 20 e away.At this distance, the moon would subtend a wild shift of 5 (vs. 2 in reality) as seen by observersplaced on opposite ends of the earths surface; this can be found by rearranging (11). If instead of si-multaneously watching the moon from two extreme points on the earths surface we watch it from apoint on the earths equator which moves with the earths rotation and gets two earth radii away every

twelve hours, we would witness the strange phenomenon of the moon looping its way across the sky(like the outer planets), as its mean sidereal motion (which amounts to about 6 in half a day) wouldadd its own touch to the shift of 5 mentioned above! 1 Since the moon travels 2 M (where M is themoons distance, taken here as 20 e) in one sidereal month (of 27.32 days), it follows that the moontravels 2 e in half a day. At the same time, our observer on the equator moves along an arc equal inlength to e (or half the earths circumference), which is more than the arc travelled by the moon. Thismeans that the equatorial observer travels faster than the moon, and Rawlins (2008:20), the first topoint this out, explains that the moon would be seen to move in retrograde against the background ofthe stars as a consequence of the observer being moved towards the moon or away from it in thecourse of half a day. 2 This happens because, though the observer moves faster than the moon, neithermoves in a straight line (while the moon may be taken to do so, certainly the observer cannot), and theresult is that in half a day the effective displacement of the equatorial observer is not e but 2 e, which

is not more but less than the moons displacement. So, at a distance of 20e

, we would see the moonfly a wide loop against the background of the stars starting at moonrise, slowing, halting, and revers-ing its motion only to resume moving forwards near moonset. 3

To add fuel to the flames, the 11 th proposition of the book On Sizes is not mathematically consistentwith the others. It states that the moon is 22 to 30 times farther away than it is wide. 4 So, by takingthe moons width from the 17 th proposition, we can easily work out the moons distance as rangingfrom 14 e (the result of 19/3022) to 23.89 e (the result of 43/5430), and averaging 19 e, instead ofthe 20 e derived from the 15 th and 17 th propositions. 5 Note that the closest of these distances wouldmake the moon fly loops so wide as could be seen even from Mediterranean climes!

None of these follies is ever mentioned by Archimedes. On the contrary, he reports different things.This very likely means that Archimedes, one of the greatest mathematical geniuses of all times and to

whom we owe the most detailed first hand account of Aristarchos theory, never read the book OnSizes , presumably because it was written after his (and Aristarchos) time. Actually, the book Ar-chimedes talks about describes an already fully blown heliocentric theory that places the stars at least a

1 Since the moon takes a mean of 27.321662 days to complete one sidereal revolution (that is, to cover 360around the earth in reference to the fixed stars), it follows that it covers about 6 in half a day.2 Note that Rawlins, following Heath, uses the lower of the moons distances derived from the 15 th proposition,that is, 19 e, instead of the mean value of 20 e used here, and therefore gets a moons parallax of 3. Note alsothat Rawlins refers to the moons (horizontal) parallax , which is the angle (of about 1 in reality) subtended atthe moons distance by the earths radius, while I use the moons shift , or angle (of about 2 in reality) subtendedat the moons distance by the earths width.3 The reader may actually see how this would happen in a free sky simulator like Stellarium : the programminginstructions are included in Appendix 1.4 The eleventh proposition of the book On Sizes and Distances states that (ed. Heath 1913:386), ` K 5; +6 , ; , : 4 4 K 5; K [4 k8, !56 4 ! z

+? , >% +Q z 5. Meaning (tr. Heath 1913:387), The diameter of the moon is less than 2/45 ths, butgreater than 1/30 th, of the distance of the centre of the moon from our eye.5 Eratosthenes is the first person ever to use a moon distance of 19 e (Rawlins 2008:6) as can be deduced from hisown value of 78 myriad stades, which comes from multiplying (and rounding) 19 by his own earths radius of40800 stades which, a hundredfold, is also the basis for his suns distance as bequeathed to us by Eusebios,Bishop of Caesarea-Palestine ( Praeparatio 15.53). Being a geocentrist, the former would prefer the 19 e value ofthe 11 th proposition to the slightly farther 20 e of the 15 th and 17 th propositions, all this assuming he derived hisown value from the book On Sizes (if indeed it had already been written), rather than the other way round.


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myriad times farther away than the sun, and the sun at least a myriad earth radii away! The book OnSizes , by contrast, though building upon some Aristarchan data and providing invaluable insight intothe 3 to half moon detail, seems the work of a geocentrist willing to bring both sun and moon as closeto earth as possible, little realizing the implications of his own constructions: a 2 wide moon flyingloops! Definitely, this doesnt sound like the Aristarchos Archimedes talks about in his Sandreckoner .

Thus, as far as we can tell, the 10c Greek copy of On Sizes kept at the Vatican, which is the oldest andbest preserved of its kind, is a fake: albeit a most valuable one. Proof of this is the most absurd of allits follies: why did Pseudoaristarchos take the suns and moons apparent size as 2 instead of asArchimedes reports Aristarchos really took? Rawlins (1991:69) suggests that Aristarchos used an an-cient unit called 4 or part , which was of a zodiacal sign. Neugebauer (1975:652, 671) attestshow the ancients used to divide the great circle of the sky in twelfths (each being a zodiacal sign of30), twenty-fourths (called steps or half signs of 15), and forty-eighths (called parts or half steps of7). Poseidonios, for one, whom we know was greatly influenced by Aristarchos, is said by Cle-omedes ( Heavens 1.10) to use exactly these. 1 So, while the real Aristarchos would say that the moonsubtends one fifteenth of a part of a sign (that is, 1/15 th of 7, or ), Pseudoaristarchos would takethis to mean that the moon subtends one fifteenth part of a sign (that is, 1/15 th of 30, or 2).

9. The solar system scale

Let us go back to the earlier Aristarchos: the one who placed the moon 60 e away, and the sun noless than 19.1 times farther away than the moon, that is, 1146 e away. Wouldnt he attempt to measurethe distance to the planets too? Why, to start with, Mercury and Venus might roughly be assumed tobe and of an AU from the sun (and this wouldnt be too far off from the truth as a first approxi-mation). Even Mars could be assumed to follow a similar pattern and be placed about 1 AU from thesun (vs. 1 AU in reality). One way to test and refine this guesstimate is to try to measure any plane-tary diurnal parallax , that is, the shift that the planets show against the background of the stars owingto the earths rotation.

If we place the sun 1146 e away in a simplified heliocentric system such as the one just mentioned,then, when at their closest to earth, both Venus and Mars should show a daily shift of about 18 to anequatorial observer, as can be found by putting w as 2 e and d as of 1146 e in (2). If we were in Alex-andria rather than at the equator, we would still witness a wild shift of 15 (comparable to half themoons disc). 2 This is plain enough to see with the naked eye. Mercury too, when closest to earth (thatis, when it is of an AU from us), would show a daily shift of about 9 to an equatorial observer (8 ifseen from Alexandria). Now let us see when the best time would be for such observations.

In the case of Mars, Jupiter, and Saturn, these observations would be best made when in opposition ,that is, when these planets are closest to earth and opposite the sun in the sky, staying visible thewhole night long, rising as the sun sets and setting as the sun rises; or, if you do not wish the planetsproper motion to bear too much on your observations during the time it takes the earths rotation towaft you over 2 e, you might prefer (as Flamsteed did on the night of October 6, 1672, when he meas-ured the parallax of Mars) to observe these planets when at a station , that is, when they seem to halttheir motion against the background of stars. There is no proper motion correction to make then.

1 In Cleomedes words (ed. Ziegler 1891:92), @+= b k %1+ B , !. . ' H +? b 4 9, H C] . 6 4 + , P ++> ', H 4 4. Meaning (tr. Bowen and Todd 2004:79), Posidonius divides the zodiacal circle(which, since it too divides the cosmos into 2 equal parts, is equal to the meridians) into 48 parts by dividingeach of its [twelve signs] into quarters.2 Rather than using just 2 e for this calculation, you may need the formula for the radius r of the parallel corre-sponding to a given latitude (that of Alexandria: 3112 50 N), which is r = r ecos( Lat )/Sqr(1-(1- r p / r e)sin( Lat )),where r e and r p are the earths equatorial and polar radii, and Lat is the latitude (Meeus 1998:83).


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The case of Mercury is most ill suited for this sort of observation, because it is always too close tothe sun and we are dazzled before we can detect any shift in the planets position. Venus is slightlybetter. When closest to us, it is said to be in inferior conjunction (that is, passing in front of the sun).This is not the best time to choose for naked eye observers, even though, today, with the help of spe-cial cameras and accurate clocks, astronomers use the rare transits of the sun by Venus to gauge its

parallax, which turns out to be about (or a whole shift of about 1 ) at this moment. Naked eye as-tronomers would do better to wait for Venus to be at one of its greatest elongations from the sun (thatis, when it remains longest in the sky as a morning or evening star), so that the earths rotation maywaft us along the greatest possible distance while it is visible.

As Rawlins (2008:27) points out, only three observations of a planet occulting a star have survivedfrom this far ago to our days: all are preserved in Ptolemys Almagest , all are Greek, all date fromwithin the period of Aristarchos lifetime, and two of them are dated according to the Dionysian cal-endar. Ptolemy doesnt give a hint as to the purpose of such unique observations, but he ( Alm . 10.4)mentions Timocharis of Alexandria (a close associate of Aristyllos of Samos) as the author of one ofthem: that of Venus passing over Zaniah ( Vir) on October 12, 271, at near dawn. This observationwas made a month and a half after Venus had passed its greatest morning elongation, so its not quitethe ideal moment to look for any possible shift in Venus position, though it still remains long in thesky and is not too bright to dazzle nearby stars and could, therefore, be used. The shift that could beexpected of Venus at this moment when seen from Alexandria in the simplified heliocentric modelmentioned earlier is about 1 .1 (This would be difficult to spot!)

Ptolemy ( Alm . 10.9) also mentions the occultation of Acrab ( B Sco) by Mars as recorded by some-one he doesnt name on January 16, 271, at near dawn. This moment is not the best we should choosefor measuring the shift of Mars, for it doesnt happen near a station. The moment chosen, instead, iswhen Mars is near its western quadrature (that is, at right angles with the earth-sun line), rising atmidnight and setting at noon. Yet, even so, given the simplified model and the early 1146 e value forthe suns distance we are considering here, even at this moment we should expect a shift of nearly 2 for Mars as seen from Alexandria. 2 (Still quite demanding for the naked eye to see!)

The last of the planet-star occultations mentioned by Ptolemy ( Alm . 11.3) is that of Jupiter passingover the Southern Ass ( + Cnc) on September 4, 240, before dawn. This observation was taken nearlya month before a station of Jupiter and, therefore, while it was still far from earth. At this moment,Jupiter should show a shift of about as seen from Alexandria, but if we assume Jupiter to be 1 AUfrom the sun (following the same pattern as assumed for the inner planets), its shift then becomes 1 .3

1 Supposing Venus to be AU from the sun and the sun to be 1146 e from earth, then Timocharis observation ismade when Venus is about 1181 e from earth. Taking the radius of the parallel at Alexandrias latitude to beabout 0.86 e (instead of 1 e at the equator), and putting twice this number in (2) together with the Earth-Venus dis-tance, will give a daily shift of 5 , but this is a maximum that can never be reached because, at this time, Venus isvisible only during 2 hours (not 12), so the only shift we can really expect to see amounts to about 1 .2 Supposing Mars to be 1 AU from the sun and the sun to be 1146 e from earth, then this observation is madewhen Mars is about 1605 e from earth. Taking the radius of the parallel at Alexandrias latitude to be as before,

and putting twice this number in (2) together with the Earth-Mars distance, will give a daily shift of about 3 ,but at quadrature Mars is visible only during 6 hours (not 12), so the only shift we can really expect to seeamounts to about 1.8 .3 Supposing the sun to be 1146 e from earth, then this observation is made when Jupiter is about 6539 e from earth.Taking the radius of the parallel at Alexandrias latitude as before, and putting twice this number in (2) togetherwith the Earth-Jupiter distance, will give a daily shift of about 1 , but on the date chosen Jupiter is visible onlyduring 4 hours (not 12), so the only shift we can really expect to see amounts to about . If we assume Jupiterto follow the same distance pattern as was assumed earlier for Mercury and Venus, that is, if we assume Jupiterto be 1 AU from the sun, or AU from earth when at its closest, things change: now Jupiter would be about2094 e from earth at this moment, which would yield a daily shift of about 2.8 , or about 1 in 4 hours.


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None of these observations is taken at a time when we should expect the greatest possible shift inplanetary positions, but they prove that Timocharis in Alexandria and perhaps someone else who had aliking for the Dionysian calendar had an interest in planet-star occultations. And why should this be?Rawlins suggests that these three unique occultations saved by Ptolemy from oblivion were, in fact,only part of a larger body of observations intended to find out whether the planets showed any daily

shift, as they surely should in the model originally proposed by Aristarchos, in which the sun is placedat least 1146 e away (let this model be called Early Aristarchan, or EA for short, from now on).Measuring these shifts would allow the suns distance to be more accurately defined. We have alreadyseen that in this system both Mars and Venus should be expected to shift about 18 (15 if observedfrom Alexandria) when at their closest to earth. We have also seen that the observation of the innerplanets Mercury and Venus is marred by their proximity to the sun. Therefore, Mars is our best hope,especially when at opposition and, best of all, when at a station. 1

When John Flamsteed measured the parallactic shift of Mars from Derby, England, on the night ofOctober 6, 1672, he found it to be 30 (vs. 38 in reality). 2 Giovanni Cassini measured it too thesame year, and found it to be 25 . Both astronomers realized that 1672 would be an excellent time forsuch an observation, for Mars would be at opposition near perihelion. Cassini used a different method:he observed Mars from Paris and sent an assistant to French Cayenne Island with instructions to recordthe position of Mars from there at a specified time, then he had to wait a year for his return before put-ting together his own observations and those of his assistant (Moreland 2002). Recently, the Marsopposition perihelion of August 8, 2003, was the closest Mars got to earth in some 60,000 years. Itgave astronomers an excellent opportunity to gauge Mars parallax, which Moreland (2003) found was22.9 2 (a shift of about 46 ) at this momentous occasion.

So, not even at their closest do any of the planets show any shift that is detectable to the naked eye.Besides, even though the typical human eye can see detail as fine as about 1 , observations of planetarydiurnal parallax are much more challenging than just trying to tell the moment of half-moon, or tryingto distinguish two bright stars against the background of the night sky. This time the problem is muchmore challenging: we have to compare the position of a planet at one moment with its position severalhours later! When Aristarchos failed to see the wild shift of 15 that his EA model predicted for Mars,

in fact, when he failed to see any shift at all, he realized that the 1146e

distance was still too short! Healso found that, if you multiply this number by 9, a curious symmetry occurs in this model, which thusbecomes what I shall call the Ultimate Aristarchan (UA) model. In the UA model the distance to thesun becomes 10314 e, which, rounded down to a myriad earth radii, is the distance Archimedes saysAristarchos finally adopted for the sun. At this distance the sun subtends a parallactic shift of exactly (which is the visual acuity Aristarchos used to determine the distance to the nearest stars). Theplanets, too, when at their closest to earth and assuming them to be regularly spaced by gaps of about AU in the UA model, are allowed a shift of exactly 1 for Mercury and 2 for Venus and Mars! Nextquestion: how big is the sun? Compute it by (13) and you get exactly 90 e wide. Now, in the same wayas the small moon moves round the big earth, lest the tail wag the dog, shouldnt the tiny earth moveround the huge sun? There is still one more piece of the puzzle to fit into place: how did Aristarchosget his 60 e distance for the Moon? Let this and more be the subject of the next part of this article.

1 The reader is encouraged to try and gauge any possible planetary shifts in this EA model, either throughmathematical formulas as I have done here or, even, by using a sky simulator like Stellarium in a similar way asindicated in Appendix 1: rather than bringing each of the heavenly bodies closer to earth in order to simulate theEA model, its much easier to simply change the earths radius from its 6378.14 km value to an imaginary sizeof 130489.127 km. This produces exactly the same effect as if the sun was 1146 e away. Try and see how muchthe planets shift against the background of the stars.2 The true value can be found by putting the earths equatorial diameter (2 e) in equation (2) together with thedistance (10678 e) from Earth to Mars at the specified moment (the night of 1672/10/6).


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Aristarchos of Samos the Polymath

Alberto Gomez [email protected]

(2011-07-26; revised on 2011-11-20)

Part 2Contents of Part 2

1. Introduction......................................23 2. The moons distance ........................24 3. Aristarchos solstice.........................29 4. Ancient year lengths.........................32 5. Precession of the equinoxes .............34

6. Clues on clay ................................... 35 7. Aristarchos Great Year.............. ..... 36 8. Tropical and sidereal years.............. 38 9. Summary ......................................... 39 10. Conclusion..................................... 39

1. Introduction

As we saw in Part 1, there are two main sources from which we learn of Aristarchos: one is thebook On Sizes , which presents him as a geocentrist ascribing to the moon a wrong apparent size of 2and a wrong mean distance of either 19 e or 20 e (as follows from Propositions 11, and 15 and 17); theother source is Archimedes, who presents him in the Sandreckoner as a heliocentrist ascribing to themoon a correct apparent size of and a correct distance of 60 e (as equation 15 proves). 1 Whether wechoose to trust one source more than the other is completely up to us, but Archimedes report showsthat Aristarchos got it all correctly, even when he said that, unless a star is less than a myriad times asfar away as the sun, we wont be able to spot any yearly shift in its position with the naked eye.

Attempts to reconcile both traditions were made, for example, when Heath (1913:312) suggested

that the book On Sizes was an early work written before Aristarchus had made the more accurate []observation recorded by Archimedes, but this cannot be so, because the 87 of the 4 th Hypothesis hadbeen computed from (and this implies that the real Aristarchos had already arrived at) a wide moon(as shown in Figure 4), and not from the 2 reported in On Sizes . Voltaire ( Phil. Dic. , System) wasthe first to suspect that this book is a sham, but he p

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Aristarco de Samos - GÓMEZ GÓMEZ, A. (2011)