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Eudoxussupposedthatthemotionofthesunorofthemoon
involves，ineithercase，threespheres，ofwhichthefirstisthe
sphereofthefixedstars，andthesecondmovesinthecirclewhich
runsalongthemiddleofthezodiac，andthethirdinthecirclewhich
isinclinedacrossthebreadthofthezodiac；butthecirclein
whichthemoonmovesisinclinedatagreateranglethanthatinwhich
thesunmoves。Andthemotionoftheplanetsinvolves，ineachcase，
fourspheres，andofthesealsothefirstandsecondarethesameas
thefirsttwomentionedabove(forthesphereofthefixedstarsis
thatwhichmovesalltheotherspheres，andthatwhichisplaced
beneaththisandhasitsmovementinthecirclewhichbisectsthe
zodiaciscommontoall)，butthepolesofthethirdsphereofeach
planetareinthecirclewhichbisectsthezodiac，andthemotionof
thefourthsphereisinthecirclewhichisinclinedatanangleto
theequatorofthethirdsphere；andthepolesofthethirdsphereare
differentforeachoftheotherplanets，butthoseofVenusand
Mercuryarethesame。
CallippusmadethepositionofthespheresthesameasEudoxus
did，butwhileheassignedthesamenumberasEudoxusdidtoJupiter
andtoSaturn，hethoughttwomorespheresshouldbeaddedtothe
sunandtwotothemoon，ifoneistoexplaintheobservedfacts；
andonemoretoeachoftheotherplanets。
Butitisnecessary，ifallthespherescombinedaretoexplain
theobservedfacts，thatforeachoftheplanetsthereshouldbeother
spheres(onefewerthanthosehithertoassigned)whichcounteract
thosealreadymentionedandbringbacktothesamepositionthe
outermostsphereofthestarwhichineachcaseissituatedbelow
thestarinquestion；foronlythuscanalltheforcesatworkproduce
theobservedmotionoftheplanets。Since，then，thespheres
involvedinthemovementoftheplanetsthemselvesare——eightfor
SaturnandJupiterandtwenty-fivefortheothers，andoftheseonly
thoseinvolvedinthemovementofthelowest-situatedplanetneed
notbecounteractedthesphereswhichcounteractthoseofthe
outermosttwoplanetswillbesixinnumber，andthesphereswhich
counteractthoseofthenextfourplanetswillbesixteen；therefore
thenumberofallthespheres——boththosewhichmovetheplanetsand
thosewhichcounteractthese——willbefifty-five。Andifonewere
nottoaddtothemoonandtothesunthemovementswementioned，
thewholesetofsphereswillbeforty-seveninnumber。
Letthis，then，betakenasthenumberofthespheres，sothatthe
unmovablesubstancesandprinciplesalsomayprobablybetakenasjust
somany；theassertionofnecessitymustbelefttomorepowerful
thinkers。Butiftherecanbenospatialmovementwhichdoesnot
conducetothemovingofastar，andiffurthereverybeingand
everysubstancewhichisimmunefromchangeandinvirtueofitself
hasattainedtothebestmustbeconsideredanend，therecanbeno
otherbeingapartfromthesewehavenamed，butthismustbethe
numberofthesubstances。Forifthereareothers，theywillcause
changeasbeingafinalcauseofmovement；buttherecannotheother
movementsbesidesthosementioned。Anditisreasonabletoinfer
thisfromaconsiderationofthebodiesthataremoved；forif
everythingthatmovesisforthesakeofthatwhichismoved，and
everymovementbelongstosomethingthatismoved，nomovementcan
beforthesakeofitselforofanothermovement，butallthe
movementsmustbeforthesakeofthestars。Forifthereistobea
movementforthesakeofamovement，thislatteralsowillhaveto
beforthesakeofsomethingelse；sothatsincetherecannotbean
infiniteregress，theendofeverymovementwillbeoneofthe
divinebodieswhichmovethroughtheheaven。
(Evidentlythereisbutoneheaven。Foriftherearemany
heavensastherearemanymen，themovingprinciples，ofwhicheach
heavenwillhaveone，willbeoneinformbutinnumbermany。But
allthingsthataremanyinnumberha
