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ZENO OF ELEA
A Texty with Translation and Notes
BY
H. D. P. LEE Fellow and Tutor of Corpus Christi
College, Cambridge
CAMBRIDGE
A T THE UNIVERSITY PRESS
42
MOTION
§ A. Place and motion
17· Diogenes, IX. 72; Diels, fro 4 Z-riv 0 ' " , - \ , , , " , ., WV E T7)V KLVTJa£v avaLpn AEyWV TO KLVOUJLEVOV OUT
, I .... , " .,. \" EUTt T07Tep KLVHTa£ OUT EV ep JLTJ EUTt.
18. Epiphanius, adv. Haer. III. I I; Diels, Dox. 590 . 20
'A' < 'Z' )" , Ka£ Eyn sC. 0 TJVWV OUTWS' TO KLVOVJLEVOV 7/TOL €V , .... """"'" T07Tep KLVE£TaL TJ EV ep OUK EUTL. Ka~ ounE €V c:; €UTL T67TW . . 5 " , 'l' ,,, ,,,
OUTE EV ep OUK EUTtv· OUK apa TL Ktv€£Tat.
§ B. The four arguments on motion
I. THE DICHOTOMY
19· Aristotle, PAys. Z 9. 239b 14~ Diels, A 25 , 0" , \' " TETTapES E£m AOYOL 7TEpL KLVTJUEWS Z1jvwvos ol TUlr.rlFVrlV..,.".
TaS OUUKoAtas TOtS AVOUUt-7TPWTOS JLEV 0 1TEp2 TOU JL~ KL·U<".fTN,u
Ott]. T" '\ " ~ - , ..I.. ' fJ '. o 7TpOTEpOV ns TO TJJLLUU OELV a'f'£KEU at TO ~Ep6JLEVOV ~ .
TO TlAO~ '.. '" ,\. , - , ~, 1TEpL OU O£E£AOJLEV EV TOts 7TpOTEpOV A6yOtS'. sc. PAys. Z 2 •.
10233 a 21 OtO Kat 0 Z1jvwvos A6yos tPEUOOS AaJL{1&'VE£ TO JL~ €VOE
XEufJaL TO. a7THpa O£EAfJEtV ~ uif1aufJa£ TWV a7TE£pwv KafJ' €KaUTOV
€V 7TE7TEpauJLlvep Xp6vep. Otxws yo.p ,\IYETat Ka2 TO JLfjKOS Ka2 <I '" , "\
XPOVOS a7TE£pov, Ka£ OAWS miv TO UUVEXls, 7/TOt KaTo. OtatpEUtV ~ TOtS €fTVc. ~'''' , " . -~ TOtS. TWV JLEV OUV KaTa 1TOUOV a7TEtpwv OVK €VOlXETaI
15 uif1aufJa£ €V 7TE1TEpauJLEvep Xpovep, TWV Oe KaTo. O£atpEu£v €volXETat. Ka" , \ r I "" ~ £ yap aUTOS 0 Xpovos OUTWS a1THpoS. WUTE €V TW a7TE£pw Kal, , , ..... .. '" OUK EV Tep 1TE7TEpauJLlvep uuJL{1a{vn OttEVat TO a7TELpOV Ka2 U7TT€ufJaL . TW " ... , I , .... v a1THpwv TOtS a1TE£pOLS, OU TOtS 1T€1T€paUJLEVOtS.
MOTION
§ A. place and motion
Diogenes, IX. 72; Diels, fro 4
43
does away with motion, saying that "what moves does not either in the place in which it is or in the place in which it is
Epiphanius, adv. Haer. III. 11; Diels, Dox. 590.20
argues as follows: what moves, moves either in the place it is, or in the place in which it is not; and it does not move
in the place in which it is or in the place in which it is not; therefore moves.
§ B. The four arguments on motion
'1. THE DICHOTOMY
. Aristotle, Phys. Z 9. 239b 14; Diels, A 25
are four arguments of Zeno about motion which give to those who try to solve the problems they involve. The
says that motion is impossible, because an object in motion 10
reach the half-way point before it gets to the end. This we discussed above. sc. PAys. Z. 2. 233a 21 Hence Zeno's argumakes a false assumption when it asserts that it is impossible to
tra.verse an infinite number of positions or to make an infinite number '-V.lUQ''-LO one by one in a finite time. For there are two senses in 15
length and time and, generally, any continuum are called namely either in respect of divisibility or of extension. $0
it is impossible to make an infinite number of contacts in a time where the infinite is a quantitative infinite, yet it is possible
where the infinite is an infinite in respect of division; for the time 20
itself is also infinite in this respect. And so we find that it is possible to traverse an infinite number of positions in a time in this sense infinite, not finite; and to make an infinite number of contacts because its moments are in this sense infinite, not finite.
44 MOTION THE DICHOTOMY 45
20. Simplicius, 1013.4, ad 2390 10
The first argument is the following: If there is motion, then a moving object must in a finite time complete an infinite number of positions, but since this is impossible there is no motion. He proves his hypothesis thus: An object in motion must move through a certain distance; but since every distance is infinitely divisible the moving object must first traverse half the distance through which it is moving, and then the whole distance; but before it traverses the whole of the half distance, it must traverse half of the half, and again half of this half. If then these halves are infinite in number, because it is always possible to halve any given length, and if it is impossible IO
to traverse an infinite number of positions in a finite time--this Zeno assumed as self-evident 1 and Aristotle has previously referred to his argument when he speaks of it being impossible to traverse an infinite number of positions or to make an infinite number of contacts in a finite time--anyhow, to resume, every magnitude has IS an infinite number of svbdivisions, and therefore it is impossible to traverse any magnitude in a finite time.
Simplicius,947. ), ad 233a 21 Zeno's argument is the following: tIf there is motion, it is possible in a finite time to traverse an infinite number of positions, making an infinite number of contacts one by 20
one; but this is impossible, and therefore there is no motion. His hypothesis he proved by means of the infinite divisibility of magnitude. For if every magnitude is infinitely divisible, it will be made up of an infinite number of parts, and so a body, moving through and traversing a distance of given magnitude, will move through and 25 complete an infinite number of positions and make an infinite number of contacts in a finite time, that is, in the time it takes to move through the whole finite distance. He says" to make an infinite
.. number of contacts one by one", because it might seem that a body traversed an infinite number of positions by passing over them without 30 making contact with each. In this way he proves the hypothesis. The minor premiss, which says "but it is impossible (I) to traverse an infinite number of positions or (2) to make an infinite number of contacts in a finite time", he proves (I) from the inte~minability of the infinite and (2) from the impossibility of making an infinite 35 number of contacts in a finite time, if the moving object makes contact with the successive parts of the distance in question at successive moments of time; for he said that it is impossible to make contact with each member of an infinite collection because the person
. making the contacts is as it were counting, and it is impossible to 40 count infinite collections.
MOTION
23. Aristotle, Phys. 0 8. 263a 5 ; \ , \ ') ,.... s
25 70VaV'TOv o€ 7P07TOV G.7TUV'TTj'TEOV KaL 7TpOS TOUS EpWTwv::a
Z~VWVos i\oyov, Kat dgwvv'Tas, d dd TO i/fLwu OWfVUL on,
THE DICHOTOMY 47
21. Philoponus, 802.31, ad 233a 21
Zeno in doing away with the reality of motion made use of the following syllogism. If there is motion, it is possible to traverse an infinity of positions in a finite time; but this is impossible; therefore there is no motion. For suppose a thing moves over the length of a cubit in one hour: then, since there are an infinite number of points 5 in every magnitude, the moving object must make contact with all these points in the course of its movement: it will therefore traverse an infinite number of positions in a finite time, which is impossible.
Philoponus, 81. 7, ad 187a I To show that this one is also unmoved, he made use of the following argument. If anything moves IO
along a given finite straight line, it must, before moving along the whole of it, move along the half of it, and, before moving along the half of the whole, it must first move along a quarter of it, and before a quarter an: eighth, and so on ad infinitum; for the continuum is infinitely divisible. So if anything moves along a finite straight line, 15 it must, before completing its movement, have moved through an infinite number of magnitudes: but if this is so, and if every movement occupies a definite finite time (for there is no motion that occupies an infinite time), then we find that in a finite time a motion through an infinite number of magnitudes has taken place, 20
which is an impossibility; for the infinite is interminable absolutely.
Themistius, 186. 30~ ad 233a 21
Of this Zeno either is or pretends to be ignorant, when he supposes he does away with motion by saying that it is impossible for a moving body to traverse an infinite number of positions in a finite time and for a body A to make an infinite number of contacts 25
by one; for the foot length is divisible into an infinite number parts and ad infinitum, hut the time taken by a motion through
distance is finite.
Aristotle, Phys. 0 8. 263a 5
The same method should be adopted in replying to those who put Zeno's puzzle, and claim that in traversing any distance we must 30
25. [Aristotle), De Lin. Insec. 968a 18
One of the arguments used by the s~ppor:er~ of the I >In OE KaTe. TOV Z~VWVOS' AOYOV avaYK'Y) TL UE''VEI10S
ypa[k[kat. E I I
, \ tvat eL7TEP dovvaTOV [kEV EV 7TE7TEpaU[kEVCf/ XpovCf/ 25 a[kEpES' E , " <;»> \ \" t
".1. eat Kae> €KaaTOV a7TT6[kEVOV, avaYK'Y) 0 E7Tt TO 'Y)[k av a~aa , 0' \' .,... '7T"CVJW..-
'rEpOV dcptKVELueat TO KtVOV[kEVOV, TOV E [k'Y) a[kEpOVS'
EaT£V 7J[k£UV,
THE DICHOTOMY 49
first traverse the half of it, that these subdivisions are infinite, and that it is impossible to complete an infinite number of distances: or, as some, who put the puzzle in a different form, claim, that in the course of its motion the moving body must, as it reaches each
. half-way point, count the half of this half, so that when it has moved 5 through the whole distance it has counted an infinite numberwhich is admittedly impossible.
The argument of Zeno, to which he now refers, was as follows: If there is motion there will be something which has traversed an ,infinite number of positions in a finite time; for, since the process 10
of dichotomy can continue infinitely, in every continuum there will be an infinite number of halves owing to every part of it having a half. A body therefore which has moved over a finite distance will have traversed an infinite number of halves in a finite time, that is,
the time which it took to traverse the finite distance in question. IS ,then goes on to assume the opposite of the consequence that
from his hypothesis, i.e:' he assumes that it is impossible to to the end of an infinite number of positions in a finite time,
it is impossible absolutely to exhaust any infinite collection, so does away with the reality of motion. So Zeno argued: but 20
Aristotle says, put ,the puzzle in a different way, as follows: is motion, since there is an infinite number of halves in any
~(Jn'timlUn[}, a body moving through a continuum should be able to each of these halves as it coines to it. But if this is so, then the moving body has traversed the finite magnitude in ques- 25
the counter will have counted an infinite number of halves. If 'n~.,~"_~' it is impossible to count an infinite number, then any
from which this follows as conclusion must be impossible; premiss from which it followed was the supposition that is real.'
they suppose that the necessary result of Zeno's argument must be some indivisible magnitude. For it is impossible
an infinite number of contacts one by one in a finite time:
30
a moving body must first reach the half-way point of any disand there is always a half of any distance not absolutely 35
4
MOTION
II. THE ACHILLES
26 Aristotle, Phys. z 9. 2396 14; Die1s, A 26 ,,\ • , ( aA' , AXLAAEtIS (tun 0' OOTOS on TO OEVrEpOS OE 0 K OUf-LEVOS e f 'elov tmo TOU TaX{U70V'
{3paoVraTov I OUOE1TOTE "...aTaA,~cp() JUH~\. WKOV oBEY WP f-LYjU€ TO " e ' 'aYKaWY E/\ HV TO U. '" '»' EfL1TpOU E;: raf, av 'V civa Kat:ov TO {3paouiaTOV. wn u~ cpEVyOV, WU7 aH n 1TP~EXH _ I' '> a"-EpH OE EV TW OtaLpELV f-L1J
'1' (' 'S TW ULXOTOfLHV, ot 't' • , _" \ " 5 Kat OUTOS 0 aUTO. , e ' nEV OOV 11:11 KaTCJ.Aaf-L-",f , \ Ravall.EVOV f-LEYE as. TO , ,"/ '>' utxa TO 1TPOUl\a f-Lp, R'R 'K au Aayou ytyvETaL OE R' e 'f3 OUTEPOV UUf-LpEpYjKEV E T , {3' \ pavEu at TO pa ." "-OTEPOLS yap UUf-L aLVE£ f-L1J
' ".,...n O,xoT0f-LLa-ev afL't' 'AA' 1Tapa TaUTO . iI \ , '0 OU lvou 1TWS TOU f-LEYE80us' a a
acpLKVEZu()aL 1TPOS TO '".,Epas, '" ~aLp' I! ~OV TETpaywO'Yl f-LEVOV EV Tep - , , TW OTt OUUE TO TaXLv. ."/ i \
10 1TpOUKHTaL EV TOU, • '" " Kat T~V AUULV E vaL T1JV OtLVKHV TO {3paoUTaToV-WUT avaYKYj
aUn}v.
27 Simplicius, IOI]. 31, ad lac. I
• , 1'" '\ , '.,...ns E1T' a;'EtpOV OWLPEUEWS E1TLKEXHp'Y)Ta! Kat aUTOS a I\oyos EK . "/ , " I \
, '" av Towuros' H EUTL KW'Y)ULS, TO KaT' ru'Y)V OLaUKEUYjV. Kat HYJ _ \ "-e f
'" f U'1T<) TOU TaX{UTOU OUOl1TOTE KaTCJ.A'Y)'t' YJUE~aL' 15 {3pauuTaTov '" (9) "-Y'uv"""
- ''> , • OUK apa EU7L KWYjULS.... 1014. f-L~v TOUTO aouvaTOV ., , ~ 'A AAEwS ... 'A' as a1TO TOU 1TapaAYjcp()lvTOS EV aUT<tJ, XL
OUV 0 01' ", A' v O'WKOJlTa OV aouvaTov CPYjUL~ 0 ~oyos T\Yj~/. JfE :;'Yj 1TPO TOU KaTaAa{3ELv
\ , , f K'YI /l.EV TO KaTaAf)'t'0f-LE , .. "" KaL yap avay "/ r- ~ Q() 'I;' 71UE TO "-EUyoy. EV <tJ UE
" 'AeELV 1TPWTOV a EV E",WPf-L"/ 't' I I
20 TO 1TEpas E , I 'OUTW TO cpEiJyov 1TponUL OtWKOV E1Tl TOVTO 1TapaYLVETaL, EV T " _ ~
, '''\ .. pO-nAeEV TO OLWKOV T<tJ '>'a' uT'YIlI·a H Kat El\aTTOV OU 1T "/ \' .. '\ 0" . "J, , , \' .... ' Kat €V W '1Ta/UV ElVaL' dAA' OOV 1TpaHULV' OU yap YjPEf-LH~ ,. I 'ALV
TOUTO OtEWL TO OLWKOV (] 1TpOij,\{}E TO" CPEUYOV, !V T~UT<tJ 1T~ f '>" , JEUYOV TOUOVTW EAaTTOV ou 1TPOT(,POV xpovw OI..€LOf.. TL TO ¥' ... '(j"
25 • '" ' " V aUTO TOV OU.vKOVTOS. Kat OUTW, EV OUW {3paUVTEpoV Eun f e \ "- _
xp6vw EV cL TO OLWKOV otHULV, 0 1TPOEAYJAU E T~ 't'EvrOV, , • • I I \ \ "-EVYOV' Kav yap aEt ~ EV TOVTW 1TPOEWL TL Kat TO 't' ~ ~ "" , ,
OV, • '" I v oAws TW UE E1T ~'\ \' ovv oiEtu{ Tt KaL aUTO KtVOV f-LEVO '> \' ,', '" pOV UI\/\ f A {3 I tv ota TI]V E1T a1TEL "AaTTOV aAAo ruou OtauTYJf-La afL ~VE\ ~ 'A AA f ,
30 E 8 ~ I OU lJovov ~EI(TWP U1TO TOU XL EWS OU f-L€YE WV T0f-LYJV, r
AYJcpe~UETaL, dAA' OUOE i] XEAWvYJ. d d th I Following the Loeb note ad Joc. I have. emen ~ 1 lIe Bekker
of Bekker's text to {3paoVTaT!JlJ, both here and m 1. 4 (a . {3PaOVTUTOV) .
THE ACHILLES
II. THE ACHILLES
26. Aristotle, Phys. z 9. 239b 14; Diels, A 26.
The second is the so-called Achilles. This is that the slowest runner will never be overtaken by the swiftest, since the pursuer must first reach the point from which the pursued started, and so the slower must always be ahead. This argument is essentially the same as that depending on dichotomy, but differs in that the successively given lengths are not divided into halves. The conclusion of the argument is that the slowest runner is not overtaken, but it proceeds on the same lines as the dichotomy argument (for in both, by dividing the distance in a given way, we conclude that the goal is not reached: only in the Achilles a dramatic effect is produced by saying that not IO
even the swiftest will be Successful in its pursuit of the slowest) and so the solution of it must be the same.
27· Simplicius, IOI]. 31, ad lac.
This argument also bases its attempted proof on infinite divisibility but is arranged differently. It runs as follows: If there is movement the slowest will never be overtaken by the swiftest: 15 but this is impossible: therefore there is no motion .... (1014. 9) The argument is called the Achilles because of the introduction into it of Achilles, who, the argument says, cannot possibly over-
the tortoise he is pursuing .. For the overtaker must, before overtakes the pursued, first come to the point from which the 20
started. But.during the time taken by the pursuer to reach point, the pursued advances a certain distance; even if this
is less than that covered by the pursuer, because the pursued the slower of the two, yet none the less it does advance, for it is
at rest. And again during the time which the pursuer takes to 25 this distance which the pursued has advanced, the pursued
covers a certain distance which is proportionately smaller than according as its speed is slower than that of the pursuer.
so, during every period of time in which the pursuer is covering distance which the pursued moving at its lower relative speed JO already advanced, the pursued advances a yet further distance; even though this distance decreases at each step, yet, since the
is also definitely in motion, it does advance some positive And so by taking distances decreasing in a given pro
ad infinitum because of the infinite divisibility of magnitudes, arrive at the conclusion that not only will Hector never be Over- J 5
by Achilles, but not even the tortoise.
MOTION
III. THE ARROW
28. Aristotle, Phys. Z 9· 239 h 30
TpETOS 0' 0 VVV p1')8ds, on ~ OLUTOS cpEpOJLEV1'} €U: KEV•
f3alvEt 8~ 7Tapa 70 AafLf3aVEW TOV XPOVOV UUYKELu~at EK
fL~ OtOOfLEVOV yap TOlhov OUK gU7aL 0 uvMoytUfLos.
30. Simplicius, 1015· 19, ad 239 h 30 fJ ,,' I -n
TO CPEPOfLEVOV f3EAOS EV T0 cpEpWe<J.L W'TaTaL, EL7TEp <J.V<J.yK,/
" ~ e ", ~ 'OE A. Epa ItEVOV dd K<J.Ta TO rUOV 10'l1 KLVEtU at YJ YJPEfLEtV, TO 'f' r A "
" " , ~ , apa , , ", a' ~t' K<J.Td. TO Zaov EaVTW ov OV Kw€tTat· 1')PEfL€t •
EU7L. 70 DE c •
31 • Simplicius, 1011. 19, ad 239 D 5 , \ , \ p' " 7Tav OT<J.V ii KaTa o oE Z~vwvos /\Oyos rrpOl\a,.,wv, on ,
"" ... ,q '~EV £V EaVTW 7) KLVELTat YJ YJPEfLH, Kat on OVO ., " .. , \., ......" ~ '" ') a7L Ka8 €KaaTOV
OTt TO CPEPOfLEVOV aEt EV Tip tuip aVTip E "
15 EWKEt uvMoyt~EaeaL OV7WS" 70 cpEpafLEVOV f3lA~S EV, 7TaVTL \ , " \ , \ TW XPOVW' TO , \ "UOV ~a.VTW~ :taTW WUTE Kat. EV 7TaVTt, • Karu 70 t c.' , " \ 8'
, ~ ~ nTa' TO LUOV EaVTw OV of; KtVEt:7at, E7TEWYJ p.1') EV EV TW VVV K~ • ,A , 0' A
W~ v' VV KLvE'irat· TO O~ fL~ KLVOVfLEVOV YJPEfLEt, E71H 1) 7Tav l' 1!! fA " A. I
K:Ve"tTat 7) ~PEfLEZ' TO upa CPEPOfLEVOV ,.,E OS, Ewe; 'f'EpETat,
20 Ka7d. 71aVTa 70V Tfj, cpopas xpavov.
I ~ -. In all the manuscripts, as in Simplicius and YJ KLVHTCL.. , ., (below, Nos. 31 ,32)' Zeller, on the ground that. Zena s premIss 15 a nirian of rest, ejected the words. Themistius omHS them (below, No.
z (OVOEV OE KLV€LTCLt) Diels.
THE ARROW
III. THE ARROW
Aristotle, Phys. Z 9. 239b 30 The third is that just given above, that the flying arrow is at rest.
conclusion follows from the assumption that time is composed of instants; for if this is not granted the conclusion cannot be inferred.
Aristotle, Phys. Z 9· 239 b 5 Zeno's argument is fallacious.F or if, he says, everything is either 5
at rest or in motion, but nothing is in motion when it occupies a space equal to itself, and what is in flight is always at any given instant occupying a space equal to itself, then the flying arrow is motionless. But this is false, for time is not composed of indivisible
·1":'l<1.l1l" any more than any other magnitude is composed of in- 10
divisibles.
3o.Simplicius, 1015.19, ad 239b 30
The flying missile is at rest during its flight, if everything must be in motion or at rest, but an object in flight always occupies
equal to itself. But what always occupies a space equal to is not in motion, it is therefore at rest.
• Simplicius, 1011. 19, ad 239 b 5 Zeno's argument after making the preliminary assumptions that
when it occupies a space equal to itselfis either in motion at rest, that nothing is in motion in the instant, and that an object flight occupies at each instant a space equal to itself, seems to
15
as follows: ,The flying missile occupies a space equal to itself 20
each instant, and so during the whole time of its flight: what ' .... UjJIC" a space equal to itself at an instant is not in motion, since
is in motion at an instant: but what is not in motion is at rest, is either in motion or at rest: therefore the fiying
while it is in flight, is at rest during the whole time of its flight. 25
E-ll TW'IIVV TW Ka.nt TO rUOV rcF: TW KCL'nt TO LUOV am. eett. Bekker. Diels (Vors. ;9. A. 27) has the follo~ng readings: ~ KtvEiTCLt (ouoEv KWE'iT<J.t) GTaV ii KUTd TO LUOV, Eun 3' dd TO f€p6f1-EVOV EV T0 vvv,
BE Kurd TO rUOV .Iv r0 VVV), K.r,/o.. (cf. Philop. 817.6, ad 239b 6 71p6u()E, 71av oE ro EV r0 vvv Ell r0 tao/ EUVTOV inrapXH r6710/).
MOTION
32- Simplicius, 1034· 4 EK Se Toth·ov Kal TOV Z1}vwvos EAvaE AOYOV TOV AEyovTa El 'TO
, ., \ ,\" ( ....' I \ 8. ~ \ rpEpOJ1-EVOV {3EAOS aEL KaTa TO Laov EaUTcp EaTL, TO E KaTa " • ~, \ '\' A \.J.. ' ov aE"o'" €w'" Laov EaVTcp XPOVOV TLva OV '1PEJ1-Et, TO 'f'EP0J1-EV P 1\ ~ . ~
KLvfjTaL ~PEJ1-Et.
33. Philoponus, 816. 30, ad 239 h 5 d:rrav, rp'Y}atv, EV 'To/ racp JaUTov TOTTCP {lTTapxov 1j ~PEJ1-E'i
, (' , (' \' A" • A A 0 ' E II.Et /ipa KLvE'iTat, aovvaTOV DE EV Tcp tacp EaVTOV K£VEta at, '1P r •
, , a " ' • , A - A' 0' TOtVVV rpEpOJ1-EVOV pEI\OS EV EKaaTcp TWV VVV TOV XPOVOV Ka
.,,, t _, (I ~ , ' 8' , K£vELTaL EV Lacp EaVTOV TOTTCP VTTapxoV 'Y}pEJ1-TJaEt, EL E EV
I .... " I '" ., .... \' , TOLS 'TOV XPOVOV ,VVV aTTELpOLS OVaLV '1PEJ1-Et, Kat EV TTaVTL
, "" A ' \ a'A " 10 J1-'1 aEt • ci.AA VTTOKEtTat KLVOVJ1-EVOV' TO pE OS apa
~pEJ1-1}aEL.
34. Themistius, 199· 4, ad 239h 1 ", .... ..1.' q f1 ~ \\Jl v
Et yap 'Y}pEJ1-Et, 'f''Y}atV, f aTTaVTa, OTav 1I Ka7'a TO tao " ~\ J' \.J.. ., \ \" ( ... StaaT'1J1-a EaTL DE aEt TO 'f'EpOJ1-EVOV KaTa TO Laov EaV7'cp OIJ'urTJllLU..
" " "'1 \.J.. ' aKtV'T]TOV avayK'Y) 'T'T]V otaTOV E vat 'T'T]V 'f'EpOJ1-EV'T]V.
Themistius, 200. 29, adds nothing.
IV. THE STADIUM
35- Aristotle, Phys. Z 9· 239 h 33 , "-" \ ~, ..- I , El:
15 TETap-ros 0 0 TTEpL TWV EV aTaotcp KtVOVJ1-EVWV !>
iawv OyKWV TTap' Zaovs, TWV J1-EV aTTo TEAOVS TOV aTa8tov TWV , \ I " , ,.,. R I " " tv (, aTTO J1-EUOV, tacp TaXEL, EV cp aVJ1-paLVELV OLETat Laov E a
I ,~ ,,~, ( . , \',..,
To/ SmAaaLcp TOV '1J1-Lavv. EaT' 0 0 TTapal\oYLaJ1-0S EV Tcp TO \ , \ ('\ " _ \., , 0 '/:. A
TTapa KLVOV}.tEVOV TOOE TTap '1PEJ1-0VV TO Laov J1-EYE OS a!>LOVV , \" .J.. ' 0 ' ~ ... ", .1. AS ,..
20 raw TaXEt TOV Laov 'f'EpEa at XpOVOV' TOUTO b EaTt 'f'EV O~. , t • "" '.J..'" \ AA t..-' '.J..' ..
EaTwuav Ot EaTWTES UTOL 0YKot E'f' WV Ta ,Ot 0 E'y WV
B " '\ A' [-AI]" \ '0\ B apxoJ1-EVOL aTTO TOV J1-Eaov TWV , Laot TOV apt }.tOV
10m. EHI, Ross. TWV A cett: cf. Simp. 1017·4·
THE ARROW 55
• Simplicius, 1034· 4
By this reasoning also he disproved Zeno's argument that if the missile always occupies a space equal to itself, and if what
t''''''''''~';A.~ for any time a space equal ~o itself is at rest, then the flying IS at rest all the, time it is in motion.
· PhilopoilUS, 816. 30~ ad 239h5 Eve~ing, ~e says, that occupies a space equal to itself is either 5 rest?r In m<>,tlOn; but it is impossible for anything to be in motion
It occuI:le~ a space equal to itself, and it is therefore at rest. . . ,mlssd,e, therefo,re, at every instant of the time during
• ,It ~s In motIOn occupIes a space equal to itself and so is at rest; Iflt IS a! res! at all the instants of this time, which are infinite IO
number, It wIll be at rest during the whole time. But it was SUlllpo:sed ~o ?e ~n motion: and our conclusion is therefore that the .. ",,,,., ..... mIssIle IS at rest.
Themistius, 199. 4~ ad 239b 1
For if, ,he says, everything is at rest, when it occupies an extension to ~tself, but an object in flight always occupies an extension 15
to Itself, then the flying arrow must be motionless.
IV. THE STADIUM
Aristotle, Phys. Z 9. 239 b 33
fourth is the one about the two rows of equal bodies which past each other in a stadium with equal velocities in opposite
. the one row o~ginally stretching from the goal (to the , of the stadlUm, the other from the middle-point (to 20
start1?g P?st>. This, ~e thinks, involves the conclusion that half · tlme ,IS equal to Its double <i.e. the whole time>. The fallacy In ~sumlng that a body takes an equal time to pass with equal
a body that is in motion and a body of equal size at rest assumptionw~ich is false. For example, let AA be the stationa~ 25
of equal SIze, letRE be the bodies, equal in number and size
" '" 0 < N '.J.' '.. 'rr" ~, , OVrES KaL ro p,(£y(£ oS", OL 0 (£'1' WV ra a7rO rou (£oxarOV,
"0'" I "'0 " rOY apL p,ov ovr(£S" rourOLS" KaL, ro p,(£y(£ OS", KaL Luorax(££S" rots-f3 ' '1>'1' - B" ., -" t ' uvp, aLV(£L 0"1 ro 7Tpwrov ap,a (£7TL rep (£oxarep (£ vaL KaL
7Tpc'lJrov r, 7Tap' aAA1JAa KLVOUp,EVWV. uup,{JaLV(£L 0~2 ro r 5 7TCl.vra ra B3 OL(£~(£A1JAvOEvaL, ro o~ B4 7Tapa ra -fJp,lU1J'
" t ' I " ,,(/ " , 1Jp,LUVV (£ vaL roy xpOVOV·- LUOV yap (£Kar(£pov (£UTL 7Tap ., 1>' f3' , - B5 " , r ap,a U(£ uvp, aLVEt ro 7Tpwrov 7Tapa 7Tavra ra 7TapEA7JAvOEVat
ap,a yap ([urat ro 7Tpwrov r Ka~ ro 7Tpwrov B E7T~ ro£S" , ., [" , , «'I' • I 6 (£uxaroLS", tUOV xpovov 7Tap EKaUTOV YLYVOP,(£VOV rwv B
10 7T(£P rwv A, wS" c/>1JUL 7,J ota ro ap'c/>6upa tuov Xp6vov 7Tapa Ta , 0 <, ... ,' .. ', f3" yLyVEU at. 0 p'EV OVV 1\0yoS" ouroS" (£UTtV, uvp, atvH 0(£ 7Tapa
• , .1.-1> Hp1Jp,EVOV 'f'EUUOS".
]6. Simplicius, 1016.9-1019. 9, ad loco \(' .... \ I "'Z" \1 KaL 0 r(£raproS" rwv 7TEpL KtV1]U(£WS" rov 7JvwvoS" IIOYWV
,~ I " \ l' '"f: I , aovvarov a7TUYWV KaL ovroS" ro (£WaL KLV1]UW TOLOth-OS" TtS" ~v.
" , ...." 0.... \' ....." ~ 15 (£UTL KLV1]ULS", rwv LUWV p,Ey(£ wv KaL Luoraxwv ro (£r(£pov (' , .... ,.... , ~" I , (£r(£pou (£V rep avrep Xpovep ot7T/\UULaV KW7JUW KtV1]U(£rat KU~
tU1Jv. Ka, ([UTL P,€V Ka, roth-o C1.ro7Tov, C1.ro7Tov U KU' TO f I "",. , " , " ~.\" E7TOp,EVOV ro TOV avrov KaL LUOV XpOVOV ap,a ut7Tl\aUtoV TE
w t I> , 1>" , <" , 7Jp,tuu (£ vat. O€tKVUUt U(£ auro op,o/\oYOVP,(£VOV /\af3c1v ro ra. , .- ,., '_II I" ~,
2otUOTaXTJ Kat tua EV,Tep tuep XPOvep tUOv utaUT1Jp,a K(£KtvfjuOat. , H , ...., .... , " '" \ \ #I Kat (£Tt P,(£vrOt rwv LUOTaxwv r(£ Kat WWV, av ro P,(£V 7Jp,tuv, r~
~\ ~ \' 'J' " ~ , . \ i \ w 0(£ ot7Tl\aUtOV n KEKtV1JP,(£VOV,(£V 7Jp,tUEL P,(£:- ,<povep ro 7JP,£uv, EV oL7TAaulep S€ ro SL7TAaULOV EtvaL KEKLV1Jp,EVOV. rourwv 7TPO)..1J-'
I o€ FHK: Simp. 1017.29, Alex. apud Simp. 1019. 27. o~ cett. , o€ EI FHK: Simp. 1017.29, Alex. apud Simp. 1019. 27. o~ cett.,
Bekker. 3 7TaVTa TO. A, FKE': Simp_ 1018. I, Alex. apud Simp. 1019.28.
7TaVTa TO. B, EI HI Bekker. - , 4 '0' B E S- Al . " TO E , : Imp_ 1018. I, ex. apud SImp. 10190 29. Ta o€ B;
FkHI Bekker. 5 TO afJ E: TO. B cett., Simp. 1019. 13, Bekker: TO B Ross. I take TO
7TPWTOV fJ to be the true reading, ,and suppose that this was at some time
---- ------~... )f
the As, stretching from the middle-point <of the stadium to the Q"",tTincypost), and CC those stretching from the goal <to the middle
being equal in number and size to the As, and moving with velocity equal to that of the Bs. Then it follows that, as Bs and Cs
past each other, the first B reaches the last C at the same time 5 the first C reaches the last B. And it follows that the first C has
passed all the Bs, the first B half that number of bodies <viz. two As>: "and so the first B has taken only half the time <that the first C has taken>, since each takes an equal time in passing each bod,y. And it
that at the same moment the first B has passed all the Cs: 10
for the first C and the first B will arrive simultaneously atthe opposite , end As, since both take an equal time passing the As. This then is , his argument, and it rests on the above-mentioned fallacy.
36. Simplicius, 1016.9-1019. 9, ad loco
The fourth of Zeno's arguments about motion, which also leads to the conclusion that it is impossible for motion to be a reality, was 15 as follows: If there is motion, of two bodies of equal size and moving with equal velocities, one will move twice as far as the other, and not the same distance, in the same time. This is of course an absurd conclusion, but so also is the conclusion that follows upon this that the time they take, which is equal and the same, is at once both 20
double and half. In his proof he assumes as admitted that bodies moving with an equal velocity and of equal size move an equal distance in equal times, and further that of such bodies, if one moves half as far as the other, then the motion of the first will occupy half the time of that of the second. This being premised he goes on to 25
written TO afJ (cf. Simp. 1017. 15 and crit. note 2, where certain manuscripts haveTov afJ instead of TOV npwTov fJ). This accounts naturally enough for the TO. fJ of the other MSS. [This reading was suggested to me by Prof. Cornford, who has incorporated it into the text of the Loeb Physics.]
6 TWV I' Loeb: TWV B codd. 7 Om.Zuov Xp6vov .. . cOS" r?TJa. as a gloss on raov yap €KaTEp6v EUTL nap'
€KaUTOV in 1. 6: Ross. 8 Kant TO A Alex. apud Simp. 1019.32. napa TO. A. codd;
MOTION
-'-() , '<;:" 'f) i' 'hE \ I 'e '/' EVTWV UTauwv VTrETt ETO OWV TO Ll , Ko.~ TEaao.po. fLEYE 7J f ....." I " " III " _ (~ ~, f ouaouv, aPTta fLoVOV, WUTE EXHV 7JfL'G'V tUooYKa WS UE 0
AAAA
BBBB +--rrrr
E
~7JUL, d{lovs) N' Jiv Ta A, WS TO fLEaov OuJ.UT7JfLa lTrEXELV ~, ~..... .... 1" rr, ..... (' 'Y \ '\
UTauLOV EaTWTa TaUTa o WV EUTWTWV 7TPWTOV OpL';,Et TO 7TpOS rfi 5 apxv TOU aTaO£OU TV KaTa TO fl, EUXaTOV O~ TO 7TPOS Tcp E, Kal
)..afL{laVEL a.:\)..OVS' TEaaapaS' 0YKOUS ~ KV{lOVS LUOUS TOLS , , I 8 ", 8 \ ',../.., 'P 'B' , Ka. TO fLEYE OS Kat TOV apt fLOV E,/, WV TO. ,o.pXOfLEVOVS
, , , A A <;:" 'A <;:" '" a7TO apX7]S TOU UTaoLOV, TEIIEVTwvras UE KaTa TO fLEaov 'T(VV E' A I ~\ , f' \ '\" T auapwv ,KLVOVfLEVOVS OE TOVTOVS ws em ro EUXaTOV TOU
10 <;: I 'E <;:, , A \ , , '" UTaoLOU TO 0 ow Kat 7TPWTOV IIEYEL TOV Ka'Ta TO fLEUOV T(VV A WS EfLTrpou8EV T(VV )..omwv Gvra EV rfj €Trt TO E KWrlUEL. OLa
..... ~ , " "\ R ,,, q" '1/
rOUTO OE apTtOUS Ellat'E TOVS 0YKOVS, tva EXWatV 7JfLtG'V' OELrUI , I • e I e <;:, \. \ A B ' yap rOUTOV, WS fLO. TJaofLE 0. 0 ow KCU TO TrPWTOV KUTa rou , A" A'8 .,. , !'"\ \ " fLEaou TWV EaTWTWV n YJaLV, ELTa Kat U/IIIOVS LuallS Tcp fLEy.E()Et
15 Kat Tcp dpt{)fLCP, TOLS B, oijAov o~ on Ka;' TOLs A, AafL{lrJ.VH E1>' d)v Ta r aVTtKLVOUfLEVOVS TOLS B. TWV yap B (lTrO TOU ji-EUOl)
..... ~I '1" \ .... A ' , ,:t " '\ JI TOU UTauwv, EV c.p Kat TWV TO fLEaov 'IV, E1T1. TO €lT)(aTOV rou
'" \ E I , r " A" I UTaowu TO KLVOVfLEVWV Ot aTrO TOV EUXaTOll fLEpOVS, EV T 'E ' \ \ A ..... " .... ,.......... ~I cP TO ,ETrt TO U KLVOUVTCU TO EV TTl apXTJ TOU UTaoLOV, Ka~
A 0'\' A I r' , 'A , 20 TrPWTOV 'I ovon TWV TEuuapwv 0 7TpOS TO U VEVEVKWS, E~' 0 -q K{VYJaLS TOLS r· r[{)YJaLV O~ TO 7TPWTOV r KaTa TOU 7TPWTOU B.
I l'......'c,.... () I , e' '\ t' rotauT7JS OUV TYJS E~ apxYJS EaEWS VTrOTE ELG'YJS Eav TWV A €UTWTWV
Ta fLdv B KwijTat ws a7To TOU ji-Eaou TWV TE A Kat TOU UTo.otOl)
" "\ A <;:, 'E \'" r' " " E7TL TO TEIIOS TOU aTaowu TO ;, TO. UE WS aTrO TOU €axo.TOV TOU
5 '" " \ , \ ,,\ , (' \ "" , 2 UTaoLOV ETrL T7JV apxYJV uYJllovon OU yap oYJ ws "am) TOU €IT)(Cl.TOV B" ct f" " _, I.J... f '\ ,
, OTrEP' WS EOLK€V EV TLaLV avnypa,/,oLS" EVPWV <> MEtavopos , 'e \1 ff n I l'
7]vayKau YJ IIEYELV, OTt 0 TrpOTEpOV HTrEV TrPWTOV B, TOVTO VUV
€UXG.TOV EKU'\WE), uUfL{latvHTO TrpCJTOV B afLa €Tr~ Tcp €UXaTCP
ElvaL TijS EaUTOU KtV~UEWS Ka~ TO TrPWTOV r, Trap' a>.>.YJAa KLVOV-
THE STADIUM 59 sut)Oc)se a stadium DE, and four bodies of equal size AA-or any
, provided it be even, so that the number of bodies (or,
D
AAAA
EEEE->-
+--- CCCC
E
as Eudemus calls them, cubes) has a half-which are stationary and are placed so as to occupy a central stretch of the stadium. Of these stationary bodies the" first" he defines as that nearest the beginning 5 of the stadium, on our diagram D, the last as that nearest E. And he supposes four other bodies or cubes EE equal in size and number to the stationary, originally stretching from the beginning of the stadium to the middle of the four As and moving towards the end of the stadium Eo And therefore he calls the E which is over against 10
the middle of the As the "first" E, since it will be ahead of the others in their motion towards Eo The reason for supposing the number of the bodies to be even is so that they should have a half: for this is necessary to the argument, as we shall see. Accordingly he places the first E over against the middle of the stationary As, and then 15 supposes another rOw of bodies CC equal in size and number to the Es, and therefore of course to the As, and moving in the opposite direction to the Es. For the Es move from the middle of the stadium, which is also the mid-point of the As, towards the end of the stadium E, while the Cs move from the end of the stadium, which we have 20
called E, towards the beginning of the stadium, D in our diagram, and so clearly the "first" of the four Cs is the one furthest advanced towards D, in the direction of which the Cs are moving, and the first C is placed adjacent to the first Eo
This then is the initial position. Then Jet the As remain stationary, 25 and let the Es move from the middle of the As and of the stadium towards the end of the stadium E, and the Cs from the end of the stadium towards the beginning (this must clearly be the meaning, and not from the end E, a reading which it seems that Alexander found in some manuscripts, and was forced to adopt: for then what 30 he previously called the first E he has now called the last). Then it results that the first E and the first C will "be at the end" of their
60 MOTION
fL.!VWV aVTwv KatlaOTaxws, 7j e7Tt Tip euxaTep lli7]AWV.
, r ' A , B" 't' A 7TpWTOV.. KaTa TOV 7TPWTOV OVTOS E!, apX7JS, aVTLlCLJi'OVILEII(
aVTWV laOTaxws Kat OL€gEAB6VTWV aAA1JAa, TO fL~V 7TPWTOV B
Tip eaxaTep €a7aL r, TO oi. 7TPWTOV r e7Tt Tip euxaTep
..... "1\ " , Q I , .... B fI " ..... 5 TOVTO av ELTJ TO aVfLfJaLV€LV TO 7TPWTOV afLa E7TL Tep
ElvaL Kat TO 7TPWTOV r 7Tap' aAATJAa KLVOVfLlvwv· .TJ yap
clAATJAa K{V7]aLS 7TOL€;: TO ev TOtS euxaTOLS lli7]AWV
a' 0:>'.1. ' , 'r ' A 0:>' I , aVfLfJaLV€L OE, 'f'TJaL, KaL TO ,TO 7TPWTOV 0TJI\OVOTL, 7Tapa
Ta A OL€ATJAvB.!vaL, TO oi. B 7Tapa Ta TJfLlU7J A. Kat OTL fL~V
A" ., "oX" ." 'rc:" A TWV TJfLLaEWV, 07Toaa av U apna, €V oaep TO . uta TWV OL~ITI\(l.aLW
B ot€Lat, OfjAOV' TO yap 7TPWTOV B a7TO TOU fLlaov TWV A
JaTWTa KLV€tTaL, TO r TO 7TPWTOV aVnKLVOVfL€VOV TOtS B
15 T€ac}'(1.pwv B ot€LaLv' at yap ovo KLv1Jo"€LS TWV aV'T,LKL,V0t1U.€V'~1
oL7TAaawv avvovaL OtaaTTJfLa TfjS fLtaS, ~v KtV€tTaL TO B
taTafL€va A. Kat TOVTO fL~V OfjAOV. 7TWS O~ TO r 7Tapa 7TaVTa Ta
OL€).:r}AvB€v; OUT€ ya.p 7Tapa. TavTa EKLV€LTO, aAAa. 7Tapa. Ta. B,
a7T' apxfjs TWV A EKLV€tTO, aAAa a7T' apxfjs TWV B, ijns ~v " AA"" "B" oX ~A ''''r 20 TO fL€aov TWV . TJ on KaL Ta Laa . 'IV TOtS • TO ovv.
oaep xp6vep 7Tapa Ta B K€KtV7]Tat, €iTJ a.v Kd 7Tapa Ta A Ta
TO;:S B K€KLVTJfL'!VOV. Kat 0 7TapaAOYLafLos EVTavfJct Eanv,
THE STADIUM 61
"",.;-.LJlY\;; motions simultaneously" as they move past each other" equal velocities. Or else we can interpret the phrase to mean
the first B will be opposite the last C, and vice-versa, at the same : for since the first C was to begin with adjacent to the first
as the two rows move past each other in opposite directions and 5 equal velocity, the first B will come opposite the last C and
first C opposite the last B. And this would be the meaning of that it results that" the first B and the first C will, as they
past one another, each be opposite the end simultaneously": their movement past each other brings each opposite the end 10
of the other row. But it further results,he says, that" the C", that is, obviously, first C, "has passed all the As, but the (first) B has only passed the As". It is of course evident that the first B, starting from
mid-point of the As, has moved past two As or through half 15 !l'hclte'ler the even number of bodies chosen, while C has passed
th~number of Bs: for the first B was supposed to start from middle of the As. Also while B moves past the two end As, which stationary, the first C, moving in the opposite direction to the has passed four Bs: for the two contrary motions, taken together, 20
the effect of doubling the distance of B's motion, taken singly, past the stationary As. So much is evident. But what is meant by saying that C has passed all the As? For it was not past (all) the As ,that it moved but past (all) the Bs, nor did it move from the begin
of the As but from the beginning of the Bs, which was adjacent 25 middle of the As. The reason must be because the Bs also are
equal to the As. Therefore during the time in which the first C moved past the (four) Bs it must have moved past (four) As, since .these are equal to the Bs.
The fallacy lies in assuming without qualification that movements 30 "\ Q (\.... , JI , I '\, " ,
€l\a fJ€V a7TI\WS €V Laep Xpovep KLVOVfL€VOV TO 7Tapa Laa KLVOV past bodies of equal size take an equal time, without taking into fL~ 7TpoaAoYLaafL€voS, on TWV rawv Ta fLi.v aVT'KLVOvfL€va ~v account the further fact that of the equal bodies some are moving in 0:>" A 'a' 0:>'" ,,'" , I B " opposite directions and some are stationary. None the less he makes
25 O€ €aTWTa. l\afJwv O€ ofLws, OTL €V Laep xpovep Ta T€ KaL Ta th th th r kith B e assumption' at e l.-S ta e an equa time to pass e s and the otELaL Ta r, e7T€LO~ ev oaep xp6vep TO 7TPWTOV B st€LaL Ta ovo As and concludes that since, during the time which the first B takes 35 EV ToaOVTep TO r Ta T.!aaapa B ';;70L Ta Tlaaapa A, vviI'II,a'll€v.a to pass two As, the first C has passed four Bs or four As, the first B, " , B " '" A r' A 'A I \ thoug.h its velocity is the same as that of the first C, yet moves only on TO KaLTOL LaOTax€s OV Tep EV;ep aVTep 'K.pOvep TO. 'YILJ~LO'II" half the distance the first C moves in the same time-which is in KLV€;:TaL, 00 TO r KLV€;:TaL, 07T€P .eaTt 7Tapa Ta 7TpOO accordance neither with the presuppositions of the argument Iwr
, \, ...... '\ \, .... 'JI " \ " 30 Ka, Ta €vapYTJ· Ta yap wOTaXTJ EV Laep Xpovep TO ,aov with common sense: for bodies moving with an equal velocity cover 40
MOTION
'\\' ff e: I " f$ "".J.. \ \ r .... (lJ\/\ OTo.V OfLOLWS EXT/> WUTE 7] ufL't'w napa TO. EaTW'TU ..... N .J.. \ \ / "~, \ ,\ 7] afL't'w 7Tapo. TO. KtVOUfL€Vo., Kat OUX OTav TO. fLEV 7Tapa TO.
WS TO B, Td. O~ 7Tapd. Td. aVTLKtVovfLEva WS TO r. €Tt O~ Xpovos, Ell iJ> K~VEG"aL TO Bod. TWV ovo A, T/fLLUVS Eun
5 XPOIlOU, €V iJ> KLVEG"at TO r OLd. TWV Twuapwv B, €t7TEP rua To'
TOLS B, Kat iuoTaxi} TO TE B Ka~ TO r. EOOKH O~ Kat tuos
.5 Xpovoc; 7JTOL 0 mh·os, €V iJ> TO B €KtVELTO OLd. nov Ova A, Kal
.,. 'r.;,' ~ , B f3' ..', 8 cp TO ULa. TWV T€UUapWV • aVfL 7]aETaL ouv Kat fLEYE OS
aUTO Etvat om'\aatov TE KaL T/fLLUV, EL7TEP €V 1'0 aVT0 xpovcp TWV
10 lUOTaxwv TO fL~V B TO. Ova A IkUn, TO O£ r TO. T€aUapa B, iuwv
OllTWV TWV B TOLS A· Ka~ Xpovov 'TOV athov Om'\aatov TE Kul " " '" ,J: < , ' .. ' B';" ~ .;,' A 7JfLWVV, H7TEp Kat 7JfL'uvs 'IV a Xpovos, EV <p TO aLa TWV UVO
OtIlH, TaU Xpovov, EV iJ> TO r OLa. TWV TEuaapwv B, Ka~ .5 am-os. . ,';'1" '" " , " .;, \ ~ " \ \ B TO OE Laov yap EKaTEpov EUTL 7Tap EKa..aTOV U1]/\OL OTt Kat TO
15 Kat TO r laOTaxfj OVTa Zaov Xp6vov 7TOLEL 'lTap'. EKaUTOV, ?h' wv KtVEG"aL TWV TE B Ka~ nov A. El O~ taOV, ofj,\ov on om'\aatos
EUTLV .5 Xpovos, €V iJ> TO r Td. T€aaapa B OlELUW, lJfLLUVS 3€, €v clJ TO B TO. Slio A, 7J fLa».ov, €V iJ> TO r 'To. T€aaapa A Stnat, TOU Ell iJ> TO B TO laOTaXES a1h0 TO. SUo A. E'tpYJTaL ya.p, on €V iJ> TO. B
20 SLEWL 'TO r, €V TOllT1p Ko.L Td. A.
THE STADIUM
distance in an equal time, but only when their relative "'UJ111~' LdllU:~ are the same and either both are moving past stationary
or both past moving, but not when some are moving past sta1tlOllarv bodies (like B) and some past bodies moving in an
direction (like C). Further, the time taken by B to pass As is half the time taken by C to pass four Bs, if the As are equal
the Bs and Band C move with equal velocity. But the time in B passes two As and that in which C passes four Bs are
ol1,.,,,,nQ,,rl to be equal or the same. It follows therefore both. that the same magnitude is double and half, since in the same time of two 10
bodies moving with equal velocity B passes two As and C passes four Bs, though the Bs are equal in size to the As: and that the same
. time is both double and half, if the time taken by B to pass two As is both half and the same as the time C took to pass four Bs .
The phrase" each takes an equal time to pass each body" means 15
that B and C, since they move with equal velocity, take an equal time in passing both each B and each A. But if this is so, then it is clear that the time taken by C to pass four Bs is double that taken by B to pass two As, or rather that the time which C takes to pass four As is double that which B, though moving with a velocity 20
equal to C's, takes to pass two As: for as was said the time C takes to pass the Bs, it will also take to pass the As.
