Archimedes' Approximation of Pi
One of the major contributions Archimedes made to mathematics
was his method for approximating the value of pi. It had long been
recognized that the ratio of the circumference of a circle to its
diameter was constant, and a number of approximations had been
given up to that point in time by the Babylonians, Egyptians, and
even the Chinese. There are some authors who claim that a biblical
passage1also implies an approximate value of 3 (and in fact there
is an interesting story2associated with that).At any rate, the
method used by Archimedes differs from earlier approximations in a
fundamental way. Earlier schemes for approximating pi simply gave
an approximate value, usually based on comparing the area or
perimeter of a certain polygon with that of a circle. Archimedes'
method is new in that it is an iterative process, whereby one can
get as accurate an approximation as desired by repeating the
process, using the previous estimate of pi to obtain a new one.
This is a new feature of Greek mathematics, although it has an
ancient tradition among the Chinese in their methods for
approximating square roots.Archimedes' method, as he did it
originally, skips over a lot of computational steps, and is not
fully explained, so authors of history of math books have often
presented slight variations on his method to make it easier to
follow. Here we will try to stick to the original as much as
possible, following essentially Heath's translation3.The
Approximation of PiThe method of Archimedes involves approximating
pi by the perimeters of polygons inscribed and circumscribed about
a given circle. Rather than trying to measure the polygons one at a
time, Archimedes uses a theorem of Euclid to develop a numerical
procedure for calculating the perimeter of a circumscribing polygon
of2nsides, once the perimeter of the polygon ofnsides is known.
Then, beginning with a circumscribing hexagon, he uses his formula
to calculate the perimeters of circumscribing polygons of 12, 24,
48, and finally 96 sides. He then repeats the process using
inscribing polygons (after developing the corresponding formula).
The truly unique aspect of Archimedes' procedure is that he has
eliminated the geometry and reduced it to a completely arithmetical
procedure, something that probably would have horrified Plato but
was actually common practice in Eastern cultures, particularly
among the Chinese scholars.The Key TheoremThe key result used by
Archimedes is Proposition 3 of Book VI of Euclid'sElements. The
full statement of the theorem is as follows:If an angle of a
triangle be bisected and the straight line cutting the angle cut
the base also, the segments of the base will have the same ratio as
the remaining sides of the triangle; and, if the segments of the
base have the same ratio as the remaining sides of the triangle,
the straight line joined from the vertex to the point of section
will bisect the angle of the triangle.4
We will just prove one direction of this theorem here, namely
that the angle bisector cuts the opposite side in the ratio
claimed. More precisely, in the diagram shown, ifADbisects
angleBAC, thenBD : CD = BA : AC.
Animated GIF Proof ofTheorem (99K)QuickTime Video Proof of
Theorem (243K)
Our proof differs from the original somewhat: the proof (and
diagram) given here makes it more clear how Archimedes will use the
theorem in his approximation scheme. For Euclid's original,
complete proof, along with averyneat interactive diagram, seeDavid
Joyce'sElementsWeb site.Archimedes' MethodHere we outline the
method used by Archimedes to approximate pi. The specific statement
of Archimedes is Proposition 3 of his treatiseMeasurement of a
Circle:The ratio of the circumference of any circle to its diameter
is less than 31/7but greater than 310/71.The proof we give below
essentially follows that of Archimedes, as set out in Heath's
translation5. Much of the text skips over steps in the proof;
rather than adding intermediate steps as Heath does6, we are
putting those in pop-up windows. Look for buttons like this:.
Clicking on these will bring up pop-up windows showing intermediate
steps that Archimedes has left out of this text (HTGT stands for
How'd They Get That?).Proof:[Note: throughout this proof,
Archimedes uses several rational approximations to various square
roots. Nowhere does he say how he got those approximations--they
are simply stated without any explanation--so how he came up with
some of these is anybody's guess.]
I.LetABbe the diameter of any circle,Oits center,ACthe tangent
atA; and let the angleAOCbe one-third of a right angle. Then
(1)OA:AC> 265 : 153 and (2)OC:AC= 306 : 153. First,
drawODbisecting the angleAOCand meetingACinD. NowCO:OA=CD:DAso that
(CO+OA):CA=OA:ADTherefore (3)OA:AD> 571 : 153.HenceOD2:AD2>
349450 : 23409so that (4)OD:DA> 5911/8: 153.Secondly,
letOEbisect the angleAOD, meetingADinE. Therefore (5)OA:AE>
11621/8: 153 Thus (6)OE:EA> 11721/8: 153.Thirdly, letOFbisect
the angleAOEand meetAEinF. We thus obtain the result that
(7)OA:AF> 23341/4: 153 Thus (8)OF:FA> 23391/4: 153.Fourthly,
letOGbisect the angleAOF, meetingAFinG. We have thenOA:AG>
46731/2: 153. Now the angleAOC, which is one-third of a right
angle, has been bisected four times, and it follows that angleAOG=
1/48 (a right angle). Make the angleAOHon the other side ofOAequal
to the angleAOG, and letGAproduced meetOHinH. Then angleGOH= 1/24
(a right angle). ThusGHis one side of a regular polygon of 96 sides
circumscribed to the given circle. And, sinceOA:AG> 46731/2:
153,while AB = 2 OA, GH = 2 AG,it follows thatAB: (perimeter of a
polygon of 96 sides) > 46731/2: 14688But
Therefore the circumference of the circle (being less than the
perimeter of the polygon) isa fortioriless than 3 1/7 times the
diameter AB.
II.Next letABbe the diameter of a circle, and letAC, meeting the
circle inC, make the angleCABequal to one-third of a right angle.
JoinBC. ThenAC:BC< 1351 : 780.First, letADbisect the angleBACand
meetBCindand the circle inD. JoinBD. ThenangleBAD= angledAC=
angledBDand the angles at D, C are both right angles. It follows
that the triangles ADB, BDd are similar. ThereforeAD:BD=BD:Dd=AB:Bd
= (AB+AC) : (Bd+Cd) = (AB+AC) :BCor(BA+AC) :BC=AD:DB.Therefore
(1)AD:DB< 2911 : 780. Thus (2)AB:BD< 30133/4: 780.Secondly,
letAEbisect the angleBAD,meeting the circle inE; and letBEbe
joined. Then we prove, in the same way as before, that (3)AE:EB<
59243/4: 780 = 1823 : 240.Therefore (4)AB:BE< 18389/11:
240.Thirdly, let AF bisect the angle BAE, meeting the circle in F.
Thus, (5)AF:FB< 36619/11x11/40: 240 x11/40 = 1007 :
66.Therefore, (6)AB:BF< 10091/6: 66. Fourthly, let the
angleBAFbe bisected byAGmeeting the circle inG. ThenAG:GB<
20161/6: 66, by (5) and (6). Therefore (7)AB:BG< 20171/4: 66.
ThereforeBGis a side of a regular inscribed polygon of 96 sides. It
follows from (7) that(perimeter of polygon) :AB> 6336 : 20171/4.
And. Much more then is the circumference to the diameter< 31/7
but> 310/71.
http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html
Paradoks Zeno - Ketakhinggaan dalam KeberhinggaanParadoks Zeno
yang paling terkenal dalam sejarah Yunani dan juga matematika
adalah paradoks Achilles dan Kura-kura. Terkenal karena orang
Yunani gagal menjelaskan paradoks ini. Walau sekarang terkesan
tidak terlalu sulit, tapi butuh waktu ribuan tahun sebelum
matematikawan dapat menjelaskannya. Paradoks Achilles dan kura-kura
kira-kira seperti ini :
Pelari tercepat (A) tidak akan bisa mendahului pelari yang lebih
lambat (B). Hal ini terjadi karena A harus berada pada titik B
mula-mula, sementara B sudah meninggalkan (berada di depan) titik
tersebut.
Zeno menganalogikan paradoks ini dengan membayangkan lomba lari
Achilles dan seekor kura-kura. Keduanya dianggap lari dengan
kecepatan konstan dan kura-kura sudah tentu jauh lebih lambat.
Untuk itu, si kura-kura diberi keuntungan dengan start awal di
depan, katakanlah 10 meter. Ketika lomba sudah dimulai, Achilles
akan mencapai titik 10 m (titik di mana kura-kura mula-mula).
Tetapi si kura ini juga pasti sudah melangkah maju, jauh lebih
lambat memang, katakanlah dia baru melangkah 1 meter. Beberapa saat
kemudian Achilles berada di titik 11m, tapi si kura lagi-lagi udah
melangkah maju 0,1 m. Demikian seterusnya, setiap kali Achilles
berada pada titik di mana kura-kura tadinya berada, si kura-kura
sudah melangkah lebih maju. Artinya, Achilles, secepat apa pun dia
berlari tidak akan bisa mendahului kura-kura (selambat apa pun dia
melangkah).
Secara konteks percakapan, kira-kira begini:
kura2: "Jika aku mulai beberapa meter di depanmu, pasti aku
menang, Achilles."
Achilles: "Hahaha... Dasar kura-kura. Kamu ingin berapa meter di
depanku?"
kura2: "10 meter"
Achilles: "Baiklah, aku dapat mencapai 10 meter dalam satu
detik"
kura2: "Tapi dalam satu detik itu aku sudah maju lagi kan?"
Achilles: "Ya, paling hanya 1 meter krn kecepatanmu 1m per
detik. sedangkan Aku dapat maju 1 meter dengan 0,1 detik"
kura2: "Tapi dlm 0,1 detik itu aku sudah maju lagi kan?"
Achilles: "Hm.. Ya" (Achilles mulai ragu)
kura2: "Ini akan terjadi terus menerus, sehingga aku terus
berada di depanmu."
Achilles: "Baiklah, kau menang kura-kura. Aku menyerah."
Achilles yang malang ... Dia terjebak dalam logika ketakhinggaan
kura-kura ...
i. The AchillesAchilles, who is the fastest runner of antiquity,
is racing to catch the tortoise that is slowly crawling away from
him. Both are moving along a linear path at constant speeds. In
order to catch the tortoise, Achilles will have to reach the place
where the tortoise presently is. However, by the time Achilles gets
there, the tortoise will have crawled to a new location. Achilles
will then have to reach this new location. By the time Achilles
reaches that location, the tortoise will have moved on to yet
another location, and so on forever. Zeno claimsAchilles will never
catch the tortoise. He might have defended this conclusion in
various waysby saying it is because the sequence of goals or
locations has no final member, or requires too much distance to
travel, or requires too much travel time, or requires too many
tasks. However, if we do believe that Achilles succeeds and that
motion is possible, then we are victims of illusion, as Parmenides
says we are.The source for Zenos views is Aristotle
(Physics239b14-16) and some passages from Simplicius in the fifth
century C.E. There is no evidence that Zeno used a tortoise rather
than a slow human. The tortoise is a commentators addition.
Aristotle spoke simply of the runner who competes with Achilles.It
wont do to react and say the solution to the paradox is that there
are biological limitations on how small a step Achilles can take.
Achilles feet arent obligated to stop and start again at each of
the locations described above, so there is no limit to how close
one of those locations can be to another. It is best to think of
the change from one location to another as a movement rather than
as incremental steps requiring halting and starting again.Zeno is
assuming that space and time are infinitely divisible; they are not
discrete or atomistic. If they were, the Paradoxs argument would
not work.One common complaint with Zenos reasoning is that he is
setting up astraw manbecause it is obvious that Achilles cannot
catch the tortoise if he continually takes a bad aim toward the
place where the tortoise is; he should aim farther ahead. The
mistake in this complaint is that even if Achilles took some sort
of better aim, it is stilltruethat he is required to goto every one
of those locations that are the goals of the so-called bad aims, so
Zenos argument needs a better treatment.The treatment called the
Standard Solution to the Achilles Paradox uses calculus and other
parts of real analysis to describe the situation. It implies that
Zeno is assuming in the Achilles situation that Achilles cannot
achieve his goal because(1) there is too far to run, or(2) there is
not enough time, or(3) there are too many places to go, or(4) there
is no final step, or(5) there are too many tasks.The historical
record does not tell us which of these was Zenos real assumption,
but they are all false assumptions, according to the Standard
Solution. Lets consider (1). Presumably Zeno would defend the
assumption by remarking that the sum of the distances along so many
of the runs to where the tortoise is must be infinite, which is too
much for even Achilles. However, the advocate of the Standard
Solution will remark, How does Zeno know what the sum of this
infinite series is? According to the Standard Solution the sum is
not infinite. Here is a graph using the methods of the Standard
Solution showing the activity of Achilles as he chases the tortoise
and overtakes it.
To describe this graph in more detail, we need to say that
Achilles path [the path of some dimensionless point of Achilles'
body] is a linear continuum and so is composed of an actual
infinity of points. (An actual infinity is also called a completed
infinity or transfinite infinity, and the word actual does not mean
real as opposed to imaginary.) Since Zeno doesnt make this
assumption, that is another source of error in Zenos reasoning.
Achilles travels a distance d1in reaching the point x1where the
tortoise starts, but by the time Achilles reaches x1, the tortoise
has moved on to a new point x2. When Achilles reaches x2, having
gone an additional distance d2, the tortoise has moved on to point
x3, requiring Achilles to cover an additional distance d3, and so
forth. This sequence of non-overlapping distances (or intervals or
sub-paths) is an actual infinity, but happily the geometric series
converges. The sum of its terms d1+ d2+ d3+ is a finite distance
that Achilles can readily complete while moving at a constant
speed.Similar reasoning would apply if Zeno were to have made
assumption (2) or (3). Regarding (4), the requirement that there be
a final step or final sub-path is simply mistaken, according to the
Standard Solution. More will be said about assumption (5) inSection
5c.By the way, the Paradox does not require the tortoise to crawl
at a constant speed but only to never stop crawling and for
Achilles to travel faster on average than the tortoise. The
assumption of constant speed is made simply for ease of
understanding.The Achilles Argument presumes that space and time
are infinitely divisible. So, Zenos conclusion may not simply have
been that Achilles cannot catch the tortoise but instead that he
cannot catch the tortoise if space and time are infinitely
divisible. Perhaps, as some commentators have speculated, Zeno used
the Achilles only to attack continuous space, and he intended his
other paradoxes such as The Moving Rows to attack discrete space.
The historical record is not clear. Notice that, although space and
time are infinitely divisible for Zeno, he did not have the
concepts to properly describe the limit of the infinite division.
Neither Zeno nor any of the other ancient Greeks had the concept of
a dimensionless point; they did not even have the concept of zero.
However, todays versions of Zenos Paradoxes can and do use those
concepts.ii. The Dichotomy (The Racetrack)In his Progressive
Dichotomy Paradox, Zeno argued that a runner will never reach the
stationary goal line of a racetrack. The reason is that the runner
must first reach half the distance to the goal, but when there he
must still cross half the remaining distance to the goal, but
having done that the runner must cover half of the new remainder,
and so on. If the goal is one meter away, the runner must cover a
distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so onad
infinitum. The runner cannot reach the final goal, says Zeno. Why
not? There are few traces of Zenos reasoning here, but for
reconstructions that give the strongest reasoning, we may say that
the runner will not reach the final goal because there is too far
to run, the sum is actually infinite. The Standard Solution argues
instead that the sum of this infinite geometric series is one, not
infinity.The problem of the runner getting to the goal can be
viewed from a different perspective. According to the Regressive
version of the Dichotomy Paradox, the runner cannot even take a
first step. Here is why. Any step may be divided conceptually into
a first half and a second half. Before taking a full step, the
runner must take a 1/2 step, but before that he must take a 1/4
step, but before that a 1/8 step, and so forthad infinitum, so
Achilles will never get going. Like the Achilles Paradox, this
paradox also concludes that any motion is impossible. The original
source is Aristotle (Physics, 239b11-13).The Dichotomy paradox, in
either its Progressive version or its Regressive version, assumes
for the sake of simplicity that the runners positions are point
places. Actual runners take up some larger volume, but assuming
point places is not a controversial assumption because Zeno could
have reconstructed his paradox by speaking of the point places
occupied by, say, the tip of the runners nose, and this assumption
makes for a strong paradox than assuming the runners position are
larger.In the Dichotomy Paradox, the runner reaches the points 1/2
and 3/4 and 7/8 and so forth on the way to his goal, but under the
influence of Bolzano and Dedekind and Cantor, who developed the
first theory of sets, the set of those points is no longer
considered to be potentially infinite. It is an actually infinite
set of points abstracted from a continuum of pointsin the
contemporary sense of continuum at the heart of calculus. And the
ancient idea that the actually infinite series of path lengths or
segments 1/2 + 1/4 + 1/8 + is infinite had to be rejected in favor
of the new theory that it converges to 1. This is key to solving
the Dichotomy Paradox, according to the Standard Solution. It is
basically the same treatment as that given to the Achilles. The
Dichotomy Paradox has been called The Stadium by some commentators,
but that name is also commonly used for the Paradox of the Moving
Rows.Aristotle, inPhysicsZ9, said of the Dichotomy that it is
possible for a runner to come in contact with a potentially
infinite number of things in a finite time provided the time
intervals becomes shorter and shorter. Aristotle said Zeno assumed
this is impossible, and that is one of his errors in the Dichotomy.
However, Aristotle merely asserted this and could give no detailed
theory that enables the computation of the finite amount of time.
So, Aristotle could not really defend his diagnosis of Zenos error.
Today the calculus is used to provide the Standard Solution with
that detailed theory.There is another detail of the Dichotomy that
needs resolution. How does Zeno complete the trip if there is no
final step or last member of the infinite sequence of steps
(intervals and goals)? Dont trips need last steps? The Standard
Solution answers no and says the intuitive answer yes is one of our
many intuitions that must be rejected when embracing the Standard
Solution.iii. The ArrowZenos Arrow Paradox takes a different
approach to challenging the coherence of our common sense concepts
of time and motion. As Aristotle explains, from Zenos assumption
that time is composed of moments, a moving arrow must occupy a
space equal to itself during any moment. That is, during any moment
it is at the place where it is. But places do not move. So, if in
each moment, the arrow is occupying a space equal to itself, then
the arrow is not moving in that moment because it has no time in
which to move; it is simply there at the place. The same holds for
any other moment during the so-called flight of the arrow. So, the
arrow is never moving. Similarly, nothing else moves. The source
for Zenos argument is Aristotle (Physics, 239b5-32).The Standard
Solution to the Arrow Paradox uses the at-at theory of motion,
which says motion is beingatdifferent placesatdifferent times and
that being at rest involves being motionlessata particular pointata
particular time. The difference between rest and motion has to do
with what is happening at nearby moments and has nothing to do with
what is happeningduringa moment. An object cannot be in
motioninorduringan instant, but it can be in motionatan instant in
the sense of having a speed at that instant, provided the object
occupies different positions at times before or after that instant
so that the instant is part of a period in which the arrow is
continuously in motion. If we dont pay attention to what happens at
nearby instants, it is impossible to distinguish instantaneous
motion from instantaneous rest, but distinguishing the two is the
way out of the Arrow Paradox. Zeno would have balked at the idea of
motionatan instant, and Aristotle explicitly denied it. The Arrow
Paradox seems especially strong to someone who would say that
motion is an intrinsic property of an instant, being some
propensity or disposition to be elsewhere.In standard calculus,
speed of an objectatan instant (instantaneous velocity) is the time
derivative of the objects position; this means the objects speed is
the limit of its speeds during arbitrarily small intervals of time
containing the instant. Equivalently, we say the objects speed is
the limit of its speed over an interval as the length of the
interval tends to zero.The derivative of position x with respect to
time t, namely dx/dt, is the arrows speed, and it has non-zero
valuesatspecific placesatspecific instants during the flight,
contra Zeno and Aristotle. The speedduringan instant orinan
instant, which is what Zeno is calling for, would be 0/0 and so be
undefined. Using these modern concepts, Zeno cannot successfully
argue thatateach moment the arrow is at rest or that the speed of
the arrow is zeroatevery instant. Therefore, advocates of the
Standard Solution conclude that Zenos Arrow Paradox has a false,
but crucial, assumption and so is unsound.Independently of Zeno,
the Arrow Paradox was discovered by the Chinese dialectician
Kung-sun Lung (Gongsun Long, ca. 325250 B.C.E.). A lingering
philosophical question about the arrow paradox is whether there is
a way to properly refute Zenos argument that motion is impossible
without using the apparatus of calculus.iv. The Moving Rows (The
Stadium)It takes a body moving at a given speed a certain amount of
time to traverse a body of a fixed length. Passing the body again
at that speed will take the same amount of time, provided the bodys
length stays fixed. Zeno challenged this common reasoning.
According to Aristotle (Physics239b33-240a18), Zeno considered
bodies of equal length aligned along three parallel racetracks
within a stadium. One track contains A bodies (three A bodies are
shown below); another contains B bodies; and a third contains C
bodies. Each body is the same distance from its neighbors along its
track. The A bodies are stationary, but the Bs are moving to the
right, and the Cs are moving with the same speed to the left. Here
are two snapshots of the situation, before and after.
Zeno points out that, in the time between the before-snapshot
and the after-snapshot, the leftmost C passes two Bs but only one
A, contradicting the common sense assumption that the C should take
longer to pass two Bs than one A. The usual way out of this paradox
is to remark that Zeno mistakenly supposes that a moving body
passes both moving and stationary objects with equal
speed.Aristotle argues that how long it takes to pass a body
depends on the speed of the body; for example, if the body is
coming towards you, then you can pass it in less time than if it is
stationary. Todays analysts agree with Aristotles diagnosis, and
historically this paradox of motion has seemed weaker than the
previous three. This paradox is also called The Stadium, but
occasionally so is the Dichotomy Paradox.Some analysts, such as
Tannery (1887), believe Zeno may have had in mind that the paradox
was supposed to have assumed that space and time are discrete
(quantized, atomized) as opposed to continuous, and Zeno intended
his argument to challenge the coherence of this assumption about
discrete space and time. Well, the paradox could be interpreted
this way. Assume the three objects are adjacent to each other in
their tracks or spaces; that is, the middle object is only one atom
of space away from its neighbors. Then, if the Cs were moving at a
speed of, say, one atom of space in one atom of time, the leftmost
C would pass two atoms of B-space in the time it passed one atom of
A-space, which is a contradiction to our assumption that the Cs
move at a rate of one atom of space in one atom of time. Or else
wed have to say that in that atom of time, the leftmost C somehow
got beyond two Bs by passing only one of them, which is also absurd
(according to Zeno). Interpreted this way, Zenos argument produces
a challenge to the idea that space and time are discrete. However,
most commentators believe Zeno himself did not interpret his
paradox this way.
Achilles, yang pelari terpantas kuno, berlumba untuk menangkap
kura-kura yang perlahan-lahan merangkak daripadanya. Kedua-duanya
bergerak di sepanjang jalan yang linear pada kelajuan malar. Dalam
usaha untuk menangkap kura-kura, Achilles perlu sampai di tempat
yang di mana kura-kura kini adalah. Walau bagaimanapun, dalam masa
yang Achilles mendapat di sana, kura-kura akan telah merangkak ke
lokasi baru. Achilles akan mempunyai untuk mencapai lokasi baru
ini. Apabila sampai Achilles lokasi itu, kura-kura akan telah
berpindah ke lokasi yang lain belum, dan sebagainya selama-lamanya.
Zeno mendakwa Achilles tidak akan menangkap kura-kura. Beliau
mungkin telah mempertahankan kesimpulan ini dalam pelbagai
cara-dengan mengatakan ia adalah kerana urutan matlamat atau lokasi
tidak mempunyai ahli akhir, atau memerlukan jarak terlalu banyak
untuk melakukan perjalanan, atau memerlukan masa perjalanan terlalu
banyak, atau memerlukan terlalu banyak tugas. Walau bagaimanapun,
jika kita percaya bahawa Achilles berjaya dan pergerakan yang
mungkin, maka kita menjadi mangsa ilusi, sebagai Parmenides kata
kita. Sumber untuk pandangan Zeno ialah Aristotle (Fizik 239b14-16)
dan beberapa petikan dari Simplicius pada abad kelima CE Tidak ada
bukti bahawa Zeno digunakan kura-kura dan bukannya manusia yang
perlahan. Kura-kura adalah tambahan yang pengulas. Aristotle
bercakap hanya dari "pelari" yang bersaing dengan Achilles. Ia
tidak akan lakukan untuk bertindak balas dan berkata penyelesaian
untuk paradoks adalah bahawa terdapat had biologi bagaimana kecil
langkah Achilles boleh mengambil. Kaki Achilles 'tidak diwajibkan
untuk berhenti dan bermula sekali lagi di setiap lokasi yang
dinyatakan di atas, maka tidak ada had berapa rapat salah satu
lokasi yang boleh kepada yang lain. Adalah lebih baik untuk
memikirkan perubahan dari satu lokasi ke lokasi lain sebagai
pergerakan bukan langkah yang memerlukan tambahan terhenti-henti
dan bermula sekali lagi. Zeno adalah menganggap bahawa ruang dan
masa adalah tak terhingga dibahagikan; mereka tidak diskret atau
atomistik. Jika mereka, hujah Paradox tidak akan bekerja. Salah
satu aduan yang sama dengan hujah Zeno ialah beliau menubuhkan
seorang lelaki jerami kerana ia adalah jelas bahawa Achilles tidak
boleh menangkap kura-kura itu jika dia terus mengambil tujuan yang
tidak baik ke arah tempat di mana kura-kura adalah; dia harus
berusaha lebih jauh ke hadapan. Kesilapan dalam aduan ini adalah
bahawa walaupun Achilles mengambil beberapa jenis matlamat yang
lebih baik, ia masih benar bahawa dia dikehendaki untuk pergi ke
setiap salah satu daripada lokasi yang matlamat yang dipanggil
"matlamat tidak baik," demikian hujah Zeno ini memerlukan rawatan
yang lebih baik. Rawatan yang dikenali sebagai "Penyelesaian
Standard" kepada Achilles Paradox menggunakan kalkulus dan
bahagian-bahagian lain analisis sebenar untuk menggambarkan
keadaan. Ia membayangkan bahawa Zeno adalah andaian dalam keadaan
Achilles Achilles yang tidak dapat mencapai matlamatnya kerana (1)
terdapat terlalu jauh untuk menjalankan, atau (2) tidak ada masa
yang cukup, atau (3) terdapat terlalu banyak tempat untuk pergi,
atau (4) tiada Langkah terakhir, atau (5) terdapat terlalu banyak
tugas.http://www.iep.utm.edu/zeno-par/