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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 passage 1 also implies an approximate value of 3 (and in fact there is an interesting story 2  associated 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 obtai n 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 o ften 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 translation 3 . The Approximation of Pi The 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 de velop a numerical  procedure for calculating the perimeter of a c ircumscribing polygon of 2n sides, once the perimeter of the polygon of n sides 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 pol ygons (after developing the corres ponding 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.

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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/