3382:, dated 7 March 1697. Either Gregory did not understand Newton's argument, or Newton's explanation was very brief. However, it is possible, with a high degree of confidence, to construct Newton's proof from Gregory's notes, by analogy with his method to determine the solid of minimum resistance (Principia, Book 2, Proposition 34, Scholium 2). A detailed description of his solution of this latter problem is included in the draft of a letter in 1694, also to David Gregory. In addition to the minimum time curve problem, there was a second problem that Newton also solved at the same time. Both solutions appeared anonymously in Philosophical Transactions of the Royal Society, for January 1697.
4548:
4534:
120:
31:
3325:, is correct but does not indicate the method by which Newton arrived at his conclusion. Bernoulli, writing to Henri Basnage in March 1697, indicated that even though its author, "by an excess of modesty", had not revealed his name, yet even from the scant details supplied it could be recognised as Newton's work, "as the lion by its claw" (in Latin,
4813:(His communication together with of two others in a report to him first from Johann Bernoulli, then from the Marquis de l'Hôpital, of reported solutions of the problem of the curve of quickest descent, publicly proposed by Johann Bernoulli, geometer — one with a solution of his other problem proposed afterward by the same .),
4811:"Communicatio suae pariter, duarumque alienarum ad edendum sibi primum a Dn. Jo. Bernoullio, deinde a Dn. Marchione Hospitalio communicatarum solutionum problematis curva celerrimi descensus a Dn. Jo. Bernoullio Geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea propositi."
199:
the time along the lesser arc DB will be less than for any other path from D to B. In fact, the quickest path from A to B or from D to B, the brachistochrone, is a cycloidal arc, which is shown in Fig. 3 for the path from A to B, and Fig.4 for the path from D to B, superposed on the respective circular arc.
3700:
Assuming now that Fig. 1 is the minimum curve not yet determined, with vertical axis CV, and the circle CHV removed, and Fig. 2 shows part of the curve between the infinitesimal arc eE and a further infinitesimal arc Ff a finite distance along the curve. The extra time, t, to traverse eL (rather than
3401:
Consider a small arc eE, which the body is ascending. Assume that it traverses the straight line eL to point L, horizontally displaced from E by a small distance, o, instead of the arc eE. Note, that eL is not the tangent at e, and that o is negative when L is between B and E. Draw the line through E
3344:
is due to David
Brewster's 1855 book on the life and works of Newton. Bernoulli's intention was, Whiteside argues, simply to indicate he could tell the anonymous solution was Newton's, just as it was possible to tell that an animal was a lion given its claw; it was not meant to suggest that Bernoulli
198:
In Fig.1, from the “Dialogue
Concerning the Two Chief World Systems”, Galileo claims that the body sliding along the circular arc of a quarter circle, from A to B will reach B in less time than if it took any other path from A to B. Similarly, in Fig. 2, from any point D on the arc AB, he claims that
3320:
is dated 1 January 1697, in the
Gregorian Calendar. This was 22 December 1696 in the Julian Calendar, in use in Britain. According to Newton's niece, Catherine Conduitt, Newton learned of the challenge at 4 pm on 29 January and had solved it by 4 am the following morning. His solution, communicated
378:
The line KNC intersects AL at N, and line Kne intersects it at n, and they make a small angle CKe at K. Let NK = a, and define a variable point, C on KN extended. Of all the possible circular arcs Ce, it is required to find the arc Mm, which requires the minimum time to slide between the 2 radii, KM
268:
found the challenge in a letter from Johann
Bernoulli. Newton stayed up all night to solve it and mailed the solution anonymously by the next post. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he "recognizes a lion from his claw mark". This story gives some
3397:
Fig. 1, shows
Gregory’s diagram (except the additional line IF is absent from it, and Z, the start point has been added). The curve ZVA is a cycloid and CHV is its generating circle. Since it appears that the body is moving upward from e to E, it must be assumed that a small body is released from Z
3308:
to pose a challenge to the international mathematical community: to find the form of the curve joining two fixed points so that a mass will slide down along it, under the influence of gravity alone, in the minimum amount of time. The solution was originally to be submitted within six months. At the
348:
Johann
Bernoulli's direct method is historically important as a proof that the brachistochrone is the cycloid. The method is to determine the curvature of the curve at each point. All the other proofs, including Newton's (which was not revealed at the time) are based on finding the gradient at each
3822:
At L the particle continues along a path LM, parallel to the original EF, to some arbitrary point M. As it has the same speed at L as at E, the time to traverse LM is the same as it would have been along the original curve EF. At M it returns to the original path at point f. By the same reasoning,
730:
Assume AMmB is the part of the cycloid joining A to B, which the body slides down in the minimum time. Let ICcJ be part of a different curve joining A to B, which can be closer to AL than AMmB. If the arc Mm subtends the angle MKm at its centre of curvature, K, let the arc on IJ that subtends the
341:
of May 1697. He wrote that this was partly because he believed it was sufficient to convince anyone who doubted the conclusion, partly because it also resolved two famous problems in optics that "the late Mr. Huygens" had raised in his treatise on light. In the same letter he criticised Newton for
327:
In a letter to L’Hôpital, (21/12/1696), Bernoulli stated that when considering the problem of the curve of quickest descent, after only 2 days he noticed a curious affinity or connection with another no less remarkable problem leading to an ‘indirect method’ of solution. Then shortly afterwards he
221:
gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise
220:
I, Johann
Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to
336:
In a letter to Henri
Basnage, held at the University of Basel Public Library, dated 30 March 1697, Johann Bernoulli stated that he had found two methods (always referred to as "direct" and "indirect") to show that the Brachistochrone was the "common cycloid", also called the "roulette". Following
292:. Four of the solutions (excluding l'Hôpital's) were published in the same edition of the journal as Johann Bernoulli's. In his paper, Jakob Bernoulli gave a proof of the condition for least time similar to that below before showing that its solution is a cycloid. According to Newtonian scholar
2480:
showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and
3693:
Therefore, the increase in time to traverse a small arc displaced at one endpoint depends only on the displacement at the endpoint and is independent of the position of the arc. However, by Newton’s method, this is just the condition required for the curve to be traversed in the minimum time
3154:
3405:
Since the displacement EL is small, it differs little in direction from the tangent at E so that the angle EnL is close to a right-angle. In the limit as the arc eE approaches zero, eL becomes parallel to VH, provided o is small compared to eE making the triangles EnL and CHV similar.
564:
This condition defines the curve that the body slides along in the shortest time possible. For each point, M on the curve, the radius of curvature, MK is cut in 2 equal parts by its axis AL. This property, which
Bernoulli says had been known for a long time, is unique to the cycloid.
1121:
4826:"Curvatura radii in diaphanis non uniformibus, Solutioque Problematis a se in Actis 1696, p. 269, propositi, de invenienda Linea Brachystochrona, id est, in qua grave a dato puncto ad datum punctum brevissimo tempore decurrit, & de curva Synchrona seu radiorum unda construenda."
123:
Balls rolling under uniform gravity without friction on a cycloid (black) and straight lines with various gradients. It demonstrates that the ball on the curve always beats the balls travelling in a straight line path to the intersection point of the curve and each straight
186:, Galileo warns of possible fallacies and the need for a "higher science". In this dialogue Galileo reviews his own work. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics.
240:
derived the same solution, but Johann's derivation was incorrect, and he tried to pass off Jakob's solution as his own. Johann published the solution in the journal in May of the following year, and noted that the solution is the same curve as
Huygens's
4782:(Given in a vertical plane two points A and B (see Figure 5), assign to the moving M, the path AMB, by means of which — descending by its own weight and beginning to be moved from point A — it would arrive at the other point B in the shortest time.)
2366:
3689:
4003:
3282:
484:, which has to be a minimum (‘un plus petit’). He does not explain that because Mm is so small the speed along it can be assumed to be the speed at M, which is as the square root of MD, the vertical distance of M below the horizontal line AL.
359:, who stated that it shows that the cycloid is the only possible curve of quickest descent. According to him, the other solutions simply implied that the time of descent is stationary for the cycloid, but not necessarily the minimum possible.
263:
Bernoulli allowed six months for the solutions but none were received during this period. At the request of Leibniz, the time was publicly extended for a year and a half. At 4 p.m. on 29 January 1697 when he arrived home from the Royal Mint,
2725:
where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is
189:
374:
A body is regarded as sliding along any small circular arc Ce between the radii KC and Ke, with centre K fixed. The first stage of the proof involves finding the particular circular arc, Mm, which the body traverses in the minimum time.
1856:
104:. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a
482:
245:. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid. However, his proof is marred by his use of a single constant instead of the three constants,
4832:
of 1696, p. 269, from which is to be found the brachistochrone line , that is, in which a weight descends from a given point to a given point in the shortest time, and on constructing the tautochrone or the wave of rays.),
2958:
3355:, who also failed to solve it. After Newton had submitted his solution, Gregory asked him for the details and made notes from their conversation. These can be found in the University of Edinburgh Library, manuscript A
2947:
2842:
4516:
4115:
3351:, who was 80 years old at the time, had learned of the problem in September 1696 from Johann Bernoulli's youngest brother Hieronymus, and had spent three months attempting a solution before passing it in December to
719:
He then proceeds with what he called his Synthetic Solution, which was a classical, geometrical proof, that there is only a single curve that a body can slide down in the minimum time, and that curve is the cycloid.
4780:"Datis in plano verticali duobus punctis A & B (vid Fig. 5) assignare Mobili M, viam AMB, per quam gravitate sua descendens & moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B."
194:
Galileo’s conjecture is that “The shortest time of all will be that of its fall along the arc ADB and similar properties are to be understood as holding for all lesser arcs taken upward from the lowest limit B.”
177:
Consequently the nearer the inscribed polygon approaches a circle the shorter the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the
4521:
Newton gives no indication of how he discovered that the cycloid satisfied this last relation. It may have been by trial and error, or he may have recognised immediately that it implied the curve was the cycloid.
4798:(On a proof the time in which a weight slides by a line joining two given points the shortest in terms of time when it passes, via gravitational force, from one of these to the other through a cycloidal arc),
2459:
1699:
used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity
990:
727:. He has little time for our new analysis, describing it as false (He claims he has found 3 ways to prove that the curve is a cubic parabola) – Letter from Johan Bernoulli to Pierre Varignon dated 27 Jul 1697.
4433:
269:
idea of Newton's power, since Johann Bernoulli took two weeks to solve it. Newton also wrote, "I do not love to be dunned and teased by foreigners about mathematical things...", and Newton had already solved
2083:
1316:
369:
4347:
810:
2169:
5246:
1214:
3885:
3817:
865:
2720:
558:
382:
Let MN = x. He defines m so that MD = mx, and n so that Mm = nx + na and notes that x is the only variable and that m is finite and n is infinitely small. The small time to travel along arc Mm is
2276:
2244:
1574:
1407:
982:
188:
1673:
If the arc, Cc subtended by the angle infinitesimal angle MKm on IJ is not circular, it must be greater than Ce, since Cec becomes a right-triangle in the limit as angle MKm approaches zero.
355:
He explained that he had not published it in 1697, for reasons that no longer applied in 1718. This paper was largely ignored until 1904 when the depth of the method was first appreciated by
3392:
1642:
4289:
4242:
4195:
3477:
1483:
2625:
922:
665:
2544:
1957:
171:
From the preceding it is possible to infer that the quickest path of all , from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle.
3579:
731:
same angle be Cc. The circular arc through C with centre K is Ce. Point D on AL is vertically above M. Join K to D and point H is where CG intersects KD, extended if necessary.
607:
2271:
3591:
4796:"De ratione temporis quo grave labitur per rectam data duo puncta conjungentem, ad tempus brevissimum quo, vi gravitatis, transit ab horum uno ad alterum per arcum cycloidis"
1756:
709:
296:, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that were refined by
3894:
3726:
3534:
3165:
711:
and which gives MN (=x) as a function of NK (= a). From this the equation of the curve could be obtained from the integral calculus, though he does not demonstrate this.
1668:
1909:
Assuming for simplicity that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after falling a vertical distance
4035:
Because eEFf is the minimum curve, (t – T) is must be greater than zero, whether o is positive or negative. It follows that the coefficient of o in (1) must be zero:
1888:
3380:
108:. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal. The problem can be solved using tools from the
4142:
4030:
3753:
3504:
752:
229:
Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time
1771:
128:
The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point
4349:, everywhere, and this condition characterises the curve that is sought. This is the same technique he uses to find the form of the Solid of Least Resistance.
368:
4622:
385:
5172:
3402:
parallel to CH, cutting eL at n. From a property of the cycloid, En is the normal to the tangent at E, and similarly the tangent at E is parallel to VH.
3149:{\displaystyle d^{2}t_{2}-d^{2}t_{1}=0={\bigg (}{\frac {1}{v_{2}}}{\frac {dx_{2}}{ds_{2}}}-{\frac {1}{v_{1}}}{\frac {dx_{1}}{ds_{1}}}{\bigg )}d^{2}x}
2848:
2743:
4438:
4581:
4040:
270:
4547:
5229:
5155:
5133:
5101:
5056:
4686:
1116:{\displaystyle {\frac {\tau }{t}}={\frac {Mm}{Ce}}.\left({\frac {CG}{MD}}\right)^{\frac {1}{2}}=\left({\frac {CG}{CF}}\right)^{\frac {1}{2}}}
3391:
2404:
4875:
4860:
4845:
4825:
4810:
4355:
2734:
4744:"Dialogue Concerning the Two Chief World Systems – Ptolemaic and Copernican translated by Stillman Drake, foreword by Albert Einstein "
3309:
suggestion of Leibniz, Bernoulli extended the challenge until Easter 1697, by means of a printed text called "Programma", published in
1969:
285:
1219:
5016:
4948:
4920:
4297:
1765:
Bernoulli noted that Snell's law of refraction gives a constant of the motion for a beam of light in a medium of variable density:
760:
2102:
4117:(2) in the limit as eE and fF approach zero. Note since eEFf is the minimum curve it has to be assumed that the coefficient of
1132:
5325:
3828:
3760:
815:
5340:
4751:
2637:
493:
4147:
Clearly there has to be 2 equal and opposite displacements, or the body would not return to the endpoint, A, of the curve.
5285:
3352:
568:
Finally, he considers the more general case where the speed is an arbitrary function X(x), so the time to be minimised is
5352:
89:
under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by
4561:
2730:(the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,
2192:
1488:
1321:
927:
289:
5280:
2375:, corresponding to the angle through which the rolling circle has rotated. For given φ, the circle's centre lies at
1585:
356:
5148:"Der Briefwechsel von Johann I Bernoulli", Vol. II: "Der Briefwechsel mit Pierre Varignon, Erster Teil: 1692-1702"
4247:
4200:
4153:
5094:
Der Briefwechsel von Johann I Bernoulli", Vol. II: "Der Briefwechsel mit Pierre Varignon, Erster Teil: 1692-1702"
3412:
1418:
5247:"Chute d'une bille le long d'une gouttière cycloïdale; Tautochrone et brachistochrone; Propriétés et historique"
5203:
4963:
Herman Erlichson (1999), "Johann Bernoulli's brachistochrone solution using Fermat's principle of least time",
5346:
5311:
4795:
5116:
Bernoulli, Johann. Mémoires de l'Académie des Sciences (French Academy of Sciences) Vol. 3, 1718, pp. 135–138
4719:
2560:
874:
612:
345:
In addition to his indirect method he also published the five other replies to the problem that he received.
3889:
The difference (t – T) is the extra time it takes along the path eLMf compared to the original eEFf :
1688:
4571:
2487:
2361:{\displaystyle {\begin{aligned}x&=r(\varphi -\sin \varphi )\\y&=r(1-\cos \varphi ).\end{aligned}}}
1898:
At the onset, the angle must be zero when the particle speed is zero. Hence, the brachistochrone curve is
1708:
311:
302:
274:
109:
55:
4150:
If e is fixed, and if f is considered a variable point higher up the curve, then for all such points, f,
3684:{\displaystyle t\propto {\frac {nL}{\sqrt {CB}}}={\frac {o.CH}{CV.{\sqrt {CB}}}}={\frac {o}{\sqrt {CV}}}}
1919:
1762:
The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement.
5388:
4701:
Hand, Louis N., and Janet D. Finch. "Chapter 2: Variational Calculus and Its Application to Mechanics."
3539:
2398:
In the brachistochrone problem, the motion of the body is given by the time evolution of the parameter:
2250:
571:
307:
4533:
161:
had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his
5275:
4972:
3998:{\displaystyle (t-T)\propto \left({\frac {DE}{eE{\sqrt {CB}}}}-{\frac {FG}{Ff{\sqrt {CI}}}}\right).o}
3277:{\displaystyle {\frac {1}{v_{2}}}{\frac {dx_{2}}{ds_{2}}}={\frac {1}{v_{1}}}{\frac {dx_{1}}{ds_{1}}}}
1725:
670:
1679:
From this he concludes that a body traverses the cycloid AMB in less time than any other curve ACB.
4828:(The curvature of rays in non-uniform media, and a solution of the problem proposed by me in the
4591:
2262:
724:
3704:
3512:
5393:
5195:
4988:
4539:
105:
1647:
5318:
4616:
3345:
considered Newton to be the lion among mathematicians, as it has since come to be interpreted.
1905:
The speed reaches a maximum value when the trajectory becomes horizontal and the angle θ = 90°.
5294:
5225:
5151:
5129:
5097:
5052:
5048:
5041:
5012:
4944:
4916:
4767:
4747:
4682:
4651:
4566:
3321:
to the Royal Society, is dated 30 January. This solution, later published anonymously in the
281:
242:
141:
97:
5366:
5006:
1873:
1851:{\displaystyle {\frac {\sin {\theta }}{v}}={\frac {1}{v}}{\frac {dx}{ds}}={\frac {1}{v_{m}}}}
352:
In 1718, Bernoulli explained how he solved the brachistochrone problem by his direct method.
5336:
5297:
5187:
4980:
4654:
3506:
and higher, which represent the error due to the approximation that eL and VH are parallel.
3358:
1696:
207:
163:
140:, or if friction is taken into account, then the curve that minimizes time differs from the
90:
5173:"The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem"
4746:(Hardback ed.). University of California Press Berkeley and Los Angeles. p. 451.
4120:
4008:
3731:
3482:
737:
167:. He draws the conclusion that the arc of a circle is faster than any number of its chords,
5360:
4715:
4674:
2477:
723:
The reason for the synthetic demonstration, in the manner of the ancients, is to convince
237:
212:
158:
113:
4976:
4553:
3288:
1692:
297:
5331:
5224:(Paperback ed.). Cambridge University Press. pp. 9–10, notes (21) and (22).
4863:(Lord Marquis de l'Hôpital's solution of the problem of the line of fastest descent),
119:
5382:
5071:
4992:
4984:
4611:
477:{\displaystyle {\frac {Mm}{MD^{\frac {1}{2}}}}={\frac {n(x+a)}{(mx)^{\frac {1}{2}}}}}
293:
5199:
4912:
280:
In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli,
265:
2952:
For the path of least times these times are equal so for their difference we get,
4906:
4902:
4518:, which was shown above to be constant, and the Brachistochrone is the cycloid.
3398:
and slides along the curve to A, without friction, under the action of gravity.
3348:
47:
4861:"Domini Marchionis Hospitalii solutio problematis de linea celerrimi descensus"
17:
4529:
3694:
possible. Therefore, he concludes that the minimum curve must be the cycloid.
2733:
34:
The curve of fastest descent is not a straight or polygonal line (blue) but a
30:
5373:
The straight line, the catenary, the brachistochrone, the circle, and Fermat
5302:
5191:
4659:
3310:
2372:
1676:
Note, Bernoulli proves that CF > CG by a similar but different argument.
4294:
But, since points, e and f are arbitrary, equation (2) can be true only if
2942:{\displaystyle d^{2}t_{2}={\frac {1}{v_{2}}}{\frac {dx_{2}}{ds_{2}}}d^{2}x}
2837:{\displaystyle d^{2}t_{1}={\frac {1}{v_{1}}}{\frac {dx_{1}}{ds_{1}}}d^{2}x}
1691:, the actual path between two points taken by a beam of light (which obeys
5150:(Hardback ed.). Springer Basel Aktiengesellschaft. pp. 117–118.
4511:{\displaystyle {\frac {DE}{eE{\sqrt {CB}}}}={\frac {CH}{CV.{\sqrt {CB}}}}}
5356:
4636:
Stewart, James. "Section 10.1 - Curves Defined by Parametric Equations."
4586:
4576:
4110:{\displaystyle {\frac {DE}{eE{\sqrt {CB}}}}={\frac {FG}{Ff{\sqrt {CI}}}}}
86:
754:
and t be the times the body takes to fall along Mm and Ce respectively.
4908:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
3409:
Also en approaches the length of chord eE, and the increase in length,
2254:
1899:
101:
43:
35:
4615:
2454:{\displaystyle \varphi (t)=\omega t\,,\omega ={\sqrt {\frac {g}{r}}}}
1891:
represents the angle of the trajectory with respect to the vertical.
3509:
The speed along eE or eL can be taken as that at E, proportional to
5372:
5359:
including two ways of derivation of the equation of geodesic with
4428:{\displaystyle {\frac {DE}{eE}}={\frac {BH}{VH}}={\frac {CH}{CV}}}
3823:
the reduction in time, T, to reach f from M rather than from F is
118:
69:
of fastest descent, is the one lying on the plane between a point
66:
29:
337:
advice from Leibniz, he included only the indirect method in the
2468:
is the time since the release of the body from the point (0,0).
2078:{\displaystyle v_{m}^{2}dx^{2}=v^{2}ds^{2}=v^{2}(dx^{2}+dy^{2})}
4770:(A new problem to whose solution mathematicians are invited.),
1963:
Rearranging terms in the law of refraction and squaring gives:
1311:{\displaystyle CH={\frac {MD.CK}{MK}}={\frac {MD.(MK+CM)}{MK}}}
4342:{\displaystyle {\frac {DE}{eE{\sqrt {CB}}}}={\text{constant}}}
4640:. 7th ed. Belmont, CA: Thomson Brooks/Cole, 2012. 640. Print.
4244:). By keeping f fixed and making e variable it is clear that
4768:"Problema novum ad cujus solutionem Mathematici invitantur."
4728:
This conclusion had appeared six years earlier in Galileo's
2732:
805:{\displaystyle \tau \propto {\frac {Mm}{MD^{\frac {1}{2}}}}}
4876:"Excerpta ex Transactionibus Philos. Anglic. M. Jan. 1697."
2164:{\displaystyle dx={\frac {v\,dy}{\sqrt {v_{m}^{2}-v^{2}}}}}
210:
posed the problem of the brachistochrone to the readers of
1209:{\displaystyle GH={\frac {MD.HD}{DK}}={\frac {MD.CM}{MK}}}
5326:
Table IV from Bernoulli's article in Acta Eruditorum 1697
4800:
Philosophical Transactions of the Royal Society of London
3304:
In June 1696, Johann Bernoulli had used the pages of the
3880:{\displaystyle T\propto {\frac {o.FG}{Ff.{\sqrt {CI}}}}}
3812:{\displaystyle t\propto {\frac {o.DE}{eE.{\sqrt {CB}}}}}
860:{\displaystyle t\propto {\frac {Ce}{CG^{\frac {1}{2}}}}}
2715:{\displaystyle {\frac {dx}{ds}}d^{2}x=d^{2}s=v\ d^{2}t}
553:{\displaystyle {\frac {(x-a)dx}{2x^{\frac {3}{2}}}}=0}
5096:(Hardback ed.). Springer Basel Ag. p. 329.
4441:
4358:
4300:
4250:
4203:
4156:
4123:
4043:
4011:
3897:
3831:
3763:
3734:
3707:
3701:
eE) is nL divided by the speed at E (proportional to
3594:
3583:
This appears to be all that Gregory’s note contains.
3542:
3515:
3485:
3415:
3361:
3168:
2961:
2851:
2746:
2640:
2563:
2490:
2407:
2274:
2195:
2105:
1972:
1922:
1876:
1774:
1728:
1650:
1588:
1491:
1421:
1324:
1222:
1135:
993:
930:
877:
818:
763:
740:
673:
615:
574:
496:
487:
It follows that, when differentiated this must give
388:
3332:
D. T. Whiteside notes that the letter in French has
4679:
Primer on Pontryagin's Principle in Optimal Control
3287:which agrees with Johann's assumption based on the
96:The brachistochrone curve is the same shape as the
5040:
4510:
4427:
4341:
4283:
4236:
4189:
4136:
4109:
4024:
3997:
3879:
3811:
3747:
3720:
3683:
3573:
3528:
3498:
3471:
3374:
3276:
3148:
2941:
2836:
2714:
2619:
2538:
2453:
2360:
2239:{\displaystyle dx={\sqrt {\frac {y}{D-y}}}\,dy\,,}
2238:
2163:
2077:
1951:
1882:
1850:
1750:
1662:
1636:
1569:{\displaystyle CG=CH-GH={\frac {MD.(MK-2CM)}{MK}}}
1568:
1477:
1402:{\displaystyle CG=CH+GH={\frac {MD.(MK+2CM)}{MK}}}
1401:
1310:
1208:
1115:
976:
916:
859:
804:
746:
703:
659:
601:
552:
476:
5314:( at MathCurve, with excellent animated examples)
4911:(First trade paperback ed.). New York City:
4791:Solutions to Johann Bernoulli's problem of 1696:
3128:
3016:
977:{\displaystyle {\frac {Mm}{Ce}}={\frac {MD}{CH}}}
5047:(2nd ed.). Addison Wesley Longman. p.
379:and Km. To find Mm Bernoulli argues as follows.
5128:, by P. Freguglia and M. Giaquinta, pp. 53–57,
4730:Dialogue Concerning the Two Chief World Systems
4705:. Cambridge: Cambridge UP, 1998. 45, 70. Print.
273:, which is considered the first of the kind in
227:
218:
5222:The Mathematical Papers of Isaac Newton Vol. 8
5126:The Early Period of the Calculus of Variations
4809:G.G.L. (Gottfried Wilhelm Leibniz) (May 1697)
1719:in a uniform gravitational field is given by:
136:. If the body is given an initial velocity at
5011:. Random House Publishing Group. p. 94.
1894:The equations above lead to two conclusions:
1637:{\displaystyle CF={\frac {CH^{2}}{MD}}>CG}
8:
4626:(11th ed.). Cambridge University Press.
4284:{\displaystyle {\frac {DE}{eE{\sqrt {CB}}}}}
4237:{\displaystyle {\frac {DE}{eE{\sqrt {CB}}}}}
4190:{\displaystyle {\frac {FG}{Ff{\sqrt {CI}}}}}
1695:) is one that takes the least time. In 1697
3472:{\displaystyle eL-eE=nL={\frac {o.CH}{CV}}}
1478:{\displaystyle CH={\frac {MD.(MK-CM)}{MK}}}
5146:Costabel, Pierre; Peiffer, Jeanne (1988).
5092:Costabel, Pierre; Peiffer, Jeanne (1988).
300:into what the latter called (in 1766) the
225:Bernoulli wrote the problem statement as:
5369:to the Brachistochrone problem in Python.
5043:A History of Mathematics: An Introduction
4934:
4932:
4848:(A solution of brother's problems, … ),
4495:
4475:
4459:
4442:
4440:
4405:
4382:
4359:
4357:
4334:
4318:
4301:
4299:
4268:
4251:
4249:
4221:
4204:
4202:
4174:
4157:
4155:
4128:
4122:
4094:
4077:
4061:
4044:
4042:
4016:
4010:
3971:
3954:
3938:
3921:
3896:
3864:
3838:
3830:
3796:
3770:
3762:
3739:
3733:
3708:
3706:
3666:
3650:
3624:
3601:
3593:
3586:Let t be the additional time to reach L,
3552:
3541:
3516:
3514:
3490:
3484:
3443:
3414:
3366:
3360:
3265:
3250:
3240:
3232:
3223:
3211:
3196:
3186:
3178:
3169:
3167:
3137:
3127:
3126:
3117:
3102:
3092:
3084:
3075:
3063:
3048:
3038:
3030:
3021:
3015:
3014:
2999:
2989:
2976:
2966:
2960:
2930:
2917:
2902:
2892:
2884:
2875:
2866:
2856:
2850:
2825:
2812:
2797:
2787:
2779:
2770:
2761:
2751:
2745:
2703:
2681:
2665:
2641:
2639:
2608:
2580:
2562:
2530:
2514:
2498:
2489:
2439:
2429:
2406:
2275:
2273:
2232:
2225:
2205:
2194:
2152:
2139:
2134:
2121:
2115:
2104:
2066:
2050:
2034:
2021:
2008:
1995:
1982:
1977:
1971:
1936:
1927:
1921:
1875:
1840:
1831:
1808:
1798:
1784:
1775:
1773:
1735:
1727:
1649:
1608:
1598:
1587:
1519:
1490:
1431:
1420:
1352:
1323:
1264:
1232:
1221:
1177:
1145:
1134:
1102:
1078:
1059:
1035:
1007:
994:
992:
954:
931:
929:
897:
887:
876:
843:
825:
817:
788:
770:
762:
739:
672:
622:
614:
575:
573:
530:
497:
495:
460:
424:
407:
389:
387:
310:did further work that resulted in modern
5171:Babb, Jeff; Currie, James (July 2008),
4941:A Source Book in Mathematics, 1200-1800
4603:
2620:{\displaystyle 2ds\ d^{2}s=2dx\ d^{2}x}
917:{\displaystyle CF={\frac {CH^{2}}{MD}}}
660:{\displaystyle X={\frac {(x+a)dX}{dx}}}
27:Fastest curve descent without friction
4697:
4695:
2175:Substituting from the expressions for
609:. The minimum condition then becomes
4874:reprinted: Isaac Newton (May 1697)
4846:"Solutio problematum fraternorum, … "
4724:Discourses regarding two new sciences
3159:And the condition for least time is,
2631:And finally rearranging terms gives,
7:
2539:{\displaystyle ds^{2}=dx^{2}+dy^{2}}
1711:, the instantaneous speed of a body
5375:Unified approach to some geodesics.
4582:Newton's minimal resistance problem
1952:{\displaystyle v_{m}={\sqrt {2gD}}}
1412:If Ce is closer to K than Mm then
271:Newton's minimal resistance problem
5254:Bulletin de l'Union des Physiciens
5180:The Montana Mathematics Enthusiast
4882:of the month of January in 1697),
3574:{\displaystyle CH={\sqrt {CB.CV}}}
2257:generated by a circle of diameter
695:
602:{\displaystyle {\frac {(x+a)}{X}}}
286:Ehrenfried Walther von Tschirnhaus
25:
4720:"Third Day, Theorem 22, Prop. 36"
4677:The Brachistochrone Paradigm, in
5363:as a special case of a geodesic.
5220:Whiteside, Derek Thomas (2008).
4859:Marquis de l'Hôpital (May 1697)
4546:
4532:
3390:
1126:Since MN = NK, for the cycloid:
367:
187:
5080:Contemporary Newtonian Research
4681:, Collegiate Publishers, 2009.
4638:Calculus: Early Transcendentals
1751:{\displaystyle v={\sqrt {2gy}}}
704:{\displaystyle X=(x+a)\Delta x}
328:discovered a ‘direct method’.
5341:Wolfram Demonstrations Project
3910:
3898:
2417:
2411:
2348:
2330:
2310:
2292:
2072:
2040:
1902:to the vertical at the origin.
1552:
1531:
1461:
1443:
1385:
1364:
1294:
1276:
692:
680:
637:
625:
590:
578:
512:
500:
457:
447:
442:
430:
1:
5322:, Whistler Alley Mathematics.
4778: : 269. From p. 269:
4766:Johann Bernoulli (June 1696)
4943:, Harvard University Press,
4824:Johann Bernoulli (May 1697)
4794:Isaac Newton (January 1697)
4592:Uniformly accelerated motion
3721:{\displaystyle {\sqrt {CB}}}
3529:{\displaystyle {\sqrt {CB}}}
3336:preceded by the French word
871:Extend CG to point F where,
5347:The Brachistochrone problem
5281:Encyclopedia of Mathematics
4844:Jacob Bernoulli (May 1697)
3386:The Brachistochrone problem
318:Johann Bernoulli's solution
203:Introduction of the problem
65: 'shortest time'), or
5410:
4985:10.1088/0143-0807/20/5/301
4880:Philosophical Transactions
4878:(Excerpt from the English
3340:. The much quoted version
3323:Philosophical Transactions
2472:Jakob Bernoulli's solution
1663:{\displaystyle \tau <t}
5298:"Brachistochrone Problem"
4742:Galilei, Galileo (1967).
4655:"Brachistochrone Problem"
4562:Aristotle's wheel paradox
1693:Snell's law of refraction
667:which he writes as :
85:, on which a bead slides
5367:Optimal control solution
5245:Dubois, Jacques (1991).
5076:Newton the Mathematician
5039:Katz, Victor J. (1998).
3536:, which is as CH, since
3306:Acta Eruditorum Lipsidae
3294:
2550:On differentiation with
2481:vertical displacements,
2471:
2088:which can be solved for
339:Acta Eruditorum Lipsidae
317:
216:in June, 1696. He said:
182:Just after Theorem 6 of
5337:Brachistochrone Problem
5192:10.54870/1551-3440.1099
4623:Encyclopædia Britannica
4617:"Brachistochrone"
3342:tanquam ex ungue Leonem
1883:{\displaystyle \theta }
1715:after falling a height
357:Constantin Carathéodory
342:concealing his method.
236:Johann and his brother
152:
4939:Struik, J. D. (1969),
4572:Calculus of variations
4512:
4429:
4343:
4285:
4238:
4197:is constant (equal to
4191:
4144:is greater than zero.
4138:
4111:
4026:
3999:
3881:
3813:
3749:
3722:
3697:He argues as follows.
3685:
3575:
3530:
3500:
3473:
3376:
3375:{\displaystyle 78^{1}}
3313:, in the Netherlands.
3278:
3150:
2943:
2838:
2737:
2716:
2621:
2540:
2455:
2362:
2240:
2165:
2079:
1953:
1884:
1852:
1752:
1709:conservation of energy
1664:
1644:, and it follows that
1638:
1570:
1479:
1403:
1312:
1210:
1117:
978:
918:
861:
806:
748:
705:
661:
603:
554:
478:
312:infinitesimal calculus
303:calculus of variations
290:Guillaume de l'Hôpital
275:calculus of variations
234:
223:
180:
125:
110:calculus of variations
81:is not directly below
39:
5335:by Michael Trott and
4513:
4430:
4344:
4286:
4239:
4192:
4139:
4137:{\displaystyle o^{2}}
4112:
4027:
4025:{\displaystyle o^{2}}
4000:
3882:
3814:
3750:
3748:{\displaystyle o^{2}}
3728:), ignoring terms in
3723:
3686:
3576:
3531:
3501:
3499:{\displaystyle o^{2}}
3474:
3377:
3279:
3151:
2944:
2839:
2736:
2717:
2622:
2541:
2456:
2363:
2251:differential equation
2241:
2166:
2080:
1954:
1885:
1853:
1753:
1665:
1639:
1571:
1480:
1404:
1313:
1211:
1118:
979:
919:
862:
807:
749:
747:{\displaystyle \tau }
706:
662:
604:
555:
479:
308:Joseph-Louis Lagrange
169:
132:and the ending point
122:
52:brachistochrone curve
33:
5186:(2&3): 169–184,
5005:Sagan, Carl (2011).
4703:Analytical Mechanics
4439:
4356:
4298:
4248:
4201:
4154:
4121:
4041:
4009:
3895:
3829:
3761:
3732:
3705:
3592:
3540:
3513:
3483:
3479:, ignoring terms in
3413:
3359:
3166:
2959:
2849:
2744:
2638:
2561:
2488:
2405:
2272:
2193:
2103:
1970:
1920:
1874:
1869:is the constant and
1772:
1726:
1648:
1586:
1489:
1419:
1322:
1220:
1133:
991:
928:
875:
816:
761:
738:
671:
613:
572:
494:
386:
63:(brákhistos khrónos)
5353:Geodesics Revisited
5319:The Brachistochrone
4977:1999EJPh...20..299E
2263:parametric equation
2144:
1987:
5355:— Introduction to
5295:Weisstein, Eric W.
4652:Weisstein, Eric W.
4540:Mathematics portal
4508:
4425:
4339:
4291:is also constant.
4281:
4234:
4187:
4134:
4107:
4022:
3995:
3877:
3809:
3745:
3718:
3681:
3571:
3526:
3496:
3469:
3372:
3274:
3146:
2939:
2834:
2738:
2712:
2617:
2536:
2451:
2371:where φ is a real
2358:
2356:
2236:
2161:
2130:
2075:
1973:
1949:
1880:
1848:
1748:
1689:Fermat’s principle
1660:
1634:
1566:
1475:
1399:
1308:
1206:
1113:
984:, it follows that
974:
914:
857:
802:
744:
715:Synthetic solution
701:
657:
599:
550:
474:
157:Earlier, in 1638,
126:
73:and a lower point
40:
5276:"Brachistochrone"
5260:(737): 1251–1289.
5231:978-0-521-20103-2
5157:978-3-0348-5068-1
5134:978-3-319-38945-5
5103:978-3-0348-5068-1
5058:978-0-321-01618-8
4890: : 223–224.
4871: : 217-220.
4856: : 211–214.
4841: : 206–211.
4821: : 201–205.
4806: : 424-425.
4687:978-0-9843571-0-9
4567:Beltrami identity
4506:
4503:
4470:
4467:
4423:
4400:
4377:
4352:For the cycloid,
4337:
4329:
4326:
4279:
4276:
4232:
4229:
4185:
4182:
4105:
4102:
4072:
4069:
4032:and higher (1)
3982:
3979:
3949:
3946:
3875:
3872:
3807:
3804:
3716:
3679:
3678:
3661:
3658:
3619:
3618:
3569:
3524:
3467:
3295:Newton's solution
3289:law of refraction
3272:
3238:
3218:
3184:
3124:
3090:
3070:
3036:
2924:
2890:
2819:
2785:
2698:
2659:
2603:
2575:
2476:Johann's brother
2449:
2448:
2223:
2222:
2159:
2158:
1947:
1846:
1826:
1806:
1793:
1746:
1623:
1579:In either case,
1564:
1473:
1397:
1306:
1259:
1204:
1172:
1110:
1096:
1067:
1053:
1025:
1002:
972:
949:
912:
855:
851:
800:
796:
655:
597:
542:
538:
472:
468:
419:
415:
363:Analytic solution
282:Gottfried Leibniz
243:tautochrone curve
153:Galileo's problem
142:tautochrone curve
98:tautochrone curve
16:(Redirected from
5401:
5332:Brachistochrones
5308:
5307:
5289:
5262:
5261:
5251:
5242:
5236:
5235:
5217:
5211:
5210:
5208:
5202:, archived from
5177:
5168:
5162:
5161:
5143:
5137:
5123:
5117:
5114:
5108:
5107:
5089:
5083:
5069:
5063:
5062:
5046:
5036:
5030:
5029:
5027:
5025:
5002:
4996:
4995:
4960:
4954:
4953:
4936:
4927:
4926:
4899:
4893:
4789:
4783:
4764:
4758:
4757:
4739:
4733:
4727:
4712:
4706:
4699:
4690:
4672:
4666:
4665:
4664:
4647:
4641:
4634:
4628:
4627:
4619:
4608:
4556:
4551:
4550:
4542:
4537:
4536:
4517:
4515:
4514:
4509:
4507:
4505:
4504:
4496:
4484:
4476:
4471:
4469:
4468:
4460:
4451:
4443:
4434:
4432:
4431:
4426:
4424:
4422:
4414:
4406:
4401:
4399:
4391:
4383:
4378:
4376:
4368:
4360:
4348:
4346:
4345:
4340:
4338:
4335:
4330:
4328:
4327:
4319:
4310:
4302:
4290:
4288:
4287:
4282:
4280:
4278:
4277:
4269:
4260:
4252:
4243:
4241:
4240:
4235:
4233:
4231:
4230:
4222:
4213:
4205:
4196:
4194:
4193:
4188:
4186:
4184:
4183:
4175:
4166:
4158:
4143:
4141:
4140:
4135:
4133:
4132:
4116:
4114:
4113:
4108:
4106:
4104:
4103:
4095:
4086:
4078:
4073:
4071:
4070:
4062:
4053:
4045:
4031:
4029:
4028:
4023:
4021:
4020:
4004:
4002:
4001:
3996:
3988:
3984:
3983:
3981:
3980:
3972:
3963:
3955:
3950:
3948:
3947:
3939:
3930:
3922:
3886:
3884:
3883:
3878:
3876:
3874:
3873:
3865:
3853:
3839:
3818:
3816:
3815:
3810:
3808:
3806:
3805:
3797:
3785:
3771:
3754:
3752:
3751:
3746:
3744:
3743:
3727:
3725:
3724:
3719:
3717:
3709:
3690:
3688:
3687:
3682:
3680:
3671:
3667:
3662:
3660:
3659:
3651:
3639:
3625:
3620:
3611:
3610:
3602:
3580:
3578:
3577:
3572:
3570:
3553:
3535:
3533:
3532:
3527:
3525:
3517:
3505:
3503:
3502:
3497:
3495:
3494:
3478:
3476:
3475:
3470:
3468:
3466:
3458:
3444:
3394:
3381:
3379:
3378:
3373:
3371:
3370:
3283:
3281:
3280:
3275:
3273:
3271:
3270:
3269:
3256:
3255:
3254:
3241:
3239:
3237:
3236:
3224:
3219:
3217:
3216:
3215:
3202:
3201:
3200:
3187:
3185:
3183:
3182:
3170:
3155:
3153:
3152:
3147:
3142:
3141:
3132:
3131:
3125:
3123:
3122:
3121:
3108:
3107:
3106:
3093:
3091:
3089:
3088:
3076:
3071:
3069:
3068:
3067:
3054:
3053:
3052:
3039:
3037:
3035:
3034:
3022:
3020:
3019:
3004:
3003:
2994:
2993:
2981:
2980:
2971:
2970:
2948:
2946:
2945:
2940:
2935:
2934:
2925:
2923:
2922:
2921:
2908:
2907:
2906:
2893:
2891:
2889:
2888:
2876:
2871:
2870:
2861:
2860:
2843:
2841:
2840:
2835:
2830:
2829:
2820:
2818:
2817:
2816:
2803:
2802:
2801:
2788:
2786:
2784:
2783:
2771:
2766:
2765:
2756:
2755:
2721:
2719:
2718:
2713:
2708:
2707:
2696:
2686:
2685:
2670:
2669:
2660:
2658:
2650:
2642:
2626:
2624:
2623:
2618:
2613:
2612:
2601:
2585:
2584:
2573:
2545:
2543:
2542:
2537:
2535:
2534:
2519:
2518:
2503:
2502:
2460:
2458:
2457:
2452:
2450:
2441:
2440:
2394:
2367:
2365:
2364:
2359:
2357:
2245:
2243:
2242:
2237:
2224:
2221:
2207:
2206:
2170:
2168:
2167:
2162:
2160:
2157:
2156:
2143:
2138:
2129:
2128:
2116:
2084:
2082:
2081:
2076:
2071:
2070:
2055:
2054:
2039:
2038:
2026:
2025:
2013:
2012:
2000:
1999:
1986:
1981:
1958:
1956:
1955:
1950:
1948:
1937:
1932:
1931:
1889:
1887:
1886:
1881:
1857:
1855:
1854:
1849:
1847:
1845:
1844:
1832:
1827:
1825:
1817:
1809:
1807:
1799:
1794:
1789:
1788:
1776:
1757:
1755:
1754:
1749:
1747:
1736:
1697:Johann Bernoulli
1669:
1667:
1666:
1661:
1643:
1641:
1640:
1635:
1624:
1622:
1614:
1613:
1612:
1599:
1575:
1573:
1572:
1567:
1565:
1563:
1555:
1520:
1484:
1482:
1481:
1476:
1474:
1472:
1464:
1432:
1408:
1406:
1405:
1400:
1398:
1396:
1388:
1353:
1317:
1315:
1314:
1309:
1307:
1305:
1297:
1265:
1260:
1258:
1250:
1233:
1215:
1213:
1212:
1207:
1205:
1203:
1195:
1178:
1173:
1171:
1163:
1146:
1122:
1120:
1119:
1114:
1112:
1111:
1103:
1101:
1097:
1095:
1087:
1079:
1069:
1068:
1060:
1058:
1054:
1052:
1044:
1036:
1026:
1024:
1016:
1008:
1003:
995:
983:
981:
980:
975:
973:
971:
963:
955:
950:
948:
940:
932:
923:
921:
920:
915:
913:
911:
903:
902:
901:
888:
866:
864:
863:
858:
856:
854:
853:
852:
844:
834:
826:
811:
809:
808:
803:
801:
799:
798:
797:
789:
779:
771:
753:
751:
750:
745:
710:
708:
707:
702:
666:
664:
663:
658:
656:
654:
646:
623:
608:
606:
605:
600:
598:
593:
576:
559:
557:
556:
551:
543:
541:
540:
539:
531:
521:
498:
483:
481:
480:
475:
473:
471:
470:
469:
461:
445:
425:
420:
418:
417:
416:
408:
398:
390:
371:
208:Johann Bernoulli
191:
184:Two New Sciences
164:Two New Sciences
91:Johann Bernoulli
61:βράχιστος χρόνος
21:
5409:
5408:
5404:
5403:
5402:
5400:
5399:
5398:
5379:
5378:
5361:brachistochrone
5312:Brachistochrone
5293:
5292:
5274:
5271:
5266:
5265:
5249:
5244:
5243:
5239:
5232:
5219:
5218:
5214:
5206:
5175:
5170:
5169:
5165:
5158:
5145:
5144:
5140:
5124:
5120:
5115:
5111:
5104:
5091:
5090:
5086:
5070:
5066:
5059:
5038:
5037:
5033:
5023:
5021:
5019:
5004:
5003:
4999:
4962:
4961:
4957:
4951:
4938:
4937:
4930:
4923:
4915:. p. 116.
4901:
4900:
4896:
4884:Acta Eruditorum
4865:Acta Eruditorum
4850:Acta Eruditorum
4835:Acta Eruditorum
4830:Acta Eruditorum
4815:Acta Eruditorum
4790:
4786:
4772:Acta Eruditorum
4765:
4761:
4754:
4741:
4740:
4736:
4716:Galileo Galilei
4714:
4713:
4709:
4700:
4693:
4673:
4669:
4650:
4649:
4648:
4644:
4635:
4631:
4610:
4609:
4605:
4600:
4552:
4545:
4538:
4531:
4528:
4485:
4477:
4452:
4444:
4437:
4436:
4415:
4407:
4392:
4384:
4369:
4361:
4354:
4353:
4311:
4303:
4296:
4295:
4261:
4253:
4246:
4245:
4214:
4206:
4199:
4198:
4167:
4159:
4152:
4151:
4124:
4119:
4118:
4087:
4079:
4054:
4046:
4039:
4038:
4012:
4007:
4006:
3964:
3956:
3931:
3923:
3920:
3916:
3893:
3892:
3854:
3840:
3827:
3826:
3786:
3772:
3759:
3758:
3735:
3730:
3729:
3703:
3702:
3640:
3626:
3603:
3590:
3589:
3538:
3537:
3511:
3510:
3486:
3481:
3480:
3459:
3445:
3411:
3410:
3388:
3362:
3357:
3356:
3334:ex ungue Leonem
3327:ex ungue Leonem
3302:
3297:
3261:
3257:
3246:
3242:
3228:
3207:
3203:
3192:
3188:
3174:
3164:
3163:
3133:
3113:
3109:
3098:
3094:
3080:
3059:
3055:
3044:
3040:
3026:
2995:
2985:
2972:
2962:
2957:
2956:
2926:
2913:
2909:
2898:
2894:
2880:
2862:
2852:
2847:
2846:
2821:
2808:
2804:
2793:
2789:
2775:
2757:
2747:
2742:
2741:
2699:
2677:
2661:
2651:
2643:
2636:
2635:
2604:
2576:
2559:
2558:
2526:
2510:
2494:
2486:
2485:
2474:
2403:
2402:
2376:
2355:
2354:
2320:
2314:
2313:
2282:
2270:
2269:
2253:of an inverted
2211:
2191:
2190:
2184:
2148:
2117:
2101:
2100:
2062:
2046:
2030:
2017:
2004:
1991:
1968:
1967:
1923:
1918:
1917:
1872:
1871:
1867:
1836:
1818:
1810:
1777:
1770:
1769:
1724:
1723:
1685:
1683:Indirect method
1646:
1645:
1615:
1604:
1600:
1584:
1583:
1556:
1521:
1487:
1486:
1465:
1433:
1417:
1416:
1389:
1354:
1320:
1319:
1298:
1266:
1251:
1234:
1218:
1217:
1196:
1179:
1164:
1147:
1131:
1130:
1088:
1080:
1074:
1073:
1045:
1037:
1031:
1030:
1017:
1009:
989:
988:
964:
956:
941:
933:
926:
925:
904:
893:
889:
873:
872:
839:
835:
827:
814:
813:
784:
780:
772:
759:
758:
736:
735:
717:
669:
668:
647:
624:
611:
610:
577:
570:
569:
526:
522:
499:
492:
491:
456:
446:
426:
403:
399:
391:
384:
383:
365:
334:
325:
320:
250:
238:Jakob Bernoulli
213:Acta Eruditorum
205:
159:Galileo Galilei
155:
150:
114:optimal control
28:
23:
22:
18:Brachistochrone
15:
12:
11:
5:
5407:
5405:
5397:
5396:
5391:
5381:
5380:
5377:
5376:
5370:
5364:
5350:
5344:
5339:by Okay Arik,
5328:
5323:
5315:
5309:
5290:
5270:
5269:External links
5267:
5264:
5263:
5237:
5230:
5212:
5163:
5156:
5138:
5118:
5109:
5102:
5084:
5078:, in Bechler,
5072:D.T. Whiteside
5064:
5057:
5031:
5017:
4997:
4971:(5): 299–304,
4955:
4949:
4928:
4921:
4913:Broadway Books
4894:
4892:
4891:
4872:
4857:
4842:
4822:
4807:
4784:
4759:
4752:
4734:
4707:
4691:
4667:
4642:
4629:
4614:, ed. (1911).
4612:Chisholm, Hugh
4602:
4601:
4599:
4596:
4595:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4558:
4557:
4554:Physics portal
4543:
4527:
4524:
4502:
4499:
4494:
4491:
4488:
4483:
4480:
4474:
4466:
4463:
4458:
4455:
4450:
4447:
4421:
4418:
4413:
4410:
4404:
4398:
4395:
4390:
4387:
4381:
4375:
4372:
4367:
4364:
4333:
4325:
4322:
4317:
4314:
4309:
4306:
4275:
4272:
4267:
4264:
4259:
4256:
4228:
4225:
4220:
4217:
4212:
4209:
4181:
4178:
4173:
4170:
4165:
4162:
4131:
4127:
4101:
4098:
4093:
4090:
4085:
4082:
4076:
4068:
4065:
4060:
4057:
4052:
4049:
4019:
4015:
4005:plus terms in
3994:
3991:
3987:
3978:
3975:
3970:
3967:
3962:
3959:
3953:
3945:
3942:
3937:
3934:
3929:
3926:
3919:
3915:
3912:
3909:
3906:
3903:
3900:
3871:
3868:
3863:
3860:
3857:
3852:
3849:
3846:
3843:
3837:
3834:
3803:
3800:
3795:
3792:
3789:
3784:
3781:
3778:
3775:
3769:
3766:
3742:
3738:
3715:
3712:
3677:
3674:
3670:
3665:
3657:
3654:
3649:
3646:
3643:
3638:
3635:
3632:
3629:
3623:
3617:
3614:
3609:
3606:
3600:
3597:
3568:
3565:
3562:
3559:
3556:
3551:
3548:
3545:
3523:
3520:
3493:
3489:
3465:
3462:
3457:
3454:
3451:
3448:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3387:
3384:
3369:
3365:
3301:
3298:
3296:
3293:
3285:
3284:
3268:
3264:
3260:
3253:
3249:
3245:
3235:
3231:
3227:
3222:
3214:
3210:
3206:
3199:
3195:
3191:
3181:
3177:
3173:
3157:
3156:
3145:
3140:
3136:
3130:
3120:
3116:
3112:
3105:
3101:
3097:
3087:
3083:
3079:
3074:
3066:
3062:
3058:
3051:
3047:
3043:
3033:
3029:
3025:
3018:
3013:
3010:
3007:
3002:
2998:
2992:
2988:
2984:
2979:
2975:
2969:
2965:
2950:
2949:
2938:
2933:
2929:
2920:
2916:
2912:
2905:
2901:
2897:
2887:
2883:
2879:
2874:
2869:
2865:
2859:
2855:
2844:
2833:
2828:
2824:
2815:
2811:
2807:
2800:
2796:
2792:
2782:
2778:
2774:
2769:
2764:
2760:
2754:
2750:
2723:
2722:
2711:
2706:
2702:
2695:
2692:
2689:
2684:
2680:
2676:
2673:
2668:
2664:
2657:
2654:
2649:
2646:
2629:
2628:
2616:
2611:
2607:
2600:
2597:
2594:
2591:
2588:
2583:
2579:
2572:
2569:
2566:
2554:fixed we get,
2548:
2547:
2533:
2529:
2525:
2522:
2517:
2513:
2509:
2506:
2501:
2497:
2493:
2473:
2470:
2462:
2461:
2447:
2444:
2438:
2435:
2432:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2369:
2368:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2321:
2319:
2316:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2283:
2281:
2278:
2277:
2247:
2246:
2235:
2231:
2228:
2220:
2217:
2214:
2210:
2204:
2201:
2198:
2182:
2173:
2172:
2155:
2151:
2147:
2142:
2137:
2133:
2127:
2124:
2120:
2114:
2111:
2108:
2086:
2085:
2074:
2069:
2065:
2061:
2058:
2053:
2049:
2045:
2042:
2037:
2033:
2029:
2024:
2020:
2016:
2011:
2007:
2003:
1998:
1994:
1990:
1985:
1980:
1976:
1961:
1960:
1946:
1943:
1940:
1935:
1930:
1926:
1907:
1906:
1903:
1879:
1865:
1860:
1859:
1843:
1839:
1835:
1830:
1824:
1821:
1816:
1813:
1805:
1802:
1797:
1792:
1787:
1783:
1780:
1760:
1759:
1745:
1742:
1739:
1734:
1731:
1684:
1681:
1671:
1670:
1659:
1656:
1653:
1633:
1630:
1627:
1621:
1618:
1611:
1607:
1603:
1597:
1594:
1591:
1577:
1576:
1562:
1559:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1471:
1468:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1430:
1427:
1424:
1410:
1409:
1395:
1392:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1304:
1301:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1263:
1257:
1254:
1249:
1246:
1243:
1240:
1237:
1231:
1228:
1225:
1202:
1199:
1194:
1191:
1188:
1185:
1182:
1176:
1170:
1167:
1162:
1159:
1156:
1153:
1150:
1144:
1141:
1138:
1124:
1123:
1109:
1106:
1100:
1094:
1091:
1086:
1083:
1077:
1072:
1066:
1063:
1057:
1051:
1048:
1043:
1040:
1034:
1029:
1023:
1020:
1015:
1012:
1006:
1001:
998:
970:
967:
962:
959:
953:
947:
944:
939:
936:
910:
907:
900:
896:
892:
886:
883:
880:
869:
868:
850:
847:
842:
838:
833:
830:
824:
821:
795:
792:
787:
783:
778:
775:
769:
766:
743:
725:Mr. de la Hire
716:
713:
700:
697:
694:
691:
688:
685:
682:
679:
676:
653:
650:
645:
642:
639:
636:
633:
630:
627:
621:
618:
596:
592:
589:
586:
583:
580:
562:
561:
560:so that x = a.
549:
546:
537:
534:
529:
525:
520:
517:
514:
511:
508:
505:
502:
467:
464:
459:
455:
452:
449:
444:
441:
438:
435:
432:
429:
423:
414:
411:
406:
402:
397:
394:
364:
361:
333:
330:
324:
321:
319:
316:
298:Leonhard Euler
248:
204:
201:
154:
151:
149:
146:
87:frictionlessly
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5406:
5395:
5392:
5390:
5387:
5386:
5384:
5374:
5371:
5368:
5365:
5362:
5358:
5354:
5351:
5348:
5345:
5342:
5338:
5334:
5333:
5329:
5327:
5324:
5321:
5320:
5316:
5313:
5310:
5305:
5304:
5299:
5296:
5291:
5287:
5283:
5282:
5277:
5273:
5272:
5268:
5259:
5255:
5248:
5241:
5238:
5233:
5227:
5223:
5216:
5213:
5209:on 2011-07-27
5205:
5201:
5197:
5193:
5189:
5185:
5181:
5174:
5167:
5164:
5159:
5153:
5149:
5142:
5139:
5135:
5131:
5127:
5122:
5119:
5113:
5110:
5105:
5099:
5095:
5088:
5085:
5081:
5077:
5073:
5068:
5065:
5060:
5054:
5050:
5045:
5044:
5035:
5032:
5020:
5018:9780307800985
5014:
5010:
5009:
5001:
4998:
4994:
4990:
4986:
4982:
4978:
4974:
4970:
4966:
4965:Eur. J. Phys.
4959:
4956:
4952:
4950:0-691-02397-2
4946:
4942:
4935:
4933:
4929:
4924:
4922:0-7679-0816-3
4918:
4914:
4910:
4909:
4904:
4898:
4895:
4889:
4885:
4881:
4877:
4873:
4870:
4866:
4862:
4858:
4855:
4851:
4847:
4843:
4840:
4836:
4831:
4827:
4823:
4820:
4816:
4812:
4808:
4805:
4801:
4797:
4793:
4792:
4788:
4785:
4781:
4777:
4773:
4769:
4763:
4760:
4755:
4749:
4745:
4738:
4735:
4731:
4726:, p. 239
4725:
4721:
4717:
4711:
4708:
4704:
4698:
4696:
4692:
4688:
4684:
4680:
4676:
4671:
4668:
4662:
4661:
4656:
4653:
4646:
4643:
4639:
4633:
4630:
4625:
4624:
4618:
4613:
4607:
4604:
4597:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4559:
4555:
4549:
4544:
4541:
4535:
4530:
4525:
4523:
4519:
4500:
4497:
4492:
4489:
4486:
4481:
4478:
4472:
4464:
4461:
4456:
4453:
4448:
4445:
4419:
4416:
4411:
4408:
4402:
4396:
4393:
4388:
4385:
4379:
4373:
4370:
4365:
4362:
4350:
4331:
4323:
4320:
4315:
4312:
4307:
4304:
4292:
4273:
4270:
4265:
4262:
4257:
4254:
4226:
4223:
4218:
4215:
4210:
4207:
4179:
4176:
4171:
4168:
4163:
4160:
4148:
4145:
4129:
4125:
4099:
4096:
4091:
4088:
4083:
4080:
4074:
4066:
4063:
4058:
4055:
4050:
4047:
4036:
4033:
4017:
4013:
3992:
3989:
3985:
3976:
3973:
3968:
3965:
3960:
3957:
3951:
3943:
3940:
3935:
3932:
3927:
3924:
3917:
3913:
3907:
3904:
3901:
3890:
3887:
3869:
3866:
3861:
3858:
3855:
3850:
3847:
3844:
3841:
3835:
3832:
3824:
3820:
3801:
3798:
3793:
3790:
3787:
3782:
3779:
3776:
3773:
3767:
3764:
3756:
3740:
3736:
3713:
3710:
3698:
3695:
3691:
3675:
3672:
3668:
3663:
3655:
3652:
3647:
3644:
3641:
3636:
3633:
3630:
3627:
3621:
3615:
3612:
3607:
3604:
3598:
3595:
3587:
3584:
3581:
3566:
3563:
3560:
3557:
3554:
3549:
3546:
3543:
3521:
3518:
3507:
3491:
3487:
3463:
3460:
3455:
3452:
3449:
3446:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3407:
3403:
3399:
3395:
3393:
3385:
3383:
3367:
3363:
3354:
3353:David Gregory
3350:
3346:
3343:
3339:
3335:
3330:
3328:
3324:
3319:
3314:
3312:
3307:
3299:
3292:
3290:
3266:
3262:
3258:
3251:
3247:
3243:
3233:
3229:
3225:
3220:
3212:
3208:
3204:
3197:
3193:
3189:
3179:
3175:
3171:
3162:
3161:
3160:
3143:
3138:
3134:
3118:
3114:
3110:
3103:
3099:
3095:
3085:
3081:
3077:
3072:
3064:
3060:
3056:
3049:
3045:
3041:
3031:
3027:
3023:
3011:
3008:
3005:
3000:
2996:
2990:
2986:
2982:
2977:
2973:
2967:
2963:
2955:
2954:
2953:
2936:
2931:
2927:
2918:
2914:
2910:
2903:
2899:
2895:
2885:
2881:
2877:
2872:
2867:
2863:
2857:
2853:
2845:
2831:
2826:
2822:
2813:
2809:
2805:
2798:
2794:
2790:
2780:
2776:
2772:
2767:
2762:
2758:
2752:
2748:
2740:
2739:
2735:
2731:
2729:
2709:
2704:
2700:
2693:
2690:
2687:
2682:
2678:
2674:
2671:
2666:
2662:
2655:
2652:
2647:
2644:
2634:
2633:
2632:
2614:
2609:
2605:
2598:
2595:
2592:
2589:
2586:
2581:
2577:
2570:
2567:
2564:
2557:
2556:
2555:
2553:
2531:
2527:
2523:
2520:
2515:
2511:
2507:
2504:
2499:
2495:
2491:
2484:
2483:
2482:
2479:
2469:
2467:
2445:
2442:
2436:
2433:
2430:
2426:
2423:
2420:
2414:
2408:
2401:
2400:
2399:
2396:
2392:
2388:
2384:
2380:
2374:
2351:
2345:
2342:
2339:
2336:
2333:
2327:
2324:
2322:
2317:
2307:
2304:
2301:
2298:
2295:
2289:
2286:
2284:
2279:
2268:
2267:
2266:
2264:
2260:
2256:
2252:
2249:which is the
2233:
2229:
2226:
2218:
2215:
2212:
2208:
2202:
2199:
2196:
2189:
2188:
2187:
2186:above gives:
2185:
2178:
2153:
2149:
2145:
2140:
2135:
2131:
2125:
2122:
2118:
2112:
2109:
2106:
2099:
2098:
2097:
2095:
2091:
2067:
2063:
2059:
2056:
2051:
2047:
2043:
2035:
2031:
2027:
2022:
2018:
2014:
2009:
2005:
2001:
1996:
1992:
1988:
1983:
1978:
1974:
1966:
1965:
1964:
1944:
1941:
1938:
1933:
1928:
1924:
1916:
1915:
1914:
1912:
1904:
1901:
1897:
1896:
1895:
1892:
1890:
1877:
1868:
1841:
1837:
1833:
1828:
1822:
1819:
1814:
1811:
1803:
1800:
1795:
1790:
1785:
1781:
1778:
1768:
1767:
1766:
1763:
1743:
1740:
1737:
1732:
1729:
1722:
1721:
1720:
1718:
1714:
1710:
1705:
1703:
1698:
1694:
1690:
1687:According to
1682:
1680:
1677:
1674:
1657:
1654:
1651:
1631:
1628:
1625:
1619:
1616:
1609:
1605:
1601:
1595:
1592:
1589:
1582:
1581:
1580:
1560:
1557:
1549:
1546:
1543:
1540:
1537:
1534:
1528:
1525:
1522:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1469:
1466:
1458:
1455:
1452:
1449:
1446:
1440:
1437:
1434:
1428:
1425:
1422:
1415:
1414:
1413:
1393:
1390:
1382:
1379:
1376:
1373:
1370:
1367:
1361:
1358:
1355:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1302:
1299:
1291:
1288:
1285:
1282:
1279:
1273:
1270:
1267:
1261:
1255:
1252:
1247:
1244:
1241:
1238:
1235:
1229:
1226:
1223:
1200:
1197:
1192:
1189:
1186:
1183:
1180:
1174:
1168:
1165:
1160:
1157:
1154:
1151:
1148:
1142:
1139:
1136:
1129:
1128:
1127:
1107:
1104:
1098:
1092:
1089:
1084:
1081:
1075:
1070:
1064:
1061:
1055:
1049:
1046:
1041:
1038:
1032:
1027:
1021:
1018:
1013:
1010:
1004:
999:
996:
987:
986:
985:
968:
965:
960:
957:
951:
945:
942:
937:
934:
908:
905:
898:
894:
890:
884:
881:
878:
848:
845:
840:
836:
831:
828:
822:
819:
793:
790:
785:
781:
776:
773:
767:
764:
757:
756:
755:
741:
732:
728:
726:
721:
714:
712:
698:
689:
686:
683:
677:
674:
651:
648:
643:
640:
634:
631:
628:
619:
616:
594:
587:
584:
581:
566:
547:
544:
535:
532:
527:
523:
518:
515:
509:
506:
503:
490:
489:
488:
485:
465:
462:
453:
450:
439:
436:
433:
427:
421:
412:
409:
404:
400:
395:
392:
380:
376:
372:
370:
362:
360:
358:
353:
350:
346:
343:
340:
332:Direct method
331:
329:
322:
315:
313:
309:
305:
304:
299:
295:
294:Tom Whiteside
291:
287:
283:
278:
276:
272:
267:
261:
259:
255:
251:
244:
239:
233:
232:
226:
222:
217:
215:
214:
209:
202:
200:
196:
192:
190:
185:
179:
175:
172:
168:
166:
165:
160:
147:
145:
143:
139:
135:
131:
121:
117:
115:
111:
107:
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
57:
56:Ancient Greek
53:
49:
45:
37:
32:
19:
5389:Plane curves
5330:
5317:
5301:
5279:
5257:
5253:
5240:
5221:
5215:
5204:the original
5183:
5179:
5166:
5147:
5141:
5125:
5121:
5112:
5093:
5087:
5079:
5075:
5067:
5042:
5034:
5022:. Retrieved
5007:
5000:
4968:
4964:
4958:
4940:
4907:
4903:Livio, Mario
4897:
4887:
4883:
4879:
4868:
4864:
4853:
4849:
4838:
4834:
4829:
4818:
4814:
4803:
4799:
4787:
4779:
4775:
4771:
4762:
4743:
4737:
4729:
4723:
4710:
4702:
4678:
4670:
4658:
4645:
4637:
4632:
4621:
4606:
4520:
4351:
4293:
4149:
4146:
4037:
4034:
3891:
3888:
3825:
3821:
3757:
3755:and higher:
3699:
3696:
3692:
3588:
3585:
3582:
3508:
3408:
3404:
3400:
3396:
3389:
3347:
3341:
3337:
3333:
3331:
3326:
3322:
3317:
3315:
3305:
3303:
3300:Introduction
3286:
3158:
2951:
2727:
2724:
2630:
2551:
2549:
2475:
2465:
2463:
2397:
2390:
2386:
2382:
2378:
2370:
2258:
2248:
2180:
2176:
2174:
2093:
2092:in terms of
2089:
2087:
1962:
1910:
1908:
1893:
1870:
1863:
1861:
1764:
1761:
1716:
1712:
1706:
1701:
1686:
1678:
1675:
1672:
1578:
1411:
1125:
870:
733:
729:
722:
718:
567:
563:
486:
381:
377:
373:
366:
354:
351:
347:
344:
338:
335:
326:
323:Introduction
301:
279:
266:Isaac Newton
262:
257:
253:
246:
235:
230:
228:
224:
219:
211:
206:
197:
193:
183:
181:
176:
173:
170:
162:
156:
137:
133:
129:
127:
95:
82:
78:
74:
70:
62:
59:
51:
41:
5349:at MacTutor
4675:Ross, I. M.
3349:John Wallis
100:; both are
48:mathematics
5383:Categories
4753:0520004493
4598:References
4435:, so that
924:and since
54:(from
5394:Mechanics
5357:geodesics
5303:MathWorld
5286:EMS Press
5082:, p. 122.
4993:250741844
4905:(2003) .
4660:MathWorld
3952:−
3914:∝
3905:−
3836:∝
3768:∝
3599:∝
3423:−
3318:Programma
3311:Groningen
3073:−
2983:−
2434:ω
2424:ω
2409:φ
2373:parameter
2346:φ
2343:
2337:−
2308:φ
2305:
2299:−
2296:φ
2216:−
2146:−
1878:θ
1786:θ
1782:
1652:τ
1541:−
1508:−
1453:−
997:τ
823:∝
768:∝
765:τ
742:τ
696:Δ
507:−
260:, below.
93:in 1696.
4732:(Day 4).
4718:(1638),
4587:Trochoid
4577:Catenary
4526:See also
4336:constant
2261:, whose
102:cycloids
77:, where
5288:, 2001
5200:8923709
4973:Bibcode
2255:cycloid
1900:tangent
1707:By the
349:point.
148:History
44:physics
36:cycloid
5228:
5198:
5154:
5132:
5100:
5055:
5024:2 June
5015:
5008:Cosmos
4991:
4947:
4919:
4750:
4685:
2697:
2602:
2574:
2464:where
1862:where
1318:, and
38:(red).
5250:(PDF)
5207:(PDF)
5196:S2CID
5176:(PDF)
4989:S2CID
3338:comme
2478:Jakob
2385:) = (
178:same.
124:line.
67:curve
58:
5226:ISBN
5152:ISBN
5130:ISBN
5098:ISBN
5053:ISBN
5026:2016
5013:ISBN
4945:ISBN
4917:ISBN
4748:ISBN
4683:ISBN
3316:The
2265:is:
2259:D=2r
2179:and
1655:<
1626:>
1485:and
734:Let
288:and
256:and
174:...
112:and
106:cusp
50:, a
46:and
5188:doi
5049:547
4981:doi
3329:).
2340:cos
2302:sin
1779:sin
1704:).
306:.
42:In
5385::
5300:.
5284:,
5278:,
5258:85
5256:.
5252:.
5194:,
5182:,
5178:,
5074:,
5051:.
4987:,
4979:,
4969:20
4967:,
4931:^
4888:19
4886:,
4869:19
4867:,
4854:19
4852:,
4839:19
4837:,
4819:19
4817:,
4804:19
4802:,
4776:18
4774:,
4722:,
4694:^
4657:.
4620:.
3819:,
3364:78
3291:.
2728:dx
2552:dy
2395:.
2389:,
2387:rφ
2381:,
2096::
2094:dy
2090:dx
1913::
1216:,
812:,
314:.
284:,
277:.
254:2g
252:,
144:.
116:.
5343:.
5306:.
5234:.
5190::
5184:5
5160:.
5136:.
5106:.
5061:.
5028:.
4983::
4975::
4925:.
4756:.
4689:.
4663:.
4501:B
4498:C
4493:.
4490:V
4487:C
4482:H
4479:C
4473:=
4465:B
4462:C
4457:E
4454:e
4449:E
4446:D
4420:V
4417:C
4412:H
4409:C
4403:=
4397:H
4394:V
4389:H
4386:B
4380:=
4374:E
4371:e
4366:E
4363:D
4332:=
4324:B
4321:C
4316:E
4313:e
4308:E
4305:D
4274:B
4271:C
4266:E
4263:e
4258:E
4255:D
4227:B
4224:C
4219:E
4216:e
4211:E
4208:D
4180:I
4177:C
4172:f
4169:F
4164:G
4161:F
4130:2
4126:o
4100:I
4097:C
4092:f
4089:F
4084:G
4081:F
4075:=
4067:B
4064:C
4059:E
4056:e
4051:E
4048:D
4018:2
4014:o
3993:o
3990:.
3986:)
3977:I
3974:C
3969:f
3966:F
3961:G
3958:F
3944:B
3941:C
3936:E
3933:e
3928:E
3925:D
3918:(
3911:)
3908:T
3902:t
3899:(
3870:I
3867:C
3862:.
3859:f
3856:F
3851:G
3848:F
3845:.
3842:o
3833:T
3802:B
3799:C
3794:.
3791:E
3788:e
3783:E
3780:D
3777:.
3774:o
3765:t
3741:2
3737:o
3714:B
3711:C
3676:V
3673:C
3669:o
3664:=
3656:B
3653:C
3648:.
3645:V
3642:C
3637:H
3634:C
3631:.
3628:o
3622:=
3616:B
3613:C
3608:L
3605:n
3596:t
3567:V
3564:C
3561:.
3558:B
3555:C
3550:=
3547:H
3544:C
3522:B
3519:C
3492:2
3488:o
3464:V
3461:C
3456:H
3453:C
3450:.
3447:o
3441:=
3438:L
3435:n
3432:=
3429:E
3426:e
3420:L
3417:e
3368:1
3267:1
3263:s
3259:d
3252:1
3248:x
3244:d
3234:1
3230:v
3226:1
3221:=
3213:2
3209:s
3205:d
3198:2
3194:x
3190:d
3180:2
3176:v
3172:1
3144:x
3139:2
3135:d
3129:)
3119:1
3115:s
3111:d
3104:1
3100:x
3096:d
3086:1
3082:v
3078:1
3065:2
3061:s
3057:d
3050:2
3046:x
3042:d
3032:2
3028:v
3024:1
3017:(
3012:=
3009:0
3006:=
3001:1
2997:t
2991:2
2987:d
2978:2
2974:t
2968:2
2964:d
2937:x
2932:2
2928:d
2919:2
2915:s
2911:d
2904:2
2900:x
2896:d
2886:2
2882:v
2878:1
2873:=
2868:2
2864:t
2858:2
2854:d
2832:x
2827:2
2823:d
2814:1
2810:s
2806:d
2799:1
2795:x
2791:d
2781:1
2777:v
2773:1
2768:=
2763:1
2759:t
2753:2
2749:d
2710:t
2705:2
2701:d
2694:v
2691:=
2688:s
2683:2
2679:d
2675:=
2672:x
2667:2
2663:d
2656:s
2653:d
2648:x
2645:d
2627:.
2615:x
2610:2
2606:d
2599:x
2596:d
2593:2
2590:=
2587:s
2582:2
2578:d
2571:s
2568:d
2565:2
2546:.
2532:2
2528:y
2524:d
2521:+
2516:2
2512:x
2508:d
2505:=
2500:2
2496:s
2492:d
2466:t
2446:r
2443:g
2437:=
2431:,
2427:t
2421:=
2418:)
2415:t
2412:(
2393:)
2391:r
2383:y
2379:x
2377:(
2352:.
2349:)
2334:1
2331:(
2328:r
2325:=
2318:y
2311:)
2293:(
2290:r
2287:=
2280:x
2234:,
2230:y
2227:d
2219:y
2213:D
2209:y
2203:=
2200:x
2197:d
2183:m
2181:v
2177:v
2171:.
2154:2
2150:v
2141:2
2136:m
2132:v
2126:y
2123:d
2119:v
2113:=
2110:x
2107:d
2073:)
2068:2
2064:y
2060:d
2057:+
2052:2
2048:x
2044:d
2041:(
2036:2
2032:v
2028:=
2023:2
2019:s
2015:d
2010:2
2006:v
2002:=
1997:2
1993:x
1989:d
1984:2
1979:m
1975:v
1959:.
1945:D
1942:g
1939:2
1934:=
1929:m
1925:v
1911:D
1866:m
1864:v
1858:,
1842:m
1838:v
1834:1
1829:=
1823:s
1820:d
1815:x
1812:d
1804:v
1801:1
1796:=
1791:v
1758:,
1744:y
1741:g
1738:2
1733:=
1730:v
1717:y
1713:v
1702:g
1658:t
1632:G
1629:C
1620:D
1617:M
1610:2
1606:H
1602:C
1596:=
1593:F
1590:C
1561:K
1558:M
1553:)
1550:M
1547:C
1544:2
1538:K
1535:M
1532:(
1529:.
1526:D
1523:M
1517:=
1514:H
1511:G
1505:H
1502:C
1499:=
1496:G
1493:C
1470:K
1467:M
1462:)
1459:M
1456:C
1450:K
1447:M
1444:(
1441:.
1438:D
1435:M
1429:=
1426:H
1423:C
1394:K
1391:M
1386:)
1383:M
1380:C
1377:2
1374:+
1371:K
1368:M
1365:(
1362:.
1359:D
1356:M
1350:=
1347:H
1344:G
1341:+
1338:H
1335:C
1332:=
1329:G
1326:C
1303:K
1300:M
1295:)
1292:M
1289:C
1286:+
1283:K
1280:M
1277:(
1274:.
1271:D
1268:M
1262:=
1256:K
1253:M
1248:K
1245:C
1242:.
1239:D
1236:M
1230:=
1227:H
1224:C
1201:K
1198:M
1193:M
1190:C
1187:.
1184:D
1181:M
1175:=
1169:K
1166:D
1161:D
1158:H
1155:.
1152:D
1149:M
1143:=
1140:H
1137:G
1108:2
1105:1
1099:)
1093:F
1090:C
1085:G
1082:C
1076:(
1071:=
1065:2
1062:1
1056:)
1050:D
1047:M
1042:G
1039:C
1033:(
1028:.
1022:e
1019:C
1014:m
1011:M
1005:=
1000:t
969:H
966:C
961:D
958:M
952:=
946:e
943:C
938:m
935:M
909:D
906:M
899:2
895:H
891:C
885:=
882:F
879:C
867:,
849:2
846:1
841:G
837:C
832:e
829:C
820:t
794:2
791:1
786:D
782:M
777:m
774:M
699:x
693:)
690:a
687:+
684:x
681:(
678:=
675:X
652:x
649:d
644:X
641:d
638:)
635:a
632:+
629:x
626:(
620:=
617:X
595:X
591:)
588:a
585:+
582:x
579:(
548:0
545:=
536:2
533:3
528:x
524:2
519:x
516:d
513:)
510:a
504:x
501:(
466:2
463:1
458:)
454:x
451:m
448:(
443:)
440:a
437:+
434:x
431:(
428:n
422:=
413:2
410:1
405:D
401:M
396:m
393:M
258:D
249:m
247:v
231:.
138:A
134:B
130:A
83:A
79:B
75:B
71:A
20:)
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