122:
29:
2961:
3363:
2625:
4727:
3056:
436:
2956:{\displaystyle {\begin{aligned}ds&={\frac {g}{\omega ^{2}}}\cos \theta \,d\theta \\{\frac {dx}{\cos \theta }}&={\frac {g}{\omega ^{2}}}\cos \theta \,d\theta \\dx&={\frac {g}{\omega ^{2}}}\cos ^{2}\theta \,d\theta \\&={\frac {g}{2\omega ^{2}}}\left(\cos 2\theta +1\right)d\theta \\x&={\frac {g}{4\omega ^{2}}}\left(\sin 2\theta +2\theta \right)+C_{x}\end{aligned}}}
480:
proved unhelpful for a number of reasons. First, the bending of the string causes friction, changing the timing. Second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the "circular error" of a pendulum decreases as length of the swing decreases, so better clock
461:
3358:{\displaystyle {\begin{aligned}ds&={\frac {g}{\omega ^{2}}}\cos \theta \,d\theta \\{\frac {dy}{\sin \theta }}&={\frac {g}{\omega ^{2}}}\cos \theta \,d\theta \\dy&={\frac {g}{\omega ^{2}}}\sin \theta \cos \theta \,d\theta \\&={\frac {g}{2\omega ^{2}}}\sin 2\theta \,d\theta \\y&=-{\frac {g}{4\omega ^{2}}}\cos 2\theta +C_{y}\end{aligned}}}
4412:
1098:
6299:
4117:
2340:
479:
would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan
Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts
140:
It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.
5053:
4722:{\displaystyle {\begin{aligned}{\frac {1}{2}}m\left({\frac {d\ell }{dt}}\right)^{2}&=mg(y_{0}-y)\\{\frac {d\ell }{dt}}&=\pm {\sqrt {2g(y_{0}-y)}}\\dt&=\pm {\frac {d\ell }{\sqrt {2g(y_{0}-y)}}}\\dt&=-{\frac {1}{\sqrt {2g(y_{0}-y)}}}{\frac {d\ell }{dy}}\,dy\end{aligned}}}
1286:
831:
3913:
6108:
3683:
2191:
507:
released from rest, regardless of its initial displacement, the time it takes to reach the lowest potential energy point is always a quarter of its period, which is independent of its amplitude. Therefore, the
Lagrangian of a simple harmonic oscillator is
178:
On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each
2530:
321:
1600:
The simplest solution to the tautochrone problem is to note a direct relation between the angle of an incline and the gravity felt by a particle on the incline. A particle on a 90° vertical incline undergoes full gravitational acceleration
4000:
5739:
6380:
2224:
801:
576:. One way the curve in the tautochrone problem can be an isochrone is if the Lagrangian is mathematically equivalent to a simple harmonic oscillator; that is, the height of the curve must be proportional to the arclength squared:
1526:
121:
5395:
4886:
1154:
23:
Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position
4169:
that specifies the total time of descent for a given starting height, find an equation of the curve that yields this result. The tautochrone problem is a special case of Abel's mechanical problem when
1851:
5544:
6113:
4417:
4005:
3818:
3575:
3061:
2630:
2447:
2229:
2093:
1433:
1159:
1093:{\displaystyle {\begin{aligned}dh&=s\,ds/(4r),\\dh^{2}&=s^{2}\,ds^{2}/(16r^{2})=h\left(dx^{2}+dh^{2}\right)/(2r),\\\left({\frac {dx}{dh}}\right)^{2}&={\frac {2r}{h}}-1\end{aligned}}}
836:
231:
6294:{\displaystyle {\begin{aligned}F(s)={\mathcal {L}}{\left}&={\sqrt {\frac {2g}{\pi }}}s^{\frac {1}{2}}{\mathcal {L}}\\&={\sqrt {\frac {2g}{\pi }}}T_{0}s^{-{\frac {1}{2}}}\end{aligned}}}
3813:
226:
3570:
5469:
4817:
3973:
3440:
2088:
325:
Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as diameter of the circle that generates the cycloid, multiplied by
3740:
6048:
1991:
6516:
A Treatise on the
Cycloid and All Forms of Cycloidal Curves, and on the Use of Such Curves in Dealing with the Motions of Planets, Comets, etc., and of Matter Projected from the Sun
3563:
5187:
1335:
4255:
2442:
5942:
638:
6101:
5263:
3394:
563:
386:
5889:
5851:
5582:
5229:
5097:
4293:
2050:
1648:
1935:
4405:
1621:, while a particle on a horizontal plane undergoes zero gravitational acceleration. At intermediate angles, the acceleration due to "virtual gravity" by the particle is
1421:
5134:
1895:
1375:
4761:
727:
5813:
5777:
5618:
5974:
4853:
4112:{\displaystyle {\begin{aligned}r&={\frac {g}{4\omega ^{2}}}\\\omega &={\frac {1}{2}}{\sqrt {\frac {g}{r}}}\\T&=\pi {\sqrt {\frac {r}{g}}}\\\end{aligned}}}
3935:
3806:
3773:
3049:
2618:
2366:
1719:
1688:
1668:
1747:
1567:
677:
351:
4340:
4313:
3705:
2077:
4196:
4167:
1779:
5625:
4879:
2335:{\displaystyle {\begin{aligned}g\cos \theta \,d\theta &=\omega ^{2}\,ds\\\Longrightarrow ds&={\frac {g}{\omega ^{2}}}\cos \theta \,d\theta \end{aligned}}}
2019:
3009:
2986:
2578:
2555:
2435:
2412:
2389:
1590:
6402:
6308:
5287:
4360:
3993:
3504:
3484:
3464:
3029:
2598:
2213:
1619:
821:
426:
406:
221:
201:
732:
1428:
5294:
5048:{\displaystyle T(y_{0})=\int _{y=y_{0}}^{y=0}\,dt={\frac {1}{\sqrt {2g}}}\int _{0}^{y_{0}}{\frac {1}{\sqrt {y_{0}-y}}}{\frac {d\ell }{dy}}\,dy}
1281:{\displaystyle {\begin{aligned}{\frac {dx}{dh}}&=-{\frac {\sqrt {2r-h}}{\sqrt {h}}},\\x&=-4r\int {\sqrt {1-u^{2}}}\,du,\end{aligned}}}
5189:, from which an equation for the curve would follow in a straightforward manner. To proceed, we note that the integral on the right is the
1670:
is measured between the tangent to the curve and the horizontal, with angles above the horizontal being treated as positive angles. Thus,
20:
471:
The tautochrone problem was studied by
Huygens more closely when it was realized that a pendulum, which follows a circular path, was not
6503:
6474:
4315:
is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height
1786:
5474:
1569:. It's interesting to note that the arc length squared is equal to the height difference multiplied by the full arch length
3908:{\displaystyle {\begin{aligned}x&=r\left(\phi +\sin \phi \right)\\y&=r\left(1-\cos \phi \right)\\\end{aligned}}}
3975:
is the time required for descent, being a quarter of a whole cycle, we find the descent time in terms of the radius
3678:{\displaystyle {\begin{aligned}x&=r\left(\sin \phi +\phi \right)+C_{x}\\y&=-r\cos \phi +C_{y}\end{aligned}}}
6384:
It can be shown that the cycloid obeys this equation. It needs one step further to do the integral with respect to
2186:{\displaystyle {\begin{aligned}-g\sin \theta &={\frac {d^{2}s}{{dt}^{2}}}\\&=-\omega ^{2}s\,\end{aligned}}}
106:
4731:
In the last equation, we have anticipated writing the distance remaining along the curve as a function of height (
1995:
It can be easily verified both that this solution solves the differential equation and that a particle will reach
2344:
This equation relates the change in the curve's angle to the change in the distance along the curve. We now use
5402:
3940:
28:
4770:
3399:
166:
130:
3710:
4763:, recognized that the distance remaining must decrease as time increases (thus the minus sign), and used the
1942:
5979:
826:
To solve for the analytical equation of the curve, note that the differential form of the above relation is
6430:
4202:
3513:
2525:{\displaystyle {\begin{aligned}ds={\frac {dx}{\cos \theta }}\\ds={\frac {dy}{\sin \theta }}\end{aligned}}}
2080:
504:
45:
6545:
6425:
5139:
1293:
488:
444:
110:
5061:
and allows us to compute the total time required for a particle to fall along a given curve (for which
316:{\displaystyle {\begin{aligned}x&=r(\theta -\sin \theta )\\y&=r(1-\cos \theta ),\end{aligned}}}
5897:
4220:
581:
5234:
4214:
3443:
680:
6053:
3370:
532:
6445:
5856:
5818:
5549:
5196:
5064:
4260:
2024:
1624:
356:
6550:
4365:
1380:
1131:
509:
465:
161:
125:
5102:
1858:
1340:
435:
4734:
686:
6499:
6470:
6420:
5782:
5746:
5587:
5266:
4133:
2079:. The problem is now to construct a curve that will cause the mass to obey the above motion.
5947:
4825:
3920:
3778:
3745:
3034:
2603:
2351:
1900:
1693:
1673:
1653:
5734:{\displaystyle {\mathcal {L}}\left={\sqrt {\frac {2g}{\pi }}}s^{\frac {1}{2}}{\mathcal {L}}}
1724:
1537:
645:
448:
429:
328:
4318:
4298:
3690:
2055:
6375:{\displaystyle {\frac {d\ell }{dy}}=T_{0}{\frac {\sqrt {2g}}{\pi }}{\frac {1}{\sqrt {y}}}}
4172:
4143:
1755:
453:
152:
5099:
would be easy to calculate). But Abel's mechanical problem requires the converse – given
4858:
1998:
6530:
2991:
2968:
2560:
2537:
2417:
2394:
2371:
1572:
1377:. This integral is the area under a circle, which can be done with another substitution
6387:
6303:
Making use again of the
Laplace transform above, we invert the transform and conclude:
5272:
4345:
4210:
3978:
3489:
3469:
3449:
3014:
2583:
2198:
1604:
806:
796:{\displaystyle T/4={\frac {\pi }{2}}{\sqrt {\frac {m}{k}}}=\pi {\sqrt {\frac {r}{g}}}.}
492:
476:
411:
391:
206:
186:
93:
to its lowest point is independent of its starting point on the curve. The curve is a
4257:, and since the particle is constrained to move along a curve, its velocity is simply
1521:{\displaystyle {\begin{aligned}x&=r(t-\sin t),\\h&=r(1+\cos t).\end{aligned}}}
6539:
5815:
is known, we can compute its
Laplace transform, calculate the Laplace transform of
2345:
76:
6514:
6469:. Ames, Iowa: Iowa State University Press. Part II, Proposition XXV, p. 69.
2083:
shows that the force of gravity and the acceleration of the mass are related by:
5190:
1130:
is counted from its vertex (the point with a horizontal tangent) instead of the
472:
102:
19:
4764:
2216:
515:
In the tautochrone problem, if the particle's position is parametrized by the
481:
5390:{\displaystyle {\mathcal {L}}={\frac {1}{\sqrt {2g}}}{\mathcal {L}}\leftF(s)}
3808:
so that the lowest point on the curve coincides with the origin. Therefore:
823:
is not clear until we determine the exact analytical equation of the curve.
516:
460:
147:
16:
Curve for which the time to roll to the end is equal for all starting points
160:
The tautochrone problem, the attempt to identify this curve, was solved by
6435:
86:
54:
6440:
3507:
1531:
1121:
171:
94:
90:
65:
105:
of the radius (of the circle which generates the cycloid) over the
4205:– since the particle is frictionless, and thus loses no energy to
529:
from the lowest point, the kinetic energy is then proportional to
459:
434:
120:
82:
18:
4206:
439:
Five isochronous cycloidal pendulums with different amplitudes
432:, or more accurately, the earth's gravitational acceleration.
6496:
Differential
Equations with Applications and Historical Notes
408:
is the radius of the circle which generates the cycloid, and
6209:
6133:
5985:
5701:
5631:
5480:
5423:
5350:
5300:
4136:
attacked a generalized version of the tautochrone problem (
1752:
The position of a mass measured along a tautochrone curve,
98:
353:. In modern terms, this means that the time of descent is
5546:, we now have an expression for the Laplace transform of
4881:
to get the total time required for the particle to fall:
1337:, and the height decreases as the particle moves forward
565:, and the potential energy is proportional to the height
443:
This solution was later used to solve the problem of the
5853:
and then take the inverse transform (or try to) to find
1846:{\displaystyle {\frac {d^{2}s}{{dt}^{2}}}=-\omega ^{2}s}
5539:{\textstyle {\mathcal {L}}{\left}={\sqrt {{\pi }/{s}}}}
6056:
5982:
5477:
4773:
4223:
3402:
359:
183:
The cycloid is given by a point on a circle of radius
85:
for which the time taken by an object sliding without
6390:
6311:
6111:
5950:
5900:
5859:
5821:
5785:
5749:
5628:
5590:
5552:
5405:
5297:
5275:
5237:
5199:
5142:
5105:
5067:
4889:
4861:
4828:
4737:
4415:
4368:
4348:
4321:
4301:
4263:
4175:
4146:
4003:
3981:
3943:
3923:
3816:
3781:
3748:
3713:
3693:
3573:
3516:
3492:
3472:
3452:
3373:
3059:
3037:
3017:
2994:
2971:
2628:
2606:
2586:
2563:
2540:
2445:
2420:
2397:
2374:
2354:
2227:
2201:
2091:
2058:
2027:
2001:
1945:
1903:
1861:
1789:
1758:
1727:
1696:
1676:
1656:
1627:
1607:
1575:
1540:
1431:
1383:
1343:
1296:
1157:
834:
809:
735:
689:
648:
584:
535:
414:
394:
331:
229:
209:
189:
170:, originally published in 1673, that the curve is a
4213:at any point is exactly equal to the difference in
6396:
6374:
6293:
6095:
6042:
5968:
5944:is constant. Since the Laplace transform of 1 is
5936:
5883:
5845:
5807:
5771:
5733:
5612:
5576:
5538:
5463:
5389:
5281:
5257:
5223:
5181:
5128:
5091:
5047:
4873:
4847:
4811:
4755:
4721:
4399:
4354:
4334:
4307:
4287:
4249:
4190:
4161:
4111:
3987:
3967:
3929:
3907:
3800:
3767:
3734:
3699:
3677:
3557:
3498:
3478:
3458:
3434:
3388:
3357:
3043:
3023:
3003:
2980:
2955:
2612:
2592:
2572:
2549:
2524:
2429:
2406:
2383:
2360:
2334:
2207:
2185:
2071:
2044:
2013:
1985:
1929:
1889:
1845:
1773:
1741:
1713:
1682:
1662:
1642:
1613:
1584:
1561:
1520:
1415:
1369:
1329:
1280:
1092:
815:
795:
721:
671:
632:
557:
420:
400:
380:
345:
315:
215:
195:
1781:, must obey the following differential equation:
4217:from its starting point. The kinetic energy is
495:provided an analytical solution to the problem.
484:could greatly reduce this source of inaccuracy.
131:Horologium oscillatorium sive de motu pendulorum
5743:This is as far as we can go without specifying
679:. Compared to the simple harmonic oscillator's
176:
138:
803:However, the physical meaning of the constant
203:tracing a curve as the circle rolls along the
4201:Abel's solution begins with the principle of
3510:), with the circle center at the coordinates
8:
6404:to obtain the expression of the path shape.
3486:are those of a point on a circle of radius
1530:This is the standard parameterization of a
109:. The tautochrone curve is related to the
5464:{\displaystyle F(s)={\mathcal {L}}{\left}}
4812:{\textstyle d\ell ={\frac {d\ell }{dy}}dy}
3968:{\displaystyle T={\frac {\pi }{2\omega }}}
3435:{\textstyle r={\frac {g}{4\omega ^{2}}}\,}
1120:. This is the differential equation for a
6389:
6360:
6345:
6339:
6312:
6310:
6275:
6271:
6261:
6240:
6221:
6208:
6207:
6196:
6175:
6143:
6138:
6132:
6131:
6112:
6110:
6085:
6080:
6072:
6055:
6035:
6030:
6023:
6018:
6003:
5984:
5983:
5981:
5961:
5956:
5951:
5949:
5933:
5927:
5911:
5899:
5873:
5868:
5860:
5858:
5835:
5830:
5822:
5820:
5796:
5784:
5760:
5748:
5719:
5700:
5699:
5688:
5667:
5640:
5630:
5629:
5627:
5601:
5589:
5566:
5561:
5553:
5551:
5529:
5524:
5519:
5517:
5501:
5496:
5491:
5485:
5479:
5478:
5476:
5447:
5442:
5434:
5428:
5422:
5421:
5404:
5359:
5349:
5348:
5333:
5318:
5299:
5298:
5296:
5274:
5248:
5243:
5238:
5236:
5213:
5208:
5200:
5198:
5171:
5166:
5158:
5141:
5125:
5116:
5104:
5081:
5076:
5068:
5066:
5038:
5018:
5003:
4993:
4985:
4980:
4975:
4956:
4946:
4934:
4927:
4916:
4900:
4888:
4860:
4839:
4827:
4783:
4772:
4736:
4708:
4688:
4670:
4651:
4613:
4589:
4552:
4537:
4507:
4488:
4462:
4438:
4420:
4416:
4414:
4382:
4367:
4347:
4326:
4320:
4300:
4277:
4272:
4264:
4262:
4241:
4224:
4222:
4174:
4145:
4093:
4064:
4054:
4031:
4018:
4004:
4002:
3980:
3950:
3942:
3922:
3817:
3815:
3786:
3780:
3753:
3747:
3712:
3692:
3665:
3623:
3574:
3572:
3546:
3524:
3515:
3491:
3471:
3451:
3431:
3422:
3409:
3401:
3372:
3345:
3317:
3304:
3280:
3259:
3246:
3229:
3203:
3194:
3170:
3153:
3144:
3114:
3103:
3086:
3077:
3060:
3058:
3036:
3016:
2993:
2970:
2943:
2896:
2883:
2826:
2813:
2796:
2784:
2772:
2763:
2739:
2722:
2713:
2683:
2672:
2655:
2646:
2629:
2627:
2605:
2585:
2562:
2539:
2495:
2459:
2446:
2444:
2419:
2396:
2373:
2353:
2321:
2304:
2295:
2268:
2262:
2244:
2228:
2226:
2200:
2195:The explicit appearance of the distance,
2178:
2169:
2144:
2136:
2125:
2118:
2092:
2090:
2063:
2057:
2031:
2026:
2000:
1965:
1944:
1902:
1881:
1860:
1855:which, along with the initial conditions
1834:
1816:
1808:
1797:
1790:
1788:
1757:
1731:
1726:
1703:
1695:
1675:
1655:
1626:
1606:
1574:
1539:
1432:
1430:
1402:
1382:
1350:
1342:
1308:
1303:
1295:
1264:
1256:
1244:
1192:
1162:
1158:
1156:
1108:, and leaves a differential equation for
1065:
1052:
1028:
999:
988:
972:
945:
930:
924:
916:
910:
893:
862:
855:
835:
833:
808:
778:
760:
750:
739:
734:
702:
688:
652:
647:
642:where the constant of proportionality is
610:
604:
583:
549:
538:
537:
534:
413:
393:
368:
363:
358:
335:
330:
230:
228:
208:
188:
3735:{\displaystyle -\pi \leq \phi \leq \pi }
2580:in the above equation lets us solve for
164:in 1659. He proved geometrically in his
27:
6457:
6043:{\textstyle {\mathcal {L}}={T_{0}}/{s}}
5269:of both sides with respect to variable
1986:{\displaystyle s(t)=s_{0}\cos \omega t}
6467:Christiaan Huygens' The Pendulum Clock
32:Objects representing tautochrone curve
5584:in terms of the Laplace transform of
7:
3558:{\displaystyle (C_{x}+r\phi ,C_{y})}
1137:To find the solution, integrate for
683:, the equivalent spring constant is
3506:rolling along a horizontal line (a
5182:{\displaystyle f(y)={d\ell }/{dy}}
2219:to obtain a more manageable form:
1330:{\displaystyle u={\sqrt {h/(2r)}}}
14:
6513:Proctor, Richard Anthony (1878).
4250:{\textstyle {\frac {1}{2}}mv^{2}}
5937:{\displaystyle T(y_{0})=T_{0}\,}
633:{\displaystyle h(s)=s^{2}/(8r),}
6096:{\textstyle f(y)={d\ell }/{dy}}
5258:{\displaystyle {1}/{\sqrt {y}}}
451:solved the problem in a paper (
6465:Blackwell, Richard J. (1986).
6227:
6214:
6125:
6119:
6066:
6060:
6012:
6009:
5996:
5990:
5917:
5904:
5802:
5789:
5766:
5753:
5728:
5725:
5712:
5706:
5607:
5594:
5415:
5409:
5384:
5378:
5327:
5324:
5311:
5305:
5152:
5146:
5122:
5109:
4906:
4893:
4750:
4747:
4741:
4682:
4663:
4625:
4606:
4564:
4545:
4500:
4481:
4394:
4375:
4215:gravitational potential energy
4185:
4179:
4156:
4150:
3552:
3517:
3389:{\displaystyle \phi =2\theta }
2965:Likewise, we can also express
2279:
1955:
1949:
1918:
1912:
1871:
1865:
1768:
1762:
1508:
1490:
1467:
1449:
1410:
1396:
1322:
1313:
1013:
1004:
951:
935:
876:
867:
716:
707:
666:
657:
624:
615:
594:
588:
558:{\displaystyle {\dot {s}}^{2}}
381:{\textstyle \pi {\sqrt {r/g}}}
303:
285:
265:
247:
1:
6050:, we find the shape function
5894:For the tautochrone problem,
5884:{\displaystyle {d\ell }/{dy}}
5846:{\displaystyle {d\ell }/{dy}}
5577:{\displaystyle {d\ell }/{dy}}
5224:{\displaystyle {d\ell }/{dy}}
5092:{\displaystyle {d\ell }/{dy}}
4288:{\displaystyle {d\ell }/{dt}}
2215:, is troublesome, but we can
2045:{\displaystyle \pi /2\omega }
1643:{\displaystyle g\sin \theta }
1124:when the vertical coordinate
729:, and the time of descent is
6446:Uniformly accelerated motion
4140:), namely, given a function
2368:to the differential lengths
4400:{\displaystyle mg(y_{0}-y)}
2052:from any starting position
1416:{\displaystyle u=\cos(t/2)}
113:, which is also a cycloid.
97:, and the time is equal to
6567:
5129:{\displaystyle T(y_{0})\,}
1890:{\displaystyle s(0)=s_{0}}
1596:"Virtual gravity" solution
1370:{\displaystyle dx/dh<0}
505:simple harmonic oscillator
487:Later, the mathematicians
70: 'equal' and
4756:{\displaystyle \ell (y))}
4138:Abel's mechanical problem
722:{\displaystyle k=mg/(4r)}
6494:Simmons, George (1972).
5808:{\displaystyle T(y_{0})}
5772:{\displaystyle T(y_{0})}
5613:{\displaystyle T(y_{0})}
5059:Abel's integral equation
4128:
167:Horologium Oscillatorium
5969:{\displaystyle {1}/{s}}
4848:{\displaystyle y=y_{0}}
3930:{\displaystyle \omega }
3801:{\displaystyle C_{y}=r}
3768:{\displaystyle C_{x}=0}
3742:. It is typical to set
3044:{\displaystyle \theta }
2613:{\displaystyle \theta }
2361:{\displaystyle \theta }
1930:{\displaystyle s'(0)=0}
1714:{\displaystyle -\pi /2}
1683:{\displaystyle \theta }
1663:{\displaystyle \theta }
117:The tautochrone problem
107:acceleration of gravity
6431:Calculus of variations
6398:
6376:
6295:
6097:
6044:
5970:
5938:
5885:
5847:
5809:
5773:
5735:
5614:
5578:
5540:
5465:
5391:
5283:
5259:
5225:
5183:
5130:
5093:
5049:
4875:
4849:
4822:Now we integrate from
4813:
4757:
4723:
4401:
4356:
4336:
4309:
4289:
4251:
4203:conservation of energy
4192:
4163:
4113:
3989:
3969:
3931:
3909:
3802:
3769:
3736:
3701:
3679:
3559:
3500:
3480:
3460:
3436:
3390:
3359:
3045:
3025:
3005:
2982:
2957:
2614:
2594:
2574:
2551:
2526:
2431:
2408:
2385:
2362:
2336:
2209:
2187:
2073:
2046:
2015:
1987:
1931:
1891:
1847:
1775:
1743:
1742:{\displaystyle \pi /2}
1715:
1684:
1664:
1644:
1615:
1586:
1563:
1562:{\displaystyle h=2r-y}
1522:
1417:
1371:
1331:
1282:
1094:
817:
797:
723:
673:
672:{\displaystyle 1/(8r)}
634:
559:
468:
440:
422:
402:
382:
347:
346:{\displaystyle \pi /2}
317:
217:
197:
181:
143:
135:
33:
25:
6426:Brachistochrone curve
6399:
6377:
6296:
6098:
6045:
5971:
5939:
5886:
5848:
5810:
5774:
5736:
5615:
5579:
5541:
5466:
5392:
5284:
5260:
5226:
5184:
5131:
5094:
5050:
4876:
4850:
4814:
4758:
4724:
4402:
4357:
4337:
4335:{\displaystyle y_{0}}
4310:
4308:{\displaystyle \ell }
4290:
4252:
4193:
4164:
4114:
3990:
3970:
3937:and remembering that
3932:
3910:
3803:
3770:
3737:
3702:
3700:{\displaystyle \phi }
3680:
3560:
3501:
3481:
3461:
3437:
3391:
3360:
3046:
3026:
3006:
2983:
2958:
2615:
2595:
2575:
2552:
2527:
2432:
2409:
2386:
2363:
2337:
2210:
2188:
2074:
2072:{\displaystyle s_{0}}
2047:
2016:
1988:
1932:
1892:
1848:
1776:
1744:
1716:
1685:
1665:
1645:
1616:
1587:
1564:
1523:
1418:
1372:
1332:
1283:
1095:
818:
798:
724:
674:
635:
560:
489:Joseph Louis Lagrange
463:
445:brachistochrone curve
438:
423:
403:
383:
348:
318:
218:
198:
124:
111:brachistochrone curve
81: 'time') is the
31:
22:
6388:
6309:
6109:
6054:
5980:
5948:
5898:
5857:
5819:
5783:
5747:
5626:
5588:
5550:
5475:
5403:
5295:
5273:
5235:
5197:
5140:
5103:
5065:
4887:
4859:
4826:
4771:
4735:
4413:
4366:
4346:
4319:
4299:
4261:
4221:
4191:{\displaystyle T(y)}
4173:
4162:{\displaystyle T(y)}
4144:
4125:, pp. 135–139)
4001:
3979:
3941:
3921:
3814:
3779:
3746:
3711:
3691:
3571:
3514:
3490:
3470:
3450:
3444:parametric equations
3442:, we see that these
3400:
3371:
3057:
3035:
3015:
2992:
2969:
2626:
2604:
2584:
2561:
2538:
2443:
2418:
2395:
2372:
2352:
2348:to relate the angle
2225:
2199:
2089:
2056:
2025:
1999:
1943:
1901:
1859:
1787:
1774:{\displaystyle s(t)}
1756:
1725:
1694:
1674:
1654:
1625:
1605:
1573:
1538:
1429:
1381:
1341:
1294:
1155:
832:
807:
733:
687:
646:
582:
533:
412:
392:
357:
329:
227:
207:
187:
4992:
4945:
4874:{\displaystyle y=0}
2081:Newton's second law
2014:{\displaystyle s=0}
499:Lagrangian solution
6394:
6372:
6291:
6289:
6093:
6040:
5966:
5934:
5881:
5843:
5805:
5769:
5731:
5610:
5574:
5536:
5461:
5387:
5279:
5265:and thus take the
5255:
5221:
5179:
5136:, we wish to find
5126:
5089:
5045:
4971:
4912:
4871:
4845:
4809:
4753:
4719:
4717:
4397:
4352:
4332:
4305:
4285:
4247:
4188:
4159:
4121:(Based loosely on
4109:
4107:
3985:
3965:
3927:
3905:
3903:
3798:
3765:
3732:
3697:
3675:
3673:
3555:
3496:
3476:
3456:
3432:
3386:
3355:
3353:
3041:
3021:
3004:{\displaystyle dy}
3001:
2981:{\displaystyle ds}
2978:
2953:
2951:
2610:
2590:
2573:{\displaystyle dx}
2570:
2550:{\displaystyle ds}
2547:
2522:
2520:
2430:{\displaystyle ds}
2427:
2407:{\displaystyle dy}
2404:
2384:{\displaystyle dx}
2381:
2358:
2332:
2330:
2205:
2183:
2181:
2069:
2042:
2011:
1983:
1927:
1887:
1843:
1771:
1739:
1711:
1680:
1660:
1640:
1611:
1585:{\displaystyle 8r}
1582:
1559:
1518:
1516:
1413:
1367:
1327:
1278:
1276:
1090:
1088:
813:
793:
719:
669:
630:
555:
469:
466:cycloidal pendulum
441:
418:
398:
378:
343:
313:
311:
213:
193:
162:Christiaan Huygens
136:
126:Christiaan Huygens
34:
26:
6421:Beltrami identity
6397:{\displaystyle y}
6370:
6369:
6358:
6354:
6330:
6283:
6255:
6254:
6204:
6190:
6189:
6161:
5696:
5682:
5681:
5658:
5534:
5506:
5369:
5368:
5346:
5345:
5282:{\displaystyle y}
5267:Laplace transform
5253:
5036:
5016:
5015:
4969:
4968:
4801:
4706:
4686:
4685:
4629:
4628:
4567:
4525:
4456:
4428:
4355:{\displaystyle y}
4232:
4134:Niels Henrik Abel
4103:
4102:
4074:
4073:
4062:
4038:
3988:{\displaystyle r}
3963:
3499:{\displaystyle r}
3479:{\displaystyle y}
3459:{\displaystyle x}
3429:
3324:
3266:
3209:
3159:
3135:
3092:
3024:{\displaystyle y}
2903:
2833:
2778:
2728:
2704:
2661:
2593:{\displaystyle x}
2516:
2480:
2310:
2208:{\displaystyle s}
2150:
1822:
1614:{\displaystyle g}
1325:
1262:
1213:
1212:
1207:
1180:
1102:which eliminates
1078:
1046:
816:{\displaystyle r}
788:
787:
770:
769:
758:
546:
421:{\displaystyle g}
401:{\displaystyle r}
376:
216:{\displaystyle x}
196:{\displaystyle r}
59: 'same'
38:tautochrone curve
6558:
6520:
6509:
6481:
6480:
6462:
6403:
6401:
6400:
6395:
6381:
6379:
6378:
6373:
6371:
6365:
6361:
6359:
6347:
6346:
6344:
6343:
6331:
6329:
6321:
6313:
6300:
6298:
6297:
6292:
6290:
6286:
6285:
6284:
6276:
6266:
6265:
6256:
6250:
6242:
6241:
6233:
6226:
6225:
6213:
6212:
6206:
6205:
6197:
6191:
6185:
6177:
6176:
6167:
6166:
6162:
6160:
6152:
6144:
6137:
6136:
6102:
6100:
6099:
6094:
6092:
6084:
6079:
6049:
6047:
6046:
6041:
6039:
6034:
6029:
6028:
6027:
6008:
6007:
5989:
5988:
5975:
5973:
5972:
5967:
5965:
5960:
5955:
5943:
5941:
5940:
5935:
5932:
5931:
5916:
5915:
5890:
5888:
5887:
5882:
5880:
5872:
5867:
5852:
5850:
5849:
5844:
5842:
5834:
5829:
5814:
5812:
5811:
5806:
5801:
5800:
5778:
5776:
5775:
5770:
5765:
5764:
5740:
5738:
5737:
5732:
5724:
5723:
5705:
5704:
5698:
5697:
5689:
5683:
5677:
5669:
5668:
5663:
5659:
5657:
5649:
5641:
5635:
5634:
5619:
5617:
5616:
5611:
5606:
5605:
5583:
5581:
5580:
5575:
5573:
5565:
5560:
5545:
5543:
5542:
5537:
5535:
5533:
5528:
5523:
5518:
5513:
5512:
5508:
5507:
5502:
5500:
5495:
5484:
5483:
5470:
5468:
5467:
5462:
5460:
5459:
5455:
5454:
5446:
5441:
5427:
5426:
5396:
5394:
5393:
5388:
5374:
5370:
5364:
5360:
5354:
5353:
5347:
5338:
5334:
5323:
5322:
5304:
5303:
5288:
5286:
5285:
5280:
5264:
5262:
5261:
5256:
5254:
5249:
5247:
5242:
5230:
5228:
5227:
5222:
5220:
5212:
5207:
5188:
5186:
5185:
5180:
5178:
5170:
5165:
5135:
5133:
5132:
5127:
5121:
5120:
5098:
5096:
5095:
5090:
5088:
5080:
5075:
5054:
5052:
5051:
5046:
5037:
5035:
5027:
5019:
5017:
5008:
5007:
4998:
4994:
4991:
4990:
4989:
4979:
4970:
4961:
4957:
4944:
4933:
4932:
4931:
4905:
4904:
4880:
4878:
4877:
4872:
4854:
4852:
4851:
4846:
4844:
4843:
4818:
4816:
4815:
4810:
4802:
4800:
4792:
4784:
4762:
4760:
4759:
4754:
4728:
4726:
4725:
4720:
4718:
4707:
4705:
4697:
4689:
4687:
4675:
4674:
4656:
4652:
4630:
4618:
4617:
4599:
4598:
4590:
4568:
4557:
4556:
4538:
4526:
4524:
4516:
4508:
4493:
4492:
4467:
4466:
4461:
4457:
4455:
4447:
4439:
4429:
4421:
4406:
4404:
4403:
4398:
4387:
4386:
4361:
4359:
4358:
4353:
4341:
4339:
4338:
4333:
4331:
4330:
4314:
4312:
4311:
4306:
4294:
4292:
4291:
4286:
4284:
4276:
4271:
4256:
4254:
4253:
4248:
4246:
4245:
4233:
4225:
4197:
4195:
4194:
4189:
4168:
4166:
4165:
4160:
4118:
4116:
4115:
4110:
4108:
4104:
4095:
4094:
4075:
4066:
4065:
4063:
4055:
4039:
4037:
4036:
4035:
4019:
3994:
3992:
3991:
3986:
3974:
3972:
3971:
3966:
3964:
3962:
3951:
3936:
3934:
3933:
3928:
3914:
3912:
3911:
3906:
3904:
3900:
3896:
3858:
3854:
3807:
3805:
3804:
3799:
3791:
3790:
3774:
3772:
3771:
3766:
3758:
3757:
3741:
3739:
3738:
3733:
3706:
3704:
3703:
3698:
3684:
3682:
3681:
3676:
3674:
3670:
3669:
3628:
3627:
3615:
3611:
3564:
3562:
3561:
3556:
3551:
3550:
3529:
3528:
3505:
3503:
3502:
3497:
3485:
3483:
3482:
3477:
3465:
3463:
3462:
3457:
3441:
3439:
3438:
3433:
3430:
3428:
3427:
3426:
3410:
3395:
3393:
3392:
3387:
3364:
3362:
3361:
3356:
3354:
3350:
3349:
3325:
3323:
3322:
3321:
3305:
3267:
3265:
3264:
3263:
3247:
3239:
3210:
3208:
3207:
3195:
3160:
3158:
3157:
3145:
3136:
3134:
3123:
3115:
3093:
3091:
3090:
3078:
3050:
3048:
3047:
3042:
3030:
3028:
3027:
3022:
3010:
3008:
3007:
3002:
2987:
2985:
2984:
2979:
2962:
2960:
2959:
2954:
2952:
2948:
2947:
2935:
2931:
2904:
2902:
2901:
2900:
2884:
2862:
2858:
2834:
2832:
2831:
2830:
2814:
2806:
2789:
2788:
2779:
2777:
2776:
2764:
2729:
2727:
2726:
2714:
2705:
2703:
2692:
2684:
2662:
2660:
2659:
2647:
2619:
2617:
2616:
2611:
2599:
2597:
2596:
2591:
2579:
2577:
2576:
2571:
2556:
2554:
2553:
2548:
2531:
2529:
2528:
2523:
2521:
2517:
2515:
2504:
2496:
2481:
2479:
2468:
2460:
2436:
2434:
2433:
2428:
2413:
2411:
2410:
2405:
2390:
2388:
2387:
2382:
2367:
2365:
2364:
2359:
2341:
2339:
2338:
2333:
2331:
2311:
2309:
2308:
2296:
2267:
2266:
2214:
2212:
2211:
2206:
2192:
2190:
2189:
2184:
2182:
2174:
2173:
2155:
2151:
2149:
2148:
2143:
2134:
2130:
2129:
2119:
2078:
2076:
2075:
2070:
2068:
2067:
2051:
2049:
2048:
2043:
2035:
2020:
2018:
2017:
2012:
1992:
1990:
1989:
1984:
1970:
1969:
1937:, has solution:
1936:
1934:
1933:
1928:
1911:
1896:
1894:
1893:
1888:
1886:
1885:
1852:
1850:
1849:
1844:
1839:
1838:
1823:
1821:
1820:
1815:
1806:
1802:
1801:
1791:
1780:
1778:
1777:
1772:
1748:
1746:
1745:
1740:
1735:
1720:
1718:
1717:
1712:
1707:
1689:
1687:
1686:
1681:
1669:
1667:
1666:
1661:
1649:
1647:
1646:
1641:
1620:
1618:
1617:
1612:
1591:
1589:
1588:
1583:
1568:
1566:
1565:
1560:
1527:
1525:
1524:
1519:
1517:
1422:
1420:
1419:
1414:
1406:
1376:
1374:
1373:
1368:
1354:
1336:
1334:
1333:
1328:
1326:
1312:
1304:
1287:
1285:
1284:
1279:
1277:
1263:
1261:
1260:
1245:
1214:
1208:
1194:
1193:
1181:
1179:
1171:
1163:
1148:
1142:
1129:
1119:
1113:
1107:
1099:
1097:
1096:
1091:
1089:
1079:
1074:
1066:
1057:
1056:
1051:
1047:
1045:
1037:
1029:
1003:
998:
994:
993:
992:
977:
976:
950:
949:
934:
929:
928:
915:
914:
898:
897:
866:
822:
820:
819:
814:
802:
800:
799:
794:
789:
780:
779:
771:
762:
761:
759:
751:
743:
728:
726:
725:
720:
706:
678:
676:
675:
670:
656:
639:
637:
636:
631:
614:
609:
608:
575:
564:
562:
561:
556:
554:
553:
548:
547:
539:
528:
449:Johann Bernoulli
430:gravity of Earth
427:
425:
424:
419:
407:
405:
404:
399:
387:
385:
384:
379:
377:
372:
364:
352:
350:
349:
344:
339:
322:
320:
319:
314:
312:
222:
220:
219:
214:
202:
200:
199:
194:
156:
6566:
6565:
6561:
6560:
6559:
6557:
6556:
6555:
6536:
6535:
6527:
6512:
6506:
6498:. McGraw–Hill.
6493:
6490:
6485:
6484:
6477:
6464:
6463:
6459:
6454:
6417:
6411:, Section 54).
6386:
6385:
6382:
6335:
6322:
6314:
6307:
6306:
6301:
6288:
6287:
6267:
6257:
6243:
6231:
6230:
6217:
6192:
6178:
6168:
6153:
6145:
6139:
6107:
6106:
6052:
6051:
6019:
5999:
5978:
5977:
5946:
5945:
5923:
5907:
5896:
5895:
5855:
5854:
5817:
5816:
5792:
5781:
5780:
5756:
5745:
5744:
5741:
5715:
5684:
5670:
5650:
5642:
5636:
5624:
5623:
5597:
5586:
5585:
5548:
5547:
5490:
5486:
5473:
5472:
5433:
5429:
5401:
5400:
5397:
5355:
5314:
5293:
5292:
5271:
5270:
5233:
5232:
5195:
5194:
5138:
5137:
5112:
5101:
5100:
5063:
5062:
5057:This is called
5055:
5028:
5020:
4999:
4981:
4923:
4896:
4885:
4884:
4857:
4856:
4835:
4824:
4823:
4793:
4785:
4769:
4768:
4733:
4732:
4729:
4716:
4715:
4698:
4690:
4666:
4641:
4632:
4631:
4609:
4591:
4579:
4570:
4569:
4548:
4527:
4517:
4509:
4504:
4503:
4484:
4468:
4448:
4440:
4434:
4433:
4411:
4410:
4378:
4364:
4363:
4344:
4343:
4322:
4317:
4316:
4297:
4296:
4259:
4258:
4237:
4219:
4218:
4198:is a constant.
4171:
4170:
4142:
4141:
4131:
4129:Abel's solution
4119:
4106:
4105:
4083:
4077:
4076:
4047:
4041:
4040:
4027:
4023:
4011:
3999:
3998:
3977:
3976:
3955:
3939:
3938:
3919:
3918:
3915:
3902:
3901:
3880:
3876:
3866:
3860:
3859:
3838:
3834:
3824:
3812:
3811:
3782:
3777:
3776:
3749:
3744:
3743:
3709:
3708:
3689:
3688:
3685:
3672:
3671:
3661:
3636:
3630:
3629:
3619:
3595:
3591:
3581:
3569:
3568:
3542:
3520:
3512:
3511:
3488:
3487:
3468:
3467:
3448:
3447:
3418:
3414:
3398:
3397:
3369:
3368:
3365:
3352:
3351:
3341:
3313:
3309:
3294:
3288:
3287:
3255:
3251:
3237:
3236:
3199:
3187:
3178:
3177:
3149:
3137:
3124:
3116:
3111:
3110:
3082:
3070:
3055:
3054:
3033:
3032:
3013:
3012:
2990:
2989:
2967:
2966:
2963:
2950:
2949:
2939:
2909:
2905:
2892:
2888:
2876:
2870:
2869:
2839:
2835:
2822:
2818:
2804:
2803:
2780:
2768:
2756:
2747:
2746:
2718:
2706:
2693:
2685:
2680:
2679:
2651:
2639:
2624:
2623:
2602:
2601:
2582:
2581:
2559:
2558:
2536:
2535:
2532:
2519:
2518:
2505:
2497:
2483:
2482:
2469:
2461:
2441:
2440:
2416:
2415:
2393:
2392:
2370:
2369:
2350:
2349:
2342:
2329:
2328:
2300:
2288:
2276:
2275:
2258:
2251:
2223:
2222:
2197:
2196:
2193:
2180:
2179:
2165:
2153:
2152:
2135:
2121:
2120:
2111:
2087:
2086:
2059:
2054:
2053:
2023:
2022:
1997:
1996:
1993:
1961:
1941:
1940:
1904:
1899:
1898:
1877:
1857:
1856:
1853:
1830:
1807:
1793:
1792:
1785:
1784:
1754:
1753:
1723:
1722:
1692:
1691:
1672:
1671:
1652:
1651:
1623:
1622:
1603:
1602:
1598:
1571:
1570:
1536:
1535:
1528:
1515:
1514:
1480:
1474:
1473:
1439:
1427:
1426:
1379:
1378:
1339:
1338:
1292:
1291:
1288:
1275:
1274:
1252:
1225:
1219:
1218:
1182:
1172:
1164:
1153:
1152:
1144:
1138:
1125:
1115:
1109:
1103:
1100:
1087:
1086:
1067:
1058:
1038:
1030:
1024:
1023:
1020:
1019:
984:
968:
964:
960:
941:
920:
906:
899:
889:
883:
882:
845:
830:
829:
805:
804:
731:
730:
685:
684:
644:
643:
640:
600:
580:
579:
566:
536:
531:
530:
519:
501:
464:Schematic of a
454:Acta Eruditorum
410:
409:
390:
389:
355:
354:
327:
326:
310:
309:
275:
269:
268:
237:
225:
224:
205:
204:
185:
184:
158:
153:Herman Melville
145:
119:
42:isochrone curve
17:
12:
11:
5:
6564:
6562:
6554:
6553:
6548:
6538:
6537:
6534:
6533:
6526:
6525:External links
6523:
6522:
6521:
6510:
6504:
6489:
6486:
6483:
6482:
6475:
6456:
6455:
6453:
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6393:
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6317:
6305:
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5604:
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5219:
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5119:
5115:
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5108:
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5011:
5006:
5002:
4997:
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4911:
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4805:
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4796:
4791:
4788:
4782:
4779:
4776:
4752:
4749:
4746:
4743:
4740:
4714:
4711:
4704:
4701:
4696:
4693:
4684:
4681:
4678:
4673:
4669:
4665:
4662:
4659:
4655:
4650:
4647:
4644:
4642:
4640:
4637:
4634:
4633:
4627:
4624:
4621:
4616:
4612:
4608:
4605:
4602:
4597:
4594:
4588:
4585:
4582:
4580:
4578:
4575:
4572:
4571:
4566:
4563:
4560:
4555:
4551:
4547:
4544:
4541:
4536:
4533:
4530:
4528:
4523:
4520:
4515:
4512:
4506:
4505:
4502:
4499:
4496:
4491:
4487:
4483:
4480:
4477:
4474:
4471:
4469:
4465:
4460:
4454:
4451:
4446:
4443:
4437:
4432:
4427:
4424:
4419:
4418:
4409:
4396:
4393:
4390:
4385:
4381:
4377:
4374:
4371:
4351:
4329:
4325:
4304:
4283:
4280:
4275:
4270:
4267:
4244:
4240:
4236:
4231:
4228:
4211:kinetic energy
4187:
4184:
4181:
4178:
4158:
4155:
4152:
4149:
4130:
4127:
4101:
4098:
4092:
4089:
4086:
4084:
4082:
4079:
4078:
4072:
4069:
4061:
4058:
4053:
4050:
4048:
4046:
4043:
4042:
4034:
4030:
4026:
4022:
4017:
4014:
4012:
4010:
4007:
4006:
3997:
3984:
3961:
3958:
3954:
3949:
3946:
3926:
3899:
3895:
3892:
3889:
3886:
3883:
3879:
3875:
3872:
3869:
3867:
3865:
3862:
3861:
3857:
3853:
3850:
3847:
3844:
3841:
3837:
3833:
3830:
3827:
3825:
3823:
3820:
3819:
3810:
3797:
3794:
3789:
3785:
3764:
3761:
3756:
3752:
3731:
3728:
3725:
3722:
3719:
3716:
3696:
3668:
3664:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3637:
3635:
3632:
3631:
3626:
3622:
3618:
3614:
3610:
3607:
3604:
3601:
3598:
3594:
3590:
3587:
3584:
3582:
3580:
3577:
3576:
3567:
3554:
3549:
3545:
3541:
3538:
3535:
3532:
3527:
3523:
3519:
3495:
3475:
3455:
3425:
3421:
3417:
3413:
3408:
3405:
3385:
3382:
3379:
3376:
3348:
3344:
3340:
3337:
3334:
3331:
3328:
3320:
3316:
3312:
3308:
3303:
3300:
3297:
3295:
3293:
3290:
3289:
3286:
3283:
3279:
3276:
3273:
3270:
3262:
3258:
3254:
3250:
3245:
3242:
3240:
3238:
3235:
3232:
3228:
3225:
3222:
3219:
3216:
3213:
3206:
3202:
3198:
3193:
3190:
3188:
3186:
3183:
3180:
3179:
3176:
3173:
3169:
3166:
3163:
3156:
3152:
3148:
3143:
3140:
3138:
3133:
3130:
3127:
3122:
3119:
3113:
3112:
3109:
3106:
3102:
3099:
3096:
3089:
3085:
3081:
3076:
3073:
3071:
3069:
3066:
3063:
3062:
3053:
3040:
3020:
3011:and solve for
3000:
2997:
2977:
2974:
2946:
2942:
2938:
2934:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2908:
2899:
2895:
2891:
2887:
2882:
2879:
2877:
2875:
2872:
2871:
2868:
2865:
2861:
2857:
2854:
2851:
2848:
2845:
2842:
2838:
2829:
2825:
2821:
2817:
2812:
2809:
2807:
2805:
2802:
2799:
2795:
2792:
2787:
2783:
2775:
2771:
2767:
2762:
2759:
2757:
2755:
2752:
2749:
2748:
2745:
2742:
2738:
2735:
2732:
2725:
2721:
2717:
2712:
2709:
2707:
2702:
2699:
2696:
2691:
2688:
2682:
2681:
2678:
2675:
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2665:
2658:
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2645:
2642:
2640:
2638:
2635:
2632:
2631:
2622:
2609:
2589:
2569:
2566:
2546:
2543:
2514:
2511:
2508:
2503:
2500:
2494:
2491:
2488:
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2484:
2478:
2475:
2472:
2467:
2464:
2458:
2455:
2452:
2449:
2448:
2439:
2426:
2423:
2403:
2400:
2380:
2377:
2357:
2327:
2324:
2320:
2317:
2314:
2307:
2303:
2299:
2294:
2291:
2289:
2287:
2284:
2281:
2278:
2277:
2274:
2271:
2265:
2261:
2257:
2254:
2252:
2250:
2247:
2243:
2240:
2237:
2234:
2231:
2230:
2221:
2204:
2177:
2172:
2168:
2164:
2161:
2158:
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2142:
2139:
2133:
2128:
2124:
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2114:
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2110:
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2104:
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2098:
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2062:
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2010:
2007:
2004:
1982:
1979:
1976:
1973:
1968:
1964:
1960:
1957:
1954:
1951:
1948:
1939:
1926:
1923:
1920:
1917:
1914:
1910:
1907:
1884:
1880:
1876:
1873:
1870:
1867:
1864:
1842:
1837:
1833:
1829:
1826:
1819:
1814:
1811:
1805:
1800:
1796:
1783:
1770:
1767:
1764:
1761:
1738:
1734:
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1710:
1706:
1702:
1699:
1679:
1659:
1639:
1636:
1633:
1630:
1610:
1597:
1594:
1581:
1578:
1558:
1555:
1552:
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1543:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1481:
1479:
1476:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1440:
1438:
1435:
1434:
1425:
1412:
1409:
1405:
1401:
1398:
1395:
1392:
1389:
1386:
1366:
1363:
1360:
1357:
1353:
1349:
1346:
1324:
1321:
1318:
1315:
1311:
1307:
1302:
1299:
1273:
1270:
1267:
1259:
1255:
1251:
1248:
1243:
1240:
1237:
1234:
1231:
1228:
1226:
1224:
1221:
1220:
1217:
1211:
1206:
1203:
1200:
1197:
1191:
1188:
1185:
1183:
1178:
1175:
1170:
1167:
1161:
1160:
1151:
1085:
1082:
1077:
1073:
1070:
1064:
1061:
1059:
1055:
1050:
1044:
1041:
1036:
1033:
1027:
1022:
1021:
1018:
1015:
1012:
1009:
1006:
1002:
997:
991:
987:
983:
980:
975:
971:
967:
963:
959:
956:
953:
948:
944:
940:
937:
933:
927:
923:
919:
913:
909:
905:
902:
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709:
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668:
665:
662:
659:
655:
651:
629:
626:
623:
620:
617:
613:
607:
603:
599:
596:
593:
590:
587:
578:
552:
545:
542:
500:
497:
493:Leonhard Euler
477:pendulum clock
417:
397:
375:
371:
367:
362:
342:
338:
334:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
276:
274:
271:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
238:
236:
233:
232:
212:
192:
179:other ...
137:
118:
115:
15:
13:
10:
9:
6:
4:
3:
2:
6563:
6552:
6549:
6547:
6544:
6543:
6541:
6532:
6529:
6528:
6524:
6518:
6517:
6511:
6507:
6505:0-07-057540-1
6501:
6497:
6492:
6491:
6487:
6478:
6476:0-8138-0933-9
6472:
6468:
6461:
6458:
6451:
6447:
6444:
6442:
6439:
6437:
6434:
6432:
6429:
6427:
6424:
6422:
6419:
6418:
6414:
6412:
6410:
6405:
6391:
6366:
6362:
6355:
6351:
6348:
6340:
6336:
6332:
6326:
6323:
6318:
6315:
6304:
6280:
6277:
6272:
6268:
6262:
6258:
6251:
6247:
6244:
6237:
6235:
6222:
6218:
6201:
6198:
6193:
6186:
6182:
6179:
6172:
6170:
6163:
6157:
6154:
6149:
6146:
6140:
6128:
6122:
6116:
6104:
6089:
6086:
6081:
6076:
6073:
6069:
6063:
6057:
6036:
6031:
6024:
6020:
6015:
6004:
6000:
5993:
5962:
5957:
5952:
5928:
5924:
5920:
5912:
5908:
5901:
5892:
5877:
5874:
5869:
5864:
5861:
5839:
5836:
5831:
5826:
5823:
5797:
5793:
5786:
5761:
5757:
5750:
5720:
5716:
5709:
5693:
5690:
5685:
5678:
5674:
5671:
5664:
5660:
5654:
5651:
5646:
5643:
5637:
5621:
5602:
5598:
5591:
5570:
5567:
5562:
5557:
5554:
5530:
5525:
5520:
5514:
5509:
5503:
5497:
5492:
5487:
5456:
5451:
5448:
5443:
5438:
5435:
5430:
5418:
5412:
5406:
5381:
5375:
5371:
5365:
5361:
5356:
5342:
5339:
5335:
5330:
5319:
5315:
5308:
5290:
5276:
5268:
5250:
5244:
5239:
5217:
5214:
5209:
5204:
5201:
5192:
5175:
5172:
5167:
5162:
5159:
5155:
5149:
5143:
5117:
5113:
5106:
5085:
5082:
5077:
5072:
5069:
5060:
5042:
5039:
5032:
5029:
5024:
5021:
5012:
5009:
5004:
5000:
4995:
4986:
4982:
4976:
4972:
4965:
4962:
4958:
4953:
4950:
4947:
4941:
4938:
4935:
4928:
4924:
4920:
4917:
4913:
4909:
4901:
4897:
4890:
4882:
4868:
4865:
4862:
4840:
4836:
4832:
4829:
4820:
4806:
4803:
4797:
4794:
4789:
4786:
4780:
4777:
4774:
4766:
4744:
4738:
4712:
4709:
4702:
4699:
4694:
4691:
4679:
4676:
4671:
4667:
4660:
4657:
4653:
4648:
4645:
4643:
4638:
4635:
4622:
4619:
4614:
4610:
4603:
4600:
4595:
4592:
4586:
4583:
4581:
4576:
4573:
4561:
4558:
4553:
4549:
4542:
4539:
4534:
4531:
4529:
4521:
4518:
4513:
4510:
4497:
4494:
4489:
4485:
4478:
4475:
4472:
4470:
4463:
4458:
4452:
4449:
4444:
4441:
4435:
4430:
4425:
4422:
4408:
4391:
4388:
4383:
4379:
4372:
4369:
4349:
4327:
4323:
4302:
4281:
4278:
4273:
4268:
4265:
4242:
4238:
4234:
4229:
4226:
4216:
4212:
4208:
4204:
4199:
4182:
4176:
4153:
4147:
4139:
4135:
4126:
4124:
4099:
4096:
4090:
4087:
4085:
4080:
4070:
4067:
4059:
4056:
4051:
4049:
4044:
4032:
4028:
4024:
4020:
4015:
4013:
4008:
3996:
3982:
3959:
3956:
3952:
3947:
3944:
3924:
3897:
3893:
3890:
3887:
3884:
3881:
3877:
3873:
3870:
3868:
3863:
3855:
3851:
3848:
3845:
3842:
3839:
3835:
3831:
3828:
3826:
3821:
3809:
3795:
3792:
3787:
3783:
3762:
3759:
3754:
3750:
3729:
3726:
3723:
3720:
3717:
3714:
3694:
3666:
3662:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3638:
3633:
3624:
3620:
3616:
3612:
3608:
3605:
3602:
3599:
3596:
3592:
3588:
3585:
3583:
3578:
3566:
3547:
3543:
3539:
3536:
3533:
3530:
3525:
3521:
3509:
3493:
3473:
3453:
3445:
3423:
3419:
3415:
3411:
3406:
3403:
3383:
3380:
3377:
3374:
3367:Substituting
3346:
3342:
3338:
3335:
3332:
3329:
3326:
3318:
3314:
3310:
3306:
3301:
3298:
3296:
3291:
3284:
3281:
3277:
3274:
3271:
3268:
3260:
3256:
3252:
3248:
3243:
3241:
3233:
3230:
3226:
3223:
3220:
3217:
3214:
3211:
3204:
3200:
3196:
3191:
3189:
3184:
3181:
3174:
3171:
3167:
3164:
3161:
3154:
3150:
3146:
3141:
3139:
3131:
3128:
3125:
3120:
3117:
3107:
3104:
3100:
3097:
3094:
3087:
3083:
3079:
3074:
3072:
3067:
3064:
3052:
3038:
3018:
2998:
2995:
2975:
2972:
2944:
2940:
2936:
2932:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2906:
2897:
2893:
2889:
2885:
2880:
2878:
2873:
2866:
2863:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2836:
2827:
2823:
2819:
2815:
2810:
2808:
2800:
2797:
2793:
2790:
2785:
2781:
2773:
2769:
2765:
2760:
2758:
2753:
2750:
2743:
2740:
2736:
2733:
2730:
2723:
2719:
2715:
2710:
2708:
2700:
2697:
2694:
2689:
2686:
2676:
2673:
2669:
2666:
2663:
2656:
2652:
2648:
2643:
2641:
2636:
2633:
2621:
2607:
2587:
2567:
2564:
2544:
2541:
2512:
2509:
2506:
2501:
2498:
2492:
2489:
2486:
2476:
2473:
2470:
2465:
2462:
2456:
2453:
2450:
2438:
2424:
2421:
2401:
2398:
2378:
2375:
2355:
2347:
2325:
2322:
2318:
2315:
2312:
2305:
2301:
2297:
2292:
2290:
2285:
2282:
2272:
2269:
2263:
2259:
2255:
2253:
2248:
2245:
2241:
2238:
2235:
2232:
2220:
2218:
2217:differentiate
2202:
2175:
2170:
2166:
2162:
2159:
2157:
2145:
2140:
2137:
2131:
2126:
2122:
2115:
2113:
2108:
2105:
2102:
2099:
2096:
2084:
2082:
2064:
2060:
2039:
2036:
2032:
2028:
2008:
2005:
2002:
1980:
1977:
1974:
1971:
1966:
1962:
1958:
1952:
1946:
1938:
1924:
1921:
1915:
1908:
1905:
1882:
1878:
1874:
1868:
1862:
1840:
1835:
1831:
1827:
1824:
1817:
1812:
1809:
1803:
1798:
1794:
1782:
1765:
1759:
1750:
1736:
1732:
1728:
1708:
1704:
1700:
1697:
1677:
1657:
1637:
1634:
1631:
1628:
1608:
1595:
1593:
1579:
1576:
1556:
1553:
1550:
1547:
1544:
1541:
1533:
1511:
1505:
1502:
1499:
1496:
1493:
1487:
1484:
1482:
1477:
1470:
1464:
1461:
1458:
1455:
1452:
1446:
1443:
1441:
1436:
1424:
1407:
1403:
1399:
1393:
1390:
1387:
1384:
1364:
1361:
1358:
1355:
1351:
1347:
1344:
1319:
1316:
1309:
1305:
1300:
1297:
1271:
1268:
1265:
1257:
1253:
1249:
1246:
1241:
1238:
1235:
1232:
1229:
1227:
1222:
1215:
1209:
1204:
1201:
1198:
1195:
1189:
1186:
1184:
1176:
1173:
1168:
1165:
1150:
1147:
1141:
1135:
1133:
1128:
1123:
1118:
1112:
1106:
1083:
1080:
1075:
1071:
1068:
1062:
1060:
1053:
1048:
1042:
1039:
1034:
1031:
1025:
1016:
1010:
1007:
1000:
995:
989:
985:
981:
978:
973:
969:
965:
961:
957:
954:
946:
942:
938:
931:
925:
921:
917:
911:
907:
903:
901:
894:
890:
886:
879:
873:
870:
863:
859:
856:
852:
849:
847:
842:
839:
827:
824:
810:
790:
784:
781:
775:
772:
766:
763:
755:
752:
747:
744:
740:
736:
713:
710:
703:
699:
696:
693:
690:
682:
663:
660:
653:
649:
627:
621:
618:
611:
605:
601:
597:
591:
585:
577:
573:
569:
550:
543:
540:
526:
522:
518:
513:
511:
506:
498:
496:
494:
490:
485:
483:
478:
475:and thus his
474:
467:
462:
458:
456:
455:
450:
446:
437:
433:
431:
415:
395:
373:
369:
365:
360:
340:
336:
332:
323:
306:
300:
297:
294:
291:
288:
282:
279:
277:
272:
262:
259:
256:
253:
250:
244:
241:
239:
234:
210:
190:
180:
175:
173:
169:
168:
163:
157:
154:
150:
149:
142:
133:
132:
127:
123:
116:
114:
112:
108:
104:
100:
96:
92:
88:
84:
80:
78:
72:
69:
67:
61:
58:
56:
50:
47:
46:Ancient Greek
43:
39:
30:
21:
6546:Plane curves
6515:
6495:
6488:Bibliography
6466:
6460:
6408:
6406:
6383:
6302:
5893:
5742:
5398:
5058:
5056:
4821:
4767:in the form
4730:
4342:to a height
4200:
4137:
4132:
4122:
4120:
3917:Solving for
3916:
3707:ranges from
3686:
3366:
3031:in terms of
2988:in terms of
2964:
2600:in terms of
2533:
2346:trigonometry
2343:
2194:
1994:
1854:
1751:
1690:varies from
1650:. Note that
1599:
1529:
1289:
1145:
1143:in terms of
1139:
1136:
1126:
1116:
1110:
1104:
1101:
825:
641:
571:
567:
524:
520:
514:
502:
486:
470:
452:
442:
324:
182:
177:
165:
159:
146:
144:
139:
129:
74:
71:
63:
60:
52:
49:
41:
37:
35:
5191:convolution
1423:and yield:
510:isochronous
482:escapements
473:isochronous
103:square root
89:in uniform
6540:Categories
6452:References
4765:chain rule
3687:Note that
2534:Replacing
681:Lagrangian
223:axis, as:
101:times the
44:(from
6551:Mechanics
6531:Mathworld
6356:π
6319:ℓ
6273:−
6252:π
6187:π
6150:ℓ
6077:ℓ
5865:ℓ
5827:ℓ
5679:π
5647:ℓ
5558:ℓ
5521:π
5439:ℓ
5205:ℓ
5163:ℓ
5073:ℓ
5025:ℓ
5010:−
4973:∫
4914:∫
4790:ℓ
4778:ℓ
4739:ℓ
4695:ℓ
4677:−
4649:−
4620:−
4596:ℓ
4587:±
4559:−
4535:±
4514:ℓ
4495:−
4445:ℓ
4389:−
4303:ℓ
4269:ℓ
4091:π
4045:ω
4029:ω
3960:ω
3953:π
3925:ω
3894:ϕ
3891:
3885:−
3852:ϕ
3849:
3840:ϕ
3730:π
3727:≤
3724:ϕ
3721:≤
3718:π
3715:−
3695:ϕ
3656:ϕ
3653:
3644:−
3609:ϕ
3603:ϕ
3600:
3537:ϕ
3420:ω
3384:θ
3375:ϕ
3336:θ
3330:
3315:ω
3302:−
3285:θ
3278:θ
3272:
3257:ω
3234:θ
3227:θ
3224:
3218:θ
3215:
3201:ω
3175:θ
3168:θ
3165:
3151:ω
3132:θ
3129:
3108:θ
3101:θ
3098:
3084:ω
3039:θ
2929:θ
2920:θ
2914:
2894:ω
2867:θ
2850:θ
2844:
2824:ω
2801:θ
2794:θ
2791:
2770:ω
2744:θ
2737:θ
2734:
2720:ω
2701:θ
2698:
2677:θ
2670:θ
2667:
2653:ω
2608:θ
2513:θ
2510:
2477:θ
2474:
2356:θ
2326:θ
2319:θ
2316:
2302:ω
2280:⟹
2260:ω
2249:θ
2242:θ
2239:
2167:ω
2163:−
2109:θ
2106:
2097:−
2040:ω
2029:π
1978:ω
1975:
1832:ω
1828:−
1729:π
1701:π
1698:−
1678:θ
1658:θ
1638:θ
1635:
1554:−
1503:
1462:
1456:−
1394:
1250:−
1242:∫
1233:−
1202:−
1190:−
1081:−
776:π
753:π
544:˙
517:arclength
457:, 1697).
361:π
333:π
301:θ
298:
292:−
263:θ
260:
254:−
251:θ
148:Moby Dick
6436:Catenary
6415:See also
5976:, i.e.,
5471:. Since
4407:, thus:
4295:, where
2021:at time
1909:′
388:, where
87:friction
24:diagram.
6441:Cycloid
6409:Simmons
5779:. Once
4123:Proctor
3508:cycloid
1532:cycloid
1122:cycloid
428:is the
172:cycloid
95:cycloid
91:gravity
77:chronos
6502:
6473:
5399:where
4209:, its
1290:where
503:For a
155:, 1851
134:, 1673
73:χρόνος
55:tauto-
5231:with
2557:with
1534:with
83:curve
66:isos-
51:ταὐτό
48:
6500:ISBN
6471:ISBN
4207:heat
3775:and
3466:and
3446:for
3396:and
2414:and
1897:and
1362:<
1132:cusp
1114:and
491:and
62:ἴσος
5193:of
4855:to
4362:is
3888:cos
3846:sin
3650:cos
3597:sin
3327:cos
3269:sin
3221:cos
3212:sin
3162:cos
3126:sin
3095:cos
2911:sin
2841:cos
2782:cos
2731:cos
2695:cos
2664:cos
2507:sin
2471:cos
2313:cos
2236:cos
2103:sin
1972:cos
1721:to
1632:sin
1500:cos
1459:sin
1391:cos
447:.
295:cos
257:sin
151:by
40:or
6542::
6103::
5891:.
5620::
5289::
4819:.
3995::
3565::
3051::
2620::
2437::
2391:,
1749:.
1592:.
1149::
1134:.
1117:dh
1111:dx
939:16
512:.
174:.
128:,
36:A
6519:.
6508:.
6479:.
6407:(
6392:y
6367:y
6363:1
6352:g
6349:2
6341:0
6337:T
6333:=
6327:y
6324:d
6316:d
6281:2
6278:1
6269:s
6263:0
6259:T
6248:g
6245:2
6238:=
6228:]
6223:0
6219:T
6215:[
6210:L
6202:2
6199:1
6194:s
6183:g
6180:2
6173:=
6164:]
6158:y
6155:d
6147:d
6141:[
6134:L
6129:=
6126:)
6123:s
6120:(
6117:F
6090:y
6087:d
6082:/
6074:d
6070:=
6067:)
6064:y
6061:(
6058:f
6037:s
6032:/
6025:0
6021:T
6016:=
6013:]
6010:)
6005:0
6001:y
5997:(
5994:T
5991:[
5986:L
5963:s
5958:/
5953:1
5929:0
5925:T
5921:=
5918:)
5913:0
5909:y
5905:(
5902:T
5878:y
5875:d
5870:/
5862:d
5840:y
5837:d
5832:/
5824:d
5803:)
5798:0
5794:y
5790:(
5787:T
5767:)
5762:0
5758:y
5754:(
5751:T
5729:]
5726:)
5721:0
5717:y
5713:(
5710:T
5707:[
5702:L
5694:2
5691:1
5686:s
5675:g
5672:2
5665:=
5661:]
5655:y
5652:d
5644:d
5638:[
5632:L
5608:)
5603:0
5599:y
5595:(
5592:T
5571:y
5568:d
5563:/
5555:d
5531:s
5526:/
5515:=
5510:]
5504:y
5498:/
5493:1
5488:[
5481:L
5457:]
5452:y
5449:d
5444:/
5436:d
5431:[
5424:L
5419:=
5416:)
5413:s
5410:(
5407:F
5385:)
5382:s
5379:(
5376:F
5372:]
5366:y
5362:1
5357:[
5351:L
5343:g
5340:2
5336:1
5331:=
5328:]
5325:)
5320:0
5316:y
5312:(
5309:T
5306:[
5301:L
5277:y
5251:y
5245:/
5240:1
5218:y
5215:d
5210:/
5202:d
5176:y
5173:d
5168:/
5160:d
5156:=
5153:)
5150:y
5147:(
5144:f
5123:)
5118:0
5114:y
5110:(
5107:T
5086:y
5083:d
5078:/
5070:d
5043:y
5040:d
5033:y
5030:d
5022:d
5013:y
5005:0
5001:y
4996:1
4987:0
4983:y
4977:0
4966:g
4963:2
4959:1
4954:=
4951:t
4948:d
4942:0
4939:=
4936:y
4929:0
4925:y
4921:=
4918:y
4910:=
4907:)
4902:0
4898:y
4894:(
4891:T
4869:0
4866:=
4863:y
4841:0
4837:y
4833:=
4830:y
4807:y
4804:d
4798:y
4795:d
4787:d
4781:=
4775:d
4751:)
4748:)
4745:y
4742:(
4713:y
4710:d
4703:y
4700:d
4692:d
4683:)
4680:y
4672:0
4668:y
4664:(
4661:g
4658:2
4654:1
4646:=
4639:t
4636:d
4626:)
4623:y
4615:0
4611:y
4607:(
4604:g
4601:2
4593:d
4584:=
4577:t
4574:d
4565:)
4562:y
4554:0
4550:y
4546:(
4543:g
4540:2
4532:=
4522:t
4519:d
4511:d
4501:)
4498:y
4490:0
4486:y
4482:(
4479:g
4476:m
4473:=
4464:2
4459:)
4453:t
4450:d
4442:d
4436:(
4431:m
4426:2
4423:1
4395:)
4392:y
4384:0
4380:y
4376:(
4373:g
4370:m
4350:y
4328:0
4324:y
4282:t
4279:d
4274:/
4266:d
4243:2
4239:v
4235:m
4230:2
4227:1
4186:)
4183:y
4180:(
4177:T
4157:)
4154:y
4151:(
4148:T
4100:g
4097:r
4088:=
4081:T
4071:r
4068:g
4060:2
4057:1
4052:=
4033:2
4025:4
4021:g
4016:=
4009:r
3983:r
3957:2
3948:=
3945:T
3898:)
3882:1
3878:(
3874:r
3871:=
3864:y
3856:)
3843:+
3836:(
3832:r
3829:=
3822:x
3796:r
3793:=
3788:y
3784:C
3763:0
3760:=
3755:x
3751:C
3667:y
3663:C
3659:+
3647:r
3641:=
3634:y
3625:x
3621:C
3617:+
3613:)
3606:+
3593:(
3589:r
3586:=
3579:x
3553:)
3548:y
3544:C
3540:,
3534:r
3531:+
3526:x
3522:C
3518:(
3494:r
3474:y
3454:x
3424:2
3416:4
3412:g
3407:=
3404:r
3381:2
3378:=
3347:y
3343:C
3339:+
3333:2
3319:2
3311:4
3307:g
3299:=
3292:y
3282:d
3275:2
3261:2
3253:2
3249:g
3244:=
3231:d
3205:2
3197:g
3192:=
3185:y
3182:d
3172:d
3155:2
3147:g
3142:=
3121:y
3118:d
3105:d
3088:2
3080:g
3075:=
3068:s
3065:d
3019:y
2999:y
2996:d
2976:s
2973:d
2945:x
2941:C
2937:+
2933:)
2926:2
2923:+
2917:2
2907:(
2898:2
2890:4
2886:g
2881:=
2874:x
2864:d
2860:)
2856:1
2853:+
2847:2
2837:(
2828:2
2820:2
2816:g
2811:=
2798:d
2786:2
2774:2
2766:g
2761:=
2754:x
2751:d
2741:d
2724:2
2716:g
2711:=
2690:x
2687:d
2674:d
2657:2
2649:g
2644:=
2637:s
2634:d
2588:x
2568:x
2565:d
2545:s
2542:d
2502:y
2499:d
2493:=
2490:s
2487:d
2466:x
2463:d
2457:=
2454:s
2451:d
2425:s
2422:d
2402:y
2399:d
2379:x
2376:d
2323:d
2306:2
2298:g
2293:=
2286:s
2283:d
2273:s
2270:d
2264:2
2256:=
2246:d
2233:g
2203:s
2176:s
2171:2
2160:=
2146:2
2141:t
2138:d
2132:s
2127:2
2123:d
2116:=
2100:g
2065:0
2061:s
2037:2
2033:/
2009:0
2006:=
2003:s
1981:t
1967:0
1963:s
1959:=
1956:)
1953:t
1950:(
1947:s
1925:0
1922:=
1919:)
1916:0
1913:(
1906:s
1883:0
1879:s
1875:=
1872:)
1869:0
1866:(
1863:s
1841:s
1836:2
1825:=
1818:2
1813:t
1810:d
1804:s
1799:2
1795:d
1769:)
1766:t
1763:(
1760:s
1737:2
1733:/
1709:2
1705:/
1629:g
1609:g
1580:r
1577:8
1557:y
1551:r
1548:2
1545:=
1542:h
1512:.
1509:)
1506:t
1497:+
1494:1
1491:(
1488:r
1485:=
1478:h
1471:,
1468:)
1465:t
1453:t
1450:(
1447:r
1444:=
1437:x
1411:)
1408:2
1404:/
1400:t
1397:(
1388:=
1385:u
1365:0
1359:h
1356:d
1352:/
1348:x
1345:d
1323:)
1320:r
1317:2
1314:(
1310:/
1306:h
1301:=
1298:u
1272:,
1269:u
1266:d
1258:2
1254:u
1247:1
1239:r
1236:4
1230:=
1223:x
1216:,
1210:h
1205:h
1199:r
1196:2
1187:=
1177:h
1174:d
1169:x
1166:d
1146:h
1140:x
1127:h
1105:s
1084:1
1076:h
1072:r
1069:2
1063:=
1054:2
1049:)
1043:h
1040:d
1035:x
1032:d
1026:(
1017:,
1014:)
1011:r
1008:2
1005:(
1001:/
996:)
990:2
986:h
982:d
979:+
974:2
970:x
966:d
962:(
958:h
955:=
952:)
947:2
943:r
936:(
932:/
926:2
922:s
918:d
912:2
908:s
904:=
895:2
891:h
887:d
880:,
877:)
874:r
871:4
868:(
864:/
860:s
857:d
853:s
850:=
843:h
840:d
811:r
791:.
785:g
782:r
773:=
767:k
764:m
756:2
748:=
745:4
741:/
737:T
717:)
714:r
711:4
708:(
704:/
700:g
697:m
694:=
691:k
667:)
664:r
661:8
658:(
654:/
650:1
628:,
625:)
622:r
619:8
616:(
612:/
606:2
602:s
598:=
595:)
592:s
589:(
586:h
574:)
572:s
570:(
568:h
551:2
541:s
527:)
525:t
523:(
521:s
416:g
396:r
374:g
370:/
366:r
341:2
337:/
307:,
304:)
289:1
286:(
283:r
280:=
273:y
266:)
248:(
245:r
242:=
235:x
211:x
191:r
99:π
79:)
75:(
68:)
64:(
57:)
53:(
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