Editing Brachistochrone curve (section)

==History==

=== Galileo's problem ===
Earlier, in 1638, [[Galileo Galilei]] had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his ''[[Two New Sciences]]''. He draws the conclusion that the arc of a circle is faster than any number of its chords,<ref>{{citation |author=Galileo Galilei |title=Discourses regarding two new sciences |page=239 |year=1638 |chapter=Third Day, Theorem 22, Prop. 36 |chapter-url=http://galileoandeinstein.physics.virginia.edu/tns_draft/tns_160to243.html |author-link=Galileo Galilei}} This conclusion had appeared six years earlier in Galileo's ''Dialogue Concerning the Two Chief World Systems'' (Day 4).</ref><blockquote>From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle.
 
...

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 same.</blockquote>
[[File:Galileo's_Shortest_Time_Curve_Conjecture.jpg|alt=Diagrams for Wikipedia entry regarding Galileo's Conjecture|center]]
Just after Theorem 6 of ''Two New Sciences'', 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.

Galileo’s conjecture is that “The shortest time of all [for a movable body] will be that of its fall along the arc ADB [of a quarter circle] and similar properties are to be understood as holding for all lesser arcs taken upward from the lowest limit B.”

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 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.
<ref>{{cite book |last1=Galilei |first1=Galileo |title="Dialogue Concerning the Two Chief World Systems – Ptolemaic and Copernican translated by Stillman Drake, foreword by Albert Einstein " |date=1967 |publisher=University of California Press Berkeley and Los Angeles |isbn=0520004493 |edition=Hardback |page=451}}</ref>

=== Introduction of the problem ===
[[Johann Bernoulli]] posed the problem of the brachistochrone to the readers of ''[[Acta Eruditorum]]'' in June, 1696.<ref>Johann Bernoulli (June 1696) [https://books.google.com/books?id=4q1RAAAAcAAJ&pg=PA269 "Problema novum ad cujus solutionem Mathematici invitantur."] (A new problem to whose solution mathematicians are invited.), ''Acta Eruditorum'', '''18''' :  269.  From p. 269:  ''"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."'' (Given in a vertical plane two points A and B (see Figure 5), assign to the moving [body] M, the path AMB, by means of which — descending by its own weight and beginning to be moved [by gravity] from point A — it would arrive at the other point B in the shortest time.)</ref><ref>Solutions to Johann Bernoulli's problem of 1696:
*Isaac Newton (January 1697) [http://rstl.royalsocietypublishing.org/content/19/215-235/424.full.pdf+html "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"] (On a proof [that] the time in which a weight slides by a line joining two given points [is] the shortest in terms of time when it passes, via gravitational force, from one of these [points] to the other through a cycloidal arc), ''Philosophical Transactions of the Royal Society of London'', '''19''' :  424-425.
*G.G.L. (Gottfried Wilhelm Leibniz) (May 1697) [https://books.google.com/books?id=aTaZHrvK6zIC&dq=Acta%20eruditorum%20Anno%20MDCXCVII&pg=PA201 "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."] (His communication together with [those] of two others in a report to him first from Johann Bernoulli, [and] then from the Marquis de l'Hôpital, of reported solutions of the problem of the curve of quickest descent, [which was] publicly proposed by Johann Bernoulli, geometer — one with a solution of his other problem proposed afterward by the same [person].), ''Acta Eruditorum'', '''19''' :  201–205.
*Johann Bernoulli (May 1697) [https://books.google.com/books?id=aTaZHrvK6zIC&dq=Acta%20eruditorum%20Anno%20MDCXCVII&pg=PA206 "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."] (The curvature of [light] rays in non-uniform media, and a solution of the problem [which was] proposed by me in the ''Acta Eruditorum'' of 1696, p. 269, from which is to be found the brachistochrone line [i.e., curve], 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 [light] rays.), ''Acta Eruditorum'', '''19''' :  206–211.
*Jacob Bernoulli (May 1697) [https://books.google.com/books?id=aTaZHrvK6zIC&dq=Acta%20eruditorum%20Anno%20MDCXCVII&pg=PA211 "Solutio problematum fraternorum, … "] (A solution of [my] brother's problems, … ), ''Acta Eruditorum'', '''19''' :  211–214.
*Marquis de l'Hôpital (May 1697) [https://books.google.com/books?id=aTaZHrvK6zIC&dq=Acta%20eruditorum%20Anno%20MDCXCVII&pg=PA217 "Domini Marchionis Hospitalii solutio problematis de linea celerrimi descensus"] (Lord Marquis de l'Hôpital's solution of the problem of the line of fastest descent), ''Acta Eruditorum'', '''19''' :  217-220.
*reprinted:  Isaac Newton (May 1697)  [https://books.google.com/books?id=aTaZHrvK6zIC&dq=Acta%20eruditorum%20Anno%20MDCXCVII&pg=PA223 "Excerpta ex Transactionibus Philos. Anglic. M. Jan. 1697."] (Excerpt from the English ''Philosophical Transactions'' of the month of January in 1697), ''Acta Eruditorum'', '''19''' :  223–224.</ref> He said:

{{Quote
|text=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 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}}

Bernoulli wrote the problem statement as:

{{Quote
|text=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.}}

Johann and his brother [[Jakob Bernoulli]] derived the same solution, but Johann's derivation was incorrect, and he tried to pass off Jakob's solution as his own.<ref>{{cite book|last=Livio|first=Mario|author-link=Mario Livio|title=The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C|orig-year=2002|edition=First trade paperback|year=2003|publisher=[[Random House|Broadway Books]]|location=New York City|isbn=0-7679-0816-3|page=116}}</ref> Johann published the solution in the journal in May of the following year, and noted that the solution is the same curve as [[Christiaan Huygens|Huygens']] [[tautochrone curve]]. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid.<ref name="Struik">{{Citation|title=A Source Book in Mathematics, 1200-1800|last=Struik|first=J. D.|year=1969|publisher=Harvard University Press|isbn=0-691-02397-2}}</ref><ref>{{citation |
title=Johann Bernoulli's brachistochrone solution using Fermat's principle of least time| author=Herman Erlichson |year=1999 |journal=Eur. J. Phys.|
pages=299–304 |doi=10.1088/0143-0807/20/5/301| volume=20 |
issue=5| bibcode=1999EJPh...20..299E | s2cid=250741844 }}</ref> However, his proof is marred by his use of a single constant instead of the three constants, ''v<sub>m</sub>'', ''2g'' and ''D'', below.

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.<ref>{{cite book|last1=Sagan|first1=Carl|title=Cosmos|date=2011|publisher=Random House Publishing Group|isbn=9780307800985|url=https://books.google.com/books?id=EIqoiww1r9sC&pg=PT94|access-date=2 June 2016|page=94}}</ref> At 4 p.m. on 29 January 1697 when he arrived home from the [[Royal Mint]], [[Isaac Newton]] found the challenge in a letter from Johann Bernoulli.<ref>{{cite book|first=Victor J.|last=Katz|title=A History of Mathematics: An Introduction|edition=2nd|year=1998|publisher=Addison Wesley Longman|isbn=978-0-321-01618-8|page=[https://archive.org/details/historyofmathema00katz/page/547 547]|url-access=registration|url=https://archive.org/details/historyofmathema00katz/page/547}}</ref> 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 idea of Newton's power, since Johann Bernoulli took two weeks to solve it.<ref name="Hand"/><ref>[[Tom Whiteside|D.T. Whiteside]], ''Newton the Mathematician'', in Bechler, ''Contemporary Newtonian Research'', p. 122.</ref> Newton also wrote, "I do not love to be dunned [pestered] and teased by foreigners about mathematical things...", and Newton had already solved [[Newton's minimal resistance problem]], which is considered the first of the kind in [[calculus of variations]]. Bernoulli had used the principle of least time in his solution but not calculus of variations, whereas Newton did to solve the problem, and as a result, pioneered the field with his work on the two problems.<ref name=":1">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA36 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=36–39 |language=en |doi=10.1142/q0108}}</ref>

In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli, [[Gottfried Leibniz]], [[Ehrenfried Walther von Tschirnhaus]] and [[Guillaume de l'Hôpital]].  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.<ref name="Struik"/> According to Newtonian scholar [[Tom Whiteside]], 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 [[Leonhard Euler]] into what the latter called (in 1766) the ''[[calculus of variations]]''.  [[Joseph-Louis Lagrange]] did further work that resulted in modern [[infinitesimal calculus]].