## Supplementum defectus geometria cartesianae circa inventionem locorum, et al. [The Brachistochrone Problem, in Acta Eruditorum]

FIRST PRINTINGS OF THE PAPERS DOCUMENTING THE PROPOSAL AND SOLUTION OF THE “BRACHISTOCHRONE PROBLEM”, ONE OF THE MOST FAMOUS MATHEMATICAL CHALLENGES, AND ONE OF THE EARLIEST PROBLEMS POSED IN THE CALCULATION OF VARIATIONS.

The challenge of the brachistochrone “began in June of 1696 when Johann Bernoulli published a challenge problem in Leibniz’s journal *Acta Eruditorum*. Obviously, a legacy of public challenge remained from the days of Fior and Tartaglia. Although contests were now conducted in the sedate pages of scholarly journals, they retained their power to make or break reputations, as Johann himself observed:

*‘… it is known with certainty that there is scarcely anything which more greatly excites noble and ingenious spirits to labors which lead to the increase of knowledge than to propose difficult and at the same time useful problems through the solution of which, as by no other means, they may attain to fame and build for themselves eternal monuments among posterity.’*

“Johann’s particular challenge was a good one. He imagined points *A* and *B* at different heights above the ground and not lying one directly above the other. There is certainly an infinitude of different curves connecting these two points, from a straight line, to an arc of a circle, to any number of other wavy, undulating paths. Now imagine a ball rolling from *A* down to *B* along such a curve. The time it take to complete the trip depends, of course, on the curve’s shape. Bernoulli challenged the mathematical world to find that one particular curve *AMB* along which the ball will roll the *shortest* time. He called this curve the ‘brachistochrone’ from the Greek words for ‘shortest’ and ‘time’.

“An obvious first guess is to take *AMB* as the straight line joining *A* and *B*. But Johann cautioned against this simplistic approach:

*‘… to forestall hasty judgment, although the straight line AB is indeed the shortest between the points A and B, it nevertheless is not the path traversed in the shortest time. However the curve AMB, whose name I shall give if no one else discovered it before the end of this year, is one well-known to geometers.’*

“Johann gave the mathematical world until January 1, 1697, to come up with a solution. However, when his deadline arrived, he had received but one solution, from the ‘celebrated Leibniz’, who:

*’has courteously asked me to extend the time limit to next Easter in order than in the interim the problem might be made public … that no one might have cause to complain of the shortness of the time allotted. I have not only agreed to this commendable request but I have decided to announce myself the prolongation and shall now see who attacks this excellent and difficult question and after so long a time finally masters it.’”*

At this point, Johann (and others) were surprised (and perhaps a little delighted) that they had not received a solution from their English rival Sir Isaac Newton. Wondering if Newton has not noticed the challenge, Johann sent Newton directly a personal letter outlining the problem. When Newton received the letter, he did not disappoint. As Newton’s niece, Catherine Conduitt explained:

*”When the problem in 1697 was sent by Bernoulli — Sir I.N. was in the midst of the hurry of the great recoinage and did not come home till four from the Tower very much tired, but did not sleep till he had solved it, which was by four in the morning.”*

“Even late in life and tired from a hectic day’s work, Isaac Newton triumphed where most of Europe had failed! It was a remarkable display of the powers of the great British genius. He had clearly felt his reputation and honor were on the line; after all, both Bernoulli and Leibniz were waiting in the wings to publish their own solutions. So Newton rose to the occasion and solved the problem in a matter of hours. Somewhat exasperated, he is reported at one point to have said, ‘I do not love … to be … teezed by foreigners about Mathematical things.’

“Back in Europe, as Easter neared, a few solutions came into the hands of Johann Bernoulli. The curve that everyone was seeking — one that ‘is well-known to geometers’ — was none other than an upside-down cycloid… [T]his important curve was studied by Pascal and Huygens, but neither of these mathematicians had realized that it would also serve as the curve of quickest descent. Johann wrote with characteristic hyperbole, ‘… you will be petrified with astonishment when I say that precisely this cycloid … of Huygens is our required brachistochrone.'

“On Easter, the challenge period had expired. All together, Johann had received five solutions. There was his own and the one from Leibniz. His brother Jakob came through (perhaps to Johann’s dismay) with a third, and the Marquis de l’Hospital added a fourth. Finally, there was a submission bearing an English postmark. Opening it, Johann found the solution correct, although anonymous. He clearly had met his match in the person of Isaac Newton. Although unsigned, the solution bore the unmistakable signs of supreme genius.

“There is a legend — probably of dubious authenticity but nonetheless of great charm — that Johann, partially chastened, partially in awe, put down the unsigned document and knowingly remarked, ‘I recognize the lion by his claw.’” (Quoted from William Dunham, *Journey Through Genius: The Great Theorems of Mathematics*, Wiley, 1990, page 199-202.)

*The Brachistochrone Papers - the proposal and the solutions included:*

Johann: Supplementum defectus geometria cartesianae circa inventionem locorum; 2. Leibniz: Communicatio suae pariter, duarumque alienarum ad edendum sibi primum a Dn. Joh. Bernoullio; 3. Johann: Curvatura radii in diaphanis non uniformibus, ... ; 4. Jakob: Solutio problematum fraternorum, ... ; 5. L'Hospital: Solutio problematis de linea celerrimi descensus; 6. Tschirnhaus: De methodo universalia theoremata eruendi, ... ; 7. Newton: Epistola missa ad praenobilem virum D. Carolum Mountague …

*Note:* Newton’s solution original appeared in the *Philosophical Transactions*.

*Provenance* With stamps and withdrawal markings (7-3-1984) from the famous John Crerar Library, Chicago.

In: Acta Eruditorum, vol. 15 and 16: no.1 in 15:264-69, 1 plate; no. 2 in 16:201-5, 1 plate; no. 3 in 16: 206-11; no. 4 in 16:211-17; no. 5 in 16: 217-20; no. 6 in 16: 220-23; no. 7 in 16: 223-24. Leipzig: Gross & Fritsch, 1696-1697. The two entire volumes offered. Quarto (208x170 mm). Two volumes in uniform contemporary three-quarter vellum over marbled boards. pp [2] 604 and 9 plates; [8] 594 and 8 plates. Some heavy worming to pp 324-42 and plate vi of volume 15 (which is not part of any of the above mentioned articles). 1697 volume with repaired gutter tear to plate 8; reinforcement to p.449/50, and minor restoration to binding. Some toning throughout as usual with the Acta. In all a very good set.

Price: $4,200 .