Institute and Museum of History of Science, Florence, ITALY
Evangelista Torricelli
3.2 Quadrature of the cycloid space

The history of the cicloid curve is not easy to summarize in a few lines. Its invention dates from the beginning of the seventeenth century, according to the affirmations of Carlo Dati in his "Lettera a Filaleti di Timauro Antiante. Della vera storia della cicloide, e della famosissima esperienza dell'argento vivo", published in Florence in 1662. According to Dati, Galileo studied this curve, and he was the first to conceive of it in the years around 1600. This priority was confirmed, as we will see, by Galileo himself: the discovery was to be ascribed to the last decade of the sixteenth century. Dati cites a work of Stefano Degli Angeli from 1661, De superficie ungulae, in which the author attributed to Galileo the merit of being the first to have imagined and studied this particular curve, and to Torricelli that of having been the first to calculate the exact value of the surface included between a complete arc of cycloid and the fixed line. Degli Angeli based his assertion on a letter (of which he possessed a copy) sent by Bonaventura Cavalieri to Galileo on 14 February 1640. "From Paris", Cavalieri wrote, "I have been sent two of those mathematical problems from which I fear I shall gain little honour. Some of the questions concern the cycloid. It was JeanFrançois Niceron who submitted them to me, on the occasion of a trip to Italy". Galileo's answer, of 24 February of the same year, clarifies some points: "I do not know", Galileo wrote, "if any of the problems sent to you from France have been demonstrated. Like you, I regard them as very difficult to resolve. More than 50 years ago I had it in mind to describe that curved line [i.e. the cycloid] […] so as to adapt it to the arches of a bridge […]. It appeared to me from the beginning that the space could be the triple of the circle that describes it, but this was not the case, although the difference is not much […] Around a year ago I received a writing of a Father Mersenne of the Minims of St. Francesco di Paola sent to me from Paris, but written in such characters that all of the Accademia of Florence were unable to understand it, to such an extent that no construct could be derived from it […] I answered the friend who sent it to me that he should let that Father know that he should write to me in more intelligible handwriting". Dati criticised the contents of a publication [due to Pascal] that appeared in France in October 1658, "l'Histoire de la Roulette, appelée autrement la Trochoide ou la Cycloide", in which he stated that Fr. Marin Mersenne was the first to imagine the cycloid curve, around 1615, en considérant le roulement des roues (from considerations of the rolling of wheels). After having asked Galileo for his solution, Mersenne turned to Gilles Personne de Roberval in 1634, who demonstrated that the cycloidal space was 3pr. According to the author of the Histoire, Roberval had asked Mersenne to write to all mathematicians to tell them that the solution had been discovered (by Roberval) without, however, informing them of its contents. Later, in 1635, Mersenne sent Roberval's solution to various mathematicians, asking them to demonstrate it. The only two responses received, which differed from each other, were those of Pierre de Fermat and René Descartes. In 1638, again according to the author of the Histoire, Jean de Beaugrand sent what he knew of the solution to Galileo, in such a way as he himself appeared to be the author of these results. After Galileo's death in January 1642 Torricelli found the solution sent by Beaugrand in 1638 amongst his master's papers. This contradicts the claims made in Galileo's letter of 24 February 1640 cited above. Finally, it should be emphasized
that Cavalieri wrote to Torricelli on 23 April 1643 and congratulated
him for the solution he had discovered. "I have finally heard in your
last letter", wrote Cavalieri, "of the measure of the cycloidal space,
to my great admiration, as this has always been judged to be a problem
of great difficulty, which surpassed even Galileo. I too left it aside,
as it appeared extremely difficult; so you will have no small praise
for this, in addition to your marvellous inventions which will give
you eternal fame. I will not refrain from telling you on this matter
that Galileo wrote to me once of having attempted it 40 years ago,
and was unable to find anything; and that he was persuaded that the
said space was the triple of the circle that generates it, but that
it then appeared to him not to be precisely the case, if I remember
well, as as much as I have searched amongst my papers I have been
unable to find such a letter. Here we give a translation
of Torricelli's preamble, based on the Italian translation by L. Belloni
(cfr. Torricelli, Opere, Torino, UTET, 1975, pp. 410412). APPENDIX ON THE MEASUREMENT OF THE CYCLOID It
pleases me to add here, as an appendix, the solution of an interesting
problem that seems extremely difficult on first inspection if one
considers its subject and its formulation. This [problem] tormented
and eluded the finest mathematicians of our century many years ago.
In fact, the demonstration that was sought in vain eluded their clutches
because of the fallaciousness of the experiment. As it happens, when
the material spaces of the figures are suspended from a balance made
by hand, I know not for what end, that proportion which is triple
in reality always results as less than triple. It happens then that,
suspecting that one was dealing with incommensurable magnitudes (so
I believe) rather than despairing of a solution, the investigation
that had been begun was abandoned by those mathematicians.
We will limit ourselves to giving the first demonstration made with the method of indivisibles, again basing our English translation on Belloni's Italian translation (cf. ibidem, pp. 412413). Theorem I The space included between the cycloid and the straight line forming its base is the triple of the generating circle or one and a half times the triangle having the same base and height. Let
there be given the cycloid ABC, described by the point C of the circle
CDEF when it rotates on the fixed base AF. Let us consider the semicycloid
and the semicircle only to avoid too much confusion in the diagram.
I say that the space ABCF is the triple of the semicircle CDEGF, or
one and a half times the triangle ACF. Take two points, H and I, on
the diameter CF, equally distant from the centre G. Trace HB, IL and
CM parallel to FA, with the semicircles OBP and MLN, passing through
the points B and L, equal to CDF, tangents to the base at the points
P and N.

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