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 Jean-François 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.
Without any doubt Torricelli was the first to publish the solution of the problem (in Opera Geometrica, "De dimensione Parabolae, solidique Hyperbolici problemata duo …", pp. 85-90). Three demonstrations are found as an appendix to the chapter indicated, "by means of which we demonstrate", wrote Torricelli, "with the help of God that [the cycloidal space] is the triple [of the generating circle]". The first and the third demonstration are carried out using the method of indivisibles, the second is made in the manner of the ancients, by double reductio ad absurdum. [in Torricelli, Opere, ed. G. Loria and G. Vassura, Faenza, 1919, I (1), pp. 163-169].
Torricelli indicates, above all, the first empirical attempts to measure the "material spaces" (spatijs figurarum materialibus) delimited by the cycloidal curve, and then the method for constructing the curve itself.

Here we give a translation of Torricelli's preamble, based on the Italian translation by L. Belloni (cfr. Torricelli, Opere, Torino, UTET, 1975, pp. 410-412).


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.

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The following suppositions are made. Imagine on a fixed line AB the circle AC, tangent to the line AB at the point A. Let the point A be fixed onto the periphery of the circle. Then let one imagine rotating the circle AC on the fixed line AB, with simultaneously circular and progressive motion towards B, in such a way that, at successive moments it always touches the straight line AB at one point, until the fixed point turns once again to make contact with the line, for example at B. It is certain that point A, fixed on the periphery of the rotating circle AC, will describe a line, first ascending from the underlying line AB, then culminating towards D and finally downturned and descending towards the point B. This line has been called the cycloid by our predecessors, especially by Galileo as long as 45 years ago. The straight line AB has been called the base of the cycloid, and the circle AC the generator of the cycloid. From the nature of the cycloid is derived the property that its base AB is equal to the periphery of the generating circle AC. And this is not so obscure. In fact, the whole of the periphery AC is co-measured, during its rotation, with the fixed line AB. One now asks what proportion the cycloidal space ADB has to its generating circle AC. We demonstrate (and may thanks be given to God) that it is triple. The demonstrations will be three, all of which are different from each other. The first and the third will proceed with the double position, according to the method of the ancients, from which those who favour either method will be satisfied. Of the rest, I say this: almost all of the principles with which one demonstrates something in the geometry of indivisibles can be reduced to the usual indirect demonstration of the ancients. That has been done by us, as in many other cases. even in the first and the third of the following theorems. In any case, so as not to abuse the patience of the reader too much, we have decided to leave many demonstrations aside and to give only three.

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. 412-413).

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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It is clear that the lines HD, IE, XB, QL are equal by proposition 14 of book III. The arcs OB and LN will also be equal. Analagously, since CH and IF are equal, CR and UA will be equal, by the properties of parallel lines. The whole of the periphery MLN, by the definition of the cycloid itself, is equal to the line AF. Analogously, the arc LN is equal to the line AN for the same reason, since the arc LN will be distended on the line AN. The remaining arc LM will thus be equal to the remaining line NF. For the same reason the arc BP will be equal to the line AP, and the arc BO to the line PF.
Now the line AN is equal to the arc LN, and to the arc BO, and to the line PF. Thus, by the properties of parallels, AT will be equal to SC. Now, because CR and AU were also equal, the remaining UT and SR will be equal. For this reason, in the equiangular triangles UTQ and RSX, the homologous sides UQ and XR will be equal. It is clear, thus, that the two straight lines LU and BR, taken together, will be equal to the two lines LQ and BX, that is to EI and DH. And this will always be true, wherever the two points H and I are taken, even if they are equidistant from the centre. All of the lines of the figure ALBCA will thus be equal to all of the lines of the semicircle CDEF. For this reason the bilinear figure ALBCA will be equal to the semicircle CDEF.
But the triangle ACF is double the semicircle CDEF. In fact the triangle ACF is the reciprocal of the triangle of proposition I of Archimedes on the measure of the circle, as the side AF is equal to the semi-circumference, and the side FC is equal to the diameter. From this it follows that triangle AFC is equal to the whole circle of diameter CF. Thus, combining, the entire cycloidal space [ALBCFA] will be one and a half times the inscribed triangle ACF and the triple of the semicircle CDEF. And this etc.

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