Adopted by Galileo. 11. Galileo, says Fabroni, trod in the steps of Kepler, and in his first dialogue on mechanics, when treating on a cylinder cut out of a hemisphere, became conversant with indivisibles (familiarem habere cœpit cum indivisibilibus usum). But in that dialogue he confused the metaphysical notions of divisible quantity, supposing it to be composed of unextended indivisibles; and not venturing to affirm that infinites could be equal or unequal to one another, he preferred to say, that words denoting equality or excess could only be used as to finite quantities. In his fourth dialogue on the centre of gravity, he comes back to the exhaustive method of Archimedes.[605]
[605] Fabroni, Vitæ Italorum, i., 272.
Extended by Cavalieri. 12. Cavalieri, professor of mathematics at Bologna, the generally reputed father of the new geometry, though Kepler seems to have so greatly anticipated him, had completed his method of indivisibles in 1626. The book was not published till 1635. His leading principle is that solids are composed of an infinite number of surfaces placed one above another as their indivisible elements. Surfaces are formed in like manner by lines, and lines by points. This, however, he asserts with some excuse and explanation; declaring that he does not use the words so strictly, as to have it supposed that divisible quantities truly and literally consist of indivisibles, but that the ratio of solids is the same as that of an infinite number of surfaces, and that of surfaces the same as of an infinite number of lines; and to put an end to cavil, he demonstrated that the same consequences would follow if a method should be adopted, borrowing nothing from the consideration of indivisibles.[606] This explanation seems to have been given after his method had been attacked by Guldin in 1640.
[606] Non eo rigore a se voces adhiberi, ac si dividuæ quantitates verè ac propriè ex indivisibilibus existerent; verumtamen id sibi duntaxat velle, ut proportio solidorum eadem esset ac ratio superficierum omnium numero inflnitarum, et proportio superficierum eadem ac illa infinitarum linearum: denique ut omnia, quæ contra dici poterant, in radice præcideret, demonstravit, easdem omnino consecutiones erui, si methodi aut rationes adhiberentur omnino diversæ, quæ nihil ab indivisibilium consideratione penderent. Fabroni.
Il n’est aucun cas dans la géometrie des indivisibles, qu’on ne puisse facilement reduire à la forme ancienne de démonstration. Ainsi, c’est s’arrêter à l’écorce que de chicaner sur le mot d’indivisibles. Il est impropre si l’on veut, mais il n’en résulte aucun danger pour la géometrie; et loin de conduire à l’erreur, cette méthode, au contraire, a été utile pour atteindre à des vérités qui avoient échappé jusqu’alors aux efforts des géométres. Montucla, vol. ii., p. 39.
Applied to the ratios of solids. 13. It was a main object of Cavalieri’s geometry to demonstrate the proportions of different solids. This is partly done by Euclid, but generally in an indirect manner. A cone, according to Cavalieri, is composed of an infinite number of circles decreasing from the base to the summit, a cylinder of an infinite number of equal circles. He seeks, therefore, the ratio of the sum of all the former to that of all the latter. The method of summing an infinite series of terms in arithmetical progression was already known. The diameters of the circles in the cone decreasing uniformly were in arithmetical progression, and the circles would be as their squares. He found that when the number of terms is infinitely great, the sum of all the squares described on lines in arithmetical progression is exactly one third of the greatest square multiplied by the number of terms. Hence, the cone is one third of a cylinder of the same base and altitude, and the same may be shown of other solids.
Problem of the cycloid. 14. This bolder geometry was now very generally applied in difficult investigations. A proof was given in the celebrated problems relative to the cycloid, which served as a test of skill to the mathematicians of that age. The cycloid is the curve described by a point in a circle, while it makes one revolution along a horizontal base, as in the case of a carriage wheel. It was far more difficult to determine its area. It was at first taken for the segment of a circle. Galileo considered it, but with no success. Mersenne, who was also unequal to the problem, suggested it to a very good geometer, Roberval, who, after some years, in 1634, demonstrated that the area of the cycloid is equal to thrice the area of the generating circle. Mersenne communicated this discovery to Descartes, who, treating the matter as easy, sent a short demonstration of his own. On Roberval’s intimating that he had been aided by a knowledge of the solution, Descartes found out the tangents of the curve, and challenged Roberval and Fermat to do the same. Fermat succeeded in this, but Roberval could not achieve the problem, in which Galileo also and Cavalieri failed; though it seems to have been solved afterwards by Viviani. “Such,” says Montucla, “was the superiority of Descartes over all the geometers of his age, that questions which most perplexed them cost him but an ordinary degree of attention.” In this problem of the tangents (and it might not, perhaps, have been worth while to mention it otherwise in so brief a sketch), Descartes made use of the principle introduced by Kepler, considering the curve as a polygon of an infinite number of sides, so that an infinitely small arc is equal to its chord. The cycloid has been called by Montucla, the Helen of geometers. This beauty was, at least, the cause of war, and produced a long controversy. The Italians claim the original invention as their own; but Montucla seems to have vindicated the right of France to every solution important in geometry. Nor were the friends of Roberval and Fermat disposed to acknowledge so much of the exclusive right of Descartes as was challenged by his disciples. Pascal, in his history of the cycloid, enters the lists on the side of Roberval. This was not published till 1658.
Progress of algebra. 15. Without dwelling more minutely on geometrical treatises of less importance, though in themselves valuable, such as that of Gregory St. Vincent, in 1647, or the Cyclometricus of Willebrod Snell, in 1621, we come to the progress of analysis during this period. The works of Vieta, it may be observed, were chiefly published after the year 1600. They left, as must be admitted, not much in principle for the more splendid generalisations of Harriott and Descartes. It is not unlikely, that the mere employment of a more perfect notation would have led the acute mind of Vieta to truths which seem to us, who are acquainted with them, but a little beyond what he discovered.
Briggs. Girard. 16. Briggs, in his Arithmetica Logarithmica, was the first who clearly showed what is called the Binomial Theorem, or a compendious method of involution, by means of the necessary order of co-efficients in the successive powers of a binomial quantity. Cardan had partially, and Vieta much more clearly, seen this, nor was it likely to escape one so observant of algebraic relations as the latter. Albert Girard, a Dutchman, in his Invention Nouvelle en Algebre, 1629, conceived a better notion of negative roots than his predecessors. Even Vieta had not paid attention to them in any solution. Girard, however, not only assigns their form, and shows, that in a certain class of cubic equations there must always be one or two of this description, but uses this remarkable expression: “A negative solution means in geometry that the minus recedes as the plus advances.”[607] It seems manifest that till some such idea suggested itself to the minds of analysts, the consideration of negative roots, though they could not possibly avoid perceiving their existence, would merely have confused their solutions. It cannot, therefore, be surprising that not only Cardan and Vieta, but Harriott himself, should have disregarded them.
[607] La solution par moins s’explique en géometrie en retrogradant, et le moins recule ou le plus avance. Montucla, p. 112.