With the new Galilean sciences, the field of mathematics was broadened and its methods were newly defined. In 1628 the volume Della misura dell'acque correnti [On the Measurement of Running Waters] was published, dedicated to Pope Urban VIII Barberini (fig. 1). It was written by Benedetto Castelli, who had been ordered by the pope to settle a controversy over water between the cities of Ferrara and Bologna. In the first part of this treatise, the engineering and political elements are so strong as to overshadow the second part, which presents a new geometry of flowing quantities and furnishes a precise statement of the law of continuity. Already in the introduction however, Castelli had noted that the subject of water was included in the general science of motion. He had also stressed its particularity and difficulty, paraphrasing the statement of Galileo, the "singular light of Philosophy in our time, and my Master", that we sometimes know more about distant things like the motion of planets and the orbital periods of stars than about nearby, readily perceptible phenomena such as the motion of rivers and seas.
River currents and fluid kinematics
In January 1631, during a debate on managing the waters of the Bisenzio River, Galileo had written a report in which he applied the laws of accelerated motion to study the flow of water in rivers. He had, however, neglected the factor of resistance deriving from the current striking against bends in the riverbank, and from the roughness of the river channel. In 1628 Castelli had dealt with this subject using a different approach, assuming that water flows at the same speed in a given section of the river channel and taking no account of transitory states (as when the flow rate abruptly increases due to heavy rainfall or the passage of a flood wave). On these bases, he had demonstrated that the mass or weight of water passing through section A of the river ABC is equal to that which passes, during the same time interval, through section B (fig. 2). Castelli had also demonstrated the existence of a relationship of inverse proportionality between the areas of the cross-sections and their respective velocities. Although this concept was certainly not a given in the philosophical culture of the time, since it assumes that the diminution in section B is due to an increase in the speed of the current and not to densification of the particles of water, it rapidly won acceptance in studies on the motion of incompressible fluids. This was not the case, instead, for Castelli's scale of velocities, published posthumously in 1660 in Book II of Della misura dell'acque correnti, which hypothesized that a river current's speed is proportional to its height or depth. Already in October 1642 Torricelli had declared that the speed of discharge from a vessel filled with water is not proportional to the lowering or height h of the orifice, but to the Öh, as in the free fall of weights. This cast doubt on Castelli's scale of velocities among those who believed that the discharge experiment constituted a mathematical model of river current speed, as hypothesized by Torricelli and other 17th-century writers on hydraulics.
A river can only be managed by studying the natural forces that direct its course. It is in this spirit that Famiano Michelini wrote his Trattato della direzione de' fiumi [Treatise on the Direction of Rivers] (1664), setting forth a mechanism to explain the formation of the bends in river channels, which was one of the sources for Guglielmini's Della natura de' fiumi [The Nature of Rivers] (1697) (fig. 3). Vincenzo Viviani's Discorso [Discourse] on the Arno, published in 1688, also championed the maxim that technology should not conflict with, but "be of aid to nature". In this case however the naturalist discourse was extended to the entire basin, aimed especially at deforesting and the "not entirely appropriate cultivations" that had led to the erosion and leaching of rocky and earthy materials. This noble civil concern was not common to Galileans alone but, at least in part, to the Florentine judiciary involved in regulating watercourses. For nearly half a century Viviani, in his capacity as engineer to the Capitani di Parte, supervised the Grand Duchy's hydraulic policy and helped train new generations of hydraulic engineers. The alliance between Galilean science and the institutions responsible for managing watercourses, already anticipated in Castelli's work, was thus implemented in full.
Engraved title page of Della misura dell’acque correnti (1628) by Benedetto Castelli
At the top of the three arches of the bridge one can see the shield and bees of the Barberini family surmounted above the keys of St. Peter and the Papal crown.
The "mathematical" river by Castelli
A and B are two transverse sections, rectangular in shape, that do not change over the course of time, in which the current is always supposed to have a given speed.