PHYSICS
57
PHYSICS
equal periods, a new and uniform impetus which,
added to that already acquired, causes the total
impetus to increase in arithmetical progression
according to the time occupied in the fall; hence the
velocity of the falling body. This argument towards
which all Parisian tradition had been tending and
which, in the last ])lace, had been broached by Sca-
liger, leads to our modern law: a constant force
produces uniformly accelerated motion. In Gali-
leo's work there is no trace either of the argument
or of the conclusion deduced therefrom; however,
the argument itself was carefully developed by
Galileo's friend, Giambattista Baliani (1582-1666).
From the very definition of velocity, Baliani en- deavoured to deduce the law according to which the space traversed by a falling body is increased pro- portionally to the time occupied in the fall. Here he was confronted by a difficulty that had also baffled Vinci; however, he eventually anticipated its solu- tion, which was given, after similar hesitation, by another of Galileo's disciples, Pierre Gassendi (1592- 1655). Galileo had reached the law connecting the time occupied in the fall with the space traversed by a falling body, by using a demonstration that became celebrated as the "demonstration of the triangle". It was textually that given by Oresme in the four- teenth century and, as we have seen, Soto had thought of using Oresme's proposition in the study of the accelerated fall of bodies. Galileo extended the laws of freely falling bodies to a fall down an inclined plane and subjected to the test of experiment the law of the motion of a weight on an inclined plane.
A body which, without friction or resistance of any kind, would describe the circumference of a circle concentric with the Earth would retain an invariable impelo or momento, as gravity would in no wise tend to increase or destroy this impcto: this principle, which belonged to the dynamics of Buridan and Albert of Saxony, was acknowledged by Galileo. On a small surface, a sphere concentric with the Earth is apparently merged into a horizontal plane; a body thrown upon a horizontal plane and free from all friction would therefore assume a motion appar- ently rectilinear and uniform. It is only under this restricted and erroneous form that Galileo recognized the law of inertia and, in this, he was the faithful disciple of the School of Paris.
If a heavy body moved by an impelo that would make it describe a circle concentric with the Earth is, moreover, free to fall, the impelo of uniform rota- tion and gravity are component forces. Over a small extent the motion produced by this impelo may be assumed to be rectilinear, horizontal, and uniform; hence the approximate law may be enun- ciated as follows: a heavy body, to which a hori- zontal initial velocity has been imparted at the very moment that it is abandoned to the action of gravity, assumes a motion which is sensibly the combination of a uniform horizontal motion with the vertical motion that it would assume without initial velocity. Galileo then demonstrated that the trajectory of this heavy body is a parabola with vertical axis. This theory of the motion of projectiles rests upon principles in no wise conformable to an exact knowledge of the law of inertia and which are, at bottom, identical with those invoked by Oresme when he wished to explain how, despite the Earth's rotation, a body seems to fall vertically. The argument employed by Galileo did not permit him to state how a projectile moves when its initial velocity is not horizontal.
Evangehsta Torricelli (1608-47), a disciple of Castelli and of Galileo, extended the latter's method to the case of a projectile whose initial velocity had a direction other than horizontal, and proved that the trajectory remained a parabola with a vertical axis. On the other hand Gassendi showed that in this problem of the motion of projectiles, the real
law of inertia which had just been formulated by
Descartes should be substituted for the principles
admitted by the Parisian dynamics of the fourteenth
century.
Mention should be made of Galileo's observations on the duration of the oscillation of the pendulum, as these observations opened up to dynamics a new field. Galileo's progress in dynamics served as a defence of the Copernican system and the discoveries which, with the aid of the telescope, he was able to make in the heavens contributed to the same end. The spots on the sun's surface and the mountains, similar to those upon the Earth, that hid from view certain portions of the lunar disc, gave ample proof of the fact that the celestial bodies were not, as Aris- totelean physics had maintained, formed of an in- corruptible substance unlike sublunary elements; moreover, the role of satellite which, in this helio- centric astronomy, the moon played in regard to the Earth was carried out in relation to Jupiter by the two "Medicean planets", which Gahleo had been the first to discover. Not satisfied with having defeated the argviments opposed to the Copernican system by adducing these excellent reasons, Galileo was eager to establish a positive proof in favour of this system. Inspired perhaps by Calcagnini, he believed that the phenomenon of the tides woulil furnish him the de- sired proof and he consequently rejected every expla- nation of ebb and flow founded on the attraction of the sun and the moon, in order to attribute the motion of the seas to the centrifugal force produced by ter- restrial rotation. Such an explanation would con- nect the period of high tide with the sidereal instead of the lunar day, thus contradicting the most ordi- nary and ancient observations. This remark alone ought to have held Galileo back and prevented him from producing an argument better calculated to overthrow the doctrine of the Earth's rotation than to establish and confirm it.
On two occasions, in 1616 and 1633, the Inquisi- tion condemned what Galileo had written in favour of the system of Copernicus. The hypothesis of the Earth's motion was declared fnha in Philosophia et ad minus erronea in fide; the hyixithesis of the sun being stationary was adjudged /V;/.sa in Philosophia el formaliler hoerelica. Ado])ting the doctrine formu- lated by Tycho Brahe in 1578, the Holy Office forbade the use of all astronomical hypotheses that did not agree both with the principles of Aristotelean physics, and with the letter of the Sacred Scriptures (see Galilei, Galileo).
XVIII. Initial Attempts in Cele.stial Mechan- ics — Gilbert — Kepler. — Copernicus had endeav- oured to describe accurately the motion of each of the celestial bodies, and Gafileo had striven to show that the views of Copernicus were correct; but neither Copernicus nor Galileo had attempted to extend to the stars, what they knew concerning the dynamics of sublunary motions, or to determine thereby the forces that sustain celestial motions. They were satisfied with holding that the daily rotation of the Earth is perpetuated by virtue of an impetus given once for all; that the various parts of an element belonging to a star tend towards the centre of this star by reason of a gravity peculiar to each of the celestial bodies through which the body is enabled to preserve its entireness. Thus, in celestial mechan- ics, these two great scientists contributed scarcely anything to what had already been taught by Buridan, Oresme, and Nicholas of Cusa. About Galileo's time we notice the first attempts to consti- tute celestial mechanics, that is to say, to explain the motion of the stars by the aid of forces analogous to those the effects of which we feel upon earth; the most important of these initial attempts were made by William Gilbert (1540-1603), and Johann Kepler (1571-1631).
