In the same treatise, he lays it down as a postulate (supposizione), that the effect of one 'heavy body to turn another round a centre of motion, is proportional to the perpendicular drawn froin that centre to the vertical passing through the body, or in general to the di. rection of the force. This proposition he states without a demonstration, and passes by means of it to the oblique lever, and thence to the inclined To speak strictly, however, the demonstrations with respect to both these last are incomplete, the preceding proposition being assumed in them without proof. It is probable that he satisfied himself of The truth of it, on the principle, that the distances of forces from the centre of motion must always be measured by lines making the same angles with their directions, and that of such lines the simplest are the perpendiculars. His demonstration is regarded by La Grange as quite satisfactory.' Galileo extended the theory of motion still farther. He had begun, while pursuing his studies at the university of Pisa, to make experiments on the descent of falling bodies, and discovered the fact, that heavy and light bodies fall to the ground from the same height in the same time, or in times so nearly the same, that the difference can only be ascribed to the resistance of the air. From observing the vibrations of the lamps in the cathedral, he also arrived at this very important conclusion in mechanics, that the great and the small vibrations of the same pendulum are performed in the same time, and that this time depends only on the length of the pendulum. The date of these observations goes back as far as the year 1583. • These experiments drew upon him the displeasure of his masters, who considered it as unbecoming of their pupil to seek for truth in the Book of Nature, rather than in the writings of Aristotle, when elucidated by their commentaries, and, from that moment be gan the persecutions with which the prejudice, the jealousy, and bigotry of his contempo raries continued to harass or afflict this great man throughout his whole life.
That the acceleration of falling bodies is uniform, or, that they receive equal increments of velocity in equal times, he appears first to have assumed as the law which they follow, . merely on account of its simplicity. Having once assumed this principle, he showed, by mathematical reasoning, that the spaces descended through must be as the squares of the times, and that the space fallen through in one second is just the .half of that which the body would have described in the same time with the velocity last acquired.
The knowledge which he already had of the properties of the inclined plane enabled him very readily to perceive, that a body descending on such a plane must be uniformly accelerated, though more slowly than when it falls directly, and is accelerated by its whole weight. By means of the inclined plane, therefore, he was able to bring the whole theory of falling bodies to the test of experiment, and to prove the truth of his original assump tion, the uniformity of their acceleration.
His next step was to determine the path of a heavy body, when obliquely projected. He showed this path to be a parabola ; and here, for the first time, occurs the use of a principle which is the same with the composition of motion in its full extent. Galileo, however, gave no name to this principle ; he did not enunciate it generally, nor did he give any demonstration of it, though he employed it in his reasonings. The inertia of
body was assumed in the same manner ; it was, indeed, involved in the uniform accelera tion of falling bodies, for these bodies did not lose in one minute the motion acquired in the preceding, but, retaining it, went on continually receiving more.
The theory of the inclined plane had.led to the knowledge of this proposition, that, if a circle be placed vertically, the chords of different arches terminating in the lowest point of the circle, are all descended through in the same space of time. This seemed to ex plain why, in a circle, the great and the small vibrations are of equal duration. Here, however, Galileo was under a mistake, as the motions in the chord and in the arch are very dissimilar. The accelerating force in the chord remains the same from the beginning to the end, but, in the arch, it varies continually, and becomes, at the lowest point, equal to nothing. The times in the chords, and in the arches, are therefore different, so that here we have a point marking the greatest distance in this quarter, to which the mechani cal discoveries of Gaol extended. The first person who investigated the exact time of a vibration in an arch of a circle was Huygens, a very profound mathematician.
To this list of mechanical discoveries, already so important and extensive, we must add, that Galileo was the first who maintained the existence of the law of continuity, and who made use of it as a principle in his reasonings on the phenomena of motion.
The vibrations of the pendulum having suggested to Galileo the means of measuring time accurately, it appears certain that the idea of applying it to the clock had also occur red to him, and of using the chronometer so formed for finding the longitude, by means of observations made on the eclipses of the satellites of Jupiter. How far he had actually proceeded in an invention which required great practical knowledge, and which afterwards did so much credit to Huygens, appears to be uncertain, and not now easy to be ascertain. ed. But that the project had occurred to him, and that he had taken some steps towards realizing it, is sufficiently established.
One forms, however, a very imperfect idea of this philosopher, from considering the dis coveries and inventions, numerous and splendid as they are, of which he was the undis puted author. It is by following his reasonings, and by pursuing the train of his thoughts in his own elegant, though somewhat diffuse exposition of them, that we become acquaint ed with the fertility of his genius, with the sagacity, penetration, and comprehensiveness of his mind. The service which he rendered to real knowledge, is to be estimated not only from the truths which he discovered, but from the errors which he detected,—not mere ly from the sound principles which he established, but from the pernicious idols which he overthrew. His acuteness was strongly displayed in the address with which he exposed the errors of his adversaries, and refuted their opinions, by comparing one part of them with another, and proving their extreme inconsistency. Of all the writers who have lived in an age, which was yet only emerging from ignorance and barbarism, Galileo has most entirely the tone of true philosophy, and is most free from any contamination of the times in taste, sentiment, and opinion.