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of the balance are equal, the two weights supposed to be in equilibrio are also of necessity equal; he then shows, that, if one of the arms be increased, the weight applied to it must be proportionally diminished. Hence he deduces the general conclusion, that two weights, suspended to the arms of a balance of unequal lengths, and remaining in equilibrio, must be reciprocally proportional to the arms of the balance. This principle we know includes all the theory of the equilibrium of the lever, and machines referrible to the lever. Archimedes having farther observed, that the two weights exert the same pressure on the fulcrum of the balance, as if they were directly applied to it, made this substitution in his mind, and combining the sum of the two weights with a third, he came to the same conclusion for the assemblage of three as for the former two, and so on for a greater number. Hence he demonstrated step by step, that there exists in every assemblage of small bodies, and in every large body considered as such an assemblage, one general centre of pressure, which we call the centre of gravity. This theory he applied to particular instances; and determined the situation of the centre of gravity in the parallelogram, the triangle, the common rectilinear trapezium, the area of the parabola, the parabolic trapezium, &c.
To him likewise are attributed the theory of the inclined plane, the pulley, and the screw. He also invented a multitude of compound machines, but neglected giving descriptions of them, so that little more than the names of them remains.
We may judge of the state in which the theory of mechanics at that time was, from the great astonishment of king Hiero, his relation, at being told by him, that, if he had a fixed point, he could move the World: 'Give me a place to stand upon,' said Archimedes, and I will move the Globe.' This proposition, however, is but a very simple consequence of the equilibrium of the lever: by lengthening one of it's arms, and proportionally dimi nishing the weight applied to the extremity of this arm, we may balance any weight applied to the shorter.
Had Archimedes been no more than the first geometrician of his age, with this great but indefinite claim to glory he might have lived and died in obscurity; but the machines he invented acquired him the greatest fame. This is the polar star, that directs the admiration of the vulgar, that is of the generality of mankind. Incapable of appreciating the speculations of genius, the multitude admires the man, who strikes it's senses and imagination by new and extraordinary spectacles. Archimedes was far from attaching the same value to his mechanical inventions. On this subject let us hear Plutarch in his life of Marcellus. After having related, that Appius, a roman engineer, at the siege of Syracuse, had brought several large machines against the walls of the city, to batter them down, he continues thus. 'But Archimedes despised all this, and confided in the superiority of his engines, though he did not think the inventing of them an object worthy of his
serious studies, but only reckoned them among the amusements of geometry. Nor had he gone so far, but at the pressing instances of king Hiero, who intreated him to turn his art from abstracted notions to matters of sense, and to make his reasonings more intelligible to the generality of mankind, by applying them to the uses of common life.' After this passage Plutarch proceeds to relate how long the capture of Syracuse was retarded by the machines of Archimedes; and then continues; Yet Archimedes had such a depth of understanding, such a dignity of sentiment, and so copious a fund of mathematical knowledge, that, though in the invention of these machines he gained the reputation of a man endowed with divine rather than human knowledge, yet he did not vouchsafe to leave any account of them in writing. For he considered all attention to mechanics, and every art that ministers to common uses, as mean and sordid; and placed his whole delight in those intellectual speculations, which, without any relation to the necessities of life, have an intrinsic excellence arising from truth and demonstration only. Indeed, if mechanical knowledge be so valuable for the curious frame and amazing power of those machines which it produces, the other infinitely excels on account of it's invincible force of conviction. And certain it is, that abstruse and profound questions in geometry are no where solved by a more simple process, or upon clearer principles, than in the writings of Archimedes,'
The judgment, which Archimedes passed on the geometry of his time, he would equally have passed
on the great discoveries of the moderus in geometry and rational mechanics. All knowledge of this kind incontestably occupies the first rank in the domains of science. We certainly must not place practical mechanics on a level with it; since a man, who was at the same time equally great as a geometrician and machinist, forbids us in such a peremptory manner: yet they sometimes require much sagacity and research; and assuredly a machinist of the first order, such as Vaucanson, is less common, and deserves higher consideration, than a geometrician merely skilled in the science, but destitute of the talent of invention.
To complete the science of statics, nothing more was wanting than to unfold and generalise the principles, which Archimedes had established for the equilibrium of the lever. We cannot doubt, that he himself extended the spirit of these principles to the numerous engines he invented, and of which he did not think proper to leave a description: his successors for a long time did nothing more, than languidly tread in his steps; and we do not find, that they enriched the science with any theoretical proposition worthy notice: yet by the combination of known principles, they produced from time to time a great number of machines very useful for various purposes.
Of the theory of motion the ancients had but the most superficial ideas. They were acquainted only with the general properties of uniform motion. They knew what a little reflection and mere common sense might teach any one, that the greater the space through which a body passes in a given time, or the
shorter the time in which a body passes through a given space, the greater it's velocity must be: or that velocity is expressed by the ratio of the number of measures of space passed through to the number of measures of time: and that the spaces uniformly passed through by two bodies are generally as the products of the times by the velocities; so that, if the times be equal, the spaces are as the velocities, or if the velocities be equal, the spaces are as the times. But such simple and easy acquirements cannot be considered as a science: the true science of mechanics is that, the subject of which is the theory of variable motion, and the laws of the communication of motion. In it's general state this was inaccessible to the geometry of the ancients; and belongs exclusively to the moderns.