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tangent or secant: he also constructed series to find directly the logarithm of the tangent or of the secant, when the arc is given; and reciprocally the logarithm of the arc by that of the tangent or secant: lastly, he applied this theory of series to the rectification of the ellipsis and hyperbola.
The use of series in geometry made some progress in Germany likewise. Leibnitz gave a method for transforming one curvilinear surface into another, the parts of which, supposed equal to those of the former, should have such a figure and position, that the methods of Mercator and Wallis would be applicable to the quadrature of this latter curve.
Progress of Mechanics.
In this period, as in the two preceding, a great number of very ingenious machines were invented; but the theory of mechanics remained in a state of stagnation till the sixteenth century. Stevinus, a flemish mathematician, who died in 1635, appears to be the first person, that made known directly, and without the assistance of the lever, the laws of equifibrium of a body placed on an inclined plane. He also examined with the same success many other ques tions in statics. The manner, in which he determined the conditions of equilibrium between several forces concurring to one point, comes in fact to the famous principle of the parallelogram of forces; but he was not aware of all it's consequences and advantages.
In 1592 Galileo composed a little treatise on statics, which he reduced to this single principle: it requires an equal power, to raise two different bodies to heights in the inverse ratio of their weights; that is to say, the same power will raise two pounds to the height of one foot, as will raise one pound to the height of two feet. Hence it was easy to infer, that, in all machines in equilibrium, the powers, which counteract each other, are inversely proportional to the spaces, which they would pass through in a given
time. The only question then is accurately to determine the spaces, from the arrangement and action of the parts of the machine. Thus, for instance, in the common screw, where the weight rises the height of one thread of the screw, while the power describes the periphery of a circle in a horizontal direction, the weight is to the power as this periphery is to the height of the thread of the screw. A long while after this des Cartes employed the same principle, to determine the equilibrium of all machines, in a small work entitled, an Explanation of Machines and Engines. He ought however to have quoted Galileo.
It does not enter into my plan, to relate the practical applications made of the principles of mechanics: yet I cannot avoid remarking here incidentally, that Claude Perrault, so much decried by Boileau, who was incapable of appreciating his talents, displayed no less mathematical and physical knowledge, than he did genius, in the machines he invented to raise the enormous stones, that compose the pediment of the colonnade of the Louvre. A description of them may be seen in his commentary on the eighteenth chapter of the tenth book of Vitru
The general theory of motion, of which the ancients were acquainted only with the particular case where it is uniform, originated with Galileo. He discovered the law of the acceleration of bodies falling freely by the power of gravity, or sliding down inclined planes; and on this subject he established the general properties of motion uniformly accelerated. The conformity of his theory with the phe
nomena of nature is one of the largest strides, that the modern science of physics ever took; it formed the first step in the system of universal gravitation.
Every person, who sees a stone fall, may suppose it's motion to be accelerated, and become so much the more rapid in proportion to the height from which it falls; since the stone, the weight of which remains the same, strikes a blow so much the harder, as the height of it's fall is greater. But in what proportiondoes this acceleration take place? This is the new problem, which Galileo solved; and he was led to it by one of those simple reflections, which may enter into any mind, but turns to account only in the mind of genius.
Since all bodies are heavy, said Galileo, and into whatever number of parts we divide any mass, whether an ingot of gold, or a block of marble, all these parts are themselves heavy bodies, it follows, that the total weight of the mass is proportional to the number of material atoms, of which it is composed. Now the weight being thus a power always uniform in quantity, and it's action never undergoing any interruption, it must in consequence be continually giving equal impulses to a body, in every equal and successive instant of time. If the body be stopped by any obstacle; as, for example, if it be placed on a horizontal table: the impulses of the weight, as they are incessantly renewed, are incessantly destroyed by the resistance of the table. But if the body fall freely, these impulses are incessantly accumulating, and remain in the body without alteration, the resistance of the air alone being deducted: whence it
follows, that the motion must be accelerated by equal degrees. Experience has fully confirmed this sound reasoning.
Fortunately Galileo brought to this question a mind free from all prejudice, and from any systematic opinion concerning the cause of gravity: for if he had supposed, for example, as some philosophers did after him, that the impulses of gravity are produced by the impulse of some subtile ambient fluid, he would have missed the truth; the impulses in question not being proportionate to the masses of the falling bodies, and decreasing continually as the velócity increases.
Among the philosophers who were the first to embrace and comment on Galileo's theory of the fall of heavy bodies, we must distinguish his pupil Torricelli, who published a very elegant work on this subject in 1644, entitled: De Motu Gravium naturaliter accelerato. He added several very curious propositions to those, which Galileo had given on the motion of projectiles.
Huygens considered the motion of heavy bodies on given curves. He demonstrated generally, that the velocity of a heavy body, which descends along any curve, is the same at every instant in the direction of the tangent, as it would have acquired by falling freely from a height equal to the correspondent vertical absciss. Then applying this principle to a reversed cycloid, the axis of which is vertical, he found, that a heavy body, from whatever part of the cycloidal arc it falls, always arrives at the lowest point, or the inferiour extremity of the arc, in the