general rule came to be more simply expressed by saying, in multiplication like signs gave plus, and that unlike signs gave minus. Hence the signs plus and minus were considered, not as merely denoting the relation of one quantity to another placed before it, but, by a kind of fiction, they were considered as denoting qualities inherent in the quantities to the names of which they were prefixed. Even the most scrupulous purist in mathematical language must admit, that no real error is ever introduced by employing the signs in this most abstract sense. If the equation 3+pr2+qr-ro, be said to have one positive and two negative roots, this is certainly as exceptionable an application of the term negative, as any that can be proposed; yet, in reality, it means nothing but this intelligible and simple truth, that x+pr2+qr-r=(x—a)(x+b)(x+c); or that the former of these quantities is produced by the multiplication of the three binomial factors, -a, x+b, x+c. We might say the same nearly as to imaginary roots; they shew that the simple factors cannot be found, but that the quadratic factors may be found; and they also point out the means of discovering them. The aptitude of these same signs to denote contrariety of position among geometric magnitudes, makes the foregoing application of them infinitely more extensive and more indispensable. In the end of the sixteenth century, the time and labour consumed in astronomical and other calculations had become excessive, and were felt as extremely burdensome by the mathematicians and astroaomers all over Europe. NAPIER of Merchiston, whose mind seems o have been peculiarly turned to arithmetical researches, and who vas also devoted to the study of astronomy, had early sought for he means of relieving himself and others from this difficulty. He aad viewed the subject in a variety of lights, and a number of ingeious devices had occurred to him, by which the tediousness of arithnetical operations might, more or less completely, be avoided. In the course of these attempts, he did not fail to observe, that whenever the numbers to be multiplied or divided were terms of a geometrical progression, the product or the quotient must also be a term of that progression, and must occupy a place in it pointed out by the places of the given numbers, so that it might be found from mere inspection, if the progression were far enough continued. If, for instance, the third term of the progression were to be multiplied by the seventh, the product must be the tenth, and if the twelfth were to be divided by the fourth, the quotient must be the eighth; so that the multiplication and division of such terms was reduced to the addition and subtraction of the numbers which indicated their places in the progression. It is plain, however, that the resource of the geometrical progression was sufficient, when the given numbers were terms of that progression; but if they were not, it did not seem that any advantage could be derived from it. Napier, however, perceived, and it was by no means obvious, that all numbers whatsoever might be inserted in the progression, and have their places assigned in it. After conceiving the possibility of this, the next difficulty was, to discover the principle, and to execute the arithmetical process, by which these places were to be ascertained. The way in which he satisfied himself that all numbers might be intercalated between the terms of the given progression, and by which he found the places they must occupy, was founded on a most ingenious supposition,-that of two points describing two different lines, the one with a constant velocity, and the other with a velocity always increasing in the ratio of the space the point had already gone over: the first of these would generate magnitudes in arithmetical, and the second magnitudes in geometrical progression; and it is plain, that all numbers whatsoever would find their places among the magnitudes so generated. The numbers which indicate the places of the terms of the geometrical progression, are called by Napier the logarithms of those terms. Various systems of logarithms, it is evident, may be constructed according to the geometrical progression assumed; and of these, that which was first contrived by Napier, though the simplest, and the foundation of the rest, was not so convenient for the purposes of calculation, as one which soon afterwards occurred, both to himself and his friend Briggs, by whom the actual calculation was performed. The first writer on the subject of Mechanics is ARCHIMEDES. He treated of the lever, and of the centre of gravity, and has shown that there will be an equilibrium between two heavy bodies connected by an inflexible rod or lever, when the point in which the lever is supported is so placed between the bodies, that their distances from it are inversely as their weights. The same great geometer gave beginning to the science of Hydrostaties, and discovered the law which determines the loss of weight sustained by a body on being immersed in water, or in any other fluid. Archimedes, therefore, is the person who first made the application of mathematics to natural philosophy. The mechanical inquiries, begun by the geometer of Syracuse, were extended by CTESIBIUS and HERO; by ANTHEMIUS of Tralles ; and, lastly, by PAPPUS ALEXANDRINUS. Ctesibius and Hero were the first who analyzed mechanical engines, reducing them all to combinations of five simple mechanical contrivances, to which they gave the name of Avvaμs, or Powers, the same which they retain at the present moment. Even in mechanics, however, the success of these investigations was limited; and failed in those cases where the resolution of forces is necessary, that principle being then entirely unknown. GALILEO was born at Pisa in the year 1564, and as early as 1592 published a treatise, della Scienza Mechanica, in which he gave the theory, not of the lever only, but of the inclined plane and the screw; and also laid down this general proposition, that mechanical engines make a small force equivalent to a great one, by making the former move over a greater space in the same time than the latter, just in proportion as it is less. He was the first person to whom the mechanical principle, since denominated that of the virtual velocities, had occurred in its full extent. The object of his consideration was the action of machines in motion, and not merely of machines in equilibrio, or at rest; and he showed, that, if the effect of a force be estimated by the weight it can raise to a given height in a given time, this effect can never be increased by any mechanical contrivance whatsoever. 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. 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. By means of the inclined plane, this illustrious Philosopher brought the whole theory of falling bodies to the test of experiment, and proved the truth of his original assumption, the uniformity of their acceleration. The theory of the inclined plane led also 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 explain 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 first addition which was made to the mechanical discoveries of Galileo was by TORRICELLI. To this ingenious man we are indebted for the discovery of a remarkable property of the centre of gravity, and a general principle with respect to the equilibrium of bodies. It is this: If there be any number of heavy bodies connected together, and so circumstanced, that by their motion their centre of gravity can neither ascend nor descend, these bodies will remain at rest. This proposition often furnishes the means of resolving very difficult questions in mechanics. The first discovery in Hydraulics, or the motion of fluids, is to be ascribed also to TORRICELLI, who, though younger than Galileo, was for some time his contemporary. He proved that water issues from a hole in the side or bottom of a vessel, with the velocity which a body would acquire, by falling from the level of the surface to the level of the orifice. Galileo had failed in assigning the reason why water cannot be raised in pumps higher than thirty-three feet, but he had remarked, that if a pump is more than thirty-three feet in length, a vacuum will be left in it. TORRICELLI, reflecting on this, conceived, that if a heavier fluid than water was used, a vacuum might be produced, in way far shorter, and more compendious. He tried mercury, therefore, and made use of a glass tube about three feet long, open at one end, and close at the other, where it terminated in a globe. He filled this tube, shut it with his finger, and inverted it in a bason of mercury. The result is well known ;-he found that a column of mercury was suspended in the tube, an effect which he immediately ascribed to the pressure of the atmosphere; and it was afterwards found, that the fall of the mercury corresponded exactly to the diminution of the length of the pressing column, so that it afforded a measure of that diminution, and, consequently, of the heights of mountains. The invention of the air-pump by Orто GUERICKE, burgomaster of Magdeburg, quickly followed that of the barometer by Torricelli, though it does not appear that the invention of the Italian philosopher was known to the German. The elasticity of the air, as well as its weight, now became known; its necessity to combustion, and the absorption of a certain proportion of it, during that process; its necessity for conveying sound:-all these things were clearly demonstrated. The necessity of air to the respiration of animals required no proof from experiment, but the sudden extinction of life, by immersion in a vacuum, was a new illustration of the fact. In what respects the theory of motion, HUYGENS has strong claim to notice, from his solution of the problem of finding the centre of oscillation of a compound pendulum, or the length of the simple pendulum vibrating in the same time with it. Without the solution of this problem, the conclusions respecting the pendulum were inapplicable to the construction of clocks, in which the pendulums used are of necessity compound. The problem was by no means easy, and Huygens was obliged to introduce a principle which had not before been recognised, that if the compound pendulum, after descending to its lowest point, was to be separated into particles distinct and unconnected with one another, and each left at liberty to continue its own vibration, the common centre of gravity of all those detached weights would ascend to the same height to which it would have ascended had they continued to constitute one body.. COPERNICUS, the celebrated astronomer, was born at Thorn in Prussia, in 1473; and was the inventor of the bo system which removes the earth from the centre of the world, and ascribes to it a twofold nation. At first, however, his system attracted little notice, and was rejected by the greater part even of astronomers. It lay fermenting in secret with other new discoveries for more than fifty years, till, by the exertions of Galileo, it was kindled into so bright aflame, as to consume the philosophy of Aristotle, to alarm the hierarchy of Rome, and to threaten the existence of every opinion not founded on experience and observation. After COPERNICUS, TYCHO BRAHE was the most distinguished astronomer of the sixteenth century. The instruments which he used, were of far greater size, more skillfully contrived, and more nicely divided, than any that had yet been directed to the heavens. By means of them, TYCHо could measure angles to ten seconds, which may be accounted sixty times the accuracy of the instruments of Ptolemy, or of any that had belonged to the school of Alexandria. He first became acquainted with the atmospherical refraction, by which the heavenly bodies are made to appear more elevated above the horizon than they really are. KEPLER was born in 1571; he discovered that all the planetary orbits are elliptical, having the sun in their common focus, that the area described by the line drawn from a planet to the sun, increased at a uniform rate, and, therefore, that any two such areas are proportional to the times in which they are described. The same great astronomer was, perhaps, the first person who conceived that there must be always a law capable of being expressed by arithmetic or geometry, which connects such phenomena as have a physical dependence on one another, and he as last found, that in any two planets, the squares of the times of the revolution, are as the cubes of their mean distances from the sun. These laws have in our own days been shewn to be necessary results of the diffusion of MOTION from a central mass through a gaseous medium. It was in the year 1609 that the news of a discovery, made in Holland, reached GALILEO, viz. that two glasses had been so combined, as greatly to magnify the objects seen through them. Galileo immediately applied himself to try various combinations of lenses, and he quickly fell on one which made objects appear greater than when seen by the naked eye, in the proportion of three to one. He soon improved on this construction, and found one which magnified thirty-two times, nearly as much as the kind of telescope he used is capable of. That telescope was formed of two lenses; the lens next the object convex, the other concave; the objects were presented upright, and magnified in their lineal dimensions in the proportion just assigned. Having tried the effect of this combination on terrestrial objects, he next directed it to the moon. What the telescope discovers on the ever-varying face of that luminary, is now well known, and needs not to be described; but the sensations which the |