Magnitude

“ Newton's law of gravitation.—"Every particle of matter in the universe attracts every other particle with a force whose direction is that of a line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other." We know that we ourselves and the things about us are pulled toward the earth by a force (weight) which is called, in the Latin that Newton wrote, gravitas, and the word marks well the true significance of the law of gravitation. ”

“ Decimals express all intermediate graduations of magnitude. Astronomers carry the telescopic magnitudes much farther, till a magnitude beyond the twentieth is reached, preserving in every case the ratio of two and one-half for each magnitude in relation to that numerically next to it. ”

“ The planets revolutions we are acquainted with, but who is able to this day, regularly to demonstrate either their magnitude or their distance? Unless he will prove, as is the usual way, the magnitude from the distance, and the distance from the magnitude. ”

“ It is manifest that the velocity of light and the ratio of the units are quantities of the same order of magnitude. Neither of them can be said to be determined as yet with such a degree of accuracy as to enable us to assert that the one is greater or less than the other. ”

“ In reference to apparent magnitudes, we scarcely ever judge any object to be so great or so small as it appears to be, or that there is so great a disparity in the visible magnitude of two equal bodies at different distances from the eye. ”

“ We cannot, for instance, annihilate the stars or the sun or the electrons. Thus, whereas a magnitude which has been established as relative will presumably remain a relative for all time to come, the same measure of assurance can no longer be claimed for our determinations of absolute quantities. ”

“ The force of attraction between two bodies. According to Newton, gravitation is directly proportional to the product of the masses of two bodies and inversely proportional to the square of the distance between them. 2. The acceleration produced by the mutual attraction of two masses, directed along the line joining their centers of mass, and of magnitude inversely proportional to the square of the distance between the two centers of mass. ”

“ And because whatsoever is real here, or there, or in any place, has dimensions, that is to say, magnitude; that which hath magnitude, whether it be visible or invisible, finite or infinite, is called by all the learned a body. ”

“ One of these characteristics is the fact that the distances between the masses which compose the system are very much greater than the dimensions of the masses themselves, of quite a higher order of magnitude. ”

“ Less definite but of a higher order of magnitude are the figures expressing the quantity of mass of the earth, or what is sometimes designated by the scientifically meaningless phrase 'the weight of the earth.' We all have a tolerably clear idea of the mass in a ton of coal, but few of us are fitted to realize the nearly equally definite quantity of the earth's mass, which is in round numbers six thousand six hundred million million million tons. ”

“ Also, magnitudes are equal, or greater, or lesser, than one another, for the same consideration, namely, when the bodies, of which they are the magnitudes, are either equal, or greater, or less, &c.
One and the same body has always one and the same magnitude. ”

“ But even when we suppose that some magnitudes can never be connected with mass, length, and time, it still holds a preponderating place, and its progress finds an echo throughout the whole domain of the natural sciences. It is therefore well, in order to give an account of the general progress of physics, to examine at the outset the improvements which have been effected in these fundamental measurements, and to see what precision these improvements have allowed us to attain. ”

“ In every case the magnitude of a force can be described by the number of pounds expressing the weight to which it is equivalent. There is another but much more difficult mode of measuring force occasionally used in the higher branches of mechanics (Art. 497) , but the simpler method is preferable for our present purpose. [Pg 5]
Fig. 1.
6. The straight line in which a force tends to move the body to which it is applied is called the direction of the force. Let us suppose, for example, that a force of 3 lbs. is applied at the point a, Fig. ”

“ It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.1 In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) ”

“ This only arose when astronomy began to be seriously studied, and the exact position of the heavenly bodies to be determined, which created the necessity for strictly accurate means of measuring linear and angular magnitudes. ”

“ Should any one inquire why we have been furnished with several senses in place of having only one, it might be answered, " that we have so been constituted in order that the sequences and common properties of bodies, as motion, magnitude, and number, may the less readily escape our notice." If vision, in fact, were our only sense and it perceptive only of whiteness, then all other qualities would more readily escape our notice and seem to be identical, on account of colour and magnitude being in an invariable sequence to one another. ”

“ A priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes z, y, x, t, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. ”

“ But if we take the number of stars down to the ninth magnitude, which are known with considerable accuracy, and find the numbers in each succeeding magnitude down to the seventeenth, according to the same ratio of increase which has been found to correspond very nearly in the case of the higher magnitudes, Mr. ”

“ We must also admit, in taking for our guide, the reasoning which discourses to us things which are invisible to the senses, that the most minute magnitudes, those which are not compound magnitudes, and which from the limit of sensible extent, are the first measure of the other magnitudes which are only called greater or less in their relation to the others. ”

“ The magnitudes, distances, and velocities, of the heavenly bodies are so much beyond every thing of this kind which belongs to things around us, from which we borrowed our first ideas of these qualities, that we can scarcely avoid looking with incredulity at the numerical results to which the unerring principles of mathematics have conducted us. ”

“ It is true that an experiment, in which the principal aim is to see what happens under certain conditions, may be regarded as an experiment of research by those who are not yet familiar with the result, but in experimental researches, strictly so called, the ultimate object is to measure something which we have already seen—to obtain a numerical estimate of some magnitude. ”

“ But how do we know the quantity of matter of a body? We may reply, that we take the weight as the measure of the quantity of matter: but we may then be again asked, how it appears that the weight is proportional to the inertia; which it must be, in order that the quantity of matter may be proportional to both one and the other. We answer, that this appears to be true experimentally, because all bodies fall with equal velocities by gravity, when the known causes of difference are removed. ”

“ Consequently gravitation is proportional to the volume, while the pressure of light is proportional to the surface of the body acted upon. Now the mass, or volume, of any body varies as the cube of its diameter, and the surface only as the square. If, then, we have two bodies, one of which has twice the diameter of the other, the mass of the second will be eight times less than that of the first, but the surface will be only four times less. If the second has only one-third the diameter of the first, then its mass will be twenty-seven times less, but its surface only nine times less. ”

“ We have made comparative estimates of the weight of the earth and planets, and have reason to believe that the force expression of gravitation attains a maximum at about one-sixth the distance toward the center of the earth, then decreases, until at the very center of our planet, matter has no weight. This, together with the rule I repeated a few moments ago, is about all we know, or think we know, of gravitation. Gravitation is the beginning and gravitation is the end; all earthly bodies kneel to gravitation. ”

“ Considered in the most general way, we may define matter as a complex of gravitational, kinetic and chemical energies, which are found to cling together in the same space. Of these energies the capacity factors, namely, weight, mass and elements, are conservative as described, while the intensity factors, potential, velocity and affinity, may change in wide limits. To explain why we find these energies constantly combined one with another, we only have to think of a mass without gravity or a ponderable body without mass. ”

“ Mass, which used to be regarded as the most indubitable of physical quantities, is now generally believed to vary according to velocity, and to be, in fact, a vector quantity which at a given moment is different in different directions. The detailed conclusions deduced from the supposed constancy of mass for such motions as used to be studied in physics will remain very nearly exact, and therefore over the field of the older investigations very little modification of the older results is required. ”

“ The astronomical unit of mass is that mass which attracts another body placed at the unit of distance so as to produce in that body the unit of acceleration.In framing a universal system of units we may either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science ”

“ The intensity of gravity, or the acceleration of a freely falling body, is an important physical quantity for the several physical sciences. The intensity of gravity determines the weight of a standard pound or kilogram as a standard or unit of force. ”

“ A force which belongs to individual atoms, is independent of chemical and physical conditions, and cannot be altered or destroyed by any known means, must be closely related to the fundamental nature of the atoms. One of the most essential parts in our concept of matter is mass, and the force of gravitation of an atom is proportional to its mass. Mass and gravitation stand, therefore, either as co-effects of the same cause or as cause and effect. ”

“ In descriptive astronomy the mass of the sun or that of the earth is sometimes taken as a unit, but in the dynamical theory of astronomy the unit of mass is deduced from the units of time and length, combined with the fact of universal gravitation. ”

“ Since energy is the same thing as measured mass, we may also take action to be measured mass multiplied by time. In classical mechanics, the “density” of matter in any region is the mass divided by the volume; that is to say, if you know the density in a small region, you discover the total amount of matter by multiplying the density by the volume of the small region. In relativity mechanics, we always want to substitute space-time for space ”

“ Let A be the weight of the globe in vacuo, B its weight in the resisting medium, D the diameter of the globe, F a space which is to 4/3D as the density of the globe to the density of the medium, that is, as A to A - B, G the time in which the globe falling with' the weight B without resistance describes the space F, and H the velocity which the body acquires by that fall. ”

“ But we prefer to conclude with a mathematical demonstration drawn from the weight of matter. The weight of a mass of matter depends on the number of material terms to which the action of gravity is applied, and it increases exactly in the same ratio as the number of the elementary terms contained in the mass. This being the case, let us assume that there is somewhere an atom of continuous matter. The action of gravity will find in it an infinite multitude of points of application; for it is of the nature of continuum to supply matter for an endless division. ”

“ For the purpose of general illustration the reader is asked to conceive a mass of heterogeneous matter, concentrated round a given point in space, forming a single body. This mass is assumed to be assembled and to obtain its coherent form in virtue of that universal and inherent property of matter, namely, gravitative or central attraction. ”

“ Since the force which pulls the stone towards the earth is one kilogram, and since in any system of units the product of mass and acceleration represents force, the mass of our stone is the 9.81th part of unit mass. ”