|Hutton, Charles Mathematical and Philosophical Dictionary 1795|
, in Chronology, the 13th month of the Jewish ecclesiastical year, answering commonly to our March; this month is intercalated, to prevent the beginning of Nisan from being removed to the end of February.
, in Mechanics, one of the simple mechanical powers, more usually called the Lever.
, or Radius Vector, in Astronomy, is a line supposed to be drawn from any planet moving round a centre, or the focus of an ellipse, to that centre, or focus. It is so called, because it is that line by which the planet seems to be carried round its centre; and with which it describes areas proportional to the times.
, or Swistness, in Mechanics, is that affection of motion, by which a moving body passes over a certain space in a certain time. It is also called celerity; and it is always proportional to the space moved over in a given time, when the Velocity is uniform, or always the same during that time.
Velocity is either uniform or variable. Uniform, or equal Velocity, is that with which a body passes always over equal spaces in equal times. And it is variable, or unequal, when the spaces passed over in equal times are unequal; in which case it is either accelerated or retarded Velocity; and this acceleration, or retardation, may also be equal or unequal, i. e. uniform or variable, &c. See Acceleration, and Motion.
Velocity is also either absolute or relative. Absolute Velocity is that we have hitherto been considering, in which the Velocity of a body is considered simply in itself, or as passing over a certain space in a certain time. But relative or respective Velocity, is that with which bodies approach to, or recede from one another, whether they both move, or one of them be at rest. Thus, if one body move with the absolute Velocity of 2 feet per second, and another with that of 6 feet per second; then if they move directly towards each other, the relative velocity with which they approach is that of 8 feet per second; but if they move both the same way, so that the latter overtake the former, then the relative Velocity with which that overtakes it, is only that of 4 feet per second, or only half of the former; and consequently it will take double the time of the former before they come in contact together.
Velocity in a Right Line.—When a body moves with a uniform Velocity, the spaces passed over by it, in different times, are proportional to the times; also the spaces described by two different uniform Velocities, in the same time, are proportional to the Velocities; and consequently, when both times and Velocities are unequal, the spaces described are in the compound ratio of the times and Velocities. That is, S <*> TV, and s <*> tv; or S : s :: TV : tv. Hence also, V : v :: S/T : s/t, or the Velocity is as the space directly and the time reciprocally.
But in uniformly accelerated motions; the last degree of Velocity uniformly gained by a body in beginning from rest, is proportional to the time; and the space described from the beginning of the motion, is as the product of the time and Velocity, or as the square of the Velocity, or as the square of the time. That is, in uniformly accelerated motions, v t, and s tv or v2 or t2. And, in fluxions, s. = vt..
Velocity of Bodies moving in Curves.—According to Galileo's system of the fall of heavy bodies, which is now universally admitted among philosophers, the Velocities of a body falling vertically are, at each moment of its fall, as the square roots of the heights from whence it has fallen; reckoning from the beginning of the descent. And hence he inferred, that if a body descend along an inclined plane, the Velocities it has, at the different times, will be in the same ratio: for since its Velocity is all owing to its fall, and it only falls as much as there is perpendicular height in the inclined plane, the Velocity should be still measured by that height, the same as if the fall were vertical.
The same principle led him also to conclude, that if a body fall through several contiguous inclined planes, making any angles with each other, much like a stick when broken, the Velocity would still be regulated after the same manner, by the vertical heights of the different planes taken together, considering the last Velocity as the same that the body would acquire by a fall through the same perpendicular height.
This conclusion it seems continued to be acquiesced in, till the year 1672, when it was demonstrated to be false, by James Gregory, in a small piece of his intitled Tentamina quædam Geometrica de Motu Penduli & Projectorum. This piece has been very little known, because it was only added to the end of an obscure and pseudonymous piece of his, then published, to expose the errors and vanity of Mr. Sinclair, professor of natural philosophy at Glasgow. This little jeu d'esprit of Gregory is intitled, The great and new Art of Weighing Vanity: or a discovery of the Ignorance and Arrogance of the great and new Artist, in his Pseudo-Philosophical writings: by M. Patrick Mathers, Arch-Bedal to the University of S. Andrews. In the Tentamina, Gregory shews what the real Velocity is, which a body acquires by descending down two contiguous inclined planes, forming an obtuse angle, and that it is different from the Velocity a body acquires by descending perpendicularly through the same height; also that the Velocity in quitting the first plane, is to that with which it enters the second, and in this latter direction, as radius to the cosine of the angle of inclination between the two planes.
This conclusion however, Gregory observes, does not apply to the motions of descent down any curve lines, because the contiguous parts of curve lines do not form any angle between them, and consequently no part of the Velocity is lost by passing from one part of the curve to the other; and hence he infers, that the Velocities acquired in descending down a continued curve line, are the same as by falling perpendicularly through the same height. This principle is then applied, by the author, to the motion of pendulums and projectiles.
Varignon too, in the year 1693, followed in the same track, shewing that the Velocity lost in passing from one right lined direction to another, becomes indefinitely small in the course of a curve line; and that therefore the doctrine of Galileo holds good for the descent of bodies down a curve line, viz, that the Velocity