THE LOST SOLAR SYSTEM OF THE ANCIENTS DISCOVERED. GRAVITATION NEAR THE EARTH'S SURFACE. CONSTRUCTION OF THE OBELISK. VARIATION OF TIME, VELOCITY, AND DISTANCE REPRESENTED BY THE ORDINATES AND AXIS OF THE OBELISK. THE OBELISCAL AND PARABOLIC AREAS COMPARED. CON- STRUCTION AND SUMMATION OF OBELISCAL SERIES OF NUMBERS, PYLONIC CURVE GENERATED BY THE OR- DINATE WHICH VARIES INVERSELY AS THE ORDINATE OF THE The Laws of Gravitation expounded by the Geometrical Pro- It was found by Galileo that a heavy body, when allowed to fall freely from a state of rest towards the earth, described distances proportionate to the square of the times elapsed during the descent; or proportionate to the square of the velocities acquired at the end of the descent. That is, at the end of the 1st second the body had de- scribed a distance of 16 feet English, which call 1 P. ΤΣ At the end of the 2nd second, from the beginning of motion, the body had described a distance of 4 P. At the end of the 3rd second, a distance of 9 P. At the end of the 4th second, a distance of 16 P. Here 1, 4, 9, 16 P are the series of distances described in 1, 2, 3, 4, seconds. 1, 3, 5, 7, the series of distances described in each second. 1, 2, 2, 2, the series of incremental distances described in each second more than was described in the preceding second. = During the first second the distance described = 1 P. If the velocity had been uniform the distance would have been described in 1 second with the mean velocity = half the extreme velocities (0+2)=1 P. So that at the end of the 1st second the acquired velocity would 2 P. The velocity acquired at the end of the 2nd second would = twice the mean velocity with which the whole distance 4 P was described in two seconds. The mean velocity will =(0+4)=2 P; therefore the velocity at the end of the 2nd second will = 4 P; at the end of the 3rd second = 6 P; at the end of the 4th second = 8 P. The velocity acquired at the end of the 1st second, if continued uniform during the 2nd second, would, of itself, have carried the body 2 P; but during the 2nd second the body received an additional accelerating velocity from gravity equal to that which caused it to describe 1 P in the 1st second. So that during the 2nd second the distance described will =2+1=3=1+2 P. In like manner, during the 3rd second, the distance described will =4+1=5=3+2 P. In the 4th second 6+1=7=5+2 P, will be described. The distances described in the successive seconds will be 1, 3, 5, 7 P. The velocities at the beginning of the 1st, 2nd, 3rd, and 4th seconds will be Generally, the distance (2n-1) P, described in the nth second with an accelerated velocity, will be uniformly described with the mean of the velocities at the beginning and end of the nth second; which mean velocity will = (2 n − 2 + 2 n ) = (2 n − 1) P. The whole distance described during n seconds will be proportionate to the square of the time, and n2 P. = At the end of the descent the acquired velocity will be proportionate to the whole time elapsed, and = 2 n p in a second. During the descent equal increments of velocity 2 P are generated during each second. Hence the effect produced by gravity may be regarded as constant for so small a distance as the body describes while falling freely near the earth's surface. To construct the Obelisk. When a body falls from a state of rest, near the earth's surface, by the action of gravity, the time elapsed and the velocity acquired at the end of the descent will vary as the square root of the distance described. A body falling from rest will describe a straight line. Let the point whence the body begins to fall be the apex of the obelisk, and the distance described be along the axis. (fig. 1.) If at the end of the descent a straight 46 Fig. 1. Ord.6 = from the apex and made 1, or 1; this ordinate will represent 1 second, the time of describing 1 P along the axis. Again when the body has fallen from the apex to a distance of 4 P, there draw an ordinate = √4 =2, which will represent the time 2 seconds, during which the body fell from rest to a distance of 4 P. When the body has fallen from the apex to a distance of 9 P, there draw an ordinate 9-3, which will represent 3 seconds, the time of falling 9 P. Thus any number of ordinates may be drawn, and each made = the axis When the extremities of these ordinates are joined by straight lines, the area included by these lines, the axis and the last ordinate will be an obeliscal area. The ordinate of an obeliscal area will in units the number of seconds elapsed during the descent from the apex to the ordinate; and the axis will in units the number of P's described during the descent from the apex to the ordinate. As the time and velocity both vary as the square root of the distance, and at the end of 1, 2, 3, 4 seconds 2, 4, 6, 8 P, are the acquired velocities, Then since ordinates made equal the square root of the axes represent the times, or number of seconds elapsed during the descent; it follows, that double ordinates, or ordinates twice the length of the corresponding time ordinates, will represent the velocity acquired in the descent from the apex to these ordinates. As the nth velocity ordinate will equal 2 n, or twice the corresponding time ordinate, so an additional ordinate like the time ordinate may be drawn on the other side of the axis; these together will represent the velocity ordinate. So that during n seconds the distance described will n2 P, and the velocity acquired at the end of the descent will =2 n P in a second. When the ordinates (fig. 6.) 1, 2, 3, 4, &c. are bisected and joined at the extremities by straight lines, an obeliscal area is formed equal to that of fig. 1. An obeliscal sectional axis is the part of the axis intercepted by two consecutive ordinates, and are as 1, 3, 5, 7. An obeliscal sectional area is the area included between two consecutive ordinates. Sum of n sectional axes = whole axis. or 1+3+5+7=n2. Sum of n ordinates=1+2+3+4=1 n + 1. n Difference-1.n. Hence the difference between the sum of the sectional axes, or whole axis of the obelisk, and the sum of the corresponding ordinates will equal axis-ordinate = n2-n. Figs. 2. and 3. will represent 1st series, 1, 2, 3, 4 time ordinates. The 2nd The 1st series represents the time ordinates. series the velocity ordinates. The 3rd series their corresponding axes, or distances D, described from the apex to the time or velocity ordinates. The 4th series the sectional axes, or dis |