The discovery of Neptune is esteemed to be the most notable discovery in mathematical astronomy. The discovery was no accident, neither was it the result of long tedious searches with the telescope.

The earth and other planets with their satellites are PROJECTILES hurled through space, yet their paths are so accurately computed and recorded that a planet may be found at any time. When one comprehends the significant fact that the astronomer is on a projectile (the earth) moving through space at the prodigious velocity of 18 miles per second (our swiftest rifle ball travels about 3200 FEET per second), that the projectile (earth) on which he is an inhabitant is rotating upon its axis with a velocity of 1000 miles per hour, and that the (earth) projectile does not travel uniformily in its path (its velocity varying) and that it is attracted by the sun and other planets so that its path is composed of many curves, the greatness of the astronomer's feat will begin to dawn upon the imagination.

Prior to the discovery of Neptune, Uranus was the most distant planet known, its distance being 1,728,100,000 miles. Uranus refused to follow the path among the stars that was plotted and laid out for it. In plain words, it had jumped its track, yet it was off its path the small, yet intolerable quantity of 2' of arc. One would think that such a minute discrepancy between observation and theory was not worth attention, yet mathematics is so exact that such a minute difference could not be overlooked.

The path of a planet is an ellipse due to its projectile force and the attraction of the sun. It is

made up of many compound curves due to the attraction of other planets, the sun and other heavenly bodies, and owing to the fact that the positions of all the planets are continually changing, the path of any planet is continually changing. With all these forces taken into consideration, the path of the planet is laid out, and unless, for some unknown force, the planet will HOLD its course for ages.

The fact that Uranus had jumped the track set the whole astronomical world thinking. Something, unknown before, had happened. Two theories were advanced. Had Newton's law of gravity become inoperative at such great distances? This theory could not be entertained. Or was there some large unknown body attracting the planet and by virtue thereof derailed it?

Working upon this theory John Couch Adams of England, and Levierre, an astronomer of France, unknown to each other, set about to solve the problem.

The history of this wonderful achievement is interesting, briefly; Adams was first to solve the problem and sent his manuscript to the Astronomer Royal of England, but Adams, being a young man of no fame, his letter was pigeon-holed. Levierre sent his solution to the French Academy of Sciences. This fact, when learned in England, brought forth the manuscript of Adams from its pigeonhole. The Astronomer Royal awoke from his lethargy, but still doubting the accuracy of the predictions, and still more indisposed to review carefully the solution made by Adams, began in his slow manner

to map a large area of the heavens. Three times he saw the tiny planet but was too careless to note its disc. In the meanwhile, Levierre had written to Galle, of Berlin, saying that he would find within one degree of a certain part of the ecliptic another planet. Galle, following his instruction, with his great telescope, within one-half hour on Sept. 23, 1846, found the planet. It was ONLY 52′ from the precise point that Levierre had stated, and in 24 hours its motion proved it to be the planet.

Wonderful, beyond the comprehension of man, that the mind of man had reached out into fathomless space 2,292,000,000 miles and by mathematics forced it to give up its secrets.

While mathematics reached out 1,982,000,000 miles and determined the path of the projectile (Uranus) with all its intricacies it could not take into consideration the influence of an unknown quantity. Experimentation, in this case observation, detected a disturbing factor, but observation could not locate it; it detected the result of its influence, but mathematics alone determined it in its entirety.

Mathematics will solve the problems of our projectiles, but it cannot take into consideration unknown forces. Experimentation may detect the influence of a disturbing factor, and practice will not agree with theory, but it will be left to mathematics to deduce, to generalize and exemplify the quantity of disturbance.

B

CHAPTER I.

Definitions.

ALLISTICS is the science and art of throwing projectiles. It is classified into Interior and Exterior Ballistics.

Exterior ballistics treats of the projectile during its flight; of the influence of gravity, atmospheric resistance and other forces acting upon it; of its form, size and weight in relation to its efficency; of its construction in relation to its destructive qualities and of those factors that determine its control: briefly, it is the science that will enable the marksman and hunter, under all conditions that may confront him, to determine with precision the elevation and correction of sights, and also the position, velocity and energy of his bullet at any point in its flight.

Interior Ballistics treats of the formation, temperature and volume of gases into which a charge of powder is converted by combustion and the work performed by these gases upon the gun and projectile.

Every particle of matter in the universe has an attraction for every other particle. It is this force that holds a planet in its course, that holds all objects on the earth and that causes any object to fall toward the earth.

Attraction between the earth and bodies upon or near its surface is called gravity. Gravitation be

tween two bodies varies directly as the product of their masses and inversely as the square of the distance between their centers.

Gravity is measured by the term we call weight. It acts uniformly and continually upon a body whether that body be in motion or at rest, and is unaffected by the interposition of any substance.

Acceleration is the rate at which the velocity of a body freely falling, due to the force of gravity, increases per unit of time. Its value is 32.16 foot seconds and is designated as g. At the end of any second the acceleration of a freely-falling body equals gt, in which t is the number of seconds falling.

The space passed over by a freely falling body is expressed by

If a projectile be fired horizontally from a rifle over a plane surface and another projectile be dropped simultaneously from the muzzle as the first projectile issues therefrom, both will strike the earth at the same time, no matter how great the muzzle velocity of the first projectile. No projectile, then, however great its velocity, moves in a straight line, except when fired vertically. Hence, if a rifle be shot horizontally at a point in line with the axis of the bore, the ball will inevitably fall below that point. This being true, to hit an object we must elevate the rear sight of the rifle, by which means