Sixth Law: And the common Velocity will be fo much less than the firft, as both the Bodies together are greater than the Body first moved. (10.) If two unequal Bodies, void of Elafticity, which are moved with equal Velocity to oppofite Parts, hit against one another, the Quantity of Motion in both, taken together after the Collifion, will be the Difference only of the former Motions. (11.) If two equal Bodies, void of Elafticity, be mov'd with unequal Velocity towards the fame Part, upon their Collifion there will remain the fame Quantity or Sum of their Motions; but the common Velocity will be only the half of both the former Velocities put to gether. (12.) If of two unequal Bodies, void of Elafticity, the Greater overtakes the Leffer, the common Velocity, after the Shock, will be greater than half the Sum of the former Velocities. And on the contrary, it will be lefs when the leffer Body overtakes the greater. (13.) If a Body perfectly Elaftic dafheth upon another Body of the fame fort which is Quiefcent and Equal; after the Collifion the Motion will be wholly transferr'd into that which was quiefcent before, and with the fame Celerity; but the Body which was mov'd before, will now reft. (14.) If two Bodies perfectly Elaftic, which are equal, but mov'd with an unequal Celerity, dafh one upon another, they, whether they were before carried to the fame part, or to the contrary, will, after the Contact,be mov'd each with that Celerity which the other had before. B 2 (15.) Any (15.) Any Body, how great foever, may be moved by any Body, how small foever, coming. with any Velocity whatsoever. (16.) When two Bodies, perfectly Elastical, are dafh'd one upon the other, they depart from one another with the fame Celerity wherewith they approach'd one to the other; that is, not with the fame abfolute, but relative Celerity. (17.) If two Bodies perfectly Elaftical, do each return to the Impulse with the fame Celerity wherewith they rebounded from it; they will each of them, after the Second Impulfe, acquire the fame Celerity as they had before the firft Meeting. (18.) If two Bodies meet one another, whether they be Elaftic or not Elaftic, there doth not always remain the fame Quantity of Motion as was before, but it may be greater or lefs. (19.) If a Body perfectly Elaftical, which is greater, hits upon a leffer one which is quiefcent, it will give a Velocity to it less than the double of its own. (20.) If two Bodies perfectly Elaftic, the Celerities whereof are in reciprocal Proportion to their Magnitudes, meet one another directly and oppofitely, they will both rebound with the fame Celerity with which they came to each other. (21.) The Celerity which a greater Body perfectly Elaftic, gives to a leffer perfectly quief cent, which is alfo perfectly Elaftic, hath that Proportion to that Velocity, which the leffer moved with the like Celerity gives to the greater when quiefcent, which the Magnitude of the greater hath to the Magnitude of the lefs. (22.) Every (22.) Every Body will in the fame Time defcribe the Diagonal of a Parallelogram with Forces conjunct, that it would do the Sides with those Forces separate. (23.) All compound Forces and Motion whatever may be reduc'd into innumerable other direct Forces and Motions; and on the contrary, all direct Forces, and rectilinear Motions, may be fuppos'd to be compounded of innumerable oblique Motions and Forces. (24.) The Quantity of Motion which is collected, by taking the Sum of the Motions to the fame Part, and the Difference of thofe to the contrary Parts, is not chang'd by the Actions of Bodies one upon another. (25.) The common Center of Gravity of a Syftem of Bodies doth not change its State either of Motion or Reft, from the Actions of the Bodies amongst themselves, (whether they be Attractions or Impulfes ;) and therefore the common Center of Gravity of all Bodies acting upon one another (Actions and Impediments, whether External or otherwise arifing, being excluded) doth either reft, or is mov'd uniformly ftraight forwards. (26.) The Motions of two Bodies included in a given Space, and partaking of the Motion thereof, are the fame amongst themselves, whether that Space refteth, or the fame is mov'd uniformly ftraight forward, without a Circular Motion. (27.) If Bodies be mov'd in any wife amongst themselves, and be preffed with equal accelerative Forces according to parallel Lines, they will all continue to be mov'd in the fame manner amongst themselves, as if they were not preffed wih thofe Forces. B 3 PROP. PROPOSITIONS. III. The Velocities of a Body accelerated by any uniform urging Force whatever, are betwixt themselves, as the Times are wherein that uniform Force is imprefs'd; that is, in Double the Time Double, in Triple the Time Triple, and in Four Times the Time Quadruple. IV. The Lines which Bodies by any urging uniform Force do defcribe, are in the duplicate proportion of the Times, i. e. if the Times be Seconds, One, Two, Three, Four, Five, &c. the whole Lines defcrib'd will be amongst themfelves, as One, Four, Nine, Sixteen, Twentyfive, &c. which are the Squares of the for mer. VII. In a Cycloid inverted,whofe Axis is erected perpendicular, the Times of the Descent wherein a Body let down from any Point whatever in it comes to the lowest Point, are always equal betwixt themselves.. VIII. All Projectiles, not perpendicular to the Horizon, defcribe Parabola's, fo far as they are not hindred by the refiftance of the Air. IX. If two Bodies do in equal Times run over Two whole unequal Circumferences, with an equable Motion, the centripetal Force in the greater Circumference will be to that which is in the lefs, as the Circumferences are one to another directly; or, which is the fame, as their Diameters, or Radii. X. If two Bodies revolve in the fame, or equal Circles with unequal Celerities, but both with an A an equable Motion, the centripetal Force of the Swifter will be to that of the Slower, in the Proportion of the Celerities duplicated; or as the Squares of the Arches defcribed together. XI. If two Bodies revolve in unequal Circles with equal Velocity, their centripetal Forces will be in the reciprocal Proportion of their Circumference or Diameters; fo that in the leffer Circumference there will be the greater centripetal Force, and in the greater the leffer. XII. If two Bodies be mov'd in unequal Circles, with an unequal Velocity, in the fub-duplicate Proportion of the Circumferences, Diameters, or Radii, the centripetal Forces will be equal every where, and neither increas'd in the Access nor Recefs. XIII. If two Bodies be mov'd in unequal Circles, with an unequal Velocity, in the fubduplicate Proportion of the Circumferences, Diameters, or Radii, reciprocally; fo that in the greater Circle the Velocity be the leffer, and in the leffer Circle the greater, and this in the faid fub-duplicate reciprocal Proportion, the centripetal Force will be reciprocally as the Squares of the Radii or Distances. XIV. If two Bodies revolve in unequal Circles with an unequal Celerity; fo that by how much greater the Radius, Diameter or Circumference is, fo much the lefs the Velocity is; and by how much the less the Radius is, fo much the greater is the Velocity, and this in the Reciprocal Proportion of the Radii, the Centri-petal Forces will be as the Cubes of the Radii reciprocally. XV. The Area's, which revolving Bodies do defcribe by Radii drawn unto the unmovable B 4 Center |