ON ACCELERATING FORCES. 25 



the spaces are equal, the forces are as the squares of the velocities. Wher- 

 ever the space and the force remain the same, whether the force be uniform 

 or not, the squares of any two velocities with which different bodies enter 

 the space, will receive equal additions while they pass through it. 



When a force acts in a direction contrary to that of the moving body, we 

 may readily determine the retardation that it produces, by comparing the 

 motion with that of a body accelerated by the same force. For the degrees 

 by which an ascending body loses its motion, are the same as those by 

 which it is again accelerated at the same points when it has acquired its 

 greatest height and again descends. We may thus calculate to what height 

 a body will rise when projected upwards with a given velocity, and retarded 

 by the force of gravitation. Since the force of gravitation produces or de- 

 stroys a velocity of 32 feet in every second, a velocity of 320 feet, for 

 instance, will be destroyed in 10 seconds ; and according to what has been 

 premised, a body will fall in 10 seconds through a hundred times 16 feet, 

 or 1600 feet, which is therefore the height to which a velocity of 320 feet 

 in a second will carry a body moving without resistance in a vertical 

 direction. We may also obtain the same result by squaring one eighth of 

 the velocity : thus one eighth of 320 is 40, of which the square is 1600, 

 the height corresponding to the given velocity ; and this velocity is some- 

 times called the velocity due to the height. 



LECT. III. ADDITIONAL AUTHORITIES. 



Galileo, Discorsi e Dimostrazioni Matematiche intorno a due Nuove Scienze,! 

 Leyd. 1638. Riccioli, Almagestum Novum, fol. 1641, ii. c. 21. Mersenni Cogi- 

 tata Physico-Mathematica, fol. Paris, 1644. Toricellius de Motu Gravium, 4to, Flor. 

 1644. Hooke on Falling Bodies, Birch, i. 195. Borellus de Motionibus a Gravi- 

 tate dependentibus, 4to, Reg. Jul. 1670. Halley on Gravity, Ph. Tr. 1686, xvi. p. 3. 

 Mariotte on the Fall of Heavy Bodies, Hist, et Mem. de FAcad. i. 249. Varignon, 

 ibid. ii. 96. See also x. 231, 242, and an. 1709, pp. 69, 267, H. 97 ; an. 1719, 

 p. 195, H. 77; an. 1720, p. 107, H. 97. Camus, an. 1726, p. 159, H. 73. Riccati 

 on the Effects of Attraction, Comm. Bon. ii. III. 143, 6 O. 138. Euler, Me- 

 chanics, 1736, Hist, et Mem. Berlin, 1748, p. 184. Nov. Comm. Petrop. ii. 144. 

 Cotes de Descensu Gravium, 4to, Camb. 1770. 



On the Laws of Motion. Hooke's Posthumous Works, p. 355. Euler, Mec. i. 8. 

 D'Alembert, Encyc. au mot Force. Laplace, Mecanique Celeste, liv. i. c. 2, 5. 

 Robison, Mech. Ph. i. 121. Playfair's Outlines, 2 vols. Edin. 1816, vol.i. Home 

 and Stewart's Lit. and Phil. Essays, i. Whewell, Edin. Journ. of Science, No. 15. 

 Trans, of the Cambridge Philos. Soc. V. Hist, of the Inductive Sciences, ii. Me- 

 chanics, Camb. 1828, 3rd ed. Poisson, Mecanique, i. 278. Powell, Nature and 

 Evidence of the Primary Laws of Motion, Oxf. 1837. 



