'4fl6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1685-6. 



AB. And, by the same reason, the areas Abe will represent the spaces descended 

 in the times Ab ; so then the spaces descended in the times AB, Ab, are as the 

 areas of the triangles ABC, Abe, which by the 20th of the 6 of Euclid, are 

 as the squares of their homologous sides AB, Ab, that is, of the times. There- 

 fore the descents of falling bodies, are as the squares of the times of their fall, 

 Q. E. D. 



Prop. III. The velocity, which a falling body acquires in any space of time, 

 is double to that, with which it would have moved the space descended by an 

 equable motion, in the same time. — For, draw the line EC parallel to AB, and 

 AE parallel to BC, in the same fig. 1, and compleat the parallelogram ABCE : 

 it is evident that its area may represent the space, a body moved equably with 

 the velocity BC, would describe in the time AB; and the triangle ABC repre- 

 sents the space described by the fall of a body, in the same time AB, by the 

 2d proposition. Now the triangle ABC, is half the parallelogram ABCE, and 

 consequently the space described by the fall is half what would have been de- 

 scribed by an equable motion with the velocity BC, in the same time ; therefore 

 the velocity BC, at the end of the fall, is double to that velocity, which in the 

 time AB would have described the space fallen, represented by the triangle 

 ABC, with an equable motion, Q. E. D. 



Prop. IV. All bodies on or near the surface of the earth, in their fall, de- 

 scend in such a manner, as at the end of the 1st second of time, to have de- 

 scribed l6 feet and 1 inch, London measure, and acquired the velocity of 32 

 feet 2 inches in a second. — This is made out from the 25th prop, of the 2d 

 part of that excellent treatise of M. Huygens de Horologio Oscillatorio ; where 

 he demonstrates the time of the least vibrations of a pendulum to be to the 

 time of the fall of a body, from the height of half the length of the pendulum, 

 as the circumference of a circle is to its diameter ; whence, as a corollary, it 

 follows, that as the square of the diameter is to the square of the circumference, 

 so is half the length of the pendulum, vibrating seconds, to the space described 

 by the fall of a body in a second of time : and the length of the pendulum, 

 vibrating seconds, being found SQ^ inches, the descent in a second will be 

 found by the aforesaid analogy to be 1 6 feet 1 inch : and by the third prop, the 

 velocity will be double thereto ; and thus nearly it has been found by several 

 experiments, which, by reason of the swiftness of the fall, cannot so exactly 

 determine its quantity. 



From these four propositions, all questions concerning the perpendicular fall 

 of bodies are easily solved; and either the time, height, or velocity being as- 

 signed, the other two may be readily found. From them likewise is the doc- 

 trine of projectiles deducible, assuming the two following axioms, viz. That 



