# Page:A short history of astronomy(1898).djvu/275

constantly to draw the body towards o, or counteracting the tendency which it has, in virtue of the First Law of Motion, to get farther and farther away from o. To express either of these tendencies numerically we want a more complex idea than that of velocity or rate of motion, namely acceleration or rate of change of velocity, an idea which Galilei added to science in his discussion of the law of falling bodies (chapter vi., §§ 116, § 133). A falling body, for example, is moving after one second with the velocity of about 32 feet per second, after two seconds with the velocity of 64, after three seconds with the velocity of 96, and so on; thus in every second it gains a downward velocity of 32 feet per second; and this may be expressed otherwise by saying that the body has a downward acceleration of 32 feet per second per second. A further investigation of the motion in a circle shews that the motion is completely explained if the moving body has, in addition to its original velocity, an acceleration of a certain magnitude directed towards the centre of the circle. It can be shewn further that the acceleration may be numerically expressed by taking the square of the velocity of the moving body (expressed, say, in feet per second), and dividing this by the radius of the circle in feet. If, for example, the body is moving in a circle having a radius of four feet, at the rate of ten feet a second, then the acceleration towards the centre is ${\displaystyle \left({\frac {10\times 10}{4}}=\right)}$ 25 feet per second per second.