MATHEMATICAL SCIENCES-SEVENTEENTH CENT. 163 



this experiment, we let the same ball descend through a fourth part 

 only of the length of the groove, and found the measured time to be 

 exactly half the former. Continuing our experiments with other por- 

 tions of the length, comparing the fall through the whole with the fall 

 through half? two-thirds, three-fourths, in short, with the fall through 

 any part, we found by many hundred experiments that the spaces 

 passed over were as the squares of the times, and that this was the 

 case in all inclinations of the plank .... As to the estimation of the 

 time, we hung up a great bucket full of water, which through a very 

 small hole pierced in the bottom squirted out a fine thread of water, 

 which we caught in a small glass during the whole time of the different 

 descents ; then weighing from time to time in an exact pair of scales 

 the quantity of water caught in this way, the differences and propor- 

 tions in their weights gave the differences and proportions of the times ; 

 and this with such exactness that, as I said before, although the ex- 

 periments were repeated again and again, they never differed in any 

 degree worth noticing." 



From the fact that the spaces fallen through are proportional to the 

 squares of the time, Galileo was able to prove that upon the principle 

 that the projectile falls independently of its forward motion, the path 

 of a projectile must be a parabola. Anticipating objections, he explains 

 that the resistance of the air and the convergence of the action of 

 gravity towards the earth's centre will cause the course of the projectile 

 to be not strictly parabolic. The latter cause will, he remarks, not 

 sensibly affect our experiments. The futility of some objections which 

 had- been urged against his experimental proof of the falsity of Aris- 

 totle's law of falling bodies is thus shown in the "Dialogue on Motion," 

 where one of the interlocutors is made to assert that if a cannon-ball 

 weighing two hundred pounds, and a musket-ball weighing half a pound, 

 be dropped together from a lofty tower, the former will not anticipate 

 the latter by so much as a foot; ''and I would not have you do as some 

 are wont, who fasten upon some saying of mine that may want a hair's 

 breadth of the truth, and under this hair they seek to hide another 

 man's blunder as by a cable. Aristotle says that an iron ball weighing 

 a hundred pounds will fall from a height of a hundred yards while 

 one weighing one pound falls but one yard : I say they will reach the 

 ground together. They find the bigger to anticipate the less by two 

 inches, and under these two inches they seek to hide Aristotle's ninety- 

 nine yards." 



Galileo's theory of falling bodies is thus expressed : "A heavy body 

 has by nature an intrinsic principle of moving towards the common 

 centre of heavy things, that is to say, towards the centre of our terres- 

 trial globe, with a motion continually accelerated in such manner that 

 in equal times there are always equal additions of velocity. This is 

 to be understood as holding true only when all accidental and ex- 

 ternal impediments are removed, amongst which is one that we cannot 



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