To Explain the World: The Discovery of Modern Science Read Online Free PDF

1295–1349), William Heytesbury (fl. 1335), Richard Swineshead (fl. 1340–1355), and John of Dumbleton (fl. 1338–1348). Their most notable achievement, known as the Merton College mean speed theorem, for the first time gives a mathematical description of nonuniform motion—that is, motion at a speed that does not remain constant.
    The earliest surviving statement of this theorem is by William of Heytesbury (chancellor of the University of Oxford in 1371), in Regulae solvendi sophismata. He defined the velocity at any instant in nonuniform motion as the ratio of the distance traveled to the time that would have elapsed if the motion had been uniform at that velocity. As it stands, this definition is circular, and hence useless. A more modern definition, possibly what Heytesbury meant to say, is that the velocity at any instant in nonuniform motion is the ratio of the distance traveled to the time elapsed if the velocity were the same as it is in a very shortinterval of time around that instant, so short that during this interval the change in velocity is negligible. Heytesbury then defined uniform acceleration as nonuniform motion in which the velocity increases by the same increment in each equal time. He then went on to state the theorem: 15
When any mobile body is uniformly accelerated from rest to some given degree [of velocity], it will in that time traverse one-half the distance that it would traverse if, in that same time, it were moved uniformly at the degree of velocity terminating that increment of velocity. For that motion, as a whole, will correspond to the mean degree of that increment of velocity, which is precisely one-half that degree of velocity which is its terminal velocity.
    That is, the distance traveled during an interval of time when a body is uniformly accelerated is the distance it would have traveled in uniform motion if its velocity in that interval equaled the average of the actual velocity. If something is uniformly accelerated from rest to some final velocity, then its average velocity during that interval is half the final velocity, so the distance traveled is half the final velocity times the time elapsed.
    Various proofs of this theorem were offered by Heytesbury, by John of Dumbleton, and then by Nicole Oresme. Oresme’s proof is the most interesting, because he introduced a technique of representing algebraic relations by graphs. In this way, he was able to reduce the problem of calculating the distance traveled when a body is uniformly accelerated from rest to some final velocity to the problem of calculating the area of a right triangle, whose sides that meet at the right angle have lengths equal respectively to the time elapsed and to the final velocity. (See Technical Note 17 .) The mean speed theorem then follows immediately from an elementary fact of geometry, that the area of a right triangle is half the product of the two sides that meet at the right angle.
    Neither any don of Merton College nor Nicole Oresme seems to have applied the mean speed theorem to the most importantcase where it is relevant, the motion of freely falling bodies. For the dons and Oresme the theorem was an intellectual exercise, undertaken to show that they were capable of dealing mathematically with nonuniform motion. If the mean speed theorem is evidence of an increasing ability to use mathematics, it also shows how uneasy the fit between mathematics and natural science still was.
    It must be acknowledged that although it is obvious (as Strato had demonstrated) that falling bodies accelerate, it is not obvious that the speed of a falling body increases in proportion to the time , the characteristic of uniform acceleration, rather than to the distance fallen. If the rate of change of the distance fallen (that is, the speed) were proportional to the distance fallen, then the distance fallen once the body starts to fall would increase exponentially with time, * just as a bank account that receives interest