The Day the Universe Changed - James Burke [76]
The basic problem was that these forces changed constantly. A planet in orbit is constantly under two influences: the inertia which propels it outwards at a tangent to its orbit, and the force pulling it inward towards the sun. The balance between these two forces is what keeps the planet in orbit. However, as Kepler had shown, in a non-circular orbit the forces change constantly as the planet alters speed through its orbit. The rate of change in planetary speed would itself change. What was needed was a way of measuring changing rates of change, instantaneously, at any point in the trajectory. The sums involved were infinitely small.
Newton developed two kinds of calculus to solve the problem. Differential calculus measured the difference in behaviour which showed as the effect of rate of change. Integral calculus showed how the rates of change varied one with the other and showed them as a ratio of one to the other. Newton called the rate-of-change units ‘fluxions’. He used them to calculate the behaviour of a universe full of falling bodies.
Whether the story of the apple falling from the tree is true or not, Newton used a falling apple to illustrate his theory that every body attracts every other body. He said that while the earth may attract the apple, so too, to an infinitesimally small extent, the apple attracts the earth. This was Kepler’s original idea of mutual attraction. But Kepler had only seen it operating to hold the moon in orbit round the earth, acting with a force relative to the masses of the two bodies and causing the tides. He had not seen the force acting universally, although in stating that planets were held by two forces, one of which was attraction towards the sun and the other a desire to leave it, Kepler prepared the way for Newton.
In his approach to all problems Newton followed Descartes’ method of thought. He used mathematics to work out the consequences of any proposed solution and then showed by experiment and observation that his conclusion was correct. Swinging a stone on the end of a string, Newton showed that the stone moved in a circle because the string held it. The moon, therefore, had to be held by the earth and the planets by the sun. The fact that they did not fly off at a tangent to their orbits had to be because the attraction inwards equalled the outward force of their own inertia.
Newton agreed with Kepler that the mutual attraction operated in relation to the distance between the planetary bodies. He theorised that the force would work at a ratio inversely proportional to their separation. In the case of the moon, at a distance of sixty times the earth’s radius, the strength of the attraction of the earth should be 1/602 of the attraction, which Galileo had shown to be 16 feet per second. The earth should therefore be attracting the moon away from her inertial path out into space at a rate of 16/602, or 0.0044 feet per second. Examination of the moon’s path second by second showed Newton to be right.
In the Principia Newton went on to explain how to use these calculations to derive the masses of all the planets from their orbital behaviour. He demonstrated that the irregularities of the moon’s behaviour were due to the pull of the sun, that the moon does indeed cause tides, that comets are part of the solar system with calculable orbits, and that the earth tilts on its axis by 66½ degrees to the plane of its orbit.
Newton’s discovery: the moon’s path would take it away from earth at a tangent to its orbit were it not for the pull of earth’s gravity which alters its path at every instant, pulling it inwards by 0.0044 feet per second, which balances