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64 THE NATURE OF MATHEMATICS 

As the units become greater, /' becomes less ; and, since the 
dimensions of F are jL-J, , the factor is obviously suggested 

to us the symbol " [2 1 ] 2 " suggesting a squaring of the number 
measuring the time. 

From Galileo's work resulted the conclusion that, where 
there is no change of velocity in a straight line, there is no force. 
The state of a body unacted upon by force is uniform rectili- 
near motion ; and rest in a special case of this motion, where 
the velocity is and remains zero. This " law of inertia " was 
exactly opposite to the philosophical opinion, derived from 
Aristotle, that force is requisite to keep up a uniform motion ; 
and may be roughly verified by noticing the behaviour of a 
body projected with a given velocity and moving under little 
resistance as a stone on a sheet of ice. Newton and his 
contemporaries saw how important this law was in the 
explanation of the motion of a planet say, about the sun. 
Think of a simple case, and imagine the orbit to be a circle. 
The planet tends to move along the tangent with uniform 
velocity, but the attraction of the sun simultaneously draws 
the planet towards itself, and the result of this continual 
combination of two motions is the circular orbit. Newton 
succeeded in calculating the shapes of the orbits for different 
laws of attraction, and found that, when attraction varies 
inversely as the square of the distance, the shapes are conic 
sections, as had been observed in the case of our solar system. 

The problem of the solar system appeared, then, in a 
mathematical dress ; various things move about in space, and 
this motion is completely described if we know the geometrical 
relations distances, positions, and angular distances between 
these things at some moment, the velocities at this moment, and 
the accelerations at every moment. Of course, if we knew all 
the positions of all the things at all the instants, our descrip- 
tion would be complete ; it happens that the accelerations are 
usually simpler to find directly than the positions : thus, in 
Galileo's case the acceleration was simply constant. Thus, 
we are given functional relations between these positions and 
their rates of change. We have to determine the positions 
from these relations.