nature must start from some assumed law of nature, such, for instance, as the assumed law of the cost of building stated above. Accordingly, however accurately we have calculated that some event must occur, the doubt always remains–-Is the law true? If the law states a precise result, almost certainly it is not precisely accurate; and thus even at the best the result, precisely as calculated, is not likely to occur. But then we have no faculty capable of observation with ideal precision, so, after all, our inaccurate laws may be good enough.

We will now turn to an actual case, that of Newton and the Law of Gravity. This law states that any two bodies attract one another with a force proportional to the product of their masses, and inversely proportional to the square of the distance between them. Thus if $m$ and $M$ are the masses of the two bodies, reckoned in lbs. say, and $d$ miles is the distance between them, the force on either body, due to the attraction of the other and directed towards it, is proportional to $\frac{m M}{d^{2}}$ ; thus this force can be written as equal to $\frac{k m M}{d^{2}}$ , where $k$ is a definite number depending on the absolute magnitude of this attraction and also on the scale by which we choose to measure forces. It is easy to see that, if we

wish to reckon in terms of forces such as the weight of a mass of $1$ lb., the number which $k$ represents must be extremely small; for when $m$ and $M$ and $d$ are each put equal to $1$ , $\frac{k m M}{d^{2}}$ becomes the gravitational attraction of two equal masses of $1$ lb. at the distance of one mile, and this is quite inappreciable.

However, we have now got our formula for the force of attraction. If we call this force $F$ , it is $F = k \frac{m M}{d^{2}}$ , giving the correlation between the variables $F$ , $m$ , $M$ , and $d$ . We all know the story of how it was found out. Newton, it states, was sitting in an orchard and watched the fall of an apple, and then the law of universal gravitation burst upon

his mind. It may be that the final formulation of the law occurred to him in an orchard, as well as elsewhere–-and he must have been somewhere. But for our purposes it is more instructive to dwell upon the vast amount of preparatory thought, the product of many minds and many centuries, which was necessary before this exact law could be formulated. In the first place, the mathematical habit of mind and the mathematical procedure explained in the previous two chapters had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses

at any distance. Again, what are the meanings

of the terms employed, Force, Mass, Distance?

Take the easiest of these terms, Distance. It seems very obvious to us to conceive all material things as forming a definite geometrical whole, such that the distances of the various parts are measurable in terms of some unit length, such as a mile or a yard. This is almost the first aspect of a material structure which occurs to us. It is the gradual outcome of the study of geometry and of the theory of measurement. Even now, in certain cases, other modes of thought are convenient. In a mountainous country distances are often reckoned in hours. But leaving distance, the other terms, Force and Mass, are much more obscure. The exact comprehension of the ideas which Newton

meant to convey by these words was of slow growth, and, indeed, Newton himself was the first man who had thoroughly mastered the true general principles of Dynamics.

Throughout the middle ages, under the influence of Aristotle, the science was entirely