Then, again, new points of view are provided by the invention of new
methods in that system of logic which we call mathematics. All that
mathematics can do is to assure us that a statement A is equivalent to
statements B, C, D, or is one of the facts expressed by the statements
B, C, D; so that we may know, if B, C, and D are true, then A is true.
To many people our inability to understand all that is contained in
statements B, C, and D, without the cumbrous process of a mathematical
demonstration, proves the feebleness of the human mind as a logical
machine. For it required the new point of view imagined by Newton's
analysis to enable people to see that, so far as planetary orbits are
concerned, Kepler's three laws (B, C, D) were identical with Newton's
law of gravitation (A). No one recognises more than the mathematical
astronomer this feebleness of the human intellect, and no one is more
conscious of the limitations of the logical process called
mathematics, which even now has not solved directly the problem of
only three bodies.

These reflections, arising from the writing of this History, go to
explain the invariable humility of the great mathematical astronomers.
Newton's comparison of himself to the child on the seashore applies to
them all. As each new discovery opens up, it may be, boundless oceans
for investigation, for wonder, and for admiration, the great
astronomers, refusing to accept mere hypotheses as true, have founded
upon these discoveries a science as exact in its observation of facts
as in theories. So it is that these men, who have built up the most
sure and most solid of all the sciences, refuse to invite others to
join them in vain speculation. The writer has, therefore, in this
short History, tried to follow that great master, Airy, whose pupil he
was, and the key to whose character was exactness and accuracy; and he