Of this we may be satisfied by comparing the subjects considered in the transactions of the various learned societies with the discussions that had occupied the attention of the middle ages.
But the use of mathematics was not limited to the verification of theories; as above indicated, it also furnished a means of predicting what had hitherto been unobserved.

In this it offered a counterpart to the prophecies of ecclesiasticism.

The discovery of Neptune is an instance of the kind furnished by astronomy, and that of conical refraction by the optical theory of undulations.
But, while this great instrument led to such a wonderful development in natural science, it was itself undergoing development--improvement.

Let us in a few lines recall its progress.
The germ of algebra may be discerned in the works of Diophantus of Alexandria, who is supposed to have lived in the second century of our era.

In that Egyptian school Euclid had formerly collected the great truths of geometry, and arranged them in logical sequence.