Comparing now the philosophical principles on which science was proceeding, with the principles on which ecclesiasticism rested, we see that, while the former repudiated tradition, to the latter it was the main support while the former insisted on the agreement of calculation and observation, or the correspondence of reasoning and fact, the latter leaned upon mysteries; while the former summarily rejected its own theories, if it saw that they could not be coordinated with Nature, the latter found merit in a faith that blindly accepted the inexplicable, a satisfied contemplation of “things above reason.”  The alienation between the two continually increased.  On one side there was a sentiment of disdain, on the other a sentiment of hatred.  Impartial witnesses on all hands perceived that science was rapidly undermining ecclesiasticism.

Mathematics.  Mathematics had thus become the great instrument of scientific research, it had become the instrument of scientific reasoning.  In one respect it may be said that it reduced the operations of the mind to a mechanical process, for its symbols often saved the labor of thinking.  The habit of mental exactness it encouraged extended to other branches of thought, and produced an intellectual revolution.  No longer was it possible to be satisfied with miracle-proof, or the logic that had been relied upon throughout the middle ages.  Not only did it thus influence the manner of thinking, it also changed the direction of thought.  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.  Archimedes, in Syracuse, had attempted the solution of the higher problems by the method of exhaustions.  Such was the tendency of things that, had the patronage of science been continued, algebra would inevitably have been invented.