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proved to be a true science. I found it a very hard and thorny path to 
acquire this knowledge. Without this discipline I consider it impossible 
to judge the pretensions of any theory of astronomy to be a demonstrable 
science. If you have not gone through such a training as this, and you would 
ask me what you must do before you can understand the reasonings on 
which Physical Astronomy is based, I tell you you must acquire knowledge 
of the whole science of pure mathematics. But this will require an exercise 
of a vast amount of patience, perseverance, and docility. As Sir John 
Herschel once so pertinently remarked, you must enter upon this subject 
through the portal of humility. And in a science of pure reasoning, founded 
professedly on pure reasoning, you must first defer to your teachers. You 
must admit the humiliating confession that you cannot at first appreciate 
the reasoning processes of your teachers. But taking on trust their superior 
power of reasoning to your own, you cannot test their accuracy till fami- 
liarity with their processes has strengthened your own powers. To take 
Geometry alone, as an instance, what does the study of Euclid require ? 
The admission, at its very commencement, of the most difficult metaphysical 
problems and paradoxes on which metaphysicians might dispute for ever. 
That this is no exaggeration on my part, I may mention that only a few days 
since I was conversing with a most distinguished mathematical professor, and 
he told me he was engaged in preparing a geometry which should be sound 
in its logic. He said that it was not till called upon to teach Euclid to others 
as it had been taught to himself, that he learned how very faulty and illogical 
that method had been. But the metaphysical difaculties of plane geometry 
sink into utter insignificance when compared with those of the higher algebra 
and mathematical analysis. (Hear, hear.) Here long familiarity with new 
processes and new methods of thought- continued drudgery in the mechani- 
cal combinations of symbols, by rules and methods-taken at first as true on 
the authority of your teachers, or that of men famous m the mathematical 
world : all this must be gone through before you are capable of compre- 
hending the reasonings, or mathematical logic, by which the pro ems o 
physical astronomy are proved. There may be mathematical geniuses who 
may perceive almost by intuition what costs so much tod and mental labour 
to others. But men of the average endowment of intellect must pass throug 
this course of mental drudgery with profound docility and humility, before 
they can feel competent to reason for themselves as to the truth or error of 
the demonstrations of physical astronomy. The task does not end here. 
Before his mathematical analysis can be applied to solve the motions of the 
heavenly bodies, “Laws of motion” must be accepted, which have been 
inferred, but not proved r from thousands of experiments, which can never ^e 
repeated by one man, and must be taken for granted on the faith of others. 
And after all, the grand problem' of celestial mechanism must be solved by 
methods admitted by no incompetent mathematical authority (M. Comte,, 
to be quite illogical, because of the insuperable difficulty of applying thos. 
that are considered strictly logical. Then, when you have mterpreted the equa- 
tion of the moon’s place, or that of a perturbed planet, you depend upon the 
