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Chandler—Transcendental Space. 
After carefully following the Non-Euclidian lines of reasoning, 
and recognizing their logical exactness, until their very rigid¬ 
ity seems to be a guaranty of a real subject of investigation, 
after all the question persistently returns, “ What does it all 
mean, or what real elements has it ? ” 
If, however, the question be asked whether mathematics has 
any right to be really transcendental, that is to deal with that 
which is beyond mental experience, the questioner may be re¬ 
minded of the historical illustrations before mentioned, showing 
that by advances into the unknown it has become the known. 
But were not these advances essentially different from those of 
the Non-Euclidians ? Negative quantities persistently presented 
themselves without invitation, and the advances made by Des¬ 
cartes and Harriot and others involved no introduction of new 
members to the mathematical state, but a bestowal of real in¬ 
telligible rights upon supposed aliens already present. The 
same was true of “ imaginary quantities ” and the body of 
allied facts. Hamilton’s extension of the powers of multiplica¬ 
tion introduced no new existences. It merely made clear the 
existing relations between conceptions already recognized. If 
we follow the Theory of Functions of a Complex Variable until 
the magnitudes represented become too complex for our clear 
comprehension, they seem none the less real; and we may still 
see that the obscurity is due merely to an increasing com¬ 
plexity differing only in degree from that which we have 
already fully grasped, and from which as a stand point we trust 
we may be able to gain clear vision of that which is still beyond 
our sight. 
But our relations to transcendental spaces are so essentially 
different from this that a belief in their existence comes near 
being a true act of faith, almost sublime in its completeness, 
reverently maintaining “ Thus say the formulae. We cannot 
understand; but we believe. ” Of course any fragments of ex¬ 
perimental evidence must be given due weight. Even though 
space of four dimensions is utterly inconceivable, yet, if cer. 
tain so-called spiritualistic phenomena persist in defying all 
other explanation, to whatever extent the fourth dimension 
makes otherwise irreconcilable facts consistent, by just so 
