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ment is more or less beneficial according as one or other effect predominates in it.

Mathematics may be loved by a sybarite, but it will never be his occupation. Yet it is a most gentlemanly amusement, pursued far from vulgar passions and applause, in the calm of a library, nothing sordid mingling with it. Those harsh and crabbed formulas which fill us outsiders with such dismay, and from which we flee in terror, doubtless become, after a while, to the devotee as musical as is Apollo's lute, and are regarded by him as affectionately as a chess-player regards his finely-carved queens and castles.

Everybody, of course, knows that the practical utility, not to say necessity, even of the highest mathematics, is immense. But we have preferred to consider it in relation to the purely theoretical or very remotely practical departments which will no doubt occupy a large portion of the American Journal of Mathematics. Even so considered, the present mathematical revival which the coming of Mr. Sylvester has occasioned, and the establishment of this journal, may be characterized as important events for the highest welfare of the country—important even for its material welfare. It is not to be expected that the actual readers of such a journal should be very numerous, and, though subscriptions will be drawn from Europe, yet the success of it must depend upon its finding subscribers who do not read it, but who appreciate its value. Whether a sufficient class exists in America which understands the importance of intellectual activity of the profoundest kind to enable such activity to exist here, is a vital question for our destiny.

Not to close this notice without a bit of criticism, we may mention that one of the editors of the Journal has by means of a "Crelle's Table" found in a few minutes that the number 191,071 is either a prime or else divisible by a factor less than 137. This is pronounced by another of the editors to be "a real stroke of genius." We should like to have the glory of it a little further elucidated. We have put Crelle's Table into the hands of a computer, and requested him to find the divisors of the number in question. Having no pretensions to genius, he did not stop at 137, but, proceeding in the ordinary humdrum way, announced absolutely, in a quarter of an hour, that the number was prime. We confidently look for an article in the coming number setting out and explaining the wonderful stroke of genius in question, the marvel of which does not seem to have lain in its reaching a very speedy or complete solution of the problem undertaken.

Democracy in Europe: A History. By Sir Thomas Erskine May. New York: W. J. Widdleton. 1878. 2 vols.

In an appropriate introductory essay to this work the author sets forth the principles that constitute its foundation. It can scarcely be said that he offers anything new here; he, however, points out truths that can never be too sedulously insisted upon. After stating that by democracy he understands "the political power or influence of the people under all forms of government," that "it denotes a principle or force and not simply an institution," he discusses the moral, social, and physical causes of freedom, supporting his assertions, the while, with historical instances. He next shows that, as the constant development of popular power is the result of the intellectual and material progress of a nation, it must be accepted as a natural law; and that, instead of striving to breast the current, statesmen should endeavor to urge it forward in the most advisable channels. As a matter of course, the author has not avoided the opportunity of commenting upon the excesses of democratic ideas, and of breaking lances with devotees of communism and socialism.

Sir Erskine May enters the historical field with a gloomy portraiture of the constitutions of the old Eastern nations, and, in order to show that freedom is a growth wholly peculiar to European soil, he endeavors to prove that in all times—even the present—the main characteristics of Eastern society have been immutability and immobility. He begins with an examination of the constitution of India, and claims that the Hindoos have never known freedom; that their ignorance has been opposed to it; that their enslavement has fostered their ignorance; and concludes that "England has already given more 'liberty' to India than ever she aspired to under her former rulers." In the histories of Persia, China, Japan,