[July 1894]
"When two reviewers hold such opposite opinions, and express their
views in a manner so radically different, as Professor Halsted and
the writer do concerning Cajori's History of Mathematics, the
question becomes merely a matter for the reader to decide. It is
submitted that Dr. Halsted's statements invalidate in no single
point the review published in the May number.
As regards the first objection, the works of Ball and Fink are
not the only, nor even the chief, ones referred to, as Dr.
Halsted's article seems to imply. Indeed the quotation from Fink
is rather to illustrate the faulty translation, that author not
being mentioned among those from whom Professor Cajori has chiefly
drawn. The works of Gow, Hankel, and Cantor, which will probably
be admitted to be "better" works, are the ones to which, with
Ball, the most prominent reference is made. But the real point
of the criticism is not as to which of any two works is the better,
but as to the way in which various works have been laid under
contribution. If Professor Cajori's book contains such close
paraphrases of inferior histories, so much the worse. While it is
foreign to the real issue, the writer would not have it inferred
from any silence of his that he for a moment subscribes to the
proposition that "Cajori is superior in point of actual accuracy"
to the second edition of Ball.
As to the second point, it is not clear why it should have
interfered with "the popular character and readableness of his
book" if such a bibliographical reference [...] had been made
more explicit, or if references had been given whenever an extract
was made. Can any one justly affirm that the references as they
stand are of any material value? Replying further to Professor
Halsted, it is reasserted that the valuable _articles_ (as
distinguished from the valuable _article_ mentioned) in Smith's
Dictionary of Biography are worthy of mention. That the author
should have "waited to obtain and study all the works of all the
men" mentioned was not asserted. The criticism distinctly refers
to the "sweeping omission" -- to the fact that not one of them
was consulted. But surely Professor Halsted, who has himself done
so much for the history of mathematics in this country, will not
contend but that every advanced scholar in this line is familiar
with substantially every author named in the general list; and the
professor must admit that one like himself could easily, with no
hesitation, add several well-known names, and many of lesser note,
to those mentioned.
The error pointed out in the _Educational Review_ can hardly, save
on the principle of _ipse dixit_, be called "the most serious error
in Cajori's book". It is, however, quite pardonable for Professor
Halsted to be partial to that article; he wrote it (Aha!, JGC). As
to the article in the _Nation_ that was a charming essay, but it
was hardly a serious review of Cajori.
Since with the exception of a few pages in Ball we have no similar
attempt at a synopsis of the whole of modern mathematical history
up to the present time, it is quite safe to make the sweeping
assertion that this portion of the work is "without a rival in the
world, in any language". Nevertheless one may close the final
chapters with disappointment.
Having no idea of the meaning of Professor Halsted's reference to
Chasles' Christian name, or of his statement "30 is perhaps a slip
for 300", the writer ventures to pass them by. He also ventures to
reassert his appreciation of the work under discussion as a popular
exposition of the historical advance of mathematical science, as set
forth in the closing paragraph of his review.
David Eugene Smith."