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Perhaps no subject has given occasion for so much unsound reasoning as the Calculus of Variations. Mr. Todhunter's learning in the subject is well known to all who have examined his very useful history. But I think that he fails to detect the faults in principle which generally disfigure the writings of mathematicians on this subject, and that in these researches he makes the errors of previous writers his own, adding original investigations nearly as faulty as what is not original. Nevertheless, the book has many merits. It is very suggestive, and the felicity of expression and clearness of illustration remind the student of Mr. Todhunter's educational works. But I maintain that unsound argument in a mathematical work is a fault so very great, that no beauty of style or other merit can compensate for it. A false theory is refuted when a single case of failure is proved. I shall state and illustrate a few elementary principles, and shew, both that in Mr. Todhunter's Researches these principles are contravened, and that what I object to is inconsistent with positions maintained by Mr. Todhunter elsewhere. -The Messenger of Mathematics

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