PHYSICAL AND MATHEMATICAL SCIENCES. 99

figures which the accuracy of our calculation may de-mand : a problem undistinguished in this respect byany peculiarity, but which stands only as one amongmany others of the same class, and in the same pre-dicament. And yet, without any conceivable practicalobject-in view, it has in all ages, since its first introduc-tion, obtained an unexampled celebrity even among themost ignorant classes ; and has called up a host of in-vestigators who have devoted themselves to its solutionwith an ardour absolutely bordering on insanity; andwhich in many instances, if it did not commence, ac-tually terminated in mental aberration. Entirely igno-rant of the real object to which such an enquiry oughtto be directed, they have pursued their pretended solu-tions by almost any path except that to which the veryconditions of the problem restrict it, viz. strictly geo-metrical demonstration by elementary methods. Thatto attain this is absolutely impossible, we have everydegree of assurance short of actual demonstration. Itcertainly has not been demonstrated that the circum-ference and diameter of a circle are incommensurable ;but every argument from analogy and probability leadsus to believe it. By other methods than those of strictgeometry the problem is solved, and the pretended me-thods are therefore superfluous.

We shall see at a future stage of our enquiry howmuch good has been done indirectly by those discussions ;in that, from the agitation of the question, real mathe-maticians have been led to examine the various formsand laws of the series in which the approximations tothese and other values of the same kind may be carriedon. But the fame which the problem has acquiredamong ignorant empirics has probably arisen from someconnection it was supposed to have with we know notwhat mystical secrets; hence the persevering diligencewith which they have been always going on proposingone after another, every variety of childish conceitsdignified with the name of demonstrations : some mea-suring the length of the line gone over during one re-h 2