I am strongly reminded of the monks in the darker part of the Middle Ages. To a certain proportion of them, perhaps two out of five, we are indebted for the preservation of literature, and their contemporaries for good teaching and mitigation of socials evils. But the remaining three were the fleas and flies and thistles and briars with whom the satirist lumps them, about a century before the Reformation: "Flen, flyys, and freris, populum domini male caedunt; Thystlis and breris crescentia gramina laedunt. Christe nolens guerras qui cuncta pace tueris, Destrue per terras breris, flen, flyys, and freris. Flen, flyys, and freris, foul falle hem thys fyften yeris, For non that her is lovit flen, flyys ne freris."[667] I should not be quite so savage with my second class. Taken together, they may be made to give useful warning to those who are engaged in learning under better auspices: aye, even useful hints; for bad things are very often only good things spoiled or misused. My plan is that of a predecessor in the time of Edward the Second: "Meum est propositum genti imperitae Artes frugi reddere melioris vitae."[668] To this end I have spoken with freedom of books as books, of opinions as opinions, of ignorance as ignorance, of {366} presumption as presumption; and of writers as I judge may be fairly inferred from what they have written. Some--to whom I am therefore under great obligation--have permitted me to enlarge my plan by assaults to which I have alluded; assaults which allow a privilege of retort, of which I have often availed myself; assaults which give my readers a right of partnership in the amusement which I myself have received. For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, with _Quousque tandem? Quamdiu nos? Nihil ne te?_[669] ending with, _In te conferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiu machinaris!_ I carry with me the reflection that I have furnished to those who need it such a magazine of warnings as they will not find elsewhere; _a signatis cavetote_:[670] and I throw back at my pursuers--_Valete, doctores sine doctrina; facite ut proxima congressu vos salvos corporibus et sanos mentibus videamus._[671] Here ends the Budget of Paradoxes. {367} * * * * * APPENDIX. I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert,[672] in the _Berlin Memoirs_ for 1761, and has been also given in the notes to Legendre's[673] Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.[674] Let a continued fraction, such as a ----- b + c ----- d + e - f + etc.,