Oct. 28, 1 875 J 



NATURE 



559 



1507 (?), and 1510-1517 editions; and also his " Gdo- 

 in^trie Practique " in the 1549 and 1555 editions ; and we 

 are disposed to think that Do Morgan (B. of P., pp. 31, 

 32) is in error, possibly in this case following Montucla 

 (for he says he has not seen the former work, and 

 he makes no mention of the second), though all the 

 copies of the " Introductorium " cited above contain the 

 De Quadratura which De Morgan states that he has 

 seen. Any how, all the constructions we have seen of 

 Bovilh's give v/io, and not 3 J. This will readily be seen 

 from the following : — Bovillus inscribes a square in a 

 circle, and then states that the quadrantal arc is equal to 

 the line drawn from an angle of the square to the middle 

 point of one of the opposite sides. In his "Gdomdtrie" 

 he says of Cusa (whose views De Morgan states him to 

 have adopted) : " II ha us^ de dimensions infinies, les- 

 quelles un gdomdtrien ne cognoist, et ne confesseroit 

 jamais estre possibles. Nonobstant, son invention est 

 bonne et approuvee, tant par raison que par experience." 

 Nor do we find any account of his quadrature agreeing 

 with that of a peasant labourer, but he states that he too 

 had attempted the problem by another method (than that 

 of Cusa), and not without success. Whilst standing on a 

 bridge at Paris he noticed the carriage-wheels passing 

 over the road ; the fact that when the wheel has per- 

 formed a revolution we have a straight line whose length 

 equals the circumference of the wheel, suggested his solu- 

 tion to him, and on his return home he easily got his 

 construction, which is this : Divide a radius of the circle 

 into four equal parts, produce this radius through a fourth 

 of its length ; join the extremity of this line with an ex- 

 tremity of the diameter at right angles to the radius, and 

 with the point as centre and this distance as radius 

 describe a circle ; the portion of the tangent at the ex- 

 tremity of the selected radius cut off by this circle, he 

 says, equals the semi-circumference. It will be seen that 

 this is the same value as that given above. Bovillus, 

 also, in a libellus de mathematicis supplementis (1509), 

 gives a third construction, which leads to the same value. 



Before leaving this writer we ought to state that he 

 attributes the first construction we have given to his 

 friend M. Achaire Barbel, a man " ingenious at new 

 inventions of use in geometry." It is with considerable 

 diffidence that we have ventured to go thus into detail, 

 but it seems to us that De Morgan had fallen into error 

 in the case of this early writer. 



We propose now to take up the subject at the point 

 where it is left in the '"' Budget," constantly regretting that 

 the hand which so vigorously lashed the offenders in this 

 line now lies cold. Here we must give place to that 

 arch circle- squarer, Mr. James Smith. We shall deal 

 tenderly, however, with his book, as we learn that he too 

 has gone over to the majority and joined his former 

 opponent. The book we have now before us is " Why 

 is Euclid unsuitable as a Text-book of Geometry ? This 

 question answered and the Propositions of Euclid 8 and 

 13, Book VI., proved to be erroneous by Heterodox 

 Geometry." (Motto — " Magna est Veritas et prasvalebit." 

 London : Simpkin, Marshall, and Co., 1871.) The editor, 

 whose name does not appear, in an address to the reader, 

 states that Geometricus, a principal correspondent in the 

 pamphlet, is " an intimate acquaintance and almost in 

 daily communication with Mr. James Smith, the well- 

 known author," &c. Geometricus became a convert to 

 Mr. Smith's views. He has no niche in the " Budget : " 

 were we not informed to the contrary, we should have 

 been disposed to say that Geometricus and Mr. James 

 Smith were one and the same person. The first fifteen 

 pages are mainly devoted to a correspondence between 

 Geometricus and' the Rev. Dr. Jones, if that can be called 

 a correspondence in which the writing on one side is 

 copious and on the other confined to simple acknowledg- 

 ments of receipts of letters. 



The doctor was singled out for this honour in conse- 



quence of his having written an able pamphlet " On the 

 unsuitableness of Euclid as a Text-book of Geometry." 

 Geometricus was delighted at the appearance of this 

 work, thinking now at last " here is a recognised mathe- 

 matician, who has got out of the groove and who can see 

 a geometrical truth by whomsoever propounded ; " but 

 alas ! he is soon disappointed, and finds that, as in Mr. 

 Smith's experience, directly a mathematician is driven 

 into a corner, he invariably gets out of it by pleading 

 pressing engagements, want of time, &c., " and so a great 

 and important scientific truth— it may be— is born to 

 blush unseen," &c. He then sends James Smith's works 

 (which we said above had converted himself), and now 

 the redoubtable champion of " tt = 3 J " himself descends 

 into the arena, and must have given the doctor a pretty 

 lively time of it, from the 13th of April to the loth of June, 

 1 87 1, as he assails him in six long letters, with diagrams, 

 occupying nearly thirty-three octavo pages of print. 

 Much of what had been written in the "Athenaeum 

 Budget of Paradoxes " is brought up and the Smithian 

 value maintained, for though this incontrovertible solu- 

 tion "may not be admitted by you or Clifford (alluding 

 to Prof. CHfford's paper ' On an unexplained contradiction 

 in Geometry,' read before the British Association), or any 

 such like mathematicians of the present age, I can afford 

 to bide my time and trust to posterity doing me justice." 



This is the main portion of the pamphlet ; there is, 

 however, occasional sparring, both on the part of Geome- 

 tricus and of Mr. Smith, with the editor of (and some 

 writers in) the Mechanics' Magazine. In an appendix a 

 correspondent recommends J. S., "now poor De Morgan 

 (who made you look so rediculous [j/^]), has departed 

 from this life, there are still some great men left — Prof. 

 Sylvester. Try him. Smith ; if you convert that gentle- 

 man to your 3j, I will give in hmnbly." Similar advice 

 is given by the same writer in a second letter. The 

 whole book is provocative of much amusement, and is 

 quite of a piece with J. Smith's previous writings. 



At the time of writing the previous remarks, we 

 were under the impression that the " Budget " had 

 exposed " Cyclometry and Circle Squaring in a Nut- 

 shell, by a member of the British Association for 

 the Advancement of Science." This we at once found 

 was not the case when the pamphlet was lent us by a 

 friend. As we have devoted sufficient attention to Mr. 

 Smith, we may shortly say that it is in octavo form, forty- 

 four pages, and contains letters written between 24th 

 October 1870 and January 1871 ; that is, immediately pre- 

 ceding the earliest dale in the work we have noticed 

 above. The correspondents are A. E. M. (is this the 

 E. M. of the " Budget " T) and S. B. J. This last is 

 another signature, we find, for the pertinacious Smith, 

 who has figured elsewhere as " Nauticus," and wherefore 

 not as " Geometricus " .? The " Budget," though it does 

 not discuss this brochure individually, has well demolished 

 it by anticipation. 



The close of the work is of a prophetical cast. " It is 

 more than sixty years ago since an astronomer of recog- 

 nised authority — who repudiated the idea that I could 

 solve the problem of ' squaring the circle ' — said to me : 

 ^ A bright day will have dawned on the astronomical 

 world if ever the exact ratio oj diameter to circumference 

 in a circle shall be discovered. The day will arrive when 

 it will be said : * In the nineteenth century of the Christian 

 era — that remarkable century of invention and discovery 

 — darkness still overshadowed the mathematical world. 

 Scientific truth is, and ever has been, a plant of slow 

 growth, but Magna est Veritas, &c.'" It is to be hoped 

 that the good man has not left his mantle behind, and 

 that " Geometricus " and he were really one and the same. 



Mr. John Davey Hailes has a place in the " Budget" 

 (pp. 339, 340). He has not, so far as we know, touched 

 upon the squaring of the circle, but possibly he is ap- 

 proaching that as the termination of his labours. We 



