Nov. 



1 1, 



1875J 



NATURE 



29 



Mr. John Hampden, but we have preserved nothing from 

 either of these paradoxers. The former has gained notice 

 in the Budi-it (we are sorry* to record the recent death of 

 another able opponent of these views, Mr. T. T. Wilkin- 

 son, F.R.A.S.) ; the latter has figured before the public in 

 the daily papers. A consequence of Mr. Wallace's accept- 

 ance of Mr. Hampden's wager is that the former genUe- 

 man has for nearly five years been the subject of con- 

 tinuous libels (see letters in Daily A'ews, March 11, also 

 March 9). It is to be hoped that an enforced retirement 

 of a twelvemonth will result in Mr. Hampden's learning 

 wisdom and the keeping of the peace towards Mr. Wallace 

 and all others. 



In De Morgan's account of Taylor the Platonist {B of 

 P. pp. 182, &c.) there is nothing said of an early work 

 of his, " The Elements of a new method of Reasoning in 

 Geometr}- applied to the rectification of the Circle" (1780), 

 " a juvenilq. performance lost or suppressed " (biographer 

 in Penny Cyclopadici). We have examined this work, 

 but it is impossible to give an account of it here ; the solu- 

 tion is approximative. 



The malady {Mains cyclometricus) is not confined to the 

 Old World ; our concluding instances will be drawn from a 

 Geometry published at New York, and from a treatise 

 specially devoted to the subject and printed at Montreal. 

 We have not a copy of Mr. Lawrence S. Benson's " The 

 Elements of Euclid and Legendre, with Elements of 

 Plane and Spherical Trigonometry," but he has sent us 

 " A Reply to Criticisms on Benson's Geometry'." This 

 will answer our purpose better, for the defence shows 

 that the malady is confirmed. The s>Tnptoms are even 

 more exaggerated than in Mr. J. Smith's case, for whereas 

 his circumference ("Budget," p. 318) shrank into exactly 

 3j times his diameter, Mr. Benson's has shrunk to only 

 3 times ! Wliere all this will end if the malady increases 

 it is hard to say ; perhaps the unfortunate circle will 

 shrink up into its own centre ! Opponents had pointed 

 out " that when the areas of polygons inscribed in the 

 circle are computed by means of plane triangles, a result 

 is obtained for the inscribed polygons greater than 3R-/' 

 and they reasoned (it seems to us irresistibly) " that it is 

 impossible for a circle to be less than a figure inscribed 

 in the circle." Mr. Benson trusts, however, that after 

 fourteen years' application to mathematics he will not be 

 thought to have committed so egregious a blunder as to 

 bring himself into direct contradiction of the self-evident 

 proposition, " A part is less than the whole." He com- 

 mences his defence with the statement that ToreUi con- 

 tends that the circle will be proved to be the square on 

 its diameter exactly as 3 to 4. He then goes on to instance 

 that Playfair (" Euchd," p. 307) demonstrates that Torelli's 

 proposition is true on two conditions. Is it credible that 

 Mr. Benson should proceed to say : " The fact that the 

 preposition is true ' on two conditions ' prevents the pro- 

 position from being false, for a false proposition can be 

 true on no condition." The conclusion of the whole 

 matter is that he replies to the inquiry, " How is it that 

 reasoning from plane triangles for the computation of the 

 areas of polygons, and reasoning from the ratios of rect- 

 angles, when they are all rectilinear magnitudes, that 

 different and conflicting results are obtained ? " that " the 

 reasoning on the ratios and rotation of surfaces involves 

 their relation to each other ; whereas the computation of 

 the plane triangles involves their bojindaries : and since 

 for the QUADRATURE OF THE CIRCLE the relation between 

 the circle and a certain rectajigular space is required, it is 

 evident that the proper mode of reasoning is by means 

 of the relation of the ratios of the small rectangles in- 

 scribed in the circular spaces to the ratios of the sums 

 of those rectangles, or of the whole rectiUnear figures ; or 

 by means of the rotation of rectilinear and curvilinear 

 surfaces around a common axis — and not by the process 

 of continually doubling the number of sides of the poly- 

 gons described about the circle ; since the sides do not 



reach the circumference, this process gives an approxi- 

 mate result only, which is inconsistent with the strictness 

 of geometrical reasoning." We do not profess to follow 

 the writers reasoning, but hold fast by the terra Jirma 

 which he appears to discard. 



" The Circle and Straight Line " is a work got up in an 

 elaborate and elegant dress : it consists of Parts I., II., 

 III., and a supplement in brown binding, and a duplicate 

 of the supplement in green (there is a portion of a flyleaf 

 additional in the former supplement, or else the two 

 copies appear to be identical). Further, there is with 

 each a book of plates, all most clearly dra%vn, and the 

 diagrams protected by slips of tissue paper. Evidently 

 the author, John Harris, or Kuklos, is not a needy man. 

 Let us gather from Mr. Harris's preface the object he has 

 in view. Deeming the solution of the geometrical prob- 

 lem which demonstrates the relation of the circle to the 

 straight line to be peculiarly of public importance, he 

 gives a statement of what he has done in the matter. 

 " The discovery of the solution was communicated by 

 letter, dated 29th of December, 1870, accompanied with 

 demonstration, &c., to the Astronomer Royal." There 

 was, the author admits, imperfection and error in the 

 case as then presented. The Astronomer Royal declined 

 to examine the case. In January 1873 the papers were 

 presented to the President of the Royal Society (still Sir 

 G. B. Airy ), " with a request (claim) in writing to have the 

 case judicially examined by that Society." The documents 

 were returned ; they met with a similar fate at the hands 

 of the Professors of M'GiU College. The subject is to 

 describe a circle (or circumference) equal in length to a 

 given straight line, and to draw a straight line equal in 

 length to the arc of a circle, " accompanied with demon- 

 stration that the conditions of the requisition have been 

 mathematically fulfilled. We publish our solution with 

 the distinct statement that it is essentially in strict 

 accordance with that scientific system known as Euclid's. 

 We claim to have our demonstration admitted or dis- 

 proved, and we challenge objection or adverse argument 

 on that system." We shall first convince our mathemati- 

 cal readers, on Kuklos's own simiming up (" Corollary," 

 p. 34), that he is wrong, and then, on the charitable sup- 

 position that he is willing to be convinced, point out where 

 we consider he has failed. We shall take the last sen- 

 tence of the Corollary cited above : " Wherefore, if a 

 square be inscribed in a circle, the ratio of the inscribed 

 square to the circle is the ratio of nine to ten." It will be 



seen that this gives for the value of tt, ^2-^' that is 



9 

 3*142696 ; not a very close approximation to the accepted 

 value. But, of course, in arguing with J^Ir. Harris we 

 must go over his work and point out, if possible, where 

 he has tripped. We commence witJi enunciating his 

 Theorem A. : " If an arc containing one-eighth of a circle 

 be applied upon a straight line, and from the terminal 

 extremity of the arc a perpendicular be drawn intercept- 

 ing the straight line, and if from the arc one-tenth thereof 

 be cut off, then, if the remaining arc (to wit, the arc con- 

 taining nine-tenths of the whole arc) be rolled upon the 

 straight line, the point of contact shall be the same point 

 on the straight line intercepted by the perpendicular 

 drawn from the terminal extremity of the whole arc." 

 B M, B n are taken to be the two arcs, and O, d are 

 taken to be the corresponding points to M, n, on the 

 tangent at B, also D is the foot of the perpendicular from 

 M on the same tangent. Mr. Harris's object is to show 

 that D d coincide : if they did, then we would admit that 

 he has proved his point ; but on p. 22, line 13 (all his pre- 

 vious working having been sound, though somewhat tedi- 

 ously put), he has '■'•cd" instead of CD (his cd \s,2l 

 misprint, we presume, for C^), and then easily gets to his 

 desired conclusion. We would ask him hoiu he gets 

 " C^" Again, on p. 24, third line from bottom of page, 

 we tell him that " Z> C? is one-tenth oi BO" is a cool 



