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Squaring the Circle ” Is Roughly That of Constructing a Square of Which the Area Is T Equal to That Enclosed by Th E Circle

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Squaring the Circle ” Is Roughly That of Constructing a Square of Which the Area Is T Equal to That Enclosed by Th E Circle PR E FACE N the Easter Term of th e present year I deli vered a sh or t course of six Professorial Lectures on the history of the problem of the in n quadrature of the circle , the hope that a short accou t of the fortun es of this celebrated problem might n ot only prove interestin g m in itself, but ight also act as a stimulant of in terest i n th e more t M m h . as me general his ory of athe atics It occurred to that, by the n L t i n publicatio of the ectures , they migh perhaps be of use, the m M sa e way, to a larger circle of students of athematics . Th e account of the problem here given is n ot the result of any f n m independent historical research, but the acts have been take fro the writings of those authors who have investigated various parts of the history of the problem. The works to which I am most indebted are th e very interesting “ A m n m P . R L book by rof F udio entitled rchi edes , Huyge s , a bert, ” L n A n n fi K n L ege dre . Vier bha dlunge ber die reismessu g ( eipzig ’ “ Th A m L. e and Sir T . Heath s treatise works of rchi edes ’ “ C m I m C n G ( a bridge , have also ade use of a tor s eschichte der ’ ” M m Vahlen s K n n n u n d A r oximat- athe atik , of o struktio e pp ionen ’ “ L Y Mikami s ( eipzig , of oshio treatise The development of M C L n athematics in hina and Japan ( eipzig, of the tra slation ’ “ T Mc r mack 1 . h M m . Co C 898 by J ( hicago , ) of H Sc ubert s athe atical ” “ R n an d an d Essays and ecreatio s , of the article The history trans ” 1 r m D . i n cen den ce of written by Prof. E S ith which appeared the “ M t P on n . A . Monographs Moder athematics edi ed by rof J W . Y . On n n oung special poi ts I have co sulted various other writings . W E . H . ’ ' CHR Isr s O LLE G E A M B R ID GE . C , C r 1 3 . Octo be , 1 9 CHAP TER I G E N E RA L A CCOU N T O F THE P RO BLE M A GE NE RAL survey of th e history of thought reveals to u s the fact of the existen ce of various questions that have occupied the almost contin uous attention of the thin kin g part of mankind for lon g series of n ce turies . Certain fundamental questions presented themselves to the human mind at the dawn of the history of speculative thought, an d h n n h have maintained t eir substa tial ide tity t roughout the centuries , although the precise terms in which such questions have been state d have varied from age to age in accordance with the ever varyin g k d m n a . ener al attitude of man ind towards fun a e t ls In g , it may be n t n maintai ed hat, to such questio s, even after thousands of years of n n o n n discussio , answers have bee give that have permanen tly satisfied n the thi king world, or that have been generall y accepted as final n h solutions of the matters concer ed . It as been said that those problems that have the longest history are the insoluble ones . If the contemplation of th is kin d of relative failure of the efforts of m t the hu an mind is calculated to produce a cer ain sense of depression, a it may be a relief to turn to cert in problems, albeit in a more restr icted u th e m domain, that have occ pied inds of men for thousands of years, n but which have at last, in the course of the nineteenth ce tury, received solutions that we have reason s of overwhelming co gency to in as . regard final Success, even a comparatively limited field, is some compensation for failure in a wider field of endeavour . Our legitimate satisfaction at such exceptional success is but slightly qualified by the fact that the answers ultimately reached are in a certain sense of a negative character. We may rest co ntented with a proofs th t these problems, in their original somewhat narrow form , n as a are insoluble, provided we attai , is actu lly the case in some e n n celebrat d i sta ces, to a complete comprehension of the grounds , l i restin g u pon a thorough y establ shed theoretical basis, upon which t our final conviction of the insolubili y of the problems is founded . H . 2 GENERAL ACCO UNT O F T HE PRO BLEM Th e three celebrated problems of the quadrature of the circle, the an trisection of angle , and the duplication of the cube, although all of them are somewhat special in character , have one great advantage for mix. the purposes of historical study, that their complete history as m u s . scientific proble s lies , in a completed form , before Taking the m first of these proble s, which will be here our special subject of study, n n n in m we possess i dicatio s of its origi re ote antiquity, we are able to follow the lin es on which the treatment of the problem proceeded and chan ged from age to age i n accordance with the progressive develop n M m o n x me t of general athe atical Science, which it e ercised a noticeable reaction . We are also able to see how the progress of en deavours towards a solution was affected by the in tervention of some M n of the greatest athematical thi kers that the world has seen, such m n A e . L as rchimedes , Huyghens , Euler, and Hermite astly, we know when and h ow the resources of modern Mathematical Science became sufficien tly powerful to make possible that resolution of the n th problem which, although egative, in that the impossibility of e m to n was proble subject the implied restrictio s proved, is far from being m n n n a ere egatio , in that the true grou ds of the impossibility have been set forth with a fin ality and completeness which is somewhat i n rare the history of Science . he an n If the question raised, why such appare tly special problem, as that of the quadrature of the circle, is deserving of th e sustain ed n an d i terest which has attached to it, which it still possesses, the answer is only to be found in a scrutiny of the history of the an d problem, especially in the closeness of the con nection of that M history with the general history of athematical Science. It would ffi m be di cult to select another special proble , an account of the history of which would afford so good an opportunity of obtaining a glimpse of so many of the main phases of the development of general Mathe an d n maties ; it is for that reason, eve more than on accoun t of the n n i tri sic interest of the problem , that I have selected it as appropriate for treatment in a short course of lectures . A n the part from , and alo gside of, scientific history of the problem , has t n an d it a his ory of another ki d, due to the fact that, at all times , as m r almost much at the present ti e as fo merly, it has attracted the n atte tion of a class of persons who have, usually with a very inadequate equipmen t of knowledge of the true nature of the problem or of its r n t histo y , devoted their attentio to it, of en with passionate enthusiasm . n th Such perso s have very frequently maintained , in e face of all efforts G E N E R AL ACCO U NT o r THE PRO BLEM 3 at refutation made by genuine Mathematicians , that they had obtained n a solutio of the problem . The solutions propounded by the circle x squarer e hibit every grade of skill , varying from the most futile i n an attempts, which the writers shew utter lack of power to reason x n t u correctly, up to appro imate solutions the co s r ction of which n n n n required much i ge uity o the part of their inventor . I some cases it requires an effort of sustained attention to find out the precise in the m n i n an point de o stration at which the error occurs, or which approximate determination is made to do duty for a theoreticall y exact o ne . Th e psychology of the scien tific cran k is a subject with which the officials of every Scien tific Society have some practical n i acquai tance . Every Sc en tific Society still receives from time to time communications from the circle squarer an d the trisector of m n m angles , who often make a usi g atte pts to disguise the real n character of their essays .
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