Jean Leurechon’s Mathematical Recreations in the Context of Early 17th Century Europe

    1   Recreations​ in the Context of Early 17th Century Europe   ​ Jean Leurechon’s Mathematical   European thinkers in the 17th century revolutionized humanity’s understanding of nature.   Differing from the humanists who focused largely on works from long ago, the new wave of   scientists looked ahead to great inventions based on discoveries  about the world. Breaking away   from the philosophy of Aristotle, thinkers  such as Francis Bacon and René Descartes introduced   ideas  that led to the modern scientific method and spurred the later  scientific revolution.   Astronomers such as Galileo Galilei and Johannes Kepler put these  ideas into practice, fundamentally altering humanity’s understanding of the cosmos. In the midst of this scientific   revolution was a humble book, Mathematical Recreations,​ originally published in French as   ​ Récréations  Mathématiques​ in 1624. The book was written by Jean Leurechon under the   pseudonym Hendrik Van Etten and first appeared in English print in 1633, translated by William   Oughtred1.  It was printed in octavo form, with pages in sets of 16, and measured 4 inches wide   and 6.5 inches tall. The book itself consisted of over 100 “problems,”  demonstrations of   scientifically  based tricks or facts. Importantly, Leurechon wrote Mathematical ​ Recreations​ for a   broad audience, likely hoping to expose people across Europe to scientific wonders. Analysis of   its contents illuminates not only the early state of scientific understanding, but also the political   and religious conditions of the world it helped influence.   For centuries, studying the natural world had been considered  a fringe interest, only   suitable  for isolated scholars. The early 17th century, however, saw the beginnings of change   from this view. The English philosopher and scientist Francis Bacon  was at the fore in   popularizing learning. In his famous ​The Advancement of Learning,​ published in 1605, Bacon   1 chaaf, William. "Number Game." Encyclopedia Britannica Online.  Encyclopedia Britannica. Web.  18 Apr. 2015.         2   defended learning for its own sake, especially study of the natural  world2.  He considered five   common worries: that learning is close to original sin, that learning  leads to worry and   unhappiness, that learning leads to atheism, that learning leads  to an unwillingness to fight, and   that learning makes people unfit for governing. Calmly but confidently,  he responded to each in   turn. The original sin  was the desire to abandon God, not the thirst for knowledge. Applications   of  knowledge would lead to improvements in life, not unhappiness.  While a small amount of   learning might tempt a man away from God, a thorough education would bring him back to   religion. If learning were so bad for fighting, how could Caesar  and Alexander the Great have   been educated but also great generals? Finally, Bacon argued that learning is essential for   governing  for the same reason that a doctor must understand the human body in order to be   effective. By clearly articulating and responding to a host of concerns against learning, Bacon   made learning feel more acceptable to Europeans.   The early 17th century also saw a move away from the ideas of Aristotle.  Refined into   scholasticism, Aristotle’s philosophical methods had dominated Europe for centuries3. acon   disliked the abstractness of scholasticism and argued instead for  greater reliance on data and   evidence. He asserted that this empiricism would allow for refutation  of incorrect theories, so   that nature could be studied inductively. Bacon also believed that  scientific advances could lead   to improvements for mankind, an idea that led to the later founding of the London Royal Society   in  1660. Though he never made any significant scientific discoveries  himself, Bacon’s advocacy   of empiricism and applicability was instrumental in the development  of modern science4.     Francis Bacon, ​The Advancement of Learning, ​ed. W. A. Wright (Oxford: Clarendon Press, 1880)   Pieper, Josef. "Scholasticism." Encyclopedia Britannica Online.  Encyclopedia Britannica.  Web. 20 Apr. 2015.     4  ​"Francis Bacon." ​Bio.​ A&E Television Networks, 2015. Web. 16 Apr. 2015.   2 3     3   The  Frenchman  René  Descartes  also  contributed  to  the  shift  away  from  Aristotelianism.   In  his  1605  work  ​Discourse  on  Method,​  he  cautioned  against  accepting  as  true  what  one  did  not   know  for  certain.  Descartes  also  criticized  existing  philosophy,  complaining  that  philosophers   debated  endlessly  but  agreed  on  nothing.  He  therefore  argued  that  philosophical  results  could  not   be  trusted,  and  he  proposed  a  system  of  slow,  careful  reasoning  in  order  to  find  statements  he   knew​  to  be  true.  Since  these  results  would  be  established  for  certain,  they  could  be  used  to   discover  new  truths  without  worry  or  hesitation5.  The  ideas  of  men  like  Bacon  and  Descartes   initiated  the  development  of  modern  science,  the  first  steps  of  the  European  scientific  revolution   that  would  see  advances  in  mechanics,  mathematics,  optics,  chemistry,  and  more  throughout  the   17th  century.   Mathematical  Recreations,​  published  after  Bacon’s  ​The  Advancement  of  Learning​  but   before  Descartes’  ​Discourse  on  Method,​  likely  played  a  similar  but  smaller  role  in  the  advance  of   science.  The  book  contains  a  vast  assortment  of  facts,  tricks,  and  techniques,  all  based  in  math   and  science.  These  “problems,”  as  Leurechon  called  them,  spanned  a  wide  range  of  topics  such   as  mathematics,  physics,  astronomy,  and  even  fireworks.  Unlike  Bacon  and  Descartes,   Leurechon  did  not  propose  new  systems  of  reasoning.  Rather,  he  helped  expose  Europeans  to   scientific  applications  and  tricks,  likely  increasing  general  scientific  awareness  among  the   educated.   Differing  from  the  writings  of  Bacon  and  Descartes,  ​Mathematical  Recreations​  was   intended  to  appeal  to  anybody  with  an  interest  in  science.  Leurechon  himself  described  the  book   as  “Fit  for  scholars,  students,  and  gentlemen  that  desire  to  know  the  philosophical  cause  of  many      Descartes,  René.  ​Discourse  on  Method  and  Meditations  on  First  Philosophy,​  4th  ed.  Indianapolis:  Hackett,  1998.  5.   Print.     5     4   admirable  conclusions”  (Leurechon,  ​Mathematical  Reflections,​  cover).  The   cover  appealed  to  a  wide  audience,  depicting  the  range  of  topics  touched  on   in  the  book  and  calling  them  “useful  and  recreative.”  Also  to  appeal  to  a   layperson,  Leurechon  made  a  point  of  omitting  detailed,  technical   explanations  for  the  problems,  stating  in  the  introductory  “By  way  of   advertisement”  section,  “Those  which  understand  the  mathematics  can   conceive  them  easily;;  others  for  the  most  part  will  content  themselves  only   ​ with  the  knowledge  of  them”  (Leurechon,  Mathematical  Reflections,​  “By   This image is in the public domain. way  of  advertisement”).  Probably  attempting  to  improve  the  casual  reader’s  experience,   Leurechon  switched  rapidly  between  topics.  Later  editions  of  the  book  also  included  a  table  of   contents,  perhaps  because  the  constant  topic-­changing  made  finding  specific  problems  difficult   otherwise.  The  reader  certainly  had  the  option  of  delving  deeper  into  the  science  behind  the   demonstration,  but  Leurechon’s  goal  seems  to  have  been  to  impress  rather  than  to  instruct.  This   appeal  to  a  wide  audience  led  to  a  commercial  success:  ​Mathematical  Recreations  ​went  through   over  30  editions  by  the  end  of  the  century6.   Handwriting  inside  the  copy  of  ​Mathematical  Recreations​  found  at  the  MIT  library  offers   another  glimpse  into  its  appeal.  We  can  tell  that  the  owner,  Thomas  Soewitt,  highly  valued  his   copy  of  the  book  because  an  early  page  prominently  displays  his  name.  Arithmetic  calculations   scribbled  on  another  page  show  he  tried  to  work  through  some  problems  for  himself.  Despite   probably  being  a  scientific  layman,  he  took  a  keen  interest  in  the  material.  It  seems  that  Thomas   Soewitt  found  reading  ​Mathematical  Recreations​  engaging  and  thoughtful  as  well  as  enjoyable.   6  Schaaf.  “Number  Game.”  Britannica.     5     This image is in the public domain. This image is in the public domain. The  book’s  introduction  also  illuminates  the  divide  at  the  time  between  science  and   humanism.  Humanism  was  an  intellectual  movement  from  the  1400’s  that  emphasized   widespread  education  and  especially  focused  on  ancient  texts.  The  humanist  ideas  still  held   strong  influence  in  Europe  in  the  early  17th  century,  and  many  prominent  scientists  were   humanists  as  well.  The  famous  astronomer  Johannes  Kepler,  also  dabbled  in  poetry  and  history,   subjects  championed  by  the  humanists7.  Despite  this,  humanists  and  scientists  took   fundamentally  different  approaches  in  their  studies.  Humanists  studied  the  past,  striving  to   improve  society  and  restore  the  glory  of  the  ancient  world.  Scientists,  by  contrast,  sought  to   invent  and  discover  machines  and  ideas  that  were  fundamentally  new.   Rather  than  take  a  moderate  stance,  Leurechon  made  it  clear  that  he  prefered  science  to   humanism.  In  his  “Epistle  to  the  Reader”  he  described  ​Mathematical  Recreations​  as  “an   invitation  and  motive  to  the  search  of  greater  matters,  and  to  employ  the  mind  in  useful   knowledge,  rather  than  to  be  bullied  in  vain  pamphlets,  play-­books,  fruitless  legends,  and   prodigious  histories  that  are  invented  out  of  fancy,  which  abuse  many  noble  spirits,  dull  their    Anthony  Grafton.  ​Defenders  of  the  Text.​  Cambridge:  Harvard  University  Press.   7     6   wits,  and  alienate  their  thoughts  from  laudable  and  honourable  studies.”  (Leurechon,   Mathematical  Reflections,​  “Epistle  to  the  Reader”)  Leurechon  scathingly  dismissed  a  great  deal   of  humanist  study  as  a  waste  of  time  and  brainpower,  asserting  the  superiority  of  “useful”   scientific  study.  His  disdain  for  these  long-­entrenched  ideas  is  similar  to  that  of  Descartes,  who   in  his  ​Discourse​  expressed  disenchantment  with  many  existing  fields  of  study.     In  the  book,  the  problems  themselves  span  a  variety  of  topics.  Many  problems  consist  of   practical,  useful  tricks  based  in  science.  Problem  48  explains  how  to  measure  the  weight  of  an   unfamiliar  object  using  a  small  set  of  known  weights  (Leurechon,  ​Mathematical  Reflections,​  71).   The  method  uses  the  base-­3  number  system  and  would  likely  have  been  useful  to  merchants.   Problem  50  consists  of  a  trick  for  lifting  bottles:  one  needs  simply  to  bend  a  straw,  insert  it  into   the  bottle,  and  pull  up  (Leurechon,  ​Mathematical  Reflections,​  74).  The  potentially  very  useful   problem  124  showcases  a  trick  to  improve  candle  lifetimes.  The  idea  is  to  float  lit  candles  on   water.  The  water  absorbs  the  fire’s  heat,  slowing  the  wax’s  melting  (Leurechon,  ​Mathematical   Reflections,​  259).  Demonstrations  like  these  would  have  been  especially  impressive  to  a   layperson,  and  helped  make  science  more  relevant  to  the  public.   Other  problems  were  likely  meant  as  mere  curiosities  without  much  significance  at  all.   Problem  1  is  a  collection  of  arithmetic  tricks.  The  first  such  trick,  for  example,  can  be  rephrased   in  modern  mathematical  language  as  the  identity   4x+2y 2 − y = 2x  (Leurechon,  ​Mathematical   Reflections,​  21).  Problem  122  simply  asserts  that  a  heated  anvil  can  be  destroyed  with  a  gunshot   (Leurechon,  ​Mathematical  Reflections,​  259).  Problem  97  points  out  the  fact  that  on  the  surface  of   a  sphere,  triangles  can  have  3  right  angles,  even  though  this  is  impossible  on  a  flat  surface   (Leurechon,  ​Mathematical  Reflections,​  234).  This  last  problem,  interestingly,  is  actually  highly       7   significant  from  today’s  perspective:  the  development  of  non-­Euclidean  geometry  in  the  19th   century  made  such  examples  of  lines  on  curved  surfaces  fundamentally  important8 .   Several  of  Leurechon’s  problems,  seen  today,  illuminate  the  fledgling  state  of  scientific   knowledge  at  the  time.  For  example,  in  problem  13,  Leurechon  explains  that  to  find  the  mass  of   the  smoke  given  off  in  a  fire,  all  one  must  do  is  measure  the  lost  mass  in  wood;;  the  mass  of  the   smoke  produced  must  equal  the  mass  of  the  wood  consumed  (Leurechon,  ​Mathematical   Reflections,​  27).  Despite  the  appealing  logic,  this  reasoning  is  invalid  because  of  the  oxygen  in   the  air,  which  contributes  to  the  reaction  of  the  burning  and  must  be  accounted  for  in  any  sort  of   calculation.  Though  it  seems  incredible  today,  scientists  at  the  time  had  no  concept  of  oxygen;;   until  over  a  century  later,  scientists  believed  that  combustible  material  contained  a  special   element,  ​phlogiston,​  that  allowed  for  burning9.  Problem  130  appears  today  as  silly  as  alchemy  or   witch-­craft.  Explaining  a  method  to  harden  metal,  it  begins,  “Quench  your  blade  or  other   instrument  seven  times  in  the  blood  of  a  male  hog  mixed  with  goose-­grease”  (Leurechon,   Mathematical  Reflections,​  263).  Such  a  procedure  seems  ridiculous  given  modern  knowledge,   but  would  have  been  seen  as  scientific  in  Leurechon’s  day.   The  politics  and  religion  of  the  time  also  affected  Leurechon’s  book,  in  particular  his   writing  on  astronomy.  For  centuries  before,  the  accepted  and  Church-­supported  position  had   been  that  the  universe  was  centered  at  the  Earth,  with  the  Sun  and  planets  orbiting  it  in  perfect   spheres.  As  early  as  1543,  Copernicus  challenged  this  idea  with  a  heliocentric  model,  arguing   that  the  Earth  revolved  around  the  Sun,  which  was  itself  stationary10.  Though  little  attention  was   8    Henderson,  David.  "Non-­Euclidean  Geometry."  Encyclopedia  Britannica  Online.  Encyclopedia  Britannica.  Web.  18   Apr.  2015.     9  "Phlogiston."  Encyclopedia  Britannica  Online.  Encyclopedia  Britannica.  Web.  18  Apr.  2015.     10    Ross,  James,  and  Mary  McLaughlin.  "The  Revolutions  of  the  Celestial  Spheres."  The  Portable  Renaissance  Reader.   New  York:  Viking.  583-­593.  Print.         8   given  to  Copernicus  at  first,  further  developments  began  to  appear  by  the  end  of  the  1500’s.  The   Danish  astronomer  Tycho  Brahe  proposed  an  intermediate  model  called  a  geo-­heliocentric   system;;  in  his  theory,  the  Moon  and  the  Sun  orbited  Earth  while  the  other  planets  orbited  the  Sun 11 .  The  Italian  Galileo  Galilei  used  a  powerful  telescope  to  observe  Venus  and  the  moons  of   Jupiter.  His  findings,  published  in  the  1610  ​Starry  Messenger,​  were  inconsistent  with   geocentrism  and  lent  support  to  Copernicus’s  heliocentric  model12.  Brahe’s  student  Johannes   Kepler  further  challenged  the  Church’s  position  with  his  explanation  of  the  strange,  sometimes   backwards  motion  of  Mars  in  the  sky.  He  suggested,  correctly,  that  this  unusual  orbit  could  be   explained  by  a  heliocentric  model  in  which  planetary  orbits  were  elliptical  rather  than  perfectly   circular13.   The  Church  responded  severely  to  these  dissenting  astronomical  views.  In  1616,  a   committee  brought  together  by  the  Church  declared  that  heliocentrism  was  “foolish  and  absurd   in  philosophy,  and  formally  heretical  since  it  explicitly  contradicts  in  many  places  the  sense  of   Holy  Scripture.”  The  Church  also  personally  warned  Galileo  to  stop  defending  heliocentrism.   Galileo  did  not  stop,  and  in  fact  published  ​Dialogue  concerning  the  two  chief  world  systems   which  subtly  mocked  both  geocentrism  and  the  Pope  himself.  Infuriated,  the  Church  brought   Galileo  before  the  Inquisition  and  threatened  him  with  torture.  Forced  to  recant,  Galileo  lived   under  house  arrest  until  his  death14.  Heliocentrism,  though  scientifically  promising,  had  become   perilous  to  support.   11    Hatch,  Robert.  "Early  Geo-­Heliocentric  Models."  The  Scientific  Revolution  Webpage.  Web.  19  Apr.  2015.        Wudka,  Jose.  "Galileo  and  the  Inquisition."  Web.  17  Apr.  2015.   13    "Johannes  Kepler:  The  Laws  of  Planetary  Motion."  Web.  20  Apr.  2015.   14  Wudka.  “Galileo”     12     9   Leurechon  touches  on  astronomy  in  problem  88.  The  problem,  broken  into  8  parts,   attempts  to  convey  the  enormity  of  several  astronomical  distances  and  speeds.  In  the  second  part,   he  asserts  that  “if  a  bird  should  fly  round  about  it  [Earth]  in  two  days,  then  must  the  motion  be   450  miles  in  an  hour”  (Leurechon,  ​Mathematical  Reflections,​  221).  By  expressing  such  a  vast   distance  as  the  Earth’s  circumference  in  terms  of  more  familiar  quantities,  Leurechon  hopes  to   impress  the  reader  with  the  vast  scale  of  the  world.  In  other  parts  of  the  same  problem,  he   explains  the  huge  distances  to  the  Sun  and  the  Moon  in  similar  terms.   In  part  7  of  problem  88,  Leurechon  describes  the  speed  of  the  Sun,  giving  us  an  indirect   glimpse  of  the  Church’s  influence  on  astronomy.  He  writes  that  “[The  Sun]  moves  more  than   seven  thousand  five  hundred  and  seventy  miles  in  one  minute  of  time”  (Leurechon,   Mathematical  Reflections,​  221).  By  taking  the  Sun’s  motion  for  granted,  Leurechon  is  siding   with  the  Church’s  geocentrism.  Later  Leurechon  dismisses  Copernicus’s  heliocentric  theory  as   well-­intentioned  but  misguided,  “This  made  Copernicus,  not  unadvisedly,  to  attribute  this   motion…to  the  earth…for  it  is  beyond  human  sense  to  apprehend  or  conceive  the  rapture  and   violence  of  that  motion  being  quicker  than  thought”  (Leurechon,  ​Mathematical  Reflections,​  222).   Being  well-­versed  in  science,  he  would  have  been  aware  of  the  substantial  evidence  for  a   heliocentric  model,  but  likely  took  the  geocentric  position  to  avoid  the  Church’s  ire.  The  contrast   with  Leurechon’s  dismissal  of  history  and  other  subjects  in  the  introduction  further  underlines   the  strength  the  Church’s  influence  must  have  had;;  Leurechon  has  demonstrated  that  he  is  not   afraid  to  anger  historians,  but  he  still  feebly  submits  to  the  Church’s  threats.  We  can  imagine  that   the  Church’s  censorship  had  similar  effects  on  a  great  deal  of  other  scientific  writing  at  the  time.       10   Despite  these  quirks,  Leurechon’s  exposition  is  generally  solid,  revealing  a  broad   understanding  of  scientific  and  mathematical  principles.  In  part  19  of  problem  85,  Leurechon   demonstrates  an  elegant  trick  to  determine  which  of  two  fluids  is  denser,  essentially  using   Archimedes’  buoyancy  principle  (Leurechon,  ​Mathematical  Reflections,​  206).  The  idea  is  to   float  a  piece  of  wax  in  the  first  liquid,  and  then  slowly  add  lead  to  the  wax  until  the  solid  just   barely  floats.  The  resulting  solid  now  has  the  same  density  as  the  first  liquid,  so  by  submerging  it   in  the  second  liquid,  one  can  compare  the  densities.  More  theoretical,  problem  63  describes  the   ancient  yet  still  fascinating  notion  of  perfect  numbers.  Perfect  numbers  are  equal  to  the  sum  of   their  factors:  6  is  perfect  as   6 = 1 + 2 + 3 ,  while  10  is  not  because   10  ≠  1 + 2 + 5 (Leurechon,   Mathematical  Reflections,​  92).  In  these  problems  and  many  more,  Leurechon’s  presentation  is   completely  correct  and  would  have  excited  many  a  curious  reader.   Leurechon’s  ​Mathematical  Recreations​  was  only  a  small  part  of  the  scientific  revolution   that  swept  through  Europe  in  the  1600’s.  The  book  was  published  at  the  dawn  of  a  plethora  of   exciting  and  new  ideas  and  exposed  many  people  across  Europe  to  its  scientific  wonders.  A   product  of  its  time,  ​Mathematical  Recreations​  also  gives  us  a  glimpse  into  the  scientific  and   political  state  of  Europe  in  the  early  17th  century.