November 25, 2005

• Holy Geometry

  What was “the holy geometry book” (“das heilige Geometrie-Büchlein,” p.
  10 in the Schilpp book below) that so impressed the young Albert
  Einstein?

  “At
  the age of 12 I experienced a second wonder of a totally different
  nature: in a little book dealing with Euclidian plane geometry, which
  came into my hands at the beginning of a schoolyear.  Here were
  assertions, as for example the intersection of the three altitudes of a
  triangle in one point, which– though by no means evident– could
  nevertheless be proved with such certainty that any doubt appeared to
  be out of the question.  This lucidity and certainty made an
  indescribable impression upon me.”
  

  (“Im
  Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener
  Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am
  Anfang eines
  Schuljahres in die Hand bekam.  Da waren Aussagen wie z.B. das
  Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die–
  obwohl an sich keineswegs evident– doch mit solcher Sicherheit
  bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein
  schien.  Diese Klarheit und Sicherheit machte einen
  unbeschreiblichen Eindruck auf mich.”)
  

  – Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp

  From a website by Hans-Josef Küpper:

  “Today
  it cannot be said with certainty which book is Einstein’s ‘holy
  geometry book.’  There are three different titles that come into
  question:

  Theodor Spieker,
  1890
  
  Lehrbuch der ebenen
  Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.
  

  Heinrich Borchert
  Lübsen, 1870
  
  Ausführliches
  Lehrbuch der ebenen und sphärischen Trigonometrie. Zum
  Selbstunterricht. Mit Rücksicht auf die Zwecke
  des praktischen Lebens.

  Adolf Sickenberger,
  1888
  
  Leitfaden der
  elementaren Mathematik.

  Young
  Albert Einstein owned all of these three books. The book by T.
  Spieker was given to him by Max Talmud (later: Talmey), a Jewish
  medic. The book by H. B. Lübsen was from the library of his uncle
  Jakob Einstein and the one of A. Sickenberger was from his parents.”
  

  Küpper does not state clearly his source for the geometry-book information.

  According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

  An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:

      “Throughout Einstein’s five and a half years at
  the Luitpold Gymnasium, he was taught mathematics from one or
  another edition of the separately published parts of
  Sickenberger’s Textbook of Elementary Mathematics. 
  When it first appeared in 1888 the book constituted a major
  contribution to reform pedagogy.  Sickenberger based his book on
  twenty years of experience that in his view necessarily took
  precedence over ‘theoretical doubts and systematic scruples.’  At
  the same time Sickenberger made much use of the recent
  pedagogical literature, especially that published in the pages of
  Immanuel Carl Volkmar Hoffmann’s Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht,
  the leading pedagogical mathematics journal of the day.  Following
  in the tradition of the reform movement, he sought to present
  everything in the simplest, most intuitive way possible.  He
  opposed introducing scientific rigour and higher approaches in an
  elementary text.  He emphasised that he would follow neither the
  synthesis of Euclidean geometry nor the so-called
  analytical-genetic approach.  He opted for a great deal of freedom in
  the form of presentation because he believed that a textbook was
  no more than a crutch for oral instruction.  The spoken word, in
  Sickenberger’s view, could infuse life into the dead forms of the
  printed text.  Too often, he insisted in the preface to his text,
  mathematics was seen and valued ‘as the pure science of
  reason.’  In reality, he continued, mathematics was also ‘an
  essential tool for daily work.’  In view of the practical
  dimension of mathematics Sickenberger sought most of all to
  present basic propositions clearly rather than to arrive at
  formal conciseness.   Numerous examples took the place of long,
  complicated, and boring generalities.  In addition to the
  usual rules of arithmetic Sickenberger introduced diophantine
  equations.  To solve three linear, homogeneous, first-order
  equations with three unknowns he specified determinants and
  determinant algebra.  Then he went on to quadratic equations and
  logarithms.  In the second part of his book, Sickenberger treated
  plane geometry.
       According to a biography of Einstein written
  by his step-son-in-law, Rudolf Kayser– one that the theoretical
  physicist described as ‘duly accurate’– when he was twelve years
  old Einstein fell into possession of the ‘small geometry book’
  used in the Luitpold Gymnasium before this subject was formally
  presented to him.  Einstein corroborated Kayser’s passage in
  autobiographical notes of 1949, when he described how at the age
  of twelve ‘a little book dealing with Euclidean plane geometry’
  came into his hands ‘at the beginning of a school year.’  The
  ‘lucidity and certainty’ of plane geometry according to this
  ‘holy geometry booklet’ made, Einstein wrote, ‘an indescribable
  impression on me.’  Einstein saw here what he found in other
  texts that he enjoyed: it was ‘not too particular’ in logical
  rigour but ‘made up for this by permitting the main thoughts to
  stand out clearly and synoptically.’  Upon working his way through
  this text, Einstein was then presented with one of the many
  editions of Theodor Spieker’s geometry by Max Talmey, a medical
  student at the University of Munich who dined with the Einsteins
  and who was young Einstein’s friend when Einstein was between the
  ages of ten and fifteen.  We can only infer from Einstein’s
  retrospective judgment that the first geometry book exerted an
  impact greater than that produced by Spieker’s treatment, by the
  popular science expositions of Aaron Bernstein and Ludwig Büchner
  also given to him by Talmey, or by the texts of Heinrich Borchert
  Lübsen from which Einstein had by the age of fourteen taught himself differential and integral calculus.
       Which text constituted the ‘holy geometry
  booklet’?  In his will Einstein gave ‘all his books’ to his
  long-time secretary Helen Dukas.  Present in this collection are
  three bearing the signature ‘J Einstein’: a logarithmic and
  trigonometric handbook, a textbook on analysis, and an
  introduction to infinitesimal calculus.  The signature is that of
  Einstein’s father’s brother Jakob, a business partner and member
  of Einstein’s household in Ulm and Munich.  He presented the books
  to his nephew Albert.  A fourth book in Miss Dukas’s collection,
  which does not bear Jakob Einstein’s name, is the second part of
  a textbook on geometry, a work of astronomer Eduard Heis’s which
  was rewritten after his death by the Cologne schoolteacher Thomas
  Joseph Eschweiler.  Without offering reasons for his choice
  Banesh Hoffmann has recently identified Heis and Eschweiler’s
  text as the geometry book that made such an impression on
  Einstein.  Yet, assuming that Kayser’s unambiguous reporting
  is correct, it is far more likely that the geometrical part of
  Sickenberger’s text was what Einstein referred to in his
  autobiographical notes.  Sickenberger’s exposition was published
  seven years after that of Heis and Eschweiler, and unlike the
  latter it appeared with a Munich press.  Because it was used in
  the Luitpold Gymnasium, copies would have been readily available
  to Uncle Jakob or to whoever first acquainted Einstein with
  Euclidean geometry.”
  

  What might be the modern version of a “holy geometry book”?

  I suggest the following,

  first published in 1940:

  

  Click on picture for details.

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