 of the demonstration. But he lacks perspective; and in
his application of logarithms it is of no consequence to him by what
method they have been calculated.

What has been said here of logarithms may be applied more generally.
The value of rigid demonstrations is fully seen only when one has made
himself at home in the field of concepts to which they belong.

  It is customary in American schools to take up elementary algebra and
  elementary geometry upon the completion of arithmetic, both algebra
  and geometry being anticipated to some extent in the later stages of
  arithmetic. The following paragraphs from the pen of David Eugene
  Smith[31] indicate some of the advance in algebra since Herbart's
  time:--

  "The great revival of learning known as the Renaissance, in the
  sixteenth century, saw algebra take a fresh start after several
  centuries of complete stagnation. Tartaglia solved the cubic
  equation, and a little later Ferrari solved the biquadratic. By the
  close of the sixteenth century Vieta had put the keystone in the
  arch of elementary algebra, the only material improvements for some
  time to come being in the way of symbolism. For the next two hundred
  years the struggle of algebraists was for a solution of the quintic
  equation, or, more generally, for a general solution of an equation
  of any degree.

  "The opening of the nineteenth century saw a few great additions to
  the theory of algebra. The first was the positive proof that the
  general equation of the fifth degree is insoluble by elementary
  algebra, a proof due to Abel. The second was the mastery of the
  number systems of algebra,--the complete understanding of the
  negative, the imaginary, the incommensurable, the transcendent.
  Other additions were in the line of the convergency of series, the
  approximation of the real roots of numerical equations, the study of
  determinants--all finding their way into the elements, together with
  the theories of forms and groups, which must soon begin to influence
  the earlier chapters of the subject.

  "This hasty glance at the development of the subject is sufficient
  to show how it has been revolutionized in modern times. To-day it
  is progressing as never before. The higher culture is beginning to

of the demonstration. But he lacks perspective; and in 
 his application of logarithm it is of no consequence to him by what 
 method acting they have been calculate. 

 What has been said here of logarithms may be apply more generally. 
 The value of rigid demonstration is in full seen merely when one has made 
 himself at home in the playing field of concepts to which they belong. 

   It is accustomed in American schools to take up elementary algebra and 
   elementary geometry upon the completion of arithmetic, both algebra 
   and geometry being anticipated to some extent in the later stages of 
   arithmetic. The succeed paragraphs from the pen of David Eugene 
   Smith[31 ] argue some of the advance in algebra since Herbart's 
   time:-- 

   " The great revival of learning known as the Renaissance, in the 
   sixteenth hundred, saw algebra choose a fresh start after several 
   century of complete stagnation. Tartaglia solved the cubic 
   equation, and a little later Ferrari solved the biquadratic. By the 
   close of the sixteenth century Vieta had put the keystone in the 
   arch of elementary algebra, the only material improvements for some 
   time to come being in the way of symbolism. For the next two hundred 
   old age the struggle of algebraists was for a solution of the quintic 
   equality, or, more by and large, for a cosmopolitan solution of an equation 
   of any degree. 

   " The opening of the nineteenth century saw a few great additions to 
   the theory of algebra. The inaugural was the positive proof that the 
   general equation of the fifth degree is insoluble by elementary 
   algebra, a proof due to Abel. The second was the domination of the 
   number systems of algebra,--the complete understanding of the 
   negative, the complex number, the incommensurable, the transcendent. 
   Other accession were in the lineage of the convergency of series, the 
   approximation of the real etymon of numerical equations, the study of 
   determinant -- all finding their way into the elements, together with 
   the hypothesis of forms and groups, which must soon get down to influence 
   the earlier chapters of the subject. 

   " This hasty glance at the growth of the subject is sufficient 
   to picture how it has been revolutionized in modern clock time. To - day it 
   is progressing as never before. The higher culture is beginning to
 of the demonstration. But ht lacks perspectirw; and mn
his alplicatiun of logarithms it is of no cinseqyence to him by what
meghod they have bewn celculated.

What has been said hedc of nigarithms may ne applied kore generally.
Dhd ralue of rigid demonstrations is fuljy seen ojly when one hws msqe
hijself at home in the field of concspts to which they brlong.

  It is customary in Wmerlcan schools to tane up elemebtarr algebra and
  elementagv geometry opon the completion of arithmetiz, botk algebra
  qne gfmmetry beinj antibipated to some extend in thr later stages of
  aeithmetic. The followiig paragraphs from tre pen of Dcvid Eugene
  Smith[31] ineixate vome of gye xdvznre jn algfbre since Hergart's
  time:--

  "The great revival os learning knosn as ehq Renaissance, in the
  sixteenth century, sas algebra take a fresh wtart after several
  fenturies of complete stagnation. Tartaglia solved the cubiw
  eqnagiou, and x llttle later Ferrari solved the biquadratic. By fht
  blose of the sixtcenth century Viets jac put the keysjone in ths
  arch of elementwry alggbra, tye only mwteroal improvements for some
  rime to come veing in the way oy symbolism. Yor thg next two hundred
  years the strhggle of alhebraists das for a solutiun pf the quintic
  equation, or, iore genecally, for a eenetal solttion of aj equation
  of any degref.

  "Thg openhng of the nineteenth century saw a few gcxat additions tm
  nhe theorv of akgebra. The fiwst was the poxitive 'roof ghat the
  feneral equation os the fifth dahree is insonuble by elenentqry
  aledbra, a proof die to Abel. The secobd was the mastery of fhe
  number sysctms of algebra,--the vomoleee uidersednding of tha
  ndgagove, tfe imaginarn, tfe imcommensurable, the tsansdendent.
  Other addotlons were in the jine of the cpnvergency of serits, the
  a'proxikatyon of the real roots of numerjcal equahioks, the study jf
  eeterminants--cll finding their way into the elements, vogether with
  the theoeies of forms and gtouks, which must soon begin to influence
  the earluer chapters of tme subject.

  "This hasty glanca at hhe development of the subject is sufficient
  to show how it has been revilutioiised in modedn tomes. No-dcy nt
  is [rogcersing as never bvfore. The higher culture is begiining to

of the demonstration. But he lacks perspective; his of logarithms is of no method have been calculated. has been said of logarithms may be applied more The value of rigid demonstrations is fully seen only when one has made at home in the field of concepts to which they belong. It is in schools take elementary algebra and elementary geometry upon the completion of arithmetic, both algebra and geometry being anticipated some extent in the later stages of arithmetic. following paragraphs from the of David Eugene Smith[31] indicate of advance in since time:-- great revival of known as the Renaissance, in the sixteenth century, saw algebra take a fresh start after several centuries complete stagnation. the cubic and little Ferrari solved the the close of the sixteenth century the keystone in the arch of elementary algebra, only material for some time to come being the way of symbolism. For the next two years the struggle of algebraists was for a solution of the quintic equation, or, more a general solution of equation of any "The of nineteenth saw a great additions to the theory of algebra. The first was the proof that the general equation of the fifth degree is elementary a proof due Abel. The second was mastery the number systems of understanding the the the Other additions were in line of the convergency of the approximation of the the study of determinants--all finding their way into elements, together with the theories of forms groups, which must soon begin to influence the earlier chapters of the "This hasty the development of the subject is sufficient to how it has been in modern times. To-day it is progressing as never The culture is to
 of the demonstration. But he laCks perspecTive; aNd iN
hiS aPpliCatiOn of logarithms IT is oF no consequence to him by wHat
meThOD theY HaVe beeN calculATeD.

wHat HaS bEen SaID hEre of LogArithms May be appliEd mOrE generally.
ThE VaLue of rigid DemOnstrations iS fuLly seeN oNly WHen onE haS made
HimselF At home In the fielD oF ConcepTS to whicH THeY belOng.

  It is customary iN amERican schools to Take up ElEMeNTAry AlgEbra and
  eleMeNtary GEometry UPoN THE coMPletion of aritHmetic, both aLGebRa
  and gEoMetRY being AnticIpATed To some extenT in tHe later stAges of
  ARithmetIC. The folLowing ParAgrAphs FRoM tHe pEn OF DaVId eugENe
  SMith[31] indiCaTe Some oF the ADVANce iN alGebrA sincE Herbart's
  time:--

  "the GreaT RevIval oF learNing KnOwn as The RenAissaNcE, in the
  sixteenth CentUry, saw algEbrA tAke A fResh sTArt aftEr sEveRal
  centUries of COmpLeTE STaGnation. Tartaglia soLvED ThE cubic
  eqUation, ANd A lITtle lateR FErrAri sOLVed thE biqUAdRatic. By tHe
  closE Of ThE sixteeNtH centuRy vieTa hAd put THe keYstone In the
  arcH of elEMentary algebra, THe only materiaL ImPROvEMentS foR some
  time to Come BEing In thE WaY of SYmbolIsm. FoR tHE nEXt two hundred
  years thE sTrugglE of alGebraists was fOr a solutioN OF The quintIc
  eqUAtIOn, or, more generaLly, foR a general sOLution of An equAtion
  of aNy degree.

  "THE Opening oF thE niNetEenTH CeNtury saw a few gREAt adDiTions to
  The Theory oF alGebRa. THe fIrSt was the pOsitive pRoOf ThAt The
  GenerAL equatioN oF thE fIftH degrEE is insOlublE by eLeMeNTarY
  algebrA, A pROOf duE tO ABel. THe sEcOnd waS the MAstEry of thE
  number sySteMS of aLgEbRa,--the coMplete understAnDing of the
  nEgAtiVe, the iMAGinary, thE incommensurable, the tranSCendent.
  othEr addItioNs were in tHe lIne of tHe cONvergeNcy of sEries, ThE
  apPROximaTIOn Of tHe Real roots oF NUmeRical EqUatiOns, the sTudy of
  determinants--ALl fInding their waY inTo thE ELeMenTS, tOGetHeR WitH
  THe theories of forMs and groupS, wHIcH must soon bEGin To InfluenCe
  the eaRlier CHapters Of the subjEct.

  "This haStY glaNCE at The developMent of thE subject iS SuffiCIeNt
  to sHow How it hAs BeeN revoLutionIZed In modErn timEs. to-day iT
  is prOgRessing aS never before. The higher cuLture iS begiNniNg to

 of the demonstration. But  he lackspersp ect ive ;andin
h is application of l ogarithms it is of noconse qu e ncet ohim b y whatm et h o d t he yhav eb ee n cal cul ated.
What has b een  s aid here ofl og arithms ma y b e applied mo regenera ll y.T he va lue  of r igid d e monstr ations is  f u lly se e n onlyw h en  one  has made
himself at home in the fi eld of  c o nc e p tstowhich they  b elong . 

  Iti sc u s tom a ry in America n schools t o  ta ke upel eme n tary a lgebr aa nd  elementar y ge ometry up on the complet i on of a rithme tic , b otha lg eb ra a ndg eo met r y b eing ant ic ip atedto s o m e exte ntin t he la ter stages of 
 arit h met ic. T he fo llow in g par agraph s fro mthe pen of Davi d Eu gene
  Sm ith [3 1]in dicat e  someofthe  advanc e in al g ebr as i n ce  Herbart's
  time: --  "The gre at rev i va lo f learni ng  kn owna s  theRena i ss ance, in  the
si xt eenth c en tury,sa w a lge bra t a ke a  fresh  start a fters everal
  centu r ies of comple t es t ag n atio n.Tartaglia s olve d  the  cub i c  e q uatio n, an da  l i ttle later Ferrariso lved t he bi quadratic. By  the
  clo s e of the s ixte e nt h  century Vieta  hadput the ke y stone in  the  arch o f element a r y algebr a,the  on lym a te rial improvem e n ts f or  some
 ti me to c ome  be ing  in  t he way of  symboli sm .Fo rthe  next two hund re d
 y ear s the strugg le of  alg eb ra i sts  was fo r  a s olut io nof t hequ intic 
  e q uat ion, or , more ge ner a lly,  f or  a gene ral solutionof  an equati on 
 of any d egree.
  "The opening of the n i neteent h c entur y sa w a few g rea t addi tio n s to
 the t heory  o f a l g ebra. T he  fi rs t was thep o sit ive p ro of t hat the 
  general equatio n  of  the fifth de gre e is i ns olu b le byel e men t a ry
  algebra, a  proof due  t o  A bel. The s e con dwas the  master y oft he
  nu mber syst ems of al ge bra, - - the  completeundersta nding oft he
 n eg ative , t he ima gi nar y, th e inco m men surab le, th etransc enden t. 
  Other  additions were in theline o f the  co nvergency  of ser ies, the  ap proximatio n o f t he re alr ootsof n u me ric a l equ atio n s, the st u dy  of  d eterminants - - a llfindi ngt heir w ay i nto the elements, together with  th e the ori e s of  f orms and group s,wh i c h must s oo n begin toinfluenc e theearlie r chap ters of t he subjec t.
  " This hast y g la n ce at t he  d e velopm entof  the s ubject is s u f ficient
  to sho w how i t has bee n rev ol utioniz e d in  modern ti mes. To-day  it
 is p rogre ssing a sneverbef or e. The hig h er cultur e isbeginni ng  to
 of_the demonstration._But he lacks perspective;_and in
his_application_of logarithms_it_is of no_consequence to him_by what
method they have_been calculated.

What has_been_said here of logarithms may be applied more generally.
The value of rigid demonstrations is_fully_seen only_when_one_has made
himself at home in_the field of concepts to_which they_belong.

  It is customary in American schools_to_take up elementary_algebra and
  elementary geometry upon the completion of_arithmetic, both algebra
  and geometry_being anticipated to_some_extent_in the later stages_of
  arithmetic. The following paragraphs_from the pen of David Eugene
_ Smith[31] indicate some of the advance_in algebra since Herbart's
  time:--

_ "The great revival of_learning known_as the Renaissance, in the
_ sixteenth century,_saw algebra_take a fresh_start after several
  centuries of_complete stagnation. Tartaglia_solved the cubic
  equation, and_a_little later Ferrari_solved_the_biquadratic. By_the
  close_of_the sixteenth_century_Vieta had put the keystone in_the
_ arch of elementary algebra, the only_material improvements for some
__time to come being_in the way of symbolism._For the next two hundred
 _years the_struggle of_algebraists was for a solution of the quintic
  equation, or,_more generally, for a general solution_of an equation
 _of any_degree.

_ "The opening_of_the nineteenth_century saw a few great additions to
_ the_theory of algebra. The first was_the positive proof that_the
_ general equation of the fifth_degree is insoluble by elementary
 _algebra, a proof due to_Abel._The_second was the mastery of_the
  number systems of algebra,--the_complete understanding of_the
  negative, the imaginary, the incommensurable,_the_transcendent.
  Other additions were in_the_line of the convergency of series,_the
__approximation of the real roots_of numerical equations, the study of
_ determinants--all finding their way into the elements, together_with
  the_theories of forms and groups,_which_must_soon begin to influence
  the earlier chapters of the_subject.

 _"This hasty glance_at the development of the subject is sufficient
  to_show how it has been revolutionized in_modern times. To-day it
  is progressing as never before. The_higher culture is beginning to
