from the ,cornell ,university ,library3 ,historical ,mathematics ,monographs collection47 ,transcriber's ,note3 ,the following codes are used for characters that are not present in the character set used for this version of the book4 a a with macron 7etc47 4g g with dot above 7etc47 's s with acute accent d4 d with dot below 7etc47 d d with line below ,h7 ,h with breve below ,,the ,,hindu-,,arabic ,,numerals ,,by ,,david ,,eugene ,,smith ,,and ,,louis ,,charles ,,karpinski ,,boston ,,and ,,london ,,ginn ,,and ,,company1 ,,publishers #aiaa ,,copyright1 #aiaa1 ,,by ,,david ,,eugene ,,smith ,,and ,,louis ,,charles ,,karpinski ,,all ,,rights ,,reserved #haa4g ,,the ,,athenaeum ,,press ,,ginn ,,and ,,company 4 ,,proprietors ,,boston 4 ,u4,s4,a4 99 99 99 99 99 iii ,,preface ,so familiar are we with the numerals that bear the misleading name of ,arabic1 and so extensive is their use in ,europe and the ,americas1 that it is difficult for us to realize that their general acceptance in the transactions of commerce is a matter of only the last four centuries1 and that they are unknown to a very large part of the human race to-day4 ,it seems strange that such a labor-saving device should have struggled for nearly a thousand years after its system of place value was perfected before it replaced such crude notations as the one that the ,roman conqueror made substantially universal in ,europe4 ,such1 however1 is the case1 and there is probably no one who has not at least some slight passing interest in the story of this struggle4 ,to the mathematician and the student of civilization the interest is generally a deep one2 to the teacher of the elements of knowledge the interest may be less marked1 but nevertheless it is real2 and even the business man who makes daily use of the curious symbols by which we express the numbers of commerce1 cannot fail to have some appreciation for the story of the rise and progress of these tools of his trade4 ,this story has often been told in part1 but it is a long time since any effort has been made to bring together the fragmentary narrations and to set forth the general problem of the origin and development of these iv numerals4 ,in this little work we have attempted to state the history of these forms in small compass1 to place before the student materials for the investigation of the problems involved1 and to express as clearly as possible the results of the labors of scholars who have studied the subject in different parts of the world4 ,we have had no theory to exploit1 for the history of mathematics has seen too much of this tendency already1 but as far as possible we have weighed the testimony and have set forth what seem to be the reasonable conclusions from the evidence at hand4 ,to facilitate the work of students an index has been prepared which we hope may be serviceable4 ,in this the names of authors appear only when some use has been made of their opinions or when their works are first mentioned in full in a footnote4 ,if this work shall show more clearly the value of our number system1 and shall make the study of mathematics seem more real to the teacher and student1 and shall offer material for interesting some pupil more fully in his work with numbers1 the authors will feel that the considerable labor involved in its preparation has not been in vain4 ,we desire to acknowledge our especial indebtedness to ,professor ,alexander ,ziwet for reading all the proof1 as well as for the digest of a ,russian work1 to ,professor ,clarence ,l4 ,meader for ,sanskrit transliterations1 and to ,mr4 ,steven ,t4 ,byington for ,arabic transliterations and the scheme of pronunciation of ,oriental names1 and also our indebtedness to other scholars in ,oriental learning for information4 ,,david ,,eugene ,,smith ,,louis ,,charles ,,karpinski 99 99 99 99 99 v ,,contents ,,chapter ,,pronunciation ,,of ,,oriental ,,names vi ,i4 ,,early ,,ideas ,,of ,,their ,,origin #a ,,ii4 ,,early ,,hindu ,,forms ,,with ,,no ,,place ,,value #ab ,,iii4 ,,later ,,hindu ,,forms1 ,,with ,a ,,place ,,value #ch ,,iv4 ,,the ,,symbol ,,zero #ea ,v4 ,,the ,,question ,,of ,,the ,,introduction ,,of ,,the ,,numerals ,,into ,,europe ,,by ,,boethius #fc ,,vi4 ,,the ,,development ,,of ,,the ,,numerals ,,among ,,the ,,arabs #ia ,,vii4 ,,the ,,definite ,,introduction ,,of ,,the ,,numerals ,,into ,,europe #ii ,,viii4 ,,the ,,spread ,,of ,,the ,,numerals ,,in ,,europe #abh ,,index #aec 99 99 99 99 99 vi ,,pronunciation ,,of ,,oriental ,,names 7,s7 in ,sanskrit names and words2 7,a7 in ,arabic names and words4 ,b1 ,d1 ,f1 ,g1 ,h1 ,j1 ,l1 ,m1 ,n1 ,p1 ,,sh 7,a71 ,t1 ,,th 7,a71 ,v1 ,w1 ,x1 ,z1 as in ,english4 ,a1 7,s7 like u in but3 thus pandit1 pronounced pundit4 7,a7 like a in ask or in man4 ,a1 as in father4 ,c1 7,s7 like ch in church 7,italian c in cento74 ,d41 ,n41 ,s41 ,t41 7,s7 d1 n1 sh1 t1 made with the tip of the tongue turned up and back into the dome of the palate4 ,d41 ,s41 ,t41 ,z41 7,a7 d1 s1 t1 z1 made with the tongue spread so that the sounds are produced largely against the side teeth4 ,europeans commonly pronounce ,d41 ,n41 ,s41 ,t41 ,z41 both 7,s7 and 7,a71 as simple d1 n1 sh 7,s7 or s 7,a71 t1 z4 ,d 7,a71 like th in this4 ,e1 7,s7 as in they4 7,a7 as in bed4 4,g1 7,a7 a voiced consonant formed below the vocal cords2 its sound is compared by some to a g1 by others to a guttural r2 in ,arabic words adopted into ,english it is represented by gh 7e4g4 ghoul71 less often r 7e4g4 razzia74 ,h preceded by b1 c1 t1 t41 etc4 does not form a single sound with these letters1 but is a more or less distinct h sound following them2 cf4 the sounds in abhor1 boathook1 etc41 or1 more accurately for 7,s71 the bhoys etc4 of ,irish brogue4 ,h 7,a7 retains its consonant sound at the end of a word4 ,h41 7,a7 an unvoiced consonant formed below the vocal cords2 its sound is sometimes compared to ,german hard ch1 and may be represented by an h as strong as possible4 ,in ,arabic words adopted into ,english it is represented by h1 e4g4 in sahib1 hakeem4 ,h4 7,s7 is final consonant h1 like final h 7,a74 ,i1 as in pin4 ,i1 as in pique4 ,k1 as in kick4 ,,kh1 7,a7 the hard ch of ,scotch loch1 ,german ach1 especially of ,german as pronounced by the ,swiss4 4,m1 4,n1 7,s7 like ,french final m or n1 nasalizing the preceding vowel4 ,n41 see ,d44 ,n1 like ng in singing4 ,o1 7,s7 as in so4 7,a7 as in obey4 ,q1 7,a7 like k 7or c7 in cook2 further back in the mouth than in kick4 ,r1 7,s7 ,english r1 smooth and untrilled4 7,a7 stronger4 ,r41 7,s7 r used as vowel1 as in apron when pronounced aprn and not apern2 modern ,hindus say ri1 hence our amrita1 ,krishna1 for a-mr4ta1 ,kr4s4n4a4 ,s1 as in same4 ,s41 see ,d44 ',s1 7,s7 ,english sh 7,german sch74 ,t41 see ,d44 ,u1 as in put4 ,u1 as in rule4 ,y1 as in you4 ,z41 see ,d44 1 7,a7 a sound kindred to the spiritus lenis 7that is1 to our ears1 the mere distinct separation of a vowel from the preceding sound1 as at the beginning of a word in ,german7 and to h44 ,the is a very distinct sound in ,arabic1 but is more nearly represented by the spiritus lenis than by any sound that we can produce without much special training4 ,that is1 it should be treated as silent1 but the sounds that precede and follow it should not run together4 ,in ,arabic words adopted into ,english it is treated as silent1 e4g4 in ,arab1 amber1 ,caaba 7,arab1 anbar1 kaabah74 7,a7 ,a final long vowel is shortened before al 7'l7 or ibn 7whose i is then silent74 ,accent3 7,s7 as if ,latin2 in determining the place of the accent 4m and 4n count as consonants1 but h after another consonant does not4 7,a71 on the last syllable that contains a long vowel or a vowel followed by two consonants1 except that a final long vowel is not ordinarily accented2 if there is no long vowel nor two consecutive consonants1 the accent falls on the first syllable4 ,the words al and ibn are never accented4 99 99 99 99 99 #a ,,the ,,hindu-,,arabic ,,numerals ,,chapter ,i ,,early ,,ideas ,,of ,,their ,,origin ,it has long been recognized that the common numerals used in daily life are of comparatively recent origin4 ,the number of systems of notation employed before the ,christian era was about the same as the number of written languages1 and in some cases a single language had several systems4 ,the ,egyptians1 for example1 had three systems of writing1 with a numerical notation for each2 the ,greeks had two well-defined sets of numerals1 and the ,roman symbols for number changed more or less from century to century4 ,even to-day the number of methods of expressing numerical concepts is much greater than one would believe before making a study of the subject1 for the idea that our common numerals are universal is far from being correct4 ,it will be well1 then1 to think of the numerals that we still commonly call ,arabic1 as only one of many systems in use just before the ,christian era4 ,as it then existed the system was no better than many others1 it was of late origin1 it contained no zero1 it was cumbersome and little used1 #b and it had no particular promise4 ,not until centuries later did the system have any standing in the world of business and science2 and had the place value which now characterizes it1 and which requires a zero1 been worked out in ,greece1 we might have been using ,greek numerals to-day instead of the ones with which we are familiar4 ,of the first number forms that the world used this is not the place to speak4 ,many of them are interesting1 but none had much scientific value4 ,in ,europe the invention of notation was generally assigned to the eastern shores of the ,mediterranean until the critical period of about a century ago1--sometimes to the ,hebrews1 sometimes to the ,egyptians1 but more often to the early trading ,phoenicians4#a ,the idea that our common numerals are ,arabic in origin is not an old one4 ,the mediaeval and ,renaissance writers generally recognized them as ,indian1 and many of them expressly stated that they were of ,hindu origin4#b #c ,others argued that they were probably invented by the ,chaldeans or the ,jews because they increased in value from right to left1 an argument that would apply quite as well to the ,roman and ,greek systems1 or to any other4 ,it was1 indeed1 to the general idea of notation that many of these writers referred1 as is evident from the words of ,england's earliest arithmetical textbook-maker1 ,robert ,recorde 7c4 #aedb73 ,in that thinge all men do agree1 that the ,chaldays1 whiche fyrste inuented thys arte1 did set these figures as thei set all their letters4 for they wryte backwarde as you tearme it1 and so doo they reade4 ,and that may appeare in all ,hebrewe1 ,chaldaye and ,arabike bookes 444 where as the ,greekes1 ,latines1 and all nations of ,europe1 do wryte and reade from the lefte hand towarde the ryghte4#c ,others1 and #d among them such influential writers as ,tartaglia#d in ,italy and ,koebel#e in ,germany1 asserted the ,arabic origin of the numerals1 while still others left the matter undecided#f or simply dismissed them as barbaric4#g ,of course the ,arabs themselves never laid claim to the invention1 always recognizing their indebtedness to the ,hindus both for the numeral forms and for the distinguishing feature of place value4 ,foremost among these writers was the great master of the golden age of ,bagdad1 one of the first of the ,arab writers to collect the mathematical classics of both the ,east and the ,west1 preserving them and finally passing them on to awakening ,europe4 ,this man was ,moh4ammed the ,son of ,moses1 from ,khowarezm1 or1 more after the manner of the ,arab1 ,moh4ammed ibn ,musa al-,khowarazmi1#h a man of great #e learning and one to whom the world is much indebted for its present knowledge of algebra#i and of arithmetic4 ,of him there will often be occasion to speak2 and in the arithmetic which he wrote1 and of which ,adelhard of ,bath#aj 7c4 #aacj7 may have made the translation or paraphrase1#aa he stated distinctly that the numerals were due to the ,hindus4#ab ,this is as plainly asserted by later ,arab #f writers1 even to the present day4#ac ,indeed the phrase ilm hindi1 ,indian science1 is used by them for arithmetic1 as also the adjective hindi alone4#ad ,probably the most striking testimony from ,arabic sources is that given by the ,arabic traveler and scholar ,mohammed ibn ,ah4med1 ,abu 'l-,rih4an al-,biruni 7#igc-ajdh71 who spent many years in ,hindustan4 ,he wrote a large work on ,india1#ae one on ancient chronology1#af the ,book of the ,ciphers1 unfortunately lost1 which treated doubtless of the ,hindu art of calculating1 and was the author of numerous other works4 ,al-,biruni was a man of unusual attainments1 being versed in ,arabic1 ,persian1 ,sanskrit1 ,hebrew1 and ,syriac1 as well as in astronomy1 chronology1 and mathematics4 ,in his work on ,india he gives detailed information concerning the language and #g customs of the people of that country1 and states explicitly#ag that the ,hindus of his time did not use the letters of their alphabet for numerical notation1 as the ,arabs did4 ,he also states that the numeral signs called a4nka#ah had different shapes in various parts of ,india1 as was the case with the letters4 ,in his ,chronology of ,ancient ,nations he gives the sum of a geometric progression and shows how1 in order to avoid any possibility of error1 the number may be expressed in three different systems3 with ,indian symbols1 in sexagesimal notation1 and by an alphabet system which will be touched upon later4 ,he also speaks#ai of #agi1 #hgf1 #gee1 expressed in ,indian ciphers1 thus again attributing these forms to ,hindu sources4 ,preceding ,al-,biruni there was another ,arabic writer of the tenth century1 ,mot4ahhar ibn ,t4ahir1#bj author of the ,book of the ,creation and of ,history1 who gave as a curiosity1 in ,indian 7,nagari7 symbols1 a large number asserted by the people of ,india to represent the duration of the world4 ,huart feels positive that in ,mot4ahhar's time the present ,arabic symbols had not yet come into use1 and that the ,indian symbols1 although known to scholars1 were not current4 ,unless this were the case1 neither the author nor his readers would have found anything extraordinary in the appearance of the number which he cites4 ,mention should also be made of a widely-traveled student1 ,al-,masudi 7#hhe8-ief71 whose journeys carried him from ,bagdad to ,persia1 ,india1 ,ceylon1 and even #h across the ,china sea1 and at other times to ,madagascar1 ,syria1 and ,palestine4#ba ,he seems to have neglected no accessible sources of information1 examining also the history of the ,persians1 the ,hindus1 and the ,romans4 ,touching the period of the ,caliphs his work entitled ,meadows of ,gold furnishes a most entertaining fund of information4 ,he states#bb that the wise men of ,india1 assembled by the king1 composed the ,sindhind4 ,further on#bc he states1 upon the authority of the historian ,moh4ammed ibn ,ali ,abdi1 that by order of ,al-,mans4ur many works of science and astrology were translated into ,arabic1 notably the ,sindhind 7,siddhanta74 ,concerning the meaning and spelling of this name there is considerable diversity of opinion4 ,colebrooke#bd first pointed out the connection between ,siddhanta and ,sindhind4 ,he ascribes to the word the meaning the revolving ages4#be ,similar designations are collected by ,sedillot1#bf who inclined to the ,greek origin of the sciences commonly attributed to the ,hindus4#bg ,casiri1#bh citing the ,tarikh al-h4okama or ,chronicles of the ,learned1#bi refers to the work #i as the ,sindum-,indum with the meaning perpetuum aeternumque4 ,the reference#cj in this ancient ,arabic work to ,al-,khowarazmi is worthy of note4 ,this ,sindhind is the book1 says ,masudi1#ca which gives all that the ,hindus know of the spheres1 the stars1 arithmetic1#cb and the other branches of science4 ,he mentions also ,al-,khowarazmi and ,h4abash#cc as translators of the tables of the ,sindhind4 ,al-,biruni#cd refers to two other translations from a work furnished by a ,hindu who came to ,bagdad as a member of the political mission which ,sindh sent to the caliph ,al-,mans4ur1 in the year of the ,hejira #aed 7,a4,d4 #gga74 ,the oldest work1 in any sense complete1 on the history of ,arabic literature and history is the ,kitab al-,fihrist1 written in the year #ihg ,a4,d41 by ,ibn ,abi ,yaqub al-,nadim4 ,it is of fundamental importance for the history of ,arabic culture4 ,of the ten chief divisions of the work1 the seventh demands attention in this discussion for the reason that its second subdivision treats of mathematicians and astronomers4#ce #aj ,the first of the ,arabic writers mentioned is ,al-,kindi 7#hjj-hgj ,a4,d471 who wrote five books on arithmetic and four books on the use of the ,indian method of reckoning4 ,sened ibn ,ali1 the ,jew1 who was converted to ,islam under the caliph ,al-,mamun1 is also given as the author of a work on the ,hindu method of reckoning4 ,nevertheless1 there is a possibility#cf that some of the works ascribed to ,sened ibn ,ali are really works of ,al-,khowarazmi1 whose name immediately precedes his4 ,however1 it is to be noted in this connection that ,casiri#cg also mentions the same writer as the author of a most celebrated work on arithmetic4 ,to ,al-,s4ufi1 who died in #ihf ,a4,d41 is also credited a large work on the same subject1 and similar treatises by other writers are mentioned4 ,we are therefore forced to the conclusion that the ,arabs from the early ninth century on fully recognized the ,hindu origin of the new numerals4 ,leonard of ,pisa1 of whom we shall speak at length in the chapter on the ,introduction of the ,numerals into ,europe1 wrote his ,liber ,abbaci#ch in #abjb4 ,in this work he refers frequently to the nine ,indian figures1#ci thus showing again the general consensus of opinion in the ,middle ,ages that the numerals were of ,hindu origin4 ,some interest also attaches to the oldest documents on arithmetic in our own language4 ,one of the earliest #aa treatises on algorism is a commentary#dj on a set of verses called the ,carmen de ,algorismo1 written by ,alexander de ,villa ,dei 7,alexandra de ,ville-,dieu71 a ,minorite monk of about #abdj ,a4,d4 ,the text of the first few lines is as follows3 ,hec algorism' ars p'sens dicit' in qua ,talib2 indorum fruim bis quinq2 figuris4#da ,this boke is called the boke of algorim or augrym after lewder use4 ,and this boke tretys of the ,craft of ,nombryng1 the quych crafte is called also ,algorym4 ,ther was a kyng of ,inde the quich heyth ,algor he made this craft4444 ,algorisms1 in the quych we use teen figurys of ,inde4 99 99 99 99 99 #ab ,,chapter ,,ii ,,early ,,hindu ,,forms ,,with ,,no ,,place ,,value ,while it is generally conceded that the scientific development of astronomy among the ,hindus towards the beginning of the ,christian era rested upon ,greek#db or ,chinese#dc sources1 yet their ancient literature testifies to a high state of civilization1 and to a considerable advance in sciences1 in philosophy1 and along literary lines1 long before the golden age of ,greece4 ,from the earliest times even up to the present day the ,hindu has been wont to put his thought into rhythmic form4 ,the first of this poetry--it well deserves this name1 being also worthy from a metaphysical point of view#dd--;consists of the ,vedas1 hymns of praise and poems of worship1 collected during the ,vedic period which dates from approximately #bjjj ,b4,c4 to #adjj ,b4,c4#de ,following this work1 or possibly contemporary with it1 is the ,brahmanic literature1 which is partly ritualistic 7the ,brahman4as71 and partly philosophical 7the ,upanishads74 ,our especial interest is #ac in the ,sutras1 versified abridgments of the ritual and of ceremonial rules1 which contain considerable geometric material used in connection with altar construction1 and also numerous examples of rational numbers the sum of whose squares is also a square1 i4e4 ,pythagorean numbers1 although this was long before ,pythagoras lived4 ,whitney#df places the whole of the ,veda literature1 including the ,vedas1 the ,brahman4as1 and the ,sutras1 between #aejj ,b4,c4 and #hjj ,b4,c41 thus agreeing with ,buerk#dg who holds that the knowledge of the ,pythagorean theorem revealed in the ,sutras goes back to the eighth century ,b4,c4 ,the importance of the ,sutras as showing an independent origin of ,hindu geometry1 contrary to the opinion long held by ,cantor#dh of a ,greek origin1 has been repeatedly emphasized in recent literature1#di especially since the appearance of the important work of ,von ,schroeder4#ej ,further fundamental mathematical notions such as the conception of irrationals and the use of gnomons1 as well as the philosophical doctrine of the transmigration of souls1--all of these having long been attributed to the ,greeks1--are shown in these works to be native to ,india4 ,although this discussion does not bear directly upon the #ad origin of our numerals1 yet it is highly pertinent as showing the aptitude of the ,hindu for mathematical and mental work1 a fact further attested by the independent development of the drama and of epic and lyric poetry4 ,it should be stated definitely at the outset1 however1 that we are not at all sure that the most ancient forms of the numerals commonly known as ,arabic had their origin in ,india4 ,as will presently be seen1 their forms may have been suggested by those used in ,egypt1 or in ,eastern ,persia1 or in ,china1 or on the plains of ,mesopotamia4 ,we are quite in the dark as to these early steps2 but as to their development in ,india1 the approximate period of the rise of their essential feature of place value1 their introduction into the ,arab civilization1 and their spread to the ,west1 we have more or less definite information4 ,when1 therefore1 we consider the rise of the numerals in the land of the ,sindhu1#ea it must be understood that it is only the large movement that is meant1 and that there must further be considered the numerous possible sources outside of ,india itself and long anterior to the first prominent appearance of the number symbols4 ,no one attempts to examine any detail in the history of ancient ,india without being struck with the great dearth of reliable material4#eb ,so little sympathy have the people with any save those of their own caste that a general literature is wholly lacking1 and it is only in the observations of strangers that any all-round view of scientific progress is to be found4 ,there is evidence that primary schools #ae existed in earliest times1 and of the seventy-two recognized sciences writing and arithmetic were the most prized4#ec ,in the ,vedic period1 say from #bjjj to #adjj ,b4,c41 there was the same attention to astronomy that was found in the earlier civilizations of ,babylon1 ,china1 and ,egypt1 a fact attested by the ,vedas themselves4#ed ,such advance in science presupposes a fair knowledge of calculation1 but of the manner of calculating we are quite ignorant and probably always shall be4 ,one of the ,buddhist sacred books1 the ,lalitavistara1 relates that when the ,bodhisattva#ee was of age to marry1 the father of ,gopa1 his intended bride1 demanded an examination of the five hundred suitors1 the subjects including arithmetic1 writing1 the lute1 and archery4 ,having vanquished his rivals in all else1 he is matched against ,arjuna the great arithmetician and is asked to express numbers greater than #ajj kotis4#ef ,in reply he gave a scheme of number names as high as #ajec1 adding that he could proceed as far as #ajdba1eg all of which suggests the system of ,archimedes and the unsettled question of the indebtedness of the ,west to the ,east in the realm of ancient mathematics4#eh ,sir ,edwin ,arnold1 #af in ,the ,light of ,asia1 does not mention this part of the contest1 but he speaks of ,buddha's training at the hands of the learned ,vis4vamitra3 ,and ,viswamitra said1 ',it is enough1 ,let us to numbers4 ,after me repeat ,your numeration till we reach the lakh1#ei ,one1 two1 three1 four1 to ten1 and then by tens ,to hundreds1 thousands4' ,after him the child ,named digits1 decads1 centuries1 nor paused1 ,the round lakh reached1 but softly murmured on1 ,then comes the koti1 nahut1 ninnahut1 ,khamba1 viskhamba1 abab1 attata1 ,to kumuds1 gundhikas1 and utpalas1 ,by pundarikas into padumas1 ,which last is how you count the utmost grains ,of ,hastagiri ground to finest dust2#fj ,but beyond that a numeration is1 ,the ,katha1 used to count the stars of night1 ,the ,koti-,katha1 for the ocean drops2 ,ingga1 the calculus of circulars2 ,sarvanikchepa1 by the which you deal ,with all the sands of ,gunga1 till we come ,to ,antah-,kalpas1 where the unit is ,the sands of the ten crore ,gungas4 ,if one seeks ,more comprehensive scale1 th' arithmic mounts ,by the ,asankya1 which is the tale ,of all the drops that in ten thousand years ,would fall on all the worlds by daily rain2 ,thence unto ,maha ,kalpas1 by the which ,the gods compute their future and their past4' #ag ,thereupon ,vis4vamitra ,acarya#fa expresses his approval of the task1 and asks to hear the measure of the line as far as yojana1 the longest measure bearing name4 ,this given1 ,buddha adds3 444 ',and master6 if it please1 ,i shall recite how many sun-motes lie ,from end to end within a yojana4' ,thereat1 with instant skill1 the little prince ,pronounced the total of the atoms true4 ,but ,viswamitra heard it on his face ,prostrate before the boy2 ',for thou1' he cried1 ',art ,teacher of thy teachers--thou1 not ,i1 ,art ,guru4' ,it is needless to say that this is far from being history4 ,and yet it puts in charming rhythm only what the ancient ,lalitavistara relates of the number-series of the ,buddha's time4 ,while it extends beyond all reason1 nevertheless it reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement4 ,to this pre-,christian period belong also the ,veda4ngas1 or limbs for supporting the ,veda1 part of that great branch of ,hindu literature known as ,smr4iti 7recollection71 that which was to be handed down by tradition4 ,of these the sixth is known as ,jyotis4a 7astronomy71 a short treatise of only thirty-six verses1 written not earlier than #cjj ,b4,c41 and affording us some knowledge of the extent of number work in that period4#fb ,the ,hindus #ah also speak of eighteen ancient ,siddhantas or astronomical works1 which1 though mostly lost1 confirm this evidence4#fc ,as to authentic histories1 however1 there exist in ,india none relating to the period before the ,mohammedan era 7#fbb ,a4,d474 ,about all that we know of the earlier civilization is what we glean from the two great epics1 the ,mahabharata#fd and the ,ramayana1 from coins1 and from a few inscriptions4#fe ,it is with this unsatisfactory material1 then1 that we have to deal in searching for the early history of the ,hindu-,arabic numerals1 and the fact that many unsolved problems exist and will continue to exist is no longer strange when we consider the conditions4 ,it is rather surprising that so much has been discovered within a century1 than that we are so uncertain as to origins and dates and the early spread of the system4 ,the probability being that writing was not introduced into ,india before the close of the fourth century ,b4,c41 and literature existing only in spoken form prior to that period1#ff the number work was doubtless that of all primitive peoples1 palpable1 merely a matter of placing sticks or cowries or pebbles on the ground1 of marking a sand-covered board1 or of cutting notches or tying cords as is still done in parts of ,southern ,india to-day4#fg #ai ,the early ,hindu numerals#fh may be classified into three great groups1 7#a7 the ,kharos4t4hi1 7#b7 the ,brahmi1 and 7#c7 the word and letter forms2 and these will be considered in order4 ,the ,kharos4t4hi numerals are found in inscriptions formerly known as ,bactrian1 ,indo-,bactrian1 and ,aryan1 and appearing in ancient ,gandhara1 now eastern ,afghanistan and northern ,punjab4 ,the alphabet of the language is found in inscriptions dating from the fourth century ,b4,c4 to the third century ,a4,d41 and from the fact that the words are written from right to left it is assumed to be of ,semitic origin4 ,no numerals1 however1 have been found in the earliest of these inscriptions1 number-names probably having been written out in words as was the custom with many ancient peoples4 ,not until the time of the powerful ,king ,a'soka1 in the third century ,b4,c41 do numerals appear in any inscriptions thus far discovered2 and then only in the primitive form of marks1 quite as they would be found in ,egypt1 ,greece1 ,rome1 or in #bj various other parts of the world4 ,these ,a'soka#fi inscriptions1 some thirty in all1 are found in widely separated parts of ,india1 often on columns1 and are in the various vernaculars that were familiar to the people4 ,two are in the ,kharos4t4hi characters1 and the rest in some form of ,brahmi4 ,in the ,kharos4t4hi inscriptions only four numerals have been found1 and these are merely vertical marks for one1 two1 four1 and five1 thus3 ,in the so-called ',saka inscriptions1 possibly of the first century ,b4,c41 more numerals are found1 and in more highly developed form1 the right-to-left system appearing1 together with evidences of three different scales of counting1--four1 ten1 and twenty4 ,the numerals of this period are as follows3 ,illustration ,there are several noteworthy points to be observed in studying this system4 ,in the first place1 it is probably not as early as that shown in the ,nana ,ghat forms hereafter given1 although the inscriptions themselves at ,nana ,ghat are later than those of the ,a'soka period4 ,the #ba four is to this system what the ,x was to the ,roman1 probably a canceling of three marks as a workman does to-day for five1 or a laying of one stick across three others4 ,the ten has never been satisfactorily explained4 ,it is similar to the ,a of the ,kharos4t4hi alphabet1 but we have no knowledge as to why it was chosen4 ,the twenty is evidently a ligature of two tens1 and this in turn suggested a kind of radix1 so that ninety was probably written in a way reminding one of the quatre-vingt-dix of the ,french4 ,the hundred is unexplained1 although it resembles the letter ta or tra of the ,brahmi alphabet with #a before 7to the right of7 it4 ,the two hundred is only a variant of the symbol for hundred1 with two vertical marks4#gj ,this system has many points of similarity with the ,nabatean numerals#ga in use in the first centuries of the ,christian era4 ,the cross is here used for four1 and the ,kharos4t4hi form is employed for twenty4 ,in addition to this there is a trace of an analogous use of a scale of twenty4 ,while the symbol for #ajj is quite different1 the method of forming the other hundreds is the same4 ,the correspondence seems to be too marked to be wholly accidental4 ,it is not in the ,kharos4t4hi numerals1 therefore1 that we can hope to find the origin of those used by us1 and we turn to the second of the ,indian types1 the ,brahmi characters4 ,the alphabet attributed to ,brahma is the oldest of the several known in ,india1 and was used from the earliest historic times4 ,there are various theories of its origin1 #bb none of which has as yet any wide acceptance1#gb although the problem offers hope of solution in due time4 ,the numerals are not as old as the alphabet1 or at least they have not as yet been found in inscriptions earlier than those in which the edicts of ,a'soka appear1 some of these having been incised in ,brahmi as well as ,kharos4t4hi4 ,as already stated1 the older writers probably wrote the numbers in words1 as seems to have been the case in the earliest ,pali writings of ,ceylon4#gc ,the following numerals are1 as far as known1 the only ones to appear in the ,a'soka edicts3#gd ,illustration ,these fragments from the third century ,b4,c41 crude and unsatisfactory as they are1 are the undoubted early forms from which our present system developed4 ,they next appear in the second century ,b4,c4 in some inscriptions in the cave on the top of the ,nana ,ghat hill1 about seventy-five miles from ,poona in central ,india4 ,these inscriptions may be memorials of the early ,andhra dynasty of southern ,india1 but their chief interest lies in the numerals which they contain4 ,the cave was made as a resting-place for travelers ascending the hill1 which lies on the road from ,kalyana to ,junar4 ,it seems to have been cut out by a descendant #bc of ,king ',satavahana1#ge for inside the wall opposite the entrance are representations of the members of his family1 much defaced1 but with the names still legible4 ,it would seem that the excavation was made by order of a king named ,vedisiri1 and the inscription contains a list of gifts made on the occasion of the performance of several yagnas or religious sacrifices1 and numerals are to be seen in no less than thirty places4#gf ,there is considerable dispute as to what numerals are really found in these inscriptions1 owing to the difficulty of deciphering them2 but the following1 which have been copied from a rubbing1 are probably number forms3#gg ,illustration ,the inscription itself1 so important as containing the earliest considerable ,hindu numeral system connected with our own1 is of sufficient interest to warrant reproducing part of it in facsimile1 as is done on page #bd4 #bd ,illustration ,the next very noteworthy evidence of the numerals1 and this quite complete as will be seen1 is found in certain other cave inscriptions dating back to the first or second century ,a4,d4 ,in these1 the ,nasik#gh cave inscriptions1 the forms are as follows3 ,illustration ,from this time on1 until the decimal system finally adopted the first nine characters and replaced the rest of the ,brahmi notation by adding the zero1 the progress of these forms is well marked4 ,it is therefore well to present synoptically the best-known specimens that have come down to us1 and this is done in the table on page #be4gi #be ,,table ,,showing ,,the ,,progress ,,of ,,number ,,forms ,,in ,,india ,,numerals #a #b #c #d #e #f #g #h #i #aj #bj #cj #dj #ej #fj #gj #hj #ij #ajj #bjj #ajjj ,a'soka#hj ,illustration ',saka#ha ,illustration ,a'soka#hb ,illustration ,nagari#hc ,illustration ,nasik#hd ,illustration ,ks4atrapa#he ,illustration ,kus4ana #hf ,illustration ,gupta#hg ,illustration ,valhabi#hh ,illustration ,nepal #hi ,illustration ,kali4nga#ij ,illustration ,vakat4aka#ia ,illustration ,most of these numerals are given by ,buehler1 loc4 cit41 ,tafel ,,ix4 #bf ,with respect to these numerals it should first be noted that no zero appears in the table1 and as a matter of fact none existed in any of the cases cited4 ,it was therefore impossible to have any place value1 and the numbers like twenty1 thirty1 and other multiples of ten1 one hundred1 and so on1 required separate symbols except where they were written out in words4 ,the ancient ,hindus had no less than twenty of these symbols1#ib a number that was afterward greatly increased4 ,the following are examples of their method of indicating certain numbers between one hundred and one thousand3 #ic ,numerals for #agd #id ,numerals for #aia #ie ,numerals for #bfi #if ,numerals for #beb #ig ,numerals for #djj #ih ,numerals for #cef #bg ,to these may be added the following numerals below one hundred1 similar to those in the table3 ,numerals#ii for #ij ,numerals#ajj for #gj ,we have thus far spoken of the ,kharos4t4hi and ,brahmi numerals1 and it remains to mention the third type1 the word and letter forms4 ,these are1 however1 so closely connected with the perfecting of the system by the invention of the zero that they are more appropriately considered in the next chapter1 particularly as they have little relation to the problem of the origin of the forms known as the ,arabic4 ,having now examined types of the early forms it is appropriate to turn our attention to the question of their origin4 ,as to the first three there is no question4 ,the #a vertical stroke or #a horizontal stroke is simply one stroke1 or one stick laid down by the computer4 ,the #b vertical strokes or #b horizontal strokes represents two strokes or two sticks1 and so for the #c vertical strokes and #c horizontal strokes4 ,from some primitive #b vertical strokes came the two of ,egypt1 of ,rome1 of early ,greece1 and of various other civilizations4 ,it appears in the three ,egyptian numeral systems in the following forms3 ,hieroglyphic #b vertical strokes ,hieratic ,hieratic #b ,demotic ,demotic #b ,the last of these is merely a cursive form as in the ,arabic ,arabic #b1 which becomes our #b if tipped through a right angle4 ,from some primitive #b horizontal strokes came the ,chinese #bh symbol1 which is practically identical with the symbols found commonly in ,india from #aej ,b4,c4 to #gjj ,a4,d4 ,in the cursive form it becomes #b horizontal strokes joined1 and this was frequently used for two in ,germany until the #ahth century4 ,it finally went into the modern form #b1 and the #c horizontal strokes in the same way became our #c4 ,there is1 however1 considerable ground for interesting speculation with respect to these first three numerals4 ,the earliest ,hindu forms were perpendicular4 ,in the ,nana ,ghat inscriptions they are vertical4 ,but long before either the ,a'soka or the ,nana ,ghat inscriptions the ,chinese were using the horizontal forms for the first three numerals1 but a vertical arrangement for four4#aja ,now where did ,china get these forms8 ,surely not from ,india1 for she had them1 as her monuments and literature#ajb show1 long before the ,hindus knew them4 ,the tradition is that ,china brought her civilization around the north of ,tibet1 from ,mongolia1 the primitive habitat being ,mesopotamia1 or possibly the oases of ,turkestan4 ,now what numerals did ,mesopotamia use8 ,the ,babylonian system1 simple in its general principles but very complicated in many of its details1 is now well known4#ajc ,in particular1 one1 two1 and three were represented by vertical arrow-heads4 ,why1 then1 did the ,chinese write #bi theirs horizontally8 ,the problem now takes a new interest when we find that these ,babylonian forms were not the primitive ones of this region1 but that the early ,sumerian forms were horizontal4#ajd ,what interpretation shall be given to these facts8 ,shall we say that it was mere accident that one people wrote one vertically and that another wrote it horizontally8 ,this may be the case2 but it may also be the case that the tribal migrations that ended in the ,mongol invasion of ,china started from the ,euphrates while yet the ,sumerian civilization was prominent1 or from some common source in ,turkestan1 and that they carried to the ,east the primitive numerals of their ancient home1 the first three1 these being all that the people as a whole knew or needed4 ,it is equally possible that these three horizontal forms represent primitive stick-laying1 the most natural position of a stick placed in front of a calculator being the horizontal one4 ,when1 however1 the cuneiform writing developed more fully1 the vertical form may have been proved the easier to make1 so that by the time the migrations to the ,west began these were in use1 and from them came the upright forms of ,egypt1 ,greece1 ,rome1 and other ,mediterranean lands1 and those of ,a'soka's time in ,india4 ,after ,a'soka1 and perhaps among the merchants of earlier centuries1 the horizontal forms may have come down into ,india from ,china1 thus giving those of the ,nana ,ghat cave and of later inscriptions4 ,this is in the realm of speculation1 but it is not improbable that further epigraphical studies may confirm the hypothesis4 #cj ,as to the numerals above three there have been very many conjectures4 ,the figure one of the ,demotic looks like the one of the ,sanskrit1 the two 7reversed7 like that of the ,arabic1 the four has some resemblance to that in the ,nasik caves1 the five 7reversed7 to that on the ,ks4atrapa coins1 the nine to that of the ,kus4ana inscriptions1 and other points of similarity have been imagined4 ,some have traced resemblance between the ,hieratic five and seven and those of the ,indian inscriptions4 ,there have not1 therefore1 been wanting those who asserted an ,egyptian origin for these numerals4#aje ,there has already been mentioned the fact that the ,kharos4t4hi numerals were formerly known as ,bactrian1 ,indo-,bactrian1 and ,aryan4 ,cunningham#ajf was the first to suggest that these numerals were derived from the alphabet of the ,bactrian civilization of ,eastern ,persia1 perhaps a thousand years before our era1 and in this he was supported by the scholarly work of ,sir ,e4 ,clive ,bayley1#ajg who in turn was followed by ,canon ,taylor4#ajh ,the resemblance has not proved convincing1 however1 and ,bayley's drawings #ca have been criticized as being affected by his theory4 ,the following is part of the hypothesis3#aji ,numeral ,hindu ,bactrian ,sanskrit #d ,symbol ,symbol ch chatur1 ,lat4 quattuor #e ,symbol ,symbol p pancha1 ,gk4 ,greek3p_/ente #f ,symbol ,symbol s s4as4 #g ,symbol ,symbol s4 sapta 7 the s and s4 are interchanged as occasionally in ,n4 ,w4 ,india7 ,buehler#aaj rejects this hypothesis1 stating that in four cases 7four1 six1 seven1 and ten7 the facts are absolutely against it4 ,while the relation to ancient ,bactrian forms has been generally doubted1 it is agreed that most of the numerals resemble ,brahmi letters1 and we would naturally expect them to be initials4#aaa ,but1 knowing the ancient pronunciation of most of the number names1#aab we find this not to be the case4 ,we next fall back upon the hypothesis #cb that they represent the order of letters#aac in the ancient alphabet4 ,from what we know of this order1 however1 there seems also no basis for this assumption4 ,we have1 therefore1 to confess that we are not certain that the numerals were alphabetic at all1 and if they were alphabetic we have no evidence at present as to the basis of selection4 ,the later forms may possibly have been alphabetical expressions of certain syllables called aks4aras1 which possessed in ,sanskrit fixed numerical values1#aad but this is equally uncertain with the rest4 ,bayley also thought#aae that some of the forms were ,phoenician1 as notably the use of a circle for twenty1 but the resemblance is in general too remote to be convincing4 ,there is also some slight possibility that ,chinese influence is to be seen in certain of the early forms of ,hindu numerals4#aaf #cc ,more absurd is the hypothesis of a ,greek origin1 supposedly supported by derivation of the current symbols from the first nine letters of the ,greek alphabet4#aag ,this difficult feat is accomplished by twisting some of the letters1 cutting off1 adding on1 and effecting other changes to make the letters fit the theory4 ,this peculiar theory was first set up by ,dasypodius#aah 7,conrad ,rauhfuss71 and was later elaborated by ,huet4#aai #cd ,a bizarre derivation based upon early ,arabic 7c4 #ajdj ,a4,d47 sources is given by ,kircher in his work#abj on number mysticism4 ,he quotes from ,abenragel1#aba giving the ,arabic and a ,latin translation#abb and stating that the ordinary ,arabic forms are derived from sectors of a circle1 circle4 ,out of all these conflicting theories1 and from all the resemblances seen or imagined between the numerals of the ,west and those of the ,east1 what conclusions are we prepared to draw as the evidence now stands8 ,probably none that is satisfactory4 ,indeed1 upon the evidence at #ce hand we might properly feel that everything points to the numerals as being substantially indigenous to ,india4 ,and why should this not be the case8 ,if the king ,srong-tsan-,gampo 7#fci ,a4,d471 the founder of ,lhasa1#abc could have set about to devise a new alphabet for ,tibet1 and if the ,siamese1 and the ,singhalese1 and the ,burmese1 and other peoples in the ,east1 could have created alphabets of their own1 why should not the numerals also have been fashioned by some temple school1 or some king1 or some merchant guild8 ,by way of illustration1 there are shown in the table on page #cf certain systems of the ,east1 and while a few resemblances are evident1 it is also evident that the creators of each system endeavored to find original forms that should not be found in other systems4 ,this1 then1 would seem to be a fair interpretation of the evidence4 ,a human mind cannot readily create simple forms that are absolutely new2 what it fashions will naturally resemble what other minds have fashioned1 or what it has known through hearsay or through sight4 ,a circle is one of the world's common stock of figures1 and that it should mean twenty in ,phoenicia and in ,india is hardly more surprising than that it signified ten at one time in ,babylon4#abd ,it is therefore quite probable that an extraneous origin cannot be found for the very sufficient reason that none exists4 ,of absolute nonsense about the origin of the symbols which we use much has been written4 ,conjectures1 #cf however1 without any historical evidence for support1 have no place in a serious discussion of the gradual evolution of the present numeral forms4#abe ,,table ,,of ,,certain ,,eastern ,,systems ,siam ,illustration3 numerals ,burma#abf ,illustration3 numerals ,malabar#abg ,illustration3 numerals ,tibet#abh ,illustration3 numerals ,ceylon#abi ,illustration3 numerals ,malayalam#abi ,illustration3 numerals #cg ,we may summarize this chapter by saying that no one knows what suggested certain of the early numeral forms used in ,india4 ,the origin of some is evident1 but the origin of others will probably never be known4 ,there is no reason why they should not have been invented by some priest or teacher or guild1 by the order of some king1 or as part of the mysticism of some temple4 ,whatever the origin1 they were no better than scores of other ancient systems and no better than the present ,chinese system when written without the zero1 and there would never have been any chance of their triumphal progress westward had it not been for this relatively late symbol4 ,there could hardly be demanded a stronger proof of the ,hindu origin of the character for zero than this1 and to it further reference will be made in ,chapter ,,iv4 99 99 99 99 99 #ch ,,chapter ,,iii ,,later ,,hindu ,,forms1 ,,with ,a ,,place ,,value ,before speaking of the perfected ,hindu numerals with the zero and the place value1 it is necessary to consider the third system mentioned on page #ai1--the word and letter forms4 ,the use of words with place value began at least as early as the #fth century of the ,christian era4 ,in many of the manuals of astronomy and mathematics1 and often in other works in mentioning dates1 numbers are represented by the names of certain objects or ideas4 ,for example1 zero is represented by the void 7'sunya71 or heaven-space 7ambara aka'sa72 one by stick 7rupa71 moon 7indu 'sa'sin71 earth 7bhu71 beginning 7adi71 ,brahma1 or1 in general1 by anything markedly unique2 two by the twins 7yama71 hands 7kara71 eyes 7nayana71 etc42 four by oceans1 five by senses 7vis4aya7 or arrows 7the five arrows of ,kamadeva72 six by seasons or flavors2 seven by mountain 7aga71 and so on4#acj ,these names1 accommodating themselves to the verse in which scientific works were written1 had the additional advantage of not admitting1 as did the figures1 easy alteration1 since any change would tend to disturb the meter4 #ci ,as an example of this system1 the date ',saka ,sam4vat1 #hfg 7,a4,d4 #ide or #idf71 is given by giri-ras4a-vasu1 meaning the mountains 7seven71 the flavors 7six71 and the gods ,vasu of which there were eight4 ,in reading the date these are read from right to left4#aca ,the period of invention of this system is uncertain4 ,the first trace seems to be in the ',srautasutra of ,katyayana and ,lat4yayana4#acb ,it was certainly known to ,varaha-,mihira 7d4 #ehg71acc for he used it in the ,br4hat-,sam4hita4#acd ,it has also been asserted#ace that ,aryabhat4a 7c4 #ejj ,a4,d47 was familiar with this system1 but there is nothing to prove the statement4#acf ,the earliest epigraphical examples of the system are found in the ,bayang 7,cambodia7 inscriptions of #fjd and #fbd ,a4,d4#acg ,mention should also be made1 in this connection1 of a curious system of alphabetic numerals that sprang up in southern ,india4 ,in this we have the numerals represented by the letters as given in the following table3 #a #b #c #d #e #f #g #h #i #j k kh g gh 4n c ch j jh n t4 t4h d4 d4h n4 t th d th n p ph b bh m y r l v 's s4 s h l #dj ,by this plan a numeral might be represented by any one of several letters1 as shown in the preceding table1 and thus it could the more easily be formed into a word for mnemonic purposes4 ,for example1 the word #b #c #a #e #f #e #a kha gont yan me s4a ma pa has the value #a1efe1acb1 reading from right to left4#ach ,this1 the oldest specimen 7#aahd ,a4,d47 known of this notation1 is given in a commentary on the ,rigveda1 representing the number of days that had elapsed from the beginning of the ,kaliyuga4 ,burnell#aci states that this system is even yet in use for remembering rules to calculate horoscopes1 and for astronomical tables4 ,a second system of this kind is still used in the pagination of manuscripts in ,ceylon1 ,siam1 and ,burma1 having also had its rise in southern ,india4 ,in this the thirty-four consonants when followed by a 7as ka 444 la7 designate the numbers #a-cd2 by a 7as ka 444 la71 those from #ce to #fh2 by i 7ki 444 li71 those from #fi to #ajb1 inclusive2 and so on4#adj ,as already stated1 however1 the ,hindu system as thus far described was no improvement upon many others of the ancients1 such as those used by the ,greeks and the ,hebrews4 ,having no zero1 it was impracticable to designate the tens1 hundreds1 and other units of higher order by the same symbols used for the units from one to nine4 ,in other words1 there was no possibility of place value without some further improvement4 ,so the ,nana ,ghat #da symbols required the writing of thousand seven twenty-four about like ,t #g1 tw1 #d in modern symbols1 instead of #gjbd1 in which the seven of the thousands1 the two of the tens 7concealed in the word twenty1 being originally twain of tens1 the -ty signifying ten71 and the four of the units are given as spoken and the order of the unit 7tens1 hundreds1 etc47 is given by the place4 ,to complete the system only the zero was needed2 but it was probably eight centuries after the ,nana ,ghat inscriptions were cut1 before this important symbol appeared2 and not until a considerably later period did it become well known4 ,who it was to whom the invention is due1 or where he lived1 or even in what century1 will probably always remain a mystery4#ada ,it is possible that one of the forms of ancient abacus suggested to some ,hindu astronomer or mathematician the use of a symbol to stand for the vacant line when the counters were removed4 ,it is well established that in different parts of ,india the names of the higher powers took different forms1 even the order being interchanged4#adb ,nevertheless1 as the significance of the name of the unit was given by the order in reading1 these variations did not lead to error4 ,indeed the variation itself may have necessitated the introduction of a word to signify a vacant place or lacking unit1 with the ultimate introduction of a zero symbol for this word4 ,to enable us to appreciate the force of this argument a large number1 #h1ddc1fhb1aee1 may be considered as the ,hindus wrote and read it1 and then1 by way of contrast1 as the ,greeks and ,arabs would have read it4 #db ,modern ,american reading1 #h billion1 #ddc million1 #fhb thousand1 #aee4 ,hindu1 #h padmas1 #d vyarbudas1 #d kot4is1 #c prayutas1 #f laks4as1 #h ayutas1 #b sahasra1 #a 'sata1 #e da'san1 #e4 ,arabic and early ,german1 eight thousand thousand thousand and four hundred thousand thousand and forty-three thousand thousand1 and six hundred thousand and eighty-two thousand and one hundred fifty-five 7or five and fifty74 ,greek1 eighty-four myriads of myriads and four thousand three hundred sixty-eight myriads and two thousand and one hundred fifty-five4 ,as ,woepcke#adc pointed out1 the reading of numbers of this kind shows that the notation adopted by the ,hindus tended to bring out the place idea4 ,no other language than the ,sanskrit has made such consistent application1 in numeration1 of the decimal system of numbers4 ,the introduction of myriads as in the ,greek1 and thousands as in ,arabic and in modern numeration1 is really a step away from a decimal scheme4 ,so in the numbers below one hundred1 in ,english1 eleven and twelve are out of harmony with the rest of the -teens1 while the naming of all the numbers between ten and twenty is not analogous to the naming of the numbers above twenty4 ,to conform to our written system we should have ten-one1 ten-two1 ten-three1 and so on1 as we have twenty-one1 twenty-two1 and the like4 ,the ,sanskrit is consistent1 the units1 however1 preceding the tens and hundreds4 ,nor did any other ancient people carry the numeration as far as did the ,hindus4#add #dc ,when the a4nkapalli1#ade the decimal-place system of writing numbers1 was perfected1 the tenth symbol was called the 'sunyabindu1 generally shortened to 'sunya 7the void74 ,brockhaus#adf has well said that if there was any invention for which the ,hindus1 by all their philosophy and religion1 were well fitted1 it was the invention of a symbol for zero4 ,this making of nothingness the crux of a tremendous achievement was a step in complete harmony with the genius of the ,hindu4 ,it is generally thought that this 'sunya as a symbol was not used before about #ejj ,a4,d41 although some writers have placed it earlier4#adg ,since ,aryabhat4a gives our common method of extracting roots1 it would seem that he may have known a decimal notation1#adh although he did not use the characters from which our numerals are derived4#adi ,moreover1 he frequently speaks of the #dd void4#aej ,if he refers to a symbol this would put the zero as far back as #ejj ,a4,d41 but of course he may have referred merely to the concept of nothingness4 ,a little later1 but also in the sixth century1 ,varaha-,mihira#aea wrote a work entitled ,br4hat ,sam4hita#aeb in which he frequently uses 'sunya in speaking of numerals1 so that it has been thought that he was referring to a definite symbol4 ,this1 of course1 would add to the probability that ,aryabhat4a was doing the same4 ,it should also be mentioned as a matter of interest1 and somewhat related to the question at issue1 that ,varaha-,mihira used the word-system with place value#aec as explained above4 ,the first kind of alphabetic numerals and also the word-system 7in both of which the place value is used7 are plays upon1 or variations of1 position arithmetic1 which would be most likely to occur in the country of its origin4#aed ,at the opening of the next century 7c4 #fbj ,a4,d47 ,ban4a#aee wrote of ,subandhus's ,vasavadatta as a celebrated work1 #de and mentioned that the stars dotting the sky are here compared with zeros1 these being points as in the modern ,arabic system4 ,on the other hand1 a strong argument against any ,hindu knowledge of the symbol zero at this time is the fact that about #gjj ,a4,d4 the ,arabs overran the province of ,sind and thus had an opportunity of knowing the common methods used there for writing numbers4 ,and yet1 when they received the complete system in #ggf they looked upon it as something new4#aef ,such evidence is not conclusive1 but it tends to show that the complete system was probably not in common use in ,india at the beginning of the eighth century4 ,on the other hand1 we must bear in mind the fact that a traveler in ,germany in the year #agjj would probably have heard or seen nothing of decimal fractions1 although these were perfected a century before that date4 ,the elite of the mathematicians may have known the zero even in ,aryabhat4a's time1 while the merchants and the common people may not have grasped the significance of the novelty until a long time after4 ,on the whole1 the evidence seems to point to the west coast of ,india as the region where the complete system was first seen4#aeg ,as mentioned above1 traces of the numeral words with place value1 which do not1 however1 absolutely require a decimal place-system of symbols1 are found very early in ,cambodia1 as well as in ,india4 ,concerning the earliest epigraphical instances of the use of the nine symbols1 plus the zero1 with place value1 there #df is some question4 ,colebrooke#aeh in #ahjg warned against the possibility of forgery in many of the ancient copper-plate land grants4 ,on this account ,fleet1 in the ,indian ,antiquary1#aei discusses at length this phase of the work of the epigraphists in ,india1 holding that many of these forgeries were made about the end of the eleventh century4 ,colebrooke#afj takes a more rational view of these forgeries than does ,kaye1 who seems to hold that they tend to invalidate the whole ,indian hypothesis4 ,but even where that may be suspected1 the historical uses of a monument fabricated so much nearer to the times to which it assumes to belong1 will not be entirely superseded4 ,the necessity of rendering the forged grant credible would compel a fabricator to adhere to history1 and conform to established notions3 and the tradition1 which prevailed in his time1 and by which he must be guided1 would probably be so much nearer to the truth1 as it was less remote from the period which it concerned4#afa ,buehler#afb gives the copper-plate ,gurjara inscription of ,cedi-sam4vat #cdf 7#eie ,a4,d47 as the oldest epigraphical use of the numerals#afc in which the symbols correspond to the alphabet numerals of the period and the place4 ,vincent ,a4 ,smith#afd quotes a stone inscription of #hae ,a4,d41 dated ,sam4vat #hgb4 ,so ,f4 ,kielhorn in the ,epigraphia ,indica#afe gives a ,pathari pillar inscription of ,parabala1 dated ,vikrama-sam4vat #iag1 which corresponds to #hfa ,a4,d41 #dg and refers also to another copper-plate inscription dated ,vikrama-sam4vat #hac 7#gef ,a4,d474 ,the inscription quoted by ,v4 ,a4 ,smith above is that given by ,d4 ,r4 ,bhandarkar1#aff and another is given by the same writer as of date ,saka-sam4vat #gae 7#gih ,a4,d471 being incised on a pilaster4 ,kielhorn#afg also gives two copper-plate inscriptions of the time of ,mahendrapala of ,kanauj1 ,valhabi-sam4vat #egd 7#hic ,a4,d47 and ,vikrama-sam4vat #ief 7#hii ,a4,d474 ,that there should be any inscriptions of date as early even as #gej ,a4,d41 would tend to show that the system was at least a century older4 ,as will be shown in the further development1 it was more than two centuries after the introduction of the numerals into ,europe that they appeared there upon coins and inscriptions4 ,while ,thibaut#afh does not consider it necessary to quote any specific instances of the use of the numerals1 he states that traces are found from #eij ,a4,d4 on4 ,that the system now in use by all civilized nations is of ,hindu origin cannot be doubted2 no other nation has any claim upon its discovery1 especially since the references to the origin of the system which are found in the nations of western ,asia point unanimously towards ,india4#afi ,the testimony and opinions of men like ,buehler1 ,kielhorn1 ,v4 ,a4 ,smith1 ,bhandarkar1 and ,thibaut are entitled to the most serious consideration4 ,as authorities on ancient ,indian epigraphy no others rank higher4 ,their work is accepted by ,indian scholars the world over1 and their united judgment as to the rise of the system with a place value--that it took place in ,india as early as the #dh sixth century ,a4,d4--must stand unless new evidence of great weight can be submitted to the contrary4 ,many early writers remarked upon the diversity of ,indian numeral forms4 ,al-,biruni was probably the first2 noteworthy is also ,johannes ,hispalensis1#agj who gives the variant forms for seven and four4 ,we insert on p4 #di a table of numerals used with place value4 ,while the chief authority for this is ,buehler1#aga several specimens are given which are not found in his work and which are of unusual interest4 ,the ',sarada forms given in the table use the circle as a symbol for #a and the dot for zero4 ,they are taken from the paging and text of ,the ,kashmirian ,atharva-,veda#agb1 of which the manuscript used is certainly four hundred years old4 ,similar forms are found in a manuscript belonging to the ,university of ,tuebingen4 ,two other series presented are from ,tibetan books in the library of one of the authors4 ,for purposes of comparison the modern ,sanskrit and ,arabic numeral forms are added4 ,sanskrit1 ,illustration ,arabic1 ,illustration #di ,,numerals ,,used ,,with ,,place ,,value #a #b #c #d #e #f #g #h #i #j a#agc ,illustration b#agd ,illustration c#age ,illustration d#agf ,illustration e#agg ,illustration f#agh ,illustration g#agi ,illustration h#ahj ,illustration i#ahj ,illustration j#aha ,illustration k#aha ,illustration l#ahb ,illustration m#ahc ,illustration n#ahd ,illustration 99 99 99 99 99 #ea ,,chapter ,,iv ,,the ,,symbol ,,zero ,what has been said of the improved ,hindu system with a place value does not touch directly the origin of a symbol for zero1 although it assumes that such a symbol exists4 ,the importance of such a sign1 the fact that it is a prerequisite to a place-value system1 and the further fact that without it the ,hindu-,arabic numerals would never have dominated the computation system of the western world1 make it proper to devote a chapter to its origin and history4 ,it was some centuries after the primitive ,brahmi and ,kharos4t4hi numerals had made their appearance in ,india that the zero first appeared there1 although such a character was used by the ,babylonians#ahe in the centuries immediately preceding the ,christian era4 ,the symbol is ,babylonian zero symbol or ,babylonian zero symbol1 and apparently it was not used in calculation4 ,nor does it always occur when units of any order are lacking2 thus #ahj is written ,babylonian numerals #ahj with the meaning three sixties and no units1 since #aha immediately following is ,babylonian numerals #aha1 three sixties and one unit4#ahf ,the main #eb use of this ,babylonian symbol seems to have been in the fractions1 #fjths1 #cfjjths1 etc41 and somewhat similar to the ,greek use of ,greek3 o1 for ,greek3 ouden1 with the meaning vacant4 ,the earliest undoubted occurrence of a zero in ,india is an inscription at ,gwalior1 dated ,samvat #icc 7#hgf ,a4,d474 ,where #ej garlands are mentioned 7line #bj71 #ej is written ,gwalior numerals #ej4 #bgj 7line #d7 is written ,gwalior numerals #bgj4ahg ,the ,bakhs4ali ,manuscript#ahh probably antedates this1 using the point or dot as a zero symbol4 ,bayley mentions a grant of ,jaika ,rashtrakuta of ,bharuj1 found at ,okamandel1 of date #gch ,a4,d41 which contains a zero1 and also a coin with indistinct ,gupta date #gjg 7#hig ,a4,d471 but the reliability of ,bayley's work is questioned4 ,as has been noted1 the appearance of the numerals in inscriptions and on coins would be of much later occurrence than the origin and written exposition of the system4 ,from the period mentioned the spread was rapid over all of ,india1 save the southern part1 where the ,tamil and ,malayalam people retain the old system even to the present day4#ahi ,aside from its appearance in early inscriptions1 there is still another indication of the ,hindu origin of the symbol in the special treatment of the concept zero in the early works on arithmetic4 ,brahmagupta1 who lived in ,ujjain1 the center of ,indian astronomy1#aij in the early part #ec of the seventh century1 gives in his arithmetic#aia a distinct treatment of the properties of zero4 ,he does not discuss a symbol1 but he shows by his treatment that in some way zero had acquired a special significance not found in the ,greek or other ancient arithmetics4 ,a still more scientific treatment is given by ,bhaskara1#aib although in one place he permits himself an unallowed liberty in dividing by zero4 ,the most recently discovered work of ancient ,indian mathematical lore1 the ,ganita-,sara-,sa4ngraha#aic of ,mahaviracarya 7c4 #hcj ,a4,d471 while it does not use the numerals with place value1 has a similar discussion of the calculation with zero4 ,what suggested the form for the zero is1 of course1 purely a matter of conjecture4 ,the dot1 which the ,hindus used to fill up lacunae in their manuscripts1 much as we indicate a break in a sentence1#aid would have been a more natural symbol2 and this is the one which the ,hindus first used#aie and which most ,arabs use to-day4 ,there was also used for this purpose a cross1 like our ,x1 and this is occasionally found as a zero symbol4#aif ,in the ,bakhs4ali manuscript above mentioned1 the word 'sunya1 with the dot as its symbol1 is used to denote the unknown quantity1 as well as to denote zero4 ,an analogous use of the #ed zero1 for the unknown quantity in a proportion1 appears in a ,latin manuscript of some lectures by ,gottfried ,wolack in the ,university of ,erfurt in #adfg and #adfh4aig ,the usage was noted even as early as the eighteenth century4#aih ,the small circle was possibly suggested by the spurred circle which was used for ten4#aii ,it has also been thought that the omicron used by ,ptolemy in his ,almagest1 to mark accidental blanks in the sexagesimal system which he employed1 may have influenced the ,indian writers4#bjj ,this symbol was used quite generally in ,europe and ,asia1 and the ,arabic astronomer ,al-,battani#bja 7died #ibi ,a4,d47 used a similar symbol in connection with the alphabetic system of numerals4 ,the occasional use by ,al-,battani of the ,arabic negative1 la1 to indicate the absence of minutes #ee 7or seconds71 is noted by ,nallino4#bjb ,noteworthy is also the use of the ,circle for unity in the ',sarada characters of the ,kashmirian ,atharva-,veda1 the writing being at least #djj years old4 ,bhaskara 7c4 #aaej7 used a small circle above a number to indicate subtraction1 and in the ,tartar writing a redundant word is removed by drawing an oval around it4 ,it would be interesting to know whether our score mark score mark1 read four in the hole1 could trace its pedigree to the same sources4 ,o',creat#bjc 7c4 #aacj71 in a letter to his teacher1 ,adelhard of ,bath1 uses ,greek3 t for zero1 being an abbreviation for the word teca which we shall see was one of the names used for zero1 although it could quite as well be from ,greek3 tziphra4 ,more rarely ,o',creat uses circle with bar1 applying the name cyfra to both forms4 ,frater ,sigsboto#bjd 7c4 #aaej7 uses the same symbol4 ,other peculiar forms are noted by ,heiberg#bje as being in use among the ,byzantine ,greeks in the fifteenth century4 ,it is evident from the text that some of these writers did not understand the import of the new system4#bjf ,although the dot was used at first in ,india1 as noted above1 the small circle later replaced it and continues in use to this day4 ,the ,arabs1 however1 did not adopt the #ef circle1 since it bore some resemblance to the letter which expressed the number five in the alphabet system4#bjg ,the earliest ,arabic zero known is the dot1 used in a manuscript of #hgc ,a4,d4#bjh ,sometimes both the dot and the circle are used in the same work1 having the same meaning1 which is the case in an ,arabic ,,ms41 an abridged arithmetic of ,jamshid1#bji #ihb ,a4,h4 7#aege ,a4,d474 ,as given in this work the numerals are symbols4 ,the form for #e varies1 in some works becoming symbol or symbol2 symbol is found in ,egypt and symbol appears in some fonts of type4 ,to-day the ,arabs use the #j only when1 under ,european influence1 they adopt the ordinary system4 ,among the ,chinese the first definite trace of zero is in the work of ,tsin#baj of #abdg ,a4,d4 ,the form is the circular one of the ,hindus1 and undoubtedly was brought to ,china by some traveler4 ,the name of this all-important symbol also demands some attention1 especially as we are even yet quite undecided as to what to call it4 ,we speak of it to-day as zero1 naught1 and even cipher2 the telephone operator often calls it ,o1 and the illiterate or careless person calls it aught4 ,in view of all this uncertainty we may well inquire what it has been called in the past4#baa #eg ,as already stated1 the ,hindus called it 'sunya1 void4#bab ,this passed over into the ,arabic as as4-s4ifr or s4ifr4#bac ,when ,leonard of ,pisa 7#abjb7 wrote upon the ,hindu numerals he spoke of this character as zephirum4#bad ,maximus ,planudes 7#accj71 writing under both the ,greek and the ,arabic influence1 called it tziphra4#bae ,in a treatise on arithmetic written in the ,italian language by ,jacob of ,florence#baf #eh 7#acjg7 it is called zeuero1#bag while in an arithmetic of ,giovanni di ,danti of ,arezzo 7#acgj7 the word appears as ceuero4#bah ,another form is zepiro1#bai which was also a step from zephirum to zero4#bbj ,of course the ,english cipher1 ,french chiffre1 is derived from the same ,arabic word1 as4-s4ifr1 but in several languages it has come to mean the numeral figures in general4 ,a trace of this appears in our word ciphering1 meaning figuring or computing4#bba ,johann ,huswirt#bbb uses the word with both meanings2 he gives for the tenth character the four names theca1 circulus1 cifra1 and figura nihili4 ,in this statement ,huswirt probably follows1 as did many writers of that period1 the ,algorismus of ,johannes de ,sacrobosco 7c4 #abej ,a4,d471 who was also known as ,john of ,halifax or ,john of ,holywood4 ,the commentary of #ei ,petrus de ,dacia#bbc 7c4 #abia ,a4,d47 on the ,algorismus vulgaris of ,sacrobosco was also widely used4 ,the widespread use of this ,englishman's work on arithmetic in the universities of that time is attested by the large number#bbd of ,,mss4 from the thirteenth to the seventeenth century still extant1 twenty in ,munich1 twelve in ,vienna1 thirteen in ,erfurt1 several in ,england given by ,halliwell1#bbe ten listed in ,coxe's ,catalogue of the ,oxford ,college ,library1 one in the ,plimpton collection1#bbf one in the ,columbia ,university ,library1 and1 of course1 many others4 ,from as4-s4ifr has come zephyr1 cipher1 and finally the abridged form zero4 ,the earliest printed work in which is found this final form appears to be ,calandri's arithmetic of #adia1bbg while in manuscript it appears at least as early as the middle of the fourteenth century4#bbh ,it also appears in a work1 ,le ,kadran des marchans1 by ,jehan #fj ,certain1#bbi written in #adhe4 ,this word soon became fairly well known in ,spain#bcj and ,france4#bca ,the medieval writers also spoke of it as the sipos1#bcb and occasionally as the wheel1#bcc circulus#bcd 7in ,german das ,ringlein#bce71 circular #fa note1#bcf theca1#bcg long supposed to be from its resemblance to the ,greek theta1 but explained by ,petrus de ,dacia as being derived from the name of the iron#bch used to brand thieves and robbers with a circular mark placed on the forehead or on the cheek4 ,it was also called omicron#bci 7the ,greek o71 being sometimes written o or ,greek3 ph to distinguish it from the letter o4 ,it also went by the name null#bdj 7in the ,latin books #fb nihil#bda or nulla1#bdb and in the ,french rien#bdc71 and very commonly by the name cipher4#bdd ,wallis#bde gives one of the earliest extended discussions of the various forms of the word1 giving certain other variations worthy of note1 as ziphra1 zifera1 siphra1 ciphra1 tsiphra1 tziphra1 and the ,greek ,greek3 tziphra4#bdf 99 99 99 99 99 #fc ,,chapter ,v ,,the ,,question ,,of ,,the ,,introduction ,,of ,,the ,,numerals ,,into ,,europe ,,by ,,boethius ,just as we were quite uncertain as to the origin of the numeral forms1 so too are we uncertain as to the time and place of their introduction into ,europe4 ,there are two general theories as to this introduction4 ,the first is that they were carried by the ,moors to ,spain in the eighth or ninth century1 and thence were transmitted to ,christian ,europe1 a theory which will be considered later4 ,the second1 advanced by ,woepcke1#bdg is that they were not brought to ,spain by the ,moors1 but that they were already in ,spain when the ,arabs arrived there1 having reached the ,west through the ,neo-,pythagoreans4 ,there are two facts to support this second theory3 7#a7 the forms of these numerals are characteristic1 differing materially from those which were brought by ,leonardo of ,pisa from ,northern ,africa early in the thirteenth century 7before #abjb ,a4,d472 7#b7 they are essentially those which #fd tradition has so persistently assigned to ,boethius 7c4 #ejj ,a4,d471 and which he would naturally have received1 if at all1 from these same ,neo-,pythagoreans or from the sources from which they derived them4 ,furthermore1 ,woepcke points out that the ,arabs on entering ,spain 7#gaa ,a4,d47 would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered1#bdh so that the numerals brought from ,spain to ,italy1 not having undergone the same modifications as those of the ,eastern ,arab empire1 would have differed1 as they certainly did1 from those that came through ,bagdad4 ,the theory is that the ,hindu system1 without the zero1 early reached ,alexandria 7say #dej ,a4,d471 and that the ,neo-,pythagorean love for the mysterious and especially for the ,oriental led to its use as something bizarre and cabalistic2 that it was then passed along the ,mediterranean1 reaching ,boethius in ,athens or in ,rome1 and to the schools of ,spain1 being discovered in ,africa and ,spain by the ,arabs even before they themselves knew the improved system with the place value4 #fe ,a recent theory set forth by ,bubnov#bdi also deserves mention1 chiefly because of the seriousness of purpose shown by this well-known writer4 ,bubnov holds that the forms first found in ,europe are derived from ancient symbols used on the abacus1 but that the zero is of ,hindu origin4 ,this theory does not seem tenable1 however1 in the light of the evidence already set forth4 ,two questions are presented by ,woepcke's theory3 7#a7 ,what was the nature of these ,spanish numerals1 and how were they made known to ,italy8 7#b7 ,did ,boethius know them8 ,the ,spanish forms of the numerals were called the h4uruf al-4gobar1 the 4gobar or dust numerals1 as distinguished from the h4uruf al-jumal or alphabetic numerals4 ,probably the latter1 under the influence of the ,syrians or ,jews1#bej were also used by the ,arabs4 ,the significance of the term 4gobar is doubtless that these numerals were written on the dust abacus1 this plan being distinct from the counter method of representing numbers4 ,it is also worthy of note that ,al-,biruni states that the ,hindus often performed numerical computations in the sand4 ,the term is found as early as c4 #iej1 in the verses of an anonymous writer of ,kairwan1 in ,tunis1 in which the author speaks of one of his works on 4gobar calculation2#bea and1 much later1 the ,arab writer ,abu ,bekr ,moh4ammed ibn ,abdallah1 surnamed al-,h4as4s4ar #ff 7the arithmetician71 wrote a work of which the second chapter was ,on the dust figures4#beb ,the 4gobar numerals themselves were first made known to modern scholars by ,silvestre de ,sacy1 who discovered them in an ,arabic manuscript from the library of the ancient abbey of ,st4-,germain-des-,pres4#bec ,the system has nine characters1 but no zero4 ,a dot above a character indicates tens1 two dots hundreds1 and so on1 #e with dot meaning #ej1 and #e with #c dots meaning #ejjj4 ,it has been suggested that possibly these dots1 sprinkled like dust above the numerals1 gave rise to the word 4gobar1#bed but this is not at all probable4 ,this system of dots is found in ,persia at a much later date with numerals quite like the modern ,arabic2#bee but that it was used at all is significant1 for it is hardly likely that the western system would go back to ,persia1 when the perfected ,hindu one was near at hand4 ,at first sight there would seem to be some reason for believing that this feature of the 4gobar system was of #fg ,arabic origin1 and that the present zero of these people1#bef the dot1 was derived from it4 ,it was entirely natural that the ,semitic people generally should have adopted such a scheme1 since their diacritical marks would suggest it1 not to speak of the possible influence of the ,greek accents in the ,hellenic number system4 ,when we consider1 however1 that the dot is found for zero in the ,bakhs4ali manuscript1#beg and that it was used in subscript form in the ,kitab al-,fihrist#beh in the tenth century1 and as late as the sixteenth century1#bei although in this case probably under ,arabic influence1 we are forced to believe that this form may also have been of ,hindu origin4 ,the fact seems to be that1 as already stated1#bfj the ,arabs did not immediately adopt the ,hindu zero1 because it resembled their #e2 they used the superscript dot as serving their purposes fairly well2 they may1 indeed1 have carried this to the west and have added it to the 4gobar forms already there1 just as they transmitted it to the ,persians4 ,furthermore1 the ,arab and ,hebrew scholars of ,northern ,africa in the tenth century knew these numerals as ,indian forms1 for a commentary on the ,sefer ,yes4irah by ,abu ,sahl ibn ,tamim 7probably composed at ,kairwan1 c4 #iej7 speaks of the ,indian arithmetic known under the name of 4gobar or dust calculation4#bfa ,all this suggests that the ,arabs may very #fh likely have known the 4gobar forms before the numerals reached them again in #ggc4bfb ,the term 4gobar numerals was also used without any reference to the peculiar use of dots4#bfc ,in this connection it is worthy of mention that the ,algerians employed two different forms of numerals in manuscripts even of the fourteenth century1#bfd and that the ,moroccans of to-day employ the ,european forms instead of the present ,arabic4 ,the ,indian use of subscript dots to indicate the tens1 hundreds1 thousands1 etc41 is established by a passage in the ,kitab al-,fihrist#bfe 7#ihg ,a4,d47 in which the writer discusses the written language of the people of ,india4 ,notwithstanding the importance of this reference for the early history of the numerals1 it has not been mentioned by previous writers on this subject4 ,the numeral forms given are those which have usually been called ,indian1#bff in opposition to 4gobar4 ,in this document the dots are placed below the characters1 instead of being superposed as described above4 ,the significance was the same4 ,in form these 4gobar numerals resemble our own much more closely than the ,arab numerals do4 ,they varied more or less1 but were substantially as follows3 #fi #abfg;,illustration #bbfh;,illustration #cbfi;,illustration #dbgj;,illustration #ebga;,illustration #fbga;,illustration ,the question of the possible influence of the ,egyptian demotic and hieratic ordinal forms has been so often suggested that it seems well to introduce them at this point1 for comparison with the 4gobar forms4 ,they would as appropriately be used in connection with the ,hindu forms1 and the evidence of a relation of the first three with all these systems is apparent4 ,the only further resemblance is in the ,demotic #d and in the #i1 so that the statement that the ,hindu forms in general came from #gj this source has no foundation4 ,the first four ,egyptian cardinal numerals#bgb resemble more the modern ,arabic4 ,illustration3 ,,demotic ,,and ,,hieratic ,,ordinals ,this theory of the very early introduction of the numerals into ,europe fails in several points4 ,in the first place the early ,western forms are not known2 in the second place some early ,eastern forms are like the 4gobar1 as is seen in the third line on p4 #fi1 where the forms are from a manuscript written at ,shiraz about #igj ,a4,d41 and in which some western ,arabic forms1 e4g4 symbol for #b1 are also used4 ,probably most significant of all is the fact that the 4gobar numerals as given by ,sacy are all1 with the exception of the symbol for eight1 either single ,arabic letters or combinations of letters4 ,so much for the ,woepcke theory and the meaning of the 4gobar numerals4 ,we now have to consider the question as to whether ,boethius knew these 4gobar forms1 or forms akin to them4 ,this large question#bgc suggests several minor ones3 7#a7 ,who was ,boethius8 7#b7 ,could he have known these numerals8 7#c7 ,is there any positive or strong circumstantial evidence that he did know them8 7#d7 ,what are the probabilities in the case8 #ga ,first1 who was ,boethius1--,divus#bgd ,boethius as he was called in the ,middle ,ages8 ,anicius ,manlius ,severinus ,boethius#bge was born at ,rome c4 #dge4 ,he was a member of the distinguished family of the ,anicii1#bgf which had for some time before his birth been ,christian4 ,early left an orphan1 the tradition is that he was taken to ,athens at about the age of ten1 and that he remained there eighteen years4#bgg ,he married ,rusticiana1 daughter of the senator ,symmachus1 and this union of two such powerful families allowed him to move in the highest circles4#bgh ,standing strictly for the right1 and against all iniquity at court1 he became the object of hatred on the part of all the unscrupulous element near the throne1 and his bold defense of the ex-consul ,albinus1 unjustly accused of treason1 led to his imprisonment at ,pavia#bgi and his execution in #ebd4bhj ,not many generations after his death1 the period being one in which historical criticism was at its lowest ebb1 the church found it profitable to look upon his execution as a martyrdom4#bha ,he was #gb accordingly looked upon as a saint1#bhb his bones were enshrined1#bhc and as a natural consequence his books were among the classics in the church schools for a thousand years4#bhd ,it is pathetic1 however1 to think of the medieval student trying to extract mental nourishment from a work so abstract1 so meaningless1 so unnecessarily complicated1 as the arithmetic of ,boethius4 ,he was looked upon by his contemporaries and immediate successors as a master1 for ,cassiodorus#bhe 7c4 #dij-;c4 #ehe ,a4,d47 says to him3 ,through your translations the music of ,pythagoras and the astronomy of ,ptolemy are read by those of ,italy1 and the arithmetic of ,nicomachus and the geometry of ,euclid are known to those of the ,west4#bhf ,founder of the medieval scholasticism1 #gc distinguishing the trivium and quadrivium1#bhg writing the only classics of his time1 ,gibbon well called him the last of the ,romans whom ,cato or ,tully could have acknowledged for their countryman4#bhh ,the second question relating to ,boethius is this3 ,could he possibly have known the ,hindu numerals8 ,in view of the relations that will be shown to have existed between the ,east and the ,west1 there can only be an affirmative answer to this question4 ,the numerals had existed1 without the zero1 for several centuries2 they had been well known in ,india2 there had been a continued interchange of thought between the ,east and ,west2 and warriors1 ambassadors1 scholars1 and the restless trader1 all had gone back and forth1 by land or more frequently by sea1 between the ,mediterranean lands and the centers of ,indian commerce and culture4 ,boethius could very well have learned one or more forms of ,hindu numerals from some traveler or merchant4 ,to justify this statement it is necessary to speak more fully of these relations between the ,far ,east and ,europe4 ,it is true that we have no records of the interchange of learning1 in any large way1 between eastern ,asia and central ,europe in the century preceding the time of ,boethius4 ,but it is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge4 #gd ,as a matter of fact there is abundant reason for believing that ,hindu numerals would naturally have been known to the ,arabs1 and even along every trade route to the remote west1 long before the zero entered to make their place-value possible1 and that the characters1 the methods of calculating1 the improvements that took place from time to time1 the zero when it appeared1 and the customs as to solving business problems1 would all have been made known from generation to generation along these same trade routes from the ,orient to the ,occident4 ,it must always be kept in mind that it was to the tradesman and the wandering scholar that the spread of such learning was due1 rather than to the school man4 ,indeed1 ,avicenna#bhi 7#ihj-ajcg ,a4,d47 in a short biography of himself relates that when his people were living at ,bokhara his father sent him to the house of a grocer to learn the ,hindu art of reckoning1 in which this grocer 7oil dealer1 possibly7 was expert4 ,leonardo of ,pisa1 too1 had a similar training4 ,the whole question of this spread of mercantile knowledge along the trade routes is so connected with the 4gobar numerals1 the ,boethius question1 ,gerbert1 ,leonardo of ,pisa1 and other names and events1 that a digression for its consideration now becomes necessary4#bij #ge ,even in very remote times1 before the ,hindu numerals were sculptured in the cave of ,nana ,ghat1 there were trade relations between ,arabia and ,india4 ,indeed1 long before the ,aryans went to ,india the great ,turanian race had spread its civilization from the ,mediterranean to the ,indus4#bia ,at a much later period the ,arabs were the intermediaries between ,egypt and ,syria on the west1 and the farther ,orient4#bib ,in the sixth century ,b4,c41 ,hecataeus1#bic the father of geography1 was acquainted not only with the ,mediterranean lands but with the countries as far as the ,indus1#bid and in ,biblical times there were regular triennial voyages to ,india4 ,indeed1 the story of ,joseph bears witness to the caravan trade from ,india1 across ,arabia1 and on to the banks of the ,nile4 ,about the same time as ,hecataeus1 ,scylax1 a ,persian admiral under ,darius1 from ,caryanda on the coast of ,asia ,minor1 traveled to #gf northwest ,india and wrote upon his ventures4#bie ,he induced the nations along the ,indus to acknowledge the ,persian supremacy1 and such number systems as there were in these lands would naturally have been known to a man of his attainments4 ,a century after ,scylax1 ,herodotus showed considerable knowledge of ,india1 speaking of its cotton and its gold1#bif telling how ,sesostris#big fitted out ships to sail to that country1 and mentioning the routes to the east4 ,these routes were generally by the ,red ,sea1 and had been followed by the ,phoenicians and the ,sabaeans1 and later were taken by the ,greeks and ,romans4#bih ,in the fourth century ,b4,c4 the ,west and ,east came into very close relations4 ,as early as #ccj1 ,pytheas of ,massilia 7,marseilles7 had explored as far north as the northern end of the ,british ,isles and the coasts of the ,german ,sea1 while ,macedon1 in close touch with southern ,france1 was also sending her armies under ,alexander#bii through ,afghanistan as far east as the ,punjab4#cjj ,pliny tells us that ,alexander the ,great employed surveyors to measure #gg the roads of ,india2 and one of the great highways is described by ,megasthenes1 who in #bie ,b4,c41 as the ambassador of ,seleucus1 resided at ,pataliput4ra1 the present ,patna4#cja ,the ,hindus also learned the art of coining from the ,greeks1 or possibly from the ,chinese1 and the stores of ,greco-,hindu coins still found in northern ,india are a constant source of historical information4#cjb ,the ,ramayana speaks of merchants traveling in great caravans and embarking by sea for foreign lands4#cjc ,ceylon traded with ,malacca and ,siam1 and ,java was colonized by ,hindu traders1 so that mercantile knowledge was being spread about the ,indies during all the formative period of the numerals4 ,moreover the results of the early ,greek invasion were embodied by ,dicaearchus of ,messana 7about #cbj ,b4,c47 in a map that long remained a standard4 ,furthermore1 ,alexander did not allow his influence on the ,east to cease4 ,he divided ,india into three satrapies1#cjd placing ,greek governors over two of them and leaving a ,hindu ruler in charge of the third1 and in ,bactriana1 a part of ,ariana or ancient ,persia1 he left governors2 and in these the western civilization was long in evidence4 ,some of the ,greek and ,roman metrical and astronomical terms #gh found their way1 doubtless at this time1 into the ,sanskrit language4#cje ,even as late as from the second to the fifth centuries ,a4,d41 ,indian coins showed the ,hellenic influence4 ,the ,hindu astronomical terminology reveals the same relationship to western thought1 for ,varaha-,mihira 7#fth century ,a4,d471 a contemporary of ,aryabhat4a1 entitled a work of his the ,br4hat-,sam4hita1 a literal translation of ,greek3 megale suntaxis of ,ptolemy2#cjf and in various ways is this interchange of ideas apparent4#cjg ,it could not have been at all unusual for the ancient ,greeks to go to ,india1 for ,strabo lays down the route1 saying that all who make the journey start from ,ephesus and traverse ,phrygia and ,cappadocia before taking the direct road4#cjh ,the products of the ,east were always finding their way to the ,west1 the ,greeks getting their ginger#cji from ,malabar1 as the ,phoenicians had long before brought gold from ,malacca4 ,greece must also have had early relations with ,china1 for there is a notable similarity between the ,greek and ,chinese life1 as is shown in their houses1 their domestic customs1 their marriage ceremonies1 the public story-tellers1 the puppet shows which ,herodotus says were introduced from ,egypt1 the street jugglers1 the games of dice1#caj the game of finger-guessing1#caa the water clock1 the #gi music system1 the use of the myriad1#cab the calendars1 and in many other ways4#cac ,in passing through the suburbs of ,peking to-day1 on the way to the ,great ,bell temple1 one is constantly reminded of the semi-,greek architecture of ,pompeii1 so closely does modern ,china touch the old classical civilization of the ,mediterranean4 ,the ,chinese historians tell us that about #bjj ,b4,c4 their arms were successful in the far west1 and that in #ahj ,b4,c4 an ambassador went to ,bactria1 then a ,greek city1 and reported that ,chinese products were on sale in the markets there4#cad ,there is also a noteworthy resemblance between certain ,greek and ,chinese words1#cae showing that in remote times there must have been more or less interchange of thought4 ,the ,romans also exchanged products with the ,east4 ,horace says1 ,a busy trader1 you hasten to the farthest ,indies1 flying from poverty over sea1 over crags1 over fires4#caf ,the products of the ,orient1 spices and jewels from ,india1 frankincense from ,persia1 and silks from ,china1 being more in demand than the exports from the ,mediterranean lands1 the balance of trade was against the ,west1 and thus ,roman coin found its way eastward4 ,in #ahih1 for example1 a number of ,roman coins dating from #aad ,b4,c4 to ,hadrian's time were found at ,pakli1 a part of the ,hazara district1 sixteen miles north of ,abbottabad1#cag and numerous similar discoveries have been made from time to time4 #hj ,augustus speaks of envoys received by him from ,india1 a thing never before known1#cah and it is not improbable that he also received an embassy from ,china4#cai ,suetonius 7first century ,a4,d47 speaks in his history of these relations1#cbj as do several of his contemporaries1#cba and ,vergil#cbb tells of ,augustus doing battle in ,persia4 ,in ,pliny's time the trade of the ,roman ,empire with ,asia amounted to a million and a quarter dollars a year1 a sum far greater relatively then than now1#cbc while by the time of ,constantine ,europe was in direct communication with the ,far ,east4#cbd ,in view of these relations it is not beyond the range of possibility that proof may sometime come to light to show that the ,greeks and ,romans knew something of the #ha number system of ,india1 as several writers have maintained4#cbe ,returning to the ,east1 there are many evidences of the spread of knowledge in and about ,india itself4 ,in the third century ,b4,c4 ,buddhism began to be a connecting medium of thought4 ,it had already permeated the ,himalaya territory1 had reached eastern ,turkestan1 and had probably gone thence to ,china4 ,some centuries later 7in #fb ,a4,d47 the ,chinese emperor sent an ambassador to ,india1 and in #fg ,a4,d4 a ,buddhist monk was invited to ,china4#cbf ,then1 too1 in ,india itself ,a'soka1 whose name has already been mentioned in this work1 extended the boundaries of his domains even into ,afghanistan1 so that it was entirely possible for the numerals of the ,punjab to have worked their way north even at that early date4#cbg ,furthermore1 the influence of ,persia must not be forgotten in considering this transmission of knowledge4 ,in the fifth century the ,persian medical school at ,jondi-,sapur admitted both the ,hindu and the ,greek doctrines1 and ,firdusi tells us that during the brilliant reign of #hb ,khosru ,i1#cbh the golden age of ,pahlavi literature1 the ,hindu game of chess was introduced into ,persia1 at a time when wars with the ,greeks were bringing prestige to the ,sassanid dynasty4 ,again1 not far from the time of ,boethius1 in the sixth century1 the ,egyptian monk ,cosmas1 in his earlier years as a trader1 made journeys to ,abyssinia and even to ,india and ,ceylon1 receiving the name ,indicopleustes 7the ,indian traveler74 ,his map 7#edg ,a4,d47 shows some knowledge of the earth from the ,atlantic to ,india4 ,such a man would1 with hardly a doubt1 have observed every numeral system used by the people with whom he sojourned1#cbi and whether or not he recorded his studies in permanent form he would have transmitted such scraps of knowledge by word of mouth4 ,as to the ,arabs1 it is a mistake to feel that their activities began with ,mohammed4 ,commerce had always been held in honor by them1 and the ,qoreish#ccj had annually for many generations sent caravans bearing the spices and textiles of ,yemen to the shores of the ,mediterranean4 ,in the fifth century they traded by sea with ,india and even with ,china1 and ,h4ira was an emporium for the wares of the ,east1#cca so that any numeral system of any part of the trading world could hardly have remained isolated4 ,long before the warlike activity of the ,arabs1 ,alexandria had become the great market-place of the world4 ,from this center caravans traversed ,arabia to ,hadramaut1 where they met ships from ,india4 ,others went north to ,damascus1 while still others made their way #hc along the southern shores of the ,mediterranean4 ,ships sailed from the isthmus of ,suez to all the commercial ports of ,southern ,europe and up into the ,black ,sea4 ,hindus were found among the merchants#ccb who frequented the bazaars of ,alexandria1 and ,brahmins were reported even in ,byzantium4 ,such is a very brief resume of the evidence showing that the numerals of the ,punjab and of other parts of ,india as well1 and indeed those of ,china and farther ,persia1 of ,ceylon and the ,malay peninsula1 might well have been known to the merchants of ,alexandria1 and even to those of any other seaport of the ,mediterranean1 in the time of ,boethius4 ,the ,brahmi numerals would not have attracted the attention of scholars1 for they had no zero so far as we know1 and therefore they were no better and no worse than those of dozens of other systems4 ,if ,boethius was attracted to them it was probably exactly as any one is naturally attracted to the bizarre or the mystic1 and he would have mentioned them in his works only incidentally1 as indeed they are mentioned in the manuscripts in which they occur4 ,in answer therefore to the second question1 ,could ,boethius have known the ,hindu numerals8 the reply must be1 without the slightest doubt1 that he could easily have known them1 and that it would have been strange if a man of his inquiring mind did not pick up many curious bits of information of this kind even though he never thought of making use of them4 ,let us now consider the third question1 ,is there any positive or strong circumstantial evidence that ,boethius did know these numerals8 ,the question is not new1 #hd nor is it much nearer being answered than it was over two centuries ago when ,wallis 7#afic7 expressed his doubts about it#ccc soon after ,vossius 7#afeh7 had called attention to the matter4#ccd ,stated briefly1 there are three works on mathematics attributed to ,boethius3#cce 7#a7 the arithmetic1 7#b7 a work on music1 and 7#c7 the geometry4#ccf ,the genuineness of the arithmetic and the treatise on music is generally recognized1 but the geometry1 which contains the ,hindu numerals with the zero1 is under suspicion4#ccg ,there are plenty of supporters of the idea that ,boethius knew the numerals and included them in this book1#cch and on the other hand there are as many who #he feel that the geometry1 or at least the part mentioning the numerals1 is spurious4#cci ,the argument of those who deny the authenticity of the particular passage in question may briefly be stated thus3 #a4 ,the falsification of texts has always been the subject of complaint4 ,it was so with the ,romans1#cdj it was common in the ,middle ,ages1#cda and it is much more prevalent #hf to-day than we commonly think4 ,we have but to see how every hymn-book compiler feels himself authorized to change at will the classics of our language1 and how unknown editors have mutilated ,shakespeare1 to see how much more easy it was for medieval scribes to insert or eliminate paragraphs without any protest from critics4#cdb #b4 ,if ,boethius had known these numerals he would have mentioned them in his arithmetic1 but he does not do so4#cdc #c4 ,if he had known them1 and had mentioned them in any of his works1 his contemporaries1 disciples1 and successors would have known and mentioned them4 ,but neither ,capella 7c4 #dge7cdd nor any of the numerous medieval writers who knew the works of ,boethius makes any reference to the system4#cde #hg #d4 ,the passage in question has all the appearance of an interpolation by some scribe4 ,boethius is speaking of angles1 in his work on geometry1 when the text suddenly changes to a discussion of classes of numbers4#cdf ,this is followed by a chapter in explanation of the abacus1#cdg in which are described those numeral forms which are called apices or caracteres4#cdh ,the forms#cdi of these characters vary in different manuscripts1 but in general are about as shown on page #hh4 ,they are commonly written with the #i at the left1 decreasing to the unit at the right1 numerous writers stating that this was because they were derived from ,semitic sources in which the direction of writing is the opposite of our own4 ,this practice continued until the sixteenth century4#cej ,the writer then leaves the subject entirely1 using the ,roman numerals for the rest of his discussion1 a proceeding so foreign to the method of ,boethius as to be inexplicable on the hypothesis of authenticity4 ,why should such a scholarly writer have given them with no mention of their origin or use8 ,either he would have mentioned some historical interest attaching to them1 or he would have used them in some discussion2 he certainly would not have left the passage as it is4 #hh ,,forms ,,of ,,the ,,numerals1 ,,largely ,,from ,,works ,,on ,,the ,,abacus#cea a#ceb ,illustration b#cec ,illustration c#ced ,illustration d#cee ,illustration e#cef ,illustration f#ceg ,illustration g#ceh ,illustration h#cei ,illustration i#cfj ,illustration #hi ,sir ,e4 ,clive ,bayley has added#cfa a further reason for believing them spurious1 namely that the #d is not of the ,nana ,ghat type1 but of the ,kabul form which the ,arabs did not receive until #ggf2cfb so that it is not likely1 even if the characters were known in ,europe in the time of ,boethius1 that this particular form was recognized4 ,it is worthy of mention1 also1 that in the six abacus forms from the chief manuscripts as given by ,friedlein1#cfc each contains some form of zero1 which symbol probably originated in ,india about this time or later4 ,it could hardly have reached ,europe so soon4 ,as to the fourth question1 ,did ,boethius probably know the numerals8 ,it seems to be a fair conclusion1 according to our present evidence1 that 7#a7 ,boethius might very easily have known these numerals without the zero1 but1 7#b7 there is no reliable evidence that he did know them4 ,and just as ,boethius might have come in contact with them1 so any other inquiring mind might have done so either in his time or at any time before they definitely appeared in the tenth century4 ,these centuries1 five in number1 represented the darkest of the ,dark ,ages1 and even if these numerals were occasionally met and studied1 no trace of them would be likely to show itself in the #ij literature of the period1 unless by chance it should get into the writings of some man like ,alcuin4 ,as a matter of fact1 it was not until the ninth or tenth century that there is any tangible evidence of their presence in ,christendom4 ,they were probably known to merchants here and there1 but in their incomplete state they were not of sufficient importance to attract any considerable attention4 ,as a result of this brief survey of the evidence several conclusions seem reasonable3 7#a7 commerce1 and travel for travel's sake1 never died out between the ,east and the ,west2 7#b7 merchants had every opportunity of knowing1 and would have been unreasonably stupid if they had not known1 the elementary number systems of the peoples with whom they were trading1 but they would not have put this knowledge in permanent written form2 7#c7 wandering scholars would have known many and strange things about the peoples they met1 but they too were not1 as a class1 writers2 7#d7 there is every reason a priori for believing that the 4gobar numerals would have been known to merchants1 and probably to some of the wandering scholars1 long before the ,arabs conquered northern ,africa2 7#e7 the wonder is not that the ,hindu-,arabic numerals were known about #ajjj ,a4,d41 and that they were the subject of an elaborate work in #abjb by ,fibonacci1 but rather that more extended manuscript evidence of their appearance before that time has not been found4 ,that they were more or less known early in the ,middle ,ages1 certainly to many merchants of ,christian ,europe1 and probably to several scholars1 but without the zero1 is hardly to be doubted4 ,the lack of documentary evidence is not at all strange1 in view of all of the circumstances4 99 99 99 99 99 #ia ,,chapter ,,vi ,,the ,,development ,,of ,,the ,,numerals ,,among ,,the ,,arabs ,if the numerals had their origin in ,india1 as seems most probable1 when did the ,arabs come to know of them8 ,it is customary to say that it was due to the influence of ,mohammedanism that learning spread through ,persia and ,arabia2 and so it was1 in part4 ,but learning was already respected in these countries long before ,mohammed appeared1 and commerce flourished all through this region4 ,in ,persia1 for example1 the reign of ,khosru ,nu'sirwan1#cfd the great contemporary of ,justinian the law-maker1 was characterized not only by an improvement in social and economic conditions1 but by the cultivation of letters4 ,khosru fostered learning1 inviting to his court scholars from ,greece1 and encouraging the introduction of culture from the ,west as well as from the ,east4 ,at this time ,aristotle and ,plato were translated1 and portions of the ,hito-pade'sa1 or ,fables of ,pilpay1 were rendered from the ,sanskrit into ,persian4 ,all this means that some three centuries before the great intellectual ascendancy of ,bagdad a similar fostering of learning was taking place in ,persia1 and under pre-,mohammedan influences4 #ib ,the first definite trace that we have of the introduction of the ,hindu system into ,arabia dates from #ggc ,a4,d41#cfe when an ,indian astronomer visited the court of the caliph1 bringing with him astronomical tables which at the caliph's command were translated into ,arabic by ,al-,fazari4#cff ,al-,khowarazmi and ,h4abash 7,ah4med ibn ,abdallah1 died c4 #hgj7 based their well-known tables upon the work of ,al-,fazari4 ,it may be asserted as highly probable that the numerals came at the same time as the tables4 ,they were certainly known a few decades later1 and before #hbe ,a4,d41 about which time the original of the ,algoritmi de numero ,indorum was written1 as that work makes no pretense of being the first work to treat of the ,hindu numerals4 ,the three writers mentioned cover the period from the end of the eighth to the end of the ninth century4 ,while the historians ,al-,ma'sudi and ,al-,biruni follow quite closely upon the men mentioned1 it is well to note again the ,arab writers on ,hindu arithmetic1 contemporary with ,al-,khowarazmi1 who were mentioned in chapter ,i1 viz4 ,al-,kindi1 ,sened ibn ,ali1 and ,al-,s4ufi4 ,for over five hundred years ,arabic writers and others continued to apply to works on arithmetic the name ,indian4 ,in the tenth century such writers are ,abdallah ibn al-,h4asan1 ,abu 'l-,qasim#cfg 7died #ihg ,a4,d47 of ,antioch1 and ,moh4ammed ibn ,abdallah1 ,abu ,nas4r#cfh 7c4 #ihb71 of ,kalwada near ,bagdad4 ,others of the same period or #ic earlier 7since they are mentioned in the ,fihrist1#cfi #ihg ,a4,d471 who explicitly use the word ,hindu or ,indian1 are ,sinan ibn al-,fath4#cgj of ,h4arran1 and ,ahmed ibn ,omar1 al-,karabisi4#cga ,in the eleventh century come ,al-,biruni#cgb 7#igc-ajdh7 and ,ali ibn ,ah4med1 ,abu 'l-,h4asan1 ,al-,nasawi#cgc 7c4 #ajcj74 ,the following century brings similar works by ,ishaq ibn ,yusuf al-,s4ardafi#cgd and ,samu'il ibn ,yah4ya ibn ,abbas al-,ma4grebi al-,andalusi#cge 7c4 #aagd71 and in the thirteenth century are ,abdallatif ibn ,yusuf ibn ,moh4ammed1 ,muwaffaq al-,din ,abu ,moh4ammed al-,ba4gdadi#cgf 7c4 #abca71 and ,ibn al-,banna4#cgg ,the ,greek monk ,maximus ,planudes1 writing in the first half of the fourteenth century1 followed the ,arabic usage in calling his work ,indian ,arithmetic4#cgh ,there were numerous other ,arabic writers upon arithmetic1 as that subject occupied one of the high places among the sciences1 but most of them did not feel it necessary to refer to the origin of the symbols1 the knowledge of which might well have been taken for granted4 #id ,one document1 cited by ,woepcke1#cgi is of special interest since it shows at an early period1 #igj ,a4,d41 the use of the ordinary ,arabic forms alongside the 4gobar4 ,the title of the work is ,interesting and ,beautiful ,problems on ,numbers copied by ,ah4med ibn ,moh4ammed ibn ,abdaljalil1 ,abu ,said1 al-,sijzi1#chj 7#iea-ajbd7 from a work by a priest and physician1 ,naz4if ibn ,yumn1#cha al-,qass 7died c4 #iij74 ,suter does not mention this work of ,naz4if4 ,the second reason for not ascribing too much credit to the purely ,arab influence is that the ,arab by himself never showed any intellectual strength4 ,what took place after ,moh4ammed had lighted the fire in the hearts of his people was just what always takes place when different types of strong races blend1--a great renaissance in divers lines4 ,it was seen in the blending of such types at ,miletus in the time of ,thales1 at ,rome in the days of the early invaders1 at ,alexandria when the ,greek set firm foot on ,egyptian soil1 and we see it now when all the nations mingle their vitality in the ,new ,world4 ,so when the ,arab culture joined with the ,persian1 a new civilization rose and flourished4#chb ,the ,arab influence came not from its purity1 but from its intermingling with an influence more cultured if less virile4 ,as a result of this interactivity among peoples of diverse interests and powers1 ,mohammedanism was to the world from the eighth to the thirteenth century what ,rome and ,athens and the ,italo-,hellenic influence generally had #ie been to the ancient civilization4 ,if they did not possess the spirit of invention which distinguished the ,greeks and the ,hindus1 if they did not show the perseverance in their observations that characterized the ,chinese astronomers1 they at least possessed the virility of a new and victorious people1 with a desire to understand what others had accomplished1 and a taste which led them with equal ardor to the study of algebra and of poetry1 of philosophy and of language4#chc ,it was in #fbb ,a4,d4 that ,moh4ammed fled from ,mecca1 and within a century from that time the crescent had replaced the cross in ,christian ,asia1 in ,northern ,africa1 and in a goodly portion of ,spain4 ,the ,arab empire was an ellipse of learning with its foci at ,bagdad and ,cordova1 and its rulers not infrequently took pride in demanding intellectual rather than commercial treasure as the result of conquest4#chd ,it was under these influences1 either pre-,mohammedan or later1 that the ,hindu numerals found their way to the ,north4 ,if they were known before ,moh4ammed's time1 the proof of this fact is now lost4 ,this much1 however1 is known1 that in the eighth century they were taken to ,bagdad4 ,it was early in that century that the ,mohammedans obtained their first foothold in northern ,india1 thus foreshadowing an epoch of supremacy that endured with varied fortunes until after the golden age of ,akbar the ,great 7#aedb-afje7 and ,shah ,jehan4 ,they also conquered ,khorassan and ,afghanistan1 so that the learning and the commercial customs of ,india at once found easy #if access to the newly-established schools and the bazaars of ,mesopotamia and western ,asia4 ,the particular paths of conquest and of commerce were either by way of the ,khyber ,pass and through ,kabul1 ,herat and ,khorassan1 or by sea through the strait of ,ormuz to ,basra 7,busra7 at the head of the ,persian ,gulf1 and thence to ,bagdad4 ,as a matter of fact1 one form of ,arabic numerals1 the one now in use by the ,arabs1 is attributed to the influence of ,kabul1 while the other1 which eventually became our numerals1 may very likely have reached ,arabia by the other route4 ,it is in ,bagdad1#che ,dar al-,salam--the ,abode of ,peace1 that our special interest in the introduction of the numerals centers4 ,built upon the ruins of an ancient town by ,al-,mans4ur#chf in the second half of the eighth century1 it lies in one of those regions where the converging routes of trade give rise to large cities4#chg ,quite as well of ,bagdad as of ,athens might ,cardinal ,newman have said3#chh ,what it lost in conveniences of approach1 it gained in its neighborhood to the traditions of the mysterious ,east1 and in the loveliness of the region in which it lay4 ,hither1 then1 as to a sort of ideal land1 where all archetypes of the great and the fair were found in substantial being1 and all departments of truth explored1 and all diversities of intellectual power exhibited1 where taste and philosophy were majestically enthroned as in a royal court1 where there was no sovereignty but that of mind1 and no nobility but that of genius1 where professors were #ig rulers1 and princes did homage1 thither flocked continually from the very corners of the orbis terrarum the many-tongued generation1 just rising1 or just risen into manhood1 in order to gain wisdom4 ,for here it was that ,al-,mans4ur and ,al-,mamun and ,harun al-,rashid 7,aaron the ,just7 made for a time the world's center of intellectual activity in general and in the domain of mathematics in particular4#chi ,it was just after the ,sindhind was brought to ,bagdad that ,moh4ammed ibn ,musa al-,khowarazmi1 whose name has already been mentioned1#cij was called to that city4 ,he was the most celebrated mathematician of his time1 either in the ,east or ,west1 writing treatises on arithmetic1 the sundial1 the astrolabe1 chronology1 geometry1 and algebra1 and giving through the ,latin transliteration of his name1 algoritmi1 the name of algorism to the early arithmetics using the new ,hindu numerals4#cia ,appreciating at once the value of the position system so recently brought from ,india1 he wrote an arithmetic based upon these numerals1 and this was translated into ,latin in the time of ,adelhard of ,bath 7c4 #aahj71 although possibly by his contemporary countryman ,robert ,cestrensis4#cib ,this translation was found in ,cambridge and was published by ,boncompagni in #aheg4cic ,contemporary with ,al-,khowarazmi1 and working also under ,al-,mamun1 was a ,jewish astronomer1 ,abu 'l-,t4eiyib1 #ih ,sened ibn ,ali1 who is said to have adopted the ,mohammedan religion at the caliph's request4 ,he also wrote a work on ,hindu arithmetic1#cid so that the subject must have been attracting considerable attention at that time4 ,indeed1 the struggle to have the ,hindu numerals replace the ,arabic did not cease for a long time thereafter4 ,ali ibn ,ah4med al-,nasawi1 in his arithmetic of c4 #ajbe1 tells us that the symbolism of number was still unsettled in his day1 although most people preferred the strictly ,arabic forms4#cie ,we thus have the numerals in ,arabia1 in two forms3 one the form now used there1 and the other the one used by ,al-,khowarazmi4 ,the question then remains1 how did this second form find its way into ,europe8 and this question will be considered in the next chapter4 99 99 99 99 99 #ii ,,chapter ,,vii ,,the ,,definite ,,introduction ,,of ,,the ,,numerals ,,into ,,europe ,it being doubtful whether ,boethius ever knew the ,hindu numeral forms1 certainly without the zero in any case1 it becomes necessary now to consider the question of their definite introduction into ,europe4 ,from what has been said of the trade relations between the ,east and the ,west1 and of the probability that it was the trader rather than the scholar who carried these numerals from their original habitat to various commercial centers1 it is evident that we shall never know when they first made their inconspicuous entrance into ,europe4 ,curious customs from the ,east and from the tropics1--concerning games1 social peculiarities1 oddities of dress1 and the like1--are continually being related by sailors and traders in their resorts in ,new ,york1 ,london1 ,hamburg1 and ,rotterdam to-day1 customs that no scholar has yet described in print and that may not become known for many years1 if ever4 ,and if this be so now1 how much more would it have been true a thousand years before the invention of printing1 when learning was at its lowest ebb4 ,it was at this period of low esteem of culture that the ,hindu numerals undoubtedly made their first appearance in ,europe4 ,there were many opportunities for such knowledge to reach ,spain and ,italy4 ,in the first place the ,moors went into ,spain as helpers of a claimant of the throne1 and #ajj remained as conquerors4 ,the power of the ,goths1 who had held ,spain for three centuries1 was shattered at the battle of ,jerez de la ,frontera in #gaa1 and almost immediately the ,moors became masters of ,spain and so remained for five hundred years1 and masters of ,granada for a much longer period4 ,until #hej the ,christians were absolutely free as to religion and as to holding political office1 so that priests and monks were not infrequently skilled both in ,latin and ,arabic1 acting as official translators1 and naturally reporting directly or indirectly to ,rome4 ,there was indeed at this time a complaint that ,christian youths cultivated too assiduously a love for the literature of the ,saracen1 and married too frequently the daughters of the infidel4#cif ,it is true that this happy state of affairs was not permanent1 but while it lasted the learning and the customs of the ,east must have become more or less the property of ,christian ,spain4 ,at this time the 4gobar numerals were probably in that country1 and these may well have made their way into ,europe from the schools of ,cordova1 ,granada1 and ,toledo4 ,furthermore1 there was abundant opportunity for the numerals of the ,east to reach ,europe through the journeys of travelers and ambassadors4 ,it was from the records of ,suleiman the ,merchant1 a well-known ,arab trader of the ninth century1 that part of the story of ,sindbad the ,sailor was taken4#cig ,such a merchant would have been particularly likely to know the numerals of the people whom he met1 and he is a type of man that may well have taken such symbols to ,european markets4 ,a little later1 #aja ,abu 'l-,h4asan ,ali al-,masudi 7d4 #ief7 of ,bagdad traveled to the ,china ,sea on the east1 at least as far south as ,zanzibar1 and to the ,atlantic on the west1#cih and he speaks of the nine figures with which the ,hindus reckoned4#cii ,there was also a ,bagdad merchant1 one ,abu 'l-,qasim ,obeidallah ibn ,ah4med1 better known by his ,persian name ,ibn ,khordad4beh1#djj who wrote about #hej ,a4,d4 a work entitled ,book of ,roads and ,provinces#dja in which the following graphic account appears3#djb ,the ,jewish merchants speak ,persian1 ,roman 7,greek and ,latin71 ,arabic1 ,french1 ,spanish1 and ,slavic4 ,they travel from the ,west to the ,east1 and from the ,east to the ,west1 sometimes by land1 sometimes by sea4 ,they take ship from ,france on the ,western ,sea1 and they voyage to ,farama 7near the ruins of the ancient ,pelusium72 there they transfer their goods to caravans and go by land to ,colzom 7on the ,red ,sea74 ,they there reembark on the ,oriental 7,red7 ,sea and go to ,hejaz and to ,jiddah1 and thence to the ,sind1 ,india1 and ,china4 ,returning1 they bring back the products of the oriental lands4444 ,these journeys are also made by land4 ,the merchants1 leaving ,france and ,spain1 cross to ,tangier and thence pass through the ,african provinces and ,egypt4 ,they then go to ,ramleh1 visit ,damascus1 ,kufa1 ,bagdad1 and ,basra1 penetrate into ,ahwaz1 ,fars1 ,kerman1 ,sind1 and thus reach ,india and ,china4 ,such travelers1 about #ijj ,a4,d41 must necessarily have spread abroad a knowledge of all number #ajb systems used in recording prices or in the computations of the market4 ,there is an interesting witness to this movement1 a cruciform brooch now in the ,british ,museum4 ,it is ,english1 certainly as early as the eleventh century1 but it is inlaid with a piece of paste on which is the ,mohammedan inscription1 in ,kufic characters1 ,there is no ,god but ,god4 ,how did such an inscription find its way1 perhaps in the time of ,alcuin of ,york1 to ,england8 ,and if these ,kufic characters reached there1 then why not the numeral forms as well8 ,even in literature of the better class there appears now and then some stray proof of the important fact that the great trade routes to the far ,east were never closed for long1 and that the customs and marks of trade endured from generation to generation4 ,the ,gulistan of the ,persian poet ,sadi#djc contains such a passage3 ,i met a merchant who owned one hundred and forty camels1 and fifty slaves and porters4444 ,he answered to me3 ',i want to carry sulphur of ,persia to ,china1 which in that country1 as ,i hear1 bears a high price2 and thence to take ,chinese ware to ,roum2 and from ,roum to load up with brocades for ,hind2 and so to trade ,indian steel 7pulab7 to ,halib4 ,from ,halib ,i will convey its glass to ,yeman1 and carry the painted cloths of ,yeman back to ,persia4'#djd ,on the other hand1 these men were not of the learned class1 nor would they preserve in treatises any knowledge that they might have1 although this knowledge would occasionally reach the ears of the learned as bits of curious information4 #ajc ,there were also ambassadors passing back and forth from time to time1 between the ,east and the ,west1 and in particular during the period when these numerals probably began to enter ,europe4 ,thus ,charlemagne 7c4 #hjj7 sent emissaries to ,bagdad just at the time of the opening of the mathematical activity there4#dje ,and with such ambassadors must have gone the adventurous scholar1 inspired1 as ,alcuin says of ,archbishop ,albert of ,york 7#gff-ghj71djf to seek the learning of other lands4 ,furthermore1 the ,nestorian communities1 established in ,eastern ,asia and in ,india at this time1 were favored both by the ,persians and by their ,mohammedan conquerors4 ,the ,nestorian ,patriarch of ,syria1 ,timotheus 7#ggh-hbj71 sent missionaries both to ,india and to ,china1 and a bishop was appointed for the latter field4 ,ibn ,wahab1 who traveled to ,china in the ninth century1 found images of ,christ and the apostles in the ,emperor's court4#djg ,such a learned body of men1 knowing intimately the countries in which they labored1 could hardly have failed to make strange customs known as they returned to their home stations4 ,then1 too1 in ,alfred's time 7#hdi-ija7 emissaries went #ajd from ,england as far as ,india1#djh and generally in the ,middle ,ages groceries came to ,europe from ,asia as now they come from the colonies and from ,america4 ,syria1 ,asia ,minor1 and ,cyprus furnished sugar and wool1 and ,india yielded her perfumes and spices1 while rich tapestries for the courts and the wealthy burghers came from ,persia and from ,china4#dji ,even in the time of ,justinian 7c4 #eej7 there seems to have been a silk trade with ,china1 which country in turn carried on commerce with ,ceylon1#daj and reached out to ,turkestan where other merchants transmitted the ,eastern products westward4 ,in the seventh century there was a well-defined commerce between ,persia and ,india1 as well as between ,persia and ,constantinople4#daa ,the ,byzantine commerciarii were stationed at the outposts not merely as customs officers but as government purchasing agents4#dab ,occasionally there went along these routes of trade men of real learning1 and such would surely have carried the knowledge of many customs back and forth4 ,thus at a period when the numerals are known to have been partly understood in ,italy1 at the opening of the eleventh century1 one ,constantine1 an ,african1 traveled from ,italy through a great part of ,africa and ,asia1 even on to ,india1 for the purpose of learning the sciences of the ,orient4 ,he spent thirty-nine years in travel1 having been hospitably received in ,babylon1 and upon his return he was welcomed with great honor at ,salerno4#dac ,a very interesting illustration of this intercourse also appears in the tenth century1 when the son of ,otto ,i #aje 7#icf-igc7 married a princess from ,constantinople4 ,this monarch was in touch with the ,moors of ,spain and invited to his court numerous scholars from abroad1#dad and his intercourse with the ,east as well as the ,west must have brought together much of the learning of each4 ,another powerful means for the circulation of mysticism and philosophy1 and more or less of culture1 took its start just before the conversion of ,constantine 7c4 #cab71 in the form of ,christian pilgrim travel4 ,this was a feature peculiar to the zealots of early ,christianity1 found in only a slight degree among their ,jewish predecessors in the annual pilgrimage to ,jerusalem1 and almost wholly wanting in other pre-,christian peoples4 ,chief among these early pilgrims were the two ,placentians1 ,john and ,antonine the ,elder 7c4 #cjc71 who1 in their wanderings to ,jerusalem1 seem to have started a movement which culminated centuries later in the crusades4#dae ,in #ccc a ,bordeaux pilgrim compiled the first ,christian guide-book1 the ,itinerary from ,bordeaux to ,jerusalem1#daf and from this time on the holy pilgrimage never entirely ceased4 ,still another certain route for the entrance of the numerals into ,christian ,europe was through the pillaging and trading carried on by the ,arabs on the northern shores of the ,mediterranean4 ,as early as #feb ,a4,d41 in the thirtieth year of the ,hejira1 the ,mohammedans descended upon the shores of ,sicily and took much spoil4 ,hardly had the wretched ,constans given place to the #ajf young ,constantine ,,iv when they again attacked the island and plundered ancient ,syracuse4 ,again in #hbg1 under ,asad1 they ravaged the coasts4 ,although at this time they failed to conquer ,syracuse1 they soon held a good part of the island1 and a little later they successfully besieged the city4 ,before ,syracuse fell1 however1 they had plundered the shores of ,italy1 even to the walls of ,rome itself2 and had not ,leo ,,iv1 in #hdi1 repaired the neglected fortifications1 the effects of the ,moslem raid of that year might have been very far-reaching4 ,ibn ,khordad4beh1 who left ,bagdad in the latter part of the ninth century1 gives a picture of the great commercial activity at that time in the ,saracen city of ,palermo4 ,in this same century they had established themselves in ,piedmont1 and in #ijf they pillaged ,turin4#dag ,on the ,sorrento peninsula the traveler who climbs the hill to the beautiful ,ravello sees still several traces of the ,arab architecture1 reminding him of the fact that about #ijj ,a4,d4 ,amalfi was a commercial center of the ,moors4#dah ,not only at this time1 but even a century earlier1 the artists of northern ,india sold their wares at such centers1 and in the courts both of ,harun al-,rashid and of ,charlemagne4#dai ,thus the ,arabs dominated the ,mediterranean ,sea long before ,venice held the gorgeous ,east in fee ,and was the safeguard of the ,west1 and long before ,genoa had become her powerful rival4#dbj #ajg ,only a little later than this the brothers ,nicolo and ,maffeo ,polo entered upon their famous wanderings4#dba ,leaving ,constantinople in #abfj1 they went by the ,sea of ,azov to ,bokhara1 and thence to the court of ,kublai ,khan1 penetrating ,china1 and returning by way of ,acre in #abfi with a commission which required them to go back to ,china two years later4 ,this time they took with them ,nicolo's son ,marco1 the historian of the journey1 and went across the plateau of ,pamir2 they spent about twenty years in ,china1 and came back by sea from ,china to ,persia4 ,the ventures of the ,poli were not long unique1 however3 the thirteenth century had not closed before ,roman missionaries and the merchant ,petrus de ,lucolongo had penetrated ,china4 ,before #acej the company of missionaries was large1 converts were numerous1 churches and ,franciscan convents had been organized in the ,east1 travelers were appealing for the truth of their accounts to the many persons in ,venice who had been in ,china1 ,tsuan-chau-fu had a ,european merchant community1 and ,italian trade and travel to ,china was a thing that occupied two chapters of a commercial handbook4#dbb #ajh ,it is therefore reasonable to conclude that in the ,middle ,ages1 as in the time of ,boethius1 it was a simple matter for any inquiring scholar to become acquainted with such numerals of the ,orient as merchants may have used for warehouse or price marks4 ,and the fact that ,gerbert seems to have known only the forms of the simplest of these1 not comprehending their full significance1 seems to prove that he picked them up in just this way4 ,even if ,gerbert did not bring his knowledge of the ,oriental numerals from ,spain1 he may easily have obtained them from the marks on merchant's goods1 had he been so inclined4 ,such knowledge was probably obtainable in various parts of ,italy1 though as parts of mere mercantile knowledge the forms might soon have been lost1 it needing the pen of the scholar to preserve them4 ,trade at this time was not stagnant4 ,during the eleventh and twelfth centuries the ,slavs1 for example1 had very great commercial interests1 their trade reaching to ,kiev and ,novgorod1 and thence to the ,east4 ,constantinople was a great clearing-house of commerce with the ,orient1#dbc and the ,byzantine merchants must have been entirely familiar with the various numerals of the ,eastern peoples4 ,in the eleventh century the ,italian town of ,amalfi established a factory#dbd in ,constantinople1 and had trade relations with ,antioch and ,egypt4 ,venice1 as early as the ninth century1 had a valuable trade with ,syria and ,cairo4#dbe ,fifty years after ,gerbert died1 in the time of ,cnut1 the ,dane and the ,norwegian pushed their commerce far beyond the northern seas1 both by caravans through ,russia to the ,orient1 and by their venturesome barks which #aji sailed through the ,strait of ,gibraltar into the ,mediterranean4#dbf ,only a little later1 probably before #abjj ,a4,d41 a clerk in the service of ,thomas a ,becket1 present at the latter's death1 wrote a life of the martyr1 to which 7fortunately for our purposes7 he prefixed a brief eulogy of the city of ,london4#dbg ,this clerk1 ,william ,fitz ,stephen by name1 thus speaks of the ,british capital3 ,aurum mittit ,arabs3 species et thura ,sabaeus3 ,arma ,sythes3 oleum palmarum divite sylva ,pingue solum ,babylon3 ,nilus lapides pretiosos3 ,norwegi1 ,russi1 varium grisum1 sabdinas3 ,seres1 purpureas vestes3 ,galli1 sua vina4 ,although1 as a matter of fact1 the ,arabs had no gold to send1 and the ,scythians no arms1 and ,egypt no precious stones save only the turquoise1 the ,chinese 7,seres7 may have sent their purple vestments1 and the north her sables and other furs1 and ,france her wines4 ,at any rate the verses show very clearly an extensive foreign trade4 ,then there were the ,crusades1 which in these times brought the ,east in touch with the ,west4 ,the spirit of the ,orient showed itself in the songs of the troubadours1 and the baudekin1#dbh the canopy of ,bagdad1#dbi became common in the churches of ,italy4 ,in ,sicily and in ,venice the textile industries of the ,east found place1 and made their way even to the ,scandinavian peninsula4#dcj ,we therefore have this state of affairs3 ,there was abundant intercourse between the ,east and ,west for #aaj some centuries before the ,hindu numerals appear in any manuscripts in ,christian ,europe4 ,the numerals must of necessity have been known to many traders in a country like ,italy at least as early as the ninth century1 and probably even earlier1 but there was no reason for preserving them in treatises4 ,therefore when a man like ,gerbert made them known to the scholarly circles1 he was merely describing what had been familiar in a small way to many people in a different walk of life4 ,since ,gerbert#dca was for a long time thought to have been the one to introduce the numerals into ,italy1#dcb a brief sketch of this unique character is proper4 ,born of humble parents1#dcc this remarkable man became the counselor and companion of kings1 and finally wore the papal tiara as ,sylvester ,,ii1 from #iii until his death in #ajjc4dcd ,he was early brought under the influence of the monks at ,aurillac1 and particularly of ,raimund1 who had been a pupil of ,odo of ,cluny1 and there in due time he himself took holy orders4 ,he visited ,spain in about #ifg in company with ,count ,borel1#dce remaining there three years1 #aaa and studying under ,bishop ,hatto of ,vich1#dcf a city in the province of ,barcelona1#dcg then entirely under ,christian rule4 ,indeed1 all of ,gerbert's testimony is as to the influence of the ,christian civilization upon his education4 ,thus he speaks often of his study of ,boethius1#dch so that if the latter knew the numerals ,gerbert would have learned them from him4#dci ,if ,gerbert had studied in any ,moorish schools he would1 under the decree of the emir ,hisham 7#ghg-hbb71 have been obliged to know ,arabic1 which would have taken most of his three years in ,spain1 and of which study we have not the slightest hint in any of his letters4#ddj ,on the other hand1 ,barcelona was the only ,christian province in immediate touch with the ,moorish civilization at that time4#dda ,furthermore we know that earlier in the same century ,king ,alonzo of ,asturias 7d4 #iaj7 confided the education of his son ,ordono to the ,arab scholars of the court of the #aab wali of ,saragossa1#ddb so that there was more or less of friendly relation between ,christian and ,moor4 ,after his three years in ,spain1 ,gerbert went to ,italy1 about #igj1 where he met ,pope ,john ,,xiii1 being by him presented to the emperor ,otto ,i4 ,two years later 7#igb71 at the emperor's request1 he went to ,rheims1 where he studied philosophy1 assisting to make of that place an educational center2 and in #ihc he became abbot at ,bobbio4 ,the next year he returned to ,rheims1 and became archbishop of that diocese in #iia4 ,for political reasons he returned to ,italy in #iif1 became archbishop of ,ravenna in #iih1 and the following year was elected to the papal chair4 ,far ahead of his age in wisdom1 he suffered as many such scholars have even in times not so remote by being accused of heresy and witchcraft4 ,as late as #aebb1 in a biography published at ,venice1 it is related that by black art he attained the papacy1 after having given his soul to the devil4#ddc ,gerbert was1 however1 interested in astrology1#ddd although this was merely the astronomy of that time and was such a science as any learned man would wish to know1 even as to-day we wish to be reasonably familiar with physics and chemistry4 ,that ,gerbert and his pupils knew the 4gobar numerals is a fact no longer open to controversy4#dde ,bernelinus and ,richer#ddf call them by the well-known name of #aac caracteres1 a word used by ,radulph of ,laon in the same sense a century later4#ddg ,it is probable that ,gerbert was the first to describe these 4gobar numerals in any scientific way in ,christian ,europe1 but without the zero4 ,if he knew the latter he certainly did not understand its use4#ddh ,the question still to be settled is as to where he found these numerals4 ,that he did not bring them from ,spain is the opinion of a number of careful investigators4#ddi ,this is thought to be the more probable because most of the men who made ,spain famous for learning lived after ,gerbert was there4 ,such were ,ibn ,sina 7,avicenna7 who lived at the beginning1 and ,gerber of ,seville who flourished in the middle1 of the eleventh century1 and ,abu ,roshd 7,averroes7 who lived at the end of the twelfth4#dej ,others hold that his proximity to #aad the ,arabs for three years makes it probable that he assimilated some of their learning1 in spite of the fact that the lines between ,christian and ,moor at that time were sharply drawn4#dea ,writers fail1 however1 to recognize that a commercial numeral system would have been more likely to be made known by merchants than by scholars4 ,the itinerant peddler knew no forbidden pale in ,spain1 any more than he has known one in other lands4 ,if the 4gobar numerals were used for marking wares or keeping simple accounts1 it was he who would have known them1 and who would have been the one rather than any ,arab scholar to bring them to the inquiring mind of the young ,french monk4 ,the facts that ,gerbert knew them only imperfectly1 that he used them solely for calculations1 and that the forms are evidently like the ,spanish 4gobar1 make it all the more probable that it was through the small tradesman of the ,moors that this versatile scholar derived his knowledge4 ,moreover the part of the geometry bearing his name1 and that seems unquestionably his1 shows the ,arab influence1 proving that he at least came into contact with the transplanted ,oriental learning1 even though imperfectly4#deb ,there was also the persistent ,jewish merchant trading with both peoples then as now1 always alive to the acquiring of useful knowledge1 and it would be very natural for a man like ,gerbert to welcome learning from such a source4 ,on the other hand1 the two leading sources of information as to the life of ,gerbert reveal practically nothing to show that he came within the ,moorish sphere of influence during his sojourn in ,spain4 ,these sources #aae are his letters and the history written by ,richer4 ,gerbert was a master of the epistolary art1 and his exalted position led to the preservation of his letters to a degree that would not have been vouchsafed even by their classic excellence4#dec ,richer was a monk at ,st4 ,remi de ,rheims1 and was doubtless a pupil of ,gerbert4 ,the latter1 when archbishop of ,rheims1 asked ,richer to write a history of his times1 and this was done4 ,the work lay in manuscript1 entirely forgotten until ,pertz discovered it at ,bamberg in #ahcc4ded ,the work is dedicated to ,gerbert as archbishop of ,rheims1#dee and would assuredly have testified to such efforts as he may have made to secure the learning of the ,moors4 ,now it is a fact that neither the letters nor this history makes any statement as to ,gerbert's contact with the ,saracens4 ,the letters do not speak of the ,moors1 of the ,arab numerals1 nor of ,cordova4 ,spain is not referred to by that name1 and only one ,spanish scholar is mentioned4 ,in one of his letters he speaks of ,joseph ,ispanus1#def or ,joseph ,sapiens1 but who this ,joseph the ,wise of ,spain may have been we do not know4 ,possibly #aaf it was he who contributed the morsel of knowledge so imperfectly assimilated by the young ,french monk4#deg ,within a few years after ,gerbert's visit two young ,spanish monks of lesser fame1 and doubtless with not that keen interest in mathematical matters which ,gerbert had1 regarded the apparently slight knowledge which they had of the ,hindu numeral forms as worthy of somewhat permanent record#deh in manuscripts which they were transcribing4 ,the fact that such knowledge had penetrated to their modest cloisters in northern ,spain--the one ,albelda or ,albaida--indicates that it was rather widely diffused4 ,gerbert's treatise ,libellus de numerorum divisione#dei is characterized by ,chasles as one of the most obscure documents in the history of science4#dfj ,the most complete information in regard to this and the other mathematical works of ,gerbert is given by ,bubnov1#dfa who considers this work to be genuine4#dfb #aag ,so little did ,gerbert appreciate these numerals that in his works known as the ,regula de abaco computi and the ,libellus he makes no use of them at all1 employing only the ,roman forms4#dfc ,nevertheless ,bernelinus#dfd refers to the nine 4gobar characters4#dfe ,these ,gerbert had marked on a thousand jetons or counters1#dff using the latter on an abacus which he had a sign-maker prepare for him4#dfg ,instead of putting eight counters in say the tens' column1 ,gerbert would put a single counter marked #h1 and so for the other places1 leaving the column empty where we would place a zero1 but where he1 lacking the zero1 had no counter to place4 ,these counters he possibly called caracteres1 a name which adhered also to the figures themselves4 ,it is an interesting speculation to consider whether these apices1 as they are called in the ,boethius interpolations1 were in any way suggested by those ,roman jetons generally known in numismatics as tesserae1 and bearing the figures ,i-,,xvi1 the sixteen referring to the number of assi in a sestertius4#dfh ,the #aah name apices adhered to the ,hindu-,arabic numerals until the sixteenth century4#dfi ,to the figures on the apices were given the names ,igin1 andras1 ormis1 arbas1 quimas1 calctis or caltis1 zenis1 temenias1 celentis1 sipos1#dgj the origin and meaning of which still remain a mystery4 ,the ,semitic origin of several of the words seems probable4 ,wahud1 thaneine1 #aai thalata1 arba1 kumsa1 setta1 sebba1 timinia1 taseud are given by the ,rev4 ,r4 ,patrick#dga as the names1 in an ,arabic dialect used in ,morocco1 for the numerals from one to nine4 ,of these the words for four1 five1 and eight are strikingly like those given above4 ,the name apices was not1 however1 a common one in later times4 ,notae was more often used1 and it finally gave the name to notation4#dgb ,still more common were the names figures1 ciphers1 signs1 elements1 and characters4#dgc ,so little effect did the teachings of ,gerbert have in making known the new numerals1 that ,o',creat1 who lived a century later1 a friend and pupil of ,adelhard #abj of ,bath1 used the zero with the ,roman characters1 in contrast to ,gerbert's use of the 4gobar forms without the zero4#dgd ,o',creat uses three forms for zero1 o1 o1 and ,greek3 t1 as in ,maximus ,planudes4 ,with this use of the zero goes1 naturally1 a place value1 for he writes ,,iii ,,iii for #cc1 ,,iccoo and ,i4 ,,ii4 tau4 tau for #abjj1 ,i4 ,o4 ,,viii4 ,,ix for #ajhi1 and ,i4 ,,iiii4 ,,iiii4 tautautautau for the square of #abjj4 ,the period from the time of ,gerbert until after the appearance of ,leonardo's monumental work may be called the period of the abacists4 ,even for many years after the appearance early in the twelfth century of the books explaining the ,hindu art of reckoning1 there was strife between the abacists1 the advocates of the abacus1 and the algorists1 those who favored the new numerals4 ,the words cifra and algorismus cifra were used with a somewhat derisive significance1 indicative of absolute uselessness1 as indeed the zero is useless on an abacus in which the value of any unit is given by the column which it occupies4#dge ,so ,gautier de ,coincy 7#aagg-abcf7 in a work on the miracles of ,mary says3 ,a horned beast1 a sheep1 ,an algorismus-cipher1 ,is a priest1 who on such a feast day ,does not celebrate the holy ,mother4#dgf ,so the abacus held the field for a long time1 even against the new algorism employing the new numerals4 #aba ,geoffrey ,chaucer#dgg describes in ,the ,miller's ,tale the clerk with ,his ,almageste and bokes grete and smale1 ,his astrelabie1 longinge for his art1 ,his augrim-stones layen faire apart ,on shelves couched at his beddes heed4 ,so1 too1 in ,chaucer's explanation of the astrolabe1#dgh written for his son ,lewis1 the number of degrees is expressed on the instrument in ,hindu-,arabic numerals3 ,over the whiche degrees ther ben noumbres of augrim1 that devyden thilke same degrees fro fyve to fyve1 and 444 the nombres 444 ben writen in augrim1 meaning in the way of the algorism4 ,thomas ,usk about #achg writes3#dgi a sypher in augrim have no might in signification of it-selve1 yet he yeveth power in signification to other4 ,so slow and so painful is the assimilation of new ideas4 ,bernelinus#dhj states that the abacus is a well-polished board 7or table71 which is covered with blue sand and used by geometers in drawing geometrical figures4 ,we have previously mentioned the fact that the ,hindus also performed mathematical computations in the sand1 although there is no evidence to show that they had any column abacus4#dha ,for the purposes of computation1 ,bernelinus continues1 the board is divided into thirty vertical columns1 three of which are reserved for fractions4 ,beginning with the units columns1 each set of #abb three columns 7lineae is the word which ,bernelinus uses7 is grouped together by a semicircular arc placed above them1 while a smaller arc is placed over the units column and another joins the tens and hundreds columns4 ,thus arose the designation arcus pictagore#dhb or sometimes simply arcus4#dhc ,the operations of addition1 subtraction1 and multiplication upon this form of the abacus required little explanation1 although they were rather extensively treated1 especially the multiplication of different orders of numbers4 ,but the operation of division was effected with some difficulty4 ,for the explanation of the method of division by the use of the complementary difference1#dhd long the stumbling-block in the way of the medieval arithmetician1 the reader is referred to works on the history of mathematics#dhe and to works relating particularly to the abacus4#dhf ,among the writers on the subject may be mentioned ,abbo#dhg of ,fleury 7c4 #igj71 ,heriger#dhh of ,lobbes or ,laubach #abc 7c4 #iej-ajjg71 and ,hermannus ,contractus#dhi 7#ajac-ajed71 all of whom employed only the ,roman numerals4 ,similarly ,adelhard of ,bath 7c4 #aacj71 in his work ,regulae ,abaci1#dij gives no reference to the new numerals1 although it is certain that he knew them4 ,other writers on the abacus who used some form of ,hindu numerals were ,gerland#dia 7first half of twelfth century7 and ,turchill#dib 7c4 #abjj74 ,for the forms used at this period the reader is referred to the plate on page #hh4 ,after ,gerbert's death1 little by little the scholars of ,europe came to know the new figures1 chiefly through the introduction of ,arab learning4 ,the ,dark ,ages had passed1 although arithmetic did not find another advocate as prominent as ,gerbert for two centuries4 ,speaking of this great revival1 ,raoul ,glaber#dic 7#ihe-;c4 #ajdf71 a monk of the great ,benedictine abbey of ,cluny1 of the eleventh century1 says3 ,it was as though the world had arisen and tossed aside the worn-out garments of ancient time1 and wished to apparel itself in a white robe of churches4 ,and with this activity in religion came a corresponding interest in other lines4 ,algorisms began to appear1 and knowledge from the outside world found #abd interested listeners4 ,another ,raoul1 or ,radulph1 to whom we have referred as ,radulph of ,laon1#did a teacher in the cloister school of his city1 and the brother of ,anselm of ,laon#die the celebrated theologian1 wrote a treatise on music1 extant but unpublished1 and an arithmetic which ,nagl first published in #ahij4dif ,the latter work1 preserved to us in a parchment manuscript of seventy-seven leaves1 contains a curious mixture of ,roman and 4gobar numerals1 the former for expressing large results1 the latter for practical calculation4 ,these 4gobar caracteres include the sipos 7zero71 ,symbol1 of which1 however1 ,radulph did not know the full significance2 showing that at the opening of the twelfth century the system was still uncertain in its status in the church schools of central ,france4 ,at the same time the words algorismus and cifra were coming into general use even in non-mathematical literature4 ,jordan #dig cites numerous instances of such use from the works of ,alanus ab ,insulis#dih 7,alain de ,lille71 ,gautier de ,coincy 7#aagg-abcf71 and others4 ,another contributor to arithmetic during this interesting period was a prominent ,spanish ,jew called variously ,john of ,luna1 ,john of ,seville1 ,johannes ,hispalensis1 ,johannes ,toletanus1 and ,johannes ,hispanensis de ,luna4#dii #abe ,his date is rather closely fixed by the fact that he dedicated a work to ,raimund who was archbishop of ,toledo between #aacj and #aaej4ejj ,his interests were chiefly in the translation of ,arabic works1 especially such as bore upon the ,aristotelian philosophy4 ,from the standpoint of arithmetic1 however1 the chief interest centers about a manuscript entitled ,joannis ,hispalensis liber ,algorismi de ,practica ,arismetrice which ,boncompagni found in what is now the ,bibliotheque nationale at ,paris4 ,although this distinctly lays claim to being ,al-,khowarazmi's work1#eja the evidence is altogether against the statement1#ejb but the book is quite as valuable1 since it represents the knowledge of the time in which it was written4 ,it relates to the operations with integers and sexagesimal fractions1 including roots1 and contains no applications4#ejc ,contemporary with ,john of ,luna1 and also living in ,toledo1 was ,gherard of ,cremona1#ejd who has sometimes been identified1 but erroneously1 with ,gernardus1#eje the #abf author of a work on algorism4 ,he was a physician1 an astronomer1 and a mathematician1 translating from the ,arabic both in ,italy and in ,spain4 ,in arithmetic he was influential in spreading the ideas of algorism4 ,four ,englishmen--,adelhard of ,bath 7c4 #aacj71 ,robert of ,chester 7,robertus ,cestrensis1 c4 #aadc71 ,william ,shelley1 and ,daniel ,morley 7#aahj7--;are known#ejf to have journeyed to ,spain in the twelfth century for the purpose of studying mathematics and ,arabic4 ,adelhard of ,bath made translations from ,arabic into ,latin of ,al-,khowarazmi's astronomical tables#ejg and of ,euclid's ,elements1#ejh while ,robert of ,chester is known as the translator of ,al-,khowarazmi's algebra4#eji ,there is no reason to doubt that all of these men1 and others1 were familiar with the numerals which the ,arabs were using4 ,the earliest trace we have of computation with ,hindu numerals in ,germany is in an ,algorismus of #aadc1 now in the ,hofbibliothek in ,vienna4#eaj ,it is bound in with a #abg ,computus by the same author and bearing the date given4 ,it contains chapters ,de additione1 ,de diminutione1 ,de mediatione1 ,de divisione1 and part of a chapter on multiplication4 ,the numerals are in the usual medieval forms except the #b which1 as will be seen from the illustration1#eaa is somewhat different1 and the #c1 which takes the peculiar shape ,symbol1 a form characteristic of the twelfth century4 ,it was about the same time that the ,sefer ha-,mispar1#eab the ,book of ,number1 appeared in the ,hebrew language4 ,the author1 ,rabbi ,abraham ibn ,meir ibn ,ezra1#eac was born in ,toledo 7c4 #ajib74 ,in #aaci he went to ,egypt1 ,palestine1 and the ,orient1 spending also some years in ,italy4 ,later he lived in southern ,france and in ,england4 ,he died in #aafg4 ,the probability is that he acquired his knowledge of the ,hindu arithmetic#ead in his native town of ,toledo1 but it is also likely that the knowledge of other systems which he acquired on travels increased his appreciation of this one4 ,we have mentioned the fact that he used the first letters of the ,hebrew alphabet1 ,hebrew3 ,a ,b ,g ,d ,h ,w ,z ,,ch ,t1 for the numerals #i #h #g #f #e #d #c #b #a1 and a circle for the zero4 ,the quotation in the note given below shows that he knew of the ,hindu origin2 but in his manuscript1 although he set down the ,hindu forms1 he used the above nine ,hebrew letters with place value for all computations4 99 99 99 99 99 #abh ,,chapter ,,viii ,,the ,,spread ,,of ,,the ,,numerals ,,in ,,europe ,of all the medieval writers1 probably the one most influential in introducing the new numerals to the scholars of ,europe was ,leonardo ,fibonacci1 of ,pisa4#eae ,this remarkable man1 the most noteworthy mathematical genius of the ,middle ,ages1 was born at ,pisa about #aage4eaf ,the traveler of to-day may cross the ,via ,fibonacci on his way to the ,campo ,santo1 and there he may see at the end of the long corridor1 across the quadrangle1 the statue of ,leonardo in scholars garb4 ,few towns have honored a mathematician more1 and few mathematicians have so distinctly honored their birthplace4 ,leonardo was born in the golden age of this city1 the period of its commercial1 religious1 and intellectual prosperity4#eag #abi ,situated practically at the mouth of the ,arno1 ,pisa formed with ,genoa and ,venice the trio of the greatest commercial centers of ,italy at the opening of the thirteenth century4 ,even before ,venice had captured the ,levantine trade1 ,pisa had close relations with the ,east4 ,an old ,latin chronicle relates that in #ajje ,pisa was captured by the ,saracens1 that in the following year the ,pisans overthrew the ,saracens at ,reggio1 and that in #ajab the ,saracens came to ,pisa and destroyed it4 ,the city soon recovered1 however1 sending no fewer than a hundred and twenty ships to ,syria in #ajii1eah founding a merchant colony in ,constantinople a few years later1#eai and meanwhile carrying on an interurban warfare in ,italy that seemed to stimulate it to great activity4#ebj ,a writer of #aaad tells us that at that time there were many heathen people--,turks1 ,libyans1 ,parthians1 and ,chaldeans--to be found in ,pisa4 ,it was in the midst of such wars1 in a cosmopolitan and commercial town1 in a center where literary work was not appreciated1#eba that the genius of ,leonardo appears as one of the surprises of history1 warning us again that we should draw no horoscope2 that we should expect little1 for what we expect will not come to pass4#ebb ,leonardo's father was one ,william1#ebc and he had a brother named ,bonaccingus1#ebd but nothing further is #acj known of his family4 ,as to ,fibonacci1 most writers#ebe have assumed that his father's name was ,bonaccio1#ebf whence filius ,bonaccii1 or ,fibonacci4 ,others#ebg believe that the name1 even in the ,latin form of filius ,bonaccii as used in ,leonardo's work1 was simply a general one1 like our ,johnson or ,bronson 7,brown's son72 and the only contemporary evidence that we have bears out this view4 ,as to the name ,bigollo1 used by ,leonardo1 some have thought it a self-assumed one meaning blockhead1 a term that had been applied to him by the commercial world or possibly by the university circle1 and taken by him that he might prove what a blockhead could do4 ,milanesi1#ebh however1 has shown that the word ,bigollo 7or ,pigollo7 was used in ,tuscany to mean a traveler1 and was naturally assumed by one who had studied1 as ,leonardo had1 in foreign lands4 ,leonardo's father was a commercial agent at ,bugia1 the modern ,bougie1#ebi the ancient ,saldae on the coast of ,barbary1#ecj a royal capital under the ,vandals and again1 a century before ,leonardo1 under the ,beni ,hammad4 ,it had one of the best harbors on the coast1 sheltered as it is by ,mt4 ,lalla ,guraia1#eca and at the close of the twelfth century it was a center of ,african commerce4 ,it was here that ,leonardo was taken as a child1 and here he went to school to a ,moorish master4 ,when he reached the years of young manhood he started on a tour of the ,mediterranean ,sea1 and visited ,egypt1 ,syria1 ,greece1 ,sicily1 and ,provence1 meeting with scholars as well as with #aca merchants1 and imbibing a knowledge of the various systems of numbers in use in the centers of trade4 ,all these systems1 however1 he says he counted almost as errors compared with that of the ,hindus4#ecb ,returning to ,pisa1 he wrote his ,liber ,abaci#ecc in #abjb1 rewriting it in #abbh4ecd ,in this work the numerals are explained and are used in the usual computations of business4 ,such a treatise was not destined to be popular1 however1 because it was too advanced for the mercantile class1 and too novel for the conservative university circles4 ,indeed1 at this time mathematics had only slight place in the newly established universities1 as witness the oldest known statute of the ,sorbonne at ,paris1 dated #abae1 where the subject is referred to only in an incidental way4#ece ,the period was one of great commercial activity1 and on this very #acb account such a book would attract even less attention than usual4#ecf ,it would now be thought that the western world would at once adopt the new numerals which ,leonardo had made known1 and which were so much superior to anything that had been in use in ,christian ,europe4 ,the antagonism of the universities would avail but little1 it would seem1 against such an improvement4 ,it must be remembered1 however1 that there was great difficulty in spreading knowledge at this time1 some two hundred and fifty years before printing was invented4 ,popes and princes and even great religious institutions possessed far fewer books than many farmers of the present age4 ,the library belonging to the ,cathedral ,church of ,san ,martino at ,lucca in the ninth century contained only nineteen volumes of abridgments from ecclesiastical commentaries4#ecg ,indeed1 it was not until the early part of the fifteenth century that ,palla degli ,strozzi took steps to carry out the project that had been in the mind of ,petrarch1 the founding of a public library4 ,it was largely by word of mouth1 therefore1 that this early knowledge had to be transmitted4 ,fortunately the presence of foreign students in ,italy at this time made this transmission feasible4 7,if human nature was the same then as now1 it is not impossible that the very opposition of the faculties to the works of ,leonardo led the students to investigate #acc them the more zealously47 ,at ,vicenza in #abji1 for example1 there were ,bohemians1 ,poles1 ,frenchmen1 ,burgundians1 ,germans1 and ,spaniards1 not to speak of representatives of divers towns of ,italy2 and what was true there was also true of other intellectual centers4 ,the knowledge could not fail to spread1 therefore1 and as a matter of fact we find numerous bits of evidence that this was the case4 ,although the bankers of ,florence were forbidden to use these numerals in #abii1 and the statutes of the university of ,padua required stationers to keep the price lists of books non per cifras1 sed per literas claros1#ech the numerals really made much headway from about #abge on4 ,it was1 however1 rather exceptional for the common people of ,germany to use the ,arabic numerals before the sixteenth century1 a good witness to this fact being the popular almanacs4 ,calendars of #adeg-adifeci have generally the ,roman numerals1 while ,koebel's calendar of #aeah gives the ,arabic forms as subordinate to the ,roman4 ,in the register of the ,kreuzschule at ,dresden the ,roman forms were used even until #aeci4 ,while not minimizing the importance of the scientific work of ,leonardo of ,pisa1 we may note that the more popular treatises by ,alexander de ,villa ,dei 7c4 #abdj ,a4,d47 and ,john of ,halifax 7,sacrobosco1 c4 #abej ,a4,d47 were much more widely used1 and doubtless contributed more to the spread of the numerals among the common people4 #acd ,the ,carmen de ,algorismo#edj of ,alexander de ,villa ,dei was written in verse1 as indeed were many other textbooks of that time4 ,that it was widely used is evidenced by the large number of manuscripts#eda extant in ,european libraries4 ,sacrobosco's ,algorismus1#edb in which some lines from the ,carmen are quoted1 enjoyed a wide popularity as a textbook for university instruction4#edc ,the work was evidently written with this end in view1 as numerous commentaries by university lecturers are found4 ,probably the most widely used of these was that of ,petrus de ,dacia#edd written in #abia4 ,these works throw an interesting light upon the method of instruction in mathematics in use in the universities from the thirteenth even to the sixteenth century4 ,evidently the text was first read and copied by students4#ede ,following this came line by line an exposition of the text1 such as is given in ,petrus de ,dacia's commentary4 ,sacrobosco's work is of interest also because it was probably due to the extended use of this work that the #ace term ,arabic numerals became common4 ,in two places there is mention of the inventors of this system4 ,in the introduction it is stated that this science of reckoning was due to a philosopher named ,algus1 whence the name algorismus1#edf and in the section on numeration reference is made to the ,arabs as the inventors of this science4#edg ,while some of the commentators1 ,petrus de ,dacia#edh among them1 knew of the ,hindu origin1 most of them undoubtedly took the text as it stood2 and so the ,arabs were credited with the invention of the system4 ,the first definite trace that we have of an algorism in the ,french language is found in a manuscript written about #abge4edi ,this interesting leaf1 for the part on algorism consists of a single folio1 was noticed by the ,abbe ,leboeuf as early as #agda1eej and by ,daunou in #ahbd4eea ,it then seems to have been lost in the multitude of ,paris manuscripts2 for although ,chasles#eeb relates his vain search for it1 it was not rediscovered until #ahhb4 ,in that year ,m4 ,ch4 ,henry found it1 and to his care we owe our knowledge of the interesting manuscript4 ,the work is anonymous and is devoted almost entirely to geometry1 only #acf two pages 7one folio7 relating to arithmetic4 ,in these the forms of the numerals are given1 and a very brief statement as to the operations1 it being evident that the writer himself had only the slightest understanding of the subject4 ,once the new system was known in ,france1 even thus superficially1 it would be passed across the ,channel to ,england4 ,higden1#eec writing soon after the opening of the fourteenth century1 speaks of the ,french influence at that time and for some generations preceding3#eed ,for two hundred years children in scole1 agenst the usage and manir of all other nations beeth compelled for to leave hire own language1 and for to construe hir lessons and hire thynges in ,frensche4444 ,gentilmen children beeth taught to speke ,frensche from the tyme that they bith rokked in hir cradell2 and uplondissche men will likne himself to gentylmen1 and fondeth with greet besynesse for to speke ,frensche4 ,the question is often asked1 why did not these new numerals attract more immediate attention8 ,why did they have to wait until the sixteenth century to be generally used in business and in the schools8 ,in reply it may be said that in their elementary work the schools always wait upon the demands of trade4 ,that work which pretends to touch the life of the people must come reasonably near doing so4 ,now the computations of business until about #aejj did not demand the new figures1 for two reasons3 ,first1 cheap paper was not known4 ,paper-making of any kind was not introduced into ,europe until #acg the twelfth century1 and cheap paper is a product of the nineteenth4 ,pencils1 too1 of the modern type1 date only from the sixteenth century4 ,in the second place1 modern methods of operating1 particularly of multiplying and dividing 7operations of relatively greater importance when all measures were in compound numbers requiring reductions at every step71 were not yet invented4 ,the old plan required the erasing of figures after they had served their purpose1 an operation very simple with counters1 since they could be removed4 ,the new plan did not as easily permit this4 ,hence we find the new numerals very tardily admitted to the counting-house1 and not welcomed with any enthusiasm by teachers4#eee ,aside from their use in the early treatises on the new art of reckoning1 the numerals appeared from time to time in the dating of manuscripts and upon monuments4 ,the oldest definitely dated ,european document known #ach to contain the numerals is a ,latin manuscript1#eef the ,codex ,vigilanus1 written in the ,albelda ,cloister not far from ,logrono in ,spain1 in #igf ,a4,d4 ,the nine characters 7of 4gobar type71 without the zero1 are given as an addition to the first chapters of the third book of the ,origines by ,isidorus of ,seville1 in which the ,roman numerals are under discussion4 ,another ,spanish copy of the same work1 of #iib ,a4,d41 contains the numerals in the corresponding section4 ,the writer ascribes an ,indian origin to them in the following words3 ,item de figuris arithmetice14 ,scire debemus in ,indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus disciplinis concedere4 ,et hoc manifestum est in nobem figuris1 quibus designant unumquemque gradum cuiuslibet gradus4 ,quarum hec sunt forma4 ,the nine 4gobar characters follow4 ,some of the abacus forms#eeg previously given are doubtless also of the tenth century4 ,the earliest ,arabic documents containing the numerals are two manuscripts of #hgd and #hhh ,a4,d4#eeh ,they appear about a century later in a work#eei written at ,shiraz in #igj ,a4,d4 ,there is also an early trace of their use on a pillar recently discovered in a church apparently destroyed as early as the tenth century1 not far from the ,jeremias ,monastery1 in ,egypt4 #aci ,a graffito in ,arabic on this pillar has the date #cdi ,a4,h41 which corresponds to #ifa ,a4,d4#efj ,for the dating of ,latin documents the ,arabic forms were used as early as the thirteenth century4#efa ,on the early use of these numerals in ,europe the only scientific study worthy the name is that made by ,mr4 ,g4 ,f4 ,hill of the ,british ,museum4#efb ,from his investigations it appears that the earliest occurrence of a date in these numerals on a coin is found in the reign of ,roger of ,sicily in #aach4efc ,until recently it was thought that the earliest such date was #abag ,a4,d4 for an ,arabic piece and #achh for a ,turkish one4#efd ,most of the seals and medals containing dates that were at one time thought to be very early have been shown by ,mr4 ,hill to be of relatively late workmanship4 ,there are1 however1 in ,european manuscripts1 numerous instances of the use of these numerals before the twelfth century4 ,besides the example in the ,codex ,vigilanus1 another of the tenth century has been found in the ,st4 ,gall ,,ms4 now in the ,university ,library at ,zuerich1 the forms differing materially from those in the ,spanish codex4 ,the third specimen in point of time in ,mr4 ,hill's list is from a ,vatican ,,ms4 of #ajgg4 ,the fourth and fifth specimens are from the ,erlangen ,,ms4 of ,boethius1 of the same #adj 7eleventh7 century1 and the sixth and seventh are also from an eleventh-century ,,ms4 of ,boethius at ,chartres4 ,these and other early forms are given by ,mr4 ,hill in this table1 which is reproduced with his kind permission4 ,,earliest ,,manuscript ,,forms ,illustration ,this is one of more than fifty tables given in ,mr4 ,hill's valuable paper1 and to this monograph students #ada are referred for details as to the development of number-forms in ,europe from the tenth to the sixteenth century4 ,it is of interest to add that he has found that among the earliest dates of ,european coins or medals in these numerals1 after the ,sicilian one already mentioned1 are the following3 ,austria1 #adhd2 ,germany1 #adhi 7,cologne72 ,switzerland1 #adbd 7,st4 ,gall72 ,netherlands1 #adgd2 ,france1 #adhe2 ,italy1 #acij4efe ,the earliest ,english coin dated in these numerals was struck in #aeea1eff although there is a ,scotch piece of #aeci4efg ,in numbering pages of a printed book these numerals were first used in a work of ,petrarch's published at ,cologne in #adga4efh ,the date is given in the following form in the ,biblia ,pauperum1#efi a block-book of #adgj1 ,illustration while in another block-book which possibly goes back to c4 #adcjegj the numerals appear in several illustrations1 with forms as follows3 ,illustration ,many printed works anterior to #adga have pages or chapters numbered by hand1 but many of these numerals are #adb of date much later than the printing of the work4 ,other works were probably numbered directly after printing4 ,thus the chapters #b1 #c1 #d1 #e1 #f in a book of #adgjega are numbered as follows3 ,capitulem ,symbol #bm41444 ,symbol #cm41444 #dm41444 v1444 vi1 and followed by ,roman numerals4 ,this appears in the body of the text1 in spaces left by the printer to be filled in by hand4 ,another book#egb of #adgj has pages numbered by hand with a mixture of ,roman and ,hindu numerals1 thus1 ,illustration for #abe ,illustration for #aej ,illustration for #adg ,illustration for #bjb ,as to monumental inscriptions1#egc there was once thought to be a gravestone at ,katharein1 near ,troppau1 with the date #ajjg1 and one at ,biebrich of #abii4 ,there is no doubt1 however1 of one at ,pforzheim of #acga and one at ,ulm of #achh4egd ,certain numerals on ,wells ,cathedral have been assigned to the thirteenth century1 but they are undoubtedly considerably later4#ege ,the table on page #adc will serve to supplement that from ,mr4 ,hill's work4#egf #adc ,,early ,,manuscript ,,forms #egg ,illustration ,twelfth century ,a4,d4 #egh ,illustration #aaig ,a4,d4 #egi ,illustration #abge ,a4,d4 #ehj ,illustration c4 #abid ,a4,d4 #eha ,illustration c4 #acjc ,a4,d4 #ehb ,illustration c4 #acfj ,a4,d4 #ehc ,illustration c4 #addb ,a4,d4 #add ,illustration ,for the sake of further comparison1 three illustrations from works in ,mr4 ,plimpton's library1 reproduced from the ,rara ,arithmetica1 may be considered4 ,the first is from a ,latin manuscript on arithmetic1#ehd of which the original was written at ,paris in #adbd by ,rollandus1 a ,portuguese physician1 who prepared the work at the command of ,john of ,lancaster1 ,duke of ,bedford1 at one time ,protector of ,england and ,regent of ,france1 to whom the work is dedicated4 ,the figures show the successive powers of #b4 ,the second illustration is from ,luca da ,firenze's ,inprencipio darte dabacho1#ehe c4 #adge1 and the third is from an anonymous manuscript#ehf of about #aejj4 ,illustration ,as to the forms of the numerals1 fashion played a leading part until printing was invented4 ,this tended to fix these forms1 although in writing there is still a great variation1 as witness the ,french #e and the ,german #g and #i4 ,even in printing there is not complete uniformity1 #ade and it is often difficult for a foreigner to distinguish between the #c and #e of the ,french types4 ,illustration ,as to the particular numerals1 the following are some of the forms to be found in the later manuscripts and in the early printed books4 #a4 ,in the early printed books one was often i1 perhaps to save types1 just as some modern typewriters use the same character for l and #a4ehg ,in the manuscripts the one appears in such forms as#ehh ,illustration #b4 ,two often appears as z in the early printed books1 #ab appearing as iz4#ehi ,in the medieval manuscripts the following forms are common3#eij ,illustration #adf ,it is evident1 from the early traces1 that it is merely a cursive form for the primitive #b horizontal strokes1 just as #c comes from #c horizontal strokes1 as in the ,nana ,ghat inscriptions4 #c4 ,three usually had a special type in the first printed books1 although occasionally it appears as ,symbol4#eia ,in the medieval manuscripts it varied rather less than most of the others4 ,the following are common forms3#eib ,illustration #d4 ,four has changed greatly2 and one of the first tests as to the age of a manuscript on arithmetic1 and the place where it was written1 is the examination of this numeral4 ,until the time of printing the most common form was ,symbol1 although the ,florentine manuscript of ,leonard of ,pisa's work has the form ,symbol2#eic but the manuscripts show that the ,florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like ,symbol#eid and ,symbol#eid or ,symbol1#eie more closely resembling our own4 ,the first printed books generally used our present form#eif with the closed top ,symbol1 the open top used in writing 7 ,symbol7 being #adg purely modern4 ,the following are other forms of the four1 from various manuscripts3#eig ,illustration #e4 ,five also varied greatly before the time of printing4 ,the following are some of the forms3#eih ,illustration #f4 ,six has changed rather less than most of the others4 ,the chief variation has been in the slope of the top1 as will be seen in the following3#eii ,illustration #g4 ,seven1 like four1 has assumed its present erect form only since the fifteenth century4 ,in medieval times it appeared as follows3#fjj ,illustration #adh #h4 ,eight1 like six1 has changed but little4 ,in medieval times there are a few variants of interest as follows3#fja ,illustration ,in the sixteenth century1 however1 there was manifested a tendency to write it ,symbol4#fjb #i4 ,nine has not varied as much as most of the others4 ,among the medieval forms are the following3#fjc ,illustration #j4 ,the shape of the zero also had a varied history4 ,the following are common medieval forms3#fjd ,illustration ,the explanation of the place value was a serious matter to most of the early writers4 ,if they had been using an abacus constructed like the ,russian chotue1 and had placed this before all learners of the positional system1 there would have been little trouble4 ,but the medieval #adi line-reckoning1 where the lines stood for powers of #aj and the spaces for half of such powers1 did not lend itself to this comparison4 ,accordingly we find such labored explanations as the following1 from ,the ,crafte of ,nombrynge3 ,euery of these figuris bitokens hym selfe no more1 yf he stonde in the first place of the rewele4444 ,if it stonde in the secunde place of the rewle1 he betokens ten tymes hym selfe1 as this figure #b here #bj tokens ten tyme hym selfe1 that is twenty1 for he hym selfe betokens tweyne1 ten tymes twene is twenty4 ,and for he stondis on the lyft side in the secunde place1 he betokens ten tyme hym selfe4 ,and so go forth4444 ,nil cifra significat sed dat signare sequenti4 ,expone this verse4 ,a cifre tokens noght1 bot he makes the figure to betoken that comes after hym more than he shuld he were away1 as thus #aj4 here the figure of one tokens ten1 yf the cifre were away no figure byfore hym he schuld token bot one1 for than he schuld stonde in the first place4444#fje ,it would seem that a system that was thus used for dating documents1 coins1 and monuments1 would have been generally adopted much earlier than it was1 particularly in those countries north of ,italy where it did not come into general use until the sixteenth century4 ,this1 however1 has been the fate of many inventions1 as witness our neglect of logarithms and of contracted processes to-day4 ,as to ,germany1 the fifteenth century saw the rise of the new symbolism2 the sixteenth century saw it slowly #aej gain the mastery2 the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business4 ,not a little of the success of the new plan was due to ,luther's demand that all learning should go into the vernacular4#fjf ,during the transition period from the ,roman to the ,arabic numerals1 various anomalous forms found place4 ,for example1 we have in the fourteenth century calpha for #ajd2fjg #ajjj4 #cjj4 #hj et #d for #achd2fjh and in a manuscript of the fifteenth century #abija for #abia4fji ,in the same century m4 cccc4 #h;,,ii appears for #adhb1faj while ,mo,c,c,c,co#ej 7#adej7 and ,,mccccxl#f 7#addf7 are used by ,theodoricus ,ruffi about the same time4#faa ,to the next century belongs the form #avojj for #aejb4 ,even in ,sfortunati's ,nuovo lume#fab the use of ordinals is quite confused1 the propositions on a single page being numbered tertia1 #d1 and ,v4 ,although not connected with the ,arabic numerals in any direct way1 the medieval astrological numerals may here be mentioned4 ,these are given by several early writers1 but notably by ,noviomagus 7#aeci71fac as follows#fad3 ,illustration #aea ,thus we find the numerals gradually replacing the ,roman forms all over ,europe1 from the time of ,leonardo of ,pisa until the seventeenth century4 ,but in the ,far ,east to-day they are quite unknown in many countries1 and they still have their way to make4 ,in many parts of ,india1 among the common people of ,japan and ,china1 in ,siam and generally about the ,malay ,peninsula1 in ,tibet1 and among the ,east ,india islands1 the natives still adhere to their own numeral forms4 ,only as ,western civilization is making its way into the commercial life of the ,east do the numerals as used by us find place1 save as the ,sanskrit forms appear in parts of ,india4 ,it is therefore with surprise that the student of mathematics comes to realize how modern are these forms so common in the ,west1 how limited is their use even at the present time1 and how slow the world has been and is in adopting such a simple device as the ,hindu-,arabic numerals4 99 99 99 99 99 #aec ,,index ,transcriber's note3 many of the entries refer to footnotes linked from the page numbers given4 ,abbo of ,fleury1 #abb ,abdallah ibn al-,h4asan1 #ib ,abdallatif ibn ,yusuf1 #ic ,abdalqadir ibn ,ali al-,sakhawi1 #f ,abenragel1 #cd ,abraham ibn ,meir ibn ,ezra1 see ,rabbi ben ,ezra ,abu ,ali al-,h4osein ibn ,sina1 #gd ,abu 'l-,h4asan1 #ic1 #ajj ,abu 'l-,qasim1 #ib ,abu 'l-,t4eiyib1 #ig ,abu ,nas4r1 #ib ,abu ,roshd1 #aac ,abu ,sahl ,dunash ibn ,tamim1 #fe1 #fg ,adelhard of ,bath1 #e1 #ee1 #ig1 #aai1 #abc1 #abf ,adhemar of ,chabanois1 #aaa ,ah4med al-,nasawi1 #ih ,ah4med ibn ,abdallah1 #i1 #ib ,ah4med ibn ,moh4ammed1 #id ,ah4med ibn ,omar1 #ic ,aks4aras1 #cb ,alanus ab ,insulis1 #abd ,al-,ba4gdadi1 #ic ,al-,battani1 #ed ,albelda 7,albaida7 ,,ms41 #aaf ,albert1 ,j41 #fb ,albert of ,york1 #ajc ,al-,biruni1 #f1 #da1 #di1 #fe1 #ib1 #ic ,alcuin1 #ajc ,alexander the ,great1 #gf ,alexander de ,villa ,dei1 #aa1 #acc ,alexandria1 #fd1 #hb ,al-,fazari1 #ib ,alfred1 #ajc ,algebra1 etymology1 #e ,algerian numerals1 #fh ,algorism1 #ig ,algorismus1 #abd1 #abf1 #ace ,algorismus cifra1 #abj ,al-,h4as4s4ar1 #fe ,ali ibn ,abi ,bekr1 #f ,ali ibn ,ah4med1 #ic1 #ih ,al-,karabisi1 #ic ,al-,khowarazmi1 #d1 #i1 #aj1 #ib1 #ig1 #ih1 #abe1 #abf ,al-,kindi1 #aj1 #ib ,almagest1 #ed ,al-,ma4grebi1 #ic ,al-,mah4alli1 #f ,al-,mamun1 #aj1 #ig ,al-,mans4ur1 #if1 #ig ,al-,masudi1 #g1 #ib ,al-,nadim1 #i ,al-,nasawi1 #ic1 #ih ,alphabetic numerals1 #ci1 #dj1 #dc ,al-,qasim1 #ib ,al-,qass1 #id ,al-,sakhawi1 #f ,al-,s4ardafi1 #ic ,al-,sijzi1 #id ,al-,sufi1 #aj1 #ib ,ambrosoli1 #aah ,a4nkapalli1 #dc ,apices1 #hg1 #aag1 #aah ,arabs1 #ia-ih ,arbuthnot1 #ada #aed ,archimedes1 #ae1 #af ,arcus ,pictagore1 #abb ,arjuna1 #ae ,arnold1 ,e41 #ae1 #ajb ,ars memorandi1 #ada ,aryabhat4a1 #ci1 #dc1 #dd ,aryan numerals1 #ai ,aschbach1 #acd ,ashmole1 #acd ,a'soka1 #ai1 #bj1 #bb1 #ha ,as4-s4ifr1 #eg1 #eh ,astrological numerals1 #aej ,atharva-,veda1 #dh1 #di1 #ee ,augustus1 #hj ,averroes1 #aac ,avicenna1 #eh1 #gd1 #aac ,babylonian numerals1 #bh ,babylonian zero1 #ea ,bacon1 ,r41 #aca ,bactrian numerals1 #ai1 #cj ,baeda1 #b1 #gb ,bagdad1 #d1 #if ,bakhs4ali manuscript1 #dc1 #di1 #eb1 #ec ,ball1 ,c4 ,j41 #ce ,ball1 ,w4 ,w4 ,r41 #cf1 #aca ,ban4a1 #dd ,barth1 ,a41 #ci ,bayang inscriptions1 #ci ,bayer1 #cc ,bayley1 ,e4 ,c41 #ai1 #bc1 #cj1 #cb1 #eb1 #hi ,beazley1 #ge ,bede1 see ,baeda ,beldomandi1 #acg ,beloch1 ,j41 #gg ,bendall1 #be1 #eb ,benfey1 ,t41 #bf ,bernelinus1 #hh1 #aab1 #aag1 #aba ,besagne1 #abh ,besant1 ,w41 #aji ,bettino1 #cf ,bhandarkar1 #ah1 #dg1 #di ,bhaskara1 #ec1 #ee ,biernatzki1 #cb ,biot1 #cb ,bjoernbo1 ,a4 ,a41 #abe1 #abf ,blassiere1 #aai ,bloomfield1 #dh ,blume1 #he ,boeckh1 #fb ,boehmer1 #adc ,boeschenstein1 #aai ,boethius1 #fc1 #gj-gc1 #hc-ij ,boissiere1 #fc ,bombelli1 #ha ,bonaini1 #abh ,boncompagni1 #e1 #f1 #aj1 #dh1 #di1 #abc1 #abe ,borghi1 #ei ,borgo1 #aai ,bougie1 #acj ,bowring1 ,j41 #ef ,brahmagupta1 #eb ,brahman4as1 #ab1 #ac ,brahmi1 #ai1 #bj1 #ca1 #hc ,brandis1 ,j41 #ed ,br4hat-,sam4hita1 #ci1 #dd1 #gh ,brockhaus1 #dc ,bubnov1 #fe1 #hd1 #aaj1 #aaf ,buddha1 education of1 #ae1 #af ,buedinger1 #aaj ,bugia1 #acj ,buehler1 ,g41 #ae1 #ai1 #bb1 #ca1 #dd1 #di ,burgess1 #be ,buerk1 #ac ,burmese numerals1 #cf ,burnell1 ,a4 ,c41 #ah1 #dj ,buteo1 #fa ,calandri1 #ei1 #ha ,caldwell1 ,r41 #ai ,calendars1 #acc ,calmet1 #cd ,cantor1 ,m41 #e1 #ac1 #cj1 #dc1 #hd #aee ,capella1 #hf ,cappelli1 #adc ,caracteres1 #hg1 #aac1 #aag1 #aai ,cardan1 #aai ,carmen de ,algorismo1 #aa1 #acd ,casagrandi1 #acb ,casiri1 #h1 #aj ,cassiodorus1 #gb ,cataldi1 #fb ,cataneo1 #c ,caxton1 #adc1 #adf ,ceretti1 #cb ,ceylon numerals1 #cf ,chalfont1 ,f4 ,h41 #bh ,champenois1 #fj ,characters1 see ,caracteres ,charlemagne1 #ajc ,chasles1 #ed1 #fj1 #he1 #aaf1 #abb1 #ace ,chassant1 ,l4 ,a41 #adb ,chaucer1 #aba ,chiarini1 #ade1 #adf ,chiffre1 #eh ,chinese numerals1 #bh1 #ef ,chinese zero1 #ef ,cifra1 #abj1 #abd ,cipher1 #eh ,circulus1 #eh1 #fj ,clichtoveus1 #fa1 #aai1 #ade ,codex ,vigilanus1 #ach ,codrington1 ,o41 #aci ,coins dated1 #ada ,colebrooke1 #h1 #bf1 #df1 #ec ,constantine1 #ajd1 #aje ,cosmas1 #hb ,cossali1 #e ,counters1 #aag ,courteille1 #h ,coxe1 #ei ,crafte of ,nombrynge1 #aa1 #hg1 #adi ,crusades1 #aji ,cunningham1 ,a41 #cj1 #ge ,curtze1 #ee1 #ei1 #abf1 #acd ,cyfra1 #ee ,dagomari1 #adf ,d',alviella1 #ae ,dante1 #gb ,dasypodius1 #cc1 #fg1 #fc ,daunou1 #ace ,delambre1 #ed ,devanagari1 #g ,devoulx1 ,a41 #fh ,dhruva1 #di ,dicaearchus of ,messana1 #gg ,digits1 #aai ,diodorus ,siculus1 #gf ,du ,cange1 #fb ,dumesnil1 #cf ,dutt1 ,r4 ,c41 #ab1 #ae1 #ah1 #ge ,dvivedi1 #dd ,east and ,west1 relations1 #gc-ha1 #ajj-aji ,egyptian numerals1 #bg ,eisenlohr1 #bh ,elia ,misrachi1 #eg ,enchiridion ,algorismi1 #eh ,enestroem1 #e1 #dh1 #ei1 #ig1 #abe1 #abh ,europe1 numerals in1 #fc1 #ii1 #abh1 #acf ,eusebius ,caesariensis1 #adb ,euting1 #ba ,ewald1 ,p41 #aaf ,fazzari1 #ec1 #ed ,fibonacci1 see ,leonardo of ,pisa ,figura nihili1 #eh ,figures1 #aai4 ,see numerals4 ,fihrist1 #fg1 #fh1 #ic ,finaeus1 #eg ,firdusi1 #ha ,fitz ,stephen1 ,w41 #aji ,fleet1 ,j4 ,c41 #ai1 #bj1 #di #aef ,florus1 #hj ,fluegel1 ,g41 #fh ,francisco de ,retza1 #adb ,francois1 #eh ,friedlein1 ,g41 #hd1 #aac1 #aaf1 #abb ,froude1 ,j4 ,a41 #abi ,gandhara1 #ai ,garbe1 #dh ,gasbarri1 #eh ,gautier de ,coincy1 #abj1 #abd ,gemma ,frisius1 #b1 #c1 #aai ,gerber1 #aac ,gerbert1 #ajh1 #aaj-abj1 #abb ,gerhardt1 ,c4 ,i41 #dc1 #ef1 #ic1 #aah ,gerland1 #hh1 #abc ,gherard of ,cremona1 #abe ,gibbon1 #gb ,giles1 ,h4 ,a41 #gi ,ginanni1 #ha ,giovanni di ,danti1 #eh ,glareanus1 #d1 #aai ,gnecchi1 #ga1 #aag 4,gobar numerals1 #fe1 #ajj1 #aab1 #abd1 #ach ,gow1 ,j41 #ha ,grammateus1 #fa ,greek origin1 #cc ,green1 ,j4 ,r41 #aji ,greenwood1 ,i41 #fb1 #aai ,guglielmini1 #abh ,gulistan1 #ajb ,guenther1 ,s41 #aca ,guyard1 ,s41 #hb ,h4abash1 #i1 #ib ,hager1 ,j4 7,g471 #bh1 #cb ,halliwell1 #ei1 #he ,hankel1 #ic ,harun al-,rashid1 #ig1 #ajf ,havet1 #aaj ,heath1 ,t4 ,l41 #abe ,hebrew numerals1 #abg ,hecataeus1 #ge ,heiberg1 ,j4 ,l41 #ee1 #he1 #adh ,heilbronner1 #e ,henry1 ,c41 #e1 #ca1 #ee1 #hg1 #abj1 #ace ,heriger1 #abb ,hermannus ,contractus1 #abc ,herodotus1 #gf1 #gh ,heyd1 #ge ,higden1 #acf ,hill1 ,g4 ,f41 #eb1 #aci1 #adb ,hillebrandt1 ,a41 #ae1 #gd ,hilprecht1 ,h4 ,v41 #bh ,hindu forms1 early1 #ab ,hindu number names1 #db ,hodder1 #fb ,hoernle1 #dc1 #di ,holywood1 see ,sacrobosco ,hopkins1 ,e4 ,w41 #ab ,horace1 #gi1 #hj ,h4osein ibn ,moh4ammed al-,mah4alli1 #f ,hostus1 ,m41 #ef ,howard1 ,h4 ,h41 #bi ,hrabanus ,maurus1 #gb ,huart1 #g ,huet1 #cc ,hugo1 ,h41 #eg ,humboldt1 ,a4 von1 #fb ,huswirt1 #eh ,iamblichus1 #ha ,ibn ,abi ,yaqub1 #i ,ibn al-,adami1 #ib ,ibn al-,banna1 #ic ,ibn ,khordad4beh1 #aja1 #ajf ,ibn ,wahab1 #ajc ,india1 history of1 #ad writing in1 #ah ,indicopleustes1 #hc ,indo-,bactrian numerals1 #ai #aeg ,indraji1 #bc ,ish4aq ibn ,yusuf al-,s4ardafi1 #ic ,jacob of ,florence1 #eg ,jacquet1 ,e41 #ch ,jamshid1 #ef ,jehan ,certain1 #ei ,jetons1 #eh1 #aag ,jevons1 ,f4 ,b41 #gf ,johannes ,hispalensis1 #dh1 #hh1 #abd ,john of ,halifax1 see ,sacrobosco ,john of ,luna1 see ,johannes ,hispalensis ,jordan1 ,l41 #eh1 #abd ,joseph ,ispanus 7,joseph ,sapiens71 #aae ,justinian1 #ajd ,kale1 ,m4 ,r41 #bf ,karabacek1 #ef ,karpinski1 ,l4 ,c41 #abf1 #acd1 #ach ,katyayana1 #ci ,kaye1 ,c4 ,r41 #f1 #af1 #dc1 #df1 #aba ,keane1 ,j41 #ge1 #hb ,keene1 ,h4 ,g41 #ae ,kern1 #dd ,kharos4t4hi1 #ai1 #bj ,khosru1 #hb1 #ia ,kielhorn1 ,f41 #df1 #dg ,kircher1 ,a41 #cd ,kitab al-,fihrist1 see ,fihrist ,kleinwaechter1 #cb ,klos1 #fb ,koebel1 #d1 #eh1 #fj1 #aai1 #abc ,krumbacher1 ,k41 #eg ,kuckuck1 #fb1 #acc ,kugler1 ,f4 ,x41 #ea ,lachmann1 #he ,lacouperie1 #cc1 #ce ,lalitavistara1 #ae1 #ag ,lami1 ,g41 #eg ,la ,roche1 #fa ,lassen1 #ci ,lat4yayana1 #ci ,leboeuf1 #ace ,leonardo of ,pisa1 #e1 #aj1 #eg1 #fd1 #gd1 #abj1 #abh-acc ,lethaby1 ,w4 ,r41 #adb ,levi1 ,b41 #ac ,levias1 #c ,libri1 #gc1 #he1 #ie ,light of ,asia1 #af ,luca da ,firenze1 #add ,lucas1 #abh ,mahabharata1 #ah ,mahaviracarya1 #ec ,malabar numerals1 #cf ,malayalam numerals1 #cf ,mannert1 #ha ,margarita ,philosophica1 #adf ,marie1 #gh ,marquardt1 ,j41 #he ,marshman1 ,j4 ,c41 #ag ,martin1 ,t4 ,h41 #cj1 #fb1 #he1 #aac ,martines1 ,d4 ,c41 #eh ,mashallah1 #c ,maspero1 #bh ,mauch1 #adb ,maximus ,planudes1 #b1 #eg1 #ff1 #ic1 #abj ,megasthenes1 #gg ,merchants1 #aad ,meynard1 #h ,migne1 #hg ,mikami1 ,y41 #ef ,milanesi1 #abh ,moh4ammed ibn ,abdallah1 #ib ,moh4ammed ibn ,ah4med1 #f ,moh4ammed ibn ,ali ,abdi1 #h ,moh4ammed ibn ,musa1 see ,al-,khowarazmi ,molinier1 #abc ,monier-,williams1 #ag #aeh ,morley1 ,d41 #abf ,moroccan numerals1 #fh1 #aai ,mortet1 ,v41 #aa ,moseley1 ,c4 ,b41 #cc ,mot4ahhar ibn ,t4ahir1 #g ,mueller1 ,a41 #fh ,mumford1 ,j4 ,k41 #aji ,muwaffaq al-,din1 #ic ,nabatean forms1 #ba ,nallino1 #d1 #ed1 #ee ,nagl1 ,a41 #ee1 #aaj1 #aac1 #abf ,nana ,ghat inscriptions1 #bj1 #bb1 #bc1 #dj ,narducci1 #abc ,nasik cave inscriptions1 #bd ,naz4if ibn ,yumn1 #id ,neander1 ,a41 #ge ,neophytos1 #eg1 #fb ,neo-,pythagoreans1 #fd ,nesselmann1 #eh ,newman1 ,cardinal1 #if ,newman1 ,f4 ,w41 #aca ,noeldeke1 ,th41 #ia ,notation1 #fa ,note1 #fa1 #aai ,noviomagus1 #de1 #fa1 #aai1 #aej ,null1 #fa ,numerals1 ,algerian1 #fh astrological1 #aej ,brahmi1 #ai-bb1 #hc early ideas of origin1 #a ,hindu1 #bf ,hindu1 classified1 #ai1 #ch ,kharos4t4hi1 #ai-bb ,moroccan1 #fh ,nabatean1 #ba origin1 #bg1 #cj1 #ca1 #cg supposed ,arabic origin1 #b supposed ,babylonian origin1 #bh supposed ,chaldean and ,jewish origin1 #c supposed ,chinese origin1 #bh1 #cb supposed ,egyptian origin1 #bg1 #cj1 #fi1 #gj supposed ,greek origin1 #cc supposed ,phoenician origin1 #cb tables of1 #bb-bg1 #cf1 #dh1 #di1 #fi1 #hh1 #adj1 #adc1 #ade-adh ,o',creat1 #e1 #ee1 #aai1 #abj ,olleris1 #aaj1 #aac ,oppert1 ,g41 #ad1 #ge ,pali1 #bb ,pancasiddhantika1 #dd ,paravey1 #cb1 #eg ,pataliput4ra1 #gg ,patna1 #gg ,patrick1 ,r41 #aai ,payne1 ,e4 ,j41 #ajf ,pegolotti1 #ajg ,peletier1 #b1 #fb ,perrot1 #hj ,persia1 #ff1 #ia1 #ajg ,pertz1 #aae ,petrus de ,dacia1 #ei1 #fa1 #fb ,pez1 ,p4 ,b41 #aag ,philalethes1 #ge ,phillips1 ,g41 #ajg ,picavet1 #aje ,pichler1 ,f41 #ada ,pihan1 ,a4 ,p41 #cf ,pisa1 #abh ,place value1 #bf1 #db1 #df1 #dh ,planudes1 see ,maximus ,planudes ,plimpton1 ,g4 ,a41 #ef1 #ei1 #he1 #adc1 #add1 #ade1 #adh ,pliny1 #gf ,polo1 ,n4 and ,m41 #ajg #aei ,praendel1 ,j4 ,g41 #ed ,prinsep1 ,j41 #bj1 #ca ,propertius1 #hj ,prosdocimo de' ,beldomandi1 #acg ,prou1 #adc ,ptolemy1 #ed1 #gh ,putnam1 #ajc ,pythagoras1 #fc ,pythagorean numbers1 #ac ,pytheas of ,massilia1 #gf ,rabbi ben ,ezra1 #fj1 #abg ,radulph of ,laon1 #fj1 #aac1 #aah1 #abd ,raets1 #fb ,rainer1 see ,gemma ,frisius ,ramayana1 #ah ,ramus1 #b1 #da1 #fj1 #fa ,raoul ,glaber1 #abc ,rapson1 #gg ,rauhfuss1 see ,dasypodius ,raumer1 ,k4 von1 #aaa ,reclus1 ,e41 #ad1 #if1 #acj ,recorde1 #c1 #eh ,reinaud1 #fg1 #gd1 #hj ,reveillaud1 #cf ,richer1 #aaj1 #aab1 #aae ,riese1 ,a41 #aai ,robertson1 #ha ,robertus ,cestrensis1 #ig1 #abf ,rodet1 #e1 #dd ,roediger1 ,j41 #fh ,rollandus1 #add ,romagnosi1 #ha ,rosen1 ,f41 #e ,rotula1 #fj ,rudolff1 #he ,rudolph1 #fb1 #fg ,ruffi1 #aej ,sachau1 #f ,sacrobosco1 #c1 #eh1 #acc ,sacy1 ,s4 de1 #ff1 #gj ,sadi1 #ajb ',saka inscriptions1 #bj ,samu'il ibn ,yah4ya1 #ic ',sarada characters1 #ee ,savonne1 #fj ,scaliger1 ,j4 ,c41 #gc ,scheubel1 #fb ,schlegel1 #ab ,schmidt1 #acc ,schonerus1 #hg1 #aai ,schroeder1 ,l4 von1 #ac ,scylax1 #ge ,sedillot1 #h1 #cd ,senart1 #bj1 #bd1 #be ,sened ibn ,ali1 #aj1 #ih ,sfortunati1 #fb1 #aej ,shelley1 ,w41 #abf ,siamese numerals1 #cf ,siddhanta1 #h1 #ah ,s4ifr1 #eg ,sigsboto1 #ee ,sihab al-,din1 #fg ,silberberg1 #fj ,simon1 #ac ,sinan ibn al-,fath41 #ic ,sindbad1 #ajj ,sindhind1 #ig ,sipos1 #fj ,sirr1 ,h4 ,c41 #ge ,skeel1 ,c4 ,a41 #gd ,smith1 ,d4 ,e41 #aa1 #ag1 #ec1 #hf1 #ada1 #adc ,smith1 ,v4 ,a41 #bj1 #ce1 #df1 #dg ,smith1 ,wm41 #ge ,smr4ti1 #ag ,spain1 #fd1 #fe1 #ajj ,spitta-,bey1 #e ,sprenger1 #id ',srautasutra1 #ci ,steffens1 ,f41 #aaf ,steinschneider1 #e1 #eg1 #fe1 #ff1 #ih1 #abf ,stifel1 #fb #afj ,subandhus1 #dd ,suetonius1 #hj ,suleiman1 #ajj ',sunya1 #dc1 #ec1 #eg ,suter1 #e1 #i1 #fh1 #fi1 #ic1 #aaf1 #aca ,sutras1 #ac ,sykes1 ,p4 ,m41 #ge ,sylvester ,,ii1 see ,gerbert ,symonds1 ,j4 ,a41 #abi ,tannery1 ,p41 #fb1 #hd1 #he ,tartaglia1 #d1 #fa ,taylor1 ,i41 #ai1 #cj ,teca1 #ee1 #fa ,tennent1 ,j4 ,e41 #ge ,texada1 #fj ,theca1 #eh1 #fa ,theophanes1 #fd ,thibaut1 ,g41 #ab1 #ac1 #af1 #dd1 #dg ,tibetan numerals1 #cf ,timotheus1 #ajc ,tonstall1 ,c41 #c1 #fa ,trenchant1 #fj ,treutlein1 #e1 #fc1 #abc ,trevisa1 #acf ,treviso arithmetic1 #ade ,trivium and quadrivium1 #gc ,tsin1 #ef ,tunis1 #fe ,turchill1 #hh1 #aah1 #abc ,turnour1 ,g41 #ge ,tziphra1 #eg1 #fb ,greek3 tziphra1 #ee1 #eg1 #fb ,tzwivel1 #fa1 #aah1 #ade ,ujjain1 #cb ,unger1 #acc ,upanishads1 #ab ,usk1 #aba ,valla1 ,g41 #fa ,van der ,schuere1 #fb ,varaha-,mihira1 #ci1 #dd1 #gh ,vasavadatta1 #dd ,vaux1 ,carra de1 #i1 #gd ,vaux1 ,w4 ,s4 ,w41 #ia ,veda4ngas1 #ag ,vedas1 #ab1 #ae1 #ag ,vergil1 #hj ,vincent1 ,a4 ,j4 ,h41 #eg ,vogt1 #ac ,voizot1 ,p41 #cf ,vossius1 #d1 #gf1 #ha1 #hd ,wallis1 #c1 #fb1 #hd1 #aaf ,wappler1 ,e41 #ed1 #abf ,waeschke1 ,h41 #b1 #ic ,wattenbach1 #adc ,weber1 ,a41 #ca ,weidler1 ,i4 ,f41 #cd1 #ff ,weidler1 ,i4 ,f4 and ,g4 ,i41 #fc1 #ff ,weissenborn1 #he1 #aaj ,wertheim1 ,g41 #eg1 #fa ,whitney1 ,w4 ,d41 #ac ,wilford1 ,f41 #ge ,wilkens1 #fb ,wilkinson1 ,j4 ,g41 #gj ,willichius1 #c ,woepcke1 #c1 #f1 #db1 #fc1 #fd1 #fe1 #fg1 #fi1 #gj1 #id1 #aac1 #ach ,wolack1 ,g41 #ed ,woodruff1 ,c4 ,e41 #cb ,word and letter numerals1 #ch1 #dd ,wuestenfeld1 #gd ,yule1 ,h41 #ajg ,zephirum1 #eg1 #eh ,zephyr1 #ei ,zepiro1 #eh ,zero1 #bf1 #ch1 #dj1 #dc1 #de1 #di1 #ea-fb1 #fg ,zeuero1 #eh 99 99 99 99 99 ,,announcements 99 99 99 99 99 ,,wentworth',s ,,college ,,algebra ,,revised ,,edition #abmo4 ,half morocco4 #ecj pages4 ,list price1 #a4ej2 mailing price1 #a4fe 99 99 99 99 99 ,this book is a thorough revision of the author's ,college ,algebra4 ,some chapters of the old edition have been wholly rewritten1 and the other chapters have been rewritten in part and greatly improved4 ,the order of topics has been changed to a certain extent2 the plan is to have each chapter as complete in itself as possible1 so that the teacher may vary the order of succession at his discretion4 ,as the name implies1 the work is intended for colleges and scientific schools4 ,the first part is simply a review of the principles of algebra preceding ,quadratic ,equations1 with just enough examples to illustrate and enforce these principles4 ,by this brief treatment of the first chapters sufficient space is allowed1 without making the book cumbersome1 for a full discussion of ,quadratic ,equations1 ,the ,binomial ,theorem1 ,choice ,chance1 ,series1 ,determinants1 and the ,general ,properties of ,equations4 ,every effort has been made to present in the clearest light each subject discussed1 and to give in matter and methods the best training in algebraic analysis at present attainable4 ,,additional ,,publications ,by ,g4 ,a4 ,,wentworth ,list ,mailing price price ,plane and ,solid ,geometry 7,revised ,edition7 #a4be #a4dj ,plane ,geometry 7,revised ,edition7 4#ge 4#he ,solid ,geometry 7,revised ,edition7 4#ge 4#he ,analytic ,geometry #a4be #a4ce ,trigonometries 7,second ,revised ,editions7 ,a list of the various editions of ,wentworth's ,trigonometries will be sent on request4 99 99 99 99 99 ,,ginn ,,and ,,company ,,publishers 99 99 99 99 99 ,,algebra ,,for ,,beginners ,by ,,david ,,eugene ,,smith1 ,professor of ,mathematics in ,teachers ,college1 ,columbia ,university 99 99 99 99 99 #abmo1 cloth1 #aed pages1 #ej cents 99 99 99 99 99 ,this work is intended to serve as an introduction to the study of algebra1 and is adapted to the needs of the seventh or eighth school year4 ,it is arranged in harmony with the leading courses of study that include algebra in the curriculum of the grades4 ,the relation of algebra to arithmetic is emphasized1 the subject is treated topically1 and each important point is touched at least twice4 ,the book begins by showing the uses of algebra1 employing such practical applications as are within the pupil's range of knowledge4 ,when an interest has thus been awakened in the subject1 the fundamental operations are presented with the simple explanations necessary to make the student independent of dogmatic rules4 ,throughout the book abundant oral and written drill exercises are provided4 ,the work includes linear equations with two unknown quantities1 and easy quadratics4 ,the leading features may be summarized as follows3 7#a7 an arrangement in harmony with existing courses of study2 7#b7 a presentation designed to awaken the interest of the pupils2 7#c7 a topical arrangement for each half year1 every important topic being repeated2 7#d7 simplicity of explanations2 7#e7 development of the relation of algebra to arithmetic both in theory and in applications2 7#f7 emphasis laid on the importance of oral as well as written algebra4 99 99 99 99 99 ,,ginn ,,company ,,publishers 99 99 99 99 99 ,,books ,,for ,,teachers 99 99 99 99 99 ,list price ,allen3 ,civics and ,health #a4be ,brigham3 ,geographic ,influences in ,american ,history #a4be ,channing and ,hart3 ,guide to the ,study of ,american ,history #b4jj ,hall3 ,aspects of ,child ,life and ,education #a4ej ,hodge3 ,nature ,study and ,life #a4ej ,johnson3 ,education by ,plays and ,games 4#ij ,johnson3 ,what to ,do at ,recess 4#be ,kern3 ,among ,country ,schools #a4be ,mace3 ,method in ,history #a4jj ,mac,vicar3 ,the ,principles of ,education 4#fj ,moral ,training in the ,public ,schools #a4be ,prince3 ,courses of ,studies and ,methods of ,teaching 4#ge ,scott3 ,social ,education #a4be ,tompkins3 ,philosophy of ,school ,management 4#ge ,tompkins3 ,philosophy of ,teaching 4#ge ,wiltse3 ,place of the ,story in ,early ,education1 and ,other ,essays4 ,a ,manual for ,teachers 4#ej ,,for ,,class ,,records ,comings3 ,complete ,record--,attendance and ,scholarship ,graded-,school ,edition 4#cj ,high-,school ,edition 4#cj ,ginn and ,company3 ,teacher's ,class ,books ,no4 ,i 4#cj ,no4 ,,ii 4#dj ,twenty ,weeks' ,class ,book 4#cj 99 99 99 99 99 ,,ginn ,,and ,,company ,publishers 99 99 99 99 99 ,,textbooks ,,on ,,mathematics ,,for ,,higher ,,schools ,,and ,,colleges ,list price ,bailey and ,woods3 ,analytic ,geometry #b4jj ,beman and ,smith3 ,academic ,algebra #a4ab ,beman and ,smith3 ,higher ,arithmetic 4#hj ,beman and ,smith3 ,new ,plane and ,solid ,geometry #a4be ,breckenridge1 ,mersereau1 and ,moore3 ,shop ,problems in ,mathematics #a4jj ,byerly3 ,differential ,calculus #b4jj ,byerly3 ,integral ,calculus 7,second ,edition4 ,revised and ,enlarged7 #b4jj ,eisenhart3 ,differential ,geometry of ,curves and ,surfaces #d4ej ,faunce3 ,descriptive ,geometry #a4be ,fine3 ,college ,algebra #a4ej ,granville3 ,plane and ,spherical ,trigonometry and ,tables #a4be ,granville3 ,plane and ,spherical ,trigonometry ,without ,tables #a4jj ,granville3 ,plane ,trigonometry and ,tables #a4jj ,granville3 ,logarithmic ,tables 4#ej ,granville and ,smith3 ,elements of ,differential and ,integral ,calculus #b4ej ,hardy3 ,analytic ,geometry #a4ej ,hardy3 ,elements of the ,calculus #a4ej ,hawkes3 ,advanced ,algebra #a4dj ,hawkes1 ,luby1 and ,touton3 ,first ,course in ,algebra #a4jj ,hedrick3 ,goursat's ,mathematical ,analysis4 ,vol4 ,i #d4jj ,hedrick and ,kellogg3 ,applications of the ,calculus to ,mechanics #a4be ,hooper and ,wells3 ,electrical ,problems #a4be ,lester3 ,integrals of ,mechanics 4#hj ,peirce1 ,b4 ,o43 ,newtonian ,potential ,function 7,third1 ,revised1 and ,enlarged ,edition7 #b4ej ,peirce1 ,j4 ,m43 ,elements of ,logarithms 4#ej ,peirce1 ,j4 ,m43 ,mathematical ,tables 4#dj ,pierpont3 ,theory of ,functions of ,real ,variables4 ,vol4 ,i #d4ej ,shepard3 ,problems in the ,strength of ,materials #a4be ,slocum and ,hancock3 ,textbook on the ,strength of ,materials #c4jj ,smith and ,gale3 ,elements of ,analytic ,geometry #b4jj ,smith and ,gale3 ,introduction to ,analytic ,geometry #a4be ,smith and ,longley3 ,theoretical ,mechanics #b4ej ,taylor3 ,elements of the ,calculus 7,revised and ,enlarged ,edition7 #b4jj ,taylor3 ,plane and ,spherical ,trigonometry #a4jj ,taylor3 ,plane ,trigonometry 4#ge ,taylor and ,puryear3 ,elements of ,plane and ,spherical ,trigonometry #a4be ,wentworth3 ,advanced ,arithmetic #a4jj ,wentworth3 ,analytic ,geometry #a4be ,wentworth3 ,college ,algebra 7,revised ,edition7 #a4ej ,wentworth3 ,elementary ,algebra #a4ab ,wentworth3 ,higher ,algebra #a4dj ,wentworth3 ,new ,school ,algebra #a4ab ,wentworth3 ,plane and ,solid ,geometry 7,revised ,edition7 #a4be ,wentworth3 ,plane ,geometry 7,revised ,edition72 ,solid ,geometry 7,revised ,edition71 each 4#ge ,wentworth3 ,trigonometries 7,second ,revised ,editions7 7,for list1 see ,descriptive ,catalogue7 ,woods and ,bailey3 ,a ,course ,in ,mathematics ,volume ,i #b4be ,volume ,,ii #b4be 99 99 99 99 99 ,,ginn ,,and ,,company ,,publishers 99 99 99 99 99 ,notes al-,mekki on a treatise on 4gobar arithmetic 7explained later7 called ,al-murshidah1 found by ,woepcke in ,paris 7,propagation1 p4 #ff71 there is mentioned the fact that there are nine ,indian figures and a second kind of ,indian figures 444 although these are the figures of the 4gobar writing4 ,so in a commentary by ,h4osein ibn ,moh4ammed al-,mah4alli 7died in #agef7 on the ,mokhtas4ar fiilm el-h4isab 7,extract from ,arithmetic7 by ,abdalqadir ibn ,ali al-,sakhawi 7died c4 #ajjj7 it is related that the preface treats of the forms of the figures of ,hindu signs1 such as were established by the ,hindu nation4 ,woepcke1 ,propagation1 p4 #fc4 which1 of course1 are interpolations4 ,an interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by ,st4 ,patrick1 granting indulgences to the benefactors of ,glastonbury1 dated ,in nomine domini nostri ,jhesu ,christi ,ego ,patricius humilis servunculus ,dei anno incarnationis ejusdem ccccxxx4 ,now if the ,benedictines are right in saying that ,dionysius ,exiguus1 a ,scythian monk1 first arranged the ,christian chronology c4 #ecb ,a4,d41 this can hardly be other than spurious4 ,see ,arbuthnot1 loc4 cit41 p4 #ch4 #a ,discipulus4 ,quis primus invenit numerum apud ,hebraeos et ,a,egyptios8 ,magister4 ,abraham primus invenit numerum apud ,hebraeos1 deinde ,moses2 et ,abraham tradidit istam scientiam numeri ad ,a,egyptios1 et docuit eos3 deinde ,josephus4 ,bede1 ,de computo dialogus 7doubtfully assigned to him71 ,opera omnia1 ,paris1 #ahfb1 ,vol4 ,i1 p4 #fej4 ,alii referunt ad ,phoenices inventores arithmeticae1 propter eandem commerciorum caussam3 ,alii ad ,indos3 ,ioannes de ,sacrobosco1 cujus sepulchrum est ,lutetiae in comitio ,maturinensi1 refert ad ,arabes4 ,ramus1 ,arithmeticae libri dvo1 ,basel1 #aefi1 p4 #aab4 ,similar notes are given by ,peletarius in his commentary on the arithmetic of ,gemma ,frisius 7#aefc ed41 fol4 #gg71 and in his own work 7#aegj ,lyons ed41 p4 #ad73 ,la valeur des ,figures commence au coste dextre tirant vers le coste senestre3 au rebours de notre maniere d'escrire par ce que la premiere prattique est venue des ,chaldees3 ou des ,pheniciens1 qui ont ete les premiers traffiquers de marchandise4 #b ,maximus ,planudes 7c4 #accj7 states that the nine symbols come from the ,indians4 ,waeschke's ,german translation1 ,halle1 #ahgh1 p4 #c4 ,willichius speaks of the ,zyphrae ,indicae1 in his ,arithmeticae libri tres 7,strasburg1 #aedj1 p4 #ic71 and ,cataneo of le noue figure de gli ,indi1 in his ,le pratiche delle dve prime mathematiche 7,venice1 #aedf1 fol4 #a74 ,woepcke is not correct1 therefore1 in saying 7,memoire sur la propagation des chiffres indiens1 hereafter referred to as ,propagation ,journal ,asiatique1 ,vol4 ,i 7#f71 #ahfc1 p4 #cd7 that ,wallis 7,a ,treatise on ,algebra1 both historical and practical1 ,london1 #afhe1 p4 #ac1 and ,de algebra tractatus1 ,latin edition in his ,opera omnia1 #afic1 ,vol4 ,,ii1 p4 #aj7 was one of the first to give the ,hindu origin4 #c ,from the #aeeh edition of ,the ,grovnd of ,artes1 fol4 ,c1 #e4 ,similarly ,bishop ,tonstall writes3 ,qui a ,chaldeis primum in finitimos1 deinde in omnes pene gentes fluxit4444 ,numerandi artem a ,chaldeis esse profectam3 qui dum scribunt1 a dextra incipiunt1 et in leuam progrediuntur4 ,de arte supputandi1 ,london1 #aebb1 fol4 ,b1 #c4 ,gemma ,frisius1 the great continental rival of ,recorde1 had the same idea3 ,primum autem appellamus dexterum locum1 eo quod haec ars vel a ,chaldaeis1 vel ab ,hebraeis ortum habere credatur1 qui etiam eo ordine scribunt2 but this refers more evidently to the ,arabic numerals4 ,arithmeticae practicae methodvs facilis1 ,antwerp1 #aedj1 fol4 #d of the #aefc ed4 ,sacrobosco 7c4 #abbe7 mentions the same thing4 ,even the modern ,jewish writers claim that one of their scholars1 ,mashallah 7c4 #hjj71 introduced them to the ,mohammedan world4 ,c4 ,levias1 ,the ,jewish ,encyclopedia1 ,new ,york1 #aije1 ,vol4 ,,ix1 p4 #cdh4 #d 444 que esto fu trouato di fare da gli ,arabi con diece figure4 ,la prima parte del general trattato di nvmeri1 et misvre1 ,venice1 #aeef1 fol4 #i of the #aeib edition4 #e ,vom welchen ,arabischen auch disz ,kunst entsprungen ist4 ,ain nerv geordnet ,rechenbiechlin1 ,augsburg1 #aead1 fol4 #ac of the #aeca edition4 ,the printer used the letters rv for w in new in the first edition1 as he had no w of the proper font4 #f ,among them ,glareanus3 ,characteres simplices sunt nouem significatiui1 ab ,indis usque1 siue ,chaldaeis asciti 4#a4b4c4d4e4f4g4h4i4 ,est item unus 4#j circulus1 qui nihil significat4 ,de ,,vi4 ,arithmeticae practicae speciebvs1 ,paris1 #aeci1 fol4 #i of the #aedc edition4 #g ,barbarische oder gemeine ,ziffern4 ,anonymous1 ,das ,einmahl ,eins cum notis variorum1 ,dresden1 #agjc1 p4 #c4 ,so ,vossius 7,de universae matheseos natura et constitutione liber1 ,amsterdam1 #afej1 p4 #cd7 calls them ,barbaras numeri notas4 ,the word at that time was possibly synonymous with ,arabic4 #h ,his full name was ,abu ,abdallah ,moh4ammed ibn ,musa al-,khowarazmi4 ,he was born in ,khowarezm1 the lowlands1 the country about the present ,khiva and bordering on the ,oxus1 and lived at ,bagdad under the caliph al-,mamun4 ,he died probably between #bbj and #bcj of the ,mohammedan era1 that is1 between #hce and #hde ,a4,d41 although some put the date as early as #hab4 ,the best account of this great scholar may be found in an article by ,c4 ,nallino1 ,al-,h7uwarizmi in the ,atti della ,r4 ,accad4 dei ,lincei1 ,rome1 #ahif4 ,see also ,verhandlungen des #e4 ,congresses der ,orientalisten1 ,berlin1 #ahhb1 ,vol4 ,,ii1 p4 #ai2 ,w4 ,spitta-,bey in the ,zeitschrift der deutschen ,morgenlaend4 ,gesellschaft1 ,vol4 ,,xxxiii1 p4 #bbd2 ,steinschneider in the ,zeitschrift der deutschen ,morgenlaend4 ,gesellschaft1 ,vol4 ,l1 p4 #bad2 ,treutlein in the ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,i1 p4 #e2 ,suter1 ,die ,mathematiker und ,astronomen der ,araber und ihre ,werke1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,x1 ,leipzig1 #aijj1 p4 #aj1 and ,nachtraege1 in ,vol4 ,,xiv1 p4 #aeh2 ,cantor1 ,geschichte der ,mathematik1 ,vol4 ,i1 #c;d ed41 pp4 #gab-gcc etc42 ,f4 ,woepcke in ,propagation1 p4 #dhi4 ,so recently has he become known that ,heilbronner1 writing in #agdb1 merely mentions him as ,ben-,musa1 inter ,arabes celebris ,geometra1 scripsit de figuris planis sphericis4 ,historia matheseos universae1 ,leipzig1 #agdb1 p4 #dch4 ,in this work most of the ,arabic names will be transliterated substantially as laid down by ,suter in his work ,die ,mathematiker etc41 except where this violates ,english pronunciation4 ,the scheme of pronunciation of oriental names is set forth in the preface4 #i ,our word algebra is from the title of one of his works1 ,al-jabr wa'l-muqabalah1 ,completion and ,comparison4 ,the work was translated into ,english by ,f4 ,rosen1 ,london1 #ahca1 and treated in ,l',algebre d'al-,kharizmi et les methodes indienne et grecque1 ,leon ,rodet1 ,paris1 #ahgh1 extract from the ,journal ,asiatique4 ,for the derivation of the word algebra1 see ,cossali1 ,scritti ,inediti1 pp4 #cha-chc1 ,rome1 #aheg2 ,leonardo's ,liber ,abbaci 7#abjb71 p4 #daj1 ,rome1 #aheg2 both published by ,b4 ,boncompagni4 ,almuchabala also was used as a name for algebra4 #aj ,this learned scholar1 teacher of ,o',creat who wrote the ,helceph 7,prologus ,n4 ,ocreati in ,helceph ad ,adelardum ,batensem magistrum suum71 studied in ,toledo1 learned ,arabic1 traveled as far east as ,egypt1 and brought from the ,levant numerous manuscripts for study and translation4 ,see ,henry in the ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,iii1 p4 #aca2 ,woepcke in ,propagation1 p4 #eah4 #aa ,the title is ,algoritmi de numero ,indorum4 ,that he did not make this translation is asserted by ,enestroem in the ,bibliotheca ,mathematica1 ,vol4 ,i 7#c71 p4 #ebj4 #ab ,thus he speaks de numero indorum per 4,,ix4 literas1 and proceeds3 ,dixit algoritmi3 ,cum uidissem yndos constituisse 4,,ix4 literas in uniuerso numero suo1 propter dispositionem suam quam posuerunt1 uolui patefacere de opera quod fit per eas aliquid quod esset leuius discentibus1 si deus uoluerit4 ,boncompagni1 ,trattati d',aritmetica1 ,rome1 #aheg4 ,discussed by ,f4 ,woepcke1 ,sur l'introduction de l'arithmetique indienne en ,occident1 ,rome1 #ahei4 #ac ,thus in a commentary by ,ali ibn ,abi ,bekr ibn al-,jamal al-,ans4ari #ad ,see also ,woepcke1 ,propagation1 p4 #eje4 ,the origin is discussed at much length by ,g4 ,r4 ,kaye1 ,notes on ,indian ,mathematics4--,arithmetical ,notation1 ,journ4 and ,proc4 of the ,asiatic ,soc4 of ,bengal1 ,vol4 ,,iii1 #aijg1 p4 #dhi4 #ae ,alberuni's ,india1 ,arabic version1 ,london1 #ahhg2 ,english translation1 ibid41 #ahhh4 #af ,chronology of ,ancient ,nations1 ,london1 #ahgi4 ,arabic and ,english versions1 by ,c4 ,e4 ,sachau4 #ag ,india1 ,vol4 ,i1 chap4 xvi4 #ah ,the ,hindu name for the symbols of the decimal place system4 #ai ,sachau's ,english edition of the ,chronology1 p4 #fd4 #bj ,litterature arabe1 ,cl4 ,huart1 ,paris1 #aijb4 #ba ,huart1 ,history of ,arabic ,literature1 ,english ed41 ,new ,york1 #aijc1 p4 #ahb seq4 #bb ,al-,masudi's ,meadows of ,gold1 translated in part by ,aloys ,sprenger1 ,london1 #ahda2 ,les prairies d'or1 trad4 par ,c4 ,barbier de ,meynard et ,pavet de ,courteille1 ,vols4 ,i to ,,ix1 ,paris1 #ahfa-ahgg4 #bc ,les prairies d'or1 ,vol4 ,,viii1 p4 #bhi seq4 #bd ,essays1 ,vol4 ,,ii1 p4 #dbh4 #be ,loc4 cit41 p4 #ejd4 #bf ,materiaux pour servir a l'histoire comparee des sciences mathematiques chez les ,grecs et les ,orientaux1 #b vols41 ,paris1 #ahde-ahdi1 pp4 #dch-dci4 #bg ,he made an exception1 however1 in favor of the numerals1 loc4 cit41 ,vol4 ,,ii1 p4 #ejc4 #bh ,bibliotheca ,arabico-,hispana ,escurialensis1 ,madrid1 #agfj-aggj1 pp4 #dbf-dbg4 #bi ,the author1 ,ibn al-,qift4i1 flourished ,a4,d4 #aaih ,colebrooke1 loc4 cit41 note ,vol4 ,,ii1 p4 #eaj4 #cj ,liber ,artis ,logisticae a ,mohamado ,ben ,musa ,alkhuarezmita exornatus1 qui ceteros omnes brevitate methodi ac facilitate praestat1 ,indorum que in praeclarissimis inventis ingenium acumen ostendit4 ,casiri1 loc4 cit41 p4 #dbg4 #ca ,macoudi1 ,le livre de l'avertissement et de la revision4 ,translation by ,b4 ,carra de ,vaux1 ,paris1 #ahif4 #cb ,verifying the hypothesis of ,woepcke1 ,propagation1 that the ,sindhind included a treatment of arithmetic4 #cc ,ah4med ibn ,abdallah1 ,suter1 ,die ,mathematiker1 etc41 p4 #ab4 #cd ,india1 ,vol4 ,,ii1 p4 #ae4 #ce ,see ,h4 ,suter1 ,das ,mathematiker-,verzeichniss im ,fihrist1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,vi1 ,leipzig1 #ahib4 ,for further references to early ,arabic writers the reader is referred to ,h4 ,suter1 ,die ,mathematiker und ,astronomen der ,araber und ihre ,werke4 ,also ,nachtraege und ,berichtigungen to the same 7,abhandlungen1 ,vol4 ,,xiv1 #aijb1 pp4 #aee-ahf74 #cf ,suter1 loc4 cit41 note #afe1 pp4 #fb-fc4 #cg ,send ,ben ,ali1444 tum arithmetica scripta maxime celebrata1 quae publici juris fecit4 ,loc4 cit41 p4 #ddj4 #ch ,scritti di ,leonardo ,pisano1 ,vol4 ,i1 ,liber ,abbaci 7#aheg72 ,vol4 ,,ii1 ,scritti 7#ahfb72 published by ,baldassarre ,boncompagni1 ,rome4 ,also ,tre ,scritti ,inediti1 and ,intorno ad ,opere di ,leonardo ,pisano1 ,rome1 #ahed4 #ci ,ubi ex mirabili magisterio in arte per novem figuras indorum introductus etc4 ,in another place1 as a heading to a separate division1 he writes1 ,de cognitione novem figurarum yndorum etc4 ,novem figure indorum he sunt #i #h #g #f #e #d #c #b #a4 #dj ,see ,an ,ancient ,english ,algorism1 by ,david ,eugene ,smith1 in ,festschrift ,moritz ,cantor1 ,leipzig1 #aiji4 ,see also ,victor ,mortet1 ,le plus ancien traite francais d'algorisme1 ,bibliotheca ,mathematica1 ,vol4 ,,ix 7#c71 pp4 #ee-fd4 #da ,these are the two opening lines of the ,carmen de ,algorismo that the anonymous author is explaining4 ,they should read as follows3 ,haec algorismus ars praesens dicitur1 in qua ,talibus ,indorum fruimur bis quinque figuris4 ,what follows is the translation4 #db ,thibaut1 ,astronomie1 ,astrologie und ,mathematik1 ,strassburg1 #ahii4 #dc ,gustave ,schlegel1 ,uranographie chinoise ou preuves directes que l'astronomie primitive est originaire de la ,chine1 et qu'elle a ete empruntee par les anciens peuples occidentaux a la sphere chinoise2 ouvrage accompagne d'un atlas celeste chinois et grec1 ,the ,hague and ,leyden1 #ahge4 #dd ,e4 ,w4 ,hopkins1 ,the ,religions of ,india1 ,boston1 #ahih1 p4 #g4 #de ,r4 ,c4 ,dutt1 ,history of ,india1 ,london1 #aijf4 #df ,w4 ,d4 ,whitney1 ,sanskrit ,grammar1 #c;d ed41 ,leipzig1 #ahif4 #dg ,das ,apastamba-',sulba-,sutra1 ,zeitschrift der deutschen ,morgenlaendischen ,gesellschaft1 ,vol4 ,,lv1 p4 #edc1 and ,vol4 ,,lvi1 p4 #cbg4 #dh ,geschichte der ,math41 ,vol4 ,i1 #b;d ed41 p4 #eie4 #di ,l4 von ,schroeder1 ,pythagoras und die ,inder1 ,leipzig1 #ahhd2 ,h4 ,vogt1 ,haben die alten ,inder den ,pythagoreischen ,lehrsatz und das ,irrationale gekannt8 ,bibliotheca ,mathematica1 ,vol4 ,,vii 7#c71 pp4 #f-bj2 ,a4 ,buerk1 loc4 cit42 ,max ,simon1 ,geschichte der ,mathematik im ,altertum1 ,berlin1 #aiji1 pp4 #acg-afe2 three ,sutras are translated in part by ,thibaut1 ,journal of the ,asiatic ,society of ,bengal1 #ahge1 and one appeared in ,the ,pandit1 #ahge2 ,beppo ,levi1 ,osservazioni e congetture sopra la geometria degli indiani1 ,bibliotheca ,mathematica1 ,vol4 ,,ix 7#c71 #aijh1 pp4 #ig-aje4 #ej ,loc4 cit42 also ,indiens ,literatur und ,cultur1 ,leipzig1 #ahhg4 #ea ,it is generally agreed that the name of the river ,sindhu1 corrupted by western peoples to ,hindhu1 ,indos1 ,indus1 is the root of ,hindustan and of ,india4 ,reclus1 ,asia1 ,english ed41 ,vol4 ,,iii1 p4 #ad4 #eb ,see the comments of ,oppert1 ,on the ,original ,inhabitants of ,bharatavars4a or ,india1 ,london1 #ahic1 p4 #a4 #ec ,a4 ,hillebrandt1 ,alt-,indien1 ,breslau1 #ahii1 p4 #aaa4 ,fragmentary records relate that ,kharavela1 king of ,kali4nga1 learned as a boy lekha 7writing71 gan4ana 7reckoning71 and rupa 7arithmetic applied to monetary affairs and mensuration71 probably in the #eth century ,b4,c4 ,buehler1 ,indische ,palaeographie1 ,strassburg1 #ahif1 p4 #e4 #ed ,r4 ,c4 ,dutt1 ,a ,history of ,civilization in ,ancient ,india1 ,london1 #ahic1 ,vol4 ,i1 p4 #agd4 #ee ,the ,buddha4 ,the date of his birth is uncertain4 ,sir ,edwin ,arnold put it c4 #fbj ,b4,c4 #ef ,i4e4 #ajj4ajg4 #eg ,there is some uncertainty about this limit4 #eh ,this problem deserves more study than has yet been given it4 ,a beginning may be made with ,comte ,goblet d',alviella1 ,ce que l',inde doit a la ,grece1 ,paris1 #ahig1 and ,h4 ,g4 ,keene's review1 ,the ,greeks in ,india1 in the ,calcutta ,review1 ,vol4 ,,cxiv1 #aijb1 p4 #a4 ,see also ,f4 ,woepeke1 ,propagation1 p4 #bec2 ,g4 ,r4 ,kaye1 loc4 cit41 p4 #dge seq41 and ,the ,source of ,hindu ,mathematics1 ,journal of the ,royal ,asiatic ,society1 ,july1 #aiaj1 pp4 #gdi-gfj2 ,g4 ,thibaut1 ,astronomie1 ,astrologie und ,mathematik1 pp4 #dc-ej and #gf-gi4 ,it will be discussed more fully in ,chapter ,,vi4 #ei ,i4e4 to #ajj1jjj4 ,the lakh is still the common large unit in ,india1 like the myriad in ancient ,greece and the million in the ,west4 #fj ,this again suggests the ,psammites1 or ,de harenae numero as it is called in the #aedd edition of the ,opera of ,archimedes1 a work in which the great ,syracusan proposes to show to the king by geometric proofs which you can follow1 that the numbers which have been named by us 444 are sufficient to exceed not only the number of a sand-heap as large as the whole earth1 but one as large as the universe4 ,for a list of early editions of this work see ,d4 ,e4 ,smith1 ,rara ,arithmetica1 ,boston1 #aiji1 p4 #bbg4 #fa ,i4e4 the ,wise4 #fb ,sir ,monier ,monier-,williams1 ,indian ,wisdom1 #dth ed41 ,london1 #ahic1 pp4 #add1 #agg4 ,see also ,j4 ,c4 ,marshman1 ,abridgment of the ,history of ,india1 ,london1 #ahic1 p4 #b4 #fc ,for a list and for some description of these works see ,r4 ,c4 ,dutt1 ,a ,history of ,civilization in ,ancient ,india1 ,vol4 ,,ii1 p4 #aba4 #fd ,professor ,ramkrishna ,gopal ,bhandarkar fixes the date as the fifth century ,b4,c4 ,consideration of the ,date of the ,mahabharata1 in the ,journal of the ,bombay ,branch of the ,r4 ,a4 ,soc41 ,bombay1 #ahgc1 ,vol4 ,x1 p4 #b44 #fe ,marshman1 loc4 cit41 p4 #b4 #ff ,a4 ,c4 ,burnell1 ,south ,indian ,palaeography1 #b;d ed41 ,london1 #ahgh1 p4 #a1 seq4 #fg ,this extensive subject of palpable arithmetic1 essentially the history of the abacus1 deserves to be treated in a work by itself4 #fh ,the following are the leading sources of information upon this subject3 ,g4 ,buehler1 ,indische ,palaeographie1 particularly chap4 vi2 ,a4 ,c4 ,burnell1 ,south ,indian ,palaeography1 #b;d ed41 ,london1 #ahgh1 where tables of the various ,indian numerals are given in ,plate ,,xxiii2 ,e4 ,c4 ,bayley1 ,on the ,genealogy of ,modern ,numerals1 ,journal of the ,royal ,asiatic ,society1 ,vol4 ,,xiv1 part #c1 and ,vol4 ,,xv1 part #a1 and reprint1 ,london1 #ahhb2 ,i4 ,taylor1 in ,the ,academy1 ,january #bh1 #ahhb1 with a repetition of his argument in his work ,the ,alphabet1 ,london1 #ahhc1 ,vol4 ,,ii1 p4 #bfe1 based on ,bayley2 ,g4 ,r4 ,kaye1 loc4 cit41 in some respects one of the most critical articles thus far published2 ,j4 ,c4 ,fleet1 ,corpus inscriptionum ,indicarum1 ,london1 #ahhh1 ,vol4 ,,iii1 with facsimiles of many ,indian inscriptions1 and ,indian ,epigraphy1 ,oxford1 #aijg1 reprinted from the ,imperial ,gazetteer of ,india1 ,vol4 ,,ii1 pp4 #a-hh1 #aijg2 ,g4 ,thibaut1 loc4 cit41 ,astronomie etc42 ,r4 ,caldwell1 ,comparative ,grammar of the ,dravidian ,languages1 ,london1 #ahef1 p4 #bfb seq42 and ,epigraphia ,indica 7official publication of the government of ,india71 ,vols4 ,i-,,ix4 ,another work of ,buehler's1 ,on the ,origin of the ,indian ,brahma ,alphabet1 is also of value4 #fi ,the earliest work on the subject was by ,james ,prinsep1 ,on the ,inscriptions of ,piyadasi or ,a'soka1 etc41 ,journal of the ,asiatic ,society of ,bengal1 #ahch1 following a preliminary suggestion in the same journal in #ahcg4 ,see also ,a'soka ,notes1 by ,v4 ,a4 ,smith1 ,the ,indian ,antiquary1 ,vol4 ,,xxxvii1 #aijh1 p4 #bd seq41 ,vol4 ,,xxxviii1 pp4 #aea-aei1 ,june1 #aiji2 ,the ,early ,history of ,india1 #b;d ed41 ,oxford1 #aijh1 p4 #aed2 ,j4 ,f4 ,fleet1 ,the ,last ,words of ,a'soka1 ,journal of the ,royal ,asiatic ,society1 ,october1 #aiji1 pp4 #iha-ajaf2 ,e4 ,senart1 ,les inscriptions de ,piyadasi1 #b vols41 ,paris1 #ahhg4 #gj ,for a discussion of the minor details of this system1 see ,buehler1 loc4 cit41 p4 #gc4 #ga ,julius ,euting1 ,nabataeische ,inschriften aus ,arabien1 ,berlin1 #ahhe1 pp4 #if-ig1 with a table of numerals4 #gb ,for the five principal theories see ,buehler1 loc4 cit41 p4 #aj4 #gc ,bayley1 loc4 cit41 reprint p4 #c4 #gd ,buehler1 loc4 cit42 ,epigraphia ,indica1 ,vol4 ,,iii1 p4 #acd2 ,indian ,antiquary1 ,vol4 ,,vi1 p4 #aee seq41 and ,vol4 ,x1 p4 #ajg4 #ge ,pandit ,bhagavanlal ,indraji1 ,on ,ancient ,nagari ,numeration2 from an ,inscription at ,naneghat1 ,journal of the ,bombay ,branch of the ,royal ,asiatic ,society1 #ahgf1 ,vol4 ,,xii1 p4 #djd4 #gf ,ib41 p4 #dje4 ,he gives also a plate and an interpretation of each numeral4 #gg ,these may be compared with ,buehler's drawings1 loc4 cit42 with ,bayley1 loc4 cit41 p4 #ccg and plates2 and with ,bayley's article in the ,encyclopaedia ,britannica1 #ith ed41 art4 ,numerals4 #gh ,e4 ,senart1 ,the ,inscriptions in the ,caves at ,nasik1 ,epigraphia ,indica1 ,vol4 ,,viii1 pp4 #ei-if2 ,the ,inscriptions in the ,cave at ,karle1 ,epigraphia ,indica1 ,vol4 ,,vii1 pp4 #dg-gd2 ,buehler1 ,palaeographie1 ,tafel ,,ix4 #gi ,see ,fleet1 loc4 cit4 ,see also ,t4 ,benfey1 ,sanskrit ,grammar1 ,london1 #ahfc1 p4 #bag2 ,m4 ,r4 ,kale1 ,higher ,sanskrit ,grammar1 #b;d ed41 ,bombay1 #ahih1 p4 #aaj1 and other authorities as cited4 #hj ,kharos4t4hi numerals1 ,a'soka inscriptions1 c4 #bej ,b4,c4 ,senart1 ,notes d'epigraphie indienne4 ,given by ,buehler1 loc4 cit41 ,tafel ,i4 #ha ,same1 ',saka inscriptions1 probably of the first century ,b4,c4 ,senart1 loc4 cit42 ,buehler1 loc4 cit4 #hb ,brahmi numerals1 ,a'soka inscriptions1 c4 #bej ,b4,c4 ,indian ,antiquary1 ,vol4 ,,vi1 p4 #aee seq4 #hc ,same1 ,nana ,ghat inscriptions1 c4 #aej ,b4,c4 ,bhagavanlal ,indraji1 ,on ,ancient ,nagari ,numeration1 loc4 cit4 ,copied from a squeeze of the original4 #hd ,same1 ,nasik inscription1 c4 #ajj ,b4,c4 ,burgess1 ,archeological ,survey ,report1 ,western ,india2 ,senart1 ,epigraphia ,indica1 ,vol4 ,,vii1 pp4 #dg-gi1 and ,vol4 ,,viii1 pp4 #ei-if4 #he ,ks4atrapa coins1 c4 #bjj ,a4,d4 ,journal of the ,royal ,asiatic ,society1 #ahij1 p4 #fci4 #hf ,kus4ana inscriptions1 c4 #aej ,a4,d4 ,epigraphia ,indica1 ,vol4 ,i1 p4 #cha1 and ,vol4 ,,ii1 p4 #bja4 #hg ,gupta ,inscriptions1 c4 #cjj ,a4,d4 to #dej ,a4,d4 ,fleet1 loc4 cit41 ,vol4 ,,iii4 #hh ,valhabi1 c4 #fjj ,a4,d4 ,corpus1 ,vol4 ,,iii4 #hi ,bendall's ,table of ,numerals1 in ,cat4 ,sansk4 ,budd4 ,,mss41 ,british ,museum4 #ij ,indian ,antiquary1 ,vol4 ,,xiii1 #abj2 ,epigraphia ,indica1 ,vol4 ,,iii1 #abg ff4 #ia ,fleet1 loc4 cit4 #ib ,bayley1 loc4 cit41 p4 #cce4 #ic ,from a copper plate of #dic ,a4,d41 found at ,karitalai1 ,central ,india4 ,fleet1 loc4 cit41 ,plate ,,xvi4 ,it should be stated1 however1 that many of these copper plates1 being deeds of property1 have forged dates so as to give the appearance of antiquity of title4 ,on the other hand1 as ,colebrooke long ago pointed out1 a successful forgery has to imitate the writing of the period in question1 so that it becomes evidence well worth considering1 as shown in ,chapter ,,iii4 #id ,from a copper plate of #eaj ,a4,d41 found at ,majhgawain1 ,central ,india4 ,fleet1 loc4 cit41 ,plate ,,xiv4 #ie ,from an inscription of #ehh ,a4,d41 found at ,bodh-,gaya1 ,bengal ,presidency4 ,fleet1 loc4 cit41 ,plate ,,xxiv4 #if ,from a copper plate of #ega ,a4,d41 found at ,maliya1 ,bombay ,presidency4 ,fleet1 loc4 cit41 ,plate ,,xxiv4 #ig ,from a ,bijayagad4h pillar inscription of #cgb ,a4,d4 ,fleet1 loc4 cit41 ,plate ,,xxxvi1 ,c4 #ih ,from a copper plate of #dcd ,a4,d4 ,indian ,antiquary1 ,vol4 ,i1 p4 #fj4 #ii ,gadhwa inscription1 c4 #dag ,a4,d4 ,fleet1 loc4 cit41 ,plate ,,iv1 ,d4 #ajj ,karitalai plate of #dic ,a4,d41 referred to above4 #aja ,it seems evident that the ,chinese four1 curiously enough called eight in the mouth1 is only a cursive #d vertical strokes4 #ajb ,chalfont1 ,f4 ,h41 ,memoirs of the ,carnegie ,museum1 ,vol4 ,,iv1 no4 #a2 ,j4 ,hager1 ,an ,explanation of the ,elementary ,characters of the ,chinese1 ,london1 #ahja4 #ajc ,h4 ,v4 ,hilprecht1 ,mathematical1 ,metrological and ,chronological ,tablets from the ,temple ,library at ,nippur1 ,vol4 ,,xx1 part ,i1 of ,series ,a1 ,cuneiform ,texts ,published by the ,babylonian ,expedition of the ,university of ,pennsylvania1 #aijf2 ,a4 ,eisenlohr1 ,ein altbabylonischer ,felderplan1 ,leipzig1 #aijf2 ,maspero1 ,dawn of ,civilization1 p4 #ggc4 #ajd ,sir ,h4 ,h4 ,howard1 ,on the ,earliest ,inscriptions from ,chaldea1 ,proceedings of the ,society of ,biblical ,archaeology1 ,,xxi1 p4 #cja1 ,london1 #ahii4 #aje ,for a bibliography of the principal hypotheses of this nature see ,buehler1 loc4 cit41 p4 #gg4 ,buehler 7p4 #gh7 feels that of all these hypotheses that which connects the ,brahmi with the ,egyptian numerals is the most plausible1 although he does not adduce any convincing proof4 ,th4 ,henri ,martin1 ,les signes numeraux et l'arithmetique chez les peuples de l'antiquite et du moyen age 7being an examination of ,cantor's ,mathematische ,beitraege zum ,culturleben der ,voelker71 ,annali di matematica pura ed applicata1 ,vol4 ,v1 ,rome1 #ahfd1 pp4 #h1 #gj4 ,also1 same author1 ,recherches nouvelles sur l'origine de notre systeme de numeration ecrite1 ,revue ,archeologique1 #aheg1 pp4 #cf1 #ee4 ,see also the tables given later in this work4 #ajf ,journal of the ,royal ,asiatic ,society1 ,bombay ,branch1 ,vol4 ,,xxiii4 #ajg ,loc4 cit41 reprint1 ,part ,i1 pp4 #ab1 #ag4 ,bayley's deductions are generally regarded as unwarranted4 #ajh ,the ,alphabet2 ,london1 #ahhc1 ,vol4 ,,ii1 pp4 #bfe1 #bff1 and ,the ,academy of ,jan4 #bh1 #ahhb4 #aji ,taylor1 ,the ,alphabet1 loc4 cit41 table on p4 #bff4 #aaj ,buehler1 ,on the ,origin of the ,indian ,brahma ,alphabet1 ,strassburg1 #ahih1 footnote1 pp4 #eb1 #ec4 #aaa ,albrecht ,weber1 ,history of ,indian ,literature1 ,english ed41 ,boston1 #ahgh1 p4 #bef3 ,the ,indian figures from #a-i are abbreviated forms of the initial letters of the numerals themselves4443 the zero1 too1 has arisen out of the first letter of the word s4unya 7empty7 7it occurs even in ,pingala74 ,it is the decimal place value of these figures which gives them significance4 ,c4 ,henry1 ,sur l'origine de quelques notations mathematiques1 ,revue ,archeologique1 ,june and ,july1 #ahgi1 attempts to derive the ,boethian forms from the initials of ,latin words4 ,see also ,j4 ,prinsep1 ,examination of the ,inscriptions from ,girnar in ,gujerat1 and ,dhauli in ,cuttach1 ,journal of the ,asiatic ,society of ,bengal1 #ahch1 especially ,plate ,,xx1 p4 #cdh2 this was the first work on the subject4 #aab ,buehler1 ,palaeographie1 p4 #ge1 gives the list1 with the list of letters 7p4 #gf7 corresponding to the number symbols4 #aac ,for a general discussion of the connection between the numerals and the different kinds of alphabets1 see the articles by ,u4 ,ceretti1 ,sulla origine delle cifre numerali moderne1 ,rivista di fisica1 matematica e scienze naturali1 ,pisa and ,pavia1 #aiji1 anno ,x1 numbers #aad1 #aah1 #aai1 and #abj1 and continuation in #aiaj4 #aad ,this is one of ,buehler's hypotheses4 ,see ,bayley1 loc4 cit41 reprint p4 #d2 a good bibliography of original sources is given in this work1 p4 #ch4 #aae ,loc4 cit41 reprint1 part ,i1 pp4 #ab1 #ag4 ,see also ,burnell1 loc4 cit41 p4 #fd1 and tables in plate ,,xxiii4 #aaf ,this was asserted by ,g4 ,hager 7,memoria sulle cifre arabiche1 ,milan1 #ahac1 also published in ,fundgruben des ,orients1 ,vienna1 #ahaa1 and in ,bibliotheque ,britannique1 ,geneva1 #ahab74 ,see also the recent article by ,major ,charles ,e4 ,woodruff1 ,the ,evolution of ,modern ,numerals from ,tally ,marks1 ,american ,mathematical ,monthly1 ,august-,september1 #aiji4 ,biernatzki1 ,die ,arithmetik der ,chinesen1 ,crelle's ,journal fuer die reine und angewandte ,mathematik1 ,vol4 ,,lii1 #aheg1 pp4 #ei-if1 also asserts the priority of the ,chinese claim for a place system and the zero1 but upon the flimsiest authority4 ,ch4 de ,paravey1 ,essai sur l'origine unique et hieroglyphique des chiffres et des lettres de tous les peuples1 ,paris1 #ahbf2 ,g4 ,kleinwaechter1 ,the ,origin of the ,arabic ,numerals1 ,china ,review1 ,vol4 ,,xi1 #ahhb-ahhc1 pp4 #cgi-cha1 ,vol4 ,,xii1 pp4 #bh-cj2 ,biot1 ,note sur la connaissance que les ,chinois ont eue de la valeur de position des chiffres1 ,journal ,asiatique1 #ahci1 pp4 #dig-ejb4 ,a4 ,terrien de ,lacouperie1 ,the ,old ,numerals1 the ,counting-,rods and the ,swan-,pan in ,china1 ,numismatic ,chronicle1 ,vol4 ,,iii 7#c71 pp4 #big-cdj1 and ,crowder ,b4 ,moseley1 ,numeral ,characters3 ,theory of ,origin and ,development1 ,american ,antiquarian1 ,vol4 ,,xxii1 pp4 #bgi-bhd1 both propose to derive our numerals from ,chinese characters1 in much the same way as is done by ,major ,woodruff1 in the article above cited4 #aag ,the ,greeks1 probably following the ,semitic custom1 used nine letters of the alphabet for the numerals from #a to #i1 then nine others for #aj to #ij1 and further letters to represent #ajj to #ijj4 ,as the ordinary ,greek alphabet was insufficient1 containing only twenty-four letters1 an alphabet of twenty-seven letters was used4 #aah ,institutiones mathematicae1 #b vols41 ,strassburg1 #aeic-aeif1 a somewhat rare work from which the following quotation is taken3 ,quis est harum ,cyphrarum autor8 ,a quibus hae usitatae syphrarum notae sint inventae3 hactenus incertum fuit3 meo tamen iudicio1 quod exiguum esse fateor3 a graecis librarijs 7quorum olim magna fuit copia7 literae ,graecorum quibus veteres ,graeci tamquam numerorum notis sunt usi3 fuerunt corruptae4 vt ex his licet videre4 ,graecorum ,literae corruptae4 ,illustration ,sed qua ratione graecorum literae ita fuerunt corruptae8 ,finxerunt has corruptas ,graecorum literarum notas3 vel abiectione vt in nota binarij numeri1 vel additione vt in ternarij1 vel inuersione vt in septenarij1 numeri nota1 nostrae notae1 quibus hodie utimur3 ab his sola differunt elegantia1 vt apparet4 ,see also ,bayer1 ,historia regni ,graecorum ,bactriani1 ,st4 ,petersburg1 #aghh1 pp4 #abi-acj1 quoted by ,martin1 ,recherches nouvelles1 etc41 loc4 cit4 #aai ,p4 ,d4 ,huet1 ,demonstratio evangelica1 ,paris1 #agfi1 note to p4 #aci on p4 #fdg3 ,ab ,arabibus vel ab ,indis inventas esse1 non vulgus eruditorum modo1 sed doctissimi quique ad hanc diem arbitrati sunt4 ,ego vero falsum id esse1 merosque esse ,graecorum characteres aio2 a librariis ,graecae linguae ignaris interpolatos1 et diuturna scribendi consuetudine corruptos4 ,nam primum #a apex fuit1 seu virgula1 nota ,greek3 monados4 #b1 est ipsum beta extremis suis truncatum4 gamma1 si in sinistram partem inclinaveris cauda mutilaveris sinistrum cornu sinistrorsum flexeris1 fiet #c4 ,res ipsa loquitur #d ipsissimum esse ,delta1 cujus crus sinistrum erigitur ,greek3 kata katheton1 infra basim descendit2 basis vero ipsa ultra crus producta eminet4 ,vides quam #e simile sit ,greek3 toi epsilon2 infimo tantum semicirculo1 qui sinistrorsum patebat1 dextrorsum converso4 ,greek3 episemon bau quod ita notabatur digamma1 rotundato ventre1 pede detracto1 peperit ,greek3 to #f4 ,ex ,zeta basi sua mutilato1 ortum est ,greek3 to #g4 ,si ,eta inflexis introrsum apicibus in rotundiorem commodiorem formam mutaveris1 exurget ,greek3 to #h4 ,at #i ipsissimum est alt theta4 ,i4 ,weidler1 ,spicilegium observationum ad historiam notarum numeralium1 ,wittenberg1 #agee1 derives them from the ,hebrew letters2 ,dom ,augustin ,calmet1 ,recherches sur l'origine des chiffres d'arithmetique1 ,memoires pour l'histoire des sciences et des beaux arts1 ,trevoux1 #agjg 7pp4 #afbj-afce1 with two plates71 derives the current symbols from the ,romans1 stating that they are relics of the ancient ,notae ,tironianae4 ,these notes were part of a system of shorthand invented1 or at least perfected1 by ,tiro1 a slave who was freed by ,cicero4 ,l4 ,a4 ,sedillot1 ,sur l'origine de nos chiffres1 ,atti dell' ,accademia pontificia dei nuovi ,lincei1 ,vol4 ,,xviii1 #ahfd-ahfe1 pp4 #caf-cbb1 derives the ,arabic forms from the ,roman numerals4 #abj ,athanasius ,kircher1 ,arithmologia sive ,de abditis ,numerorum1 mysterijs qua origo1 antiquitas fabrica ,numerorum exponitur1 ,rome1 #affe4 #aba ,see ,suter1 ,die ,mathematiker und ,astronomen der ,araber1 p4 #ajj4 #abb ,et hi numeri sunt numeri ,indiani1 a ,brachmanis ,indiae ,sapientibus ex figura circuli secti inuenti4 #abc ,v4 ,a4 ,smith1 ,the ,early ,history of ,india1 ,oxford1 #b;d ed41 #aijh1 p4 #ccc4 #abd ,c4 ,j4 ,ball1 ,an ,inscribed ,limestone ,tablet from ,sippara1 ,proceedings of the ,society of ,biblical ,archaeology1 ,vol4 ,,xx1 p4 #be 7,london1 #ahih74 ,terrien de ,lacouperie states that the ,chinese used the circle for #aj before the beginning of the ,christian era4 ,catalogue of ,chinese ,coins1 ,london1 #ahib1 p4 xl4 #abe ,for a purely fanciful derivation from the corresponding number of strokes1 see ,w4 ,w4 ,r4 ,ball1 ,a ,short ,account of the ,history of ,mathematics1 #ast ed41 ,london1 #ahhh1 p4 #adg2 similarly ,j4 ,b4 ,reveillaud1 ,essai sur les chiffres arabes1 ,paris1 #ahhc2 ,p4 ,voizot1 ,les chiffres arabes et leur origine1 ,la ,nature1 #ahii1 p4 #bbb2 ,g4 ,dumesnil1 ,de la forme des chiffres usuels1 ,annales de l'universite de ,grenoble1 #aijg1 ,vol4 ,,xix1 pp4 #feg-fgd1 also a note in ,revue ,archeologique1 #ahij1 ,vol4 ,,xvi 7#c71 pp4 #cdb-cdh2 one of the earliest references to a possible derivation from points is in a work by ,bettino entitled ,apiaria universae philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus eximios traducta1 et facillimis demonstrationibus confirmata1 ,bologna1 #aede1 ,vol4 ,,ii1 ,apiarium ,,xi1 p4 #e4 #abf ,alphabetum ,barmanum1 ,romae1 ,,mdcclxxvi1 p4 #ej4 ,the #a is evidently ,sanskrit1 and the #d1 #g1 and possibly #i are from ,india4 #abg ,alphabetum ,grandonico-,malabaricum1 ,romae1 ,,mdcclxxii1 p4 #ij4 ,the zero is not used1 but the symbols for #aj1 #ajj1 and so on1 are joined to the units to make the higher numbers4 #abh ,alphabetum ,tangutanum1 ,romae1 ,,mdcclxxiii1 p4 #ajg4 ,in a ,tibetan ,,ms4 in the library of ,professor ,smith1 probably of the eighteenth century1 substantially these forms are given4 #abi ,bayley1 loc4 cit41 plate ,,ii4 ,similar forms to these here shown1 and numerous other forms found in ,india1 as well as those of other oriental countries1 are given by ,a4 ,p4 ,pihan1 ,expose des signes de numeration usites chez les peuples orientaux anciens et modernes1 ,paris1 #ahfj4 #acj ,buehler1 loc4 cit41 p4 #hj2 ,j4 ,f4 ,fleet1 ,corpus inscriptionum ,indicarum1 ,vol4 ,,iii1 ,calcutta1 #ahhh4 ,lists of such words are given also by ,al-,biruni in his work ,india2 by ,burnell1 loc4 cit42 by ,e4 ,jacquet1 ,mode d'expression symbolique des nombres employe par les ,indiens1 les ,tibetains et les ,javanais1 ,journal ,asiatique1 ,vol4 ,,xvi1 ,paris1 #ahce4 #aca ,this date is given by ,fleet1 loc4 cit41 ,vol4 ,,iii1 p4 #gc1 as the earliest epigraphical instance of this usage in ,india proper4 #acb ,weber1 ,indische ,studien1 ,vol4 ,,viii1 p4 #aff seq4 #acc ,journal of the ,royal ,asiatic ,society1 ,vol4 ,i 7,n4,s471 p4 #djg4 #acd ,,viii1 #bj1 #ba4 #ace ,th4 ,h4 ,martin1 ,les signes numeraux 4441 ,rome1 #ahfd2 ,lassen1 ,indische ,alterthumskunde1 ,vol4 ,,ii1 #b;d ed41 ,leipzig and ,london1 #ahgd1 p4 #aaec4 #acf ,but see ,burnell1 loc4 cit41 and ,thibaut1 ,astronomie1 ,astrologie und ,mathematik1 p4 #ga4 #acg ,a4 ,barth1 ,inscriptions ,sanscrites du ,cambodge1 in the ,notices et extraits des ,mss4 de la ,bibliotheque nationale1 ,vol4 ,,xxvii1 ,part ,i1 pp4 #a-ahj1 #ahhe2 see also numerous articles in ,journal ,asiatique1 by ,aymonier4 #ach ,buehler1 loc4 cit41 p4 #hb4 #aci ,loc4 cit41 p4 #gi4 #adj ,buehler1 loc4 cit41 p4 #hc4 ,the ,hindu astrologers still use an alphabetical system of numerals4 ,burnell1 loc4 cit41 p4 #gi4 #ada ,well could ,ramus say1 ,quicunq2 autem fuerit inventor decem notarum laudem magnam meruit4 #adb ,al-,biruni gives lists4 #adc ,propagation1 loc4 cit41 p4 #ddc4 #add ,see the quotation from ,the ,light of ,asia in ,chapter ,,ii1 p4 #af4 #ade ,the nine ciphers were called a4nka4 #adf ,zur ,geschichte des indischen ,ziffernsystems1 ,zeitschrift fuer die ,kunde des ,morgenlandes1 ,vol4 ,,iv1 #ahdb1 pp4 #gd-hc4 #adg ,it is found in the ,bakhs4ali ,,ms4 of an elementary arithmetic which ,hoernle placed1 at first1 about the beginning of our era1 but the date is much in question4 ,g4 ,thibaut1 loc4 cit41 places it between #gjj and #ijj ,a4,d42 ,cantor places the body of the work about the third or fourth century ,a4,d41 ,geschichte der ,mathematik1 ,vol4 ,i 7#c71 p4 #eih4 #adh ,for the opposite side of the case see ,g4 ,r4 ,kaye1 ,notes on ,indian ,mathematics1 ,no4 #b4--;,aryabhat4a1 ,journ4 and ,proc4 of the ,asiatic ,soc4 of ,bengal1 ,vol4 ,,iv1 #aijh1 pp4 #aaa-ada4 #adi ,he used one of the alphabetic systems explained above4 ,this ran up to #ajah and was not difficult1 beginning as follows3 ,illustration the same letter 7ka7 appearing in the successive consonant forms1 ka1 kha1 ga1 gha1 etc4 ,see ,c4 ,i4 ,gerhardt1 ,ueber die ,entstehung und ,ausbreitung des dekadischen ,zahlensystems1 ,programm1 p4 #ag1 ,salzwedel1 #ahec1 and ,etudes historiques sur l'arithmetique de position1 ,programm1 p4 #bd1 ,berlin1 #ahef2 ,e4 ,jacquet1 ,mode d'expression symbolique des nombres1 loc4 cit41 p4 #ig2 ,l4 ,rodet1 ,sur la veritable signification de la notation numerique inventee par ,aryabhata1 ,journal ,asiatique1 ,vol4 ,,xvi 7#g71 pp4 #ddj-dhe4 ,on the two ,aryabhat4as see ,kaye1 ,bibl4 ,math41 ,vol4 ,x 7#c71 p4 #bhi4 #aej ,using kha1 a synonym of 'sunya4 ,bayley1 loc4 cit41 p4 #bb1 and ,l4 ,rodet1 ,journal ,asiatique1 ,vol4 ,,xvi 7#g71 p4 #ddc4 #aea ,varaha-,mihira1 ,pancasiddhantika1 translated by ,g4 ,thibaut and ,m4 ,s4 ,dvivedi1 ,benares1 #ahhi2 see ,buehler1 loc4 cit41 p4 #gh2 ,bayley1 loc4 cit41 p4 #bc4 #aeb ,br4hat ,sam4hita1 translated by ,kern1 ,journal of the ,royal ,asiatic ,society1 #ahgj-ahge4 #aec ,it is stated by ,buehler in a personal letter to ,bayley 7loc4 cit41 p4 #fe7 that there are hundreds of instances of this usage in the ,br4hat ,sam4hita4 ,the system was also used in the ,pancasiddhantika as early as #eje ,a4,d4 ,buehler1 ,palaeographie1 p4 #hj1 and ,fleet1 ,journal of the ,royal ,asiatic ,society1 #aiaj1 p4 #hai4 #aed ,cantor1 ,geschichte der ,mathematik1 ,vol4 ,i 7#c71 p4 #fjh4 #aee ,buehler1 loc4 cit41 p4 #gh4 #aef ,bayley1 p4 #ch4 #aeg ,noviomagus1 in his ,de numeris libri duo1 ,paris1 #aeci1 confesses his ignorance as to the origin of the zero1 but says3 ,d4 ,henricus ,grauius1 vir ,graece ,hebraice exime doctus1 ,hebraicam originem ostendit1 adding that ,valla ,indis ,orientalibus gentibus inventionem tribuit4 #aeh ,see ,essays1 ,vol4 ,,ii1 pp4 #bhg and #bhh4 #aei ,vol4 ,,xxx1 p4 #bje seqq4 #afj ,loc4 cit41 p4 #bhd seqq4 #afa ,colebrooke1 loc4 cit41 p4 #bhh4 #afb ,loc4 cit41 p4 #gh4 #afc ,hereafter1 unless expressly stated to the contrary1 we shall use the word numerals to mean numerals with place value4 #afd ,the ,gurjaras of ,rajputana and ,kanauj1 in ,journal of the ,royal ,asiatic ,society1 ,january and ,april1 #aiji4 #afe ,vol4 ,,ix1 #aijh1 p4 #bdh4 #aff ,epigraphia ,indica1 ,vol4 ,,ix1 pp4 #aic and #aih4 #afg ,epigraphia ,indica1 ,vol4 ,,ix1 p4 #a4 #afh ,loc4 cit41 p4 #ga4 #afi ,thibaut1 p4 #ga4 #agj ,est autem in aliquibus figurarum istaram apud multos diuersitas4 ,quidam enim septimam hanc figuram representant1 etc4 ,boncompagni1 ,trattati1 p4 #bh4 ,enestroem has shown that very likely this work is incorrectly attributed to ,johannes ,hispalensis4 ,bibliotheca ,mathematica1 ,vol4 ,,ix 7#c71 p4 #b4 #aga ,indische ,palaeographie1 ,tafel ,,ix4 #agb ,edited by ,bloomfield and ,garbe1 ,baltimore1 #aija1 containing photographic reproductions of the manuscript4 #agc ,bakhs4ali ,,ms4 ,see page #dc2 ,hoernle1 ,r41 ,the ,indian ,antiquary1 ,vol4 ,,xvii1 pp4 #cc-dh1 #a plate2 ,hoernle1 ,verhandlungen des ,,vii4 ,internationalen ,orientalisten-,congresses1 ,arische ,section1 ,vienna1 #ahhh1 ,on the ,bakshali ,manuscript1 pp4 #abg-adg1 #c plates2 ,buehler1 loc4 cit4 #agd #c1 #d1 #f1 from ,h4 ,h4 ,dhruva1 ,three ,land-,grants from ,sankheda1 ,epigraphia ,indica1 ,vol4 ,,ii1 pp4 #ai-bd with plates2 date #eie ,a4,d4 #g1 #a1 #e1 from ,bhandarkar1 ,daulatabad ,plates1 ,epigraphia ,indica1 ,vol4 ,,ix1 part ,v2 date c4 #gih ,a4,d4 #age #h1 #g1 #b1 from ,buckhala ,inscription of ,nagabhatta1 ,bhandarkar1 ,epigraphia ,indica1 ,vol4 ,,ix1 part ,v2 date #hae ,a4,d4 #e from ,the ,morbi ,copper-,plate1 ,bhandarkar1 ,the ,indian ,antiquary1 ,vol4 ,,ii1 pp4 #beg-beh1 with plate2 date #hjd ,a4,d4 ,see ,buehler1 loc4 cit4 #agf #h from the above ,morbi ,copper-,plate4 #d1 #e1 #g1 #i1 and #j1 from ,asni ,inscription of ,mahipala1 ,the ,indian ,antiquary1 ,vol4 ,,xvi1 pp4 #agd-age2 inscription is on red sandstone1 date #iag ,a4,d4 ,see ,buehler4 #agg #h1 #i1 #d1 from ,rashtrakuta ,grant of ,amoghavarsha1 ,j4 ,f4 ,fleet1 ,the ,indian ,antiquary1 ,vol4 ,,xii1 pp4 #bfc-bgb2 copper-plate grant of date c4 #igb ,a4,d4 ,see ,buehler4 #g1 #c1 #e1 from ,torkhede ,copper-,plate ,grant of the ,time of ,govindaraja of ,gujerat1 ,fleet1 ,epigraphia ,indica1 ,vol4 ,,iii1 pp4 #ec-eh4 ,see ,buehler4 #agh ,from ,a ,copper-,plate ,grant of ,king ,tritochanapala ,chanlukya of ,lat4ade'sa1 ,h4,h4 ,dhruva1 ,indian ,antiquary1 ,vol4 ,,xii1 pp4 #aif-bje2 date #ajej ,a4,d4 ,see ,buehler4 #agi ,burnell1 ,a4 ,c41 ,south ,indian ,palaeography1 plate ,,xxiii1 ,telugu-,canarese numerals of the eleventh century4 ,see ,buehler4 #ahj ,from a manuscript of the second half of the thirteenth century1 reproduced in ,della vita e delle opere di ,leonardo ,pisano1 ,baldassare ,boncompagni1 ,rome1 #aheb1 in ,atti dell' ,accademia ,pontificia dei nuovi ,lincei1 anno ,v4 #aha ,from a fourteenth-century manuscript1 as reproduced in ,della vita etc41 ,boncompagni1 loc4 cit4 #ahb ,from a ,tibetan ,,ms4 in the library of ,d4 ,e4 ,smith4 #ahc ,from a ,tibetan block-book in the library of ,d4 ,e4 ,smith4 #ahd ',sarada numerals from ,the ,kashmirian ,atharva-,veda1 reproduced by chromophotography from the manuscript in the ,university ,library at ,tuebingen1 ,bloomfield and ,garbe1 ,baltimore1 #aija4 ,somewhat similar forms are given under ,numeration ,cachemirienne1 by ,pihan1 ,expose etc41 p4 #hd4 #ahe ,franz ,x4 ,kugler1 ,die ,babylonische ,mondrechnung1 ,freiburg i4 ,br41 #aijj1 in the numerous plates at the end of the book2 practically all of these contain the symbol to which reference is made4 ,cantor1 ,geschichte1 ,vol4 ,i1 p4 #ca4 #ahf ,f4 ,x4 ,kugler1 ,sternkunde und ,sterndienst in ,babel1 ,i4 ,buch1 from the beginnings to the time of ,christ1 ,muenster i4 ,westfalen1 #aijg4 ,it also has numerous tables containing the above zero4 #ahg ,from a letter to ,d4 ,e4 ,smith1 from ,g4 ,f4 ,hill of the ,british ,museum4 ,see also his monograph ,on the ,early ,use of ,arabic ,numerals in ,europe1 in ,archaeologia1 ,vol4 ,,lxii 7#aiaj71 p4 #acg4 #ahh ,r4 ,hoernle1 ,the ,bakshali ,manuscript1 ,indian ,antiquary1 ,vol4 ,,xvii1 pp4 #cc-dh and #bge-bgi1 #ahhh2 ,thibaut1 ,astronomie1 ,astrologie und ,mathematik1 p4 #ge2 ,hoernle1 ,verhandlungen1 loc4 cit41 p4 #acb4 #ahi ,bayley1 loc4 cit41 ,vol4 ,,xv1 p4 #bi4 ,also ,bendall1 ,on a ,system of ,numerals used in ,south ,india1 ,journal of the ,royal ,asiatic ,society1 #ahif1 pp4 #ghi-gib4 #aij ,v4 ,a4 ,smith1 ,the ,early ,history of ,india1 #b;d ed41 ,oxford1 #aijh1 p4 #ad4 #aia ,colebrooke1 ,algebra1 with ,arithmetic and ,mensuration1 from the ,sanskrit of ,brahmegupta and ,bhascara1 ,london1 #ahag1 pp4 #cci-cdj4 #aib ,ibid41 p4 #ach4 #aic ,d4 ,e4 ,smith1 in the ,bibliotheca ,mathematica1 ,vol4 ,,ix 7#c71 pp4 #ajf-aaj4 #aid ,as when we use three dots 744474 #aie ,the ,hindus call the nought explicitly 'sunyabindu 'the dot marking a blank1' and about #ejj ,a4,d4 they marked it by a simple dot1 which latter is commonly used in inscriptions and ,,mss4 in order to mark a blank1 and which was later converted into a small circle4 ,buehler1 ,on the ,origin of the ,indian ,alphabet1 p4 #ec1 note4 #aif ,fazzari1 ,dell' origine delle parole zero e cifra1 ,naples1 #aijc4 #aig ,e4 ,wappler1 ,zur ,geschichte der ,mathematik im #ae4 ,jahrhundert1 in the ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,xlv1 ,hist4-lit4 ,abt41 p4 #dg4 ,the manuscript is ,no4 ,c4 #hj1 in the ,dresden library4 #aih ,j4 ,g4 ,praendel1 ,algebra nebst ihrer literarischen ,geschichte1 p4 #egb1 ,munich1 #agie4 #aii ,see the table1 p4 #bc4 ,does the fact that the early ,european arithmetics1 following the ,arab custom1 always put the #j after the #i1 suggest that the #j was derived from the old ,hindu symbol for #aj8 #bjj ,bayley1 loc4 cit41 p4 #dh4 ,from this fact ,delambre 7,histoire de l'astronomie ancienne7 inferred that ,ptolemy knew the zero1 a theory accepted by ,chasles1 ,apercu historique sur l'origine et le developpement des methodes en geometrie1 #ahge ed41 p4 #dgf2 ,nesselmann1 however1 showed 7,algebra der ,griechen1 #ahdb1 p4 #ach71 that ,ptolemy merely used ,greek3 o for ,greek3 ouden1 with no notion of zero4 ,see also ,g4 ,fazzari1 ,dell' origine delle parole zero e cifra1 ,ateneo1 ,anno ,i1 ,no4 #aa1 reprinted at ,naples in #aijc1 where the use of the point and the small cross for zero is also mentioned4 ,th4 ,h4 ,martin1 ,les signes numeraux etc41 reprint p4 #cj1 and ,j4 ,brandis1 ,das ,muenz-1 ,mass- und ,gewichtswesen in ,vorderasien bis auf ,alexander den ,grossen1 ,berlin1 #ahff1 p4 #aj1 also discuss this usage of ,greek3 o1 without the notion of place value1 by the ,greeks4 #bja ,al-,battani sive ,albatenii opus astronomicum4 ,ad fidem codicis escurialensis arabice editum1 latine versum1 adnotationibus instructum a ,carolo ,alphonso ,nallino1 #ahii-aijg4 ,publicazioni del ,r4 ,osservatorio di ,brera in ,milano1 ,no4 ,,xl4 #bjb ,loc4 cit41 ,vol4 ,,ii1 p4 #bga4 #bjc ,c4 ,henry1 ,prologus ,n4 ,ocreati in ,helceph ad ,adelardum ,batensem magistrum suum1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,iii1 #ahhj4 #bjd ,max4 ,curtze1 ,ueber eine ,algorismus-,schrift des ,,xii4 ,jahrhunderts1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,viii1 #ahih1 pp4 #a-bg2 ,alfred ,nagl1 ,ueber eine ,algorismus-,schrift des ,,xii4 ,jahrhunderts und ueber die ,verbreitung der indisch-arabischen ,rechenkunst und ,zahlzeichen im christl4 ,abendlande1 ,zeitschrift fuer ,mathematik und ,physik1 ,hist4-lit4 ,abth41 ,vol4 ,,xxxiv1 pp4 #abi-adf and #afa-agj1 with one plate4 #bje ,byzantinische ,analekten1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,ix1 pp4 #afa-ahi4 #bjf symbol or symbol for #j4 symbol also used for #e4 symbols for #ac4 ,heiberg1 loc4 cit4 #bjg ,gerhardt1 ,etudes historiques sur l'arithmetique de position1 ,berlin1 #ahef1 p4 #ab2 ,j4 ,bowring1 ,the ,decimal ,system in ,numbers1 ,coins1 ,accounts1 ,london1 #ahed1 p4 #cc4 #bjh ,karabacek1 ,wiener ,zeitschrift fuer die ,kunde des ,morgenlandes1 ,vol4 ,,xi1 p4 #ac2 ,fuehrer durch die ,papyrus-,ausstellung ,erzherzog ,rainer1 ,vienna1 #ahid1 p4 #baf4 #bji ,in the library of ,g4 ,a4 ,plimpton1 ,esq4 #baj ,cantor1 ,geschichte1 ,vol4 ,i 7#c71 p4 #fgd2 ,y4 ,mikami1 ,a ,remark on the ,chinese ,mathematics in ,cantor's ,geschichte der ,mathematik1 ,archiv der ,mathematik und ,physik1 ,vol4 ,,xv 7#c71 pp4 #fh-gj4 #baa ,of course the earlier historians made innumerable guesses as to the origin of the word cipher4 ,e4g4 ,matthew ,hostus1 ,de numeratione emendata1 ,antwerp1 #aehb1 p4 #aj1 says3 ,siphra vox ,hebraeam originem sapit refertque3 ut docti arbitrantur1 a verbo saphar1 quod ,ordine numerauit significat4 ,unde ,sephar numerus est3 hinc ,siphra 7vulgo corruptius74 ,etsi vero gens ,iudaica his notis1 quae hodie ,siphrae vocantur1 usa non fuit3 mansit tamen rei appellatio apud multas gentes4 ,dasypodius1 ,institutiones mathematicae1 ,vol4 ,i1 #aeic1 gives a large part of this quotation word for word1 without any mention of the source4 ,hermannus ,hugo1 ,de prima scribendi origine1 ,trajecti ad ,rhenum1 #agch1 pp4 #cjd-cje1 and note1 p4 #cje2 ,karl ,krumbacher1 ,woher stammt das ,wort ,ziffer 7,chiffre781 ,etudes de philologie neo-grecque1 ,paris1 #ahib4 #bab ,buehler1 loc4 cit41 p4 #gh and p4 #hf4 #bac ,fazzari1 loc4 cit41 p4 #d4 ,so ,elia ,misrachi 7#adee-aebf7 in his posthumous ,book of ,number1 ,constantinople1 #aecd1 explains sifra as being ,arabic4 ,see also ,steinschneider1 ,bibliotheca ,mathematica1 #ahic1 p4 #fi1 and ,g4 ,wertheim1 ,die ,arithmetik des ,elia ,misrachi1 ,programm1 ,frankfurt1 #ahic4 #bad ,cum his novem figuris1 et cum hoc signo #j1 quod arabice zephirum appellatur1 scribitur quilibet numerus4 #bae ,greek3 tziphra1 a form also used by ,neophytos 7date unknown1 probably c4 #accj74 ,it is curious that ,finaeus 7#aeee ed41 f4 #b7 used the form tziphra throughout4 ,a4 ,j4 ,h4 ,vincent ,sur l'origine de nos chiffres1 ,notices et ,extraits des ,,mss41 ,paris1 #ahdg1 pp4 #adc-aej says3 ,ce cercle fut nomme par les uns1 sipos1 rota1 galgal 4442 par les autres tsiphra 7de ,hebrew3 ,,tspr1 couronne ou diademe7 ou ciphra 7de ,hebrew3 ,,spr1 numeration74 ,ch4 de ,paravey1 ,essai sur l'origine unique et hieroglyphique des chiffres et des lettres de tous les peuples1 ,paris1 #ahbf1 p4 #afe1 a rather fanciful work1 gives vase1 vase arrondi et ferme par un couvercle1 qui est le symbole de la #aj;e ,heure1 symbol1 among the ,chinese2 also ,tsiphron ,zeron1 ou tout a fait vide en arabe1 ,greek3 tziphra en grec 444 d'ou chiffre 7qui derive plutot1 suivant nous1 de l',hebreu ,sepher1 compter47 #baf ,compilatus a ,magistro ,jacobo de ,florentia apud montem pesalanum1 and described by ,g4 ,lami in his ,catalogus codicum manuscriptorum qui in bibliotheca ,riccardiana ,florentiae adservantur4 ,see ,fazzari1 loc4 cit41 p4 #e4 #bag ,et doveto sapere chel zeuero per se solo non significa nulla ma e potentia di fare significare1 444 ,et decina o centinaia o migliaia non si puote scrivere senza questo segno #j4 la quale si chiama zeuero4 ,fazzari1 loc4 cit41 p4 #e4 #bah ,ibid41 p4 #f4 #bai ,avicenna 7#ihj-ajcf71 translation by ,gasbarri et ,francois1 piu il punto 7gli ,arabi adoperavano il punto in vece dello zero il cui segno #j in arabo si chiama zepiro donde il vocabolo zero71 che per se stesso non esprime nessun numero4 ,this quotation is taken from ,d4 ,c4 ,martines1 ,origine e progressi dell' aritmetica1 ,messina1 #ahfe4 #bbj ,leo ,jordan1 ,materialien zur ,geschichte der arabischen ,zahlzeichen in ,frankreich1 ,archiv fuer ,kulturgeschichte1 ,berlin1 #aije1 pp4 #aee-aie1 gives the following two schemes of derivation1 7#a7 zefiro1 zeviro1 zeiro1 zero1 7#b7 zefiro1 zefro1 zevro1 zero4 #bba ,koebel 7#aeah ed41 f4 ,a#d7 speaks of the numerals in general as die der gemain man ,zyfer nendt4 ,recorde 7,grounde of ,artes1 #aeeh ed41 f4 ,b#f7 says that the zero is called priuatly a ,cyphar1 though all the other sometimes be likewise named4 #bbb ,decimo ,x #j theca1 circulus cifra sive figura nihili appelat'4 ,enchiridion ,algorismi1 ,cologne1 #aeja4 ,later1 quoniam de integris tam in cifris quam in proiectilibus1--the word proiectilibus referring to markers thrown and used on an abacus1 whence the ,french jetons and the ,english expression to cast an account4 #bbc ,decima vero o dicitur teca1 circulus1 vel cyfra vel figura nichili4 ,maximilian ,curtze1 ,petri ,philomeni de ,dacia in ,algorismum ,vulgarem ,johannis de ,sacrobosco commentarius1 una cum ,algorismo ipso1 ,copenhagen1 #ahig1 p4 #b4 ,curtze cites five manuscripts 7fourteenth and fifteenth centuries7 of ,dacia's commentary in the libraries at ,erfurt1 ,leipzig1 and ,salzburg1 in addition to those given by ,enestroem1 ,oefversigt af ,kongl4 ,vetenskaps-,akademiens ,foerhandlingar1 #ahhe1 pp4 #ae-bg1 #fe-gj2 #ahhf1 pp4 #eg-fj4 #bbd ,curtze1 loc4 cit41 p4 ,,vi4 #bbe ,rara ,mathematica1 ,london1 #ahda1 chap1 i1 ,joannis de ,sacro-,bosco ,tractatus de ,arte ,numerandi4 #bbf ,smith1 ,rara ,arithmetica1 ,boston1 #aiji4 #bbg ,in the #adhd edition1 ,borghi uses the form cefiro3 ouero nulla3 while in the #adhh edition he uses zefiro3 ouero nulla1 and in the #aedj edition1 f4 #c1 appears ,chiamata zero1 ouero nulla4 ,woepcke asserted that it first appeared in ,calandri 7#adia7 in this sentence3 ,sono dieci le figure con le quali ciascuno numero si puo significare3 delle quali n'e una che si chiama zero3 et per se sola nulla significa4 7f4 #d74 ,see ,propagation1 p4 #ebb4 #bbh ,boncompagni ,bulletino1 ,vol4 ,,xvi1 pp4 #fgc-fhe4 #bbi ,leo ,jordan1 loc4 cit4 ,in the ,catalogue of ,,mss41 ,bibl4 de l',arsenal1 ,vol4 ,,iii1 pp4 #aed-aef1 this work is ,no4 #bijd 7#ahd ,s4,a4,f471 ,bibl4 ,nat41 and is also called ,petit traicte de algorisme4 #bcj ,texada 7#aedf7 says that there are nueue letros yvn zero o cifra 7f4 #c74 #bca ,savonne 7#aefc1 #agea ed41 f4 #a73 ,vne ansi formee 7o7 qui s'appelle nulle1 entre marchans zero1 showing the influence of ,italian names on ,french mercantile customs4 ,trenchant 7,lyons1 #aeff1 #aegh ed41 p4 #ab7 also says3 ,la derniere qui s'apele nulle1 ou zero2 but ,champenois1 his contemporary1 writing in ,paris in #aegg 7although the work was not published until #aegh71 uses cipher1 the ,italian influence showing itself less in this center of university culture than in the commercial atmosphere of ,lyons4 #bcb ,thus ,radulph of ,laon 7c4 #aajj73 ,inscribitur in ultimo ordine et figura symbol sipos nomine1 quae1 licet numerum nullum signitet1 tantum ad alia quaedam utilis1 ut insequentibus declarabitur4 ,der ,arithmetische ,tractat des ,radulph von ,laon1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,v1 p4 #ig1 from a manuscript of the thirteenth century4 ,chasles 7,comptes rendus1 t4 #af1 #ahdc1 pp4 #acic1 #adjh7 calls attention to the fact that ,radulph did not know how to use the zero1 and he doubts if the sipos was really identical with it4 ,radulph says3 444 figuram1 cui sipos nomen est symbol in motum rotulae formatam nullius numeri significatione inscribi solere praediximus1 and thereafter uses rotula4 ,he uses the sipos simply as a kind of marker on the abacus4 #bcc ,rabbi ben ,ezra 7#ajib-aafh7 used both ,hebrew3 ,,glgl1 galgal 7the ,hebrew for wheel71 and ,hebrew3 ,,spr'1 sifra4 ,see ,m4 ,steinschneider1 ,die ,mathematik bei den ,juden1 in ,bibliotheca ,mathematica1 #ahic1 p4 #fi1 and ,silberberg1 ,das ,buch der ,zahl des ,r4 ,abraham ibn ,esra1 ,frankfurt a4 ,m41 #ahie1 p4 #if1 note #bc2 in this work the ,hebrew letters are used for numerals with place value1 having the zero4 #bcd ,e4g41 in the twelfth-century ,liber aligorismi 7see ,boncompagni's ,trattati1 ,,ii1 p4 #bh74 ,so ,ramus 7,libri ,,ii1 #aefi ed41 p4 #a7 says3 ,circulus quae nota est ultima3 nil per se significat4 7,see also the ,schonerus ed4 of ,ramus1 #aehf1 p4 #a47 #bce ,und wirt das ringlein o4 die ,ziffer genant die nichts bedeut4 ,koebel's ,rechenbuch1 #aedi ed41 f4 #aj1 and other editions4 #bcf ,i4e4 circular figure1 our word notation having come from the medieval nota4 ,thus ,tzwivel 7#aejg1 f4 #b7 says3 ,nota autem circularis 4o4 per se sumpta nihil vsus habet4 alijs tamen adiuncta earum significantiam et auget et ordinem permutat quantum quo ponit ordinem4 vt adiuncta note binarij hoc modo #bj facit eam significare bis decem etc4 ,also 7ibid41 f4 #d71 figura circularis1 circularis nota4 ,clichtoveus 7#aejc ed41 f4 ,,xxxvii7 calls it nota aut circularis o1 circularis nota1 and figura circularis4 ,tonstall 7#aebb1 f4 ,b#c7 says of it3 ,decimo uero nota ad formam symbol litterae circulari figura est3 quam alij circulum1 uulgus cyphram uocat1 and later 7f4 ,c#d7 speaks of the circulos4 ,grammateus1 in his ,algorismus de integris 7,erfurt1 #aebc1 f4 ,a#b71 speaking of the nine significant figures1 remarks3 ,his autem superadditur decima figura circularis ut #j existens que ratione sua nihil significat4 ,noviomagus 7,de ,numeris libri ,,ii1 ,paris1 #aeci1 chap4 xvi1 ,de notis numerorum1 quas zyphras vocant7 calls it circularis nota1 quam ex his solam1 alij sipheram1 ,georgius ,valla zyphram4 #bcg ,huswirt1 as above4 ,ramus 7,scholae mathematicae1 #aefi ed41 p4 #aab7 discusses the name interestingly1 saying3 ,circulum appellamus cum multis1 quam alii thecam1 alii figuram nihili1 alii figuram privationis1 seu figuram nullam vocant1 alii ciphram1 cum tamen hodie omnes hae notae vulgo ciphrae nominentur1 his notis numerare idem sit quod ciphrare4 ,tartaglia 7#aeib ed41 f4 #i7 says3 si chiama da alcuni tecca1 da alcuni circolo1 da altri cifra1 da altri zero1 da alcuni altri nulla4 #bch ,quare autem aliis nominibus vocetur1 non dicit auctor1 quia omnia alia nomina habent rationem suae lineationis sive figurationis4 ,quia rotunda est1 dicitur haec figura teca ad similitudinem tecae4 ,teca enim est ferrum figurae rotundae1 quod ignitum solet in quibusdam regionibus imprimi fronti vel maxillae furis seu latronum4 ,loc4 cit41 p4 #bf4 ,but in ,greek theca 7,,theke1 ,greek3 theke7 is a place to put something1 a receptacle4 ,if a vacant column1 e4g4 in the abacus1 was so called1 the initial might have given the early forms symbol and symbol for the zero4 #bci ,buteo1 ,logistica1 ,lyons1 #aeei4 ,see also ,wertheim in the ,bibliotheca ,mathematica1 #aija1 p4 #bad4 #bdj #j est appellee chiffre ou nulle ou figure de nulle valeur4 ,la ,roche1 ,l'arithmetique1 ,lyons1 #aebj4 #bda ,decima autem figura nihil uocata1 figura nihili 7quam etiam cifram uocant74 ,stifel1 ,arithmetica integra1 #aedd1 f4 #a4 #bdb ,zifra1 ,nulla uel figura ,nihili4 ,scheubel1 #aede1 p4 #a of ch4 #a4 ,nulla is also used by ,italian writers4 ,thus ,sfortunati 7#aede ed41 f4 #d7 says3 et la decima nulla e chiamata questa decima zero2 ,cataldi 7#afjb1 p4 #a73 ,la prima1 che e o1 si chiama nulla1 ouero zero1 ouero niente4 ,it also found its way into the ,dutch arithmetics1 e4g4 ,raets 7#aegf1 #aehj ed41 f4 ,a#c73 ,nullo dat ist niet2 ,van der ,schuere 7#afjj1 #afbd ed41 f4 #g72 ,wilkens 7#affi ed41 p4 #a74 ,in ,germany ,johann ,albert 7,wittenberg1 #aecd7 and ,rudolff 7#aebf7 both adopted the ,italian nulla and popularized it4 7,see also ,kuckuck1 ,die ,rechenkunst im sechzehnten ,jahrhundert1 ,berlin1 #ahgd1 p4 #g2 ,guenther1 ,geschichte1 p4 #caf47 #bdc ,la dixieme s'appelle chifre vulgairement3 les vns l'appellant zero3 nous la pourrons appeller vn ,rien4 ,peletier1 #afjg ed41 p4 #ad4 #bdd ,it appears in the ,polish arithmetic of ,klos 7#aech7 as cyfra4 ,the ,ciphra #j augmenteth places1 but of himselfe signifieth not1 ,digges1 #aegi1 p4 #a4 ,hodder 7#ajth ed41 #afgb1 p4 #b7 uses only this word 7cypher or cipher71 and the same is true of the first native ,american arithmetic1 written by ,isaac ,greenwood 7#agbi1 p4 #a74 ,petrus de ,dacia derives cyfra from circumference4 ,vocatur etiam cyfra1 quasi circumfacta vel circumferenda1 quod idem est1 quod circulus non habito respectu ad centrum4 ,loc4 cit41 p4 #bf4 #bde ,opera mathematica1 #afie1 ,oxford1 ,vol4 ,i1 chap4 ix1 ,mathesis universalis1 ,de figuris numeralibus1 pp4 #df-di2 ,vol4 ,,ii1 ,algebra1 p4 #aj4 #bdf ,martin1 ,origine de notre systeme de numeration ecrite1 note #adi1 p4 #cf of reprint1 spells ,greek3 tsiphra from ,maximus ,planudes1 citing ,wallis as an authority4 ,this is an error1 for ,wallis gives the correct form as above4 ,alexander von ,humboldt1 ,ueber die bei verschiedenen ,voelkern ueblichen ,systeme von ,zahlzeichen und ueber den ,ursprung des ,stellenwerthes in den indischen ,zahlen1 ,crelle's ,journal fuer reine und angewandte ,mathematik1 ,vol4 ,,iv1 #ahbi1 called attention to the work ,greek3 arithmoi ,indikoi of the monk ,neophytos1 supposed to be of the fourteenth century4 ,in this work the forms ,greek3 tzuphra and ,greek3 tzumphra appear4 ,see also ,boeckh1 ,de abaco ,graecorum1 ,berlin1 #ahda1 and ,tannery1 ,le ,scholie du moine ,neophytos1 ,revue ,archeologique1 #ahhe1 pp4 #ii-ajb4 ,jordan1 loc4 cit41 gives from twelfth and thirteenth century manuscripts the forms cifra1 ciffre1 chifras1 and cifrus4 ,du ,cange1 ,glossarium mediae et infimae ,latinitatis1 ,paris1 #ahdb1 gives also chilerae4 ,dasypodius1 ,institutiones ,mathematicae1 ,strassburg1 #aeic-aeif1 adds the forms zyphra and syphra4 ,boissiere1 ,l'art d'arythmetique contenant toute dimention1 tres-singulier et commode1 tant pour l'art militaire que autres calculations1 ,paris1 #aeed3 ,puis y en a vn autre dict zero lequel ne designe nulle quantite par soy1 ains seulement les loges vuides4 #bdg ,propagation1 pp4 #bg1 #bcd1 #ddb4 ,treutlein1 ,das ,rechnen im #af4 ,jahrhundert1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,i1 p4 #e1 favors the same view4 ,it is combated by many writers1 e4g4 ,a4 ,c4 ,burnell1 loc4 cit41 p4 #ei4 ,long before ,woepcke1 ,i4 ,f4 and ,g4 ,i4 ,weidler1 ,de characteribus numerorum vulgaribus et eorum aetatibus1 ,wittenberg1 #agbg1 asserted the possibility of their introduction into ,greece by ,pythagoras or one of his followers3 ,potuerunt autem ex oriente1 uel ex phoenicia1 ad graecos traduci1 uel ,pythagorae1 uel eius discipulorum auxilio1 cum aliquis eo1 proficiendi in literis causa1 iter faceret1 et hoc quoque inuentum addisceret4 #bdh ,e4g41 they adopted the ,greek numerals in use in ,damascus and ,syria1 and the ,coptic in ,egypt4 ,theophanes 7#geh-hah ,a4,d471 ,chronographia1 ,scriptores ,historiae ,byzantinae1 ,vol4 ,,xxxix1 ,bonnae1 #ahci1 p4 #ege1 relates that in #fii ,a4,d4 the caliph ,walid forbade the use of the ,greek language in the bookkeeping of the treasury of the caliphate1 but permitted the use of the ,greek alphabetic numerals1 since the ,arabs had no convenient number notation3 ,greek3 kai ekoluse graphesthai ,hellenisti tous demosious ton logothesion kodikas1 all' ,arabiois auta parasemainesthai1 choris ton psephon1 epeide adunaton tei ekeinon glossei monada e duada e triada e okto hemisu e tria graphesthai2 dio kai heos semeron eisin sun autois notarioi ,christianoi4 ,the importance of this contemporaneous document was pointed out by ,martin1 loc4 cit4 ,karabacek1 ,die ,involutio im arabischen ,schriftwesen1 ,vol4 ,,cxxxv of ,sitzungsberichte d4 phil4-hist4 ,classe d4 k4 ,akad4 d4 ,wiss41 ,vienna1 #ahif1 p4 #be1 gives an ,arabic date of #hfh ,a4,d4 in ,greek letters4 #bdi ,the ,origin and ,history of ,our ,numerals 7in ,russian71 ,kiev1 #aijh2 ,the ,independence of ,european ,arithmetic 7in ,russian71 ,kiev4 #bej ,woepcke1 loc4 cit41 pp4 #dfb1 #bfb4 #bea ,woepcke1 loc4 cit41 p4 #bdj4 ,h4isab-al-4,gobar1 by an anonymous author1 probably ,abu ,sahl ,dunash ibn ,tamim1 is given by ,steinschneider1 ,die ,mathematik bei den ,juden1 ,bibliotheca ,mathematica1 #ahif1 p4 #bf4 #beb ,steinschneider in the ,abhandlungen1 ,vol4 ,,iii1 p4 #aaj4 #bec ,see his ,grammaire arabe1 ,vol4 ,i1 ,paris1 #ahaj1 plate ,,viii2 ,gerhardt1 ,etudes1 pp4 #i-aa1 and ,entstehung etc41 p4 #h2 ,i4 ,f4 ,weidler1 ,spicilegium observationum ad historiam notarum numeralium pertinentium1 ,wittenberg1 #agee1 speaks of the figura cifrarum ,saracenicarum as being different from that of the characterum ,boethianorum1 which are similar to the vulgar or common numerals2 see also ,humboldt1 loc4 cit4 #bed ,gerhardt mentions it in his ,entstehung etc41 p4 #h2 ,woepcke1 ,propagation1 states that these numerals were used not for calculation1 but very much as we use ,roman numerals4 ,these superposed dots are found with both forms of numerals 7,propagation1 pp4 #bdd-bdf74 #bee ,gerhardt 7,etudes1 p4 #i7 from a manuscript in the ,bibliotheque ,nationale4 ,the numeral forms are symbols1 #bj being indicated by symbol with dot and #bjj by symbol with #b dots4 ,this scheme of zero dots was also adopted by the ,byzantine ,greeks1 for a manuscript of ,planudes in the ,bibliotheque ,nationale has numbers like pi alpha with #d dots for #h1ajj1jjj1jjj4 ,see ,gerhardt1 ,etudes1 p4 #ai4 ,pihan1 ,expose etc41 p4 #bjh1 gives two forms1 ,asiatic and ,maghrebian1 of ,ghobar numerals4 #bef ,see ,chap4 ,,iv4 #beg ,possibly as early as the third century ,a4,d41 but probably of the eighth or ninth4 ,see ,cantor1 ,i 7#c71 p4 #eih4 #beh ,ascribed by the ,arabic writer to ,india4 #bei ,see ,woepcke's description of a manuscript in the ,chasles library1 ,recherches sur l'histoire des sciences mathematiques chez les orientaux1 ,journal ,asiatique1 ,,iv 7#e71 #ahei1 p4 #ceh1 note4 #bfj ,p4 #ef4 #bfa ,reinaud1 ,memoire sur l',inde1 p4 #cii4 ,in the fourteenth century one ,sihab al-,din wrote a work on which1 a scholiast to the ,bodleian manuscript remarks3 ,the science is called ,algobar because the inventor had the habit of writing the figures on a tablet covered with sand4 ,gerhardt1 ,etudes1 p4 #aa1 note4 #bfb ,gerhardt1 ,entstehung etc41 p4 #bj4 #bfc ,h4 ,suter1 ,das ,rechenbuch des ,abu ,zakarija el-,h4as4s4ar1 ,bibliotheca ,mathematica1 ,vol4 ,,ii 7#c71 p4 #ae4 #bfd ,a4 ,devoulx1 ,les chiffres arabes1 ,revue ,africaine1 ,vol4 ,,xvi1 pp4 #dee-deh4 #bfe ,kitab al-,fihrist1 ,g4 ,fluegel1 ,leipzig1 ,vol4 ,i1 #ahga1 and ,vol4 ,,ii1 #ahgb4 ,this work was published after ,professor ,fluegel's death by ,j4 ,roediger and ,a4 ,mueller4 ,the first volume contains the ,arabic text and the second volume contains critical notes upon it4 #bff ,like those of line #e in the illustration on page #fi4 #bfg ,woepcke1 ,recherches sur l'histoire des sciences mathematiques chez les orientaux1 loc4 cit42 ,propagation1 p4 #eg4 #bfh ,al-,h4as4s4ar's forms1 ,suter1 ,bibliotheca ,mathematica1 ,vol4 ,,ii 7#c71 p4 #ae4 #bfi ,woepcke1 ,sur une donnee historique1 etc41 loc4 cit4 ,the name 4gobar is not used in the text4 ,the manuscript from which these are taken is the oldest 7#igj ,a4,d47 ,arabic document known to contain all of the numerals4 #bgj ,silvestre de ,sacy1 loc4 cit4 ,he gives the ordinary modern ,arabic forms1 calling them ,indien4 #bga ,woepcke1 ,introduction au calcul ,gobari et ,hawai1 ,atti dell' accademia pontificia dei nuovi ,lincei1 ,vol4 ,,xix4 ,the adjective applied to the forms in #e is gobari and to those in #f indienne4 ,this is the direct opposite of ,woepcke's use of these adjectives in the ,recherches sur l'histoire cited above1 in which the ordinary ,arabic forms 7like those in row #e7 are called indiens4 ,these forms are usually written from right to left4 #bgb ,j4 ,g4 ,wilkinson1 ,the ,manners and ,customs of the ,ancient ,egyptians1 revised by ,s4 ,birch1 ,london1 #ahgh1 ,vol4 ,,ii1 p4 #dic1 plate ,,xvi4 #bgc ,there is an extensive literature on this ,boethius-,frage4 ,the reader who cares to go fully into it should consult the various volumes of the ,jahrbuch ueber die ,fortschritte der ,mathematik4 #bgd ,this title was first applied to ,roman emperors in posthumous coins of ,julius ,caesar4 ,subsequently the emperors assumed it during their own lifetimes1 thus deifying themselves4 ,see ,f4 ,gnecchi1 ,monete romane1 #b;d ed41 ,milan1 #aijj1 p4 #bii4 #bge ,this is the common spelling of the name1 although the more correct ,latin form is ,boetius4 ,see ,harper's ,dict4 of ,class4 ,lit4 and ,antiq41 ,new ,york1 #ahig1 ,vol4 ,i1 p4 #bac4 ,there is much uncertainty as to his life4 ,a good summary of the evidence is given in the last two editions of the ,encyclopaedia ,britannica4 #bgf ,his father1 ,flavius ,manlius ,boethius1 was consul in #dhg4 #bgg ,there is1 however1 no good historic evidence of this sojourn in ,athens4 #bgh ,his arithmetic is dedicated to ,symmachus3 ,domino suo patricio ,symmacho ,boetius4 ,friedlein ed41 p4 #c4 #bgi ,it was while here that he wrote ,de consolatione philosophiae4 #bhj ,it is sometimes given as #ebe4 #bha ,there was a medieval tradition that he was executed because of a work on the ,trinity4 #bhb ,hence the ,divus in his name4 #bhc ,thus ,dante1 speaking of his burial place in the monastery of ,st4 ,pietro in ,ciel d',oro1 at ,pavia1 says3 ,the saintly soul1 that shows ,the world's deceitfulness1 to all who hear him1 ,is1 with the sight of all the good that is1 ,blest there4 ,the limbs1 whence it was driven1 lie ,down in ,cieldauro2 and from martyrdom ,and exile came it here4--,paradiso1 ,canto ,x4 #bhd ,not1 however1 in the mercantile schools4 ,the arithmetic of ,boethius would have been about the last book to be thought of in such institutions4 ,while referred to by ,baeda 7#fgb-gce7 and ,hrabanus ,maurus 7c4 #ggf-hef71 it was only after ,gerbert's time that the ,boetii de institutione arithmetica libri duo was really a common work4 #bhe ,also spelled ,cassiodorius4 #bhf ,as a matter of fact1 ,boethius could not have translated any work by ,pythagoras on music1 because there was no such work1 but he did make the theories of the ,pythagoreans known4 ,neither did he translate ,nicomachus1 although he embodied many of the ideas of the ,greek writer in his own arithmetic4 ,gibbon follows ,cassiodorus in these statements in his ,decline and ,fall of the ,roman ,empire1 chap4 xxxix4 ,martin pointed out with positiveness the similarity of the first book of ,boethius to the first five books of ,nicomachus4 ,les signes numeraux etc41 reprint1 p4 #d4 #bhg ,the general idea goes back to ,pythagoras1 however4 #bhh ,j4 ,c4 ,scaliger in his ,poetice also said of him3 ,boethii ,severini ingenium1 eruditio1 ars1 sapientia facile provocat omnes auctores1 sive illi ,graeci sint1 sive ,latini ,heilbronner1 ,hist4 math4 univ41 p4 #chg4 ,libri1 speaking of the time of ,boethius1 remarks3 ,nous voyons du temps de ,theodoric1 les lettres reprendre une nouvelle vie en ,italie1 les ecoles florissantes et les savans honores4 ,et certes les ouvrages de ,boece1 de ,cassiodore1 de ,symmaque1 surpassent de beaucoup toutes les productions du siecle precedent4 ,histoire des mathematiques1 ,vol4 ,i1 p4 #gh4 #bhi ,carra de ,vaux1 ,avicenne1 ,paris1 #aijj2 ,woepcke1 ,sur l'introduction1 etc42 ,gerhardt1 ,entstehung etc41 p4 #bj4 ,avicenna is a corruption from ,ibn ,sina1 as pointed out by ,wuestenfeld1 ,geschichte der arabischen ,aerzte und ,naturforscher1 ,goettingen1 #ahdj4 ,his full name is ,abu ,ali al-,h4osein ibn ,sina4 ,for notes on ,avicenna's arithmetic1 see ,woepcke1 ,propagation1 p4 #ejb4 #bij ,on the early travel between the ,east and the ,west the following works may be consulted3 ,a4 ,hillebrandt1 ,alt-,indien1 containing ,chinesische ,reisende in ,indien1 ,breslau1 #ahii1 p4 #agi2 ,c4 ,a4 ,skeel1 ,travel in the ,first ,century after ,christ1 ,cambridge1 #aija1 p4 #adb2 ,m4 ,reinaud1 ,relations politiques et commerciales de l'empire romain avec l',asie orientale1 in the ,journal ,asiatique1 ,mars-,avril1 #ahfc1 ,vol4 ,i 7#f71 p4 #ic2 ,beazley1 ,dawn of ,modern ,geography1 a ,history of ,exploration and ,geographical ,science from the ,conversion of the ,roman ,empire to ,a4,d4 #adbj1 ,london1 #ahig-aijf1 #c vols42 ,heyd1 ,geschichte des ,levanthandels im ,mittelalter1 ,stuttgart1 #ahig2 ,j4 ,keane1 ,the ,evolution of ,geography1 ,london1 #ahii1 p4 #ch2 ,a4 ,cunningham1 ,corpus inscriptionum ,indicarum1 ,calcutta1 #ahgg1 ,vol4 ,i2 ,a4 ,neander1 ,general ,history of the ,christian ,religion and ,church1 #eth ,american ed41 ,boston1 #ahee1 ,vol4 ,,iii1 p4 #hi2 ,r4 ,c4 ,dutt1 ,a ,history of ,civilization in ,ancient ,india1 ,vol4 ,,ii1 ,bk4 ,v1 chap1 ii2 ,e4 ,c4 ,bayley1 loc4 cit41 p4 #bh et seq42 ,a4 ,c4 ,burnell1 loc4 cit41 p4 #c2 ,j4 ,e4 ,tennent1 ,ceylon1 ,london1 #ahei1 ,vol4 ,i1 p4 #aei2 ,geo4 ,turnour1 ,epitome of the ,history of ,ceylon1 ,london1 n4d41 preface2 ,philalethes1 ,history of ,ceylon1 ,london1 #ahaf1 chap1 i2 ,h4 ,c4 ,sirr1 ,ceylon and the ,cingalese1 ,london1 #ahej1 ,vol4 ,i1 chap4 ix4 ,on the ,hindu knowledge of the ,nile see ,f4 ,wilford1 ,asiatick ,researches1 ,vol4 ,,iii1 p4 #bie1 ,calcutta1 #agib4 #bia ,g4 ,oppert1 ,on the ,ancient ,commerce of ,india1 ,madras1 #ahgi1 p4 #h4 #bib ,gerhardt1 ,etudes etc41 pp4 #h1 #aa4 #bic ,see ,smith's ,dictionary of ,greek and ,roman ,biography and ,mythology4 #bid ,p4 ,m4 ,sykes1 ,ten ,thousand ,miles in ,persia1 or ,eight ,years in ,iran1 ,london1 #aijb1 p4 #afg4 ,sykes was the first ,european to follow the course of ,alexander's army across eastern ,persia4 #bie ,buehler1 ,indian ,brahma ,alphabet1 note1 p4 #bg2 ,palaeographie1 p4 #b2 ,herodoti ,halicarnassei historia1 ,amsterdam1 #agfc1 ,bk4 ,,iv1 p4 #cjj2 ,isaac ,vossius1 ,periplus ,scylacis ,caryandensis1 #afci4 ,it is doubtful whether the work attributed to ,scylax was written by him1 but in any case the work dates back to the fourth century ,b4,c4 ,see ,smith's ,dictionary of ,greek and ,roman ,biography4 #bif ,herodotus1 ,bk4 ,,iii4 #big ,rameses ,,ii7871 the ,sesoosis of ,diodorus ,siculus4 #bih ,indian ,antiquary1 ,vol4 ,i1 p4 #bbi2 ,f4 ,b4 ,jevons1 ,manual of ,greek ,antiquities1 ,london1 #ahie1 p4 #chf4 ,on the relations1 political and commercial1 between ,india and ,egypt c4 #gb ,b4,c41 under ,ptolemy ,auletes1 see the ,journal ,asiatique1 #ahfc1 p4 #big4 #bii ,sikandar1 as the name still remains in northern ,india4 #cjj ,harper's ,classical ,dict41 ,new ,york1 #ahig1 ,vol4 ,i1 p4 #gbd2 ,f4 ,b4 ,jevons1 loc4 cit41 p4 #chi2 ,j4 ,c4 ,marshman1 ,abridgment of the ,history of ,india1 chaps4 i and ii4 #cja ,oppert1 loc4 cit41 p4 #aa4 ,it was at or near this place that the first great ,indian mathematician1 ,aryabhat4a1 was born in #dgf ,a4,d4 #cjb ,buehler1 ,palaeographie1 p4 #b1 speaks of ,greek coins of a period anterior to ,alexander1 found in northern ,india4 ,more complete information may be found in ,indian ,coins1 by ,e4 ,j4 ,rapson1 ,strassburg1 #ahih1 pp4 #c-g4 #cjc ,oppert1 loc4 cit41 p4 #ad2 and to him is due other similar information4 #cjd ,j4 ,beloch1 ,griechische ,geschichte1 ,vol4 ,,iii1 ,strassburg1 #aijd1 pp4 #cj-ca4 #cje ,e4g41 the denarius1 the words for hour and minute 7,greek3 hora1 lepton71 and possibly the signs of the zodiac4 ,r4 ,caldwell1 ,comparative ,grammar of the ,dravidian ,languages1 ,london1 #ahef1 p4 #dch4 ,on the probable ,chinese origin of the zodiac see ,schlegel1 loc4 cit4 #cjf ,marie1 ,vol4 ,,ii1 p4 #gc2 ,r4 ,caldwell1 loc4 cit4 #cjg ,a4 ,cunningham1 loc4 cit41 p4 #ej4 #cjh ,c4 ,a4 ,j4 ,skeel1 ,travel1 loc4 cit41 p4 #ad4 #cji ,inchiver1 from inchi1 the green root4 ,indian ,antiquary1 ,vol4 ,i1 p4 #ceb4 #caj ,in ,china dating only from the second century ,a4,d41 however4 #caa ,the ,italian morra4 #cab ,j4 ,bowring1 ,the ,decimal ,system1 ,london1 #ahed1 p4 #b4 #cac ,h4 ,a4 ,giles1 lecture at ,columbia ,university1 ,march #ab1 #aijb1 on ,china and ,ancient ,greece4 #cad ,giles1 loc4 cit4 #cae ,e4g41 the names for grape1 radish 7la-po1 ,greek3 rhaphe71 water-lily 7si-kua1 west gourds2 ,greek3 sikua1 gourds71 are much alike4 ,giles1 loc4 cit4 #caf ,epistles1 ,i1 #a1 #de-df4 ,on the ,roman trade routes1 see ,beazley1 loc4 cit41 ,vol4 ,i1 p4 #agi4 #cag ,am4 ,journ4 of ,archeol41 ,vol4 ,,iv1 p4 #cff4 #cah ,m4 ,perrot gives this conjectural restoration of his words3 ,ad me ex ,india regum legationes saepe missi sunt numquam antea visae apud quemquam principem ,romanorum4 ,m4 ,reinaud1 ,relations politiques et commerciales de l'empire romain avec l',asie orientale1 ,journ4 ,asiat41 ,vol4 ,i 7#f71 p4 #ic4 #cai ,reinaud1 loc4 cit41 p4 #ahi4 ,florus1 ,,ii1 #cd 7,,iv1 #ab71 refers to it3 ,seres etiam habitantesque sub ipso sole ,indi1 cum gemmis et margaritis elephantes quoque inter munera trahentes nihil magis quam longinquitatem viae imputabant4 ,horace shows his geographical knowledge by saying3 ,not those who drink of the deep ,danube shall now break the ,julian edicts2 not the ,getae1 not the ,seres1 nor the perfidious ,persians1 nor those born on the river ,tanais4 ,odes1 ,bk4 ,,iv1 ,ode #ae1 #ba-bd4 #cbj ,qua virtutis moderationisque fama ,indos etiam ac ,scythas auditu modo cognitos pellexit ad amicitiam suam populique ,romani ultro per legatos petendam4 ,reinaud1 loc4 cit41 p4 #ahj4 #cba ,reinaud1 loc4 cit41 p4 #ahj4 #cbb ,georgics1 ,,ii1 #agj-agb4 ,so ,propertius 7,elegies1 ,,iii1 #d73 ,arma deus ,caesar dites meditatur ad ,indos ,et freta gemmiferi findere classe maris4 ,the divine ,caesar meditated carrying arms against opulent ,india1 and with his ships to cut the gem-bearing seas4 #cbc ,heyd1 loc4 cit41 ,vol4 ,i1 p4 #d4 #cbd ,reinaud1 loc4 cit41 p4 #cic4 #cbe ,the title page of ,calandri 7#adia71 for example1 represents ,pythagoras with these numerals before him4 ,smith1 ,rara ,arithmetica1 p4 #df4 ,isaacus ,vossius1 ,observationes ad ,pomponium ,melam de situ orbis1 #afeh1 maintained that the ,arabs derived these numerals from the west4 ,a learned dissertation to this effect1 but deriving them from the ,romans instead of the ,greeks1 was written by ,ginanni in #agec 7,dissertatio mathematica critica de numeralium notarum minuscularum origine1 ,venice1 #agec74 ,see also ,mannert1 ,de numerorum quos arabicos vocant vera origine ,pythagorica1 ,nuernberg1 #ahja4 ,even as late as #ahbg ,romagnosi 7in his supplement to ,ricerche storiche sull' ,india etc41 by ,robertson1 ,vol4 ,,ii1 p4 #ehj1 #ahbg7 asserted that ,pythagoras originated them4 ,r4 ,bombelli1 ,l'antica numerazione italica1 ,rome1 #ahgf1 p4 #ei4 ,gow 7,hist4 of ,greek ,math41 p4 #ih7 thinks that ,iamblichus must have known a similar system in order to have worked out certain of his theorems1 but this is an unwarranted deduction from the passage given4 #cbf ,a4 ,hillebrandt1 ,alt-,indien1 p4 #agi4 #cbg ,j4 ,c4 ,marshman1 loc4 cit41 chaps4 i and ii4 #cbh ,he reigned #fca-egi ,a4,d42 called ,nu'sirwan1 the holy one4 #cbi ,j4 ,keane1 ,the ,evolution of ,geography1 ,london1 #ahii1 p4 #ch4 #ccj ,the ,arabs who lived in and about ,mecca4 #cca ,s4 ,guyard1 in ,encyc4 ,brit41 #ith ed41 ,vol4 ,,xvi1 p4 #eig4 #ccb ,oppert1 loc4 cit41 p4 #bi4 #ccc ,at non credendum est id in ,autographis contigisse1 aut vetustioribus ,codd4 ,,mss4 ,wallis1 ,opera omnia1 ,vol4 ,,ii1 p4 #aa4 #ccd ,in ,observationes ad ,pomponium ,melam de situ orbis4 ,the question was next taken up in a large way by ,weidler1 loc4 cit41 ,de characteribus etc41 #agbg1 and in ,spicilegium etc41 #agee4 #cce ,the best edition of these works is that of ,g4 ,friedlein1 ,anicii ,manlii ,torquati ,severini ,boetii de institutione arithmetica libri duo1 de institutione musica libri quinque4 ,accedit geometria quae fertur ,boetii4444 ,leipzig4444 ,,mdccclxvii4 #ccf ,see also ,p4 ,tannery1 ,notes sur la pseudo-geometrie de ,boece1 in ,bibliotheca ,mathematica1 ,vol4 ,i 7#c71 p4 #ci4 ,this is not the geometry in two books in which are mentioned the numerals4 ,there is a manuscript of this pseudo-geometry of the ninth century1 but the earliest one of the other work is of the eleventh century 7,tannery71 unless the ,vatican codex is of the tenth century as ,friedlein 7p4 #cgb7 asserts4 #ccg ,friedlein feels that it is partly spurious1 but he says3 ,eorum librorum1 quos ,boetius de geometria scripsisse dicitur1 investigare veram inscriptionem nihil aliud esset nisi operam et tempus perdere4 ,preface1 p4 v4 ,n4 ,bubnov in the ,russian ,journal of the ,ministry of ,public ,instruction1 #aijg1 in an article of which a synopsis is given in the ,jahrbuch ueber die ,fortschritte der ,mathematik for #aijg1 asserts that the geometry was written in the eleventh century4 #cch ,the most noteworthy of these was for a long time ,cantor 7,geschichte1 ,vol4 ,i41 #c;d ed41 pp4 #ehg-ehh71 who in his earlier days even believed that ,pythagoras had known them4 ,cantor says 7,die roemischen ,agrimensoren1 ,leipzig1 #ahge1 p4 #acj73 ,uns also1 wir wiederholen es1 ist die ,geometrie des ,boetius echt1 dieselbe ,schrift1 welche er nach ,euklid bearbeitete1 von welcher ein ,codex bereits in ,jahre #hba im ,kloster ,reichenau vorhanden war1 von welcher ein anderes ,exemplar im ,jahre #ihb zu ,mantua in die ,haende ,gerbert's gelangte1 von welcher mannigfache ,handschriften noch heute vorhanden sind4 ,but against this opinion of the antiquity of ,,mss4 containing these numerals is the important statement of ,p4 ,tannery1 perhaps the most critical of modern historians of mathematics1 that none exists earlier than the eleventh century4 ,see also ,j4 ,l4 ,heiberg in ,philologus1 ,zeitschrift f4 d4 klass4 ,altertum1 ,vol4 ,,xliii1 p4 #ejh4 ,of ,cantor's predecessors1 ,th4 ,h4 ,martin was one of the most prominent1 his argument for authenticity appearing in the ,revue ,archeologique for #ahef-aheg1 and in his treatise ,les signes numeraux etc4 ,see also ,m4 ,chasles1 ,de la connaissance qu'ont eu les anciens d'une numeration decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de position1 ,comptes rendus1 ,vol4 ,,vi1 pp4 #fgh-fhj2 ,sur l'origine de notre systeme de numeration1 ,comptes rendus1 ,vol4 ,,viii1 pp4 #gb-ha2 and note ,sur le passage du premier livre de la geometrie de ,boece1 relatif a un nouveau systeme de numeration1 in his work ,apercu historique sur l'origine et le developpement des methodes en geometrie1 of which the first edition appeared in #ahcg4 #cci ,j4 ,l4 ,heiberg places the book in the eleventh century on philological grounds1 ,philologus1 loc4 cit42 ,woepcke1 in ,propagation1 p4 #dd2 ,blume1 ,lachmann1 and ,rudorff1 ,die ,schriften der roemischen ,feldmesser1 ,berlin1 #ahdh2 ,boeckh1 ,de abaco graecorum1 ,berlin1 #ahda2 ,friedlein1 in his ,leipzig edition of #ahfg2 ,weissenborn1 ,abhandlungen1 ,vol4 ,,ii1 p4 #ahe1 his ,gerbert1 pp4 #a1 #bdg1 and his ,geschichte der ,einfuehrung der jetzigen ,ziffern in ,europa durch ,gerbert1 ,berlin1 #ahib1 p4 #aa2 ,bayley1 loc4 cit41 p4 #ei2 ,gerhardt1 ,etudes1 p4 #ag1 ,entstehung und ,ausbreitung1 p4 #ad2 ,nagl1 ,gerbert1 p4 #eg2 ,bubnov1 loc4 cit4 ,see also the discussion by ,chasles1 ,halliwell1 and ,libri1 in the ,comptes rendus1 #ahci1 ,vol4 ,,ix1 p4 #ddg1 and in ,vols4 ,,viii1 ,,xvi1 ,,xvii of the same journal4 #cdj ,j4 ,marquardt1 ,la vie privee des ,romains1 ,vol4 ,,ii 7,french trans471 p4 #eje1 ,paris1 #ahic4 #cda ,in a ,plimpton manuscript of the arithmetic of ,boethius of the thirteenth century1 for example1 the ,roman numerals are all replaced by the ,arabic1 and the same is true in the first printed edition of the book4 7,see ,smith's ,rara ,arithmetica1 pp4 #dcd1 #be-bg47 ,d4 ,e4 ,smith also copied from a manuscript of the arithmetic in the ,laurentian library at ,florence1 of #acgj1 the following forms1 ,forged numerals #cdb ,halliwell1 in his ,rara ,mathematica1 p4 #ajg1 states that the disputed passage is not in a manuscript belonging to ,mr4 ,ames1 nor in one at ,trinity ,college4 ,see also ,woepcke1 in ,propagation1 pp4 #cg and #db4 ,it was the evident corruption of the texts in such editions of ,boethius as those of ,venice1 #adii1 ,basel1 #aedf and #aegj1 that led ,woepcke to publish his work ,sur l'introduction de l'arithmetique indienne en ,occident4 #cdc ,they are found in none of the very ancient manuscripts1 as1 for example1 in the ninth-century 787 codex in the ,laurentian library which one of the authors has examined4 ,it should be said1 however1 that the disputed passage was written after the arithmetic1 for it contains a reference to that work4 ,see the ,friedlein ed41 p4 #cig4 #cdd ,smith1 ,rara ,arithmetica1 p4 #ff4 #cde ,j4 ,l4 ,heiberg1 ,philologus1 ,vol4 ,,xliii1 p4 #ejg4 #cdf ,nosse autem huius artis dispicientem1 quid sint digiti1 quid articuli1 quid compositi1 quid incompositi numeri4 ,friedlein ed41 p4 #cie4 #cdg ,de ratione abaci4 ,in this he describes quandam formulam1 quam ob honorem sui praeceptoris mensam ,pythagoream nominabant 444 a posterioribus appellabatur abacus4 ,this1 as pictured in the text1 is the common ,gerbert abacus4 ,in the edition in ,migne's ,patrologia ,latina1 ,vol4 ,,lxiii1 an ordinary multiplication table 7sometimes called ,pythagorean abacus7 is given in the illustration4 #cdh ,habebant enim diverse formatos apices vel caracteres4 ,see the reference to ,gerbert on p4 #aag4 #cdi ,c4 ,henry1 ,sur l'origine de quelques notations mathematiques1 ,revue ,archeologique1 #ahgi1 derives these from the initial letters used as abbreviations for the names of the numerals1 a theory that finds few supporters4 #cej ,e4g41 it appears in ,schonerus1 ,algorithmus ,demonstratus1 ,nuernberg1 #aecd1 f4 ,a#d4 ,in ,england it appeared in the earliest ,english arithmetical manuscript known1 ,the ,crafte of ,nombrynge3 fforthermore ye most vndirstonde that in this craft ben vsid teen figurys1 as here bene writen for ensampul1 ,numerals 444 in the quych we vse teen figurys of ,inde4 ,questio4 why ten fyguris of ,inde8 ,solucio4 for as ,i have sayd afore thei were fonde fyrst in ,inde of a kynge of that ,cuntre1 that was called ,algor4 ,see ,smith1 ,an ,early ,english ,algorism1 loc4 cit4 #cea ,friedlein ed41 p4 #cig4 #ceb ,carlsruhe codex of ,gerlando4 #cec ,munich codex of ,gerlando4 #ced ,carlsruhe codex of ,bernelinus4 #cee ,munich codex of ,bernelinus4 #cef ,turchill1 c4 #abjj4 #ceg ,anon4 ,,ms41 thirteenth century1 ,alexandrian ,library1 ,rome4 #ceh ,twelfth-century ,boethius1 ,friedlein1 p4 #cif4 #cei ,vatican codex1 tenth century1 ,boethius4 #cfj a1 h1 i1 are from the ,friedlein ed42 the original in the manuscript from which a is taken contains a zero symbol1 as do all of the six plates given by ,friedlein4 b-e from the ,boncompagni ,bulletino1 ,vol4 ,x1 p4 #eif2 f ibid41 ,vol4 ,,xv1 p4 #ahf2 g ,memorie della classe di sci41 ,reale ,acc4 dei ,lincei1 ,an4 ,,cclxxiv 7#ahgf-ahgg71 ,april1 #ahgg4 ,a twelfth-century arithmetician1 possibly ,john of ,luna 7,hispalensis1 of ,seville1 c4 #aaej71 speaks of the great diversity of these forms even in his day1 saying3 ,est autem in aliquibus figuram istarum apud multos diuersitas4 ,quidam enim septimam hanc figuram representant ,symbol alii autem sic ,symbol1 uel sic ,symbol4 ,quidam vero quartam sic ,symbol4 ,boncompagni1 ,trattati1 ,vol4 ,,ii1 p4 #bh4 #cfa ,loc4 cit41 p4 #ei4 #cfb ,ibid41 p4 #aja4 #cfc ,loc4 cit41 p4 #cif4 #cfd ,khosru ,i1 who began to reign in #eca ,a4,d4 ,see ,w4 ,s4 ,w ,vaux1 ,persia1 ,london1 #ahge1 p4 #afi2 ,th4 ,noeldeke1 ,aufsaetze zur persichen ,geschichte1 ,leipzig1 #ahhg1 p4 #aac1 and his article in the ninth edition of the ,encyclopaedia ,britannica4 #cfe ,colebrooke1 ,essays1 ,vol4 ,,ii1 p4 #ejd1 on the authority of ,ibn al-,adami1 astronomer1 in a work published by his continuator ,al-,qasim in #ibj ,a4,d42 ,al-,biruni1 ,india1 ,vol4 ,,ii1 p4 #ae4 #cff ,h4 ,suter1 ,die ,mathematiker etc41 pp4 #d-e1 states that ,al-,fazari died between #gif and #hjf4 #cfg ,suter1 loc4 cit41 p4 #fc4 #cfh ,suter1 loc4 cit41 p4 #gd4 #cfi ,suter1 ,das ,mathematiker-,verzeichniss im ,fihrist4 ,the references to ,suter1 unless otherwise stated1 are to his later work ,die ,mathematiker und ,astronomen der ,araber etc4 #cgj ,suter1 ,fihrist1 p4 #cg1 no date4 #cga ,suter1 ,fihrist1 p4 #ch1 no date4 #cgb ,possibly late tenth1 since he refers to one arithmetical work which is entitled ,book of the ,cyphers in his ,chronology1 ,english ed41 p4 #acb4 ,suter1 ,die ,mathematiker etc41 pp4 #ih-ajj1 does not mention this work2 see the ,nachtraege und ,berichtigungen1 pp4 #agj-agb4 #cgc ,suter1 pp4 #if-ig4 #cgd ,suter1 p4 #aaa4 #cge ,suter1 p4 #abd4 ,as the name shows1 he came from the ,west4 #cgf ,suter1 p4 #ach4 #cgg ,hankel1 ,zur ,geschichte der ,mathematik1 p4 #bef1 refers to him as writing on the ,hindu art of reckoning2 ,suter1 p4 #afb4 #cgh ,greek3 ,psephophoria kat' ,indous1 ,greek ed41 ,c4 ,i4 ,gerhardt1 ,halle1 #ahfe2 and ,german translation1 ,das ,rechenbuch des ,maximus ,planudes1 ,h4 ,waeschke1 ,halle1 #ahgh4 #cgi ,sur une donnee historique relative a l'emploi des chiffres indiens par les ,arabes1 ,tortolini's ,annali di scienze mat4 e fis41 #ahee4 #chj ,suter1 p4 #hj4 #cha ,suter1 p4 #fh4 #chb ,sprenger also calls attention to this fact1 in the ,zeitschrift d4 deutschen morgenlaend4 ,gesellschaft1 ,vol4 ,,xlv1 p4 #cfg4 #chc ,libri1 ,histoire des mathematiques1 ,vol4 ,i1 p4 #adg4 #chd ,dictant la paix a l'empereur de ,constantinople1 l',arabe victorieux demandait des manuscrits et des savans4 ,libri1 loc4 cit41 p4 #ajh4 #che ,persian bagadata1 ,god-given4 #chf ,one of the ,abbassides1 the 7at least pretended7 descendants of ,al-,abbas1 uncle and adviser of ,moh4ammed4 #chg ,e4 ,reclus1 ,asia1 ,american ed41 ,n4 ,y41 #ahia1 ,vol4 ,,iv1 p4 #bbg4 #chh ,historical ,sketches1 ,vol4 ,,iii1 chap4 iii4 #chi ,on its prominence at that period see ,villicus1 p4 #gj4 #cij ,see pp4 #d-e4 #cia ,smith1 ,d4 ,e41 in the ,cantor ,festschrift1 #aiji1 note pp4 #aj-aa4 ,see also ,f4 ,woepcke1 ,propagation4 #cib ,enestroem1 in ,bibliotheca ,mathematica1 ,vol4 ,i 7#c71 p4 #dii2 ,cantor1 ,geschichte1 ,vol4 ,i 7#c71 p4 #fga4 #cic ,cited in ,chapter ,i4 ,it begins3 ,dixit algoritmi3 laudes deo rectori nostro atque defensori dicamus dignas4 ,it is devoted entirely to the fundamental operations and contains no applications4 #cid ,m4 ,steinschneider1 ,die ,mathematik bei den ,juden1 ,bibliotheca ,mathematica1 ,vol4 ,,viii 7#b71 p4 #ii4 ,see also the reference to this writer in ,chapter ,i4 #cie ,part of this work has been translated from a ,leyden ,,ms4 by ,f4 ,woepcke1 ,propagation1 and more recently by ,h4 ,suter1 ,bibliotheca ,mathematica1 ,vol4 ,,vii 7#c71 pp4 #aac-aai4 #cif ,a4 ,neander1 ,general ,history of the ,christian ,religion and ,church1 #eth ,american ed41 ,boston1 #ahee1 ,vol4 ,,iii1 p4 #cce4 #cig ,beazley1 loc4 cit41 ,vol4 ,i1 p4 #di4 #cih ,beazley1 loc4 cit41 ,vol4 ,i1 pp4 #ej1 #dfj4 #cii ,see pp4 #g-h4 #djj ,the name also appears as ,moh4ammed ,abu'l-,qasim1 and ,ibn ,hauqal4 ,beazley1 loc4 cit41 ,vol4 ,i1 p4 #de4 #dja ,kitab al-masalik wa'l-mamalik4 #djb ,reinaud1 ,mem4 sur l',inde2 in ,gerhardt1 ,etudes1 p4 #ah4 #djc ,born at ,shiraz in #aaic4 ,he himself had traveled from ,india to ,europe4 #djd ,gulistan 7,rose ,garden71 ,gateway the third1 ,,xxii4 ,sir ,edwin ,arnold's translation1 ,n4 ,y41 #ahii1 p4 #agg4 #dje ,cunningham1 loc4 cit41 p4 #ha4 #djf ,putnam1 ,books1 ,vol4 ,i1 p4 #bbg3 ,non semel externas peregrino tramite terras ,jam peragravit ovans1 sophiae deductus amore1 ,si quid forte novi librorum seu studiorum ,quod secum ferret1 terris reperiret in illis4 ,hic quoque ,romuleum venit devotus ad urbem4 7,more than once he has traveled joyfully through remote regions and by strange roads1 led on by his zeal for knowledge and seeking to discover in foreign lands novelties in books or in studies which he could take back with him4 ,and this zealous student journeyed to the city of ,romulus47 #djg ,a4 ,neander1 ,general ,history of the ,christian ,religion and ,church1 #eth ,american ed41 ,boston1 #ahee1 ,vol4 ,,iii1 p4 #hi1 note #d2 ,libri1 ,histoire1 ,vol4 ,i1 p4 #adc4 #djh ,cunningham1 loc4 cit41 p4 #ha4 #dji ,heyd1 loc4 cit41 ,vol4 ,i1 p4 #d4 #daj ,ibid41 p4 #e4 #daa ,ibid41 p4 #ba4 #dab ,ibid41 p4 #bc4 #dac ,libri1 ,histoire1 ,vol4 ,i1 p4 #afg4 #dad ,picavet1 ,gerbert1 un pape philosophe1 d'apres l'histoire et d'apres la legende1 ,paris1 #ahig1 p4 #ai4 #dae ,beazley1 loc4 cit41 ,vol4 ,i1 chap1 i1 and p4 #ed seq4 #daf ,ibid41 p4 #eg4 #dag ,libri1 ,histoire1 ,vol4 ,i1 p4 #aaj1 n41 citing authorities1 and p4 #aeb4 #dah ,possibly the old tradition1 ,prima dedit nautis usum magnetis ,amalphis1 is true so far as it means the modern form of compass card4 ,see ,beazley1 loc4 cit41 ,vol4 ,,ii1 p4 #cih4 #dai ,r4 ,c4 ,dutt1 loc4 cit41 ,vol4 ,,ii1 p4 #cab4 #dbj ,e4 ,j4 ,payne1 in ,the ,cambridge ,modern ,history1 ,london1 #aijb1 ,vol4 ,i1 chap4 i4 #dba ,geo4 ,phillips1 ,the ,identity of ,marco ,polo's ,zaitun with ,changchau1 in ,t'oung pao1 ,archives pour servir a l'etude de l'histoire de l',asie orientale1 ,leyden1 #ahij1 ,vol4 ,i1 p4 #bah4 ,w4 ,heyd1 ,geschichte des ,levanthandels im ,mittelalter1 ,vol4 ,,ii1 p4 #baf4 ,the ,palazzo dei ,poli1 where ,marco was born and died1 still stands in the ,corte del ,milione1 in ,venice4 ,the best description of the ,polo travels1 and of other travels of the later ,middle ,ages1 is found in ,c4 ,r4 ,beazley's ,dawn of ,modern ,geography1 ,vol4 ,,iii1 chap1 ii1 and ,part ,,ii4 #dbb ,heyd1 loc4 cit41 ,vol4 ,,ii1 p4 #bbj2 ,h4 ,yule1 in ,encyclopaedia ,britannica1 #ith 7#ajth7 or #aath ed41 article ,china4 ,the handbook cited is ,pegolotti's ,libro di divisamenti di paesi1 chapters i-ii1 where it is implied that #fj1jjj would be a likely amount for a merchant going to ,china to invest in his trip4 #dbc ,cunningham1 loc4 cit41 p4 #aid4 #dbd ,i4e4 a commission house4 #dbe ,cunningham1 loc4 cit41 p4 #ahf4 #dbf ,j4 ,r4 ,green1 ,short ,history of the ,english ,people1 ,new ,york1 #ahij1 p4 #ff4 #dbg ,w4 ,besant1 ,london1 ,new ,york1 #ahib1 p4 #dc4 #dbh ,baldakin1 baldekin1 baldachino4 #dbi ,italian ,baldacco4 #dcj ,j4 ,k4 ,mumford1 ,oriental ,rugs1 ,new ,york1 #aija1 p4 #ah4 #dca ,or ,girbert1 the ,latin forms ,gerbertus and ,girbertus appearing indifferently in the documents of his time4 #dcb ,see1 for example1 ,j4 ,c4 ,heilbronner1 ,historia matheseos universae1 p4 #gdj4 #dcc ,obscuro loco natum1 as an old chronicle of ,aurillac has it4 #dcd ,n4 ,bubnov1 ,gerberti postea ,silvestri ,,ii papae opera mathematica1 ,berlin1 #ahii1 is the most complete and reliable source of information2 ,picavet1 loc4 cit41 ,gerbert etc42 ,olleris1 ,oeuvres de ,gerbert1 ,paris1 #ahfg2 ,havet1 ,lettres de ,gerbert1 ,paris1 #ahhi 2 ,h4 ,weissenborn1 ,gerbert2 ,beitraege zur ,kenntnis der ,mathematik des ,mittelalters1 ,berlin1 #ahhh1 and ,zur ,geschichte der ,einfuehrung der jetzigen ,ziffern in ,europa durch ,gerbert1 ,berlin1 #ahib2 ,buedinger1 ,ueber ,gerberts wissenschaftliche und politische ,stellung1 ,cassel1 #ahea2 ,richer1 ,historiarum liber ,,iii1 in ,bubnov1 loc4 cit41 pp4 #cgf-cha2 ,nagl1 ,gerbert und die ,rechenkunst des #aj4 ,jahrhunderts1 ,vienna1 #ahhh4 #dce ,richer tells of the visit to ,aurillac by ,borel1 a ,spanish nobleman1 just as ,gerbert was entering into young manhood4 ,he relates how affectionately the abbot received him1 asking if there were men in ,spain well versed in the arts4 ,upon ,borel's reply in the affirmative1 the abbot asked that one of his young men might accompany him upon his return1 that he might carry on his studies there4 #dcf ,vicus ,ausona4 ,hatto also appears as ,atton and ,hatton4 #dcg ,this is all that we know of his sojourn in ,spain1 and this comes from his pupil ,richer4 ,the stories told by ,adhemar of ,chabanois1 an apparently ignorant and certainly untrustworthy contemporary1 of his going to ,cordova1 are unsupported4 7,see e4g4 ,picavet1 p4 #cd47 ,nevertheless this testimony is still accepted3 ,k4 von ,raumer1 for example 7,geschichte der ,paedagogik1 #fth ed41 #ahij1 ,vol4 ,i1 p4 #f71 says ,mathematik studierte man im ,mittelalter bei den ,arabern in ,spanien4 ,zu ihnen gieng ,gerbert1 nachmaliger ,pabst ,sylvester ,,ii4 #dch ,thus in a letter to ,aldaberon he says3 ,quos post repperimus speretis1 id est ,,viii volumina ,boeti de astrologia1 praeclarissima quoque figurarum geometriae1 aliaque non minus admiranda 7,epist4 #h74 ,also in a letter to ,rainard 7,epist4 #acj71 he says3 ,ex tuis sumptibus fac ut michi scribantur ,m4 ,manlius 7,manilius in one ,,ms47 de astrologia4 #dci ,picavet1 loc4 cit41 p4 #ca4 #ddj ,picavet1 loc4 cit41 p4 #cf4 #dda ,havet1 loc4 cit41 p4 vii4 #ddb ,picavet1 loc4 cit41 p4 #cg4 #ddc ,con sinistre arti conseguri la dignita del ,pontificato4444 ,lasciato poi l' abito1 e 'l monasterio1 e datosi tutto in potere del diavolo4 ,quoted in ,bombelli1 ,l'antica numerazione ,italica1 ,rome1 #ahgf1 p4 #da n4 #ddd ,he writes from ,rheims in #ihd to one ,lupitus1 in ,barcelona1 saying3 ,itaque librum de astrologia translatum a te michi petenti dirige1 presumably referring to some ,arabic treatise4 ,epist4 no4 #bd of the ,havet collection1 p4 #ai4 #dde ,see ,bubnov1 loc4 cit41 p4 x4 #ddf ,olleris1 loc4 cit41 p4 #cfa1 l4 #ae1 for ,bernelinus2 and ,bubnov1 loc4 cit41 p4 #cha1 l4 #d1 for ,richer4 #ddg ,woepcke found this in a ,paris ,,ms4 of ,radulph of ,laon1 c4 #aajj4 ,propagation1 p4 #bdf4 ,et prima quidem trium spaciorum superductio unitatis caractere inscribitur1 qui chaldeo nomine dicitur igin4 ,see also ,alfred ,nagl1 ,der arithmetische ,tractat des ,radulph von ,laon 7,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,v1 pp4 #he-acc71 p4 #ig4 #ddh ,weissenborn1 loc4 cit41 p4 #bci4 ,when ,olleris 7,oeuvres de ,gerbert1 ,paris1 #ahfg1 p4 cci7 says1 ,c'est a lui et non point aux ,arabes1 que l',europe doit son systeme et ses signes de numeration1 he exaggerates1 since the evidence is all against his knowing the place value4 ,friedlein emphasizes this in the ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,xii 7#ahfg71 ,literaturzeitung1 p4 #gj3 ,fuer das ,system unserer ,numeration ist die ,null das wesentlichste ,merkmal1 und diese kannte ,gerbert nicht4 ,er selbst schrieb alle ,zahlen mit den roemischen ,zahlzeichen und man kann ihm also nicht verdanken1 was er selbst nicht kannte4 #ddi ,e4g41 ,chasles1 ,buedinger1 ,gerhardt1 and ,richer4 ,so ,martin 7,recherches nouvelles etc47 believes that ,gerbert received them from ,boethius or his followers4 ,see ,woepcke1 ,propagation1 p4 #da4 #dej ,buedinger1 loc4 cit41 p4 #aj4 ,nevertheless1 in ,gerbert's time one ,al-,mans4ur1 governing ,spain under the name of ,hisham 7#igf-ajjb71 called from the ,orient ,al-,be4gani to teach his son1 so that scholars were recognized4 ,picavet1 p4 #cf4 #dea ,weissenborn1 loc4 cit41 p4 #bce4 #deb ,ibid41 p4 #bcd4 #dec ,these letters1 of the period #ihc-iig1 were edited by ,havet1 loc4 cit41 and1 less completely1 by ,olleris1 loc4 cit4 ,those touching mathematical topics were edited by ,bubnov1 loc4 cit41 pp4 #ih-ajf4 #ded ,he published it in the ,monumenta ,germaniae historica1 ,scriptores1 ,vol4 ,,iii1 and at least three other editions have since appeared1 viz4 those by ,guadet in #ahde1 by ,poinsignon in #ahee1 and by ,waitz in #ahgg4 #dee ,domino ac beatissimo ,patri ,gerberto1 ,remorum archiepiscopo1 ,richerus ,monchus1 ,gallorum congressibus in volumine regerendis1 imperii tui1 pater sanctissime ,gerberte1 auctoritas seminarium dedit4 #def ,in epistle #ag 7,havet collection7 he speaks of the ,de multiplicatione et divisione numerorum libellum a ,joseph ,ispano editum abbas ,warnerius 7a person otherwise unknown74 ,in epistle #be he says3 ,de multiplicatione et divisione numerorum1 ,joseph ,sapiens sententias quasdam edidit4 #deg ,h4 ,suter1 ,zur ,frage ueber den ,josephus ,sapiens1 ,bibliotheca ,mathematica1 ,vol4 ,,viii 7#b71 p4 #hd2 ,weissenborn1 ,einfuehrung1 p4 #ad2 also his ,gerbert2 ,m4 ,steinschneider1 in ,bibliotheca ,mathematica1 #ahic1 p4 #fh4 ,wallis 7,algebra1 #afhe1 chap4 #ad7 went over the list of ,spanish ,josephs very carefully1 but could find nothing save that ,josephus ,hispanus seu ,josephus sapiens videtur aut ,maurus fuisse aut alius quis in ,hispania4 #deh ,p4 ,ewald1 ,mittheilungen1 ,neues ,archiv d4 ,gesellschaft fuer aeltere deutsche ,geschichtskunde1 ,vol4 ,,viii1 #ahhc1 pp4 #ced-cfd4 ,one of the manuscripts is of #igf ,a4,d4 and the other of #iib ,a4,d4 ,see also ,franz ,steffens1 ,lateinische ,palaeographie1 ,freiburg 7,schweiz71 #aijc1 pp4 xxxix-xl4 ,the forms are reproduced in the plate on page #adj4 #dei ,it is entitled ,constantino suo ,gerbertus scolasticus1 because it was addressed to ,constantine1 a monk of the ,abbey of ,fleury4 ,the text of the letter to ,constantine1 preceding the treatise on the ,abacus1 is given in the ,comptes rendus1 ,vol4 ,,xvi 7#ahdc71 p4 #bie4 ,this book seems to have been written c4 #ihj ,a4,d4 ,bubnov1 loc4 cit41 p4 #f4 #dfj ,histoire de l',arithmetique1 ,comptes rendus1 ,vol4 ,,xvi 7#ahdc71 pp4 #aef1 #bha4 #dfa ,loc4 cit41 ,gerberti ,opera etc4 #dfb ,friedlein thought it spurious4 ,see ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,xii 7#ahfg71 ,hist4-lit4 suppl41 p4 #gd4 ,it was discovered in the library of the ,benedictine monastry of ,st4 ,peter1 at ,salzburg1 and was published by ,peter ,bernhard ,pez in #agba4 ,doubt was first cast upon it in the ,olleris edition 7,oeuvres de ,gerbert74 ,see ,weissenborn1 ,gerbert1 pp4 #b1 #f1 #afh1 and ,picavet1 p4 #ha4 ,hock1 ,cantor1 and ,th4 ,martin place the composition of the work at c4 #iif when ,gerbert was in ,germany1 while ,olleris and ,picavet refer it to the period when he was at ,rheims4 #dfc ,picavet1 loc4 cit41 p4 #ahb4 #dfd ,who wrote after ,gerbert became pope1 for he uses1 in his preface1 the words1 a domino pape ,gerberto4 ,he was quite certainly not later than the eleventh century2 we do not have exact information about the time in which he lived4 #dfe ,picavet1 loc4 cit41 p4 #ahb4 ,weissenborn1 ,gerbert1 p4 #bbg4 ,in ,olleris1 ,liber ,abaci 7of ,bernelinus71 p4 #cfa4 #dff ,richer1 in ,bubnov1 loc4 cit41 p4 #cha4 #dfg ,weissenborn1 ,gerbert1 p4 #bda4 #dfh ,writers on numismatics are quite uncertain as to their use4 ,see ,f4 ,gnecchi1 ,monete ,romane1 #b;d ed41 ,milan1 #aijj1 cap4 ,,xxxvii4 ,for pictures of old ,greek tesserae of ,sarmatia1 see ,s4 ,ambrosoli1 ,monete ,greche1 ,milan1 #ahii1 p4 #bjb4 #dfi ,thus ,tzwivel's arithmetic of #aejg1 fol4 #b1 v41 speaks of the ten figures as characteres sive numerorum apices a diuo ,seuerino ,boetio4 #dgj ,weissenborn uses sipos for #j4 ,it is not given by ,bernelinus1 and appears in ,radulph of ,laon1 in the twelfth century4 ,see ,guenther's ,geschichte1 p4 #ih1 n42 ,weissenborn1 p4 #aa2 ,pihan1 ,expose etc41 pp4 xvi-xxii4 ,in ,friedlein's ,boetius1 p4 #cif1 the plate shows that all of the six important manuscripts from which the illustrations are taken contain the symbol1 while four out of five which give the words use the word sipos for #j4 ,the names appear in a twelfth-century anonymous manuscript in the ,vatican1 in a passage beginning ,ordine primigeno sibi nomen possidet igin4 ,andras ecce locum mox uendicat ipse secundum ,ormis post numeros incompositus sibi primus4 ,boncompagni ,buttetino1 ,,xv1 p4 #acb4 ,turchill 7twelfth century7 gives the names ,igin1 andras1 hormis1 arbas1 quimas1 caletis1 zenis1 temenias1 celentis1 saying3 ,has autem figuras1 ut donnus dominus ,gvillelmus ,rx testatur1 a pytagoricis habemus1 nomina uero ab arabibus4 7,who the ,william ,r4 was is not known4 ,boncompagni ,bulletino ,,xv1 p4 #acf47 ,radulph of ,laon 7d4 #aaca7 asserted that they were ,chaldean 7,propagation1 p4 #dh n474 ,a discussion of the whole question is also given in ,e4 ,c4 ,bayley1 loc4 cit4 ,huet1 writing in #afgi1 asserted that they were of ,semitic origin1 as did ,nesselmann in spite of his despair over ormis1 calctis1 and celentis2 see ,woepcke1 ,propagation1 p4 #dh4 ,the names were used as late as the fifteenth century1 without the zero1 but with the superscript dot for #aj's1 two dots for #ajj's1 etc41 as among the early ,arabs4 ,gerhardt mentions having seen a fourteenth or fifteenth century manuscript in the ,bibliotheca ,amploniana with the names ,ingnin1 andras1 armis1 arbas1 quinas1 calctis1 zencis1 zemenias1 zcelentis1 and the statement ,si unum punctum super ingnin ponitur1 ,x significat4444 ,si duo puncta super 444 figuras superponunter1 fiet decuplim illius quod cum uno puncto significabatur1 in ,monatsberichte der ,k4 ,p4 ,akad4 d4 ,wiss41 ,berlin1 #ahfg1 p4 #dj4 #dga ,a chart of ten numerals in #bjj tongues1 by ,rev4 ,r4 ,patrick1 ,london1 #ahab4 #dgb ,numeratio figuralis est cuiusuis numeri per notas1 et figuras numerates descriptio4 ,clichtoveus1 edition of c4 #aejg1 fol4 ,c ii1 v4 ,aristoteles enim uoces rerum ,greek3 sumbola uocat3 id translatum1 sonat notas4 ,noviomagus1 ,de ,numeris ,libri ,,ii1 cap4 vi4 ,alphabetum decem notarum4 ,schonerus1 notes to ,ramus1 #aehf1 p4 #c seq4 ,richer says3 novem numero notas omnem numerum significantes4 ,bubnov1 loc4 cit41 p4 #cha4 #dgc ,il y a dix ,characteres1 autrement ,figures1 ,notes1 ou ,elements4 ,peletier1 edition of #afjg1 p4 #ac4 ,numerorum notas alij figuras1 alij signa1 alij characteres uocant4 ,glareanus1 #aede edition1 f4 #i1 r4 ,per figuras 7quas zyphras uocant7 assignationem1 quales sunt hae notulae1 #a4 #b4 #c4 #d4444 ,noviomagus1 ,de ,numeris ,libri ,,ii1 cap4 vi4 ,gemma ,frisius also uses elementa and ,cardan uses literae4 ,in the first arithmetic by an ,american 7,greenwood1 #agbi7 the author speaks of a few ,arabian ,charecters or ,numeral ,figures1 called ,digits 7p4 #a71 and as late as #agij1 in the third edition of ,j4 ,j4 ,blassiere's arithmetic 7#ast ed4 #agfi71 the name characters is still in use1 both for de ,latynsche en de ,arabische 7p4 #d71 as is also the term ,cyfferletters 7p4 #f1 n474 ,ziffer1 the modern ,german form of cipher1 was commonly used to designate any of the nine figures1 as by ,boeschenstein and ,riese1 although others1 like ,koebel1 used it only for the zero4 ,so zifre appears in the arithmetic by ,borgo1 #aeej ed4 ,in a ,munich codex of the twelfth century1 attributed to ,gerland1 they are called characters only3 ,usque ad ,,viiii4 enim porrigitur omnis numerus et qui supercrescit eisdem designator ,karacteribus4 ,boncompagni ,bulletino1 ,vol4 ,x4 p4 #fjg4 #dgd ,the title of his work is ,prologus ,n4 ,ocreati in ,helceph 7,arabic al-qeif1 investigation or memoir7 ad ,adelardum ,batensem magistrum suum4 ,the work was made known by ,c4 ,henry1 in the ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,xxv1 p4 #abi1 and in the ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,iii2 ,weissenborn1 ,gerbert1 p4 #ahh4 #dge ,the zero is indicated by a vacant column4 #dgf ,leo ,jordan1 loc4 cit41 p4 #agj4 ,chifre en augorisme is the expression used1 while a century later giffre en argorisme and cyffres d'augorisme are similarly used4 #dgg ,the ,works of ,geoffrey ,chaucer1 edited by ,w4 ,w4 ,skeat1 ,vol4 ,,iv1 ,oxford1 #ahid1 p4 #ib4 #dgh ,loc4 cit41 ,vol4 ,,iii1 pp4 #agi and #ahj4 #dgi ,in ,book ,,ii1 chap1 vii1 of ,the ,testament of ,love1 printed with ,chaucer's ,works1 loc4 cit41 ,vol4 ,,vii1 ,london1 #ahig4 #dhj ,liber ,abacci1 published in ,olleris1 ,oeuvres de ,gerbert1 pp4 #ceg-djj4 #dha ,g4 ,r4 ,kaye1 ,the ,use of the ,abacus in ,ancient ,india1 ,journal and ,proceedings of the ,asiatic ,society of ,bengal1 #aijh1 pp4 #bic-big4 #dhb ,liber ,abbaci1 by ,leonardo ,pisano1 loc4 cit41 p4 #a4 #dhc ,friedlein1 ,die ,entwickelung des ,rechnens mit ,columnen1 ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,x1 p4 #bdg4 #dhd ,the divisor #f or #af being increased by the difference #d1 to #aj or #bj respectively4 #dhe ,e4g4 ,cantor1 ,vol4 ,i1 p4 #hhb4 #dhf ,friedlein1 loc4 cit42 ,friedlein1 ,gerbert's ,regeln der ,division and ,das ,rechnen mit ,columnen vor dem #aj4 ,jahrhundert1 ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,ix2 ,bubnov1 loc4 cit41 pp4 #aig-bde2 ,m4 ,chasles1 ,histoire de l'arithmetique4 ,recherches des traces du systeme de l'abacus1 apres que cette methode a pris le nom d',algorisme4--,preuves qu'a toutes les epoques1 jusq'au ,,xvie siecle1 on a su que l'arithmetique vulgaire avait pour origine cette methode ancienne1 ,comptes rendus1 ,vol4 ,,xvii1 pp4 #adc-aed1 also ,regles de l'abacus1 ,comptes rendus1 ,vol4 ,,xvi1 pp4 #bah-bdf1 and ,analyse et explication du traite de ,gerbert1 ,comptes rendus1 ,vol4 ,,xvi1 pp4 #bha-bii4 #dhg ,bubnov1 loc4 cit41 pp4 #bjc-bjd1 ,abbonis abacus4 #dhh ,regulae de numerorum abaci rationibus1 in ,bubnov1 loc4 cit41 pp4 #bje-bbe4 #dhi ,p4 ,treutlein1 ,intorno ad alcuni scritti inediti relativi al calcolo dell' abaco1 ,bulletino di bibliografia e di storia delle scienze matematiche e fisiche1 ,vol4 ,x1 pp4 #ehi-fdg4 #dij ,intorno ad uno scritto inedito di ,adelhardo di ,bath intitolato ',regulae ,abaci1' ,b4 ,boncompagni1 in his ,bulletino1 ,vol4 ,,xiv1 pp4 #a-acd4 #dia ,treutlein1 loc4 cit42 ,boncompagni1 ,intorno al ,tractatus de ,abaco di ,gerlando1 ,bulletino1 ,vol4 ,x1 pp4 #fdh-fef4 #dib ,e4 ,narducci1 ,intorno a due trattati inediti d'abaco contenuti in due codici ,vaticani del secolo ,,xii1 ,boncompagni ,bulletino1 ,vol4 ,,xv1 pp4 #aaa-afb4 #dic ,see ,molinier1 ,les sources de l'histoire de ,france1 ,vol4 ,,ii1 ,paris1 #aijb1 pp4 #b1 #c4 #did ,cantor1 ,geschichte1 ,vol4 ,i1 p4 #gfb4 ,a4 ,nagl in the ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,v1 p4 #he4 #die #ajcj-aaag4 #dif ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,v1 pp4 #he-acc4 ,the work begins ,incipit ,liber ,radulfi laudunensis de abaco4 #dig ,materialien zur ,geschichte der arabischen ,zahlzeichen in ,frankreich1 loc4 cit4 #dih ,who died in #abjb4 #dii ,cantor1 ,geschichte1 ,vol4 ,i 7#c71 pp4 #hjj-hjc2 ,boncompagni1 ,trattati1 ,part ,,ii4 ,m4 ,steinschneider 7,die ,mathematik bei den ,juden1 ,bibliotheca ,mathematica1 ,vol4 ,x 7#b71 p4 #gi7 ingeniously derives another name by which he is called 7,abendeuth7 from ,ibn ,daud 7,son of ,david74 ,see also ,abhandlungen1 ,vol4 ,,iii1 p4 #aaj4 #ejj ,john is said to have died in #aaeg4 #eja ,for it says1 ,incipit prologus in libro alghoarismi de practica arismetrice4 ,qui editus est a magistro ,johanne yspalensi4 ,it is published in full in the second part of ,boncompagni's ,trattati d'aritmetica4 #ejb ,possibly1 indeed1 the meaning of libro alghoarismi is not to ,al-,khowarazmi's book1 but to a book of algorism4 ,john of ,luna says of it3 ,hoc idem est illud etiam quod 444 alcorismus dicere videtur4 ,trattati1 p4 #fh4 #ejc ,for a resume1 see ,cantor1 ,vol4 ,i 7#c71 pp4 #hjj-hjc4 ,as to the author1 see ,enestroem in the ,bibliotheca ,mathematica1 ,vol4 ,,vi 7#c71 p4 #aad1 and ,vol4 ,,ix 7#c71 p4 #b4 #ejd ,born at ,cremona 7although some have asserted at ,carmona1 in ,andalusia7 in #aaad2 died at ,toledo in #aahg4 ,cantor1 loc4 cit42 ,boncompagni1 ,atti d4 ,r4 ,accad4 d4 n4 ,lincei1 #ahea4 #eje ,see ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,xiv1 p4 #adi2 ,bibliotheca ,mathematica1 ,vol4 ,,iv 7#c71 p4 #bjf4 ,boncompagni had a fourteenth-century manuscript of his work1 ,gerardi ,cremonensis artis metrice practice4 ,see also ,t4 ,l4 ,heath1 ,the ,thirteen ,books of ,euclid's ,elements1 #c vols41 ,cambridge1 #aijh1 ,vol4 ,i1 pp4 #ib-id 2 ,a4 ,a4 ,bjoernbo1 ,gerhard von ,cremonas ,uebersetzung von ,alkwarizmis ,algebra und von ,euklids ,elementen1 ,bibliotheca ,mathematica1 ,vol4 ,,vi 7#c71 pp4 #bci-bdh4 #ejf ,wallis1 ,algebra1 #afhe1 p4 #ab seq4 #ejg ,cantor1 ,geschichte1 ,vol4 ,i 7#c71 p4 #ijf2 ,a4 ,a4 ,bjoernbo1 ,al-,chwarizmi's trigonometriske ,tavler1 ,festskrift til ,h4 ,g4 ,zeuthen1 ,copenhagen1 #aiji1 pp4 #a-ag4 #ejh ,heath1 loc4 cit41 pp4 #ic-if4 #eji ,m4 ,steinschneider1 ,zeitschrift der deutschen morgenlaendischen ,gesellschaft1 ,vol4 ,,xxv1 #ahga1 p4 #ajd1 and ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,xvi1 #ahga1 pp4 #cib-cic2 ,m4 ,curtze1 ,centralblatt fuer ,bibliothekswesen1 #ahii1 p4 #bhi2 ,e4 ,wappler1 ,zur ,geschichte der deutschen ,algebra im #ae4 ,jahrhundert1 ,programm1 ,zwickau1 #ahhg2 ,l4 ,c4 ,karpinski1 ,robert of ,chester's ,translation of the ,algebra of ,al-,khowarazmi1 ,bibliotheca ,mathematica1 ,vol4 ,,xi 7#c71 p4 #abe4 ,he is also known as ,robertus ,retinensis1 or ,robert of ,reading4 #eaj ,nagl1 ,a41 ,ueber eine ,algorismus-,schrift des ,,xii4 ,jahrhunderts und ueber die ,verbreitung der indisch-arabischen ,rechenkunst und ,zahlzeichen im christl4 ,abendlande1 in the ,zeitschrift fuer ,mathematik und ,physik1 ,hist4-lit4 ,abth41 ,vol4 ,,xxxiv1 p4 #abi4 ,curtze1 ,abhandlungen zur ,geschichte der ,mathematik1 ,vol4 ,,viii1 pp4 #a-bg4 #eaa ,see line a in the plate on p4 #adc4 #eab ,sefer ha-,mispar1 ,das ,buch der ,zahl1 ein hebraeisch-arithmetisches ,werk des ,r4 ,abraham ibn ,esra1 ,moritz ,silberberg1 ,frankfurt a4 ,m41 #ahie4 #eac ,browning's ,rabbi ben ,ezra4 #ead ,darum haben auch die ,weisen ,indiens all ihre ,zahlen durch neun bezeichnet und ,formen fuer die #i ,ziffern gebildet4 ,sefer ha-,mispar1 loc4 cit41 p4 #b4 #eae ,f4 ,bonaini1 ,memoria unica sincrona di ,leonardo ,fibonacci1 ,pisa1 #aheh1 republished in #ahfg1 and appearing in the ,giornale ,arcadico1 ,vol4 ,,cxcvii 7,n4,s4 ,,lii72 ,gaetano ,milanesi1 ,documento inedito e sconosciuto a ,lionardo ,fibonacci1 ,roma1 #ahfg2 ,guglielmini1 ,elogio di ,lionardo ,pisano1 ,bologna1 #ahab1 p4 #ce2 ,libri1 ,histoire des sciences mathematiques1 ,vol4 ,,ii1 p4 #be2 ,d4 ,martines1 ,origine e progressi dell' aritmetica1 ,messina1 #ahfe1 p4 #dg2 ,lucas1 in ,boncompagni ,bulletino1 ,vol4 ,x1 pp4 #abi1 #bci2 ,besagne1 ibid41 ,vol4 ,,ix1 p4 #ehc2 ,boncompagni1 three works as cited in ,chap4 ,i2 ,g4 ,enestroem1 ,ueber zwei angebliche mathematische ,schulen im christlichen ,mittelalter1 ,bibliotheca ,mathematica1 ,vol4 ,,viii 7#c71 pp4 #beb-bfb2 ,boncompagni1 ,della vita e delle opere di ,leonardo ,pisano1 loc4 cit4 #eaf ,the date is purely conjectural4 ,see the ,bibliotheca ,mathematica1 ,vol4 ,,iv 7#c71 p4 #bae4 #eag ,an old chronicle relates that in #ajfc ,pisa fought a great battle with the ,saracens at ,palermo1 capturing six ships1 one being full of wondrous treasure1 and this was devoted to building the cathedral4 #eah ,heyd1 loc4 cit41 ,vol4 ,i1 p4 #adi4 #eai ,ibid41 p4 #baa4 #ebj ,j4 ,a4 ,symonds1 ,renaissance in ,italy4 ,the ,age of ,despots4 ,new ,york1 #ahhc1 p4 #fb4 #eba ,symonds1 loc4 cit41 p4 #gi4 #ebb ,j4 ,a4 ,froude1 ,the ,science of ,history1 ,london1 #ahfd4 ,un brevet d'apothicaire n'empecha pas ,dante d'etre le plus grand poete de l',italie1 et ce fut un petit marchand de ,pise qui donna l'algebre aux ,chretiens4 ,libri1 ,histoire1 ,vol4 ,i1 p4 xvi4 #ebc ,a document of #abbf1 found and published in #aheh1 reads3 ,leonardo bigollo quondam ,guilielmi4 #ebd ,bonaccingo germano suo4 #ebe ,e4g4 ,libri1 ,guglielmini1 ,tiraboschi4 #ebf ,latin1 ,bonaccius4 #ebg ,boncompagni and ,milanesi4 #ebh ,reprint1 p4 #e4 #ebi ,whence the ,french name for candle4 #ecj ,now part of ,algiers4 #eca ,e4 ,reclus1 ,africa1 ,new ,york1 #ahic1 ,vol4 ,,ii1 p4 #bec4 #ecb ,sed hoc totum et algorismum atque arcus pictagore quasi errorem computavi respectu modi indorum4 ,woepcke1 ,propagation etc41 regards this as referring to two different systems1 but the expression may very well mean algorism as performed upon the ,pythagorean arcs 7or table74 #ecc ,book of the ,abacus1 this term then being used1 and long afterwards in ,italy1 to mean merely the arithmetic of computation4 #ecd ,incipit liber ,abaci a ,leonardo filio ,bonacci compositus anno #abjb et correctus ab eodem anno #abbh4 ,three ,,mss4 of the thirteenth century are known1 viz4 at ,milan1 at ,siena1 and in the ,vatican library4 ,the work was first printed by ,boncompagni in #aheg4 #ece ,i4e4 in relation to the quadrivium4 ,non legant in festivis diebus1 nisi ,philosophos et rhetoricas et quadrivalia et barbarismum et ethicam1 si placet4 ,suter1 ,die ,mathematik auf den ,universitaeten des ,mittelalters1 ,zuerich1 #ahhg1 p4 #ef4 ,roger ,bacon gives a still more gloomy view of ,oxford in his time in his ,opus minus1 in the ,rerum ,britannicarum medii aevi scriptores1 ,london1 #ahei1 ,vol4 ,i1 p4 #cbg4 ,for a picture of ,cambridge at this time consult ,f4 ,w4 ,newman1 ,the ,english ,universities1 translated from the ,german of ,v4 ,a4 ,huber1 ,london1 #ahdc1 ,vol4 ,i1 p4 #fa2 ,w4 ,w4 ,r4 ,ball1 ,history of ,mathematics at ,cambridge1 #ahhi2 ,s4 ,guenther1 ,geschichte des mathematischen ,unterrichts im deutschen ,mittelalter bis zum ,jahre #aebe1 ,berlin1 #ahhg1 being ,vol4 ,,iii of ,monumenta ,germaniae paedagogica4 #ecf ,on the commercial activity of the period1 it is known that bills of exchange passed between ,messina and ,constantinople in #aafa1 and that a bank was founded at ,venice in #aagj1 the ,bank of ,san ,marco being established in the following year4 ,the activity of ,pisa was very manifest at this time4 ,heyd1 loc4 cit41 ,vol4 ,,ii1 p4 #e2 ,v4 ,casagrandi1 ,storia e cronologia1 #c;d ed41 ,milan1 #aija1 p4 #ef4 #ecg ,j4 ,a4 ,symonds1 loc4 cit41 ,vol4 ,,ii1 p4 #abg4 #ech ,i4 ,taylor1 ,the ,alphabet1 ,london1 #ahhc1 ,vol4 ,,ii1 p4 #bfc4 #eci ,cited by ,unger's ,history1 p4 #ae4 ,the ,arabic numerals appear in a ,regensburg chronicle of #aafg and in ,silesia in #acdj4 ,see ,schmidt's ,encyclopaedie der ,erziehung1 ,vol4 ,,vi1 p4 #gbf2 ,a4 ,kuckuk1 ,die ,rechenkunst im sechzehnten ,jahrhundert1 ,festschrift zur dritten ,saecularfeier des ,berlinischen ,gymnasiums zum grauen ,kloster1 ,berlin1 #ahgd1 p4 #d4 #edj ,the text is given in ,halliwell1 ,rara ,mathematica1 ,london1 #ahci4 #eda ,seven are given in ,ashmole's ,catalogue of ,manuscripts in the ,oxford ,library1 #ahde4 #edb ,maximilian ,curtze1 ,petri ,philomeni de ,dacia in ,algorismum ,vulgarem ,johannis de ,sacrobosco commentarius1 una cum ,algorismo ipso1 ,copenhagen1 #ahig2 ,l4 ,c4 ,karpinski1 ,jordanus ,nemorarius and ,john of ,halifax1 ,american ,mathematical ,monthly1 ,vol4 ,,xvii1 pp4 #ajh-aac4 #edc ,j4 ,aschbach1 ,geschichte der ,wiener ,universitaet im ersten ,jahrhunderte ihres ,bestehens1 ,wien1 #ahfe1 p4 #ic4 #edd ,curtze1 loc4 cit41 gives the text4 #ede ,curtze1 loc4 cit41 found some forty-five copies of the ,algorismus in three libraries of ,munich1 ,venice1 and ,erfurt 7,amploniana74 ,examination of two manuscripts from the ,plimpton collection and the ,columbia library shows such marked divergence from each other and from the text published by ,curtze that the conclusion seems legitimate that these were students' lecture notes4 ,the shorthand character of the writing further confirms this view1 as it shows that they were written largely for the personal use of the writers4 #edf ,quidam philosophus edidit nomine ,algus1 unde et ,algorismus nuncupatur4 ,curtze1 loc4 cit41 p4 #a4 #edg ,sinistrorsum autera scribimus in hac arte more arabico sive iudaico1 huius scientiae inventorum4 ,curtze1 loc4 cit41 p4 #g4 ,the ,plimpton manuscript omits the words sive iudaico4 #edh ,non enim omnis numerus per quascumque figuras ,indorum repraesentatur1 sed tantum determinatus per determinatam1 ut #d non per #e1444 ,curtze1 loc4 cit41 p4 #be4 #edi ,c4 ,henry1 ,sur les deux plus anciens traites francais d'algorisme et de geometrie1 ,boncompagni ,bulletino1 ,vol4 ,,xv1 p4 #di2 ,victor ,mortet1 ,le plus ancien traite francais d'algorisme1 loc4 cit4 #eej ,l',etat des sciences en ,france1 depute la mort du ,roy ,robert1 arrivee en #ajca1 jusqu'a celle de ,philippe le ,bel1 arrivee en #acad1 ,paris1 #agda4 #eea ,discours sur l'etat des lettres en ,france au ,,xiiie siecle1 ,paris1 #ahbd4 #eeb ,apercu historique1 ,paris1 #ahgf ed41 p4 #dfd4 #eec ,ranulf ,higden1 a native of the west of ,england1 entered ,st4 ,werburgh's monastery at ,chester in #abii4 ,he was a ,benedictine monk and chronicler1 and died in #acfd4 ,his ,polychronicon1 a history in seven books1 was printed by ,caxton in #adhj4 #eed ,trevisa's translation1 ,higden having written in ,latin4 #eee ,an illustration of this feeling is seen in the writings of ,prosdocimo de' ,beldomandi 7b4 c4 #acgj-achj1 d4 #adbh73 ,inveni in quam pluribus libris algorismi nuncupatis mores circa numeros operandi satis varios atque diversos1 qui licet boni existerent atque veri erant1 tamen fastidiosi1 tum propter ipsarum regularum multitudinem1 tum propter earum deleationes1 tum etiam propter ipsarum operationum probationes1 utrum si bone fuerint vel ne4 ,erant et etiam isti modi interim fastidiosi1 quod si in aliquo calculo astroloico error contigisset1 calculatorem operationem suam a capite incipere oportebat1 dato quod error suus adhuc satis propinquus existeret2 et hoc propter figuras in sua operatione deletas4 ,indigebat etiam calculator semper aliquo lapide vel sibi conformi1 super quo scribere atque faciliter delere posset figuras cum quibus operabatur in calculo suo4 ,et quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt1 disposui libellum edere in quo omnia ista abicerentur3 qui etiam algorismus sive liber de numeris denominari poterit4 ,scias tamen quod in hoc libello ponere non intendo nisi ea quae ad calculum necessaria sunt1 alia quae in aliis libris practice arismetrice tanguntur1 ad calculum non necessaria1 propter brevitatem dimitendo4 ,quoted by ,a4 ,nagl1 ,zeitschrift fuer ,mathematik und ,physik1 ,hist4-lit4 ,abth41 ,vol4 ,,xxxiv1 p4 #adc2 ,smith1 ,rara ,arithmetica1 p4 #ad1 in facsimile4 #eef ,p4 ,ewald1 loc4 cit42 ,franz ,steffens1 ,lateinische ,palaeographie1 pp4 xxxix-xl4 ,we are indebted to ,professor ,j4 ,m4 ,burnam for a photograph of this rare manuscript4 #eeg ,see the plate of forms on p4 #hh4 #eeh ,karabacek1 loc4 cit41 p4 #ef2 ,karpinski1 ,hindu ,numerals in the ,fihrist1 ,bibliotheca ,mathematica1 ,vol4 ,,xi 7#c71 p4 #aba4 #eei ,woepcke1 ,sur une donnee historique1 etc41 loc4 cit41 and ,essai d'une restitution de travaux perdus d',apollonius sur les quantites irrationnelles1 d'apres des indications tirees d'un manuscrit arabe1 ,tome ,,xiv des ,memoires presentes par divers savants a l',academie des sciences1 ,paris1 #ahef1 note1 pp4 #f-ad4 #efj ,archeological ,report of the ,egypt ,exploration ,fund for #aijh-aiji1 ,london1 #aiaj1 p4 #ah4 #efa ,there was a set of astronomical tables in ,boncompagni's library bearing this date3 ,nota quod anno dni nri ihu xpi4 #abfd4 perfecto4 ,see ,narducci's ,catalogo1 p4 #acj4 #efb ,on the ,early use of ,arabic ,numerals in ,europe1 read before the ,society of ,antiquaries ,april #ad1 #aiaj1 and published in ,archaeologia in the same year4 #efc ,ibid41 p4 #h1 n4 ,the date is part of an ,arabic inscription4 #efd ,o4 ,codrington1 ,a ,manual of ,musalman ,numismatics1 ,london1 #aijd4 #efe ,see ,arbuthnot1 ,the ,mysteries of ,chronology1 ,london1 #aijj1 pp4 #ge1 #gh1 #ih2 ,f4 ,pichler1 ,repertorium der steierischen ,muenzkunde1 ,graetz1 #ahge1 where the claim is made of an ,austrian coin of #adeh2 ,bibliotheca ,mathematica1 ,vol4 ,x 7#b71 p4 #abj1 and ,vol4 ,,xii 7#b71 p4 #abj4 ,there is a ,brabant piece of #adgh in the collection of ,d4 ,e4 ,smith4 #eff ,a specimen is in the ,british ,museum4 ,arbuthnot1 p4 #gi4 #efg ,ibid41 p4 #gi4 #efh ,liber de ,remediis utriusque fortunae ,coloniae4 #efi ,fr4 ,walthern et ,hans ,hurning1 ,noerdlingen4 #egj ,ars ,memorandi1 one of the oldest ,european block-books4 #ega ,eusebius ,caesariensis1 ,de praeparatione evangelica1 ,venice1 ,jenson1 #adgj4 ,the above statement holds for copies in the ,astor ,library and in the ,harvard ,university ,library4 #egb ,francisco de ,retza1 ,comestorium vitiorum1 ,nuernberg1 #adgj4 ,the copy referred to is in the ,astor ,library4 #egc ,see ,mauch1 ,ueber den ,gebrauch arabischer ,ziffern und die ,veraenderungen derselben1 ,anzeiger fuer ,kunde der deutschen ,vorzeit1 #ahfa1 columns #df1 #ha1 #aaf1 #aea1 #ahi1 #bbi1 and #bfh2 ,calmet1 ,recherches sur l'origine des chiffres d'arithmetique1 plate1 loc4 cit4 #egd ,guenther1 ,geschichte1 p4 #age1 n42 ,mauch1 loc4 cit4 #ege ,these are given by ,w4 ,r4 ,lethaby1 from drawings by ,j4 ,t4 ,irvine1 in the ,proceedings of the ,society of ,antiquaries1 #aijf1 p4 #bjj4 #egf ,there are some ill-tabulated forms to be found in ,j4 ,bowring1 ,the ,decimal ,system1 ,london1 #ahed1 pp4 #bc1 #be1 and in ,l4 ,a4 ,chassant1 ,dictionnaire des abreviations latines et francaises 444 du moyen age1 ,paris1 ,,mdccclxvi1 p4 #aac4 ,the best sources we have at present1 aside from the ,hill monograph1 are ,p4 ,treutlein1 ,geschichte unserer ,zahlzeichen1 ,karlsruhe1 #ahge2 ,cantor's ,geschichte1 ,vol4 ,i1 table2 ,m4 ,prou1 ,manuel de paleographie latine et francaise1 #b;d ed41 ,paris1 #ahib1 p4 #afd2 ,a4 ,cappelli1 ,dizionario di abbreviature latine ed italiane1 ,milan1 #ahii4 ,an interesting early source is found in the rare ,caxton work of #adhj1 ,the ,myrrour of the ,world4 ,in ,chap4 ,x is a cut with the various numerals1 the chapter beginning ,the fourth scyence is called arsmetrique4 ,two of the fifteen extant copies of this work are at present in the library of ,mr4 ,j4 ,p4 ,morgan1 in ,new ,york4 #egg ,from the twelfth-century manuscript on arithmetic1 ,curtze1 loc4 cit41 ,abhandlungen1 and ,nagl1 loc4 cit4 ,the forms are copied from ,plate ,,vii in ,zeitschrift fuer ,mathematik und ,physik1 ,vol4 ,,xxxiv4 #egh ,from the ,regensburg chronicle4 ,plate containing some of these numerals in ,monumenta ,germaniae historica1 ,scriptores ,vol4 ,,xvii1 plate to p4 #ahd2 ,wattenbach1 ,anleitung zur lateinischen ,palaeographie1 ,leipzig1 #ahhf1 p4 #ajb2 ,boehmer1 ,fontes rerum ,germanicarum1 ,vol4 ,,iii1 ,stuttgart1 #aheb1 p4 lxv4 #egi ,french ,algorismus of #abge2 from an unpublished photograph of the original1 in the possession of ,d4 ,e4 ,smith4 ,see also p4 #ace4 #ehj ,from a manuscript of ,boethius c4 #abid1 in ,mr4 ,plimpton's library4 ,smith1 ,rara ,arithmetica1 ,plate ,i4 #eha ,numerals in a #acjc manuscript in ,sigmaringen1 copied from ,wattenbach1 loc4 cit41 p4 #ajb4 #ehb ,from a manuscript1 ,add4 ,manuscript #bg1ehi1 ,british ,museum1 #acfj ,a4,d4 ,the work is a computus in which the date #acfj appears1 assigned in the ,british ,museum catalogue to the thirteenth century4 #ehc ,from the copy of ,sacrabosco's ,algorismus in ,mr4 ,plimpton's library4 ,date c4 #addb4 ,see ,smith1 ,rara ,arithmetica1 p4 #dej4 #ehd ,see ,rara ,arithmetica1 pp4 #ddf-ddg4 #ehe ,ibid41 pp4 #dfi-dgj4 #ehf ,ibid41 pp4 #dgg-dgh4 #ehg ,the i is used for one in the ,treviso arithmetic 7#adgh71 ,clichtoveus 7c4 #aejg ed41 where both i and j are so used71 ,chiarini 7#adha71 ,sacrobosco 7#adhh ed471 and ,tzwivel 7#aejg ed41 where jj and jz are used for #aa and #ab74 ,this was not universal1 however1 for the ,algorithmus linealis of c4 #adhh has a special type for #a4 ,in a student's notebook of lectures taken at the ,university of ,wuerzburg in #affj1 in ,mr4 ,plimpton's library1 the ones are all in the form of i4 #ehh ,thus the date ,numerals #aehj1 for #aehj1 appears in a ,,ms4 in the ,laurentian library at ,florence4 ,the second and the following five characters are taken from ,cappelli's ,dizionario1 p4 #chj1 and are from manuscripts of the twelfth1 thirteenth1 fourteenth1 sixteenth1 seventeenth1 and eighteenth centuries1 respectively4 #ehi ,e4g4 ,chiarini's work of #adha2 ,clichtoveus 7c4 #aejg74 #eij ,the first is from an algorismus of the thirteenth century1 in the ,hannover ,library4 ,see ,gerhardt1 ,ueber die ,entstehung und ,ausbreitung des dekadischen ,zahlensystems1 loc4 cit41 p4 #bh4 ,the second character is from a ,french algorismus1 c4 #abge4 ,boncompagni ,bulletino1 ,vol4 ,,xv1 p4 #ea4 ,the third and the following sixteen characters are given by ,cappelli1 loc4 cit41 and are from manuscripts of the twelfth 7#a71 thirteenth 7#b71 fourteenth 7#g71 fifteenth 7#c71 sixteenth 7#a71 seventeenth 7#b71 and eighteenth 7#a7 centuries1 respectively4 #eia ,thus ,chiarini 7#adha7 has ,symbol for #bc4 #eib ,the first of these is from a ,french algorismus1 c4 #abge4 ,the second and the following eight characters are given by ,cappelli1 loc4 cit41 and are from manuscripts of the twelfth 7#b71 thirteenth1 fourteenth1 fifteenth 7#c71 seventeenth1 and eighteenth centuries1 respectively4 #eic ,see ,nagl1 loc4 cit4 #eid ,hannover algorismus1 thirteenth century4 #eie ,see the ,dagomari manuscript1 in ,rara ,arithmetica1 pp4 #dce1 #dcg-ddj4 #eif ,but in the woodcuts of the ,margarita ,philosophica 7#aejc7 the old forms are used1 although the new ones appear in the text4 ,in ,caxton's ,myrrour of the ,world 7#adhj7 the old form is used4 #eig ,cappelli1 loc4 cit4 ,they are partly from manuscripts of the tenth1 twelfth1 thirteenth 7#c71 fourteenth 7#g71 fifteenth 7#f71 and eighteenth centuries1 respectively4 ,those in the third line are from ,chassant's ,dictionnaire1 p4 #aac1 without mention of dates4 #eih ,the first is from the ,hannover algorismus1 thirteenth century4 ,the second is taken from the ,rollandus manuscript1 #adbd4 ,the others in the first two lines are from ,cappelli1 twelfth 7#c71 fourteenth 7#f71 fifteenth 7#ac7 centuries1 respectively4 ,the third line is from ,chassant1 loc4 cit41 p4 #aac1 no mention of dates4 #eii ,the first of these forms is from the ,hannover algorismus1 thirteenth century4 ,the following are from ,cappelli1 fourteenth 7#c71 fifteenth1 sixteenth 7#b71 and eighteenth centuries1 respectively4 #fjj ,the first of these is taken from the ,hannover algorismus1 thirteenth century4 ,the following forms are from ,cappelli1 twelfth1 thirteenth1 fourteenth 7#e71 fifteenth 7#b71 seventeenth1 and eighteenth centuries1 respectively4 #fja ,all of these are given by ,cappelli1 thirteenth1 fourteenth1 fifteenth 7#b71 and sixteenth centuries1 respectively4 #fjb ,smith1 ,rara ,arithmetica1 p4 #dhi4 ,this is also seen in several of the ,plimpton manuscripts1 as in one written at ,ancona in #afhd4 ,see also ,cappelli1 loc4 cit4 #fjc ,french algorismus1 c4 #abge1 for the first of these forms4 ,cappelli1 thirteenth1 fourteenth1 fifteenth 7#c71 and seventeenth centuries1 respectively4 ,the last three are taken from ,byzantinische ,analekten1 ,j4 ,l4 ,heiberg1 being forms of the fifteenth century1 but not at all common4 ,symbol3 ,qoppa was the old ,greek symbol for #ij4 #fjd ,for the first of these the reader is referred to the forms ascribed to ,boethius1 in the illustration on p4 #hh2 for the second1 to ,radulph of ,laon1 see p4 #fj4 ,the third is used occasionally in the ,rollandus 7#adbd7 manuscript1 in ,mr4 ,plimpton's library4 ,the remaining three are from ,cappelli1 fourteenth 7#b7 and seventeenth centuries4 #fje ,smith1 ,an ,early ,english ,algorism4 #fjf ,kuckuck1 p4 #e4 #fjg ,a4 ,cappelli1 loc4 cit41 p4 #cgb4 #fjh ,smith1 ,rara ,arithmetica1 p4 #ddc4 #fji ,curtze1 ,petri ,philomeni de ,dacia etc41 p4 ,,ix4 #faj ,cappelli1 loc4 cit41 p4 #cgf4 #faa ,curtze1 loc4 cit41 pp4 ,,viii-,,ix1 note4 #fab ,edition of #aedd-aede1 f4 #eb4 #fac ,de numeris libri ,,ii1 #aedd ed41 cap4 ,,xv4 ,heilbronner1 loc4 cit41 p4 #gcf1 also gives them1 and compares this with other systems4 #fad ,noviomagus says of them3 ,de quibusdam ,astrologicis1 sive ,chaldaicis numerorum notis4444 ,sunt aliae quaedam notae1 quibus ,chaldaei ,astrologii quemlibet numerum artificiose argute describunt1 scitu periucundae1 quas nobis communicauit ,rodolphus ,paludanus ,nouiomagus4