THE HISTORY OF ARITHMETIC RECKONING WITH THE PEN RECKONING WITH COUNTERS The old and the new systems of computation depicted in an early encyclopedic work, entitled Margarita Philosophica by Gregorius Reisch, published first in 1503. THE HISTORY OF ARITHMETIC By LOUIS CHARLES KARPINSKI Professor of Mathematics, University of Michigan RAND McNALLY & COMPANY CHICAGO NEW YORK Copyright, 1925, by Rand McNally & Company All rights reserved Made in U. S. A A-25 THE PREFACE The purpose of this book is to present the development of arithmetic as a vital and an integral part of the history of civilization* Particular attention is paid to the material of arithmetic which continues to be taught in our elementary schools and to the historical phases of that work with which the teacher of arithmetic should be familiar. Particular attention is given, also, to the early American textbooks of arithmetic, printed before 1800, and to the popular treatises on the subject used in England which were the direct source of the American arithmetic. To understand the progress of arithmetic in America is to understand more fully the whole history of the New World. In this progress the arithmetic of England most directly influenced American arithmetic; but the science of Germany and Italy and Spain and France, the science of the Arabs and the Hindus, and the beginnings of the science of the Egyptians and the Babylonians, all had a working part in the development of pur modern science and, in particular, of arithmetic. The modern tendency in arithmetic is to provide contact with life at as many points as possible. In The History of Arithmetic it is shown that arithmetic connects intimately with the early civilization of America and the Orient, that it is associated directly with the progress of the art of printing, and that illuminating contact is made with the development of the English language. All of this contributes to effective teaching in the largest sense. The study of the history of arithmetic enables the teacher to discriminate between the essential and.the nonessential, a particularly important point at the present time when the content of American arithmetic in the schools is changing rapidly. v VI THE PREFACE The author is indebted to a number of librarians and scholars at home and abroad, who have generously contributed information upon many points. To the publishers the writer's thanks are due, as they have not hesitated at the expense involved in the many illustrations, which stimulate interest in the subject and which, by themselves, practically carry the story of the development of arithmetic in its progress from the Old World to the New World. Louis C. Karpinski Ann Arbor, Michigan the contents CHAPTER PAGE I. Early Forms of Numerals and Early Arithmetic . 1 Egyptian Numerals and Arithmetic...... 1 Babylonian Numerals and Arithmetic...... 7 Greek Numerals............ 11 Roman Numerals and Roman Arithmetic..... 19 The Fingers and the Abacus......... 23 Chinese, Japanese, and Korean Arithmetic .... 28 Native American Numeral Systems...... 30 European Developments of the Abacus...... 33 Bibliography for Supplementary Reading . . . . .36 II. The Numerals Which We Use Today......38 Hindu Origin............. 38 The Arabic Development of the Hindu Arithmetic . . 47 European Translators of Arabic Scientific Works ... 52 Early English Arithmetic......... 56 Bibliography for Supplementary Reading..... 60 III. The Textbooks of Arithmetic........ 61 Egyptian.............. 61 Greek Arithmetic............ 61 Hindu and Arabic Texts.......... 64 European Textbooks after 1200........ 65 American Works............ 78 List of Arithmetics and Arithmetical Works Published in America before 1800.......... 85 Early Canadian Arithmetics......... 97 Kentucky and West of the Mississippi River .... 98 Bibliography for Supplementary Reading..... 99 IV. The Fundamental Operations in Early Arithmetic Employing Numerals......... 100 Addition and Subtraction.......�. . 102 Multiplication............. 106 vii viii THE CONTENTS CHAPTER , PAGE Division..............112 Bibliography for Supplementary Reading . . . .120 V. Fractions.............. 121 Common Fractions............ 121 Decimal Fractions........... 127 Bibliography for Supplementary Reading..... 136 VI. Business Arithmetic...........137 Bibliography for Supplementary Reading.....145 VII. The Terminology of Arithmetic....... 146 The Origin of Number Names........ 147 Operative Terms and Symbols........ 150 Bibliography for Supplementary Reading..... 155 VIII. Denominate Numbers.......... 156 General Considerations.......... 156 Time............... 156 Mensuration............. 157 The Metric System of Weights and Measures . . .164 Money............... 167 Bibliography for Supplementary Reading..... 168 IX. The Teacher and the Teaching of Arithmetic . .169 Bibliography for Supplementary Reading . . . .178 The Index................179 LIST OF ILLUSTRATIONS Gregorius Reisch, ^Margarita philosophical 1503 . . Frontispiece PAGE Aztec hieroglyphics...........Facing 1 Egyptian hieroglyphics............. 2 Egyptian hieroglyphic numerals.......... 3 Egyptian hieroglyphic numerals from a royal tomb..... 4 Egyptian hieratic numerals........... 5 Algebraic problem from the Ahmes papyrus....... 6 Egyptian problem.............. 7 Babylonian cuneiform numerals.......... 8 Two Babylonian tablets............. 9 Curvilinear numerals............. 10 Babylonian curvilinear numerals.......... 10 Minoan or Cretan numeral forms of hieroglyphic type .... 11 Ancient symbol for li Drachmas 1000"........ 12 Ordinary Roman numeral forms.......... 20 Roman numeral forms............. 21 30,000 sestertii as found on an early Roman monument . . . .21 Milestone and signboard giving distances from Rhegium, 130 B.C. . 22 Finger reckoning.............. 24 Modern finger reckoning............ 25 Roman abacus with little stones.......... 26 Chinese " suan-pan" Japanese " soroban," and Russian abacus . 27 Chinese and Japanese li sangi" or number rods...... 29 Korean number rods.............. 29 Aztec numerals............... 31 Maya codex................ 31 Maya numerals............... 31 Maya hieroglyphic number symbols....... . 32 Penman "quipu".............. 33 KobeVs "Arithmetic upon Lines and with Numerals" .... 34 Robert Recorders explanation of subtraction....... 35 Modern Sanskrit numerals........... 43 Bakhshali arithmetical manuscript......... 44 IX X LIST OF ILLUSTRATIONS PAGE Arabic manuscript on arithmetic.......... 47 Twelfth-century algorism in Latin.......... 48 Arabic manuscript on Hindu arithmetic........ 51 Thirteenth-century algorism........... 52 Thirteenth-century arithmetic........... 55 "The Crafte of Nombrynge" . ......... 57 Brief discussion of arithmetic based upon work by Vincent de Beauvais............... 58 One of the earliest printed arithmetic texts in English, 1539 . . 59 Symbols for addition and subtraction......... 61 Arithmetic of Boethius from his "Opera Omnia"..... 63 The "Carmen de Algorismo" of Alexandre de Ville Dieu ... 65 First printed arithmetic............'. 66 Title page of Recorde's arithmetic......."... 67 Illustrated problems "on reversed questions," from KdbeVs arithmetic 68 Arithmetic by Gemma Frisius........... 69 Louvain arithmetic.............. 70 Arithmetic by Peter Ramus........... 71 Two pages from Baker's arithmetic......... 72 Work by Christopher Clavius........... 73 Two works widely popular in England........ 74 Cocker's " Arithmetick"............ 75 Mather's "Young Man's Companion"........ 76 William Leybourn's " Arithmetick"......... 77 Title page and colophon of Freyle's Mexican work..... 79 First known American arithmetic in English, by Bradford ... 80 The first separate treatises to appear in colonial America, by Eodder and by Greenwood............. 82 Dutch arithmetic, by Peter Venema, New York, 1730 .... 83 The first popular American arithmetics appearing after the Revolution, by Pike.............. 84 Fisher's compendium for self-instruction, widely popular in England and America............. 87 Late edition of Dilworth............ 88 Bonny castle's arithmetic............ 91 German arithmetic by Rauch, a Pennsylvania product .... 97 LIST OF ILLUSTRATIONS XI PAGE "Jealousy" multiplication, 14&4, Paciuolo....... 99 Numeration in a fifteenth-century manuscript...... 100 Duplication in a twelfth-century manuscript....... 102 Addition and subtraction in "The Art of Nombryng" .... 104 Multiplication and division in "The Art of Nombryng" . . . 107 Juan Diez Freyle's "Sumario".......... 109 Robert Recorde's "The Grounde of Artes"....... 112 "The Grounde of Artes"........... 113 Clavius'" Epitome arithmeticae practicae".......114 Bradford's "The Young Man's Companion"...... 116 Long division as taught by Isaac Greenwood....... 117 Hodder's"Arithmetick"............ 118 Long division with check from Hodder........ 119 Gemma Frisius............... 123 Roman fractions.............. 125 Riese, "Rechnung auff der Linien und Fedem"...... 128 Simon Stevin, "La Thiende"........... 129 English translation of the first work on decimal fractions . . .132 Decimal fractions as presented by Greenwood....... 134 Partnership as presented by Recorde......... 139 Contents of Greenwood's "Arithmetick"......140,141 Compound interest as treated in the colonial arithmetic of Hodder . 143 Benjamin Dearborn, "The PupiVs Guide"....... 148 Pike's" Arithmetick"............. 159 Currency troubles from Pike's " Abridgement"...... 160 Labored explanation of federal money in Pike's "Abridgement" . 161 Chauncey Lee, "The American Accomptant"...... 163 Daboll's "Schoolmaster's Assistant"......... 166 "Mr. Blundevil his exercises contayning eight treatises" . . . 172 The presidents of Harvard, Yale, and Dartmouth recommend Pike's arithmetic............... 175 Early New England copybook of arithmetic....... 176 AZTEC HIEROGLYPHICS Showing the education of the Aztec boy and girl, age u to age 14. The food allowance is indicated by the elliptical disks, representing tortillas or cornbread. THE HISTORY OF ARITHMETIC CHAPTER I EARLY FORMS OP NUMERALS AND EARLY ARITHMETIC Egyptian Numerals and Arithmetic Picture writing. Before an alphabet was invented our savage ancestors used pictures instead of words to represent ideas. If a single lion was to be represented, a picture of a lion was drawn; later only the head was drawn. To represent three lions, three of those pictures were made. Early American Indians used this form of writing, known as picture writing, or hieroglyphics. Later some man conceived the bright idea of representing three lions by one lion's head with three strokes under it; five lions by the head with five strokes. Picture writing is particularly adapted to the representation of numbers. This type of numeral was most highly developed in early Egypt, as much as four thousand years ago. Egyptians. The Egyptian symbol for 100 may be a surveyor's chain, one hundred units in length. The symbol for 1000 represents the lotus flower, of which there were so many in Egyptian fields. For 10,000 a pointed finger was drawn; and for 100,000 a tadpole was depicted. There were not more tadpoles than lotus l 2 THE HISTORY OF ARITHMETIC flowers, but probably they seemed like more. For a million the picture of a man with his hands outstretched, j$ f EGYPTIAN HIEROGLYPHICS �^ Q Prom the Ahamesu papy- gfnnn rus, c. 1700 B.C. Repre- ^ sents probably illustration to a problem dealing with grains of corn, sheaf, mouse, cat, old woman. These words, cat, mouse, Jtt ^ ^- sheaf, grain, represent '1"'.'"1 ��� *m also the second, third, fourth, and fifth powers of a quantity. apparently in amazement at so large a number, was used. These symbols are all called hieroglyphics, since numbers are represented by pictures of objects. Decimal system. The Egyptian system proceeds by powers of ten, and so is called a decimal system. This is a natural system to use, since man has ten fingers. As Egyptian civilization progressed men found the need of some more rapid method of representing numbers and ideas. This advance was made largely by the Egyptian priests, who had time to think about such things. A so-called priestly (hieratic) writing was developed in which shorter ways of writing numerals were used. An Egyptian arithmetical work on papyrus, employing these hieratic numerals, was found in Egypt about seventy years ago. This papyrus is in the British EARLY FORMS OF NUMERALS 3 Museum; it is known by the name of the finder as the Rhind papyrus, and also by the name of the Egyptian scribe as the Ahamesu or Ahmes papyrus. It is our chief source of information, but confirmed as representative by numerous other documents. Fundamental operations. The operations of addition and subtraction in the Egyptian arithmetic offer no peculiarities beyond those made necessary by the symbols used. Multiplication and division, however, were effected 123 456 7 8 9 10 20 30 100 200 1000 2000 10000 EGYPTIAN HIEROGLYPHIC NUMERALS not at all as we do but rather by repeated doubling. Thus to multiply 37 by 11 the Egyptian wrote with his symbols: 37 V 74 2' 148 4 296 8' 407 The number 407 is obtained as 8X37+2X37+37 or (8+2+1) times 37. Any product of integers can be obtained in this way; one multiplies by repeated doubling and summation of r THE HISTORY OF ARITHMETIC ^'yi&lLi&n �&3> .fcN^JII?*� ail^MfJ EGYPTIAN HIEROGLYPHIC NUMERALS Prom a roya tomb. These numerals, above the animals, give the total number of animals of each kind. The "polywog" represents 100,000; the "finger," 10,000; the "lotus flower," 1000; the "coiled rope," 100; the "arch," 10; and a single stroke, 1. 123,440 cattle; 223,400 donkeys; 232,413 goats; 243,688 animals of the kind represented on the lowest line. EARLY FORMS OF NUMERALS 5 the necessary products. Occasionally, but not always, the tenfold product was introduced with the doubles. Division of 407 by 37 would be obtained by the same sequence, giving 8+2 + 1 as the quotient. In the case of division the doubles are written until the next double I l( 111 _ "I � t =. 0L 123 456789 100 200 300 400 500 600 700 800 900 f4^ i ^ 1000 2000 3000 4000 10,000 100,000 T S K a| \ %l =! ^C \ Y* M K H H Vi Vs H % EGYPTIAN HIERATIC NUMERALS would exceed the dividend. Then partial products above are added to produce the dividend. Fractional parts are treated in a similar manner. In Greece and in Europe until the sixteenth century doubling and halving remained in arithmetic as separate operations, due undoubtedly to the Egyptian influence. Egyptian fractions. The second outstanding peculiarity of Egyptian arithmetic is the use of fractions with 6 THE HISTORY OF ARITHMETIC numerator unity, together with f. Thus f- would be written as % ^ or as � yV; f would be written ^ -�-. To multiply 37 by llf, or 11^ -�-, the same sequence is written as before, together with -J- and i of -J- (or -��) of 37; J of 37 is 18 J; t of this is ^ of -J of 37 and equals 4 J -g-; the sum Kibitz, tt'] �&*,*: ALGEBRAIC PROBLEM FROM THE AHMES PAPYRUS Read from right to left, to the central blank space. First line: unknown and its seventh makes 19. Just at the right of the central blank space 2 J \ is multiplied by 7, by taking the number itself, the double (4^ f), and the double of the double (9�). These added give the value of the unknown. At the extreme top is the Egyptian symbol for the unknown quantity, our x. Under it is 16-| � which when � of it is added, 21 ^, gives 19. total of llf times 37 is obtained then as 296+74+37 + 18J+4Hor430i. 37 1' 18J i' 74 2' 4*i i' 148 4 296 8' 430i, the product The Egyptian frequently used approximations in this process; the numbers used in the final result were indicated by strokes as we have indicated above. Practical arithmetic. The Egyptian arithmetic had the further great merit that it took its problem material directly from the life of the common people, so that we have problems on the baking of bread, stuffing of geese for market, on mensuration, and numerous problems on the pyramids, which were quite as much a wonder to EARLY FORMS OF NUMERALS 7 them as to us. The Egyptians were clever with numbers and with geometrical figures. The pyramids and the obelisks could never have been built as fine as they were without the aid of numbers and geometry. In arithmetic, and, it may be added, in geometry and algebra, the Egyptians made noteworthy progress, establishing the foundations upon which Greek mathematical science rose. Babylonian Numerals and Arithmetic Babylonians. The other ancient civilization with which we are most familiar and of which we are reminded several times every day is the Babylonian. Whenever you tell the time of day you pay an unconscious tribute to the ancient Babylonians, for they first divided the day into twenty-four hours and they were the first to divide the hour into sixty minutes. So also the degrees which we use to measure angles, to measure latitude and longitude, all go back to ancient Babylon. Cuneiform writing. The Babylonians wrote on soft clay with.a pointed stick called a stylus. Tablets to be kept were baked after the writing was placed upon them. The wedge-shaped characters made in the clay constitute what is called cuneiform writing. One hundred years ago nobody could read either Egyptian hieroglyphics or EGYPTIAN PROBLEM On the distribution of 100 loaves of bread in arithmetical progression among 5 people. Reads from right to left. First line mentions 100 loaves of bread among five people. At the extreme left, in a column, is the arithmetical series, reading down, beginning on second line, 23, 17*. 12, 6#, 1. 6 7 8 9 00 ( = 60+40) g w YTY 9 8 THE HISTORY OF ARITHMETIC Babylonian cuneiform writing. Today, however, there are many scholars who know how to read these languages that long might properly have been termed "dead." The story of the unraveling of these mysteries records one of the great triumphs of the human intellect. r yy m jjr 12 3 4.5 < <1 �� W 10 11 30 81 BABYLONIAN CUNEIFORM NUMERALS The sexagesimal system � our minutes and seconds. Probably for a long time the Babylonian system of numerals did not go beyond 60. At Senkereh on the Euphrates some old clay tablets were found upon which a Babylonian had written the squares of numbers up to 30. The tablets read easily up to 72 is 49. Then the tablet gives for the square of eight: 1-4; since we know that 82 is 64, the 1 must stand for 60. The same system was followed throughout these tablets of squares and cubes which furnish a check. Following 82, 92 is given as 1.21, in the cuneiform characters; the first unit again must represent 60, just as with us a unit in the second place represents 10. Since that time many old Babylonian documents have been found containing these numerals. From the use of the sixty or sexagesimal system we get our minutes and seconds, both in the measurement of time and of angles. The Babylonians were the earliest scientific astronomers, and it was through EARLY FORMS OF NUMERALS 9 astronomy that the degrees and minutes were transmitted to Greece and thus to all of Europe. Babylonian multiplication. We know less about Babylonian arithmetic than about Egyptian because of the early Egyptian, textbook of arithmetic, written on papyrus, found in an Egyptian tomb, while no similar treatise has been found among the many clay tablets deciphered. However, parts of an old Babylonian multiplication table have been found and many hundreds of tablets going back as far as 3000 b.c. to 3500 b.c. which contain numerals. The Babylonian multiplication table, since the system is a 60 system, extended up I to 59 times 59. However, the tables did involve some simplifications. The table of 18 begins 18X1, 18X2, and so on to 18X19, 18X20; then the u e , tv-n,3M tables give 18X30, 18X40, and 18X50. TJy.(JffipJ Evidently 18X58 would have been obtained as 18X50 added to 18X8. Even with these simplifications the table was difficult, and the series of tablets needed by a computer were too heavy and awkward to be carried about in the pocket. BABYLONIAN TABLET Prom Temple Library at Nippur, c. 1350 B.C. Multiplication table of 18X1. 18X2, etc. In the central column 2, 3, 4,5, down to 11 in the lowest line; at the right, in column form, 18, 36,54,72(60,10,2), . . . BABYLONIAN TABLET Table of squares, c. 2200 B.C., from Nippur. Central column gives 30, 31,32___to 39. At the left are the squares 15 for 900 (15x60);961insecond Iine(16.1orl6x60 + 1). 10 THE HISTORY OF ARITHMETIC Babylonian curvilinear numerals. A second place system of Babylonian numerals was devised, using the blunt circular end of the stylus; in this the crescent was used for a unit with the complete circle for 10. These curvilinear numeral forms were used more than five thousand years ago in the same documents with cuneiform characters, somewhat as we use Roman and Hindu-Arabic CURVILINEAR NUMERALS Sumerian clay tablet, c. 2500 B.C., in the Harvard Semitic Museum. In the center the number 6; below this 24 indicated by two circles and four half circles. numerals. The cuneiform type of numerals was always used for the number of the year, for the age of an animal, j n >� il ff> � 8� > 00^> 12 3 4 5' 10 30 81( = 60+21) BABYLONIAN CURVILINEAR NUMERALS and in stating that a second or third payment has been made; it was regularly used for the number of animals in accounts concerning the allotment of food. In tablets EARLY FORMS OF NUMERALS 11 giving wages it appears that those actually paid were written in curvilinear and wages due in cuneiform. The cuneiform characters are also found placed horizontally. Furthermore, on some ancient tablets a system of representation of 100 appears, and on other tablets separate symbols for 600, 3600, 216,000. It must be remembered that what we have somewhat loosely designated as Babylonian civilization covers a period of more than four thousand years and includes at least three historically distinct civilizations. Babylonian interests. The Babylonians were the most careful bookkeepers of antiquity. The detailed records of ordinary things bought and sold, together with the wages of laborers, including men, women, and children, give us a somewhat comprehensive idea of economic conditions in ancient Babylon. The Babylonians were interested in the mysticism of numbers and in astrology. These interests stimulated them to study both arithmetic and astronomy, so that their priests were able to teach science as well as mysticism to the Greek students who came to them. Greek Numerals Archaic Greek numerals. The earliest Greek numerals do not come from Greece proper but are found in excavations on the island of Crete. These antedate by five or six hundred years the Golden Age of Greece. 0 \\// *:* * �� "a y. i v 1000 200 200 50 10 4 4 1 1 }i M1NOAN OR CRETAN NUMERAL FORMS OF HIEROGLYPHIC TYPE The illustration follows the forms given in the work by Sir Arthur Evans, The Palace of Minos (London, 1921), p. 279. 12 THE HISTORY OF ARITHMETIC Initial letter numerals. In the time of Thales (624-547 b.c), the first Greek mathematician known to us by name, and for centuries thereafter, the initial letters of the words for five, ten, one hundred, one thousand, and ten thousand were used to represent the corresponding M" X 1 H A 7T�r 10,000 1000 ] L00 10 5 livpia XiXia Ikoltov deKa irevre English words: . , kilometer hectare decimal pentagon myriad 7 ,7 kilogram Four straight lines were used to represent 4; four symbols A for 40. The 5 symbol was combined with the higher symbols to give 50, p or (^ 500, 5000, and 50,000. r phh rxxx r 50 700 8000 50,000 Alphabet numerals. About five hundred years before the Christian Era, 500 b.c, a new and more compact system of number symbols was intro-. duced into Greece. The first nine letters of the Greek alphabet were used for 1 to 9; the next nine letters were used to represent 10, 20, etc., to 90; the final group of nine letters was used for 100, 200, 300, etc., to 900. Three older letters not found in the present Greek alphabet were introduced to make Y fa ANCIENT SYMBOL For "Drachmas 1000" in Elephantine papyrus, c. 300 B.C. EARLY FORMS OF NUMERALS 13 the necessary 27 letters. By placing a stroke before a letter it multiplied the number represented by 1000; thus $ represented not 2, but 2000. The older myriad symbol (for 10,000) was frequently used with this system; thus A for 20,000. a '0 7 0 � F I f 0 1 2 3 4 5 6 7 8 9 t K X M f * 0 IT 9 10 20 30 40 50 60 70 80 90 P a T u 100 200 300 400 500 600 700 800 900 Hebrew letter numerals. The Hebrews used this same system with Hebrew letters, and the Arabs continued its use up to 800 or 900 a.d. This system is more compact than is the initial letter (or Attic) system. But the multiplication table is much longer than with our numerals. Thus 2X3 = 6 2X30 = 60 20X3 = 60 20X30 = 600 would be j8 7w 7 f j8 t&v X J k tQv y � k t&v X % The numerical connection of these products is not evident in the letter products, making each one a separate thing to remember. Commonly when letters were written with numerical value a bar was placed over the letters to show that a number and not a word was intended. Occasionally a play on this system was used by giving instead of a name the number made by the letters. In the Bible in Revelation "the number of the beast" is given as 666, or more correctly 616; this refers probably to "Nero Caesar," spelled in Hebrew letters which can be made to total 14 THE HISTORY OF ARITHMETIC 666 (616). Thus the letters nar would have the number 20+1+300, or 321 would be the number of "kat." Greek tablets. The Greeks frequently wrote on a wax tablet with a sharp-pointed stick, the stylus. The ancient tablet resembles an old slate. Upon such a tablet a Greek child would write his letters; at least one tablet has been found on which probably a child wrote the multiplication table, beginning a a a, a (3 /3, . . . . Sometimes Greek geometers used a board covered with sand, in which figures were easily drawn and easily erased by smoothing out the sand. The great Archimedes is reported to have been engaged in drawing a diagram on the sand when he was killed by a soldier of Marcellus, after the fall of Syracuse. Arithmetic and logistic. Concerning the operations of addition, subtraction, multiplication, and division with Greek numerals no treatise has come down to us. The Greeks divided the subject of arithmetic into two parts. The one subdivision called arithmetica was purely theoretical arithmetic corresponding to our modern number theory; the other, called logistica, was devoted to computation. Greek practical arithmetic. An ancient scholion (commentary) on a work of Plato informs us that logistica "is useful in the relations of life and business"; also that it treats "the methods called Greek and Egyptian for multiplication and division, as well as the summation and decomposition of fractions.'' Similar information is given by Proclus in the fifth century a.d. The indication of continued Egyptian influence is seen both in the EARLY FORMS OF NUMERALS 15 multiplication and in the reference to fractions. The Greek letter numerals made computation difficult, which may explain the fact that the Greeks had no fondness for computing. Greek children had some drill in the multiplication table and in addition with their numerals, and also undoubtedly drill upon representing numbers with the fingers and upon an abacus with little stones. Speculative arithmetic. The speculative arithmetic of the Greeks engaged the attention of students over such a long period of time that it is worthy of attention as interesting from the pedagogical point of view. The terminology of present-day arithmetic and some current phrases bear evidence of the continued influence of the mystic element in numbers. Such expressions as "luck in odd numbers," "lucky seven," "come seven, come eleven," and "all good things are three" carry us back in spirit and even in content to the mysticism of numbers as practiced first in Babylon and then in Greece and Rome. Among the Greeks two divergent methods of treating the same arithmetical facts were followed even from the time of Pythagoras. On the one PCCCC 100 400 500 900 CO CD do k W 1000 1000 10,000 50,000 100,000 ORDINARY ROMAN NUMERAL FORMS The other tens and hundreds are built up in precisely the same manner by-addition, as indicated above for 60, 90, 400, 900. form was made because centum begins with the letter. However, in the early Latin the initial letter of centum was rather "K." The student does well to remember the connection between centum and such words as "cent," "century," and "centimeter." The ancient numeral for one thousand in Latin inscriptions resembles a Greek � placed horizontally. M is used occasionally as symbol for milia in the expression milia passuum, from which the word "mile" is derived. M is EARLY FORMS OF NUMERALS 21 not used in early inscriptions as a numeral in combination with the other symbols. The symbol for fifty in the early f|X IX XvL XL L XXC XC 8 9 40 40 50 80 90 CD D CC<$> VTI JxJ 400 500 800 7000 1,000,000 ROMAN NUMERAL FORMS The forms in the above two lines are rather less commonly used than those on page 20. These forms and those on page 20 have been copied directly from photographs of Roman monumental inscriptions. forms is only rarely an L; similarly, the symbol for five hundred is not a D but rather one-half the symbol for one thousand. |\\ � �i XH.CJV'UMu 30,000 SESTERTII AS FOUND ON AN EARLY ROMAN MONUMENT Variant Roman forms. Thousands were occasionally indicated by a bar above a given numeral; XVIII for eighteen thousand. From the time of Hadrian inscriptions are found which indicate thousands by a bar above and vertical bars at the side, but in general this notation was used for hundred thousands. Thus! �>/ for one million or ten times one hundred thousand. The method of writing millions and other large numbers in the time of Caesar and Cicero varied; in general, writers employed the words in full, in the form of "thou-ands of thousands" and the like. 22 THE HISTORY OF ARITHMETIC Subtraction in symbols. The subtractive principle was employed by the Romans as a convenience when space MILESTONE AND SIGNBOARD GIVING DISTANCES FROM RHEGIUM, 13U B.C. In the fourth line is our word "miles" in the ancient spelling meilia, followed by 51. The numeral at the end of this line is 83, written in the subtractive form because there was not room for LXXXIII. In the fifth line in the numerals for 74 and in the numeral forms which follow the subtractive principle is not employed. limitations necessitated abbreviations. The full forms, like XXXX for 40, are the common Roman forms. It is worth noting that occasionally the Babylonians used the subtractive principle in writing numbers like 18 and 19. The Latin words for 18 and 19 suggest the forms, reading duo de viginti and unus de viginti or "two from twenty" and "one from twenty." Roman arithmetic. Concerning Latin instruction in arithmetic in classical times we have nothing more definite than concerning the Greek. Again the use of fingers and EARLY FORMS OF NUMERALS 23 abacus were quite certainly taught to children. However, this instruction was not considered a fundamental part of the education, and contemporary discussion of it is only accidental and incidental. Roman fractions. The Roman fractions, as we shall note later, did leave an impress, albeit an unfortunate one, upon our system of weights and measures. Aside from this the civilization of Rome exerted only indirect influence upon mathematical science. The technical vocabulary of mathematics (see Chapter VII) largely traces back to Latin, but primarily because Latin continued so long the language of the schools. The early commercial arithmetic which reached its highest development in Italy certainly owes some of its practices and its attention to detail to the legal genius of the Romans which so profoundly affected European institutions. The Fingers and the Abacus Finger reckoning. An entirely different system of representing numbers is by use of the fingers (Latin, digiti). The Greeks used this type of representation and it is still used by savage races of Africa, by Arabs, and by Persians. In North and South America the native Indian and Esquimo tribes use the fingers, and many of their words for numbers refer to fingers and hands, just as the word "digits" traces to the use of fingers. In medieval Latin the word articuli was used to indicate pure tens or hundreds, referring to the joints of the fingers employed in the representation of tens. The illustration on page 24 shows the use of the fingers as taught in the early printed books on arithmetic. 24 THE HISTORY OF ARITHMETIC FINGER RECKONING This illustration is from the works of Noviomagus (Bronkhurst), De Numeris (Cologne, 1544). Similar illustrations appear in some editions of Recorde's arithmetic and in Paciuolo's great Italian treatise of 1494. The Venerable Bede wrote, probably early in the eighth century A.D., a treatise on the subject, explaining this system. Hundreds on the right hand follow the tens on the left, and thousands are like units on the left. EARLY FORMS OF NUMERALS 25 Two peculiarities are worth noting. Practically all people begin with the little finger of the left hand to repre- MODERN FINGER RECKONING The fingers of the left hand are used in this way on the floor of the greatest grain market in the world, the ^Chicago Board of Trade. Price is indicated always with reference to the last sale, by the hand held with palm toward the buyer and with palm outward for a broker trying to sell. A broker wishing to buy 5000 bushels of wheat at 106f holds his five fingers outstretched, palm in. Another broker nods acceptance; the buyer indicates 5000 by a single finger held vertically. sent one unit. Further, the American Indians generally count up to twenty using both fingers and toes, which when you are barefooted is quite easy. Among the Mayas of Yucatan twenty was the base of their number system, used similarly to the way that we use ten. The Esquimos and the Indian tribes along the west coast of North America use twenty generally as the base, but they do not carry the system as far as did the Mayas. Abacus reckoning. For large numbers the fingers were found inconvenient. A system of recording numbers by using small stones on an abacus was used by early Egyptians, Greeks, and Romans, and a variation continues in use today in China, Russia, and Persia. The Chinaman in America who runs a laundry or a store generally uses an abacus in the form of beads on a wire frame, 3 26 THE HISTORY OF ARITHMETIC MCXI f37^ ^o\< ^ �J o 1� \� o o o\ 1 llll J The Romans used on the abacus little stones or calculi from which we have the word "calculate"; similarly in Greek the words for "stone" and "to calculate'' have a common stem. A stone in the first column represents simply one unit; a stone placed in the second column represents ten; in the third column it represents one hundred; a stone in any column represents ten stones in the next column to the right. In adding two numbers on the abacus when you have ten stones in any one column you take them up and carry one stone over to the next column. Or in subtraction you may borrow one stone from the next column to the left to make ten in the right-hand column. On the Chinese suan-pan, meaning "reckoning board," and the Russian abacus, and the Japanese machine for computation are found beads strung on wires. In the Chinese form a rod separates each wire into two parts, one with five beads and one with one bead; the Japanese use five beads and two beads. A single bead from the two represents five units; thus 781 is represented by one bead from the five on the units' wire; one bead from the two and three from the five on the tens' wire; one from the two and two of the five on the hundreds' wire. ROMAN ABACUS WITH LITTLE STONES On this abacus is represented the number 3621. This abacus could be made with as many columns as one pleased. Pope Ger-bert (Sylvester II, c. 1000 a.d.) is said to have had one made with 27 columns. Counters marked with numerals 1 to 9 were sometimes used. 7 8 1 Only the beads employed are shown. EARLY FORMS OF NUMERALS 27 CHINESE SUAN-PAN Fundamental operations on the abacus. The earliest treatises discussing the fundamental operations upon the abacus date from the tenth century a.d. The details of ancient usage are not known. Division is the difficult operation; during the Middle Ages a method of division on the board by completing the divisor to one hundred or to a multiple of ten was introduced. This was called "golden division" as opposed to the ordinary, called the The number 27'091 is presented '' iron division." It is worthy of note that Chinese computers become so expert with the abacus that they can carry through long computations more rapidly than an expert computer can in writing. Decimal system universal. Herodotus pointed out that the use of ten quite universally as the base of number systems is undoubtedly due to the fact that we have ten fingers. In his day and even to the present day children have used their fingers for computation purposes, with this difference that for centuries among the Greeks and Romans formal instruction was given in the use of the fingers. Tangible arithmetic. Stones on an abacus were used as we have indicated by the Latins and by the Greeks, JAPANESE "SOROBAN" 1987654321 is represented at the right; 16 ill the two columns to the left of the center. RUSSIAN ABACU 28 THE HISTORY OF ARITHMETIC by Chinese and Russians in the form of beads on a wire, by medieval Europe in the form of counters thrown on lines, and such devices continue in wide use today among the Chinese, Persians, and Russians. Certainly more individuals are doing arithmetical sums by machinery today than by the processes of our arithmetic; possibly we might even exclude from the comparison the more recent widely used mechanical computers, which have taken such a burden from the shoulders of individuals condemned to endless computation. Possibly the schools of tomorrow will teach the use of machines to eliminate the work spent on multiplication tables and on long division. Chinese, Japanese, and Korean Arithmetic Ancient systems. The Chinese arithmetic lays*claim to an antiquity which approximates that of Egypt and Babylon. Although the dates are not established as precisely as might be desired, nevertheless the development of arithmetic centuries before the Christian Era appears to be certain. In large measure the early arithmetic of Korea and Japan was based upon the Chinese. Tangible arithmetic. The most striking fact concerning the early development of operations with numbers is that tangible methods of representing numbers were employed. Before the abacus or suan-pan appeared, the Chinese used little rods, called sangi, to represent numbers. These rods were adopted by the Japanese and the Koreans, who continue to employ the computing rods to the present day. Decimal system. The numerals were represented in powers of ten upon a checkered board, later replaced EARLY FORMS OF NUMERALS 29 by lines. After the zero was introduced zeros were used with the rods to indicate vacant places. In algebraic processes a stroke or rod placed diagonally across a number indicates that the given number is to be subtracted. The abacus. About the twelfth century the abacus was introduced into China, and that continues in use there until the present day. The Japanese later modified the Chinese suan-pan, making a more logically correct I W III till Hill T T � 1 - 123456 7 8 9 10 I IiT-T 19 8 6 17 CHINESE AND JAPANESE "SANGI," OR NUMBER RODS in \\\ mixxi xnxm xmi 123456 7 8 9 KOREAN NUMBER RODS Sometimes made of bones and called "Korean bones." instrument. The Japanese soroban is made with a sufficient number of columns so that two numbers can be represented upon the instrument simultaneously, as in multiplication or division; any column can be taken to represent units. Practical problems. The problems of the early Chinese arithmetic as found in the classical work entitled Chiu-chang are similar to those of -early India and to those of the Arabs in that mensuration and commerce and alligation and other practical topics receive attention in 30 THE HISTORY OF ARITHMETIC addition to abstract problems whose nature is concealed under the oriental phrasing. "If 5 oxen and 2 sheep cost 10 taels of gold, and 2 -oxen and 8 sheep cost 8 taels, what is the price of each? "1 A partnership problem is the following: "When buying things in companionship, if each gives 8 pieces, the surplus is 3; if each gives 7, the deficiency is 4. It is required to know the number of persons and price of the things bought."2 Oriental source of European problems. Unfortunately the date of composition ascribed by Chinese scholars, the first century b.c, appears to be based upon insufficient evidence. In any event, however, the Chinese had a real gift for numerical problems quite analogous to that, displayed by the Hindus and Arabs. The oriental source of many problems which appeared in Europe in 1202 in Leonard of Pisa's voluminous work on arithmetic is not to be denied. Not only the same type of problems as in the early Chinese and Hindu works are given by Leonard, but frequently precisely the same series of numbers, so that the oriental origin is evident. These problems were taken over by Italian arithmeticians and then from them by other Europeans. By this route the problems of ancient India and China found their way into* American textbooks. Native American Numeral Systems Maya twenty system. The Mayas of Yucatan had a highly developed twenty system. The Mayas had separate words for 20, 400 or 20X20, and for 8000 or 20X20X20. 1 Smith and Mikami, A History of Japanese Mathematics (Chicago, Open Court Pub. Co., 1924), p. 13. 2Mikami, The Development of Mathematics in China and Japan (Leipzig, Teubner, 1913), p. 16. O oo ooo oooo ooooo oooooo 12 3 4 5 6 0 oooo oooo ooooo oooooo oooo ooooo ooooo 7 8 9 10 AZTEC NUMERALS Primitive American numeral forms as found among the Aztecs in Mexico. (See the frontispiece.)......... __ _____ MAYA CODEX In column form in the lower right-hand corner is found the Maya numeral representing 8 years; 8 in the third line represents 8X360; 2 in the second line represents 2)<20 or 40; together this makes 8 X (-360+ 5) or8X365. At the left this is'doubled, giving 16:4:0; then added again, giving 24:6:0 which is written 1:4:6:0, in a vertical column. In the next numeral three dots were left out of the original, which should be 1:12:8:0. This continues above at the right. The other symbols represent months and days. 10 11 15 18 20 123 (6X20+3) 360 MAYA NUMERALS The Maya symbols proceed by powers of 20 except the third place: which represents not 20X20 but 18X20 or 360, which was more convenient for calendar purposes. 31 32 THE HISTORY OF ARITHMETIC In some early Mexican languages the word for 20 is "man"; for 10 it is "two hands," and for 5, "one hand." MAYA HIEROGLYPHIC NUMBER SYMBOLS The flag represents 20; thus 200 jars of honey are indicated. The spiked leaf represents/400; 2000 blankets are indicated; 400 covered baskets; 1200 open baskets. The epaulet-like symbol at the bottom of the second column stands for 8000 or 20X20X20; 8000 pellets of copal, used as incense, are indicated. All from the Mayan Codex of Mendoza, now in the Royal Library at Dresden. The Mayas paid a great deal of attention to the calendar, and so most of their numerals are in connection with months and days and years. The year consisted of eighteen months of twenty days each; five extra holidays were included at the end of each year. EARLY FORMS OF NUMERALS 33 Maya picture numerals. The Maya picture numerals suggest the hieroglyphics; indeed, their writing is probably a picture writing. Twenty baskets are represented by a basket with a flag flying from it; the flag is the symbol for 20. For 400 the symbol is a spiked leaf; for 8000 another symbol, representing something like an epaulet. Peruvian knots. The Peruvians used knots upon strings of different colors. A large knot represented ten; small knots represented units. This instrument is called a quipu. The American Indians did not live in great cities like the Aztecs, Mayas, and Peruvians, nor did they reach as high a degree of civilization. Hence the Indians did not develop any very extensive systems of representation of numbers. European Developments of the Abacus "Reckoning on lines." The use of the abacus led to another type of computation, called reckoning on lines. We sometimes speak today of "casting an account," which refers, as we shall see, to this type of representation of numbers. PERUVIAN QUIPU This numerical record represents probably a census of some district. Occasionally variously colored cords were used to represent men, women, and children respectively. Each cord here may represent some district or family. The final sum commonly appears on a single major strand. The writer is indebted to Dr. L. Leland Locke for the illustration. 34 THE HISTORY OF ARITHMETIC Little round markers or counters were made to use upon lines drawn upon a table called in German ein knf)u�/ Stofflfttfeh fcttf> Mem/ fut? We im$e anemic &wkv. SJJif tintm fetc^fen 93ifnfotcfycm I ^fa* �flwjtod 93flcpi/mi|f affe SBajfc imD�>Ktftier.jHcfcciu ^df Robert Recorde *s explanation of subtraction, employing counters. At the right is represented 8746; 1 is on the lowest line and 5 in the space above it, represented by one counter there; four counters on the second line give 40; two on the third line for 200, with one above for 500; three counters on the fourth line for 3000, and a counter in the space above for 5000. At the left, 2892; below, the first step in the subtraction has been performed. -���- -����- � - &!>wt&ali3(bcBmtQfubtraa: tytgrcatctt l�jf numbers flrtt (conttfitp to t&S fcfc Of tie pen) !ljat is tfy thoufand M i&te cpample: t%m> foje 31 ftiiDataottgC tfre choufends i,fo;toWcb 3 toWfipQto to manp from tfce fccond fummc (totere ate 8) ano (o rtmatatty t$tte tf, a* ftj(� example fljefofty. �----- from which we get the words "exchequer" and "check" or "cheque," and the expression "to check an account." 36 THE HISTORY OF ARITHMETIC The "reckoning on lines'' began in the thirteenth century and extended over all of Europe. Long after the invention of printing treatises on this subject continued to appear, usually together with the written arithmetic. The two methods were frequently contrasted in illustrations in early arithmetics as in the title page of Kobel's arithmetic of 1584 on page 34. The details of the operations involve largely the inevitable peculiarities due to the type of notation. Undoubtedly the work seems much more awkward to us than it actually was to one experienced with this form of representation of numbers. The long-continued use of the abacus and of this visual form of representation of numbers testifies to the usefulness of tangible and visual aids in instruction. Teachers do well to use such methods wherever possible in instruction. Bibliography for Supplementary Reading T. Eric Peet, The Rhind Mathematical Papyrus. Liverpool, University Press, 1923. H. V. Hilprecht, Mathematical, Meteorological and Chronological Tablets from Nippur. University of Pennsylvania Studies, Vol. 19. T. L. Heath, A History of Greek Mathematics. 2 vols. Oxford, 1921. J. F. Payne, "Natural History and Science/' in A Companion to Latin Studies, edited by Sir J. E. Sandys. Cambridge, 1921. Third edition. F. P. Barnard, The Casting-Counter and the Counting Board. Oxford, Clarendon Press, 1916. D. E. Smith, Computing Jetons. New York, American Numismatic Society, 1921. Yoshu Mikami, "The Development of Mathematics in China and EARLY FORMS OF NUMERALS 37 Japan." Abhandlungen zur Geschichte der Mathematischen Wissen. Vol. XXX. Leipzig, Teubner, 1913. D. E. Smith and Y. Mikami, A History of Japanese Mathematics. Chicago, Open Court Pub. Co., 1914. C. P. Bowditch, The Numeration, Calendar Systems and Astronomical Knowledge of the Mayas, Cambridge, University Press, 1910. L. Leland Locke, The Ancient Quipu or Peruvian Knot Record. New York, American Museum of Natural History, 1923. Cyrus Thomas, "Mayan Calendar Systems." I: U. S. Bureau of Ethnology, 19th Report (Washington, 1900), pp. 693-820; II: U. S. Bureau of Ethnology, 22d Report (Washington, 1904), pp. 197-305. Cyrus Thomas, "Numeral Systems of Mexico and Central America." U. S. Bureau of Ethnology, 19th Report, pp. 853-955. W. J. McGee, "Primitive Numbers." U. S. Bureau of Ethnology, 19th Report, pp. 821-851. See encyclopedias under Babylon, Cuneiform, Egypt, Greece, Hieroglyphic, Maya, Mexico, Peru. CHAPTER II THE NUMERALS WHICH WE USE TODAY Hindu Origin "And Viswamitra said, 'It is enough, Let us to numbers. After me repeat Your numeration till we reach the Lakh, One, two, three, four, to ten, and then by tens To hundreds, thousands.' After him the child Named digits, decads, centuries; nor paused, The round lakh reached, but softly murmured on, 'Then comes the koti, nahut, ninnahut, Khamba, viskhamba, abab, attata, To kumuds, gundhikas, and utpalas, By pundarikas unto padumas, Which last is how you count the utmost grains Of Hastagiri ground to finest dust; But beyond that a numeration is, The Katha, used to count the stars of night; The Koti-Katha, for the ocean drops; Ingga, the calculus of circulars; Sarvanikchepa, by the which you deal With all the sands of Gunga, till we come To Antah-Kalpas, where the unit is The sands of the ten crore Gungas. If one seeks More comprehensive scale, th' arithmic mounts By the Asankya, which is the tale Of all the drops that in ten thousand years Would fall on all the worlds by daily rain; Thence unto Maha Kalpas, by the which The gods compute their future and their past.' " .... " 'And, Master! if it please, I shall recite how many sun-motes lie 38 THE NUMERALS WHICH WE USE TODAY 39 From end to end within a yojana.' Thereat, with instant skill, the little Prince Pronounced the total of the atoms true. But Viswamitra heard it on his face Prostrate before the boy; 'For thou/ he cried, 'Art Teacher of thy teachers � thou, not I, Art Guru.' "* Reading of large numbers. Centuries before the Christian Era, Hindu writers showed a great fondness for calculating with large numbers. The Hindus carried the decimal numeration, naming of the successive powers of ten, far beyond that of any other people. In different parts of India varying names were used for some of the higher powers, but knowing the complete sequence the value of any unit was given by its place in the sequence; thus with units, tens, hundreds, thousands, the fourth name applies to the third power of 10, one thousand equals 103. The seventh name applies to the sixth power of ten, and similarly with every other name. In reading a large number in which every place is represented the Hindu read the name of each place in turn as indicated in the verses at the beginning of this chapter. In our reading of numbers we follow the early Arabs and the later Germans in grouping with reference to thousands and powers of one thousand; the Greeks grouped by myriads or ten thousands; the Hindu way calls attention to the place, or sequence value. A large number, like 8,443,682,155, is read according to these different systems in the English, Sanskrit, Arabic 1 This quotation from Sir Edwin Arnold's The Light of Asia indicates the prominent place given to arithmetic and numbers in the education of the Buddha. The passage suggests the problem discussed by Archimedes to indicate the number of grains of sand in the seashore. 40 THE HISTORY OF ARITHMETIC and early German, and in the Greek language as follows: English: 8 billion, 443 million, 682 thousand, 155. Sanskrit or Hindu: 8 padmas, 4 vyarbudas, 4 kotis, 3 prayutas, 6 laksas, 8 ayutas, 2 sahasra, 1 sata, 5 dasan, 5. Arabic and early German: Eight thousand thousand thousand and four hundred thousand thousand and forty-three thousand thousand, and six hundred thousand and eighty-two thousand and one hundred fifty-five (or five and fifty). Greek: Eighty-four myriads of myriads and four thousand three hundred sixty-eight myriads and two thousand and one hundred fifty-five. When every place is present we would know the number if we read (beginning with the units), simply: five, five, one, two, eight, six, three, four, four, eight. When certain powers of ten are not present, as in 3,080,046, we could read the number six, four, vacant, vacant, eight, vacant, three. Development of the zero. This idea of using a word and finally a symbol for a vacant order of numbers was most highly developed in India. However, quite early in Greece the initial letter of the Greek word for "vacant" was occasionally used in writing degrees, minutes, and seconds; in Babylon a zero symbol -fc. was introduced several centuries before the Christian Era. Even the Mayas of Yucatan had a similar idea with their twenty system. But only with the Hindus was the idea carried to its full logical development, to a place system of numbers in which any number, however great, can be expressed with the symbols for the first nine integers and a zero, and with all computations reduced to combinations indicated by these symbols. To make the system easily THE NUMERALS WHICH WE USE TODAY 41 applicable to computation it is essential that the nine unit symbols in a decimal system, or the nineteen in a twenty system, should be independent like the letter symbols, and not compounded. Thus ]f jf in Babylonian symbols or Z -- in Maya represent 2 and 10; not 61 and 105 (5 twenties and five) as they would in a pure place system. Each of these symbols occupies two places instead of one. The Sanskrit word for vacant is sunya, and this word was used in numeration with this idea. Later a symbol was developed for this, a dot or a small circle. The Arabs about 800 a.d., in arithmetical treatises, translated sunya, writing the Arabic word sifr, meaning vacant. This word was transliterated about 1200 a.d. into Latin, the sound and not the sense being kept, becoming cyfra and tziphra and zephirum. The difficulties in transliteration from a different alphabet are indicated by the "c," "tz,M "z," "ch" as in chiffre, "g" in gero, and "s" in sifr, all to represent the same Arabic letter. Various progressive changes of these forms have given us our words, "cipher'' and "zero." In early English and American schools "ciphering" meant computation; in French the word for digits is chiffres, and similarly in some other of the Romance languages. This double meaning of "cipher" appears also in the early printed works explaining our numerals, but not in the original Hindu works. Hindu word and letter systems. The Hindus were accustomed to put history and even astronomy and mathematics into verse. For the sake of the rime it was necessary to have several alternate ways of giving the numbers which were involved in the verses. The Hindus developed, probably before 600 a.d., a word-system of 4 42 THE HISTORY OF ARITHMETIC recording numbers in which the place idea is prominent. For one was used "Buddha,'' "sun," or "moon"; for two was used "twins" (jama), "eyes" (nayana), or "hands" (kara); "oceans" for four; "senses" (visaya) or "arrows" (the five arrows of Kamadeva) for five; seven by "mountain" (aga); eight by Vasu, the eight gods; and so on. In this system zero is given by ' * point" or " vacant'' (sunya), or by " heaven-space'' [ambara, akasa). To give the number two thousand eight hundred and five, the words senses, vacant, Vasu, twins could be used; or arrows, heaven-space, Vasu, hands; note that the units are given first. Another similar play on the place idea is found in a letter system used in southern India. The illustration will be with our letters, although the Sanskrit is better for this purpose, as it contains more consonants. In this system the consonants are given the values from 1 to 9 and 0 in turn, as follows: bcdfghjklm 1234567890 npqrstvwxy 1234567890 z 1 To give 1492 by words, you pick out of the consonants for 1, 4, 9, and 2, a group such that by inserting only . Q vowels you get full words. Thus "near ? lace " or " by real ice'' would represent 1492. J Apparently those Hindus who used this sys- * tern had a great deal of time to spend on thinking up words. THE NUMERALS WHICH WE USE TODAY 43 Modern Sanskrit numerals. The Sanskrit numerals as used now in India are of the following form: 1234567890 MODERN SANSKRIT NUMERALS An arithmetical work on bark, probably a thousand years old, found in India, contains numeral forms analogous to the Sanskrit. Although the date is somewhat uncertain it is without serious question the oldest Hindu arithmetical document extant. Included are a large number of problems requiring algebraic processes for their solutions and similar to later Arabic and European problems, particularly like many in Leonard of Pisa's work of 1202 a.d. The fraction forms are suggestive of Arabic. One problem reads: "One who purchases 7 for 2, sells 6 for 3. 18 is his profit. Say now, what was his capital?"1 Hindu processes and problems. The details of the fundamental operations as practiced by the Hindus will be touched upon in the later systematic discussion of the fundamental operations. No early Hindu treatise gives us detailed and clear accounts of the processes of the arithmetical operations. Hence we are not able to connect the methods of Arabic arithmeticians directly with Hindu sources to which the Arabs give credit. However, the explanations as far as given, and more particularly the topics and the content of the Hindu arithmetic, correspond to the work of the Arabs. iRudolph Hoernle, "The Bakhshali Manuscript," Indian Antiquary, Vol. XVIII (1888), pp. 33-48, 275-279, three plates. 44 THE HISTORY OF ARITHMETIC Even as early as Brahmagupta (seventh century a.d.) the systematization of arithmetic had attained to a high stage of development. Brahmagupta states: "He, who distinctly and severally knows addition and the rest of the twenty logistics, and the eight determinations including measurement by shadow, is a mathematician."1 BAKHSHALI ARITHMETICAL MANUSCRIPT The earliest Hindu manuscript on arithmetic employing the numerals with place value. The symbols within the checkered diagram represent: 13 6 30 1 45 1 1 3 2 30 1 A later commentator gives the list as follows: "Addition, subtraction, multiplication, division, square, square root, cube, cube root, five (should be, six) rules, of reduction of fractions, rule of three terms (direct and inverse), of five terms, seven terms, nine terms, eleven terms, and barter, are twenty arithmetical operations. Mixture, progression, plane figure, excavation, stack, saw, mound, and shadow are eight determinations." 2 1H. T. Colebrooke, Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhaskara (London, 1817), p. 277. 2 Colebrooke, loc. cit., p. 277. THE NUMERALS WHICH WE USE TODAY 45 Mahavir's arithmetic. The treatise by Mahavir some two centuries later (c. 925 a.d.) takes largely similar topics, but the number and variety of the numerical illustrations is greatly increased. Mahavir states the topics of arithmetic most engagingly, as follows: "With the help of the accomplished holy sages, who are worthy to be worshiped by the lords of the world, and of their disciples and disciples' disciples, who constitute the well-known jointed series of preceptors, I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are (picked up) from the sea, gold is from the stony rock and the pearl from the oyster shell; and give out, according to the power of my intelligence, the Sarasangraha, a small work on arithmetic which is not small in value. "Accordingly, from this ocean of Sarasangraha, which is filled with the water of terminology and has the arithmetical operations for its bank; which is full of the bold rolling fish represented by the operations relating to fractions, and is characterized by the great crocodile represented by the chapter of miscellaneous examples; which is possessed of the waves represented by the chapter on the rule-of-three, and is variegated in splendor through the luster of the gems represented by the excellent language relating to the chapter on mixed problems; and which possesses the extensive bottom represented by the chapter on area problems, and has the sands represented by the chapter on the cubic contents of excavations; and wherein shines forth the advancing tide represented by the chapter on shadows, which is related to the department of practical calculation in astronomy � (from this ocean) arithmeticians possessing the necessary qualifications in abundance will, through the instrumentality of calculation, obtain such pure gems as they desire."1 Bhaskara's arithmetic. The following problems from Bhaskara's Lilavati of the twelfth century have a truly familiar sound: l_M. Rahgacarya, The Ganita-Sara-Sangraha of Mahaviracarya, with English translation and notes (Madras, 1912), pp. 3-4. 46 THE HISTORY OF ARITHMETIC "Say quickly, friend, in what portion of a day will (four) fountains, being let loose together, fill a cistern, which, if severally opened, they would fill in one day, half a day, the third, and the sixth part, respectively?"1 "If three and a half mdnas (a measure) of rice may be had for one dramma, and eight of kidney beans for the like price, take these thirteen cdcintSy merchant, and give me quickly two parts of rice with one of kidney beans; for we must make a hasty meal and depart, since my companion will proceed onwards."2 Mahavir has numerous problems resulting in strings of units or zeros, with a kind of play on the zero. Thus the problems: "In this (problem) write down 3, 4, 1, 7, 8, 2, 4, and 1 (in order from the units' place upwards), and multiply by 7; and then say that it is the necklace of precious gems." Ans. 100,010,001. "Write down (the number) 142857143, and multiply it by 7; and then say that it is the royal necklace." A ns. 100,000,001.3 Hindu interests. Not only the topics but the methods and terminology of arithmetic are highly developed in all of the early Hindu arithmetical works. In all fundamental aspects the Hindu arithmetic corresponds to the modern subject much more closely than the same subject as developed by any other people before the year 800 a.d. Delight in computation in and for itself is evident throughout the Hindu arithmetics. This love for computation led them to expand the subject of business arithmetic by inverse problems under which our children continue to suffer even today. But the love of computing led the Hindu into other problems of algebraic and iColebrooke, loc. cit., p. 42. 2Colebrooke, loc. cit., p. 43. 3Rahgacarya, loc. cit., p. 11, THE NUMERALS WHICH WE USE TODAY 47 trigonometric nature. These developments were destined to play a great part in the further progress of mathematical science. 'n � mammBBBmmammammmtmmmmm V �' 1 � "f'r.:;.--?��.�'.� m;'.V*.'~�.�.?it e�4>JHiiw.w..�S>r. a?, a^^ �> c. -N".j .\\i. -��� ."^a-.-~- aj>?ni. ,V:ii- .�> -A..-.nj^.i lixi. MS. Royal, 15 B IX, Britiah Museum TWELFTH-CENTURY ALGORISM IN LATIN This is part of a single page containing a complete discussion of the new numerals, with the fundamental operations. The opening line reads: " Intencio Algorismi est (-5-) in hoc opere doctrinam praestare, procedendi, addendi, minuendi duplan-"; the second line continues: "-di, et mediandi multiplicand! et diuidendi per x karacteres indorum." In the fourth line the numerals appear: 0, 9, 8, 7, 6, 5 (Roman V) and below 4, 3, 2, 1. As in most manuscripts of this period, many abbreviations are used; the symbol like our division sign stands for est, a line through the stem of a "p" makes it per, while a line over a letter usually stands for "n." the ninth century wrote treatises on the Hindu art of reckoning. The earliest reference outside of India to the numerals which we now use is by a famous Syrian monk, Severus Sebokht, living in a monastery at Quenesre on the banks of the Euphrates. This Syrian was attempting to show that all science was not due to Greece. He said: "I will not now speak of the science of the Hindus . . . and of the easy method of their calculations and of their computations which surpasses words. I mean that made with nine symbols."1 Early Arabic arithmetics. The earliest systematic treatise on the new arithmetic which has come down to us is 1 Karpinski, "The Hindu-Arabic Numerals," Science, 1912, Vol. 35, pp. 969-970. THE NUMERALS WHICH WE USE TODAY 49 that by Al-Khowarizmi, a Persian scientist who lived in the ninth century of the Christian Era. Even his work has not been found in Arabic, but is preserved only in a Latin translation made quite certainly in the twelfth century. The English word "algorism" comes from the Latin form of his name. This word was long used to mean arithmetic with the Hindu-Arabic numerals; in French the form of the word was augrim, and this form of the word is used by Chaucer about 1350 a.d. and appears in English literature from the thirteenth century. The Latin translation of Al-Khowarizmi's arithmetic begins, "Dixit algorithm''; this word algorithm later came to be used as title for the subject. Arabic and Hebrew arithmetics. Numerous Arabic works on the new arithmetic appeared from the ninth to the fifteenth centuries. Several of these have been translated from Arabic into modern languages. The Arabs were good traders, so that the practical applications of the Hindu arithmeticians particularly appealed to them. The Arabs systematized their knowledge, making excellent textbooks on arithmetic and algebra which continued to influence European mathematics for several centuries. Jewish students in Spain were also quick to learn the new numerals. Jewish and Arabic writers show greater appreciation of the possibilities of the new arithmetic than the writers of the early European treatises. Among the famous Jewish arithmeticians was the Rabbi Ben Esra (Ibrahim ibn Ezra), whose fame was sung by Robert Browning; his treatise on arithmetic, preserved 50 THE HISTORY OF ARITHMETIC in a Hebrew manuscript, has been published with a German translation.1 Several of these early Arabic arithmetics have been preserved among the great European collections of oriental manuscripts. The Arabic arithmetics of Al-Nasawi2 and Al-Karkhi3 of the eleventh century a.d., of Al-Hassar4 of the twelfth, of Ibn Al-Benna5 of the thirteenth, and of Al-Kalgadi6 of the fifteenth century have been made accessible in translation and a few others have been summarized. A twelfth-century Arabic arithmetic. The introduction to the work of Al-Hassar is characteristically Arabic: '' In the name of God, merciful and compassionate. My Lord! Make easy (my task), oh thou Beneficent One. Speaks the teacher Abu Zakarija Mohammed Abdallah, known by the name Al-Hassar. Praise be to God, etc. . . . . " Al-Hassar continued with further calls upon Allah and also with interjected statements concerning the dependence of his work upon the writings of older scientists. Integers and fractions are treated in separate sections. The work on integers falls under ten subheads: numeration, notation, addition, subtraction, multiplication, denomination including the check by nines, division, halving, doubling, and extraction of roots. The fractions involve numerous complications peculiar to the. Arabs 1Silberberg, Das Buck der Zahl des R. Abraham ibn Esra. Frankfort, 1895. 2Woepcke, Journal Asiatique, I 6, pp. 491-500. 3Hochheim, Programm. Halle, 1878-1879. 4Suter, Bibliotheca Mathematica, Q 3, pp. 12-40. 5A. Marre, Atti delV Accad. Pont, de nuovi Lincei, XVII, pp. 289-319. 6Woepcke, Atti delV Accad. Pont, de nuovi Lincei, XII, pp. 230-275, 399-438. THE NUMERALS WHICH WE USE TODAY 51 which fortunately found little favor with their European translators. Arabic shortcuts. An interesting innovation is found in Al-Karkhi's Kafi fil Hisab, or "Sufficient concerning Arithmetic" (c. 1010 a.d.)- The author gives a number of shortcuts in multiplication when the multiplier is either an aliquot part of 100 or 1000 or near to such an aliquot part. In this text we find considerable material on mensuration of surfaces and solids, algebra with applied problems, but few applied arithmetical problems. ARABIC MANUSCRIPT ON HINDU ARITHMETIC This is a section from a treatise on compound proportion by the Arabic scientist and traveler, Albiruni, written about 1000 a.d. In the first column are the numerals 4, 2, 5, 30, 60; at the extreme right, 8, 6, 2, 20. Arabic business arithmetic. A more complete business arithmetic was written by Abu '1 Wefa (990-998 a.d.) which unfortunately has not yet been published. This work in seven sections of seven chapters each is entitled "Treatise of that which is necessary in regard to calculation for collectors and clerks." The chapter headings indicate a comprehensive commercial arithmetic with mensuration, exchange, and denominate numbers. The one striking omission in his work appears to be the subject of interest. 52 THE HISTORY OF ARITHMETIC European Translators op Arabic Scientific Works The Arabs in Spain. The Arabs entered Spain in the year 772 a.d. and continued in parts of Spain until shortly before the discovery of America. During all of this time there was much contact between the Arabs and the MS. Egerton 2261, British Museum Europeans. At the beginning of the twelfth century the fame of the Moslem schools at Toledo in Spain and in other Spanish cities had spread over all of Europe. The Crusades, also, aroused new interest in the Arabs and their teachings. In the twelfth century activity in the translation into Latin and Hebrew of Arabic scientific works reached its highest point. The greatest of the THIRTEENTH-CENTURY ALGORISM End of a work on the Computus (on the calendar) and beginning of an early explanation of our numerals. The first three lines of the algorism read: " Intendit algorismus in hoc opere primum docere, pro-cedere addere, subtrahere, dupli-care, mediare, multiplicare, diuidere per ix figuras yndorum que sunt huius modi 098765432 1." In the first line only one word is written out in full; all of the others have the abbreviations characteristic of Latin manuscripts made during the three or four centuries preceding the invention of printing. THE NUMERALS WHICH WE USE TODAY 53 translators was Gerard of Cremona, an Italian. This student of astronomy heard that there was a copy of the great Greek work on astronomy, Ptolemy's Almagest, to be had in Spain. He made the long and difficult journey from Italy to Spain in order to get this work. When he arrived in Spain he found the work, but it was in Arabic. Not to be daunted, Gerard with the help of a Jewish scientist who knew Arabic and Spanish, in which language they probably conversed, translated the Almagest into Latin. While doing this he learned of the great number of Arabic works of science, and he determined to devote his life to the translation into Latin of such works. Gerard spent nearly fifty years in Spain, translating medical, astronomical, philosophical, and mathematical treatises. Latin translations from Arabic. The earliest European treatises on the new arithmetic and on algebra were Latin translations made during the twelfth century. Robert of Chester, an Englishman, made one translation of the algebra of Al-Khowarizmi; another translation of the algebra was probably made by Gerard of Cremona. The arithmetic by the same author was twice translated in the twelfth century, possibly by Adelard of Bath and John of Luna, a Spaniard. Several other Latin treatises directly related to the translations appeared during the twelfth century. The total number of complete explanations of the new arithmetic written before 1300 a.d. and available today probably does not exceed twenty, published and still in manuscript. The early translations were exceedingly concise and required considerable explanation for people who were 54 THE HISTORY OF ARITHMETIC accustomed only to the Roman (rarely to Greek) numerals and the abacus. Early in the thirteenth century at least three more extended treatises on arithmetic appeared. This subject became a part of the curriculum in the universities which were just being formed in France, Italy, England, and Germany. Thirteenth-century arithmeticians. The Algorismus Vulgaris (common algorism) of John of Holywood or Halifax or Sacrobosco was the most widely used of these three treatises, and copies of it written by students of mathematics in the thirteenth to fifteenth centuries are found in many European libraries. The Liber Abbaci of Leonard of Pisa was the most extensive treatment; it was printed in Italy in 1852, more than six hundred years after it was written, and the sections on arithmetic cover some two hundred pages. The length explains why it never became popular. A Frenchman, Alexandre de Ville Dieu, wrote a treatise in Latin verse, Carmen de Algorismo, which was second in popularity only to Sacro-bosco's Algorismus. A German, Jordanus Nemorarius, wrote an explanation of the new arithmetic with demonstrations in Euclidean form, Demonstratio Jordani in Algorismo, but it did not become widely popular. All of these works were in Latin, which was the language of the universities and the language in which European scientific works were commonly written up to the eighteenth century. Three of the works cited above have the word algorism involved in the title, and Leonard of Pisa also uses the word. In the thirteenth and fourteenth centuries there were thousands of students in the European universities who THE NUMERALS WHICH WE USE TODAY 55 became familiar with the new arithmetic. Commonly the teacher would read Sacrobosco's treatise, line by 1$� toe �ft>fir>i1fetifte�ii mmam �& �J�o�itm �> waa� mtwi �.�{$� *�of tj qactitte ftjpt� p&a w <�Mj�oto^ te^usmtttasua., f*�eeiomti to� yftm �j$ tsaymw �% �gMWi,�jie #�i /figmft m? ft wro&gi , �a�. ��b low $>ftta.>,ft|lf xsam futtta %K�srf3?�r wtutftw. iitmw .����& :tv wme &w flaw otstne^^iBp wMfV trome*n,e>�tt t&M? ftm aotomo, SUtstufrnfo maunt avn^<^,^.h^1 � THIRTEENTH-CENTURY ARITHMETIC This is Sacrobosco's Algorismus as found in Codex Arundel 332, British Museum. The decorated column has the heading at the right, "De ad- dicione." The text of this reads: "Addicio est numeri ad numerum aggregatio . . . . " In the eighth line note the symbol like a -|- sign for et or "and." line, explaining each line as he went along. One of these extended commentaries made in 1292 by Petrus de Dacia, lecturing at Paris, has been published. The commentary covers about four times as many pages as does the text, which occupies some twenty printed pages. 56 THE HISTORY OF ARITHMETIC Early English Arithmetic Early treatises in the vulgar tongues. Latin continued until well into the eighteenth century as the language of instruction in European schools. However, occasionally treatises appeared in the vulgar tongues. The new arithmetic appeared in the fourteenth century in Icelandic, a translation of Sacrobosco; in the fourteenth century a discussion based on the Carmen by Alexandre de Ville Dieu appeared in French; in the next century a German treatise appeared. In English brief and incomplete discussions appeared in the fourteenth century, but no complete work is known earlier than the two fifteenth-century treatises discussed below. Even in general English literature of the thirteenth and fourteenth centuries references to the new arithmetic are extremely rare. The Ancren Riwle refers to the "nombres of augrym" and Langlois, in Rechard Redeles of 1399, mentions: "as siphre doth in awgrym, that noteth a place and no thing availith." First English arithmetics. The Crafte of Nombrynge (fifteenth century) is found in a single manuscript, Eger-ton MS. 2622, in the British Museum.1 This is a running commentary on the Carmen de Algorismo. It begins: "This boke is called the boke of algorym, or Augrym after lewder vse .... " , . . . fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, 0987654321."2 The other fifteenth-century treatise is a translation with commentary of Sacrobosco's Algorismus Vulgaris. This 1 Recently published by Robert Steele, The Earliest Arithmetics in English, Oxford Univ. Press, 1922. 2Steele, loc. tit., p. 3. THE NUMERALS WHICH WE USE TODAY 57 Art of Nombryng is found in another of the great English repositories of manuscripts, the Bodleian Library at Oxford, MS. Ashmole 396. (See illustration, page 107.) The opening words are: "Boys seying in the begyn-nyng of his Arsemetrike:�Allethynges .... ofnom- f$~*(*bif fmb&u. Awiii ***? *f*"*ff fignptt "THE CRAFTE OF NOMBRYNGE," C. I45O A.D. In this earliest treatise on arithmetic in English the first two lines are in Latin, a quotation from the verses of Alexandre de Ville Dieu. The Latin begins: " Hec algorismus ars praesens dicitur . . . ." The English begins: "This boke is called the boke of algorym," etc. The last line reads: "for ye latyn word of hit is Algorismus comes." bre ben formede." A little farther on the writer says: "So algorisme is clepede the art of Nombryng . . . . " "Boys" stands for Boethius, whose arithmetic, largely a translation of the Greek work of Nicomachus, was in wide use both before and after the introduction of the Hindu-Arabic numerals. Printing of arithmetical works in England. In English the earliest printed discussion of arithmetic was published by the first English printer, William Caxton, in the first 5 58 THE HISTORY OF ARITHMETIC English work containing illustrations, The Mirrour of the World. On the page discussing arithmetic all of the new This brief discussion of arithmetic is based upon a thirteenth-century encyclopedic work by Vincent de Beauvais. "The fourth science is called arsmetrique this science cometh after rethoryque, and is sette in the myddle of the vii sciences. And wythout her may none of the vii sciences parfyghtly ne weel and entierly be knowen." The numerals are indicated on the tablets or scrolls, and a heap of counters is on the table. Caxton was the first printer in England. forms, apparently not fully comprehended by the illustrator, appear on a kind of horn book, but they are not placed so as to make a rational problem; some forms are reversed. No reference is made to these forms in the text, nor is there anything except the most general praise of arithmetic. "The fourth scyence is called arsmetrique .... Who that knewe wel the science of arsmetrique he myght see thordynance of alle thynges." The seven liberal arts THE NUMERALS WHICH WE USE TODAY 53 include grammar, logic, and rhetoric, together called the trivium; and arithmetic, "the fourth science," with geometry, music, and astronomy, called the quadrivium. ONE OF THE EARLIEST PRINTED ARITHMETIC TEXTS IN ENGLISH, 1539 There was printed in 1537 a somewhat similar work. Reeorde's arithmetic appeared five years later and became popular. The earliest printed works in English which explain the new numerals are two anonymous treatises of 1537 and 1539. The earlier one is entitled: '' An Introduction for to lerne to reckon with the Pen and with the Counters after the true cast of Arsmetyke, or Awgrym"; it was published at St. Albans. The other work, published in London, is entitled, as may be read above: "An Introduction for to learne to recken with the pen, or with the 60 THE HISTORY OF ARITHMETIC counters.*' The latter of these was published in a second edition in London in 1546, and there were later editions. The English treatise which is primarily concerned with the popularization of our system of numerals and computation is Robert Recorde's The Grounde of Artes, which appeared first about 1542 and in twenty-seven further editions up to 1699. So far as America is concerned English texts were long imported, as well as Spanish, Dutch, French, and German. The first separate English text on arithmetic in the United States appeared in Boston in 1719, but it was preceded by Spanish works by more than a century. Bibliography for Supplementary Reading D. E. Smith and L. C. Karpinski, The Hindu-Arabic Numerals. Boston, Ginn and Co., 1911. G. F. Hill, The Development of Arabic Numerals in Europe. Oxford, Clarendon Press, 1915. G. R. Kaye, Indian Mathematics. Calcutta, 1915. H. T. Colebrooke, Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhaskara. London, 1817. M. Rangacarya, The Ganita-Sara-Sangraha of Mahaviracarya. Madras, Government Press, 1912. L. C. Karpinski, " Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi." University of Michigan, Humanistic Series, Vol. XI, Part I. New York, 1912. Augustus De Morgan, Arithmetical Books from the Invention of Printing to the Present Time. London, 1847. J. O. Halliwell, Rara Mathematical or, A Collection of Treatises on the Mathematics. London, 1839. D. E. Smith, Rara Arithmetica. Boston, Ginn and Co., 1908. Bibliography of printed arithmetics to 1601 with special reference to those in the library of Mr. George A. Plimpton in New York. Charles H. Haskins, Studies in the History of Medieval Science. Cambridge, Harvard University Press, 1924. CHAPTER III THE TEXTBOOKS OF ARITHMETIC Egyptian Egyptian textbook. The first systematic treatise on mathematics is the Ahmes papyrus, which represents the type of instruction given in Egypt nearly four thousand years ago. The work was evidently designed as a textbook; the occasional use of red ink suggests, in fact, a modern teacher's corrections. Both the problems and the methods employed in this ancient manual continued to appear in Egypt *v ��� for centuries, in Greek arithmetic up symbols for addition , 1nAA j � T ,. , ,. AND SUBTRACTION to 1000 a.d., and even m Latin treatises ^ . ,, , These symbols repre- of the thirteenth century. Between sent a pair of legs waik- . ing in the direction of the Egyptian work and the Greek the writing for addition, , ,. .... ,. .. nr, and reversed for sub- treatises on arithmetic nearly fifteen traction. (From Rhind hundred years intervene. The separa- papyrus,) tion of arithmetical material and geometrical material on mensuration into a single work as done by the Egyptians constitutes a notable step in the progress of science and civilization. Greek Arithmetic Euclid's "Elements." The first great mathematical textbook of the Greeks is Euclid's Thirteen Books of the Elements, the Geometry written about 320 B.C. and continuing in active use almost to the present day. This work contains in books seven, eight, and nine a treatise on theoretical arithmetic, numbers being represented by 61 62 THE HISTORY OF ARITHMETIC geometrical lines. In this work no explanation is given of the fundamental operations but rather properties of numbers now treated in number theory {see Chapter I). The proofs are by the rigid logical processes of the Greek geometry. Problems analogous to finding the greatest common divisor of two or more numbers and the least common multiple are treated incidentally, but not applied to fractions as in our arithmetic. The fraction idea is treated under proportion and found application in the theory of music, long regarded as a mathematical science. Speculative or theoretical arithmetic. The Greek arithmetical treatise by Nicomachus, translated into Latin by Boethius, continued in active use well into the seventeenth century, and was used in European church schools almost exclusively for the subject of arithmetic in the tenth, eleventh, and twelfth centuries. Two distinct types of speculative arithmetic were current, the Boethian and a mystical arithmetic involving contemplation of the numbers appearing in the Bible. Long after the invention of printing both types continued to flourish. The Boethian type in diluted form, more verbose and even less mathematical than the original, is represented by the treatises of the other Romans, Martianus Gapella (c. 410 a.d.) and Cassiodorus {c. 470-c. 564 a.d.). Slightly better is the Arithmetica Speculative!, published in 1495, written by Thomas Bradwardine {c. 1290-1349), professor at Oxford and later Archbishop of Canterbury, and the Tractatus proportionum of Albert of Saxony (c. 1330), published in 1470. In many later textbooks of practical arithmetic the Boethian number theory was given as introductory to the practical work. THE TEXTBOOKS OF ARITHMETIC 63 The chapter headings indicate the speculative character of the material. Chapter eight takes up the division ^ of even numbers (Diuisio parts numeri) followed by a discussion in chapters nine, ten, and eleven of "evenly-even" numbers, "evenly-odd," and of "oddly-even" numbers. "Evenly-even" are numbers of the form 2n; that is, 32; when divided repeatedly by 2 these lead finally to unity. THE ARITHMETIC OF BOETHIUS FROM HIS "OPERA OMNIA, PUBLISHED AT BASLE IN 1493 64 THE HISTORY OF ARITHMETIC Mystical arithmetic. The second type of speculative arithmetic is well represented by the Mysticae Numerorum by Petrus Bongus of Bergamo, Italy, which appeared in 1583-84 and enjoyed seven editions. Some 400 pages are devoted to the discussion of the numbers from one to ten, in each case with copious references to Biblical material. A beginning was made along the same line about a thousand years earlier by Isidore of Seville, born in the year 570 a.d., who wrote further a short extract along Boethian lines in his Origines or Etymologies. Hindu and Arabic Texts Hindu texts. The Hindu treatise on arithmetic by Aryabhata consists of 108 couplets of verse, intended to be memorized and requiring extensive interpretive explanation. Brahmagupta in the seventh century, Maha-viracarya in the ninth, and Bh&skarain the eleventh wrote quite extensive systematic treatises including arithmetical and algebraical material and mensuration. The Arabs drew both method and content of arithmetic and algebra from Hindu sources, but the precise texts which they utilized are not known. Arabic texts. The Arabs rendered a great service to the progress of civilization by writing admirable textbooks on arithmetic, algebra, and trigonometry, and on other subjects as well. Between 750 a.d. and 1450 a.d., some five hundred Arabs wrote treatises, whose fame at least has survived, on mathematical and astronomical subjects. So far as service to elementary mathematics is concerned the Persian Mohammed ibn Musa al-Khowarizmi, who lived at Bagdad in the ninth century, enjoys the greatest THE TEXTBOOKS OF ARITHMETIC 65 distinction. His arithmetic in Latin translation, with his name Algorismus concealed as a title, continued in direct use until the fourteenth century, while his algebra was used in Latin translation until the sixteenth century. The Arabs had real ability in devising proper problems and methods for instruction. Extended lists of their algebraical and arithmetical problems appeared in the work of Leonard of Pisa in 1202, revised by him in 1228, THE "CARMEN DE ALGORISMO" OF ALEXANDRE DE VILLE DIEU Verses from the Carmen of Alexandre de Ville Dieu: "Hec algorisimus ars praesens dicitur in qua . ...0"9-8"7-6-6'4-3-2-l. The following section gives a fourteenth-century commentary. and also in the Summa d'Arithmetica of 1494 by Luca Paciuolo, which includes the first printed algebra. For further centuries numerous other mathematical texts continued to present the Arabic material, frequently quite unaltered. European Textbooks after 1200 Two popular texts. The two most popular European textbooks on the new arithmetic with the zero were undoubtedly the Algorismus Vulgaris of John of Halifax (Sacrobosco) and the Carmen de Algorismo of Alexandre FIRST PRINTED ARITHMETIC The Treviso Arithmetic of 1478. The first separate treatise on practical arithmetic was printed at Treviso in Italy in 1478. The author is not known. The blank spaces are left for ornamental initials. The author states that he writes the work at the earnest solicitation of numerous students. 66 THE HISTORY OF ARITHMETIC de Ville Dieu, both written during the first half of the thirteenth century. Until the invention of printing THE TEXTBOOKS OF ARITHMETIC 67 countries. Undoubtedly commercial activities stimulated both the appearance and the use of these textbooks. "SJECOTtjys ARITHMETICS OR, The Ground of Arts: TEACHING The perfed work and pra&ife of Arithmetic^, both in whole Numbers and Fractions, afcer a more eafie and exact form then in former time hatli been fee forth: Made by Mr. Robert Record, D.inPhyiick. j4fterrvard augmented by Mr. John Dee And fince enlarged with a Third part of Rules ofPra*. ftife, abridged m:o a briefer method then hitherto hath been publt-Xied, with divers neceflary Rates incident to the Trade 6f Merchnr* dife: wich Tables of the Valuation of all Coynes, as rh�-y are currant at this prelent time. By John MellU. And now diligently perufed, Corrected, Illuftrated and Enlarged J with sn appendix of figurative Numbers, and the extra�i-on of their Roots, according to the method of ClniflianX/ifiiM: yi'nh Tables of Board" and Timber meafurcj and new Tablej of Iritereft, after to. and 8. per too. withthetiue value of Annuities to be bought or fold, prefent, Refpitcd,or in Reverfioa : The fmi cajculated by R.c. but corrected .And the latter diligently calculated by Ro. H&tmll. thilomathmat. Scicntia mnhabct in'mkitm wp itnorantcm. Fide--------fed--------Vide L O tt T> O U Printed by James Fiefhcr, and are to be fold by F.dw.trd Dad X �h�" figneof the Gunin Ivi�-fane. 16^. The first widely popular compendium of mathematics is the so-called Summa d* Arithmetica of Luke Paciuolo of 1494; the Liber Abbaci by Leonard of Pisa, written in 1202, is of similar nature, and there were several others written in Latin in the fourteenth and fifteenth centuries. In the encyclopedic works of the same period the treatment of mathematics is generally quite fragmentary, not sufficient to be counted as texts. The material on arithmetic in Caxton's M in our of the World {see page 58) TITLE PAGE OF RECORDE'S ARITHMETIC The first edition of Recorde's popular arithmetic appeared about 1542. Authors had a weakness for the type of title here shown. Recorde entitled his astronomical work of 1556 The Castle of Knowledge; his geometry was called The Pathwaie to Knowledge; his algebra was entitled The Whetstone of Witte (London, 1557). Editions of the arithmetic continued to appear until 1700 a.d. 68 THE HISTORY OF ARITHMETIC ILLUSTRATED PROBLEMS, "ON REVERSED QUESTIONS," FROM KOBEL'S ARITHMETIC OF 1584 The first problem concerns a barrel of herring bought for 18 "alb," containing 180 herring. How many herring fori "alb"? The second problem involves the Rule of Three, or "Regel de Tri." is an excellent illustration. The Margarita Philosophica by Gregorius Reisch (1501), a sixteenth-century encyclopedia often reprinted, contains an excellent treatise on arithmetic. Between the invention of printing and 1500 a.d. there appeared some thirty practical arithmetics of which more than one-half were in Latin, seven in Italian, four in German, and one in French. During this period there appeared about twenty-six editions in Latin of the theoretical arithmetics, along Boethian lines. Up to 1514 the arithmetics in Latin greatly exceeded all others, THE TEXTBOOKS OP ARITHMETIC 69 but in 1514 Kobel's Rechenbiechlin in German marks the opening of the era in which the mother tongue began to be used in instruction. There were over thirty sixteenth-century editions in German of Kobel's three books on arithmetic. The Italian commercial arithmetic of Borghi of 1484 passed through seventeen editions up to 1577, while the work of 1515 in Italian by Girolamo and Giannan-tonio Tagliente achieved thirty-six editions up to 1586. Adam Riese's arithmetics beginning in 1522 eclipsed all others in the vernacular, having more than forty sixteenth-century editions and several of the seventeenth century. Strangely enough, the arithmetic to enjoy the greatest number of editions in the sixteenth century, over sixty, was a Latin treatise by Gemma Rainer Frisius (1508-1555), a Dutch physician; this Arithmeticae practicae methodus facilis appeared first in 1540, with some later editions modified by a Frenchman, Jacques Peletier of Mans, and other editions modified further by the German, Jacob Stein. TITLE PAGE This arithmetic by Gemma Frisius was the most popular treatise in Latin during the sixteenth century. 70 THE HISTORY OF ARITHMETIC Fifteenth-century arithmetics. The first printed arithmetic is an anonymous work of the commercial type which appeared in Tre- '�.......................~.........�"" - '��."�� ARITHMETIC!: 5 PRAXIS, AD t)VAM '.I TtRVM I'RKMVi.'i A 1 X- 1478, writ- s'. ,i >!' �,' \K* �!. i ;� M �lu Jolt 't �*- ' ? , ^, , ���;�' - � \, �> ) t mm viso, Italy, in ten in Italian. The first printed Latin treatise on the new numerals was written by Pros-docimo de Beldamandi' (died 1428), a professor at the University of Padua, Italy. This was printed. in 1483 at Padua with the title: Prosdocimi de belda-mandis algorismi tractatus perutilis, etc. The work includes the first treatise in print on fractions, by Johannes de Liveriis, a Sicilian astronomer (c. 1300-1350); here is found the designation "de minutijs tarn vulgaribus quam physicis,'' having reference to vulgar fractions and astronomical minutes and seconds. Preceding this work by one year comes a German treatise, printed at Bamberg; the author was Ulrich Wagner, a Nurnberg Rechenmeister, or professional teacher of reckoning. The first illustrated arithmetic appears to be by Johann Widman of Eger, a German commercial work published at Pforzheim in 1489. LDV 'in i T:\ TITLE PAGE This arithmetic appeared at Louvain one of the centers of mathematical learning during the sixteenth century. THE TEXTBOOKS OF ARITHMETIC 71 ,F. RA'MI ARITHMETICS LIBRI DVO, Cuiw Cdmrncnwriis � Sixteenth-century arithmetics. About nine hundred arithmetics appeared during the sixteenth century; they are listed and described in David Eugene Smith's Rara Arithmetica, which is a descriptive catalogue copiously illustrated of the great collection of arithmetics gathered by Mr. George A. Plimpton of New York.1 Of the nine hundred arithmetics about four hundred are in Latin, and of those one-quarter are of the Boethian type of number theory, or on mysticism of numbers. About two hundred commercial arithmetics appeared in German, followed closely in number and content by Italian arithmetics. The German writers OFJnCJKA PtAHTtK tAWjT ; HJP.tfBLE N Gtr,: ,'� "'- .. M. D. C XII if' V' TITLE PAGE The widely popular arithmetic of Peter Ramus, with commentary by Snell, who first stated the law of refraction. The work is the product of the , , . 1 <� i rrn �. famous Plantin Press; the compasses date largely from 155U tO constitute a part of the printer's device 1600, whereas the Italian ortrademark-writers are largely of the earlier period. Similarly the English (45), French (40), Spanish (25), and Dutch (15) arithmetics are almost exclusively from the second l Mr. Plimpton with great generosity has placed this wonderful library at the service of many workers in the field of the history of science; the writer is deeply indebted to Mr. Plimpton for many courtesies over a long period of years. 72 THE HISTORY OF ARITHMETIC half of the sixteenth century. During this latter period the production of arithmetics in Latin dropped to less than one-quarter of the total number as opposed to one-half the total during the preceding part of the century. 168 TWO PAGES FROM BAKER'S ARITHMETIC Baker's arithmetic was second in popularity to Recorde's treatise, works were largely based on Italian commercial arithmetics. Both In the seventeenth century the arithmetics in Latin diminished in use and in number published. The total number of arithmetics published during the seventeenth century, including editions, would doubtless approximate two thousand. THE TEXTBOOKS OF ARITHMETIC 73 ARITMETICA PRATTICA COMPOSTA DAL MOLTO Reueren. Padre Chriftoforo Clauio Bamberger^ nella Corapa- gniadiGIESV. E* tradotta da Latino in Ualiano dalS.Lorenz9 Carte llano Patritio Romano* Rcuifta dal medefimo Padre Clauio conalcuneaggiunte. Seventeenth and eighteenth centuries. Popular interest in arithmetic and general instruction in the subject increased so rapidly after the sixteenth century that hundreds of books appeared to supply the new demand. The continental texts were, on the whole, more scientific than those employed in England. The encyclopedic works touching all subjects remotely connected with mathematics were common. In England one popular compendium for self-instruction included arithmetic with reading, writing, bookkeeping, with instructions for carpenters, bricklayers, and the like, and with dialing, pickling, and a list of fairs in England. The English texts directly fashioned the American arithmetic; in fact, the majority of texts used in America up to 1810 were either imported from England or were American reprints of English works. Recorde and Baker continued in use during the seventeenth century, being gradually supplanted by treatises 6 IN ROMA, Per Guglielmo Facciotti. M. D C. X111. CON LICENZADE'SVBfiiUORU TITLE PAGE The Jesuit, Christopher Clavius, wrote a whole series of treatises on elementary mathematics and astronomy. This work appeared first in Latin at Rome in 1583, and enjoyed numerous editions. 74 THE HISTORY OF ARITHMETIC devoting more attention to decimal fractions and to new commercial problems, and less attention to counters. Among the early successful texts of this class may be placed that with the title: Mr. Blundevil: His Exercises THE T Vulgar and Decimal,in all its Ud-ful Rules; With general Method of Eurafting the Roots of all Single Powers. II. 3Igebjfl, or Arithmettck in Species; wherein the Method of Rating and Relblving Equations is rendered Ealie > and lllu* ftrated with Variety of Examples, and Numerical Queftions. Alfo the Whole Bufinelsof Intereft and Annuities, ere fully and plainly Handled, with feveral New Improvements* III. The Clement* of Ctomctrp, ContracM, and Anatyti,. cally Demonftrated; With a New and Ealie Method of finding the Circle's Periphery and Area to any afligned EKactnefs, by one /Equation only; Alfo a New Way of making Sines and Tangents. IV. CarUtfejfeeCtionS, wherein the Chief Properties &e. of the EUiplis, Parabola, and Hyperbola, are Clearly Demonftrated. V. The arithmetic!* of anfinttPfl Explain'd, and render'd Eafie; with its Application to Superficial, and Solid Geometry. With an Appendix of tactical <&Xttgtog� By JOHN WARD. Teacher of the ^afhematitba. Heretofore Chief Surveyor and Cauger Gener/il in the Excife. LONDON: Ptinied by Edvo. Midwinter, for John Taylor at the SbJj> in St. PauriQburcb-Tard. 1707. ____________________________Trice Bound 61. Mr. Wingate's Arithmetic^' CONTAINING A PLAIK and FAMILIAR METHOD! For Attaining the Knowledge and Pra&ice O F Common Arithmetick< Compofcd by ED MV ND IVI NO ATE of Grays-lnrt, Efcj, And upon his Requeft Enlarged in his Life-rime, alio fince hisDeceafe carefully Rerifed and much Improv'd, as will appear by the Preface and Table of Contents. By fOJ/N I^E^SEr, late Teacher of the Ma-theroa ticks. title Clrfaenn) (Eaitfom With a nrto feupplenWIlt, Of EafwContractu* Id the liccdiary Parts of Arithmetic!:, ufefttlj Tables of Intereft and Flcmijl) Exchanges, as alfo Practical Me rijurat ton. By GEORGE SHELLET, Writing-Mailer at the Hand and Pen ill Warwick-Lam. near St. P&als. LONDON: Printed for J. Philipt, at the Kings-Asms J. Taylor, at the Ship ; and J. Kimfton, a the Crown, in St. JWs Chorch-yard. 17^4' TWO TITLE PAGES Two works widely popular in England and imported in large quantities by New England booksellers. In Harvard University Ward's treatise was used for a time as a textbook. contayning Eight Treatises, and Edmund Wingate's Arith-metique made Easie. Of the former treatise the seventh edition of 1636 was " corrected and somewhat enlarged by R. Hartwell, Philomathematicus.'' Wingate's work appeared about 1629, and the edition of 1650 was a popular THE TEXTBOOKS OF ARITHMETIC 75 revision by John Kersey. Probably the most significant addition by Kersey was the introduction of decimal TITLE PAGE No arithmetic in the English language has had as many editions as Cocker, and only Recorde was used over as long a period of time. "George Fisher" is a pseudonym for one Mrs. Slack, concerning whom we know only that she wrote three or four treatises in the nature of compendiums which were among the most widely read books for instruction of the eighteenth century, both in England and in America. The expressions, "Printed . . . at the Bible and Sun in Amen-Corner," . . . "at the Red-Lion," . . . "at the Looking-Glass," refer to signboards placed outside of the book shops similar to signs employed then by taverns. fractions. Practical problems on tare, trett loss, gain, and barter were also added. Wingate continued to appear until late in the eighteenth century. The most popular arithmetic, at least in number of editions, appears to have been Edward Cocker's Ariih-metick, "Perused and published by J. Hawkins" in 1678, three years after the death of Cocker. Approximately one hundred editions were published in the British Isles, but strange to say only one edition (at most) appeared in 76 THE HISTORY OF ARITHMETIC America. Cocker dropped the old terminology, condensed the presentation of most topics, particularly exchange, TITLE PAGE � The type of textbook represented by Mather's Young Man's Companion was responsible in some measure for the introduction into arithmetic of problems on plastering, carpeting, masonry, and allied topics. This work, like Fisher's Instructor and Bradford's The Secretary's Guide, was designed for self-instruction. dropped the work on counters, and introduced many lists of problems. Another popular book which appeared in the seventeenth century and continued in wide use during the eighteenth century was William Mather's The Young Man's Companion, or Arithmetick Made Easy. Ley-bourn's treatise on Arithmetick, vulgar, decimal, instrumental, algebraical, was more scientific but not so widely THE TEXTBOOKS OF ARITHMETIC 71 used. Both texts were used by William Bradford in the preparation of the first American arithmetic in English. TITLE PAGE Leybourn was a well-known teacher and surveyor. Leybourn's numerous treatises included one on recreations, long a favorite topic with mathematicians. Among the problems given in his Pleasure with Profit (London, 1693) is that "to put five odd numbers together to make 20," which Ley-bourn designates as a Falacy; the answer is, "Three nines turned upside down, and two vinits." Augustus De Morgan thinks "the question more than answered, viz., in very odd numbers." Among the earliest Ready-Reckoners was the one by Leybourn entitled: Pan-Arithmologia; Being a Mirror for Merchants, a Breviate for Bankers, a Treasure for Tradesmen, a Mate for Mechanics, . . . (London, 1694). Iarithmetick;|- ~VV LQAK )DECfMALy l~)lN$TRVM'�*NT'ALf f \Jalg:ehr-aicai. x:\ �_ � .':. ���' ,...........-..--LiJ-l i Jri'-Four Paktc-s.- i WILLIAM LETMOVRN. .�rtiUv Corr,;cl.;d; and very much En- ,.,�..!,.... AUTHOR. Account-wjiurcof is pven in tia Prtf.us totl'c READER. L 0 iv D 0 A , inial by ''i. M'tllartx, kw.Jliwjiitm'-iivi �~j:,h,< Ci. A. M. 5 FOR THE U S E OF S C H O O L S1 \ 2 \ i AND WILL BE FOUND TO BE � ^�ID Muxns mirviucn majos milivsve Arrexus fos- 2 An Eafy and Sure Guide to the Scholar, HUNTED IN NEWBURYPORT, BY J. MYCALL, ion ISAIAH THOMAS, $ N E IV B U R Y-P 0 R T. SOLD by him in WORCESTER, ldrifobyh.dT.ioMA,, andAKDR.ws, in Bostox ; and by �A Pmbted and Sold bv J O H N M Y C A L L. � fiid Thomas, and Carhsle, irtWAi-FOtr, Noy�J)�J�pOu�| &^ MDCCLXXXVIII. �1 sind kv the Bookfellers in the United Sutei. *fi� g% d fcy the fiookfcllcrs in the United States, MDCCXCIII. THE FIRST POPULAR AMERICAN ARITHMETICS APPEARING AFTER THE REVOLUTION Pike continued in use until the middle of the nineteenth century. the fact that arithmetic and mathematics are developments intimately connected with the whole history of culture. How much of the history of the New World is reflected in the story of its arithmetic? The Spanish discoverers THE TEXTBOOKS OF ARITHMETIC 85 are represented by the early Mexican works in Spanish; the English colonizers are represented by Hodder and the Dutch by Peter Venema; the French are represented by a Canadian arithmetic of 1809 in French; the native sons appear with William Bradford in 1705 and Isaac Greenwood in 1729, and come to a dominating place with Pike in 1788. Other great historical movements of American history are reflected in the arithmetics, worthy of somewhat attentive study by teachers who seek through arithmetic and other studies to educate the American youth. List of Arithmetics and Arithmetical Works Published in America before 1800 1556 Juan Diez Freyle, Sumario Compendioso de las quentas de plata y oro .... Con algunas reglas tocantes al Arith-metica. Spanish. Mexico. Printed by Juan Pablos of Brescia. Copies in British Museum and in the Escorial. Photographic cqpy in the University of Michigan Library and library of David Eugene Smith, Columbia University. 1623 Pedro Paz, Arte menor aprender todo el menor del Arithmetical sin Maestro. Mexico. Printed by Joan Ruyz. 2 1.+181 numbered folios+3 1. of tables; 21 chapters. 1649 Atanasius Reaton (Pasamonte), Arte menor de Arismetrica. Printed by Viuda de B. Calderon. 3 1.+78 numbered folios; 14 chapters. 1675 Benito Fernandez de Belo, Breve arithmetica por el mas sucinto modo, que hasta oy se ha visto. Mexico. Printed by Viuda de B. Calderon. John Carter Brown Library. 1705 William Bradford, The Young Man's Companion. New York. No copy known. 1710 William Bradford, The Young Man's Companion. In four parts. Part II. Arithmetick made easie, and the rules thereof Explained and made familiar to the Capacity of those that desire to learn in a little time.....Printed 86 THE HISTORY OF ARITHMETIC and Sold by William and Andrew Bradford, at the Bible in New York, 1710. Two imperfect copies in private hands; photographic copy in New York Public Library. 1719 William Bradford, The Secretary's Guide, or Young Man's Companion. 1719 Hodder, Boston. Reprint of an English text. Printer: J. Franklin. 2 1., viii+216 pp. L. C; A. A. 1728 William Bradford, The Secretary's Guide. Part II, Arith- metick made easie. New York. W. Bradford. Pp. (2), (2), (6), 192. L. C. 1729 Isaac Greenwood, Arithmetick, Vulgar and Decimal. Boston. Printers: S. Kneeland and T. Green. First separate text by a native of colonial America. Title, 158 pp., 4 pp. Index, and 4 pp. Adv. L. C. William Bradford, The Secretary's Guide. New York. Printer: Wm. Bradford. Pp. (2), (2), (6), 192. 1730 Peter Venema, Arithmetica of Cyfer Konst. Dutch, New York. Printer: J. Peter Zenger. 120 pp. N.Y.H.S. 1737 Wm. Bradford, The Secretary's Guide. New York. Printer: Wm. Bradford. Pp. (2), (8), 248. N. Y. P. 1738 Wm. Bradford, The Secretary's Guide. Philadelphia. Printer: Andrew Bradford. Pp. (2), (8), 248. P. 1748 Jonathan Burnham, Arithmetick for the use of farmers and Country people. New London. Printer: T. Green. 1748 George Fisher, The American Instructor; or, Young Man's Best Companion, containing Spelling, Reading, Writing, and Arithmetick, in an easier Way than any yet published. Reprint of an English work. Philadelphia. Printer: Benjamin Franklin and D. Hall. Pp. v, 378; 5 plates. N. Y. P., P. 1749 Same. Boston. 1753 Fisher, Young Man's Best Companion. Philadelphia. Printer: Benj. Franklin and D. Hall. Pp. v, 384 (2); 6 plates. Hist. Soc. Penn. 1758 Conclusiones Mathematicas .... por Don Fernando de The American str.uct.or: ' � ' OR, ' � ' � ! Young Mans Bejt Companion. :��:���� CO NT A IN INC, Spelling, Reading, Writing, and Arithmetick, in ati eafi<*r Way than any yet pubtifheu s-atid how to qua ' lify anyPerfori for Bufmefi, without the Hclp'-of a Matter ,Inftfusions to write Variety of Hands, with Copies �" both in Pro fc and Vcrfe. IIow to write Letters on Bit-fmeft or Friendihip. Forms of Indentures, Bond;., Bills of Sale, Receipts, Wais.Ixafes, Rekal'es, �^f. Alfo Merchants Accompts, and a fliort and eafy Me- thalW Shftp and Boek-keeoifla. j with a Defoijmoo of the kvctd America*Jl^lcmlrs. '1" .��-,' ' Tog#K�^iMth the Carpenter'* Plain and Exaft Rule : Shew-in0>6&$i> mi>(unC*rf>t Defection of the Sliding-Rub, ' _,. '�.'�-;'.���.'�. iLikewiie the Practicai. Gavoe* at.tje Eaftrj the Art of Bialli'g, **' *>8W $e!t^ �rsd ft* �oy ZJf'flf; with loftrailiOM fee ty**Zi C�^M",'ff�* and miking Cekuru � .. :' ^ V ^-. Tt vthiih h oMe$t -~- . . The Boor Planters Physician. t With InftroQions for Marking on tinnen*', how to Phife i�'�'. ini frrfityt j to make etirera Suits of Wine j and many-txctUmt \ I'laiftm, aid MtJichti, ncccfTaiy in/l! Ftmilti, (' An4 �Mb r/udent Advjcd to young Tradtfmcn and tyeehrs, \1he whole bettir adapted H ibefe American' CAtes, titan I ��)( Cwr irJiWi? ofjielste Kind., ' By GEORGE'&ISHER, ArJ^mntT >' �/ . The Mntk Edition Rcvifed and Caneite4. ���.......'��� PHI HAD EtF-VLfJi iPfhttcdi%y B. F r .-. �s 1.1 n and V. Hai. l. at the'ftew-taatiBg-Qffice, SoMtukhSitait r/<0� COMPENDIUM FOR SELF-INSTRUCTION, WIDELY POPULAR IX ENGLAND AND AMERICA 87 THE HISTORY OF ARITHMETIC THE SCHOOLMASTER'S ASSISTANT, BEING A COMPENDIUM OF ARITHMETIC, BOTH PRACTICAL AND THEORETICAL. IN FIVE PARTS. CONTAINING* I. Arithmetic in whole Numbers, . IV. A large collection of Que* wherein all the common rules, r tions with their Answers, sciv-having each of them a suffi- 5 ing to exercise the fbregoing cient Number of Question*, i Utiles, together with a few with their answers, are metho- \ others, both pleasant and di-dically and briefly handled. I verting II. Vulg'ar Fractions, wherein > V. Duodtcimals. commonly cal-several Things, not commonly > led Cross Multiplication; met with, are distinctly treat- * wherein that sort of ArHhme-ed of and laid down in the J tic is thoroughly considered, most plain and er.sv manner. \ and rendered very plain and III. Decimals, m which, among � easy, together wfth the Me-other Thtngs, are considered > thocl of proving all the forego-the Extraction of Roots ; In- � ing* operations at once, by Divi-terests, both simple and com-1 sion of several Denominations, pound; Annuities, Rebate and > without Reducing them into Eq&ation of Payments. 5 the lowest Terms mentioned. The whole being delivered in the most familiar way of Question and Answer, is recommended by several eminent Mathematicians, Accomptants, and Schoolmasters, as necessary to be used in Schools by all teachers, who would have their Scholars thoroughly understand, and make a quick progress in Arithmetic. TO WHICH IS PREFIXED, AN ESSAY OjY THE EDUCATION OF YOUTH RUttBLT OFFERED TO THE COXSIDEIIATION OP PARENTS. BY THOMAS^DILWORTH, . hllhov of the JVevo Guide to tlvc English Tongue, Young- Jloek keeper** jitistant, &c. NEWWORK: tfclJmD BY T. & W. M2-UCEIX, 93 HOLD STREET LATE EDITION OF DILWORTH The most popular arithmetic in the United States during and directly following the War of the Revolution. THE TEXTBOOKS OF ARITHMETIC 89 < Araya. Manila. A mathematical thesis defended at the University of the Society of Jesus in Manila. M. 1760 George Fisher, The American Instructor. New York. Printer: Gaine. 12th ed. Pp. v, (1), 378. N. Y. H. S. 1766 Fisher, same. New York. Printer: Gaine. 1770 Fisher, same. Philadelphia. Printer: Dunlap. Pp. v, 390, plate. 15th ed. A. A., N. Y. P. Anon., The Youth's Instructor . . . 777 Rules in Arithmetick. Boston. Printer: Mein and Fleming. 152 pp. P. 1773 Thomas Dilworth, The Schoolmaster's Assistant; being a Com- pendium of Arithmetic both practical and theoretical. Philadelphia. Printer: J. Crukshank. Pp. (2), xiv, (10), 192; folded leaf; portrait. Hist. Soc. Penn. 1774 Daniel Fenning, The Ready Reckoner. American edition of an English work. Germantown. Printer: Chr. Saur. A. A. Daniel Fenning, Der Geschwinde Rechner, oder; des Handler's nutzlicher Gehillfe in Kauffung, etc. Germantown. Printer: Christoph Saur. Largely tables, but strictly arithmetical. N. Y. P. 1775 Thises de Mathematiques qui seront soutenues au Seminaires de Quebec .... par MM. .... Panet, Perrault, Chauveaux. Quebec. N. Y. P. 1775 Fisher, Instructor, cr Young Man's Best Companion. Burlington, N. J. Printer: Isaac Collins. Pp. xii, 372. Hist. Soc. Penn. 1778 Fisher, same. New York. Printer: Hugh Gaine. 1779 Cocker's Arithmetick. Philadelphia. Reprint of an English work. Fisher, American Instructor. Boston. Printer: Boyle and M'Dougall. Pp. vi, 378; folded plate. A. A., N. Y. P. 1781 Dilworth, The Schoolmaster's Assistant. Philadelphia. A reprint of an English work. Apparently two editions: one printed and sold by R. Aitken, 220 pp., and the other by Jos. Crukshank, xiv, 5 1., 192 pp.; 1 folded leaf; 1 portrait. A. A. 7 90 THE HISTORY OF ARITHMETIC 1782 Dearborn, The Pupil's Guide. Portsmouth. Printer: Daniel Fowle. 2 1. + 16 pp. A. A. 1783 Dearborn, reprint of above. Boston. Printer: B. Edes and Sons. Pp. vi, 26. M. 1784 Dilworth, The Schoolmaster's Assistant. New York. Printer: Gaine. xiv, 51., 192 pp.; folded leaf; portrait. A. A. Dilworth, same. Philadelphia. Printer: Joseph Cruk- shank. Pp. (2), xiv, (10), 192; 1 folded leaf; 1 portrait. A. A.; Hist. Soc. Penn. 1784 An Arithmetical Card. Philadelphia. Sold by William Poyntell. 1785 Thomas Dilworth, The Schoolmaster's Assistant. Hartford. Printer: N. Patten. 200 pp. N. Y. P. Alexander McDonald, The Youth's Assistant. Norwich. Printer: John Trumbull. 102 pp. A. A. Fisher, The American Instructor. New York. Printer: Hugh Gaine. Fisher, The Instructor; or, American Young Man's Best Companion. Worcester. Printer: Isaiah Thomas. Pp. 384; frontispiece. A. A., N. Y. P. 1786 Method of teaching rudiments of arithmetic. Broadside {i.e., single page). Providence. John Carter Brown Library. 1786 Bonnycastle, The Scholar's Guide to Arithmetic. Boston. Reprinted by John West Folsom. Pp. 200+2 leaves. M. Dilworth, The Schoolmaster's Assistant. Hartford. Printer: N. Patten. 9 1., pp. 9-199+1 bl.; portrait; folded leaf. A. A. 1786 Ludwig Hoecker, Rechenbuechlein. Ephrata, Penn. Daniel Fenning, The New Ready Reckoner. Reprint of an English text. New York. Printer: Samuel Campbell. Fisher, Instructor, etc. Worcester. Printer: Isaiah Thomas. 384 pp., 6 plates. N. Y. P. Fisher, American Instructor. Philadelphia. Printer: J. Cruk- shank. Pp. xii, 372; 6 plates. A. A. 1787 Dilworth, Schoolmaster's Assistant. Hartford. Printer: N. Patten. Pp. 384+frontispiece. Advertised May 14, 1787, THE TEXTBOOKS OF ARITHMETIC 91 THE SCHOLAR'S GUIDE T O ARITHMETIC: O R A complete Exercife-Book, FOR THE USE of SCHOOL WITH NOTES, CONTAINING The Reason of every Rule, demonftrated from the moft Ample and jevident Principle? ; TOGETHER WITH General THEOREMS for the more extenfive USE of the S CIENC E. By JOHN BQNNYCASTLE, Private Teacher of the Mathematics. the second edi.tioj, corrected. LONDON, Printed. BOSTON: Re-printed by John West FolsOm, at fhe Corner of Jinn-Jlreet* mdcclxxxvi. bonnycastle's arithmetic - Bonnycastle's treatises, popular in America, covered'practically the whole range of elementary mathematics, 92 THE HISTORY OF ARITHMETIC Fisher. American Instructor. Philadelphia. Printer: J. Cruk-shank. Pp. xii, 372+frontispiece. A. A. 1788 Nicholas Pike. A New and Complete System of Arithmetic, composed for the use of the citizens of the United States. Newburyport. Printed and sold by John Mycall. 512 pp. A. A., M., N. Y. P. 1788 Thomas Sargeant, Elementary Principles of Arithmetic, with their application to the trade and commerce of the United States of America. Philadelphia. Printer: Dobson and Lang. Pp. (4), 95, (1). A. A. "A few copies intended for the use of young ladies are printed on fine paper and in a more elegant binding." A. A. 1788 Sargeant, A Synopsis of logarithmical arithmetic. Philadel- phia. Printer: Dobson and Lang. Sargeant, Select Arithmetical Tables. Philadelphia. Printer: Dobson and Lang. Single card. Gough, A Treatise of Arithmetic. Ed. by Benj. Workman. Philadelphia. Printer: McCulloch. Pp. 370+1 leaf. A. A. 1789 Benjamin Workman, The American Accoiintant. Philadel- phia. Printer: John McCulloch. Based on the English work by John Gough. Pp. (2), (2), 224. A. A. Alexander McDonald, The Youth's Assistant. 2nd ed. Litchfield. Printer: Thomas Collier. 104 pp. P. Fenning, Ready Reckoner. Reading. Printer: Benj. Johnson. 195 pp. A. A. Fenning, Der Geschwinde Rechner. Reading. Printers: Johnson, Barton und Jungman. 195 pp. In collection of Mr. Wilberforce Eames. John Gough, Treatise on Arithmetic. Ed. by Benj. Workman. Boston. 1790 Consider and John Sterry, The American Youth: being a new and complete course of introductory mathematics. Providence. Printer: John Wheeler. Pp. 387, (1). A. A.; M. Dilworth, Schoolmaster's Assistant. Philadelphia. Printer: J. Crukshank. Pp. xiv, (10), 192; portrait. A. A, THE TEXTBOOKS OF ARITHMETIC 93 Bonnycastle, The Scholar's Guide. New York. Daniel Fenning, The Ready Reckoner. Reading. 1791 Alexander McDonald, The Youth's Assistant. Providence. Dilworth, Schoolmaster's Assistant. Wilmington. Printer: Andrews, Craig, and Brynberg. Pp. xiv, (6), (2), (2), 192; folded table; portrait. A. A. 1792 Fenning, Ready Reckoner. Worcester. Printer: Thomas. Pp. (2), (2), 166. A. A. 1792 Gordon Johnson, An Introduction to Arithmetic. Springfield. Printer: E. W. Weld. John Vinall, The Preceptor's Assistant, or Student's Guide: being a systematical treatise of arithmetic. Boston. P. Edes, for Thomas and Andrews. 288 pp. A. A. Thomas Dilworth, Schoolmaster's Assistant. New York. Printer: Gaine. Pp. xiv, (6), (2), (2), 192; folded table. A. A. 1793 Thomas Sargeant, The Federal Arithmetician. Philadelphia. Printer: Thomas Dobson. 264 pp.; 2 folded leaves. A. A.; M. Erastus Root, Introduction to Arithmetic. Norwich, Conn, Nicholas Pike, Abridgement of the new and complete system of arithmetic. Newburyport. Printed by J. Mycall for Isaiah Thomas. Pp. i-iv, 5-371. A. A., M. Another Copy . . . Printed for Isaiah Thomas, Worcester. Composed for the use and adapted to the commerce of the citizens of the United States ... for the use of schools, and will be found to be an easy and sure guide to the scholar. A. A. Pike's arithmetics were the first widely popular American works. 1793 Dilworth, Schoolmaster's Assistant. "Latest Edition.'' New York. Printed by J. Allen, xvi, 3 1., 192 pp.; portrait. A. A. Dilworth, Schoolmaster's Assistant. "Latest Edition." New York. Printed by John Buel. xvi, 3 1., 192 pp.; portrait; folded leaf. A. A. Dilworth, Schoolmaster's Assistant. Philadelphia. Printer: Crukshank. xiv, 5 1., 192 pp.; portrait; folded leaf. A. A. 94 THE HISTORY OF ARITHMETIC Gordon Johnson, Introduction to Arithmetic. 2nd ed. Springfield. Printer: J. R. Hutchins. 27 pp. + l bl. A. A. Phinehas Merrill, The Scholar's Guide to Arithmetic. Exeter. Gough, American Accountant. 2nd ed. by Benj. Workman, revised by R. Patterson. Philadelphia. Printer: Wm. Young. Pp. 220, (4). A. A. Anon., The Federal or New Ready Reckoner. Chestnut Hill. Printer: S. Sower. 2 1.+56 1.+4 folded leaves+4 1. P. 1794 Phinehas Merrill, The Scholar's Guide to Arithmetic. 2nd ed. Exeter. Printer: H. Ranlet. Pp. iv, 107. A. A. Fisher, The Instructor. Walpole, N. H. Printer: Thomas and Carlisle. 384 pp., 5 plates. A. A. 1794 Daniel Fenning. The Ready Reckoner. Newburyport. 11th ed. Printer: E. M. Blunt. 160 pp. N. Y. P. John Todd, Zachariah Jess, William Waring and Jeremiah Paul, American Tutor's Assistant. Philadelphia. 3rd ed. Printer: Z. Poulson, Jr. 2 1. + 196 pp. + (imperfect copy). Hist. Soc. Penn. 1794 D. Gregorius Bedoya. Selectiones ex Elementis etiam turn Arithmeticae turn Algebrae utrius que etiam planae ac solidae Geometriae. Sectionum Conicarum denum Trigonometriae, ac practicae Geometricae quas publico .... exhibet D. Gregorius Bedoya, Lima, Peru. There were numerous other theses defended at the University in Lima, printed between 1768 and 1800. Anon., A short and easy guide to arithmetick. Boston. Printer: Samuel Hall. John Carter Brown Library 1795 Joseph Chaplin, The Trader's best companion. Newburyport. Printer: Wm. Barrett. 36 pp. P. Pike. Abridgement, 2nd ed. Worcester. Printed by Isaiah Thomas. Apparently there was a second imprint of the same year. 348 pp. A. A., M. Anon.> The Stranger's Assistant and Schoolboy's Instructor. New York. A. A. Fenning, American Youth's Instructor. Dover, N. H. Printer: Samuel Bragg. iv? 260 pp. A. A, THE TEXTBOOKS OF ARITHMETIC 95 Erastus Root, An Introduction to Arithmetick for the use of Common Schools. Norwich. Printer: Thomas Hubbard. 105pp. + lbl. A. A., P. 1795 Consider and John Sterry, A Complete Exercise Book in Arithmetic. Norwich. Printer: John Sterry & Co. 120 pp. + 11. A. A. Wm. Wilkinson, The Federal Calculator and American Ready Reckoner. Providence. Printer: Carter and Wilkinson. 64 pp. A. A. 1796 Donald Fraser, The Young Gentleman's and Lady's Assistant . . . including . . > Practical Arithmetic. New York. Printer: T. &. J. Swords. 216 pp. A. A. Workman, American Accountant. Revised by Patterson. Philadelphia. 3rd ed. Printer: Wm. Young. 220 pp., 2 1. A. A. Gough, A Treatise of Arithmetic. Ed. by Benj. Workman. Philadelphia. Printer: Wm. Young. 376 pp. A. A., M. Erastus Root, An Introduction to Arithmetick for the Use of common schools. Norwich. 2nd ed. Printer: Thomas Hubbard. 106 pp. A.A. Erastus Root, An Introduction to Arithmetick for the Use of common schools. Boston. Harvard College Library. Dilworth, Schoolmaster's Assistant. New York. Printer: Mott and Lyon, xii, 3 1., 192 pp.; frontispiece. A. A., N. Y. P. Dilworth, Schoolmaster's Assistant. Philadelphia. Printer: Crukshank. xiv, 4 1., 192 pp.; portrait; folded leaf. A.A. Dilworth, Schoolmaster's Assistant. Wilmington. Printed by Peter Brynberg. 3 1., 192 pp. A. A. 1796(?) Dilworth, Schoolmaster's Assistant. Wilmington. Printer: Brynberg. 3 1., 192 pp. A. A. 1797 Dilworth, Schoolmaster's Assistant. New London. Printer: Samuel Green, xvi, 3 1., 192 pp., 1 1.; 1 folded leaf; 1 portrait. A. A. 1797 Todd, Jess, Waring, and Paul, American Tutor's Assistant. 3rded, Philadelphia. Printer: Z. Poulson. 200 pp. A.A, 96 THE HISTORY OF ARITHMETIC 1797 ChaunceyLee, American Accountant. Lansingburg. Printer: W. W, Wands. 300+12 pp. list of subscribers + 1 plate. First to use the dollar sign in print. A. A., M# Fisher, The Instructor, etc. Wilmington. Printer: Brynberg. xii, 360 pp. A. A. James Noyes, The Federal Arithmetic. Exeter. David Kendall, The Young Lady's Arithmetic. Leominster, Mass. Printer: C. Prentiss. 44 pp. A. A. William Milns, American Accountant. New York. Printer: J. S. Mott. Pp. (8)+320. A. A., M. F. Nichols, A Treatise of Practical Arithmetic and Bookkeeping. Boston. Printer: Mann and Loring. 104 pp. A. A. Nicholas Pike, A New and Complete System of Arithmetic. 2nd ed. Worcester. Printer: Isaiah Thomas. 516 pp.+ an extra leaf numbered 5-6 with Preface by Isaiah Thomas. A. A. Francis Walkingame, The Tutor's Assistant. A new edition. Gainsborough. Printer: H. Mozley & Co. 180 pp.+ folded leaf. A. A. 1798 Peter Tharp, New and Complete System of Federal Arithme- tic. Newburgh. Printer: D. Denniston. 138 pp.+l leaf. A. A. Nicholas Pike. Abridgement. Worcester. 3rd ed. Printer: Thomas. 352 pp. M. Samuel Temple, A Concise Introduction to Practical Arithmetic. Boston. Printer: Hall. 2nd ed., 118 pp. P# 1799 David Cook. American Arithmetic. New Haven. Printer: T. Green. Ed. of 1800 in Library of Congress may be first edition. Ezekiel Little, The Usher, etc. Exeter. Printer: H. Ranlet. 240 pp. A. A. DanielYenning, The Ready Reckoner. Wilmington. Printer: Brynberg. 189+3 pp. 12th ed. Hist. Soc. Penn. Zachariah Jess, The American Tutor's Assistant. Baltimore. Printer: Bonsai and Niles. 2 1., 204 pp., 1 leaf errata. L. C. THE TEXTBOOKS OF ARITHMETIC 97 1799 (?) Dilworth, The Schoolmaster's A ssistant. Printed and sold by Bonsai and Niles. Also sold at their book store, Baltimore. 3 1., 192 pp.; folded leaf; portrait. A. A. Set Sfutfcfcctt Bauer* unfc ianbmamtf GERMAN ARITHMETIC Strange title for an American arithmetic: The German Peasant's and Farmer' s A rithmetic and Schoolmaster's Assistant. Arithmetics in German were published in St. Louis from 1850 to 1860, and doubtless more recently in other parts of the United State?. $ecl)ro&uclj, � u t t c $ c*cv tin furjcv Suksvijf fccv prcif tifc^e n Sftccfocnf unff. G it 11; a it c it & in funf Cap iff f a, O 2>ic funf �pfcietf unti 4.) �>U �anfmanure<$� a.) Die 9t<�iiU to $vi. 4 ?.) 2>�c tjc(;cru Staffs 3on(�6rtfcJ6.2Rau* 6 a |! 0 it/ 3totucfi tr ifti an 3 ft c. ^nfnr. Early Canadian Arithmetics 1809 Jean Antoine Bouthillier, Traite d'Arithmetique pour Vusage des Ecoles. Quebec. Printer: John Neilson. 3 1., 144 pp. 2nd ed. 1829. P., L. C. Rev. John Strachan, A concise introduction to practical arithmetic for the use of Schools, Montreal, 1809. Toronto Public Library. 98 THE HISTORY OF ARITHMETIC 1816 MrcHEL Bibaud, L'arithmetique en quatre parties . . . vulgaire} . . . marchande . . . scientifique . . . curieuse . . . Montreal. Printer: N. Mower, iv, 199 pp. 2nd ed. widely used, 1832. 1836 Casimir Ladreyt, Nouvelle arithmetique. Montreal. Printed by author, viii, 120 pp. 1836 Jos. Laurin, Traite d'Arithmetique et d'Algebre. Quebec. 206 pp. Kentucky and West of the Mississippi River 1804 Jesse Gtjthrie, The American Schoolmaster's Assistant. Lexington, Kentucky. 231 pp. A copy of the 1810 edition is in the Library of Congress. 1822 Rene Paul, Elements of Arithmetic. St. Louis, Mo. Printer: Ford andOrr. 18 mo., 160 pp. 1826 J. Stockton, The Western Calculator. Fourth edition, 1826. Pittsburgh. Printer: Eichbaum and Johnston. 204 pp. L. C. 1828 Anon., He helu kamalii, o ke aritemetika. Ohau, Hawaii, 1828. (Missionary Press.) Second edition, 1832. 1836 Tabla para los ninos que empiezan contar. Broadside. Monterey, California. 1836 Tobias para los ninos, etc. Monterey. Printer: A. Zamorano. The largest collections of early American arithmetics are to be found in the Library of Congress (L. C), New York Public Library (N. Y. P.), American Antiquarian Society (A. A.), and in the private library of Mr. George A. Plimpton, New York City. The University of Michigan (M.) has a fair collection. The writer is particularly indebted to Dr. Clarence S. Brigham of the American Antiquarian Society for suggestions and notes on books; to Dr. Wilberf orce Eames of the New York Public Library for information touching many points of this bibliography; to Dr. Lawrence C. THE TEXTBOOKS OF ARITHMETIC 99 Wroth of the John Carter Brown Library for notes; and also to the librarians at the other places mentioned. The Hawaiian arithmetic appears to be the first published in the New World west of St. Louis. The information concerning the first edition was very kindly given by the Reverend Howard M. Ballou of Honolulu. Bibliography tor Supplementary Reading D. E. Smith, Rara Arithmetica. See Bibliography, Chapter II. Augustus De Morgan, Arithmetical Books. See Chapter II. J. O. Halliwell, Rara Mathematica. See Chapter II. Florian Cajori, "The Teaching and History of Mathematics in the United States." Washington, Bureau of Education, Circular No. 3, 1890. Walter S. Monroe, " Development of Arithmetic as a School Subject." Washington, Bureau of Education, Bulletin No. 10, 1917. David Eugene Smith, The Sumario Compendioso of Brother Juan Diez. Boston, Ginn and Co., 1921. Robert Steele, " The Earliest Arithmetic in English." London, Early English Text Society, Extra Series No. cxviii, 1922. George E. Littlefield, Early Schools and Schoolbooks of New England. Boston, The Club of Odd Volumes, 1904. Susan R. Benedict, " A Comparative Study of the Early Treatises introducing into Europe the Hindu Art of Reckoning." Thesis, University of Michigan. Published privately, 1916. CHAPTER IV THE FUNDAMENTAL OPERATIONS IN EARLY ARITHMETIC EMPLOYING NUMERALS Fundamental operations. Today we speak of the four fundamental operations of arithmetic without hesitation as to the number of the operations. The early works on our system of arithmetic include frequently seven, eight, or nine subdivisions. Numeration or notation continued to be called a fundamental operation until the nineteenth century; doubling and halving as separate operations appear in nearly all the treatises before the fifteenth century, and in many up to the seventeenth century; extraction of roots, square and cube root, was regarded as fundamental in the Arabic treatises up to the ... .. .. . .. ^ ....... !J*-3r I ; r � � ( ;l|� tf�it*t� loco i>��K*|.�*P*ft'->j�, .|>(na (u.xi^vtw^ ..� .,,.ffcfici� <35 .*�� &�� :. |^a� JS*>o,�:# tmflfc^t�2 ^r St- : I /,p�^ku*^ri$ -"1*/**** /tfWtfirsfttas ti'^tc !t!nHi i^i/'f,') j , v }ic�vi�i sr'�w**"~] i-#IL: �_:,..-- � . L'........i.......___......................... _..... .... ............M BritiBh Museum. Add. MS. 24059 fol. 22b. NUMERATION IN A FIFTEENTH-CENTURY MANUSCRIPT The first line of the text begins with the numeral? 0, 9, 8, . . . 1, in reverse order, possibly due to the fact that Arabic writing proceeds from right to left. The text continues: "Quelibet figura primo loco posita significat seipsam, tamen in secundo loco Decies seipsam . . . . " This means that any digit in the first or units' place signifies so many units, in the second place so many tens, and so on. The title is "Informacio ad computandum Algorismi," or "Information concerning the Computation by Algorism." twelfth century, and in the popular works of Alexandre de Ville Dieu and Sacrobosco; progressions also are 100 FUNDAMENTAL OPERATIONS 101 included by these two writers, by Recorde, and by many others. The Hindu arithmeticians do not include doubling and halving, but several list among the fundamental operations many topics on application to practical affairs. Mahavir in the ninth century lists only eight operations: squaring, square root, cubing, cube root, summation (addition), multiplication, division, and subtraction, the final two applied to series. "Septem sunt partes, non plures, istius artis: Addere, subtrahere, duplare, dimidiare, Sextaque diuidere, sed quinta multiplicare; Radicem extrahere pars septima dicitur esse. "Here telles that ther ben 7 spices or partes of this craft. The first is called addicion, the secunde is called subtraccion. The thryd is called duplacion. The 4 is called dimydicion. The 5 is called multiplicacion. The 6 is called diuision. The 7 is called extraccion of the Rote. What all these spices bene hit schalle be tolde singillatim in here caputule." This is Alexandre de Ville Dieu's notion of the '' species'' or operations, and also the notion of his English commentator. Robert Recorde, three centuries later, says " There are reckoned commonly seven parts or works of it. "Numeration, Addition, Subtraction, Multiplication, Division, Progression, and Extraction of roots: to these men adde Duplication, Triplation, and Mediation.'' Nicholas Pike states that there are "five principal or fundamental Rules, viz. Notation or Numeration, Addition, Subtraction, Multiplication and Division/* With the occasional inclusion of the extraction of roots and later with the exclusion of notation, this list is fairly typical of the" fundamental operations as listed in works in English from the eighteenth century to the present day. 102 THE HISTORY OF ARITHMETIC Addition and Subtraction Addition. The operations of addition and substraction are so elementary that much variation in procedure from ' 'ifw^Mii' �?- cfr 'h* Oman* Vniftvt n�mit "f&�� A$ihg*in nil* I �^tlrtwM*. JHi? i���;'ntfft the place of the omyst shalt thow write the digit excrescying, as thus: The resultant 2 To whom it shal be addede 1 The nombre to be addede 1" * The Latin and other early versions almost invariably indicate to "add a number to a number,'' and many of them indicate also that in the process both the numbers vanish, leaving the sum in the place of the larger number. To begin at the left to add was almost as common as to begin at the right. The successive stages of the addition of 826 and 483 are represented graphically, as follows: 826 829 909 1309 483 48 4 Leonard of Pisa (1202) departs from the custom of only two addends, and also introduces the novelty of writing the sum above the addends. With the appearance of printed arithmetics in the fifteenth century addition assumed the modern form, 1 Steele, loc. cit., p. 35. ADDITION AND SUBTRACTION IN "THE ART OF NOMBRYNO" Above at the right is an addition cvitniic in whirl] the sum. "The resultant." appears at the top; "To whom it shall bo added!.-'' is S, and "The norabre to With the decorative S bcHiii:; the treatment of subtraction, as follows: "Subtraceioun is of 2 proimccdc nrmitji-es 'he fyndlnir ni the excesse of the more to the lesse. Other subtraceioun is abiacioun of o oombre fro another The remainder, "remanent," is written above in the two problems at the foot of the page. FUNDAMENTAL OPERATIONS 105 Subtraction. Subtraction in the early algorisms reveals also peculiarities which follow the procedure on an abacus. The remainder replaces the minuend, both minuend and subtrahend disappearing. Even Pike in 1788 retains two peculiarities found in several of the algorisms: "If the lower figure be greater than the upper, borrow ten, and subtract the lower figure therefrom: to this difference add the upper figure, which, being set down, you must add one to the tens' place of the lower line for that which you borrowed." This method of "borrowing above" and "paying back below," common in England from Shakespeare's day (Recorde, Baker) almost to the present time, has its advocates in America today. Austrian method. The strictly addition procedure in subtraction is mentioned in the Handbuch der Mathe-matik by Bittner, published at Prague in 1821; the method is explained in Solomon's Lehrbuch der Arithmetik und Algebra, Vienna, 1849. In America this has been known as the Austrian method, and its use is recommended to primary teachers in the courses of study of several large school systems. The procedure is as follows: Think of the number which added to 483 will give 826; 3 added to 3 gives 6; 4 added to 8 gives �- 12; write down 4 and mentally carry 1 to the next 4, making 5; 3 added to 5 makes 8. 343 is the number which added to 483 gives 826. The "check by nines." The check upon subtraction by addition, and vice versa, is particularly recommended by the earliest writers on arithmetic. The further ' * check 8 106 THE HISTORY OF ARITHMETIC by nines" upon addition, subtraction, and other operations was quite as common among earlier writers. The "check by nines'' depends upon the fact that 10, 100, 1000, . . . each when divided by 9 has 1 as remainder; in consequence a number like 6724 when divided by 9 will have the remainder 6+7+2+4 or "casting out" 9 twice the remainder will be 1. In other words, the remainder when dividing a number by 9 is obtained by taking the sum of the digits and, if necessary, casting out any multiple of 9. If a number whose remainder when divided by 9 is 1 is added to one whose remainder is 6, it is obvious that the resulting number when divided by 9 will have the remainder 1+6 or 7. Thus 6724 rem. 1 1221 rem. 6 7945 rem. 7 This is indicated in many old texts by a diagram at the side. Obviously, there are somewhat analogous rules, with the proper changes, to apply the "check by nines" to the other fundamental operations. Multiplication Early methods of multiplication. The operation of multiplication invites a variety of methods of treatment. Several of the early methods employed are instructive and worthy of somewhat detailed treatment. The method in most common use in Europe in the early algorisms corresponds precisely to the procedure indicated, but not fully explained, in the early Hindu treatises, FUNDAMENTAL OPERATIONS 107 MULTIPLICATION AND DIVISION IN "THE ART OF NOMBRYNG" The product or "resultant" is placed above in the two illustrative problems in multiplication. In the final example on this page, under division, four separate problems are given: 680 * 32; 66 -s- 3; 342 -5- 63; and 332 + 34, which last has the quotient 9, and the "residuum," or remainder, 26. portions1 in the multiplier, and is severally multiplied by them." Mahavir says that multiplier and multiplicand are placed "in the manner of the hinges of a 1 Meaning digits. Brahmagupta states that "the multiplicand is repeated like a string for cattle as often as there are integrant 108 THE HISTORY OF ARITHMETIC door," and Sridharacarya adds to this "multiply in order, directly or inversely, repeating the multiplier each time." The method indicated is explained in The Crafte of Nombrynge, one of the two earliest discussions in English. "Here begynnes the Chaptre of multiplication, in the quych thou must know four thynges. First, qwat is multiplicacion. The secunde, how mony cases may hap in multiplicacion. The thryde, how mony rewes of figures there most be. The 4 what is the profet of this craft."1 Before giving the complete explanation, the writer interjects the multiplication table in triangular form from 1X1 up to 9X9, as being necessary in the multiplication. "Lo an Ensampul here folowynge."2 1 82 2465 464465 464865 a. 232 b. 232 c. 232 11 11 110 121 1211 828 8285 d. 464825 e. 464820 232 2465 is to be multiplied by 232 (multiplier). In figure b 232 has been multiplied by 2 (for 2000), and the product 464 (for 464000) written in the same line with 2465, of which the 2 has disappeared in the final step. Next 232 is moved over one space and multiplication of 232 by 4 (400) follows. The partial products (8 for 80000, 12 for 12000) are written above, and finally 8 for 800 (400 X 2) takes the place of the multiplying digit 4. Then i Steele, loc. cit., p. 21. 2 Steele, loc. cit., pp. 24-25. FUNDAMENTAL OPERATIONS 109 232 is moved again one place, until the unit 2 falls under the 6 of the multiplicand. The partial products again are written above. Finally 232 is moved again so that the units' place 2 comes under the sole remaining digit of the multiplicand, 5, and the partial products are again written above. After the partial products are written they are summed, JUAN DIEZ FREYLE'S "SUMARIO," MEXICO, 1556 Multiplication of 875 by 978 with all partial products completely written in column form. At the right, division by the "scratch method" of 432175 by 124. This method is explained below, under Division. here from left to right, and 571880 appears as the final result. Variations. Some writers of the twelfth century combine as they go along; in some the figures do not disappear but are "scratched out," giving rise to the name "scratch method." A variation of the above process is given by Juan Diez Freyle in the first multiplication example to appear in the New "World. Freyle herein probably follows Spanish arithmeticians of the early sixteenth century; the variation is found in Paciuolo's work and in numerous others. Another method involving all separate partial products is introduced by Paciuolo as the "lattice work" or 110 THE HISTORY OF ARITHMETIC "jealousy" method. The jalouse or "lattice work" is that screen behind which ladies are accustomed to stand to observe without being observed. The derivation from the same word as our "jealous " is obvious. In this device as applied to multiplication the units, tens, and hundreds appear in the same diagonal row and are combined diagonally. Napier, the inventor of logarithms, introduced a variation of this method by having the multiplication table written on' * rods'' of wood or bone, whence called "Napier's bones." Recently an English writer again suggests the use of Napier's rods in arithmetic. Beginnings of modern methods. Our present method of multiplication appears in the Treviso arithmetic of 1470, in Calandri in 1491, in Paciuolo, and in printed arithmetics from that day to this. Early German and Italian arithmetics sometimes presented a "lightning" method of multiplication in which only the digits of the product are written in one line below multiplier and multiplicand. The illustration given by Paciuolo in 1494 indicates that the units' digit from the product of the units is written down, the tens' digit being kept "in the heart" or "in the hand" as Leonard of Pisa expressed it in 1202; then the two products giving tens are combined with this possible tens' digit from the product of the units; the three products involving hundreds {e.g. units by hundreds, twice, and tens by tens) are combined, and so on. The following illustrates the "lightning" method: 456 \0/ ___456 y^ 207936 FUNDAMENTAL OPERATIONS 111 The only units' digit is obtained from 6X6, giving 6 to be put down and 3 to be carried; the tens' digit is obtained from 6X5, twice, to which 3 is to be added; the hundreds' digit is obtained from 6X4, 6X4, and 5X5 with 6 carried over, etc. A German writer adds: "It takes much head." At the right is indicated the check by nines, 6 and 6 being the remainders when the factors are divided by 9; 0 that of the product. A notable desirable modern innovation consists in beginning multiplication with the highest and thus most significant figures of the multiplier. This is particularly useful in computations with decimals when only a limited number of places in the product are significant. As early as 1592 this method was used by Jobst Biirgi in an unpublished manuscript on arithmetic. The only necessary variation from our method is the reversal of the order of multiplication; however, with decimal fractions this method permits one more easily to neglect the nonsignificant digits. 232 2.32 624 6.24 1392 1392 464 46 928_____9^ 144768 14.47 If 2.32 and 6.24 represent measurements, 14.47 is as accurate as the measurements justify; equally, if 232 and 624 represent the dimensions of a rectangle, the final 68 in 144,768 has no real significance in the area, 112 THE HISTORY OF ARITHMETIC Division "Scratch" method of division. The Hindus undoubtedly worked arithmetical problems on a board strewn with sand. With this arrangement it was more convenient ROBERT RECORDE'S "THE GROUNDE OF ARTES," C. 1542 An explanation of the scratch method, 7656^-29. to erase the intermediate digits involved in any calculation, replacing them gradually by the figures of the final result. Some of the methods of multiplication explained in the preceding section doubtless had their origin in the sand table. In division a method bespeaking FUNDAMENTAL OPERATIONS 113 33)7$$o(2 THE GROUNDE OF ARTES Our present method of division as taught by Recorde. The remainders and partial products were frequently printed in vertical columns as they appear here. It is customary for authors to attach the blame for this type of error to the printer. 33)7^93 (2 59 {j 66 129 99 3 anDfab&cct it 0tnofttfIju0. &f)nifejfogffjeiieri figure 9 DOtDHC, 3RD Ut It foitfc tfce Rcmaiucr 12, it mafeeff? 129, and r*moutog tjje Diuifor 33 thereto, *n* quire ftcfD often 35 to tctirafacD in uq> and 3 3 finoc a but t&tfce, (t!)Dug?) at t&e fitfi it tm&eaftgtoof mojejt&ctefcje (it $in tU Quotient, ano mtiUiplymQ 33 &P 3� ht W produftftioer up, fabputtmgt^at produft out of tfje namber abase, ano pjoceeo as before. %tyn ftaii poo ffnbe tfje diuifor 9 times in f&e Rcmoincr, t^ercfoic fctrfng 9 in t$z Quoticnt,maU Firft, I fet down the Dividend 398, & 6 (the Divifor) under the 9 thus, becaufe I cannot take 6 out of ?� 398(6 6 Chap. V* EtiviJioTt. gp in the Quotient beyond the crooked tine, faying,^ times 6 is 365 now 36 from 39* and there remains 3, which I fet down o-ver the 9, and cancel the 39 & 6 my Divifor, thus, 3 ^8(6 Again, 1 remove my Divifor to the next Place under 8, and feek how many times 6 I can have in 38, which is alfo 6 times, I fet 6 in the Quotient, faying, o times * is 3^, 36 from 38, and there remains 2, wbich 2 I fet over the 8, and cancel the 6 thus �, *(* 39* (66 So that every man muft have 661. and 2 /. over, which I may turn inro Pence,and divide alfo by 6, and the Quotient will be 80 Pence, which is in all 66 pound 6 Shillings and 8 Pence a-piece. This order Iobferve to Divide hy one Figure �, but if the Divifor do confift of more Figures than one, I muft take the firft Figure of the Divifor no oftnerout of the Divideni D 2 than hodder's "arithmetick," boston, 1719 A labored explanation of division of 4648 by 2 by the scratch method. Then I try how many times 6 I can hsve in *?* which is 6 times, I placed in mentary school,the method has much to commend it. 264 29|7656 18 11 0 2X9, 18, 8+18, 26; put down 8 and carry mentally 2; 2X2, 4; 4+2 = 6; 6 + 1 = 7. The remainder 18 appears FUNDAMENTAL OPERATIONS 119 below. 6X9,54; +1, 55; 6X2 is 12; +5 is 17; + lisl8; 4X9 is 36; +0 gives 6; 4X2 is 8; +3 is 11; no remainder. Note that first the �. fr n. .r _ Chap. v. Vivificm number is found which added to 58(2X29) will give 76; this is 18, first remainder. Secondly, the number is found which added to 6 times 29 (6X9 + 6X20) will give 185; this gives 11 as second remainder. Finally 4X29 (as 4X9 and 4 X 20) exactly equals 116. Other variations. Many of the older arithmetics recommend writing first in a column the first multiples of the divisor, for use in division. Occasionally the printed form gives all the remainders in a vertical column, not preserving the decimal order; Jflmllnot, 1(hope)needto trouble myfejfjr Learner,to fhevo theWorking of thisSum^or a~ ny other J)avingyn<>iB(a9 Ifuppefe Sufficiently treated ofUtv'ifwn\ but mil leave it to theCen-fure of the experienced to judge, whether this Manner of dividing be not plain/incalfi? to be wrought voithfewcrFigures than any which is commonly faught: As for Example appeared?, (8 �07 (5 &6$ (o 0704*8* (o 40&&rM$*fy6g (4 XZM%6?%90*?&y$Z?x $%70$4$�xxx'xtixx-?x- 0%7&Mmm% 0%702$4Mm 0*777 $$% 0 (U4999999 9S75?4?2,r 1*4999999 149999998a 37^9999974 4999999966 6i<{99999$8 7499999940 87.499999}$ 9999999910 11*4999991$ 8 Proaf E 2 IZM*678?987<5543** CHAP, hodder's arithmetic, boston, 1719 Long division with check 123456789987654321-5-987654321 this is probably frequently the printer's error. Cocker's Decimal Arithmetic shows this peculiarity in several division examples. Another peculiarity which connects with the older process of division consists in repeating the divisor under the proper place in the dividend, and then under it the product by the corresponding digit of the 120 THE HISTORY OF ARITHMETIC quotient. Humphrey Baker teaches a modification of this procedure, subtracting above, but both Baker and Recorde constantly employ the scratch method. For several centuries one who could' perform long division was considered an expert mathematician. Today in oriental countries, even in Arabia and in India, one would with difficulty find a native who has this ability. Bibliography for Supplementary Reading Tropfke, Geschichte der Elementar-M athematik in systematischer Darsiellung. Vol. I, Rechnen. Second ed. Leipzig, 1921. This work in German takes up the fundamental operations and other topics of arithmetic systematically in the historical development. There is no work in English which corresponds in completeness to this work. L. L. Jackson, "The educational significance of sixteenth-century arithmetic from the point of view of the present time." New York. Teachers College, Columbia University, 1906. Susan R. Benedict, "A Comparative Study of the Early Treatises introducing into Europe the Hindu Art of Reckoning." Thesis, University of Michigan. Privately published, 1916. L. C. Karpinski, "Two Twelfth-Century Algorisms." I sis, Vol. Ill, pp. 396-413, 1921. Florian Cajori, History of Elementary Mathematics. New York. Macmillan. D. E. Smith, History of Mathematics. Boston. Ginn & Co., 1921 Vol. I, 1923. Vol. II, promised for early publication, will furnish desirable additional material on arithmetic. D. E. Smith's Rara Arithmetica (Boston, 1908) contains many illustrations showing the fundamental operations as given in the early texts. In almost every large library some of the early arithmetics mentioned in this and the preceding chapter may be found. The examination of an original copy of this kind is an interesting and profitable exercise for any teacher of arithmetic. CHAPTER V FRACTIONS Common Fractions Multiplication is vexation, Division is as bad; The Rule of Three perplexes me, And Fractions drive me mad. Egyptian and Babylonian fractions. Fractions have always occasioned difficulty for teachers of the art of arithmetic. A fundamental part of the early Egyptian arithmetic consists in the explanation of operations with fractions. With the contemporary student of arithmetic in Babylon, fractions also constituted an important part of the arithmetic. Both types of treatment profoundly influenced arithmetic for three thousand years, and the Babylonian peculiarities confront us hourly, whenever we note the time of day. The concept of a numerator and a denominator, combined in a particular way to form a fraction, is obviously complicated. The Egyptians sought to escape the difficulty by confining their attention to unit fractions, having the numerator unity, with the single exception of two-thirds. This required the writing of long series of fractions. Thus seven-eighths was written as K, 34, % or as %, %, }{2- The illustration reveals one weakness of this system, namely, that the separation is not unique. The earliest numerical table which has come down to us consists of the Egyptian conversion into unit fractions of fractions with numerator 2 and odd denominators from 9 121 122 THE HISTORY OF ARITHMETIC 5 to 99. Herein % is given as %f Ms; Ks as Mo, %o, and so on, occasionally employing 4 or 5 unit fractions. These Egyptian fractions occupied a large place in later Greek arithmetic, wherein also an exception is made in a special symbol and special treatment for %. Egyptian problems on fractions appear in Greek papyri from the fourth to the ninth centuries of the Christian Era. The Ahmes papyrus contains problems concerning the division of loaves of bread, not exceeding nine, among ten people. The results are expressed in unit fractions. The Greek papyri, many centuries later, have the same problems couched in abstract terms. Minutes and seconds. The Babylonian sought escape from fractional difficulties by confining his attention to sexagesimal fractions, that is, with denominators 60 and powers of 60. This is precisely our present method with decimal fractions, using however the denominator 10. In particular the Babylonians applied these fractions to the circle and to the measurement of time; from this they progressed to the application of these subdivisions to weights and other measures. This logical and psychological advance with decimal subdivisions is today being advocated by many scholars and manufacturers. The burden which our awkward system of weights and measures places upon our school children is tremendous; the bad effect upon our export trade is serious. Teachers should be conscious of the great advantages offered by the metric system. Babylonian influence on Greek astronomy. The Babylonian astronomy exerted from early times great influence upon the Greek astronomy. In the second century B.C. FRACTIONS 123 Hipparchus, father of astronomy, introduced Babylonian fractions into Greek astronomy. From that time to this they have remained in astronomical computations. During the Middle Ages these fractions were applied to GEMMA FR1SIUS, COLOGNE, 1576 Treatment of "astronomical" fractions in the arithmetic of Gemma Frisius (.see page 69). The various subdivisions are signs of the zodiac (S.), 30�, degrees (g.), minutes (m.), seconds (2), thirds or -5^3 (3), and fourths or In the first addition problem wchave: S. g. m. 2. 3. 4. 1 16 25 17 21 27 2 20 18 22 30 12 3 39 43 39 51 39 wherein the author suggests the use of a "sign" of 60�. In the problem below the ordinary sign of 30� is used. S. g. m. 2. 3. 4. 2 16 25 17 21 27 4 20 18 22 30 12 7 6 43 39 51 39 In the second column the 30� have made one sign to be carried. Such explanations were common up to 1650 a.d. all computations, replacing in large measure the unit fractions of Egypt. The designation astronomical or physical fractions was applied to them. Leonard of Pisa expresses the approximate root of a third-degree equation in sexagesimal fractions carried to the eighth place. This corresponds to our solution by Horner's method to the twelfth place, using decimal fractions. Our words 124 THE HISTORY OF ARITHMETIC "minutes" and "seconds" go back to Latin forms minutiae primae, minutiae secundae, meaning first fractions, second fractions, and so on. Roman fractions; apothecary tables. The Romans simplified the fractions following the Babylonian pattern. The base was chosen as 12, which appears also to be an original Babylonian subdivision. This was applied in Rome first to the unit of weight, the as; the twelfth of this as was the uncia, from which we get our words "ounce" and "inch." Note that again after beginning with the concrete, the fractional numbers used are made abstract to apply to other measurements. Roman fractions had a further complication in that special symbols were devised and used for K2 to 1;Hl2, for }� as one and one-half twelfths, 3^4,3^6, 34s, He, H44, and on to 3/576 and to even smaller fractions. In spite of the great difficulty of operating with these symbols, they continued in arithmetical instruction during the tenth to the thirteenth centuries. Our present apothecary weight symbols trace back to these Roman devices. Upon the Roman abacus separate little bars were provided for certain of the more common fractions. Greek fractions. The Greeks employed common fractions as well as unit and sexagesimal, writing the numerator with one accent mark and the denominator written twice with two accent marks. Unit fractions were indicated simply by the denominator with one accent, quite similar to the Egyptian procedure, which was to write the denominator surmounted by a heavy dot. Occasionally the Greeks wrote the numerator with the denominator in the position where we write an exponent. FRACTIONS 125 As Dcunx Dccunx wlDexuns Dodrans Bisse S^ptunx Semis Qaincmn X W w tt 1 1 288 scripulus 1 1 1 1 216 tremissis 576 1152 obolus . cerates .1728 siliqua 2304 chalcus SMALLER SUBDIVISIONS OF THE "uncia" 126 THE HISTORY OF ARITHMETIC Hindu and Arabic fractions. The Hindu fractional forms were similar to the form which we use, without the bar. Mixed numbers were written with the integral part o above the fraction; thus 8%i was written 3. Brahma- gupta gave systematic rules for the fundamental operations with fractions. Extensive treatment, including negative forms, is giv6n by Mahavir and also by Bhas-kara. The latter writer says: "After reversing the numerator and denominator of the divisor, the remaining process for division of fractions is that of multiplication "; Brahmagupta reduces dividend and divisor to a common denominator, before inverting; and Aryabhata indicates the same procedure. The complications of unit fractions, common fractions, and sexagesimal fractions were augmented by an Arabic device which fortunately made little impression on European writers beyond Leonard of Pisa. This Arabic device consisted in writing a fractional form -J- -�- to mean H+H of }i or TV t\ to mean Hs+Hi of Ms; an elaborate treatment of such forms was given by Al-Hassar, probably in the twelfth century, translated into Hebrew by Moses ben Tibbon in the thirteenth. The word "fraction." "Our notation of fractions is quite certainly based upon Arabic forms without the bar, these being derived from the Hindu. The Arabic word for fraction, al-kasr, is derived from the stem of the verb, meaning "to break." The early writers on algorism commonly used fractio, while Leonard of Pisa and John of Meurs (fourteenth century) use both fractio and minutum ruptus or ruptus. Early writers in English FRACTIONS 127 frequently used the corresponding expression, " broken numbers.'' The two earliest English algorisms, mentioned above, do not contain any discussion of fractions, except incidentally one-half; in this these manuscripts follow Sacrobosco and Alexandre de Ville Dieu. Common or vulgar fractions. The modern treatment and the terminology of common fractions appears in Recorded Grounde of Artes, with the exception only of the expression "common" fraction or "vulgar" fraction. This latter designation was used after the introduction of decimal fractions to distinguish the ordinary from the decimal fractions. Continental writers, like Peurbach of the fifteenth century in his arithmetic printed in 1534 and earlier, used the Latin expression fractiones vulgares or minutiae vulgares to distinguish these from the sexagesimal fractions. In English the word "fraction" appears to have been used first by Chaucer (1321). In early American arithmetics the designation "broken numbers" was used as an alternate for "fractions." "Vulgar" was applied to common fractions to distinguish them from "decimal fractions" or "decimals" (Pike, 1788); the treatment involved few modifications from present procedure or terminology. Decimal Fractions The forerunners of decimal fractions. A thousand years intervened between the discovery of the simple device for representing all integers in a decimal scale by nine symbols with a zero and the extension of the same principle to fractions in a decimal scale downward. Numerous approaches to the fundamental principles of 128 THE HISTORY OF ARITHMETIC decimal fractions were made. The summary of these steps leading to the development of decimal fractions RIESE, " RECHNUNG AUFF DER LINIEN UND FEDERN, LEIPZIG, 1559 The multiplication and division of common fractions in the popular German arithmetic of Adam Riese. The German word Bruch, or eine gebrochene Zahl (ZaV), corresponds to the English " broken number," in use at this time by Humphrey Baker in England. well illustrates the naturally slow progress in effecting changes in the symbols of number and of measure which the workaday world employs. The Babylonian system of fractions, to which we owe our minutes and seconds, corresponds in the scale of 60 to decimal fractions in the scale of 10. The same symbolism of primes was later used by some writers for decimal fractions. In approximations of square root and cube FRACTIONS 129 root results were frequently given in sexagesimal fractions. However, Johannis de Muris in the fourteenth SIMON STEVIN, "LATHIENDE," LEYDEN, 1585 First printed work to give a discussion of decimal fractions, appearing both in Flemish and in French (Disme) in 1585. At the top of the right-hand page in the third line 89.46 is indicated by 89 followed by 0 within a circle, 4 followed by 1 within a circle, and 6 followed by 2 within a circle. Incidentally it may be mentioned that this same notation was used for algebraic purposes by Stevin; with two plus signs interjected this expression just given would represent 89+4a;+63C2. The decimal point appears to have been used first in 1616 in the works of John Napier, inventor of logarithms. century gave the square root of 2 as 1-4-1-4, saying that the 1 represented units, the first 4 tenths,' the second 1 "tenths of tenths," and the second 4 "tenths of tenths of 130 THE HISTORY OF ARITHMETIC tenths." However, he then extended this, writing the result also to twentieths of twentieths of twentieths, finally giving the result in sexagesimal fractions. The square root of 2 was obtained by writing 2,000,000, extracting the root of this, dividing by 1000, and reducing the result to sexagesimal fractions. This method appears in manuscripts of the twelfth century and in printed books of the sixteenth century. Special rules for division by integral multiples of 10, 100, and the like appeared before the invention of printing. This led to an actual decimal point in one problem in the treatise by Pellizzati (1492) but the author makes no further use of the device. Interest problems brought Christian Rudolff to a practical use of decimal fractions in computing compound interest. His mark of separation is a vertical bar, which was quite frequently used by later writers who gave an explanation of decimal fractions. The trigonometric functions had a large part in emphasizing the practical necessity of some simple device for extended computations. The Greeks gave a table of chords in a circle with radius 60; for any refined computations this required the use of primes and seconds or more. Peurbach about the middle of the fifteenth century determined to use in a table of sines the radius 60,000 or 600,000; his able pupil Regiomontanus extended this to 6,000,000, and finally to 10,000,000. The Hindus and the Arabs gave the shadow function, or cotangent, with a stick of length 12. Regiomontanus here also adopted a decimal base 100,000. In both cases only a decimal point was necessary with a unit radius to give the modern tables. FRACTIONS 131 Simon Stevin discovers decimal fractions. The first systematic discussion of decimal fractions with full appreciation of their significance was given by Simon Stevin of Bruges in 1585. His work in Flemish, entitled La Thiende, was published at Leyden by the famous Plantin press. This was republished again in 1585 in French with the title La Disme; in 1608 an English translation by Robert Norton, The Art of Tenths or Decimall Arithnietike, appeared in London. This work is addressed to astronomers, surveyors, masters of money (of the mint), and to all merchants. Stevin says, of this work, that it treats of "something so simple, that it hardly merits the name of invention.,, He adds: "We will speak freely of the great utility of this invention; I say great, much greater than I judge any of you will suspect, and this without at all exalting my own opinion .... For the astronomer knows the difficult multiplications and divisions which proceed from the progression with degrees, minutes, seconds and thirds .... the surveyor, he will recognize the great benefit which the world would receive from this science, to avoid .... the tiresome multiplications in Verges, feet and often inches, which are notably awkward, and often the cause of error. The same of the masters of the mint, merchants, and others .... But the more that these things mentioned are worth while, and the ways to achieve them more laborious, the greater still is this discovery disme, which removes all these difficulties. But how? It teaches (to tell much in one word) to compute easily, without fractions, all computations which are encountered in the affairs of human beings, in such a way that the four principles of arithmetic which are called addition, subtraction, multiplication and division, are able to achieve this end, causing also similar facility to those who use the casting-board (jetons). Now if by this means will be gained precious time; . . . . if by this means labor, annoyance, error, damage, and other accidents commonly joined with these 132 THE HISTORY OF ARITHMETIC DISME: The Art of Tenths, OR, rtrait!oi!jMulHpliuuon, and DJiufioa. Inutntii hjtht exctfieut Mtthtmsticim, Simon Steuin. Publiftied in Engblhwkb fome additions Imprinted at London byS.S. iotMuqb ^///fy.aridareiobc/oldatliisfi'opac Saint MJgnut ccrutr. I 6 o 2, ENGLISH TRANSLATION OF THE FIRST WORK ON DECIMAL FRACTIONS The notation used in this text is the same as that employed in La Disme by Stevin in 1585. From disme we have the word "dime." FRACTIONS 133 computations, be avoided, then I submit this plan voluntarily to your judgment. "* What can one add to these words of the first writer on the subject, and an independent discoverer of decimal fractions? All that Stevin says applies today, hardly with the change of a letter. The genius of Stevin is evident in the comprehensive grasp which he had of the universal application of decimal fractions to affairs. Much of the benefit of this invention is lost to us in America, because we persist in using non-decimal weights and measures. Evolution of the decimal point. The symbolism of Stevin consisted in marking the place of units by a zero within a circle, and each decimal place by the digit corresponding to the number of the decimal place inclosed within a circle, following or above or below the digit of the numerator of the decimal fraction to which it appertained. Such an awkward notation could not survive, for even the author tried three variations of this symbolism on one page. The transition to the decimal point as now used in America and England, or the comma as used on the Continent, was independently effected by several early writers. The immediate application of decimal fractions was made particularly to the trigonometric functions and to logarithms. The decimal point in print appears in 1616 in the English translation of Napier's fundamental work on logarithms. Many writers to the end of the seventeenth century used awkward notations; thus Milliet de Chales, in his encyclopedic work on the mathematical 1Simon Stevin, La Disme, 1585. THE HISTORY OF ARITHMETIC Vulgar and Decimal. ^ NOTATION. Tb$ TABLE. | wbotc Numticis i Decimal farts. 1 \6 S 4 3 2 I , i 2 3 4 5 <* | r&~tf~3 hs S 5 s� *T3 �t> �"& & r r t undreds, at si 5* ? �-� 51" rts of a Hundred S ^ 5 �5, *, *. * ft! ' 5* *� 3 1 �*- � 1 ** 1 The V S E. Op thislWir will appear in the" following Qbffrw* tions. I. Tha.t Decimal TraRions are always feperated from whole Numbers by fome diltinguifhing Mark,asa Comma* z%Period9or the like. So 654321, are Integers-, aod ,123456 Decimal Parts. And from hence_is derived a Untverfal Rub to diftinguifll Integers from Decimals, in any mtx'd Sums whatfoever,viz. That the Integers always lay on the leftf and the Decimals on4he Right hand oj the Separatrix. II. The Denominator ?s always omitted in the Notation �F Decimal VraQions. \ ?lws, , 1 is the Notation of 40* Q 2 DECU DECIMAL FRACTIONS AS PRESENTED BY GREENWOOD IN 1729 FRACTIONS 135 sciences, published in 1690 after the death of the author, used the left half of a pair of brackets. Cavalerius in his Trigonometria (Bologna, 1643) uses the decimal point and gives a full explanation of the subject. During the seventeenth century the arithmetics which avoided any mention of decimal fractions were about as numerous as those which gave some treatment of them. Advantages of decimal fractions. By the eighteenth century the utility of these geometrical fractions, as they were sometimes termed, had been demonstrated so often and so clearly that the treatment of this subject became a regular part of arithmetic. English texts of the early eighteenth century commonly treated the decimal arithmetic extensively. The American texts of the eighteenth century included full discussion of decimals, using the word separatrix to designate the decimal point. Isaac Greenwood in his Ariihmetick Vulgar and Decimal of 1729, and Nicholas Pike in 1788 not only give a modern treatment of the subject, but both include the abbreviated process to obtain the product to any given number of places by reversing the multiplier and the abbreviated process in division. To such complete explanations of decimal notation is undoubtedly due the adoption of decimal coinage in 1785 by the Continental Congress. The development of decimal fractions illustrates the process of evolution in the realm of mathematical ideas. As we trace the steps culminating in this useful device it is perfectly evident that a succession of thinkers made possible this attainment. Similarly in practically every advance in arithmetic, algebra, and trigonometry, as in 136 THE HISTORY OF ARITHMETIC all science, a host of intellectual workers have participated to make the advance possible. In the field of science we are truly the heirs of all the ages past. Bibliography for Supplementary Reading Sir William Ridgeway, Roman Measures and Weights, and Roman Money, in Sandys, "A Companion to Latin Studies/' 3d ed., Cambridge University Press, 1921. Aubrey Drury et al.> World Metric Standardization. San Francisco, 1922. L. L. Jackson, loc. cit., pp. 85-110. See the references to Chapter I. Consult the encyclopedias and dictionaries under Decimal, Fraction, Minute, Second, Time (measurement of), and Trigonometry (angles). CHAPTER VI BUSINESS ARITHMETIC Applied arithmetic. The application of arithmetic to commercial problems extends from earliest times of historical record to the present day. However, among the Greeks and in Europe until towards the end of the fourteenth century these applications did not become the material for written exposition. The early European algorisms present simply the technique of arithmetic, without any practical applications. The single exception to the rule is the work of Leonard of Pisa (1202 a.d.) which directly under Arabic influence gave a sufficiently extensive treatment of arithmetic to permit inclusion of problems on business. Everyday problems of the Egyptians. The Egyptian arithmetic might well be taken as a model today in that the problems of that ancient day are concerned with the daily life of the people. How much corn does it take to stuff a goose? How many sacks of flour in a given granary? What is the cost of manufacturing a certain fine piece of jewelry? What is the cost of making 20 gallons of beer? For the preceding, the formula is given. The pedagogical soundness of Egyptian procedure was appreciated by Plato, who said {Laws, 819): "All freemen, I conceive, should learn as much of these various disciplines as every child in Egypt is taught when he learns his alphabet. In that country, systems of calculation have been actually invented for the use of children, which they learn as a pleasure and amusement. They have to distribute apples and garlands, adapting the same number either to a larger or less number of persons. . , , 10 137 138 THE HISTORY OF ARITHMETIC Another mode of amusing them is by taking vessels of gold, and brass, and silver, and the like, and mingling them or distributing them without mingling; as I was saying, they adapt to their amusement the numbers in common use, and in this way make more intelligible to their pupils the arrangements and movements of armies and expeditions, and in the management of a household they make people more useful to themselves, and more wide awake; and again in measurements of things which have length, and breadth, and depth, they free us from that ludicrous and disgraceful ignorance of all things which is natural to man." Practical arithmetic of the Hindus. The Hindus, no less than the more ancient Egyptians, had a fondness for arithmetic as applied to the world of affairs. Brah-magupta, following Aryabhata, gives the Rule of Three, applied to ''the barter of commodities,'' employing a terminology later adopted in translation by the Arabs. Principal and interest, partnership, and gain in trade also make their initial appearance in systematic form in India with Aryabhata and Brahmagupta, and continue in prominence in the Hindu treatises of Mahavir, Sridhara, and Bhaskara. Mensuration as an arithmetical exercise is also a favorite Hindu topic, but this appeared earlier in Greece in the works of Heron of Alexandria (c. 50 a.d.). The work on arithmetical and other series is more detailed in India than in Greece, following algebraic lines* Arabic business arithmetic. The practical work of the Hindus was utilized by several Arabic writers, and particularly the work on series. A systematic exposition of Arabic commercial arithmetic was written by Abu '1 Wefa (940-998 a.d.) of Bagdad, one of the most famous astronomers of his day. His arithmetic includes such topics as duties, exchange, bookkeeping, commercial BUSINESS ARITHMETIC 139 operations, mensuration, and weights and measures. The Hindu Rule of Three is found generally in Arabic alge- 302 The Rule of Fellowftiip. The RuM of Fellowftiip wiih� out time � Ve now will I fhew you 0f the Rule of Fcllowfhipor Company, which hath fundry opcration$,a;cor. ding to the diuer; number of the Company.This Rule is fotperime without difference of time, and foraetimes there is k it difference of time Firfl I will fpeake of cdac without difference of time, of which let this be an example. Aqoeftion foure CMcrchtntt of one fimpanj made 4 tfcopany. y^^f^^y dmerfy ifor thefirft byed in $0 pound, tho fecond 50 pound, the third 60 poMlid^ud the fourth 100 pound, whichftocfy they occupy ft iongf till it was increaftdto 3000 pound. Now I demand efjwrvbtt (hould etch recline at the parting of this money\ Scholar. Jpettcfo* frjat ttys Rule i'3 lift* tbc otfter, bat pet t&ere (0 a Difference Uiijicfr 3jperce(oenot. Maftcr, �J)M feill 3 $�to it to pOtl, fitft bn Addition, von tysll b;mg sU tbe particular famines of t\jc Merchant* into one fumme, tofcfeb Gjal be t&e fir ft fumme in poor tooling bpt^eGolden Rule, an&tfjc tofjole fumme of r&e gafocs bp ebat ftocke ftalt b$ f&e fecond furarae, $3tt> foj tye third fumme braic works, particularly in Al-Khowarizmi's algebra (c. 825 a.d.), in Al-Karkhi's arithmetic {c. 1010 a.d), in the arithmetic by Al-Kalasadi, a Spanish Arab of the fifteenth century, and in the sixteenth-century work by Beha Eddin (1547-1622). partnership as presented in recorde's "grounde of artes," London, c. 1542. {Illustration from a later edition) The rule of fellowship was a continuation in English of the Italian problems of a similar nature. The Italian commercial arithmetics of the late fifteenth and the sixteenth centuries greatly influenced European and British arithmetic. In Italy this type of work is found first in the treatise of Leonard of Pisa, written in 1202 A.D. 140 THE HISTORY OF ARITHMETIC Italian commercial problems. Greek and Roman use of the principles of business arithmetic are indicated by numerous early references to the subjects of interest, inheritance, and mixture or alligation. Leonard of Pisa, mmmmmmmmm THE CONTENTS. Page T.he Introduction, i �S�*�33hr Arabian CharaElers. I ,_' vsj Arithmetical CharaQers, � tf^3$K CHAP I. NvMERition, The table. Mr. Lock's Method of Numeration. The Roman Notation, CHAP. II. Addition. Of Integers. The Proof. . Of Divsrfe 'Denminattbns; with Tabus* va. Of Coin. M , . Of Weights; Troy, Apothecaries, Averdupots. Of Meafures : long, Z)ry, Cloth, Wm, Seer & Ale. to, it Of7�4. ' ** Of Time. CHAP. III. Suoa-raACTiOM. Of Integers' The "Proof. .,�*,, 8( diverse 'Denominations ; with Tables, sat f Hebrew Coin. Of the Roman Money mentioned in ScrtptuT*. Qh�erip*re-Long-Meafure. Of the Scripture-Meafures of Capacity. II �4 �4� �J a, �* CHA1 Page �! tO US' 17, CO NT E NTS. CHAP. IV. MBlTIHICiTlON The Table. Abbreviations. The 'Proof. A Method otfSTlcttnhg Multiplicationby Addition; C H A P. V. Division. The Italian Method, Long and Short. Abbreviations. Several Wcys of 'Proof A Method otperfurming Divifion by Tabling the Divifor. a 8, 19 CHAP- VI. Reduction. 30 3* to 5* r-� 35 33.3* CHAP. VII. VvteAa Fractions. 3> Nutation. i > Reduction. Abbreviations in Rcdutlion. Addition. Subfttaflion, a General Rule. Particular Rules. Multiplication. Vivifinn. UhfeenAing. Afcending. Mifcella'neous Examples, 55. to 41 43 44, 45 4<5 4*, 47 CHAP. VIII. Decimal Fa actions. 48 Notation, with Mr. Ougbtrtd's Table. 49, 50 Addition, and Subjiratlion. 51 Multiplication. ji, 55 A.Method o(cot\tr*&\r\g2)ccimal Multiplication. J4� to 5* tDiviJion, General Rule>. 56, to 59 'Partwlar Rules. jy, to 10 CHAP. IX Rootj, and Powers. n CONTENTS. Involution. Evolution, with a General Table. Of the Square-Root. A Ufeful ContraBion of Sir. Jfaae Newton's. Sir Jfaac Newton's Univerfal Rule, for. all Roots, viz. The Cube,�iquadrate, Suifolid and timber'Powers. 78 to C H A P. X. Continue? Proportion. Arithmetical 'Progrrffion. Geometrical Progrefliih. The Method of Changing the Order of Things. 7? 73 to 78 8> 8* to 89 CHAP. XI. Disjunct Proportion. 91 to 94 94 to 98 p8, 99 The Single Rule of Three, in General. "Direfi, Three Rules with Examples. Inverfe- The 'Double Rule of Three, Two Methods, viz. ,T The �#�*/ Method, by TwOperations in the fitogk fl�fe. 99, rco The Arew Method, by One Operation. ioi to tej The Method of varying Proportional Terms. io* yro/7orr/,d<;pending on the '7Joftri�e tfRootstii'Prxers 104,10s II. Barter. Two Examples. Ill' EQUATION Of PAVMINU The Common Rule. A Neceffary Obfervation. IV. Loss, and Gain. Tour Cafes. V. AttlCiTION. Alternate ? 2**)W C4/&1. Allegation "Partial **S w Stf Allegation Total. VI Position, or the RutE cf Fat$E, VII. Interest. &�/>/* ; JKwr 'Proportions. Compound j /^r 'Proportions, VIII. RjktATE, or Disccunt Compound. IX. Annvities. Page 12? J*3, 124 114 tO 125 US t%6, U7 128 128, \2f 119 to 155 131 1,4 �34 134 to 137 138 to I4� 140 to 142 143, 144 144 CHAP. XII. Practice ic� A Table of the Aliquot "Parts of a Shilling. 108 A Table of the Aliquot "Parts in the Uneven "Part\ of a Shilling. to� A Table of the Aliquot Parts of-i �ityfi. tti A Table of the Aliquot "Parts in the few; y<*W of a Pound Hi Concerning Tir*, Trett & Clough. 117 to 119 CHAP. XIII- R u t e s relating to T r a r> *, and Commerce. is� J. Fellowship. izo <&��/& no, tif ^Double. ISI) i:. An Abbreviation, m j. Annuities, at A'wp/e Interefl. 144 to 149 Annuities in Arrears; Four "Proportions. 14410 147 The "Prefent Worth of Annuities; iw -PropofititfH. 147 to 149 i* Annuities, at Compound Interefi. 149 to 154 Annuities in Arrears ; Four "Proportions. 149 to ij� The "PrefentWorth of Annuities; Four "Proportions. 151 to 154 X Of Reversions, and F&EtHOis Estates. 154 to ij# Annuities in Reverfion; Two "Proportions. 154 to 15* Tht'Purrhafe of Freehold Eflates, FivtPropofitions- 156 to 158 Dj. Hal kfs Table of the fWw of Annuities upon Z/fc j j8 S5SK 383� CONTENTS OF GREENWOOD'S " ARITHMETICS," 1729 The topics of business arithmetic are indicated in the headings. compound proportion, on the rule of false position, on capital and gain, and on miscellaneous problems constituted for centuries the storehouse from which other writers drew material. The words "capital" (Leonard of Pisa), "percent" and the symbol, "debit," and "credit" are all original with Italian writers, the final 142 THE HISTORY OF ARITHMETIC two words being found in Paciuolo's Summa d'Arithmetica of 1494, which includes the first treatise on bookkeeping. Italy was a center of trade from the thirteenth to the sixteenth centuries. Leonard of Pisa's father was an agent, a factor, in charge of a distribution center, a fac-torie, in northern Africa. There it was that Leonard learned the new numerals "instructed by a grocer.'' It is interesting to note that American arithmetics as late as 1870 used the word "factor" as meaning "agent." In consequence of their commerce, Italians were for centuries particularly interested in arithmetic. Manuscript treatises of the fourteenth century give numerous problems on equation of payments and other topics of commercial arithmetic. The Treviso arithmetic of 1478, the widely popular and oft reprinted work by Borghi entitled "la nobel opera de arithmethica" of 1484, Calandri's illustrated arithmetic of 1491, Pellizzati's Art de arithmeticha of 1492, and Paciuolo's Summa of 1494 are the great commercial arithmetics published in Italy during the fifteenth century (incunabula); only one other commercial arithmetic was published during this period, Widman's Behennd und hupsch Rechnung uff alien Kauffmanschqfften, Leipzig, 1489. Partnership with time, barter, interest, alligation, and a host of other topics taught in many continental schools and in American schools even in the twentieth century are included in these early commercial arithmetics. Such words as merchant, company, tariff, duty, payment, as well as debit, credit, percent, factor, and capital, are directly traceable to the popularity of the Italian arithmetics. American arithmetic exhibits today the influence BUSINESS ARITHMETIC 143 COMPOUND INTEREST AS TREATED IN THE COLONIAL ARITHMETIC OF HODDER, 1719 The multiplication by 20 gives shillings, by 12 gives pence, and by 4 gives farthings. Note that the interest on 356 pounds for 1 year at 6% is 21 � 7s. 2Kd. The fa. is abbreviation for the Latin facit, *'it makes." of the Italian civilization of the fifteenth century, which contributed so largely to the discovery and early explora- Chap. XVII. ,57 CHAP. XVII. A mofl brief & compendious way of working all manner of^ueftionsoflnterettuponlntereft Example. FIrft, State your Queftion thus: If ioo /. gain 61 what the Principal? 2. Multiply the fecond and third Numbers together, and divide by your firir, which is done by cutting off two firft Figures of the Pounds with a line, 3. Multiply them by 20, by 12, and 4, and all above 2 figures in each Multiplication carry over the line unto the left,as you fee in thefe following Examples. If 100/. in 12 Months gain 6 L what will # 5 61. gain in 18 Months? If, oo L--------6/.�-----�356 I. 6 I. s. d. 11' Mon.fa. 21-�07-�2-6 Max. fa. 10�13�-7 92-----00-�9 K3 ?7?� tion of the New World which now bears an Italian's name. Amerigo Vespucci made his first journeys to Spain while engaged in Italian commercial enterprises. It w&s not only the Italian commerce but also the interest in navigation and discovery which proved a fine stimulus to the study of mathematics. Recorded list of applied topics is largely self-explanatory: the Golden Rule, or Rule of Proportion direct, called 144 THE HISTORY OF ARITHMETIC the Rule of Three; the Golden Rule Reverse, and Double, and Compound; the Rule of Fellowship; the Rule of Alligation; and the Rule of Falsehood. Humphrey Baker includes, as do the seventeenth-century editions of Recorde, exchange and weights and measures as well as the topics mentioned above. American commercial arithmetic. Our American Pike (1789) apparently determined to include all possible applications of arithmetic. The Table of Contents covers six pages, touching more than one hundred separate arithmetical topics, as well as numerous others. The inclusion of annuities and of the tables of the compound interest functions is particularly worthy of note, as these functions are now returning to the American college texts on freshman mathematics. The reason why the early arithmetics contained such complete discussions of commercial arithmetic lies undoubtedly in the fact that this topic was studied largely by adults, and not by children. In American schools until long after the Civil War, and in rural schools of the nineteenth century, pupils of the seventh and eighth grades were mature; the great majority passed from the eighth grade into active business life. At the present time even in rural schools children of twelve to fourteen years of age are found in the seventh and eighth grades. For these children elementary algebra and constructive geometry are much more suitable than commercial arithmetic, which might well be presented in the tenth and eleventh grades, when the pupils are mature enough to be interested in topics which relate directly to business. The complexities of modern business arithmetic are greater BUSINESS ARITHMETIC 145 than those of elementary algebra and constructive geometry, which subjects are now found in seventh- and eighth-grade textbooks replacing the material which represented bygone days and conditions. Bibliography for Supplementary Reading Clive Day, A History of Commerce. Rev. ed. New York, Longmans Green & Co., 1922. C. A. Herrick, History of Commerce and Industry. New York, Macmillan Co., 1917. L. L. Jackson, The Educational Significance of Sixteenth Century Arithmetic. New York, 1906. F. Cajori, A History of Elementary Mathematics. Rev. ed. New York, 1917. D. E. Smith, History of Mathematics. Ginn and Co., 1923. A rapid survey of the history of Europe from 1000 a.d. to 1600 a.d., as found in any General History, will give to the teacher the necessary background for the work of this chapter. At the same time the development of the arithmetic will illuminate the general history of the period. Consult also encyclopedias under Commerce. CHAPTER VII THE TERMINOLOGY OF ARITHMETIC The progress of arithmetic. The words used in arithmetic reflect in some measure the historical progress of arithmetical science. The Egyptians, the Babylonians (indirectly), the Greeks, the Romans, the Europeans of the Middle Ages, the Normans and the Anglo-Saxons, the French, the Hindus and the Arabs, and even the Americans are represented. To assign each word to its proper place is no easy task; indeed, in several instances authorities would disagree as to whether a given word entered, for example, through the French or through the Latin. However, the purpose in this chapter is to present in broad outline the fundamental facts concerning the terminology found in American arithmetics, with indication of the genesis and development of the terms. Greek and Roman influence. The Greek influence is seen in words relating to mensuration, due to the Greek devotion to geometry. To the Greeks we owe the separation of mathematics into the four great fields of arithmetic, geometry, astronomy, and music. This philosophical tendency of the Greeks is reflected in the fact that the words mentioned, and the word "mathematics'' as well, are Greek. The Latin words in arithmetic are rather difficult to classify since the preponderance of Latin terms corresponds not to Roman interest in arithmetic but rather to the use of Latin as the universal language of educated people in Europe until the eighteenth century or 146 THE TERMINOLOGY OF ARITHMETIC 147 later. Even the early American universities continued for a time the use of textbooks in Latin. Both Greek and Latin forms are used to construct technical terms which are entirely modern. The word "telephone'' is based on Greek stems; similarly the word " trigonometry " is Greek in form, but it is a constructed word appearing first in a Latin work of 1590. In arithmetic there are several such constructed terms. Thus "fraction'' was used by Leonard of Pisa in 1202, and in two twelfth-century Latin translations of Arabic arithmetical works, being a translation into Latin of the Arabic term. Neither Caesar nor Cicero knew the term, and the fundamental concept back of the word is Arabic and not Latin. The words which represent ideas not technical but necessary in everyday affairs are largely from a parent language preceding Greek and Latin. Several Old English (Anglo-Saxon, old Saxon) terms are included in this group. Influence of other races. The Hindu and Arabic terms are few in number but significant in meaning and in import. Spanish or Dutch influence is evident only rarely, if at all, in arithmetical terminology, and German rarely. French terminology connects most closely with the English, while the Italian offers quite a few terms in business arithmetic. The Origin of Number Names Number words. The word "number" comes to us through the Latin numerate, meaning "to count," and more directly through the French nombre. The stem connects with a Greek word having the same significance, 148 THE HISTORY OF. ARITHMETIC and probably both have a common origin. The separate words designating numbers from one to one thousand connect directly, so far as we know, with the earliest forms which were used to designate numbers in the � n ] Tare and Tret. Tare is an allowance made to the buyer for the Weight oF.the hogfhead, barrel, box, or whatever elfe Contains thegoods bought, and is calculated at fo much per hogftead, barrel &'C. or at fo muc'i per cent, or at fo much in the grofs weight. Tret is an allowance made to the buyer of 4 pound* in 104. for wafle and duft in fome forts of goods. 1 ir pounds weight is call'd a grofs hundred, and J 00 pounds a neat hundred ; fome forts of goods arc foici by ore weight and-Jage by the other. When an article is fold by grofs figHffrcds, the price is generally fpecifiedat fo much per hundred, and the tare per cent. is upon 112 pounds. When an article is fold by ceat hundreds, the price is generally fpecified at fo much per pcund, and the tare per cent, is upon 100 pounds. The whole weight of ah aiticle, and the hogfhead or whatever contains it, being weighed together* is called the grofs weight, whether the article be fold by grefi. hundreds or neat hundreds. The weight of the article itftlf, after all allowance! are drafted, is called the neat weight,whether the article be fold by grofs hundreds or neat. Caft tft. "When the tare is at lo much per hogfheacL barrrel, &c. multiply the number oF hogfheads or b*r-�els by the tare.and the produft will be neat hundreds � reduce this produft to grofi hundreds if the article is fpecify'd in grofs hundreds, and fubtraft it from the grofs weight $ the remainder is the neat weight. Cafe td. When *the tare is at fo much per cent*, and is the aliquot part or parts of an hundred weight, divide the whole grofs by the faid part or parts which the tare is of an hundred weight� tht quotient thence arifiog girca I �4 1 Rebate or Difcount. Rebate or difcount is when a fum of money due at aqy time to come, is fatisfy'd by paying fo much pre� fcnt money, as being put out to inlereft, would amount iq the given furn, in ihe fame fpace of time. Find the amount of j�ioo for the time and rata per cent, given, which intereft add t# �100 * ^lcn by * ft** ting in the ruleof three fay, as that lum is to �.100 fo is. the debt or fum propofed to the prefent worth required. The difference between the prefent worth and the given fum is the rebate. Equation of Payments. When feveral Aims of money are to be paid at Jif-ferent times, and it is required at what time the whole ihall be paid together, without lofs to debtor or creditor j this is called equatbn of payments, or equating the time of payment. Multiply each payment by its lime, add the produils together, and divide this fum by thft whole deb:, the quotient is the equated time. Fellowfhip. By Fellowship the accompts of feveral partners, �**� ^ing in a company are fo adjufted or made up, that every partner may have his juft part of the gain, From the Arabic business arithmetic, tare J azimuth From Arabic astronomical works. Translation of an Arabic word gidr, which in turn is the radix) translation of the Sanskrit word mtila, meaning "root" root ) (vegetable) and "square root'' of a number. Used also by the Arabs in the sense of root of an equation, radical Appears in Clavius, Algebra, 1608; probably Arabic influence, fraction Used in Latin translations of Arabic works; translation of Arabic word meaning "broken number," which latter term was long used in English texts as explanatory to "fractions." Chaucerian usage. The first writer to introduce into English a large number of technical terms was Chaucer in the second half of the fourteenth century. In his treatise on the Astrolabe and in the popular Canterbury Tales Chaucer uses many technical terms, largely following French forms. Among these words introduced or made popular by Chaucer are the following: adden degree adding diameter altitude diminucioun (1303) angle . divisioun calculinge doseyn (1300) centre double circle egal (equal) circumscryve emispherie compasse encrees (increase) consentrik Of the above words many are also used in English by Gower and Wiclif in the same century; the Chaucerian mfinit latitude longitude millioun multiplicacioun' perpendiculeer proporcioun serie divyde 11 154 THE HISTORY OF ARITHMETIC spellings are retained above. Chaucer used also the following terms of measurement: mesure (1200 in English), galourj (1300), ounce, quart, busshel (1300), myle, barel, minutes, and secoundes. Norman French influence. Many words of Latin and Greek origin were introduced into the English language through the mediation of French from the time of the Norman conquest well into the fifteenth century. In England during this period French was the common language of the educated classes. In the universities Latin was used for a longer period of time, but the terminology of arithmetic in English was largely fixed by the end of the fifteenth century. In algebra and geometry, texts in English did not appear until in the sixteenth century, and the common terms in- these subjects date from that time. The printed works on arithmetic by Recorde, Humphrey Baker, and Leonard Digges probably reflect a terminology already well established in mathematical circles, particularly in the separate "ciphering" schools. Attention has been called {see page 142) to the fact that many of the business terms of arithmetic are Italian in origin. The use of the dictionary. The terms of mathematics may well be used by a teacher in instructing high-school children in the extended use of the dictionary. The history of mathematics in its major outlines is reflected in the terminology. Under the wise direction of a good teacher a child may himself discover much of this history through the aid of the dictionary. Real appreciation of the historical development of the mathematical sciences can be obtained by this simple method, THE TERMINOLOGY OF ARITHMETIC 155 Bibliography for Supplementary Reading James A. H. Murray (editor), A New English Dictionary, on historical principles. Oxford, 1884 to date. This remarkable historical dictionary is the achievement of a large group of scholars. Under each word are found illustrations of the use of the word from its first known appearance down to recent times. The teacher of mathematics, no less than the teacher of English and history, will obtain a new appreciation of the development of the English scientific vocabulary by examining this work. The preface and the chapter, "A Brief History of the English Language," in Webster's New International Dictionary (1917), should be read by the teacher using this excellent dictionary. Greenough and Kittredge, Words and their Ways in English Speech. New York, Macmillan Co., 1901. W. W. Skeat, An Etymological Dictionary of the English Language. Numerous editions. The introductory material is particularly worth reading in connection with the etymology of mathematical terms. CHAPTER VIII DENOMINATE NUMBERS General Considerations From concrete to abstract arithmetic. The use of numbers with concrete objects in measuring and weighing and counting doubtless constitutes the earliest step in the development of mathematical ideas. So far as integers are concerned, the step from the abstract to the concrete precedes any historical record. However, in the development of fractions among the Babylonians and among the Romans the abstract fractions were directly derived from concrete units which continued in use. The fractions of the Babylonians survive in our minutes and seconds; those of the Romans survive in our inches and ounces and in apothecary weights and measures. Time Beginning of the day. The most natural unit for measurement of time is the length of the day, from sun to sun. This day the Babylonians took from sunrise to sunrise, while the Hebrews and the Greeks reckoned the day from sunset to sunset. Modern business and law follow Roman procedure in beginning the day at midnight, while astronomers find it more convenient to follow the Arabs in taking high noon as the starting point. Subdivision of the day. The Babylonians divided the day into twelve hours as recorded by equal divisions on a sundial; the night was conceived as divided into twelve 156 DENOMINATE NUMBERS 157 corresponding parts. This twenty-four-hour day is Babylonian in origin; twenty-four divisions equal in length were first established by the Greeks. To the Babylonians is due also the division of the hour into sixty minutes and of the minute into sixty seconds. In circular measurement the degree was correspondingly subdivided, and the Latin translations used the terms partes minutae (or minutiae) primae and partes minutae secundae, whence our minutes and seconds. The use of terms corresponding to these does not appear however before the ninth century among the Arabs, and doubtless appeared in Europe first in Latin translations of Arabic works. The week and the month. To the Babylonians is traced also our week of seven days, corresponding respectively to the seven planets: Sun, Moon, Mars, Mercury, Jupiter, Venus, and Saturn. The origin of our words, "Sunday," "Monday," and "Saturday," is evident. The month, as the word indicates, is connected with the passage of the moon about the earth. The word is an old Anglo-Saxon one. Mensuration Origin of units of measure. The primitive measures of length are those derived from the human body; the foot, the digit, the palm, the span, the ell (elbow) or cubit, the pouce or thumb, and the pace appear as units of measure in the earliest records of civilized peoples, and similar natural units are in use among primitive peoples today. Scientific systems of measures and of weights and moneys, as we shall see, began with these primitive 158 THE HISTORY OF ARITHMETIC forms, modifying them in accordance with developments requiring greater precision. Babylonian linear units. The earliest scientific units of linear measure known to us are the Babylonian units, which correspond in idea most remarkably to the metric system of the French. The Babylonians, as we have shown in a preceding chapter, based their number system upon sixty. This same base appears constantly in their systems of weights and measures. Babylonian Table 3 lines =1 sossus 10 sossus = 1 palm 3 palms = 1 small ell (or cubit) 5 palms = 1 large ell 6 large ells = 30 palms = 1 large seed 60 palms = 1 gar 60 gar = 1 ush (or stadion) 30 ush = 1 kask or parasang The palm appears to have measured approximately four inches; the twelfth of a palm was also used as a measure, and also the third of a palm, or a digit. The measures of area were based upon the squares of the above units. Most striking is the fact that the cubic palm, or ka, was taken as a measure of capacity, closely approximating the liter and quart, while the weight of one ka of water was taken as the unit of weight, one mina, French development parallels the Babylonian. This ancient Babylonian procedure corresponds precisely to the French procedure, 3500 years later, in establishing a connection between linear and cubic measure, and a unit of weight. The parallel is one of the most striking to be found in the development of scientific ideas. Later DENOMINATE NUMBERS A Comparifon of the American foot with the feet 6f other Countries. The American foot being divided into 1000 parts, or into i inches, the feet of feveral other Countries will be as follow. America � London �� Antwerp � Bologna � Bremen � Cologne � Copenhagen �*-Amfrerdam � Dantzick � Dort � Frankfort on the main The Greek � Lorrain � Mantua � Meckltn � Middleburg � France � Prague � RhynelandorLeyden Riga � Roman � Old Roman � Scotch �- Strafburgh � Toledo �. Turin � Venice �- Parts. IOOO iooo 946 1204 964 954 965 942 944 1184 94.8 1007 958 1569 919 991 938 1026 1033 1831 967 970 1005 920 809 1062 1162 IncbJin. 12 12 II H II II .11 II II �4 n iz 11 18 n 11 11 12 12 points. 0 O 4 >>32 5 2>*S 6 4,89 6 5,76 3 3.8* 3 $M 2 2,97 4 3>�7 1 0,04 5 5>7i 9 5,61 O 2,01' IO 4,22 3 �>4J 3 4>40 4 4>S* � 21 11 3,98 7 i>48 8 o 11 11 12 11 IO 12 '3 o 4,32 o 2,88 9 2 73 8 5.66 11 1,96 A Ta b l f reprefentirg the conformity of tbe wr gl ts of the principal trading Cities of Euiope with thofe of America of America. lOClb.Ooz, 109 8 103 12 113 14 ft 100 of England, Scotland atd heUnd � � I qual 100 of dmflerdam, Paris, Lourdeuuy, Lrc, �� 100 0/* Antwert, or Braiant � � 100 of Rouen, /A* Vifounty �� �� 100 3/* Zy$�x, /ta G/y � � � 94 3 100 of Rocbelk �'�� � � � ...... no 9 100 of Toulouje, and upper Longucaoc �- - 92 6 loo of Marjcillei and Frotfenct � � 88 ir 100 of Geneva �� ' �. 123 100 of Hamburg � � � 107 5 100 of Francfort � � �� m jx 100 of Leigfic �� ..... � 104 5 pike's "arithmetick," 1788 ]3arly difficulties with weights and measures. 160 THE HISTORY OF ARITHMETIC among the Egyptians and among the Greeks the attempt was made to establish similar connections for unity of weights and measures. The mina in Babylon was subdivided into sixty shekels, and each of these into 360 she, or grains of corn. Origi- 17.2 REDUCTION of COINS. Roles For reducing jhe Federal Coin, and the'Cur-rencics of the ieveral United Sutas �, alfo Englifl), Jnfh, Canada, Nova-Scotia, Livres Tournois and Spamjh milled Dollars, each, to the par of all the others. I. To reduce Nixv-Hawp- 3. To South-Carolina and fiire^Mnffachufelt^Rhode- Georgia currency IJland^ Conneclicut, and Rule.�-Multiply the gi� Virginia currency. ven fum by 7, and divide t. To New-York and the product by 9. North-Carolina currency. Reduce � too New- Rule.�Add one third Hampfhirc&c. to South-to the given fum. Carolina, &c. Reduce �. too New- ico Hampshire; &c. to New- 7 York,&c. ------- �' 9)700 2)ico ,------------ 4- 33'6 8 �77 15 6} anfw. ...... 4. To EngHJh Monty. � 13} 6 8 anfw. Rule.------Deduct one �,-----*------ fourth from the given 2. To Penitfylvaniay fum. Nciv-Jerfey, Delaware and Reduce � too New-Maryhmdcurrency* Hampshire, j&c. \Q �ii- Rult.~-Add one fourth glifti Money. to the given fum. 4)100 Reduce /ido New- �� 25 Hampfhire, &c. to Penn- ------ fylVania, &c. � 75 jmfw. 4)100 5. To Irijh Money. �+� 25 Rule,�Multiplythegi- � ----- ven fum by 13, and divide �125 anfw. the produft by 16. Reduce nally the shekel was a weight, but in Babylon, as later in Greece and Rome, the term soon became employed as a unit of money. Quite probably the measures and weights of Egypt were based in ancient times upon those of Babylon. Certain it is that these orientals directly influenced the Greek, and thus the Roman, systems of weights and measures. currency troubles from pike's "abridgement" of 1793 In 1788 Pike printed ^.100 instead of ;6100 as here shown. It is evident that the early colonists had currency-difficulties somewhat analogous to those troubling Europe today. George Washington wrote in praise of Pike's work of 1788 as follows: "The handsome manner in which that work is printed and the elegant manner in which it is bound, are pleasing proof of the progress which the Arts are making in this Country. . . . The investigation of mathematical truths accustoms the mind to method and correctness in reasoning and is an employment peculiarly worthy of rational beings." DENOMINATE NUMBERS 161 The mile. The Roman foot, pes, with plural, pedes, is now determined as having been slightly less than the English foot; 5 feet gave the Roman passus or pace, and milia pasuum, or 1000 paces, gave the Roman mile, about 95 yards shorter than our mile. FEDERAL MON.EY. *<>} LABORED EXPLANATION OF FEDERAL MONEY IN PIKE'S * 'ABRIDGEMENT" OF 1793 There are only slight variations in this discussion from that given in the "Complete System of Arithme-tick" of 1788, which quotes the Act of Congress "the 8th of August 1786" establishing the federal money. The Coinage Act of April 2, 1792, formally established our present system in the essential details concerning the units. The silver dollar was first coined under the Act in 1794. A s the Money of Account proceeds in 3 decuple, or tenfold, proportion, fo, any number of Dollars, Dimes, Cents and Mills, is, firnplv the exprefiion of Dollars and Decimal parts of a Dollar :�Thus g Dollars and .8 Dimes are expreflkii.cjji =: qf^doU.� 12 Dollars, 4, Dimes and 7 Cents, .thus, 112,47�12 -.Vcr d�U' 20 Dollar*,.3^Dimes, 4 Cents and 5 Mills, thus 2C,345=rio7\\^ �nrt8 � Did. cm-=�5^o\-Do"arJ5=;4ST'o95Ur'a?Je$:5E:+ >S ** 9 9 7- Also, the names of the Coin3, le.6 th'ao a dollar* are fignitkant of their values. For the Mi}/, which ftands in the 3d place, at the right hand of the comma, or. place of thousandths, ia contracted' from Milk, the Latin for Tbtuftmd :�Cen(, which occupies tht fecond place, or place of Hundredths, is an abbreviation of Centum, the Latin for Hundred i�aVd Dime, which is in the fir ft place, or place of. tenths, U derived frc>iu Dif-mi, the French for Ttntbi- Natural measures of weight. The primitive system of weights takes as its ultimate unit the barleycorn or the grain of wheat, or the seed of the carob (a plant), whence we obtain the "carat" used in weighing gold and diamonds. Dry and liquid measure. The Romans employed different systems for dry and liquid measure which, in 162 THE HISTORY OF ARITHMETIC altered form, continue with us to the present day. The cubic foot, termed an amphora, was the fundamental unit of liquid measure, and one-third of it, or modius, the unit of dry measure. The congius, of which eight make an amphora, is about three-quarters of a gallon; the modius is very nearly one peck. The systems of avoirdupois and Troy weights are directly French in origin, based on Roman and modified by early Saxon forms. In particular Troy weight probably referred originally to weights used by jewelers in the French city of Troyes. Origin of inches and ounces. The Roman foot or pes was divided into twelve unciae, whence our inches. The Roman unit of weight, a bar one foot in length, was divided into sixteen smaller units also called unciae (uncia), whence our ounces. Varying standards. In early England up to 1400 a.d. there were, as on the continent of Europe, varying standards for the foot* and for the pound. However, the most common foot was probably that which measures 13.22 of our inches. By the statute of the Assize of Bread and Ale in 1266 the following table was established: "An English penny called a sterling, round and without any clipping, shall weigh thirty-two wheat corns in the midst of the ear; and twenty pence do make an ounce, and twelve ounces do make a pound; and eight pounds do make a gallon of wine, and eight gallons of wine do make a bushel/' The American colonies continued largely the use of the English units. However, after the Revolutionary War no less personages than Washington and Jefferson DENOMINATE NUMBERS CHAUNCEY LEE, "THE AMERICAN ACCOMPTANT," LANSINGBURGH, 1797 First appearance in print of the dollar sign. 164 THE HISTORY OF ARITHMETIC attempted to introduce decimal systems of weights and measures as well as of moneys. In linear and square measure and in weights our units correspond to the English. However, the English imperial gallon and quart are fully twenty percent larger than the American gallon and quart; the imperial gallon contains 277.274 cubic inches as opposed to 231 in an American gallon. Similarly, the English imperial bushel contains 2718.192 cubic inches as opposed to 2150.42 in the American legal bushel. The Metric System of Weights and Measures Metric system. The desirability is evident of some unit of length established with reference to some fairly unchangeable natural distance upon the earth's surface or in nature. More than one hundred years before the French Revolution Gabriel Mouton of Lyons, France, proposed that the arc of one minute of a great circle of the earth should be taken as the mile, of which the thousandth part should be the unit of length. Shortly afterwards the famous astronomers Picard (in 1671) and Huygens (in 1673) proposed the length of a pendulum beating seconds as the universal yard. In the eighteenth century numerous proposals along these general lines were made, particularly in France. Long before Mouton's proposals of 1670, the Flemish mathematician and scientist, Simon Stevin of Bruges, published, first in Flemish and in the same year, 1585, in French, a treatise in which the first explanation of decimal fractions is given. In this same treatise Stevin DENOMINATE NUMBERS 165 proposes that not only weights and moneys but also linear, square, and cubic measure and even degrees and minutes should be reduced to a decimal system. This proposal, together with the explanation of the decimal fractions, establishes for Simon Stevin a proud place in the history of the development of scientific systems of measurement. Talleyrand in 1790 brought the matter of uniform systems of weights and measures to the attention of the French National Assembly. In October of 1790 a committee qn which mathematicians were represented by Lagrange and Laplace reported favorably upon the desirability of a decimal system of weights, measures, and moneys. In 1791 it was agreed that the unit of length should be one ten-millionth part of the quadrant of a meridian. In 1795 the meter, as unit of length, the are (100 square meters) as unit of area, the liter as unit of volume, and the gramme or gram as unit of weight were formally adopted, together with the franc as monetary unit. In the course of the scientific determination of the unit of length and of weight the great scholars of France were assisted by Danish, Swiss, Spanish, and other European scholars. Incidental to the final determination of the gram the physicists Lefevre-Gineau and Fabbroni discovered that the maximum density of water is reached at 4� Centigrade. Uniform measures desired. In America Washington recognized in two annual messages to Congress the great desirability of uniformity in currency, weights, and measures. Thomas Jefferson, in a "Report on Weights, Measures and Coinage" submitted in 1790, urged the 166 THE HISTORY OF ARITHMETIC Schoolm'after's Affiftant. 12$ 3. Required the Intereft of 94I. 7s. 6d. for one year, five months and a half, at # per cent per annum. Ans. SI. 5s. id. 3t5qr*\ 4. What is the intereft or 12I. 18s. for one third of a month; at 6 per cent I Ans, 5,06^. 2. For Federal Money. RULE. Divide the principal by 2, placing the feparatrix as nfuar?, and the quotient will be the intereft for one month, in cents, and decimals of a cent ; that is, the figures at the left of the {epardtrix will be cents, and thofe on the right decimals of a cent. 2. Multiply the intereft of one month by the given number of months, or months and decimal parts thereof, or for the days lake the even parts of a.monih, Sec. EXAMPLES. i. What is the intereft of 341 dols. 52 cts. for 7 J months? ------s� Or thus, 170,76 Int. for 1 mo, 170,76 Int. for x month. x 7,5 months. 7� 85380 1195,52 ditto for 7 months. * 19532 85,33 ditto for � month, ------------ D.dcn;, ----------- 1280,700 cts.=12,807 1280,70 Ans. 1280,7 cts. = udoh. Zocts. ^m. 2. Required the intereft of jo dols. 44 cts. for 3 years, t months and 10 days, 2)10,44 xo days = \) 5^2 Intereft for 1 month* 41 months 2i4;0* ditto for 41 months, i,74 ditto for 10 days. 215,76 as. Ans. � idols, i$cts* 7**.*f> L 2 DABOLL'S "SCHOOLMASTER'S ASSISTANT," NEW LONDON, 1802 First appearance in print of the six-percent rule. DENOMINATE NUMBERS 167 change to a decimal system. James Madison in his "Annual Message of 1816" and John Quincy Adams in a "Report on Weights and Measures" presented in 1821 urged the extension of the decimal system of coinage to similar decimal weights and measures. Not until 1866, however, was the metric system made legal; at that time the yard was replaced by the meter, as official standard, in that the yard was legally defined as bearing the ratio 3600 to 3937 to the meter. Every year sees further progress towards the general adoption of the metric system, so that one may reasonably hope that before another century appears the prophecy of John Quincy Adams in 1821 will be fulfilled that "the meter will surround the globe in use as well as in multiplied extension; and one language of weights and measures will be spoken from the equator to the poles." Money Federal money. The dollar currency was established by an Act of Congress of 1786. However, before 1775 the dollar was frequently employed, having reference to the Spanish dollar, which was widely used even as late as 1850. So far as the textbooks of arithmetic were concerned, the bulk of the problems involving money continued to use pounds well to the end of the eighteenth century. American dollars were first coined in 1794. Our symbol for the dollar has been shown very clearly by Professor Cajori to be a transformation of the symbol for pesos employed in dealing with Mexico and South America and long current in colonial America as Spanish dollars. 168 THE HISTORY OF ARITHMETIC In early colonial America English pounds and guineas were more or less standard. However, French guineas, Dutch or German ducats, Spanish pistoles and pesos, and a dozen other types of coins were common. The early American almanacs frequently include tables of the value and weight of coins that were common in the colonies. Bibliography for Supplementary Reading Florian Cajori, "The Evolution of the Dollar Mark," Popular Science Monthly, December, 1912 (Vol. 81, pp. 521-530). W. H. Hallock, Outline of the Evolution of Weights and Measures and the Metric System. New York, Macmillan Co., 1906. See also the references at the end of Chapter VI. See encyclopedias under Calendar, Weights and Measures, Metric System, Dollar, and Money. CHAPTER IX THE TEACHER AND THE TEACHING OP ARITHMETIC Egyptian priests. The Egyptian priests were devoted students of the mathematical sciences. Undoubtedly to them was confided the instruction of the Egyptian children in arithmetic and geometry, as Greek writers indicate. The methods of the Egyptians are given high praise by Plato, who states (see page 137) that the Egyptians taught their children arithmetic by means of games, with apples and nuts and bowls. This testimony of Plato bears witness to the ability of the Egyptian teachers, and to this pedagogical gift the practical problems of the Ahmes papyrus and other ancient documents from the land of the Nile bear witness. Classical pedagogues. The pedagogue in Greece was the slave who accompanied the child to and from school. Both in Greece and in Rome elementary instruction including numbers was frequently given by such a slave. However, the teacher of arithmetic and more particularly of geometry enjoyed a higher status. In two points the classical tradition has continued practically to the present day: the teachers of Greece and Rome were poorly paid and their instruction was supplemented by a liberal use of the rod (or ferrule). Illustrations from the classical period represent the unfortunate subjects of instruction receiving corporal punishment at the hands of the teacher. Theoretical arithmetic was studied in Greece by adults as preparatory to philosophy. Our knowledge of the 12 169 170 THE HISTORY OF ARITHMETIC instruction of children in computation is fragmentary, depending upon chance references and not upon any systematic Greek account of the subject. Medieval instruction. In the church schools of the Middle Ages arithmetic was included largely for the computation of Easter; the technical treatise on this latter topic was called a computus. So far as arithmetic itself is concerned Boethius was the author whose text was widely used. In general this meager instruction in the mathematical sciences was given as an extra on feast days and on holidays. The most famous teacher of the early Anglo-Saxons was the Venerable Bede (c. 673-735), to whom is credited a computus dealing with the determination of Easter, and a treatise on reckoning with the fingers. In his course of study for priests arithmetic is given a proper place. Public education had a faint beginning in the church schools which are connected with the names of the educator Alcuin, born in 735, and the Emperor Charles the Great whom Alcuin greatly influenced. The Capitulary of 789 a.d. designates arithmetic as one of the subjects to be taught to children in the schools attached to religious foundations, and in such a way some instruction in elementary arithmetic was given to children in many parts of Europe from the ninth to the fifteenth centuries. In spite of this beginning, practical arithmetic from the twelfth to the sixteenth century was commonly taught by laymen outside the schools, in a way similar to modern instruction in music and dancing. In a few universities lectures on the new Hindu-Arabic arithmetic were given, commonly following the algorisms of Sacrobosco or that THE TEACHING OF ARITHMETIC 171 of Alexandre de Ville Dieu. The lecturer would read a few lines of the text, following this with a long disquisition upon the passage read. Reckoning schools. In Germany and Holland the Rechenmeister was appointed by the city to act as town clerk and was given a practical monopoly of the business of instruction in arithmetic. Frequently it was contracted that the Rechenmeister and city clerk should give instruction in arithmetic in the Latin schools where Latin and Greek constituted the principal subjects of instruction. The salary for instruction in reading, writing, and reckoning was frequently paid "in kind," and the teacher eked out an existence by supplementary tasks as sexton, bell ringer, or by attendance at the house of the wealthy or noble. These unfortunate characteristics of the profession marked also, as we shall note, the status of the teacher in early American schools. Instruction in England. In English grammar schools arithmetic was rarely taught, but appears in private schools for a separate fee. Humphrey Baker presents in his arithmetic of 1562 a typical advertisement: "Such as are desirous, eyther themselves to learn or to have theyr children or servants instructed in any of these Arts and Faculties heere under named: It may please them to repay re unto the house of Humphrey Baker, dwelling on the North side of the Roy all Exchange, next adjoyn-ing to the signe of the shippe. Where they shall fynde the Professors of the said Artes, etc., Readie to doe their diligent endeavours for a reasonable consideration. Also if any be minded to have their children boorded at the said house, for the speedier expedition of their learning, 172 THE HISTORY OF ARITHMETIC wmm, ARTS J*CD SCIENCES MATHEMATICAL!, TAughtin Fetter-lane neare the golden Lyon, op privatly abroad ac convenient houres, by Rob$rt liartmlt Teacher of the Mathcmaticki* f Arithmetic!^ 1 ^ f Cosmography 1 I | Geography, r.v I 1 Q I Navigation, I V%<\ r Si Architecture, f i ^ I Fortification, I Geometry, j [Horologxography^&oj Meafuring of Land. rSines5 Tan- The do&rineSptaine and^the ufeof the J gents, Se- of Trhnglcstfphericall.i Tables of ^ cants^ Lo- � garithmes. Accompts for Merchants by order of Debitor and Creditor* Fide Sed Vidt* VilUt %�*� :e, may meet with much for tficir enteitainment at aleiiure hour. We are happy to fee fo ufefui an American production, wh.ch, if it mould meet with the cncouiag^ment it deferves, among the tnhabi'ants of the United State?, will fave much money in the country, which-would otherwife.be fent to �urope� for-publications of this kind. We heartily recommend it to fthools, and tcvhe Community atlarge, and wim that the in-�uf>ry ani Ikill of th2 Author may be rewarded, for fo beneficial a work, by meeting with the general avtyrobation and encouragement cf the Publick. JOSEPH WILLARD, D. D. Pref dent of the Univerfity. E. WIGGLESWORTH, S. T. P. HolliS. S� WILLIAMS, L. L. D. Math, et Phil. Nat. Prof. Hollis. . . ^. -~ . ... rale College* 1786. TTPON examining M". Pike's Syftem cf Arithmetick and Geometry in Manufcript, 1 find VJ �t to be ^ Work of fuch Mathematical Ingenuity, that I efteem myfelf honoured injom-��.$ with the Reverend Prefident Willard, ar.d other learned Gentlemen, in recommending : -o the Publick a* a Production of Ger.ius, interfperfed with Originality in this Part of )-q*Tpiiij.y and as a Brsde fuitable to be taught in Schools�of Utflity to the Merchant, an* wetj Htia^tel even for the Univerftty Inftruirion.�I conf.der it of i'uch Merit, as that it wiH *5r�bv.fcy sain a very general deception and Ufe thrcw$h*ut the Republick of Letters. �ZRA UTILES, Prefidcr.W THE PRESIDENTS OF HARVARD, YALE, AND DARTMOUTH RECOMMEND PIKE'S ARITHMETIC THE HISTORY OF ARITHMETIC EARLY NEW ENGLAND COPY BOOK OF ARITHMETIC, 1784 176 THE TEACHING OF ARITHMETIC 177 The Detroit Gazette of Friday, October 31 1823. and in other issues contains an advertisement of the projected opening on November 3rd of that year of "a Classical School at the Academy." The terms of tuition were fixed at five dollars per quarter for Greek and Latin, the same for trigonometry with mensuration, surveying and navigation, whereas " Reading, Writing, English Grammar, Geography and vulgar Arithmetick', were given at two dollars and fifty cents per quarter. The final lines of the advertisement read: "N.B. Each scholar will be required to furnish one load of wood, delivered at the Academy within one week after their admission." In America the newspaper was, as we have seen, the medium chosen by the early teachers to reach the "patrons of science/' The textbooks of the colonies did not, so far as I have been able to find, include advertisements of schools, as did many of the English texts. Methods of instruction. Copy books were commonly employed in the schools of colonial days. At the top of a page the teacher wrote the topic under consideration; below this the pupil worked examples with a neatness not often found in the exercises of today. There is, of course, the possibility that only the finer specimens are preserved to the present day. Arithmetic a college study. Arithmetic continued to be taught in American universities until the end of the eighteenth century, while elementary algebra continued as a college subject throughout the first half of the nineteenth century. At Yale College Jeremiah Day, who taught mathematics there in the period from 1798 to 1817, was the author of an algebra which enjoyed 178 THE HISTORY OF ARITHMETIC long-continued popularity. Up to this time elementary arithmetic was practically a college-entrance subject. Recognition of the teacher. The high standing of the colonial teacher of arithmetic is indicated by the distinguished supporters whose commendations are recorded in early American textbooks (see page 175). Though the teacher was compelled to supplement his meager earnings by outside tasks, yet in numerous ways aside from the necessary adequate financial support the community expressed its interest in the fundamentally important task of teaching arithmetic. Bibliography for Supplementary Reading Foster Watson, The Beginnings of the Teaching of Modern Subjects in England. London, Putnam and Sons, 1909. Clifton Johnson, Old-Time Schools and School-Books. New York, Macmillan Co., 1904. George E. Littlefield, The Early Schools and Schoolbooks of New England. Boston, Club of Odd Volumes, 1904. W. H. Kilpatrick, "The Dutch Schools of New Netherland and Colonial New York." U. S. Bureau of Education, Bulletin No. 12, 1912. E. E. Brown, The Making of Our Middle Schools. New York Longmans, Green and Co., 1918. E. G. Dexter, A History of Education in the United States. New York, Macmillan Co., 1919. Paul Monroe (editor), A Cyclopedia of Education. New York, Macmillan Co., 1919. See articles under Arithmetic, School, and Teacher. THE INDEX Abacus: ancient use of, 25, 26 Greek children taught on, 10 influence of, on methods, 102 modern use of, 25, 26, 27 (ill.), 29,33 Roman children taught on, 23 Roman fractions worked on, 124 See also Soroban; Suan pan Abstract arithmetic, derived from concrete, 156 Abu 1 Wefa of Bagdad, commercial arithmetic by, 51, 138 Abu Zakarija Mohammed Abdallah. See Al-Hassar Addition: abacus used for, 26, 102 and subtraction, 102 ff., 104 (ill.) Egyptian method of, 3 Egyptian symbol for, 61 (ill.) in early copybook, 176 (ill.) methods of, 102, 103, 104 (ill.) Adelard of Bath, translator of Al-Khowarizmi, 53 Ahamesu papyrus. See Ahmes papyrus Ahmes or Rhind papyrus: algebraic problem from, 6 (ill.) designed as textbook, 61, 169 finding of, 2, 3, fractions appear in, 122 hieratic numerals used in, 2 practical problem from, 2 (ill.) Albert of Saxony {Tractatus propor- tionum), 62 Albiruni: problems involving proportion by, 47 (ill.) treatise on compound proportion by, 51 (ill.) Alcuin, church schools founded by, 170 Algebra: Arabian, 64 problems in, 65 earliest European treatises on, 53 early college subject, 177 Egyptian problems in, 6 , English terms established for, 154 Hindu adaptation of, 138 in text by John Ward, 74 (ill.) problem from papyrus, 6 (ill.) terms in Recorde's work, 151 Algebra of Al-Khowarizmi (L. C. Karpinski), 60 Algebras: Arabian, 49, 51, 64, 138, 139 bibliography of, 44 n, 60, 94,105 f. in Dutch, 81, 83 (ill.) of Oughtred, 77 twelfth-century translations, 53 Algorisms: bibliography of, 54, 55, 55 (ill.), 56, 57, 57 (ill.), 59, 65 (ill.), 66, 70, 120 early European, 137 early operations in, 105, 106 teaching of, in Middle Ages, 170 thirteenth-century, 52 (ill.) treatises on, 54, 56, 65, 70 twelfth-century, in Latin, 48 cm.) Algorisms t Two Twelfth-Century (L. C. Karpinski), 120 Al-Hassar: Arabic arithmetic of, 50 quotation from introduction of, 50 treatment of fractions in, 126 Al-Kalasadi, Aboul-Hassan AH ben Mohammed, Rule of Three, 139 Al-Kalcadi, Arabic arithmetic by, 50 Al-Karkhi: arithmetic by, 50, 139 Kafi fil Hisabt shortcuts in, 51 Al-Khowarizmi, Mohammed ibn Musa: fractions treated by, 151 influence of, on Latin texts, 64, 65 John of Spain's translation of arithmetic of, 151 180 THE INDEX Al-Khowarizmi� Continued: Robert of Chester's translation of algebra of, 53 Rule of Three in algebra by, 139 work of, preserved in Latin translations, 49 Al-Khowarizmi (L. C. Karpinski), 60 Alligation: Greek and Roman references to, 140 in colonial arithmetics, 144 in fifteenth century, 142 Al-Nasawi, Arabic arithmetic of, 50 Alphabet numerals of ancient nations, 12, 13, 19, 20, 21 America: colonial texts in, 85 ff. early instruction in, 173 first arithmetic printed in, 78, 79 (ill.) first known English arithmetic printed in, 80 (ill.) first university founded in, 78 native numerical systems in, 30 units of weights and measures in colonies of, 162 American Antiquarian Society Library, collection of arithmetics in, 98 American arithmetical work, list of early, 85 ff. American Indian, writing of, 1 Amphora, capacity of, 162 Angles, Babylonian system for measuring, 7, 8 Apothecary tables and symbols, origin of, 124, 156 Arabic arithmetics: bibliography of, 36, 44 n, 45 n, 49, 60 development of, 47 early, 48, 49, 64 Latin translations from, 53 twelfth-century, 50 See also Arabs, texts by Arabic fractions, writing of, 126 Arabic manuscript: � on arithmetic, 47 (ill.) on Hindu arithmetic, 51 (ill.) Arabic numerals, 13 bibliography of, 60 Arabic scientific works, European translators of, 52 Arabic shortcuts, 51 Arabic terms in present use, 147, 152, 153 Arabs: cotangent given by, 130 finger reckoning of, 23 hour for beginning day fixed by, 156 in Spain, 52 terms from, 146, 147, 152, 153 texts by, 50, 51, 64, 65, 138 use of large numbers by, 39, 40 Araya of Manila {Conclusiones Mathematicas), 86, 89 Archimedes: death of, 14 reference to problem of, 39 n Areas: Babylonian measurement of, 158 unit of, 165 Arithmetica and logista, 14 Arnold, Sir Edwin {Light of Asia), 39 n Art of Nombrynge, 57, 104 (ill.) Art of Tenths {La Disme, La Thi-ende), 131, 132 (ill.). See also Norton; Stevin Articuli for reckoning, 23 Aryabhata: Hindu text of, 64 practical use of Rule of Three by, 138 treatment of fractions by, 126 Astrology, interest of Babylonians in, 11 Astronomers: hour for beginning day fixed by, 150 La Thiende addressed to, 131 Astronomy: earliest scientific treatment, 8, 9 Gerard of Cremona, student of, 53 Greek, Babylonian influence on, 122, 123 Hindu versification of, 41 THE INDEX 181 Astronomy � Continued in early American universities, 81 mysticism and, 11 Attic system of numerals, 13 Augrim, or Augrym, 49, 56, 59 Austrian method: in texts, 105, 117 (ill.), 118 (ill.), 119 (ill.) of division, 116 of subtraction, 105 Avoirdupois weight, origin of, 162 Aztec hieroglyphics, facing p. 1 (ill.) Aztec numerals, 31 (ill.) Babylonian arithmetic, 7 ff. Babylonian fractions: basis of Roman, 124 in astronomical computations, 123 sexagesimal system in, 7, 122, 128 Babylonian numerals, 7 ff. Babylonians: bookkeeping done by, 11 circle as measured by, 122 cuneiform numerals of, 8 (ill.) curvilinear numerals of, 10, 10 (ill.) divisions of day and week fixed by, 157 hour for beginning day fixed by, 156 influence of, on Greek astronomy, 122 linear units of, 158 mathematics of, 7 multiplication table of, 9 sexagesimal system of, 8, 123, 157 tablets of, 9 (ill.), 10 (ill.) twenty system of, 41 zero symbol introduced by, 40 Baker, Humphrey: advertisement in textbook of, 171 commercial topics treated by, 171 modern terms used by, 151, 154 pages from arithmetic of, 72 (ill.) procedure in division by, 120 procedure in subtraction by, 105 Wellspring of Sciences, 72 (ill.), 151 Bakhshali manuscript, 43 n, 44 (ill.), 45 Ballou, Rev. Howard M., of Honolulu, 99 Bank, banker, bankrupt, origin of, 34 Barleycorn as unit of weight, 161 Barnard, F. P. {Casting Counter and Counting Board), 36 Beausard, Peter {Ariihmetices praxis), 70 (ill.) Beauvais, Vincent de, discussion from cyclopedia by, 58 (ill.) Bede, The Venerable: treatise by, 24 n use of computus by, 170 Bedoya of Peru {Selectiones ex Ele- mentis.....Arithmeticae .... Algebrae .... Geo-metriae . . . .), 94 Beha Eddin. See Eddin, Beha Beldamandi, Prosdocimo de, author of first printed book on new numerals, 70 Belleveder, Juan, author of mathematical work, 78 Belo, Benito Fernandez de {Breve Ariihmetica . . . .), 85 Ben Esra, Rabbi, treatise by, 59, 50 n Benedict, Susan R. (. . . . Hindu Art of Reckoning), 99, 120 Bh&skara, 44 n commercial terms used by, 138 Hindu text of, 64 Lilavati, 45 treatment of fractions by, 126 Bibaud, Michel {V arithmetique, in quatre parties . . . .), 98 Bible, mystic numbers in, 64 Bibliographies on: business arithmetic, 145 denominate numbers, 168 early textbooks, 85 supplementary reading, 99 fractions, 136 fundamental operations, 120 numerals: early forms, 36 used today, 60 teachers and teaching, 178 terminology, 155 182 THE INDEX Billingsley's Euclid, 152 Billion, amounts meant by, 149 Bittner (Handbuch der Mathematik), 105 Blundevil, Mr., His Exercises, 74, 172 (ills.) Board of Trade, finger numerals used by, 25 (ill.) Bodleian Library, Art of Nombryng in, 57 Boethius: arithmetic of, taught in Middle Ages, 170 chapter titles from, 62 (ill.) figurate numbers taught by, 18 Greek arithmetic translated by, 57,62 number theory of, 62 terms from, 150, 151 treatise after type of, 62, 68 Bongus of Bergamo, Petrus (Mys- ticae Numerorum), 64 Bonnycastle, John (Scholar's Guide to Arithmetic), 90, 91 (ill.) Bookkeeping: Babylonian, 11 early text on, 96 first treatise on, 142 problems, by Abu '1 Wefa, 138 Borghi, Piero: A La Nobel Opera . . . .: a commercial treatise, 142 first use of million in, 149 numerous editions of, 69 Bouthillier, Jean Antoine (Traite d1 Arithmetique . . . .), 97 Bowditch, C. P. {Numeration .... of the Mayas), 37 Bradford, William: first American arithmetic in English by, 77 "scratch" method used by, 116 (ill.) Secretary's Guide, 80 (ill.), 81, 85, 86 texts used by, 77 Young Man's Companion, 80 (ill.), 81, 85, 86 Bradwardine, Thomas (Arithmetica Speculativa), 62 Brahmagupta: Hindu text of, 64 practical problems by, 138 quoted, 44 rules for fractions by, 126 Brigham, Dr. Clarence S., of American Antiquarian Society, 98 Brown, E. E. (The Making of Our Middle Schools), 178 Browning, Robert, reference to Ibrahim ibn Ezra, 49 Buddha, numerals of, 38, 39 Biirgi, Jobst, method of multiplication, 111 Burnham, John (Arithmetick for Farmers), 82, 86 Bushel: American and imperial, 164 twelfth-century standard, 162 Business arithmetics. See Commercial arithmetics Cajori, Florian: Evolution of Dollar Mark, 168 History of Elementary Mathematics, 120, 145 .... History of Mathematics in the United States, 99 symbol for dollar explained by, 167 Calandri: illustrated commercial work of, 142 modern method of multiplication used by, 110, 115 Calculate, origin of, 26 Calculi on abacus, 26 Calendar, Mayan, division of year and months, 32 Canadian arithmetical works, list of, 97 f. Capella, Martianus, treatise by, 62 Capital, origin of, 141, 142 Carat, origin of, 161 Cassiodorus, Flavius, treatise by, 62 Casting accounts: on checkered board, 35 origin of term, 33 teaching in colonial America, 173 THE INDEX 183 Cavalerius (Trigonometria), explanation of decimal point, 135 Caxton, William (Mirrour of the World), 57, 58 (ill.), 67 Centimeter, origin of, 20 Chalcidian alphabet, earliest number forms from, 19 Chales, Milliet de, marking of decimal by, 133, 135 Chaplin, Joseph (Trader's Best Companion), 94 Charles the Great, influence of, on public instruction, 170 Chaucer, Geoffrey, terms used by, 49, 127, 153 Chauveaux, et al. (Thhes de Maihe- matiques . . . . ), 89 "Check by nines," 105, 106 Check, origin of, 35 Children: Egyptian, practical arithmetic taught, 137 Greek, education of, 10, 15, 169 Roman, taught on abacus, 23 Chinese abacus, 25, 26, 27 (ill.) Chinese arithmetic, 28 Chinese commercial problems used by Leonard of Pisa, 140 Chords, table used by Greeks, 130 Chuquet, Nicholas, use of " billion'' and "million" by, 149 Cipher, meaning of, 41, 149 Circle, division by Babylonians, 122 Clavius, Christopher: Algebra, use of "radical" in, 153 Arithmetica Prattica, 73 (ill.) Austrian method of division, 116 Epitome Arithmeticae Practicae, 114 "scratch" method of division used by, 114 (ill.), 116 Cocker, Edward: Arithmetick, 75 (ill.), 81, 89 Decimal Arithmetic, Austrian method of division in, 119 Codex, Maya, 31 (ill.) Coinage: decimal, influence behind, 135 weights, measures and, Jefferson's report on, 165, 167 Colebrooke, H. T. {Algebra .... from Sanskrit of Brahmagupta and Bhdskara), 44 n, 60 Collections of arithmetics, in: American Antiquarian Society Library, 98 Library of Congress, 98 New York Public Library, 98 Private librarv of Mr. George A. Plimpton, 98 University of Michigan Library, 98 Colleges, colonial, arithmetic taught in, 177 Colonial arithmetics, early, 78 ff. Commercial arithmetic: ancient and modern, 72 (ill.), 75 (ill.), 82 (ill.), 84 (ill.), 137 ff. Arabic, terms used in, 153 bibliography of, 51, 83, 84, 114 (ill.), 116, 138, 142, 144 fifteenth-century, 70, 142 medieval instruction in, 170 of Abu 1 Wefa, 51, 138 popularized by printing, 66 Commercial operations, treatise by Abu 1 Wefa on, 138 Commercial problems, Italian, 140 ff. Commercial terms: Italian origin of, 141, 142, 154 modern, used by Pacinolo, 141 f. used by Dearborn, 148 (ill.) Common fractions, 121 distinguished from decimal, sexagesimal, and unit, 127 Greek, 124 modern treatment of, 127 Computation, old and new, systems of (frontispiece) Computer, mechanical. See Aba-. cus; Computing rods; Counters Computing rods, 28 Computus: manuscript showing, 52 (ill.) reckoning Easter by, 170 used by the Venerable Bede, 170 Concrete arithmetic, early development of, 156 184 THE INDEX Congius, Roman measure, 162 Cook, David (American Arithmetic), 96 Copy books, early New England, 176 (ill.), 177 Cotangent of Arabs and Hindus, 130 Counters (jetons): bibliography of, 36 on Rechenbanck, 34, 35 used by Recorde, 35 n Crafte of Nombrynge, 56, 57 (ill.), 108 Credit and debit, first use of, 141,142 Cretan numerals, 11 (ill.) Crusades, interest aroused in Arabian teaching by, 52 Cubes, sums that make, 18 Cubic foot, Roman liquid unit, 162 Cubic palm, water content as unit of weight, 158 Cubit, meaning of, 157 Cuneiform numerals: applied use of, 10, 11 Babylonian, 8 (ill.) Cyclopedic works, 58 (ill.), 68, 178 Dacia, Petrus de, commentary by, 55 Day: hours for beginning, 156 subdivisions of, 156 Day, Clive (History of Commerce), 145 Day, Jeremiah, algebra by, 177, 178 Dearborn, Benjamin: PupiVs Guide, 90 commercial terms used in, 148 (ill.) Debit and credit, first use of, 141, 142 Decimal coinage, 135 Decimal fractions, 127 ff. and common, 127, 128 forerunners of, 127 See also Decimals Decimal numerals: finger reckoning in, 23, 24 Roman, 20 (ill.) trigonometric computations, 130 various systems of, 129 Decimal point: evolution of, 133 explanation of, by CaValerius, 135 first use of, 129 original symbols and words used for, 133, 150 Decimal system: among early peoples, 2 f., 39, 40 Jefferson's recommendation on, 165, 167 on checkered board, 28 recommended by Madison and Adams, 167 universal origin of, 27 Washington and Jefferson's attempt to establish, 162, 164 See also Metric system Decimals: and sexagesimals, 128 by Greenwood, 86, 134 (ill.), 135 English texts on, 74, 75, 77 English translation of first work on, 132 (ill.) first printed work on, 129 (ill.) first use of, in computing compound interest, 130 texts treating, 76, 77 (ill.), 86 See also Decimal fractions Dee, John, revision of Recorde's arithmetic by, 67 (ill.) Degrees of angles, origin of, 7 De Morgan, Augustus (Arithmetical Books from Invention of Printing . . . .), 60, 99 Denominate numbers, 156 ff. Denominator: in Egyptian unit system, 121 in Greek fractions, 124 Dexter, E. G. (History of Education in the United States), 178 Dictionary, use of, in mathematics, 154 Digges, Leonard, terminology of, 154 Digit: length of, 158 unit of measurement, 157 See also Finger numerals Digit, origin and meaning, 23, 151, 157 THE INDEX 185 Dilworth, Thomas: Schoolmaster's Assistant, 90, 92, , 93, 95, 97 advertisement in, 173 popularity of, 78, 83 title page and contents of, 88 (ill.) Dime, origin of, 132 n Discount in Dearborn's work, 148 (ill.) Division, 112 ff. as treated by Boethius, 63 Austrian method of, 116, 117 (ill.) early methods in, 114, 115, 116, 119 Egyptain operations in, 3 in Art of Nornbryng, 107 (ill.) in Grounde of Artes, 114 long, 113 (ills.), 117 (ill.), 119 (ill.) of fractions by Reise, 128 (ill.) on abacus, 27 ''scratch" method of, 109 (ill.), used by Hindus, 112, 112 (ill.), 114 (ill.),. 116 (ill.), 118 (ill.) Dollar, American: first, 167 symbol, 167 Dollar sign, first printed, 96, 163 (ill.) D'Ooge, M. L. (Nichomachus of Gerasa), 17 n Doubling and halving, 100 Drachmas 1000 in papyrus, 12 (ill.) Drury, Aubrey, et al. (World Metric Standardization), 136 Dry measure, systems of, 161, 162 Ducats in American colonies, 168 Duplication^ 101, 102 (ill.) Dutch colonies represented by work of Pieter Venema, 85 Duties, Arabic problems in, 138 Eames, Dr. Wilberforce, of New York Public Library, 98 Easter, determination in Computus, 170 Eddin, Beha, work by, 139 Egypt: children taught by games in, 137, 169 first textbook in, 61 fundamental operations in, 3 mathematical progress in, 7 weights and measures used in, 160 Egyptian arithmetic, 1 ff. Egyptian fractions, 5, 6 (ill.), 121, 122 Egyptian hieratic numerals, 5 (ill.) Egyptian hieroglyphic numerals, 2 (ill.), 3 (ill.), 4 (ill.) Egyptian problems: algebraic, 6 (ill.) arithmetical progression in, 7 (ill.) practical nature of, 6, 137 Eighteenth-century texts, 73 Elements (Euclid), 15, 61 Elephantine papyrus, 12 (ill.) England- first English printer in, 57 medieval, instruction in, 171 printing arithmetics in, 57 standards of measure in, 162 English: first extended treatise by American in, 80 (ill.) first manuscript arithmetics in, 56, 57 (ill.) first printed texts in, 57 f., 59 (in.) first separate American text in, 60 first use of fraction in, 127 first work on decimals translated into, 132 (ill.) reading of large numbers in, 39 f. English penny, twelfth-century, 162 Eskimos, number systems of, 23, 25 Euclid {Elements), 15, 61 algebraic terms in, 152 classification of numbers in, 16 treatment of fractions in, 62 European translators of Arabic scientific works, 52 Evans, Sir Arthur (The Palace of Minos), 11 n Even numbers. See Odd and even numbers r 186 THE INDEX Exchange of moneys; by Leonard of Pisa, 141 exposition by Ab 1 Wefa, 138 "reduction of coin" by Pike, 160 an.) Exchequer; origin of, 35 Fabbroni, Giovanni Valentino Ma-thias, maximum density of water determined by, 165 Farner, Jacob, arithmetician, 174 Federal money: adoption of standard, 167 problem, 163 (ill.) table by Pike, 161 (ill.) Fellowship. See Partnership Fenning, Daniel: American Youth's Instructor, 94 Der Geschwinde Rechner (The Ready Reckoner), 89, 90, 93, 94, 96 Fifteenth-century arithmetics, 70 Fifteenth-century commercial arithmetics, 142 Fifteenth-century manuscriot, numeration in, 100 (ill.) Figurate numbers, 18 Figure, probable origin of, 19 Finger numerals: ancient and medieval, 23, 24 (ill.) early Mexican, 32 modern (board of trade), 25 (ill.) See also Digit Finger reckoning: from Noviomagus, 24 (ill.) medieval treatise on, 170 peoples using, 23 Fink, first use of radius, 151 First printed texts in America: arithmetic of Venema, 81, 83 (ill.) arithmetic of Freyle, 78 English: extended treatise in, 77, 80 (ill.), 81 separate treatise by native colonial author, 82 (ill.), 86 separate treatise in, 82 (ill.), 86 First printed texts in German: commercial work of Widman, 70 on new numerals, 70 First printed texts on mathematics: algebra, by Paciuolo, 65 arithmetic of Treviso, 66 (ill.) bookkeeping, treatise by Paciuolo, 142 decimals, treatises on, 129 (ill.), 131, 132 (ill.) English: earliest on new numerals, 59, 60 first printed, in, ^7 f., 59 (ill.) illustrated arithmetic, 70 on fractions, 70 on new numerals, 70 First texts on mathematics : Egyptian, 61 English, two earliest, 56, 57 First use of: coefficient, 151 debit and credit, 141, 142 decimal point, 129 n decimals in computing compound interest, 130 fraction, 127 million, 149 percent, 141 problems in interest, 138 problems in partnership, 138 n problems in profit, 138 radius, 151 rectangle, 151 surd, 151 symbol for percent, 141 Fisher, George, pseud.: American Instructor, 78, 81, 82, 83, 86, 87 (ill.), 89, 90, 94, 96 Cocker's Arithmetick, revision by, 75 (ill.) treatises by, 75 Young Man's Best Companion, 81, 86, 89, 90 See also Slack, Mrs. Foot: American, compared, 159 (ill.) cubic, Roman liquid measure, 162 Roman, length of, 161 varying lengths for, 162 Fraction: Arabic and Latin words for, 126 first use of, 151 original concept, 126, 147, 153 THE INDEX 187 Fractions, 121 ff. Arabic device for, 126 Arabic treatment of, 50 astronomical, in Middle Ages, 123 Babylonian system of, 7,121,128, 156 common. See Common fractions decimal. See Decimal fractions Egyptian, 5, 6 (ill.), 121, 122 Euclid's treatment of, 62 first printed work on, 70 Greek: Babylonian influence on, 123 common, sexigesimal, unit, 124 treatment of % in, 122 Hindu, forms of, 126 in Greek astronomy, 123 physical, in Middle Ages, 123 Roman: Babylonian influence on, 124 in works of vSylvester II, 125 (ill.) influence of, 23 survivors of, 156 symbols for, 124 Roman abacus made for, 124 sexagesimal, 122 treated by Adam Riese, 128 (ill.) unit, 6, 121, 122, 124 Franc, monetary unit, 165 Fraser, Donald (. . . . Practical Arithmetic), 95 French development of weights and measures, 158 French, works in, 56, 129 (ill.) Freyle, Juan Diez: Sumario Compendioso,85,109 (ill.) colophon and title page of, 79 an.) commentary on, by D. E. Smith, 99 material on arithmetic and algebra in, 78 multiplication and division in, 109 Frisius, Gemma Rainier: Arithmeticae Practicae . . . .): many editions of, 69 title page of, 69 (ill.) treatment of astronomic fractions in, 123 (ill.) Frontius, modern terms used by, 150 Fundamental operations in early arithmetics, 100 ff. Gallon � American and imperial, 164 twelfth-century standard for, 162 Games, Egyptian children taught by, 169 Garner, Joseph, arithmetician, 174 Geometry: Peruvian text on, 94 practical, of Egyptians, 7 terms adopted in English, 154 texts on, in American colonies, 81 treated by John Ward, 74 (ill.) Gerard of Cremona: first use of surd by, 151 translation of Al-Khowarizmi by, 53 translation of Ptolemy by, 53 German: first arithmetic in, 69 numeration in early, 40 texts in, 34 (ill.), 50 n, 69 (ill.), 70, 105, 120, 128 (ill.), 142 German arithmetics: early, in America, 68, 69, 89, 92, 97 (ill.) "lightning" method used in, 110 published in St. Louis, 97 Germany, medieval, reckoning schools, in, 171 Golden Rule. See Rule of Three Golden Rule Reverse, commercial treatment by Recorde, 144 Gough, John: American Accountant, 94 Treatise on Arithmetic, 92, 95 Gower, John, terms used by, 153 Grain of wheat, unit of weight, 161 Gram unit of weight, 165 Greek: alphabet numerals in, 20, 21 first mathematical textbook in, 61 mathematical terms from the, 152 zero symbol in the, 40 188 THE INDEX Greek arithmetics: bibliography of, 36 Egyptian influence on, 14 Euclid's Elements, 61 figurate numbers in, 18 fractions in, 122, 123, 124 mensuration in early, 138 practical, 14 children taught, 15, 169 speculative, 15 theoretic, 169 treatises on, 16, 17 used by Leonard of Pisa, 140 Greek numerals, 11, 12 (ill.), 13, 14 Greek papyri, fractions in, 122 Greek tablets, 14 Greek weights and measures, 160 Greeks: decimal numeration, of, 40 hour for beginning day fixed by, 156 table of chords used by, 130 Greenough and Kittredge {Words . ... in English Speech), 155 Greenwood, Isaac: {Arithmetic!*, Vulgar and Decimal), 81, 86 Austrian method given in, 116, 117 (ill.) commercial topics, 141 (ill.) contents of book, 140 (ill.), 141 (ill.) decimals presented in, 134 (ill.), 135 first text in English by native colonial author, 82 (ill.), 86 term for decimal coint in, 150 terms in "-illion" in, 149 title page of, 82 (ill.) Guineas, French, in colonial America, 168 Guthrie, Jesse {The American Schoolmaster's Assistant), 98 Halifax, John of. See Sacrobosco Halliwell, J. O. {Rara Mathe- matica), 60, 99 Hallock, W. H. {Evolution of Weights and Measures), 168 Halving and doubling, 100 Hartwell, R.: "Philomathematicus," 67 (ill.), 74 revisions by, 67 (ill.), 172 (ill.) Harvard University, early texts used in, 81 Haskins, Charles H. {Studies in the History of Medieval Science), 60 Hawaiian arithmetic: early appearance of, 99 He helu kamalii, o ke aritemetika, 98 Hawkins, John, work on Cocker's ^ Arithmetick, 75 (ill.) Heath, T. L. {History of Greek Mathematics), 36 Hebrew, Arabic works translated into, 52 Hebrew arithmetics, 49 Hebrew numerals, 13 Hebrews, hour of beginning day, fixed by, 156 Hectare, origin of, 12 Heron of Alexandria, mensuration treated by, 138 Herrick, C. A. {History of Commerce and Industry), 145 Hieratic numerals, 2, 5 (ill.) Hieroglyphic numerals: Archaic Greek, 11 (ill.) Egyptian, 2 (ill.), 3 (ill.), 4 (ill.) Hieroglyphics: American Indian, 1 Egyptian, 1, 2 (ill.), 3 (ill.) Mayan, 32 (ill.) Hill, G. F. {The Development of Arabic Numerals in Europe), 60 Hilprecht, H. V. {Mathematical, .... Tablets from Nippur), 36 Hindu-Arabic arithmetic, teaching of, in Middle Ages, 170 Hindu-Arabic numerals, treatises on, 48 n, 60, 99, 120 Hindu arithmetic application of, 49 Arabic development of, 47 Arabic manuscript on, 51 (ill.) bibliography of, 48 n, 60, 99, 120 methods of, 46 THE INDEX 189 Hindu arithmetic � Continued oldest, 43 practical, 138 similar to modern, 46 Hindu fractions, forms of, 126 Hindu numerals, 38 ff. letter and word systems in, 41 Hindu problems used by Leonard of Pisa, 140 Hindus: Arab traders and methods of, 47 cotangent used by, 130 decimal numeration of, 39, 40 early texts by, 64 ff. multiplication by, 106, 107, 108 operations listed by, 101 processes and problems of, 43 "scratch" method used by, 112 zero symbols of, 40 Hipparchus, Babylonian fractions introduced by, 123 History, Hindu custom in writing, 41 of mathematics reflected in terminology, 154 reflected in arithmetic, 83 Hochheim (Progranim), 50 n Hodder, James: advertisement in arithmetic, 173 Arithmetick, 81, 82 (ill.) Austrian method taught by, 118 (ill.), 119 (ill.) English colonies represented by works of, 85 first separate arithmetic, 81, 116 reprint of English text, 86, 173 "scratch" method taught by, 116 treatment of compound interest by, 143 (ill.) Hoecker of Ephrata, Penn., Ludwig (Rechenbuechlein), 90 Hoernle, Rudolph (The Bakhshali Manuscript), 43 n Holland, medieval reckoning, schools in, 171 Holywood, John of. See Sacrobosco Horn book, 58 (ill.) Horner, approximate root, 123 Huygens, Christian, length of universal yard proposed by, 164 Ibn Al-Benna, Arabic arithmetic by, 50 Ibrahim ibn Ezra. See Ben Esra, Rabbi Icelandic, Sacrobosco's arithmetic translated into, 56 Inch, origin of, 124, 156, 162 India, letter system for numbers, 42 Indians, American: limited number system of, 33 number system of, 23, 25 picture writing used by, 1 Inheritance, Greek and Roman, 140 Initial letter numerals, 12, 12 (ill.) Integers, in Arabic arithmetic, 50 Interest: compound, Hodder's treatment of, 143 (ill.) fifteenth-century treatment of, 142 first use of decimals for, 130 first use of problems in, 138 Greek and Roman references to, 140 Isadore of Seville, mystic numbers in Origines by, 64 Italian, printed texts before 1500 A.D., 68 Italian problems: commercial, 140 ff. "lightning" method used in, 110 partnership treated in, 139 n Italy, decimal point used in, in 1492, 130 Jackson, L. L. (. . . . Sixteenth Century Arithmetic), 120, 136, 145 Japanese abacus, 26, 27 (ill.) Japanese arithmetic, 28 bibliography of, 30 n, 36, 37 Japanese soroban, 27 (ill.), 29 " Jealousy method " (Paciuolo), 110 Jefferson, Thomas: decimal systems recommended by, 162, 164 "Report on Weights, Measures and Coinage," 165 Jess, Zachariah, et al. (American Tutor's Assistant), 94, 95, 96 190 THE Jetons (counters), 35 Jewish arithmeticians in Spain, 49 John of Holywood. See Sacrobosco John of Luna, translation of Al- Khowarizmi, 53 John of Meurs, words used for fraction, 126 John of Spain, translation of Al- Khowarizmi, 151 Johnson, Clifton {Old-Time Schools and School-Books), 178 Johnson, Gordon: early colonial arithmetic, 81 Introduction to Arithmetic, 93, 94 Jordanus, Nemorarius {Demonstra- tio Jordani in Algorismo), 54 Karpinski, L. C: "Hindu-Arabic Numerals," in Science, 48 Robert of Chester's Latin Trans- lation, 60 Two Twelfth-Century Algorisms, 120 Karpinski, L. C, and D. E. Smith {Hindu-Arabic Numerals), 60 Kaye, G. R. {Indian Mathematics), 60 Kendall, David {Young Lady's Arithmetic), 82, 96 Kersey, John, revision of Wingate's Arithmetick, 74 (ill.), 75 Kilogram, origin of, 12 Kilometer, origin of, 12 Kilpatrick, W. H. {Dutch. Schools of New Netherland), 178 Kittredge. See Greenough and Kittredge Klau. See Clavius, Christopher Knots, Peruvian, records by, 33 Kobel, Jacob: problems from text by, 68 (ill.) Rechenbiechlin, by, 69 Rechenbuch, by, 34 (ill.) Korean arithmetic, 28 Korean number rods, 29 (ill.) La Disme. See Art of Tenths Ladreyt, Casimir {Nouvelle arith-metique), 98 INDEX Lagrange, Joseph Louis, decimal system proposed by, 165 Langlois {Rechard Redeles), 56 Laplace, Pierre Simon, decimal sys- m tern proposed by, 165 Latin: Arabic treatises translated into, 52,53 common fractions in, 127 first arithmetic printed in, 70 printed texts in, before 1500 A.D., 68 terms from, 150, 151 texts in Harvard and Yale, 81 thirteenth-century algorism in, 52 (ill.) thirteenth-century arithmetics in, 54 twelfth-century algorism in, 48 (ill.) works in, 54, 55 (ill.), 62, 65 (ill.), 67, 68, 69, 71 (ill.), 94, 142 Latin schools, medieval, arithmetic tanght in, 171 Latin versions, arithmetical operations in, 103 Laurin, Jos. {TraitS d'Arithmetique et d'Alglbre), 98 Lee, Chauncey {American Accountant), first to use dollar sign, 96, 163 (ill.) Lefevre-Gineau, on density of water 165 Length, metric unit of, 165 Leonard of Pisa: Arabic problems revised by, 65 business terminology from, 141 commercial problems treated by, 140 father of, a factor, 142 "fraction" from Arabic, 126, 147, 151 Hindu and Arabic problems by, 140 Liber Abbaci, 54, 67 "lightning" method by, 110 method of addition of, 103 oriental source of work of, 30 partnership treated by, 139 n practical work of, 137 THE INDEX 191 Leonard of Pisa � Continued problems from other languages treated by, 140 sexagesimal fractions used by, 123, 126 Letter systems for numbers, 42 Leybourn, William {Arithmetick ....), 76, 77 (ill.) Library of Congress, collections of arithmetics in, 98' Linear units, Babylonian, 158 Liquid measure, systems of, 161,162 Liter, unit of volume, 165 Little, Ezekiel {The Usher ....), 96 Littlefield, George E. {Earlv Schools .... of New England), 99,178 Liveriis, Johannes de, of Sicily, first treatise in print on fractions by, 70 Locke, L. Leland {Ancient Quipu or Peruvian Knot Record), 33, 37 Logarithms, inventor of, 110 Logistica, arithmetica and, 14 McDonald, Alexander (Youth's Assistant), 90, 92, 93 McGee, W. J. {Primitive Numbers), 37 Mahavir, or Mahaviracarya: arithmetic of, 45 commercial terms used by, 138 method of multiplication used by, 107 ninth-century Hindu text of, 64 operations listed in, 101 treatment of fractions in, 126 Manuscript: addition and subtraction in The Art of Nombryng, 104 (ill.) Arabic on arithmetic, 47 (ill.) Arabic, on Hindu arithmetic, 51 (ill.) Bakhshali arithmetical, 43 n, 44 (ill.) Carmen de Algorismo of Alexandre de Ville Dieu, 65 (ill) Crafte of Nombrynge, 56, 57 (ill.) fifteenth-century, 100 (ill.), 102 (ill.) Sacrobosco's Algorismus, 55 (ill.) The Art of Nombryng, 107 (ill.) thirteenth-century algorism, 52 (ill.) twelfth-century, 48 (ill.), 102 (ill.) Marre, Eugene Aristides, cited, 50 n Mathematics: first American lecturer on, 78 Hindu versification of, 41 history, reflected in terms of. 154 in early American universities, 81, 177 f. in schools, 173 ff. Mather, William {Young Man's Companion), 76 Mayas of Yucatan: codex of, 31 (ill.) hieroglyphic number symbols used by, 32 (ill.), 33 numerals used 037-, 31 (ill.), 40, 41 twenty system of, 25, 30, 40, 41 Measures: Babylonian, French parallels, 158 decimal system recommended for United States, 167 dry and liquid, systems of, 161, 162 Egyptian, attempt at unity, 160 Greek, attempt at unity, 160 origin of units of, 157 Roman, unit of money, 160 treated by Abu '1 Wefa, 138, 139 uniform, attempt to secure, 165 varying standards of, 162 weights, coinage and, Jefferson's report on, 165, 167 Mechanical arithmetic. See Abacus; Computing rods; Counters Medieval instruction, 170 Medieval Latin, terms from, 131 Mellis, John: work on decimals by, 77 work on Recorde, 67 (ill.) Mensuration, 157 ff. favorite Hindu topic, 138 treated by Abu '1 Wefa, 139 See also Measures Merchandise prices, illustrations by Leonard of Pisa, 141 192 THE INDEX Merrill, Phinehas (Scholar's Guide), 94 Mersenne, Marin, use of rectangle in work of, 151 Metcalfe, Michael, teacher who used Recorde's arithmetic, 174 Metric system: advantage of, 122 legal adoption of, in the United States, 167 units proposed for, 164, 165 Mexican arithmetics, bibliography of, 37, 78, 79 (ill.), 85, 109 (ill.) Mexican works, early, Spanish discoverers represented by, 85 Mikami, Yoshi (Development of Mathematics in China and Japan), 30 n, 36 Mikami, Yoshu, and D. E. Smith (History of Japanese Mathematics), 30 n, 37 Mile: length proposed by Moutons, 164 Roman, measure of, 20, 161 Milestone, Roman, 22 (ill.) Million, meanings of, 21, 149 Milns, William (American Accountant), 96 Mina, Babylonian weight, 160 Minoan numerals, 11 (ill.) Minutes and seconds: Babylonian sexagesimal system survives in, 8, 122, 128, 156, 157 Latin terms for, 124, 157 Mixture. See Alligation Modius, Roman unit of measure, 162 Money. See Federal money Monroe, Paul (Cyclopedia of Education), 178 Monroe, Walter S. (Development of Arithmetic), 99 Mose, Henry, revision of Hodder's Arithmetick by, 82 (ill.) Moutons, Gabriel, proposed measure of mile, 164 Mula, Arabic, concept taken from, 153y Multiplication: Arabic short cuts in, 51 early methods of, 101, 106 fT. Egyptian operations, in, 3 in Art of Nombryng, 107 (ill.) in Crafte of Nombrynge, 108, 109 "jealousy" method of, 110 method: of Brahmagupta, 107 of Freyle, 109 (ill.) of Napier, 110 modern methods of, 110 of fractions by Riese, 128 (ill.) reverse of division, 115 Multiplication table: Babylonian, 9 (ill.) Hebrew, 13 % Muris, Johannis de, square root expressed by, 129, 130 Murray, James A. H. (New English Dictionary), 155 Mysticism: of numbers, 11, 15 speculative arithmetic from, 11, 15,62 Napier, John: decimal point, idea of, 129, 133 decimals applied to logarithms by, 133 method of multiplication used by, 110 "Napier's bones," computing by, 110 Navigation, early works on, 81 Negrete, Juan, first American lecturer on mathematics, 78 New York Public Library, collections of arithmetics in, 98 Nichols, P. (. . . . Practical Arithmetic and Bookkeeping), 96 Nicomachus of Gerasa: Greek treatise by, 15, 16 quoted, 16, 17 translation by Boethius, 57, 62 Nines, casting out, 105, 106 Norman French: influence of, on English terms, 154 THE INDEX 193 Norman French � Continued Latin and Greek terms from, 154 terminology from, 146 Norton, Robert, Art of Tenths or Decimall Arithmetike, 131, 132 (ill.) Norwood: Navigation, 81 Trigonometry, 81 Noviomagus (De Numeris), 24 n Noyes, James {Federal Arithmetic), 96 "Number of the beast" in Revelation, 13 Number rods, 29 (ill.) Numbers: Boethian theory of, 62 "broken," origin of concept, 127, 153 deficient, 16 denominate, 156 ff. even, 15 figurate, 18 Greek, series of, 18 in Bible, theory of, 13, 14, 62, 64 Mayan symbols for, 32 (ill.) odd, 15 pentagonal, 19 peffect, superabundant and deficient, 16 prime, used as symbols for fractions, 128 square, 19 symbolic, 13, 14 triangular, 18 Numeral systems, native American, 30 Numerals: alphabet or initial letter (Attic system), 12, 13 Arabic use of alphabet, 13 Aztec (frontispiece), 31 Babylonian, 7 ff. Cretan, 11 (ill.) cuneiform, Babylonian, 8 (ill.) special use of, 10, 11 curvilinear, special use of, 10, 11 tablet showing, 10 (ill.) earliest English printed works explaining, 59 Egyptian, 1 ff., 3 (ill.), 4 (ill.), 5 (ill.) finger: ancient, 23, 24 (ill.) modern, 25 (ill.) Greek, 11, 12 (ill.), 13, 14 hieroglyphic type, 11 (ill.) Hebrew, 13 hieroglyphic, frontispiece, 3 (ill.), 4 (ill.), 11 (ill.), 32 (ill.) Hindu origin of, 38 Mayan twenty system, 30 ff. Minoan, 11 (ill.) modern, 38 ff. Roman, 19, 20 (ill.), 21 (ill.) Numeration: in early arithmetics, 100, 101 in fifteenth-century manuscript, 100 (ill.) Numerator of fractions: Egyptian treatment of, 121 Greek treatments of, 124 Odd and even numbers, ancient divisions into, 15 Operations, fundamental, in early arithmetics, 100 ff. Oriental arithmetic, 28. See also Arabs; Chinese; Hindus; Japanese Oughtred, William {Key to Mathe-maticks), 77 Ounce, origin of, 124, 162 Pablos, Juan, first American printer, 79 Paciuolo, Luca di: Summa d}Arithmetical Arabian problems used in, 65 commercial terms from, 141 f. finger reckoning treated in, 24 n first work on bookkeeping, 142 "jealousy" or "lattice work" method of multiplication named by, 109, 110 "lightning" method of multiplication in, 110 popularity of work, 67 "scratch" method of division in, 115 194 THE Palm: Babylonian, length of, 158 cubic, water capacity as unit of weight, 158 origin of, as unit of measure, 157 Panet, et al. (Theses de Mathemat- iques . . . .), 89 Papyrus: Ahmes: algebraic problem on, 6 (ill.) as textbook, 61, 169 commercial problems from, 2 (ill.), 122 Egyptian hieroglyphics shown on, 2 (ill.) names of, 3 Elephantine, 12 (ill.) Greek, Egyptian problems in fractions used in, 122 Partnership: "fellowship/' 139 (ill.), 141 (ill.), 148 (ill.) fifteenth-century works on, 142 first use of problems in, 138 illustrations on, by Leonard of Pisa, 141 principles presented by Recorde, 139 (ill.) Patterson, R.: Gough's arithmetic, work on, 94 Workman's arithmetic, work on, 95 Paul, Jeremiah, et al. {American Tutor's Assistant), 94, 95 Paul, Rene (Elements of Arithmetic), 98 Payne, J. F. (Natural History and Science), 36 Paz, Pedro (Arte menor del Arith- metica Practica), 78, 85 Pedagogues, classical, 169 Peet, T. Eric (The Rhind Mathematical Papyrus), 36 Peletier, Jacques, of Mans, revision of Frisius by, 69 Pellizzati, Francesco: Art de Arithmeticha: commercial work in, 142 first use of decimal point in, 129 INDEX Pentagon, origin of, 12 Pentagonal numbers, 19 Perfect numbers, 16 Perrault, et al. (Theses de Mathemat-ique . . . .), 89 Persians, finger reckoning of, 23 Peruvian knots or quipu, 33 record kept by, 33 (ill.) Peruvian, mathematics, bibliography of, 33 n, 37, 94 Pesos, currency of, 167, 168 Peurbach: table of sines compiled by, 130 terminology of fractions by, 127 " Philomathematicus," title of Hart-well, 74 Physical fractions. See Fractions, astronomical Picard, Jean, universal yard proposed by, 164 Picture numerals, Mayan, 33 Picture writing, 1 Pike, Nicholas: arithmetic by, 83, 92, 93, 94, 96, 161 Arithmetick: federal money, table in, 161 (ill.) fractions treated in, 127, 135 fundamental rules of, 101 one hundred topics treated in, 144 1' recommendations " in, 175 (ill.) "reduction of coin," in, 160 cm.) term for decimal point in, 150 terms in'' -illion,'' defined in, 149 title pages of, 84 (ill.) weights and measures compared in, 159 (ill.) Pistoles, Spanish, currency in American colonies, 168 Plato: Egyptian instruction mentioned by (quoted), 137, 169 quoted, 14 Plimpton, George A., collection of arithmetics in library of, 71, 98 Pound, various standards for, 162 Pounds sterling, in colonial America, 167, 168 THE INDEX 195 Poyntell, William, Arithmetical Card sold by, 90 Practical arithmetic. See Commercial arithmetic Prince, Nathan, colonial teacher, advertisement of, 174 Principal and interest, first use of problems in, 138 Printing, arithmetics popularized by, 66, 67 Problems: algebraic and. arithmetical: by Leonard of Pisa, 65 on Bakhshali manuscript, 43 arithmetical: by Euclid, 61, 62 by Mahavir, 46 Egyptian: algebraic, 6 (ill.) in arithmetical progression, 7 (ill.) European, oriental source of, 30 from Kobel's arithmetic, 68 (ill.) Hindu, 43 practical, in Orient, 29, 30 Proclus, quoted, 14 Profit, first problems in, 138 Proportion: application of, by Leonard of Pisa, 141, 144 Latin treatise on, 62 Ptolemy {Almagest), 53 Quadrivium, liberal arts included in, 59 Quart, English and American compared, 164 Quenesre, Syrian monk of, 48 Quipu, Peruvian, 33 (ill.) Ramus, Peter {Arithmeticae), 71 (ill.) Rafigacarya, M. {The Ganita-Sara- Sangraha of Mahaviracarya), 45 n, 60 Rauch, Christ. H. {Rechenbuch), 97 (ill.) Reaton, Don Atanasius {Arte menor de Arismetrica), 78, 85 Rebate, used by Dearborn, 148 (ill.) Reckoning: finger: ancient, 23, 24 (ill.) medieval treatise on, 170 modern, 25 (ill.) on lines, 33. See also Abacus. See also Numbers, and topics on fundamental operations Reckoning schools, German, 171 Recorde, Robert: algebraic and geometric terms used by, 152 colonial use of works by, 81, 174 commercial topics treated by, 140, 141, 143, 144 counters used by, 35 (ill.) finger reckoning used by, 24 n Grounde of Artes, 67 (ill.) division in, 113 (ill.) interest in revision of, 77 many editions of, 60 modern terms in, 127, 151 partnership as presented in, 139 n "scratch" method in, 112 (ill.) 114, 120 method of subtraction used by, 105 Pathwaie to Knowledge by 152 progressions treated by, 100, 101 separatrix (term for decimal point) used by, 150 "seven fundamental operations" used by, 101 seventeenth-centurv authority, 73 Whetstone of Witte" 151 Regiomontanus, table of sines compiled by, 130 Reisch, Gregorius {Margarita Phil- osophica), 68 Revolutionary War, textbooks in period of, 83 Rhind Papyrus, 3. See also Ahmes papyrus Ridgeway, Sir William {Roman Measures and Weights), 136 Riese, Adam: Rechnung auff der Linien und Federn, 128 (ill.) many editions of, 69 196 THE INDEX Robbins, F. E., and L. C. Kar-pinski, studies in Greek arithmetic, 17 n Robert of Chester, translation of Arabic algebra by, 53 Roman abacus: fractions shown on, 124 reckoning on, 25, 26 (ill.) Roman arithmetic, 19, 22 Roman fractions: Babylonian base of, 124 influence of, upon system of weights and measures, 23 of Pope Sylvester II, 125 (ill.) survival of, in measures, 23, 156 symbols used for, 124 Roman mile, length of, 161 Roman milestone, 22 (ill.) Roman numerals, forms of, 19, 20 (ill.), 21 (ill.) Roman weights and measures, 124, 160 Romans: children taught on abacus by, 23 hour for beginning the day fixed by, 156 Root: cube, taught by Hindus, 101 extraction, in Arabic works, 100 square, taught by Hindus, 101 Root, Erastus (An Introduction to Arithmetick), 95 Rudolff, Christian, practical use of decimals by, 130 Rule of Three: Arabian algebras including, 139 commercial treatment of, by Recorde, 144 practical use of, by Aryabhata, 138 Russian abacus, forms of, 26, 27 (ill.) Ruyz, Joan, of Mexico, printer of Paz's text, 85 Sacrobosco, John of: Algorismus Vulgaris: fifteenth-century translation of, 56 Icelandic translations, 56 medieval teaching from, 170 method of addition used in, 103 section of manuscript of, 55 an.) thirteenth-century authority, 54,65 treatment of fractions in, 127 treatment of roots in, 10Q Sangi, 28, 29 (ill.) Sanskrit: decimal numeration in, 40 letter system for numbers in, 42 modern, numerals in, 43 (ill.) Sargeant, Thomas, works of: Elementary Principles, 92 Federal Arithmetic, 93 Schools for mathematics: advertisements in colonial mathematics listed in, 174 methods of instruction in, 177 tuition in, 177 Schoy of Essen, Karl, cited, 47 "Scratch" method of division, 109 (ill.), 112 (ill.), 113, 114 (ill.), 118 (ill.), 119 (ill.), 120 Sebokht, Severus, numerals discussed by, 48 Seconds. See Minutes and seconds Seed of carob, unit of weight, 161 Seller {Navigation), 81 Separatrix, 150 early name for decimal point, 150 in early arithmetic, 166 (ill.) Series: Arabian problems in, 138 number, 18 Seventeenth-century arithmetics: continental texts, 73 English texts, 73 number published, 72 Sexagesimal system: fractions, principles used in decimals, 122, 128 in Greek fractions, 124 linear table in, 158 present use of, 8, 122 She, Babylonian unit of weight, 160 Shekel, unit of weight, 160 # Shelley, George, work on Wingate's Arithmetick, 74 (ill.) THE INDEX 197 Signboard, Roman, 22 (ill.) Silberberg {Das Buck der Zahl des Rabbi Ibrahim ibn Esra), 50 n Sines, table of, used by Peurbach, 130 Six-percent rule, first appearance in print of, 166 (ill.) Sixteenth-century arithmetics: editions of, 68, 69 number published, 71 Skeat, W. W. {An Etymological Dictionary of the English Language), 155 Slack, Mrs. See Fisher, George, pseud. Slide, rule, inventor of, 77 Smith, David Eugene: and L. C. Karpinski {The Hindu-Arabic Numerals), 60 and Yoshu Mikami (^4 History of Japanese Mathematics), 30 n, 37 Computing Jetons, 36 History of Mathematics, 120, 145 Rara Arithmetica, 60, 71, 99, 120 "Sumario Compendioso of Brother Juan Diez, 99 Snell, Wilebrordus, commentary on the Arithmeticae of Ramus, 71 (ill.) Solomon {Lehrbuch der Arithmetik und Algebra), 105 Soroban, Japanese, 27 (ill.), 29 Sower, S., printer of The Federal Reckoner, 94 Spain: Arabs in, 52 Moslem schools in, 52 work of Gerard of Cremona in, 53 Spanish arithmetical texts, 64, 78, 85, 86, 89, 98, 142 Speculative arithmetic: of Greeks, 15, 62 two types of, 62 Square numbers, 19 Squares, Babylonian table of, 9 (ill.) Sridhara, commercial terms of, 138 Sridharacarya, method of multiplication by, 108 Steele, Robert: Earliest Arithmetics in English, 99 quoted, 56, 108 translation of Sacrobosco by, 103 Stein, Jacob, revisions of Frisius by, 69 Sterry, Consider and John, textbooks by, 92, 95 Stevin of Bruges, Simon: decimal fractions first treated by, 131 discussion of decimals, 129 (ill.) English translation of treatise by, 132 (ill.) La Disme, 131, 132 (ill.) La Thiende, 129 (ill.), 131 proposed decimal systems, 165 quoted, 131, 133 symbols for decimals used by, 133 treatises on decimals by, 164 See also Art of Tenths; Norton, Robert Stockton, F. {The Western Calculator), 98 -Strachan, Rev. John {Introduction to Practical Arithmetic), 97 Stylus, use of, 14 Suan-pan: Chinese abacus, 27 (ill.), 28 Japanese modification of, 27 (ill.), 29 meaning of name, 26 Subtraction: addition and, 102 ff., 104 (ill.) Austrian method of, 105 counters used for, 35 (ill.) Egyptian operations in, 3 Egyptian symbol for, 61 (ill.) in early algorisms, 105 in symbols, 22 influence of abacus on method, 105 procedure from Shakespeare's day, 105 procedure on abacus, 26, 102 Sumerian clay tablet, 10 (ill.) Superabundant numbers, 16 Surveying, works used by American colonists, 81 Suter, cited, 50 198 THE Sylvester II (Pope Gerbert): abacus made by, 26 n Roman fractions from, 125 (ill.) Symbol for "percent," first use of, 141 Symbols: and terms, operative, 150 ff. Egyptian, for addition and subtraction, 61 (ill.) Egyptian, numerical, 1 ff. in numbers, 13, 14 Mayan hieroglyphic numbers, 32 (m.) _ subtraction in, 22 zero, of ancients, 40 Syrian monk of Quenesre, 48 Tables: Babylonian, 9 (ill.), 158 from Pike's Arithmetick, 159 Hebrew, 13 Tablet, clay, Sumerian, 10 (ill.) Tablets: Babvlonian, 9 (ill.), 10 (ill.) Greek, 14 Tagliente, Girolamo and Giannan-tonio, arithmetical works by, 69 Tallyrand-Periogord, Charles Maurice de, uniform systems of weights and measures proposed by, 165 Tare and trett: origin of terms, 148 used by Dearborn, 148 (ill.) Teacher: colonial, advertisements by, listing mathematics, 173, 174 colonial, regard for, 178 medieval, salary of, 171 Temple, Samuel {Introduction to Practical Arithmetic), 96 Terminology of arithmetic: Arabic words given to, 49, 147, 152, 153 English, establishment of, 154 English, of Chaucer, 153 French contribution to, 146 Greek influence on, 146, 147, 152 Hindu concepts in, 138, 147 INDEX history of mathematics reflected in, 154 Italian contribution to, 141, 142, 147, 149 Latin influence on, 20, 146, 149, 150,151 Norman French, a medium to, 154 number words, 147 ff. progress of arithmetic marked by, 146 Roman influence on, 146, 147j 161 technical origin of, 150 Terms and symbols, operative. 150 ff. Textbooks, 61 ff. American, 78 ff. bibliography of, 85 ff. European, after 1200 a.d., 65 ff. Texts; Hindu and Arabic, 64 ff. Thales, numerals used by, 12 Tharp, Peter {Federal Arithmetic), 96 Theon of Smyrna {Arithmetica), 16 Theoretical arithmetic, 62 Thirteenth-century algorism, 52 (ill.) Thirteenth-century arithmetic, 55 (in.) Thirteenth-century arithmeticians, 54 Thomas, Cyrus: Mayan Calendar Systems, 37 Numeral Systems of Mexico and Central America, 37 Thomas, Isaiah, preface to Pike's arithmetic, 96 Thumb, primitive unit of measure, 157 Tibbon, Moses ben, thirteenth-century translation of Al-Hassar by, 126 Time, 156 ff. Babylonian division of, 7, 122 fifteenth-century works with problems involving, 142 Todd, John, etal. {American Tutor's Assistant), 94, 95 Toledo, Moslem schools in, 52 THE INDEX 199 Treviso arithmetic: commercial character of, 142 first printed arithmetic, 70 method of multiplication in, 110 page from, 66 (ill.) Triangular number, 18 Trigonometry: development of tables of sines for, 130 early texts on, 81, 135 Latin origin of terms in, 150 . of Arabs, 64 Trivium, liberal arts in, 59 Tropfke, J. (Geschichte der Elemental Mathematik) , 120 Troy weight, origin of term, 162 " Twelfth-Century Algorisms, Two" in Isis (L. C. Karpinski), 120 Twelfth-century Arabic arithmetic, 50 Twelfth-century arithmetics, variations in multiplication and division in, 109 Twelfth-century treatise giving duplication, 102 (ill.) Twenty in number system, 25 Twenty system of Mayas, 30 Uncia: derivatives from, 124, 162 tables of, 125 (ill.) Unit, in finger reckoning, 25 Unit fractions: Egyptian system of, 5, 6, 121, 122 Greek, 124 in Ahmes papyrus, 122 Units expressed by counter, 34, 35 University, first founded in America, 78 University of Michigan Library, collection of arithmetics in, 98 Venema, Pieter: Arithmetica of Cyffer-Konst, 83 (ill.), 86 Dutch colonies represented by work of, 85 Vespucci, Amerigo, commercial enterprises of, 143 Viete, Frangois, first use of coefficient, 151 Ville Dieu, Alexandre de: Carmen de Algorismo: in Latin verse, 54 popularity of text, 54 transcription of, by students, 66 translated into French and German, 56 verses from, 65 (ill.) fractions not discussed by, 127 n medieval teaching from, 171 quoted in Crafte of Nombrynge, 57 (ill.) roots treated by, 100 seven operations, quoted, 101 Vinall, John (Preceptor's Assistant), 93 Vitruvius, modern terms in work of, 150 Vulgar fractions. See Fractions, common Wagner of Nurnberg, first German treatise on new numerals by, 70 Walkingame, Francis (Tutor's Assistant), 96 Ward, John: Arithmetic^, 81 Introduction to Mathematicks, 74 (in.) Waring, William, et al. (American Tutor's Assistant), 94, 95 Washington, George: attempt by, to introduce decimal system, 162, 164 quoted, praising Pike's work, 160 n uniform weights and measures recommended by, 165 Watson, Foster (Teaching of Modern Subjects in England), 178 Weights: American, standardized, 162, 163 and measures, treated by Abu '1 Wefa, 138, 139 avoirdupois, origin of, 162 Babylonian, unit of, 158 200 THE INDEX Weights � Continued: decimal system recommended for, 162, 163, 167 Egyptian, basis of, 160 equivalent, of various countries, 159 (ill.) French development of, 158 Greek, basis of, 160 measures, coinage and, Jefferson's report on, 167 natural measures of, 161 primitive system of, 161 Roman, basis of, 160 tables of, from Lee's Accomptant, 163 (ill.) Troy, origin of, 162 varying standards for, 162 Wiclif, John, technical terms used by, 153 Widman of Eger, Johann: Behennd und hiipsch Rechnung uff alien Kauffmanschafften (Leip-rig), 142 first illustrated arithmetic by (Pforzheim), 70 Wilkinson, William {Federal Calculator) , 95 Wingate, Edmund: Arithmetique made Easie, 74, 81 title page of revision by John Kersey, 74 (ill.) Woepcke, Franz (cited), 50 n Workman, Benjamin: American Accountant, 92, 95 editor of Gough's works, 92, 94, 95 Writing, cuneiform, 7 Wroth, Dr. Lawrence C, of John Carter Brown Library, 99 Yale University: algebra issued by early professor, 177 colonial texts used in, 81 Yard, universal, length proposed by Picard and Huygens, 164 Zamorano, A., publisher of Spanish text, 98 Zero: Babylonian symbol for, 40 development of, 40 Zero, origin and meaning of, 41, 149, 152