# Popular Science Monthly/Volume 77/September 1910/A Unique Collection of Arithmetics

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* A UNIQUE COLLECTION OF ARITHMETICS
By Dr. LOUIS C. KARPINSKI
UNIVERSITY OF MICHIGAN
RECENT visitors to the Metropolitan Museum of Art have been impressed by the wealth of the loan collections standing in names comparatively unknown to the general public. A two-million-dollar sale of works of art lately excited only passing comment—in spite of the fact that many priceless treasures were forever lost to America. The existence of this private gallery was made widely known only through the dispersal of its paintings—and the unfortunate story of its loss to New York City. There are many other storehouses of those things which we human beings prize in this great city. Fortunately not all of them need to be destroyed as collections before their significance and charm receive adequate recognition. So the Morgan library in its own somewhat permanent home is now numbered among the city's choicest possessions.
The existence in the metropolis of an absolutely unrivaled collection of fifteenth and sixteenth century arithmetics has been brought to the attention of the scientific world by the publication of David Eugene Smith's "Rara Arithmetica" While the work purports to be a mere descriptive catalogue of the arithmetical books of the period mentioned which are in the library of G. A. Plimpton, it is in fact a comparatively complete bibliography of the subject, since this library contains practically all the arithmetic books published in the first hundred and fifty years of printing. As the third, and by far the most complete, collection of arithmetical works of international fame the Plimpton books take a high place among modern private libraries.
George A. Plimpton's interest in arithmetics grew out of his business as a publisher of text-books. The historical development of the school curriculum is exhibited by his library. Included are geographies from the invention of printing up to modern times, spellers, writing books with wonderful specimens of writing from all the world, geometries, reading books and representatives of the other subjects of the ordinary school program. But the gems of the collection are doubtless the mathematical works, for in these Mr. Plimpton's interest has been stimulated by Professor David Eugene Smith, himself an enthusiastic bibliophile. The bookshops of all the world have yielded their treasures to these indefatigable searchers. Professor Smith's recent trip ​around the world brought mathematical finds to the Plimpton and Smith collections in the shape of Arabic and Persian, Hindu and Chinese and Japanese manuscripts and rolls.
Mention should also be made of the medallions of mathematicians, on exhibition in Teachers College, extending back nearly to the time of Pythagoras. The Smith collection of portraits of the devotees of numbers is without parallel and the autograph letters and documents are priceless. Here is an original note-book from the hand of Newton, and the more prosaic receipt for his semi-annual annuity of fifty pounds granted by parliament. The diploma of the great physiologist E. H. Weber, signed by Carl Friedrich Gauss, probably the greatest mathematician of all time, will interest especially those who are familiar with the labors of these men.
The invention of printing gave a tremendous stimulus to all scientific work by making possible the wide diffusion of knowledge, as well as by facilitating the intercourse of scholars. A potent indication of the really scientific spirit of the learned men of that day is the fact that the newly discovered art was used to give the older classics a wider circulation. Thus it need not surprise us to find in this bibliography of the fifteenth and sixteenth centuries the names of the more ancient writers.
Archimedes (287–212 B.C.), whom we ordinarily recall as a geometer tracing figures in the sand and incidentally being killed while engaged in this harmless occupation, or as a master of applied mechanics defending Syracuse with his catapults and burning glasses, appears in the "Rara Arithmetica" as the author of a work on numeration. Archimedes explains how it is possible to obtain numbers sufficient to express the grains of sand in a sand-heap as large as the world and even as large as the universe, a problem which is also found in India.
The arithmetic of Boethius (c. 480–524) involving that of Nicomachus of Gerasa (fl. c. A.D. 100) was the most widely used text-book in the monastic schools of the middle ages. Doubtless never again will any text-book be kept in use for approximately a thousand years, and yet an examination of the content of this text reveals not science, but hair-splitting philosophical discussions and extreme poverty of ideas. Boethius might have been expected to be a more practical philosopher, for he wrote his "Consolations of Philosophy" while he was in prison.
The exchange of professors by the leading universities was more common in the early days of these institutions than it is even now. Thus the Englishman, John of Halifax, or Holywood, who was known in the middle ages by the Latin form of his name Sacrobosco, studied and probably lectured at Oxford before settling in Paris about 1250. Sacrobosco's "Algorism," while by no means the first European work on the Hindu art of reckoning, was one of the most widely used and ​served largely to spread the knowledge of the numerals which we now employ. This "Algorism" was first published at Strassburg in 1488 and at least thirteen other editions followed before fifty years had elapsed. In the first edition it appeared with a computus, the title applied to works on the arithmetic of the church calendar. The Latin version of our rhyme "Thirty days hath September," etc., appears in this "Compotus Manualis" (in verse) and was written by Anianus, a Strassburg astronomer and poet. The name algorism was applied for some five hundred years to the arithmetic which explained the method of reckoning with the Hindu-Arabic numerals. The word is a corruption from the name of Mohammed ben Musa, al-Khowarazmi, whose Arabic work on this subject was translated into Latin in the early twelfth century. Early manuscripts of Sacrobosco's classic are found in the Columbia Library as well as in the Plimpton collection.
Many theologians and churchmen, among the earliest of these may be mentioned the Venerable Bede (c. A.D. 700), and Cassiodorus (c. A.D. 550), amused themselves by writing arithmetics, but this was inevitable in the period when learning was so largely confined to church institutions. Thomas Bradwardin (c. 1290–1349), who was professor of theology at Oxford and later archbishop of Canterbury, wrote extensively on mathematics. His name suffered, as did many others, at the hands of transcribers, being found as Bragwardine, Brandnardinus, Bredwardyn, Bradwardyn, de Bradwardina and de Bredwardina. Another of these professors of theology was Christian Ursinus (also known as Allassiderus, Allassisiderus, Wursteisen or Urstis) who published in 1579 at Basel an arithmetic entitled "Elementa Arithmeticæ."
The surnames, as noted above, were rather shabbily treated from the modern point of view, since the first names were regarded as the important ones. It was common, too, for scholars to Latinize their names, or more rarely to give the Greek equivalent. The reformer Melanchthon, who appears as a writer on the nature and value of mathematics, was baptized Schwarzerd. Schreiber (c. 1525) became as a writer of school texts Grammateus, but was also known as Scriptor. Melanchthon's friend, Camerarius, who was also a classical scholar, was born as Liebhard. Camerarius wrote a commentary on the arithmetic of Nicomachus. Conrad Dasypodius, whose family name was originally Rauchfuss or Hasenfuss, wrote two works which should have been included in this catalogue. Copies of these rare books, both published at Strassburg in 1567–1570 and 1593–1596, respectively, are found in the Astor Library. The older one is entitled "First and Simplest Mathematics," and is partly in Greek and partly in Latin, treating of geometry, logistic (a Greek name for practical arithmetic), astronomy and geography. The writer was professor of mathematics at Strassburg ​towards the end of the sixteenth century and he designed the famous clock of the Strassburg cathedral.
The unusually large number of physicians (eleven) appearing in the "Rara Arithmetica" is at first sight rather surprising, until we recollect that the scientific training of the time was largely confined to medicine. Some of these men might be counted among the best mathematicians of their day, notably the Italian Hieronymus Cardan (1501–1576) who attained fame as an algebraist, and the German Johann Widmann (fl. c. 1490), who wrote one of the first arithmetics in the German language. An English goldsmith is the author of a practical arithmetic, of which there were many designed especially for merchants and tradespeople. Jurists and numerous professors of Greek and Hebrew mingle here with priests and bishops and even two cardinals, Petrus de Alliaco and Nicolaus Cusa. The reckoning masters so frequently mentioned as authors remind us that for many years arithmetic had no place in the schools, and that the reckoning masters taught the art of reckoning outside of school hours very much as music and dancing are taught to-day.
Especial interest attaches, of course, to the first arithmetic to appear in print, the anonymous Treviso arithmetic of 1478. While there is no proper title page, the first page begins as follows: "Here commences a practical treatise, very good and very useful for any one who wishes to learn the art of merchants, vulgarly called the art of the abacus." The last page states that it was printed at Treviso (just north of Venice) on the tenth day of December, 1478. There are 124 unnumbered pages, running about 32 lines each. The first page is reproduced in the "Rara Arithmetica" in facsimile, together with three other pages. The author was evidently a teacher in Treviso, as he states that the book is written at the oft-repeated solicitation of his students; the printer's name is also unknown. Peculiarly enough this practical arithmetician applies four different names to the science, two as in the above title and further the art of "arismetrica" and algorism. This particular copy was in the Pinelli collection, and was acquired in 1790 by a Mr. Wodhull. Later it found its way into the library of Brayton Ives and at the sale of that library became the property of Mr. Plimpton. The work is strictly speaking an "algorism" since that title implied the use of the Hindu-Arabic numerals for practical computation, whereas "arithmetica" designated a theoretical treatise based largely on the work of Nicomachus and Boethius. An "abacus," strictly speaking, would be a work involving the use of some ruled surface or device to separate by columns (or rows) the units, tens, hundreds and thousands, etc., from each other. However these terms were not strictly applied, Leonard of Pisa's extended explanation of the Hindu reckoning appearing under the title "Liber Abbaci" or ​"Book of the Abacus," while "Algorithmus Linealis" was applied to numerous works explaining the reckoning on lines which was a slight variation of the abacus idea.
The beautifully printed Calandri arithmetic of 1491, the first in De Morgan's list, differs from its predecessors in having the traditional problems copiously illustrated. The slow-moving snail, who climbs up by day one seventh of a foot and slides back by night one ninth of a foot, is seen here with his head just emerging from the fifty-foot well and looking remarkably active after a climb of 1,575 days. The title page presents Pythagoras as "Pictagoras arithmetice introductor," the wholly erroneous but wide-spread notion being that this philosopher was the originator of the science of numbers.
Of the eight or ten arithmetics (two being parts of compendiums) given by this catalogue as preceding Philippi Calandri's treatise the three following are of general interest. "Prosdocimi de beldamandes algorismi tractatus" (Padua, 1483) contains probably the first reference to a slate; Pietro Borghi, one of these successful text-book writers, wrote the most elaborate of the early books on the subject and more than any other set a standard for the arithmetics of the succeeding century. This text-book for the use of merchants, written in Italian, appeared at Venice in 1484 from the press of Ratdolt. Widman's German text of 1489 employs for the first time the $+$ and $-$ signs, but simply as warehouse symbols of excess or deficiency.
One of the rarest of the catalogued treasures is the "Arithmetica" or "Compendium of the Abacus" (Turin, 1492) of Francesco Pellos (Pellizzati). It appears that this native of Nice came very near to the invention of decimal fractions, writing almost a hundred years before the first complete explanation of the subject in "La Disme" or "The Decimal" by Simon Stevin the Hollander. Pellos actually used a decimal point to indicate division by such numbers as 100, but its full significance did not dawn on him.
The first printed discussion of arithmetic in the English language is a chapter of Caxton's "The Mirrour of the World or Thymage of the same" (London, 1480); the section begins "And after that of Arsmetrike and whereof it proceedeth." An interesting sidelight is thrown on early American history by the announcement of the discovery in Madrid of the first arithmetic printed in the western hemisphere. Extensive printing was done in Mexico in the second half of the sixteenth century, and it was here that Juan Diaz Freyle published in 1556 the Spanish "Compendium. . . with Certain Rules of Arithmetic."
Of necessity many works apparently unrelated to arithmetic are introduced. The fine distinctions between the sciences did not then exist, so that an astronomer, a geometer, a philosopher or a writer on the Church calendar would not hesitate to bring into his subject a ​discussion of arithmetic. Finger reckoning and a number game called Rithmomachia are other related subjects which received elaborate treatment. The first modern encyclopedia to appear in print is the "Epitome of all Philosophy" by the Carthusian monk Gregorius Reisch, the publication appearing in Strassburg in 1503. "Pythagoras" and "Boethius" adorn the first page of the part devoted to arithmetic.
It would appear that scientists have, in the course of centuries, grown more modest in their published claims. Borghi's "Noble work of arithmetic treating all those things which are requisite for merchants" sounds like a boast. More seductive are "The Ground of Artes," "The Castle of Knowledge," "The Pathway of Knowledge" and "The Whetstone of Witte," mathematical works by Robert Recorde, the royal physician to Edward VI. and Queen Mary. Recorde was the first to use the present equality sign, stating that no two things can be more equal than two such lines. His were the most influential English mathematical publications of the sixteenth century. Equally enticing as the titles of Recorde was Humphrey Baker's "The Well spring of Sciences, Which teacheth the perfect work and practise of Arithmetick, both in whole Numbers and Fractions" (London, 1562).
The most fitting name with which to terminate a discussion of the printed arithmetics of the sixteenth century is that of Adam Riese. So-widely were his books used and so deep the impression which they made that even to-day, nearly four centuries after he wrote, the expression to reckon "nach Adam Riese" is common in Germany. Riese's works quite supplanted the numerous editions of the Rechenbuch by the versatile Jakob Köbel, who was Reichenmeister, printer, engraver, woodcarver, public official, as well as a successful text-book writer. Köbel's "Rechenbuch" of 1514 bears silent but eloquent testimony to the tremendous inertia that must be overcome by any new system that revolutionizes the common processes of thought. Köbel's arithmetic, four hundred years after the Hindu-Arabic numerals had been explained in Europe, is wholly in Roman numerals, even to the fractions. Riese's work made the publication of any other arithmetic in Roman numerals impossible.
Part II. of the "Rara Arithmetica" treats of the rich collection of mathematical manuscripts in the Plimpton library. The oldest of these is a beautifully written Latin Euclid (about A.D. 1260). This manuscript appears to be the copy given by the translator Campanus to Jacques Pantaleon when he was Patriarch of Jerusalem. Campanus was chaplain to Pantaleon both in Jerusalem and later when that churchman became Pope Urban IV.
An arithmetic written about 1339 by Paolo Dagomari, also known as Paul of the Abacus, furnishes the clue to the derivation of our per cent, symbol. The sign is derived from the abbreviation c° for cento ​(hundred), and its evolution is traced through later manuscripts. As interesting, but not as conclusive, is an illustration from a fifteenth-century manuscript containing the possible progenitor of the dollar sign.
A beautifully written and illuminated copy of the Boethius arithmetic, written on vellum about 1294, is one of the most valuable pieces; the pigskin binding is of about the same date as the text. Just as valuable, because of the rarity of the material, is the copy of al-Khowarazmi's Algebra, a Latin manuscript of 1456. The title is "Book of Mohammed on Algebra and Almuchabala, or Restoration and Opposition." The word "algebra," like the words alchemy and almanac, is of Arabic origin, having the meaning "to restore." So a surgeon, restorer of broken bones, is called in Don Quixote an "algebrista." The word "almuchabala" contains the idea of balance. Both of these terms were applied to early algebras appearing in Europe.
That no expense has been spared in the preparation of the "Rara Arithmetica" is shown by the 255 photographic reproductions, largely full-page, which constitute one of the most valuable features for bibliophiles and librarians. The tremendous labor involved in searching out twelve hundred printed works, as opposed to De Morgan's one hundred, can be understood only by one who has tried to make a complete bibliography of any subject. The citations and references which have been given are sufficient to indicate the fundamental importance of the "Rara Arithmetica" in the history of the development of arithmetic. The actual additions in the notes, to our present knowledge, are entirely too numerous to mention. They show that the library offers a rich field for research in the history of mathematics. Bibliographically the "Rara Arithmetica" will always be an authority in so far as arithmetical books of the period treated are concerned and Americans may justly be proud that this work, which in the nature of the subject might have been considered more properly the field of a European scholar, has been so ably and finally done by a Columbia professor.
The first of the great collections of mathematical works at all to be compared with Mr. Plimpton's was made by Guillaume Libri, the author of the "History of the Mathematical Sciences in Italy." The first volume of his great work was just off the press at the time of the great fire in Paris in 1835. Libri, who had been at the printer's, took a few copies home under his arm; the rest were destroyed. One of the copies preserved, to which Libri made corrections for the second edition of 1838, is on exhibition in the museum of Teachers College, having been bought in Italy by Professor Smith.
Libri began his mathematical career as a boy prodigy, for at the early age of fifteen he was in correspondence with famous mathematicians, and at the age of twenty he was appointed professor of mathematics in the University of Pisa. Being exiled from Italy for political ​reasons when he was twenty-eight years old, he took up his residence in Paris and later became a French citizen. His remarkable ability won him in the brief space of three years the chair of mathematics in the College of Prance and admission to the Academy of Sciences as successor to the great French geometer, Legendre. His activity extended to the political field as inspector-general of public instructon and later as inspector-general of the libraries of France. Soon difficulties of another nature overtook him, as he was accused of appropriating books and manuscripts from French libraries to his own use, in spite of the fact that he had previously offered his valuable collection as a whole, consisting of some 30,000 books and 2,000 manuscripts to the Royal Library of Paris on the rejected condition that it be kept intact as the Libri Collection. His conviction of the misuse of the national libraries occurred, many say unjustly, in 1805 and he was again an exile, living in England as a fugitive from the law; we will not say justice. His library was sold at auction in England, many of the works finding their way into the hands of Prince Boncompagni and after the dispersal of his library into the Plimpton collection and the private library of David Eugene Smith.
Prince Baldassarre Boncompagni, who gathered together a second famous collection of mathematical books and manuscripts, came naturally by his interest in scientific work, as he belonged to that same princely family as Pope Gregory XIII., who revised the calendar. While eminent as a contributor to mathematical literature, Boncompagni' s greater service was as a patron of the science. At his own expense he published the "Bulletin of the Bibliography and History of the Mathematical and Physical Sciences," running through twenty volumes, with many valuable contributions by German, French and Italian scholars to the history of mathematics and astronomy. Even more important were his numerous publications in regard to Leonard of Pisa, who flourished at the beginning of the thirteenth century and to whom was due in a large measure the spread of the Arabic numerals in Italy and Europe. The publications of Boncompagni included two large volumes of the writings of Leonard of Pisa and two Latin versions of the Arabic work of Mohammed ben Musa, al-Khowarazmi, who made the Hindu art of reckoning known to the Arabs in the early ninth century; these Latin versions were made by a Spaniard and an Englishman, both of whom studied at that Moslem center of learning, Toledo, in the early twelfth century. Prince Boncompagni's magnificent collection was offered, on certain mild conditions, to the city of Rome, but was refused. While in printed works of the fifteenth and sixteenth centuries this library was not as complete as is the Plimpton, yet the equal of this collection of old mathematical manuscripts will doubtless never again be held by any private library. The sale at auction of these books took place as recently as 1898.
​The small collection of Augustus De Morgan is worthy of note, as it furnished the stimulus for the publication of the first work dealing wholly with the bibliograhy of arithmetic, De Morgan's "Arithmetical Books," published in London in 1847. Of the fifteenth and sixteenth centuries De Morgan described some seventy arithmetics, while the "Rara Arithmetica" describes well over four hundred. A quotation from the prefatory letter by the great English mathematician in which the book is inscribed to the Rev. George Peacock, a writer on the history of arithmetic, is worth giving: "The most worthless book of a bygone day is a record worthy of preservation. Like a telescopic star, its obscurity may render it unavailable for most purposes; but it serves, in hands which know how to use it, to determine the place of more important bodies." De Morgan's felicity of expression in his numerous publications—he was an extensive contributor to encyclopedias—suggests his kinship to the present popular novelist, William Frend De Morgan, his son.
While the "Arithmetical Books" by De Morgan dealt wholly with arithmetical works, many others have treated the bibliography of mathematics. One of the earliest to give fairly extensive bibliographical references to mathematical literature is the "Kitab al-Fihrist," or "Book of Records," an Arabic treatise written in A.D. 987. The mathematical section of this large book was translated into German by H. Suter and appeared in Leipzig in 1892. The author, who went by the melodious name of Abou'l-Faradsch Mohammed ibn Ishak, or more commonly by the name Ibn Abi Ja'kub al-Nadim, included all the writers known to him, of whatever nationality. The Kitab al-Fihrist is of the greatest importance in the history of mathematics, as it is, indeed, in the history of the development of Christianity, for the writer describes various early sects of the christians. An appreciably large part of our knowledge of Greek mathematics comes from such Arabic sources, for the Arabs kept the spark of Greek learning alive while Europe was in the darkest of the dark ages.
Our interest, however, is in the bibliographers who treated the early printed works. Gerard Joannis Vossius in 1650 published in Amsterdam his work, "On the Four Arts," which is an unreliable mixture of bibliographical and historical material. Naturally many histories of mathematics treated also the bibliography of the subject. The first German work to attempt a somewhat complete list of early printed books in mathematics was the "Einleitung zur mathematischen Bücherkentnis," which J. E. Scheibel completed in 1769 and of which at least two editions appeared. Other German publications, purely bibliographical, are F. G. A. Murhard's "Literatur der mathematischen Wissenschaften" of 1797 and J. Rogg's "Handbuch der mathematischen Literatur," which catalogued and described books from the invention of ​printing up to 1830. The more general treatises on bibliography like those by Graesse and Hain and Copinger also touch this field, although of necessity only incidentally.
Aside from these there have been some purely national works like the "Bibliography of the Lowlands" of the mathematical and physical sciences by Bierens de Haan and the "Biblioteca Mathematica Italiana," by Pietro Riccardi. Professor Smith's "Rara Arithmetica" contains, for the period which it treats, more titles of Italian works than does Riccardi and more German than does Murhard or Eogg. In general, we may say, it is more complete for its specialty than any of the bibliographies hitherto published. The "Rara Arithmetica" may be said to be, with the exception of slight additions, the final bibliography of this field. It may safely be predicted that for centuries to come no other authority will appear to contest its claim to first place.