From Cave Paintings to the Internet A Chronological and Thematic Database on the History of Information and Media Mathematics / Logic Timeline

The Lebombo bone, the oldest known mathematical artifact, is a tally stick with 29 distinct notches that were deliberately cut into a baboon's fibula. It was discovered within the Border Cave in the Lebombo Mountains of Swaziland.

The Lebombo bone resembles the calendar sticks still used by Bushmen in Namibia.

Much of the earliest recorded information consists of paleolithic cave paintings and Cro-Magnon mobiliary art, including bones with talley marks. The purposes of this art may never be fully understood.

[In 1970 Alexander Marshack published his innovative Notation dans les gravures du Paléolithique Supérieur. He argued that talley marks on certain bones represented a system of proto-writing, and proposed the controversial theory that notches and lines carved on certain Upper Paleolithic bone plaques were in fact notation systems, specifically lunar calendars notating the passage of time. Using microscopic analysis, Marshack showed that seemingly random or meaningless notches on bone were sometimes interpretable as structured series of numbers.  Marshack expanded upon these ideas in his book, The Roots of Civilization (1972).]

♦ The oldest cave paintings confirmed by radiocarbon dating are in the Chauvet Cave discovered in the Ardèche region of France in 1994. Paintings in the Chauvet Cave date as early as 30,000 BCE. Because many cave paintings are in deep caves, often in inaccessible locations, it has been suggested that they may not have been for public display, but might have been revealed to cognoscenti by elders of a tribal community.

Mathematics began with the earliest records of attempts to quantify time. The Ishango Bone, a notched talley stick discovered in the Congo (Zaire) in 1960 by Jean de Heinzelin de Braucourt, and now preserved in the Royal Belgian Institute of Natural Sciences, represents, according to Alexander Marschak, a six-month lunar calendar. It is among the earliest known mathematical objects. Other lunar calendars from about the same date have been discovered on other bones such as the Isturitz Baton, and possibly in cave paintings in Lascaux and elsewhere.

According to one theory about the origins of counting and writing developed by Denise Schmand-Besserat, around 8000 BCE the Palaeolithic notched tallies representing the simplest form of counting — in one-to-one correspondence — were superseded by Neolithic tokens in various geometric forms suited for concrete counting. This invention is thought to have been used for about 5000 years prior to the use of abstract numbers which led to writing about 3500 BCE, and then to mathematics about 2600 BCE. Tokens followed basic geometric forms, such as spheres, tetrahedrons, cones, cylinders, discs, quadrangles, triangles. They were first kept in baskets, leather pouchs, clay bowls, and later within clay bullas. 

"The first securely datable mathematical table in world history comes from the Sumerian city of Shuruppag, c. 2600 BCE. The table is ruled into three columns on each side with ten rows on the front or obverse side. The first columns of the obverse list length measures from c. 3.6km to 360 m in descending units of 360 m, followed by the Sumerian word sa ('equal' and/ or 'opposite') while the final column gives their products in area measure. Only six rows are extant or partially preserved on the reverse. They continue the table in smaller units, from 300 to 60 m in 60 m steps, and then perhaps (in the damaged and missing lower half) from 56 to 6 m in 6 m steps. While the table is organized along two axes, there is just one axis of calculation, namely, the horizontal multiplications. Around a thousand tablets were excavated from Shuruppage, almost all of them from houses and buildings which burned down in a city-wide fire in about 2600 BCE, but sadly we have no detailed context for this table because its excavation number was lost or never recorded." (Eleanor Robson, "Tables and tabular formatting in Sumer, Babylonia, and Assyria, 2500 BCE-50," Campbell-Kelly et al [eds]. The History of Mathematical Tables from Sumer to Spreadsheets [2003] 27-29).

The Moscow Mathematical Papyrus, the older of the two best-known mathematical papyri along with the larger Rhind Mathematical Papyrus (noticed in this database), is also called the Golenischev Mathematical Papyrus after its first owner, Egyptologist Vladimir Goleniščev, who in 1909 sold his huge collection of Egyptian artifacts to  Pushkin State Museum of Fine Arts in Moscow, where the papyrus is preserved today.

"Based on the palaeography of the hieratic text, it probably dates to the Eleventh dynasty of Egypt. Approximately 18 feet long and varying between 1 1/2 and 3 inches wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930" (Wikipedia article on Moscow Mathematical Papyrus, accessed 09-11-2009).

Probably the most famous original document of Babylonian mathematics is Plimpton 322, a partly broken clay tablet, approximately 13cm wide, 9cm tall, and 2cm thick. New York publisher George A. Plimpton purchased the tablet from archaeological dealer, Edgar J. Banks about 1922, and bequeathed it with the rest of his collection to Columbia University in the mid 1930s. According to Banks, the tablet came from Senkereh, a site in sourthern Iraq, corresponding to the ancient city of Larsa. 

This tablet has a table of four columns and 15 rows of numbers in cuneiform script, and has been called the only true mathematical table surviving from the period.

Though the tablet was formerly thought to have been a listing of Pythagorean triples, Eleanor Robson rejected earlier mathematical misconceptions of the tablet and pointed out that historical, cultural and linguistic evidence all reveal that the tablet is more likely "a list of regular reciprocal pairs."

Robson, "Words and Pictures. New Light on Plimpton 322," American Mathematical Monthly 109 (2001) 105-121.

In contrast to the scarcity of original sources for Egyptian mathematics, preserved on the relatively fragile medium of papyrus, our knowledge of Babylonian mathematics is derived from several thousand extremely durable clay tablets written in Cuneiform script excavated since the beginning of the nineteenth century.  "The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, the Pythagorean theorem, the calculation of Pythagorean triples and possibly trigonometric functions."

Dating from the Second Intermediate Period of Egypt, the Rhind Mathematial Papyrus is the most significant document of Egyptian mathematics. It was copied by the scribe Ahmes from a now-lost text from the reign of Amenemhat III (12th dynasty). The manuscript  is 33 cm tall and over 5 meters long, and is written in hieratic script. It is dated  Year 33 of the Hyksos king Apophis and also contains a separate later Year 11 on its verso likely from his successor, Khamudi.

"In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving 'Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets'."

Alexander Henry Rhind, a Scottish antiquarian, purchased the papyrus in 1858 in Luxor, Egypt.  It was apparently found during illegal excavations in or near the Ramesseum. The British Museum acquired it in 1864 along with the Egyptian Mathematical Leather Roll, also owned by Rhind.

In Chhandah-shastra, a Sansrit book on meters, or long syllables, “Pingala presents the first known description of a binary numeral system. He described the binary numeral system in connection with the listing of Vedic meters with short and long syllables. His work also contains the basic ideas of maatraameru (Fibonacci number) and meruprastaara (Pascal’s triangle.)”

Because the numbering systems of the Mesopotamians, Babylonians, Egyptians, Greeks and Romans are not convenient for extensive calculation, it is believed that they used some sort of mechanical calculating device. The simplest form of calculating device is a kind of table or tablet on which calculation can be written in sand or dust, and then easily erased. This is the "sandboard abacus". One derivation of the Latin word abacus comes from the Greek abakos from the Hebrew word abaq, meaning dust.

In his Histories Herodotus of Halicarnassus, written about this time, stated that the Egyptians "write their characters and reckon with pebbles, bringing their hand from right to left, while the Greeks go from left to right." D.E. Smith, in his History of Mathematics II, p. 160 quotes this statement by Herodotus and writes, "Right to left order was that of the hieratic script and there is probably some relation between this script and the abacus. No wall pictures thus far discovered give any evidence of the use of the abacus, but in any collection of Egyptian antiquities there may be found disks of various sizes which may have been used as counters."

What we call Arabic numerals were invented in India by the Hindus. Because the Arabs transmitted this system to the West after the Hindu numerical system found its way to Persia, the numeral system became known as Arabic numerals, though Arabs call the numerals they use “Indian numerals”, أرقام هندية, arqam hindiyyah.

Euclid of Alexandria, a teacher at the Alexandrian Library under the reign of Ptolemy I, wrote the Elements during this time, “in which he summarized the preceding two centuries of mathematical research. Now known as the founding document of mathematics, the Elements was the standard textbook for mathematical education in ancient times, in the Islamic world, and in Europe through the Middle Ages, the Renaissance, and until almost the present day. The system of thought presented by the Elements, in which knowledge was distilled in the form of theorems and then given a written proof, inspired fields as diverse as law and physics. Indeed, Newton’s Principia, which marked the beginning of modern physics, took Euclid’s work as its intellectual and stylistic model.”

Excluding counting on the fingers, counting boards are the earliest known counting device, and a precursor of the abacus. They were made from stone or wood and the counting was done on the board with beads or pebbles or or sand or dust.  These devices have also been called the "sandboard abacus." The earliest surviving example of a counting board or a gaming board may be a tablet found about 1850 CE on the Greek island of Salamis which dates back to about 300 BCE. It is preserved in the Greek National Museum at Athens. 

"It is a slab of white marble 149 cm long, 75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. In the center of the tablet is a set of 5 parallel lines equally divided by a vertical line, capped with a semi-circle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semi-circle at the top of the intersection; the third, sixth and ninth of these lines are marked with a cross where they intersect with the vertical line."  Three sets of Greek symbols (numbers from the acrophonic system) are arranged along the left, right and bottom edges of the tablet.

The Mawangdui Silk Texts (Chinese: 馬王堆帛書; pinyin: Mǎwángduī Bóshū), texts of Chinese philosophical and medical works written on silk, were found buried in Tomb no. 3 at Mawangdui, in the city of Changsha, Hunan, China in 1973. 

"They include the earliest attested manuscripts of existing texts such as the I Ching, two copies of the Tao Te Ching, one similar copy of Strategies of the Warring States and a similar school of works of Gan De and Shi Shen. Scholars arranged them into silk books of 28 kinds. Together they count to about 120,000 words covering military strategy, mathematics, cartography and the six classical arts of ritual, music, archery, horsemanship, writing and arithmetic" (Wikipedia article on Mawangdui Silk Texts, accessed 01-31-2010).

Most of the Mawangdui Silk Texts are preserved in the Hunan Provincial Museum.

The Antikythera Mechanism discovered off Antikythera, Greece in 1901, includes the only specimen preserved from antiquity of a scientifically graduated instrument, and also may also be thought of as the earliest extant mechanical calculator. "The Antikythera mechanism must therefore be an arithmetical counterpart of the much more familiar geometrical models of the solar system which were known to Plato and Archimedes and evolved into the orrery and the planetarium. The mechanism is like a great astronomical clock without an escapement, or like a modern analogue computer which uses mechanical parts to save tedious calculation . . . . It is certainly very similar to the great astronomical cathedral clocks that were built. . . ." in Europe beginning in the fourteenth century.

Applying high-resolution imaging systems and three-dimensional X-ray tomography, in 2008 experts deciphered inscriptions and reconstructed functions of the bronze gears on the mechanism. The results of this research, illustrated in a video available at this link, revealed details of dials on the instrument’s back side, including the names of all 12 months of an ancient calendar. Scientists found that the device not only predicted solar eclipses but also organized the calendar in the four-year cycles of the Olympiad, forerunner of the modern Olympic Games.

In December 2008, Michael Wright described a more complete reconstruction of the device which he built, in a video available at this link.

The new findings also suggested that the mechanism’s concept originated in the colonies of Corinth, possibly Syracuse, in Sicily. The scientists said this implied a likely connection with Archimedes, who lived in Syracuse and died in 212 B.C. Archimedes invented a planetarium calculating motions of the moon and the known planets, and wrote a lost manuscript on astronomical mechanisms. Some evidence had previously linked the complex device of gears and dials to the island of Rhodes and the astronomer Hipparchos, who had made a study of irregularities in the Moon’s orbital course.

Hellenistic astronomer, geographer, and mathematician, Hipparchos of Rhodes, produces a table of chords, an early example of a trigonometric table. 

". . .some historians go so far as to say that trigonometry was invented by him. The purpose of this table of chords was to give a method for solving triangles which avoided solving each triangle from first principles. He also introduced the division of a circle into 360 degrees into Greece" (Mactutor biography of Hipparchus, accessed 11-27-2008).

The rudimentary astrolabe was invented in the Hellenistic world and is often attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was effectively an analog calculator capable of working out several different kinds of problems in spherical astronomy.

Julius Caesar introduces the Julian calendar.

The Julian Calendar has a regular year of 365 days divided into 12 months, and a leap day is added every four years, so the average Julian year is 365.25 days. The calendar remained in use into the 20th century in some countries and is still used by many national Orthodox churches. "However with this scheme too many leap days are added with respect to the astronomical seasons, which on average occur earlier in the calendar by about 11 minutes per year, causing it to gain a day about every 128 years. It is said that Caesar was aware of the discrepancy, but felt it was of little importance."

Caesar planned to establish a public library to equal or surpass the one at Alexandria. He appointed Marcus Terentius Varro, a noted scholar and book collector, to gather copies of the best-known literature for a Roman public library. However these plans were shelved when Caesar was assassinated in 44 BCE.

The first census of which records are preserved is taken in China during the Han Dynasty. At that time there are 57.5 million people living in Han China—the world’s largest population.

Date of one of the oldest and most complete diagrams from Euclid’s Elements—a fragment of papyrus found among the rubbish piles of Oxyrhynchus in 1896-97 by the expedition of B. P. Grenfell and A. S. Hunt. It is preserved at the University of Pennsylvania.

"The diagram accompanies Proposition 5 of Book II of the Elements, and along with other results in Book II it can be interpreted in modern terms as a geometric formulation of an algebraic identity - in this case, that ab + (a-b)2/4 = (a+b)2/4 (although the relationship between Euclid's propositions and algebra, which he did not possess, is controversial)."

On charges of treason, Theodoric the Great, Ostrogothic ruler of Italy, executes Hellenist and philosopher Anicius Manlius Severinus Boëthius, who had risen to the office of Magister officiorum (head of all government and court services) in Theodoric's court.

The execution took place in 524 or 525,  possibly because Theodoric suspected Boëthius's involvement in a plot with the Byzantine Emperor Justin I, whose religious orthodoxy, in contrast to Theodoric's Arian opinions, increased their political rivalry.

♦ The date of Boëthius's execution is often used as a reckoning of the onset of the Middle Ages.

"Boethius's most popular work is the Consolation of Philosophy, which he wrote in prison while awaiting his execution, but his lifelong project was a deliberate attempt to preserve ancient classical knowledge, particularly philosophy. He intended to translate all the works of Aristotle and Plato from the original Greek into Latin. His completed translations of Aristotle's works on logic were the only significant portions of Aristotle available in Europe until the 12th century. However, some of his translations (such as his treatment of the topoi in The Topics) were mixed with his own commentary, which reflected both Aristotelian and Platonic concepts.

"Boethius also wrote a commentary on the Isagoge by Porphyry, which highlighted the existence of the problem of universals: whether these concepts are subsistent entities which would exist whether anyone thought of them, or whether they only exist as ideas. This topic concerning the ontological nature of universal ideas was one of the most vocal controversies in medieval philosophy.

"Besides these advanced philosophical works, Boethius is also reported to have translated important Greek texts for the topics of the quadrivium.His loose translation of Nicomachus's treatise on arithmetic (De institutione arithmetica libri duo) and his textbook on music (De institutione musica libri quinque, unfinished) contributed to medieval education. His translations of Euclid on geometry and Ptolemy on astronomy, if they were completed, no longer survive.

"In his "De Musica", Boethius introduced the threefold classification of music:
1. Musica mundana - music of the spheres/world
2. Musica humana - harmony of human body and spiritual harmony
3. Musica instrumentalis - instrumental music (incl. human voice)" (Wikipedia article on Anicius Manlius Severinus Boethius, accessed 11-28-2008).

Note: "Boëthius" has four syllables; the o and e  are pronounced separately. This was traditionally written with a diæresis, viz. "Boëthius," a spelling which has been disappearing due to the limitations of word processors.

Dionysius Exiguus, a computist, uses a true zero in tables alongside Roman numerals, but he uses the zero as a word, nulla meaning nothing, not as a symbol. "When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future computists (calculators of Easter). 

"Computus (Latin for computation) is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age."

♦ This is the root of the modern word "computer."

Brahmagupta writes Brahmasphutasiddhanta (The Opening of the Universe). "It contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both negative and positive, a method for computing aquare roots, methods of solving linear and some quadratic equations, and rules for summing series, Brahamgupta's identity, and the Brahmagupta's theorem."

By this time a base 10 numeral system with nine symbols is widely used in India, and the concept of zero (represented by a dot) is known.

Balthild, widow of Clovis II, and her son Clotaire III, found Corbie Abbey.

The first monks at Corbie came from Luxeuil Abbey, which had been founded by Saint Columbanus in 590, and the Irish respect for classical learning fostered at Luxeuil was carried forward at Corbie. The rule of these founders was based on the Benedictine rule, as modified by Columbanus.

"Above all, Corbie was renowned for its library, which was assembled from as far as Italy, and for its scriptorium. In addition to its patristic writings, it is recognized as an important center for the transmission of the works of Antiquity to the Middle Ages. An inventory (of perhaps the 11th century) lists the church history of Hegesippus, now lost, among other extraordinary treasures. In the scriptorium at Corbie the clear and legible hand known as Carolingian minuscule was developed, in about 780, as well as a distinctive style of illumination.

"Three of Corbie's ninth-century scholars were Ratramnus (died ca. 868), Radbertus Paschasius (died 865) and the shadowy figure of Hadoard. Jean Mabillon, the father of paleography, had been a monk at Corbie.

"Among students of Tertullian, the library is of interest as it contained a number of unique copies of Tertullian's works, the so-called corpus Corbiense and included some of his unorthodox Montanist treatises, as well as two works by Novatian issued pseudepigraphically under Tertullian's name. The origin of this group of non-orthodox texts has not satisfactorily been identified.

"Among students of medieval architecture and engineering, such as are preserved in the notebooks of Villard de Honnecourt, Corbie is of interest as the center of renewed interest in geometry and surveying techniques, both theoretical and practical, as they had been transmitted from Euclid through the Geometria of Boëthius and works by Cassiodorus (Zenner).

"In 1638, 400 manuscripts were transferred to the library of the monastery of St. Germain des Prés in Paris. In the French Revolution, the library was closed and the last of the monks dispersed: 300 manuscripts still at Corbie were moved to Amiens, 15 km to the west. Those at St-Germain des Prés were loosed on the market, and many rare manuscripts were obtained by a Russian diplomat, Petrus Dubrowsky, and sent to St. Petersburg. Other Corbie manuscripts are at the Bibliothèque Nationale. Over two hundred manuscripts from the great library at Corbie are known to survive" (Wikipedia article on Corbie Abbey, accessed 08-20-2009).

"The earliest reference in the Mediterranean world to the Indian system of numeration [Arabic numerals] dated from the mid-seventh century, just after the rise of Islam. In a fragment, dated 662, of a work by Severus Sebokht, the learned bishop of the monastery of Quinnasrin (located on the Euphrates in Syria), the bishop expresses his admiration for the Indians because of their valuable method of computation 'done by means of nine signs.' Severus had probably learned about the system from Eastern merchants active in Syria. This ingenious and eminently simple system of representing any quantity by using nine symbols in decimal place value (there was orignally no zero) arose in India perhaps as early as the fifth century. The indian system seems to have been known in Baghdad as early as 770, or less than a decade after its founding, but it was principally diffused through the writings of the Abbasid mathematician and geographer Muhmmad ibn Musa al-Khwarizmi (al-Khwarazmi) who died around 846" (Bloom, Paper Before Print. The History and Impact of Paper on the Islamic World [2001] 129).

A manuscript entitled Romana computatio, dated 688, appears to be the earliest extant document on ancient Roman techniques of finger reckoning. It was probably used as a source by the Venerable Bede for his De tempore ratione liber (725).

Sherman, Writing on Hands. Memory and Knowledge in Early Modern Europe (2000) 28.

Northumbrian Anglo-Saxon monk, the Venerable Bede, writes De temporum ratione (On The Reckoning Of Time). 

"The noted historian of science, George Sarton, called the eighth century 'The Age of Bede'. Bede wrote several major scientific works: a treatise On the Nature of Things, modeled in part after the work of the same title by Isidore of Seville; a work On Time, providing an introduction to the principles of Easter computus; and a longer work on the same subject; On the Reckoning of Time, which became the cornerstone of clerical scientific education during the so-called Carolingian renaissance of the ninth century. He also wrote several shorter letters and essays discussing specific aspects of computus and a treatise on grammar and on figures of speech for his pupils.

"On the Reckoning of Time (De temporum ratione) included an introduction to the traditional ancient and medieval view of the cosmos, including an explanation of how the spherical earth influenced the changing length of daylight, of how the seasonal motion of the Sun and Moon influenced the changing appearance of the New Moon at evening twilight, and a quantitative relation between the changes of the Tides at a given place and the daily motion of the moon. Since the focus of his book was calculation, Bede gave instructions for computing the date of Easter and the related time of the Easter Full Moon, for calculating the motion of the Sun and Moon through the zodiac, and for many other calculations related to the calendar. He gives some information about the months of the Anglo-Saxon calendar in chapter XV. Any codex of Bede's Easter cycle is normally found together with a codex of his 'De Temporum Ratione' " (Wikipedia article on Bede, accessed on 11-22-2008).

The first chapter of Bede's De temporum ratione liber entitled "De computo et loquela digitorum" (On computing and speaking with the fingers) explained the method of finger reckoning which had evolved since the ancient world, as a reliable method, especially when a writing surface or writing implements were not available. Though the method was mentioned by classical authors such as Herodotus, no treatises on the topic survived, and it is thought that the technique was passed down mainly through oral tradition.  Bede described "upwards of fifty finger symbols, the numbers extending through one million" (Smith, History of Mathematics [1925] II, 200).  Undoubtedly Bede's text, of which numerous medieval manuscripts survived, was influential on conveying the method during the Middle Ages. Bede's text on finger reckoning was first published by Johannes Aventinus in Abacus atque vetustissima veterum Latinorum per digitos manusque numerandi (1522).

For a discussion of Bede's manual calculating methods see Sherman, Writing on Hands. Memory and Knowledge in Early Modern Europe (2000) 28-30.

Regarding the transmission of Hindu numbers to the Arabs, al-Qifti's "Chronology of the Scholars," written around the end of the 12th century but quoting earlier sources, stated:

". . . a person from India presented himself before the Caliph al-Mansur in the year 776 who was well versed in the siddhanta method of calculation related to the movement of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ... Al-Mansur ordered this book to be translated into Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets. . ." (Mactutor article on The Arabic numeral system, accessed 01-16-2009).

The book from which the early Indian scholar presented may have been the Brahmasphutasiddhanta (The Opening of the Universe), written in 628 by the Indian mathematician Brahmagupta, which had used Hindu Numerals with the zero sign.

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, a Persian mathematician, astronomer, and geographer at the House of Wisdom (Arabic: بيت الحكمة‎; Bait al-Hikma) in Baghdad, develops the concept of a written process to be followed to achieve some goal.

Al-Khwarizmi wrote a book on Hindu-Arabic numerals, giving the name algorithm to this process through the Latinization of his last name:

"The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum (in English Al-Khwarizmi on the Hindu Art of Reckoning) gave rise to the word algorithm deriving from his name in the title. Unfortunately the Latin translation . . . .  is known to be much changed from al-Khwarizmi's original text (of which even the title is unknown). The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of zero as a place holder in positional base notation was probably due to al-Khwarizmi in this work. Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version" (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Khwarizmi.html, accessed 01-23-2010).

Information in Al-Khwarizmi's work eventually reached Europe in books on Algorithmus by other authors that were distributed by manuscript copying, and eventually by print . . . .  Allard, "La diffusion en occident des premières oeuvres latines issues de l'arithmétique perdue d'al-Khwarizmi," J. Hist. Arabic Sci. 9 (1-2) (1991), 101-105, discusses seven twelfth century Latin treatises based on this lost Arabic treatise by al-Khwarizmi on arithmetic.

Persian mathematician, astronomer and geographer Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī publishes Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious Book on Calculation by Completion and Balancing.”

It was written "with the encouragement of the Caliph Al-Ma'mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance. The term algebra is derived from the name of one of the basic operations with equations (al-jabr) described in this book. It provided an exhaustive account of solving polynomial equations up to the second degree, and introduced the fundamental methods of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation" (Wikipedia article on Muhammad ibn Mūsā al-Khwārizmī, accessed 01-23-2010).

The work was translated in Latin as Liber algebrae et almucabala by Robert of Chester (Segovia, circa 1145) from which our word "algebra" originates, and also by Gerard of Cremona. Robert of Chester's translation was translated into English as Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi with an Introduction, Critical Notes and an English Version by Louis Charles Karpinski (1915). Karpinski included a survey of the manuscripts of Chester's text available to him.

The Vatican Euclid (Vat. gr. 190, called P), a version of the Greek text dating from the ninth century, and excluding the addendum to the final proposition of book VI by the fourth century editor, Theon of Alexandria, has been called "the single most important manuscript of the Elements" (N. M. Swerlow, "The Recovery of the Exact Sciences of Antiquity: Mathematics, Astronomy, Geography," Grafton (ed.) Rome Reborn. The Vatican Library and Renaissance Culture (1993) 128-29 & plates 101-102).

"The event, however, that had the most enduring effect within the Greek phase of the transmission of the Elements was the edition and slight emendation it underwent at the hands of Theon of Alexandria (fourth century; not to be confused with the second century Neoplatonist, Theon of Smyrna). The result of Theon's efforts funished the text for every Greek edition of Euclid until the nineteenth century. Fortunately, in his commentary to Ptolemy's Almagest, Theon indicates that he was responsible for an addendum to the final proposition of book VI in his 'edition (ekdosis) of the Elements'; for it was this confession that furnished scholars with their first clue in unraveling the problem of the pre-Theonine, 'pristine' Euclid. In 1808 François Peyrard noted that a Vatican manuscript (Vat. graec. 190) which Napoleon had appropriated for Paris did not contain the addition Theon had referred to. This, coupled with other notable differences from the usual Theonine editions of the Elements, led Peyrard to conclude that he had before him a more ancient version of Euclid's text. Accordingly, he employed the Vatican codex, as well as several others, in correcting the text presented by the editio princeps of Simon Grynaeus (Basel, 1533). Others, utilizing occasional additional (but always Theonine) manuscripts or earlier editions, continued to improve Peyrard's text, but it was not until J. L. Heiberg began the reconstruction of the text anew on the basis of the Vatican and almost all other known manuscripts that a critical edition of Elements was finally (1883-1888) established. Heiberg not only in great measure succeeded in getting behind the numerous Theonine alterations and additions, but also was able to sift out a considerable number of pre-Theonine interpolations. In addition to the authority of the non-Theonine Vatican manuscript, he culled papyri framents, scholia, and every known ancient quotation of, or reference to, the Elements for evidence in his construction of the 'original' Euclid. The result still stands" (John Murdoch, "Euclid: Transmission of the Elements," Dictionary of Scientific Biography IV [1971] 437-38).

The d'Orville Euclid is the earliest "complete" manuscript of Euclid's Elements, and,  according to the Bodleian Library exhibition catalogue, The Survival of Greek Literature, it is the oldest manuscript of a classical Greek author to bear a date.

MS. d’Orville 301, which has been preserved in the Bodleian Library, Oxford, since 1804, was written on parchment in Constantinople by Stephanus clericus, and bought by Arethas of Patrae, later Bishop of Caesarea in Cappadocia, for 14 nomismata (gold coins).

"The hand of Stephanus is pure minuscule; Arethas added the scholia and some additional mater in small uncials."

From the death of Arethas (c. 939) the ownership of the manuscript is unknown until the seventeenth century, when it was acquired by the Dutch classicist J. P. D’Orville, most of whose collection was eventually purchased by the Bodleian Library.

Hunt, R.W., The Survival of Ancient Literature, Oxford: Bodleian Library, 1975,  no. 55,

You can page through digital images of the entire manuscript at http://librarieswithoutwalls.org/bookviewer/?src=%3Fsrc%3D001eucmsd27.jpg&jump=006&zmnu=1&src=006eucmsd27.jpg&zoom=1&old_x=0&old_y=0&zdir=in&zbut=&pan=&flip=prev&jact=&width=900∏=orf&capt=&fwin=1 (accessed 07-12-2009).

The astrolabe, an astronomical instrument used for observing planetary movements, was indispensable for navigation. A type of analog calculator, brass astrolabes were developed in the medieval Islamic world, and were also used to determine the location of the Kaaba in Mecca, in which direction all Muslims face during prayer. Planispheric, or flat, astrolabes, were more common than the linear or spherical types. In planispheric astrolabes the celestial sphere was drawn on a flat surface and represented on one plate.

The earliest known dated astrolabe is of the planispheric type. Made of cast bronze, it bears the name of its maker. The inscription at the back of the kursi, or throne, is written in Kufic , the oldest calligraphic form of the various Arabic scripts, and states that the astrolabe was made by Nastulus (or Bastulus) and gives the date, which corresponds to 927/28. The date is rendered in Arabic letters, whose numerical values total 315, signifying the year in the Islamic calendar in which the astrolabe was made. It is preserved in the School of Oriental and African Studies at the University of London.

The 10th century manuscript of the Synagoge or Collection of Pappus of Alexandria, written on parchment and preserved in the Vatican Library, reached the papal library in the thirteenth century. It is the earliest surviving copy of the text, and the basis for all later versions, of which none is earlier than the sixteenth century.

Pappus  (c. 290 – c. 350) was one of the last great Greek mathematicians of antiquity. In addition to his Synagoge or Collection, Pappus is known for Pappus's Theorem in projective geometry. Nothing is known of his life, except that he had a son named Hermodorus, and was a teacher in Alexandria.

The so-called Arabic numerals were invented in India and tranferred to the Arabs who developed the system in in the moorish empire of Al-Andalus in the Iberian peninsula. The oldest record of the use of Arabic numerals in Europe is a leaf in the codex Virgilianus, ms. lat. DI.2f.9v preserved in Madrid at the Biblioteca S. Lorenzo del Escorial.

Frugon, Inventions of the Middle Ages (2007) 52, figure 36, & footnote 95.

Gerbert d'Aurillac, scholar, teacher, tutor and counsellor to Otto III, and Pope Sylvester II (or Silvester II) from 999 till his death in 1002, is considered influential in introducing Arabic knowledge of arithmetic, mathematics, and astronomy to Europe, reintroducing the abacus and armillary sphere which had been lost to Europe since the end of the Greco-Roman era.

"According to William of Malmesbury (c.1080 – c.1143), Gerbert stole the idea of the computing device of the abacus from a Spanish Arab. The abacus that Gerbert reintroduced into Europe had its length divided into 27 parts with 9 number symbols (this would exclude zero, which was represented by an empty column) and 1,000 characters in all, crafted out of animal horn by a shieldmaker of Rheims. According to his pupil Richer, Gerbert could perform speedy calculations with his abacus that were extremely difficult for people in his day to think through in using only Roman numerals. Due to Gerbert's reintroduction, the abacus became widely used in Europe once again during the 11th century" (Wikipedia article on Pope Sylvester II, accessed 11-24-2008).

Gerard of Cremona, in Toledo, Spain, translated Ptolemy's Almagest from Arabic into Latin. He also edited for Latin readers the Tables of Toledo, the most accurate compilation of astronomical data available in Europe at the time. The Tables were partly the work of Al-Zargali, known to the West as Arzachel, a mathematician and astronomer who flourished in Cordoba in the eleventh century.

Persian mathematician and astronomer of the Islamic Golden Age Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī originates the concept of mathematical function. 

"In his analysis of the equation x3 + d = bx2 for example, he begins by changing the equation's form to x2(b − x) = d. He then states that the question of whether the equation has a solution depends on whether or not the 'function' on the left side reaches the value d. To determine this, he finds a maximum value for the function. Sharaf al-Din then states that if this value is less than d, there are no positive solutions; if it is equal to d, then there is one solution; and if it is greater than d, then there are two solutions" (Wikipedia article on Function (mathematics), accessed 03-26-2009)

A version of the abacus appears in China, called suanpan in Chinese. On each rod this abacus has 2 beads on the upper deck and 5 on the lower deck.

The suanpan style of abacus is also referred to as a 2/5 abacus. The 2/5 style survived unchanged until about 1850, at which time the 1/5 (one bead on the top deck and five beads on the bottom deck) abacus appeared.

♦ "In the famous long scroll Along the River During Qing Ming Festival painted by Zhang Zeduan (1085-1145) during the Song Dynasty (960-1279), a 15 column suanpan is clearly seen lying beside an account book and doctor's prescriptions on the counter of an apothecary).

"The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, as there is some evidence of a trade relationship between the Roman Empire and China. However, no direct connection can be demonstrated, and the similarity of the abaci may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model and Chinese model (like most modern Japanese) has 4 plus 1 bead per decimal place, the old version of the Chinese suanpan has 5 plus 2, allowing less challenging arithmetic algorithms, and also allowing use with a hexadecimal numeral system. Instead of running on wires as in the Chinese and Japanese models, the beads of Roman model run in grooves, presumably making arithmetic calculations much slower.

"Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of a zero as a place holder. The zero was probably introduced to the Chinese in the Tang Dynasty (618-907) when travel in the Indian Ocean and the Middle East would have provided direct contact with India and Islam allowing them to acquire the concept of zero and the decimal point from Indian and Islamic merchants and mathematicians."

Leonardo of Pisa, later known by his nickname Fibonacci, writes Liber Abaci or The Book of the Abacus or The Book of Calculation.

In Liber Abaci Fibonacci introduced Arabic numerals to the European public. These Fibonacci had learned while in Africa with his father who wanted him to become a merchant.

"Liber Abaci was not the first Western book to describe Arabic numerals, but by addressing tradesmen rather than academics, it was the book that convinced the public of the superiority of the new system. The first section introduces the Arabic numeral system. The second section presents examples from commerce, such as conversions of currency and measurements, and calculations of profit and interest. The third section discusses a number of mathematical problems. One example, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots. The book also includes Euclidean geometric proofs and a study of simultaneous linear equations."

The European table abacus or reckoning table  became standardized to some extent by this time. The pebbles previously used as counters were replaced by specially minted coin-like objects that were cast, thrown, or pushed on the abacus table. They were called jetons from jeter (to throw) in France, and werpgeld for “thrown money” in Holland.

Majorcan writer and philosopher Ramon Llull (Ramon Lull) publishes in his Ars generalis ultima or Ars magna  (the "The Ultimate General Art") a method of combining religious and philosophical attributes selected from a number of lists, which he invented about 1275. It is believed that Llull's inspiration for the Ars magna came from observing Arab astrologers using a mechanical device called a zairja to calculate ideas.

Llull's method

"was intended as a debating tool for winning Muslims to the Christian faith through logic and reason. Through his detailed analytical efforts, Llull built an in-depth theological reference by which a reader could enter in an argument or question about the Christian faith. The reader would then turn to the appropriate index and page to find the correct answer.

"Llull also invented numerous 'machines' for the purpose. One method is now called the Lullian Circle, each of which consisted of two or more paper discs inscribed with alphabetical letters or symbols that referred to lists of attributes. The discs could be rotated individually to generate a large number of combinations of ideas. A number of terms, or symbols relating to those terms, were laid around the full circumference of the circle. They were then repeated on an inner circle which could be rotated. These combinations were said to show all possible truth about the subject of the circle. Llull based this on the notion that there were a limited number of basic, undeniable truths in all fields of knowledge, and that we could understand everything about these fields of knowledge by studying combinations of these elemental truths.

"The method was an early attempt to use logical means to produce knowledge. Llull hoped to show that Christian doctrines could be obtained artificially from a fixed set of preliminary ideas. For example, one of the tables listed the attributes of God: goodness, greatness, eternity, power, wisdom, will , virtue, truth and glory. Llull knew that all believers in the monotheistic religions - whether Jews, Muslims or Christians - would agree with these attributes, giving him a firm platform from which to argue.

"The idea was developed further by Giordano Bruno in the 16th century, and by Gottfried Leibniz in the 17th century for investigations into the philosophy of science.

"Leibniz gave Llull's idea the name ars combinatoria, by which it is now often known. Some computer scientists have adopted Llull as a sort of founding father, claiming that his system of logic was the beginning of information science" (Wikipedia article on Ramon Llull, accessed 04-02-2009).

The Aztec calendar stone or Aztec Sunstone Calendar, carved in basalt, is 3.6 meters (12 feet) in diameter and weighs about 24 metric tons. It was originally placed atop the main temple in Tenochtitlan, the capital of the Aztec empire, facing south in a vertical position and was painted a vibrant red, blue, yellow and white.

When the Spaniards conquered Tenochtitlan in 1521 they buried the stone, and built the cathedral of Mexico City on the site. For over 250 years the stone was lost until December of 1790 when it was excavated by accident during repair work on the cathedral. Today it is located in the  Museo Nacional de Antropologia, Mexico City.

"The stone was first described by the Mexican astronomer, anthropologist and writer, Antonio de León y Gama in Descripción histórica y cronológica de las dos piedras: que con ocasión del empedrado que se está formando en la plaza Principal de México, se hallaron en ella el año de 1790. Impr. de F. de Zúñiga y Ontiveros, 1792. "In it Leon y Gama described the discovery in 1790 of two of the most important pieces of aztec art in the Zócalo, main plaza of the city of Mexico: the sun stone and a statue of Coatlicue, an aztec goddess. Leon y Gama also included in it most of his knowledge and theories on how aztecs measured time. The work, as opposed to authors of previous centuries, praised Aztec society and their scientific and artistic achievements in line with the growing Mexican nationalism in the late 18th century. It was published by Felipe de Zúñiga y Ontiveros, [scientist and cartographer and ] owner of one of the most important printing establishments in America at the time. In addition to print the book had three folded manuscript watercolor drawings[presumably hand-colored engravings.] Thanks to the publication of the book Leon y Gama is considered by many the first Mexican archeologist" (Wikipedia article on Antonio de León y Gama, accessed 01-01-2010).

Michael of Rhodes, a Venetian galley commander, writes a manuscript describing his knowledge of mathematics, ships and shipbuilding, navigation, and time reckoning. It contains some of the earliest surviving portolan aids to navigation and the world's first known treatise on shipbuilding.

The anonymous Arte dell’Abbaco . . . on the operation of the abacus, printed in Treviso, Italy, probably by Gerardus de Lisa, de Flandria, is the first dated book on arithmetic. It is possible that some undated pamphlets on Algorithmus may predate this work.

"Frank J. Swetz translated the complete work using Smith's notes in 1987 in his Capitalism & Arithmetic: The New Math of the 15th Century. Swetz used a copy of the Treviso housed in the Manuscript Library at Columbia University. The volume found its way to this collection via a curious route. Maffeo Pinelli (1785), an Italian bibliophile, is the first known owner. After his death his library was purchased by a London book dealer and sold at auction on February 6, 1790. The book was obtained for three shillings by Mr. [Michael] Wodhull. About 100 years later the Arithmetic appeared in the library of Brayton Ives, a New York lawyer. When Ives sold the collection of books at auction, George [Arthur] Plimpton, a New York publisher, acquired the Treviso and made it an acquisition to his extensive collection of early scientific [i.e. mathematics] texts. Plimpton donated his library to Columbia University in 1936. Original copies of the Treviso Arithmetic are extremely rare" (Wikipedia article Treviso Arithmetic, accessed 01-10-2009).

ISTC no. ia01141000.

Erhard Ratdolt of Venice issues the first printed edition (editio princeps) of Euclid's Elements—Praeclarissimus liber elementorum Euclidis in artem geometriae.

Ratdolt's text was based upon a translation from Arabic to Latin, presumably made by Abelard of Bath in the 12th century, edited and annotated by Giovanni Compano (Campanus of Novara) in the 13th century. The first printed edition of Euclid was the first substantial book to contain geometrical figures, of which it included over 400.

Ratdolt printed several copies with a dedicatory epistle in gold letters, including a dedication copy to the Doge of Venice. Of these, seven copies are preserved. To accomplish this technical feat:

"Ratdolt developed an innovative technique derived from the methods used by bookbinders to stamp gold on leather. This involved strewing a powdered bonding agent (either resin or dried albumen) on the page and probably heating the metal types so that the gold-leaf would stick to the paper. For his 1488 edition of the 'Chronica Hungarorum', Ratdolt employed a simpler method using golden printing ink. His technique of printing in golden letters was first copied in 1499 by the Venetian printer Zacharias Kallierges" (Wagner, Als die Lettern laufen lernten. Inkunabeln aus der Bayerischen Staatsbibliothek München [2009] no. 20).

In order to print the unusually large number of complex geometrical diagrams, usually containing type, in the margins Ratdolt used printer's "rules," i.e. thin strips of metal, type high, which he bent and cut and adjusted and set into a substance that would both hold them (and pieces of type) in place, and could itself be incised with the design as a guide to modelling and assembly.

Renzo Baldasso, "La stampa dell'editio princeps degli Elementi di Euclide (Venezia, Erhard Ratdolt, 1482)", The Books of Venice/Il libro veneziano, ed. Lisa Pon and Craig Kallendorf (2009) 61-100.

There are two distinct states of the first edition. The second state has leaves a1-a9 set differently from the first state: the heading on a1v is in two lines rather than three and is set in the same type as the text rather than heading type; the three-sided woodcut border and woodcut initial P are added to a2r; the headline in red on a2r begins "Preclarissimus liber elementorum"; and headlines do not begin until a10r. "The two outer pages of sheet c1 also differ, having been evidently reprinted owing to errors in the text and the diagram. . . of the 12th proposition of the 4th book" (B.M.C. vol. 5, 285-286.). See Horblit, One Hundred Books Famous in Science (1964) no. 27. for a detailed illustrated comparison of the two states. Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 729.

♦ Characterized as the most famous textbook ever published, Euclid's Elements was one of the most widely printed and studied texts for the next 500 years. It is also considered to the most widely printed text after the Bible, with more than 1000 editions issued.

♦ You can view a digital facsimile of one of the copies with the dedication printed in gold from the website of the Bayerische Staatsbibliothek, Munich, at this link: http://daten.digitale-sammlungen.de/~db/0003/bsb00037426/images/index.html?id=00037426&fip=67.164.64.97&no=4&seite=6, accessed 04-24-2010.

Based on the unusually large number of surviving copies, Ratdolt printed an edition considerably larger than the 300 copies considered average for a 15th century print run. You can view the long list of institutions which hold a copy at ISTC no. ie00113000.

German printer Erhard Ratdolt working in Venice publishes Tabulae Alphonsinae or the Alphonsine Tables, a compilation of astronomical data tabulating the positions and movements of the planets.

The Alphonsine Tables were among the first mathematical tables printed. The tables were computed at Toledo, Spain, from 1262 to 1272 by about 50 astronomers (human computers) assembled for the purpose by King Alfonso X of Castile and León, known as el Sabio, "the learned."  They were a revision and improvement of the Tables of the Cordoban mathematician/astronomer Abū Ishāq Ibrāhīm al-Zarqālī, retaining the Ptolemaic system for explaining celestial motion. The original Spanish version was lost, and the tables became known through Latin translation.

ISTC no. ia00534000.

Luca Bartolomeo de Pacioli publishes at the press of Paganinus de Paganinis in Venice Summa de arithmetica geometria, proporzioni et proporzionalita.

This was “the first great general work on mathematics printed” (Smith, Rara arithmetica, 56).

“[The Summa] contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid’s geometry. . . . Although it lacked originality, the Summa was widely circulated and studied by the mathematicians of the sixteenth century. Cardano, while devoting a chapter of his Practica arithmetice (1539) to correcting the errors in the Summa, acknowledged his debt to Pacioli. Tartaglia’s General trattato de’ numeri et misure (1556-1560) was styled on Pacioli’s Summa. In the introduction to his Algebra, Bombelli says that Pacioli was the first mathematician after Leonardo Fibonacci to have thrown light on the science of algebra. . . . Pacioli’s treatise on bookkeeping, ‘De computis et scripturis,’ contained in the Summa, was the first printed work setting out the ‘method of Venice,’ that is, double-entry bookkeeping. [Richard] Brown has said [in his History of Accounting and Accountants, 1905] that ‘The history of bookkeeping during the next century consists of little else than registering the progress of the De computis through the various countries of Europe” (Dictionary of Scientific Biography).

ISTC no. il00315000.

Printer Johannes Herwagen (Hervagius) of Basel publishes Eukleidou Stoicheion biblon . . . , the first printed edition of the Greek text of Euclid's Elements.

Herwagen's edition was an international project. The Greek text was edited by the German theologian and philologist Simon Grynaeus (Grynäus), using the first Latin translation made directly from the Greek by Bartolomeo Zamberti published in print in 1505, and two Greek manuscripts supplied by Lazarus Bayfius and Joannes Ruellius  (Jean Ruel). To this volume Grynaeus appended the first publication of the four books of Proclus's Commentary on the first book of Euclid's Elements, taken from a manuscript provided by John Claymond, the first President of Corpus Christi College, Oxford. In a long introduction Grynaeus dedicated his translation to Cuthbert Tunstall, Bishop of Durham, England, and author of the first arithmetic book printed in English (London, 1522).

In the history of the very numerous editions of Euclid, the most widely-used of all textbooks for 500 years, Herwagen's edition stands out as the first edition to print the geometrical diagrams within the text.

The commentary on Euclid's first book of the Elements by the fifth century Greek neoplatonist philosopher Proclus is one of the most valuable sources for the history of Greek mathematics, and is considered the earliest contribution to the philosophy of mathematics.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 730.

Just before his death Nicolaus Copernicus publishes De revolutionibus orbium coelestium in Nuremberg.

De revolutionibus set out Copernicus's revolutionary theory of the heliocentric universe—that the earth and other planets revolve around the sun. The Copernican Revolution, however, was not completed until about one hundred years after the publication of De revolutionibus.

Because of the long delay between the publication of the Copernican theory and its acceptance by the scientific community, historians long believed that the book was not widely read at the time of its first publication. However, "Owen Gingerich, a widely recognized authority on both Nicolaus Copernicus and Johannes Kepler, disproved that belief after a 35-year project to examine every surviving copy of the first two editions. Gingerich showed that nearly all the leading mathematicians and astronomers of the time owned and read De revolutionibus; however, his analysis of the marginalia shows that they almost all ignored the cosmology at the beginning of the book and were only interested in Copernicus' new equant-free models of planetary motion in the later chapters" (Wikipedia article on De revolutionibus accessed 11-20-2008).

Up until the second decade of the seventeenth century the Church ignored the revolutionary implications of Copernicus's heliocentric theory of the solar system, partly because his system was useful for calendrical purposes, partly because of Andreas Osiander's anonymous and unauthorized preface "Ad lectorem" (long thought to be by Copernicus himself) presenting the heliocentric system as no more than a convenient calculating device, and partly because Copernicus himself "was annoyingly vague concerning whether or not he believed in the reality of his system" (Gingerich, p. 49).  However, Kepler's insistence in his Astronomia nova (1609) on the possible physical reality of Copernicus's system and his revelation of Osiander as the true author of "Ad lectorem," coupled with Galileo's public support of Copernicanism and his attacks on the Aristotelian-Catholic view of the heavens (beginning with his Letter on sunspots [1613]), alerted the ecclesiastical establishment to the dangers to its own authority inherent in the new system.  In 1616 the Church placed De revolutionibus on the Index librorum prohibitorum "until suitably corrected," and, for the only time in its history, spelled out the expected alterations to be made in the text.  This belated attempt at censorship was a failure, however: the census of copies published by Owen Gingerich shows that only one copy in twelve contains the prescribed changes, and that copies in France, Spain and Protestant Europe largely escaped correction.

Gingerich, "The Censorship of Copernicus's De revolutionibus," Annali dell'Istituto e Museo di Storia della Scienza di Firenze, Fasicolo2 (1981). Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 516.

Brother Juan Diez, a companion of Hernando Cortès (Hernán) in the conquest of New Spain, publishes the Sumario Compendioso in Mexico City at the press of Juan Pablos.

The Sumario Compendioso was the earliest treatise on mathematics published in the western hemisphere, and also the first textbook on any subject besides religious instruction to be printed outside of Europe.

In his introduction to The Sumario Compendioso of Brother Juan Diez, the Earliest Mathematical Work of the New World (1921), a facsimile and translation, David Eugene Smith writes of the existence of possibly four copies including one (incomplete) in the Biblioteca Nacional at Madrid, which he used for his edition, and a copy in the British Library.

"Not again in the sixteenth century did the Mexican printers publish any work on mathematics, except for a brief Instrucción Nautica which appeared in 1587. The press was generally true to its early purpose to issue only books relating to the conversion of the native inhabitants to the way of the cross" (Smith, introduction cited above, 6).

French painter, sculptor, etcher, engraver, and geometrician, Jean Cousin the Elder, publishes Livre de perspective in Paris at the press of Jean Le Royer. The folio volume includes a woodcut title device, a frontispiece of platonic solids and 58 geometrical diagrams (16 full-page, 5 double-page) by Jean Le Royer and Aubin Olivier. The frontispiece of the platonic solids is one of the finest examples of mannerist book illustration.

“According to the printer’s introduction, leaf A3v, Le Royer received from Cousin the text and ‘les figures pour l’intelligence d’iceluy necessaries, portraittes de sa main sus planches de bois,’ and he himself cut most of Cousin’s blocks and completed others which his brother-in-law, Aubin Olivier, had started. Several of the diagrams are extended into landscapes with figures. . . . Le Royer held the title of king’s printer for mathematics. Cousin is known to have been a successful painter and designer of stained glass windows. . . . His considerable reputation as a designer of woodcuts for the Paris printers has been developed chiefly by comparison of details from this volume” (Mortimer, Harvard College Library Department of Printing and Graphic Arts, Catalogue of Books and Manuscripts Part I. French Sixteenth Century Books (1964) no. 157, quote from pp. 195-97). 

English merchant, and later Lord Mayor of London Henry Billingsley publishes The Elements of Geometrie of the Most Ancient Philosopher Euclide of Megara.

Billingsley's work was the first English translation of Euclid. The title confused Euclid of Alexandria with the Greek Socratic philosopher, Euclid of Megara. The two were frequently confused during the Renaissance. Billingsley's translation included a lengthy preface by the mathematician, astronomer, astrologer, occultist, navigator, imperialist, consultant to Queen Elizabeth I. John Dee, which surveyed all the branches of pure and applied mathematics of the time. Dee also provided copious notes and other supplementary material.

Billingsley's translation, renowned for its clarity and accuracy, was made from the Greek rather than the well-known Latin translation by Adelard of Bath and Campanus of Novara.  Victorian mathematician, bibliographer and historian of mathematics Augustus De Morgan suggested that the translation was solely the work of Dee, but in his correspondence Dee stated specifically that only the introduction and the supplementary material were his. Proof that Billingsley made the translation himself is available in Billingsley's copy of the 1533 Greek editio princeps of Euclid bound with the 1558 Basel edition printed by Hervagius which reprints the Adelard-Companus Latin translation from the Arabic first printed in 1482 and the Zamberti Latin translation from the Greek first printed in 1505. Billingsley's copy is preserved in Princeton University Library.

"On the title-page is the autograph signature 'Henricus Billingsley,' in a most beautiful antique hand. Throughout the volume are very numerous corrections, additions and marginal notes, all in Billingsley's peculiar and beautiful writing. I dare hazard that no Lord Mayor, since his time, has ever written so charming a hand. By reading what he has done, it immediately appears that though he had the Adelard-Campanus Latin before him, yet he gave his special work to a careful comparison of Zamberti's Translation with the original Greek, and the corrections he has actually made sufficiently prove his scholarship and render entirely unnecessary De Morgan's suppositious aid from Dr. Dee, while, on the other hand, they establish the conclusion about the translation to which De Morgan's sagacity had led him, that 'It was certainly made from the Greek, and not from any of the Arabico-Latin versions' (Halsted, "Note on the First English Euclid," American Journal of Mathematics II [1879] 46-48).

♦ A special feature of Billingsley's edition are pasted flaps of paper that can be folded up to produce three dimensional models of the propositions in Book XI, making it one of the oldest "pop-up" books.

French lawyer, Conseil du Roi (privy councillor), and mathematician François Viète (Franciscus Vieta) publishes Canon mathematicus seu ad triangula. Cum adpendicibus.

Viète's numerous mathematical works were written during two brief periods of leisure from his career as a lawyer to the French courts of Henry III and Henry IV. His Canon mathematicus, the earliest of his published mathematical works, was the first of his studies on trigonometry.

"Here he gathered together the formulas for the solution of right and oblique plane triangles, including his own contribution, the law of tangents. . . . For spherical right triangles he gave the complete set of formulas needed to calculate any one part in terms of two other known parts, and the rule for remembering this collections of formulas, which we now call Napier's rule. He also contributed the law of cosines involving the angles of an oblique spherical triangle" (Kline, Mathematical Thought from Ancient to Modern Times [1972] 239-240).

In addition, Viète called for a reform in the expression of fractions, in which decimal fractions would replace the sexagesimal fractions then used in astronomy, physics and mathematics.

Viète's work consists of two parts: "Canon mathematicus," containing a table of trigonometric lines with some additional tables; and "Universalium inspectionum ad canonem mathematicum" (with separate title), giving the computational methods used in the construction of the canon and explaining the computation of plane and spherical triangles. Viète had originally planned to include two more parts devoted to astronomy, but these were never published.

Canon mathematicus was remarkably advanced typographically for its time. It is also very rare: privately printed in a small edition, its scarcity was compounded by Viète's displeasure over its many misprints, which caused him to withdraw from circulation all the copies he could recover.

Dibner, Heralds of Science, no. 105.  Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 2151.

Pope Gregory XIII issues a papal bull, Inter gravissimas, the founding document of the Gregorian calendar. It will be printed on March 1.

In 1599 Galileo Galilei developed his geometric and military compass into a general-purpose mechanical analog calculator, later known in English as the sector. 

As an instruction manual for purchasers of the compass, and to establish his priority for the invention, in 1606 Galileo published Le Operazioni del Compasso Geometrico et Militare in an edition of only sixty copies. To avoid having the compass pirated, Galileo had no illustrations of the device included in the pamphlet, which may be considered the first "computer manual."

During the seventeenth century the sector became one of the most widely used mechanical calculators for scientific purposes.

You may view a digital copy of Galileo's Compasso at this link.

John Napier of Scotland publishes his Mirifici logarithmorum canonis descriptio, announcing his invention of logarithms, with the goal of increasing calculating speed and reducing drudgery.

Scotish mathematician John Napier publishes Rabdologiae describing two calculating devices: “Napier’s bones,” and the Multiplicationis promptuarium, or the lightning calculator.

Johannes Kepler publishes Chilias Logarithmorum (1624) and Supplementum (1625), creating his logarithmic tables by a new geometrical procedure, the form thus differing from both Napier and Briggs.

Adriaan Vlacq, a bookseller, publisher, and human computer, computes and issues the first complete set of modern logarithms.

In a letter to theologian, philosopher, and mathematician Marin Mersenne, philosopher, mathematician and physicist René Descartes proposes an artificial universal language, with equivalent ideas in different tongues sharing one symbol:

"Et si quelqu’un avait bien expliqué quelles sont les idées simples qui sont en l’imagination des hommes, desquelles se compose tout ce qu’ils pensent, et que cela fût reçu par tout le monde, j’oserais espérer ensuite une langue universelle, fort aisée à apprendre, à prononcer et à écrire."

"The notion of a universal language was based upon the idea of precisely cataloging the elements of the human imagination. The great advantage of such a language would be that it would represent everything 'distinctement.' Yet, the great problem faced by someone who wanted to create such a language was the nature of the human imagination itself. Although separate from the mind and reason, which were the foundations of Cartesian thought, the imagination nevertheless played an important role for Descartes. As he wrote elsewhere in the Meditations, the imagination not only conceptualized external things but also considers them, 'as being present by the power and internal application of my mind.' Imagination, in other words, produced the illusion of presence, figures appearing so that can the person can 'look upon them as present with the eyes of my mind.' As a result, Descartes remains highly suspicious of the imagination because it can produce appearances that have no corresponding reality. Descartes concluded his letter to Mersenne by dismissing hopes for a universal language or a real character as only being possible in a 'terrestrial paradise' or 'fairyland' because of the confused nature of signification and the variation of human understanding.

"Mais n’espérez pas de la voir jamais en usage; cela présuppose de grands changements en l’ordre des choses, et il faudrait que tout le Monde ne fût qu’un paradis terrestre, ce qui n’est bon à proposer que dans le pays des romans.

 "A universal language that would work at the level of the imagination, describing the actual 'things' of the external world, could only produce uniform results in the perfection of Eden or the ideal of fiction. One should, instead, stick with the institution of geometry as a method of rationalizing nature, a divine language grounded upon the cogito’s transmission of being. Descartes ultimately remains skeptical about any possibility of using alternative language games aside from mathematics in the project of rationalizing the world" (Batchelor, The Republic of Codes: Cryptographic Theory and Scientific Networks in the Seventeenth Century [1999] http://www.stanford.edu/dept/HPS/writingscience/Cryptography.html, accessed 01-22-2010).

William Oughtred invents the circular form of slide rule. He publishes Circles of Proportion and the Horizontal Instrument in 1632 describing slide rules and sundials.

French philosopher, mathematician, and scientist René Descartes issues his Discours de la méthode pour bien conduire sa raison, & chercher la verité‚ dans les sciences. As Descartes spent much of his life in the Dutch Republic, he had the work published in Leiden.

Descartes's Discours presented an outline of Cartesian scientific method, summed up in the famous Four Rules presented in Book 2, together with scientific treatises intended to illustrate the method's range. The four rules may be stated as :

 1. "The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.

2. "The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.

3. "The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.

4.  "And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

"The enumerations have in time developed into many forms. He suggested drawing boxes on a paper, and connecting them. This idea has led to a multitude of graphic thinking aids that we use today" (Wikipedia article on Discourse on the Method, accessed 03-03-2009).

The work includes three scientific treatises: Dioptrique, containing Descartes's derivation of the law of refraction; Météores; and Géométrie. The work included his invention of the Cartesian coordinate system and the foundation of analytic geometry, the bridge between algebra and geometry, crucial to the invention of calculus and analysis. Though Descartes' most  famous statement is best known by its Latin translation, it was first published in the Discours as "Je pense, donc je suis," and later translated into Latin in his Principia philosophiae as "Cogito, ergo sum."

Carter & Muir, Printing and the Mind of Man (1967) no. 129. Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 621.

Mathematician and philosopher Blaise Pascal invents an adding machine, the Pascaline.

In 1645 Pascal published an eighteen-page pamphlet describing his calculating machine. It was called Lettre dédicatoire à Monseigneur le Chancelier sur le sujet de la machine nouvellement inventée par le Sieur B. P. pour faire toutes sortes d’opérations d’arithmétique, par un mouvement reglé, sans plume ny jettons avec un advis necessaire à ceux qui auront curiosité de voir ladite machine. . . . The pamphlet does not identify a place of printing or a printer’s name, so we may assume that Pascal paid for its printing. When we published Origins of Cyberspace OCLC cited only two copies of this pamphlet in one French library and no copies in North America.

Pascal's pamphlet was reprinted along with additional material related to the Pascaline in his Oeuvres (1779), vol. 4, 7-30. The additional material consisted of Pascal's 1650 letter describing the machine that he presented to Queen Christina of Sweden; the privilege for its construction and sale issued in 1649, and Denis Diderot's description of the machine published in the Encyclopédie.

Hook & Norman, Origins of Cyberspace (2002) no. 13.

The sliding-stick form of the slide rule is developed.

Gaspard Schott's posthumous Organum Mathematicum is published, in which he describes his “mathematical organ,” and his calculating machine based on Napier’s rods.

Samuel Morland publishes The Description and Use of Two Arithmetic Instruments, the first monograph on a calculating machine published in English. The book describes modifications to the Pascaline.

Gottfried Wilhelm Leibniz makes a drawing of his calculating machine mechanism.

Using a stepped drum, the Leibniz Stepped Reckoner, or step reckoner, mechanized multiplication as well as addition by performing repetitive additions. Leibniz had only a wooden model and two brass examples of the machine constructed. These would have been seen by relatively few people. However, because of descriptions published from 1710 onward, the machine was well-enough known to have great influence. The stepped-drum gear was the only workable solution to certain calculating machine problems until about 1875.

Leibniz first published a brief illustrated description of his machine in "Brevis descriptio machinae arithmeticae, cum figura. . . ," Miscellanea Berolensia ad incrementum scientiarum (1710) 317-19, figure 73. The lower portion of the frontispiece of the journal volume also shows a a tiny model of Leibniz's calculator.

"Leibniz got the idea for a calculating machine in 1672 in Paris, from a pedometer. Later he learned about Pascal's machine when he read Pascal's Pensées. He concentrated on expanding Pascal's mechanism so it could multiply and divide. He presented a wooden model to the Royal Society of London on February 1, 1673, and received much encouragement. In a letter of March 26, 1673 to Johann Friedrich, where he mentioned the presentation in London, Leibniz described the purpose of the "arithmetic machine" as making calculations "leicht, geschwind, gewiß" [sic], i.e. easy, fast, and reliable. Leibniz also added that theoretically the numbers calculated might be as large as desired, if the size of the machine was adjusted; quote: "eine zahl von einer ganzen Reihe Ziphern, sie sey so lang sie wolle (nach proportion der größe der Machine)" [sic]. In English: "a number consisting of a series of figures, as long as it may be (in proportion to the size of the machine)". His first preliminary brass machine was built 1674 - 1685. His so-called 'older machine' was built 1686 - 1694. The 'younger machine', the surviving machine, was built from 1690 to 1720.

"In 1775 the 'younger machine' was sent to Göttingen University for repair, and was forgotten. In 1876 a crew of workmen found it in an attic room of a Göttingen University building. It was returned to Hannover in 1880. In 1894-1896 Artur Burkhardt, founder of a major German calculator company restored it, and it has been kept in the Niedersaächsischen Landesbibliothek ever since" (Wikipedia article on Stepped Reckoner, accessed 05-25-2009).

Tomash & Williams, The Erwin Tomash Library on the History of Computing (2009) L69 (p. 772-73).

Dutch mathematician, astronomer, physicist and horologist Christiaan Huygens publishes Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstationes geometricae in Paris.

Depite the reference to time-measurement in its title, this work is a general treatise on dynamics of bodies in motion, with an emphasis on the motion of the pendulum. It contains the first mathematical analysis of pendulum motion, including the formula for the relation between the period and the time of free fall from rest, the rule for deriving the center of oscillation for both simple and compound pendulums, and proof of the tautochronism of the cycloid (the arc traced by a point on a circle when the circle is rolled along a flat plane), which made possible Huygens's invention of the first reliable pendulum clock in 1656. Also included are Huygens's theories of the evolutes of curves, descriptions of his marine clocks and their trials, the first value for the force of gravity (which he derived using a simple pendulum), and the most important of his studies of centrifugal force; these last were used by Newton in his determination of universal gravitation.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1137.

A dated manuscript by Gottfried Wilhelm Leibniz, preserved in the Niedersachsische Landesbibliothek, Hannover, “includes a brief discussion of the possibility of designing a mechanical binary calculator which would use moving balls to represent binary digits.”

Though Leibniz thought of the application of binary arithmetic to computing in 1679, the machine he outlined was never built, and he published nothing on the subject until his Explication de l'arithmétique binaire, qui se sert des seuls caracteres 0 & 1; avec des remarques sur son utilité, & sur ce qu'elle donne le sens des anciens figues Chinoises de Fohy' published in Histoire de l'Académie Royale des Sciences année MDCCIII. Avec les mémoires de mathématiques which appeared in print in 1705.

"The publication of the Explication was prompted by Leibniz's correspondence with Joachim Bouvet, a member of the Jesuit Mission in China. Leibniz had developed an interest in China, and in April 1697 he edited a collection of letters and essays by members of the Mission, entitled Novissima Sinica. A copy of this came into the hands of Bouvet, who wrote to Leibniz on 18 October 1697 expressing his commendation of the work. Thus began an extended correspondence between the two men which proved to be very important for the dissemination of Leibniz's ideas about binary arithmetic. The crucial exchange began on 15 February 1701, when Leibniz wrote to Bouvet describing for his correspondent the principles of his binary arithmetic, including the analogy of the formation of all the numbers from 0 and 1 with the creation of the world by God out of nothing. Bouvet immediately recognised the relationship between the hexagrams of the I ching and the binary numbers and he communicated his discovery in a letter written in Peking on 4 November 1701. This reached Leibniz, after a detour through England, on 1 April 1703. With this letter, Bouvet enclosed a woodcut of the arrangement of the hexagrams attributed to Fu-Hsi, the mythical founder of Chinese culture, which holds the key to the identification. Within a week of receiving Bouvet's letter, Leibniz had sent to Abbé Bignon for publication in the Mémoires of the Paris Academy his Explication de l'Arithmétique binaire,... & sue ce qu'elle donne le sens des anciens figures Chinoises de Fohy. Ten days later he sent a brief account to Hans Sloane, the Secretary of the Royal Society. Leibniz viewed binary arithmetic less as a computational tool than as a means of discovering mathematical, philosophical and even theological truths. He remarked to Tschirnhaus in 1682 that he anticipated from the use of binary numbers discoveries in number theory that other progressions could not reveal. It was at the same time a candidate for the characteristica generalis, his long sought-for alphabet of human thought. With base 2 numeration Leibniz witnessed a confluence of several intellectual strands in his world view, including theological and mystical ideas of order, harmony and creation. Fontanelle, secretary of the Paris Academy, wrote the unsigned review of Liebniz's paper for the Mémoires section of the volume. He noted that arithmetic could have different bases besides ten; bases such as 12, and two as in the case of Leibniz's binary system. He also noted that although the binary system was not practical for common use Leibniz thought that it would be of advantage in advanced mathematics" (W.P. Watson, antiquarian book description, http://www.ilabdatabase.com/db/detail.php?booknr=360538539, accessed 01-21-2010).

This manuscript was first published, along with as well as facsimiles of Leibniz's "Explication de l'arithmétique binaire" (1705) and his two letters to Johann Christian Schulenberg on binary arithmetic (March 29 and May 17, 1698), published in the Opera Omnia of 1768, with historical articles and translations in German, to commemorate the 250th anniversary of Leibniz's death as Herrn von Leibniz' Rechnung mit Null und Eins (1966).

Gottfried Wilhelm Leibniz publishes "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, & singulare pro illi calculi genus" in the periodical, Acta eruditorum. This was his first paper on the differential calculus, published nine years after he had independently discovered it.  Although Newton had probably discovered the calculus earlier than Leibniz, Leibniz was the first to publish his method, which employed a notation superior to that used by Newton.  The priority dispute between Newton and Leibniz over the calculus is one of the most famous controversies in the history of science; it led to a breach between English and Continental mathematics that was not healed until the early nineteenth century.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1326. Carter & Muir, Printing and the Mind of Man (1967) no. 160.

Isaac Newton publishes Philosophia naturalis principia mathematica.

We probably know as much about the printing history of Newton's Principia mathematica as of any book of the seventeenth century.  The definitive scholarship on the writing and printing of the Principia appears in I. B. Cohen's Introduction to Newton's "Principia" (1971), and in Koyré‚ and Cohen's variorum edition of the Principia (1972), which also contains William Todd's definitive bibliography of the first three editions.  Other useful research on this work was conducted by A. N. L. Munby nearly forty years ago.  Munby's and Todd's observations may be summarized here. The original printer's manuscript in the hand of Newton's amanuensis, Humphrey Newton, still exists, as do various copies of the first edition with Isaac Newton's autograph corrections.  The expenses of publication of the first edition were borne by Edmond Halley, as neither Newton nor the Royal Society had sufficient funds, and booksellers, who in those days often acted as publishers, typically refused to risk their own money on esoteric scientific books.  Halley also edited the work and saw it through the press, reporting his progress to Newton in a series of letters which are preserved at Cambridge. 

Having paid for the edition himself, Halley sent out presentation copies at Newton's direction and also sent Newton twenty copies for his personal use.  Halley decided to market the book by placing copies on consignment with various booksellers, and he sent Newton forty copies, some bound, some in sheets, which he asked Newton to "place in the hands of one or more of your ablest booksellers to dispose of them." Munby observed that many of the bindings of the two-line imprint issue were similar, suggesting that Halley may have had many of the copies bound at one shop.

Munby researched the significance of the two states of the title page of the Principia, concluding that the more commonly found state, with the title page uncancelled and the so-called two-line imprint, reflects Halley's initial sales strategy of placing the work on consignment with many booksellers ("apud plures Bibliopolas").  The state with the three-line imprint, including the name of the bookseller, Samuel Smith, reflects Halley's decision to turn over a significant portion of the edition to Smith, probably for foreign distribution.  The bookseller Heinrich Zeitlinger, of Henry Sotheran Ltd., first made the useful observation that many of the copies with the three-line "Smith" imprint were exported to the Continent.  Smith was known to be very active in the import and export of books, and Munby stated that he knew of only two "Smith" copies in contemporary English bindings.  The contemporary binding on the Norman copy is clearly French.

From his bibliographical analysis of the first edition Todd concluded that the edition was divided between two compositors, one setting the first two books, the other setting the third.  "The first compositor, however, was allowed too few sheets and too many foliations, a circumstance which necessitated his signing a supplementary gathering *** and paging it 377-383, 400."  Todd identified typographical variants which seem to be randomly distributed throughout the edition and are thus not indicative of any priority.

Todd also described the distribution of watermarks in the Principia: "The text paper exhibits a water-mark of a fleur-de-lis within a coat of arms (Heawood 626) only in preliminaries and certain sections in the earlier portion of the books, indicating perhaps that the signatures so distinguished are of later, revised settings printed off at the same time.  All copies have this water-mark in P-2K; some have it also in A, F-G, M-O, 2M-2N."  The distribution of watermarks appears to have nothing to do with the distribution of the variants listed above.

In estimating the size of the first edition Munby acknowledged that the work went out of print quickly and was already difficult to obtain in December 1691, when Nicholas Fatio de Duillier discussed a new edition in a letter to Christiaan Huygens.  Extrapolating from the partial census figures available in 1952, Munby conjectured that at least 150 copies of the work were then extant, concluding from this and from the book's relatively common appearances in the sale rooms that "the whole edition cannot have comprised less than three hundred copies, and the figure may well have been a hundred more than this."  The plentiful sales records in the forty years since Munby's account would certainly corroborate the higher estimate. Copies with the three-line imprint are much rarer than those with the two-line, suggesting that the so-called "Smith" copies may only have comprised  between seventeen and thirty-three percent of the edition. 

Newton's personal copy of the first edition of the Principia, with Newton's autograph corrections for the second edition, is preserved at the Wrenn Library, Trinity College, Cambridge.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1586. Cohen, Introduction to Newton's Principia, ch. IV.  Munby, "The two titlepages of the distribution of the first edition of Newton's Principia," Notes and Records of the Royal Society of London 10 (October 1952).  Todd, "A bibliography of the Principia.  Part I: The three substantive editions," in Koyré‚ & Cohen, Isaac Newton's Philosophiae naturalis principia mathematica II,  851-853.

Cornelis a Beughem issues Bibliotheca mathematica et artificosa novissima. . . conspectus primus. This was the first independently published and comprehensive bibliography of mathematics, limited to books published from 1551 onward. Pages 465-526 contain a bibliography of atlases.

Isaac Newton publishes Opticks: Or a Treatise of the Reflexions, Refractions, Inflexions and Colours of light.  Also Two Treatises of the Species and Magnitude of Curvilinear Figures.

Unlike most of Newton's works, Opticks was originally published in English, with the Latin version following in 1706.  The work summarized Newton's discoveries and theories concerning light and color: the spectrum of the sunlight, the degrees of refraction associated with different colors, the color circle (the first in the history of color theory), the invention of the reflecting telescope; the first workable theory of the rainbow, and experiments on what would later be called "interference effects" in conjunction with Newton's rings.  His discovery of periodicity in Newton's rings, which would later prove to be so useful to Thomas Young, led Newton to postulate that periodicity was a fundamental property either of light waves or of waves associated with light.  Nevertheless, Newton preferred the corpuscular theory of light, with which he is usually associated, because of its explanatory value for certain optical phenomena and because it a llowed him to link the action of gross bodies with the action of light. The first edition of the Opticks ends with two mathematical treatises in Latin, written to establish his priority over Gottfried Wilhelm Leibniz in the invention of the calculus.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1588. Carter & Muir, Printing and the Mind of Man (1967) no. 172.

Isaac Newton publishes Analysis per quantitatum series, fluxiones, ac differentias cum enumeratione linearum tertii ordinis, edited by William Jones.

This was the first printing of Newton's tracts De analysi per aequationes numero terminorum infinitas" and Methodus differentialis, together with reprints of the tracts on quadratures and cubics first published in Opticks (1704).  De analysi, Newton's first independent treatise on higher mathematics, was written in 1669 to protect his priority in the invention of the calculus. It contains the earliest printed account of Newton's generalized binomial theorem.  In 1711, Newton permitted mathematician William Jones (one of the few allowed access to Newton's manuscripts) to publish these four tracts. Aside from his association with Newton, Jones is chiefly remembered for having introduced the symbol  Π into mathematical notation.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1590.

In response to Leibniz’s appeal to the Royal Society for a fair hearing concerning the dispute over the invention of the differential calculus between Newton and himself, the Royal Society issues Commercium epistolicum D. Johannis Collins, et aliorum de analysi promota: Jussu Societatis Regiae in lucem editum.

The report was hardly impartial, however, because Newton, as the president of the Royal Society, hand-picked a committee of supporters to review the case and composed its favorable findings himself.  The John Collins mentioned in the title was a bookseller, amateur mathematician and member of the Royal Society. In 1669, Collins was sent a copy of Newton's manuscript on the calculus, De analysi, portions of which Leibniz transcribed in 1676.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1591.

In Gulliver's Travels Jonathan Swift describes a fictional device called The Engine, which generates permutations of word sets. It is possibly the earliest reference to a device resembling aspects of a modern computer.  Though Swift does not reference the medieval Ars combinatoria of the Spanish philosopher Ramon Llull (Lull),  the passage is considered a parody of his method.

Swift wrote:

“... Every one knew how laborious the usual method is of attaining to arts and sciences; whereas, by his contrivance, the most ignorant person, at a reasonable charge, and with a little bodily labour, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study.” He then led me to the frame, about the sides, whereof all his pupils stood in ranks. It was twenty feet square, placed in the middle of the room. The superfices was composed of several bits of wood, about the bigness of a die, but some larger than others. They were all linked together by slender wires. These bits of wood were covered, on every square, with paper pasted on them; and on these papers were written all the words of their language, in their several moods, tenses, and declensions; but without any order. The professor then desired me “to observe; for he was going to set his engine at work.” The pupils, at his command, took each of them hold of an iron handle, whereof there were forty fixed round the edges of the frame; and giving them a sudden turn, the whole disposition of the words was entirely changed. He then commanded six-and-thirty of the lads, to read the several lines softly, as they appeared upon the frame; and where they found three or four words together that might make part of a sentence, they dictated to the four remaining boys, who were scribes. This work was repeated three or four times, and at every turn, the engine was so contrived, that the words shifted into new places, as the square bits of wood moved upside down."

French mathematician and statistician Antoine Deparcieux publishes Essai sur les probabilités de la durée de la vie humaine.  He published a supplement to this work entitled Addition à l'Essai sur les probabilités de la durée de la vie humaine in 1760.

These works on annuities and mortality were the first correct "life tables."

English mathematician Thomas Simpson publishes "On the Advantage of Taking the Mean of a Number of Observations, in Practical Astronomy" in the Philosophical Transactions of the Royal Society 49, part 1, 82-93.

This paper is "a milestone in statistical inference, as well as the earliest formal treatment of any data-processing practice" (Hook & Norman, Origins of Cyberspace [2002] no. 16).

Two years after his death English clergyman and mathematician Thomas Bayes's "An Essay Towards Solving a Problem in the Doctrine of Chances" is published in the Philosophical Transactions of the Royal Society 53 (1763) 370-418.

Bayes's paper enunciated Bayes's Theorem for calculating "inverse probabilities”—the basis for methods of extracting patterns from data in decision analysis, data mining, statistical learning machines, Bayesian networks, Bayesian inference.

Hook & Norman, Origins of Cyberspace (2002) no. 1.

The British Government sanctions Nevil Maskelyne, the Astronomer Royal, to produce each year a set of navigational tables, to be called the Nautical Almanac.

This was the first permanent table-making project in the world.

Known as the "Seaman's Bible," the Nautical Almanacs greatly improved the accuracy of navigation. However, the accuracy of the tables in the Nautical Almanacs was dependent upon the accuracy of the human computers producing them—human computers who worked by hand, and were separated geographically. During the time of Charles Babbage these tables became notorious for their errors, providing Babbage the incentive to develop mechanical systems, which he called calculating engines, to improve their accuracy.

Though Adrien-Marie Legendre was the first to publish the method of least squares in 1805, Carl Friedrich Gauss is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen.

"An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres. On January 1, 1801, the Italian astronomer Giuseppe Piazzi discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, it was desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated Kepler's nonlinear equations of planetary motion. The only predictions that successfully allowed Hungarian astronomer Franz Xaver von Zach to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.

"Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium" (Wikipedia article on Least squares, accessed 08-24-2009).

"The long 's' is derived from the old Roman cursive medial s, which was very similar to an elongated check mark. When the distinction between upper case (capital) and lower case (small) letter-forms became established, towards the end of the eighth century, it developed a more vertical form. At this period it was occasionally used at the end of a word, a practice which quickly died out but was occasionally revived in Italian printing between about 1465 and 1480. The short 's' was also normally used in the combination 'sf', for example in 'ſatisfaction'. In German written in Blackletter, the rules are more complicated: short 's' also appears at the end each word within a compound word.

"The long 's' is subject to confusion with the lower case or minuscule 'f', sometimes even having an 'f'-like nub at its middle, but on the left side only, in various kinds of Roman typeface and in blackletter. There was no nub in its italic typeform, which gave the stroke a descender curling to the left—not possible with the other typeforms mentioned without kerning.

"The nub acquired its form in the blackletter style of writing. What looks like one stroke was actually a wedge pointing downward, whose widest part was at that height (x-height), and capped by a second stroke forming an ascender curling to the right. Those styles of writing and their derivatives in type design had a cross-bar at height of the nub for letters 'f' and 't', as well as 'k'. In Roman type, these disappeared except for the one on the medial 's'.

"The long 's' was used in ligatures in various languages. Three examples were for 'si', 'ss', and 'st', besides the German 'double s' 'ß'.

"Long 's' fell out of use in Roman and italic typography well before the middle of the 19th century; in French the change occurred from about 1780 onwards, in English in the decades before and after 1800, and in the United States around 1820. This may have been spurred by the fact that long 's' looks somewhat like 'f' (in both its Roman and italic forms), whereas short 's' did not have the disadvantage of looking like another letter, making it easier to read correctly, especially for people with vision problems.

"Long 's' survives in German blackletter typefaces. The present-day German 'double s' 'ß' (das Eszett "the ess-zed" or scharfes-ess, the sharp S) is an atrophied ligature form representing either 'ſz' or 'ſs' (see ß for more). Greek also features a normal sigma 'σ' and a special terminal form 'ς', which may have supported the idea of specialized 's' forms. In Renaissance Europe a significant fraction of the literate class was familiar with Greek.The long 's' survives in elongated form, and with an italic-style curled descender, as the integral symbol ∫ used in calculus; Gottfried Wilhelm von Leibniz based the character on the Latin word summa (sum), which he wrote ſumma. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June, 1686, but he had been using it in private manuscripts since at least 1675" (Wikipedia article on Long s, accessed 09-11-2009).

♦ According to R. B. McKerrow, An Introduction to Bibliography for Literary Students (1927), the effective introduction of the reform in England was credited to the printer and publisher John Bell who in his British Theatre of 1791 used s throughout.  "In London printing the reform was adopted very rapidly, and save in work of an intentionally antiquarian character, we do not find much use of [long] s in the better kind of printing after 1800" (McKerrow p. 309).  Though it would be amusing to do so, there seems to be no reason to accept the legend that  Bell initiated the change in his edition of Shakespeare because of his dismay at the appearance of the long s in Ariel's song in The Tempest: "Where the bee sucks, there suck I."

Gaspard Riche de Prony completes two manuscript sets of massive logarithmic and trigonometric tables calculated by employing systematic division of mental labor, including the use of mathematically untrained hairdressers unemployed after the French Revolution.

The method of production of the tables inspired Charles Babbage in the design of his Difference Engine No. 1 in 1822.

Portions of de Prony's tables were published for the first time in 1891.

At the age of 24 Carl Friedrich Gauss publishes Disquisitiones arithmeticae, revolutionizing number theory.

"In this book [Gauss] standardized the notation; he systematized the existing theory and extended it; and he classified the problems to be studied and the known methods of attack and introduced new methods. . . . [The Disquisitiones] not only began the modern theory of numbers but determined the directions of work in the subject up to the present time" (Kline, Mathematical Thought from Ancient to Modern Times [1972] 813).

The typesetters of this work had difficulty understanding Gauss's new and difficult mathematics, creating numerous elaborate mistakes which Gauss was unable to correct in proof. After the book was printed Gauss insisted that, in addition to an unusually lengthy four-page errata, the worst mistakes be corrected by cancel leaves to be inserted in the copies before sale. Copies vary in the number of cancel leaves—a topic about which I have never seen a comprehensive bibliographical analysis.

The difficulty of understanding Gauss's highly technical work was hardly alleviated by the sloppy typesetting.  The few mathematicians who were able to read the Disquisitiones immediately hailed Gauss as their prince, but the full understanding required for further development did not occur until publication in 1863 of Johan Peter Gustav Lejeune Dirichlet's less austere exposition in his Vorlesungen über Zahlentheorie.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 878. Carter & Muir, Printing and the Mind of Man (1967) no. 257.

Adrien-Marie Legendre publishes Nouvelles méthodes pour la détermination des orbites des comètes. His appendix to this work, “Sur la Méthode des moindres quarrés,” represents the first publication of the method of least squares, the earliest form of regression analysis.

Charles Babbage starts on a model of the Difference Engine, a special-purpose machine that links adding and subtracting mechanisms to one another to calculate the values of more complex mathematical functions.

Babbage's goal was to produce more accurate mathematical tables, the most widely-used calculating aids in his day. Babbage announced his plan to build the Difference Engine No. 1 in an open letter to Sir Humphry Davy, president of the Royal Society, and received government funding. (See Reading 4.1)

French mathematician and physicist Jean Baptiste Joseph Fourier publishes Théorie analytique de la chaleur.

Fourier’s application of new methods of mathematical analysis to the study of heat extended rational mechanics to fields outside of those defined in Newton’s Principia, enabling the systematization of a wide range of phenomena. To further his study of heat, Fourier introduced the Fourier series and Fourier integrals.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 824.

Nicolai Ivanovitch Lobachevskii (Lobachevsky) publishes "O nachalakh geometrii"  in Kazanskii vestnik, izdavaemyi pri Imperatorskom Kazamskom Universitete nos. 25, parts 1-2, 27, and 28, parts 1-2 (1829-1830), pp. 178-224, 228-241, 227-243, 251-283, and 571-636. This was the first published work on non-Euclidean geometry. It appeared in the Messenger of the University of Kazan  as a series of five papers beginning three years after Lobachevskii read the text of the first and fundamental paper to his colleagues at the University.

Lobachevskii's geometry represented the culmination of two thousand years of criticism of Euclid's Elements, most particularly Euclid's fifth, or parallel, postulate, which stated that given a line and a point not on the line, there can be drawn through the point one and only one coplanar line not intersecting the given line. As this postulate had stubbornly resisted all attempts (including Lobachevskii's) to prove it as a theorem, Lobachevskii came to the realization that it was possible to construct a logically consistent geometry in which the Euclidean postulate represented a special case of a more general system that allowed for the possibility of hyperbolically curved space. Lobachevskii's system refuted the unique applicability of Euclidean geometry to the real world, and pointed the way to the Einsteinian concept of variably curved space-- "the most consequential and revolutionary step in mathematics since Greek times" (Kline, Mathematical Thought from Ancient to Modern Times [1972] 879).

Lobachevskii was not alone in his efforts to develop a non-Euclidean geometry; indeed, its creation is an example of how the same idea can occur independently to different people at about the same time. Janos Bolyai, who published his own system a few years later has traditionally shared credit with Lobachevskii for the invention of the new geometry. However, the work of both men in this area was anticipated by that of Carl Friedrich Gauss, which, although unpublished, may possibly have been familiar to them.

Despite this confluence of mathematical thought, non-Euclidean geometry went largely ignored until the 1860s, when it was rediscovered and elaborated upon by a new generation of mathematicians including Jules Hoüel, Eugenio Beltrami and Bernhard Riemann.

The Extreme Rarity of this Publication

One reason that the writings of Lobachevskii and János Bolyai may have received little attention from the scientific community is that both works were published in very small and obscure editions. The periodical Kazanskii vestnik, in which Lobachevskii's work was originally published, seems to have had minimal circulation even within Russia. For the Grolier Club exhibition (1958) on which Horblit's One Hundred Books Famous in Science was based, it was necessary to borrow a set of the journal issues from a Soviet library (either the A.M. Gorki Library of Science or the Moscow University Library), while the Printing and the Mind of Man exhibition in London (1963) found the original edition "unprocurable" and displayed only the 1887 German translation. In 2010 no copies of the original printing were recorded in North American or European institutional libraries. Two copies were held in private collections in America (1 incomplete).

Carter & Muir, Printing and the Mind of Man (1967) No. 293a. Hook & Norman, The Haskell F. Norman Library of Science & Medicine (1991) no. 1379.

 János Bolyai publishes "Appendix scientiam spatii absolute veram exhibens: a veritate aut falsitate axiomatis xi Euclidei (a priori haud unquam decidenda) independentem. . . ." appended to a textbook by his father Farkas, entitled Tentamen juventutem studiosam in elementa matheseos purae I pp. [2] [1]-26 [2] pp. (second series). The two volumes appeared in Maros Vasarhelyini, Hungary, printed  by Joseph and Simon Kali, at the press of the Reform College.

Although the idea of a non-Euclidean geometry had occured independently to several nineteenth-century mathematicians, János Bolyai was one of the first to publish an organized, deductive and logically based system that was avowedly non-Euclidean. He was preceded only by Lobachevskii, whose "O nachalakh geometrii"  (On the Foundations of Geometry) had been published in the obscure periodical, Kazanskii vestnik, izdavaemyi pri Imperatorskom Kazamskom Universitete in Kazan in 1829-30, but Bolyai remained unaware of the Russian's work until 1848, when he came across the German translation Lobachevskii's Geometrische Untersuchungen (1840). Bolyai and Lobachevskii are generally given equal credit for the invention of non-Euclidean geometry.

János Bolyai began developing his new geometry in 1820, and completed it five years later. He undertook this task despite the warnings of his father, who discouraged his son in the strongest terms from trying to prove or refute Euclid's parallel axiom; in a letter written in 1820, Farkas told his son not to "tempt the parallels" and to "shy away from it as from lewd intercourse, it can deprive you of all your leisure, your health, your peace of mind and your entire happiness." The elder Bolyai found his son's new geometry of "absolute space" unacceptable, but finally, in the summer of 1831, decided to send János's manuscript to his old friend Carl Friedrich Gauss. Neither of the Bolyais knew that Gauss had been working for thirty years on developing his own non-Euclidean geometry, so János was dreadfully shocked to read in Gauss's reply that he [Gauss] could not praise János's system since to do so would be to praise himself! Despite this blow, János agreed to let his paper be published as an appendix to his father's obscure mathematics textbook printed in a small edition by an equally obscure Hungarian school publisher.

Unsurprisingly, Bolyai's paper failed to attract the attention of contemporary mathematicians, and his new geometry remained almost completely unknown until 1867, when Heinrich Richard Baltzer publicized the achievements of Bolyai and Lobachevskii in his Elemente der Mathematik.

Bibliographical Comments

The Tentamen was very crudely or printed at a school press; copies exhibit the earmarks of non-professional or inexperienced publishing, particularly in the clumsy typography and numerous errata and corrigenda leaves, which must have made the Tentamen extremely difficult to use. These leaves were printed on different paper stocks and were obviously added after the original printing. Copies seem to incorporate other bibliographical variations; however, a thorough analysis of the extant copies remains to be done. Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) No. 259 includes a collation and discussion of tentative issue points. The subscribers' lists in Vol. i (1r+v) and Vol. ii (266v) indicate that 156 copies were subscribed for, and the edition was probably not much larger than this. In 2010 less than 20 copies were recorded. 

Kline, Mathematical Thought from Ancient to Modern Times (1972) 873-880.

From a reproduction of just five pages of the Dresden Codex, a pre-Columbian Maya book of the eleventh or twelfth century of the Yucatecan Maya in Chichén Itzá, European-American autodidact polymath, mathematician, botanist, zoologist, and malachologist Constantine Samuel Rafinesque  deciphers the Maya's system of numerals.

"In 1832, Rafinesque declared in his newsletter, the Atlantic Journal and Friend of Knowledge, that the dots and bars seen in Maya glyphs represented simple numbers—a dot equaled one and a bar five. Later findings proved him right and also revealed that the Maya even had a symbol for zero, which appeared on Mesoamerican carvings as early as 36 B.C. (Zero didn't appear in Western Europe until the 12th century)"  (http://www.pbs.org/wgbh/nova/mayacode/time-flash.html, accessed 10-10-2009).

In Note sur un moyen de tracer des courbes données par des équations différentielles The french physicist Gaspard-Gustave Coriolis desribes a mechanical device to integrate differential equations of the first order.This is the beginning of researches on solution of differential equations using mechanical devices.

Mathematician Pierre François Verhulst publishes "Notice sur la loi que la population suit dans son accrossement" in Correspondance mathématique et physique X, 113–121.

In this paper Verhulst constructed the simplest mathematical model of a continuously growing population with an upper limit to its size. "The concept of r/K selection theory derives its name from the competing dynamics of exponential growth and environmental limitation introduced here" (Wikipedia article on Pierre François Verhulst, accessed 01-13-2009).

Mathematician Luigi Federico Menabrea publishes "Notions sur la machine analytique de M. Charles Babbage" in Bibliothèque universelle de Genève, nouvelle série 41 (1842): 352–76.

This was the first published account of Charles Babbage’s Analytical Engine and the first account of its logical design, including the first examples of computer programs ever published. As is well known, Babbage’s conception and design of his Analytical Engine—the first general purpose programmable digital computer—were so far ahead of the imagination of his mathematical and scientific colleagues that few expressed much curiosity regarding it. The only presentation that Babbage made concerning the design and operation of the Analytical Engine was to a group of Italian scientists.

In 1840 Babbage traveled to Torino to make a presentation on the Analytical Engine. Babbage’s talk, complete with charts, drawings, models, and mechanical notations, emphasized the Engine’s signal feature: its ability to guide its own operations—what we call conditional branching. In attendance at Babbage’s lecture was the young Italian mathematician Luigi Federico Menabrea (later prime minister of Italy), who prepared from his notes an account of the principles of the Analytical Engine. Reflecting a lack of urgency regarding radical innovation unimaginable to us today, Menabrea did not get around to publishing his paper until two years after Babbage made his presentation, and when he did so he published it in French in a Swiss journal. Shortly after Menabrea’s paper appeared Babbage was refused government funding for construction of the machine.

"In keeping with the more general nature and immaterial status of the Analytical Engine, Menabrea’s account dealt little with mechanical details. Instead he described the functional organization and mathematical operation of this more flexible and powerful invention. To illustrate its capabilities, he presented several charts or tables of the steps through which the machine would be directed to go in performing calculations and finding numerical solutions to algebraic equations. These steps were the instructions the engine’s operator would punch in coded form on cards to be fed into the machine; hence, the charts constituted the first computer programs [emphasis ours]. Menabrea’s charts were taken from those Babbage brought to Torino to illustrate his talks there"(Stein, Ada: A Life and Legacy, 92).

Menabrea’s 23-page paper was translated into English the following year by Lord Byron’s daughter, Augusta Ada, Countess of Lovelace, who, in collaboration with Babbage, added a series of lengthy notes enlarging on the intended design and operation of Babbage’s machine. Menabrea’s paper and Ada Lovelace’s translation represent the only detailed publications on the Analytical Engine before Babbage’s account in his autobiography (1864). Menabrea himself wrote only two other very brief articles about the Analytical Engine in 1855, primarily concerning his gratification that Countess Lovelace had translated his paper.

Hook & Norman, Origins of Cyberspace (2002) no. 60.

Augusta Ada King, Countess of Lovelace, daughter of Lord Byron, translates Menabrea’s paper, "Notions sur la machine analytique de M. Charles Babbage".

Ada expanded her translation with annotations and software examples that provided further insight into Babbage's proposed Analytical Engine: Sketch of the Analytical Engine Invented by Charles Babbage . . . with Notes by the Translator. (See Reading 6.1.)

George Boole publishes a pamphlet entitled The Mathematical Analysis of Logic -- a preliminary version of what eventually will be called Boolean algebra.

Years later, in 1938, Claude Shannon in his master’s thesis recognized that the true/false values in Boole’s two-valued logic are analogous to the open and closed states of electric circuits

Mathematician, logician and pioneer collector of the history of mathematics, Augustus de Morgan publishes Arithmetical Books from the Invention of Printing to the Present Time, being Brief Notices of a Large Number of Works Drawn up from Actual Inspection.

De Morgan's work was first separately published bibliography on the history of science. The bulk of the book consisted of an extensively annotated list of treatises on arithmetic from 1491 to 1846, arranged in chronological order; de Morgan claimed that he had personally examined every book. Most of the books described were from de Morgan’s own library, which he acquired at relatively low cost because of the obscurity of the subjects involved. A few of the books he described came from the libraries of collector friends, and a few from the library of the British Museum. There is an index of 1,580 entries.  In The History and Bibliography of Science in England (1968) A. N. L. Munby stated that “only in the physical descriptions of books cited is De Morgan’s great work disappointing.”

De Morgan was an eloquent exponent of the value of collecting the history of science. He wrote on p. ii his prefatory letter to Arithmetical Books:

“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 places of more important bodies.”

Even though de Morgan’s library was not kept together when it was transferred to the University of London, his books were separately identified in the printed catalogue of the University of London Library published in 1876. Thus it is still possible to study one of the pioneering collections of books formed in England not just on mathematics, but on a wide range of the physical sciences.

English mathematician and philosopher George Boole publishes An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities. This work contains the full expression of the first practical system of logic in algebraic form.

"He [Boole] did not regard logic as a branch of mathematics, as the title of his earlier pamphlet [The Mathematical Analysis of Logic (1847)] might be taken to imply, but he pointed out such a deep analogy between the symbols of algebra and those which can be made, in his opinion, to represent logical forms and syllogisms, that we can hardly help saying that (especially his) formal logic is mathematics restricted to the two quantities, 0 and 1. By unity Boole denoted the universe of thinkable objects; literal symbols, such as x, y, z, v, u, etc., were used with the elective meaning attaching to common adjectives and substantives. Thus, if x=horned and y=sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraic symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. Thus, (1 - x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 - x) (1 - y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.

"Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, formed a general symbolic method of logical inference. Given any propositions involving any number of terms, Boole showed how, by the purely symbolic treatment of the premises, to draw any conclusion logically contained in those premises. The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events" (Wikipedia article on George Boole, accessed 01-09-2008).

Though the audience for Boole's highly specialized work would have been judged to be small, and the edition size reduced accordingly, the existence of three issues of the first edition, all dated 1854, would suggest that the edition may have required several years to sell. The points of the issues are as follows:

1. Probable first issue: London: Walton and Maberly, Upper Gower-Street, and Ivy Lane, Paternoster-Row. Cambridge: Macmilan and Co., errata leaf bound in the back, and binding of black zigzag cloth with blindstamped border, panel, central lozenge and corner and side ornaments.

2. Probable second issue: London: Walton and Maberly as above, but with the errata after the last numbered leaf of preliminaries, an additional printed "Note" leaf following 2E4 concerning a more complex error, an eight-page Walton and Maberly catalogue of "Educational Works and Works in Science and General Literature" and a binding of black blind-panelled zigzag cloth without the central lozenge.

3. Third issue: London: Macmillan and Co. Errata on recto of last unsigned leaf, and bound in green cloth, gilt-lettered spine. This may be a later, or remainder binding

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 266.

George Parker Bidder, an engineer and one of the most remarkable human computers of all time, publishes his paper on Mental Calculation. (See Reading 3.1)

English mathematician, engineer and computer designer Charles Babbage publishes his autobiography, Passages from the Life of a Philosopher, in which he presents the most detailed descriptions of his Difference and Analytical Engines published during his lifetime, and writes about his struggles to have his highly futuristic inventions appreciated by society.

In the wording of his title Babbage used the word philosopher in its now obsolete sense of what we call a "scientist." The word scientist coined by William Whewell was not widely used until the end of the 19th or early 20th century. (See Reading 6.2.)

James Clerk Maxwell publishes "A Dynamical Theory of the Electro-Magnetic Field" in the Transactions of the Royal Society of London.

The paper provided a theoretical framework, based on experiment and a few general dynamical principles, for the propagation of electromagnetic waves through space.

William Stanley Jevons constructs his “logical piano,” the first logic machine to solve complicated problems with superhuman speed.

Belgian astronomer, mathematician, statistician and sociologist Lambert Adolphe Jacques Quetelet publishes Anthropométrie ou mesure des différentes facultés de l'homme.

In Anthropmétrie and in  Physique sociale ou essai sur le developpement des facultés de l'homme (1869), Quetelet established the basis for mathematical study of anthropological data. "Quetelet showed that if a series of anthropological measurements of either physical or intellectual qualities were plotted on squared paper, allowing x to be the measurements and y to be their frequency, they formed a curve like that representing the expansion of the binomial, or like that formed by plotting the errors of a great number of observers [i.e., the Gaussian curve]" (Penniman, 105). By applying the mathematics of the Gaussian curve to anthropological data, it became possible to plot the average or "standard" deviation from the statistical average, and thus to interpret anthropological data with greater exactness.

Martin Wiberg uses his difference engine to produce Tables de Logarithms Calculées et Imprimées au Moyen de la Machine à Calculer du M. Wiberg. This set of tables of seven-place logarithms from 1 to 100,000 is the first logarithmic table produced by a calculating machine.

Bruno Abdank-Abakanowicz, a mathematician, inventor and electrical engineer, invents the integraph, a form of integrator.

"The integraph is an elaboration and extension of the planimeter, an earlier, simpler instrument used to measure area. It is a mechanical instrument capable of deriving the integral curve corresponding to a given curve. Hence, it is capable of solving graphically a simple differential equation.

"Sets of partial differential equations are commonly encountered in mathematical physics. Most branches of physics such as aerodynamics, electricity, acoustics, plasma physics, electron-physics and nuclear energy involve complex flows, motions and rates of change which may be described mathematically by partial differential equations. A well-established example from electromagnetics is the set of partial differential equations known as Maxwell's equations.

"In practice, differential equations can be difficult to integrate, that is to solve. The integraph is capable of solving only simple differential equations. The need to handle sets of more complex non-linear differential equations, led Vannevar Bush to develop the Differential Analyzer at MIT in the early 1930s. In turn, limitations in speed, capacity and accuracy of the Bush Differential Analyzer provided the impetus for the development of the ENIAC during World War II.

"Abdank-Abakanowicz’s instrument could produce solutions to a commonly encountered class of simple differential equations of the form dy/dx = F(x) so that y = ò F(x)dx. The basic approach was to draw a graph of the function F and then use the pointer on the device to trace the contour of the function. The value of the integral could then be read from the dials. The concept of the instrument was taken up and soon put into production by such well known instrument makers as the Swiss firm of Coradi in Zurich." From Gordon Bell's website, accessed 09-01-2010.

Friedrich Ludwig Gottlob Frege publishes in Halle, Germany his Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens.

“. . . although a mere booklet of eighty-eight pages, it is perhaps the most important single work ever written in logic. Its fundamental contributions, among lesser points, are the truth-functional propositional calculus, the analysis of the proposition into function and argument(s) instead of subject and predicate, the theory of quantification, a system of logic in which derivations are carried out exclusively according to the form of the expressions, and a logical definition of the notion of mathematical sequence. Any single one of these achievements would suffice to secure the book a permanent place in the logician’s library” (Van Heijenoort, From Frege to Gödel (1967) 1).

“In his attempt to give a satisfactory definition of number and a rigorous foundation to arithmetic, Frege found ordinary language insufficient. To overcome the difficulties involved, he devised his Begriffschrift as a tool for analyzing and representing mathematical proofs completely and adequately. This tool has gradually developed into modern mathematical logic, of which Frege may justly be considered the creator“ (Dictionary of Scientific Biography article on Frege).

English clergyman and headmaster Edwin A. Abbott publishes a work of scientific fantasy entitled Flatland: A Romance of Many Dimensions. With illustrations by the Author, A SQUARE.

"It is a charming, slightly pedestrian tale of imaginary beings; polygons who live in a two-dimensional universe of the Euclidean plane. Just below the surface, though, it is a biting satire on Victorian values--especially as regards women and social status-- and an accomplished and original piece of scientific popularization about the fourth dimension. And, perhaps, an allegory of a spiritual journey" (Ian Stewart, editor, The Annotated Flatland [2002] ix).

♦ In 2008 Ladd Ehlinger Jr. issued an excellent computer-animated film of Flatland, which he characterizes as a tale of "math, physics, dimensionality, philosophy, religion and war." You can view clips from the film on Ehlinger's website and also order autographed copies of the DVD directly from the site.

The logarithmic and trigonometric tables of Gaspard Riche de Prony, compiled in 19 volumes of manuscript, mostly by hairdressers unemployed after the French Revolution, are finally published in an abbreviated form in one volume. They are the most monumental work of calculation ever carried out by human computers.

French engineer and applied mathematician Philbert Maurice d'Ocagne publishes Nomographie, les calculs usuels effectués au moyen des abaques. In this work on nomograms or nomographs he

"presented the first outline of a rationally ordered discipline embracing all the individual procedures of nomographical calulation then known. Pursuing this subject, he succeeded in defining and classifying the most general modes of representation applicable to equations with an arbitrary number of variables. The results of all these investigations, along with a considerable number of applications . . .  [he] set forth in Traité de nomographie (1899), which was followed by other more or less developed expositions. This material appeared in fifty-nine partial or entire translations in fourteen languages" (Dictionary of Science Biography X [1974] 170). 

The recently established Deutsche Mathematiker-Vereinigung holds an exhibition in Munich of Mathematical and Mathematical-Physical Models, Apparatus, and Instruments.

This was the first international exhibition limited to mathematical devices, including calculating instruments; it reflected the huge growth in the field since the London exposition of 1876. The exhibition had been planned for the previous year but was canceled because of an outbreak of cholera in northern Germany.

German mathematician and physicist David Hilbert publishes in Mathematische Probleme a list of twenty-three problems that he predicts will be of central importance to the advance of mathematics in the twentieth century.

In the second of these problems Hilbert called for a mathematical proof of the consistency of the arithmetic axioms—a question that influenced both the development of mathematical logic and computing.

Hilbert's paper was first published in Nachrichten der Königliche Gesellschaft zur Wissenschaften zu Göttingen, Mathematische-physikalischen Klasse, 3 (1900).

Hook & Norman, Origins of Cyberspace (2002) no. 320.

Historian of Mathematics David Eugene Smith publishes Rara arithmetica: A Catalogue of the Arithmetics Written Before the Year MDC! with a Description of Those in the Library of George Arthur Plimpton of New York. This two-volume work, issued by Plimpton's textbook publishing company, Ginn & Company., described and illustrated Plimpton's library of early mathematical books and medieval manuscripts before 1601.  Two versions of the catalogue were published:

1. A deluxe numbered edition limited to 151 copies printed on handmade paper and bound in full vellum, elaborately gilt, in two volumes, with the plates printed in color on Japan vellum, enclosed in a slipcase
2. A trade edition of indeterminate number, printed on regular paper and bound in one volume in cloth-backed boards. 
   

Plimpton’s mathematical library, preserved at Columbia University, is the first specialized private collection of antiquarian scientific books formed by an American for which we have an annotated bibliographical catalogue.  Smith also discussed some of Plimpton’s early manuscripts in his History of Mathematics (Boston: Ginn & Co., 1923–25), and issued a pamphlet addendum to his catalogue of Plimpton’s library in 1939 (Rara arithmetica: Addenda to “Rara arithmetica" [Boston: Ginn & Co.]).

Plimpton did not comment on his library in any of Smith’s works, all, or nearly all of which were published by Plimpton's Ginn & Company. The only place where I find published remarks by Plimpton on his mathematical library is in “The History of Elementary Mathematics in the Plimpton Library", Atti del Congresso Internazionale dei Matematici Bologna 3–10 Settembre 1928, VI (1932) 433–42.

Bertrand Russell and Alfred North Whitehead publish Principia mathematica in three volumes, taking up the task — first attempted in Russell's uncompleted Principles of Mathematics (1903) — of proving the logical basis of all mathematics by deducing the whole body of mathematical doctrine from a small number of primitive ideas and principles of logical inference. To do so Russell and Whitehead devised a complex but precise system of symbols that enabled them to sidestep the ambiguities of ordinary language, and to give an outstanding exposition of sentential logic.  Russell and Whitehead did not entirely achieve their goal -- certain of their theories and axioms were found to be unsatisfactory-- but their failures inspired further investigation of both their own and rival theories, and possibly contributed more to the development of mathematical logic than their complete success would have done.

Cambridge University Press issued 750 copies of the first volume of this work. Disappointed with the sales of that volume, the publishers reduced the printings of Volumes II and III to 500 copies. Thus the complete set is more difficult to find than copies of Volume I.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1868.

Anthropologist Leslie Leland Locke publishes "The Ancient Quipu, A Peruvian Knot Record," American Anthropologist, New Series I4 (1912) 325-332.

This was the first work to show how the Inca (Inka) Empire and its predecessor societies used the quipu (Khipu) for mathematical and accounting records in the decimal system. Locke stated his conclusions as follows:

"1. These knots were used purely for numerical purposes.

"2. Distances from the main cord were used roughly to locate the orders, which were on a decimal scale.

"3. The quipu was not used for counting or calculating but for record keeping. The mode of tying the knots was not adapted to counting, and there was ne need of its use for such a purpose, as the Quichua language contained a complete and adequate system of numeration.

"4. Other specimens examined contain the same types of knots there being but ten variations in all, two forms for the single knot and eight long knots. These eight differen from each other and from the single knot only in the number of turns taken in tying. There is nothing about any specimen examined to give the slightest suggesion that it was used for any other than numerical purposes.

"5. If the hypothesis that this quipu is a record of the same classes of objects be correct, it would seem to indicate the colors in this case have no special significance, but were taken according to the fancy or convenience of the maker. This does not signify that there was not a rough color scheme in sue for some purposes.

"6. These specimens confirm in a remarkable way the accuracy with which [the Inca] Garcilasso [de la Vega] described the manners and customs of his people."

In 1923 Locke published an expanded version of his research in a monograph entitled The Ancient Quipu or Peruvian Knot Record.

Research on this topic was further advanced by mathematician Marcia Ascher and anthropologist Robert Ascher in Code of the Quipu. A Study of Media, Mathematics, and Culture (1981).

German mathematician Leopold Löwenheim publishes Über Möglichkeiten im Relativkalkül, containing the first appearance of what is now known as the Löwenheim-Skolem theorem, the first theorem of modern logic, anticipating Kurt Gödel’s completeness theorem of 1930.

Löwenheim's paper was first published in Mathematischen Annalen 76 (1915) 447-470. A summary and English translation are in van Heijenoort, From Frege to Gödel (1967)228-51.

Austrian mathematician Johann Radon demonstrates that the image of a three-dimensional object can be constructed from an infinite number of two-dimensional images of the object.

About sixty-five years later Radon's work was applied in the invention of computed tomography.

Norwegian mathematician Albert Skolem proves the Lowenheim-Skolem theorem, a landmark in mathematical logic.

Skolem's paper was first published as "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen", Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse 6 (1920) 1–36.

Hook & Norman, Origins of Cyberspace (2002) no. 365. An English translation of Skolem's paper appears in van Heijenoort, From Frege to Gödel (1967) 254-63.

Mathematician and mathematical biologist Vito Volterra publishes "Varizioni e fluttuazioni del numero d'individui in specie animali conviventi" in Mem. R. Acad. Naz. dei Lincei (ser.6) II, 31-113.

This work was translated into English and published in the journal Nature the same year as "Fluctuations in the abundance of a species considered mathematically". In this paper Volterra created the basic equations for two species interactions.

At the International Congress of Mathematicians held in Bologna, Italy, mathematician and physicist David Hilbert returned to the second of the twenty-three problems posed in his 1900 paper, asking, is mathematics complete, is it consistent, and is it decidable?

Three years later, the first two of these questions were answered in the negative by Kurt Gödel. Working independently, Alonzo Church, Alan Turing, and Emil Post published answers to the third question in 1936.

Hilbert's paper was first published in Atti del Congresso Internazionale dei Matematici, Bologna 3-10 settembre 1928 (VI) I (1929) 135-41.

Hook & Norman, Origins of Cyberspace (2002) no. 320.

Leslie J. Comrie discovers how to use a commercial accounting machine as a difference engine.

With this technique Comrie reformed the production of the Nautical Almanac.

Ralph V. R. Hartley publishes “Transmission of Information,” in which he proves "that the total amount of information that can be transmitted is proportional to frequency range transmitted and the time of the transmission."

Hartley's law eventually became one of the elements of  Claude Shannon's  mathematical theory of communication.

Mathematician, physicist, and economist John von Neumann publishes "Zur Theorie der Gesellschaftsspiele" in Mathematische Annalen, 100, 295–300. This paper "On the Theory of Parlor Games" propounds the minimax theorem, inventing the theory of games.

French mathematician Jacques Herbrand publishes Thèses. . . 1re thèse. Recherches sur la théorie de la démonstration, printed in Warsaw, Poland.

 “The main product of Herbrand’s short life (he died in a skiing accident [at the age of 23]) was his thesis, in which he found two ways of proving that tautologies are provable. One was based upon a means of matching any quantified formula with a quantifier-free mate and proving that each was derivable; it reversed the handling of quantifications in Principia mathematica, *9, and also its systematic application in the second edition. The other method drew on model theory and normal forms, as developed by Leopold Löwenheim and Thoraf Skolem. A highlight was a result which became known as ‘the deduction theorem’; it took the form that if the premises of a theory were stated as a single conjunction H, then a proposition P was true within it if and only if ‘H ∩ P be a propositional identity’ . . . In effect though not in intention, he clarified some of Bertrand Russell’s conflations and implication and inference, and also removed a standard sloppiness among mathematicians when (not) relating a proof to its theorem. While several proofs were unclear and even defective, the thesis inspired important new lines of research” (Grattan-Guinness, The Search for Mathematical Roots 1870-1940 [2000] 550).

Van Heijenoort, From Frege to Gödel. A Source Book in Mathematical Logic (1967) 525-81.

Kurt Gödel proves the incompleteness and inconsistency of arithmetic, and invents the theory of recursive functions.

Vannevar Bush begins the Rapid Arithmetical Machine Project at MIT.

In a paper called "Instrumental Analysis", he suggested how an electromechanical machine might be built to accomplish Charles Babbage’s goals for the Analytical Engine. This was almost exactly one hundred years after Babbage began designing his Analytical Engine.

In the same paper Bush wrote that four billion punched cards were being used annually in electric tabulating machines. This amounted to ten thousand tons of punched cards.

Alonzo Church publishes his logical proof of the undecidability of arithmetic, using his lambda calculus.

Alan Turing spends more than a year at Princeton University to study mathematical logic with Alonzo Church, who is pursuing research in recursion theory.

Alan Turing publishes On Computable Numbers, a mathematical description of what he calls a universal machine that can, in principle, solve any mathematical problem that can be presented to it in symbolic form.

Turing modeled the universal machine processes after the functional processes of a human carrying out mathematical computation. (See Reading 7.1.)

Alonzo Church calls Alan Turing’s universal machine a Turing Machine. (See Reading 7.2.)

Independently of Alan Turing, Emil Post develops a mathematical model of computation that is essentially equivalent to the Turing machine. "Intending this as the first of a series of models of equivalent power but increasing complexity he titles his paper Formulation 1. This model is sometime's called "Post's machine" or a Post-Turing machine."

At Princeton University  Alan Turing and John von Neumann have their first discussions about computing and what will later be called “artificial intelligence” (AI).

George Stibitz, a research mathematician at Bell Telephone Labs in New York City, builds a binary adder out of a few light bulbs, batteries, relays and metal strips cut from tin cans on his kitchen table.

This device was similar to a theoretical design described by Claude Shannon in his master's thesis. Stibitz's "Model K" (for “Kitchen”) was the first electromechanical computer built in America.

Konrad Zuse completes his Z1 mechanical computer in his parents’ Berlin apartment.

Independently of Claude Shannon, Zuse developed a form of symbolic logic to assist in the design of the binary circuits. With Helmut Schreyer, he began work on the Z2.

Alan Turing reports to the Government Code and Cypher School, Bletchley Park, in the town of Bletchley, England.

Having collaborated with Julian Bigelow, an engineer, Norbert Wiener publishes, as a classified document, The Extrapolation, Interpretation and Smoothing of Stationery Time Series.

According to Claude Shannon , this work contained “the first clear-cut formulation of communication theory as a statistical problem, the study of operations on time series.”

Logician and cognitive psychologist Walter Pitts, an autodidact without a high school or college diploma, accepts a position at MIT to work with Norbert Wiener.

Mathematician, physicist, and economist John von Neumann and economist Oskar Morgenstern publish The Theory of Games and Economic Behavior.

Quantitative mathematical models for games such as poker or bridge at one time appeared impossible, since games like these involve free choices by the players at each move, and each move reacts to the moves of other players. However, in the 1920s John von Neumann single-handedly invented game theory, introducing the general mathematical concept of "strategy" in a paper on games of chance (Mathematische Annalen 100 [1928] 295-320). This contained the proof of his "minimax" theorem that says "a strategy exists that guarantees, for each player, a maximum payoff assuming that the adversary acts so as to minimize that payoff." The "minimax" principle, a key component of the game-playing computer programs developed in the 1950s and 1960s by Arthur Samuel, Allen Newell, Herbert Simon, and others was more fully articulated and explored in The Theory of Games and Economic Behavior, co-authored by von Neumann and Morgenstern.

Game theory, which draws upon mathematical logic, set theory and functional analysis, attempts to describe in mathematical terms the decision-making strategies used in games and other competitive situations. The Von Neumann-Morgenstern theory assumes (1) that people's preferences will remain fixed throughout; (2) that they will have wide knowledge of all available options; (3) that they will be able to calculate their own best interests intelligently; and (4) that they will always act to maximize these interests. Attempts to apply the theory in real-world situations have been problematical, and the theory has been criticized by many, including AI pioneer Herbert Simon, as failing to model the actual decision-making process, which typically takes place in circumstances of relative ignorance where only a limited number of options can be explored.

Von Neumann revolutionized mathematical economics. Had he not suffered an early death from cancer in 1957, most probably he would have received the first Nobel Prize in economics. (The first Nobel prize in economics was awarded in 1969; it cannot be awarded posthumously.) Several mathematical economists influenced by von Neumann's ideas later received the Nobel Prize in economics. 

Hook & Norman, Origins of Cyberspace (2002) no. 953.

Howard Aiken publishes Tables of the Modified Hankel Functions of Order One-Third and of Their Derivatives.

These tables, calculated by the Harvard Mark I, were the first published mathematical tables calculated by a programmed automatic computer, finally fulfilling the dream of Charles Babbage, which he first expressed in 1822. Calculating these tables required the equivalent of forty-five days of computer processing time. Prior to the Mark I calculating the tables would have required years of human computation.

A contest is held in Tokyo between the Japanese soroban, used by Kiyoshi Matsuzaki, a champion operator in the Savings Bureau of the Japanese postal administration, and an electric calculator, operated by US Army Private Thomas Nathan Wood of the 240th Finance Distributing Section of General MacArthur's headquarters, who was the most experienced calculator operator in Japan at the time. The bases for scoring in the contest were speed and accuracy of results in all four basic arithmetic operations and a problem which combines all four. The soroban won 4 to 1, with the electric calculator prevailing in multiplication.

"About the event, the Nippon Times newspaper reported that "Civilization ... tottered" that day, while the Stars and Stripes newspaper described the soroban's "decisive" victory as an event in which "the machine age took a step backward. . . ."

"The breakdown of results is as follows:

"* Five additions problems for each heat, each problem consisting of 50 three- to six-digit numbers. The soroban won in two successive heats.

"* Five subtraction problems for each heat, each problem having six- to eight-digit minuends and subtrahends. The soroban won in the first and third heats; the second heat was a no contest.

"* Five multiplication problems, each problem having five- to 12-digit factors. The calculator won in the first and third heats; the soroban won on the second.

"* Five division problems, each problem having five- to 12-digit dividends and divisors. The soroban won in the first and third heats; the calculator won on the second.

"* A composite problem which the soroban answered correctly and won on this round. It consisted of:

"o An addition problem involving 30 six-digit numbers

"o Three subtraction problems, each with two six-digit numbers o Three multiplication problems, each with two figures containing a total of five to twelve digits

"o Three division problems, each with two figures containing a total of five to twelve digits" (Wikipedia article on Soroban, accessed 04-15-2009).

Mathematician John von Neumann delivers lectures at the University of Illinois on The Theory of Self-Reproducing Automata. In these lectures von Neumann showed that in theory a program could reproduce itself. The lectures were edited by A. W. Burks and published in 1966.

Years later one application of this plausibility result in computability theory was the development of what came to be known as malware.

The ENIAC is turned off for the last time.

It was estimated that this single machine did more computation during the ten years of its operation than the entire human race had done up till the time of its invention.

Because of failing health, John von Neumann does not finish his last book, The Computer and the Brain, in which he compares the functions of computers and the human brain.

John von Neumann dies of cancer at the age of fifty-four.

Having been computed by human computers since 1767, the Nautical Almanac is finally produced by an electronic computer.

"The computation of the data for the almanacs involved a considerable amount of effort. As late as the mid-20th century, HMNAO employed a small army of human computers to carry out this work. They used the latest technology available at the time: logarithm tables, mechanical calculating machines and electro-mechanical calculating machines. In 1959 the Office obtained its own electronic computer, making it the first part of the RGO to use this emerging technology."

Reflecting the obsolescence of mathematical tables as a result of the development of electronic computing,  Mathematical Tables and Other Aids to Computation (MTAC), the first computing journal, changes its name to Mathematics of Computation.

Philosopher, mathematician and computer scientist John Alan Robinson publishes "A Machine-Oriented Logic Based on the Resolution Principle", Communications of the ACM, 5:23–41.

This paper introduced the resolution principle, a standard of logical deduction in AI applications.

James Cooley and John Tukey publish the Cooley-Tukey FFT algorithm, the most common fast Fourier transform algorithm .

Italian-American electrical engineer and businessman Andrew Viterbi develops the Viterbi algorithm,  "as an error-correction scheme for noisy digital communication links, finding universal application in decoding the convolutional codes used in both CDMA and GSM digital cellular, dial-up modems, satellite, deep-space communications, and 802.11 wireless LANs. It is now also commonly used in speech recognition, keyword spotting, computational linguistics, and bioinformatics. For example, in speech-to-text (speech recognition), the acoustic signal is treated as the observed sequence of events, and a string of text is considered to be the "hidden cause" of the acoustic signal. The Viterbi algorithm finds the most likely string of text given the acoustic signal" (Wikipedia article on Viterbi algorithm, accessed 12-29-2009).

Benoit Mandelbrot, a researcher at IBM, develops fractal geometry in his book, Les objets fractals, forme, hasard et dimension, building on the concept that seemingly irregular shapes can have identical structure at all scales.

Mandelbrot's new geometry made it possible to describe mathematically the kinds of irregularities existing in nature.

At Lehigh University, Frederick Cohen demonstrates a virus-like program on a VAX11/750 system. The program is able to install itself to, or infect, other system objects.

In 1984 Cohen used the phrase "computer virus" – as suggested by his teacher Leonard Adleman – to describe the operation of such programs in terms of "infection". He defined a 'virus' as "a program that can 'infect' other programs by modifying them to include a possibly evolved copy of itself.”

Physicist and mathematician Stephen Wolfram and Wolfram Research introduce Mathematica 1.0, "a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing" with powerful two dimensional and three dimensional visualization tools.

Mathematica evolved from Symbolic Manipulation Program, usually called SMP, "a computer algebra system designed by Chris A. Cole and Stephen Wolfram at Caltech circa 1979 and initially developed in the Caltech physics department under Wolfram's leadership . . . . It was first sold commercially in 1981 by the Computer Mathematics Corporation of Los Angeles which later became part of Inference Corporation; Inference Corp. further developed the program and marketed it commercially from 1983 to 1988. SMP was essentially Version Zero of the more ambitious Mathematica system.

"SMP was influenced by the earlier computer algebra systems Macsyma (of which Wolfram was a user) and Schoonschip (whose code Wolfram studied)" (Wikipedia article on Symbolic Manipulation Program, accessed 05-16-2009).

Anthropologists Gary Urton and Carrie Brezine publish "Khipu Accounting in Ancient Peru," Science 309 (2005) 1065 - 1067.

"Khipu [quipu] are knotted-string devices that were used for bureaucratic recording and communication in the Inka [Inca] Empire. We recently undertook a computer analysis of 21 khipu from the Inka administrative center of Puruchuco, on the central coast of Peru. Results indicate that this khipu archive exemplifies the way in which census and tribute data were synthesized, manipulated, and transferred between different accounting levels in the Inka administrative system" (Science).

"Researchers in the US believe they have come closer to solving a centuries-old mystery - by deciphering knotted string used by the ancient Incas.

"Experts say one bunch of knots appears to identify a city, marking the first intelligible word from the extinct South American civilisation.

"The coloured, knotted pieces of string,known as khipu, are believed to have been used for accounting information.

"The researchers say the finding could unlock the meaning of other khipu.

"Harvard University researchers Gary Urton and Carrie Brezine used computers to analyse 21 khipu.

"They found a three-knot pattern in some of the strings which they believe identifies the bunch as coming from the city of Puruchuco, the site of an Inca palace.

" 'We hypothesize that the arrangement of three figure-eight knots at the start of these khipu represented the place identifier, or toponym, Puruchuco,' they wrote in their report, published in the journal Science.

" 'We suggest that any khipu moving within the state administrative system bearing an initial arrangement of three figure-eight knots would have been immediately recognisable to Inca administrators as an account pertaining to the palace of Puruchuco.' (http://news.bbc.co.uk/2/hi/americas/4143968.stm, accessed 04-28-2009).

Dirk Brockmann, L. Hufnagel, and T. Geisel publish "The scaling laws of human travel," Nature 439 (2006) 46265. 

Using statistical data from the American currency tracking website, Where's George?, the paper described statistical laws of human travel in the United States, and developed a mathematical model of the spread of infectious disease.

[By January 31, 2009, Where's George? tracked over 149 million bills totaling more than $810 million. (Wikipedia).]

The Walters Art Museum reports through The New York Times that the Archimedes Palimpsest, the unique tenth century source for two treatises by Archimedes: The Method and Stomachion, and the unique source for the Greek text of On Floating Bodies, also contains ten pages of previously unknown speeches by Hyperides, "one of the foundational figures of Greek democracy," "illuminating some fascinating, time-shrouded insights into Athenian law and social history." The palimpsest includes parchment from seven texts including two texts which remain to be identified.

This manuscript was purchased by a private collector at an auction at Christie's in New York on October 28, 1998. It has been characterized as one of the most important scientific manuscripts ever to appear on the market.

Petr Sojka of the Department of Computer Graphics and Design of Faculty of Informatics, Masaryk University, Czech Republic, organizes the first conference, held at the University of Birmingham, entitled DML 2008 Towards a Digital Mathematics Library as part of the Conferences on Intelligent Computer Mathematics (CICM) and Mathematics Knowledge Management (MKM).

"Mathematicians dream of a digital archive containing all peer-reviewed mathematical literature ever published, properly linked and validated/verified. It is estimated that the entire corpus of mathematical knowledge published over the centuries does not exceed 100,000,000 pages, an amount easily manageable by current information technologies.

"The workshop's objectives are to formulate the strategy and goals of a global mathematical digital library and to summarize the current successes and failures of ongoing technologies and related projects, asking such questions as:

"* What technologies, standards, algorithms and formats should be used and what metadata should be shared?

"* What business models are suitable for publishers of mathematical literature, authors and funders of their projects and institutions?

"* Is there a model of sustainable, interoperable, and extensible mathematical library that mathematicians can use in their everyday work?

* What is the best practice for

"o retrodigitized mathematics (from images via OCR to MathML and/or TeX);

"o retro-born-digital mathematics (from existing electronic copy in DVI, PS or PDF to MathML and/or TeX);

"o born-digital mathematics (how to make needed metadata and file formats available as a side effect of publishing workflow [CEDRAM model])?"

Dirk Brockmann, and the epidemic modeling team at the Northwestern Institute on Complex Systems, use air traffic and commuter traffic patterns for the entire country, and data from the American currency tracking website, Where’s George?, to predict the spread of the H1N1 flu or "swine flu" across the United States.